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Nuclear Physics A 922 (2014) 126–139

www.elsevier.com/locate/nuclphysa

Strange and non-strange sea quark–gluon effectsin nucleons

M. Batra, A. Upadhyay ∗

School of Physics and Material Science, Thapar University, Patiala, Punjab 147004, India

Received 21 February 2013; received in revised form 20 November 2013; accepted 21 November 2013

Available online 6 December 2013

Abstract

Within a statistical approach, strange and non-strange quark–gluon Fock state contributions are analyzedfor their low energy properties. A suitable wave function is written for a nucleon that consists of threevalence quarks (qqq) and the sea (g, qq). Expansion of the nucleonic system in terms of Fock states thatcontain (g, qq) is assumed and the probabilities of all possible Fock states, that lead to such a wave-functioncontaining strange and non-strange quark–gluon contents in the sea are determined. Various approxima-tions are entertained to validate the authenticity of the model used. The statistically determined coefficientsstrongly favor a vector-dominated sea where the sea includes ss pairs. Additionally, the sea is constrainedto have a limited number of components. The phenomenological implications that affect the low energyproperties are discussed. The obtained results are compared to existing theoretical models and experimentaldata.© 2013 Elsevier B.V. All rights reserved.

Keywords: Strange quark; Statistical model; Spin distribution

1. Introduction

Quantum Chromodynamics successfully describes hadronic phenomenon at very short dis-tance scales in terms of the interaction of quarks and gluons. At long distances, because ofthe non-perturbative nature of their structures, many particles and their internal details remaina mystery. In particular, to bridge the gap between QCD and current description of hadronsstructure, discrete models have been suggested in the literature. The simple quark model sug-gests that baryons have three valence quarks and a sea that contain gluons and quark–antiquark

* Corresponding author.

0375-9474/$ – see front matter © 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.nuclphysa.2013.11.008

M. Batra, A. Upadhyay / Nuclear Physics A 922 (2014) 126–139 127

condensates. The effect of the sea on nucleon properties such as weak decay coupling ratios,magnetic moment and masses can be studied via these models. The spin structure of the protonis one of the key examples that confirm the presence of gluons which provide a non-negligiblecontribution to spin of proton. The latest studies show that only 30% of the total spin of the nu-cleon is carried by quarks. In the literature, many phenomenological models have been presentedin studies to solve these types of puzzles [1–3]. The static properties of the hadrons are studiedthrough various experiments [4–6]. The well known experiments are the EMC (Electron–MuonCollaboration) and the SMC (Spin-Muon Collaboration). A wide range of data is also availablefrom experiments at SLAC [7–9]. The NuTeV Collaboration at FermiLab predicted the non-zerovalue of the quark contribution to the spin of nucleon [10,11] via the strange quark content ratio,which is the fraction of nucleon momentum that is carried by strange quark to non-strange quark:

2(s+s)

(u+u+d+d)= 0.477±0.063±0.053 [12]. This ratio implies the existence of strange quarks in the

sea. Recent experimental collaborations such as HAPPEX and GO [13–16] as well as theoreticalstudies [17] suggest that the strange quark contribution to the nucleonic form factors is negligi-ble. R. Bijker et al. [17] concluded that a strange quark–antiquark pair contributes very slightly(−0.0004μN) to magnetic moment of the nucleon and also have suggested that the inclusion ofhigher Fock components is very small. Despite much experimental and theoretical data, the spincontent of nucleon is still not clearly understood. At present, there are many phenomenologicalmodels at hand whose domains of validity must be checked using the available experimentaldata.

In this article, we analyze the stability of the statistical approach for nucleonic system. Weextend the statistical model using the detailed balance principle to study the proton includingstrange and non-strange quark–antiquark pairs and gluons. The model assumes that the protonhas valence quarks (uud) and a virtual sea where quark–antiquark pairs are multi-connectedthrough gluons. In this paper, it is shown explicitly that a vector sea that contains strange quarksis favored by the available experimental results and dominates over the scalar and tensor contri-butions. Section 2 discusses the details of the wave-function for the nucleon. In Section 3 and4, a brief formalism of detailed balance and the statistical approach are presented. The low en-ergy properties and their dependence on various coefficients (calculated from multiplicities oflimited number of Fock states) are mentioned in Section 5. Section 6 is dedicated to a suitablediscussion of the various results. In Section 7, we conclude with the statement that the presenceof strange quark–antiquark pairs in the sea has a very small effect on the low energy propertiesof the proton.

2. Antisymmetric wave-function for nucleon

Our goal is to write the wave function for a spin up proton consisting of three quarks in thecore and quark–antiquark pairs and gluons in the sea in such a way that it maintains the totalantisymmetry of the proton.

|p〉 =∑

i,j,k,l

Ci,j,k|uud, i, j, k, l〉 (2.1)

where i is the number of uu pair, j is the number of dd pair, l is the number of ss pairs and k

is the number of gluons. The probability of finding the proton in the Fock state |uud, i, j, k, l〉 isρi,j,k,l = |Ci,j,k,l |2 and satisfies the following normalization condition:

∑ρi,j,k,l = 1.

i,j,k,l

128 M. Batra, A. Upadhyay / Nuclear Physics A 922 (2014) 126–139

Let us consider an example in which the proton has two gluons in the sea in addition to thethree valence quarks. Now, two gluons each with spin 1, in spin space will yield the followingpossibilities:

Spin: gg: 1 ⊗ 1 = 0s ⊕ 1a ⊕ 2s.

