Exclusive photon-photon processes

SLAC-PUB-7604 arXiv:hep-ph/9708345v1 13 Aug 1997 July 1997 Exclusive Photon-Photon Processes ∗ S. J. Brodsky Stanford Linear Accelerator Center, Stanford University, Stanford, California 94309 Invited Talk given at the International Conference on the Structure and the Interactions of the Photon (Photon97) including the 11th International Workshop on Photon-Photon Collisions Egmond ann Zee, The Netherlands May 10–15, 1997 ∗ Work supported by the Department of Energy, contract DE–AC03–76SF00515. Abstract Exclusive γγ → hadron pairs are among the most fundamental processes in QCD, providing a detailed examination of Compton scattering in the crossed channel. In the high momentum transfer domain (s, t, large, θcm for t/s fixed), these processes can be computed from first principles in QCD, yielding important information on the nature of the QCD coupling αs and the form of hadron distribution amplitudes. Similarly, the transition form factors γ ∗ γ, γ ∗ γ → π 0 , η 0 , η ′, ηc . . . provide rigorous tests of QCD and definitive determinations of the meson distribution amplitudes φH (x, Q). We show that the assumption of a frozen coupling at low momentum transfers can explain the observed scaling of two-photon exclusive processes. 1 Introduction Exclusive two-photon processes provide highly valuable probes of coherent effects in quantum chromodynamics. For example, in the case of exclusive final states at high momentum transfer and fixed θcm such as γγ → pp or meson pairs, photon- photon collisions provide a timelike microscope for testing fundamental scaling laws of PQCD and for measuring distribution amplitudes, the fundamental wavefunctions of hadrons. [1] At very high energies s >> −t , diffractive processes such as γγ → neutral vector (or pseudoscalar) meson pairs with real or virtual photons can test the QCD Pomeron (or the C = −1 exchange Odderon) in a detailed way utilizing the simplest possible initial state. [2] In the case of low momentum transfer processes, the comparison of the two-photon decay width for a given C = + resonance with its inferred two-gluon width provides an indirect discovery tool for gluonium. As discussed at this conference by H. Paar, [3] CLEO has reported a very small upper limit for the coupling Γ(γγ → fJ0 (1220) due to the absence of a signal for Ks Ks decays, whereas a large gg → fJ0 (1220) coupling is inferred from Mark III and BES observations of J/ψ → γfJ0 decays. Using Chanowitz’s “stickiness” criteria, [4] this points to a gluonium interpretation of the fJ0 . Traditionally, γγ data has come from the annihilation of Weisäcker–Williams effective photons emitted in e− e± collisions. Data for γγ → hadrons from ep → e′ p′ R0 2 events at HERA has also now become available. The HERA diffractive events will allow studies of photon and pomeron interference effects in hadron-induced amplitudes. As emphasized by Klein, [5] nuclear-coherent γγ → hadrons reactions can be observed in heavy-ion collisions at RHIC or the LHC, e.g. Z1 Z2 → Z1 Z2 π + π − . Even- tually γγ collisions will be studied at TeV energies with back-scattered laser beams, allowing critical probes of Standard Model and supersymmetric processes with polarized photons in exclusive channels such as Higgs production γγ → W + W − , and γγ → W + W − W + W − . [6] 2 Hard Exclusive Two-Photon Reactions Exclusive two-photon processes such as γγ → hadron pairs and the transition form factor γ ∗ γ → neutral mesons play a unique role in testing quantum chromodynamics because of the simplicity of the initial state. [1] At large momentum transfer the direct point-like coupling of the photon dominates at leading twist, leading to highly specific predictions which depend on the shape and normalization of the hadron distribution amplitudes φH (xi , Q) the basic valence bound state wavefunctions. The most recent exclusive two-photon process data from CLEO [7] provides stringent tests of these fundamental QCD predictions. Exclusive processes are particularly challenging to compute in QCD because of their sensitivity to the unknown non-perturbative bound state dynamics of the hadrons. However, in some important cases, the leading power-law behavior of an exclusive amplitude at large momentum transfer can be computed rigorously via a factorization theorem which separates the soft and hard dynamics. The key ingredient is the factorization of the hadronic amplitude at leading twist. As in the case of inclusive reactions, factorization theorems for exclusive processes [1, 8, 9] allow the analytic separation of the perturbatively-calculable