riewarticle drell-yan process in tmd...

Review Article120587 minus 119873Drell-Yan Process in TMD Factorization

XiaoyuWang 1 and Zhun Lu 2

1School of Physics and Engineering Zhengzhou University Zhengzhou Henan 450001 China2School of Physics Southeast University Nanjing 211189 China

Correspondence should be addressed to Zhun Lu zhunluseueducn

Received 16 November 2018 Accepted 9 January 2019 Published 22 January 2019

Guest Editor Zhongbo Kang

Copyright copy 2019 Xiaoyu Wang and Zhun Lu This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited The publication of this article was funded by SCOAP3

This article presents the review of the current understanding on the pion-nucleon Drell-Yan process from the point of view of theTMD factorization Using the evolution formalism for the unpolarized and polarized TMD distributions developed recently weprovide the theoretical expression of the relevantphysical observables namely the unpolarized cross section the Sivers asymmetryand the cos 2120601 asymmetry contributed by the double Boer-Mulders effectsThe corresponding phenomenology particularly at thekinematical configuration of the COMPASS 120587119873 Drell-Yan facility is displayed numerically

1 Introduction

After the first observation of the 120583+120583minus lepton pairs producedin 119901119873 collisions [1] the process was interpreted that a quarkand an antiquark from each initial hadron annihilate into avirtual photon which in turn decays into a lepton pair [2]This explanation makes the process an ideal tool to explorethe internal structure of both the beam and target hadronsSince then a wide range of studies on this (Drell-Yan) processhave been carried out In particular the120587119873 Drell-Yanprocesshas the unique capability to pin down the partonic structureof the pion which is an unstable particle and therefore cannotserve as a target in deep inelastic scattering processes Severalpion-induced experiments have been carried out such as theNA10 experiment at CERN [3ndash6] the E615 [7] E444 [8]and E537 [9] experiments at Fermilab three decades agoThese experimental measurements have provided plenty ofdata which have been used to considerably constrain thedistribution function of the pion meson Recently a newpion-induced Drell-Yan program with polarized target wasalso proposed [10] at the COMPASS of CERN and the firstdata using a high-intensity 120587 beam of 190 GeV colliding on aNH3 target has already come out [11]

Bulk of the events in the Drell-Yan reaction are from theregion where the transverse momentum of the dilepton 119902perp ismuch smaller than the mass 119876 of the virtual vector boson

thus the intrinsic transverse momenta of initial partonsbecome relevant It is also the most interesting regime wherea lot of intriguing physics arises Moreover in the small 119902perpregion (119902perp sim ΛQ119862119863) the fixed-order calculations of the crosssections in the collinear picture fail leading to large doublelogarithms of the type 120572ln2(1199022perp1198762) It is necessary to resumsuch logarithmic contributions to all orders in the strongcoupling 120572119904 to obtain a reliable result The standard approachfor such resummation is the Collins-Soper-Sterman (CSS)formalism [12] originated from previous work on the Drell-Yan process and the 119890+119890minus annihilation three decades ago Inrecent years the CSS formalism has been successfully appliedto develop a factorization theorem [13ndash15] in which thegauge-invariant [16ndash19] transverse momentum dependent(TMD) parton distribution functions or fragmentation func-tions (collectively called TMDs) [20 21] play a central roleFrom the point of view of TMD factorization [12 13 15 22]physical observables can bewritten as convolutions of a factorrelated to hard scattering and well-defined TMDs Aftersolving the evolution equations the TMDs at fixed energyscale can be expressed as a convolution of their collinearcounterparts and perturbatively calculable coefficients in theperturbative region and the evolution from one energy scaleto another energy scale is included in the exponential factorof the so-called Sudakov-like form factors [12 15 23 24]TheTMD factorization has been widely applied to various high

HindawiAdvances in High Energy PhysicsVolume 2019 Article ID 6734293 20 pageshttpsdoiorg10115520196734293

2 Advances in High Energy Physics

energy processes such as the semi-inclusive deep inelasticscattering (SIDIS) [14 15 22 23 25 26] 119890+119890minus annihilation[15 27 28] Drell-Yan [15 29] andWZ production in hadroncollision [12 15 30] The TMD factorization can be alsoextended to themoderate 119902perp regionwhere an equivalence [3132] between the TMD factorization and the twist-3 collinearfactorization is found

One of the most important observables in the polarizedDrell-Yan process is the Sivers asymmetry It is contributedby the so-called Sivers function [33] a time-reversal-odd(T-odd) distribution describing the asymmetric distribu-tion of unpolarized quarks inside a transversely polarizednucleon through the correlation between the quark trans-verse momentum and the nucleon transverse spin Remark-ably QCD predicts that the sign of the Sivers functionchanges in SIDIS with respect to the Drell-Yan process [1634 35] The verification of this sign change [36ndash41] is oneof the most fundamental tests of our understanding of theQCD dynamics and the factorization schemes and it isalso the main pursue of the existing and future Drell-Yanfacilities [10 11 42ndash45] The advantage of the 120587119873 Drell-Yan measurement at COMPASS is that almost the samesetup [11 46] is used in SIDIS and Drell-Yan processeswhich may reduce the uncertainty in the extraction of theSivers function In particular the COMPASS Collaborationmeasured for the first time the transverse-spin-dependentazimuthal asymmetries [11] in the 120587minus119873 Drell-Yan process

Another important observable in the Drell-Yan process isthe cos 2120601 angular asymmetry where 120601 corresponds to theazimuthal angle of the dilepton The fixed-target measure-ments from the NA10 and E615 collaborations showed thatthe unpolarized cross section possesses large cos 2120601 asymme-try which violates the Lam-Tung relation [47] Similar vio-lation has also been observed in the 119901119901 colliders at Tevatron[48] and LHC [49] It has been explained from the viewpointsof higher-twist effect [50ndash53] the noncoplanarity effect [3054] and the QCD radiative effects at higher order [55 56]Another promising origin [57] for the violation of the Lam-Tung relation at low transverse momentum is the convolutionof the two Boer-Mulders functions [58] from each hadronThe Boer-Mulders function is also a TMD distribution Asthe chiral-odd partner of the Sivers function it describesthe transverse-polarization asymmetry of quarks inside anunpolarized hadron [57 58] thereby allowing the probe ofthe transverse spin physics from unpolarized reaction

This article aims at a review on the current status ofour understanding on the Drell-Yan dilepton production atlow transverse momentum especially from the 120587119873 collisionbased on the recent development of the TMD factorizationWe will mainly focus on the phenomenology of the Siversasymmetry as well as the cos 2120601 asymmetry from the doubleBoer-Mulders effect In order to quantitatively understandvarious spinazimuthal asymmetries in the 120587119873 Drell-Yanprocess a particularly important step is to know in highaccuracy the spin-averaged differential cross section of thesame process with azimuthal angles integrated out sinceit always appears in the denominator of the asymmetriesrsquodefinition Thus the spin-averaged cross section will be alsodiscussed in great details

The remained content of the article is organised as fol-lows In Section 2 we will review the TMD evolution formal-ism of the TMDs mostly following the approach establishedin [15] Particularly we will discuss in detail the extraction ofthe nonperturbative Sudakov form factor for the unpolarizedTMD distribution of the protonpion as well as that for theSivers function In Section 3 putting the evolved result of theTMD distributions into the TMD factorization formulae wewill present the theoretical expression of the physical observ-ables such as the unpolarized differential cross section theSivers asymmetry and the cos 2120601 asymmetry contributedby the double Boer-Mulders effect In Section 4 we presentthe numerical evolution results of the unpolarized TMDdistributions and the Boer-Mulders function of the pionmeson as well as that of the Sivers function of the protonIn Section 5 we display the phenomenology of the physicalobservables (unpolarized differential cross section the Siversasymmetry and the cos 2120601 asymmetry) in the 120587119873 Drell-Yan with TMD factorization at the kinematical configurationof the COMPASS experiments We summarize the paper inSection 6

2 The TMD Evolution of theDistribution Functions

In this section we present a review on the TMD evolution ofthe distribution functions Particularly we provide the evo-lution formalism for the unpolarized distribution function1198911 transversity ℎ1 Sivers function 119891perp1 and the Boer-Muldersfunction ℎperp1 of the proton as well as 1198911 and ℎperp1 of the pionmeson within the Collins-11 TMD factorization scheme [15]

In general it is more convenient to solve the evolutionequations for the TMD distributions in the coordinate space(b space) other than that in the transverse momentum kperpspace with b conjugate to kperp via Fourier transformation[12 15] The TMD distributions 119865(119909 119887 120583 120577119865) in b spacehave two kinds of energy dependence namely 120583 is therenormalization scale related to the corresponding collinearPDFs and 120577119865 is the energy scale serving as a cutoff toregularize the light-cone singularity in the operator defini-tion of the TMD distributions Here 119865 is a shorthand forany TMD distribution function and the tilde denotes thatthe distribution is the one in b space If we perform theinverse Fourier transformation on 119865(119909 119887 120583 120577119865) we recoverthe distribution function in the transverse momentum space119865119902119867(119909 119896perp 120583 120577119865) which contains the information about theprobability of finding a quark with specific polarizationcollinear momentum fraction 119909 and transverse momentum119896perp in a specifically polarized hadron119867

21 TMD Evolution Equations The energy evolution for the120577119865 dependence of the TMD distributions is encoded in theCollins-Soper (CS) [12 15 63] equation

120597 ln119865 (119909 119887 120583 120577119865)120597radic120577119865 = (119887 120583) (1)

while the 120583 dependence is driven by the renormalizationgroup equation as

Advances in High Energy Physics 3

119889119889 ln 120583 = minus120574119870 (120572119904 (120583)) (2)

119889 ln119865 (119909 119887 120583 120577119865)119889 ln 120583 = 120574119865(120572119904 (120583) 12057721198651205832) (3)

with 120572119904 being the strong coupling at the energy scale 120583 being the CS evolution kernel and 120574119870 120574119865 being theanomalous dimensions The solutions of these evolutionequations were studied in detail in [15 63 64] Here wewill only discuss the final result The overall structure of thesolution for 119865(119909 119887 120583 120577119865) is similar to that for the Sudakovform factor More specifically the energy evolution of TMDdistributions from an initial energy 120583 to another energy 119876 isencoded in the Sudakov-like form factor 119878 by the exponentialform exp(minus119878)

119865 (119909 119887 119876) = F times 119890minus119878 times 119865 (119909 119887 120583) (4)

whereF is the factor related to the hard scattering Hereafterwe will set 120583 = radic120577119865 = 119876 and express 119865(119909 119887 120583 = 119876 120577119865 = 1198762)as 119865(119909 119887 119876)

As the 119887-dependence of the TMDs can provide veryuseful information regarding the transverse momentumdependence of the hadronic 3D structure through Fouriertransformation it is of fundamental importance to study theTMDs in 119887 space In the small 119887 region the 119887 dependenceis perturbatively calculable while in the large 119887 region thedependence turns to be nonperturbative andmay be obtainedfrom the experimental data To combine the perturbativeinformation at small 119887 with the nonperturbative part atlarge 119887 a matching procedure must be introduced with aparameter 119887max serving as the boundary between the tworegions The prescription also allows for a smooth transitionfrom perturbative to nonperturbative regions and avoids theLandau pole singularity in 120572119904(120583119887) A 119887-dependent function 119887lowastis defined to have the property 119887lowast asymp 119887 at low values of 119887 and119887lowast asymp 119887max at large 119887 values In this paper we adopt the originalCSS prescription [12]

119887lowast = 119887radic1 + 11988721198872max

119887max lt 1ΛQCD (5)

The typical value of 119887max is chosen around 1 GeVminus1 to guar-antee that 119887lowast is always in the perturbative region Besides theCSS prescription there were several different prescriptionsin literature In [65 66] a function 119887min(119887) decreasing withincreasing 1119876 was also introduced to match the TMDfactorization with the fixed-order collinear calculations in thevery small 119887 region

In the small 119887 region 1119876 ≪ 119887 ≪ 1ΛQCD theTMD distributions at fixed energy 120583 can be expressed asthe convolution of the perturbatively calculable coefficientsand the corresponding collinear PDFs or the multipartoncorrelation functions [22 67]

119865119902119867 (119909 119887 120583) = sum119894

119862119902larr997888119894 otimes 119865119894119867 (119909 120583) (6)

Hereotimes stands for the convolution in the momentum fraction119909

119862119902larr997888119894 otimes 1198911198941198671 (119909 120583)equiv int1

119909

119889120585120585 119862119902larr997888119894 (119909120585 119887 120583)1198911198941198671 (120585 120583) (7)

and 119891119894119867(119909 120583) is the corresponding collinear counterpart offlavor 119894 in hadron119867 at the energy scale 120583The latter one couldbe a dynamic scale related to 119887lowast by 120583119887 = 1198880119887lowast with 1198880 = 2119890minus120574119864and the Euler Constant 120574119864 asymp 0577 [22]Theperturbative hardcoefficients 119862119902larr997888119894 independent of the initial hadron typehave been calculated for the parton-target case [23 68] as theseries of (120572119904120587) and the results have been presented in [67](see also Appendix A of [23])

22 Sudakov Form Factors for the Proton and the Pion TheSudakov-like form factor 119878 in (4) can be separated into theperturbatively calculable part 119878P and the nonperturbative part119878NP

119878 = 119878P + 119878NP (8)

According to the studies in [26 39 69ndash71] the perturbativepart of the Sudakov form factor 119878119875 has the same resultamong different kinds of distribution functions ie 119878119875 isspin-independent It has the general form

119878P (119876 119887lowast)= int1198762

1205832119887

11988912058321205832 [119860 (120572119904 (120583)) ln 1198762

1205832 + 119861 (120572119904 (120583))] (9)

The coefficients 119860 and 119861 in(9) can be expanded as the seriesof 120572119904120587

119860 = infinsum119899=1

119860(119899) (120572119904120587 )119899 (10)

119861 = infinsum119899=1

119861(119899) (120572119904120587 )119899 (11)

Here we list119860(119899) to119860(2) and 119861(119899) to 119861(1) up to the accuracy ofnext-to-leading-logarithmic (NLL) order [12 23 26 69 7273]

119860(1) = 119862119865 (12)

119860(2) = 1198621198652 [119862119860(6718 minus 12058726 ) minus 109 119879119877119899119891] (13)

119861(1) = minus32119862119865 (14)

For the nonperturbative form factor 119878NP it can not beanalytically calculated by the perturbative method whichmeans it has to be parameterized to obtain the evolutioninformation in the nonperturbative region

The general form of 119878NP(119876 119887) was suggested as [12]

119878NP (119876 119887) = 1198922 (119887) ln 1198761198760 + 1198921 (119887) (15)

The nonperturbative functions 1198921(119887) and 1198922(119887) are functionsof the impact parameter 119887 and depend on the choice of 119887max


To be more specific 1198922(119887) contains the information on thelarge 119887 behavior of the evolution kernel Also accordingto the power counting analysis in [74] 1198922(119887) shall followthe power behavior as 1198872 at small-119887 region which can bean essential constraint for the parameterization of 1198922(119887)The well-known Brock-Landry-Nadolsky-Yuan (BLNY) fitparameterizes 1198922(119887) as 11989221198872 with 1198922 a free parameter [72]We note that 1198922(119887) is universal for different types of TMDsand does not depend on the particular process which is animportant prediction of QCD factorization theorems involv-ing TMDs [15 23 39 75]The nonperturbative function 1198921(119887)contains information on the intrinsic nonperturbative trans-verse motion of bound partons namely it should dependon the type of hadron and the quark flavor as well as 119909 forTMDdistributions As for theTMD fragmentation functionsitmay depend on 119911ℎ the type of the produced hadron and thequark flavor In other words 1198921(119887) depends on the specificTMDs

There are several extractions for 119878NP in literature wereview some often-used forms below

The original BLNY fit parameterized 119878NP as [72]

(1198921 + 1198922 ln( 11987621198760) + 11989211198923ln (10011990911199092)) 1198872 (16)

where 1199091 and 1199092 are the longitudinal momentum fractionsof the incoming hadrons carried by the initial state quarkand antiquark The BLNY parameterization proved to bevery reliable to describe Drell-Yan data and 119882plusmn 119885 bosonproduction [72] However when the parameterization isextrapolated to the typical SIDIS kinematics in HERMES andCOMPASS the transversemomentumdistribution of hadroncan not be described by the BLNY-type fit [76 77]

Inspired by [72 78] a widely used parameterization of119878NP for TMD distributions or fragmentation functions wasproposed [39 67 72 78ndash80]

119878pdfffNP = 1198872 (119892pdfff1 + 11989222 ln 1198761198760) (17)

where the factor 12 in front of 1198922 comes from the factthat only one hadron is involved for the parameterizationof 119878pdfffNP while the parameter in [78] is for 119901119901 collisionsThe parameter 119892pdfff

1 in (17) depends on the type of TMDswhich can be regarded as the width of intrinsic transversemomentum for the relevant TMDs at the initial energy scale1198760 [23 73 81] Assuming a Gaussian form one can obtain

119892pdf1 = ⟨1198962perp⟩11987604

119892ff1 = ⟨1199012119879⟩119876041199112

(18)

where ⟨1198962perp⟩1198760 and ⟨1199012119879⟩1198760 represent the relevant averagedintrinsic transverse momenta squared for TMD distributionsand TMD fragmentation functions at the initial scale 1198760respectively

Since the original BLNY fit fails to simultaneouslydescribe Drell-Yan process and SIDIS process in [77] theauthors proposed a new form for 119878NP which releases thetension between the BLNY fit to the Drell-Yan (such as 119882119885 and low energy Drell-Yan pair productions) data and thefit to the SIDIS data from HERMESCOMPASS in the CSSresummation formalism In addition the 119909-dependence in(16) was separated with a power law behavior assumption(1199090119909)120582 where 1199090 and 120582 are the fixed parameters as 1199090 =001 and 120582 = 02 The two different behaviors (logarithmicin (16) and power law) will differ in the intermediate 119909regime Reference [76] showed that a direct integration of theevolution kernel from low119876 to high119876 led to the form of ln(119876)term as ln(119887119887lowast)ln(119876) and could describe the SIDIS andDrell-Yan data with 119876 values ranging from a few GeV to 10 GeVThus the 1198922(119887) term was modified to the form of ln(119887119887lowast)and the functional form of 119878NP extracted in [77] turned tothe form

11989211198872 + 1198922 ln( 119887119887lowast) ln( 1198761198760)+ 11989231198872 ((11990901199091)

120582 + (11990901199092)120582)

(19)

At small 119887 region (119887 is much smaller than 119887max) the parame-terization of the 1198922(119887) term 1198922ln(119887119887lowast) can be approximatedas 1198872(21198872max) which satisfied the constraint of the 1198872 behaviorfor 1198922(119887) However at large 119887 region the logarithmic behaviorwill lead to different predictions on the 1198762 dependencesince the Gaussian-type parameterization suggests that it isstrongly suppressed [82] This form has been suggested inan early research by Collins and Soper [83] but has notyet been adopted in any phenomenological study until thestudy in [77] The comparison between the original BLNYparameterization and this form with the experimental dataof Drell-Yan type process has shown that the new form of119878NP can fit with the data as equally well as the original BLNYparameterization

In [66] the 1198922(119887) function was parameterized as 11989221198872 fol-lowing the BLNY convention Furthermore in the function1198921(119887) the Gaussian width also depends on 119909 The authorssimultaneously fit the experimental data of SIDIS processfromHERMES and COMPASSCollaborations the Drell-Yanevents at low energy and the119885 boson production with totally8059 data points The extraction can describe the data well inthe regions where TMD factorization is supposed to hold

To study the pion-nucleon Drell-Yan data it is alsonecessary to know the nonperturbative Sudakov form factorfor the pion meson In [59] we extended the functional formfor the proton TMDs [77] to the case of the pion TMDs

1198781198911199021205871NP = 1198921199021205871 1198872 + 1198921199021205872 ln 119887119887lowast ln1198761198760 (20)

with 1198921199021205871 and 1198921199021205872 the free parameters Adopting the func-tional form of 119878NP in (20) for the first time we performedthe extraction [59] of the nonperturbative Sudakov formfactor for the unpolarized TMD PDF of pion meson using


the experimental data in the 120587minus119901 Drell-Yan process collectedby the E615 Collaboration at Fermilab [7 84]The data fittingwas performed by the package MINUIT [85 86] through aleast-squares fit

