QCD Sum Rule and Perturbative QCD Approaches to Pion Compton Scattering
arXiv:hep-ph/9302231v1 8 Feb 1993
Claudio Corian`? and Hsiang-nan Li? o
Institute for Theoretical Physics, University of Stockholm S 113 Vanadisvagen, Stockholm, Sweden and Dipartimento di Fisica, Universita’ degli Studi di Parma and INFN Gruppo Collegato di Parma, Italy
Institute of Physics, Academia Sinica, Taipei, Taiwan 11529, R.O.C.
February 1, 2008
Abstract We compare two approaches to the description of pion Compton scattering at moderate momentum transfer, one being based on local duality QCD sum rules for the invariant amplitudes of the process, which have been derived recently, and the other on the modi?ed factorization formula with Sudakov e?ects included. We ?nd that perturbative QCD predictions are dominant over those from QCD sum rules only for the scattering angle greater than 90o .
1. Introduction. The method of QCD sum rules  has been very successful in describing many of the static and dynamical properties of hadrons at intermediate energy, a regime where non-perturbative e?ects are important. Based on this semi-phenomenological approach, important information on the quark distribution functions of hadrons [2-4] and on their electromagnetic form factors at momentum transfer Q about 1 to 2 GeV has been obtained . On the other hand, the factorization formula, with Sudakov suppression taken into account for soft gluon exchange, has enlarged the applicability of perturbative QCD (pQCD) to elastic hadron form factors down to Q ? 2-3 GeV, and produces reliable results in agreement with experimental data . The above energy scales indicate the transition from non-pQCD to pQCD at Q2 around 4 GeV2 . Then, natural questions to ask are whether these two approaches can be generalized to other more complicated processes, and on what energy scale they give comparable predictions. Recently, progress has been made in extending the QCD sum rule analysis to a four-point amplitude in pion Compton scattering [9, 10]. A crossed version of Compton scattering, p? → γγ, has p been investigated based on the modi?ed factorization formula . In this letter we will study pion Compton scattering following the same approach, and compare the results with those from QCD sum rules. Their behavior at moderate momentum transfer may give hints about the above questions. Besides, the additional angular dependence of Compton scattering, which is absent in elastic form factors, will exhibit more information on the transition to pQCD. Hence, Compton scattering provides a nontrivial comparision between the two methods, and experiments will determine which one gives a better description of the process. In the sum rule approach, the lowest-order diagrams for pion Compton scattering are those without virtual gluons, as shown in ?g. 1, which give real contributions [9, 10]. The basic diagrams considered in pQCD have one extra exchanged gluon as in ?g. 2. An imaginary contribution appears when the exchanged gluons in ?g. 2d and e (with two photons attaching to di?erent quark lines) go on-shell, and thus brings in a phase , which distinguishes the lowest-order sum rule and pQCD approaches. Since the singularity due to this on-shell gluon is not removed by pion wave functions as in the case of form factors , Sudakov corrections are more important here in the sense of improving the self-consistency of the perturbative formula. Therefore, our work can be regarded as an attempt to incorporate Sudakov e?ects into 1
Compton scattering. Since the main purpose is to compare the two methods, we do not compute the full cross section of the process, but concentrate on one of the invariant amplitudes contained in the scattering matrix element. New sum rules for the invariant amplitude have been derived in ref. . A modi?ed perturbative expression for the same invariant amplitude will be given below. We study the process in a “brick-wall” frame, where the incoming pion momentum p1 and outgoing one p2 are assumed to have large “+” and “?” components respectively, and pT1 = pT2 = 0. The incoming and outgoing photons, with 2 2 momenta q1 and q2 respectively, are on-shell (q1 = q2 = 0). The Mandelstam 2 2 2 invariants s = (p1 + q1 ) , t = (p1 ? p2 ) = ?Q and u = (p1 ? q2 )2 are