Sigma meson cloudSigma meson cloudand Proton’s light flavor seaand Proton’s light flavor sea

quarksquarks

Peking University, China (Peking University, China ( 北京大北京大学学 ))

Feng Huang (Feng Huang ( 黄 峰）黄 峰）Supervisor: Bo-Qiang Ma Supervisor: Bo-Qiang Ma （马伯强）（马伯强）

outlineoutline

IntroductionIntroduction

Meson cloud modelMeson cloud model

Adding sigma meson in this modelAdding sigma meson in this model

SummarySummary

IntroductionIntroduction Gottfried sum rule (GSR)Gottfried sum rule (GSR)

NMC results in 1994 (first in 1991) NMC results in 1994 (first in 1991)

imply imply 3

1026.0235.0 GI

3

1)]()([

3

2)]()([

3

1 1

0

1

0 xdxudxxdxudxI vvG

x

dxxFxFI np

G 1

0 22 )]()([

du

Light flavor sea quarks in protonLight flavor sea quarks in proton

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

d(x

)-u

(x)

X

E866/NuSea HERMES

Explanations for this asymmetryExplanations for this asymmetry

Chiral quark modelChiral quark model

Meson cloud modelMeson cloud model (directly including (directly including

mesons)mesons)

Lattice gauge Lattice gauge approachapproach

Instanton induced Instanton induced interactioninteraction

Statistical model Statistical model

Chiral quark soliton Chiral quark soliton modelmodel

(not directly including (not directly including mesons)mesons)

Meson cloud modelMeson cloud model

The nucleon was viewed as a bare nucleon The nucleon was viewed as a bare nucleon plus a series of baryon-meson Fock states plus a series of baryon-meson Fock states which result from the fluctuation of nucleon to which result from the fluctuation of nucleon to baryon plus meson then the physical proton baryon plus meson then the physical proton wave function iswave function is

So the quark distribution functions So the quark distribution functions qq((xx) in the proton ) in the proton isis

the splitting function isthe splitting function is

)()()(1

y

xqyf

y

dyxq MMB

M x

We use time-ordered perturbation theory in the infinite We use time-ordered perturbation theory in the infinite momentum frame to calculate this functionmomentum frame to calculate this function

the quark distribution functions of the quark distribution functions of PiPi meson meson

with the splitting function convolutionwith the splitting function convolution

In a full calculation, we should include all kinds of In a full calculation, we should include all kinds of mesons and baryons. While the probability of mesons and baryons. While the probability of baryon-meson fluctuation should baryon-meson fluctuation should decreasedecrease with with the invariant mass of the baryon-meson Fock the invariant mass of the baryon-meson Fock state increasing, we can state increasing, we can neglectneglect the effects of the effects of Fock states with Fock states with higherhigher invariant massinvariant mass

PionPion, the lightest meson, plays the , the lightest meson, plays the dominant dominant role. role. Difference between u_bar and d_bar in Difference between u_bar and d_bar in virtual virtual pion cloudspion clouds can provide the large can provide the large lightlight flavor flavor asymmetryasymmetry in proton naturally in proton naturally

Pi Pi mesonic contribution mesonic contribution

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.4

0.8

1.2

1.6

2.0

= 1.0 = 0.88 = 0.83

E866/NuSea HERMES

d

(x)-

u(x

)

x

)()( xuxd

Problem in description forProblem in description for )(/)( xuxd

The dominant The dominant role is played role is played by theby the pion pion, , while itwhile it provides the provides the ratio either ratio either increases increases monotonicallymonotonically with with x x or turns or turns back back towards towards unity too slowly.unity too slowly.

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

d(x

)/u

(x)

X

only E866/NuSea

Flavor symmetric sea contributionFlavor symmetric sea contribution

bare nucleon sea quarksbare nucleon sea quarks(M. Alberg, E. M. Henley, Nucl. Phys. (M. Alberg, E. M. Henley, Nucl. Phys. A663A663 (2000) 301) (2000) 301)

isoscalar meson, such as omega, isoscalar meson, such as omega, sigmasigma

(M. Alberg, E. M. Henley, G. A. Miller, Phys. Lett. B471 (2000) (M. Alberg, E. M. Henley, G. A. Miller, Phys. Lett. B471 (2000) 396) 396)

Using different bare sea quarksUsing different bare sea quarks

Comparison of a Comparison of a harder harder bare bare nucleon sea quarks nucleon sea quarks (thick solid line) (thick solid line) with a with a traditionaltraditional bare sea quarks bare sea quarks (dashed line) .(dashed line) .

The thin solid line is The thin solid line is only Pi contribution only Pi contribution to the ratioto the ratio0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

0

1

2

3

4

5

6

only N Harder N

E866/NuSea

d(x

)/u

(x)

X

Adding sigma meson in this modelAdding sigma meson in this model

Sigma meson Sigma meson

a) isoscalar scalar meson a) isoscalar scalar meson

b) chiral partner of Pib) chiral partner of Pi

c) not well established in c) not well established in experiment experiment

The splitting functionThe splitting function Nf

MeVm 600 GeV3.10.1

6.134/4/ 22 gg

the quark distribution functions of sigma the quark distribution functions of sigma mesonmeson

We assume the valence and sea quark We assume the valence and sea quark distribution of mesons related here are the distribution of mesons related here are the same.same.

Adding sigma effects hereAdding sigma effects here

Our calculations Our calculations illustrate that illustrate that the larger value the larger value of tends to of tends to give small give small values of the values of the ratio and ratio and decreasing decreasing causes the causes the maximum value maximum value of the ratio to of the ratio to be large and to be large and to appear at higher appear at higher value of xvalue of x

Together with omega mesonTogether with omega meson

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

0.00.51.01.52.02.53.03.54.04.5

E866/NuSea

=0.88 (only ) =1.0 (no ) =1.0, =1.0 (upper) =1.5, =1.0 (bottom)

d(x

)/u

(x)

X0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

0.00.51.01.52.02.53.03.54.04.5

E866/NuSea

=0.88 (only ) =1.0 (no ) =1.0, =1.3 (upper) =1.5, =1.3 (bottom)

d(x

)/u

(x)

X

compare pion+omega compare pion+omega (dashed curve, Alberg-(dashed curve, Alberg-Henley’s work)Henley’s work) to pion+omega+sigma to pion+omega+sigma..

Parameters related omega Parameters related omega mesonmeson

1.82 g GeV5.10.1

Lambda_omega as 1.5GeV could cover Lambda_omega as 1.5GeV could cover over the sigma contributionover the sigma contribution

we present the number of the related we present the number of the related mesons in the proton with the different mesons in the proton with the different cutoff valuescutoff values

Parameters and meson numbers in the protonParameters and meson numbers in the proton

The larger cutoff value leads to a large meson The larger cutoff value leads to a large meson number in protonnumber in proton

A reasonable picture of proton favors smaller A reasonable picture of proton favors smaller cutoff of omegacutoff of omega

dyyfn BMM )(1

0

SummarySummary The inclusion of the sigmaThe inclusion of the sigma meson cloud meson cloud

effects has an improvement of the effects has an improvement of the description for light flavor sea quarks description for light flavor sea quarks in the proton in the proton

We also provides a picture of a We also provides a picture of a reasonable small n_omega with a reasonable small n_omega with a smaller cutoff in the proton. smaller cutoff in the proton.

Sigma meson may play an important Sigma meson may play an important role in meson cloud modelrole in meson cloud model

n