Factorization Properties of Soft Graviton Amplitudes

We apply recently developed path integral resummation methods to perturbative quantum gravity. In particular, we provide supporting evidence that eikonal graviton amplitudes factorize into hard and soft parts, and confirm a recent hypothesis that soft gravitons are modelled by vacuum expectation values of products of certain Wilson line operators, which differ for massless and massive particles. We also investigate terms which break this factorization, and find that they are subleading with respect to the eikonal amplitude. The results may help in understanding the connections between gravity and gauge theories in more detail, as well as in studying gravitational radiation beyond the eikonal approximation.Comment: 35 pages, 5 figure

The singular behavior of massive QCD amplitudes

We discuss the structure of infrared singularities in on-shell QCD amplitudes with massive partons and present a general factorization formula in the limit of small parton masses. The factorization formula gives rise to an all-order exponentiation of both, the soft poles in dimensional regularization and the large collinear logarithms of the parton masses. Moreover, it provides a universal relation between any on-shell amplitude with massive external partons and its corresponding massless amplitude. For the form factor of a heavy quark we present explicit results including the fixed-order expansion up to three loops in the small mass limit. For general scattering processes we show how our constructive method applies to the computation of all singularities as well as the constant (mass-independent) terms of a generic massive n-parton QCD amplitude up to the next-to-next-to-leading order corrections.Comment: version to appear in JHEP (sec. 3 with expanded discussion and appendix with added results

The C parameter distribution in e+e- annihilation

We study perturbative and non-perturbative aspects of the distribution of the C parameter in e+e- annihilation using renormalon techniques. We perform an exact calculation of the characteristic function, corresponding to the C parameter differential cross section for a single off-shell gluon. We then concentrate on the two-jet region, derive the Borel representation of the Sudakov exponent in the large-beta_0 limit and compare the result to that of the thrust T. Analysing the exponent, we distinguish two ingredients: the jet function, depending on Q^2C, summarizing the effects of collinear radiation, and a function describing soft emission at large angles, with momenta of order QC. The former is the same as for the thrust upon scaling C by 1/6, whereas the latter is different. We verify that the rescaled C distribution coincides with that of 1-T to next-to-leading logarithmic accuracy, as predicted by Catani and Webber, and demonstrate that this relation breaks down beyond this order owing to soft radiation at large angles. The pattern of power corrections is also similar to that of the thrust: corrections appear as odd powers of Lambda/(QC). Based on the size of the renormalon ambiguity, however, the shape function is different: subleading power corrections for the C distribution appear to be significantly smaller than those for the thrust.Comment: 24 pages, Latex (using JHEP3.cls), 1 postscript figur

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.Comment: 60 pages, 15 figure

Rational approximations in Analytic QCD

We consider the ``modified Minimal Analytic'' (mMA) coupling that involves an infrared cut to the standard MA coupling. The mMA coupling is a Stieltjes function and, as a consequence, the paradiagonal Pade approximants converge to the coupling in the entire $Q^2$-plane except on the time-like semiaxis below the cut. The equivalence between the narrow width approximation of the discontinuity function of the coupling, on the one hand, and this Pade (rational) approximation of the coupling, on the other hand, is shown. We approximate the analytic analogs of the higher powers of mMA coupling by rational functions in such a way that the singularity region is respected by the approximants.Several comparisons, for real and complex arguments $Q^2$, between the exact and approximate expressions are made and the speed of convergence is discussed. Motivated by the success of these approximants, an improvement of the mMA coupling is suggested, and possible uses in the reproduction of experimental data are discussed.Comment: 12 pages,9 figures (6 double figures); figs.6-8 corrected due to a programming error; analysis extended to two IR cutoffs; Introduction rewritten; to appear in J.Phys.

