Maximal Transcendentality Research Articles

Fermionic contribution to the anomalous dimension of twist-2 operators in [formula omitted] SYM theory, critical indices and integrability

We compute the contribution to the anomalous dimension of the twist-2 operators in N=4 SYM theory, which is proportional to the number of fermion loops inside Feynman diagrams or, formally, to the number of fermions. The result was obtained by the method based on the calculation of critical indices at the critical point by analogy with previous similar computations in scalar theories and in QCD. The obtained result is much simpler with compare to analogous results in QCD and almost satisfies the maximal transcendentality principle. A possible relation between the obtained result and integrability is discussed.

Four-loop QCD cusp anomalous dimension at small angle

We calculate the small angle expansion of the four-loop QCD cusp anomalous dimension. As a byproduct of our calculation, we also obtain the four-loop anomalous dimension of the heavy-quark field in HQET. The validity of the calculational setup is cross-checked by the independent calculation of the four-loop QCD beta-function from heavy-quark-gluon vertex renormalization in HQET. We check the obtained results for the cusp anomalous dimension and heavy-quark field anomalous dimension against available analytical and numerical results. Finally, we find that the maximal transcendentality contribution to the QCD Bremsstrahlung function coincides, up to a factor 3/2, with the Bremsstrahlung function in mathcal{N} = 4 supersymmetric Yang-Mills theory, at least, through 4 loops.

A derivation of the planar limit of mathcal{N} = 2 chiral correlators

We derive analytically the terms of maximal transcendentality of the planar 2- and 3-point functions of single-trace chiral primary operators of mathcal{N} = 2 SQCD on ℝ4, to all orders in the ’t Hooft coupling. These results prove two conjectures we formulated in previous work. Furthermore, we also provide an explicit expression for the terms in the planar 2-point functions of these operators that contain products of two values of the ζ function.

Deciphering the maximal transcendentality principle via bootstrap

We prove the principle of maximal transcendentality for a class of form factors, including the general two-loop minimal form factors, the two-loop three-point form factor of tr(F2), and the two-loop four-point form factor of tr(F3). Our proof is based on a recently developed bootstrap method using the representation of master integral expansions, together with some unitarity cuts that are universal in general gauge theories. The maximally transcendental parts of the two-loop four-gluon form factor of tr(F3) are obtained for the first time in both planar mathcal{N} = 4 SYM and pure YM theories. This form factor can be understood as the Higgs-plus-four-gluon amplitudes involving a dimension-seven operator in the Higgs effective theory. In this case, we find that the maximally transcendental part of the mathcal{N} = 4 SYM result is different from that of pure YM, and the discrepancy is due to the gluino-loop contributions in mathcal{N} = 4 SYM. In contrast, the scalar-loop contributions have no maximally transcendental parts. Thus, the maximal transcendentality principle still holds for the form factor results in mathcal{N} = 4 SYM and QCD, after a proper identification of the fundamental quarks and adjoint gluinos as nf→ 4Nc. This seems to be the first example of the maximally transcendental principle that involves fermion-loop contributions. As another intriguing observation, we find that the four-point form factor of the half-BPS tr(ϕ3) operator is precisely a building block in the form factor of tr(F3).

Bootstrapping a stress-tensor form factor through eight loops

We bootstrap the three-point form factor of the chiral stress-tensor multiplet in planar mathcal{N} = 4 supersymmetric Yang-Mills theory at six, seven, and eight loops, using boundary data from the form factor operator product expansion. This may represent the highest perturbative order to which multi-variate quantities in a unitary four-dimensional quantum field theory have been computed. In computing this form factor, we observe and employ new restrictions on pairs and triples of adjacent letters in the symbol. We provide details about the function space required to describe the form factor through eight loops. Plotting the results on various lines provides striking numerical evidence for a finite radius of convergence of perturbation theory. By the principle of maximal transcendentality, our results are expected to give the highest weight part of the gg → Hg and H → ggg amplitudes in the heavy-top limit of QCD through eight loops. These results were also recently used to discover a new antipodal duality between this form factor and a six-point amplitude in the same theory.

Form factors with two operator insertions and the principle of maximal transcendentality

We present the first calculations of two-point two-loop form factors (FFs) with a two identical operators insertion in maximally supersymmetric Yang-Mills theory. In this article, we consider the supersymmetry protected half-BPS primary and unprotected Konishi operators. Unlike the FFs of a single operator insertion of the half-BPS primary, the FFs involving two half-BPS operators are found to contain lower transcendentality weight terms in addition to the highest ones. Moreover, in contrast to Sudakov FFs, the highest weight terms of the FFs of a double half-BPS no longer match with that of a double Konishi. We also find that the principle of maximal transcendentality, which dictates the presence of identical highest weight terms in the scalar FFs of half-BPS and quark/gluon FFs in QCD, does not hold true anymore for insertions of two identical operators. We discover the absence of any additional ultraviolet counterterm that could arise from the contact interaction between two composite operators.

