BFKL Equation Research Articles

Conformal BK equation at QCD Wilson-Fisher point

High-energy scattering in pQCD in the Regge limit is described by the evolution of Wilson lines governed by the BK equation [1, 2]. In the leading order, the BK equation is conformally invariant and the eigenfunctions of the linearized BFKL equation are powers. It is a common belief that at d ≠ 4 the BFKL equation is useless since unlike d = 4 case it cannot be solved by usual methods. However, we demonstrate that at critical Wilson-Fisher point of QCD the relevant part of NLO BK restores the conformal invariance so the solutions are again powers. As a check of our approach to high-energy amplitudes at the Wilson-Fisher point, we calculate the anomalous dimensions of twist-2 light-ray operators in the Regge limit j → 1.

A collinear perspective on the Regge limit

The high energy (Regge) limit provides a playground for understanding all loop structures of scattering amplitudes, and plays an important role in the description of many phenomenologically relevant cross-sections. While well understood in the planar limit, the structure of non-planar corrections introduces many fascinating complexities, for which a general organizing principle is still lacking. We study the structure of multi-reggeon exchanges in the context of the effective field theory for forward scattering, and derive their factorization into collinear operators (impact factors) and soft operators. We derive the structure of the renormalization group consistency equations in the effective theory, showing how the anomalous dimensions of the soft operators are related to those of the collinear operators, allowing us to derive renormalization group equations in the Regge limit purely from a collinear perspective. The rigidity of the consistency equations provides considerable insight into the all orders organization of Regge amplitudes in the effective theory, as well as its relation to other approaches. Along the way we derive a number of technical results that improve the understanding of the effective theory. We illustrate this collinear perspective by re-deriving all the standard BFKL equations for two-Glauber exchange from purely collinear calculations, and we show that this perspective provides a number of conceptual and computational advantages as compared to the standard view from soft or Glauber physics. We anticipate that this formulation in terms of collinear operators will enable a better understanding of the relation between BFKL and DGLAP in gauge theories, and facilitate the analysis of renormalization group evolution equations describing Reggeization beyond next-to-leading order.

Radiative corrections for factorized jet observables in heavy ion collisions

I look at the renormalization of the medium structure function and a medium induced jet function in a factorized cross section for jet substructure observables in Heavy Ion collisions. This is based on the formalism developed in [1], which uses an Open quantum system approach combined with the Effective Field Theory (EFT) for forward scattering to derive a factorization formula for jet observables which work as hard probes of a long lived dilute Quark Gluon Plasma (QGP) medium. I show that the universal medium structure function that captures the observable independent physics of the QGP has both rapidity and UV anomalous dimensions that appear due to medium induced Bremsstrahlung. The resulting Renormalization Group (RG) equations correspond to the BFKL equation and the running of the QCD coupling respectively. I present the first results for the numerical impact of resummation using these RG equations on the mean free path of the jet in the medium. I also briefly discuss the prospects of extending this formalism for a short lived dense medium.

Dipole-dipole scattering amplitude in the CGC approach

In this paper we propose recurrence relations for the dipole densities in QCD, which allows us to find these densities from the solution to the BFKL equation. We resolve these relations in the diffusion approximation for the BFKL kernel. Based on this solution, we found the sum of large Pomeron loops. This sum generates the scattering amplitude that decreases at large values of rapidity $Y$. It turns out that such behaviour of the scattering amplitudes is an artifact of diffusion approximation. This approximation leads to the unitarization without saturation both in deep inelastic scattering and in dipole-dipole interaction at high energies.

One-loop central-emission vertex for two gluons in mathcal{N} = 4 super Yang-Mills theory

A necessary ingredient for extending the BFKL equation to next-to-next-to-leading logarithmic (NNLL) accuracy is the one-loop central emission vertex (CEV) for two gluons which are not strongly ordered in rapidity. Here we consider the one-loop six-gluon amplitude in mathcal{N} = 4 super Yang-Mills (SYM) theory in a central next-to-multi-Regge kinematic (NMRK) limit, we show that its dispersive part factorises in terms of the two-gluon CEV, and we use it to extract the one-loop two-gluon CEV for any helicity configuration within this theory. This is a component of the two-gluon CEV in QCD. Although computed in the NMRK limit, both the colour structure and the kinematic dependence of the two-gluon CEV capture much of the complexity of the six-gluon amplitudes in general kinematics. In fact, the transcendental functions of the latter can be conveniently written in terms of impact factors, trajectories, single-emission CEVs and a remainder, which is a function of the conformally invariant cross ratios which characterise the six-gluon amplitudes in planar mathcal{N} = 4 SYM. Finally, as expected, in the MRK limit the two-gluon CEV neatly factorises in terms of two single-emission CEVs.

