Nuclear parton distribution functions and their effects on sin2W anomaly

Shunzo Kumano, Saga University

kumanos@cc.saga-u.ac.jp, http://hs.phys.saga-u.ac.jp

12th International Workshops on Deep Inelastic Scattering (DIS04)

Strbske Pleso, Slovakia, April 14-18, 2004

Refs. npdf: (1) M. Hirai, SK, M. Miyama, Phys. Rev. D64 (2001) 034003 (2) M. Hirai, SK, T.-H. Nagai, hep-ph/0404093

sin2W : (1) SK, Phys. Rev. D64 (2001) 034003

(2) research in progress (T.-H. Nagai)

April 15, 2004

Contents

PurposesDetermination of Nuclear Parton Distribution Functions (NPDFs)

(1) used data, 2 analysis method (2) results

Nuclear modification effects on NuTeV sin2W

(1) Paschos-Wolfenstein (PW) relation (2) valence-quark modification effects

on the PW relation and sin2W

Why nuclear parton distribution functions?

(1)Basic interest to understand nuclear structure in the high-energy region, Determination of sin2W

perturbative & non-perturbative QCD

sin2W in neutrino scattering (NuTeV)

(2) Practical purpose to describe hadron cross sections precisely

heavy-ion reactions: quark-gluon plasma signature

long-baseline neutrino experiments: nuclear effects in + 16O

Parametrization of

Nuclear Parton Distribution Functions

Code for the obtained NPDFs

could be obtained from

http://hs.phys.saga-u.ac.jp/nuclp.html

Nuclear modificationNuclear modification of F2

A / F2D is

well known in electron/muon scattering.

F2

A = e i2

ix q i(x) + q i(x)

A

0.7

0.8

0.9

1

1.1

1.2

0.001 0.01 0.1 1

EMC

NMC

E139

E665

shadowingoriginal

EMC finding

Fermi motion

x sea quark valence quark

Functional form of wi(x,A)

f iA(x) = wi(x,A) f i(x), i = uv, dv, q, g

first, assume the A dependence as 1/A

then, use

wi(x,A) = 1 + (1–1 /A) a i+b i x+c i x2+d i x

3

(1 – x ) i

a i, b i, c i, d i, i: parameters to be determined by 2 analysis

Fermi motion: 1

(1 – x ) i as x 1 if i > 0

Shadowing: wi(x 0, A) = 1 + (1–1 /A) a i < 1

Fine tuning: b i, c i, d i

Experimental data

(1) F2A / F2

D

NMC: He, Li, C, Ca

SLAC: He, Be, C, Al, Ca, Fe, Ag, Au EMC: C, Ca, Cu, Sn E665: C, Ca, Xe, Pb BCDMS: N, Fe HERMES: N, Kr

(2) F2A / F2

A’

NMC: Be / C, Al / C, Ca / C, Fe / C, Sn / C, Pb / C, C / Li, Ca / Li

(3) DYA /

A’

E772: C / D, Ca / D, Fe / D, W / D E866: Fe / Be, W / Be

1

10

100

500

0.001 0.01 0.1 1

Q2 (

GeV

2 )

x

NMC (F2A/F2

D)

SLAC

EMC

E665

BCDMS

HERMES

NMC (F2A/F2

A')

E772/E886 DY

Analysis conditions

parton distributionsin the nucleon

MRST01 (QCD=220

MeV)

Q2 point at whichtheparametrizedPDFs are defined: Q2=1Ge V2

usedexperimentaldata: Q21Ge V2

totalnumber of data: 951

606(F2A /F2

D) +293(F2A /F2

A') + 52(Drell–Yan )

