Hooft-Veltman Scheme Research Articles

Spurious gauge-invariance and γ5 in dimensional regularization

Dimensional regularization is arguably the most popular and efficient scheme for multi-loop calculations. Yet, when applied to chiral (gauge) theories like the Standard Model and its extensions, one is forced to deal with the infamous “γ5 problem”. The only formulation that has been demonstrated to be consistent at all orders in perturbation theory, known as Breiteinlohner-Maison-’t Hooft-Veltman scheme, is rather cumbersome because of the lack of manifest chiral gauge-invariance. In this paper we point out that this drawback can be alleviated by the introduction of auxiliary fields that restore a spurious version of gauge-invariance. If combined with the background field method, all 1PI amplitudes and the associated counterterms are formally covariant and thus severely constrained by the symmetries. As an illustration we evaluate the symmetry-restoring counterterms at 1-loop in the most general renormalizable gauge theory with Dirac fermions and scalar fields, the Standard Model representing a particular example.

Finite basis topologies for multiloop high-multiplicity Feynman integrals

In this work, we systematically analyze Feynman integrals in the ‘t Hooft-Veltman scheme. We write an explicit reduction resulting from partial fractioning the high-multiplicity integrands to a finite basis of topologies at any given loop order. We find all of these finite basis topologies at two loops in four external dimensions. Their maximal cut and the leading singularity are expressed in terms of the Gram determinant and Baikov polynomial. By performing an integration-by-parts reduction without any cut constraint on a numerical probe for one of these topologies, we show that the computational complexity drops significantly compared to the conventional dimensional regularization scheme. Formally, our work implies an upper bound on the rigidity of special functions appearing in the iterated integral solutions at each loop order in perturbative quantum field theory. Phenomenologically, the integrand-level reduction we present will substantially simplify the task of providing high-precision predictions for future high-multiplicity collider observables. Published by the American Physical Society 2024

Two-loop mixed QCD-electroweak amplitudes for Z+jet production at the LHC: bosonic corrections

We present a calculation of the bosonic contribution to the two-loop mixed QCD-electroweak scattering amplitudes for Z-boson production in association with one hard jet at hadron colliders. We employ a method to calculate amplitudes in the ’t Hooft-Veltman scheme that reduces the amount of spurious non-physical information needed at intermediate stages of the computation, to keep the complexity of the calculation under control. We compute all the relevant Feynman integrals numerically using the Auxiliary Mass Flow method. We evaluate the two-loop scattering amplitudes on a two-dimensional grid in the rapidity and transverse momentum of the Z boson, which has been designed to yield a reliable numerical sampling of the boosted-Z region. This result provides an important building block for improving the theoretical modelling of a key background for monojet searches at the LHC.

Five-parton scattering in QCD at two loops

We compute all helicity amplitudes for the scattering of five partons in two-loop QCD in all the relevant flavor configurations, retaining all contributing color structures. We employ tensor projection to obtain helicity amplitudes in the ’t Hooft-Veltman scheme starting from a set of primitive amplitudes. Our analytic results are expressed in terms of massless pentagon functions, and are easy to evaluate numerically. These amplitudes provide important input to investigations of soft-collinear factorization and to studies of the high-energy limit. Published by the American Physical Society 2024

Full three-loop renormalisation of an abelian chiral gauge theory with non-anticommuting γ5 in the BMHV scheme

In this work we present a complete three-loop renormalisation of an abelian chiral gauge theory within the Breitenlohner-Maison/’t Hooft-Veltman (BMHV) scheme of dimensional regularisation (DReg). In this scheme the γ5-matrix appearing in gauge interactions is a non-anticommuting object, leading to a breaking of gauge and BRST invariance. Employing an efficient method based on the quantum action principle, we obtain the complete three-loop counterterm action which serves both to render the theory finite and to restore gauge and BRST invariance. The UV singular counterterms involve not only higher order ϵ-poles but also new counterterm structures emerging at the three-loop level for the first time; the finite symmetry-restoring counterterms are restricted to the same structures as at lower loop orders, just with different coefficients, aligning with our expectations. Both the singular and the finite counterterms include structures which cannot be obtained by the standard multiplicative renormalisation. Our results demonstrate that a rigorous treatment of chiral gauge theories with γ5 defined in the BMHV scheme at the multi-loop level is possible and that the obtained counterterm action is suitable for computer implementations, allowing automated calculations without ambiguities caused by γ5.