restricted in the physical region of s > 0, t < 0, and u < 0. To simplify the analysis, we assume that s is larger than ?t, but not so large that higher-order corrections proportional to ln(?t/s) diverge. It then su?ces to consider the contributions which are not suppressed by s or u. The ratio ?t/s characterizes the scattering angle θ of the photon, sin2 (θ/2) = ?t/s. The condition ?t/s → 0 corresponds to forward scattering of the photon, while ?t/s → 1 to backward scattering. It is observed that the transition scale, on which the perturbative contributions become comparable to non-perturbative contributions, moves to large ?t as θ decreases. For forward scattering (θ → 0) non-perturbative contributions always dominate. Only for ?t/s > 0.5 (θ > 90o ) are pQCD predictions important at moderate momentum transfer. 2. QCD Sum Rule Approach. A detailed QCD sum rule analysis of pion Compton scattering has been developed in [9, 10]. In this section we will emphasize the essential aspects of the analysis and quote the results. We start with the correlator of 4-currents Γα?νβ (p2 , p2 , s, t) = i 1 2 d4 x d4 y d4 z exp(?ip1 · x + ip2 · y ? iq1 · z) (1)
? × 0|T ηβ (y)Jν (0)J? (z)ηα (x) |0 ,
where ? J? = eu uγ? u + ed dγ? d, ? ηα = uγ5 γα d ? (2)
are the electromagnetic and axial currents respectively in terms of quark 2
?elds u, d and charges eu = 2/3, ed = ?1/3. The pions are assumed to be o?-shell p2 = s1 = 0, p2 = s2 = 0 with s1 and s2 their virtualities. 1 2 Γα?νβ can be expressed in terms of many structures, each of which is associated with an invariant amplitude. We study the invariant ampltudes H1 and H2 with the structure proportional to p1α p2β 
2 Γα?νβ ∝ fπ p1α p2β H1 e(1) e(1) + H2 e(2) e(2) + ... , ? ν ? ν
where the helicity vectors e(1) and e(2) are de?ned by N? , e(1)? = √ ?N 2 with N ? = ??λσρ Pλ qσ Kρ , P ? = p? + νp? ? K ? 1 2 ν= p1 · K + νp2 · K , K2 P? e(2)? = √ ?P 2 (4)
p1 · p2 ? s1 , p1 · p2 ? s2 K = q1 + q2 , q = q2 ? q1 .
The structure p1α p2β comes from the insertion of pion states into the correlator in eq. (1) and the substitution
? π(p1 )|ηα (0)|0 = ?ifπ p1α , 0|ηβ (0)|π(p2) = ifπ p2β ,
which interpolates between the vacuum and single pion state. We concentrate on the combination H = H1 + H2 , which is extracted by contracting Γα?νβ with ?g ?ν nα nβ , where the circular polarization n? = (e(2) ± ie(1) )? satis?es the conditions n · q1 = n · (p1 ? p2 ) = n2 = 0 . (8) (7)
These conditions, in full analogy with those in the case of form factors , are crucial for obtaining the correct asymptotic behavior of the sum rules, which 3
is consistent with quark counting rules [14, 15]. Note that the expression for e(2) given in eq. (4) is slightly di?erent from that in ; the vector P ? is de?ned including dependence on the extra factor ν due to the o?-shell pions. This modi?ed de?nition approaches the standard one given by ν = 1 at the pion pole, and guarantees the existence of n? with the requirements of eq. (8). To evaluate the helicity form factor H(s, t), it is simplest to consider its spectral function ?(s1 , s2 , s, t) which is de?ned by the double discontinuity of H on the cuts 0 ≤ s1 , s2 ≤ λ2 ≈ (s + t)/4. This choice of λ is to avoid the inclusion of u-channel resonances. The sum rule for H(s, t), which is closely related to the ?nite energy sum rule in , has been formulated using the argument of analyticity for the correlator in eq. (1) in the ?nite region of the s1 and s2 complex planes . The explicit form of ? is essential both in the derivation of the sum rule and in the calculation of power corrections . We will evaluate ? using the standard cutting rules as developed in . It is impossible to render all internal lines on-shell because of the o?-shell assumption for the external pions. Only the case with all lines on-shell except the upper line contributes. We refer to  for a comprehensive derivation for the sum rule, and quote the ?nal expression here
2 fπ H(s, t) = s0 0 s0
ds2 ρ(s1 , s2 , s, t) exp(?
s1 + s2 ) M2 (9)
0|G?ν G?ν |0 0|(?q)2 |0 q +c1 + c2 , 4 M M6
fπ = 133 MeV being the pion decay constant. The spectral density ρ is given by ρ(s1 , s2 , s, t) = 1 ?(s1 , s2 , s, t) (n · p1 )(n · p2 ) 10 2s ? δ 2u ? δ ≈ [(s1 ? s2 )2 ? t(s1 + s2 )] + 2δ3 3(2π) s u (s1 + s2 ? t)2 ? 4s1 s2 .