short-distance contributions from the longdistance non-perturbative dynamics associated with hadronic binding. For example, the amplitude γγ → π + π − factorizes in the form Mγγ→π+ π− = Z 1 0 dx Z 0 1 e T (x, y, Q) e φ (y, Q) e dy φπ (x, Q) H π (1) e is in the pion distribution amplitude and contains all of the soft, nonwhere φπ (x, Q) perturbative dynamics of the pion qq wavefunction integrated in relative transverse 3 2 e 2 , and T is the quark/gluon hard momentum up to the separation scale k⊥ < Q H scattering amplitude for γγ → (qq)(qq) where the outgoing quarks are taken collinear with their respective pion parent. To lowest order in αs , the hard scattering amplitude is linear in αs . The most convenient definition of the coupling is the effective charge αV (Q2 ), defined from the potential for the scattering of two infinitely heavy test charges, in analogy to the definition of the QED running coupling. Another possible choice is the effective charge αR (s), defined from the QCD correction to the annihilation cross section: Re+ e− →hadrons (s) ≡ R0 (1 + αR (s)/π). One can relate αV and αR to αM S to NNLO using commensurate scale relations [10]. The contributions from non-valence Fock states and the correction from neglecting the transverse momentum in the subprocess amplitude from the non-perturbative region are higher twist, i.e., power-law suppressed. The transverse momenta in the perturbative domain lead to the evolution of the distribution amplitude and to nextto-leading-order (NLO) corrections in αs . The contribution from the endpoint regions of integration, x ∼ 1 and y ∼ 1, are power-law and Sudakov suppressed and thus can only contribute corrections at higher order in 1/Q. [1] e is boost and gauge invariant and evolves in The distribution amplitude φ(x, Q) e through an evolution equation [1]. It can be computed from the integral over ln Q transverse momenta of the renormalized hadron valence wavefunction in the light-cone gauge at fixed light-cone time [1]: e = φ(x, Q) Z   ~⊥ 2 k e 2 ~ 2 e −  ψ (Q) d k⊥ θ Q (x, k~⊥ ). x(1 − x) (2) e The natural A physical amplitude must be independent of the separation scale Q. variable in which to make this separation is the light-cone energy, or equivalently 2 the invariant mass M2 = k~⊥ /x(1 − x), of the off-shell partonic system [11, 1]. Any e for the distribution amplitude will be comresidual dependence on the choice of Q pensated by a corresponding dependence of the NLO correction in TH . In general, the NLO prediction for exclusive amplitude depends strongly on the form of the pion distribution amplitude as well as the choice of renormalization scale µ and scheme. The QCD coupling is typically evaluated at quite low scales in exclusive processes since the momentum transfers has to be divided among several constituents. In the BLM procedure, the scale of the coupling is evaluated by absorbing all vacuum 4 polarization corrections with the scale of the coupling or by taking the experimental value integrating over the gluon virtuality. Thus, in the case of the (timelike) pion 1 M2π+ π− assuming the form factor the relevant scale is of order Q∗2 ∼ e−3 M2ππ− ∼ = 20 √ asymptotic form of the pion distribution amplitude φasympt = 3 fπ x(1 − x). At such π low scales, it is likely that the coupling is frozen or relatively slow varying. In the BLM procedure, the renormalization scales are chosen such that all vacuum polarization effects from the QCD β function are re-summed into the running couplings. The coefficients of the perturbative series are thus identical to the perturbative coefficients of the corresponding conformally invariant theory with β = 0. The BLM method has the important advantage of “pre-summing” the large and strongly divergent terms in the PQCD series which grow as n!(αs β0 )n , i.e., the infrared renormalons associated with coupling constant renormalization [12, 13]. Furthermore, the renormalization scales Q∗ in the BLM method are physical in the sense that they reflect the mean virtuality of the gluon propagators [13, 14, 15, 16]. In fact, in the αV (Q) scheme, where the QCD coupling is defined from the heavy quark potential, the renormalization scale is by definition the momentum transfer caused by the gluon. Because the renormalization scale is small in the exclusive γγ processes discussed here, we will argue that the effective coupling is nearly constant, thus accounting for the nominal scaling behavior of the data [17, 18]. The heavy-quark potential V (Q2 ) can be identified via the two-particle-irreducible scattering amplitude of test charges, i.e., the scattering of an infinitely