2ε Dimensions Research Articles

Dimensional regularization and two-loop vacuum polarization operator: Master integrals, analytic results, and energy shifts

We present a complete reevaluation of the irreducible two-loop vacuum-polarization correction to the photon propagator in quantum electrodynamics, i.e. with an electron-positron pair in the fermion propagators. The integration is carried out by reducing the integrations to a limited set of master integrals, which are calculated using integration-by-parts identities. Dimensional regularization is used in D=4−2ε dimensions, and on-mass shell renormalization is employed. The one-loop effect is given to order ε, to be combined with the 1/ε divergence of the two-loop amplitude. Master integrals are given. Final evaluations of two-loop energy shifts for 1S, 2S, and 2P states are done analytically, and results are presented, with an emphasis on muonic hydrogen. For relativistic Dirac-Coulomb reference states, higher-order coefficients are obtained for the Zα-expansion. We compare the results obtained to the existing literature. Published by the American Physical Society 2024

Non-linear non-renormalization theorems

We study the mixing of operators under renormalization group flow in quantum theories, and prove a non-renormalization theorem at non-linear order. It dictates zeros up to a certain number of loops in anomalous dimension tensors that control, for example, the mixing of operators at order dimension six squared into dimension eight. We obtain new results at up to three loops for the mass dimension eight anomalous dimension tensor of ϕ4 theory in D = 4 − 2ε dimensions and verify the zeros predicted by the theorem.

The massless non-adjacent double off-shell scalar box integral — branch cut structure and all-order epsilon expansion

We generalize the result of our recent paper on the massless single off-shell scalar box integral to the case of two non-adjacent end points off the light cone. An analytic result in d = 4 − 2ε dimensions is established in terms of four Gauss hypergeometric 2F1 functions respectively their single-valued counterparts. This allows for an explicit splitting of real and imaginary parts, as well as an all-order ε-expansion in terms of single-valued polylogarithms.

Gluonic evanescent operators: two-loop anomalous dimensions

Evanescent operators are a special class of operators that vanish in four-dimensional spacetime but are non-zero in d = 4−2ϵ dimensions. In this paper, we continue our systematic study of the evanescent operators in the pure Yang-Mills theory and focus on their two-loop renormalization. We develop an efficient strategy to compute the two-loop divergences of form factors of high-dimensional and high-length operators by combining the d-dimensional unitarity method and the improved tensor reduction techniques. Two-loop anomalous dimensions are obtained for the dimension-10 basis in the planar YM theory, for which both the overline{textrm{MS}} scheme and the finite-renormalization scheme are used. We verify that the two-loop anomalous dimensions are the same in these two schemes at the Wilson-Fisher conformal fixed point. Our computation shows that the evanescent operators are indispensable in order to obtain the correct two-loop anomalous dimensions. This work provides a first computation of the two-loop anomalous dimensions of the complete set of dimension-10 operators. The method we use is also expected to provide an efficient strategy for the two-loop renormalization of general high-dimensional operators.

Master integrals for mathcal{O} (ααs) corrections to H → ZZ*

We present analytic results for all the Feynman integrals relevant for mathcal{O} (ααs) virtual corrections to H → ZZ* decay. We use the method of differential equations to solve the master integrals while keeping the full dependence on the masses of all the particles, including those propagating in the loop. Due to the presence of four mass scales, we encounter multiple square roots. We argue that all the occurring square roots can not be rationalized at the same time as a simultaneous rationalization brings us to integrals over CY3 manifolds. Hence we rationalize only three square roots simultaneously and construct suitable ansätze to obtain dlog-forms containing the square roots after obtaining an epsilon-factorised form for the differential equations in D = 4 − 2ϵ dimensions. We present the alphabet and the analytic form of all the boundary constants that appear in the solutions of the differential equations. The results for master integrals are expressed in terms of Chen’s iterated integrals with dlog one-forms.

Critical exponents from five-loop scalar theory renormalization near six-dimensions

We present five-loop results for the renormalization of various models with a cubic interaction (in d=6−2ε dimensions). For the scalar model and its O(n)-symmetric extension we provide renormalization constants, anomalous dimensions and critical exponents. We discuss in detail the method of calculation, and provide all counterterms up to five loops. This allows one to consider generalizations of the φ3 theory to other symmetries.

Massive kite diagrams with elliptics

We present the results for two-loop massive kite master integrals with elliptics in terms of iterated integrals with algebraic kernels. The key ingredients are new integral representations for sunset subgraphs in d=4−2ε and d=2−2ε dimensions together with differential equations for considered kite master integrals in A+Bε form. The obtained results can be easily generalized to all orders in ε-expansion and show that the class of functions defined as iterated integrals with algebraic kernels may be large enough for writing down results for a large class of massive Feynman diagrams.

