arie bodek, univ. of rochester1 modeling -(nucleon/nucleus) cross sections at all energies - from...

Arie Bodek, Univ. of Rochester 1

Modeling Modeling of (e// -(Nucleon/Nucleus) -(Nucleon/Nucleus) Cross Sections at all Energies - from the Cross Sections at all Energies - from the

Few GeV to the Multi GeV RegionFew GeV to the Multi GeV Region

Arie BodekUniv. of Rochester

andUn-Ki Yang

Univ. of Chicago

July 1-6, 2002NuFact02 - 4th International Workshop

Imperial College, London


Arie Bodek- Univ. of RochesterNuFact02 - 4th International Workshop - Imperial College, London

This presentation contains slides for two talks:

(1)Working Group 3 - Non-Oscillation Neutrino Physics: Physics and Detector: Wednesday (3rd July, 2002)Session: Correlations Between HT and Higher-Order Perturbative QCD Corrections(Wed. 11:30-13:00) - Chairman: S. Kumano

Talk 1- Wed. 11:30 AM : A. Bodek - Next-to-next-to-leading Order Fits with HT

Corrections and Modeling (e/ / Cross Sections from DIS to Resonance - (25 min + 5 min discussion)- Long talk

(2) Working Group 2 - Neutrino Oscillation: Physics and DetectorThursday (4th July, 2002) - (Joint Session with WG3)

Cross Sections, Detector Issues and Beam Systematics Issues Chairs: Kevin McFarland and Debbie Harris Session: Cross Sections: from Quasielastics to DIS (Thursday 10-11, 11:30-1)

Talk 2 - Thu. 10:15 AM : A. Bodek- Modeling(e/ / Cross Sections from DIS

to Resonance Region 15 min: Short Talk - summary


Neutrino cross sections at low energy?

Neutrino oscillation experiments (K2K, MINOS, CNGS, MiniBooNE, and future experiments with Superbeams at JHF,NUMI, CERN) are in the few GeV region

Important to correctly model neutrino-nucleon and neutrino-nucleus reactions at 0.5 to 4 GeV region (essential for precise next generation neutrino oscillation experiments with super neutrino beams ) as well as at the 15-30 GeV region (for future factories)

The very high energy region in neutrino-nucleon scatterings (50-300 GeV) is well understood at the few percent level in terms QCD and Parton Distributions Functions (PDFs) within the framework of the quark-parton model (data from a series of e// DIS experiments)

However, neutrino differential cross sections and final states in the few GeV region are poorly understood. ( especially, resonance and low Q2 DIS contributions). In contrast, there is enormous amount of e-N data from SLAC and Jlab in this region.


How are PDFs Extracted from High Q2 Deep Inelastic e// Data

€

uV +dVfrom ⏐ → ⏐ ⏐ F2

ν ≈x(u+u_

)+x(d+d)_

xF3ν ≈x(u−u

_

)+x(d−d)_

u+u_

from ⏐ → ⏐ ⏐ μF2p ≈

49

x(u+u_

)+19

x(d+d)_

d+d_

from ⏐ → ⏐ ⏐ μF2n ≈

19

x(u+u_

)+49

x(d+d)_

nucleareffects

typicallyignored

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪ μF2

n =2μF2

d

μF2d

d/u from ⏐ → ⏐ ⏐ pp_

W Asymmetry≈d/u(x1)−d/u(x2)d/u(x1)+d/u(x2)

At high x, deuteron binding effects introduce an uncertainty in the d distribution extracted from F2d data (but not from the W asymmetry data).

MRSR2 PDFs


Neutrino cross sections

• Neutrino interactions Quasi-Elastic / Elastic (W=Mp)

– + n --> - + p (x =1, W=Mp) well measured and described by form

factors (but need to account for Fermi Motion/binding effects in nucleus)

Bodek and Ritchie (Phys. Rev. D23, 1070 (1981)

Resonance (low Q2, W< 2) + p --> - + p +

• Poorly measured and only 1st resonance described by Rein and Seghal

Deep Inelastic– + p --> - + X (high Q2, W> 2)

– well measured by high energy experiments and well described by quark-parton model (pQCD with NLO PDFs), but doesn’t work well at low Q2

region.

(e.g. JLAB data at Q2=0.22) Issues at few GeV : Resonance production and

low Q2 DIS contribution meet. The challenge is to describe

both processes at a given neutrino (or electron) energy.

GRV94 LO

1st resonance


Building up a model for all Q2 region..

Can we build up a model to describe all Q2 region from high down to very low energies ?

[resonance, DIS, even photo production]

Advantage if we describe it in terms of the quark-parton model.

- then it is straightforward to convert charged-lepton scattering cross sections into neutrino cross section. (just matter of different couplings)

Understanding of high x PDFs at very low Q2?

- There is a of wealth SLAC, JLAB data, but it requires understanding of non-perturbative QCD effects.

- Need better understanding of resonance scattering in terms of the quark-parton model? (duality works, many studies by JLAB)

Challenges


What are Higher Twist Effects- page 1 Higher Twist Effects are terms in the structure functions that behave like a

power series in (1/Q2 ) or [Q2/(Q4+A)],… (1/Q4 ) etc….

While pQCD predicts terms in s2 ( ~1/[ln(Q2/ 2 )] )… s

4 etc…(i.e. LO, NLO, NNLO etc.)

In the few GeV region, the two power series cannot be distinguished

Higher Twist: Interaction between interacting and Spectator quarks via gluon exchange at Low Q2 affects the Cross Section. Terms are like (1/Q2 ) or [Q2/(Q4+A)],… (1/Q4 )

In NNLO p-QCD more

gluons emission affects

the cross section like

s2 ( ~1/[ln(Q2/ 2 )] )… s

4

Spectator quarks are not

Involved.


What are Higher Twist Effects - Page 2 Higher Twist Effects are terms in the structure functions that behave like a power

series in (1/Q2 ) or [Q2/(Q4+A)],… (1/Q4 ) etc…. While pQCD predicts terms in s

2 ( ~1/[ln(Q2/ 2 )] )… s4 (i.e. LO, NLO, NNLO etc.) --

>In the few GeV region, the two power series cannot be distinguished Nature has “evolved” the high Q2 PDF from the low Q2 PDF, therefore, the high

Q2 PDF include the information about the higher twists . High Q2 manifestations of higher twist/non perturbative effects include: difference

between u and d, the difference between d-bar, u-bar and s-bar etc. High Q2 PDFs “remember” the higher twists, which originate from the non-perturbative QCD terms.

