Regge Limit of Gauge Theory Amplitudes beyond Leading Power Approximation

We study the high-energy small-angle {\it Regge} limit of the fermion-antifermion scattering in gauge theories and consider the part of the amplitude suppressed by a power of the scattering angle. For abelian gauge group all-order resummation of the double-logarithmic radiative corrections to the leading power-suppressed term is performed. We find that when the logarithm of the scattering angle is comparable to the inverse gauge coupling constant the asymptotic double-logarithmic enhancement overcomes the power suppression, a formally subleading term becomes dominant, and the small-angle expansion breaks down. For the nonabelian gauge group we show that in the color-singlet channel for sufficiently small scattering angles the power-suppressed contribution becomes comparable to the one of BFKL pomeron. Possible role of the subleading-power effects for the solution of the unitarity problem of perturbative Regge analysis in QED and QCD is discussed. An intriguing relation between the asymptotic behavior of the power-suppressed amplitudes in Regge and Sudakov limits is discovered.

Small-angle or Regge limit of high-energy scattering describes a kinematical configuration with vanishing ratio of the characteristic momentum transfer to the total energy of the process. The asymptotic behavior of the gauge theory amplitudes in Regge limit remains in the focus of the theoretical studies since the early days of QED and QCD [1][2][3][4][5][6][7]. Despite a crucial simplification due to decoupling of the light-cone and transversal degrees of freedom, the gauge interactions in this limit possess highly nontrivial dynamics giving a rigorous quantum field theory realization of Regge concept for high-energy scattering [8]. Major progress has been achieved in the analysis of the leading-power amplitudes which scale at small momentum transfer as a ratio of the Mandelstam variables s/t ∼ 1/θ 2 , where θ is the scattering angle. It culminated in the evaluation of the next-to-leading QCD corrections to the theory of BFKL pomeron [9]. At the same time very little is known about the asymptotic behavior of the amplitudes suppressed by a small ratio τ = |t/s|. In contrast to the leading-power amplitudes it is determined by the double-logarithmic radiative corrections which include the second power of the large logarithm ln τ per each power of the gauge coupling constant α. This type of corrections has been discussed so far in QED only for the amplitudes which do not have the leading-power contribution and remain finite in the small-angle limit, such as the electron-to-muon pair forward annihilation [10]. However for the general scattering case the logarithmically-enhanced power-suppressed contributions can also play a crucial role. Indeed a recent study of the mass-suppressed amplitudes in the highenergy fixed-angle Sudakov limit [11][12][13][14][15] revealed that for some processes the double-logarithmic corrections result in strong enhancement of the power-suppressed terms which asymptotically makes them comparable to the leading-power contributions, i.e. formally lead to breakdown of the small-mass expansion. If a similar scenario is realised in the small-angle scattering, it may signifi-cantly alter our understanding of the gauge theory dynamics in Regge limit. Thus a systematic renormalization group analysis of subleading-power amplitudes is of primary theoretical interest.
In this Letter we make the first step toward this goal and discuss the double-logarithmic behaviour of the leading power-suppressed contribution to the fermionantifermion scattering amplitude. First we consider an abelian gauge theory and set up an effective theory framework for the analysis of the Born scattering in Regge limit. We apply it to the calculation of the one-loop power-suppressed double-logarithmic contribution and then perform the resummation of the double-logarithmic corrections to all orders of perturbation theory to find the asymptotic behavior of the amplitude. Finally we discuss the qualitative features and the impact of the power-suppressed terms in the theory of Regge limit in QED and QCD.
We start with the scattering of a fermion with the initial momentum p and the final momentum p ′ = p + q by an external abelian field A µ in the limit q → 0. We choose the reference frame in a such way that the incoming fermion momentum p µ = (ε, −ε, 0, 0) has only one lightcone component p − = p + = √ 2ε, while p + = p − = 0. In the high-energy limit one can neglect the fermion mass and the on-shell conditions p 2 = p ′ 2 = 0 imply the following scaling of the momentum transfer components By expanding a solution of the free Dirac equation ψ(p ′ ) in q we obtain the following series for the scattering amplitude where the currents read j µ = gψ(p ′ )γ µ ψ(p) , j + = gψ(p)γ + ψ(p) , with the initial momenta p µ 1 = (ε, −ε, 0, 0) and p µ 2 = −(ε, ε, 0, 0). The corresponding Born amplitude reads where the vector and axial component are defined as follows In close analogy with the nonrelativistic expansion, the leading order amplitude is mediated by a static potential 1 and the power corrections describe a contact fermionantifermion interaction. Note that if we perform the expansion in u-channel rather than in s-channel i.e. about the momentum p 3 instead of p 2 , the sign of the axial term in Eq. (4) changes.
