Three-loop Anomalous Dimension Research Articles

The four-loop β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}-function from vacuum supergraphs and the NSVZ relation for N=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{{\\mathcal {N}}}}=1$$\\end{document} SQED regularized by higher derivatives

In N=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{{\\mathcal {N}}}}=1$$\\end{document} SQED with Nf\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_f$$\\end{document} flavors regularized by higher derivatives we obtain the four-loop beta function using a method based on calculating vacuum supergraphs. For this purpose we use a special C++ program which obtain contributions to the β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}-function from supergraphs without external legs in the form of integrals of double total derivatives. After that the result was compared with the three-loop anomalous dimension calculated earlier. We explicitly check the NSVZ relation in this order.

Infrared Singularities of Multileg QCD Amplitudes with a Massive Parton at Three Loops

We derive the structure of three-loop anomalous dimensions governing infrared singularities of QCD amplitudes with one massive and an arbitrary number of massless external partons. The contributions of tripole and quadrupole correlations involving a massive parton are studied in detail. The analytical expression of tripole correlations between one massive and two massless partons is obtained at three loops for the first time. We regularize the infrared divergences in the soft matrix element in a novel approach, where no extra scale dependence is involved, and the calculation can be performed in momentum space. Our results are essential to improve the theoretical predictions of single top and top quark pair productions at hadron colliders.

The massless three-loop Wilson coefficients for the deep-inelastic structure functions F2, FL, xF3 and g1

We calculate the massless unpolarized Wilson coefficients for deeply inelastic scattering for the structure functions F2(x, Q2), FL(x, Q2), xF3(x, Q2) in the overline{textrm{MS}} scheme and the polarized Wilson coefficients of the structure function g1(x, Q2) in the Larin scheme up to three-loop order in QCD in a fully automated way based on the method of arbitrary high Mellin moments. We work in the Larin scheme in the case of contributing axial-vector couplings or polarized nucleons. For the unpolarized structure functions we compare to results given in the literature. The polarized three-loop Wilson coefficients are calculated for the first time. As a by-product we also obtain the quarkonic three-loop anomalous dimensions from the O(1/ε) terms of the unrenormalized forward Compton amplitude. Expansions for small and large values of the Bjorken variable x are provided.

Factorization and Sudakov resummation in leptonic radiative B decay — a reappraisal

The B-meson light-cone distribution amplitude is an important non- perturbative quantity arising in the factorization of the amplitudes for many exclusive decays of B mesons, such as B− → γℓ− overline{nu} . We reconsider the renormalization-group (RG) equation satisfied by this function and present its solution at next-to-leading order (NLO) in RG-improved perturbation theory in Laplace space and, for the first time, in momentum space and the so-called diagonal (or dual) space. Since the information needed to describe the B decay processes at leading order in ΛQCD/mb is most directly contained in the distribution amplitude in Laplace space evaluated near the origin, we propose a convenient parameterization of this object in terms of a small set of uncorrelated hadronic parameters. Using recent results on the three-loop anomalous dimension for heavy-light current operators, we derive an expression for the convolution integral appearing in the B− → γℓ− overline{nu} factorization formula that is explicitly scale independent, and we evaluate this formula at (approximate) NNLO.

Three-loop soft anomalous dimensions in QCD

I present results for soft anomalous dimensions through three loops for many QCD processes. In particular, I give detailed expressions for soft anomalous dimensions in various processes with electroweak and Higgs bosons as well as single top quarks and top-antitop pairs.

The three-loop anomalous dimension and the four-loop β-function for mathcal{N} = 1 SQED regularized by higher derivatives

For mathcal{N} = 1 SQED with Nf flavors regularized by higher derivatives in the general ξ-gauge we calculate the three-loop anomalous dimension of the matter superfields defined in terms of the bare coupling constant and demonstrate its gauge independence. After this the four-loop β-function defined in terms of the bare coupling constant is obtained with the help of the NSVZ equation, which is valid for these renormalization group functions in all loops. Next, we calculate the three-loop anomalous dimension and the four-loop β-function defined in terms of the renormalized coupling constant for an arbitrary subtraction scheme supplementing the higher derivative regularization. Then we construct a renormalization prescription for which the results coincide with the ones in the overline{mathrm{DR}} -scheme and describe all NSVZ schemes in the considered approximation. Also we demonstrate the existence of a subtraction scheme in which the anomalous dimension does not depend on Nf, while the β-function contains only terms of the first order in Nf. This scheme is obtained with the help of a finite renormalization compatible with a structure of quantum corrections and is NSVZ. The existence of such an NSVZ scheme is also proved in all loops.