Anomalous dimension of transverse momentum broadening in planar 𝒩 = 4 SYM

The typical transverse momentumQs(t) (or "saturation" momentum) acquired by a hard particle propagating through aN= 4 SYM plasma increases over time liketγ, with an anomalous exponentγ> 1/2 characteristic of super-diffusion. This anomalous exponent is a function of the ’t Hooft couplingλ=g2Nc. Recently, a method has been proposed to systematically compute the perturbative series ofγ(λ) at weak coupling. This method relies on the traveling wave interpretation of the time evolution ofQs(t) and on the dominance of softcollinear radiative corrections at large times. In this paper, we computeγ(λ) up to 𝒪(λ2) using the double logarithmic behaviour of the BFKL equation in planar 𝒩 = 4 SYM at three loops. This calculation allows us to discuss the transition towards the strong coupling regime where AdS/CFT calculations predictγ→1.

Non-linear equation in the re-summed next-to-leading order of perturbative QCD: the leading twist approximation

In this paper, we use the re-summation procedure, suggested in Ducloué et al. (JHEP 1904:081, 2019), Salam (JHEP 9807:019 1998), Ciafaloni et al. (Phys Rev D 60:1140361999) and Ciafaloni et al. (Phys Rev D 68:114003, 2003), to fix the BFKL kernel in the NLO. However, we suggest a different way to introduce the non-linear corrections in the saturation region, which is based on the leading twist non-linear equation. In the kinematic region: tau ,equiv ,r^2 Q^2_s(Y),le ,1, where r denotes the size of the dipole, Y its rapidity and Q_s the saturation scale, we found that the re-summation contributes mostly to the leading twist of the BFKL equation. Assuming that the scattering amplitude is small, we suggest using the linear evolution equation in this region. For tau ,>,1 we are dealing with the re-summation of (bar{alpha }_S,ln tau )^n and other corrections in NLO approximation for the leading twist. We find the BFKL kernel in this kinematic region and write the non-linear equation, which we solve analytically. We believe the new equation could be a basis for a consistent phenomenology based on the CGC approach.

Two-parton scattering amplitudes in the Regge limit to high loop orders

We study two-to-two parton scattering amplitudes in the high-energy limit of perturbative QCD by iteratively solving the BFKL equation. This allows us to predict the imaginary part of the amplitude to leading-logarithmic order for arbitrary t-channel colour exchange. The corrections we compute correspond to ladder diagrams with any number of rungs formed between two Reggeized gluons. Our approach exploits a separation of the two-Reggeon wavefunction, performed directly in momentum space, between a soft region and a generic (hard) region. The former component of the wavefunction leads to infrared divergences in the amplitude and is therefore computed in dimensional regularization; the latter is computed directly in two transverse dimensions and is expressed in terms of single-valued harmonic polylogarithms of uniform weight. By combining the two we determine exactly both infrared-divergent and finite contributions to the two-to-two scattering amplitude order-by-order in perturbation theory. We study the result numerically to 13 loops and find that finite corrections to the amplitude have a finite radius of convergence which depends on the colour representation of the t-channel exchange.

BFKL Equation: Status and Problems

The BFKL equation was derived for description of high energy behaviour of scattering amplitudes in non-Abelian gauge theories. Now it is widely known and used in quantum chromodynamics in the leading and next-to-leading logarithmic approximations. Its derivation is based on the pole Regge form of amplitudes with gluon exchanges in cross channels. In higher approximations this form is violated by contributions of Regge cuts, which complicates the derivation of the equation in the BFKL approach.