Those terms suppressed by extra powers of t or s have been neglected. The gluon and quark condensates 0|G?ν G?ν |0 and 0|(?q)2 |0 , arising from the q operator product expansion of the correlator in eq. (1), give the power corrections which are suppressed by factors (1/M 2 )n , M 2 being the Borel mass. 4
The coe?cients c1 and c2 can be obtained through the operator product expansion. The exact value of the quark-hadron duality interval s0 is determined by demanding that the right-hand side of eq. (9) have the weakest sensitivity to the variation of M 2 [2-6]. Once s0 is speci?ed, the large M 2 limit is applied to diminish the power corrections, and the ?nal expression for H is written as a local duality approximation
2 fπ H(s, t) = s0 0 s0
ds2 ρ(s1 , s2 , s, t) ,
which behaves like 1/Q4 asymptotically. We will address the calculation of the power corrections in a seperate work , and assign an approximate value to s0 here. It is also possible to derive a similar sum rule for the single invariant amplitude H1 . 3. Perturbative QCD approach. Part of the basic diagrams for the perturbative QCD approach are shown in ?g. 2. Each represents a class of diagrams which can be transformed to each other by permuting the incoming and outgoing pions, or the two quark lines. They di?er from ?g. 1 by an extra exchanged gluon. We will incorporate Sudakov e?ects for soft gluon exchange, the case in which the running coupling constant αs with its argument set to the gluon energy is large and lowest-order pQCD does not make sense. The method to calculate pion Compton scattering based on these diagrams is similar to that developed for electromagnetic form factors . Following the same reasoning and procedures, the modi?ed factorization formula for H(s, t) is given by
H(s, t) =
dx1 dx2 d2 kT1 d2 kT2 ψ(x2 , kT2 , p2 ) (12)
×THl (xi , s, t, kTi ) ψ(x1 , kT1 , p1 ) .
The additional dependence on transverse momentum kT carried by a valence quark has been included into the pion wave function ψ and the hardscattering kernel TH . Eq. (12) can be understood as an intermediate step in the standard factorization program , where kT in TH is assumed to give higher-power (k2 /Q2 ) correction and thus ignored. In fact, this approximaT tion is not proper when the exchanged gluon becomes soft. The contributions to the hard scatering from each diagram in ?g. 2, obtained by contracting 5
the two photon vertices with ?g ?ν , are given in table. 1. The transverse momenta on the virtual quark lines have been neglected in the derivation since they are associated with linear, instead of quadratic, divergences in xi , and hence less important than that on the gluon line . The contributions from ?g. 2d and e, like ?g. 1c, are suppressed by s as stated in the previous section. All contributions are grouped into only two terms (l = 1, 2) using the permutative symmetry. Rewriting eq. (12) in terms of the Fourier transformed functions, and inserting the large-b asymptotic behavior of the wave function , we have
H(s, t) =
dx1 dx2 φ(x1 )φ(x2 )
? bdbTHl (xi , s, t, b, wl ) (13)
where b, introduced by Fourier transformation, is the seperation between the two valence quark lines. Note the extra Sudakov factor exp(?S) compared to the standard factorization formula, which arises from an all-order summation of the collinear enhancements of radiative corrections to ?g. 2. The exponent S is written as 
× exp[?S(xi , b, Q, wl )] .
S(x1 , x2 , b, Q, w) =
(s(xi , b, Q) + s(1 ? xi , b, Q)) ?
