arXiv subjects

Einan Gardi

Publications and source records attributed to Einan Gardi.

At least 73 records · Page 4Linked to original sources

arXiv2011

On soft singularities at three loops and beyond

We report on further progress in understanding soft singularities of massless gauge theory scattering amplitudes. Recently, a set of equations was derived based on Sudakov factorization, constraining the soft anomalous dimension matrix of multi-leg scattering amplitudes to any loop order, and relating it to the cusp anomalous dimension. The minimal solution to these equations was shown to be a sum over color dipoles. Here we explore potential contributions to the soft anomalous dimension that go beyond the sum-over-dipoles formula. Such contributions are constrained by factorization and invariance under rescaling of parton momenta to be functions of conformally invariant cross ratios. Therefore, they must correlate the color and kinematic degrees of freedom of at least four hard partons, corresponding to gluon webs that connect four eikonal lines, which first appear at three loops. We analyze potential contributions, combining all available constraints, including Bose symmetry, the expected degree of transcendentality, and the singularity structure in the limit where two hard partons become collinear. We find that if the kinematic dependence is solely through products of logarithms of cross ratios, then at three loops there is a unique function that is consistent with all available constraints. If polylogarithms are allowed to appear as well, then at least two additional structures are consistent with the available constraints.

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arXiv2011

On the renormalization of multiparton webs

We consider the recently developed diagrammatic approach to soft-gluon exponentiation in multiparton scattering amplitudes, where the exponent is written as a sum of webs - closed sets of diagrams whose colour and kinematic parts are entangled via mixing matrices. A complementary approach to exponentiation is based on the multiplicative renormalizability of intersecting Wilson lines, and their subsequent finite anomalous dimension. Relating this framework to that of webs, we derive renormalization constraints expressing all multiple poles of any given web in terms of lower-order webs. We examine these constraints explicitly up to four loops, and find that they are realised through the action of the web mixing matrices in conjunction with the fact that multiple pole terms in each diagram reduce to sums of products of lower-loop integrals. Relevant singularities of multi-eikonal amplitudes up to three loops are calculated in dimensional regularization using an exponential infrared regulator. Finally, we formulate a new conjecture for web mixing matrices, involving a weighted sum over column entries. Our results form an important step in understanding non-Abelian exponentiation in multiparton amplitudes, and pave the way for higher-loop computations of the soft anomalous dimension.

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arXiv2011

General properties of multiparton webs: proofs from combinatorics

Recently, the diagrammatic description of soft-gluon exponentiation in scattering amplitudes has been generalized to the multiparton case. It was shown that the exponent of Wilson-line correlators is a sum of webs, where each web is formed through mixing between the kinematic factors and colour factors of a closed set of diagrams which are mutually related by permuting the gluon attachments to the Wilson lines. In this paper we use replica trick methods, as well as results from enumerative combinatorics, to prove that web mixing matrices are always: (a) idempotent, thus acting as projection operators; and (b) have zero sum rows: the elements in each row in these matrices sum up to zero, thus removing components that are symmetric under permutation of gluon attachments. Furthermore, in webs containing both planar and non-planar diagrams we show that the zero sum property holds separately for these two sets. The properties we establish here are completely general and form an important step in elucidating the structure of exponentiation in non-Abelian gauge theories.

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arXiv2010

Webs in multiparton scattering using the replica trick

Soft gluon exponentiation in non-abelian gauge theories can be described in terms of webs. So far this description has been restricted to amplitudes with two hard partons, where webs were defined as the colour-connected subset of diagrams. Here we generalise the concept of webs to the multi-leg case, where the hard interaction involves non-trivial colour flow. Using the replica trick from statistical physics we solve the combinatorial problem of non-abelian exponentiation to all orders. In particular, we derive an algorithm for computing the colour factor associated with any given diagram in the exponent. The emerging result is exponentiation of a sum of webs, where each web is a linear combination of a subset of diagrams that are mutually related by permuting the eikonal gluon attachments to each hard parton. These linear combinations are responsible for partial cancellation of subdivergences, conforming with the renormalization of a multi-leg eikonal vertex. We also discuss the generalisation of exponentiation properties to beyond the eikonal approximation.

