Does the E791 experiment have measured the pion wave function. Victor Chernyak. Budker Institute of Nuclear Physics, Novosibirsk, Russia

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1 Does the E791 experiment have measured the pion wave function profile? Victor Chernyak Budker Institute of Nuclear Physics, Novosibirsk, Russia Abstract The cross section of hard diffractive dissociation of the pion into two jets is calculated. It is obtained that the distribution of jets in longitudinal momenta 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 form for the asymptotic and CZ wave functions, and the ratio of cross sections differs from unity only within a factor of two. We argue therefore that, unfortunately, the E791-experiment have not measured yet the profile of the pion wave function. For this, the experimental accuracy has to be increased essentially. 1. The E791 experiment at Fermilab [1] has measured recently the cross section of hard diffractive dissociation of the pion into two jets. In particular, the distribution of jets in longitudinal momentum fractions y 1 and y 2, (y 1 + y 2 )=1, 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 momentum fractions x 1 and x 2, (x 1 + x 2 )=1. The hope was based on the theoretical calculations of this cross section [2]. It has been obtained in these papers that the cross section is simply proportional to the pion wave function squared: dσ/dy 1 φ π (y 1 ) 2.Insuch a case, it would be sufficient to measure only the gross features of dσ/dy 1

2 x 1 p π p 1 p π φ π M x 2 p π q 1 q 2 p 2 g p ξ (u) p Figure 1: Kinematics and notations 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 and that, unfortunately, 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. The upper blob M represents the hard kernel of the amplitude which includes the hard gluon exchange. The two soft external gluons are attached in all possible ways to the hard kernel, this gives 31 Feynman diagrams on the whole. Two of them are represented explicitly in fig. 2 and fig. 3. The lower blob represents the (skewed) gluon structure function of the target, for which we take the nucleon. The hard part of the amplitude can be considered as scattering of the pion on the gluon: π + g 1 ψ 1 + ψ 2 + g 2. In the c.m.s. and to the leading twist accuracy, the initial and final gluons can be considered to be on shell, with transverse polarizations, carrying fractions (u + ξ) and(u ξ) ofthemean nucleon momentum P. 1 The nucleon can be considered as being spinless, 1 The small skewedness, ξ 1, is always implied. It is typically: ξ 10 2,inthe Fermilab experiment. 2

3 and its skewed gluon distribution g ξ (u, t) =g ξ ( u, t); 1 <u<1(see[3] and [4]) is defined as: P A a, λ (z 1) A b, ρ (z 2) P = gλρ δ ab du g ξ (u, t) (u + ξ + iɛ)(u ξ iɛ) 1 ( e i(u+ξ)(pz 1 )+i(u ξ)(pz 2 ) + e ) i(u ξ)(pz 1)+i(u+ξ)(Pz 2 ). (1) 2 This is equivalent to the standard definition: P G a µλ (z 1) G a λν (z 2) P =2P µ P ν 1 1 du g ξ (u, t) 1 ( e i(u+ξ)(pz 1 )+i(u ξ)(pz 2 ) + e ) i(u ξ)(pz 1)+i(u+ξ)(Pz 2 ). (2) 2 The kinematical variables are defined as: q 1 =(u + ξ) P, q 2 =(u ξ) P, P =(P + P )/2, =(q 1 q 2 )=2ξ P, ξ = k2 y 1 y 2 s, z 1 = u + ξ 2 ξ, z 2 = u ξ, (3) 2 ξ z 1 z 2 =1, 2(p π ) = M 2 = k2, y 1 y 2 the final quarks are on shell, carry the fractions y 1 and y 2 of the initial pion energy, and their transverse momenta are: (k +(q /2)) and ( k + (q /2)), q k,whereq is the small final transverse momentum of the nucleon, while k is large. According to the well developed approach to description of hard exclusive processes in QCD [5-8] (see [9] for a review), the hard exchanged gluon in all diagrams (see figs.2 and 3) has to be considered as a part of the hard kernel, not as a tail of the pion wave function or of the (skewed) structure function. 2 2 In [2] 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 2,µ) φasy π (x, µ)/k 2. It remains unclear for us how it is possible to obtain such 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. 3

