Operator Tr Research Articles

Double copy for tree-level form factors. Part II. Generalizations and special topics

Both the Bern, Carrasco, and Johansson (BCJ) and the Kawai, Lewellen, and Tye (KLT) double-copy formalisms have been recently generalized to a class of scattering matrix elements (so-called form factors) that involve local gauge-invariant operators. In this paper, we continue the study of double copy for form factors. First, we generalize the double-copy prescription to form factors of higher-length operators tr(ϕm) with m ≥ 3. These higher-length operators introduce new non-trivial color identities, but the double-copy prescription works perfectly well. The closed formulae for the CK-dual numerators are also provided. Next, we discuss the υ→\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overrightarrow{\\upsilon} $$\\end{document} vectors which are central ingredients appearing in the factorization relations of both the KLT kernels and the gauge form factors. We present a general construction rule for the υ→\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overrightarrow{\\upsilon} $$\\end{document} vectors and discuss their universal properties. Finally, we consider the double copy for the form factor of the tr(F2) operator in pure Yang-Mills theory. In this case, we propose a new prescription which involves a gauge invariant decomposition for the form factor and a mixture of different CK-dual numerators appearing in the expansion. The new prescription for the more complicated double copy has been verified up to five external gluons.

Exploring the ground state spectrum of γ-deformed N = 4 SYM

We study the γ-deformation of the planar mathcal{N} = 4 super Yang-Mills theory which breaks all supersymmetries but is expected to preserve integrability of the model. We focus on the operator Tr (ϕ1ϕ1) built from two scalars, whose integrability description has been questioned before due to contributions from double-trace counterterms. We show that despite these subtle effects, the integrability-based Quantum Spectral Curve (QSC) framework works perfectly for this state and in particular reproduces the known 1-loop prediction. This resolves an earlier controversy concerning this operator and provides further evidence that the γ-deformed model is an integrable CFT at least in the planar limit. We use the QSC to compute the first 5 weak coupling orders of the anomalous dimension analytically, matching known results in the fishnet limit, and also compute it numerically all the way from weak to strong coupling. We also utilize this data to extract a new coefficient of the beta function of the double-trace operator couplings.

On the algebraic structure of polycyclic codes

In this paper, we are interested in the study of the right polycyclic codes as invariant subspaces of Fnq by a fixed operator TR. This approach has helped in one hand to connect them to the ideals of the polynomials ring Fq [x]/?f)X)?, where f (x) is the minimal polynomial of TR. On the other hand, it allows to prove that the dual of a right polycyclic code is invariant by the adjoint operator of TR. Hence, when TR is normal we prove that the dual code of a right polycyclic code is also a right polycyclic code. However, when TR isn?t normal the dual code is equivalent to a right polycyclic code. Finally, as in the cyclic case, the BCH-like and Hartmann-Tzeng-like bounds for the right polycyclic codes on Hamming distance are derived.

Towards stability of NLO corrections in high-energy factorization via modified multi-Regge kinematics approximation

The perturbatively-stable scheme of Next-to-Leading order (NLO) calculations of cross-sections for multi-scale hard-processes in DIS-like kinematics is developed in the framework of High-Energy Factorization. The evolution equation for unintegrated PDF, which resums log 1/z-corrections to the coefficient function in the Leading Logarithmic approximation together with a certain subset of Next-to-Leading Logarithmic and Next- to-Leading Power corrections, necessary for the perturbative stability of the formalism, is formulated and solved in the Doubly-Logarithmic approximation. An example of DIS-like process, induced by the operator tr [GμνGμν ], which is sensitive to gluon PDF already in the LO, is studied. Moderate (O(20%)) NLO corrections to the inclusive structure function are found at small xB< 10−4, while for the pT -spectrum of a leading jet in the considered process, NLO corrections are small (< O(20%)) and LO of kT -factorization is a good approximation. The approach can be straightforwardly extended to the case of multi-scale hard processes in pp-collisions at high energies.

Computing one-loop corrections to effective vertices with two scales in the EFT for Multi-Regge processes in QCD

The computation of one-loop corrections to Reggeon-Particle-Particle effective vertices with two scales of virtuality is considered in the framework of gauge-invariant effective field theory for Multi-Regge processes in QCD. Rapidity divergences arising in loop integrals are regulated by “tilted Wilson lines” prescription. General analysis of rapidity divergences at one loop is given and necessary scalar integrals with one and two scales of virtuality are computed. Two examples of effective vertices at one loop are considered: the effective vertex of interaction of (space-like) virtual photon with one Reggeized and one Yang-Mills quark and the effective vertex of Reggeized gluon to Yang-Mills gluon transition with an insertion of the operator tr[GμνGμν] carrying the (space-like) off-shell momentum. All terms ∼r±ϵ in the rapidity-regulator variable r cancel between diagrams and only log⁡r-divergence is left. It is checked on several examples, that obtained results indeed allow one to reproduce Regge limit of one-loop QCD scattering amplitudes.

