non-Abelian Exponentiation Research Articles

NNLO soft function for threshold single inclusive jet production

We present a computation of the global soft function for single-inclusive hadronic jet production at next-to-next-to-leading order (NNLO) in the strong coupling constant involving four light-like Wilson lines. Our soft function belongs to SCET$_{\text I}$ observables obeying the non-Abelian exponentiation theorem. We provide the calculation of the soft function involving four collinear light-cone directions for all $2\rightarrow2$ processes extending the previous result. We perform the threshold resummation for these processes at the next-to-next-to-leading logarithmic accuracy and present the corresponding numerical results.

Generic dijet soft functions at two-loop order: uncorrelated emissions

We extend our algorithm for automating the calculation of two-loop dijet soft functions to observables that do not obey the non-Abelian exponentiation theorem, i.e. to those that require an independent calculation of the uncorrelated-emission contribution. As the singularity structure of uncorrelated double emissions differs substantially from the one for correlated emissions, we introduce a novel phase-space parametrisation that isolates the corresponding divergences. The resulting integrals are implemented in SoftSERVE 1.0, which we release alongside of this work, and which we supplement by a regulator that is consistent with the rapidity renormalisation group framework. Using our automated setup, we confirm existing results for various jet-veto observables and provide a novel prediction for the soft-drop jet-grooming algorithm.

Manifestly soft gauge invariant formulation of vNRQCD

Homogeneous power counting in Non-Relativistic QCD (NRQCD) implies the simultaneous existence of both soft and ultrasoft gluons. In the velocity renormalization group (vNRQCD) formalism we show that operators involving soft fields interacting with heavy potential quarks can be put in a manifestly gauge invariant form by utilizing gluon and quark building blocks which contain Wilson lines, and are analogous to those used in the soft collinear effective theory. This leads to several simplifications, in particular significantly reducing the size of the operator basis, which stream-lines matching and anomalous dimension calculations at subleading order in the velocity expansion. Also, soft ghosts no longer couple via potential like interactions to the heavy quark fields, and hence do not appear until two loops. Furthermore, the color structures that appear at each order in αs in the static potential are only those which should arise according to non-Abelian exponentiation. We also discuss how zero-bin subtractions clarify the role of iε terms in the heavy quark propagator poles carrying soft momenta. Even though the choice of the direction of the soft Wilson lines in our building blocks does not matter, there is still a limited set of consistent choices, and the predicted form of the eikonal poles does influence results at two-loops and beyond.

Rapidity renormalized TMD soft and beam functions at two loops

We compute the transverse momentum dependent (TMD) soft function for the production of a color-neutral final state at the LHC within the rapidity renormalization group (RRG) framework to next-to-next-to-leading order (NNLO). We use this result to extract the universal renormalized TMD beam functions (aka TMDPDFs) in the same scheme and at the same order from known results in another scheme. We derive recurrence relations for the logarithmic structure of the soft and beam functions, which we use to cross check our calculation. We also explicitly confirm the non-Abelian exponentiation of the TMD soft function in the RRG framework at two loops. Our results provide the ingredients for resummed predictions of pT-differential cross sections at NNLL' in the RRG formalism. The RRG provides a systematic framework to resum large (rapidity) logarithms through (R)RG evolution and to assess the associated perturbative uncertainties.

Position-space cuts for Wilson line correlators

We further develop the formalism for taking position-space cuts of eikonal diagrams introduced in [Phys.Rev.Lett. 114 (2015), no. 18 181602, arXiv:1410.5681]. These cuts are applied directly to the position-space representation of any such diagram and compute its discontinuity to the leading order in the dimensional regulator. We provide algorithms for computing the position-space cuts and apply them to several two- and three-loop eikonal diagrams, finding agreement with results previously obtained in the literature. We discuss a non-trivial interplay between the cutting prescription and non-Abelian exponentiation. We furthermore discuss the relation of the imaginary part of the cusp anomalous dimension to the static interquark potential.

Generating function for web diagrams

We present the description of the exponentiated diagrams in terms of generating function within the universal diagrammatic technique. In particular, we show the exponentiation of the gauge theory amplitudes involving products of an arbitrary number of Wilson lines of arbitrary shapes, which generalizes the concept of web diagrams. The presented method gives a new viewpoint on the web diagrams and proves the non-Abelian exponentiation theorem.

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.