Similarly in color space, each gluon being a color octet ‘8’ will yield the following symmetricand antisymmetric states:

Color: gg: 8 ⊗ 8 = 1s ⊕ 8s ⊕ 8a ⊕ 10a ⊕ 10a ⊕ 27s.

The subscript s and a represents the symmetric and antisymmetric combinations of the states,respectively. The elaborated form of Eq. (2.1) includes various possible combinations of valenceand sea such that φ represents the wave function of the three quarks and the (H,G) indicesrepresent the spin and color of the sea part. All possible combinations that can yield proton thatis spin 1

2 , a flavor octet and a color singlet’s are as follows:

φ( 1

2 )↑1 H0G1, φ

( 12 )↑

8 H0G8, φ( 1

2 )↑10 H0G10,

[φ

121 ⊗ H1

]↑G1,(

φ128 ⊗ H1

)↑G8,

(φ

1210 ⊗ H1

)↑G10,

(φ

328 ⊗ H1

)↑G8 and

(φ

328 ⊗ H2

)↑G8.

The total flavor–spin–color wave function of a spin-up proton is written as follows [3]:

∣∣Φ↑12

⟩ = 1

N

[φ

( 12 )↑

1 H0G1 + a8φ( 1

2 )↑8 H0G8 + a10φ

( 12 )↑

10 H0G10 + b1[φ

121 ⊗ H1

]↑G1

+ b8(φ

128 ⊗ H1

)↑G8 + b10

(φ

1210 ⊗ H1

)↑G10 + c8

(φ

328 ⊗ H1

)↑G8

+ d8(φ

328 ⊗ H2

)↑G8

](2.2)

where

N2 = 1 + a28 + a2

10 + b21 + b2

8 + b210 + c2

8 + d28

Here, N is the normalization constant.The above wave-function includes all possible states with definite spin and color quantum

numbers. The valence quarks with spin 3/2 and flavor ‘8’ that form the φ328 part of the intrinsic

wave-function can give rise to spin 3/2 for the total wave function of the proton, only when thesea has spin 1 or 2. Similarly, a single gluon in the sea contributes only spin 1 and a color octet,i.e. H1G8 to the valence part. The contributions from the H0G10 and H1G1 sea for two gluonsare excluded, because H0 and G1 are symmetric under the exchange of two gluons while H1 andG10 are anti-symmetric making their product anti-symmetric. A similar treatment is extended tothe multiple gluon cases. The maximum numbers of partons in a protonic system is consideredto be up to five. All other possibilities such as H2G1, H2G10 are also excluded because theycannot lead to a color singlet state. Contributions from states such as H0G27, H2G27 are ignoredbecause of the suppression of higher multiplicities.

The coefficients mentioned in the wave function, that are associated with each state providethe relative contributions for all the Fock states. Our intent is to determine statistically the prob-abilities associated with each Fock state in the flavor, spin and color spaces individually. Todetermine the set of probability sets of each Fock state in flavor space, the principle of detailedbalance is applied.

M. Batra, A. Upadhyay / Nuclear Physics A 922 (2014) 126–139 129

3. Principle of detailed balance

This principle was proposed by Zhang et al. [18–20] and has been successful in explainingthe flavor asymmetry of hadrons [21]. This principle calculates d − u = 0.124 which is in closeagreement with the experimental value 0.118 ± 0.01 [30]. Recent applications of this principleinclude the computation of parton distribution functions [31]. We apply the principle of detailedbalance and assume the presence of quark–antiquark pairs that are multi-connected to gluons.The splitting and recombination between any two sub-processes balance each other. The principleapplied in Ref. [18] considers the balancing of Fock states with lower mass components (u, d)

and gluons. Moreover, the inclusion of strange quark–antiquark pairs in the Fock states requiresthe modification of the general expressions [18], where the non-negligible mass of the strangequarks put constrains on the free energy of the gluon. To accommodate the strange quark and toallow processes such as g ⇔ ss, we should have a system with energy greater than twice that ofthe strange quark mass. For this, we first extend the principle of detailed balance to obtain theprobability of each Fock state that include ss content with due consideration of the mass of thestrange quark. The details are given below.

Let us consider the equilibrium between the splitting and recombination of Fock states viathree main processes (q ⇔ qg,g ⇔ gg,g ⇔ ss).

(i) When q ⇔ qg is considered: The general expression of the probability for this sub-processis written as follows:∣∣{q}, {i, j, l, k − 1}⟩ 3+2i+2j+2l⇐⇒

(3+2i+2j+2l)k

∣∣{q}, {i, j, l, k}⟩ (3.1)

where i refer to uu pairs, j refers to dd , l refers to ss and k refers to the number of gluonssuch that the total numbers of partons is 3 + 2i + 2j + 2k + 2l in the final state.