1205942 (120572) = 119872sum119894=1

119873119894sum119895=1

(theo (119902perp119894119895120572) minus data119894119895)2err2119894119895

(21)

The total number of data in our fit is 119873 = sum8119894 119873119894 = 96 Sincethe TMD formalism is valid in the region 119902perp ≪ 119876 we dida simple data selection by removing the data in the region119902perp gt 3 GeV We performed the fit by minimizing the chi-square in (21) and we obtained the following values for thetwo parameters

1198921199021205871 = 0082 plusmn 00221198921199021205872 = 0394 plusmn 0103 (22)

with 1205942dof = 164Figure 1 plots the 119902perp-dependent differential cross section

(solid line) calculated from the fitted values for 1198921199021205871 and 1198921199021205872in (22) at the kinematics of E615 at different 119909119865 bins The fullsquares with error bars denote the E615 data for comparisonAs Figure 1 demonstrates a good fit is obtained in the region119909119865 lt 08

From the fitted result we find that the value of theparameter 1198921199021205871 is smaller than the parameter 1198921199021199011 extractedin [77] which used the same parameterized form For theparameter 1198921199021205872 we find that its value is very close to that ofthe parameter 1198921199021199012 for the proton [77] (here 1198921199021199012 = 11989222 =042) This may confirm that 1198922 should be universal eg 1198922is independent on the hadron type Similar to the case of theproton for the pion meson 1198921205872 is several times larger than 1198921205871 We note that a form of 1198781198911119902120587NP motivated by the NJLmodel wasgiven in [87]

23 Solutions for Different TMDs After solving the evolutionequations and incorporating the Sudakov form factor thescale-dependent TMD distribution function 119865 of the protonand the pion in 119887 space can be rewritten as

119865119902119901 (119909 119887 119876)= 119890minus(12)119878P(119876119887lowast)minus119878119865119902119901NP (119876119887)F (120572119904 (119876))sum

119894

119862119902larr997888119894otimes 119865119894119901 (119909 120583119887)

(23)


119894

119862119902larr997888119894otimes 119865119894120587 (119909 120583119887)

(24)

Here 119865119894119867(119909 120583119887) is the corresponding collinear distributionsat the initial energy scale 120583119887 To be more specific for the

unpolarized distribution function 1198911119902119867 and transversity dis-tribution function ℎ1119902119867 the collinear distributions 119865119894119867(119909120583119887) are the integrated distribution functions 1198911119902119867(119909 120583119887) andℎ1119902119867(119909 120583119887) As for the Boer-Mulders function and Siversfunction the collinear distributions are the correspondingmultiparton correlation functions Thus the unpolarizeddistribution function of the proton and pion in 119887 space canbe written as

1198911119902119901 (119909 119887 119876)= 119890minus(12)119878pert(119876119887lowast)minus1198781198911119902119901NP (119876119887)

F (120572119904 (119876))sum119894

119862119902larr997888119894otimes 1198911119894119901 (119909 120583119887)

(25)


F (120572119904 (119876))sum119894

119862119902larr997888119894otimes 1198911119894120587 (119909 120583119887)

(26)

If we perform a Fourier transformation on the 1198911119902119867(119909 119887 119876)we can obtain the distribution function in 119896perp space as

1198911119902119901 (119909 119896perp 119876) = intinfin0

1198891198871198872120587 1198690 (119896perp119887) 1198911119902119901 (119909 119887 119876) (27)

1198911119902120587 (119909 119896perp 119876) = intinfin0

1198891198871198872120587 1198690 (119896perp119887) 1198911119902120587 (119909 119887 119876) (28)

where 1198690 is the Bessel function of the first kind and 119896perp = |kperp|Similarly the evolution formalism of the proton transver-

sity distribution in 119887 space and 119896perp-space can be obtained as[75]

ℎ1119902119901 (119909 119887 119876)= 119890minus(12)119878P(119876119887lowast)minus119878ℎ1NP(119876119887)H (120572119904 (119876))sum

119894

120575119862119902larr997888119894otimes ℎ1119894119901 (119909 120583119887)

(29)

ℎ1119902119901 (119909 119896perp 119876) = intinfin0

1198891198871198872120587 1198690 (119896perp119887) ℎ1119902119901 (119909 119887 119876) (30)

whereH is the hard factor and 120575119862119902larr997888119894 is the coefficient con-voluted with the transversity The TMD evolution formalismin (30) has been applied in [75] to extract the transversitydistribution from the SIDIS data

The Sivers function and Boer-Mulders functionwhich are T-odd can be expressed as follows in 119887-space[39]

119891perp120572(DY)1119879119902119867 (119909 119887 120583 120577119865)= int1198892kperp119890minus119894997888rarrk perpsdot997888rarrb 119896120572perp119872119901

119891perp(DY)1119879119902119867 (119909 kperp 120583)


01

001

1E-300 06 12 18 24 30

Ｋperp (GeV)00 06 12 18 24 30

Ｋperp (GeV)

00 06 12 18 24 30

Ｋperp (GeV)00 06 12 18 24 30

Ｋperp (GeV)

00 06 12 18 24 30

Ｋperp (GeV)00 06 12 18 24 30

Ｋperp (GeV)

00 06 12 18 24 30

Ｋperp (GeV)00 06 12 18 24 30

Ｋperp (GeV)

＞2

＞Ｋperp＞Ｒ

(nb

GeV

nuc

leon

)

01

001

1E-3

＞2

＞Ｋperp＞Ｒ

(nb

GeV

nuc

leon

)

01

001

1E-3

＞2

＞Ｋperp＞Ｒ

(nb

GeV

nuc

leon

)

01

001

1E-3

01

001

1E-3

01

001

1E-3

01

001

1E-3

01

001

1E-3

＞2

＞Ｋperp＞Ｒ

(nb

GeV

nuc

leon

)

0ltＲlt01 01ltＲlt02




Figure 1 The fitted cross section (solid line) of pion-nucleon Drell-Yan as functions of 119902perp compared with the E615 data (full square) fordifferent 119909119865 bins in the range 0 lt 119909119865 lt 08 The error bars shown here include the statistical error and the 16 systematic error Figure from[59]

ℎperp120572(DY)1119902119867 (119909 119887 120583 120577119865)= int 1198892kperp119890minus119894997888rarrk perpsdot997888rarrb 119896120572perp119872119901

ℎperp(DY)1119902119867 (119909 kperp 120583) (31)

Here the superscript ldquoDYrdquo represents the distributions inthe Drell-Yan process Since QCD predicts that the sign ofthe distributions changes in the SIDIS process and Drell-Yanprocess for the distributions in SIDIS process there has to bean extra minus sign regard to 119891perp(DY)1119879119902119867 and ℎperp(DY)1119902119867


Similar to what has been done to the unpolarized dis-tribution function and transversity distribution function inthe low 119887 region the Sivers function 119891perp120572(DY)

1119879119902119867can also be

expressed as the convolution of perturbatively calculablehard coefficients and the corresponding collinear correlationfunctions as [69 88]

119891perp120572(DY)1119879119902119867 (119909 119887 120583)= (minus1198941198871205722 )sum

119894

Δ119862119879119902larr997888119894 otimes 119891(3)119894119901 (1199091015840 11990910158401015840 120583) (32)

Here119891(3)119894119901

(1199091015840 11990910158401015840) denotes the twist-three quark-gluon-quarkor trigluon correlation functions amongwhich the transversespin-dependent Qiu-Sterman matrix element 119879119902119865(1199091015840 11990910158401015840)[89ndash91] is the most relevant one Assuming that the Qiu-Sterman function 119879119902119865(119909 119909) is the main contribution theSivers function in 119887-space becomes

119891perp120572(DY)1119879119902119867 (119909 119887 119876) = (minus1198941198871205722 )FSiv (120572119904 (119876))sum119894

Δ119862119879119902larr997888119894otimes 119879119894119867119865 (119909 119909 120583119887) 119890minus119878sivNPminus(12)119878P

(33)

where FSiv is the factor related to the hard scattering TheBoer-Mulders function in 119887-space follows the similar resultfor the Sivers function as

ℎ120572perp(DY)1119902119867 (119909 119887 119876) = (minus1198941198871205722 )HBM (120572119904 (119876))sum119894

119862BM119902larr997888119894

otimes 119879(120590)119894119867119865 (119909 119909 120583119887) 119890minus119878BMNP minus(12)119878P (34)

Here 119862BM119902larr997888119894 stands for the flavor-dependent hard coefficients

convoluted with 119879(120590)119894119867119865

HBM the hard scattering factor and119879(120590)119894119867119865

(119909 119909 120583119887) denotes the chiral-odd twist-3 collinear corre-lation function After performing the Fourier transformationback to the transverse momentum space one can get theSivers function and the Boer-Mulders function as

119896perp119872119867

119891perp1119879119902119867 (119909 119896perp 119876)= intinfin

0119889119887( 11988722120587) 1198691 (119896perp119887)FSiv (120572119904 (119876))sum

119894

Δ119862119879119902larr997888119894otimes 119891perp(1)1119879119894119867 (119909 120583119887) 119890minus119878sivNPminus(12)119878P

(35)

119896perp119872119867

ℎperp1119902119867 (119909 119896perp 119876)= intinfin

0119889119887( 11988722120587) 1198691 (119896perp119887)HBM (120572119904 (119876))sum

119894

119862BM119902larr997888119894

otimes ℎperp(1)1119894119867 (119909 120583119887) 119890minus119878BMNP minus(12)119878P (36)

and 119879119902119865(119909 119909 120583119887) and 119879(120590)119894119867119865

(119909 119909 120583119887) are related to Siversfunction and Boer-Mulders function as [69 88]

119879119902119867119865 (119909 119909) = int1198892119896perp100381610038161003816100381610038161198962perp10038161003816100381610038161003816119872119867

119891perp(DY)1119879119902119867 (119909 119896perp)= 2119872119867119891perp(1)(DY)1119879119902119867 (119909)

(37)

119879(120590)119902119867119865 (119909 119909) = int1198892119896perp100381610038161003816100381610038161198962perp10038161003816100381610038161003816119872119867

ℎperp(DY)1119902119867 (119909 119896perp)= 2119872119867ℎperp(1)(DY)1119902119867 (119909)

(38)

The TMD evolution formalism in (35) has been appliedto extract [39 70 81 92 93] the Sivers function The similarformalism in (36) could be used to improve the previousextractions of the proton Boer-Mulders function [62 94ndash96]and future extraction of the pion Boer-Mulders function

3 Physical Observables in 120587119873 Drell-YanProcess within TMD Factorization

In this section we will set up the necessary framework forphysical observables in 120587-119873 Drell-Yan process within TMDfactorization by considering the evolution effects of the TMDdistributions following the procedure developed in [15]

In Drell-Yan process

119867119860 (119875120587) + 119867119861 (119875119873) 997888rarr120574lowast (119902) + 119883 997888rarr

119897+ (ℓ) + 119897minus (ℓ1015840) + 119883(39)

119875120587119873 and 119902 denote the momenta of the incoming hadron 120587119873and the virtual photon respectively 119902 is a time-like vectornamely 1198762 = 1199022 gt 0 which is the invariant mass square ofthe final-state lepton pair One can define the following usefulkinematical variables to express the cross section

119904 = (119875120587 + 119875119873)2 119909120587119873 = 11987622119875120587119873 sdot 119902 119909119865 = 2119902119871119904 = 119909120587 minus 119909119873120591 = 1198762119904 = 119909120587119909119873119910 = 12 ln

119902+119902minus = 12 ln 119909120587119909119873

(40)

where 119904 is the center-of-mass energy squared 119909120587119873 is thelight-front momentum fraction carried by the annihilatingquarkantiquark in the incoming hadron 120587119873 119902119871 is thelongitudinal momentum of the virtual photon in the cmframe of the incident hadrons 119909119865 is the Feynman 119909 variable


which corresponds to the longitudinal momentum fractioncarried by the lepton pair and 119910 is the rapidity of the leptonpairThus 119909120587119873 is expressed as the function of 119909119865 120591 and of 119910120591

119909120587119873 = plusmn119909119865 + radic1199092119865 + 41205912

119909120587119873 = radic120591119890plusmn119910(41)

31 Differential Cross Section for Unpolarized Drell-Yan Pro-cess The differential cross section formulated in TMD fac-torization is usually expressed in the 119887-space to guaranteeconservation of the transverse momenta of the emitted softgluons Later on it can be transformed back to the transversemomentum space to represent the experimental observablesWe will introduce the physical observables in the followingpart of this section

The general differential cross section for the unpolarizedDrell-Yan process can be written as [12]

119889412059011988011988011988911987621198891199101198892qperp = 1205900 int 1198892119887(2120587)2 119890119894

997888rarrq perp sdot997888rarrb 119880119880 (119876 119887)

+ 119884119880119880 (119876 119902perp)(42)

where 1205900 = 4120587120572211989011989831198731198621199041198762 is the cross section at treelevel with 120572119890119898 the fine-structure constant (119876 119887) is thestructure function in the 119887-space which contains all-orderresummation results and dominates in the low 119902perp region(119902perp ≪ 119876) and the 119884 term provides necessary correction at119902perp sim 119876 In this workwewill neglect the119884-term whichmeansthat we will only consider the first term on the rhs of (42)

In general TMD factorization [15] aims at separatingwell-defined TMD distributions such that they can be usedin different processes through a universal way and expressingthe schemeprocess dependence in the corresponding hardfactors Thus (119876 119887) can be expressed as [97]

119880119880 (119876 119887) = 119867119880119880 (119876 120583)sdot sum119902119902

1198902119902119891sub119902120587 (119909120587 119887 120583 120577119865) 119891sub

119902119901 (119909119901 119887 120583 120577119865) (43)

where 119891sub119902119867 is the subtracted distribution function in the119887 space and 119867119880119880(119876 120583) is the factor associated with hard

scattering The superscript ldquosubrdquo represents the distributionfunction with the soft factor subtracted The subtractionguarantees the absence of light-cone singularities in theTMDs and the self-energy divergencies of the soft factors[15 22] However the way to subtract the soft factor inthe distribution function and the hard factor 119867119880119880(119876 120583)depends on the scheme to regulate the light-cone singularityin the TMD definition [12 14 15 22 98ndash103] leading to thescheme dependence in the TMD factorization In literatureseveral different schemes are used [97] the CSS scheme[12 22] the Collins-11 (JCC) scheme [15] the Ji-Ma-Yuan(JMY) scheme [13 14] and the lattice scheme [103] Althoughdifferent schemes are adopted the final results of the structure

functions (119876 119887) as well as the differential cross sectionshould not depend on a specific scheme In the following wewill apply the JCC and JMY schemes to display the scheme-independence of the unpolarized differential cross section

The hard 119867119880119880(119876 120583) have different forms in the JCC andJMY schemes

119867JCC (119876 120583) = 1 + 120572119904 (120583)2120587 119862119865 (3 ln 11987621205832 minus ln211987621205832 + 1205872

minus 8) (44)

119867JMY (119876 120583 120588) = 1 + 120572119904 (120583)2120587 119862119865 ((1 + ln 1205882) ln 11987621205832minus ln 1205882 + ln2120588 + 21205872 minus 4)

(45)

Like 120577119865 here 120588 is another variable to regulate the light-conesingularity of TMD distributions The scheme dependenceof the distribution function is manifested in the hard fac-tor F(120572119904(119876)) which has the following forms in differentschemes

FJCC (120572119904 (119876)) = 1 + O (1205722119904 ) (46)

FJMY (120572119904 (119876) 120588)= 1 + 1205721199042120587119862119865 [ln 120588 minus 12 ln2120588 minus 12058722 minus 2] (47)

The119862 coefficients in (25) and (26) do not depend on the typesof initial hadrons and are calculated for the parton-target case[23 68] with the results presented in [67] (see also AppendixA of [23])

119862119902larr9978881199021015840 (119909 119887 120583 120577119865)= 1205751199021199021015840 [120575 (1 minus 119909) + 120572119904120587 (1198621198652 (1 minus 119909))] (48)

119862119902larr997888119892 (119909 119887 120583 120577119865) = 120572119904120587 119879119877119909 (1 minus 119909) (49)

where 119862119865 = (1198732119862 minus 1)(2119873119862) 119879119877 = 12One can absorb the scheme-dependent hard factors119867119880119880(119876 120583) and F of the TMD distributions into the 119862-

functions using

1198621015840119895larr997888119894 = 119862119895larr997888119894 timesF times radic119867119880119880 (119876 120583 = 119876) (50)

The results for the splitting to quark are

1198621015840119902larr9978881199021015840 (119909 119887 120583119887) = 1205751199021199021015840 [120575 (1 minus 119909)+ 120572119904120587 (1198621198652 (1 minus 119909) + 1198621198654 (1205872 minus 8) 120575 (1 minus 119909))]

(51)

1198621015840119902larr997888119892 (119909 119887 120583119887) = 120572119904120587 119879119877119909 (1 minus 119909) (52)

The new 119862-coefficients turn out to be scheme independent(independent on 120588) [104] but process dependent [105 106]


With the new 119862-coefficients in hand one can obtain thestructure functions 119880119880(119876 119887) in 119887-space as

119880119880 (119876 119887) = 119890minus119878pert(1198762 119887)minus1198781198911199021205871NP (1198762 119887)minus1198781198911199021199011NP (1198762 119887)timessum119902119902

11989021199021198621015840119902larr997888119894 otimes 119891119894120587minus (1199091 120583119887)1198621015840119902larr997888119895otimes 119891119895119901 (1199092 120583119887)

(53)

After performing the Fourier transformation we can get thedifferential cross section as

119889412059011988911987621198891199101198892qperp = 1205900 intinfin0

1198891198871198872120587 1198690 (119902perp119887) times 119880119880 (119876 119887) (54)

where 1198690 is the zeroth order Bessel function of the first kind

32 The Sivers Asymmetry In the Drell-Yan process with a 120587beam colliding on the transversely polarized nucleon targetan important physical observable is the Sivers asymmetry asit can test the sign change of the Sivers function betweenSIDIS and Drell-Yan processes a fundamental predictionin QCD The future precise measurement of the Siversasymmetry in 120587119873 Drell-Yan in a wide kinematical regioncan be also used to extract the Sivers function The Siversasymmetry is usually defined as [39]

119860119880119879 = 1198894Δ12059011988911987621198891199101198892qperp119889412059011988911987621198891199101198892qperp (55)

where 119889412059011988911987621198891199101198892qperp and 1198894Δ12059011988911987621198891199101198892qperp are the spin-averaged (unpolarized) and spin-dependent differential crosssection respectivelyThe latter one has the general form in theTMD factorization [39 69 88]

1198894Δ12059011988911987621198891199101198892qperp = 1205900120598120572120573perp 119878120572perp int 1198892119887(2120587)2 119890119894

997888rarrq perp sdot997888rarrb 120573

119880119879 (119876 119887)+ 119884120573119880119879 (119876 119902perp)

(56)

Similar to (42) 119880119879(119876 119887) denotes the spin-dependent struc-ture function in the 119887-space and dominates at 119902perp ≪ 119876and 119884120573119880119879 provides correction for the single-polarized processat 119902perp sim 119876 The antisymmetric tensor 120598120572120573perp is defined as120598120572120573120583]119875120583120587119875]

119901119875120587 sdot119875119901 and 119878perp is the transverse-polarization vectorof the proton target

The structure function 119880119879(119876 119887) can be written in termsof the unpolarized distribution function of pion and Siversfunction of proton as

120572119880119879 (119876 119887) = 119867119880119879 (119876 120583)sdot sum119902119902

11989021199021198911119902120587 (119909120587 119887 120583 120577119865) 119891perp120572(DY)1119879119902119901 (119909119901 119887 120583 120577119865) (57)

with 119891perp120572(DY)1119879119902119901

(119909119901 119887 120583 120577119865) given in (33) Similar to the unpo-larized case the scheme-dependent hard factors can beabsorbed into the 119862-coefficients leading to [69 88]