Evolving back the high Q2 PDFs to low Q2 (e.g. NLO-QCD) and comparing to low Q2 data is one way to check for the effects of higher order terms.

What do these higher twists come from? Kinematic higher twist – initial state target mass binding (Mp), initial state and

final state quark masses (e.g. charm production) Dynamic higher twist – correlations between quarks in initial or final state.==>

Examples : Initial or final state multiquark correlations: diquarks, elastic scattering, excitation of quarks to higher bound states e.g. resonance production, exchange of many gluons

Non-perturbative effects to satisfy gauge invariance and connection to photo-production [e.g. F2(Q2 =0) = 0]]


Duality between fixed W and DIS

OLD Picture fixed W: Elastic Scattering, Resonance ProductionElectric and Magnetic Form Factors (GE and GM) versus Q2 measure size of object (the electric charge and magnetization distributions).

Elastic scattering W = Mp = M, single nucleon in final state: Form factor measures size of nucleon.Matrix element squared | <p f | V(r) | p i > |2 between initial and final state lepton plane waves. Which becomes:

| < e -i k2. r | V(r) | e +i k1 . r > | 2 q = k1 - k2 = momentum transfer GE (q) = INT{e i q . r (r) d3r } = Electric form

factor is the Fourier transform of the charge

distribution. In the language of structure functions for example: 2xF1(x ,Q2)elastic = x2 GM

2 elastic

x-1)

Resonance Production, W=MR, Measure transition form factor between a quark in the ground state and a quark in the first excited state. For the Delta 1.238 GeV first resonance, we have a Breit-Wigner instead of x-1). 2xF1(x ,Q2) resonance ~ x2 GM

2 Resonance transition

BWW-1.238)

e +i k2 . r

e +i k1.r

rMp Mp

q

Mp

MR

e +i k1 . r

e +i k2 . r


Duality: Parton Model Pictures of Elastic and Resonance Production

Elastic Scattering, Resonance Production Scatter from one quark with the correct parton momentum , and

the two spectator are just right such that a final state interaction makes up a proton, or a resonance.

Elastic scattering W = Mp = M, single nucleon in final state.

The scattering is from a quark with a very high value of , is such that one cannot produce a single pion in the final state and the final state interaction makes a proton.

Resonance Production, W=MR, e.g. delta 1.238 resonance The scattering is from a quark with a high value of , is

such that that the final state interaction makes a low mass resonance.

Therefore, with the correct scaling variable, and if we account for low W and low Q2 higher twist effects, the prediction using QCD PDFs q (, Q2) should give an average of F2 in the elastic scattering and in the resonance region. (including both resonance and continuum contributions). If we modulate the PDFs with a final state interaction A(W, Q2 Q2) we could also reproduce the various Breit-Wigners.

X= 1.0

=0.95

Mp

Mp

q

X= 0.95

=0.90

Mp

MR


Photo-production Limit Q2=0Non-Perturbative - QCD evolution freezes

Photo-production Limit: Transverse Virtual and Real Photo-production cross sections must be equal at Q2=0.

There are no longitudinally polarized photons at Q2=0 -proton, ) = 0.112 mb F2 (, Q2) / Q2 limit as Q2 -->0-proton, ) = 0.112 mb 2xF1 (, Q2) / Q2 limit as Q2 -->0

-proton, ) = T (, Q2) limit as Q2 -->0.

• F2 (, Q2) ~ Q2 / [Q2 +C] --> 0 limit as Q2 -->0 R (, Q2) = L/T ~ Q2 / [Q2 +const] --> 0 limit as Q2 -->0

If we want PDFs to work down to Q2 = 0 they must be multiplied by a factor Q2 / [Q2 +C] (where C is a small number).

In addition, the scaling variable x does not work since the photo-production cross section is a function of Since at Q2 = 0

F2 (, Q2) = F2 (x , Q2) with x = Q2 /( 2Mreduces to one point x=0

However, a scaling variable c= (Q2 +B) /( 2M works at Q2 = 0 since F2 (, Q2) = F2 (, Q2) becomes a a function of B/ (2M.


How do we “measure” higher twist (HT)

Take a set of QCD PDF which were fit to high Q2 (e// data (in Leading Order-LO, or NLO, or NNLO)

Evolve to low Q2 (NNLO, NLO to Q2=1 GeV2) (LO to Q2=0.24) Include the “known” kinematic higher twist from initial target mass

(proton mass) and final heavy quark masses (e.g. charm production). Compare to low Q2data in the DIS region (e.g. SLAC) The difference between data and QCD+target mass predictions is the

extracted “effective” dynamic higher twists. Describe the extracted “effective” dynamic higher twist within a specific

HT model (e.g. QCD renormalons, or a purely empirical model). Obviously - results will depend on the QCD order LO, NLO, NNLO

(since in the 1 GeV region 1/Q2and 1/LnQ2 are similar). In lower orders, the “effective higher twist” will also account for missing QCD higher order terms. The question is the relative size of the terms.

Studies in NLO Yang and Bodek: Phys. Rev. Lett 82, 2467 (1999) ;ibid 84, 3456 (2000) Studies in NNLO - Yang and Bodek: Eur. Phys. J. C13, 241 (2000) Studies in LO - Bodek and Yang: hep-ex/0203009 (2002)


Lessons from previous “NLO QCD” study

Our previous studies of comparing NLO PDFs to DIS data: SLAC, NMC, and BCDMS e/scattering data on H and D targets shows that.. [ref:Yang and Bodek: Phys. Rev. Lett 82, 2467 (1999) ] Kinematic Higher Twist (target mass ) effects are large and

important at large x, and must be included in the form of Georgi & Politzer TM scaling.

Dynamic Higher Twist effects are smaller, but need to be included.

The ratio of d/u at high x must be increased if nuclear binding effects in the deuteron are taken into account.

The Very high x (=0.9) region - is described by NLO QCD (if target mass and higher twist effects are included) to

better than 10% Resonance region: NLO pQCD + Target mass + Higher Twist

describes average F2 in the resonance region (duality works) Also, in a subsequent study in NNLO QCD we find that the “empirically

measured Dynamic Higher Twist effects in the NLO study come from the missing NNLO QCD terms.