ρi,j,l,k

ρi,j,l,k−1= 1

k

(ii) When both the processes g ⇔ gg and q ⇔ qg are included: Similarly,

∣∣{q}, {i, j, l, k − 1}⟩ 3+2i+2j+2l+k−1−−−−−−−−−−−−−−⇀↽−−−−−−−−−−−−−−(3+2i+2j+2l)k+ k(k−1)

2

∣∣{q}, i, j, l, k⟩

ρi,j,l,k

ρi,j,l,k−1= (3 + 2i + 2j + 2l + k − 1)

(3 + 2i + 2j + 2l)k + k(k−1)2

(3.2)

(iii) When g ⇔ qq is considered: For gluons to undergo the process g ⇔ ss, the free energymust be at least, εg > 2Ms where Ms is the mass of the strange quark. The generation ofan ss pair from gluons is restricted by applying a constraint in the form k(1 − Cl)

n−1 [19]where n is the total number of partons that are present in the Fock state. This factor isintroduced from the gluon free energy distribution and the application of certain constraintson the momentum and total energy of the partons present in the proton. In all cases, Cl−1 =

2Ms

MP −(2l−1)Ms, Ms is the mass of s-quark and MP is the mass of the proton.

∣∣{q}, g⟩ 1(1−C0)3

−−−−−−⇀↽−−−−−−2.1

∣∣{q}, ss⟩∣∣{q}, uug

⟩ 1(1−C0)5

−−−−−−⇀↽−−−−−−∣∣{q}, uuss

⟩

1.2

130 M. Batra, A. Upadhyay / Nuclear Physics A 922 (2014) 126–139

Generalizing to a number of gluons ‘k’ and numbers of quark–antiquark pairs ‘i’, ‘j ’ and ‘l’ forproton.

{∣∣{q}, i, j, l − 1, k⟩} k(1−Cl−1)

n−1

−−−−−−−−−⇀↽−−−−−−−−l(l+1)

{∣∣{q}, i, j, l, k⟩}

ρi,j,l,k

ρi,j,l−1,k

= k(1 − Cl−1)n−1

l(l + 1)(3.3)

Suppose that no ss pair is present initially and that the generation of one pair requires gluon tohave sufficient energy then the condition becomes:

∣∣{q}, i, j,0, k⟩ k(1−C0)

n−2l−1

−−−−−−−−−⇀↽−−−−−−−−−l(l+1)

∣∣{q}, i, j,1, k − 1⟩

where n = 3 + 2i + 2j + 2l + k

ρi,j,1,k−1

ρi,j,0,k

= k(1 − C0)n−2l−1

1(1 + 1)

∣∣{q}, i, j,1, k − 1⟩ (k−1)(1−C1)

n−2l

−−−−−−−−−−−⇀↽−−−−−−−−−−2(2+1)

∣∣{q}, i, j,2, k − 2⟩

ρi,j,2,k−2

ρi,j,1,k−1= (k − 1)(1 − C1)

n−2l

2(2 + 1)(3.4)

The go-out probability depends upon the number of partons that are present at that time. Weproceed in this manner until all gluons have been converted into strange pairs.

∣∣{q}, i, j, k − 1,1⟩ 1(1−Ck−1)

n−k−2

−−−−−−−−−−⇀↽−−−−−−−−−−k(k+1)

∣∣{q}, i, j, k, o⟩

|ρi,j,k,0〉1(1−Ck−1)

n−k−2

−−−−−−−−−−⇀↽−−−−−−−−−−k(k+1)

|ρi,j,k−1,1〉

Thus,

ρi,j,l,0

ρi,j,0,l

= (k(k − 1)(k − 2)(k − 3) − 1(1 − C0)n−2l−1(1 − C1)

n−2l (1 − C2)n−2l+1(1 − Cl−1)

n−k−2)

k!(k + 1)!∣∣{q}, i, j, l + k − 1,1

⟩ 1(1−Cl−1)n−k−2

−−−−−−−−−−⇀↽−−−−−−−−−−l+k(l+k+1)

∣∣{q}, i, j, l + k,0⟩.

Generalizing it to k number of gluons, the ratio becomes:

ρi,j,l,k

ρi,j,l+k,0= (k(k − 1)(k − 2)(k − 3) − 1(1 − C0)

n−2l−1(1 − C1)n−2l (1 − C2)

n−2l+1 − (1 − Cl−1)n+k−2)

(l + 1)(l + 2) · · · (l + k)(l + k + 1)

(3.5)

The condition of normalization,∑

i,j,k,l ρi,j,k,l = 1 is used to determine the individual probabil-ities of the Fock states which requires ρi,j,l,k to be expressed in terms of ρ0000.

ρi,j,l+k,0

ρ0,0,0,0= 2

i!i + 2!j !(j + 1)!(l + k)!(l + k)! (3.6)

Using Eqs. (2.7) and (2.8) along with the condition of normalization, the sets of the individualprobabilities of all Fock states can be computed.