Δ119862119879119902larr9978881199021015840 (119909 119887 120583119887) = 1205751199021199021015840 [120575 (1 minus 119909)+ 120572119904120587 (minus 14119873119888 (1 minus 119909) + 1198621198654 (1205872 minus 8) 120575 (1 minus 119909))]

(58)

The spin-dependent differential cross section in (56) thushas the form

1198894Δ12059011988911987621198891199101198892qperp = 12059004120587 intinfin0

11988911988711988721198691 (119902perp119887) sum119902119894119895

1198902119902Δ119862119879119902larr997888119894otimes 119879119894119865 (119909119901 119909119901 120583119887)119862119902larr997888119895otimes 1198911119895120587 (119909120587 120583119887) 119890minus(119878SivNP+1198781198911119902120587NP +119878P)

(59)

Combing (55) (54) (59) one can get the Sivers asymmetryin the Drell-Yan process with a 120587 beam colliding on atransversely polarized proton target

33 The cos 2120601 Asymmetry in the Unpolarized Drell-Yan fromDouble Boer-Mulders Effect The angular differential crosssection for unpolarized Drell-Yan process has the followinggeneral form

1120590 119889120590119889Ω = 34120587 1120582 + 3 (1 + 120582 cos2120579 + 120583 sin 2120579 cos 120601+ ]2 sin2120579 cos 2120601)

(60)

where 120579 is the polar angle and 120601 is the azimuthal angle ofthe hadron plane with respect to the dilepton plane in theCollins-Soper (CS) frame [107]The coefficients 120582 120583 ] in (60)describe the sizes of different angular dependencies Partic-ularly ] stands for the asymmetry of the cos 2120601 azimuthalangular distribution of the dilepton

The coefficients 120582 120583 ] have been measured in the process120587minus119873 997888rarr 120583+120583minus119883 by the NA10 Collaboration [5 6] and theE615 Collaboration [7] for a 120587minus beam with energies of 140194 286 GeV and 252 GeV with119873 denoting a nucleon in thedeuterium or tungsten target The experimental data showeda large value of ] near 30 in the region 119876119879 sim 3 GeV Thisdemonstrates a clear violation of the Lam-Tung relation [47]In the last decade the angular coefficients were also measuredin the 119901119873 Drell-Yan processes in both the fixed-target mode[108 109] and collider mode [48 49] The origin of largecos 2120601 asymmetryndashor the violation of the Lam-Tung relationndashobserved in Drell-Yan process has been studied extensivelyin literature [30 50ndash57 110ndash114] Here we will only considerthe contribution from the coupling of two Boer-Muldersfunctions from each hadron denoted by ]BM It might bemeasured through the combination 2]BM asymp 2] + 120582 minus 1 inwhich the perturbative contribution is largely subtracted

The cos 2120601 asymmetry coefficient ]BM contributed by theBoer-Mulders function can be written as]BM

= 2sum119902F [(2h sdot kperph sdot pperp minus kperp sdot pperp) (ℎperp1119902120587ℎperp1119902119901119872120587119872119901)]sum119902F [11989111199021205871198911119902119901] (61)


1

01

001

1E-3

1E-4

1Ｏ

(xＥ

perpQ

)

0 1 2 3 4

b (Ge６-1)

Ｒ = 01

１2 = 24Ge６2

１2 = 10Ge６2

１2 = 1000Ge６2

１2 = 24Ge６2

１2 = 10Ge６2

１2 = 1000Ge６2

1 2 3 4 5 60

Ｅperp (GeV)

0

1

2

3

4

5

＜Ｍ

Ｏ＜

1Ｏ

(xb

Q)

Figure 2 Subtracted unpolarized TMD distribution of the pion meson for valence quarks in 119887-space (left panel) and 119896perp-space (right panel)at energies1198762 = 24 GeV2 (dotted lines) 1198762 = 10 GeV2 (solid lines) and 1198762 = 1000 GeV2 (dashed lines) From [59]

where the notation

F [120596119891119891] = 1198902119902 int1198892kperp1198892pperp1205752 (kperp + pperp minus qperp)sdot 120596119891 (119909120587 k2perp)119891 (119909119901 p2perp)

(62)

has been adopted to express the convolution of transversemomenta qperp kperp and pperp are the transverse momenta ofthe lepton pair quark and antiquark in the initial hadronsrespectively h is a unit vector defined as h = qperp|qperp| =qperp119902perp One can perform the Fourier transformation from qperpspace to b space on the delta function in the notation of (62)to obtain the denominator in (61) as

F [11989111199021205871198911119902119901] = sum119902

1198902119902 int 1198892119887(2120587)2

sdot int 1198892kperp1198892pperp119890119894(q119879minuskperpminuspperp)sdotb1198911119902120587 (119909120587 k2perp)sdot 1198911119902119901 (119909119901 p2perp) = sum

119902

1198902119902sdot int 1198892119887

(2120587)2 119890119894qperp sdotb1198911119902120587 (119909120587 119887 119876) 1198911119902119901 (119909119901 119887 119876)

(63)

where the unpolarized distribution function in 119887 space isgiven in (25) and (26) Similar to the treatment of the denom-inator using the expression of the Boer-Mulders function in(34) the numerator can be obtained as

F[(2h sdot kperph sdot pperp minus kperp sdot pperp) ℎperp1119902120587ℎperp1119902119901119872120587119872119901

] = sum119902

1198902119902

sdot int 1198892119887(2120587)2 int1198892kperp1198892pperp119890119894(q119879minuskperpminuspperp)sdot119887 [(2h sdot kperph

sdot pperp minus kperp sdot pperp) ℎperp1119902120587ℎperp1119902119901119872120587119872119901

] = sum119902

1198902119902

sdot int 1198892119887(2120587)2 119890119894qperpsdotb (2ℎ120572ℎ120573 minus 119892perp120572120573) ℎ120572perp1119902120587 (119909120587 119887 119876)

sdot ℎ120573perp1119902119901

(119909119901 119887 119876) = sum119902

1198902119902 intinfin0

11988911988711988738120587 1198692 (119902perp119887)sdot 119879(120590)119902120587119865 (119909120587 119909120587 120583119887) 119879(120590)119902119901119865 (119909119901 119909119901 120583119887)sdot 119890minus(1198781198911119902119901NP +119878

1198911119902120587NP +119878P)

(64)

Different from the previous two cases the hard coefficients119862BM119894 and HBM for the Boer-Mulders function have not

been calculated up to next-to-leading order (NLO) and stillremain in leading order (LO) as 119862119902larr997888119894 = 120575119902119894120575(1 minus 119909) andH = 14 Numerical Estimate for theTMD Distributions

Based on the TMD evolution formalism for the distributionsset up in Section 2 we will show the numerical results forthe TMD distributions Particularly attention will be paid onthose of the pion meson as those of the proton have beenstudied numerically in [23 71 77]

41The Unpolarized TMDDistribution of the PionMeson In[59] the authors applied (26) and the extracted parameters1198921205871 and 1198921205872 to quantitatively study the scale dependence ofthe unpolarized TMD distributions of the pion meson withthe JCC Scheme For the collinear unpolarized distributionfunction of the pionmeson theNLOSMRSparameterization[115] was chosen The results are plotted in Figure 2 with theleft and right panels showing the subtracted distribution in119887 space and 119896perp space for fixed 119909120587 = 01 at three differentenergy scales 1198762 = 24 GeV2 (dotted line) 10 GeV2 (solidline) 1000 GeV2 (dashed line) From the 119887-dependent plots


one can see that at the highest energy scale 1198762 = 1000 GeV2the peak of the curve is in the low 119887 region where 119887 lt119887max since in this case the perturbative part of the Sudakovform factor dominates However at lower energy scales eg1198762 = 10 GeV2 and 1198762 = 24 GeV2 the peak of the 119887-dependent distribution function moves towards the higher119887 region indicating that the nonperturbative part of theTMD evolution becomes important at lower energy scalesFor the distribution in 119896perp space at higher energy scale thedistribution has a tail falling off slowly at large 119896perp while atlower energy scales the distribution function falls off rapidlywith increasing 119896perp It is interesting to point out that the shapesof the pion TMD distribution at different scales are similar tothose of the proton namely Fig 8 in [77]

42 The Sivers Function of the Proton The scale dependenceof the T-odd distributions such as the Sivers function andthe Boer-Mulders function is more involved than that of theT-even distributions This is because their collinear counter-parts are the twist-3 multiparton correlation functions [3960 61 69 88] for which the exact evolution equations are farmore complicated than those for the unpolarized distributionfunction In numerical calculation some approximations onthe evolution kernels are usually adopted

In [39] the Qiu-Sterman function was assumed to beproportional to 1198911 namely it follows the same evolutionkernel as that for 1198911 A different choice was adopted in [60]where the homogenous terms of the exact evolution kernelfor the Qiu-Sterman function [88 116ndash124] were included todeal with the scale dependence of Qiu-Sterman function

119875QS119902119902 asymp 1198751198911119902119902 minus 1198731198882 1 + 11991121 minus 119911 minus 119873119888120575 (1 minus 119911) (65)

with 1198751198911119902119902 the evolution kernel of the unpolarized PDF

1198751198911119902119902 = 43 ( 1 + 1199112(1 minus 119911)+ + 32120575 (1 minus 119911)) (66)

To solve the QCD evolution numerically we resort to theQCD evolution package HOPPET [125] and we customthe code to include the splitting function in (65) For acomparison in Figure 3 we plot the TMD evolution of theSivers function for proton in 119887 space and the 119896perp space usingthe above-mentioned two approaches [60] In this estimatethe next leading order 119862-coefficients Δ119862119879119902larr997888119894 was adoptedfrom [69 88] and the nonperturbative Sudakov form factorfor the Sivers function of proton was adopted as the formin (17) The Sivers functions are presented at three differentenergy scales 1198762 = 24 GeV2 1198762 = 10 GeV2 and 1198762 =100 GeV2 Similar to the result for 1198911 one can concludefrom the curves that the perturbative Sudakov form factordominated in the low 119887 region at higher energy scales andthe nonperturbative part of the TMD evolution becamemoreimportant at lower energy scales However the 119896perp tendencyof the Sivers function in the two approaches is differentwhich indicates that the scale dependence of theQiu-Stermanfunction may play a role in the TMD evolution

43 The Boer-Mulders Function of the Pion Meson Theevolution of the Boer-Mulders function for the valence quarkinside120587meson has been calculated from (34) and (36) in [61]in which the collinear twist-3 correlation function 119879(120590)119902119865 at theinitial energy scalewas obtained by adopting amodel result oftheBoer-Mulders function of the pionmeson calculated fromthe light-cone wave functions [126] For the scale evolution of119879(120590)119902119865 the exact evolution effect has been studied in [116] Forour purpose we only consider the homogenous term in theevolution kernel

119875119879(120590)119902119865119902119902 (119909) asymp Δ119879119875119902119902 (119909) minus 119873119862120575 (1 minus 119909) (67)

with Δ119879119875119902119902(119909) = 119862119865[2119911(1 minus 119911)+ + (32)120575(1 minus 119909)] beingthe evolution kernel for the transversity distribution functionℎ1(119909) We customize the original code of QCDNUM [127] toinclude the approximate kernel in (67) For the nonpertur-bative part of the Sudakov form factor associated with Boer-Mulders function the information still remains unknownThe assumption that 119878NP for Boer-Mulders function is sameas that for 1198911 can be a practical way to access the informationof TMD evolution for Boer-Mulders function

We plot the 119887-dependent and 119896119879-dependent Boer-Mulders function at 119909 = 01 in the left and right pan-els of Figure 4 respectively In calculating ℎperp1119902120587(119909 119887 119876) inFigure 4 we have rewritten the Boer-Mulders function in 119887space as

ℎperp1119902120587 (119909 119887 119876) = 119894119887120572120587 ℎ120572perp1119902120587 (119909 119887 119876) (68)

The three curves in each panel correspond to three differentenergy scales 1198762 = 025GeV2 (solid lines) 1198762 = 10GeV2(dashed lines) 1198762 = 1000GeV2 (dotted lines) From thecurves we find that the TMD evolution effect of the Boer-Mulders function is significant and should be considered inphenomenological analysis The result also indicates that theperturbative Sudakov form factor dominates in the low 119887region at higher energy scales and the nonperturbative part ofthe TMD evolution becomes more important at lower energyscales

In conclusion we find that the tendency of the distri-butions is similar the distribution is dominated by pertur-bative region in 119887 space at large 1198762 while at lower 1198762 thedistribution shifts to the large 119887 region indicating that thenonperturbative effects of TMDevolution become importantFor the distributions in 119896perp space as the value of1198762 increasesthe distributions become wider with a perturbative tail whileat low values of 1198762 the distributions resemble Gaussian-type parameterization However the widths of the transversemomentum differ among different distributions

5 Numerical Estimate for the PhysicalObservables in 120587-119873 Drell-Yan Process

Based on the general TMD factorization framework providedin Section 3 we present several physical observables in 120587-119873Drell-Yan process in this section


１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

x=01

x=01x=01

x=01

1

01

001

1E-3

1E-4

Ｅperp

－Ｊ

perp 1４ＫＪ

(xＥ

perpQ

)

1

01

001

1E-3

1E-4

Ｅperp

－Ｊ

perp 1４ＫＪ

(xＥ

perpQ

)

00

02

04

06

08

10

perp 1４ＫＪ

(xb

Q)

00

02

04

06

08

10 perp 1

４ＫＪ

(xb

Q)

1 2 3 4 50 6

b (Ge６-1)1 2 3 4 50 6

b (Ge６-1)

05 10 15 20 2500

Ｅperp (GeV)05 10 15 20 2500

Ｅperp (GeV)

Figure 3 Subtracted Sivers function for the up quarks in Drell-Yan in 119887-space (upper panels) and 119896perp-space (lower panels) at energies1198762 = 24 GeV2 (solid lines) 1198762 = 10 GeV2 (dashed lines) and 1198762 = 100 GeV2 (dotted lines) The left panel shows the result from the sameevolution kernel as that for 1198911 in (66) the right panel shows the result from an approximate evolution kernel for the Qiu-Sterman functionin (65) respectively Figure from [60]

QCD predicts that the T-odd PDFs present generalizeduniversality ie the sign of the Sivers function measured inDrell-Yan process should be opposite to its sign measuredin SIDIS [16 34 35] process The verification of this signchange [36ndash41] is one of the most fundamental tests of ourunderstanding of the QCD dynamics and the factorizationscheme and it is also the main pursue of the existing andfuture Drell-Yan facilities [10 11 42ndash45]TheCOMPASS Col-laboration has reported the first measurement of the Siversasymmetry in the pion-induced Drell-Yan process in whicha 120587minus beam was scattered off the transversely polarized NH3target [11] The polarized Drell-Yan data from COMPASStogether with the previous measurement of the Sivers effectin the 119882- and 119885-boson production from 119901uarr119901 collision atRHIC [45] will provide the first evidence of the sign change of

the Sivers function As COMPASS experiment has almost thesame setup [11 46] for SIDIS and Drell-Yan process it willprovide the unique chance to explore the sign change sincethe uncertainties in the extraction of the Sivers function fromthe two kinds of measurements can be reduced

51 The Normalized Cross Section for Unpolarized 120587-119873 Drell-Yan Process The very first step to understand the Siversasymmetry in the 120587-119873 Drell-Yan process is to quantitativelyestimate the differential cross section in the same processfor unpolarized nucleon target with high accuracy sinceit always appears in the denominator of the asymmetrydefinition The differential cross section for unpolarizedDrell-Yan process has been given in (54) Applying theextracted nonperturbative Sudakov form factor for pion


1

01

001

1E-3

１2=025 Ge６2

１2=10 Ge６2

１2=1000 Ge６2

１2=025 Ge６2

１2=10 Ge６2

１2=1000 Ge６2

x=01 x=01

Ｅ４

－

Ｂperp 1Ｋ

(xＥ

４Q

)

1 2 3 4 50

Ｅ４ (GeV)

000

005

010

015

020

Ｂperp 1Ｋ

(xb

Q)

1 2 3 4 50

b (Ge６-1)

Figure 4The Boer-Mulders function for 119906-quark in 119887 space (left panel) and 119896119879 space (right panel) considering three different energy scales1198762 = 24GeV2 (solid lines)1198762 = 10GeV2 (dashed lines) and 1198762 = 1000GeV2 (dotted lines) Figures from [61]

02

00

06

08

04

1

＞

＞Ｋperp

(Ge６

-1)

Experiment measurementTheoretical estimate

COMPASS 2015 minus-N（3 data43 GeV lt M lt85 GeV

05 1000 20 25 3015

Ｋperp (GeV)

Figure 5 The transverse spectrum of lepton pair production in the unpolarized pion-nucleon Drell-Yan process with an NH3 target atCOMPASSThe dashed line represents the theoretical calculation in [59] The solid line shows the experimental measurement at COMPASS[11] Figure from [59]

meson in [59] and the extracted 119878NP for nucleon in [77] weestimated the normalized transverse momentum spectrumof the dimuon production in the pion-nucleon Drell-Yanprocess at COMPASS for different 119902perp bins with an intervalof 02 GeV The result is plotted in Figure 5 From the curvesone can find the theoretical estimate on the 119902perp distributionof the dimuon agreed with the COMPASS data fairly well inthe small 119902perp region where the TMD factorization is supposedto hold The comparison somehow confirms the validity ofextraction of the nonperturbative Sudakov form factor forthe unpolarized distribution 1198911120587 of pion meson within theTMD factorizationThismay indicate that the framework canalso be extended to the study of the azimuthal asymmetries inthe 120587119873 Drell-Yan process such as the Sivers asymmetry and

Boer-Mulders asymmetry We should point out that at larger119902perp the numerical estimate in [59] cannot describe the dataindicating that the perturbative correction from the119884119880119880 termmay play an important role in the region 119902perp sim 119876 Furtherstudy on the 119884 term is needed to provide a complete pictureof the 119902perp distribution of lepton pairs from 120587119873 Drell-Yan inthe whole 119902perp range52 The Sivers Asymmetry In [39] the authors adopted theGaussian form of the nonperturbative Sudakov form factor119878NP in (17) and the leading order 119862 coefficients to performa global fit on the Sivers function from the experimentaldata at HERMES [128] COMPASS [129 130] and JeffersonLab (JLab) [131] With the extracted Sivers function from


03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

01

ＲＪ

01 1

x

４Ｋ(xxQ) evolves with ０１３ＫＫ

４Ｋ(xxQ) evolves with ０1ＫＫ

COMPASS 2015 data



COMPASS 2015 data

05 10 15 20 2500

Ｋperp (GeV)0600 02 04 08 10minus02minus04

Ｒ

Figure 6The Sivers asymmetry within TMD factorization for a120587minus scattering off transversely polarized protonDrell-Yan process as functionsof 119909119901 (upper left) 119909120587 (upper right) 119909119865 (lower left) and 119902perp (lower right) compared with the COMPASS data [11] Figure from [60]

SIDIS process at hand they made predictions for the Siversasymmetry in Drell-Yan lepton pair and 119882 production atfuture planned Drell-Yan facilities at COMPASS [10] Fermi-lab [42 43] and RHIC [44 132] which can be comparedto the future experimental measurements to test the signchange of the Sivers functions between SIDIS and Drell-Yanprocesses The predictions were presented in Fig 12 and 13 of[39]

The TMD evolution effect of the Sivers asymmetry inSIDIS and119901119901Drell-Yan at low transversemomentumhas alsobeen studied in [88] in which a frameworkwas built tomatchSIDIS and Drell-Yan and cover the TMD physics with 1198762from several GeV2 to 104GeV2 (for119882119885 boson production)It has shown that the evolution equations derived by a directintegral of the CSS evolution kernel from low to high 119876 candescribe well the transverse momentum distribution of theunpolarized cross sections in the 1198762 range from 2 to 100GeV2 With this approach the transverse moment of thequark Sivers functions can be constrained from the combinedanalysis of the HERMES and COMPASS data on the Sivers

asymmetries in SIDIS Based on this result [88] provided thepredictions for the Sivers asymmetries in 119901119901 Drell-Yan aswell as in 120587minus119901 Drell-Yan The latter one has been measuredby theCOMPASSCollaboration and the comparison showedthat the theoretical result is consistent with data (Fig 6 in [11])within the error bar