2 w ? , ln β1 ?? b
A(1) A(2) q A(1) q ? ? s(ξ, b, Q) = + ?1 ? q ln ? (? + ? q b) 2 2β1 4β1 ?? 2β1 ?? b b A(1) β2 ln(?2? + 1 ln(2?) + 1 q b) ? ? q ? 3 16β1 q ? ?? b ? e2γ?1 A(2) A(1) ? ln 2 4β1 4β1 2 ln A(1) β2 ln2 (2?) ? ln2 (?2? . q b) ? 3 32β1 The variables q , ? and w are de?ned by ? b ? √ q ≡ ln ξQ/( 2Λ) ? (16) q ? ?? b
? ≡ ln(bΛ) b w ≡ ln(w/Λ) , ? 6
where the scale parameter Λ ≡ ΛQCD will be set to 0.1 GeV. The coe?cients βi and A(i) are β1 = A(1) 153 ? 19nf 33 ? 2nf , β2 = , 12 24 4 67 π 2 10 8 eγ = , A(2) = ? ? nf + β1 ln 3 9 3 27 3 2
with nf = 3 the number of quark ?avors and γ the Euler constant. The Sudakov factor is always less than 1 as explained in ref. , and decreases quickly in the large-b region. The function φ, obtained by factoring the Q and b dependences from the transformed wave function into Sudakov logarithms, is taken as the Chernyak and Zhitnitsky model  15fπ φCZ (x) = √ x(1 ? x)(1 ? 2x)2 , 2Nc where Nc = 3 is the number of colors. ? The transformed hard scatterings THl are given by [(1 ? x1 )t + u][(1 ? x2 )t + u] 16πCF (e2 + e2 )αs (w1 ) u d ? |r1 |b K0 TH1 = (1 ? x1 )(1 ? x2 ) s2 [(1 ? x1 )t + s][(1 ? x2 )t + s] ? 4(1 ? x2 ) (19) + u2 from the classes of ?g. 1a-c, and ? TH2 = 32πCF eu ed αs (w2 ) θ(?r2 )K0 iπ (1) √ H ( r2 b) 2 0 1 (1 + x2 ? x1 x2 )t2 + (1 + x2 ? x1 )ut × ? x1 (1 ? x1 ) x2 (1 ? x1 )s2 (1 + x1 ? x1 x2 )t2 + (1 + x1 ? x2 )st 1 ? (20) + x2 (1 ? x2 ) x1 (1 ? x2 )u2 |r2 |b ? θ(r2 ) (18)
from the class ?g. 1d-e with r1 = x1 x2 t, r2 = x1 x2 t + x1 u + x2 s . (21)
K0 and H0 in eqs. (19) and (20) are the Bessel functions in the standard notation. The imaginary contribution comes from eq. (20). The argument wl of αs is de?ned by the largest mass scale in the hard scattering, w1 = max |r1 |, 1 , b w2 = max |r2 |, 1 b . (22)
As long as b is small, soft rl does not lead to large αs . Therefore, the nonperturbative region in the modi?ed factorization is characterized by large-b, the region which is suppressed by Sudakov e?ects. Eq. (13), as a perturbative expression, is thus relatively self-consistent compared to the leading-power factorization . 4. Numerical Analysis. To evaluate the invariant amplitude H(s, t) from the sum rule, the value of s0 must be determined through a stability analysis on the M 2 dependence of eq. (9) as in [2-6]. However, we know that s0 must take a value between the masses of pion (m2 = 0) and A1 -meson (m2 1 = π A 1.2 GeV2 ). Hence, we estimate that s0 resides in the reasonable range 0.6 < s0 < 0.8 GeV2 , as referred to the case of pion form factor. H(s, t) with s0 = 0.6, 0.7 and 0.8 GeV2 are computed and found to vary slowly in the above range of s0 for Q2 > 4 GeV2 . Therefore, s0 will be set to 0.7 GeV2 in eq. (11) for the sum rule analysis. Results of Q2 H for di?erent ?t/s are shown in ?g. 3 with 4 < Q2 < 25 GeV2 . It is observed that H decreases monotonously following the 1/Q4 asymptotical behavior, and has a weak angular dependence. Results derived from the modi?ed expression eq. (13) in the same range of 2 Q are also shown in ?g. 3, where |H| denotes the magnitude of H. It is found that |H| has only little (logarithmical) deviation from the expected 1/Q2 behavior, and drops quickly with decreasing ?t/s. That is, pQCD predictions have a stronger angular dependence. The curve from eq.(13) denoted by ?t/s = 0.6 (θ = 100o) is always above the corresponding curve derived from the sum rule for 4 < Q2 < 25 GeV2 . It indicates the dominance of perturbative contribution in large-angle Compton scattering. For ?t/s = 0.5 (θ = 90o ) sum rule and pQCD predictions are comparable, and the transition scale is about 6 GeV2 . As ?t/s drops, the transition scale increases, and the non-perturbative contribution becomes dominant when ?t/s goes down to 0.2 (θ = 50o ). Based on the above analysis, we conjecture that the present pQCD calculation for proton Compton scattering  or its crossed version 8
