Strangeness Contribution to the Vector and Axial Form Factors

of the Nucleon

Stephen Pate,

Glen MacLachlan, David McKee, Vassili Papavassiliou

New Mexico State University

JPARC Workshop on Hadron Structure

KEK, Tsukuba, 1-December-2005

A combined analysis of HAPPEx, G0, and BNL E734 data

Outline• Program of parity-violating electron-nucleon

elastic scattering experiments will measure the strange vector (electromagnetic) form factors of the nucleon --- but these experiments are insensitive to the strange axial form factor

• Use of neutrino and anti-neutrino elastic scattering data brings in sensitivity to the strange axial form factor as well

• Combination of forward PV data with neutrino and anti-neutrino data allows extraction of vector and axial form factors over a broad Q2 range

• With better neutrino data, a determination of s from the strange axial form factor is possible

Elastic Form Factors in Electroweak Interactions

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JμEM =

N′ p Jμ

EM pN

= u ′ p ( ) γ μ F1γ ,N q2

( ) + iσ μν qν

2MF2

γ ,N q2( )

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥u p( )

for two nucleon states of momentum p and ′ p . q2 = ′ p − p( )2 ⎡

⎣ ⎢ ⎤ ⎦ ⎥

• Elastic: same initial and final state particles, but with some momentum transfer q between them

• Electroweak: photon-exchange or Z-exchange

The photon exchange (electromagnetic) interaction involves two vector operators, and thus two vector form factors, called F1 and F2, appear in the hadronic electromagnetic current:

Elastic Form Factors in Electroweak Interactions

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JμNC =

N′ p Jμ

NC pN

= u ′ p ( ) γ μ F1Z ,N q2

( ) + iσ μν qν

2MF2

Z ,N q2( )

⎡

⎣ ⎢ ⎢

+γ μγ5GAZ ,N q2

( ) +qμ

Mγ5GP

Z ,N q2( )

⎤

⎦ ⎥u p( )

The Z-exchange (neutral current weak) interaction involves those same vector operators, but since it does not conserve parity it also includes axial-vector and pseudo-scalar operators. So, there are two additional form factors, GA and GP, in the hadronic weak current:

(The pseudo-scalar form factor GP does not contribute to either PVeN scattering or to neutral-current elastic scattering, so we will ignore it hence.)

Conversion to Sachs Form Factors

The vector form factors F1 and F2 are called respectively the Dirac and Pauli form factors. It is customary in low-energy hadronic physics to use instead the Sachs electric and magnetic form factors, GE and GM:

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GEp,n = F1

p,n − τ F2p,n GM

p,n = F1p,n + F2

p,n

τ = Q2 4M 2( )

Flavor Decomposition of Form FactorsBecause the interaction between electrons or neutrinos and the quark constituents of the nucleon is point-like, we can re-write these nucleon form factors in terms of individual quark contributions. For example, the proton electric form factors:

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GEγ ,p =

2

3GE

u −1

3GE

d −1

3GE

s

GEZ ,p = 1− 8

3sin2 θW( )GE

u + −1+ 43sin2 θW( )GE

d + −1+ 43sin2 θW( )GE

s

Because of the way we defined the form factors, the underlying quark form factors are defined by the tensor properties of the current operator and not the specific interaction. The interaction is represented by the multiplying coupling constants (electric or weak charges).

Charge SymmetryWe will assume charge symmetry: that is, we assume that the

transformation between proton and neutron

is a rotation of in isospin space.

We also assume that the strange form factors in each nucleon are the same.

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Then, for example :

GEγ ,p =

2

3GE

u −1

3GE

d −1

3GE

s

GEγ ,n =

2

3GE

d −1

3GE

u −1

3GE

s

The individual quark form factors are then global properties of the nucleons.

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u ↔ d( )

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p ↔ n( )

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p ↔ n( )

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u ↔ d( )

Strange Form Factors

Our charge-symmetric expressions for the form factors allow us to solve for the up and down contributions:

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GEu = 2GE

γ ,p + GEγ ,n + GE

s

GEd = GE

γ ,p + 2GEγ ,n + GE

s

This is useful because the electromagnetic form factors of the proton and neutron are well known at low Q2.

Then we may eliminate the up and down form factors from all other formulae and focus on the strange form factors.

A QCD Relation for the Axial Current

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The axial current γ μγ5 is not only the underlying basis of the axial form

factor, but is also at the heart of the asymmetric part of the virtual Compton

amplitude at work in polarized deep - inelastic scattering. QCD relates the

polarized quark distribution functions Δq(x,Q2) (q = u, d, or s) with the

corresponding quark contribution to the axial form factor GAq (Q2).