4 So, the structure of the amplitude is (simbolically): T P A A P ( ψ 1 Mψ 2 ) 0 ū d π, (4) where the first matrix element introduces the skewed gluon distribution of the target, ψ 1 and ψ 2 are the free spinors of final quarks, M is the hard kernel, i.e. the product of all vertices and hard propagators, the last matrix element introduces the pion wave function, and means the appropriate convolution. As an example, let us consider the diagram in fig. 2. Proceeding in the above described way, one obtains the contribution to the amplitude (the Feynman gauge is used for the hard gluon): T 2 = 16 9 ω 1 o dx 1 φ π (x) 1 du g ξ (u) y 2 0 x 1 x 2 1 (u ξ)(u + ξ) (5) ω o = δ ij (4πα s ) 2 24 f π (ψ 1 ˆ γ5 ψ 2 ) (y 1y 2 ) 2, ˆ = µ γ µ, (6) where ψ 1 and ψ 2 are the free spinors of the final quarks, Σ spins ψ 1 ˆ γ5 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 have calculated only its value. For the diagram in fig. 2 this gives: ImT 2 = 16 9 πsω o y 1 k 2 k 4 1 dx 1 φ π (x) g ξ (ξ). (7) 0 x 1 x 2 As a final example, let us consider also the diagram in fig. 3, as it gives (together with the mirror diagram obtained by q 1 q 2 ) the logarithmically enhanced contribution ln(s/k 2 ): T 3 = ω o y 1 y dx 1 φ π (x) x 1 x du g ξ (u) N (u ξ)(u + ξ)[z 1 (x 1 y 1 ) x 1 y 2 ], (8) N =[ 8 z 1 z 2 ]+(8y 1 y 2 3 x 1 y 1 x 2 y 2 )+(z 1 + z 2 )(x 1 y 1 ), (9) ImT 3 = 4 πsω o k dx 1 φ π (x)g ξ (ū) Θ(ξ < ū < 1) +, (10) x 1 x 2 x 1 y 1 4

5 p π Φ π x 1 p π p 1 q 1 q 2 p 2 g p ξ (u) p Figure 2: One of the diagrams p π Φ π x 1 p π p 1 p 2 q 1 q 2 g p ξ (u) p Figure 3: The diagram giving the logarithmically enhanced contribution 5

6 ( ) x1 y 2 + x 2 y 1 ū = ξ, (11) x 1 y 1 where only the logarithmically enhanced term is shown explicitly in eq.(10), it originates from the term in square brackets in eq.(9) Proceeding in the above described way, one obtains for the cross section: dσ N = 1 1 8(2π) 5 s T dy d 2 k d 2 q, y 1 y 2 T iimt = i πsω o k 2 g ξ (ξ)ω, (12) Σ 1 = [ Ω= 1 0 dx 1 φ π (x 1 )(Σ 1 +Σ 2 +Σ 3 +Σ 4 ), (13) ] 4 g ξ (ū) Θ(ξ < ū < 1) +(y 1 y 2 ), (14) x 1 x 2 x 1 y 1 g ξ (ξ) Σ 2 = 1 x 2 1 x 2 2 y 1 y 2 { (x 1 x 2 + y 1 y 2 ) [ x 1 y 1 (x 1 y 2 ) 2 g ]} ξ(ū) g ξ (ξ) Θ(ξ < ū < 1) + (y 1 y 2 ), (15) Σ 3 = 1 { [ 9 (x 1x 2 + y 1 y 2 ) 1 x x 2 1 x 2 1 y 1 g ]} ξ(ū) 2 y 1 y 2 g ξ (ξ) Θ(ξ < ū < 1) + (y 1 y 2 ),(16) Σ 4 = ξ dg ξ(u)/du u=ξ. (17) x 1 x 2 y 1 y 2 g ξ (ξ) The expressions (12)-(17) constitute the main result of this paper. 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 3 It is not difficult to see that the Θ - function in eq.(10) excludes the region of x 1 too close to y 1, so that the integral is convergent and only the logarithmic enhancement ln(s/k 2 ) remains. 6