With the numerical results of the TMD distributionsin (57) the Sivers asymmetry 119860Siv

119880119879 as function of 119909119901 119909120587119909119865 and 119902perp in 120587minus119901uarr 997888rarr 120583+120583minus + 119883 in the kinematics ofCOMPASS Collaboration was calculated in [60] as shown inFigure 6 The magnitude of the asymmetry is around 005 divide010 which is consistent with the COMPASS measurement(full squares in Figure 6) [11] within the uncertainties of theasymmetry The different approaches dealing with the energydependence of Qiu-Sterman function lead to different shapesof the asymmetry Furthermore the asymmetry from theapproximate evolution kernel has a fall at larger 119902perp which ismore compatible to the shape of 119902perp-dependent asymmetryof measured by the COMPASS Collaboration The study mayindicate that besides the TMD evolution effect the scale


020

015

010

005

000

]－

020

015

010

005

000

]－

020

015

010

005

000

]－

020

015

010

005

000

]－

01 02 03 0400

ＲＪ

02 04 06 0800

Ｒ

05 10 15 2000

Ｋ４ (GeV)00 02 04 06 08minus02

Ｒ

Figure 7The cos 2120601 azimuthal asymmetries ]119861119872 as the functions of 119909119901 (upper left) 119909120587 (upper right) 119909119865 (lower left) and 119902perp (lower right) forthe unpolarized 120587119901Drell-Yan process considering the TMD evolution in the kinematical region of COMPASSThe shadow areas correspondto the uncertainty of the parameters in the parameterization of the Boer-Mulders function for proton in [62] Figures from [61]

dependence of the Qiu-Sterman function will also play a rolein the interpretation of the experimental data

53The cos 2120601Azimuthal Asymmetry Using (64) the cos 2120601azimuthal asymmetry contributed by the double Boer-Mulders effect in the 120587119873 Drell-Yan process was analyzedin [61] in which the TMD evolution of the Boer-Muldersfunction was included In this calculation the Boer-Muldersfunction of the proton was chosen from the parameterizationin [62] at the initial energy 11987620 = 1GeV2 As mentionedin Section 43 the Boer-Mulders function of the pion wasadopted from themodel calculation in [126] Here we plot theestimated asymmetry ]119861119872 as function of 119909119901 119909120587 119909119865 and 119902perp inthe kinematical region of COMPASS in Figure 7 The bandscorrespond to the uncertainty of the parameterization of theBoer-Mulders function of the proton [62] We find fromthe plots that in the TMD formalism the cos 2120601 azimuthalasymmetry in the unpolarized 120587minus119901 Drell-Yan process con-tributed by the Boer-Mulders functions is around severalpercent Although the uncertainty from the proton Boer-Mulders functions is rather large the asymmetry is firmlypositive in the entire kinematical region The asymmetriesas the functions of 119909119901 119909120587 119909119865 show slight dependence onthe variables while the 119902perp dependent asymmetry showsincreasing tendency along with the increasing 119902perp in thesmall 119902perp range where the TMD formalism is valid The

result in Figure 7 indicates that precise measurements on theBoer-Mulders asymmetry ]119861119872 as functions of 119909119901 119909120587 119909119865 and119902perp can provide an opportunity to access the Boer-Muldersfunction of the pion meson Furthermore the work may alsoshed light on the proton Boer-Mulders function since theprevious extractions on it were mostly performed withoutTMD evolution

6 Summary and Prospects

It has been a broad consensus that the study on the TMDobservables will provide information on the partonsrsquo intrinsictransverse motions inside a hadron In the previous sectionswe have tried to substantiate this statement mainly focusingon the unpolarized and single-polarized120587119901 Drell-Yanprocesswithin the TMD factorization In particular we reviewed theextraction of the nonperturbative function from the Drell-Yan and SIDIS data in the evolution formalism of the TMDdistributions We also discussed the further applications ofthe TMD factorization in the phenomenology of unpolarizedcross section the Sivers asymmetry and the cos 2120601 azimuthalasymmetry in the 120587119901Drell-Yan process In summary we havethe following understanding on the 120587119873 Drell-Yan from theviewpoint of the TMD factorization

(i) The extraction of nonperturbative Sudakov formfactor from the 120587119873 Drell-Yan may shed light on the


evolution (scale dependence) of the pion TMD dis-tribution The prediction on transverse momentumdistribution of the dilepton in the small 119902perp regionis compatible with the COMPASS measurement andmay serve as a first step to study the spinazimuthalasymmetry in the 120587119873 Drell-Yan process at COM-PASS

(ii) The precise measurement on the single-spin asym-metry in the kinematical region of COMPASS canprovide great opportunity to access the Sivers func-tion Besides the TMD evolution effect the choice ofthe scale dependence of the Qiu-Sterman functioncan affect the shape of the asymmetry and shouldbe considered in the future extraction of the Siversfunction

(iii) Sizable cos 2120601 asymmetry contributed by the con-volution of the Boer-Mulders functions of the pionmeson and the proton can still be observed at COM-PASS after the TMD evolution effect is consideredFuture data with higher accuracy may provide furtherconstraint on the Boer-Mulders function of the pionmeson as well as that of the proton

Although a lot of progress on the theoretical frameworkof the TMD factorization and TMD evolution has beenmadethe improvement is still necessary both from the perturbativeand nonperturbative aspects In the future the study of 119878NPbased on more precise experimental data is needed suchas including the flavor dependence and hadron dependenceon the functional form for 119878NP From the viewpoint of theperturbative region higher-order calculation of the hardfactors and coefficients will improve the accuracy of thetheoretical framework Moreover most of the numericalcalculations are based on the approximation that the 119884-termcorrection is negligible in the small transverse momentumregion the inclusion of this term in the future estimate couldbe done to test the magnitude of the term In addition theTMD factorization is suitable to describe the small trans-verse momentum physics while the collinear factorization issuitable for the large transverse momentum or the integratedtransverse momentum The matching between the two fac-torization schemes to study the unpolarized and polarizedprocess over the whole transverse momentum region may bealso necessary [65 133]

Conflicts of Interest

The authors declare that they have no conflicts of inter-est

Acknowledgments

This work is partially supported by the NSFC (China)Grant 11575043 and by the Fundamental Research Fundsfor the Central Universities of China X Wang is sup-ported by the NSFC (China) Grant 11847217 and theChina Postdoctoral Science Foundation under Grant no2018M640680

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[39] M G Echevarrıa A Idilbi Z B Kang and I Vitev ldquoQCDevolution of the Sivers asymmetryrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 89 no 7 Article ID074013 2014

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[69] Z B Kang B W Xiao and F Yuan ldquoQCD resummationfor single spin asymmetriesrdquo Physical Review Letters vol 107Article ID 152002 2011

[70] S M Aybat J C Collins J W Qiu and T C Rogers ldquoQCDevolution of the Sivers functionrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 85 no 3 Article ID034043 2012

[71] M G Echevarria A Idilbi and I Scimemi ldquoUnified treatmentof the QCD evolution of all (un-)polarized transverse momen-tum dependent functions Collins function as a study caserdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 90 no 1 Article ID 014003 2014

[72] F Landry R Brock P M Nadolsky and C Yuan ldquoFermilabTevatron run-1rdquoPhysical Review D Particles Fields Gravitationand Cosmology vol 67 no 7 Article ID 073016 2003

[73] J Qiu and X Zhang ldquoQCD Prediction for Heavy BosonTransverse Momentum Distributionsrdquo Physical Review Lettersvol 86 no 13 pp 2724ndash2727 2001

[74] G P Korchemsky and G F Sterman ldquoNonperturbative Correc-tions in Resummed Cross Sectionsrdquo Nuclear Physics B vol 437p 415 1995

[75] Z B Kang A Prokudin P Sun and F Yuan ldquoExtractionof quark transversity distribution and Collins fragmentationfunctions with QCD evolutionrdquo Physical Review D vol 93Article ID 014009 2016

[76] P Sun andF Yuan ldquoEnergy evolution for the Sivers asymmetriesin hard processesrdquoPhysical ReviewD vol 88 Article ID 0340162013

[77] P Sun J Isaacson C-P Yuan and F Yuan ldquoNonperturbativefunctions for SIDIS and DrellndashYan processesrdquo InternationalJournal of Modern Physics A vol 33 no 11 Article ID 18410062018

[78] A V Konychev and P M Nadolsky ldquoUniversality of theCollinsndashSoperndashSterman nonperturbative function in vectorboson productionrdquo Physics Letters B vol 633 p 710 2006

[79] C THDavies B RWebber andW J Stirling ldquoDrell-Yan crosssections at small transverse momentumrdquoNuclear Physics B vol256 p 413 1985

[80] R K Ellis D A Ross and S Veseli ldquoVector boson productionin hadronic collisionsrdquo Nuclear Physics B vol 503 p 309 1997

[81] M AnselminoM Boglione and S Melis ldquoStrategy towards theextraction of the Sivers function with transverse momentumdependent evolutionrdquo Physical Review D vol 86 Article ID014028 2012

[82] C A Aidala B Field L P Gamberg and T C RogersldquoLimits on TMDEvolution From Semi-Inclusive Deep InelasticScattering at Moderaterdquo Physical Review D vol 89 Article ID094002 2014

[83] J C Collins and D E Soper ldquoThe two-particle-inclusive crosssection in e+e- annihilation at PETRA PEP and LEP energiesrdquoNuclear Physics B vol 284 p 253 1987

[84] W J Stirling and M R Whalley ldquoA compilation of Drell-Yancross sectionsrdquo Journal of Physics G vol 19 p D1 1993

[85] F James and M Roos ldquoMinuit - a system for function mini-mization and analysis of the parameter errors and correlationsrdquoComputer Physics Communications vol 10 no 6 pp 343ndash3671975

[86] F James ldquoMINUITmdashFunctionMinimization and Error Analy-sisrdquo CERN Program Library Long Writeup D506 1994

[87] F A Ceccopieri A Courtoy S Noguera and S Scopetta ldquoPionnucleusDrellndashYan process and parton transversemomentum inthe pionrdquoThe European Physical Journal C vol 78 p 644 2018

[88] P Sun and F Yuan ldquoTransverse momentum dependent evo-lution Matching semi-inclusive deep inelastic scattering pro-cesses to Drell-Yan and WZ boson productionrdquo PhysicalReview D vol 88 Article ID 114012 2013

[89] J-W Qiu and G F Sterman ldquoSingle transverse spin asymme-triesrdquoPhysical Review Letters vol 67 no 17 pp 2264ndash2267 1991


[90] JWQiu andG F Sterman ldquoSingle transverse spin asymmetriesin direct photon productionrdquo Nuclear Physics B vol 378 p 521992

[91] J W Qiu and G F Sterman ldquoSingle Transverse-Spin Asymme-tries in Hadronic Pion Productionrdquo Physical Review D vol 59Article ID 014004 1999

[92] M Anselmino M Boglione and S Melis ldquoPhenomenologyof Sivers Effect with TMD Evolutionrdquo International Journal ofModern Physics Conference Series vol 20 pp 145ndash152 2012

[93] M Boglione U DrsquoAlesio C Flore and J O Gonzalez-Hernandez ldquoAssessing signals of TMD physics in SIDISazimuthal asymmetries and in the extraction of the Siversfunctionrdquo Journal of High Energy Physics vol 07 p 148 2018

[94] B Zhang Z Lu B Q Ma and I Schmidt ldquoExtracting Boer-Mulders functions from p+D Drell-Yan processesrdquo PhysicalReview D vol 77 Article ID 054011 2008

[95] V Barone S Melis and A Prokudin ldquoBoer-Mulders effect inunpolarized SIDIS An analysis of the COMPASS andHERMESdata on therdquoPhysical ReviewD Particles Fields Gravitation andCosmology vol 81 no 11 Article ID 114026 2010

[96] V Barone S Melis and A Prokudin ldquoAzimuthal asymmetriesin unpolarized Drell-Yan processes and the Boer-Muldersdistributions of antiquarksrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 82 no 11 Article ID 1140252010

[97] A Prokudin P Sun and F Yuan ldquoScheme dependenceand transverse momentum distribution interpretation ofCollinsndashSoperndashSterman resummationrdquo Physics Letters B vol750 p 533 2015

[98] J C Collins and A Metz ldquoUniversality of Soft and CollinearFactors in Hard-Scattering Factorizationrdquo Physical Review Let-ters vol 93 Article ID 252001 2004

[99] S Mantry and F Petriello ldquoFactorization and resummation ofHiggs boson differential distributions in soft-collinear effectivetheoryrdquo Physical Review D vol 81 Article ID 093007 2010

[100] T Becher and M Neubert ldquoDrellndashYan production at small q119879 transverse parton distributions and the collinear anomalyrdquoEuropean Physical Journal C vol 71 p 1665 2011

[101] M G Echevarria A Idilbi and I Scimemi ldquoFactorizationtheorem for Drell-Yan at low q119879 and transverse-momentumdistributions on-the-light-conerdquo Journal of High Energy Physicsvol 07 p 002 2012

[102] J Y Chiu A Jain D Neill and I Z Rothstein ldquoA Formalism forthe Systematic Treatment of Rapidity Logarithms in QuantumField Theoryrdquo Journal of High Energy Physics vol 05 p 0842012

[103] X Ji P Sun X Xiong and F Yuan ldquoSoft factor subtraction andtransverse momentum dependent parton distributions on thelatticerdquo Physical Review D vol 91 Article ID 074009 2015

[104] S Catani D de Florian andM Grazzini ldquoUniversality of non-leading logarithmic contributions in transverse-momentumdistributionsrdquo Nuclear Physics B vol 596 p 299 2001

[105] P Nadolsky D R Stump and C Yuan ldquoErratum Semi-inclusive hadron production at DESY HERA The effect ofQCD gluon resummationrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 64 no 5 Article ID 0599032001 Erratum [Physical Review D vol 64 article 0599032001]

[106] Y Koike J Nagashima and W Vogelsang ldquoResummation forPolarized Semi-Inclusive Deep-Inelastic Scattering at SmallTransverseMomentumrdquoNuclear Physics B vol 744 p 59 2006

[107] J C Collins and D E Soper ldquoAngular distribution of dileptonsin high-energy hadron collisionsrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 16 no 7 pp 2219ndash22251977

[108] L Y Zhu J C Peng P E Reimer et al ldquoMeasurement of Angu-lar Distributions of Drell-Yan Dimuons in p+d Interactions at800GeVcrdquo Physical Review Letters vol 99 Article ID 0823012007

[109] L Y Zhu J C Peng P E Reimer et al ldquoMeasurement of Angu-lar Distributions of Drell-Yan Dimuons in p+p Interactions at800 GeVcrdquo Physical Review Letters vol 102 Article ID 1820012009

[110] J C Collins ldquoSimple Prediction of QuantumChromodynamicsfor Angular Distribution of Dileptons in Hadron CollisionsrdquoPhysical Review Letters vol 42 p 291 1979

[111] P Chiappetta and M Le Bellac ldquoAngular distribution of leptonpairs in Drell-Yan-like processesrdquo Zeitschrift fur Physik CParticles and Fields vol 32 no 4 pp 521ndash526 1986

[112] M Blazek M Biyajima and N Suzuki ldquoAngular distributionof muon pairs produced by negative pions on deuterium andtungsten in terms of coherent statesrdquo Zeitschrift fur Physik CParticles and Fields vol 43 p 447 1989

[113] J Zhou F Yuan and Z T Liang ldquoDrellndashYan lepton pairazimuthal asymmetry in hadronic processesrdquo Physics Letters Bvol 678 p 264 2009

[114] Z Lu ldquoSpin physics through unpolarized processesrdquo Frontiersof Physics (Beijing) vol 11 no 1 Article ID 111204 2016

[115] P J Sutton A D Martin R G Roberts and W J StirlingldquoParton distributions for the pion extracted from Drell-Yanand prompt photon experimentsrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 45 no 7 pp 2349ndash23591992

[116] Z B Kang and J W Qiu ldquoQCD evolution of naive-time-reversal-odd parton distribution functionsrdquo Physics Letters Bvol 713 p 273 2012

[117] Z B Kang and J W Qiu ldquoEvolution of twist-3 multi-partoncorrelation functions relevant to single transverse-spin asym-metryrdquo Physical Review D vol 79 Article ID 016003 2009

[118] J Zhou F Yuan and Z T Liang ldquoQCD Evolution of the Trans-verse MomentumDependent Correlationsrdquo Physical Review Dvol 79 Article ID 114022 2009

[119] WVogelsang and F Yuan ldquoNext-to-leading order calculation ofthe single transverse spin asymmetry in the Drell-Yan processrdquoPhysical Review D vol 79 Article ID 094010 2009

[120] VM Braun A NManashov and B Pirnay ldquoScale dependenceof twist-three contributions to single spin asymmetriesrdquo Physi-cal Review D Particles Fields Gravitation and Cosmology vol80 Article ID 114002 2009 Erratum [Physical Review D vol86 pp 119902 2012]

[121] J P Ma and H Z Sang ldquoSoft-gluon-pole contribution in singletransverse-spin asymmetries of Drell-Yan processesrdquo Journal ofHigh Energy Physics vol 04 p 062 2011

[122] A Schafer and J Zhou ldquoNote on the scale evolution ofthe Efremov-Teryaev-Qiu-Sterman function T119865(xx)rdquo PhysicalReview D vol 85 Article ID 117501 2012

[123] J PMa andQWang ldquoScale dependence of twist-3 quarkndashgluonoperators for single spin asymmetriesrdquo Physics Letters B vol715 p 157 2012

[124] J Zhou ldquoNote on the scale dependence of the Burkardt sumrulerdquo Physical Review D Particles Fields Gravitation andCosmology vol 92 no 7 Article ID 074016 2015


[125] G P Salam and J Rojo ldquoA Higher Order Perturbative PartonEvolution Toolkit (HOPPET)rdquo Computer Physics Communica-tions vol 180 no 1 pp 120ndash156 2009

[126] Z Wang X Wang and Z Lu ldquoBoer-Mulders function of thepion and the qT-weighted cos2120593 asymmetry in the unpolarized120587minusp Drell-Yan process at COMPASSrdquo Physical Review D vol95 Article ID 094004 2017

[127] M Botje ldquoQCDNUM Fast QCD evolution and convolutionrdquoComputer Physics Communications vol 182 no 2 pp 490ndash5322011

[128] A Airapetian N Akopov Z Akopov et al ldquoObservation ofthe Naive-T-Odd Sivers Effect in Deep-Inelastic ScatteringrdquoPhysical Review Letters vol 103 Article ID 152002 2009

[129] M Alekseev V Y Alexakhin Y Alexandrov et al ldquoCollinsand Sivers asymmetries for pions and kaons in muonndashdeuteronDISrdquo Physics Letters B vol 673 p 127 2009

[130] C Adolph M G Alekseev V Y Alexakhin et al ldquoII ndashExperimental investigation of transverse spin asymmetries in120583-p SIDIS processes Sivers asymmetriesrdquo Physics Letters B vol717 p 383 2012

[131] X Qian K Allada C Dutta et al ldquoSingle Spin Asymme-tries in Charged Pion Production from Semi-Inclusive DeepInelastic Scattering on a Transversely Polarized 3He Target atQ2=14ndash27GeV2rdquo Physical Review Letters vol 107 Article ID072003 2011

[132] E C Aschenauer A Bazilevsky K Boyle et al ldquoTheRHIC Spin Program Achievements and Future Opportunitiesrdquohttpsarxivorgabs13040079

[133] L Gamberg A Metz D Pitonyak and A Prokudin ldquoConnec-tions between collinear and transverse-momentum-dependentpolarized observables within the CollinsndashSoperndashSterman for-malismrdquo Physics Letters B vol 781 p 443 2018

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energy processes such as the semi-inclusive deep inelasticscattering (SIDIS) [14 15 22 23 25 26] 119890+119890minus annihilation[15 27 28] Drell-Yan [15 29] andWZ production in hadroncollision [12 15 30] The TMD factorization can be alsoextended to themoderate 119902perp regionwhere an equivalence [3132] between the TMD factorization and the twist-3 collinearfactorization is found