Operator product expansion of the non-local gluon condensate

We consider the short-distance expansion of the product of two gluon field strength tensors connected by a straight-line-ordered Wilson line. The vacuum expectation value of this nonlocal operator is a common object in studies of the QCD vacuum structure, whereas its nucleon expectation value is known as the gluon quasi-parton distribution and is receiving a lot of attention as a tool to extract gluon distribution functions from lattice calculations. Extending our previous study [1], we calculate the three-loop coefficient functions of the scalar operators in the operator product expansion up to dimension four. As a by-product, the three-loop anomalous dimension of the nonlocal two-gluon operator is obtained as well.

Groomed jet mass at high precision

We present predictions of the distribution of groomed heavy jet mass in electron-positron collisions at the next-to-next-to-leading order accuracy matched with the resummation of large logarithms to next-to-next-to-next-to-leading logarithmic accuracy. Resummation at this accuracy is possible through extraction of necessary two-loop constants and three-loop anomalous dimensions from fixed-order codes.

Two- and three-loop anomalous dimensions of Weinberg’s dimension-sixCP-odd gluonic operator

We apply a fully automated extension of the $R^*$-operation capable of calculating higher-loop anomalous dimensions of n-point Green's functions of arbitrary, possibly non-renormalisable, local Quantum Field Theories. We focus on the case of the CP-violating Weinberg operator of the Standard Model Effective Field Theory whose anomalous dimension is so far known only at one loop. We calculate the two-loop anomalous dimension in full QCD and the three-loop anomalous dimensions in the limit of pure Yang-Mills theory. We find sizeable two-loop and large three-loop corrections, due to the appearance of a new quartic group invariant. We discuss phenomenological implications for electric dipole moments and future applications of the method.

The polarized three-loop anomalous dimensions from on-shell massive operator matrix elements

We calculate all contributions ∝TF to the polarized three–loop anomalous dimensions in the M–scheme using massive operator matrix elements and compare to results in the literature. This includes the complete anomalous dimensions γqq(2),PS and γqg(2). We also obtain the complete two–loop polarized anomalous dimensions in an independent calculation. While for most of the anomalous dimensions the usual direct computation methods in Mellin N–space can be applied since all recurrences factorize at first order, this is not the case for γqg(2). Due to the necessity of deeper expansions of the master integrals in the dimensional parameter ε=D−4, we had to use the method of arbitrary high moments to eliminate elliptic contributions in intermediate steps. 4000 moments were generated to determine this anomalous dimension and 2640 moments turned out to be sufficient. As an aside, we also recalculate the contributions ∝TF to the three–loop QCD β–function.

Soft-gluon and Coulomb corrections to hadronic top-quark pair production beyond NNLO

We construct a resummation at partial next-to-next-to-next-to-leading logarithmic accuracy for hadronic top-quark pair production near partonic threshold, including simultaneously soft-gluon and Coulomb corrections, and use this result to obtain approximate next-to-next-to-next-to-leading order predictions for the total top-quark pair-production cross section at the LHC. We generalize a required one-loop potential in non-relativistic QCD to the colour-octet case and estimate the remaining unknown twoloop potentials and three-loop anomalous dimensions. We obtain a moderate correction of 1.5% relative to the next-to-next-to-leading order prediction and observe a reduction of the perturbative uncertainty below ±5%.

The a-function for gauge theories

The a-function is a proposed quantity defined for quantum field theories which has a monotonic behaviour along renormalisation group flows, being related to the beta-functions via a gradient flow equation involving a positive definite metric. We construct the a-function at four loop order for a general gauge theory with fermions and scalars, using only one and two loop beta-functions; we are then able to provide a stringent consistency check on the general three-loop gauge beta-function. In the case of an N=1 supersymmetric gauge theory, we present a general condition on the chiral field anomalous dimension which guarantees an exact all-orders expression for the a-function; and we verify this up to fifth order (corresponding to the three-loop anomalous dimension).

Renormalization group analysis for quantum gravity with a single dimensionless coupling

We study the quantum conformal gravity whose dynamics is governed by a single dimensionless gravitational coupling with negative beta function. Since the Euler term is not dynamical classically, the constant in front of it is not an independent coupling. Quantum mechanically, however, it induces the Riegert conformal-factor dynamics with BRST conformal symmetry representing background free nature. In this paper, we propose how to handle the Euler term systematically incorporating such dynamics on the basis of renormalization group analysis using dimensional regularization. As a non-trivial test of renormalization, we explicitly calculate the three-loop anomalous dimension of the cosmological constant operator and show that it agrees with the exact expression derived using the BRST conformal symmetry. The physical significance to inflation and CMB is also discussed.

