Has the E791 experiment measured the pion wave function profile ?

The cross section of hard diffractive dissociation of the pion into two jets is calculated. It is obtained that the distribution of longitudinal momenta for jets is not simply proportional to the profile of the pion wave function, but depends on it in a complicated way. In particular, it is shown that, under the conditions of the E791 experiment, the momentum distribution of jets is similar in its shape for the asymptotic and CZ wave functions, and even the ratio of the differential cross sections is not far from unity.


1.
The E791 experiment at Fermilab [1] has recently measured the cross section of the hard diffractive dissociation of the pion into two jets. In particular, the distribution of the total pion longitudinal momentum into fractions y 1 and y 2 , (y 1 + y 2 ) = 1, between jets has been measured. The main purpose was to obtain in this way the information about the leading twist pion wave function φ π (x 1 , x 2 ), which describes the distribition of quarks inside the pion in the longitudinal momentum fractions x 1 and x 2 , (x 1 +x 2 ) = 1.
The hope was based on the theoretical calculations of this cross section in [2]- [4]. It has been obtained in all these papers that the cross section is simply proportional to the pion wave function squared: dσ/dy 1 ∼ |φ π (y 1 )| 2 . In such a case, it would be sufficient to measure only the gross features of dσ/dy to reveal the main characteristic properties of φ π (x), and to discriminate between various available models of φ π (x).
The purpose of this paper is to show that this is not the case. The real situation is much more complicated, with dσ/dy depending on φ π (x) in a highly nontrivial way. We give below (in a short form) the results of our calculation of this cross section.