Safe glueballs and baryons

We consider a non-Abelian gauge theory with Nf fermions and discuss the possible existence of a non-trivial UV fixed point at large Nf . Specifically, we study the anomalous dimension of the (rescaled) glueball operator Tr F2 to first order in 1/Nf by relating it to the derivative of the beta function at the fixed point. At the fixed point the anomalous dimension violates its unitarity bound and so the (rescaled) glueball operator is either decoupled or the fixed point does not exist. We also study the anomalous dimensions of the two spin-1/2 baryon operators to first order in 1/Nf for an SU(3) gauge theory with fundamental fermions and find them to be relatively small and well within their associated unitarity bounds.

Tr(F3) 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 Tr(F3), 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 ζ2, ζ3 and simple powers of logarithms, for which we provide explicit expressions.

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.

A limit for large R-charge correlators in mathcal{N} = 2 theories

Using supersymmetric localization, we study the sector of chiral primary operators (Tr ϕ2)n with large R-charge 4n in mathcal{N} = 2 four-dimensional superconformal theories in the weak coupling regime g → 0, where λ ≡ g2n is kept fixed as n → ∞, g representing the gauge theory coupling(s). In this limit, correlation functions G2n of these operators behave in a simple way, with an asymptotic behavior of the form {G}_{2n}approx {F}_{infty}left(uplambda right){left(frac{uplambda}{2pi e}right)}^{2n} nα, modulo O(1/n) corrections, with alpha =frac{1}{2} dim left(mathfrak{g}right) for a gauge algebra mathfrak{g} and a universal function F∞(λ). As a by-product we find several new formulas both for the partition function as well as for perturbative correlators in mathcal{N}=2kern0.5em mathfrak{s}mathfrak{u}(N) gauge theory with 2N fundamental hypermultiplets.

A piece of cake: the ground-state energies in γ i -deformed N $$ \mathcal{N} $$ = 4 SYM theory at leading wrapping order

In the non-supersymmetric gamma_i-deformed N=4 SYM theory, the scaling dimensions of the operators tr[Z^L] composed of L scalar fields Z receive finite-size wrapping and prewrapping corrections in the 't Hooft limit. In this paper, we calculate these scaling dimensions to leading wrapping order directly from Feynman diagrams. For L>=3, the result is proportional to the maximally transcendental `cake' integral. It matches with an earlier result obtained from the integrability-based Luescher corrections, TBA and Y-system equations. At L=2, where the integrability-based equations yield infinity, we find a finite rational result. This result is renormalization-scheme dependent due to the non-vanishing beta-function of an induced quartic scalar double-trace coupling, on which we have reported earlier. This explicitly shows that conformal invariance is broken - even in the 't Hooft limit.

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.

On super form factors of half-BPS operators in $ \mathcal{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(\phi^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.

Three-point correlators of twist-2 operators in N=4 SYM at Born approximation

Abstract We calculate two different types of 3-point correlators involving twist-2 operators in the leading weak coupling approximation and all orders in N c in N=4 SYM theory. Each of three operators in the first correlator can be any component of twist-2 supermultiplet, though the explicit calculation was done for a particular component which is an SU(4) singlet. It is calculated in the leading, Born approximation for arbitrary spins j 1 , j 2 , j 3. The result significantly simplifies when at least one of the spins is large or equal to zero and the coordinates are restricted to the 2d plane spanned by two light-rays. The second correlator involves two twist-2 operators Tr(X∇ j1 X) + . . ., Tr(Z∇ j2 Z) + . . . and one Konishi operator $ \mathrm{Tr}{{\left[ {\overline{Z},\overline{X}} \right]}^2} $ . It vanishes in the lowest g 0 order and is computed at the leading g 2 approximation.

