w asymmetry in rhic - sinap.ac.cn · w§ asymmetry in rhic xun chen (Ł˚), yajun mao (kæ),...

W± Asymmetry in RHIC

Xun Chen (�Ê), Yajun Mao (kæ�), Bo-Qiang Ma (êËr)

School of Physics, Peking University

QCD and RHIC Physics, CCAST, China, 2004 – p. 1/16

OutlineQuark Distribution

W± in RHIC

Theoretical Models

Numerical Calculation

Conclusion


Quark Distribution FuntionsAt leading-twist level, the quark structure of hadrons isdescribed by three distributions.

The Unpolarized distribution, f(x) or q(x),

The Helicity distribution, g(x) or ∆q(x),

The Transvercity distribution, h(x) or δq(x).






p

p′

p′/p = x






ps

s s






p

s

s

s


Quark Distribution, cont.

Polarization Quarks

unpolarized q ≡ q++ + q−+ ≡ q↑

↑+ q↓

↑

long. polarized ∆q = q++ − q−+

transversity δq = q↑↑− q↓

↑

Table 1: The quark distribution functions. Labels +,− denote he-licities, and ↑, ↓ denote transverse polarizations. Superscripts referto quarks and subscripts to the parent hadron.


Experiments on Spin PhysicsInclusive Polarized DIS experimentsAmazing Results – Total quark-plus-antiquark spin is muchsmaller than the proton’s spin. –spin crisis?There are still some uncertainties concerning the valenceflavor-spin stucture as x → 1.Can’t provide information on the polarized quark and antiquarkseparately.

Semi-inclusive DIS experimentsDepends on the detail of fragmentation functions.At RHIC the polarization of the u, u, d and d in the proton will bemeasured directly in ud → W+ and du → W−. (G. Bunce, N.Saito, J. Soffer and W. Vogelsang,Ann.Rev.Nucl.Part.Sci.50,525(2000))


The production of W

Why use W?Maximum violation of parity symmetryThe helicity of the participating quark and antiquark arefixed in the reaction.


The production of W

Why use W?Maximum violation of parity symmetryThe helicity of the participating quark and antiquark arefixed in the reaction.The longitudinally polarized proton collides with anunpolarized proton.At RHIC the polarized protons will be in brunches,alternately right-(+) and left-(−) handed.The definition of Parity-Violating Asymmetry

AWL =

1

P× N−(W ) − N+(W )

N−(W ) + N+(W ).


The production of W

The leading order production of W+.Proton helicity=”+”

u−+(x1)

d(x2)

W+

ν

l+

(a) Proton helicity = ”+”

Figure 1: A possible production of W +

in p-p collision at lowest order. The u quarkcomes from the polarized proton and the d

quark comes from the unpolarized proton.


The production of W

The leading order production of W+.Proton helicity=”-”

u−−(x1)

d(x2)

W+

ν

l+

(b) Proton helicity=”−”


in p-p collision at lowest order. The u quarkcomes from the polarized proton and the d


AW+

L=

u−

−(x1)d(x2) − u−

+(x1)d(x2)

u−

−(x1)d(x2) + u−

+(x1)d(x2)

=∆u(x1)

u(x1)


The production of W

The leading order production of W+.Proton helicity=”+”

d++(x1)

u(x2)

W+

ν

l+

(a) Proton helicity = ”+”


in p-p collision at lowest order. The d quarkcomes from the polarized proton and the u



The production of W

The leading order production of W+.Proton helicity=”-”

d+−(x1)

u(x2)

W+

ν

l+

(b) Proton helicity=”−”


in p-p collision at lowest order. The d quarkcomes from the polarized proton and the u


AW+

L=

d+

−(x1)u(x2) − d+

+(x1)u(x2)

d+

−(x1)u(x2) + d+

+(x1)u(x2)

= −∆d(x1)

d(x1)


The production of W

In general, the asymmetry is a superposition of the twocases:

AW+

L =∆u(x1)d(x2) − ∆d(x1)u(x2)

u(x1)d(x2) + d(x1)u(x2).

By interchanging u and d, we can get the parity-violatingasymmetry of W−.

AW−

L =∆d(x1)u(x2) − ∆u(x1)d(x2)

d(x1)u(x2) + u(x1)d(x2)

Higher-order corrections change the asymmetries only alittle.B. Kamal, Phys. Rev. D 57, 6663 (1998) and T. Gehrmman Nucl.Phys. B 534, 21 (1998)


Previous StudiesC. Bourrely and J. Soffer, Nucl. Phys. B 423, 329 (1994)Polarized quark distributions are constructed from unpolarizedquark distributions based on the extensive use of the Pauliexclusion principle.C. Bourrely and J. Soffer, Phys. Lett B 314, 132 (1993)

∆uv(x) = uv(x) − dv(x) (1)

∆dv(x) = −1/4dv(x), (2)

∆u(x) = u(x) − d(x) ≡ δq(x), (3)

∆d(x) ≡ 0. (4)

Calculated at two different energies.