heavy quark and antiquark at momentum transfer t = −Q2 . The relation V (Q2 ) = − 4πCF αV (Q2 ) , Q2 (3) with CF = (NC2 − 1)/2NC = 4/3, then defines the effective charge αV (Q). This coupling provides a physically-based alternative to the usual MS scheme. As in the corresponding case of Abelian QED, the scale Q of the coupling αV (Q) is identified with the exchanged momentum. The scale-fixed relation between αV and the conventional MS coupling is −5/6 αV (Q) = αM S (e   2CA αM S Q) 1 − +··· , 3 π (4) above or below any quark mass threshold. The factor e−5/6 ≃ 0.4346 is the ratio of commensurate scales between the two schemes to this order. It arises because of the 5 conventions used in defining the modified minimal subtraction scheme. The scale in the MS scheme is thus a factor ∼ 0.4 smaller than the physical scale. The coefficient 2CA /3 in the NLO term is a feature of the non-Abelian couplings of QCD; the same coefficient would occur even if the theory were conformally invariant with β0 = 0. Recent lattice calculations have provided strong constraints on the normalization and shape of αV (Q2 ). [19] The J/ψ and Υ spectra have been used to determine the normalization: (3) αV (8.2 GeV) = 0.196(3), (5) where the effective number of light flavors is nf = 3. The corresponding modified minimal subtraction coupling evolved to the Z mass using Eq. (4) is given by (5) αM S (MZ ) = 0.115(2). (6) This value is consistent with the world average of 0.117(5), but is significantly more precise. These results are valid up to NLO. Ji, Pang, Robertson, and I [20] have recently analyzed the pion transition form factor F γ ∗γ → π 0 obtained from eγ → e′ π 0 , the timelike pion form obtained from e+ e− → π + π, and the γγ → π + π − processes, all at NLO in αV . The assumption of a nearly constant coupling in the hard scattering amplitude at low scales provides an explanation for the phenomenological success of dimensional counting rules for exclusive processes; i.e., the power-law fall-off follows the nominal scaling of the hard scattering amplitude Mhad ∼ TH ∼ [pT ]4−n where n is in the total number of incident and final fields entering TH . The transition form factor has now been measured up to Q2 < 8 GeV2 in the tagged two-photon collisions eγ → e′ π 0 by the CLEO and CELLO collaborations. In this case the amplitude has the factorized form 4 Z1 H FγM (Q ) = √ dxφM (x, Q2 )Tγ→M (x, Q2 ), 3 0 2 (7) where the hard scattering amplitude for γγ ∗ → qq is H TγM (x, Q2 ) = 1 (1 + O(αs )) . (1 − x)Q2 (8) The leading QCD corrections have been computed by Braaten [21]; however, the NLO corrections are necessary to fix the BLM scale at LO. Thus it is not yet possible to rigorously determine the BLM scale for this quantity. We shall here assume that 6 this scale is the same as that occurring in the prediction for Fπ . For the asymptotic distribution amplitude we thus predict 2 2 Q Fγπ (Q ) = 2fπ ! 5 αV (Q∗ ) . 1− 3 π (9) As we shall see, given the phenomenological form of αV we employ (discussed below), this result is not terribly sensitive to the precise value of the scale. An important prediction resulting from the factorized form of these results is that the normalization of the ratio Rπ (Q2 ) ≡ Fπ (Q2 ) 4πQ2 |Fπγ (Q2 )|2  (10)  α Q) 1 − 0.56 M S = αM S (e π   α V = αV (e−3/2 Q) 1 + 1.43 π   αR 5/12−2ζ3 = αR (e Q) 1 − 0.65 π −14/6 (11) (12) (13) is formally independent of the form of the pion distribution amplitude. The αM S correction follows from combined references [21, 22, 23]. The next-to-leading correction given here assumes the asymptotic distribution amplitude. We emphasize that when we relate Rπ to αV we relate observable to observable and thus there is no scheme ambiguity. Furthermore, effective charges such as αV are defined from physical observables and thus must be finite even at low momenta. A number of proposals have been suggested for the form of the QCD coupling in the low-momentum regime. For example, Petronzio and Parisi [24] have argued that the coupling must freeze at low momentum transfer in order that perturbative QCD loop integrations be well defined. Mattingly and Stevenson [25] have incorporated such behavior into their parameterizations of αR at low scales. Gribov [26] has presented novel dynamical arguments related to the nature of confinement for a fixed coupling at low scales. Zerwas [27] has noted the heavy quark potential must saturate to a Yukawa form since the light-quark production processes will screen the linear confining potential at large distances. Cornwall [28] and others [29, 30] have argued that the gluon propagator will