M. Batra, A. Upadhyay / Nuclear Physics A 922 (2014) 126–139 131

4. Statistical model

Recent experimental and theoretical studies [12] show that the strange quark is one of theessential components of the intrinsic sea that contributes to the spin distribution among quarksand gluons within nuclei. Therefore, we extend our original work [21,37] by including Fockstates with ss. The Fock states without ss constitutes 86% of the total Fock states while theinclusion of strange quark content reduces the value to 80%. The relative probability in spin andcolor space is found for the case in which the core part should have an angular momentum j1,the sea should have j2 and the total angular momentum should come out as spin 1/2 (may bej1+j2 = 1/2) with a color singlet state. The comparison of such probabilities can be shown as:

ρj1j2

ρj ′1j

′2

= x(r1)y(r2)z(r3)

x′(r ′1)y

′(r ′2)z

′(r ′3)

where (x, y, zx′, y′, z′) are the corresponding multiplicities and r1 is the probability for q3 coreto have spin j1, similarly r2 is the probability for sea part to have spin j2. r3 represents theprobability that the total angular momentum is 1/2. For instance: in model C, for a two gluonsea,

ρ 12 ,0

ρ 32 ,1

= 2,ρ 1

2 ,0

ρ 32 ,2

= 2,ρ1,1

ρ8,8= 1

4,

ρ1,1

ρ10,10= 1

Thus, we have different ratios for all Fock states in sea with specific spin and color quantum num-bers. The product of the relative probabilities can be expressed in terms of a common multiplier“c” as shown below:

ρ 12 0[ρ11s , ρ88s ], ρ 1

2 1a[ρ88a , ρ1010], ρ 3

2 1a[ρ88a ]

ρ 32 2s

[ρ88s ] = 2c(1,2;2,1;1;1) (4.1)

The numbers shown in the indices on the right hand side represents the multiplicities for eachparticular Fock state. The Table 4 lists these multiplicities for all Fock states in model C. Weapply a few modifications to previous calculations [21] by assuming the contributions from thethree gluon states to be zero for H0G10 and H1G10. Additionally, the states which are consideredto have spin 3/2 for the valence part and to form only a color-singlet state yield zero contributionin terms of probability. Using the flavor probabilities given by principle of detailed balance, thenumber “c” for each Fock state is calculated. The sum of the total probabilities will provide thecoefficients a0, a8, a10, b1, b8, b10, c8, d8 to the total wave-function. For instance, a8 can bewritten as sum of all “nc” for all Fock states.

a8 = (n08csea)|gg〉 + (n08csea)|uug〉 + (n08csea)|ddg〉 + (n08csea)|ssg〉 + · · ·The coefficients in the wave function are then expressed in terms of the parameters definedbelow: [3]

a = 1

2

(1 − b2

1

3

), b = 1

4

(a2

8 − b28

3

), c = 1

2

(a2

10 − b210

3

)

d = 1

18

(5c2

8 − 3d28

), e =

√2

3(b8c8)

The above defined parameters can be further be related to the coefficients α and β as follows:

132 M. Batra, A. Upadhyay / Nuclear Physics A 922 (2014) 126–139

α = 2(6 + 3a28 − 2b2

1 − b28 + 4b8c8 + 5c2

8 − 3d28 )

27(1 + a210 + a2

8 + b21 + b2

10 + b28 + c2

8 + d28 )

(4.2)

β = 3 − 9a210 − 3a2

8 − b21 + 3b2

10 + b28 + 8b8c8 − 5c2

8 + 3d28

27(1 + a210 + a2

8 + b21 + b2

10 + b28 + c2

8 + d28 )

(4.3)

Properties such as magnetic moment, spin distribution and axial coupling constant ratios arecalculated using the nucleon wave function at low energies typically on the scale of 1 GeV scalewith the aid of the coefficients defined above.

The confining forces among the constituents also play a significant role in determining thelow energy observables and this fact forces us to check the stability of the statistical approachwith respect to certain modifications.

Model P is based on the constraint that a quark–antiquark pair can reside in the form of col-orless pseudo-scalar Goldstone bosons which may appear because of the soft gluonic exchangeinteractions and spin-flip process. These internal pseudoscalar bosons contribute some additionalsymmetry in the quark–gluon Fock states and the sea is no longer assumed to be an active par-ticipant in the nucleons now. In case of |ggqq〉 state, to compensate for the odd parity of theqq pair, one of the gluons is assumed to be in TE (Transverse Electric) mode while the other isassumed to be in TM (Transverse Magnetic) mode [33]. In this model, the decomposition of theFock states leads to equivalence among a single gluon state and the states with any number of(qq) with g. The assumption for two gluon states is similar. A state with single quark–antiquarkpair is assumed to be a color singlet state carrying spin 1 and states with two qq pairs have zerospin and are color singlet states.