One of the most important observables in the polarizedDrell-Yan process is the Sivers asymmetry It is contributedby the so-called Sivers function [33] a time-reversal-odd(T-odd) distribution describing the asymmetric distribu-tion of unpolarized quarks inside a transversely polarizednucleon through the correlation between the quark trans-verse momentum and the nucleon transverse spin Remark-ably QCD predicts that the sign of the Sivers functionchanges in SIDIS with respect to the Drell-Yan process [1634 35] The verification of this sign change [36ndash41] is oneof the most fundamental tests of our understanding of theQCD dynamics and the factorization schemes and it isalso the main pursue of the existing and future Drell-Yanfacilities [10 11 42ndash45] The advantage of the 120587119873 Drell-Yan measurement at COMPASS is that almost the samesetup [11 46] is used in SIDIS and Drell-Yan processeswhich may reduce the uncertainty in the extraction of theSivers function In particular the COMPASS Collaborationmeasured for the first time the transverse-spin-dependentazimuthal asymmetries [11] in the 120587minus119873 Drell-Yan process

Another important observable in the Drell-Yan process isthe cos 2120601 angular asymmetry where 120601 corresponds to theazimuthal angle of the dilepton The fixed-target measure-ments from the NA10 and E615 collaborations showed thatthe unpolarized cross section possesses large cos 2120601 asymme-try which violates the Lam-Tung relation [47] Similar vio-lation has also been observed in the 119901119901 colliders at Tevatron[48] and LHC [49] It has been explained from the viewpointsof higher-twist effect [50ndash53] the noncoplanarity effect [3054] and the QCD radiative effects at higher order [55 56]Another promising origin [57] for the violation of the Lam-Tung relation at low transverse momentum is the convolutionof the two Boer-Mulders functions [58] from each hadronThe Boer-Mulders function is also a TMD distribution Asthe chiral-odd partner of the Sivers function it describesthe transverse-polarization asymmetry of quarks inside anunpolarized hadron [57 58] thereby allowing the probe ofthe transverse spin physics from unpolarized reaction

This article aims at a review on the current status ofour understanding on the Drell-Yan dilepton production atlow transverse momentum especially from the 120587119873 collisionbased on the recent development of the TMD factorizationWe will mainly focus on the phenomenology of the Siversasymmetry as well as the cos 2120601 asymmetry from the doubleBoer-Mulders effect In order to quantitatively understandvarious spinazimuthal asymmetries in the 120587119873 Drell-Yanprocess a particularly important step is to know in highaccuracy the spin-averaged differential cross section of thesame process with azimuthal angles integrated out sinceit always appears in the denominator of the asymmetriesrsquodefinition Thus the spin-averaged cross section will be alsodiscussed in great details

The remained content of the article is organised as fol-lows In Section 2 we will review the TMD evolution formal-ism of the TMDs mostly following the approach establishedin [15] Particularly we will discuss in detail the extraction ofthe nonperturbative Sudakov form factor for the unpolarizedTMD distribution of the protonpion as well as that for theSivers function In Section 3 putting the evolved result of theTMD distributions into the TMD factorization formulae wewill present the theoretical expression of the physical observ-ables such as the unpolarized differential cross section theSivers asymmetry and the cos 2120601 asymmetry contributedby the double Boer-Mulders effect In Section 4 we presentthe numerical evolution results of the unpolarized TMDdistributions and the Boer-Mulders function of the pionmeson as well as that of the Sivers function of the protonIn Section 5 we display the phenomenology of the physicalobservables (unpolarized differential cross section the Siversasymmetry and the cos 2120601 asymmetry) in the 120587119873 Drell-Yan with TMD factorization at the kinematical configurationof the COMPASS experiments We summarize the paper inSection 6

2 The TMD Evolution of theDistribution Functions

In this section we present a review on the TMD evolution ofthe distribution functions Particularly we provide the evo-lution formalism for the unpolarized distribution function1198911 transversity ℎ1 Sivers function 119891perp1 and the Boer-Muldersfunction ℎperp1 of the proton as well as 1198911 and ℎperp1 of the pionmeson within the Collins-11 TMD factorization scheme [15]

In general it is more convenient to solve the evolutionequations for the TMD distributions in the coordinate space(b space) other than that in the transverse momentum kperpspace with b conjugate to kperp via Fourier transformation[12 15] The TMD distributions 119865(119909 119887 120583 120577119865) in b spacehave two kinds of energy dependence namely 120583 is therenormalization scale related to the corresponding collinearPDFs and 120577119865 is the energy scale serving as a cutoff toregularize the light-cone singularity in the operator defini-tion of the TMD distributions Here 119865 is a shorthand forany TMD distribution function and the tilde denotes thatthe distribution is the one in b space If we perform theinverse Fourier transformation on 119865(119909 119887 120583 120577119865) we recoverthe distribution function in the transverse momentum space119865119902119867(119909 119896perp 120583 120577119865) which contains the information about theprobability of finding a quark with specific polarizationcollinear momentum fraction 119909 and transverse momentum119896perp in a specifically polarized hadron119867

21 TMD Evolution Equations The energy evolution for the120577119865 dependence of the TMD distributions is encoded in theCollins-Soper (CS) [12 15 63] equation

120597 ln119865 (119909 119887 120583 120577119865)120597radic120577119865 = (119887 120583) (1)

while the 120583 dependence is driven by the renormalizationgroup equation as


119889119889 ln 120583 = minus120574119870 (120572119904 (120583)) (2)

119889 ln119865 (119909 119887 120583 120577119865)119889 ln 120583 = 120574119865(120572119904 (120583) 12057721198651205832) (3)





119887lowast = 119887radic1 + 11988721198872max




119865119902119867 (119909 119887 120583) = sum119894

119862119902larr997888119894 otimes 119865119894119867 (119909 120583) (6)



119909

119889120585120585 119862119902larr997888119894 (119909120585 119887 120583)1198911198941198671 (120585 120583) (7)



119878 = 119878P + 119878NP (8)


119878P (119876 119887lowast)= int1198762

1205832119887

11988912058321205832 [119860 (120572119904 (120583)) ln 1198762

1205832 + 119861 (120572119904 (120583))] (9)


119860 = infinsum119899=1

119860(119899) (120572119904120587 )119899 (10)

119861 = infinsum119899=1

119861(119899) (120572119904120587 )119899 (11)


119860(1) = 119862119865 (12)

119860(2) = 1198621198652 [119862119860(6718 minus 12058726 ) minus 109 119879119877119899119891] (13)

119861(1) = minus32119862119865 (14)



119878NP (119876 119887) = 1198922 (119887) ln 1198761198760 + 1198921 (119887) (15)






(1198921 + 1198922 ln( 11987621198760) + 11989211198923ln (10011990911199092)) 1198872 (16)



119878pdfffNP = 1198872 (119892pdfff1 + 11989222 ln 1198761198760) (17)



119892pdf1 = ⟨1198962perp⟩11987604

119892ff1 = ⟨1199012119879⟩119876041199112

(18)



11989211198872 + 1198922 ln( 119887119887lowast) ln( 1198761198760)+ 11989231198872 ((11990901199091)

120582 + (11990901199092)120582)

(19)




1198781198911199021205871NP = 1198921199021205871 1198872 + 1198921199021205872 ln 119887119887lowast ln1198761198760 (20)




1205942 (120572) = 119872sum119894=1

119873119894sum119895=1


(21)


1198921199021205871 = 0082 plusmn 00221198921199021205872 = 0394 plusmn 0103 (22)






119894

119862119902larr997888119894otimes 119865119894119901 (119909 120583119887)

(23)


119894

119862119902larr997888119894otimes 119865119894120587 (119909 120583119887)

(24)




F (120572119904 (119876))sum119894

119862119902larr997888119894otimes 1198911119894119901 (119909 120583119887)

(25)


F (120572119904 (119876))sum119894

119862119902larr997888119894otimes 1198911119894120587 (119909 120583119887)

(26)


1198911119902119901 (119909 119896perp 119876) = intinfin0

1198891198871198872120587 1198690 (119896perp119887) 1198911119902119901 (119909 119887 119876) (27)

1198911119902120587 (119909 119896perp 119876) = intinfin0

1198891198871198872120587 1198690 (119896perp119887) 1198911119902120587 (119909 119887 119876) (28)




119894

120575119862119902larr997888119894otimes ℎ1119894119901 (119909 120583119887)

(29)

ℎ1119902119901 (119909 119896perp 119876) = intinfin0

1198891198871198872120587 1198690 (119896perp119887) ℎ1119902119901 (119909 119887 119876) (30)




119891perp(DY)1119879119902119867 (119909 kperp 120583)


01

001

1E-300 06 12 18 24 30

Ｋperp (GeV)00 06 12 18 24 30

Ｋperp (GeV)

00 06 12 18 24 30

Ｋperp (GeV)00 06 12 18 24 30

Ｋperp (GeV)

00 06 12 18 24 30

Ｋperp (GeV)00 06 12 18 24 30

Ｋperp (GeV)

00 06 12 18 24 30

Ｋperp (GeV)00 06 12 18 24 30

Ｋperp (GeV)

＞2

＞Ｋperp＞Ｒ

(nb

GeV

nuc

leon

)

01

001

1E-3

＞2

＞Ｋperp＞Ｒ

(nb

GeV

nuc

leon

)

01

001

1E-3

＞2

＞Ｋperp＞Ｒ

(nb

GeV

nuc

leon

)

01

001

1E-3

01

001

1E-3

01

001

1E-3

01

001

1E-3

01

001

1E-3

＞2

＞Ｋperp＞Ｒ

(nb

GeV

nuc

leon

)







ℎperp(DY)1119902119867 (119909 kperp 120583) (31)




1119879119902119867can also be


119891perp120572(DY)1119879119902119867 (119909 119887 120583)= (minus1198941198871205722 )sum

119894

Δ119862119879119902larr997888119894 otimes 119891(3)119894119901 (1199091015840 11990910158401015840 120583) (32)

Here119891(3)119894119901




(33)



119862BM119902larr997888119894



convoluted with 119879(120590)119894119867119865



119896perp119872119867


0119889119887( 11988722120587) 1198691 (119896perp119887)FSiv (120572119904 (119876))sum

119894


(35)

119896perp119872119867


0119889119887( 11988722120587) 1198691 (119896perp119887)HBM (120572119904 (119876))sum

119894

119862BM119902larr997888119894


and 119879119902119865(119909 119909 120583119887) and 119879(120590)119894119867119865


119879119902119867119865 (119909 119909) = int1198892119896perp100381610038161003816100381610038161198962perp10038161003816100381610038161003816119872119867


(37)

119879(120590)119902119867119865 (119909 119909) = int1198892119896perp100381610038161003816100381610038161198962perp10038161003816100381610038161003816119872119867


(38)





119867119860 (119875120587) + 119867119861 (119875119873) 997888rarr120574lowast (119902) + 119883 997888rarr

119897+ (ℓ) + 119897minus (ℓ1015840) + 119883(39)


119904 = (119875120587 + 119875119873)2 119909120587119873 = 11987622119875120587119873 sdot 119902 119909119865 = 2119902119871119904 = 119909120587 minus 119909119873120591 = 1198762119904 = 119909120587119909119873119910 = 12 ln

119902+119902minus = 12 ln 119909120587119909119873

(40)




119909120587119873 = plusmn119909119865 + radic1199092119865 + 41205912

119909120587119873 = radic120591119890plusmn119910(41)



119889412059011988011988011988911987621198891199101198892qperp = 1205900 int 1198892119887(2120587)2 119890119894


+ 119884119880119880 (119876 119902perp)(42)



119880119880 (119876 119887) = 119867119880119880 (119876 120583)sdot sum119902119902

1198902119902119891sub119902120587 (119909120587 119887 120583 120577119865) 119891sub

119902119901 (119909119901 119887 120583 120577119865) (43)





119867JCC (119876 120583) = 1 + 120572119904 (120583)2120587 119862119865 (3 ln 11987621205832 minus ln211987621205832 + 1205872

minus 8) (44)


(45)


FJCC (120572119904 (119876)) = 1 + O (1205722119904 ) (46)



119862119902larr9978881199021015840 (119909 119887 120583 120577119865)= 1205751199021199021015840 [120575 (1 minus 119909) + 120572119904120587 (1198621198652 (1 minus 119909))] (48)

119862119902larr997888119892 (119909 119887 120583 120577119865) = 120572119904120587 119879119877119909 (1 minus 119909) (49)


functions using




(51)

1198621015840119902larr997888119892 (119909 119887 120583119887) = 120572119904120587 119879119877119909 (1 minus 119909) (52)






(53)


119889412059011988911987621198891199101198892qperp = 1205900 intinfin0

1198891198871198872120587 1198690 (119902perp119887) times 119880119880 (119876 119887) (54)



119860119880119879 = 1198894Δ12059011988911987621198891199101198892qperp119889412059011988911987621198891199101198892qperp (55)




119880119879 (119876 119887)+ 119884120573119880119879 (119876 119902perp)

(56)




120572119880119879 (119876 119887) = 119867119880119879 (119876 120583)sdot sum119902119902

11989021199021198911119902120587 (119909120587 119887 120583 120577119865) 119891perp120572(DY)1119879119902119901 (119909119901 119887 120583 120577119865) (57)

with 119891perp120572(DY)1119879119902119901



(58)


1198894Δ12059011988911987621198891199101198892qperp = 12059004120587 intinfin0

11988911988711988721198691 (119902perp119887) sum119902119894119895


(59)




(60)






1

01

001

1E-3

1E-4

1Ｏ

(xＥ

perpQ

)

0 1 2 3 4

b (Ge６-1)

Ｒ = 01

１2 = 24Ge６2

１2 = 10Ge６2

１2 = 1000Ge６2

１2 = 24Ge６2

１2 = 10Ge６2

１2 = 1000Ge６2

1 2 3 4 5 60

Ｅperp (GeV)

0

1

2

3

4

5

＜Ｍ

Ｏ＜

1Ｏ

(xb

Q)


where the notation


(62)


F [11989111199021205871198911119902119901] = sum119902

1198902119902 int 1198892119887(2120587)2


119902

1198902119902sdot int 1198892119887

(2120587)2 119890119894qperp sdotb1198911119902120587 (119909120587 119887 119876) 1198911119902119901 (119909119901 119887 119876)

(63)



] = sum119902

1198902119902



] = sum119902

1198902119902


sdot ℎ120573perp1119902119901

(119909119901 119887 119876) = sum119902

1198902119902 intinfin0


1198911119902120587NP +119878P)

(64)











1198751198911119902119902 = 43 ( 1 + 1199112(1 minus 119911)+ + 32120575 (1 minus 119911)) (66)






ℎperp1119902120587 (119909 119887 119876) = 119894119887120572120587 ℎ120572perp1119902120587 (119909 119887 119876) (68)






１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

x=01

x=01x=01

x=01

1

01

001

1E-3

1E-4

Ｅperp

－Ｊ

perp 1４ＫＪ

(xＥ

perpQ

)

1

01

001

1E-3

1E-4

Ｅperp

－Ｊ

perp 1４ＫＪ

(xＥ

perpQ

)

00

02

04

06

08

10

perp 1４ＫＪ

(xb

Q)

00

02

04

06

08

10 perp 1

４ＫＪ

(xb

Q)

1 2 3 4 50 6

b (Ge６-1)1 2 3 4 50 6

b (Ge６-1)

05 10 15 20 2500

Ｅperp (GeV)05 10 15 20 2500

Ｅperp (GeV)






1

01

001

1E-3

１2=025 Ge６2

１2=10 Ge６2

１2=1000 Ge６2

１2=025 Ge６2

１2=10 Ge６2

１2=1000 Ge６2

x=01 x=01

Ｅ４

－

Ｂperp 1Ｋ

(xＥ

４Q

)

1 2 3 4 50

Ｅ４ (GeV)

000

005

010

015

020

Ｂperp 1Ｋ

(xb

Q)

1 2 3 4 50

b (Ge６-1)


02

00

06

08

04

1

＞

＞Ｋperp

(Ge６

-1)



05 1000 20 25 3015

Ｋperp (GeV)





03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

01

ＲＪ

01 1

x



COMPASS 2015 data



COMPASS 2015 data

05 10 15 20 2500


Ｒ








020

015

010

005

000

]－

020

015

010

005

000

]－

020

015

010

005

000

]－

020

015

010

005

000

]－

01 02 03 0400

ＲＪ

02 04 06 0800

Ｒ

05 10 15 2000

Ｋ４ (GeV)00 02 04 06 08minus02

Ｒ















Acknowledgments


References













































































































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






119889119889 ln 120583 = minus120574119870 (120572119904 (120583)) (2)

119889 ln119865 (119909 119887 120583 120577119865)119889 ln 120583 = 120574119865(120572119904 (120583) 12057721198651205832) (3)





119887lowast = 119887radic1 + 11988721198872max




119865119902119867 (119909 119887 120583) = sum119894

119862119902larr997888119894 otimes 119865119894119867 (119909 120583) (6)



119909

119889120585120585 119862119902larr997888119894 (119909120585 119887 120583)1198911198941198671 (120585 120583) (7)



119878 = 119878P + 119878NP (8)


119878P (119876 119887lowast)= int1198762

1205832119887

11988912058321205832 [119860 (120572119904 (120583)) ln 1198762

1205832 + 119861 (120572119904 (120583))] (9)


119860 = infinsum119899=1

119860(119899) (120572119904120587 )119899 (10)

119861 = infinsum119899=1

119861(119899) (120572119904120587 )119899 (11)


119860(1) = 119862119865 (12)

119860(2) = 1198621198652 [119862119860(6718 minus 12058726 ) minus 109 119879119877119899119891] (13)

119861(1) = minus32119862119865 (14)



119878NP (119876 119887) = 1198922 (119887) ln 1198761198760 + 1198921 (119887) (15)






(1198921 + 1198922 ln( 11987621198760) + 11989211198923ln (10011990911199092)) 1198872 (16)



119878pdfffNP = 1198872 (119892pdfff1 + 11989222 ln 1198761198760) (17)



119892pdf1 = ⟨1198962perp⟩11987604

119892ff1 = ⟨1199012119879⟩119876041199112

(18)



11989211198872 + 1198922 ln( 119887119887lowast) ln( 1198761198760)+ 11989231198872 ((11990901199091)

120582 + (11990901199092)120582)

(19)




1198781198911199021205871NP = 1198921199021205871 1198872 + 1198921199021205872 ln 119887119887lowast ln1198761198760 (20)




1205942 (120572) = 119872sum119894=1

119873119894sum119895=1


(21)


1198921199021205871 = 0082 plusmn 00221198921199021205872 = 0394 plusmn 0103 (22)






119894

119862119902larr997888119894otimes 119865119894119901 (119909 120583119887)

(23)


119894

119862119902larr997888119894otimes 119865119894120587 (119909 120583119887)

(24)




F (120572119904 (119876))sum119894

119862119902larr997888119894otimes 1198911119894119901 (119909 120583119887)

(25)


F (120572119904 (119876))sum119894

119862119902larr997888119894otimes 1198911119894120587 (119909 120583119887)

(26)


1198911119902119901 (119909 119896perp 119876) = intinfin0

1198891198871198872120587 1198690 (119896perp119887) 1198911119902119901 (119909 119887 119876) (27)

1198911119902120587 (119909 119896perp 119876) = intinfin0

1198891198871198872120587 1198690 (119896perp119887) 1198911119902120587 (119909 119887 119876) (28)




119894

120575119862119902larr997888119894otimes ℎ1119894119901 (119909 120583119887)

(29)

ℎ1119902119901 (119909 119896perp 119876) = intinfin0

1198891198871198872120587 1198690 (119896perp119887) ℎ1119902119901 (119909 119887 119876) (30)




119891perp(DY)1119879119902119867 (119909 kperp 120583)


01

001

1E-300 06 12 18 24 30

Ｋperp (GeV)00 06 12 18 24 30

Ｋperp (GeV)

00 06 12 18 24 30

Ｋperp (GeV)00 06 12 18 24 30

Ｋperp (GeV)

