Polarized light flavor sea–quark asymmetry in polarized semi–inclusive processes

We propose new formulas for extracting a difference of the polarized light sea–quark density, ∆d̄(x)−∆ū(x), from polarized deep–inelastic semi–inclusive data. We have estimated the value of it from the present experimental data measured by SMC and HERMES groups. Although the data might suggest a violation of polarized light flavor sea–quark symmetry, the precision of the present data is not enough for confirming it. PACS number(s): 13.88.+e, 13.85.Ni, 14.65.Bt Traditionally, the light sea-quark distributions, ū(x) and d̄(x), have been taken to be flavor symmetric in the phenomenological analysis of structure functions of nucleons with an expectation that the strong interaction does not depend on the quark flavor for light quark–pair creations from gluons. However, the NMC experiment in 1991[1], which precisely measured the structure functions of the proton and neutron, F p 2 (x) and F n 2 (x), for a wide region of Bjorken’s x, revealed that it was not the case: the experimental result was given as follows, ∫ 1 0 [F p 2 (x)− F n 2 (x)] dx x = 1 3 − 2 3 ∫ 1 0 [d̄(x)− ū(x)]dx, = 0.235± 0.026, (1) which resulted in ∫ 1 0 [d̄(x)− ū(x)]dx = 0.147± 0.039, (2) where d̄(x) and ū(x) represent the d̄ quark and ū quark densities in the proton, respectively. Hence, we see a considerable excess of the d̄ quark density relative to the ū quark density, contrary to the flavor symmetry prediction leading the integral value of the left–hand side of eq.(1) to be 1/3, which is called the Gottfried sum rule[2]. Furthermore, from the measurement of the Drell–Yan cross section ratio, σ(p + d)/σ(p + p), the E866 collaboration[3] provided an independent confirmation of the violation of the light flavor sea-quark symmetry, though the violation of the Gottfried sum rule is smaller than reported by the NMC. Now, study on the origin of the flavor asymmetry of light sea–quarks has been a challenging subject in particle and nuclear physics because it is closely related to the dynamics of nonperturbative QCD[4]. Several approaches such as chiral quark model, Skyrme model, Pauli blocking effects, etc., have been proposed so far to understand its origin[5]. However, the discussions are still under going. For spin–dependent parton distributions, is the polarized sea–quark density, ∆ū and ∆d̄, also asymmetric at an inital value ofQ0 in the nonperturbative region? In these years, measurement of the polarized structure function of the nucleon in polarized deep–inelastic scatterings have shown that the nucleon spin is carried by quarks a little and the strange sea–quark is negatively polarized in quite large. The results were not anticipated by conventional theories and often referred as ‘the proton spin crisis’[6]. By using many data with high precision on the polarized structure functions of the proton, neutron and deuteron accumulated so far, good parametrization models of polarized parton distribution functions have been proposed at the next–to–leading order(NLO) of QCD[7]. The behavior of polarized valence u and d quarks has been well–known from such analyses. However, the knowledge of polarized sea–quarks and gluons is still poor. Although people usually assume the symmetric light sea–quark polarized distribution, i.e. ∆ū(x) = ∆d̄(x), in analyzing the polarized structure functions of nucleons, there is no physical ground of such an assumption. In order to understand the nucleon spin structure, it is very important to know if the light sea–quark flavor symmetry is broken even for polarized distributions and to determine how ∆ū(x) and ∆d̄(x) behave in the nucleon. Related to these subjects, it is interesting to know that even if we start with the symmetric distributions for the polarized light sea–quarks, ∆ū = ∆d̄, at an initial Q0, the symmetry can be violated for higher Q 2 regions, if the