[ref: Yang and Bodek Eur. Phys. J. C13, 241 (2000) ]


F2, R comparison with QCD+TM vs. NLO QCD+TM+HT (use QCD Renormalon Model for HT)

PDFs and QCD in NLO + TM + QCD Renormalon Model for DynamicHigher Twist describe the F2 and R data reasonably well, with only 2 parameters


F2, R comparison with QCD-only vs. NLO QCD+TM+HT (use QCD Renormalon Model for HT)

PDFs and QCD in NLO + TM + QCD Renormalon Model for DynamicHigher Twist describe the F2 and R data reasonably well. TM Effects are LARGE


F2 comparison with QCD+TM vs. NLO QCD+TM+HT (use Empirical Model for Dynamic HT)

PDFs and QCD in NLO + TM + Empirical Model for DynamicHigher Twist describe the data for F2 (only) reasonably well with 3 paremeters

Here we used an Empirical form for Dynamic HT. Three parameters a, b, c.

F2 theory (x,Q2) =

F2 PQCD+TM [1+ h(x)/ Q2] f(x)

f(x) = floating factor, should be 1.0 if PDFs have the correct x dependence.h(x) = a (xb/(1-x) -c)


Kinematic Higher-Twist (GP target mass:TM) Goergi and Politzer Phys. Rev. D14, 1829 (1976)

TM= { 2x / [1 + k ] } [1+ Mc2 / Q2 ] (last term only for heavy charm product)

k= ( 1 +4x2 M2 / Q2) 1/2

For Q2 large (valence) F2=2 F1= F3

F2 pQCD+TM(x,Q2) =F2

pQCD (, Q2) x2 / [k32] +J1* (6M2x3 / [Q2k4] ) + J2*(12M4x4 / [Q4k5] )

2F1 pQCD+TM(x,Q2) =2F1

pQCD (, Q2) x / [k] +J1 * (2M2x2 / [Q2k2 ] ) + J2*(4M4x4 / [Q4k5] )

F3 pQCD+TM(x,Q2) =F3

pQCD(, Q2) x / [k2] +J1F3 * (4M2x2 / [Q2k3 ])

€

J 1 = duF 2pQCD(u,Q2

ξ

1

∫ )/u2

J 1F3 = duF 3pQCD(u,Q2

ξ

1

∫ )/u

J 2 = du dV F 2pQCD(V,Q2

u

1

∫ )/V2

ξ

1

∫

Ratio F2 (pQCD+TM)/F2pQCD

At very large x, factors

of 2-50 increase at

Q2=15 GeV2


Kinematic Higher-Twist (target mass:TM)

• The Target Mass Kinematic Higher Twist effects comes from the fact that the quarks are bound in the nucleon. They are important at low Q2 and high x. They involve change in the scaling variable from x to TM and various kinematic factors and convolution integrals in terms of the PDFs for xF1, F2 and xF3

• Above x=0.9, this effect is mostly explained by a simple rescaling in TM

F2pQCD+TM(x,Q2)

=F2pQCD(TMQ2

Compare complete Target-Mass calculation to simple rescaling in TM

Ratio F2 (pQCD+TM)/F2pQCD

Q2=15 GeV2


Dynamic Higher Twist

• Use: Renormalon QCD model of Webber&Dasgupta- Phys. Lett. B382, 272 (1996), Two parameters a2 and a4.

• F2 theory (x,Q2) = F2 PQCD+TM [1+ D2 (x,Q2)/ Q2 + D4 (x,Q2)/ Q4 ]

D2 (x,Q2) = [ a2 / q (x,Q2) ] ∫ (dz/z) c2(z) q(x/z, Q2)

D4 (x,Q2) = [ a4 times function of x)

In this model, the higher twist effects are different for 2xF1, xF3 ,F2. With complicated x dependences which are defined by only two parameters a2 and a4 .

Fit a2 and a4 to experimental data for F2 and R=FL/2xF1.

F2 data (x,Q2) = [ F2 measured + F2

syst ] ( 1+ N ) : 2 weighted by errors

where N is the fitted normalization (within errors) and F2 syst is the is the

fitted correlated systematic error BCDMS (within errors).

1


Very high x F2 proton data (DIS + resonance)(not included in the original fits Q2=1. 5 to 25 GeV2)

NLO pQCD + TM + higher twist describes very high x DIS F2 and resonance F2 data well. (duality works) Q2=1. 5 to 25 GeV2

Q2= 25 GeV2 Ratio F2data/F2pQCD

Q2= 25 GeV2 Ratio F2data/ F2pQCD+TM

Q2= 25 GeV2 Ratio F2data/F2pQCD+TM+HT

F2 resonance Data versus F2pQCD+TM+HT

pQCDONLY

pQCD+TM

pQCD+TM+HT

pQCD+TM+HT

Q2= 25 GeV2

Q2= 15 GeV2Q2= 9 GeV2

Q2= 3 GeV2

Q2= 1. 5 GeV2


Look at Q2= 8, 15, 25 GeV2 very high x data

• Pion production threshold• Now Look at lower Q2 (8,15 vs 25)

DIS and resonance data for the ratio of

F2 data/( NLO pQCD +TM +HT} • High x ratio of F2 data to NLO

pQCD +TM +HT parameters extracted from lower x data. These high x data were not included in the fit. The Very high x(=0.9) region:

It is described by NLO pQCD (if target mass and higher twist effects are included) to better than 10%

Ratio F2data/F2pQCD+TM+HT

Q2= 25 GeV2

Q2= 15 GeV2

Q2= 9 GeV2


F2, R comparison with NNLO QCD=> NLO HT mostly missing NNLO terms

Size of the higher twist effect with NNLO analysis is really small (a2=-0.009(NNLO) vs –0.1(NLO)


“Converting” NLO PDFs to NNLO PDFs• f(x) = the fitted floating factor, which

is the fitted ratio of the data to theory . Note f(x) =1.00 if pQCD PDFs describe the data.

• fNLO : Here the theory is pQCD(NLO)+TM+HT using NLO PDFs.

• fNNLO : Here the theory is pQCD(NNLO)+TM+HT also using NLO PDFs

• Therefore fNNLO / fNLO is the factor to “convert” NLO PDFs to NNLO PDFs (NNLO PDFs are not yet available.

• NNLO PDFs are lower at high x and higher at low x.

• Use f(x) in NNLO calculation of QCD processes (e.g. hardon colliders)

• Recently MRST did a similar analysis including NNLO gluons.