For example :

Δs = GAs (Q2 = 0) = Δs(x,Q2 = ∞)dx

0

1

∫

Thus, the value of the strange axial form factor at Q2 = 0 is equal to the

integral of the polarized strange quark distribution measured at high Q2.

Therefore a measurement of the strange axial form factor can lead to an understanding of a portion of the nucleon spin puzzle --- a measurement of s.

Features of parity-violating forward-scattering ep data

• measures linear combination of form factors of interest

• axial terms are doubly suppressed

(1 4sin2W) ~ 0.075

kinematic factor ”' ~ 0 at forward angles

• significant radiative corrections exist, especially in the axial term

parity-violating data at forward angles are mostly sensitive to the strange electric and magnetic form factors

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For a hydrogen target, the asymmetry as a linear combination of GEs , GM

s , GACC and GA

s is :

A p = A0p + AE

pGEs + AM

p GMs + AAIV

p GACC + AA

pGAs

where A0p = −K p

1− 4sin2 θW( ) 1+ RVp

( ) εGEp 2

+ τGMp 2 ⎛

⎝ ⎜

⎞ ⎠ ⎟

− 1+ RVn

( ) εGEpGE

n + τGMp GM

n( )

− ′ ε GMp 1− 4sin2 θW( ) 3RA

T = 0GA8

[ ]

⎧

⎨

⎪ ⎪

⎩

⎪ ⎪

⎫

⎬

⎪ ⎪

⎭

⎪ ⎪

AEp = K p εGE

p 1+ RV0

( ){ }

AMp = K p τGM

p 1+ RV0

( ){ }

AAIVp = K p ′ ε GM

p 1− 4sin2 θW( ) 1+ RAT =1

( ){ }

AAp = K p ′ ε GM

p 1− 4sin2 θW( ) 1+ RA0

( ){ }

K p =GFQ2

4π 2α

1

εGEp 2

+ τGMp 2

Full Expression for the PV ep Asymmetry

Note suppression of axial terms by ( sinW and ”'

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τ = Q2

4M 2

ε = 1+ 2(1+ τ )tan2 θ /2( )[ ]−1

′ ε = (1−ε 2)τ (1+ τ )

Things known and unknown in the PV ep Asymmetry

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GACC =

gA

1+ Q2 M A2

( )2

GA8 =

1

2 3

3F − D( )

1+ Q2 M A2

( )2

M A =1.001± 0.020 GeV Budd, Bodek and Arrington : hep - ex/0308005 and 0410055[ ]

gA =1.2695 ± 0.0029 Particle Data Group 2005[ ]

3F − D = 0.585 ± 0.025 Goto et al. PRD 62 (2000) 034017[ ]

use of 3F − D implies use of flavor - SU(3), but GA8 is suppressed by ′ ε and 1− 4sin2 θW( )[ ]

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The R's are radiative corrections calculated at Q2 = 0 in the formalism of

Zhu et al. PRD 62 (2000) 033008[ ]. The Q2 - dependence is unknown,

and so we have assigned a 100% uncertainty to the values.

RVp = −0.045 RV

n = −0.012 RV0 = −0.012

RAT =1 = −0.173 RA

T =0 = −0.253 RA0 = −0.552

from evaluation of Arvieux et al., to be published[ ]

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GE ,Mp,n = Kelly parametrization PRC 70 (2004) 068202[ ]

with G0 uncertainties http : //www.npl.uiuc.edu/exp/G0/Forward[ ]

Features of elastic p data

• measures quadratic combination of form factors of interest

• axial terms are dominant at low Q2

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dσ

dQ2νp →νp( ) Q2 →0

⏐ → ⏐ ⏐ GF2

128π

M p2

Eν2

−GAu + GA

d + GAs

( )2

+ 1− 4sin2 θW( )2 ⎡

⎣ ⎢ ⎤ ⎦ ⎥

• radiative corrections are insignificant

[Marciano and Sirlin, PRD 22 (1980) 2695]

neutrino data are mostly sensitive to the strange axial form factor

Elastic NC neutrino-proton cross sections

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dσ

dQ2νp →νp( ) =

GF2

2π

Q2

Eν2

A ± BW + CW 2( )

+ ν

− ν

W = 4 Eν M p − τ( ) τ = Q2 4M p2

A =1

4GA

Z( )

21+ τ( ) − F1

Z( )

2− τ F2

Z( )

2 ⎛ ⎝ ⎜

⎞ ⎠ ⎟1− τ( ) + 4τF1

Z F2Z ⎡

⎣ ⎢ ⎤

⎦ ⎥

B = −1

4GA

Z F1Z + F2

Z( )

C =1

64τGA

Z( )

2+ F1

Z( )

2+ τ F2

Z( )

2 ⎡ ⎣ ⎢

⎤ ⎦ ⎥

Dependence on strange form factors is buried in the weak (Z) form factors.

