What is the twist of TMDs?

Como, June 12, 2013

Oleg TeryaevJINR, Dubna

Outline Definitions of twist TMDs as infinite towers of twists Quarks in vacuum and inside the

hadrons: TMDs vs non-local condensates

HT resummation and analyticity in DIS

HT resummation and scaling variables: DIS vs SIDIS

HT resummation in DIS

Higher Twists in spin-dependent DIS: GDH sum rule – finite sum of infinite number of divergent terms

Resummation of HT and analyticity

Comparing modified scaling variables

What is twist? Power corrections ~1/Q2

------//------------- ~ M2

DIS – it’s the same (~ M2 /Q2)

TMD – usually ~1/Q2 - (M2 /kT

2 )i attributed to Leading Twist

However – tracing the powers of M is helpful for studying HT in coordinate (~impact parameter) space

Collins FF and twist 3 x(T) –space : qq correlator ~ M - twist 3

Cf to momentum space (kT/M ) – M in denominator – “LT”

x <-> kT spaces Moment – twist 3 (for Sivers – Boer,

Mulders, Pijlman) Higher (2D-> Bessel) moments – infinite

tower of twists (for Sivers - Ratcliffe,OT)

Resummation in x-space (DY) Full x/kT – dependence

DY weighted cross-section

Similarity with non-local quark condensate: quarks in vacuum ~ transverse d.o.f. of quarks in hadrons (Euclidian!) ?! –cf with Radyushkin et al

Universal hadron(type-dependent)/vacuum functions?!

Hadronic vs vacuum matrix elements

Hadron-> (LC) momentum; dimension-> twist; quark virtuality -> TM; (Euclidian) space separation -> impact parameter

D-term ~ Cosmological constant in vacuum; Negative D-> negative CC in space-like/positive in time-like regions: Annihilation~Inflation!

Spin dependent DIS Two invariant tensors

Only the one proportional to contributes for transverse (appears in Born approximation of PT)

Both contribute for longitudinal Apperance of only for longitudinal case –result

of the definition for coefficients to match the helicity formalism

g1

gggT 21

Generalized GDH sum rule Define the integral – scales

asymptotically as

At real photon limit (elastic contribution subtracted) – - Gerasimov-Drell-Hearn SR

Proton- dramatic sign change at low Q2!

...1142 QQ

Q2

1

Finite limit of infinite sum of inverse powers?!

How to sum ci (- M2 /Q2 )i ?!

May be compared to standard twist 2 factorization

Light cone: Lorentz invariance Summed by

representing

Summation and analyticity? Justification (in addition to nice parton

picture) - analyticity! Correct analytic properties of virtual

Compton amlitude Defines the region of x Require: Analyticity of first moment in Q2

Strictly speaking – another integration variable (Robaschik et al, Solovtsov et al)

Summation and analyticity! Parton model with |x| < 1 – transforms poles to

cuts! – justifies the representation in terms of moments

For HT series ci = <f(x) xi> - moments of HT “density”- geometric series rather than exponent: Σ ci (- M2

/Q2 ) = < M2 f(x)/(x

M2 + Q2 )>

Like in parton model: pole -> cut Analytic properties proper integration region

(positive x, two-pion threshold) Finite value for Q2 =0: -< f(x)/x> - inverse

moment!

Summation and analyticity “Chiral” expansion: - (- Q2/M2 )i <f(x)/x i+1> “Duality” of chiral and HT expansions:

analyticity allows for EITHER positive OR negative powers (no complete series!)

Analyticity – (typically)alternating series Analyticity of HT analyticity of pQCD series

– (F)APT Finite linit -> series starts from 1/Q2 unless the

density oscillates Annihilation – (unitarity - no oscillations)

justification of “short strings”?