Floating factors

NNLO

NLO


Lessons from the NNLO pQCD analysis The origin of the

“empirically measured dynamic higher twist effects” is from the missing NNLO QCD terms.

Both TM and Dynamic higher twists effects should be similar in electron and neutrino reactions (aside from known mass differences, e.g. charm production)

The NNLO pQCD corrections and the Dynamic Higher Twist effects in NLO both have the same Q2 dependence at fixed x.

(F2NNL0/F2NLO)-1


In the few GeV region, At low x, “dynamic higher twist” look similar to “kinematic final state mass higher twist” -->both look like “enhanced” QCD

At low Q2, the final state u and d quark effective mass is not zero

uu

M* (final state interaction)Production of pons etc

C= [Q2+M*2 ] / [ 2M] (final state M* mass))

versus for mass-less quarks 2x q.P= Q2

x = [Q2] / [2M] (compared to x]

(Pi + q)2 = Pi2 + 2q.Pi + q2 = Pf2 = M*2

Ln Q2

F2

Lambda QCD

Low x QCD evolution

C slow rescaling looks like faster evolving QCDSince QCD and slow rescaling are bothpresent at the same Q2

Charm production s to c quarks in neutrino scattering-slow rescaling

sc

Mc (final state quarkmass

2 C q.P = Q2 + Mc2 (Q2 = -q2 )

2 C M= Q2 + Mc2 Cslow re-scaling

C = [Q2+Mc2 ] / [ 2M] (final state charm mass

(Pi + q)2 = Pi2 + 2q.Pi + q2 = Pf2 = Mc2

At Low x, low Q2

Cx (slow rescaling C(and the PDF is smaller at high x, so the low Q2 cross section is suppressed - threshold effect.

Final state mass effect


At high x, “dynamic higher twists” have a similar form to the “kinematic Goergi-Politzer proton target

mass effects” --> both look like “enhanced” QCDTarget Mass (G-P): tgt mass

2 TM

M + 2 TM q.P - Q2 = 0 (Q2 = -q2 )

solve quadratic equation

TM = Q2/[M (1+ (1+Q2/2 ) 1/2 ] proton target mass effect in Denominator)

Versus : Numerator in

C= [Q2+M*2 ] / [ 2M] (final state M* mass)

Combine both target mass and final state mass:

C+TM = [Q2+M*2 ] / [ M (1+(1+Q2/2) 1/2 ] - includes both initial state target proton mass and final state M* mass effect)

(Pi + q)2 = Pi2 + 2qPi + Q2 = Pf2

Ln Q2

F2

Mproton

High x

At high x, low Q2

TMx (tgt mass (and the PDF is higher at lower x, so the low Q2 cross section is enhanced .

F2 fixed Q2

X=1X=0

TM< x x < C

Final state

mass

Initial state target mass

QCD evolution

Target mass effects

[Ref:Goergi and Politzer GPPhys. Rev. D14, 1829 (1976)]]


Towards a unified model

We learned that the NNLO+ TM describes the DIS and resonance data very well.

• Theoretically, this breaks down at low Q2

• Practically, no way to implement it in MC

HT takes care of the NNLO term. So what about NLO + TM + HT?

• Still, it break down at very low Q2

• No way to implement photo-production limit.

Well, can we do something with the LO QCD and PDFs ? YES

Different scaling variables

Resonance, higher twist, and TM

=0.9X=-0.95

M* (final state interaction)

C=[Q2+M *2] / ( 2M) (quark final state M* mass)

TM= Q2/[M (1+ (1+Q2/2) 1/2 ] (initial proton mass)

= [Q2+M *2 ] / [ M (1+(1+Q2/2) 1/2 ] combined

= x [2Q2+2M *2 ] / [Q2 + (Q4 +4x2 M2 Q2) 1/2 ]

(Pi + q)2 = Pi2 + 2qPi + Q2 = M*2

Ln Q2

F2

Lambda QCD

Low x

High x

Xw

Photoproduction limit-Need to multiply byQ2/[Q2+C]

TRY: Xw = [Q2+B] /[2M + A] = x [Q2+B] / [Q2 + Ax]

(used in early fits to SLAC data in 1972)

And then follow up by trying

w= [Q2+B ] / [ M (1+(1+Q2/2) 1/2 + A]

(w works better and is theoretically motivated)

Xw worked in 1972 because it approximates w

M

B term (M *)

A term (tgt mass)

q


Modified LO PDFs for all Q2 region?

1. We find that NNLO QCD+tgt mass works very well for Q2 > 1 GeV2.

2. That target mass and missing NNLO terms “explain” what we extract as higher twists in a NLO analysis.

2. However, we want to go down all the way to Q2=0. All NNLO and NLO terms blow up. However, higher twist formalism in terms of initial state target mass binding and final state mass are valid below Q2=1, and mimic the higher order QCD terms for Q2>1 (in terms of effective masses due to gluon emission).

3. While the original approach was to explain the “empirical higher twists” in terms of NNLO QCD at low Q2 (and extract NNLO PDFs), we can reverse the approach and have “higher twist” model non-perturbative QCD, down to Q2=0, by using LO PDFs and “effective target mass and final state masses”

to account for initial target mass, final target mass, and missing NLO and NNLO terms.

[Ref:Bodek and Yang hep-ex/0203009]

Philosophy


Modified LO PDFs for all Q2 region?

1. Start with GRV94 LO (Q2min=0.23 GeV2 )- describe F2 data at high Q2

2. Replace X with a new scaling, Xw x= [Q2] / [2M Xw=[Q2+B] / [2M+A]= x[Q2+B]/[Q2+ Ax] A: initial binding/target mass effect

( but also higher twist and NLO effect)

B: final state mass effect(but also photo production

limit)

3. Multiply all PDFs by a factor of Q2/[Q2+C] for photo prod. limit and higher twist

4. Freeze the evolution at Q2 = 0.24 GeV2

- F2(x, Q2 < 0.24) = Q2/[Q2+C] F2(Xw, Q2=0.24)

Do a fit to SLAC/NMC/BCDMS H, D data, allow the normalization of the experiments and the BCDMS major

systematic error to float within errors. HERE INCLUDE DATA WITH Q2<1

which is not in the resonance region

Modified LO GRV94 PDFs with three parameters (a new scaling variable, Xw) describe DIS F2 H, D data (SLAC/BCDMS/NMC) very well.