Let us now discuss the one-loop amplitude. The corrections enhanced by a power of ln τ can only be produced by the one-particle irreducible box diagrams in Fig. 1(a,b). Hence, we do not consider the factorizable correction where a gauge boson is emitted and absorbed by the same fermion line. The double-logarithmic contribution comes from the virtual momentum region where l ≪ √ s and the virtual fermion or antifermion is close to its mass-shell [17]. Thus we can use the same scaling rules for the components of l as for the components of the momentum transfer q. Then it is sufficient to consider the eikonal approximation for the (anti)fermion propagator and the static approximation for the Glauber gauge boson propagator D µν (l) → −g +− /l 2 ⊥ . Let us first briefly discus the corresponding leading-power contribution. In this case the O(1/ √ s) terms in the eikonal propagators can be neglected and each gauge boson exchange is described by the leading term of Eq. (4) with the corresponding momentum transfer. Then the planar box diagram Fig. 1(a) is given by the amplitude M v with a factor where α = g 2 /(4π). The expression for the nonplanar diagram Fig. 1(b) differs only by the sign of l + and in the sum the antifermion propagators add up . By symmetrization the fermion propagator can be reduced to −iπδ(l − ) in the same way. After trivial integration over the lightcone components the total leading-power contribution is purely imaginary and can be written as iφM v with the Glauber phase given by an infrared divergent transverse momentum integral where µ is an infrared regulator. The Glauber phase is known to exponentiate so that the all-order leadingpower amplitude is given by e iφ M v and has no ln τ terms in any order of perturbation theory [1,2]. 2 The calculation of the double-logarithmic one-loop O(τ ) amplitude can be performed in a similar way. However, in this case the power suppressed term in the eikonal propagators should be kept and for one of the gauge boson exchanges the contact O(τ ) part of the amplitude Eq. (4) should be taken. The full effective theory oneloop expression includes also the diagrams with a local O(g 2 τ ) vertex quadratic in the gauge field which accounts for the off-shell virtual (anti)fermion contribution. Such diagrams however do not have the double-logarithmic scaling and are omitted. The planar diagram is then proportional to M v + M a i.e. contributes only to the same-helicity fermion scattering. The corresponding coefficient is infrared finite and with the double-logarithmic accuracy is given by the integral represented by the effective theory Feynman diagram in Fig. 1(c). Note that in Eq. (9) we put q = 0 and this expression is valid only for l ≫ q, which is sufficient for the calculation of the double-logarithmic terms. Integration over l + in Eq. (9) can be performed by using Cauchy theorem and taking the residue of the antifermion propagator pole, which gives l + = l 2 ⊥ /( √ 2s) and l − > 0. Then the remaining integral has the double-logarithmic scaling in the interval l 2 ⊥ / √ s < l − < √ s and q 2 ⊥ < l 2 ⊥ < s. Thus Eq. (9) with the double-logarithmic accuracy can be rewritten as follows where we introduce the variables ξ = ln(l 2 ⊥ /s)/ ln τ , η = ln(l − / √ s)/ ln τ , and z = α 2π ln 2 τ . As it was discussed before the nonplanar u-channel diagram is proportional to M v − M a and contributes only to the opposite helicity fermion scattering. In the sum of the planar and nonplanar contributions the axial part cancel and the total one-loop amplitude takes the form in agreement with the expansion of the exact result for the chiral amplitudes (see, e.g. [18]). In the near-forward scattering the charged particles almost do not accelerate and the usual double-logarithmic corrections associated with the soft gauge boson emission are suppressed [17]. Thus the higher-order doublelogarithmic corrections are due to multiple leading-power Glauber gauge boson exchanges which produce infraredfinite terms of the form (α ln 2 τ ) n . To determine the factorization structure of these corrections let us consider the two-loop power-suppressed contribution obtained by dressing the one-loop diagrams with a Glauber gauge boson of the momentum l ′ . The corresponding effective theory expression can be directly obtained by expanding the full theory diagrams in l and l ′ and keeping only the power suppressed terms linear in l ± ∼ τ and quadratic in l ⊥ ∼ τ 1/2 which give rise to the effective contact interaction as in Fig. 1c. Then in the sum of all the diagrams one can separate the part proportional to the product δ(l ′ − )δ(l ′ + ). As for the leading-power contribution it gives the infrared-divergent logarithmic Glauber phase Eq. (8), which factors out with respect to the one-loop O(τ ) amplitude. The remaining contribution reduces to the effective theory diagrams with the characteristic structure shown in Fig. 2. In the sum over all the permutations of the gauge boson vertex along the eikonal antifermion line, Fig. 2(a-c), the dependence on the light-cone component of its momentum l ′ can be factored out into δ(l ′ + ). At the same time the Clauber gauge boson is attached only to the virtual fermion line carrying the loop momentum l and the integration over l ′ − remains logarithmic. The logarithmic integration intervals can be easily identified as q 2 ⊥ < l ′ 2 ⊥ < l 2 ⊥ , l − < l ′ − < √ s and after integration over l ′ + the