High-Energy Limit of Quantum Electrodynamics beyond Sudakov Approximation

We study the high-energy behavior of the scattering amplitudes in quantum electrodynamics beyond the leading order of the small electron mass expansion in the leading logarithmic approximation. In contrast to the Sudakov logarithms, the mass-suppressed double-logarithmic radiative corrections are induced by a soft electron pair exchange and result in enhancement of the power-suppressed contribution. Possible applications of our result to the analysis of the high-energy processes in quantum chromodynamics is also discussed.

We study the high-energy behavior of the scattering amplitudes in quantum electrodynamics beyond the leading order of the small electron mass expansion in the leading logarithmic approximation. In contrast to the Sudakov logarithms, the mass-suppressed double-logarithmic radiative corrections are induced by a soft electron pair exchange and result in enhancement of the powersuppressed contribution, which dominates the amplitudes at extremely high energies. Possible applications of our result to the analysis of the high-energy processes in quantum chromodynamics is also discussed. In a renowned paper [1] V.V. Sudakov derived the leading asymptotic behavior of an electron scattering amplitude in quantum electrodynamics (QED) at high energy. It is determined by the "Sudakov" radiative corrections, which include the second power of the large logarithm of the electron mass m e divided by a characteristic momentum transfer of the process per each power of the fine structure constant α. Sudakov double logarithms exponentiate and result in a strong universal suppression of any electron scattering amplitude with a fixed number of emitted photons in the limit when all the kinematic invariants of the process are large. This result plays a fundamental role in particle physics. Within different approaches it has been extended to the nonabelian gauge theories and to the subleading logarithms [2][3][4][5][6][7], which is crucial for a wide class of applications from deep inelastic scattering to Drell-Yan processes and the Higgs boson production. At the same time no significant progress has been achieved in the study of the logarithmically enhanced corrections to the subleading contributions suppressed by a power of electron mass at high energies. However, the power-suppressed contributions are of great interest. They can become asymptotically dominant at very high energies due to Sudakov suppression of the leading terms. At the intermediate energies the power corrections in many cases are phenomenologically important [8][9][10][11]. Moreover, in contrast to the Euclidean operator product expansion [12] or nonrelativistic threshold dynamics [13] very little is known about the general all-order structure of the large logarithms beyond the leading-power approximation in the high-energy limit, which is a real challenge for the effective field theory approach. This problem is now actively discussed in various contexts (see e.g. [14][15][16][17][18][19]). In this Letter we make the first step toward the solution of the problem and generalize the result of Ref. [1] to the leading power-suppressed contribution. We present a detailed analysis of the electron scattering in the external field and later discuss the extension of the result to more complex processes.
The amplitude F of the high-energy electron scattering in an external field with the double-logarithmic accuracy can be parameterized by the Dirac form factor F 1 We consider the limit of the on-shell electron p 2 1 = p 2 2 = m 2 e and the large Euclidean momentum transfer Q 2 = −(p 2 − p 1 ) 2 when the ratio ρ ≡ m 2 e /Q 2 is positive and small. The Dirac form factor can then be expanded in an asymptotic series in ρ where F (n) 1 are given by the power series in α with the coefficients depending on ρ only logarithmically. The factor S λ = exp − α 2π B(ρ) ln λ 2 /m 2 e with B(ρ) = ln ρ+O(1) accounts for the universal singular dependence of the amplitude on the auxiliary photon mass λ introduced to regulate the infrared divergences [20]. In the doublelogarithmic approximation the leading term is given by the Sudakov exponent F . To analyze the power-suppressed double-logarithmic contribution we follow a two-step strategy: (i) the expansion by regions method [22][23][24] is applied to get a systematic expansion of the Feynman integrals in ρ, (ii) the divergences of a single-scale contribution of different regions giving rise to the double-logarithmic corrections are determined. Sudakov logarithms are produced by the soft virtual photons, which are collinear to either p 1 or p 2 . By using the approach outlined above we have found that such a configuration of virtual momenta does not produce double logarithms in the first order in ρ. This observation agrees with the analysis [25] of the cusp anomalous dimension, which determines the double-logarithmic corrections to the light-like Wilson line with a cusp. For the large cusp angle corresponding to the limit ρ → 0 from the result of Ref. [25] one gets with vanishing first-order term in ρ. Nevertheless, the O(ρ) double-logarithmic contribution does exist but orig- inates from completely different virtual momentum configuration described below. Let us consider an electron where l is the momentum of a virtual photon with the propagator D = −gµν l 2 −λ 2 . In the soft-photon limit l → 0 the electron propagator becomes eikonal S ≈ − / p i +me 2pil and develops a collinear singularity when l is parallel to p i . Alternatively, we may consider the soft-electron limit l ′ → 0, where l ′ = p i − l. Then the electron propagator becomes scalar S ≈ me Thus the roles of the electron and photon propagators are exchanged. The existence of non-Sudakov doublelogarithmic contributions due to soft electron exchange has actually been known for a long time [26,27]. However in our case this virtual momentum configuration does not produce a double-logarithmic contribution in one loop because the momentum shift distorts the eikonal structure of the second electron propagator and removes the soft singularity at small l ′ necessary to get the second power of the large logarithm. This may be avoided only in the two-loop diagram of nonplanar topology, Fig. 1(a). After shifting the photon virtual momenta by p 1 and p 2 the diagram can be twisted into the shape of Fig. 1(b,c) with soft electron pair exchange between the eikonal lines. The corresponding contribution has an explicit suppression factor m 2 e from two soft electron propagators. Hence the integration over the virtual momenta can be performed in the leading order of the small electron mass