Anomalous dimensions of gauge fields and gauge coupling beta-functions in the Standard Model at three loops

We present the results for three-loop gauge field anomalous dimensions in the SM calculated in the background field gauge within the unbroken phase of the model. The results are valid for the general background field gauge parameterized by three independent parameters. Both quantum and background fields are considered. The former are used to find three-loop anomalous dimensions for the gauge-fixing parameters, and the latter allow one to obtain the three-loop SM gauge beta-functions. Independence of beta-functions of gauge-fixing parameters serves as a validity check of our final results.

The 16th moment of the three-loop anomalous dimension of the non-singlet transversity operator in QCD

We present the result of the three-loop anomalous dimension of non-singlet transversity operator in QCD for the Mellin moment N=16. The obtained result coincides with the prediction from arXiv:1203.1022 and can serve as a confirmation of the correctness of the general expression for three-loop anomalous dimension of non-singlet transversity operator in QCD for the arbitrary Mellin moment.

Three-loop anomalous dimension of the non-singlet transversity operator in QCD

We calculate the three-loop anomalous dimension of the non-singlet transverse operator from N=1 to N=15. Using some guess we have reconstructed a general form of three-loop anomalous dimension for arbitrary Mellin moment N. Obtained result is transformed into Bjorken-x space by an inverse Mellin transformation. The final expressions are presented in both Mellin-N and Bjorken-x space.

Hidden symmetry of four-point correlation functions and amplitudes in [formula omitted] SYM

We study the four-point correlation function of stress-tensor supermultiplets in N=4 SYM using the method of Lagrangian insertions. We argue that, as a corollary of N=4 superconformal symmetry, the resulting all-loop integrand possesses an unexpected complete symmetry under the exchange of the four external and all the internal (integration) points. This alone allows us to predict the integrand of the three-loop correlation function up to four undetermined constants. Further, exploiting the conjectured amplitude/correlation function duality, we are able to fully determine the three-loop integrand in the planar limit. We perform an independent check of this result by verifying that it is consistent with the operator product expansion, in particular that it correctly reproduces the three-loop anomalous dimension of the Konishi operator. As a byproduct of our study, we also obtain the three-point function of two half-BPS operators and one Konishi operator at three-loop level. We use the same technique to work out a compact form for the four-loop four-point integrand and discuss the generalisation to higher loops.

Three-loop anomalous dimensions for squarks in supersymmetric QCD

In this Letter we evaluate the renormalization constants and anomalous dimensions for the squark wave function and mass within supersymmetric QCD. These results complement the ones obtained in Harlander et al. (2009) [1] and thus provide further confirmation on the applicability of dimensional reduction to supersymmetric QCD at three-loop order. The three-loop anomalous dimension constitute important input to precision predictions of the supersymmetric mass spectrum as obtained from the evolution from the GUT to the TeV energy scale.

ϵKat next-to-next-to-leading order: The charm-top-quark contribution

We perform a next-to-next-to-leading order (NNLO) QCD analysis of the charm-top-quark contribution eta_ct to the effective Delta S = 2 Hamiltonian in the Standard Model. eta_ct represents an important part of the short distance contribution to the parameter epsilon_K. We calculate the three-loop anomalous dimension of the leading operator Q_S2, the three-loop mixing of the current-current and penguin operators into Q_S2, and the corresponding two-loop matching conditions at the electroweak, the bottom-quark, and the charm-quark scale. As our final numerical result we obtain eta_ct = 0.496 +/- 0.047, which is roughly 7% larger than the next-to-leading-order (NLO) value eta_ct(NLO) = 0.457 +/- 0.073. This results in a prediction for epsilon_K = (1.90 +/- 0.26) x 10^(-3), which corresponds to an enhancement of approximately 3.3% with respect to the value obtained using eta_ct(NLO).

Analytic three-loop solutions for [formula omitted] SYM twist operators

We introduce a method to obtain the analytic solution of the higher-order Baxter equation for twist-two and twist-three operators of planar N = 4 SYM. Our result proofs the conjectured formula for the three-loop anomalous dimension of twist-two operators. As such we derive the maximally transcendental part of the corresponding three-loop QCD result from the maximal supersymmetric gauge theory in four dimension purely by methods of integrability.