subroutinefor the 2 analysis: CERN– Minuit

2 = (Ridata –R i

calc)2

( idata)2

i

R = F2A

F2D, F2

A

F2A' ,

DYpA

DYpA ' , i

data = (isys )2 + (i

stat)2

Comparison with F2Ca/F2

D & DYpCa/ DY

pD data

(Rexp-Rtheo)/Rtheo at the same Q2 points

0.7

0.8

0.9

1

1.1

1.2

0.001 0.01 0.1 1

F2C

a /F2D

x

EMC

NMC

E136

E665

Q2= 5 GeV2

-0.2

0

0.2

0.001 0.01 0.1 1

x

EMC

NMC

E139

E665

Ca/D

0.7

0.8

0.9

1

1.1

1.2

0.03 0.1 1

x

E772

Q2= 50 GeV2

-0.2

0

0.2

0.03 0.1 1

x

E772

Ca/D DY

Comparison with R=F2A/F2

A’ data: (Rexp-Rtheo)/Rtheo are shown

-0.2

0

0.2

0.001 0.01 0.1 1

NMC

E139

-0.2

0

0.2

0.001 0.01 0.1 1

NMC

-0.2

0

0.2

0.001 0.01 0.1 1

E139

-0.2

0

0.2

0.001 0.01 0.1 1

x

EMC

NMC

E139

E665

He/D

Be/D

Li/D

C/D

-0.2

0

0.2

0.001 0.01 0.1 1

BCDMS HERMES

-0.2

0

0.2

0.001 0.01 0.1 1

E139

E49

-0.2

0

0.2

0.001 0.01 0.1 1

EMC

NMC

E139

E665

-0.2

0

0.2

0.001 0.01 0.1 1

x

BCDMS

E87

E139

E140

Ca/D

Al/D

N/D

Fe/D

Comparison with R=DYpA/ DY

pA’ (Rexp-Rtheo)/Rtheo are shown

-0.1

0.1

0.3

0.01 0.1 1

E772

-0.2

0

0.2

0.01 0.1 1

-0.2

0

0.2

0.01 0.1 1

x

E866

C/D DY

Ca/D DY

Fe/Be DY

-0.2

0

0.2

0.01 0.1 1

-0.2

0

0.2

0.01 0.1 1

-0.3

-0.1

0.1

0.01 0.1 1

x

Fe/D DY

W/D DY

W/Be DY

Nuclear corrections of PDFs with uncertainties

0.7

0.8

0.9

1

1.1

1.2

0.001 0.01 0.1 1

x

Ca

Q2 = 1 GeV2

valence-quark

0.4

0.6

0.8

1

1.2

1.4

0.001 0.01 0.1 1

x

antiquark

0

0.5

1

1.5

2

0.001 0.01 0.1 1

x

gluon

Nuclear Effects on sin2W

Nuclear modification difference

between uvA and dv

A

sin2W anomaly

Paschos-Wolfenstein relation R– =

NCN – NC

N

CCN – CC

N = 12

– sin2W

Difference between nuclear modifications of uV and dV: v(x) =

wdV(x) – wuV

(x)

wdV(x) + w uV

(x)

RA

– =NCA – NC

A

CCA – CC

A=

{1 – (1 – y)2} [ (uL2 – uR

2){uvA(x) + cv

A(x)} + (dL2 – dR

2){dvA(x) + sv

A(x)}]

dvA(x) + sv

A(x) – (1 – y)2{uvA(x) + cv

A(x)}

Others: sin2W = 1 mW2/mZ

2 = 0.2227 0.0004

NuTeV: sin2W = 0.2277 0.0013 (stat) 0.0009 (syst)

N = isoscalar nucleon

Nuclear effects are in the weight functions: wuvand wdv

uVA(x) = wuV

(x)Z uV(x) + N dV(x)

A, dV

A(x) = wdV(x)

Z dV(x) + N uV(x)A

NuTeV target: 56Fe (Z = 26, N = 30), not isoscalar nucleus nuclear effects should be carefully taken into account

Neutron excess and a related function: n= N – Z

A, n(x) = n

uV(x) – dV(x)

uV(x) + dV(x)

Expand in v, n, s, c << 1 not so obvious: SK, PRD 66 (2002) 111301, research in progress

R A– =

(12

– sin2 W) {1 + v(x) n(x)} + 13

sin 2 W{v(x) + n(x)}

+ (12

– 23

sin 2 W) s(x) + (12

– 43

sin 2 W) c(x)

1 + v(x)

n(x) +

1 + (1 – y) 2

1 – (1 – y)2{

v(x) +

n(x)} +

2{s(x) – (1 – y)2 c(x)}

1 – (1 – y) 2

RA

– = 12

– sin2W + O(v) + O( n) + O( s) + O( c)