A=1.735, B=0.624, and C=0.188

Compare with resonance data (not used in our fit)

Compare with photo production data (not used in our fit)

Compare with medium neutrino data (not used in our fit)- except to the extent that GRV94 originally included very high energy data on xF3,

Construction Results

[Ref:Bodek and Yang hep-ex/0203009]


Comparison of Xw Fit and w Fit

Try Also

w= [Q2+B ] / [ M (1+(1+Q2/2) 1/2 + A]

Fitted normalizationsHT fitting with Xw

p d

SLAC 0.979 +-0.0024 0.967 +- 0.0025

NMC 0.993 +-0.0032 0.990 +- 0.0028

BCDMS 0.956 +-0.0015 0.974 +- 0.0020

BCDMS Lambda = 1.01 +-0.156

HT fitting with wp d

SLAC 0.982 +-0.0024 0.973 +- 0.0025

NMC 0.995 +-0.0032 0.994 +- 0.0028

BCDMS 0.958 +-0.0015 0.975 +- 0.0020

BCDMS Lambda = 0.976 +- 0.156.

Modified LO GRV94 PDFs with three parameters (a new scaling variable, Xw) describe DIS F2 H, D data (SLAC/BCDMS/NMC) very well.

A=1.735, B=0.624, and C=0.188

(+-0.022) (+-0.014) ( +-0.004) 2 = 1555 /958 DOF Modified LO GRV94 PDFs with three

parameters (a new scaling variable, w) describe DIS F2 H, D data (SLAC/BCDMS/NMC) EVEN BETTER

A=0.700, B=0.327, and C=0.197 (+-0.020) (+-0.012) ( +-0.004) 2 = 1351 /958 DOF Note: No systematic errors (except for

normalization and BCDMS B field error were included)

Same construction for Xw and w fits Comparison


LO+HT fit Comparison with DIS F2 (H, D) data [These SLAC/BCDMS/NMC are used in the Xw fit ]

ProtonDeuteron


LO+HT fit Comparison with DIS F2 (H, D) data[Try again with w: These SLAC/BCDMS/NMC are used in the w fit ]

ProtonDeuteron


Comparison of LO+HT to neutrino data on Iron [CCFR] (not used in the fit)

Apply nuclear corrections using e/ scattering data.

Calculate F2 and xF3 from the modified PDFs with Xw

Use R=Rworld fit to get 2xF1 from F2

Implement charm mass effect through a slow rescaling algorithm, for F2 2xF1, and XF3

The modified GRV94 LO PDFs with a new scaling variable, Xw describe the CCFR diff. cross section data (E=30–300 GeV) well.

Construction


Comparison with F2 resonance data [ SLAC/ Jlab] (These data were not included in the fit)

The modified LO GRV94 PDFs with a new scaling variable, Xw describe the SLAC/Jlab resonance data very well (on average). Even down to Q2 = 0.07

GeV2

Duality works: The DIS curve describes the average over resonance region


Comparison with photo production data(not included in the Xw fit)

-proton) = Q2=0, Xw) = 0.112 mb 2xF1/( KQ2 ) K depends on definition of virtual photon flux

for usual definition K= [1 - Q2/ 2M = 0.112 mb F2(x, Q2) D( , Q2) /( KQ2 ) D = (1+ Q2/ 2 )/(1+R)

• F2(x, Q2 ) limit as Q2 -->0

• = Q2/(Q2+0.188) * F2-GRV94 (Xw, Q2 =0.24)

• Try: R = 0

• R= Q2/ 2 ( evaluated at Q2 =0.24)

• R = Rw (evaluated at Q2 =0.24)

• Note: Rw=0.034 at Q2 =0.24

The modified LO GRV94 PDFs with a new scaling variable, Xw also describe photo production data (Q2=0) to within 25%: To get better agreement at high 100 GeV (very low Xw), the GRV94 need to be updated to fit latest HERA data at very low x and low Q2. If we include these photoproduction data in the fit, we will get C of about 0.22, and agreement at the few percent level. To evaluate D = (1+ Q2/ 2 )/(1+R) more precisely, we also need to compare measured Jlab R data in the Resonance Region at Q2 =0.24 to the Rw parametrization.

mb


Comparison with photo production data(not included in the w fit)

-proton) = Q2=0, Xw) = 0.112 mb 2xF1/( KQ2 ) K depends on definition of virtual photon flux

for usual definition K= [1 - Q2/ 2M = 0.112 mb F2(x, Q2) D( , Q2) /( KQ2 ) D = (1+ Q2/ 2 )/(1+R)

• F2(x, Q2 ) limit as Q2 -->0

• = Q2/(Q2+0.188) * F2-GRV94 ( w, Q2 =0.24)

• Try: R = 0

• R= Q2/ 2 ( evaluated at Q2 =0.24)

• R = Rw (evaluated at Q2 =0.24)

• Note: Rw=0.034 at Q2 =0.24

The modified LO GRV94 PDFs with a new scaling variable, w also describe photo production data (Q2=0) to within 15%: To get better agreement at high 100 GeV (very low Xw), the GRV94 need to be updated to fit latest HERA data at very low x and low Q2. If we include these photoproduction data in the fit, we will get C of about 0.22, and agreement at the few percent level. To evaluate D = (1+ Q2/ 2 )/(1+R) more precisely, we also need to compare measured Jlab R data in the Resonance Region at Q2 =0.24 to the Rw parametrization.

mb


The GRV LO need to be updated to fit latest HERA data at very low x and low Q2.

We use GRV94 since they are the only PDFs to evolve down to Q2=0.24 GeV2 . All other PDFs (LO) e.g. GRV98 stop at 1 GeV2 or 0.5 GeV2.

GRV94 LO PDFs need to be updated.at very low x, but this is not important in the few GeV region

Comparison of u quark PDF for GRV94 and CTEQ4L and CTEQ6L (more modern PDFs)

Q2=10 GeV2Q2=1 GeV2Q2=0.5 GeV2

GRV94

CTEQ6L

CTEQ4L CTEQ6L

GRV94

CTEQ4L

GRV94

CTEQ6L

CTEQ4L

X=0.01X=0.0001


Summary

Our modified GRV94 LO PDFs with a modified scaling variable, Xw describe all SLAC/BCDMS/NMC DIS data. (We will investigate other variables also)

The modified PDFs also yields the average value over the resonance region as expected from duality argument, ALL THE WAY TO Q2 = 0

Also good agreement with high energy neutrino data.