Model D assumes the suppression of Fock states with higher multiplicities. A sea with greatercolor multiplicity has a lower probability of survival because of the larger likelihood of interac-tion. This model can be assumed to be a special case of model C in which the suppression isachieved by assuming that the probability of a system in a spin and color sub-state is inverselyproportional to the multiplicity of the state. For example for the state, |u,u, s,1,0,1,0〉:

ρ 12 0[ρ11,ρ88, ρ1010], ρ 1

2 1[ρ11,ρ88, ρ1010]

ρ 32 1[ρ88], ρ 3

2 2[ρ88] = 2d

(2,

1

8,

1

50; 2

3,

1

24,

1

150; 1

96; 1

160

)

We would like to note that models P and D differ from the model C only in their evaluation ofthe multiplicities only.

5. Nucleonic parameters

The low energy properties of nucleonic system can be estimated using various models, par-ticularly in the SU(6) based Naïve Quark Model. This model explains the magnetic momentof all baryons to a better extent than it explains the spin distribution among baryons. Althoughexperimentalists and phenomenologist’s have attempted to solve the puzzles regarding spin dis-tribution, the complete picture remains clear. Most properties can be calculated by integrating g1over a four momentum variable x for the partons and called the first moment:

Γ1 =1∫g1(x) dx = 1

2

1∫ ∑f

e2f qf (x) dx (5.1)

0 0

M. Batra, A. Upadhyay / Nuclear Physics A 922 (2014) 126–139 133

Under the assumption of SU(3) symmetry, the axial vector current in the spin 12 baryon octet can

be rewritten in the form of matrix elements a0, a3 and a8 as follows:

Γ 1(p,n) = 1

12

(±a3 + 1√

3a8

)+ 1

9(a0)

= ± 1

12(u − d) + 1

36(u + d − 2s) + 1

9(u + d + s) (5.2)

Here u, d and s are the polarized quark and antiquark densities that are to be calculated todetermine the nucleonic properties.

5.1. Spin distribution for partons

Nucleon spin is said to be distributed among the valence and sea quarks and gluons and somepart of it is carried by their orbital angular momentum. Polarized deep inelastic scattering exper-iments are the key tool for probing the internal structure of the nucleon. A measurement of spindistribution function was performed by the EMC (European Muon Collaboration) experiment in1988 and Γ P

1 was measured to be 0.126 ± 0.018 [24]. Experimentally, the SMC (Spin MuonCollaboration) [25] predicted a very small contribution from the intrinsic spin of the quarks tothe proton spin. The COMPASS and HERMES collaborations focused on the measurement ofΣ (= u + d + s) [26,27] at Q2 = 3 GeV2 and 5 GeV2. The total contribution of the spinamong quarks inside the proton can be measured via the first moment of the structure function g1,which can further be expressed in terms of the polarized quark and antiquark distribution. QCDcorrections involve the Q2 dependence which replaces g1 with g1(x,Q2). The spin structurefunction Γ

p

1 is also important, as it could further be utilized to obtain the three matrix elementsa0, a3 and a8 under SU(3) flavor symmetry. It is also written in terms of the polarized quarkand antiquark densities. The matrix element a0 gives the contribution Σ = u + d + s.The spin distribution also includes the gluonic spins and momentum of quarks and gluons as perthe helicity sum rule [24]. Ellis and Karliner [22] measured the spin polarization densities usingperturbative QCD corrections. Larin et al. [36] studied next to next leading order corrections toBjorken sum rule and applied to study spin polarizations.

In our models, the total spin distribution is determined using the charged squared spin projec-tion operator

Ip

1 = 1

2

⟨∑i

e2i σ

iz

⟩p

and In1 = 1

2

⟨∑i

e2i σ

iz

⟩n

(5.3)

leading to

Ip

1 = 1

6(4α − β) and In

1 = 1

6(α − 4β). (5.4)

The operator here provides information in terms of the quark and gluon Fock states. Furthermore,the contribution of the orbital angular momentum is not considered because of the minimizedoverlapping of the valence and sea momentum distributions [38].

5.2. Axial coupling constant ratio and matrix elements

The matrix element of the quark currents between proton and neutron states in beta decaymay be calculated using isospin symmetry. From the semileptonic decay B → B ′eν, the vectorand axial coupling constants gv and gA can be determined. The matrix element a3 represents the

134 M. Batra, A. Upadhyay / Nuclear Physics A 922 (2014) 126–139

weak decay coupling constant ratio for the proton and neutron. The operator for this ratio can betaken to be: Oi

f = 2I i3, which in terms of the coefficients α and β leads to gA/gV = 3(α +β) for

a neutron decaying into proton. Moreover, QCD corrections give

Γ1(Q2) =

1∫0

g1(x,Q2)dx = 1

18(4u + d + s)

(1 − αs(π)

)(5.5)

where (1 − αsπ) is the first order corrections derived from the Bjorken sum rule [27]. On thebasis of SU(3) symmetry, all baryon decay rates depend upon two universal parameters F andD where F and D are proportional to the structure constant fijk and the antisymmetric invarianttensor dijk . The ratio of F to D is interpreted as

F

D= α

α + 2β. (5.6)

Experimentally, F and D have been determined from hyperon β-decays to be F = 0.463 ± 0.08and D = 0.804 ± 0.08 [28].