Analytical solution to DGLAP integro-differential equation via complex maps in domains of contour integrals

A simple model for QCD dynamics in which the DGLAP integro-differential equation may be solved analytically has been considered in our previous papers arXiv:1611.08787 [hep-ph] and arXiv:1906.07924 [hep-ph]. When such a model contains only one term in the splitting function of the dominant parton distribution, then Bessel function appears to be the solution to this simplified DGLAP equation. To our knowledge, this model with only one term in the splitting function for the first time has been proposed by Blümlein in arXiv:hep-ph/9506403. In arXiv:1906.07924 [hep-ph] we have shown that a dual integro-differential equation obtained from the DGLAP equation by a complex map in the plane of the Mellin moment in this model may be considered as the BFKL equation. Then, in arXiv:1906.07924 we have applied a complex diffeomorphism to obtain a standard integral from Gradshteyn and Ryzhik tables starting from the contour integral for parton distribution functions that is usually taken by calculus of residues. This standard integral from these tables appears to be the Laplace transformation of Jacobian for this complex diffeomorphism. Here we write up all the formulae behind this trick in detail and find out certain important points for further development of this strategy. We verify that the inverse Laplace transformation of the Laplace image of the Bessel function may be represented in a form of Barnes contour integral.

A semi-infinite matrix analysis of the BFKL equation

The forward BFKL equation is discretised in virtuality space and it is shown that the diffusion into infrared and ultraviolet momenta can be understood in terms of a semi-infinite matrix. The square truncation of this matrix can be exponentiated leading to asymptotic eigenstates sharing many features with the BFKL gluon Green’s function in the limit of large matrix size. This truncation is closely related to a representation of the XXX Heisenberg spin = - frac{1}{2} chain with SL(2) invariance where the Hamiltonian acts on a symmetric double copy of the harmonic oscillator. A simple modification of the BFKL matrix suppressing the infrared modes generates evolution more compatible with the Froissart bound.

Soft fragmentation on the celestial sphere

We develop two approaches to the problem of soft fragmentation of hadrons in a gauge theory for high energy processes. The first approach directly adapts the standard resummation of the parton distribution function’s anomalous dimension (that of twist-two local operators) in the forward scattering regime, using kT -factorization and BFKL theory, to the case of the fragmentation function by exploiting the mapping between the dynamics of eikonal lines on transverse-plane to the celestial-sphere. Critically, to correctly resum the anomalous dimension of the fragmentation function under this mapping, one must pay careful attention to the role of regularization, despite the manifest collinear or infra- red finiteness of the BFKL equation. The anomalous dependence on energy in the celestial case, arising due to the mismatch of dimensionality between positions and angles, drives the differences between the space-like and time-like anomalous dimension of parton densities, even in a conformal theory. The second approach adapts an angular-ordered evolution equation, but working in 4 − 2ϵ dimensions at all angles. The two approaches are united by demanding that the anomalous dimension in 4 − 2ϵ dimensions for the parton distribution function determines the kernel for the angular-ordered evolution to all orders.

Taming of preasymptotic small x evolution within resummation framework

It is well understood that the leading logarithmic approximation for the amplitudes of high energy processes is insufficient and that the next-to-leading logarithmic effects are very large and lead to instability of the solution. The resummation at low x, which includes kinematical constraints and other corrections leads to stable result. Using previously established resummation procedure we study in detail the preasymptotic effects which occur in the solution to the resummed BFKL equation when the energy is not very large. We find that in addition to the well known reduction of the intercept, which governs the energy dependence of the gluon Green’s function, resummation leads to the delay of the onset of its small x growth. Moreover the gluon Green’s function develops a dip or a plateau in wide range of rapidities, which increases for large scales. The preasymptotic region in the gluon Green’s function extends to about 8 units in rapidity for the transverse scales of the order of 30–100 GeV. To visualize the expected behavior of physical processes with two equal hard scales we calculate the cross section of the process gamma ^{*}+gamma ^{*}rightarrow X to be probed at future very high-energy electron-positron colliders. We find that at gamma ^*gamma ^* energies below 100 ; mathrm{{GeV}} the BFKL Pomeron leads to smaller value of the cross section than the Born approximation, and only starts to dominate at energies about 100 ; mathrm{{GeV}}. This pattern is significantly different from the one which we find using LLx approximation. We also analyze the transverse momentum contributions to the cross section for different virtualities of the photons and find that the dominant contributions to the integral over the transverse momenta comes from lower values than the the external scales in the process under consideration.