The BNL E734 Experiment

• performed in mid-1980’s

• measured neutrino- and antineutrino-proton elastic scattering

• used wide band neutrino and anti-neutrino beams of <E>=1.25 GeV

• covered the range 0.45 < Q2 < 1.05 GeV2

• large liquid-scintillator target-detector system

• still the only elastic neutrino-proton cross section data available

Q2 (GeV)2

d/dQ2(p) (fm/GeV)2

d/dQ2(p) (fm/GeV)2

correlation coefficient

0.45 0.165 0.033 0.0756 0.0164 0.134

0.55 0.109 0.017 0.0426 0.0062 0.256

0.65 0.0803 0.0120 0.0283 0.0037 0.294

0.75 0.0657 0.0098 0.0184 0.0027 0.261

0.85 0.0447 0.0092 0.0129 0.0022 0.163

0.95 0.0294 0.0074 0.0108 0.0022 0.116

1.05 0.0205 0.0062 0.0101 0.0027 0.071

Uncertainties shown are total (stat and sys).

Correlation coefficient arises from systematic errors.

E734 Results

Forward-Scattering Parity-Violating ep Data

These data must be in the same range of Q2 as the E734 experiment.

• The original HAPPEx measurement: Q2 = 0.477 GeV2 [PLB 509 (2001) 211 and PRC 69 (2004) 065501]

• The recent G0 data covering the range 0.1 < Q2 < 1.0 GeV2 [PRL 95 (2005) 092001]

E734 Q2 range

Combination of the ep and p data sets

Since the neutrino data are quadratic in the form factors, then there will be in general two solutions when these data sets are combined.

Fortunately, the two solutions are very distinct from each other, and other available data can select the correct physical solution.

General Features of the two Solutions

There are three strong reasons to prefer Solution 1:

• GAs in Solution 2 is inconsistent with DIS estimates for s

• GMs in Solution 2 is inconsistent with the combined

SAMPLE/PVA4/HAPPEx/G0 result of GMs = ~+0.6 at Q2 = 0.1 GeV2

• GEs in Solution 2 is inconsistent with the idea that GE

s should be small, and conflicts with expectation from recent G0 data that GE

s may be negative near Q2 = 0.3 GeV2

GEs Consistent with zero

(with large uncertainty)Large and positive

GMs Consistent with zero

(with large uncertainty)Large and negative

GAs Small and negative Large and positive

Solution 1 Solution 2

I only present Solution 1 in what follows.

HAPPEx, SAMPLE & PVA4 combined (nucl-ex/0506011)

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GAs

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GMs

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GEs

G0 Projected

HAPPEx, SAMPLE & PVA4 combined (nucl-ex/0506011)

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GAs

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GMs

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GEs

G0 Projected

HAPPEx & E734 [Pate, PRL 92 (2004) 082002]

HAPPEx, SAMPLE & PVA4 combined (nucl-ex/0506011)

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GAs

€

GMs

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GEs

HAPPEx, SAMPLE & PVA4 combined (nucl-ex/0506011)

HAPPEx & E734 [Pate, PRL 92 (2004) 082002]

G0 & E734 [to be published]

G0 Projected

First determination of the strange axial form factor.

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GAs

€

GMs

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GEs

HAPPEx & E734 [Pate, PRL 92 (2004) 082002]

G0 & E734 [to be published]

Q2-dependence suggests s < 0 !

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GAs

HAPPEx & E734 [Pate, PRL 92 (2004) 082002]

G0 & E734 [to be published] Recent calculation by Silva, Kim,

Urbano, and Goeke (hep-ph/0509281 and Phys. Rev. D 72 (2005) 094011) based on chiral quark-soliton model is in rough agreement with the data.

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GAs

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GAs

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These results on GAs ...

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combined with world data

on GEs + ηGM

s ...

Look for Riska and Zou paper in the archive in a few weeks. [This follows work already presented by An, Riska and Zou, hep-ph/0511223.]

determine a unique uudss configuration, in which the uuds system is radially excited and the s is in the ground state.