expansion. Note that in the case under consideration the electron mass regulates both soft and collinear divergences and we can put λ = 0. The calculation is conveniently performed by using the lightcone coordinates where p 1 ≈ p 1− and p 2 ≈ p 2+ . In this representation only the interaction of the transverse photons to soft electrons is not mass-suppressed and we can use g ⊥ kl 2pil for the eikonal photon propagators. To get the double-logarithmic part of the correction we use Sudakov parametrization of a virtual photon momentum l = up 1 + vp 2 + l ⊥ . After integrating over the transverse components l ⊥ we get the following representation of the two-loop power-suppressed form factor where η = ln v/ ln ρ, ξ = ln u/ ln ρ, the integration goes over the four-dimensional cube 0 < η i , ξ i < 1, and the kernel selects the kinematically allowed region of doublelogarithmic integration. This gives F [8,28]. The higher-order doublelogarithmic corrections are generated in a usual way through the exchange of soft photons with the propagator −g+− l 2 −λ 2 . A key observation here is that an exchange of a soft photon between an eikonal and a soft electron line does not produce double logarithms. The reason for this is that such a loop is always separated from the second eikonal line by a scalar electron propagator, which does not communicate any information on the second external momentum. Hence the loop integral cannot depend on the scalar product p 1 p 2 , which is the only large scale in the problem. Thus it is sufficient to consider only the diagrams of the topologies given in Fig. 2. By using the factorization properties of the soft photon contribution [1] we find the following representation of the all-order double-logarithmic result where the Sudakov correction factors corresponding to Fig. 2(a-c) are respectively, and the new kernels read (8) with K = K 1 + K 2 . We are not able to find the result for the four-fold integral (6) in a closed analytic form. However, the coefficients of the series can in principle be analytically computed for any given n. The first seven coefficients of the series are listed in Table I. At the same time in the large-n limit we get an approximate result where C = 1.1994 . . . is a numerical constant. Eqs. (9,10) give the following asymptotic behavior of the form factor at large x i.e. the power-suppressed amplitude is enhanced by the double-logarithmic corrections. A similar effect has been observed before e.g. for the electron-muon backward scattering [26]. In the case under consideration the positive sign of the exponent may be related to a specific structure of the process with the soft electron pair exchange. Through the pair emission an eikonal electron is converted into an eikonal positron with approximately the same momentum but opposite electric charge, Fig. 1(c). As a result the double-logarithmic contribution of the topology of Fig. 2(c) has an opposite sign with respect to the one of Fig. 1(a,b) and actually determines the behavior of the exponent in Eq. (11). It is interesting to compare the high-energy asymptotic behavior of the leading and subleading terms of Eq. (2). In the limit ρ → 0 we get Thus above the energy corresponding to | ln ρ| = 2π α the originally power-suppressed term exceeds the leading contribution of F 1 . Note that at this energy the pure QED running coupling α(Q 2 ) ≈ 3α and we are still in a weak coupling regime. However, this energy is too high to be phenomenologically relevant and this result is likely to be of pure theoretical interest.
Unlike the Sudakov double logarithms, the leading power-suppressed double-logarithmic corrections depend  not only on the charges of the initial and final states but also on the details of the scattering process. For example, the O(ρ) double-logarithmic corrections to the scalar form factor vanish to all orders in α due to a specific Lorentz and Dirac structure of the soft electron pair interaction with the eikonal electrons and positrons. However the O(ρ) corrections are universally related to the soft electron pair exchange and can be be obtained as a straightforward generalization of our analysis for more complicated processes such as Bhabha scattering, where only the leading result of the small electron mass expansion is available in two loops [29,30]. Moreover, up to two loops the structure of the O(ρ) double-logarithmic correction in quantum chromodynamics (QCD) is identical to the one in QED. In particular, the double-logarithmic power-suppressed term in two-loop corrections to the heavy-quark vector form factor differs from the QED result only by the C 2 F − C A C F /2 color factor of the diagram in Fig. 1. Thus our method can be applied to the calculation of the dominant two-loop power-suppressed corrections to the high-energy processes involving heavy quarks. For the energies ranging from approximately 10 to 100 times the heavy-quark mass we have | ln ρ| ≫ 1 and ρ ln 4 ρ ∼ 1, i.e. the double-logarithmic terms saturate the power-suppressed contribution and are comparable in magnitude to the nonlogarithmic leading-power corrections in the strong coupling constant, which are phenomenologically significant. Beyond the two-loop approximation our result is not directly applicable to the QCD amplitudes since the eikonal gluons in Fig. 1(b) can radiate soft gluons producing additional doublelogarithmic corrections. As a consequence, the leading power-suppressed double-logarithmic corrections to the heavy-quark vector form factor get a nonabelian contribution in every order of perturbation theory in contrast to the purely abelian Sudakov double logarithms.
To summarize, we generalized the result of Sudakov [1] to the leading power-suppressed contribution. This is an important step towards a systematic renormalization group analysis of the high energy behavior of the gauge theory amplitudes beyond the leading power approximation. The leading power-suppressed double-logarithmic corrections reveal a few characteristic features which distinguish them from the Sudakov double logarithms. In particular, they are induced by a soft electron pair exchange and result in a strong enhancement of the powersuppressed contribution. In QCD our method can be used for the analysis of the high-energy processes involv-ing heavy quarks up to two loops. Extending the analysis to the higher orders of perturbative QCD and to subleading logarithms is a very interesting problem which is beyond the scope of this Letter.
I would like to thank R. Bonciani, K. Melnikov, V. Smirnov, and N. Zerf for useful discussions and collaboration. This research was supported in part by NSERC and Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.