0

0.02

0.04

0.06

0.08

0.001 0.01 0.1 1x

20 GeV2

1 GeV2 56

Fe

prelim

inar

y

G. P. Zeller et al. Phys. Rev. D65 (2002) 111103.

NuTeV kinematics

at Q2=20 GeV2

PDFs NuTeV PDFs (*)

xuVA = wu v

Z xuv + N xdv

A=

Z uvp* + N uvn

*

A

xdVA = wd v

Z xdv + N xuv

A=

Z dvp* + N dvn

*

A

uvp* =wuv

xu v , u vn* =wu v

xdv , dvp* =wdv

xd v , d vn* =wd v

xuv

uv* = uvp

* – dvn* = – v (wu v

+ wd v) xuv

dv* = dvp

* – uvn* = + v (wu v

+ wd v) xdv

sin2W = – dx { F [uv*, x] uv

* + F [dv*, x] dv

* } 0.01 large error (preliminary!)

Summary

(1) 2 analysis for the nuclear PDFs, and their uncertainties. Valence quark: well determined except for the small-x regio

n. Antiquark: determined at small x, large uncertainties at medium and large x. Gluon: large uncertainties in the whole-x region.

(2) We provide nuclear PDFs for general users. http://hs.phys.saga-u.ac.jp/nuclp.html.

(3) Effects on NuTeV sin2W progress (esp. error estimate) possibly (sin2W ) = 0.00X 0.00X with a large error

Supplements

NuMI

J. G. Morfin at NuFact02/NuInt02

12 [F3

N F3N ]CC uvdv

test of shadowing models F3 (valence) shadowing vs. F2 (sea) shadowing

accurate determination of nuclear PDFs

F3

F2

Fe/D ratios

S. A. Kulagin

A dependence

Ref. I. Sick and D. Day, Phys. Lett. B 274 (1992)

R= r0A1/3

“volume” “surface” roughly speaking A = A V + A2/3 S

A

A = V + 1A1/3 S

~ 1A1/3 dependence

Au

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

0.3 0.4 0.5 0.6 0.7 0.8 0.9

F2A

/ F

2D

1-1/A1/3

x=0.5, Q2=5 GeV2

He

C

Be

Al

CaFe

Ag

Nuclear parton distributions (per nucleon)

if there were no modification

A uA = Z up + N un, A dA = Z dp + N dn

Isospin symmetry: un = dpd , dn = upu

uA = Z u + N dA

, dA = Z d + N uA

Take into accont the nuclear modificationby the factors wi(x,A)

uVA(x) = wuV

(x,A)Z uV(x) + N dV(x)

A

dV

A(x) = wdV(x,A)

Z dV(x) + N uV(x)A

qA(x) = wq(x,A) q(x)

gA(x) = wg(x,A) g(x)

Constraints

Nuclear charge

Z = A dx [ 23

(u A–u A) – 13

(d A–dA) – 1

3(s A–s A)]

= A dx ( 23

uVA – 1

3dV

A)

Baryon number: A = A dx 13 (uV

A + dVA)

Momentum: A = A dx x (uVA + dV

A + 6 q A + gA )

Three parameters can be determined by these conditions.

F2

F2

p = 49

x (u+u) + 19

x (d+d+s+s)

F2n = 4

9x (d+d) + 1

9x (u+u+s+s)

F2

N =F2

p+F2n

2= 5

18x (u+u+d+d) + 2

18x (s+s)

= 5

18x V + 4

18x S if flavor symmetric sea

large x small x

Drell-Yan p1 + p 2 +– + X

(q1)

q1

q2

–

+

(q2)

d q(x1)q(x2) + q(x1)q(x2)

at large xF = x1 – x2

projectile target

d qv(x1)q(x2)

q(x2) can be obtained if qv(x1) is known

Error estimation Hessian method

2() is expanded around its minimum 0 ( =parameter)

where the Hessian matrix is defined by

2(0 ) 2(0) 2(0) i

ii

12

2 2 (0 ) i j

i ji, j

In the 2 analysis, 1 standard error is

The error of a distribution F(x) is given by

H ij 1

22 2(a0) i j

2 = 2(0 ) 2(0) i Hij ji, j

P(s) N: 2(= s) distribution with N degrees of freedom

ds0

2

P(s) N = 9 = 0.6826 2=10.427 (N = 1 case, 2= 1)

F( x)

2 2 F( x) i

ij– 1 F( x)

ji, j

2 values

Small nuclei are notwell explained.