BFKL equation in the next-to-leading order: solution at large impact parameters

In this paper, we show (1) that the NLO corrections do not change the power-like decrease of the scattering amplitude at large impact parameter (b^2 ,>,r^2 exp ( 2{bar{alpha }}_Seta (1 + 4 {bar{alpha }}_S) ), where r denotes the size of scattering dipole and eta = ln (1/x_{Bj} ) for DIS), and, therefore, they do not resolve the inconsistency with unitarity; and (2) they lead to an oscillating behaviour of the scattering amplitude at large b, in direct contradiction with the unitarity constraints. However, from the more practical point of view, the NLO estimates give a faster decrease of the scattering amplitude as a function of b, and could be very useful for description of the experimental data. It turns out, that in a limited range of b, the NLO corrections generates the fast decrease of the scattering amplitude with b, which can be parameterized as N, propto ,exp ( -,mu ,b ) with mu , propto ,1/r in accord with the numerical estimates in Cepila et al. (Phys Rev D 99(5):051502, https://doi.org/10.1103/PhysRevD.99.051502, arXiv:1812.02548 [hep-ph], 2019).

Regge Cuts and NNLLA BFKL

In the leading and next-to-leading logarithmic approximations, QCD amplitudes with gluon quantum numbers in cross-channels and negative signature have the pole form corresponding to a reggeized gluon. The famous BFKL equation was derived using this form. In the next-to-next-to-leading approximation (NNLLA), the pole form is violated by contributions of Regge cuts. We discuss these contributions and their impact on the derivation of the BFKL equation in the NNLLA.

BFKL Equation and Regge Cuts

The BFKL equation is based on the gluon yReggeization. In the leading and next-to-leading logarithmic approximations it is derived using the pole Regge form of QCD amplitudes with gluon quantum numbers in cross-channels and negative signature. This form is violated in the next-to-next-to-leading approximation. In two and three loops the observed violation can be explained by the presence of the three-Reggeon cut. Contributions of this cut to elastic scattering amplitudes up to four loops is discussed.

On the different forms of the kinematical constraint in BFKL

We perform a detailed analysis of the different forms of the kinematical constraint imposed on the low x evolution that appear in the literature. We find that all of them generate the same leading anti-collinear poles in Mellin space which agree with BFKL up to NLL order and up to NNLL in N=4 sYM. The coefficients of subleading poles vanish up to NNLL order for all constraints and we prove that this property should be satisfied to all orders. We then demonstrate that the kinematical constraints differ at further subleading orders of poles. We quantify the differences between the different forms of the constraints by performing numerical analysis both in Mellin space and in momentum space. It can be shown that in all three cases BFKL equation can be recast into the differential form, with the argument of the longitudinal variable shifted by the combination of the transverse coordinates.

Studies on gluon evolution and geometrical scaling in kinematic constrained unitarized BFKL equation: application to high-precision HERA DIS data

We suggest a modified form of a unitarized BFKL equation imposing the so-called kinematic constraint on the gluon evolution in multi-Regge kinematics. The underlying nonlinear effects on the gluon evolution are investigated by solving the unitarized BFKL equation analytically. We obtain an equation of the critical boundary between dilute and dense partonic system, following a new differential geometric approach and sketch a phenomenological insight on geometrical scaling. Later we illustrate the phenomenological implication of our solution for unintegrated gluon distribution f(x,k_T^2) towards exploring high precision HERA DIS data by theoretical prediction of proton structure functions (F_2 and F_L) as well as double differential reduced cross section (sigma _r). The validity of our theory in the low Q^2 transition region is established by studying virtual photon–proton cross section in light of HERA data.

Reflection Identities of Harmonic Sums of Weight Four

In attempt to find a proper space of function expressing the eigenvalue of the color-singlet BFKL equation in N = 4 SYM, we consider an analytic continuation of harmonic sums from positive even integer values of the argument to the complex plane. The resulting meromorphic functions have pole singularities at negative integers. We derive the reflection identities for harmonic sums at weight four decomposing a product of two harmonic sums with mixed pole structure into a linear combination of terms each having a pole at either negative or non-negative values of the argument. The pole decomposition demonstrates how the product of two simpler harmonic sums can build more complicated harmonic sums at higher weight. We list a minimal irreducible set of bilinear reflection identities at weight four, which represents the main result of the paper. We also discuss how other trilinear and quadlinear reflection identities can be constructed from our result with the use of well known quasi-shuffle relations for harmonic sums.

Reggeon cuts in QCD amplitudes with negative signature

Reggeon cuts in QCD amplitudes with negative signature are discussed. These cuts appear in the next-to-next-to-leading logarithmic approximation and greatly complicate the derivation of the BFKL equation. Feynman diagrams responsible for appearance of these cuts are indicated and the cut contributions are calculated in the two- and three-loop approximations.