Hidden analytic relations for two-loop Higgs amplitudes in QCD

We compute the Higgs plus two-quark and one-gluon amplitudes () and Higgs plus three-gluon amplitudes () in the Higgs effective theory with a general class of operators. By changing the quadratic Casimir C F to C A , the maximally transcendental parts of the amplitudes turn out to be equivalent to that of the amplitudes, which also coincide with the counterparts in SYM. This generalizes the so-called maximal transcendentality principle to the Higgs amplitudes with external quark states, thus the full QCD theory. We further verify that the correspondence applies also to two-loop form factors of more general operators, in both QCD and scalar-YM theory. Another interesting relation is also observed between the planar amplitudes and the minimal density form factors in SYM.

Two-Loop QCD Corrections to the Higgs Plus Three-parton Amplitudes with Top Mass Correction

We obtain the two-loop QCD corrections to the Higgs plus three-parton amplitudes with dimension-seven operators in Higgs effective field theory. This provides the two-loop S-matrix elements for Higgs plus one-jet production at the LHC with top-mass correction. We apply efficient unitarity plus IBP methods which are described in detail. We also study the color decomposition of the fermion cuts and find a connection between fundamental and adjoint representations which can be used to reduce non-planar to planar unitarity cuts in the Higgs to three-gluon amplitudes. We obtain final results in simple analytic form which exhibits intriguing hidden structures. The principle of maximal transcendentality is found to be satisfied for all results. The lower transcendentality parts also contain universal building blocks and can be written in compact analytic form, suggesting further hidden structures.

Toward an Analytic Perturbative Solution for the Abjm Quantum Spectral Curve

Recently we showed how non-homogeneous second-order difference equations appearing within ABJM quantum spectral curve description could be solved using Mellin space technique. In particular we provided explicit results for anomalous dimensions of twist 1 operators in sl(2) sector at arbitrary spin values up to four loop order. It was shown that the obtained results may be expressed in terms of harmonic sums decorated by fourth root of unity factors, so that maximum transcendentality principle holds. In this note we show that the same result could be also obtained by direct solution of the mentioned equations in spectral parameter u-space. The solution involves new highly nontrivial identities between hypergeometric functions, which may have various other applications. We expect this method to be more easily generalizable to higher loop orders as well as to other theories, such as N=4 SYM.

Perturbative four-point functions in planar mathcal{N}=4 SYM From hexagonalization

We use hexagonalization to compute four-point correlation functions of long BPS operators with special R-charge polarizations. We perform the computation at weak coupling and show that at any loop order our correlators can be expressed in terms of single value polylogarithms with uniform and maximal transcendentality. As a check of our computation we extract nine-loop OPE data and compare it against sum rules of (squared) structures constants of non-protected exchanged operators described by hundreds of Bethe solutions.

Tr(F3) 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 Tr(F3), up to two loops and three gluons. Setting up the calculation in the maximally supersymmetric theory requires identification of an appropriate supersymmetric completion of Tr(F3), which we recognise as a descendant of the Konishi operator. We provide detailed computations for both this operator and the component operator Tr(F3), 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.

Analytic Two-Loop Higgs Amplitudes in Effective Field Theory and the Maximal Transcendentality Principle.

We obtain, for the first time, the two-loop amplitudes for Higgs plus three gluons in Higgs effective field theory including dimension-seven operators. This provides the S-matrix elements for the top mass corrections for Higgs plus a jet production at LHC. The computation is based on the on-shell unitarity method combined with integration by parts reduction. We work in conventional dimensional regularization and obtain analytic expressions renormalized in the modified minimal subtraction scheme. The two-loop anomalous dimensions present operator mixing behavior. The infrared divergences agree with that predicted by Catani and the finite remainders take remarkably simple forms, where the maximally transcendental parts are identical to the corresponding results in N=4 super-Yang-Mills theory. The parts of lower transcendentality turn out to be also largely determined by the N=4 results.

ABJM quantum spectral curve and Mellin transform

The present techniques for the perturbative solution of quantum spectral curve problems in mathcal{N}=4 SYM and ABJM models are limited to the situation when the states quantum numbers are given explicitly as some integer numbers. These techniques are sufficient to recover full analytical structure of the conserved charges provided that we know a finite basis of functions in terms of which they could be written explicitly. It is known that in the case of mathcal{N}=4 SYM both the contributions of asymptotic Bethe ansatz and wrapping or finite size corrections are expressed in terms of the harmonic sums. However, in the case of ABJM model only the asymptotic contribution can still be written in the harmonic sums basis, while the wrapping corrections part can not. Moreover, the generalization of harmonic sums basis for this problem is not known. In this paper we present a Mellin space technique for the solution of multiloop Baxter equations, which is the main ingredient for the solution of corresponding quantum spectral problems, and provide explicit results for the solution of ABJM quantum spectral curve in the case of twist 1 operators in sl(2) sector for arbitrary spin values up to four loop order with explicit account for wrapping corrections. It is shown that the result for anomalous dimensions could be expressed in terms of harmonic sums decorated by the fourth root of unity factors, so that maximum transcendentality principle holds.

