arXiv subjects

Brenda Penante

Publications and source records attributed to Brenda Penante.

At least 19 recordsLinked to original sources

arXiv2019

An analytic solution for the equal-mass banana graph

We present fully analytic results for all master integrals for the three-loop banana graph with four equal and non-zero masses. The results are remarkably simple and all integrals are expressed as linear combinations of iterated integrals of modular forms of uniform weight for the same congruence subgroup as for the two-loop equal-mass sunrise graph. We also show how to write the results in terms of elliptic polylogarithms evaluated at rational points.

hep-th↗

arXiv2019

Elliptic polylogarithms and Feynman parameter integrals

In this paper we study the calculation of multiloop Feynman integrals that cannot be expressed in terms of multiple polylogarithms. We show in detail how certain types of two- and three-point functions at two loops, which appear in the calculation of higher order corrections in QED, QCD and in the electroweak theory (EW), can naturally be expressed in terms of a recently introduced elliptic generalisation of multiple polylogarithms by direct integration over their Feynman parameter representation. Moreover, we show that in all examples that we considered a basis of pure Feynman integrals can be found.

hep-ph↗

arXiv2018

The SU(2|3) dynamic two-loop form factors

We compute two-loop form factors of operators in the $SU(2|3)$ closed subsector of $N=4$ supersymmetric Yang-Mills. In particular, we focus on the non-protected, dimension-three operators $\mathrm{Tr} (X[Y,Z])$ and $\mathrm{Tr} ( ψψ)$ for which we compute the four possible two-loop form factors, and corresponding remainder functions, with external states $\langle \bar{X} \bar{Y} \bar{Z}|$ and $\langle \barψ \barψ|$. Interestingly, the maximally transcendental part of the two-loop remainder of $\langle\bar{X} \bar{Y} \bar{Z}| \mathrm{Tr} (X[Y,Z]) |0\rangle$ turns out to be identical to that of the corresponding known quantity for the half-BPS operator $\mathrm{Tr} (X^3)$. We also find a surprising connection between the terms subleading in transcendentality and certain a priori unrelated remainder densities introduced in the study of the spin chain Hamiltonian in the SU(2) sector. Next, we use our calculation to resolve the mixing, recovering anomalous dimensions and eigenstates of the dilatation operator in the SU(2|3) sector at two loops. We also speculate on potential connections between our calculations in $N=4$ super Yang-Mills and Higgs + multi-gluon amplitudes in QCD in an effective Lagrangian approach.

hep-th↗

arXiv2018

$\text{Tr}(F^3)$ supersymmetric form factors and maximal transcendentality Part II: $0<\mathcal{N}<4$ super Yang-Mills

The study of form factors has many phenomenologically interesting applications, one of which is Higgs plus gluon amplitudes in QCD. Through effective field theory techniques these are related to form factors of various operators of increasing classical dimension. In this paper we extend our analysis of the first finite top-mass correction, arising from the operator ${\rm Tr} (F^3)$, from $\mathcal{N}=4$ super Yang-Mills to theories with $\mathcal{N}<4$, for the case of three gluons and up to two loops. We confirm our earlier result that the maximally transcendental part of the associated Catani remainder is universal and equal to that of the form factor of a protected trilinear operator in the maximally supersymmetric theory. The terms with lower transcendentality deviate from the $\mathcal{N}=4$ answer by a surprisingly small set of terms involving for example $ζ_2$, $ζ_3$ and simple powers of logarithms, for which we provide explicit expressions.

hep-th↗

arXiv2018

$\text{Tr}(F^3)$ supersymmetric form factors and maximal transcendentality Part I: $\mathcal{N}=4$ super Yang-Mills