Therefore, this model should also describe a low energy neutrino cross sections reasonably well.


Future Work Implement A e/(W,Q2) resonances into the model for F2 . For this need to fit all DIS and SLAC and JLAB resonance date and Photo-production

H and D data and CCFR neutrino data. Implement differences between and e/final state resonance masses in terms

of A , bar(w) {See Appendix)

Here Include Jlab and SLAC heavy target data for possible Q2 dependence of nuclear dependence on Iron.

Investigate different scaling variables for different flavor quark masses (u, d, s, uv, dv, usea, dsea in initial and final state) for F2. , Xw=[Q2+B] / [2M+A] versus:

w = [Q2+B ] / [ M (1+ (1+Q2/2) 1/2 +A ] (see Appendix)

or more sophisticated Genral expression: w’ =[ Q’ 2+B ] / [M (1+ (1+Q2/2) 1/2 +A] with 2Q’2 = [Q2+ m *2 - m I

2 ] + [ Q4 +2 Q2(m *2 + m I 2 ) + (m *2 - m I

2 ) 2 ] 1/2

Check other forms of scaling e.g. F2=(1+ Q2/ W2 Implementation for R (and 2xF1) is done exactly - use empirical fits to R (agrees with

NNLO+GP tgt mass for Q2>1); Need to compare to Jlab R data in resonance region. Compare our model prediction with the Rein and Seghal model for the 1st resonance

(in neutrino scattering) Compare to low-energy neutrino data (only low statistics data, thus new

measurements of neutrino differential cross sections at low energy are important). Do analysis with Xw with other LO PDFs like NuTeV Buras-Gamers LO PDFs etc. (as

an alternative LO model) - Would like to have GRV02 LO (now using GRV94)


Future Neutrino Experiments -JHF,NUMI

• Need to know the properties of neutrino interactions (both structure functions AND detailed final states on nuclear targets (e.g. Carbon, Oxygen (Water), Iron).

• Need to understand differences between neutrino and electron data for H, D and nuclear effects for the structure functions and the final states.

• Need to understand neutral current structure functions and final states.

• Need to understand implementation of Fermi motion for quasielastic scattering and the identification of Quasielastic and Inelastic processes in neutrino detectors (subject of another talk).

• A combined effort in understanding electron, muon, photoproduction and neutrino data of all these processes within a theoretical framework is needed for future precision neutrino oscillations experiments in the next decade.


Nuclear effects on heavy targets

F2(iron)/(deuteron) F2(deuteron)/(free N+P)

What are nuclear effects for F2 versus XF3; what are they at low Q2; possible differences between Electron, Neutrino CC and Neutrino NC at low Q2 (Vector dominance effects).


Current understanding of R

We find for R (Q2>1 GeV2)

1. Rw empirical fit works well (down to Q2=0.35 GeV2)

2. Rqcd (NNLO) + tg-tmass also works well (HT are small in NNLO)

3. Rqcd(NLO) +tgt mass +HT works well

(since HT in NLO mimic missing NNLO terms)

4. Need to constrain R to zero at Q2=0.

>> Use Rw for Q2 > 0.35 GeV2

For Q2 < 0.35 use: R(x, Q2 ) =

= 3.207 {Q2 / [Q4 +1) } R(x, Q2=0.35 GeV2 )

* Plan to compare to recent Jlab Data for R in the Resonance region at low Q2.


Different Scaling variables for u,d,s,c in initial and u, d, s,c in final states and valence vs. sea

Bodek and Yang hep-ex/0203009; Goergi and Politzer Phys. Rev. D14, 1829 (1976)

For further study

=[Q2+m *2] / ( 2M) (quark final state m* mass)

= Q2/[M (1+ (1+Q2/2) 1/2 ] (initial proton mass)

= [Q2+m *2 ] / [ M (1+ (1+Q2/2) 1/2 ] combined

= x [Q2+m *2 ] 2 / [Q2 + (Q4 +4x2 M2 Q2) 1/2 ]

(Pi + q)2 = Pi2 + 2qPi + q2 = m*2

M

m * 2

We Use: Xw = [Q2+B] / [2M + A]

= x [Q2+B] / [Q2 + Ax]

Could also try:

w = [Q2+B ] / [ M (1+ (1+Q2/2) 1/2 +A ]

Or fitted effective initial and final state quark masses that mimic higher twist (NLO+NNLO QCD), binding effects, +final state intractions could be different for initial and final state u,d,s, and valence vs sea? Can try ??

w’ =[ Q’ 2+B ] / [M (1+ (1+Q2/2) 1/2 +A]In general GP derive for initial quark mass m I and final mass m * bound in a proton of mass M

(at high Q2 these are current quark masses, but at low Q2maybe constituent masses? )

GP get 2Q’2 = [Q2+ m *2 - m I 2 ] + [ Q4 +2 Q2(m *2 + m I

2 ) + (m *2 - m I 2 ) 2 ] 1/2

= x [2Q’2] / [Q2 + (Q4 +4x2 M2 Q2) 1/2 ] - note masses may depend on Q2

= [Q’2] / [ M (1+(1+Q2/2) 1/2 ] (equivalent form)

m I 2

Can try to models quark masses, binding effects Higher twist, NLO and NNLO terms- All in terms of effective initial and final state quark masses and different target mass M with more complex form.

q


One Page Derivation: In general GP derive for initial quark mass m I and final mass m * bound in a proton of mass M

Goergi and Politzer Phys. Rev. D14, 1829 (1976)]- GP

(Pi + q)2 = m I 2 + 2qPi + q2 = m *2 and q = (q3) in lab

2qPi = 2 [ Pi0 + q3 Pi3] = [Q2+ m *2 - m I 2 ] (eq. 1)

( note + sign since q3 and Pi3 are pointing at each other)

= [Pi0 +Pi3] /[P0 + P3] --- frame invariant definition

= [Pi0 +Pi3] /M ---- In lab or [Pi0 +Pi3] = M --- in lab

[Pi0 -Pi3] = [Pi0 +Pi3] [Pi0 -Pi3] / M -- multiplied by [Pi0 -Pi3]

Pi0 -Pi3 = [ (Pi0 ) 2 - ( Pi3 ) 2 ]/ M = m I 2 /[ M] or [Pi0 -Pi3] = m I

2 /[ M] --- in lab

Get 2 Pi0 = M + m I 2 /[ M]