00 06 12 18 24 30

Ｋperp (GeV)00 06 12 18 24 30

Ｋperp (GeV)

00 06 12 18 24 30

Ｋperp (GeV)00 06 12 18 24 30

Ｋperp (GeV)

＞2

＞Ｋperp＞Ｒ

(nb

GeV

nuc

leon

)

01

001

1E-3

＞2

＞Ｋperp＞Ｒ

(nb

GeV

nuc

leon

)

01

001

1E-3

＞2

＞Ｋperp＞Ｒ

(nb

GeV

nuc

leon

)

01

001

1E-3

01

001

1E-3

01

001

1E-3

01

001

1E-3

01

001

1E-3

＞2

＞Ｋperp＞Ｒ

(nb

GeV

nuc

leon

)







ℎperp(DY)1119902119867 (119909 kperp 120583) (31)




1119879119902119867can also be


119891perp120572(DY)1119879119902119867 (119909 119887 120583)= (minus1198941198871205722 )sum

119894

Δ119862119879119902larr997888119894 otimes 119891(3)119894119901 (1199091015840 11990910158401015840 120583) (32)

Here119891(3)119894119901




(33)



119862BM119902larr997888119894



convoluted with 119879(120590)119894119867119865



119896perp119872119867


0119889119887( 11988722120587) 1198691 (119896perp119887)FSiv (120572119904 (119876))sum

119894


(35)

119896perp119872119867


0119889119887( 11988722120587) 1198691 (119896perp119887)HBM (120572119904 (119876))sum

119894

119862BM119902larr997888119894


and 119879119902119865(119909 119909 120583119887) and 119879(120590)119894119867119865


119879119902119867119865 (119909 119909) = int1198892119896perp100381610038161003816100381610038161198962perp10038161003816100381610038161003816119872119867


(37)

119879(120590)119902119867119865 (119909 119909) = int1198892119896perp100381610038161003816100381610038161198962perp10038161003816100381610038161003816119872119867


(38)





119867119860 (119875120587) + 119867119861 (119875119873) 997888rarr120574lowast (119902) + 119883 997888rarr

119897+ (ℓ) + 119897minus (ℓ1015840) + 119883(39)


119904 = (119875120587 + 119875119873)2 119909120587119873 = 11987622119875120587119873 sdot 119902 119909119865 = 2119902119871119904 = 119909120587 minus 119909119873120591 = 1198762119904 = 119909120587119909119873119910 = 12 ln

119902+119902minus = 12 ln 119909120587119909119873

(40)




119909120587119873 = plusmn119909119865 + radic1199092119865 + 41205912

119909120587119873 = radic120591119890plusmn119910(41)



119889412059011988011988011988911987621198891199101198892qperp = 1205900 int 1198892119887(2120587)2 119890119894


+ 119884119880119880 (119876 119902perp)(42)



119880119880 (119876 119887) = 119867119880119880 (119876 120583)sdot sum119902119902

1198902119902119891sub119902120587 (119909120587 119887 120583 120577119865) 119891sub

119902119901 (119909119901 119887 120583 120577119865) (43)





119867JCC (119876 120583) = 1 + 120572119904 (120583)2120587 119862119865 (3 ln 11987621205832 minus ln211987621205832 + 1205872

minus 8) (44)


(45)


FJCC (120572119904 (119876)) = 1 + O (1205722119904 ) (46)



119862119902larr9978881199021015840 (119909 119887 120583 120577119865)= 1205751199021199021015840 [120575 (1 minus 119909) + 120572119904120587 (1198621198652 (1 minus 119909))] (48)

119862119902larr997888119892 (119909 119887 120583 120577119865) = 120572119904120587 119879119877119909 (1 minus 119909) (49)


functions using




(51)

1198621015840119902larr997888119892 (119909 119887 120583119887) = 120572119904120587 119879119877119909 (1 minus 119909) (52)






(53)


119889412059011988911987621198891199101198892qperp = 1205900 intinfin0

1198891198871198872120587 1198690 (119902perp119887) times 119880119880 (119876 119887) (54)



119860119880119879 = 1198894Δ12059011988911987621198891199101198892qperp119889412059011988911987621198891199101198892qperp (55)




119880119879 (119876 119887)+ 119884120573119880119879 (119876 119902perp)

(56)




120572119880119879 (119876 119887) = 119867119880119879 (119876 120583)sdot sum119902119902

11989021199021198911119902120587 (119909120587 119887 120583 120577119865) 119891perp120572(DY)1119879119902119901 (119909119901 119887 120583 120577119865) (57)

with 119891perp120572(DY)1119879119902119901



(58)


1198894Δ12059011988911987621198891199101198892qperp = 12059004120587 intinfin0

11988911988711988721198691 (119902perp119887) sum119902119894119895


(59)




(60)






1

01

001

1E-3

1E-4

1Ｏ

(xＥ

perpQ

)

0 1 2 3 4

b (Ge６-1)

Ｒ = 01

１2 = 24Ge６2

１2 = 10Ge６2

１2 = 1000Ge６2

１2 = 24Ge６2

１2 = 10Ge６2

１2 = 1000Ge６2

1 2 3 4 5 60

Ｅperp (GeV)

0

1

2

3

4

5

＜Ｍ

Ｏ＜

1Ｏ

(xb

Q)


where the notation


(62)


F [11989111199021205871198911119902119901] = sum119902

1198902119902 int 1198892119887(2120587)2


119902

1198902119902sdot int 1198892119887

(2120587)2 119890119894qperp sdotb1198911119902120587 (119909120587 119887 119876) 1198911119902119901 (119909119901 119887 119876)

(63)



] = sum119902

1198902119902



] = sum119902

1198902119902


sdot ℎ120573perp1119902119901

(119909119901 119887 119876) = sum119902

1198902119902 intinfin0


1198911119902120587NP +119878P)

(64)











1198751198911119902119902 = 43 ( 1 + 1199112(1 minus 119911)+ + 32120575 (1 minus 119911)) (66)






ℎperp1119902120587 (119909 119887 119876) = 119894119887120572120587 ℎ120572perp1119902120587 (119909 119887 119876) (68)






１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

x=01

x=01x=01

x=01

1

01

001

1E-3

1E-4

Ｅperp

－Ｊ

perp 1４ＫＪ

(xＥ

perpQ

)

1

01

001

1E-3

1E-4

Ｅperp

－Ｊ

perp 1４ＫＪ

(xＥ

perpQ

)

00

02

04

06

08

10

perp 1４ＫＪ

(xb

Q)

00

02

04

06

08

10 perp 1

４ＫＪ

(xb

Q)

1 2 3 4 50 6

b (Ge６-1)1 2 3 4 50 6

b (Ge６-1)

05 10 15 20 2500

Ｅperp (GeV)05 10 15 20 2500

Ｅperp (GeV)






1

01

001

1E-3

１2=025 Ge６2

１2=10 Ge６2

１2=1000 Ge６2

１2=025 Ge６2

１2=10 Ge６2

１2=1000 Ge６2

x=01 x=01

Ｅ４

－

Ｂperp 1Ｋ

(xＥ

４Q

)

1 2 3 4 50

Ｅ４ (GeV)

000

005

010

015

020

Ｂperp 1Ｋ

(xb

Q)

1 2 3 4 50

b (Ge６-1)


02

00

06

08

04

1

＞

＞Ｋperp

(Ge６

-1)



05 1000 20 25 3015

Ｋperp (GeV)





03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

01

ＲＪ

01 1

x



COMPASS 2015 data



COMPASS 2015 data

05 10 15 20 2500


Ｒ








020

015

010

005

000

]－

020

015

010

005

000

]－

020

015

010

005

000

]－

020

015

010

005

000

]－

01 02 03 0400

ＲＪ

02 04 06 0800

Ｒ

05 10 15 2000

Ｋ４ (GeV)00 02 04 06 08minus02

Ｒ















Acknowledgments


References













































































































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery









(1198921 + 1198922 ln( 11987621198760) + 11989211198923ln (10011990911199092)) 1198872 (16)



119878pdfffNP = 1198872 (119892pdfff1 + 11989222 ln 1198761198760) (17)



119892pdf1 = ⟨1198962perp⟩11987604

119892ff1 = ⟨1199012119879⟩119876041199112

(18)



11989211198872 + 1198922 ln( 119887119887lowast) ln( 1198761198760)+ 11989231198872 ((11990901199091)

120582 + (11990901199092)120582)

(19)




1198781198911199021205871NP = 1198921199021205871 1198872 + 1198921199021205872 ln 119887119887lowast ln1198761198760 (20)




1205942 (120572) = 119872sum119894=1

119873119894sum119895=1


(21)


1198921199021205871 = 0082 plusmn 00221198921199021205872 = 0394 plusmn 0103 (22)






119894

119862119902larr997888119894otimes 119865119894119901 (119909 120583119887)

(23)


119894

119862119902larr997888119894otimes 119865119894120587 (119909 120583119887)

(24)




F (120572119904 (119876))sum119894

119862119902larr997888119894otimes 1198911119894119901 (119909 120583119887)

(25)


F (120572119904 (119876))sum119894

119862119902larr997888119894otimes 1198911119894120587 (119909 120583119887)

(26)


1198911119902119901 (119909 119896perp 119876) = intinfin0

1198891198871198872120587 1198690 (119896perp119887) 1198911119902119901 (119909 119887 119876) (27)

1198911119902120587 (119909 119896perp 119876) = intinfin0

1198891198871198872120587 1198690 (119896perp119887) 1198911119902120587 (119909 119887 119876) (28)




119894

120575119862119902larr997888119894otimes ℎ1119894119901 (119909 120583119887)

(29)

ℎ1119902119901 (119909 119896perp 119876) = intinfin0

1198891198871198872120587 1198690 (119896perp119887) ℎ1119902119901 (119909 119887 119876) (30)




119891perp(DY)1119879119902119867 (119909 kperp 120583)


01

001

1E-300 06 12 18 24 30

Ｋperp (GeV)00 06 12 18 24 30

Ｋperp (GeV)

00 06 12 18 24 30

Ｋperp (GeV)00 06 12 18 24 30

Ｋperp (GeV)

00 06 12 18 24 30

Ｋperp (GeV)00 06 12 18 24 30

Ｋperp (GeV)

00 06 12 18 24 30

Ｋperp (GeV)00 06 12 18 24 30

Ｋperp (GeV)

＞2

＞Ｋperp＞Ｒ

(nb

GeV

nuc

leon

)

01

001

1E-3

＞2

＞Ｋperp＞Ｒ

(nb

GeV

nuc

leon

)

01

001

1E-3

＞2

＞Ｋperp＞Ｒ

(nb

GeV

nuc

leon

)

01

001

1E-3

01

001

1E-3

01

001

1E-3

01

001

1E-3

01

001

1E-3

＞2

＞Ｋperp＞Ｒ

(nb

GeV

nuc

leon

)







ℎperp(DY)1119902119867 (119909 kperp 120583) (31)




1119879119902119867can also be


119891perp120572(DY)1119879119902119867 (119909 119887 120583)= (minus1198941198871205722 )sum

119894

Δ119862119879119902larr997888119894 otimes 119891(3)119894119901 (1199091015840 11990910158401015840 120583) (32)

Here119891(3)119894119901




(33)



119862BM119902larr997888119894



convoluted with 119879(120590)119894119867119865



119896perp119872119867


0119889119887( 11988722120587) 1198691 (119896perp119887)FSiv (120572119904 (119876))sum

119894


(35)

119896perp119872119867


0119889119887( 11988722120587) 1198691 (119896perp119887)HBM (120572119904 (119876))sum

119894

119862BM119902larr997888119894


and 119879119902119865(119909 119909 120583119887) and 119879(120590)119894119867119865


119879119902119867119865 (119909 119909) = int1198892119896perp100381610038161003816100381610038161198962perp10038161003816100381610038161003816119872119867


(37)

119879(120590)119902119867119865 (119909 119909) = int1198892119896perp100381610038161003816100381610038161198962perp10038161003816100381610038161003816119872119867


(38)





119867119860 (119875120587) + 119867119861 (119875119873) 997888rarr120574lowast (119902) + 119883 997888rarr

119897+ (ℓ) + 119897minus (ℓ1015840) + 119883(39)


119904 = (119875120587 + 119875119873)2 119909120587119873 = 11987622119875120587119873 sdot 119902 119909119865 = 2119902119871119904 = 119909120587 minus 119909119873120591 = 1198762119904 = 119909120587119909119873119910 = 12 ln

119902+119902minus = 12 ln 119909120587119909119873

(40)




119909120587119873 = plusmn119909119865 + radic1199092119865 + 41205912

119909120587119873 = radic120591119890plusmn119910(41)



119889412059011988011988011988911987621198891199101198892qperp = 1205900 int 1198892119887(2120587)2 119890119894


+ 119884119880119880 (119876 119902perp)(42)



119880119880 (119876 119887) = 119867119880119880 (119876 120583)sdot sum119902119902

1198902119902119891sub119902120587 (119909120587 119887 120583 120577119865) 119891sub

119902119901 (119909119901 119887 120583 120577119865) (43)





119867JCC (119876 120583) = 1 + 120572119904 (120583)2120587 119862119865 (3 ln 11987621205832 minus ln211987621205832 + 1205872

minus 8) (44)


(45)


FJCC (120572119904 (119876)) = 1 + O (1205722119904 ) (46)



119862119902larr9978881199021015840 (119909 119887 120583 120577119865)= 1205751199021199021015840 [120575 (1 minus 119909) + 120572119904120587 (1198621198652 (1 minus 119909))] (48)

119862119902larr997888119892 (119909 119887 120583 120577119865) = 120572119904120587 119879119877119909 (1 minus 119909) (49)


functions using




(51)

1198621015840119902larr997888119892 (119909 119887 120583119887) = 120572119904120587 119879119877119909 (1 minus 119909) (52)






(53)


119889412059011988911987621198891199101198892qperp = 1205900 intinfin0

1198891198871198872120587 1198690 (119902perp119887) times 119880119880 (119876 119887) (54)



119860119880119879 = 1198894Δ12059011988911987621198891199101198892qperp119889412059011988911987621198891199101198892qperp (55)




119880119879 (119876 119887)+ 119884120573119880119879 (119876 119902perp)

(56)




120572119880119879 (119876 119887) = 119867119880119879 (119876 120583)sdot sum119902119902

11989021199021198911119902120587 (119909120587 119887 120583 120577119865) 119891perp120572(DY)1119879119902119901 (119909119901 119887 120583 120577119865) (57)

with 119891perp120572(DY)1119879119902119901



(58)


1198894Δ12059011988911987621198891199101198892qperp = 12059004120587 intinfin0

11988911988711988721198691 (119902perp119887) sum119902119894119895


(59)




(60)






1

01

001

1E-3

1E-4

1Ｏ

(xＥ

perpQ

)

0 1 2 3 4

b (Ge６-1)

Ｒ = 01

１2 = 24Ge６2

１2 = 10Ge６2

１2 = 1000Ge６2

１2 = 24Ge６2

１2 = 10Ge６2

１2 = 1000Ge６2

1 2 3 4 5 60

Ｅperp (GeV)

0

1

2

3

4

5

＜Ｍ

Ｏ＜

1Ｏ

(xb

Q)


where the notation


(62)


F [11989111199021205871198911119902119901] = sum119902

1198902119902 int 1198892119887(2120587)2


119902

1198902119902sdot int 1198892119887

(2120587)2 119890119894qperp sdotb1198911119902120587 (119909120587 119887 119876) 1198911119902119901 (119909119901 119887 119876)

(63)



] = sum119902

1198902119902



] = sum119902

1198902119902


sdot ℎ120573perp1119902119901

(119909119901 119887 119876) = sum119902

1198902119902 intinfin0


1198911119902120587NP +119878P)

(64)











1198751198911119902119902 = 43 ( 1 + 1199112(1 minus 119911)+ + 32120575 (1 minus 119911)) (66)






ℎperp1119902120587 (119909 119887 119876) = 119894119887120572120587 ℎ120572perp1119902120587 (119909 119887 119876) (68)






１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

x=01

x=01x=01

x=01

1

01

001

1E-3

1E-4

Ｅperp

－Ｊ

perp 1４ＫＪ

(xＥ

perpQ

)

1

01

001

1E-3

1E-4

Ｅperp

－Ｊ

perp 1４ＫＪ

(xＥ

perpQ

)

00

02

04

06

08

10

perp 1４ＫＪ

(xb

Q)

00

02

04

06

08

10 perp 1

４ＫＪ

(xb

Q)

1 2 3 4 50 6

b (Ge６-1)1 2 3 4 50 6

b (Ge６-1)

05 10 15 20 2500

Ｅperp (GeV)05 10 15 20 2500

Ｅperp (GeV)






1

01

001

1E-3

１2=025 Ge６2

１2=10 Ge６2

１2=1000 Ge６2

１2=025 Ge６2

１2=10 Ge６2

１2=1000 Ge６2

x=01 x=01

Ｅ４

－

Ｂperp 1Ｋ

(xＥ

４Q

)

1 2 3 4 50

Ｅ４ (GeV)

000

005

010

015

020

Ｂperp 1Ｋ

(xb

Q)

1 2 3 4 50

b (Ge６-1)


02

00

06

08

04

1

＞

＞Ｋperp

(Ge６

-1)



05 1000 20 25 3015

Ｋperp (GeV)





03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

01

ＲＪ

01 1

x



COMPASS 2015 data



COMPASS 2015 data

05 10 15 20 2500


Ｒ








020

015

010

005

000

]－

020

015

010

005

000

]－

020

015

010

005

000

]－

020

015

010

005

000

]－

01 02 03 0400

ＲＪ

02 04 06 0800

Ｒ

05 10 15 2000

Ｋ４ (GeV)00 02 04 06 08minus02

Ｒ















Acknowledgments


References













































































































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery







1205942 (120572) = 119872sum119894=1

119873119894sum119895=1


(21)


1198921199021205871 = 0082 plusmn 00221198921199021205872 = 0394 plusmn 0103 (22)






119894

119862119902larr997888119894otimes 119865119894119901 (119909 120583119887)

(23)


119894

119862119902larr997888119894otimes 119865119894120587 (119909 120583119887)

(24)




F (120572119904 (119876))sum119894

119862119902larr997888119894otimes 1198911119894119901 (119909 120583119887)

(25)


F (120572119904 (119876))sum119894

119862119902larr997888119894otimes 1198911119894120587 (119909 120583119887)

(26)


1198911119902119901 (119909 119896perp 119876) = intinfin0

1198891198871198872120587 1198690 (119896perp119887) 1198911119902119901 (119909 119887 119876) (27)

1198911119902120587 (119909 119896perp 119876) = intinfin0

1198891198871198872120587 1198690 (119896perp119887) 1198911119902120587 (119909 119887 119876) (28)




119894

120575119862119902larr997888119894otimes ℎ1119894119901 (119909 120583119887)

(29)

ℎ1119902119901 (119909 119896perp 119876) = intinfin0

1198891198871198872120587 1198690 (119896perp119887) ℎ1119902119901 (119909 119887 119876) (30)




119891perp(DY)1119879119902119867 (119909 kperp 120583)


01

001

1E-300 06 12 18 24 30

Ｋperp (GeV)00 06 12 18 24 30

Ｋperp (GeV)

00 06 12 18 24 30

Ｋperp (GeV)00 06 12 18 24 30

Ｋperp (GeV)

00 06 12 18 24 30

Ｋperp (GeV)00 06 12 18 24 30

Ｋperp (GeV)

00 06 12 18 24 30

Ｋperp (GeV)00 06 12 18 24 30

Ｋperp (GeV)

＞2

＞Ｋperp＞Ｒ

(nb

GeV

nuc

leon

)

01

001

1E-3

＞2

＞Ｋperp＞Ｒ

(nb

GeV

nuc

leon

)

01

001

1E-3

＞2

＞Ｋperp＞Ｒ

(nb

GeV

nuc

leon

)

01

001

1E-3

01

001

1E-3

01

001

1E-3

01

001

1E-3

01

001

1E-3

＞2

＞Ｋperp＞Ｒ

(nb

GeV

nuc

leon

)







ℎperp(DY)1119902119867 (119909 kperp 120583) (31)