In the large top-mass limit, Higgs plus multi-gluon amplitudes in QCD can be computed using an effective field theory. This approach turns the computation of such amplitudes into that of form factors of operators of increasing classical dimension. In this paper we focus on the first finite top-mass correction, arising from the operator ${\rm Tr}(F^3)$, up to two loops and three gluons. Setting up the calculation in the maximally supersymmetric theory requires identification of an appropriate supersymmetric completion of ${\rm Tr}(F^3)$, which we recognise as a descendant of the Konishi operator. We provide detailed computations for both this operator and the component operator ${\rm Tr}(F^3)$, preparing the ground for the calculation in $\mathcal{N}<4$, to be detailed in a companion paper. Our results for both operators are expressed in terms of a few universal functions of transcendental degree four and below, some of which have appeared in other contexts, hinting at universality of such quantities. An important feature of the result is a delicate cancellation of unphysical poles appearing in soft/collinear limits of the remainders which links terms of different transcendentality. Our calculation provides another example of the principle of maximal transcendentality for observables with non-trivial kinematic dependence.

hep-th↗

arXiv2018

All two-loop MHV remainder functions in multi-Regge kinematics

We introduce a method to extract the symbol of the coefficient of $(2πi)^2$ of MHV remainder functions in planar N=4 Super Yang-Mills in multi-Regge kinematics region directly from the symbol in full kinematics. At two loops this symbol can be uplifted to the full function in a unique way, without any beyond-the-symbol ambiguities. We can therefore determine all two-loop MHV amplitudes at function level in all kinematic regions with different energy signs in multi-Regge kinematics. We analyse our results and we observe that they are consistent with the hypothesis of a contribution from the exchange of a three-Reggeon composite state starting from two loops and eight points in certain kinematic regions.

hep-th↗

arXiv2018

Elliptic Feynman integrals and pure functions

We propose a variant of elliptic multiple polylogarithms that have at most logarithmic singularities in all variables and satisfy a differential equation without homogeneous term. We investigate several non-trivial elliptic two-loop Feynman integrals with up to three external legs and express them in terms of our functions. We observe that in all cases they evaluate to pure combinations of elliptic multiple polylogarithms of uniform weight. This is the first time that a notion of uniform weight is observed in the context of Feynman integrals that evaluate to elliptic polylogarithms.

hep-th↗

arXiv2018

Elliptic polylogarithms and two-loop Feynman integrals

We review certain classes of iterated integrals that appear in the computation of Feynman integrals that involve elliptic functions. These functions generalise the well-known class of multiple polylogarithms to elliptic curves and are closely related to the elliptic multiple polylogarithms (eMPLs) studied in the mathematics literature. When evaluated at certain special values of the arguments, eMPLs reduce to another class of special functions, defined as iterated integrals of Eisenstein series. As a novel application of our formalism, we illustrate how a class of special functions introduced by Remiddi and one of the authors can always naturally be expressed in terms of either eMPLs or iterated integrals of Eisenstein series for the congruence subgroup Gamma(6).

hep-ph↗

arXiv2018

From modular forms to differential equations for Feynman integrals

In these proceedings we discuss a representation for modular forms that is more suitable for their application to the calculation of Feynman integrals in the context of iterated integrals and the differential equation method. In particular, we show that for every modular form we can find a representation in terms of powers of complete elliptic integrals of the first kind multiplied by algebraic functions. We illustrate this result on several examples. In particular, we show how to explicitly rewrite elliptic multiple zeta values as iterated integrals over powers of complete elliptic integrals and rational functions, and we discuss how to use our results in the context of the system of differential equations satisfied by the sunrise and kite integrals.

hep-th↗

arXiv2018

Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series

We present a generalization of the symbol calculus from ordinary multiple polylogarithms to their elliptic counterparts. Our formalism is based on a special case of a coaction on large classes of periods that is applied in particular to elliptic polylogarithms and iterated integrals of modular forms. We illustrate how to use our formalism to derive relations among elliptic polylogarithms, in complete analogy with the non-elliptic case. We then analyze the symbol alphabet of elliptic polylogarithms evaluated at rational points, and we observe that it is given by Eisenstein series for a certain congruence subgroup. We apply our formalism to hypergeometric functions that can be expressed in terms of elliptic polylogarithms and show that they can equally be written in terms of iterated integrals of Eisenstein series. Finally, we present the symbol of the equal-mass sunrise integral in two space-time dimensions. The symbol alphabet involves Eisenstein series of level six and weight three, and we can easily integrate the symbol in terms of iterated integrals of Eisenstein series.