Plug into (eq. 1) and 2 Pi3 = M - m I 2 /[ M]

{ M + m I 2 /[ M] } + {q3 M - q3 m I

2 /[ M] } - [Q2+ m *2 - m I 2 ] = 0

a b c

2 M 2 (q3) - M [Q2+ m *2 - m I 2 ] + m I

2 (q3) = 0 >>> = [-b +(b 2 - 4ac) 1/2 ] / 2a => solution

use (2q3 2) = q 2 = -Q 2 and (q3) = [+Q 2/ 2 ] 1/2 = [+4M2 x2/ Q 2 ] 1/2

Get = [Q’2] / [ M (1+(1+Q2/2) 1/2 ] = [Q’2] / [M [+4M2 x2/ Q 2 ] 1/2) ]

(equivalent form)

(at high Q2 these are current quark masses, but at low Q2 maybe constituent masses?)

where 2Q’2 = [Q2+ m *2 - m I 2 ] + [ Q4 +2 Q2(m *2 + m I

2 ) + (m *2 - m I 2 ) 2 ] 1/2

or = x [2Q’2] / [Q2 + (Q4 +4x2 M2 Q2) ] 1/2 (equivalent form)

P= P0 + P3,MPf, m*

Pi= Pi0,Pi3,mI

q


In general GP derive for initial quark mass m I and final mass ,mF=m * bound in a proton of mass M -- Page 1

Is the correct variable which is Invariant in any frame : q3 and P in opposite directions.

P= P0 + P3,M

PF= PI0,PI

3,mI

ξ =PI

0 +PI3

PP0 +PP

3

PI ,P0

quark ⏐ → ⏐ ⏐ ⏐ q3,q0

photon← ⏐ ⏐ ⏐ ⏐

In−LAB−Frame:→ PP0 =M,PP

3 =0

ξ =PI−LAB

0 +PI −LAB3

M→ PI−LAB

0 +PI−LAB3 =ξM

ξ =PI

0 +PI3

( ) PI0 −PI

3( )

M(PI0 −PI

3)=

PI0

( )2− PI

3( )

2

M(PI0 −PI

3)

ξM(PI0 −PI

3) =mI2 → PI

0 −PI3 =mI

2 / ξM( )

(1): PI0 −PI

3 =mI2 / ξM( )

(2): PI0 +PI

3 =ξM

2PI0 =ξM +mI

2 / ξM( )mI → 0 ⏐ → ⏐ ⏐ ξM

2PI3 =ξM −mI

2 / ξM( )mI → 0 ⏐ → ⏐ ⏐ ξM

q+PI( )2

=PF2 → q2 +2PI ⋅q+PI

2 =mF2

2(PI0q0 +PI

3q3) =Q2 +mF2 −mI

2 Q2 =−q2 =(q3)2 −(q0)2

In−LAB−Frame: → Q2 =−q2 =(q3)2 −ν2

[ξM +mI2 / ξM( )]ν +[ξM −mI

2 / ξM( )]q3 =Q2 +mF2 −mI

2 : General

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

Set:mI2 =0 ( fornow)

ξMν +ξMq3 =Q2 +mF2

ξ =Q2 +mF

2

M(ν +q3)=

Q2 +mF2

Mν(1+q3 /ν)formI

2 =0

ξ =Q2 +mF

2

Mν[1+ (1+Q2 /ν2)]formI

2 =0

PF= PF0,PF

3,mF=m*

q=q3,q0


In general GP derive for initial quark mass m I and final mass mF=m* bound in a

proton of mass M -- Page 2

For the case of non zero mI

(note P and q3 are opposite)

P= P0 + P3,M

PF= PI0,PI

3,mI

ξ =PI

0 +PI3

PP0 +PP

3

PI ,P0

quark ⏐ → ⏐ ⏐ ⏐ ⏐

q3,q0

photon← ⏐ ⏐ ⏐ ⏐ ⏐

In−LAB−Frame:→ PP0 =M,PP

3 =0

(1): 2PI0 =ξM +mI

2 / ξM( ) →→→→→→→

(1): 2PI3 =ξM −mI

2 / ξM( ) →→→→→→→

q+PI( )2

=PF2 → q2 +2PI ⋅q+PI

2 =mF2

Q2 =−q2 =(q3)2 −ν2

[ξM +mI2 / ξM( )]ν +[ξM −mI

2 / ξM( )]q3 =Q2 +mF2 −mI

2

PF= PF0,PF

3,mF=m*

-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Keep terms with mI : multiply by M and group terms in qnd 2 2 M 2 (q3) - M [Q2+ m *2 - m I

2 ] + m I 2 (q3) = 0 General Equation

a b c => solution of quadratic equation = [-b +(b 2 - 4ac) 1/2 ] / 2ause (2q3 2) = q 2 = -Q 2 and (q3) = [+Q 2/ 2 ] 1/2 = [+4M2 x2/ Q 2 ] 1/2 Get = [Q’2] / [ M (1+(1+Q2/2) 1/2 ] = [Q’2] / [M [+4M2 x2/ Q 2

] 1/2) ]

or = x [2Q’2] / [Q2 + √ (Q4 +4x2 M2 Q2) ] (equivalent form)


2 ) + (m *2 - m I 2 ) 2 ] 1/2

(at high Q2 these are current quark masses, but at low Q2 maybe constituent masses?)

q=q3,q0


In general GP derive for initial quark mass m I and final mass mF=m* bound in a

proton of mass M --- Page 3

Get = [Q’2 +B ] / [ M (1+(1+Q2/2) 1/2 +A ]

= [Q’2 +B ] / [M [+4M2 x2/ Q 2 ] 1/2) +A]

or = x [2Q’2 +2B ] / [Q2 + (Q4 +4x2 M2 Q2) 1/2 +2Ax ] (equivalent form)


2 ) + (m *2 - m I 2 ) 2 ] 1/2

Numerator: = x [2Q’2]: Special cases for 2Q’2 +2B

m * =m I =0: 2Q’2 = 2 Q2 +2B : current quarks (mi =mF =0)

m * =m I : 2Q’2 = Q2+ [ Q4 +4 Q2 m *2 ] 1/2 +2B: constituent mass (mi =mF =0.3 GeV)

m I =0 : 2Q’2 = 2Q2 + 2 m *2 +2B : final state mass (mi =0,mF =charm)

m * =0 : 2Q’2 = 2Q2 +2B -----> : initial constituent(mi=0.3); final state current (mF =0)

denominator : [Q2 + (Q4 +4x2 M2 Q2) 1/2 +2Ax ] at large Q2 ---> 2 Q2 + 2 M2x2 +2Ax

Or [ M (1+(1+Q2/2) 1/2 +A ] which at small Q2 ---> 2MM2 /x +A

Therefore A = M2 x at large Q2 and M2 /x at small Q2 ->A=constant is approximate;y OK

versus Xw = [Q2+B] / [2M + A] = x [2Q2+2B] / [2Q2 + 2Ax]

In future try to include fit using above form with floating masses and B and A. Expect A,B to be much smaller than for Xw. C is well determined if we include photo-production in the fit.