7 cancel in the sum, so that the integral is finite. And the same 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. We present in this section 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 =2GeV for calculations, supposing that the higher twist effects are not of great importance at this value of k, and to have a possibility to compare with the E791 data. b) For the skewed gluon distribution g ξ (u, t) in 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): g ξ (u, t =0,µ k 2 GeV ) u ξ u 0.3 (1 u) 5. (18) This form agrees reasonably well (numerically) with calculated in [10] the ordinary, g o (u, µ 2 GeV ), and in [11] the skewed, g ξ (u, t =0,µ 2 GeV ), gluon distributions in the nucleon (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 this results mainly in an overall factor (see [2]): dσ A (t q 2 ) dσ N(t =0) AF A (t) 2, (19) where F A (t) =exp{bt/2}, b= RA 2 /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 [12]. This last has the form: φ CZ π (x, µ o 0.5 GeV )=30x 1 x 2 (x 1 x 2 ) 2, at the very low normalization point. Being renormalized to the point µ k 2GeV, it looks as (see fig. 4): φ CZ π (x, µ 2 GeV )=15x 1x 2 [ (x1 x 2 ) ]. (20) The results of these numerical calculations are compared then with the E791 data, see fig. 5. 7

8 Figure 4: Profiles of the pion wave functions: a) φ CZ π (x, µ 0.5 GeV )=30x 1 x 2 (x 1 x 2 ) 2 - dotted line; b) φ CZ π (x, µ 2 GeV )= 15 x 1 x 2 [0.2+(x 1 x 2 ) 2 ] - solid line; c) φ asy π (x) =6x 1 x 2 - dashed line. 8

9 Figure 5: The y-distribution of jets calculated for k =2GeV, 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 the two curves is as calculated. The data points are from the E791 experiment [1]. 9

10 It is seen that, unfortunately, while the 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 cross sections is not much different from unity: dσ asy /dσ CZ 1.2 aty 1 =0.5, and the same ratio is 0.7 aty 1 =0.25. In such an unhappy situation, the theoretical calculations have also to be performed with a maximal possible accuracy (higher twist corrections, hard loops corrections, the quark structure function contribution, nuclear effects, etc.). We show also in fig. 6 the same distributions with the pion energy ten times larger, E π =5TeV, k =2GeV. It is seen that even this does not help much (as the form of the distribution depends on the pion energy only logarithmically). The same ratios of the cross sections are here 1.7and 0.7 respectively. Recently the Coulomb contribution to the cross section have been calculatedin[13]. 4 Its value for Pt is: dσpt electr /dk 2 dy < 10 5 mbarn GeV 2, for E π = 500 GeV, k =2GeV. Using the above given formulae, one obtains for the strong cross section at the middle point y =0.5: dσpt CZ /dk dy mbarn GeV 2 with the same parameters, and 1 mbarn GeV 2 at E π =5TeV. It is seen that the electromagnetic contribution is small. Note added: After this work has been finished, there appeared the paper [14] on the same subject. Comparison shows that, although the qualitative conclusions are similar, the analytic expressions for the scattering amplitude differ essentially in this paper and in [14]. Acknowlegements I am grateful to V.S. Fadin for useful discussions and critical remarks. I thank also A.E. Bondar and B.I. Khazin for explaining me some details of the E791 experiment. 4 Because the electromagnetic contribution is real, while the strong one is mainly imaginary, they do not interfere. 10

11 Figure 6: The same as in Fig.5, but with E π =5TeV. 11

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