Semi-numerical resummation of event shapes

For many event-shape observables, the most difficult part of a resummation in the Born limit is the analytical treatment of the observable's dependence on multiple emissions, which is required at single logarithmic accuracy. We present a general numerical method, suitable for a large class of event shapes, which allows the resummation specifically of these single logarithms. It is applied to the case of the thrust major and the oblateness, which have so far defied analytical resummation and to the two-jet rate in the Durham algorithm, for which only a subset of the single logs had up to now been calculated.Comment: 29 pages, 7 figures. Version 2 adds some clarifications, a reference, as well as corrections to the subleading fixed-order coefficients and to figures 4 and

On the Structure of Infrared Singularities of Gauge-Theory Amplitudes

A closed formula is obtained for the infrared singularities of dimensionally regularized, massless gauge-theory scattering amplitudes with an arbitrary number of legs and loops. It follows from an all-order conjecture for the anomalous-dimension matrix of n-jet operators in soft-collinear effective theory. We show that the form of this anomalous dimension is severely constrained by soft-collinear factorization, non-abelian exponentiation, and the behavior of amplitudes in collinear limits. Using a diagrammatic analysis, we demonstrate that these constraints imply that to three-loop order the anomalous dimension involves only two-parton correlations, with the possible exception of a single color structure multiplying a function of conformal cross ratios depending on the momenta of four external partons, which would have to vanish in all two-particle collinear limits. We argue that such a function does not appear at three-loop order, and that the same is true in higher orders. Our formula predicts Casimir scaling of the cusp anomalous dimension to all orders in perturbation theory, and we explicitly check that the constraints exclude the appearance of higher Casimir invariants at four loops. Using known results for the quark and gluon form factors, we derive the three-loop coefficients of the 1/epsilon^n pole terms (with n=1,...,6) for an arbitrary n-parton scattering amplitude in massless QCD. This generalizes Catani's two-loop formula proposed in 1998.Comment: 46 pages, 9 figures; v2: improved treatment of collinear limits, references added; v3: improved discussion of non-abelian exponentiation, references updated; v4: typo in eq. (17) fixed, references updated; v5: additional term in (17

Eikonal methods applied to gravitational scattering amplitudes

We apply factorization and eikonal methods from gauge theories to scattering amplitudes in gravity. We hypothesize that these amplitudes factor into an IR-divergent soft function and an IR-finite hard function, with the former given by the expectation value of a product of gravitational Wilson line operators. Using this approach, we show that the IR-divergent part of the n-graviton scattering amplitude is given by the exponential of the one-loop IR divergence, as originally discovered by Weinberg, with no additional subleading IR-divergent contributions in dimensional regularization.Comment: 16 pages, 3 figures; v2: title change and minor rewording (published version); v3: typos corrected in eqs.(3.2),(4.1

The infrared structure of e+ e- --> 3 jets at NNLO reloaded

This paper gives detailed information on the structure of the infrared singularities for the process e+ e- --> 3 jets at next-to-next-to-leading order in perturbation theory. Particular emphasis is put on singularities associated to soft gluons. The knowledge of the singularity structure allows the construction of appropriate subtraction terms, which in turn can be implemented into a numerical Monte Carlo program.Comment: 59 pages, additional comments added, version to be publishe

From Webs to Polylogarithms

We compute a class of diagrams contributing to the multi-leg soft anomalous dimension through three loops, by renormalizing a product of semi-infinite non-lightlike Wilson lines in dimensional regularization. Using non-Abelian exponentiation we directly compute contributions to the exponent in terms of webs. We develop a general strategy to compute webs with multiple gluon exchanges between Wilson lines in configuration space, and explore their analytic structure in terms of $\alpha_{ij}$, the exponential of the Minkowski cusp angle formed between the lines $i$ and $j$. We show that beyond the obvious inversion symmetry $\alpha_{ij}\to 1/\alpha_{ij}$, at the level of the symbol the result also admits a crossing symmetry $\alpha_{ij}\to -\alpha_{ij}$, relating spacelike and timelike kinematics, and hence argue that in this class of webs the symbol alphabet is restricted to $\alpha_{ij}$ and $1-\alpha_{ij}^2$. We carry out the calculation up to three gluons connecting four Wilson lines, finding that the contributions to the soft anomalous dimension are remarkably simple: they involve pure functions of uniform weight, which are written as a sum of products of polylogarithms, each depending on a single cusp angle. We conjecture that this type of factorization extends to all multiple-gluon-exchange contributions to the anomalous dimension.Comment: 64 pages, 8 figure