The Chern-Simons diffusion rate in improved holographic QCD

Abstract In (3 + 1)-dimensional SU(N c) Yang-Mills (YM) theory, the Chern-Simons diffusion rate, ΓCS, is determined by the zero-momentum, zero-frequency limit of the retarded two-point function of the CP-odd operator tr [F ∧ F ], with F the YM field strength. The Chern-Simons diffusion rate is a crucial ingredient for many CP-odd phenomena, including the chiral magnetic effect in the quark-gluon plasma. We compute ΓCS in the high-temperature, deconfined phase of Improved Holographic QCD, a refined holographic model for large-N c YM theory. Our result for ΓCS/(sT ), where s is entropy density and T is temperature, varies slowly at high T and increases monotonically as T approaches the transition temperature from above. We also study the retarded two-point function of tr [F ∧ F ] with non-zero frequency and momentum. Our results suggest that the CP-odd phenomena that may potentially occur in heavy ion collisions could be controlled by an excitation with energy on the order of the lightest axial glueball mass.

Integrability properties of Hurwitz partition functions. II. Multiplication of cut-and-join operators and WDVV equations

Correlators in topological theories are given by the values of a linear form on the products of operators from a commutative associative algebra (CAA). As a corollary, partition functions of topological theory always satisfy the generalized WDVV equations. We consider the Hurwitz partition functions, associated in this way with the CAA of cut-and-join operators. The ordinary Hurwitz numbers for a given number of sheets in the covering provide trivial (sums of exponentials) solutions to the WDVV equations, with finite number of time-variables. The generalized Hurwitz numbers from arXiv:0904.4227 provide a non-trivial solution with infinite number of times. The simplest solution of this type is associated with a subring, generated by the dilatation operators tr X(d/dX).

The gluonic field of a heavy quark in conformal field theories at strong coupling

We determine the gluonic field configuration sourced by a heavy quark undergoing arbitrary motion in N=4 super-Yang-Mills at strong coupling and large number of colors. More specifically, we compute the expectation value of the operator tr[F^2+...] in the presence of such a quark, by means of the AdS/CFT correspondence. Our results for this observable show that signals propagate without temporal broadening, just as was found for the expectation value of the energy density in recent work by Hatta et al. We attempt to shed some additional light on the origin of this feature, and propose a different interpretation for its physical significance. As an application of our general results, we examine <tr[F^2+...]> when the quark undergoes oscillatory motion, uniform circular motion, and uniform acceleration. Via the AdS/CFT correspondence, all of our results are pertinent to any conformal field theory in 3+1 dimensions with a dual gravity formulation.

Eigenfrequencies of nonisotropic fractal drums

Let Γ be a regular nonisotropic fractal. The aim of the paper is to investigate the distribution of the eigenvalue of the fractal differential operator ( - Δ ) - 1 ∘ tr Γ acting in the anisotropic Sobolev space W 2 1 , α ( R 2 ) , where ( - Δ ) - 1 is the inverse of the Dirichlet Laplanian in R 2 and tr Γ is closely related to the trace operator tr Γ .

Wrapping at four loops in [formula omitted] SYM

We present the planar four-loop anomalous dimension of the composite operator tr(ϕ[Z,ϕ]Z) in the flavour SU(2) sector of the N=4 SYM theory. At this loop order wrapping interactions are present: they give rise to contributions proportional to ζ(5) increasing the level of transcendentality of the anomalous dimension. In a sequel of this Letter all the details of our calculation will be reported.

Study of the nonlocal gauge invariant mass operator Tr∫d4xFμν(D2)−1Fμν in the maximal Abelian gauge

The nonlocal gauge invariant mass operator Tr∫d4xFμν(D2)−1Fμν is investigated in Yang–Mills theories in the maximal Abelian gauge. By means of the introduction of auxiliary fields a local action is achieved, enabling us to use the algebraic renormalization in order to prove the renormalizability of the resulting local model to all orders of perturbation theory.

Traces and extensions of matrix-weighted Besov spaces

Let V be a matrix weight on ℝ n+1 and let W be a matrix weight on ℝ n , satisfying, for example, the matrix A p condition. Define the trace, or restriction, operator Tr by Tr (f)(x ′ )=f(x ′ , 0), where x ′ ∈ℝ n and f is a function on ℝ n+1 . If α−1/p>n (1/p−1) + +(β−n)/p, where β is the doubling exponent of W, then the trace operator is bounded from into (matrix-weighted Besov spaces) if and only if the weights V and W uniformly satisfy an estimate controlling the average of on any dyadic cube I ⊆ ℝ n by the average of on Q(I)=I×[0, l(I)], for all . If V and W satisfy the converse inequality, then there exists a continuous linear map . If both inequalities hold, then Tr ○ Ext is the identity on .