The hydrogen atom in D = 3 − 2ϵ dimensions

The nonrelativistic hydrogen atom in D=3−2ϵ dimensions is the reference system for perturbative schemes used in dimensionally regularized nonrelativistic effective field theories to describe hydrogen-like atoms. Solutions to the D-dimensional Schrödinger–Coulomb equation are given in the form of a double power series. Energies and normalization integrals are obtained numerically and also perturbatively in terms of ϵ. The utility of the series expansion is demonstrated by the calculation of the divergent expectation value 〈(V′)2〉.

Higher spin currents in the critical O(N) vector model at 1/N2

We calculate the anomalous dimensions of higher spin singlet currents in the critical O(N ) vector model at order 1/N2. The results are shown to be in agreement with the four-loop perturbative computation in ϕ4 theory in 4 − 2ϵ dimensions. It is known that the order 1/N anomalous dimensions of higher-spin currents happen to be the same in the Gross-Neveu and the critical vector model. On the contrary, the order 1/N2 corrections are different. The results can also be interpreted as a prediction for the two-loop computation in the dual higher-spin gravity.

Explicit calculation of multi-fold contour integrals of certain ratios of Euler gamma functions. Part 1

In this paper, we proceed to study properties of Mellin–Barnes (MB) transforms of Usyukina–Davydychev (UD) functions. In our previous papers (Allendes et al., 2013 [13], Kniehl et al., 2013 [14]), we showed that multi-fold Mellin–Barnes (MB) transforms of Usyukina–Davydychev (UD) functions may be reduced to two-fold MB transforms and that the higher-order UD functions may be obtained in terms of a differential operator by applying it to a slightly modified first UD function. The result is valid in d=4 dimensions, and its analog in d=4−2ε dimensions exits, too (Gonzalez and Kondrashuk, 2013 [6]). In Allendes et al. (2013) [13], the chain of recurrence relations for analytically regularized UD functions was obtained implicitly by comparing the left-hand side and the right-hand side of the diagrammatic relations between the diagrams with different loop orders. In turn, these diagrammatic relations were obtained using the method of loop reduction for the triangle ladder diagrams proposed in 1983 by Belokurov and Usyukina. Here, we reproduce these recurrence relations by calculating explicitly, via Barnes lemmas, the contour integrals produced by the left-hand sides of the diagrammatic relations. In this a way, we explicitly calculate a family of multi-fold contour integrals of certain ratios of Euler gamma functions. We make a conjecture that similar results for the contour integrals are valid for a wider family of smooth functions, which includes the MB transforms of UD functions.

ϕ 3 theory with F4 flavor symmetry in 6 − 2ϵ dimensions: 3-loop renormalization and conformal bootstrap

We consider $\phi^3$ theory in $6-2\epsilon$ with $F_4$ global symmetry. The beta function is calculated up to 3 loops, and a stable unitary IR fixed point is observed. The anomalous dimensions of operators quadratic or cubic in $\phi$ are also computed. We then employ conformal bootstrap technique to study the fixed point predicted from the perturbative approach. For each putative scaling dimension of $\phi$ ($\Delta_{\phi})$, we obtain the corresponding upper bound on the scaling dimension of the second lowest scalar primary in the ${\mathbf 26}$ representation $(\Delta^{\rm 2nd}_{{\mathbf 26}})$ which appears in the OPE of $\phi\times\phi$. In $D=5.95$, we observe a sharp peak on the upper bound curve located at $\Delta_{\phi}$ equal to the value predicted by the 3-loop computation. In $D=5$, we observe a weak kink on the upper bound curve at $(\Delta_{\phi},\Delta^{\rm 2nd}_{{\mathbf 26}})$=$(1.6,4)$.

Fun with higher-loop Feynman diagrams

We review recent progress that we have achieved in evaluating the class of fully massive vacuum integrals at five loops. After discussing topics that arise in classification, evaluation and algorithmic codification of this specific set of Feynman integrals, we present some selected new results for their expansions around 4 — 2ε dimensions.

QCD Evolution Equations from Conformal Symmetry

QCD evolution equations in MS-like schemes can be recovered from the same equations in a modified theory, QCD in non-integer d = 4 - 2ϵ dimensions, which enjoys exact scale and conformal invariance at the critical point. Restrictions imposed by the conformal symmetry of the modified theory allow one to obtain complete evolution kernels in integer (physical) dimensions at the given order of perturbation theory from the spectrum of anomalous dimensions added by the calculation of the special conformal anomaly at one order less. We use this technique to derive two-loop evolution equations for flavor-nonsinglet quark-antiquark light-ray operators that encode the scale dependence of generalized hadron parton distributions.