polarized distributions are perturbatively evolved in NLO calculations of QCD[8]. In addition, some people have estimated the amount of its violation at an initial Q0 using some effective models. However, their results do not agree with each other[9, 10]. Therefore, it is interesting to extract the value of ∆d̄(x)−∆ū(x) from the experimental data and test the flavor symmetry of ∆d̄(x) and ∆ū(x) experimentally. Recently, using longitudinal polarized lepton beams and longitudinal polarized fixed targets, SMC group at CERN[11] and HERMES group at DESY[12, 13] observed the cross sections of the following semi–inclusive processes, ~l + ~ N → l + h+X , (3) and obtained the data on spin asymmetries for proton, deuteron and He targets, where h is a created charged hadron or one of π, K, p and p̄. A created hadron depends on the flavor of a parent quark and thus properly combining these data it is possible to decompose polarized quark distributions into the ones with individual flavor[14]. These data provide a good material to test the light flavor symmetry of polarized sea–quark distributions and it might be timely to test the symmetry by using the present data. In this letter, we propose new formulas for extracting a difference, ∆d̄ −∆ū, from the data of the above–mentioned semi–inclusive processes and estimate the value of it from the present data in order to test if the light flavor symmetry of polarized sea–quark distributions is originally violated. Let us start with the semi–inclusive asymmetry for the process of eq.(3) with proton targets, which is written by[11] Ah1p(x,Q ) = ∑ q,H e 2 q {∆q(x,Q ) D q (Q ) + ∆q̄(x,Q) D q̄ (Q )} ∑ q,H eq {q(x,Q 2) DH q (Q 2) + q̄(x,Q2) DH q̄ (Q 2)} × {1 +R(x,Q)} , (4) in the leading order(LO) of QCD[15], where ∆q(x,Q)(∆q̄(x,Q))and q(x,Q)(q̄(x,Q))are the spin–dependent and spin–independent quark distribution functions at some values of x and Q, respectively, and R(x,Q) is a ratio of the absorption cross section of longitudinally and transversely polarized virtual photons by the nucleon, R(x,Q) = σL/σT . D H q (Q ) is given by integration of the fragmentation function, D q (z,Q ), over the measured kinematical region of z, i. e. D q (Q ) = ∫ 1 zmin dz D q (z,Q ), where D q (z,Q ) represents the probability of producing a hadron H carrying momentum fraction z at some Q from a struck quark with flavor q. h is the observed hadron concerned with here. When h is h, the fragmentation function of, for example, u–quark decaying into h is given by D + u (z,Q ) = D + u (z,Q ) +D + u (z,Q ) +D u(z,Q ) , (5) because h is dominantly composed of π, K and p. Assuming the reflection symmetry along the V–spin axis, the isospin symmetry and charge conjugation invariance of the fragmentation functions, many fragmentation functions can be classified into the following 6 functions[4], D ≡ D + u = D π d̄ = D π d = D π ū , D̃ ≡ D + d = D π ū = D π u = D π d̄ = D π s = D π s̄ = D π s = D π s̄ , D ≡ D + u = D K s̄ = D K ū = D K s , (6) D̃K ≡ D + d = D K s = D K ū = D K d̄ = D K u = D K d = D K d̄ = D K s̄ , D ≡ D u = D p d = D p̄ ū = D p̄ d̄ , D̃p ≡ D s = D p ū = D p d̄ = D s̄ = D p̄ u = D p̄ d = D p̄ s = D p̄ s̄ , where D and D̃H are called favored and unfavored fragmentation functions, respectively. Here we follow the commonly taken assumption on the fragmentation functions, for simplicity. Now, we can rewrite eq.(4) as ∑ q,H eq {∆q(x,Q ) D q (Q ) + ∆q̄(x,Q) D q̄ (Q )} = Ah1p(x,Q ) [ ∑ q,H e 2 q {q(x,Q ) D q (Q ) + q̄(x,Q) D q̄ (Q )}] {1 +R(x,Q2)} = ∆N p (x,Q ) , (7) where ∆N p (x,Q ) is reffered to the spin–dependent production processes of charged hadrons with proton targets. From a combination of ∆N ,h p,n (x,Q ) for proton and neutron targets, we can obtain the following formula, ∆d̄(x,Q)−∆ū(x,Q) = ∆N + p (x,Q )−∆N + n (x,Q )−∆N − p (x,Q ) + ∆N − n (x,Q ) 2 I1(Q) − ∆N + p (x,Q )−∆N + n (x,Q ) + ∆N − p (x,Q )−∆N − n (x,Q ) 2 I2(Q) ,(8)