hep-th↗

arXiv2017

A note on NMHV form factors from the Graßmannian and the twistor string

In this note we investigate Graßmannian formulas for form factors of the chiral part of the stress-tensor multiplet in $\mathcal{N}=4$ superconformal Yang-Mills theory. We present an all-$n$ contour for the $G(3,n+2)$ Graßmannian integral of NMHV form factors derived from on-shell diagrams and the BCFW recursion relation. In addition, we study other $G(3,n+2)$ formulas obtained from the connected prescription introduced recently. We find a recursive expression for all $n$ and study its properties. For $n \geq 6$, our formula has the same recursive structure as its amplitude counterpart, making its soft behaviour manifest. Finally, we explore the connection between the two Graßmannian formulations, using the global residue theorem, and find that it is much more intricate compared to scattering amplitudes.

hep-th↗

arXiv2017

Higgs amplitudes from $\mathcal{N}=4$ super Yang-Mills theory

Higgs plus multi-gluon amplitudes in QCD can be computed in an effective Lagrangian description. In the infinite top-mass limit, an amplitude with a Higgs and $n$ gluons is computed by the form factor of the operator ${\rm Tr}\, F^2$. Up to two loops and for three gluons, its maximally transcendental part is captured entirely by the form factor of the protected stress tensor multiplet operator $\mathcal{T}_2$ in $\mathcal{N}=4$ supersymmetric Yang-Mills theory. The next order correction involves the calculation of the form factor of the higher-dimensional, trilinear operator ${\rm Tr}\, F^3$. We present explicit results at two loops for three gluons, including the subleading transcendental terms derived from a particular descendant of the Konishi operator that contains ${\rm Tr}\, F^3$. These are expressed in terms of a few universal building blocks already identified in earlier calculations. We show that the maximally transcendental part of this quantity, computed in non-supersymmetric Yang-Mills theory, is identical to the form factor of another protected operator, $\mathcal{T}_3$, in the maximally supersymmetric theory. Our results suggest that the maximally transcendental part of Higgs amplitudes in QCD can be entirely computed through $\mathcal{N}=4$ super Yang-Mills.

hep-th↗

arXiv2016

On-shell methods for off-shell quantities in N=4 Super Yang-Mills: from scattering amplitudes to form factors and the dilatation operator

Planar maximally supersymmetric Yang-Mills theory (N=4 SYM) is a special quantum field theory. A few of its remarkable features are conformal symmetry at the quantum level, evidence of integrability and, moreover, it is a prime example of the AdS/CFT duality. Triggered by Witten's twistor string theory, the past 15 years have witnessed enormous progress in reformulating this theory to make as many of these special features manifest, from the choice of convenient variables to recursion relations that allowed new mathematical structures to appear, like the Grassmannian. These methods are collectively referred to as on-shell methods. The ultimate hope is that, by understanding N=4 SYM in depth, one can learn about other, more realistic quantum field theories. The overarching theme of this thesis is the investigation of how on-shell methods can aid the computation of quantities other than scattering amplitudes. In this spirit we study form factors and correlation functions, said to be partially and completely off-shell quantities, respectively. More explicitly, we compute form factors of half-BPS operators up to two loops, and study the dilatation operator in the SO(6) and SU(2|3) sectors using techniques originally designed for amplitudes. A second part of the work is dedicated to the study of scattering amplitudes beyond the planar limit, an area of research which is still in its infancy, and not much is known about which special features of the planar theory survive in the non-planar regime. In this context, we generalise some aspects of the on-shell diagram formulation of Arkani-Hamed et al. to take into account non-planar corrections.