The principle of maximal transcendentality and the four-loop collinear anomalous dimension

We use the principle of maximal transcendentality and the universal nature of subleading infrared poles to extract the analytic value of the four-loop collinear anomalous dimension in planar mathcal{N} = 4 super-Yang-Mills theory from recent QCD results, obtaining {widehat{mathcal{G}}}_0^{(4)}=-300{zeta}_7-256{zeta}_2{zeta}_5-384{zeta}_3{zeta}_4 . This value agrees with a previous numerical result to within 0.2%. It also provides the Regge trajectory, threshold soft anomalous dimension and rapidity anomalous dimension through four loops.

Finite remainders of the Konishi at two loops in mathcal{N}=4 SYM

We present three point form factors (FF) in mathcal{N}=4 Super Yang Mills theory for both the half-BPS and the Konishi operators at two loop level in the ‘t Hooft coupling using Feynman diagrammatic approach. We have chosen on shell final states consisting of gϕϕ and ϕλλ, where ϕ, λ, g are scalar, Majorana fermion and gauge fields respectively. The computation is done both in the modified dimensional reduction as well as in the four dimensional helicity scheme. We have studied the universal structure of infrared (IR) singularities in these FFs using Catani’s IR subtraction operators. Exploiting the factorisation property of the IR singularities and following BDS like ansatz for the IR sensitive terms in FFs, we determine the finite remainders of them. We find that the finite remainders of FFs of the half-BPS for both the choices of final states give not only identical results but also contain terms of uniform transcendentality of weight two and four at one and two loop levels, respectively. In the case of the Konishi operator, the finite remainders depend on the external states and do not exhibit uniform transcendentality. However, surprisingly, the leading transcendental terms for gϕϕ agree with that of the half-BPS. We have demonstrated the role of on shell external states for the FFs in the context of maximum transcendentality principle.

The property of maximal transcendentality: Calculation of Feynman integrals

We consider examples of two-loop two- and three-point Feynman integrals for which the calculation results have the property of maximal transcendentality.

Two-loop SL(2) form factors and maximal transcendentality

Form factors of composite operators in the SL(2) sector of N=4 SYM theory are studied up to two loops via the on-shell unitarity method. The non-compactness of this subsector implies the novel feature and technical challenge of an unlimited number of loop momenta in the integrand's numerator. At one loop, we derive the full minimal form factor to all orders in the dimensional regularisation parameter. At two loops, we construct the complete integrand for composite operators with an arbitrary number of covariant derivatives, and we obtain the remainder functions as well as the dilatation operator for composite operators with up to three covariant derivatives. The remainder functions reveal curious patterns suggesting a hidden maximal uniform transcendentality for the full form factor. Finally, we speculate about an extension of these patterns to QCD.

The two-loop symbol of all multi-Regge regions

We study the symbol of the two-loop n-gluon MHV amplitude for all Mandelstam regions in multi-Regge kinematics in N=4 super Yang-Mills theory. While the number of distinct Mandelstam regions grows exponentially with n, the increase of independent symbols turns out to be merely quadratic. We uncover how to construct the symbols for any number of external gluons from just two building blocks which are naturally associated with the six- and seven-gluon amplitude, respectively. The second building block is entirely new, and in addition to its symbol, we also construct a prototype function that correctly reproduces all terms of maximal functional transcendentality.

Soft-virtual corrections to Higgs production atN3LO

In this paper, we compute the soft-virtual corrections to Higgs boson production in gluon fusion for infinite top quark mass at next-to-next-to-next-to-leading order in QCD. In addition, we present analogous soft-virtual terms for both Drell-Yan lepton production in QCD and scalar pair production in N = 4 super Yang-Mills theory. The result for Drell-Yan lepton production is derived from the result for Higgs boson production using Casimir scaling arguments together with well-known results available in the literature. For scalar pair production in the N = 4 model, we show by explicit calculation that the result is equal to the part of the Higgs boson soft-virtual term which is of maximal transcendentality weight.

Four-point amplitudes in N = 2 $$ \mathcal{N}=2 $$ SCQCD

We compute four-point scattering amplitudes in $$ \mathcal{N}=2 $$ SCQCD with general external matter configurations using $$ \mathcal{N}=1 $$ superspace Feynman diagrams, at one loop in the general case and up to two loops in the fundamental sector. In the pure adjoint sector at one loop we confirm exact agreement with the corresponding amplitudes in $$ \mathcal{N}=4 $$ SYM theory, supporting the idea that a closed subsector of the SCQCD might be exactly integrable. External matter in the fundamental representation breaks dual conformal invariance already at one-loop order while the principle of maximal transcendentality is respected up to two loops.