New four-dimensional integrals by Mellin–Barnes transform

This paper is devoted to the calculation of a special class of integrals by Mellin–Barnes transform. It contains double integrals in the position space in d=4−2ϵ dimensions, where ϵ is parameter of dimensional regularization. These integrals contribute to the effective action of the N=4 supersymmetric Yang–Mills theory. The integrand is a fraction in which the numerator is the logarithm of the ratio of space-time intervals, and the denominator is the product of powers of space-time intervals. According to the method developed in the previous papers, in order to make use of the uniqueness technique for one of two integrations, we shift exponents in powers in the denominator of integrands by some multiples of ϵ. As the next step, the second integration in the position space is done by Mellin–Barnes transform. For normalizing procedure, we reproduce first the known result obtained earlier by Gegenbauer polynomial technique. Then, we make another shift of exponents in powers in the denominator to create the logarithm in the numerator as the derivative with respect to the shift parameter δ. We show that the technique of work with the contour of the integral modified in this way by using Mellin–Barnes transform repeats the technique of work with the contour of the integral without such a modification. In particular, all the operations with a shift of contour of integration over complex variables of twofold Mellin–Barnes transform are the same as before the δ modification of indices, and even the poles of residues coincide. This confirms the observation made in the previous papers that in the position space all the Green’s function of N=4 supersymmetric Yang–Mills theory can be expressed in terms of Usyukina–Davydychev functions.

Analytic result for the one-loop scalar pentagon integral with massless propagators

The method of dimensional recurrences proposed by Tarasov (1996, 2000) [1,2] is applied to the evaluation of the pentagon-type scalar integral with on-shell external legs and massless internal lines. For the first time, an analytic result valid for arbitrary space–time dimension d and five arbitrary kinematic variables is presented. An explicit expression in terms of the Appell hypergeometric function F3 and the Gauss hypergeometric function F12, both admitting one-fold integral representations, is given. In the case when one kinematic variable vanishes, the integral reduces to a combination of Gauss hypergeometric functions F12. For the case when one scalar invariant is large compared to the others, the asymptotic values of the integral in terms of Gauss hypergeometric functions F12 are presented in d=2−2ε, 4−2ε, and 6−2ε dimensions. For multi-Regge kinematics, the asymptotic value of the integral in d=4−2ε dimensions is given in terms of the Appell function F3 and the Gauss hypergeometric function F12.

Where do the tedious products of ζ's come from?

Lamentably, the full analytical content of the ε-expansion of the master two-loop two-point function, with arbitrary self-energy insertions in 4−2ε dimensions, is still unknown. Here we show that multiple zeta values (MZVs) of weights up to 12 suffice through O( ε 9). Products of primitive MZVs are generated by a processes of “pseudo-exponentiation” whose combinatorics faithfully accord with expectations based on Kreimer's modified shuffle product and on the Drinfeld-Deligne conjecture. The existence of such a mechanism, relating thousands of complicated rational numbers, enables us to identify precise and simple combinations of MZVs specific to quantum field theories in even numbers of spacetime dimensions.

QCD corrections to the production of a heavy quark pair plus a hard photon in e+e− annihilation

We present complete results for the O( α s ) corrections to the production of a heavy quark pair plus a hard photon, in e + e − annihilation, including both photon and Z intermediate states. Virtual corrections are calculated analytically in 4 - 2ϵ dimensions, together with the soft approximation to the real emission diagrams. After cancellation of infrared divergences, real and virtual contributions are combined numerically to obtain the cross section for the production of two heavy quark jets plus a hard photon in next-to-leading order. The resulting fully differential cross section can be used to construct arbitrary distributions with the desired experimental cuts.

Dimensionally regulated one-loop integrals

Abstract We describe methods for evaluating one-loop integrals in 4−2ϵ dimensions. We give a recursion relation that expresses the scalar n-point integral as a cyclicly symmetric combination of (n−1)-point integrals. The computation of such integrals thus reduces to the calculation of box diagrams (n=4). The tensor integrals required in gauge theory may be obtained by differentiating the scalar integral with respect to certain combinations of the kinematic variables. Such relations also lead to differential equations for scalar integrals. For box integrals with massless internal lines these differential equations are easy to solve.

New approach to evaluation of multiloop Feynman integrals: The Gegenbauer polynomial x-space technique

We formulate the Gegenbauer polynomial x-space technique for evaluating dimensionally regularized massless Feynman diagrams. We show that the analytical evaluation of counterterms in the minimal subtraction scheme at the 3-loop level in any model and for any composite operator is really possible. We calculate exactly in 4 − 2ε dimensions the 2-loop massless propagator-type integrals needed in computing 3-loop counterterms. At the 4-loop level the larger part of diagrams, consisting of those with a propagator-type subgraph and some others, has proved no more difficult. We also explain how calculations can be strikingly simplified in a certain subtraction scheme - the G-scheme. As an application of the developed methods we calculate the 3-loop contribution to σ tot(e +e − →hadrons) and briefly discuss the results together with their implications for the QED β-function.