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arXiv2010

All-order results for infrared and collinear singularities in massless gauge theories

We review recent results concerning the all-order structure of infrared and collinear divergences in massless gauge theory amplitudes. While the exponentiation of these divergences for nonabelian gauge theories has been understood for a long time, in the past couple of years we have begun to unravel the all-order structure of the anomalous dimensions that build up the perturbative exponent. In the large-Nc limit, all infrared and collinear divergences are determined by just three functions; one of them, the cusp anomalous dimension, plays a key role also for non-planar contributions. Indeed, all infrared and collinear divergences of massless gauge theory amplitudes with any number of hard partons may be captured by a surprisingly simple expression constructed as a sum over color dipoles. Potential corrections to this expression, correlating four or more hard partons at three loops or beyond, are tightly constrained and are currently under study.

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arXiv2009

Infrared singularities in QCD amplitudes

We review recent progress in determining the infrared singularity structure of on-shell scattering amplitudes in massless gauge theories. We present a simple ansatz where soft singularities of any scattering amplitude of massless partons, to any loop order, are written as a sum over colour dipoles, governed by the cusp anomalous dimension. We explain how this formula was obtained, as the simplest solution to a newly-derived set of equations constraining the singularity structure to all orders. We emphasize the physical ideas underlying this derivation: the factorization of soft and collinear modes, the special properties of soft gluon interactions, and the notion of the cusp anomaly. Finally, we briefly discuss potential multi-loop contributions going beyond the sum-over-dipoles formula, which cannot be excluded at present.

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arXiv2009

Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes

We study the factorization of soft and collinear singularities in dimensionally-regularized fixed-angle scattering amplitudes in massless gauge theories. Our factorization is based on replacing the hard massless partons by light-like Wilson lines, and defining gauge-invariant jet and soft functions in dimensional regularization. In this scheme the factorized amplitude admits a powerful symmetry: it is invariant under rescaling of individual Wilson-line velocities. This symmetry is broken by cusp singularities in both the soft and the eikonal jet functions. We show that the cancellation of these cusp anomalies in any multi-leg amplitude imposes all-order constraints on the kinematic dependence of the corresponding soft anomalous dimension, relating it to the cusp anomalous dimension. For amplitudes with two or three hard partons the solution is unique: the constraints fully determine the kinematic dependence of the soft function. For amplitudes with four or more hard partons we present a minimal solution where the soft anomalous dimension is a sum over colour dipoles, multiplied by the cusp anomalous dimension. In this case additional contributions to the soft anomalous dimension at three loops or beyond are not excluded, but they are constrained to be functions of conformal cross ratios of kinematic variables.

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arXiv2008

On the determination of |V_{ub}| from inclusive semileptonic B decays

Precision tests of the CKM mechanism and searches for new physics in the flavour sector require dedicated QCD calculations of decay widths and spectra. Significant progress has been achieved in recent years in computing inclusive B decay spectra into light energetic partons. I briefly review different theoretical approaches to this problem focusing on the determination of |V_{ub}| from inclusive semileptonic decays and show that this determination is robust. The largest uncertainty is associated with the value of the b quark mass. Finally I present new numerical results in the DGE resummation-based approach, now including O(beta_0 alpha_s^2) corrections. The results are presented for all relevant experimental cuts, from which a preliminary average is derived: |V_{ub}| = (4.30 \pm 0.16 (exp) ^{+0.09}_{-0.13} (th) ^{+0.39}_{-0.34} (m_b)) 10^{-3}, where the PDG value of the b quark mass, m_b^{\MSbar} = 4.20 \pm 0.07 GeV, is assumed.

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arXiv2008

Distribution Amplitudes of the Lambda_b Baryon in QCD

The QCD description of exclusive decays of the $Λ_b$ baryon involves hadronic matrix elements of non-local light ray operators, the light-cone distribution amplitudes. We introduce the complete set of three--quark distribution amplitudes and calculate the renormalization scale dependence for the leading twist. At leading order in the strong coupling the evolution is driven by pairwise two--quark interactions: heavy--light involving Sudakov logarithms as in the B-meson case, and light--light as in light mesons. We solve the evolution equation and show that its main effect is to generate a radiative tail extending to high energies. Finally, we present simple models for the distribution amplitudes based on QCD sum rules, and study the effect of the evolution on these models.