hep-th↗

arXiv2016

Non-Planar On-Shell Diagrams

We initiate a systematic study of non-planar on-shell diagrams in N=4 SYM and develop powerful technology for doing so. We introduce canonical variables generalizing face variables, which make the dlog form of the on-shell form explicit. We make significant progress towards a general classification of arbitrary on-shell diagrams by means of two classes of combinatorial objects: generalized matching and matroid polytopes. We propose a boundary measurement that connects general on-shell diagrams to the Grassmannian. Our proposal exhibits two important and non-trivial properties: positivity in the planar case and it matches the combinatorial description of the diagrams in terms of generalized matroid polytopes. Interestingly, non-planar diagrams exhibit novel phenomena, such as the emergence of constraints on Plucker coordinates beyond Plucker relations when deleting edges, which are neatly captured by the generalized matching and matroid polytopes. This behavior is tied to the existence of a new type of poles in the on-shell form at which combinations of Plucker coordinates vanish. Finally, we introduce a prescription, applicable beyond the MHV case, for writing the on-shell form as a function of minors directly from the graph.

hep-th↗

arXiv2015

Integrability and unitarity

We show how generalised unitarity can be used to determine the one-loop dilatation operator in N=4 super Yang-Mills. Our analysis focuses on two sectors, namely the bosonic SO(6) sector and the SU(2|3) sector. The calculation is performed on shell, with no off-shell information introduced at any stage. In this way, we establish a direct connection between scattering amplitudes and the dilatation operator of the N=4 theory.

hep-th↗

arXiv2015

Integrability and MHV diagrams in N=4 supersymmetric Yang-Mills theory

We apply MHV diagrams to the derivation of the one-loop dilatation operator of N=4 super Yang-Mills in the SO(6) sector. We find that in this approach the calculation reduces to the evaluation of a single MHV diagram in dimensional regularisation. This provides the first application of MHV diagrams to an off-shell quantity. We also discuss other applications of the method and future directions.

hep-th↗

arXiv2014

On super form factors of half-BPS operators in N=4 super Yang-Mills

We compute form factors of half-BPS operators in N=4 super Yang-Mills dual to massive Kaluza-Klein modes in supergravity. These are appropriate supersymmetrisations T_k of the scalar operators Tr(ϕ^k) for any k, which for k=2 give the chiral part of the stress-tensor multiplet operator. Using harmonic superspace, we derive simple Ward identities for these form factors, which we then compute perturbatively at tree level and one loop. We propose a novel on-shell recursion relation which links form factors with different numbers of fields. Using this, we conjecture a general formula for the n-point MHV form factors of T_k for arbitrary k and n. Finally, we use supersymmetric generalised unitarity to derive compact expressions for all one-loop MHV form factors of T_k in terms of one-loop triangles and finite two-mass easy box functions.

hep-th↗

arXiv2014

The last of the simple remainders

We compute the n-point two-loop form factors of the half-BPS operators Tr(phi_{AB}^n) in N=4 super Yang-Mills for arbitrary n >2 using generalised unitarity and symbols. These form factors are minimal in the sense that the n^{th} power of the scalar field in the operator requires the presence of at least n on-shell legs. Infrared divergences are shown to exponentiate as for amplitudes, reproducing the known cusp and collinear anomalous dimensions at two loops. We define appropriate infrared-finite remainder functions and compute them analytically for all n. The results obtained by using the known expressions of the integral functions involve complicated combinations of Goncharov multiple polylogarithms, but we show that much simpler expressions can in fact be derived using the symbol of transcendental functions. For n=3 we find a very compact remainder function expressed in terms of classical polylogarithms only. For arbitrary n>3 we are able to write all the remainder functions in terms of a single compact building block, expressed as a sum of classical polylogarithms augmented by two multiple polylogarithms. The decomposition of the symbol into specific components is crucial in order to single out a natural combination of multiple polylogarithms. Finally, we analyse in detail the behaviour of these minimal form factors in collinear and soft limits, which deviates from the usual behaviour of amplitudes and non-minimal form factors.

hep-th↗