2.
The kinematics of the process is shown in fig. 1. We take the nucleon as a target, and the initial and final nucleons are substituted by two soft gluons with momenta q 1 and q 2 . The lower blob in fig.1 represents the skewed gluon distribution of the nucleon, g ξ (u).
The final quarks are on shell, carry the fractions y 1 and y 2 of the initial pion momentum, and their transverse momenta are: (k ⊥ + (q ⊥ /2)) and (−k ⊥ + (q ⊥ /2)), q ⊥ ≪ k ⊥ , where q ⊥ is the small final transverse momentum of the target, while k ⊥ is large.
The upper blob M in fig.1 represents the hard kernel of the amplitude which includes all hard propagators. For calculation of M in the leading twist approximation and in the lowest order in α s , the massless pion can be substituted in all diagrams by two massless on shell quarks with the collinear momenta x 1 p π and x 2 p π and with zero transverse momenta, as account of primordial virtualities and transverse momenta results only in higher twist corrections to M. The leading twist pion wave function φ π (x, µ o ) describes the distribution of these quarks in momentum fractions x 1 and x 2 . 1 The hard kernel M is proportional to the scattering amplitude of two initial collinear and on shell quarks of the pion on the on shell gluon: In lowest order in α s (k ⊥ ) M consists of 31 connected Born diagrams, each one is ∼ O(α 2 s (k ⊥ )) and contains exactly three hard propagators (except for one diagram with the 4-gluon vertex, which has only two). Two diagrams are represented explicitly in fig. 2 and fig. 3.
In the c.m.s. and to the leading twist accuracy, the initial and final soft gluons can be considered to be on shell, with transverse polarizations, carrying fractions (u + ξ) and (u − ξ) of the mean nucleon momentum P . The nucleon can be considered as being massless, spinless, and its skewed gluon distribution g ξ (u, t, µ o ) = g ξ (−u, t, µ o ), −1 < u < 1, is defined as 3 : This is equivalent to the standard definition (see [5][6][7]): The kinematical variables are defined as: The small skewedness, ξ ≪ 1, is always implied. It is typically: ξ ∼ 10 −2 , in the Fermilab experiment. 3 Here and below the gluon string is always implied in gauge invariant definitions of bilocal operators, both for the gluon distribution of the nucleon g ξ (u), and for the pion wave function φ π (x).
According to the well developed approach to description of hard exclusive processes in QCD [8]- [11] (see [12] for a review), all hard gluon and quark lines in all diagrams (see e.g. figs.2 and 3) have to be written down explicitly and substituted by their perturbative propagators. In other words, the hard momentum flow have to be made completely explicit and these hard lines of diagrams constitute the hard kernel M. They should not be hidden as (a derivatives of) "the tails" of the unintegrated pion wave function Ψ π (x, l ⊥ ), or of the "unintegrated gluon distribution". This is, first of all, what differs our approach from previous calculations of this process in [2][3][4] where, besides, most of diagrams were either ignored or calculated erroneously. 4 Due to this the function Ψ π (x, l ⊥ ) ( and the unintegrated gluon distribution) never appears explicitly in our calculations. It always enters the answer for each of 31 Born diagrams implicitly only through the function φ π (x, µ), i.e. only in the standard integrated form: φ π (x, µ) ∼ µ d 2 l ⊥ Ψ π (x, l ⊥ ). 5 So, the structure of the amplitude is (symbolically): where the first matrix element introduces the skewed gluon distribution of the nucleon g ξ (u),ψ 1 and ψ 2 are the free spinors of final quarks, "M" is the 4 While a small number out the whole set of 31 diagrams can possibly be reinterpreted through "the tails" of the unintegrated pion wave function Ψ π (x, l ⊥ ) or of the "unintegrated gluon distribution", this is definitely not the case for most of diagrams. So, we see no much meaning to proceed in this way.
In a sense, the contribution of each diagram is essential in obtaining the numerical description for the "y"-distribution of jets in sect.4. In our approach all 31 diagrams are treated on equal footing. 5 In [3] the authors tried to use the evolution equation for the pion wave function to obtain its "tail". For instance, for the asymptotic wave function it was obtained in this way: Ψ asy π (x, k ⊥ ) ∼ φ asy π (x, µ)/k 2 ⊥ . It remains unclear for us how it is possible to obtain such a result from the evolution equation for the asymptotic wave function which looks as: dφ asy π (x, µ)/d ln µ = dy V (x, y)φ asy π (y, µ) = 0.
hard kernel, i.e. the product of all vertices and hard propagators, the last matrix element introduces the pion wave function φ π (x), and ⊗ means the appropriate convolution.
As an example, let us consider the diagram in fig. 2. Proceeding in the above described way (see the appendix), one obtains the contribution to the amplitude (the Feynman gauge is used for the hard gluon): where ψ 1 and ψ 2 are the free spinors of the final quarks, Σ spins |ψ 1∆ γ 5 ψ 2 | 2 = 2 k 4 ⊥ /(y 1 y 2 ), "ij" are their colour indices, and f π ≃ 130 MeV is the pion decay constant.
As it is expected that, for this process, the imaginary part of the amplitude is the main one at high energy, we show explicitly in ch.3 only its value. For the diagram in fig. 2 this gives (in all diagrams the terms iǫ are introduced into denominators through s → s + iǫ, i.e. ξ → ξ − iǫ): As a final example, let us also consider the diagram in fig. 3, as it gives (together with the mirror diagram obtained by q 1 ↔ q 2 ) the enhanced contribution 6 : 6 The analytic form of the enhancement depends on the behaviour of the gluon distribution g ξ (u), see eq.(10). With g ξ (u) ∼ const the enhancement is logariphmic, ∼ ln(s/k 2 ⊥ ), but with g ξ (u) from eq.(18) it is only numerical.
where only the enhanced term is shown explicitly in eq.(10), which originates from the term in square brackets in eq.(9). 7 3. Proceeding in the above described way and summing up the contributions of all 31 Born diagrams, one obtains for the cross section: The expressions (12)- (17) constitute the main result of this paper. 7 As the Θ -function in eq.(10) excludes the region of x 1 too close to y 1 , the integral is convergent and only some enhancement remains (see the footnote 6).
At the conditions of the E791 experiment (and even at much larger energies, see fig.6), this enhancement is by no means sufficient to neglect contributions of all other diagrams.
Let us note that while the separate terms in dx φ π (x)Σ 2 are logarithmically divergent at x 1,2 → 0, it is not difficult to see that the divergences cancel in the sum, so that the integral is finite. And the same is valid for Σ 3 . This is an important point as it shows that the whole approach is selfconsistent, i.e. the hard kernel remains hard and the soft end point regions x 1,2 → 0 give only power suppressed corrections.