Low-energy effective field theory below the electroweak scale: one-loop renormalization in the ’t Hooft-Veltman scheme

The low-energy effective field theory below the electroweak scale (LEFT) describes the effects at low energies of both the weak interaction and physics beyond the Standard Model. We study the one-loop renormalization of the LEFT in the ’t Hooft-Veltman scheme, which offers an algebraically consistent definition of the Levi-Civita symbol and γ5 in dimensional regularization. However, in connection with minimal subtraction this scheme leads to a spurious breaking of chiral symmetry in intermediate steps of the calculation. Based on the ’t Hooft-Veltman prescription, we define a renormalization scheme that restores chiral symmetry by including appropriate finite counterterms. To this end, we extend the physical LEFT operator basis by a complete set of off-shell and one-loop-evanescent operators and we perform the renormalization at one loop. We determine the finite counterterms to the physical parameters that compensate both the insertions of evanescent operators, as well as the chiral-symmetry-breaking terms from the renormalizable part of the Lagrangian in D dimensions. Our results can be applied in next-to-leading-log calculations in the ’t Hooft-Veltman scheme: using our renormalization scheme instead of pure minimal subtraction separates the physical sector from the unphysical evanescent sector and leads to results that are manifestly free of spurious chiral-symmetry-breaking terms.

One-loop matching of the CP-odd three-gluon operator to the gradient flow

The calculation of the neutron electric dipole moment within effective field theories for physics beyond the Standard Model requires non-perturbative hadronic matrix elements of effective operators composed of quark and gluon fields. In order to use input from lattice computations, these matrix elements must be translated from a scheme suitable for lattice QCD to the minimal-subtraction scheme used in the effective-field-theory framework. The accuracy goal in the context of the neutron electric dipole moment necessitates at least a one-loop matching calculation. Here, we provide the one-loop matching coefficients for the CP-odd three-gluon operator between two different minimally subtracted 't Hooft–Veltman schemes and the gradient flow. This completes our program to obtain the one-loop gradient-flow matching coefficients for all CP-violating and flavor-conserving operators in the low-energy effective field theory up to dimension six.

Non-Abelian Carroll–Field–Jackiw term in a Rarita-Schwinger model

In this paper, we demonstrate the possibility of generating a non-Abelian Carroll–Field–Jackiw (CFJ) term in the theory of a non-Abelian gauge field coupled to a spin-3/2 field in the presence of the constant axial vector field. Applying two regularization schemes, we prove that this term is finite and ambiguous, particularly vanishing within the 't Hooft-Veltman scheme.

Three-loop helicity amplitudes for photon+jet production

We present three-loop helicity amplitudes for the production of a single photon in association with one jet in Quantum Chromodynamics, a final state which provides a standard candle of the Standard Model at the Large Hadron Collider. We employ a recently-proposed variation of the so-called tensor projection method in the 't Hooft-Veltman scheme (tHV) which avoids the computation of contributions due to unphysical (-2$\epsilon$)-dimensional polarisations of the external states. We obtain compact analytic results expressed in terms of harmonic polylogarithms.

Three-Loop Gluon Scattering in QCD and the Gluon Regge Trajectory.

We compute the three-loop helicity amplitudes for the scattering of four gluons in QCD. We employ projectors in the 't Hooft-Veltman scheme and construct the amplitudes from a minimal set of physical building blocks, which allows us to keep the computational complexity under control. We obtain relatively compact results that can be expressed in terms of harmonic polylogarithms. In addition, we consider the Regge limit of our amplitude and extract the gluon Regge trajectory in full three-loop QCD. This is the last missing ingredient required for studying single-Reggeon exchanges at next-to-next-to-leading logarithmic accuracy.

Three-loop helicity amplitudes for diphoton production in gluon fusion

We present a calculation of the helicity amplitudes for the process gg → γγ in three-loop massless QCD. We employ a recently proposed method to calculate scattering amplitudes in the ’t Hooft-Veltman scheme that reduces the amount of spurious non-physical information needed at intermediate stages of the computation. Our analytic results for the three-loop helicity amplitudes are remarkably compact, and can be efficiently evaluated numerically. This calculation provides the last missing building block for the computation of NNLO QCD corrections to diphoton production in gluon fusion.

Dimensional regularization and Breitenlohner-Maison/’t Hooft-Veltman scheme for γ5 applied to chiral YM theories: full one-loop counterterm and RGE structure

We study the application of the Breitenlohner-Maison-’t Hooft-Veltman (BMHV) scheme of Dimensional Regularization to the renormalization of chiral gauge theories, focusing on the specific counterterm structure required by the non-anticommuting Dirac γ5 matrix and the breaking of the BRST invariance. Calculations are performed at the one-loop level in a massless chiral Yang-Mills theory with chiral fermions and real scalar fields. We discuss the setup and properties of the regularized theory in detail. Our central results are the full counterterm structures needed for the correct renormalization: the singular UV-divergent counterterms, including evanescent counterterms that have to be kept for consistency of higher-loop calculations.We find that the required singular, evanescent counterterms associated with vector and scalar fields are uniquely determined but are not gauge invariant. Furthermore, using the framework of algebraic renormalization, we determine the symmetry-restoring finite counterterms, that are required to restore the BRST invariance, central to the consistency of the theory. These are the necessary building blocks in one-loop and higher-order calculations.Finally, renormalization group equations are derived within this framework, and the derivation is compared with the more customary calculation in the context of symmetry-invariant regularizations. We explain why, at one-loop level, the extra BMHV-specific counterterms do not change the results for the RGE. The results we find complete those that have been obtained previously in the literature in the absence of scalar fields.