acquire an effective gluon mass mg from nonperturbative dynamics, which again will regulate the form of the effective couplings 7 at low momentum. We shall adopt the simple parameterization 4π αV (Q) = β0 ln  Q2 +4m2g Λ2V , (14) which effectively freezes the αV effective charge to a finite value for Q2 ≤ 4m2g . We can use the non-relativistic heavy quark lattice results [19, 31] to fix the parameters. A fit to the lattice data of the above parameterization gives ΛV = 0.16 GeV if we use the well-known momentum-dependent nf [32]. Furthermore, the value m2g = 0.19 GeV2 gives consistency with the frozen value of αR advocated by Mattingly and Stevenson [25]. Their parameterization implies the approximate √ constraint αR (Q)/π ≃ 0.27 for Q = s < 0.3 GeV, which leads to αV (0.5 GeV) ≃ 0.37 using the NLO commensurate scale relation between αV and αR . The resulting form for αV is shown in Fig. 1. The corresponding predictions for αR and αM S using the CSRs at NLO are also shown. Note that for low Q2 the couplings, although frozen, are large. Thus the NLO and higher-order terms in the CSRs are large, and inverting them perturbatively to NLO does not give accurate results at low scales. In addition, higher-twist contributions to αV and αR , which are not reflected in the CSR relating them, may be expected to be important for low Q2 [33]. It is clear that exclusive processes such as the photon to pion transition form factors can provide a valuable window for determining the magnitude and the shape of the effective charges at quite low momentum transfers. In particular, we can check consistency with the αV prediction from lattice gauge theory. A complimentary method for determining αV at low momentum is to use the angular anisotropy of e+ e− → QQ at the heavy quark thresholds [34]. It should be emphasized that the parameterization (14) is just an approximate form. The actual behavior of αV (Q2 ) at low Q2 is one of the key uncertainties in QCD phenomenology. As we have emphasized, exclusive processes are sensitive to the magnitude and −3 2 2 shape of the QCD couplings at quite low momentum transfer: Q∗2 V ≃ e Q ≃ Q /20 2 and Q∗2 R ≃ Q /50 [35]. The fact that the data for exclusive processes such as form factors, two photon processes such as γγ → π + π − , and photoproduction at fixed θc.m. are consistent with the nominal scaling of the leading-twist QCD predictions (dimensional counting) at momentum transfers Q up to the order of a few GeV can be immediately understood if the effective charges αV and αR are slowly varying 8 0.6 αV αM S αR 0.5 0.4 0.3 0.2 0.1 0 0 5 10 15 20 Q2 (GeV2 ) Figure 1: The coupling function αV (Q2 ) as given in Eq. (14). Also shown are the corresponding predictions for αM S and αR following from the NLO commensurate scale relations. at low momentum. The scaling of the exclusive amplitude then follows that of the subprocess amplitude TH with effectively fixed coupling. Note also that the Sudakov effect of the end point region is the exponential of a double log series if the coupling is frozen, and thus is strong. In Fig. 2, we compare the recent CLEO data [7] for the photon to pion transition form factor with the prediction 2 2 Q Fγπ (Q ) = 2fπ ! 5 αV (e−3/2 Q) . 1− 3 π (15) The flat scaling of the Q2 Fγπ (Q2 ) data from Q2 = 2 to Q2 = 8 GeV2 provides an important confirmation of the applicability of leading twist QCD to this process. The magnitude of Q2 Fγπ (Q2 ) is remarkably consistent with the predicted form, assuming the asymptotic distribution amplitude and including the LO QCD radiative correction with αV (e−3/2 Q)/π ≃ 0.12. Radyushkin [36], Ong [37] and Kroll [38] have also noted that the scaling and normalization of the photon-to-pion transition form factor tends 9 0.25 0.2 r Q2 Fγπ (Q2 ) (GeV) 0.15 r r r r r r r r r r r r r r 0.1 0.05 0 0 2 4 6 8 10 Q2 (GeV2 ) Figure 2: The γ → π 0 transition form factor. The solid line is the full predic- tion including the QCD correction [Eq. (15)]; the dotted line is the LO prediction Q2 Fγπ (Q2 ) = 2fπ . to favor the asymptotic form for the pion distribution amplitude and rules out broader distributions such as the two-humped form suggested by QCD sum rules [39]. One cannot obtain a unique solution for the non-perturbative wavefunction from the Fπγ data alone. However, we have the constraint that # " 1 1 5 αV (Q∗ ) ≃ 0.8 h i 1− 3 1−x 3 π (16) (assuming the renormalization scale we have chosen in Eq. (9) is approximately correct). Thus one could allow for some broadening of the distribution amplitude with a corresponding increase in the value of αV at low scales. We have also analyzed