5.3. Magnetic moment ratio

The magnetic moment of any baryon can be related to the spin distribution of quarks(u,d,s) without giving explicit wave-functions. It can be written as follows:

μB =∑q

(q)Bμq (q = u,d, s) (5.7)

Gupta et al. [29] determined the magnetic moments of all baryons in their three and four param-eter fits of the magnetic moments of quarks. In our model, the magnetic moment ratio of protonand neutron is described either in terms of quark magnetic moment operator or in terms of thetwo parameters α and β as follows:

μp

μn

= −2α + β

α + 2β(5.8)

which can be calculated from the probabilities. The operator for the magnetic moment ratio ofthe proton and neutron is given by:

μ =∑q

eiq

2mσ

qz , q = (u, d, s) (5.9)

where μp = 3(μuα − μdβ) and μn = 3(μdα − μuβ).

6. Discussion and results

We study a statistical model that assumes that the sea contains an admixture of gluons andquark–antiquark pairs in addition to the three valence quarks. This statistical approach is basedon the principle of detailed balance [18]. The statistical model and several approaches to it areused to identify the separate contributions from the scalar, vector and tensor sea. The results ofthe quark model [3] are also modified by the inclusion of the so called negligible contributions,

M. Batra, A. Upadhyay / Nuclear Physics A 922 (2014) 126–139 135

where the non-zero contributions from the scalar and tensor sea are determined using the statis-tical approach. The new coefficients obtained without ss and with the strange quark condensateswithin these approaches are shown in Table 3.

The sea contributions in different models and their approximations are presented in Table 3 tocheck the validity of these models and compare them to experimental results thereby demonstrat-ing the importance of the vector, tensor and scalar sea in determining these properties. Each termhas a non-negligible dependence on almost all nucleonic parameters. To check the contributionsfrom the scalar sea, we suppress the vector and tensor sea contributions and we use a similarapproach to find the individual contribution from the vector and tensor sea. As the sea part isdominated by the emission of virtual gluons, we can expect b8 and c8 to be more dominant.If only the vector sea is assumed to contribute, then nucleonic properties such as the couplingconstant and F/D ratio are primarily affected by parameters b8 and c8.

It can be seen from Table 3 that for the simple quark model, if we consider non-zero scalar andtensor sea contributions from the available statistical data, then the percentage error increases to7–8% except in case of the spin distribution of the nucleons for which deviation decreases from∼ 58% to 6–7% approximately. The tensor sea appears to be less dominating because of quarkspin-flip processes but cannot be neglected in all cases. The vector sea plays an important role indetermining values that are close to those found in experiments. Some of the properties such asthe spin distribution and gA/gV ratio seem to be more strongly affected by changes in the valuesof these coefficients. The extent, to which the sea contribution affects the nucleon properties, isillustrated in Table 3.

For instance, when the scalar and tensor sea are neglected in statistical model (C, P, D) and thesimple quark model, the magnetic moment ratio deviates by 6–10% from experimental data [32]shown in Table 3. The weak decay matrix element ratio deviates by 30–40% from the originalvalue when the sea contribution is included in statistical model as well as in simple quark model.However in this case, when the sea is excluded, the ratio is fairly similar to the experimentalresults, the deviation is 0–17%. The F/D ratio is closer to the experimental data when the seais excluded in all the approaches but when the sea is taken into account, the statistical approachyields much better results, the C model in particular performs better than the other. As we gofrom the simple quark model to statistical, the sea becomes the dominant contributor to the spindistribution, and therefore, the results are better matched to the experimental results in the lattercase. A similar observation holds when we move from the exclusion of the sea to the inclusionof the sea in C model. The inclusion of strange quark condensates in the sea produces results thatare more similar to experimental observations. The extension of the principle of detailed balancecan be compared to asymmetry with a value 0.71 and 0.124 for u/d and u − d asymmetry,respectively [30] which have been found to be agree well with the experiments and with othertheoretical models. Also, Gao and Ma [23] showed that neutrino induced DIS experiments canbe more sensitive towards strange quark–antiquark asymmetry. The principle of detailed balancefinds the values of strange quark content ratio to be 2s

(u+d)= 0.37; 2s

(u+d)= 0.03 which are found

to be agree well with results of the NuTeV Collaboration [12].

7. Conclusion

Based on the statistical approach, using the detailed balance principle, strange sea contribu-tions to the low energy properties of the nucleon were calculated. The two parameters α and β

directly relates these probabilities to a sea that contains strange quarks in terms of eight coeffi-cients that represents the scalar, vector and tensor sea. The calculation is based on the application

136 M. Batra, A. Upadhyay / Nuclear Physics A 922 (2014) 126–139

Table 1Various Fock states and contribution to their probabilities in models C and D.