Webs and posets

The non-Abelian exponentiation theorem has recently been generalised to correlators of multiple Wilson line operators. The perturbative expansions of these correlators exponentiate in terms of sets of diagrams called webs, which together give rise to colour factors corresponding to connected graphs. The colour and kinematic degrees of freedom of individual diagrams in a web are entangled by mixing matrices of purely combinatorial origin. In this paper we relate the combinatorial study of these matrices to properties of partially ordered sets (posets), and hence obtain explicit solutions for certain families of web-mixing matrix, at arbitrary order in perturbation theory. We also provide a general expression for the rank of a general class of mixing matrices, which governs the number of independent colour factors arising from such webs. Finally, we use the poset language to examine a previously conjectured sum rule for the columns of web-mixing matrices which governs the cancellation of the leading subdivergences between diagrams in the web. Our results, when combined with parallel developments in the evaluation of kinematic integrals, offer new insights into the all-order structure of infrared singularities in non-Abelian gauge theories.

ABJM amplitudes and WL at finite N

We evaluate ABJM observables at two loops, for any value of the rank N of the gauge group. We compute the color subleading contributions to the four-point scattering amplitude in ABJM at two loops. Contrary to the four dimensional case, IR divergent N-subleading contributions are proportional to leading poles in the regularization parameter. We then exploit the non-planar calculation for the amplitude to derive an expression for the two-loop Sudakov form factor at any N. In the planar limit the result coincides with the one recently obtained in literature by using Feynman diagrams and unitarity. Finally, we analyze the subleading contributions to the light-like four-cusps Wilson loop and interpret the result in terms of the non-abelian exponentiation theorem. All these perturbative results satisfy the uniform transcendentality principle, hinting at its validity in ABJM beyond the planar limit.

The non-Abelian exponentiation theorem for multiple Wilson lines

We study the structure of soft gluon corrections to multi-leg scattering amplitudes in a non-Abelian gauge theory by analysing the corresponding product of semi-infinite Wilson lines. We prove that diagrams exponentiate such that the colour factors in the exponent are fully connected. This completes the generalisation of the non-Abelian exponentiation theorem, previously proven in the case of a Wilson loop, to the case of multiple Wilson lines in arbitrary representations of the colour group. Our proof is based on the replica trick in conjunction with a new formalism where multiple emissions from a Wilson line are described by effective vertices, each having a connected colour factor. The exponent consists of connected graphs made out of these vertices. We show that this readily provides a general colour basis for webs. We further discuss the kinematic combinations that accompany each connected colour factor, and explicitly catalogue all three-loop examples, as necessary for a direct computation of the soft anomalous dimension at this order.

The NNLO soft function for the pair invariant mass distribution of boosted top quarks

At high values of the pair invariant mass the differential cross section for top-quark pair production at hadron colliders factorizes into soft, hard, and fragmentation functions. In this paper we calculate the next-to-next-to-leading-order (NNLO) corrections to the soft function appearing in this factorization formula, thus providing the final piece needed to evaluate at NNLO the differential cross section in the virtual plus soft approximation in the large invariant-mass limit. Technically, this amounts to evaluating the vacuum expectation value of a soft Wilson loop operator built out of light-like Wilson lines for each of the four partons participating in the hard scattering process, with a certain constraint on the total energy of the soft radiation. Our result turns out to be surprisingly simple, because in the sum of all graphs the three and four parton contributions multiply color structures whose coefficients are governed by the non-abelian exponentiation theorem.

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.

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.

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.

Two-loop jet function and jet mass for top quarks

We compute the two-loop heavy-quark jet function in the heavy-quark limit. This is one of the key ingredients in next-to-next-to-leading order and next-to-next-to-leading-log order computations of the invariant mass distribution of top-jets at a future e{sup +}e{sup -} collider. The shape of the top-invariant mass distribution is affected by large logs which we compute at next-to-next-to-leading-log order. Exploiting the non-Abelian exponentiation theorem, a definition of the top jet mass is given which is transitive and whose renormalization group evolution is determined by the cusp-anomalous dimension to all orders in perturbation theory. Relations of the jet mass to the pole, MS, and 1S masses are presented at two-loop order.

Refining threshold resummations

We describe some recent refinements of the techniques of threshold resummation, with emphasis on the usefulness of dimensional regularization when applied to nonabelian exponentiation. Threshold resummation is now under theoretical control for DIS and electroweak annihilation cross sections all the way to the fourth tower of logarithms and up to corrections suppressed by powers of the threshold variable.

Electron-positron annihilation and non-abelian eikonal exponentiation

Using the non-abelian exponentiation theorem with eikonalized quarks, we have calculated to fourth order in perturbative QCD the exponent which contains the contribution of the annihilation process into a quark-antiquark pair, including soft gluon production up to a maximum energy Δ. We suggest a generalization of these results to all orders, which contains all leading as well as non-leading logarithms of m/ S , where m denotes the quark mass and √ S represents the center of mass energy of the electron-positron pair. We show in particular that the dependence of the exponent on these parameters is governed, in leading order, by the effective coupling g(Δm/μ√S) as given by the renormalization group.