the γγ → π + π − , K + K − data. These data exhibit true leading-twist scaling (Fig. 3), so that one would expect this process to be a good test of theory. One can show that to LO dσ dt dσ dt (γγ → π + π − ) 4|Fπ (s)|2 = 1 − cos4 θc.m. (γγ → µ+ µ− ) 10 (17) in the CMS, where dt = (s/2)d(cos θc.m. ) and here Fπ (s) is the time-like pion form factor. The ratio of the time-like to space-like pion form factor for the asymptotic distribution amplitude is given by |Fπ(timelike) (−Q2 )| |αV (−Q∗2 )| . = (spacelike) αV (Q∗2 ) Fπ (Q2 ) (18) If we simply continue Eq. (14) to negative values of Q2 then for 1 < Q2 < 10 GeV2 , and hence 0.05 < Q∗2 < 0.5 GeV2 , the ratio of couplings in Eq. (18) is of order 1.5. Of course this assumes the analytic application of Eq. (14). Thus if we assume the asymptotic form for the distribution amplitude, then we predict Fπ(timelike) (−Q2 ) ≃ (0.3 GeV2 )/Q2 and hence dσ dt dσ dt (γγ → π + π − ) .36 1 ≃ 2 . + − s 1 − cos4 θc.m. (γγ → µ µ ) (19) The resulting prediction for the combined cross section σ(γγ → π + π − , K + K − )† is shown in Fig. 3, along with CLEO data [7]. Considering the possible contribution of the resonance f2 (1270), the agreement is reasonable. We also note that the normalization of αV could be larger at low momentum than our estimate. This would also imply a broadening of the pion distribution amplitude compared to its asymptotic form since one needs to raise the expectation value of 1/(1 −x) in order to maintain consistency with the magnitude of the Fγπ (Q2 ) data. A full analysis will then also require consideration of the breaking of scaling from the evolution of the distribution amplitude. In any case, we find no compelling argument for significant higher-twist contributions in the few GeV regime from the hard scattering amplitude or the endpoint regions, since such corrections violate the observed scaling behavior of the data. The analysis we have presented here suggests a systematic program for estimating exclusive amplitudes in QCD (including exclusive B-decays) which involve hard scattering. The central input is αV (0), or 1 αV = 2 Q0 † Z 0 Q20 2 2 dQ′ αV (Q′ ), Q20 ≤ 1 GeV2 , (20) The contribution from kaons is obtained at this order simply by rescaling the prediction for pions by a factor (fK /fπ )4 ≃ 2.2. 11 100 r r r 10 r r r r r σ (nb) 1 r r r 0.1 r 0.01 1 2 3 4 M (GeV/c2 ) Figure 3: Two-photon annihilation cross section σ(γγ → π + π − , K + K − ) as a function of CMS energy, for | cos θ∗ | < 0.6. which largely controls the magnitude of the underlying quark-gluon subprocesses for hard processes in the few-GeV region. In this work, the mean coupling value for Q20 ≃ 0.5 GeV2 is αV ≃ 0.38. The main focus will then be to determine the shapes and normalization of the process-independent meson and baryon distribution amplitudes. 3 Conclusions The leading-twist scaling of the observed cross sections for exclusive two-photon processes and other fixed θcm reactions can be understood if the effective coupling αV (Q∗ ) is approximately constant in the domain of Q∗ relevant to the underlying hard scattering amplitudes. In addition, the Sudakov suppression of the long-distance contributions is strengthened if the coupling is frozen because of the exponentiation of a double log series. We have also found that the commensurate scale relation connecting the heavy quark potential, as determined from lattice gauge theory, to the photon-to-pion transition form factor is in excellent agreement with γe → π 0 e data assuming that 12 the pion distribution amplitude is close to its asymptotic form √ 3fπ x(1 − x). We also reproduce the scaling and approximate normalization of the γγ → π + π − , K + K − data at large momentum transfer. However, the normalization of the space-like pion form factor Fπ (Q2 ) obtained from electroproduction experiments is somewhat higher than that predicted by the corresponding commensurate scale relation. This discrepancy may be due to systematic errors introduced by the extrapolation of the γ ∗ p → π + n electroproduction data to the pion pole. Acknowledgments Much of this talk is based on collaborations with Peter Lepage, Hung Jung Lu, Chueng Ji, Dave Robertson, and Alex Pang, and I thank them for helpful conversations. This work is supported in part by the U.S. Department of Energy under contract no. DE–AC03–76SF00515. References [1] S. J. Brodsky and G. P. Lepage, Phys. Rev. Lett. 53, 545 (1979); Phys. Lett. 87B, 359 (1979); G. P. Lepage and S. J. Brodsky, Phys. Rev. D22, 2157 (1980). [2] I.F. Ginzburg, D.Yu. Ivanov, and V.G. 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