Medium and large nucleiare reproduced.

Drell-Yan dataare reproduced.

F2A/F2

D data

-0.2

0

0.2

0.001 0.01 0.1 10.001 0.01 0.1 1

EMC

-0.3

-0.1

0.1

0.001 0.01 0.1 1

HERMES

-0.2

0

0.2

0.001 0.01 0.1 1

E139

-0.2

0

0.2

0.001 0.01 0.1 1

x

EMC

Cu/D

Kr/D

Ag/D

Sn/D

-0.2

0

0.2

0.001 0.01 0.1 1

E665

-0.2

0

0.2

0.001 0.01 0.1 1

E139 E140

-0.2

0

0.2

0.001 0.01 0.1 1

x

E665

Xe/D

Au/D

Pb/D

F2A/F2

A’ data

-0.2

0

0.2

0.01 0.1 1

NMC

-0.2

0

0.2

0.01 0.1 1

-0.2

0

0.2

0.01 0.1 1

Be/C

Al/C

Ca/C

-0.2

0

0.2

0.01 0.1 1

x

C/Li

-0.2

0

0.2

0.01 0.1 1

-0.2

0

0.2

0.01 0.1 1

-0.2

0

0.2

0.01 0.1 1

-0.2

0

0.2

0.01 0.1 1

x

Fe/C

Sn/C

Pb/C

Ca/Li

Q2 dependence

0.8

1

1.2

0.8

1

1.2

0.8

1

1.2

1 10 1 10

HERMES

1 10

x=0.035 x=0.045 x=0.055

x=0.07 x=0.09 x=0.125

x=0.175 x=0.25 x=0.35

Q2 ( GeV2 )

0.8

1

1.2

0.8

1

1.2

0.8

1

1.2

1 10 1 10

HERMES

1 10

x=0.035 x=0.045 x=0.055

x=0.07 x=0.09 x=0.125

x=0.175 x=0.25 x=0.35

Q2 ( GeV2 )

0.75

1

0.75

1

0.75

1

0.75

1

0.75

1

1 10 100 1 10 100

NMC

x=0.0125 x=0.0175 x=0.025

x=0.035 x=0.045 x=0.055

x=0.07 x=0.09 x=0.125

x=0.175 x=0.25 x=0.35

x=0.45 x=0.55

1 10 100

x=0.7

Q2 ( GeV2 )

uV

A(x) = wuV(x,A)

Z uV(x) + N dV(x)A

dV

A(x) = wdV(x,A)

Z dV(x) + N uV(x)A

qA(x) = wq(x,A) q(x), gA(x) = wg(x,A) g(x)

in the NPDF analysis

wuv = 1 + (1–1 /A) auv+b v x+c v x2+d v x3

(1 – x ) v

wdv = 1 + (1–1 /A) adv+b v x+c v x2+d v x3

(1 – x ) v

in the current analysis

wuv + wdv = 1 + (1–1 /A) av+b v x+c v x2+d v x3

(1 – x ) v

wuv – wdv = 1 + (1–1 /A) av'+b v

' x+c v' x2+d v

' x3

(1 – x ) v

Global analysis for F2 and Drell-Yan data for v(x)

RA

– = 12

– sin2W

– v(x) {(1

2 – sin2W)1 + (1 – y)2

1 – (1 – y) 2– 1

3 sin2W} + O(v2)

0

0.02

0.04

0.06

0.08

0.001 0.01 0.1 1x

20 GeV2

1 GeV2 56

Fe

CTEQ, hep-ph/0312322, 0312323

s – s effects on sin2W

of the order of the NuTeV deviation could not be “anomalous”

0.001 dx x(s s ) 0.004

0.005 (sin2W ) 0.001

–

s (d)

W +

c

s (d)W +

+

Global fit to the data including

CCFR - NuTeV dimuon data with s - s 0

s – s

x

Attempt to describe DIS & resonance region

F2(x) Q2

Q2 0.188 F2 (xw)

where xw x Q2 0.624

Q2 1.735 x

GRV94

Bodek-Yang NP B 112 (2002) 70

Empirical formula

We need similar analysis

for the NPDFs.