like p? → γγ  should be complemented by the QCD sum rule description p at small angles. The phase φ of H obtained from eq. (13) is also studied, and its Q2 dependence for di?erent ?t/s is shown in ?g. 4. It is found that φ is very stable at intermediate angles, but changes drastically at small angles. The corresponding results from the standard factorization formula with kT neglected in TH and αs = 0.3 in eq. (12) are exhibited for reference, and are similar to those from the modi?ed one. This choice of αs has been used in order to ?t experimental data from proton Compton scattering . However, |H| shows an exact 1/Q2 behavior and φ is independent of Q2 at a ?xed angle. Note the di?erence between the standard and modi?ed factorizations that αs is e?ectively regarded as a free parameter in the former case. 5. Conclusion. In this letter we have shown that the extension of both the QCD sum rule method and modi?ed factorization formula to pion Compton scattering is possible, because their agreement at intermediate scattering angles is explicit. We believe that this coincidence is not trivial due to the very di?erent theoretical bases of the two approaches. The transition scale from QCD sum rules to pQCD is found to vary with scattering angles. QCD sum rules give important contribution at small angles, and pQCD becomes dominant only for θ > 90o from our analysis. Our results have suggested a new and interesting interplay between factorization theorems and sum rule methods in Compton scattering. All these observations need to be compared with experiments. To examine the conclusions more exactly, it is necessary to determine a precise value for s0 , and thus the calculation for gluon and quark condensates, along with a detailed discussion on the stability of the sum rule, is involved . It is also of interest to generalize the two methods to more complicated case like proton Compton scattering , for which experimental data are available. We thank Prof. G. Sterman for suggesting this investigation, and Profs. T.H. Hansson and H. Rubinstein for clarifying discussions. C. C. thanks Prof. G. Marchesini and Dr. F. Fiorani for stimulating conversations and for their hospitality at the Physics Department of Parma University, and Profs. N. Isgur and A.V. Radyushkin for their hospitality at CEBAF. This work was supported in part by the National Science Council of R.O.C. under Grant No. NSC82-0112-C001-017 and by INFN of Italy. 9
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Table 1. The expressions of the hard scattering TH corresponding to the diagrams in ?g. 2.
TH /(64παs ) ?e2 [(1 ? x1 )t + u][(1 ? x2 )t + u] u (1 ? x1 )(1 ? x2 )s2 [x1 x2 t ? (kT1 ? kT2 )2 ] e2 u (1 ? x1 )[x1 x2 t ? (kT1 ? kT2 )2 ] e2 u (1 ? x1 )[x1 x2 t ? (kT1 ? kT2 )2 ] ?eu ed x1 (1 ? x1 )[x1 x2 t + x1 u + x2 s ? (kT1 ? kT2 )2 ] eu ed [(1 + x2 ? x1 x2 )t2 + (1 + x2 ? x1 )ut] x2 (1 ? x1 )s2 [x1 x2 t + x1 u + x2 s ? (kT1 ? kT2 )2 ]
Fig. 1. Diagrams for the pion Comptom scattering in the QCD sum rule analysis. Fig. 2. Part of the diagrams for the pion Comptom scattering in the perturbative QCD analysis. Fig. 3. Dependence of Q2 |H(s, t)| on Q2 derived from QCD sum rules (dashed lines), from the modi?ed factorization (solid lines) and from the leading-power factorization (dotted lines) for (a) ?t/s = 0.6 (θ = 100o), (b) ?t/s = 0.5 (θ = 90o ) and (c) ?t/s = 0.2 (θ = 50o ). Fig. 4. Dependence of the phase φ on Q2 from the modi?ed factorization (solid lines) and from the leading-power factorization (dotted lines) for (a) ?t/s = 0.6, (b) ?t/s = 0.5, (c) ?t/s = 0.2 and (d) ?t/s = 0.02.