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arXiv2007

A dispersive approach to Sudakov resummation

We present a general all-order formulation of Sudakov resummation in QCD in terms of dispersion integrals. We show that the Sudakov exponent can be written as a dispersion integral over spectral density functions, weighted by characteristic functions that encode information on power corrections. The characteristic functions are defined and computed analytically in the large-beta_0 limit. The spectral density functions encapsulate the non-Abelian nature of the interaction. They are defined by the time-like discontinuity of specific effective charges (couplings) that are directly related to the familiar Sudakov anomalous dimensions and can be computed order-by-order in perturbation theory. The dispersive approach provides a realization of Dressed Gluon Exponentiation, where Sudakov resummation is enhanced by an internal resummation of running-coupling corrections. We establish all-order relations between the scheme-invariant Borel formulation and the dispersive one, and address the difference in the treatment of power corrections. We find that in the context of Sudakov resummation the infrared-finite-coupling hypothesis is of special interest because the relevant coupling can be uniquely identified to any order, and may have an infrared fixed point already at the perturbative level. We prove that this infrared limit is universal: it is determined by the cusp anomalous dimension. To illustrate the formalism we discuss a few examples including B-meson decay spectra, deep inelastic structure functions and Drell-Yan or Higgs production.

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arXiv2007

Inclusive B decays from resummed perturbation theory

I review the recent progress in computing inclusive B decay widths and spectra and its implications on the interpretation of measurements from the B factories. I discuss the inclusive charmless semileptonic decay, B -> X_u l nu, which provides the most robust determination of the CKM parameter |V_{ub}|, and the rare radiative decay, B -> X_s gamma, which constrains flavor violation beyond the Standard Model. I demonstrate that precise predictions for the experimentally-relevant branching fractions can be derived from resummed perturbation theory and explain the way in which the resummation further provides guidance in parametrizing non-perturbative Fermi-motion effects. Finally I address the comparison between theory and data and discuss future prospects.

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arXiv2006

Radiative B decay spectrum: DGE at NNLO

We compute the differential B -> X_s gamma decay width in the Standard Model as a function of the photon energy using Dressed Gluon Exponentiation (DGE). The resummed spectrum is matched with the fixed-order expansion, making use of the next-to-next-to-leading order (NNLO) results for the matrix element of the magnetic dipole interaction O_7 and NLO ones for other operators in the effective Weak Hamiltonian. We develop a new technique to implement constraints on the analytic structure of the Sudakov factor in moment space. This improves the behavior of the resummed spectrum away from the Sudakov region. We also derive an analytic expression for the Borel transform of the perturbative series for the O_7 spectrum in the large-beta_0 limit. Using this example we demonstrate that exponentiation in moment space is necessary for the calculation of the spectrum for E_gamma > 2 GeV. Finally, we investigate numerically the relation between renormalons, power corrections and support properties. We present predictions for the branching fraction and the first few spectral moments as a function of a cut E_gamma>E_0 and estimate the theoretical uncertainty.

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arXiv2006

Running-coupling effects in the triple-differential charmless semileptonic decay width

We compute the fully-differential B -> X_u l nu decay width to all orders in perturbation theory in the large-beta0 limit. Each of the five structure functions that build the hadronic tensor is expressed as a Borel integral, summing up O(CF beta0^{n-1} alpha_s^n) corrections for any n. We derive analytic expressions for the Borel transforms of both real and virtual diagrams with a single dressed gluon, and perform an all-order infrared subtraction, where the Borel parameter serves also as an infrared regulator. Expanding the result we recover the known triple-differential NLO coefficient, and obtain an explicit expression for the O(CF beta0 alpha_s^2) triple-differential NNLO correction. This result can be used to improve the determination of |V_{ub}| from inclusive B -> X_u l nu measurements at the B factories with a variety of kinematic cuts.

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arXiv2006

Running coupling and power corrections in nonlinear evolution at the high--energy limit

A main feature of high-energy scattering in QCD is saturation in the number density of gluons. This phenomenon is described by non-linear evolution equations, JIMWLK and BK, which have been derived at leading logarithmic accuracy. In this paper we generalize this framework to include running coupling corrections to the evolution kernel. We develop a dispersive representation of the dressed gluon propagator in the background of Weiszacker Williams fields and use it to compute O(beta_0^{n-1} alpha_s^n) corrections to the kernel to all orders in perturbation theory. The resummed kernels present infrared-renormalon ambiguities, which are indicative of the form and importance of non-perturbative power corrections. We investigate numerically the effect of the newly computed perturbative corrections as well as the power corrections on the evolution and find that at present energies they are both significant.