Previous StudiesC. Bourrely and J. Soffer, Nucl. Phys. B 423, 329 (1994)

(a)√

s = 350 GeV (b)√

s = 500 GeV

Figure 3: AL in C. Bourrely and J. Soffer, Nucl. Phys. B 423, 329 (1994)


Nucleon Structure ModelsBag-like models

Chiral modelsChiral quark modelChiral quark soliton odel

Light-cone models

Spectator models

pQCD based analysis

Non-perturbative QCD calculationsQCD sum rulesLattice


SU(6) ModelLight-cone SU(6) quark-spectator-diquark model(B.-Q. Ma,Phys.Lett. B375, 320 (1996));

Begin from the three quark SU(6) quark model wavefunction of the baryon.

Effective particle

Ψ↑,↓p (qD) = sin θφV |qV 〉↑,↓ + cos θφS |qS〉↑,↓

with

|qV 〉↑,↓ = ±1

3[V0(ud)u↓,↑ −

√2V0(uu)d↑,↓ + 2V±1(uu)d↓,↑];

|qS〉↑,↓ = S(ud)u↑,↓


SU(6) ModelLight-cone SU(6) quark-spectator-diquark model(B.-Q. Ma,Phys.Lett. B375, 320 (1996));

Begin from the three quark SU(6) quark model wavefunction of the baryon.

Effective particle

Unpolarized quark distribution

q(x) = cSq aS(x) + cV

q aV (x),

∆q(x) = cSq aS(x) + cV

q aV (x)


SU(6) ModelLight-cone SU(6) quark-spectator-diquark model(B.-Q. Ma,Phys.Lett. B375, 320 (1996));where

aD(x) ∝

Z

[d2k⊥]|φ(x,k⊥)|2 (D = S or V )

aD(x) =

Z

[d2k⊥]WD(x,k⊥)|φ(x,k⊥)|2 (D = S or V ),

and (Brodsky-Huang-Lepage prescription)

φ(x,k⊥) = AD exp{− 1

8α2D

[m2

q + k2⊥

x+

m2D + k

2⊥

1 − x]},

WD(x,k⊥) =(k+ + mq)

2 − k2⊥

(k+ + mq)2 + k2⊥

k+ = xM

M2 =m2

q + k2⊥

x+

m2D + k

2⊥

1 − x


SU(6) ModelLight-cone SU(6) quark-spectator-diquark model(B.-Q. Ma,Phys.Lett. B375, 320 (1996));As a conclusion of this model, the helicity distribution of u and d quark inproton can be written as

∆uv(x)=[uv(x) − 1

2dv(x)]WS(x) − 1

6dv(x)WV (x),

∆dv(x)=−1

3dv(x)WV (x).


pQCD based ModelS. J. Brodsky, M. Burkardt, I. Schmidt, Nucl.Phys. B 441,197 (1995)The leading order contributions in αs(k

2F ) to the quark distributions

at x → 1 can be computed in perturbative QCD from minimallyconnected tree graphs.The limiting power-law behavior at x → 1 of the helicity-depententdistributions derived from the minimally-connected graphs is

qh(x) ∼ (1 − x)p,

where p = 2n − 1 + 2∆Sz, and ∆Sz = |Sqz − SH

z |.

∆Sz = 0 ∆Sz = 1


pQCD based ModelS. J. Brodsky, M. Burkardt, I. Schmidt, Nucl.Phys. B 441,197 (1995)In this model, the helicity-dependent distributions of thevalence quarks can be written as

q+v (x) =

Aqv

B3x− 1

2 (1 − x)3;

q−v (x) =Cqv

B5x− 1

2 (1 − x)5,

where Aq + Cq = Nq is the valence quark number for quarkq, Bn = B(1/2, n + 1) is the β-function defined byB(1 − α, n + 1) =

∫ 10 x−α(1 − x)n d x for α = 1/2, and

B3 = 32/35 and B5 = 512/693.


About the two ModelsBoth models have theiradvantages and playedimportant roles in theinvestigation of variousnucleon structure functions.They all predict that∆u(x)/u(x) → 1 at x → 1.

Different prediction on the ratio∆d(x)/d(x) at x → 1:

SU(6) model:∆d(x)/d(x) → −1/3

pQCD based model:∆d(x)/d(x) → 1

x0 0.2 0.4 0.6 0.8 1

u/u

∆

0.4

0.5

0.6

0.7

0.8

0.9

1

x0 0.2 0.4 0.6 0.8 1

u/u

∆

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 4: Predictions onthe ratio ∆u(x)/u(x). Thered solid curve and the bluedashed curve correspond tothe SU(6) model and pQCDbased model, respectively.