second loop results in a factor 2α π (12) It is straightforward to check that for an arbitrary number n of the leading-power Glauber gauge bosons with the momenta l ′ i the factorization structure remains the same: the Glauber phase exponentiates and the doublelogarithmic contribution is determined by the sum of the diagrams with all the leading-power vertices attached to the virtual fermion line carrying the loop momentum l, as in Fig. 2. Again the dependence on l ′ i+ factors out to δ(l ′ i+ ) and the logarithmic integration intervals are q 2 ⊥ < l ′ i 2 ⊥ < l 2 ⊥ and l − < l ′ 1− < . . . < l ′ n− < √ s, i.e. the light-cone components are ordered along the eikonal fermion line. The integration over l ′ i then gives (4zη(1 − ξ)) n /n! and after summation over n we get an exponential factor e 4zη(1−ξ) to be inserted into the remaining loop integral Eq. (10). Thus in the leading (double) logarithmic approximation the scattering amplitude reads where is the generalized hypergeometric function with the Taylor series expansion g(z) = 1 + z/6 + z 2 /45 + . . .. The O(z 2 ) contribution to Eq. (13) agrees with the smallangle expansion of the two-loop result for Bhabha scattering [20][21][22], which is a nontrivial check of our calculation. It is interesting to note that exactly the same i ii iii iv γ α 2π ln |s/t| 2α π 1/2 1 + 11π 36 α 2 1 + 4 ln 2 π Ncαs TABLE I. The exponent γ defined in the text for (i) the subleading-power scattering, Eq. (16); (ii) electron-to-muon pair forward annihilation amplitude [10]; (iii) QED Regge cut contribution [3,4]; (iv) BFKL pomeron contribution [6,7].
function g(z) appears in the analysis of the doublelogarithmic asymptotic behavior of the mass-suppressed QED and QCD amplitudes in the Sudakov limit. Though it is fairly easy to notice a similar factorization structure in both cases, such a universality is rather surprising since in Sudakov limit the double-logarithmic corrections are associated with the exchange of the on-shell rather than Glauber gauge bosons. The function g(z) has the following asymptotic behavior at z → ∞ Thus in Regge limit we have which is the main result of this Letter. To characterize the asymptotic behavior of the amplitudes at high energy it is convenient to introduce an exponent γ so that M ∼ s γ for s → ∞. Since M (0) ∼ s, our result Eq. (16) corresponds to γ = α 2π ln |s/t|. The values of γ for a number of different amplitudes and gauge groups are collected in Table I. In particular it includes the results for the leading-power amplitudes corresponding to the contribution of the rightmost singularities in the complex angular momentum plane: the light-by-light scattering induced Regge cut in QED [3,4] and the BFKL pomeron Regge pole in QCD [6,7]. The common feature of all these examples is that γ > 0 and the radiative corrections result in the asymptotic enhancement of the amplitudes. At the same time only for the subleading-power scattering the exponent depends on τ , which is characteristic to the double-logarithmic corrections. By contrast the leading-power effects are single-logarithmic and give energy-independent γ. On the other hand the QED forward annihilation amplitude [10] is power-suppressed and does get the double-logarithmic corrections. However, at z → ∞ this amplitude becomes an exponential function of z 1/2 rather than z so that the corresponding exponent γ ∼ √ α is nonanalytic in coupling constant but does not depend on τ and is consistent with the square root branch point in the complex angular momentum plane.
We can apply the above result to describe the powersuppressed effect in the color-singled channel of quarkaniquark scattering in QCD. In this case in the expression for γ one should use α = is the number of colors, α s is the strong coupling constant, and the prefactor arises from projecting on the colorsinglet configuration. In principle in the color-singlet channel one should retain only the contribution of even number of gluons i.e. only even powers of z in the Taylor series for g(z). This however does not change the value of γ in the asymptotic formula. Though this result clearly does not give a complete account of subleading-power effects in Regge limit of QCD, it can be considered as a power correction to the BFKL pomeron contribution to the elastic parton scattering [23].
In the high-energy limit for a given γ the total cross section has the following scaling σ ∼ s 2(γ−1) . For γ > 1 such an asymptotic behavior violates the Froissart bound [24] and hence the S-matrix unitarity. This constitutes the unitarity problem of the perturbative Regge analysis, which gives γ > 1 both in QED and QCD, cf. Table I. Though a precise mechanism of unitarity restoration is not yet known, our result sets a perturbative bound on the energy scale where the Regge analysis of the leadingpower amplitudes can be trusted. Indeed, for |ln τ | ∼ 1/α or more precisely for the formally power-suppressed contribution to the cross section becomes comparable to the leading-power one and the small-angle expansion breaks down. Interestingly this is roughly the scale at which the resummation of the single-logarithmic corrections responsible for the Regge behavior of the leading-power amplitudes becomes mandatory. I would like to thank Kirill Melnikov for discussions and collaboration and Tao Liu for important crosschecks. This work is supported in part by NSERC and Perimeter Institute for Theoretical Physics.