States Without ss With ss

Probability Value of c Value of d Probability Value of c Value of d

|gg〉 0.082 0.00511 0.03887 0.070302 0.00439 0.03374|uug〉 0.055 0.00171 0.01900 0.009118 0.0028 0.00315|ddg〉 0.083 0.00258 0.02858 0.013678 0.00042 0.004727|ssg〉 – – – 0.09582 0.00994 0.004727|uudd〉 0.0293 0.00091 0.01013 0.01597 0.00049 0.005519|dddd〉 0.0145 0.00091 0.00692 0.03194 0.01368 0.003790|ggg〉 0.0320 0.00458 0.02358 0.044768 0.00639 0.0037903|uuuuuu〉 – – – 0.000266 0.00003 0.0001958|dddddd〉 – – – 0.000665 0.00005 0.000494|uuddg〉 0.0306 0.00048 0.01289 0.002761 0.00099 0.001161|uussg〉 – – – 0.03194 0.00049 0.020151|ddssg〉 – – – 0.04791 0.00074 0.001580|ddddg〉 0.0153 0.00013 0.00322 0.01380 0.00011 0.006752|uugg〉 0.0304 0.00025 0.01900 0.014193 0.00011 0.030007|ddgg〉 0.045 0.00038 0.00961 0.02129 0.00017 0.045010|ssgg〉 – – – 0.052265 0.00043 0.011049|uuuu〉 0.0072 0.00045 0.00034 0.003992 0.00912 0.001895|uuuug〉 0.0076 0.00003 0.00162 0.000691 0.0000057 0.0001452|uuddss〉 – – – 0.015197 0.00148 0.054700|g〉 + |qq〉′s 0.3483 – – 0.2208 – –

Table 2Computed values of coefficients in the three models and in two forms (with ss and without ss).

Sr.No.

Coefficients Statistical model without ss Statistical model with ss

C P D C D

1. a0 1 1 1 1 12. a8 0.517 0.470 0.2017 0.972 0.233. a10 0.0825 0.147 0.0759 0.402 0.09124. b1 0.1201 0.206 0.467 0.494 0.6075. b8 1.760 1.119 0.050 1.579 0.05886. b10 0.1984 0.261 0.0641 0.589 0.0737. d8 0.8503 0.792 0.055 0.679 0.0748. c8 0.2439 0.299 0.3494 1.092 0.406

Model P assumes that qq is an inactive contributor, so the inclusion of strange quark does not modify the results.

of a constraint to the strange quark with a mass of 110 MeV and the analysis is summarized inTables 1, 2, 3 and 4 for three different versions of the statistical model with different versions C,P and D. Our calculations hold true for scale of the order of 1 GeV2. The comparison of ourdata in different cases to the corresponding experimental results demonstrates that although thestrange contribution in sea is negligible, yet its effects can still be seen in the data of the staticproperties. The inclusion of strange sea yields results that are closer to the experimental observa-tions. The sum of the probabilities of Fock states in flavor, spin and color space (model D) withthe strange sea is

∑ssg+ssgg+ss P (flavor, spin, color) = 0.099. This small value suits the recent

experimental data [13–15] that indicates the strange sea contribution is very small. Moreover,the suppression of higher Fock components in the sea is applied in model D. Model D includesmore dynamism than the other considered models and produces data in a more authentic manner.

M.B

atra,A.U

padhyay/N

uclearP

hysicsA

922(2014)

126–139137

n different approaches (C,P,D) of the statistical model, to

odel

da

Statistical modelwith strange sea(at most one ss)

Experimentalresults

C D

0.19 0.2590.0607 0.0732

−1.41 −1.46 −1.46 [34]0.744 1.02 1.257 ± 0.03 [34]0.607 0.639 0.575 ± 0.016 [32,34]0.115 0.160 0.127 ± 0.004 [35,8]

−0.010 −0.0064 −0.030 [35,8]

Table 3Comparison of the calculated magnetic moment ratio, spin distribution and weak decay coupling constant for nucleons ithe simple quark model.

Parameter Statistical modelwith all sea

Statistical modelwithout scalarand tensor

Quark modelwith all sea

Quark mwithoutscalar antensor seModels with

modificationsModel withmodifications

Model withmodifications

C P D C P D C P D

α 0.216 0.297 0.299 0.255 0.285 0.378 0.214 0.299 0.328 0.3415B

0.071 0.080 0.081 0.092 0.100 0.107 0.0395 0.0587 0.0728 0.0774μpμn

−1.40 −1.47 −1.46 −1.37 −1.38 −1.46 −1.59 −1.57 −1.55 −1.53gA/gV 0.863 1.133 1.143 1.04 1.16 1.45 0.761 1.074 1.20 1.26F/D 0.611 0.647 0.646 0.581 0.587 0.637 0.730 0.718 0.693 0.688Ip1 0.132 0.184 0.185 0.155 0.173 0.233 0.136 0.189 0.207 0.215

In1 −0.011 −0.043 −0.047 −0.018 −0.019 −0.008 0.009 −0.010 0.006 0.0052

138 M. Batra, A. Upadhyay / Nuclear Physics A 922 (2014) 126–139

Table 4Various multiplicities of all the Fock states.