Short strings Confinement term in the heavy

quarks potential – dimension 2 (GI OPE – 4!) scale ~ tachyonic gluon mass

Effective modification of gluon propagator

Decomposition of (J. Soffer, OT ‘92) Supported by the fact

that

Linear in , quadratic term from

Natural candidate for NP, like QCD SR analysis – hope to get low energy theorem via WI (C.f. pion F.F. – Radyushkin) - smooth model

For -strong Q – dependence due to Burkhardt-Cottingham SR

gggT 21

g2

g2

Models for :proton Simplest - linear

extrapolation – PREDICTION (10 years prior to the data) of low (0.2 GeV) crossing point

Accurate JLAB data – require model account for PQCD/HT correction – matching of chiral and HT expansion

HT – values predicted from QCD SR (Balitsky, Braun, Kolesnichenko)

Rather close to the data

gT

For Proton

The model for transition to small Q (Soffer, OT ’04)

Models for :neutron and deuteron Access to the

neutron – via the (p-n) difference – linear in ->

Deuteron – refining the model eliminates the structures

gT

for neutron and deuteron

Duality for GDH – resonance approach

Textbook (Ioffe, Lipatov. Khoze) explanation of proton GGDH structure –contribution of dominant magnetic transition form factor

Is it compatible with explanation?! Yes!– magnetic transition contributes

entirely to and as a result to

)1232(

g2

g2

gggT 21

Bjorken Sum Rule – most clean test

Strongly differs from smooth interpolation for g1

(Ioffe,Lipatov,Khoze) Scaling

down to 1 GeV

New option: Analytic Perturbation Theory Shirkov, Solovtsov: Effective coupling – analytic in Q2

Generic processes: FAPT (BMS) Does not include full NPQCD dynamics (appears at ~

1GeV where coupling is still small) –> Higher Twist Depend on (A)PT

Low Q – very accurate data from JLAB

Bjorken Sum Rule-APT Accurate data + IR stable coupling ->

low Q region

PT/HT duality

Matching in PT and APT

Duality of Q and 1/Q expansions

4-loop corrections included V.L. Khandramai, R.S. Pasechnik, D.V. Shirkov, O.P. Solovtsova, O.V. Teryaev. Jun

2011. 6 pp. e-Print: arXiv:1106.6352 [hep-ph]

HT decrease with PT order and becomes compatible to zero (V.I. Zakharov’s duality)

Analog for TMD – intrinsic/extrinsic TM duality!?

Asymptotic series and HT Duality: HT can be eliminated at all (?!)

May reappear for asymptotic series - the contribution which cannot be described by series due to its asymptotic nature.

Another version of IR stable coupling – “gluon mass” – Cornwall,.. Simonov,.. Shirkov(NLO) arXiv:1208.2103v2 [hep-th] 23 Nov 2012

HT – in the “VDM” form M2/(M2

+ Q2 ) Corresponds to f(x) ~ Possible in principle to

go to arbitrarily small Q BUT NO matching with

GDH achieved Too large average

slope – signal for transverse polarization (cf Ioffe e.a. interpolation)!

)1( x

Account for transverse polarization -> descripyion in the whole Q region (Khandramai, OT, in progress)

1-st order – LO coupling with (P) gluon mass + (NP) “VDM”

GDH – relation between P and NP masses

NP vs P masses

Non-monotonic!

“Phase diagram”

P/NP masses

Data at LO

NLO

Data vs NLO

Modification of spectral function for HT Add const × Q2/(M22+Q2) 2 -> First of second

derivative of delta-function appear – double and triple poles (single – almost cancelled)

Masses: P= 0.68 NP=0.76 Expansion at

low Q2

Real scale – pion mass?!

HT – modifications of scaling variables (L-T relations) Various options since Nachtmann ~ Gluon mass

-//- new (spectrality respecting) modification

JLD representation

Resummed twists: Q->0 (D. Kotlorz, OT)

Modified scaling variable for TMD First appeared in P. Zavada model

XZ =

Suggestion – also (partial) HT resummation(M goes from denominator to numerator in cordinate/impact parameter space)?!

Conclusions/Discussion TMD – infinite towers of twists Similar to non-local quark condensates –

vacuum/hadrons universality?! Infinite sums of twists – important for DIS at Q-

>0 Representation for HT similar to parton model:

preserves analyticity changing the poles to cuts Modified scaling variables – models for twists

towers at DIS and (TMD) SIDIS Good description of the data at all Q2 with the

single scale parameter