4.
In this section we present some numerical estimates of the cross section, based on the above expressions (12)- (17). Our main purpose here is to trace the distribution of jets in longitudinal momentum fractions y 1 , y 2 depending on the profile of the pion wave function φ π (x). a) As the calculations were performed in the leading twist approximation which becomes applicable at sufficiently large k ⊥ only, we take k ⊥ = 2 GeV . This assumes that the higher twist effects are not of great importance at such a value of k ⊥ , and gives a possibility of comparison with the E791-data.
b) For the skewed gluon distribution g ξ (u, t, µ) of the nucleon at t ≃ −q 2 ⊥ ≃ 0 we use the simple form (as we need it at |u| ≥ ξ only, and because g ξ (u) → g o (u) at |u| ≫ ξ): This form agrees numerically reasonably well with the ordinary, g o (u, µ ≃ 2 GeV ), and skewed, g ξ (u, t = 0, µ ≃ 2 GeV ), gluon distributions of the nucleon calculated in [13] and [14] respectively (in the typical region of the E791 experiment: |u| ≥ ξ ∼ 10 −2 ). The detailed consideration of nuclear effects is out the scope of this paper. So, we simply assume that these effects result mainly in an overall factor (see [3]): where F A (t) = exp{bt/2}, b = R 2 A /3, is the nuclear form factor. c) As for the pion leading twist wave function, φ π (x, µ), we compare two model forms: the asymptotic form, φ asy π (x, µ) = 6x 1 x 2 , and the CZ-model [15]. The latter has the form: φ CZ π (x, µ o ≃ 0.5 GeV ) = 30 x 1 x 2 (x 1 − x 2 ) 2 , at the low normalization point. Being evolved up to the characteristic scale µ ≃ k ⊥ ≃ 2 GeV, it looks as 8 (see fig. 4): Figure 4: Profiles of the pion wave functions: a) φ CZ π (x, µ ≃ 0.5 GeV ) = 30 The results of these numerical calculations are then compared with the E791-data, see fig. 5.
It is seen that, unfortunately, while two pion wave functions are quite different, the resulting distributions of jets in longitudinal momenta are similar and, it seems, the present experimental accuracy is insufficient to distinguish clearly between them. Moreover, even the ratio of the differential cross sections is not much different from unity: dσ asy /dσ CZ ≃ 1.2 at y 1 = 0.5, and the same ratio is ≃ 0.7 at y 1 = 0.25.
In such an unhappy situation, the theoretical calculations should be also performed with a maximal possible accuracy (to account additionally for: the quark distributions, higher twist corrections, hard loop corrections, nuclear effects etc.).
We also show in fig. 6 the same distributions with the pion energy ten times larger, E π = 5 T eV, k ⊥ = 2 GeV . It is seen that even this does not help much (as the form of the distribution weakly depends on the pion energy). The same ratios of the cross sections are here ≃ 1.7 and ≃ 0.7 respectively.
Recently the Coulomb contribution to the cross section has been calculated in [16]. 9 Its value for Pt is: dσ electr P t /dk 2 ⊥ dy < 10 −5 mbarn · GeV −2 , for E π = 500 GeV, k ⊥ = 2 GeV . Using the above given formulae, one obtains for the strong cross section at the middle point y = 0.5: dσ CZ P t /dk 2 ⊥ dy ≃ 2.5·10 −2 mbarn·GeV −2 with the same parameters, and ≃ 0.25 mbarn·GeV −2 at E π = 5 T eV . It is seen that the electromagnetic contribution is small.

Note added:
After this work has been completed, the paper [17] on the same subject appeared. Comparison shows that, although the qualitative conclusions are similar, the analytic expressions for the scattering amplitude differ essentially in this paper and in [17]. 10 Figure 5: The y-distribution of jets calculated for k ⊥ = 2 GeV, E π = 500 GeV and with the pion wave functions: φ CZ π (x, µ ≃ 2 GeV ) -solid line, φ asy π (x) -dashed line. The overall normalization is arbitrary, but the relative normalization of two curves is as calculated. The data points are from the E791 experiment [1]. Figure 6: The same as in Fig.5, but with E π = 5 T eV .