1119879119902119867can also be


119891perp120572(DY)1119879119902119867 (119909 119887 120583)= (minus1198941198871205722 )sum

119894

Δ119862119879119902larr997888119894 otimes 119891(3)119894119901 (1199091015840 11990910158401015840 120583) (32)

Here119891(3)119894119901




(33)



119862BM119902larr997888119894



convoluted with 119879(120590)119894119867119865



119896perp119872119867


0119889119887( 11988722120587) 1198691 (119896perp119887)FSiv (120572119904 (119876))sum

119894


(35)

119896perp119872119867


0119889119887( 11988722120587) 1198691 (119896perp119887)HBM (120572119904 (119876))sum

119894

119862BM119902larr997888119894


and 119879119902119865(119909 119909 120583119887) and 119879(120590)119894119867119865


119879119902119867119865 (119909 119909) = int1198892119896perp100381610038161003816100381610038161198962perp10038161003816100381610038161003816119872119867


(37)

119879(120590)119902119867119865 (119909 119909) = int1198892119896perp100381610038161003816100381610038161198962perp10038161003816100381610038161003816119872119867


(38)





119867119860 (119875120587) + 119867119861 (119875119873) 997888rarr120574lowast (119902) + 119883 997888rarr

119897+ (ℓ) + 119897minus (ℓ1015840) + 119883(39)


119904 = (119875120587 + 119875119873)2 119909120587119873 = 11987622119875120587119873 sdot 119902 119909119865 = 2119902119871119904 = 119909120587 minus 119909119873120591 = 1198762119904 = 119909120587119909119873119910 = 12 ln

119902+119902minus = 12 ln 119909120587119909119873

(40)




119909120587119873 = plusmn119909119865 + radic1199092119865 + 41205912

119909120587119873 = radic120591119890plusmn119910(41)



119889412059011988011988011988911987621198891199101198892qperp = 1205900 int 1198892119887(2120587)2 119890119894


+ 119884119880119880 (119876 119902perp)(42)



119880119880 (119876 119887) = 119867119880119880 (119876 120583)sdot sum119902119902

1198902119902119891sub119902120587 (119909120587 119887 120583 120577119865) 119891sub

119902119901 (119909119901 119887 120583 120577119865) (43)





119867JCC (119876 120583) = 1 + 120572119904 (120583)2120587 119862119865 (3 ln 11987621205832 minus ln211987621205832 + 1205872

minus 8) (44)


(45)


FJCC (120572119904 (119876)) = 1 + O (1205722119904 ) (46)



119862119902larr9978881199021015840 (119909 119887 120583 120577119865)= 1205751199021199021015840 [120575 (1 minus 119909) + 120572119904120587 (1198621198652 (1 minus 119909))] (48)

119862119902larr997888119892 (119909 119887 120583 120577119865) = 120572119904120587 119879119877119909 (1 minus 119909) (49)


functions using




(51)

1198621015840119902larr997888119892 (119909 119887 120583119887) = 120572119904120587 119879119877119909 (1 minus 119909) (52)






(53)


119889412059011988911987621198891199101198892qperp = 1205900 intinfin0

1198891198871198872120587 1198690 (119902perp119887) times 119880119880 (119876 119887) (54)



119860119880119879 = 1198894Δ12059011988911987621198891199101198892qperp119889412059011988911987621198891199101198892qperp (55)




119880119879 (119876 119887)+ 119884120573119880119879 (119876 119902perp)

(56)




120572119880119879 (119876 119887) = 119867119880119879 (119876 120583)sdot sum119902119902

11989021199021198911119902120587 (119909120587 119887 120583 120577119865) 119891perp120572(DY)1119879119902119901 (119909119901 119887 120583 120577119865) (57)

with 119891perp120572(DY)1119879119902119901



(58)


1198894Δ12059011988911987621198891199101198892qperp = 12059004120587 intinfin0

11988911988711988721198691 (119902perp119887) sum119902119894119895


(59)




(60)






1

01

001

1E-3

1E-4

1Ｏ

(xＥ

perpQ

)

0 1 2 3 4

b (Ge６-1)

Ｒ = 01

１2 = 24Ge６2

１2 = 10Ge６2

１2 = 1000Ge６2

１2 = 24Ge６2

１2 = 10Ge６2

１2 = 1000Ge６2

1 2 3 4 5 60

Ｅperp (GeV)

0

1

2

3

4

5

＜Ｍ

Ｏ＜

1Ｏ

(xb

Q)


where the notation


(62)


F [11989111199021205871198911119902119901] = sum119902

1198902119902 int 1198892119887(2120587)2


119902

1198902119902sdot int 1198892119887

(2120587)2 119890119894qperp sdotb1198911119902120587 (119909120587 119887 119876) 1198911119902119901 (119909119901 119887 119876)

(63)



] = sum119902

1198902119902



] = sum119902

1198902119902


sdot ℎ120573perp1119902119901

(119909119901 119887 119876) = sum119902

1198902119902 intinfin0


1198911119902120587NP +119878P)

(64)











1198751198911119902119902 = 43 ( 1 + 1199112(1 minus 119911)+ + 32120575 (1 minus 119911)) (66)






ℎperp1119902120587 (119909 119887 119876) = 119894119887120572120587 ℎ120572perp1119902120587 (119909 119887 119876) (68)






１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

x=01

x=01x=01

x=01

1

01

001

1E-3

1E-4

Ｅperp

－Ｊ

perp 1４ＫＪ

(xＥ

perpQ

)

1

01

001

1E-3

1E-4

Ｅperp

－Ｊ

perp 1４ＫＪ

(xＥ

perpQ

)

00

02

04

06

08

10

perp 1４ＫＪ

(xb

Q)

00

02

04

06

08

10 perp 1

４ＫＪ

(xb

Q)

1 2 3 4 50 6

b (Ge６-1)1 2 3 4 50 6

b (Ge６-1)

05 10 15 20 2500

Ｅperp (GeV)05 10 15 20 2500

Ｅperp (GeV)






1

01

001

1E-3

１2=025 Ge６2

１2=10 Ge６2

１2=1000 Ge６2

１2=025 Ge６2

１2=10 Ge６2

１2=1000 Ge６2

x=01 x=01

Ｅ４

－

Ｂperp 1Ｋ

(xＥ

４Q

)

1 2 3 4 50

Ｅ４ (GeV)

000

005

010

015

020

Ｂperp 1Ｋ

(xb

Q)

1 2 3 4 50

b (Ge６-1)


02

00

06

08

04

1

＞

＞Ｋperp

(Ge６

-1)



05 1000 20 25 3015

Ｋperp (GeV)





03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

01

ＲＪ

01 1

x



COMPASS 2015 data



COMPASS 2015 data

05 10 15 20 2500


Ｒ








020

015

010

005

000

]－

020

015

010

005

000

]－

020

015

010

005

000

]－

020

015

010

005

000

]－

01 02 03 0400

ＲＪ

02 04 06 0800

Ｒ

05 10 15 2000

Ｋ４ (GeV)00 02 04 06 08minus02

Ｒ















Acknowledgments


References













































































































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






01

001

1E-300 06 12 18 24 30

Ｋperp (GeV)00 06 12 18 24 30

Ｋperp (GeV)

00 06 12 18 24 30

Ｋperp (GeV)00 06 12 18 24 30

Ｋperp (GeV)

00 06 12 18 24 30

Ｋperp (GeV)00 06 12 18 24 30

Ｋperp (GeV)

00 06 12 18 24 30

Ｋperp (GeV)00 06 12 18 24 30

Ｋperp (GeV)

＞2

＞Ｋperp＞Ｒ

(nb

GeV

nuc

leon

)

01

001

1E-3

＞2

＞Ｋperp＞Ｒ

(nb

GeV

nuc

leon

)

01

001

1E-3

＞2

＞Ｋperp＞Ｒ

(nb

GeV

nuc

leon

)

01

001

1E-3

01

001

1E-3

01

001

1E-3

01

001

1E-3

01

001

1E-3

＞2

＞Ｋperp＞Ｒ

(nb

GeV

nuc

leon

)







ℎperp(DY)1119902119867 (119909 kperp 120583) (31)




1119879119902119867can also be


119891perp120572(DY)1119879119902119867 (119909 119887 120583)= (minus1198941198871205722 )sum

119894

Δ119862119879119902larr997888119894 otimes 119891(3)119894119901 (1199091015840 11990910158401015840 120583) (32)

Here119891(3)119894119901




(33)



119862BM119902larr997888119894



convoluted with 119879(120590)119894119867119865



119896perp119872119867


0119889119887( 11988722120587) 1198691 (119896perp119887)FSiv (120572119904 (119876))sum

119894


(35)

119896perp119872119867


0119889119887( 11988722120587) 1198691 (119896perp119887)HBM (120572119904 (119876))sum

119894

119862BM119902larr997888119894


and 119879119902119865(119909 119909 120583119887) and 119879(120590)119894119867119865


119879119902119867119865 (119909 119909) = int1198892119896perp100381610038161003816100381610038161198962perp10038161003816100381610038161003816119872119867


(37)

119879(120590)119902119867119865 (119909 119909) = int1198892119896perp100381610038161003816100381610038161198962perp10038161003816100381610038161003816119872119867


(38)





119867119860 (119875120587) + 119867119861 (119875119873) 997888rarr120574lowast (119902) + 119883 997888rarr

119897+ (ℓ) + 119897minus (ℓ1015840) + 119883(39)


119904 = (119875120587 + 119875119873)2 119909120587119873 = 11987622119875120587119873 sdot 119902 119909119865 = 2119902119871119904 = 119909120587 minus 119909119873120591 = 1198762119904 = 119909120587119909119873119910 = 12 ln

119902+119902minus = 12 ln 119909120587119909119873

(40)




119909120587119873 = plusmn119909119865 + radic1199092119865 + 41205912

119909120587119873 = radic120591119890plusmn119910(41)



119889412059011988011988011988911987621198891199101198892qperp = 1205900 int 1198892119887(2120587)2 119890119894


+ 119884119880119880 (119876 119902perp)(42)



119880119880 (119876 119887) = 119867119880119880 (119876 120583)sdot sum119902119902

1198902119902119891sub119902120587 (119909120587 119887 120583 120577119865) 119891sub

119902119901 (119909119901 119887 120583 120577119865) (43)





119867JCC (119876 120583) = 1 + 120572119904 (120583)2120587 119862119865 (3 ln 11987621205832 minus ln211987621205832 + 1205872

minus 8) (44)


(45)


FJCC (120572119904 (119876)) = 1 + O (1205722119904 ) (46)



119862119902larr9978881199021015840 (119909 119887 120583 120577119865)= 1205751199021199021015840 [120575 (1 minus 119909) + 120572119904120587 (1198621198652 (1 minus 119909))] (48)

119862119902larr997888119892 (119909 119887 120583 120577119865) = 120572119904120587 119879119877119909 (1 minus 119909) (49)


functions using




(51)

1198621015840119902larr997888119892 (119909 119887 120583119887) = 120572119904120587 119879119877119909 (1 minus 119909) (52)






(53)


119889412059011988911987621198891199101198892qperp = 1205900 intinfin0

1198891198871198872120587 1198690 (119902perp119887) times 119880119880 (119876 119887) (54)



119860119880119879 = 1198894Δ12059011988911987621198891199101198892qperp119889412059011988911987621198891199101198892qperp (55)




119880119879 (119876 119887)+ 119884120573119880119879 (119876 119902perp)

(56)




120572119880119879 (119876 119887) = 119867119880119879 (119876 120583)sdot sum119902119902

11989021199021198911119902120587 (119909120587 119887 120583 120577119865) 119891perp120572(DY)1119879119902119901 (119909119901 119887 120583 120577119865) (57)

with 119891perp120572(DY)1119879119902119901



(58)


1198894Δ12059011988911987621198891199101198892qperp = 12059004120587 intinfin0

11988911988711988721198691 (119902perp119887) sum119902119894119895


(59)




(60)






1

01

001

1E-3

1E-4

1Ｏ

(xＥ

perpQ

)

0 1 2 3 4

b (Ge６-1)

Ｒ = 01

１2 = 24Ge６2

１2 = 10Ge６2

１2 = 1000Ge６2

１2 = 24Ge６2

１2 = 10Ge６2

１2 = 1000Ge６2

1 2 3 4 5 60

Ｅperp (GeV)

0

1

2

3

4

5

＜Ｍ

Ｏ＜

1Ｏ

(xb

Q)


where the notation


(62)


F [11989111199021205871198911119902119901] = sum119902

1198902119902 int 1198892119887(2120587)2


119902

1198902119902sdot int 1198892119887

(2120587)2 119890119894qperp sdotb1198911119902120587 (119909120587 119887 119876) 1198911119902119901 (119909119901 119887 119876)

(63)



] = sum119902

1198902119902



] = sum119902

1198902119902


sdot ℎ120573perp1119902119901

(119909119901 119887 119876) = sum119902

1198902119902 intinfin0


1198911119902120587NP +119878P)

(64)











1198751198911119902119902 = 43 ( 1 + 1199112(1 minus 119911)+ + 32120575 (1 minus 119911)) (66)






ℎperp1119902120587 (119909 119887 119876) = 119894119887120572120587 ℎ120572perp1119902120587 (119909 119887 119876) (68)






１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

x=01

x=01x=01

x=01

1

01

001

1E-3

1E-4

Ｅperp

－Ｊ

perp 1４ＫＪ

(xＥ

perpQ

)

1

01

001

1E-3

1E-4

Ｅperp

－Ｊ

perp 1４ＫＪ

(xＥ

perpQ

)

00

02

04

06

08

10

perp 1４ＫＪ

(xb

Q)

00

02

04

06

08

10 perp 1

４ＫＪ

(xb

Q)

1 2 3 4 50 6

b (Ge６-1)1 2 3 4 50 6

b (Ge６-1)

05 10 15 20 2500

Ｅperp (GeV)05 10 15 20 2500

Ｅperp (GeV)






1

01

001

1E-3

１2=025 Ge６2

１2=10 Ge６2

１2=1000 Ge６2

１2=025 Ge６2

１2=10 Ge６2

１2=1000 Ge６2

x=01 x=01

Ｅ４

－

Ｂperp 1Ｋ

(xＥ

４Q

)

1 2 3 4 50

Ｅ４ (GeV)

000

005

010

015

020

Ｂperp 1Ｋ

(xb

Q)

1 2 3 4 50

b (Ge６-1)


02

00

06

08

04

1

＞

＞Ｋperp

(Ge６

-1)



05 1000 20 25 3015

Ｋperp (GeV)





03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

01

ＲＪ

01 1

x



COMPASS 2015 data



COMPASS 2015 data

05 10 15 20 2500


Ｒ








020

015

010

005

000

]－

020

015

010

005

000

]－

020

015

010

005

000

]－

020

015

010

005

000

]－

01 02 03 0400

ＲＪ

02 04 06 0800

Ｒ

05 10 15 2000

Ｋ４ (GeV)00 02 04 06 08minus02

Ｒ















Acknowledgments


References













































































































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery







1119879119902119867can also be


119891perp120572(DY)1119879119902119867 (119909 119887 120583)= (minus1198941198871205722 )sum

119894

Δ119862119879119902larr997888119894 otimes 119891(3)119894119901 (1199091015840 11990910158401015840 120583) (32)

Here119891(3)119894119901




(33)



119862BM119902larr997888119894



convoluted with 119879(120590)119894119867119865



119896perp119872119867


0119889119887( 11988722120587) 1198691 (119896perp119887)FSiv (120572119904 (119876))sum

119894


(35)

119896perp119872119867


0119889119887( 11988722120587) 1198691 (119896perp119887)HBM (120572119904 (119876))sum

119894

119862BM119902larr997888119894


and 119879119902119865(119909 119909 120583119887) and 119879(120590)119894119867119865


119879119902119867119865 (119909 119909) = int1198892119896perp100381610038161003816100381610038161198962perp10038161003816100381610038161003816119872119867


(37)

119879(120590)119902119867119865 (119909 119909) = int1198892119896perp100381610038161003816100381610038161198962perp10038161003816100381610038161003816119872119867


(38)





119867119860 (119875120587) + 119867119861 (119875119873) 997888rarr120574lowast (119902) + 119883 997888rarr

119897+ (ℓ) + 119897minus (ℓ1015840) + 119883(39)


119904 = (119875120587 + 119875119873)2 119909120587119873 = 11987622119875120587119873 sdot 119902 119909119865 = 2119902119871119904 = 119909120587 minus 119909119873120591 = 1198762119904 = 119909120587119909119873119910 = 12 ln

119902+119902minus = 12 ln 119909120587119909119873

(40)




119909120587119873 = plusmn119909119865 + radic1199092119865 + 41205912

119909120587119873 = radic120591119890plusmn119910(41)



119889412059011988011988011988911987621198891199101198892qperp = 1205900 int 1198892119887(2120587)2 119890119894


+ 119884119880119880 (119876 119902perp)(42)



119880119880 (119876 119887) = 119867119880119880 (119876 120583)sdot sum119902119902

1198902119902119891sub119902120587 (119909120587 119887 120583 120577119865) 119891sub

119902119901 (119909119901 119887 120583 120577119865) (43)





119867JCC (119876 120583) = 1 + 120572119904 (120583)2120587 119862119865 (3 ln 11987621205832 minus ln211987621205832 + 1205872

minus 8) (44)


(45)


FJCC (120572119904 (119876)) = 1 + O (1205722119904 ) (46)



119862119902larr9978881199021015840 (119909 119887 120583 120577119865)= 1205751199021199021015840 [120575 (1 minus 119909) + 120572119904120587 (1198621198652 (1 minus 119909))] (48)

119862119902larr997888119892 (119909 119887 120583 120577119865) = 120572119904120587 119879119877119909 (1 minus 119909) (49)


functions using




(51)

1198621015840119902larr997888119892 (119909 119887 120583119887) = 120572119904120587 119879119877119909 (1 minus 119909) (52)






(53)


119889412059011988911987621198891199101198892qperp = 1205900 intinfin0

1198891198871198872120587 1198690 (119902perp119887) times 119880119880 (119876 119887) (54)



119860119880119879 = 1198894Δ12059011988911987621198891199101198892qperp119889412059011988911987621198891199101198892qperp (55)




119880119879 (119876 119887)+ 119884120573119880119879 (119876 119902perp)

(56)




120572119880119879 (119876 119887) = 119867119880119879 (119876 120583)sdot sum119902119902

11989021199021198911119902120587 (119909120587 119887 120583 120577119865) 119891perp120572(DY)1119879119902119901 (119909119901 119887 120583 120577119865) (57)

with 119891perp120572(DY)1119879119902119901



(58)


1198894Δ12059011988911987621198891199101198892qperp = 12059004120587 intinfin0

11988911988711988721198691 (119902perp119887) sum119902119894119895


(59)




(60)






1

01

001

1E-3

1E-4

1Ｏ

(xＥ

perpQ

)

0 1 2 3 4

b (Ge６-1)

Ｒ = 01

１2 = 24Ge６2

１2 = 10Ge６2

１2 = 1000Ge６2

１2 = 24Ge６2

１2 = 10Ge６2

１2 = 1000Ge６2

1 2 3 4 5 60

Ｅperp (GeV)

0

1

2

3

4

5

＜Ｍ

Ｏ＜

1Ｏ

(xb

Q)


where the notation


(62)


F [11989111199021205871198911119902119901] = sum119902

1198902119902 int 1198892119887(2120587)2


119902

1198902119902sdot int 1198892119887

(2120587)2 119890119894qperp sdotb1198911119902120587 (119909120587 119887 119876) 1198911119902119901 (119909119901 119887 119876)

(63)



] = sum119902

1198902119902



] = sum119902

1198902119902


sdot ℎ120573perp1119902119901

(119909119901 119887 119876) = sum119902

1198902119902 intinfin0


1198911119902120587NP +119878P)

(64)











1198751198911119902119902 = 43 ( 1 + 1199112(1 minus 119911)+ + 32120575 (1 minus 119911)) (66)






ℎperp1119902120587 (119909 119887 119876) = 119894119887120572120587 ℎ120572perp1119902120587 (119909 119887 119876) (68)