DIS vs. Form Factors

The HERMES [PRL 92 (2004) 012005] result indicates that the strange quark helicity distribution s(x) ~ 0 for x > 0.023, and the integral over their measured kinematics is also zero:

One explanation: If these two results are both true, then the average value of s(x) in the range 0 < x < 0.023 must be ~ 5. That’s not impossible, as s(x) is ~20-300 in the range x~10-2 to 10-3 (CTEQ6).

Are these results in contradiction? Not necessarily.€

"Δs" = Δs(x) dxx = 0.023

0.30

∫ = + 0.03 ± 0.03(stat) ± 0.01(sys)

At the same time, we have an indication from the analysis of PV ep and elastic p data that the full integral s is negative.

A topological “x0” contribution to the singlet axial charge?

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gA(0) = Δq

q

∑ − 3α s

2πΔg

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟partons

g A(0)

DIS

1 2 4 4 4 3 4 4 4

+ C∞

Accessible in a form factor measurement

“subtraction at infinity” term from dispersion relation integration [Steven Bass, hep-ph/0411005]

Accessible in deep-inelastic measurements

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The C∞ term would correspond to a zero - momentum

"polarized condensate" contribution to the proton spin,

observable as a difference between the DIS and form

factor measurements.

A future experiment to determine the three strange form factors and s

The program I have described determines the strange axial form factor down to Q2 = 0.45 GeV2 successfully, but it does not determine the Q2-dependence sufficiently for an extrapolation down to Q2 = 0.

A new experiment has been proposed to measure elastic and quasi-elastic neutrino-nucleon scattering to sufficiently low Q2 to measure s directly.

A better neutrino experiment is needed, with a focus on determining these form factors. The large uncertainties in the E734 data limit their usefulness beyond what I have shown here.

FINeSSE* Determination of s

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Measure ratio of NC to CC neutrino scattering from nucleons:

RNC/CC =σ νp →νp( )

σ νn → μ− p( )

⇒ Numerator is sensitive to −GACC + GA

s( )

⇒ Denominator is sensitive to GACConly

⇒ Both processes have unique charged particle

final state signatures

⇒ Ratio largely eliminates uncertainties in neutrino flux,

detector efficiency, and (we expect) nuclear target effects

* B. Fleming (Yale) and R. Tayloe (Indiana), spokespersons

FINeSSE Determination of s

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If the ratio of NC to CC processes is measured

for both neutrino and anti - ncutrino scattering

RNC/CC =σ νp →νp( )

σ νn → μ− p( ) R NC/CC =

σ ν p →ν p( )

σ ν p → μ +n( )

and combined with forward PV r e p scattering data,

the uncertainty in the strange axial form factor can be pushed

down to ± 0.02.

However, these uncertainty estimates do not include any contributions from nuclear initial state and final state effects.

The calculation and understanding of these effects is a critical component of the FINeSSE physics program.

Theoretical Effort Related to FINeSSE• Meucci, Giusti, and Pacati at INFN-Pavia [1]

• van der Ventel and Piekarewicz [2,3]

• Maieron, Martinez, Caballero, and Udias [4,5]

• Martinez, Lava, Jachowicz, Ryckebusch, Vantournhout, and Udias [6]

[1] Nucl. Phys. A 744 (2004) 307.

[2] Phys. Rev. C 69 (2004) 035501

[3] nucl-th/0506071, submitted to Phys. Rev. C

[4] Phys. Rev. C 68 (2003) 048501

[5] Nucl. Phys. Proc. Suppl. 139 (2005) 226

[6] nucl-th/0505008, submitted to Phys. Rev. C

These groups generally employ a relativistic PWIA for the baseline calculation, and use a variety of models to explore initial and final state nuclear effects (relativistic optical model or relativistic Glauber approximation, for example).

Preliminary indications from [1] and [6] are that nuclear effects cancel very nicely in the ratios to be measured in FINeSSE.

HAPPEx, SAMPLE & PVA4 combined (nucl-ex/0506011)

G0 Projected

FINeSSE (& G0) [exp. proposal: no nuclear initial or final state effects included in errors]

HAPPEx & E734 [Pate, PRL 92 (2004) 082002]

G0 & E734 [to be published]

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GAs

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GMs

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GEs

In conclusion…

Recent data from parity-violating electron-nucleon scattering experiments has brought the discovery of the strange vector form factors from the future into the present. Additional data from these experiments in the next few years will add to this new information about the strangeness component of the nucleon.

However, an even richer array of results, including also the strange axial form factor and the determination of s, can be produced if we can bring neutrino-proton scattering data into the analysis.

The E734 data have insufficient precision and too narrow a Q2 range to achieve the full potential of this physics program. The FINeSSE project can provide the necessary data to make this physics program a success.