About the two ModelsBoth models have theiradvantages and playedimportant roles in theinvestigation of variousnucleon structure functions.They all predict that∆u(x)/u(x) → 1 at x → 1.

Different prediction on the ratio∆d(x)/d(x) at x → 1:

SU(6) model:∆d(x)/d(x) → −1/3

pQCD based model:∆d(x)/d(x) → 1

x0 0.2 0.4 0.6 0.8 1

d/d

∆

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x0 0.2 0.4 0.6 0.8 1

d/d

∆

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Figure 5: Predictions onthe ratio ∆d(x)/d(x). Thered solid curve and the bluedashed curve correspond tothe SU(6) model and pQCDbased model, respectively.


About the two Models

The Jefferson Lab HallA Collaboration has givena precision results onthe neutron spin asymme-try, which provide a pre-liminary examination onthese two quark distribu-tion models.(X. Zheng, et al, Phy. Rev.Lett. 92, 012004 (2004))

(∆u

+ ∆

u)/(

u +

u)

(∆d

+ ∆

d)/(

d +

d)

1

0.5

0

0.5

−0.5

0

1 This workHERMES

x0 0.2 0.4 0.6 0.8 1

RCQM(∆qv/qv)LSS2001StatisticalLSS(BBS)


Some Variable Used

x1 =MW√

seyW , x2 =

MW√s

e−yW ,

MW = 80.419 GeV.Q2 = M2

W = 80.4192 GeV2.√s = 200 GeV and

√s = 500 GeV.

Other parameters in the two models are taken fromB.-Q. Ma, I. Schmidt, J. Soffer and J.-J. Yang, Phys. Rev. D62, 114009(2000)Terms with ∆d(x) and ∆u(x) are omitted, due to the value of∆q(x) is much smaller than the value of ∆q(x).


Calculation Result (1)

Wy-0.5 0 0.5

+W L

A

00.10.20.30.40.50.60.70.80.9

Wy-0.5 0 0.5

+W L

A

00.10.20.30.40.50.60.70.80.9

Wy-0.5 0 0.5

-W L

A

-0.2

0

0.2

0.4

0.6

0.8

Wy-0.5 0 0.5

-W L

A

-0.2

0

0.2

0.4

0.6

0.8

Figure 6: The value of AWL (yW ) in different models. The red solid

curves correspond to the SU(6) quark-spectator-diquark model, and theblue dashed curves correspond to the pQCD based analysis, for

√s =

200 GeV.



Wy-1 -0.5 0 0.5 1

+W L

A

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Wy-1 -0.5 0 0.5 1

+W L

A

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Wy-1 -0.5 0 0.5 1

-W L

A

-0.25

-0.2

-0.15

-0.1

-0.05

Wy-1 -0.5 0 0.5 1

-W L

A

-0.25

-0.2

-0.15

-0.1

-0.05

Figure 7: The value of AWL (yW ) in different models. The red solid


√s =

500 GeV.


Simple Analysis

x1 =MW√

seyW , x2 =

MW√s

e−yW ,

yW ↑ then x1 ↑

AW+

L = ∆u(x1)d(x2)u(x1)d(x2)+d(x1)u(x2)

,

AW−

L = ∆d(x1)u(x2)d(x1)u(x2)+u(x1)d(x2)

.

u(x) and d(x) is small when x is large. AW+

L approaches∆u(x1)/u(x1), and AW−

L appoaches ∆d(x1)/d(x1) whenx1 → 1.



1x0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

+W L

A

00.10.20.30.40.50.60.70.80.9

1x0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

+W L

A

00.10.20.30.40.50.60.70.80.9

1x0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-W L

A

-0.2

0

0.2

0.4

0.6

0.8

1x0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-W L

A

-0.2

0

0.2

0.4

0.6

0.8

Figure 8: The value of AWL (x1) in different models. The red solid


√s =

200 GeV.



1x0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

+W L

A

0

0.2

0.4

0.6

0.8

1

1x0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

+W L

A

0

0.2

0.4

0.6

0.8

1

1x0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

-W L

A

-0.2

0

0.2

0.4

0.6

0.8

1x0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

-W L

A

-0.2

0

0.2

0.4

0.6

0.8

Figure 9: The value of AWL (x1) in different models. The red solid


√s =

500 GeV.


ConclusionAW+

L are nearly equal in these two models.

AW−

L are quite different in these two models.

The difference of AW−

L comes from the differentpredictions on ∆d(x)/d(x).

The parity-violating asymme-try of W− in RHIC is a goodtool to examine these models.


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