１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

x=01

x=01x=01

x=01

1

01

001

1E-3

1E-4

Ｅperp

－Ｊ

perp 1４ＫＪ

(xＥ

perpQ

)

1

01

001

1E-3

1E-4

Ｅperp

－Ｊ

perp 1４ＫＪ

(xＥ

perpQ

)

00

02

04

06

08

10

perp 1４ＫＪ

(xb

Q)

00

02

04

06

08

10 perp 1

４ＫＪ

(xb

Q)

1 2 3 4 50 6

b (Ge６-1)1 2 3 4 50 6

b (Ge６-1)

05 10 15 20 2500

Ｅperp (GeV)05 10 15 20 2500

Ｅperp (GeV)






1

01

001

1E-3

１2=025 Ge６2

１2=10 Ge６2

１2=1000 Ge６2

１2=025 Ge６2

１2=10 Ge６2

１2=1000 Ge６2

x=01 x=01

Ｅ４

－

Ｂperp 1Ｋ

(xＥ

４Q

)

1 2 3 4 50

Ｅ４ (GeV)

000

005

010

015

020

Ｂperp 1Ｋ

(xb

Q)

1 2 3 4 50

b (Ge６-1)


02

00

06

08

04

1

＞

＞Ｋperp

(Ge６

-1)



05 1000 20 25 3015

Ｋperp (GeV)





03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

01

ＲＪ

01 1

x



COMPASS 2015 data



COMPASS 2015 data

05 10 15 20 2500


Ｒ








020

015

010

005

000

]－

020

015

010

005

000

]－

020

015

010

005

000

]－

020

015

010

005

000

]－

01 02 03 0400

ＲＪ

02 04 06 0800

Ｒ

05 10 15 2000

Ｋ４ (GeV)00 02 04 06 08minus02

Ｒ















Acknowledgments


References













































































































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery







119909120587119873 = plusmn119909119865 + radic1199092119865 + 41205912

119909120587119873 = radic120591119890plusmn119910(41)



119889412059011988011988011988911987621198891199101198892qperp = 1205900 int 1198892119887(2120587)2 119890119894


+ 119884119880119880 (119876 119902perp)(42)



119880119880 (119876 119887) = 119867119880119880 (119876 120583)sdot sum119902119902

1198902119902119891sub119902120587 (119909120587 119887 120583 120577119865) 119891sub

119902119901 (119909119901 119887 120583 120577119865) (43)





119867JCC (119876 120583) = 1 + 120572119904 (120583)2120587 119862119865 (3 ln 11987621205832 minus ln211987621205832 + 1205872

minus 8) (44)


(45)


FJCC (120572119904 (119876)) = 1 + O (1205722119904 ) (46)



119862119902larr9978881199021015840 (119909 119887 120583 120577119865)= 1205751199021199021015840 [120575 (1 minus 119909) + 120572119904120587 (1198621198652 (1 minus 119909))] (48)

119862119902larr997888119892 (119909 119887 120583 120577119865) = 120572119904120587 119879119877119909 (1 minus 119909) (49)


functions using




(51)

1198621015840119902larr997888119892 (119909 119887 120583119887) = 120572119904120587 119879119877119909 (1 minus 119909) (52)






(53)


119889412059011988911987621198891199101198892qperp = 1205900 intinfin0

1198891198871198872120587 1198690 (119902perp119887) times 119880119880 (119876 119887) (54)



119860119880119879 = 1198894Δ12059011988911987621198891199101198892qperp119889412059011988911987621198891199101198892qperp (55)




119880119879 (119876 119887)+ 119884120573119880119879 (119876 119902perp)

(56)




120572119880119879 (119876 119887) = 119867119880119879 (119876 120583)sdot sum119902119902

11989021199021198911119902120587 (119909120587 119887 120583 120577119865) 119891perp120572(DY)1119879119902119901 (119909119901 119887 120583 120577119865) (57)

with 119891perp120572(DY)1119879119902119901



(58)


1198894Δ12059011988911987621198891199101198892qperp = 12059004120587 intinfin0

11988911988711988721198691 (119902perp119887) sum119902119894119895


(59)




(60)






1

01

001

1E-3

1E-4

1Ｏ

(xＥ

perpQ

)

0 1 2 3 4

b (Ge６-1)

Ｒ = 01

１2 = 24Ge６2

１2 = 10Ge６2

１2 = 1000Ge６2

１2 = 24Ge６2

１2 = 10Ge６2

１2 = 1000Ge６2

1 2 3 4 5 60

Ｅperp (GeV)

0

1

2

3

4

5

＜Ｍ

Ｏ＜

1Ｏ

(xb

Q)


where the notation


(62)


F [11989111199021205871198911119902119901] = sum119902

1198902119902 int 1198892119887(2120587)2


119902

1198902119902sdot int 1198892119887

(2120587)2 119890119894qperp sdotb1198911119902120587 (119909120587 119887 119876) 1198911119902119901 (119909119901 119887 119876)

(63)



] = sum119902

1198902119902



] = sum119902

1198902119902


sdot ℎ120573perp1119902119901

(119909119901 119887 119876) = sum119902

1198902119902 intinfin0


1198911119902120587NP +119878P)

(64)











1198751198911119902119902 = 43 ( 1 + 1199112(1 minus 119911)+ + 32120575 (1 minus 119911)) (66)






ℎperp1119902120587 (119909 119887 119876) = 119894119887120572120587 ℎ120572perp1119902120587 (119909 119887 119876) (68)






１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

x=01

x=01x=01

x=01

1

01

001

1E-3

1E-4

Ｅperp

－Ｊ

perp 1４ＫＪ

(xＥ

perpQ

)

1

01

001

1E-3

1E-4

Ｅperp

－Ｊ

perp 1４ＫＪ

(xＥ

perpQ

)

00

02

04

06

08

10

perp 1４ＫＪ

(xb

Q)

00

02

04

06

08

10 perp 1

４ＫＪ

(xb

Q)

1 2 3 4 50 6

b (Ge６-1)1 2 3 4 50 6

b (Ge６-1)

05 10 15 20 2500

Ｅperp (GeV)05 10 15 20 2500

Ｅperp (GeV)






1

01

001

1E-3

１2=025 Ge６2

１2=10 Ge６2

１2=1000 Ge６2

１2=025 Ge６2

１2=10 Ge６2

１2=1000 Ge６2

x=01 x=01

Ｅ４

－

Ｂperp 1Ｋ

(xＥ

４Q

)

1 2 3 4 50

Ｅ４ (GeV)

000

005

010

015

020

Ｂperp 1Ｋ

(xb

Q)

1 2 3 4 50

b (Ge６-1)


02

00

06

08

04

1

＞

＞Ｋperp

(Ge６

-1)



05 1000 20 25 3015

Ｋperp (GeV)





03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

01

ＲＪ

01 1

x



COMPASS 2015 data



COMPASS 2015 data

05 10 15 20 2500


Ｒ








020

015

010

005

000

]－

020

015

010

005

000

]－

020

015

010

005

000

]－

020

015

010

005

000

]－

01 02 03 0400

ＲＪ

02 04 06 0800

Ｒ

05 10 15 2000

Ｋ４ (GeV)00 02 04 06 08minus02

Ｒ















Acknowledgments


References













































































































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery









(53)


119889412059011988911987621198891199101198892qperp = 1205900 intinfin0

1198891198871198872120587 1198690 (119902perp119887) times 119880119880 (119876 119887) (54)



119860119880119879 = 1198894Δ12059011988911987621198891199101198892qperp119889412059011988911987621198891199101198892qperp (55)




119880119879 (119876 119887)+ 119884120573119880119879 (119876 119902perp)

(56)




120572119880119879 (119876 119887) = 119867119880119879 (119876 120583)sdot sum119902119902

11989021199021198911119902120587 (119909120587 119887 120583 120577119865) 119891perp120572(DY)1119879119902119901 (119909119901 119887 120583 120577119865) (57)

with 119891perp120572(DY)1119879119902119901



(58)


1198894Δ12059011988911987621198891199101198892qperp = 12059004120587 intinfin0

11988911988711988721198691 (119902perp119887) sum119902119894119895


(59)




(60)






1

01

001

1E-3

1E-4

1Ｏ

(xＥ

perpQ

)

0 1 2 3 4

b (Ge６-1)

Ｒ = 01

１2 = 24Ge６2

１2 = 10Ge６2

１2 = 1000Ge６2

１2 = 24Ge６2

１2 = 10Ge６2

１2 = 1000Ge６2

1 2 3 4 5 60

Ｅperp (GeV)

0

1

2

3

4

5

＜Ｍ

Ｏ＜

1Ｏ

(xb

Q)


where the notation


(62)


F [11989111199021205871198911119902119901] = sum119902

1198902119902 int 1198892119887(2120587)2


119902

1198902119902sdot int 1198892119887

(2120587)2 119890119894qperp sdotb1198911119902120587 (119909120587 119887 119876) 1198911119902119901 (119909119901 119887 119876)

(63)



] = sum119902

1198902119902



] = sum119902

1198902119902


sdot ℎ120573perp1119902119901

(119909119901 119887 119876) = sum119902

1198902119902 intinfin0


1198911119902120587NP +119878P)

(64)











1198751198911119902119902 = 43 ( 1 + 1199112(1 minus 119911)+ + 32120575 (1 minus 119911)) (66)






ℎperp1119902120587 (119909 119887 119876) = 119894119887120572120587 ℎ120572perp1119902120587 (119909 119887 119876) (68)






１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

x=01

x=01x=01

x=01

1

01

001

1E-3

1E-4

Ｅperp

－Ｊ

perp 1４ＫＪ

(xＥ

perpQ

)

1

01

001

1E-3

1E-4

Ｅperp

－Ｊ

perp 1４ＫＪ

(xＥ

perpQ

)

00

02

04

06

08

10

perp 1４ＫＪ

(xb

Q)

00

02

04

06

08

10 perp 1

４ＫＪ

(xb

Q)

1 2 3 4 50 6

b (Ge６-1)1 2 3 4 50 6

b (Ge６-1)

05 10 15 20 2500

Ｅperp (GeV)05 10 15 20 2500

Ｅperp (GeV)






1

01

001

1E-3

１2=025 Ge６2

１2=10 Ge６2

１2=1000 Ge６2

１2=025 Ge６2

１2=10 Ge６2

１2=1000 Ge６2

x=01 x=01

Ｅ４

－

Ｂperp 1Ｋ

(xＥ

４Q

)

1 2 3 4 50

Ｅ４ (GeV)

000

005

010

015

020

Ｂperp 1Ｋ

(xb

Q)

1 2 3 4 50

b (Ge６-1)


02

00

06

08

04

1

＞

＞Ｋperp

(Ge６

-1)



05 1000 20 25 3015

Ｋperp (GeV)





03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

01

ＲＪ

01 1

x



COMPASS 2015 data



COMPASS 2015 data

05 10 15 20 2500


Ｒ








020

015

010

005

000

]－

020

015

010

005

000

]－

020

015

010

005

000

]－

020

015

010

005

000

]－

01 02 03 0400

ＲＪ

02 04 06 0800

Ｒ

05 10 15 2000

Ｋ４ (GeV)00 02 04 06 08minus02

Ｒ















Acknowledgments


References













































































































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






1

01

001

1E-3

1E-4

1Ｏ

(xＥ

perpQ

)

0 1 2 3 4

b (Ge６-1)

Ｒ = 01

１2 = 24Ge６2

１2 = 10Ge６2

１2 = 1000Ge６2

１2 = 24Ge６2

１2 = 10Ge６2

１2 = 1000Ge６2

1 2 3 4 5 60

Ｅperp (GeV)

0

1

2

3

4

5

＜Ｍ

Ｏ＜

1Ｏ

(xb

Q)


where the notation


(62)


F [11989111199021205871198911119902119901] = sum119902

1198902119902 int 1198892119887(2120587)2


119902

1198902119902sdot int 1198892119887

(2120587)2 119890119894qperp sdotb1198911119902120587 (119909120587 119887 119876) 1198911119902119901 (119909119901 119887 119876)

(63)



] = sum119902

1198902119902



] = sum119902

1198902119902


sdot ℎ120573perp1119902119901

(119909119901 119887 119876) = sum119902

1198902119902 intinfin0


1198911119902120587NP +119878P)

(64)











1198751198911119902119902 = 43 ( 1 + 1199112(1 minus 119911)+ + 32120575 (1 minus 119911)) (66)






ℎperp1119902120587 (119909 119887 119876) = 119894119887120572120587 ℎ120572perp1119902120587 (119909 119887 119876) (68)






１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

x=01

x=01x=01

x=01

1

01

001

1E-3

1E-4

Ｅperp

－Ｊ

perp 1４ＫＪ

(xＥ

perpQ

)

1

01

001

1E-3

1E-4

Ｅperp

－Ｊ

perp 1４ＫＪ

(xＥ

perpQ

)

00

02

04

06

08

10

perp 1４ＫＪ

(xb

Q)

00

02

04

06

08

10 perp 1

４ＫＪ

(xb

Q)

1 2 3 4 50 6

b (Ge６-1)1 2 3 4 50 6

b (Ge６-1)

05 10 15 20 2500

Ｅperp (GeV)05 10 15 20 2500

Ｅperp (GeV)






1

01

001

1E-3

１2=025 Ge６2

１2=10 Ge６2

１2=1000 Ge６2

１2=025 Ge６2

１2=10 Ge６2

１2=1000 Ge６2

x=01 x=01

Ｅ４

－

Ｂperp 1Ｋ

(xＥ

４Q

)

1 2 3 4 50

Ｅ４ (GeV)

000

005

010

015

020

Ｂperp 1Ｋ

(xb

Q)

1 2 3 4 50

b (Ge６-1)


02

00

06

08

04

1

＞

＞Ｋperp

(Ge６

-1)



05 1000 20 25 3015

Ｋperp (GeV)





03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

01

ＲＪ

01 1

x



COMPASS 2015 data



COMPASS 2015 data

05 10 15 20 2500


Ｒ








020

015

010

005

000

]－

020

015

010

005

000

]－

020

015

010

005

000

]－

020

015

010

005

000

]－

01 02 03 0400

ＲＪ

02 04 06 0800

Ｒ

05 10 15 2000

Ｋ４ (GeV)00 02 04 06 08minus02

Ｒ















Acknowledgments


References













































































































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery











1198751198911119902119902 = 43 ( 1 + 1199112(1 minus 119911)+ + 32120575 (1 minus 119911)) (66)






ℎperp1119902120587 (119909 119887 119876) = 119894119887120572120587 ℎ120572perp1119902120587 (119909 119887 119876) (68)






１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

x=01

x=01x=01

x=01

1

01

001

1E-3

1E-4

Ｅperp

－Ｊ

perp 1４ＫＪ

(xＥ

perpQ

)

1

01

001

1E-3

1E-4

Ｅperp

－Ｊ

perp 1４ＫＪ

(xＥ

perpQ

)

00

02

04

06

08

10

perp 1４ＫＪ

(xb

Q)

00

02

04

06

08

10 perp 1

４ＫＪ

(xb

Q)

1 2 3 4 50 6

b (Ge６-1)1 2 3 4 50 6

b (Ge６-1)

05 10 15 20 2500

Ｅperp (GeV)05 10 15 20 2500

Ｅperp (GeV)






1

01

001

1E-3

１2=025 Ge６2

１2=10 Ge６2

１2=1000 Ge６2

１2=025 Ge６2

１2=10 Ge６2

１2=1000 Ge６2

x=01 x=01

Ｅ４

－

Ｂperp 1Ｋ

(xＥ

４Q

)

1 2 3 4 50

Ｅ４ (GeV)

000

005

010

015

020

Ｂperp 1Ｋ

(xb

Q)

1 2 3 4 50

b (Ge６-1)


02

00

06

08

04

1

＞

＞Ｋperp

(Ge６

-1)



05 1000 20 25 3015

Ｋperp (GeV)





03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

01

ＲＪ

01 1

x



COMPASS 2015 data



COMPASS 2015 data

05 10 15 20 2500


Ｒ








020

015

010

005

000

]－

020

015

010

005

000

]－

020

015

010

005

000

]－

020

015

010

005

000

]－

01 02 03 0400

ＲＪ

02 04 06 0800

Ｒ

05 10 15 2000

Ｋ４ (GeV)00 02 04 06 08minus02

Ｒ















Acknowledgments


References













































































































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

１2=24 Ge６2

１2=10 Ge６2

１2=100 Ge６2

x=01

x=01x=01

x=01

1

01

001

1E-3

1E-4

Ｅperp

－Ｊ

perp 1４ＫＪ

(xＥ

perpQ

)

1

01

001

1E-3

1E-4

Ｅperp

－Ｊ

perp 1４ＫＪ

(xＥ

perpQ

)

00

02

04

06

08

10

perp 1４ＫＪ

(xb

Q)

00

02

04

06

08

10 perp 1

４ＫＪ

(xb

Q)

1 2 3 4 50 6

b (Ge６-1)1 2 3 4 50 6

b (Ge６-1)

05 10 15 20 2500

Ｅperp (GeV)05 10 15 20 2500

Ｅperp (GeV)






1

01

001

1E-3

１2=025 Ge６2

１2=10 Ge６2

１2=1000 Ge６2

１2=025 Ge６2

１2=10 Ge６2

１2=1000 Ge６2

x=01 x=01

Ｅ４

－

Ｂperp 1Ｋ

(xＥ

４Q

)

1 2 3 4 50

Ｅ４ (GeV)

000

005

010

015

020

Ｂperp 1Ｋ

(xb

Q)

1 2 3 4 50

b (Ge６-1)


02

00

06

08

04

1

＞

＞Ｋperp

(Ge６

-1)



05 1000 20 25 3015

Ｋperp (GeV)





03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

01

ＲＪ

01 1

x



COMPASS 2015 data



COMPASS 2015 data

05 10 15 20 2500


Ｒ








020

015

010

005

000

]－

020

015

010

005

000

]－

020

015

010

005

000

]－

020

015

010

005

000

]－

01 02 03 0400

ＲＪ

02 04 06 0800

Ｒ

05 10 15 2000

Ｋ４ (GeV)00 02 04 06 08minus02

Ｒ















Acknowledgments


References













































































































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






1

01

001

1E-3

１2=025 Ge６2

１2=10 Ge６2

１2=1000 Ge６2

１2=025 Ge６2

１2=10 Ge６2

１2=1000 Ge６2

x=01 x=01

Ｅ４

－

Ｂperp 1Ｋ

(xＥ

４Q

)

1 2 3 4 50

Ｅ４ (GeV)

000

005

010

015

020

Ｂperp 1Ｋ

(xb

Q)

1 2 3 4 50

b (Ge６-1)


02

00

06

08

04

1

＞

＞Ｋperp

(Ge６

-1)



05 1000 20 25 3015

Ｋperp (GeV)





03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

01

ＲＪ

01 1

x



COMPASS 2015 data



COMPASS 2015 data

05 10 15 20 2500


Ｒ








020

015

010

005

000

]－

020

015

010

005

000

]－

020

015

010

005

000

]－

020

015

010

005

000

]－

01 02 03 0400

ＲＪ

02 04 06 0800

Ｒ

05 10 15 2000

Ｋ４ (GeV)00 02 04 06 08minus02

Ｒ















Acknowledgments


References













































































































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

03

02

01

00

minus01

minus02

３ＣＰ５４

01

ＲＪ

01 1

x



COMPASS 2015 data



COMPASS 2015 data

05 10 15 20 2500


Ｒ








020

015

010

005

000

]－

020

015

010

005

000

]－

020

015

010

005

000

]－

020

015

010

005

000

]－

01 02 03 0400

ＲＪ

02 04 06 0800

Ｒ

05 10 15 2000

Ｋ４ (GeV)00 02 04 06 08minus02

Ｒ















Acknowledgments


References













































































































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






020

015

010

005

000

]－

020

015

010

005

000

]－

020

015

010

005

000

]－

020

015

010

005

000

]－

01 02 03 0400

ＲＪ

02 04 06 0800

Ｒ

05 10 15 2000

Ｋ４ (GeV)00 02 04 06 08minus02

Ｒ















Acknowledgments


References













































































































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery












Acknowledgments


References













































































































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery































































































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery





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