P= P0 + P3,M

PF= PI0,PI

3,mI PF= PF

0,PF3,mF=m*

q=q3,q0Plan to try to fit for different initial/final quark masses for

u-I, u-F, d-I,d-F, s-I,s-F,c-F. and A and B for HT


Initial quark mass m I and final mass mF=m* bound in a proton of mass M

Page 4

At High Q 2, we expect that the initial and final state quark masses are the current quark masses:(e.g. u=5 MeV, d=9 MeV, s=170 MeV, c=1.35 GeV, b=4.4 GeV.

For massive final state quarks, this is known as slow-rescaling:At low Q 2, Donnachie and Landshoff (Z. Phys. C. 61, 145 (1994)] say thatThe effective final state mass should reflect the true threshold conditions as follows:

m*2= (Wthreshold) 2 -Mp 2 . Probably not exactly true since A(w) should take care of it if target mass effects are included. Nonetheless it is indicative of the order of the final state interaction.

(This is known as fast rescaling - I.e. introducing a function to account for threshold)

Reaction initial state final state final state Wthreshold m*2 m* mF1 +mF2

quark quark threshold GeV GeV 2 GeV quarks

e-P u or d u or d Mp+Mpion 1.12 0.29 0.53 0.01

e-P s s+sbar MLambdS+MK 1.61 1.63 1.28 0.34

-N d c+u MLamdaC+Mpion 2.42 4.89 2.21 1.36

-N s c+sbar MLamdaC+MK 2.78 6.76 2.60 1.52

e-P c c+cbar MLamdaC+MD 4.15 16.29 4.04 2.70

e-P c c+cbar Mp +MD+MD 4.68 20.90 4.57 2.70

P= P0 + P3,M

PF= PI0,PI

3,mI PF= PF

0,PF3,mF=m*

q=q3,q0Plan to try to fit for different intial/fnal quark masses

for u-I, u-F, d-I,d-F, s-I,s-F,c-F. and A and B


Initial quark mass m I and final mass mF=m* bound in a proton of mass M

Page 4

At High Q 2, we expect that the initial and final state quark masses are the current quark masses:

(e.g. u=5 MeV, d=9 MeV, s=170 Mev, c=1.35 GeV, b=4.4 GeV.

However, ( m)*2 = m*2 - mud*2 is more relevant since mud*2 is already included in the scaling violation fits for the u and d PDFs

Reaction initial state final state final state Wthreshold (m)*2 m* mF1 +mF2

quark quark threshold GeV GeV 2 GeV quarks

e-P u or d u or d Mp+Mpion 1.12 0.00 0.00 0.00

e-P s s+sbar MLambda+MK 1.61 1.34 1.16 0.34

-N d c+u MLamdaC+Mpion 2.42 4.61 2.15 1.36

N s c+sbar MLamdaC+MK 2.78 6.47 2.54 1.52

e-P c c+cbar MLamdaC+MD 4.15 16.00 4.00 2.70

e-P c c+cbar Mp +MD+MD 4.68 20.62 4.54 2.70

FAST re-scaling ( function ) is a crude implementation of A(W,Q2).

P= P0 + P3,M

PF= PI0,PI

3,mI PF= PF

0,PF3,mF=m*

q=q3,q0Plan to try to fit for different intial/fnal quark masses for

u-I, u-F, d-I,d-F, s-I,s-F,c-F. and A and B


Examples of Low Energy Neutrino Data: Total (inelastic and quasielastic) cross section


Examples of Current Low Energy Neutrino Data: Single charged and neutral pion production


Examples of Current Low Energy Neutrino Data: Quasi-elastic cross section


Appendix B - Studies of d/u

Comparison of electron, muon, neutrino and e-p, and hadron

collider data.


Status of d/u a few years ago with CTEQ3and MRS A, G, R PDFs

In 1997 PDFs did not describe the new CDF W Asymmetry Data. We decided to look at F2N/F2P

Correcting for Nuclear Effects in the deuteron resulted in even poorer agreement of PDFs with the corrected F2n/F2p data


The d/u correction to MRSR2 PDFs (which were originally constrained to have d/u --> 0 at x=1)

We parametrized a correction to the d/u ratio in the MRSR2 PDFs such that they would agree with the NMC experimentally measured F2N/F2P ratio (corrected for nuclear binding effects in the deuteron).


Comparison to CDF Pbar-PTevatron W asymmetry

The modified PDFs (After d/u correction) are in better agreement with the CDF Tevatron very high energy data on unbound protons. Yang and Bodek PRL 82, 2467 (1999)


Comparison of modified PDFs to HERA charged current data for e+P and e-P

• After d/u correction • Comparison of modified PDFs to HERA data for the ratio of e+P and e-P charged current data. Modified PDFs are in better agreement with very high energy data on unbound protons.

Yang and Bodek , Phys. Rev. Lett. 82, 2467 (1999)


Comparison to CHDS Neutrino/Antineutrino Ratio on H2 and to QCD predictions

The d/u from CDHS neutrino and antineutrino data on hydrogen versus MRSR2 PDFs with and without our d/u corrections.

Note, QCD predicts d/u=0.2 as x-> 1.

The modified PDFs agree with CDHS data on unbound protons and with the QCD prediction.


Comparison to CDF and Dzero High Pt Jet Data at Higher Pt (i.e higher x)

A higher d quark density at high x yields better agreement with the rate of high Pt jets in CDF and Dzero on unbound protons and antiprotons

arie bodek, univ. of rochester1 modeling -(nucleon/nucleus) cross sections at all energies - from...

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