Parton distributions in the proton and pion

We use detailed balance for a hadron composed of quark and gluon Fock states to obtain parton distributions in the proton and pion on the basis of a simple statistical model.


Introduction: parton distributions in the proton
There has been considerable interest in the flavor dependence of the proton's quark and antiquark distributions. The first measurement of the (d −ū) asymmetry was made by the NMC group [1]. The integral of this distribution showed a violation of the Gottfried sum rule. Later, Drell-Yan [2,3] and deep inelastic scattering [4] experiments determined the Bjorken-x dependence of the asymmetry. The meson cloud model and the Sullivan process were used to explain the momentum fraction distribution of (d−ū); many other models have been proposed [5]. Most recently, Zhang and collaborators [6,7,8] have used a simple statistical model to calculate the (d −ū) distribution in the proton. They consider the proton to be an ensemble of quark-gluon Fock states, and use detailed balance [6,7] or "the principle of balance" [8] to determine the distribution functions for all partons of the proton. Despite its simplicity, the model does reasonably well in predicting the distributions of partons, as well as for that of (d −ū). The excess ofd overū comes about from the 2:1 ratio of u:d, which provides an excess of u quarks for the annihilation ofū's.
Zhang, Zou, and Yang (ZZY) [7] write a general Fock state expansion for the proton as with i the number ofūu pairs, j the number ofdd pairs and k the number of gluons. The states are normalized such that the sum of the probabilities ρ ijk = |c ijk | 2 of finding a proton in the state |{uud}{ijk} > summed over all i, j, and k is unity [6], In ZZY's statistical model, detailed balance between any two Fock states requires that in which N(A → B)) is the transfer rate of state A into state B. Transfer rates between states are assumed to be proportional to the number of partons that can split or recombine. Taking into account two processes, q ↔ q g and g ↔ qq, ZZY find that This equation, together with the normalization condition (2), determines all the ρ ijk . It is clear from this equation that uū states, labelled by i, are suppressed relative to dd states, labelled by j. Summing over all states, Zhang et al. [6] find (d −ū) ≈ 0.124, remarkably close to the experimental value of 0.118 ± 0.012 [3]. ZZY determined parton distribution functions for the proton by using a Monte Carlo simulation of the distribution of momenta among the n partons in each Fock state. The phase space volume f F n for n free partons is determined by with P and p i the 4-momenta of the proton and the i-th parton, respectively. The masses of the partons are neglected so that E i = | p i |, and ZZY argue that this free parton phase space distribution should be multiplied by E −1 i because partons with smaller momenta spend more time at the center of the proton where they are almost free; these partons are thus weighted with a higher probability. Then df n , the distribution for confined partons, is: We have found that the effect of the weighting factor is quite small, except for the very lowest and highest parton momenta. From the Monte Carlo distribution of parton momenta p i , the parton distributions can be found in terms of the light cone variable Bjorken x, in which M is the proton mass. We used RAMBO [9] for our Monte Carlo event generator. Then for an n-parton state, for which n = 3 + 2(i + j) + k, the x-distributions forū andd arē for u and d are and for the gluons is Thus, we find, in accord with ZZY, and corresponding equations ford(x), u(x), d(x) and g(x), normalized so that The average number of partons in the proton,n, is given bȳ This sets the scale µ 0 ≈Ē = M/n ≈ 0.17 GeV for Q 2 0 = µ 2 0 at which the distributions are calculated. The plot in Fig. 1, which reproduces ZZY's results, shows that the experimentally deduced (d −ū) is fit qualitatively in this model , but is low at small x and high at large x. The discrepancy with experiment shows up more starkly in Fig. 2, our plot ofd(x)/ū(x). We find that these results are changed very little if the phase space weighting factors E −m i are varied from m = 1 to m = 3. Nevertheless, we believe that it is remarkable that such a simple model does so well.
There are, of course, other explanations of the excess ofd overū, which have been considered previously, particularly the pion cloud [10,11,12,13,14] and ω mesons [15]. The leading term in the Fock state expansion of the pion cloud model is a "bare" proton which consists of valence quarks plus qq pairs due to gluon splitting. It can be argued that this gluon splitting is what is being considered by ZZY and thus the perturbative sea in the "bare" proton should not be symmetric, i.e.d =ū. The pion cloud, represented by the higher-order terms in the expansion, would then be an additional effect. If this is the case, the pions must have a larger role at high x where the fall-off of the ratiod/ū is not reproduced by the statistical model. We will not pursue this argument further here.