States Values of n for different values of c

H0G1 H0G8 H0G10 H1G1 H1G8 H1G10 H2G8 H1G328

|gg〉 2 4 0 0 4 2 2 2|ddg〉 2 8 2 2 8 2 4 4|ssg〉 2 8 2 2 8 2 4 4|uudd〉 2 8 2 2 8 2 4 4|dddd〉 2 8 2 2 8 2 4 4|ssss〉 2 8 2 2 8 2 4 4|dddddd〉 2 4 0 0 4 2 2 2|uuuuuu〉 2 4 0 0 4 2 2 2|ggg〉 1 4 0 1 2 0 0 0|uuddg〉 1 2 2 3 24 6 8 12|uussg〉 1 2 2 3 24 6 8 12|ddssg〉 1 2 2 3 24 6 8 12|ddddg〉 2 16 4 6 48 12 8 24|uugg〉 2 16 4 6 48 12 8 24|ddgg〉 2 16 4 6 48 12 8 24|ssgg〉 2 16 4 6 48 12 8 24|uuuu〉 2 4 0 0 4 2 2 2|uuuug〉 2 16 4 6 48 12 8 24|uug〉 2 8 2 2 8 2 4 4|uuddss〉 1 16 0 3 48 0 24 0

The uniqueness of the statistical framework lies in the fact that the same model is expected towork well for all spin 1/2 strange baryons. Additionally, the individual probabilities from allFock states can be obtained allowing to check the impact of the presence of various possibilitiesof sea configurations to be investigated individually.

References

[1] G. Karl, Phys. Rev. D 45 (1992) 247.[2] A. Manohar, H. Georgi, Nucl. Phys. B 259 (1991) 335.[3] X. Song, V. Gupta, Phys. Rev. D 49 (1994) 2211–2220.[4] M. Arnedo, et al., EMC Collaboration, Nucl. Phys. B 321 (1989) 541.[5] L.D. Nardo, et al., HERMES Collaboration, AIP Proc. 915 (2007) 507.[6] A. Airapetian, et al., HERMES Collaboration, Phys. Rev. D 75 (2007) 012007.[7] J. Ashman, et al., Phys. Lett. B 206 (1988) 364.[8] K. Abe, et al., Phys. Rev. Lett. 74 (1995) 346.[9] D. Adams, et al., Phys. Lett. B 329 (1994) 399–406.

[10] B. Adeva, et al., SMC Spin Muon Collaboration, Phys. Lett. B 420 (1998) 180–190.[11] K. Ackerstaff, et al., HERMES Collaboration, Phys. Lett. B 464 (1999) 123–134.[12] M. Goncharov, et al., Phys. Rev. D 64 (2001) 112006.[13] A. Acha, et al., HAPPEX Collaboration, Phys. Rev. Lett. 98 (2007) 032301.[14] S. Baunack, et al., Phys. Rev. Lett. 102 (2009) 151803.[15] D. Androic, et al., G0 Collaboration, Phys. Rev. Lett. 104 (2010) 012001.[16] H. Chen, et al., J. Phys. G, Nucl. Part. Phys. 37 (2010) 105006.[17] R. Bijker, et al., Phys. Rev. C 85 (2012) 035204.[18] Y.J. Zhang, W.Z. Deng, B.Q. Ma, Phys. Lett. B 523 (2002) 260.[19] Y.J. Zhang, B. Zhang, B.Q. Ma, Phys. Rev. D 65 (2002) 114005.[20] Y.J. Zhang, B.Q. Ma, L. Yang, Int. J. Mod. Phys. A 18 (2003) 1465–1468.

M. Batra, A. Upadhyay / Nuclear Physics A 922 (2014) 126–139 139

[21] J.P. Singh, A. Upadhyay, J. Phys. G, Nucl. Part. Phys. 30 (2004) 881–894.[22] J. Ellis, M. Karliner, Phys. Lett. B 341 (1994) 397.[23] P. Gao, B.Q. Ma, Eur. Phys. J. C 44 (2005) 63–68.[24] J. Ashman, et al., Nucl. Phys. B 328 (1989) 1.[25] B. Adeva, et al., SMC Collaboration, Phys. Rev. D 70 (2004) 012002.[26] E.S. Ageev, et al., COMPASS Collaboration, Phys. Lett. B 647 (2007) 8.[27] D. Hasch, HERMES Collaboration, Nucl. Phys. B, Proc. Suppl. 191 (2009) 79–87.[28] Y. Goto, et al., Asymmetry Analysis Collaboration, Phys. Rev. D 62 (2000) 034017.[29] V. Gupta, et al., Int. J. Mod. Phys. A 12 (1997) 1861–1874.[30] R.S. Towell, et al., NuSea Collaboration, Phys. Rev. D 64 (2001) 052002.[31] Y.J. Zhang, et al., Phys. Lett. B 528 (2002) 228–232.[32] T.P. Cheng, Ling-Fong Li, arXiv:hep-ph/9811279.[33] Z. Li, G. Karl, Phys. Rev. D 49 (1994) 2620.[34] K. Nakamura, et al., Particle Data Group, J. Phys. G 37 (2010) 075021.[35] P.L. Anthony, et al., Phys. Rev. Lett. 74 (1995) 346.[36] S.A. Larin, J.A.M. Vermaseren, Phys. Lett. B 259 (1991) 345.[37] M. Batra, A. Upadhyay, Nucl. Phys. A 889 (2012) 18–28.[38] H. Lipkin, Phys. Lett. B 256 (1991) 284.