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arXiv2006

Inclusive distributions near kinematic thresholds

The main challenge in computing inclusive cross sections and decay spectra in QCD is posed by kinematic thresholds. The threshold region is characterized by stringent phase-space constraints that are reflected in large perturbative corrections due to soft and collinear radiation as well as large non-perturbative effects. Major progress in addressing this problem was made in recent years by Dressed Gluon Exponentiation (DGE), a formalism that combines Sudakov and renormalon resummation in moment space. DGE has proven effective in extending the range of applicability of perturbation theory well into the threshold region and in identifying the relevant non-perturbative corrections. Here we review the method from a general perspective using examples from deep inelastic structure functions, event-shape distributions, heavy-quark fragmentation and inclusive decay spectra. A special discussion is devoted to the applications of DGE to radiative and semileptonic B decays that have proven valuable for the interpretation of data from the B factories.

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arXiv2006

Progress in computing inclusive B decay spectra

We review the progress in the QCD calculation of inclusive decay spectra. It was recently shown that the inherent infrared finiteness of inclusive spectra extends beyond the level of logarithms. Dressed Gluon Exponentiation makes practical use of this property by computing the Sudakov exponent as a Borel sum. Based on renormalon analysis, infrared sensitivity in the exponent is reflected in power corrections that are inversely proportional to the third power of the mass. Therefore, the parametric enhancement of non-perturbative corrections near the phase--space boundary is effective only in a small region. Consequently, the on-shell decay spectrum provides a good approximation to the meson decay. In particular, it facilitates a precise determination of V_ub from present measurements of inclusive charmless semileptonic widths without involving a non-perturbative ``shape function''.

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arXiv2005

Inclusive spectra in charmless semileptonic B decays by Dressed Gluon Exponentiation

The triple differential spectrum in \bar{B} -> X_u l \barν is computed by Dressed Gluon Exponentiation (DGE). In this framework the on-shell calculation, converted into hadronic variables, can be directly used as an approximation to the meson decay spectrum, without involving a leading-power non-perturbative function. Sudakov resummation for the fully differential \bar{B} -> X_u l \barν width is formulated in moment space, where moments are defined using the ratio between the lightcone momentum components of the partonic jet p^+/p^- and the hard scale is p^-. In these variables the correspondence with the \bar{B} -> X_s γcase is transparent. The Sudakov exponent is known to next-to-next-to-leading logarithmic accuracy. Further constraints are put on its Borel sum using the cancellation of the leading renormalon ambiguity and the absence of the next-to-leading one, which was proven in the large-beta_0 limit and assumed here to be general. Based on the resummed spectrum, matched to the fully differential NLO result, we calculate the event fraction associated with experimental cuts on the hadronic mass (or the small lightcone component) as well as on the lepton energy. Finally, we extract |V_ub| from recent measurements by Belle and analyze the theoretical uncertainty.

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arXiv2005

Taming the B --> X(s) gamma spectrum by Dressed Gluon Exponentiation

We show that the B \to X(s) gamma photon energy (E_gamma) spectrum can be reliably computed by resummed perturbation theory. Our calculation is based on Dressed Gluon Exponentiation (DGE) incorporating Sudakov and renormalon resummation. It is shown that the resummed spectrum does not have the perturbative support properties: it smoothly extends to the non-perturbative region E_gamma > m/2, where m is the quark pole mass, and tends to zero near the physical endpoint. The calculation of the Sudakov factor, which determines the shape of the spectrum in the peak region, as well as that of the pole mass, which sets the energy scale, are performed using Principal-Value Borel summation. By using the same prescription in both, the cancellation of the leading renormalon ambiguity is respected. Furthermore, in computing the Sudakov exponent we go beyond the formal next-to-next-to-leading logarithmic accuracy using the large-order asymptotic behavior of the series, which is accurately determined from the relation with the pole mass. Upon matching the resummed result with the next-to-leading order expression we compute the spectrum, obtain its moments as a function of a minimum photon energy cut, analyze sources of uncertainty and show that our predictions are in good agreement with Belle data.

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