Parton distributions in the pion
If the statistical model has some validity, then it should not only work for the proton, but also for the pion. This distribution function is, in fact, required in the pion cloud model. We have therefore investigated the valence and sea quark distributions for the π + in the statistical model. The formulas are similar to those for the proton, but because there is only one valence quark of each flavor in the π + , the sea is flavor symmetric.
We write the Fock state expansion for the pion as The analysis of section 1 is unchanged, except that n = 2 + 2(i + j) + k, and the ratio of probabilities for different Fock states (4) is now so that the π + sea is symmetric, i.e.ū(x) = d(x). We findn = 4.5 in the pion.
The quark distributions for a specific n-parton state arē and for the gluons is The parton distributions are found by summing these distribution functions over all values of {ijk} and The valence quark distribution function is Our results are shown in Fig. 3. The valence quark distributions are too high x q x ,x g x x v x Figure 4: Our calculation of the valence quark distribution x v(x) in the pion, compared to the experimental results of Conway et al. [16]. The dashed curve shows our results without any evolution, as in Fig. 3, which correspond to a scale of Q 2 0 = 1.96 GeV 2 . The solid curve shows our results evolved to Q 2 = 16 GeV 2 of the E615 experiment.
for large x because of the dominant contribution of the n = 2, {ijk} = {000} state, the leading term in the Fock expansion, for which f 2 (x) is a uniform distribution in x. The sea quark distribution is flavor symmetric, as noted above. In order to compare our valence quark distributions to experiment, we carried out an evolution in Q 2 . We determined the starting scale of our distributions by requiring that the first and second moments of our valence quark distribution at Q 2 = 4 GeV 2 be equal to those found by Sutton et al. [17]. This gave us a starting scale of Q 2 0 = 1.96 GeV 2 . We used Miyama and Kumano's code BF1 [18] for the DGLAP [19] evolution. We compare our pion valence quark distribution with that obtained by E615 from pion scattering on tungsten [16] in Fig. 4. The dashed curve shows our results without any evolution, as in Fig. 3. The solid curve shows our results evolved to Q 2 = 16 GeV 2 of the E615 experiment. The agreement between theory and experiment is good. Other theoretical calculations of pion parton distribution functions have used constituent quark models, the Nambu-Jona-Lasinio model, instantons, or the Dyson Schwinger equations. Moments of the distributions can be calculated in lattice QCD, from which particular forms of the distributions can be reconstructed. For references see the recent papers of Hecht, Roberts and Schmidt [20] and Detmold, Melnitchouk and Thomas [21]. In Fig. 5 we compare our valence quark distribution to the Dyson-Schwinger calculation of Hecht et al. and to experiment. Both distributions were evolved to Q 2 = 16 GeV 2 . We find it remarkable that our simple statistical model agrees with experiment as well as the covariant, QCD-based model.

Conclusions
The calculation of parton distribution functions is an important goal of nonperturbative QCD. We have used the statistical model of Zhang et al., developed for the calculation of parton distribution functions in the proton, to calculate the parton distribution functions of the pion. We find that this simple model, with no free parameters, is in good agreement with experiment and other calculations. x v x in the pion, compared to the calculation of Hecht, Roberts and Smith [20] (dashed curve) and the experimental results of Conway et al. [16]. Both calculations were evolved to Q 2 = 16 GeV 2 of the E615 experiment.