Subleading power corrections to the pion-photon transition form factor in QCD

We reconsider QCD factorization for the leading power contribution to the γ*γ → π0 form factor Fγ*γ→π0(Q2) at one loop using the evanescent operator approach, and demonstrate the equivalence of the resulting factorization formulae derived with distinct prescriptions of γ5 in dimensional regularization. Applying the light-cone QCD sum rules (LCSRs) with photon distribution amplitudes (DAs) we further compute the subleading power contribution to the pion-photon form factor induced by the “hadronic” component of the real photon at the next-to-leading-order in mathcal{O}left({alpha}_sright) , with both naive dimensional regularization and ’t Hooft-Veltman schemes of γ5. Confronting our theoretical predictions of Fγ*γ→π0 (Q2) with the experimental measurements from the BaBar and the Belle Collaborations implies that a reasonable agreement can be achieved without introducing an “exotic” end-point behaviour for the twist-2 pion DA.

Four dimensional helicity scheme beyond one loop

I describe a procedure by which one can transform scattering amplitudes computed in the four dimensional helicity scheme into properly renormalized amplitudes in the 't Hooft-Veltman scheme. I describe a new renormalization program, based upon that of the dimensional reduction scheme and explain how to remove both finite and infrared-singular contributions of the evanescent degrees of freedom to the scattering amplitude.

B-meson semi-inclusive decay to2−+charmonium in nonrelativistic QCD andX(3872)

The semi-inclusive B-meson decay into spin-singlet D-wave $2^{-+}$ charmonium, $B\to \eta_{c2}+X$, is studied in nonrelativistic QCD (NRQCD). Both color-singlet and color-octet contributions are calculated at next-to-leading order (NLO) in the strong coupling constant $\alpha_s$. The non-perturbative long-distance matrix elements are evaluated using operator evolution equations. It is found that the color-singlet $^1D_2$ contribution is tiny, while the color-octet channels make dominant contributions. The estimated branching ratio $B(B\to \eta_{c2}+X)$ is about $0.41\,\times10^{-4}$ in the Naive Dimensional Regularization (NDR) scheme and $1.24\,\times10^{-4}$ in the t'Hooft-Veltman (HV) scheme, with renormalization scale $\mu=m_b=4.8$\,GeV. The scheme-sensitivity of these numerical results is due to cancelation between ${}^1S_0^{[8]}$ and ${}^1P_1^{[8]}$ contributions. The $\mu$-dependence curves of NLO branching ratios in both schemes are also shown, with $\mu$ varying from $\frac{m_b}{2}$ to $2m_b$ and the NRQCD factorization or renormalization scale $\mu_{\Lambda}$ taken to be $2m_c$. Comparison of the estimated branching ratio of $B\to \eta_{c2}+X$ with the observed branching ratio of $B \to X(3872)+K$ may lead to the conclusion that X(3872) is unlikely to be the $2^{-+}$ charmonium state $\eta_{c2}$.

Two-loop splitting functions in QCD

We present the universal two-loop splitting functions that describe the limits of two-loop $n$-point amplitudes of massless particles when two of the momenta are collinear. To derive the splitting amplitudes, we take the collinear limits of explicit two-loop four-point helicity amplitudes computed in the 't Hooft-Veltman scheme. The $g \to gg$ splitting amplitude has recently been computed using the unitarity sewing method and we find complete agreement with the results of Ref. \cite{Bern:2lsplit}. The two-loop $q \to qg$ and $g \to q\bar q$ splitting functions are new results. We also provide an expression for the two-loop soft splitting function.

Two-loop QCD helicity amplitudes for massless quark-quark scattering

We present the two-loop helicity amplitudes for the scattering of massless quarks in QCD. We use projector techniques to compute the coefficients of the general four-quark amplitude at up to two-loops in conventional dimensional regularisation and use these coefficients to derive the helicity amplitudes in the 't Hooft-Veltman scheme. The structure of the infrared divergences agrees with that predicted by Catani while expressions for the finite remainders for $q \bar q \to Q\bar Q$ and the processes related by crossing symmetry are given in terms of logarithms and polylogarithms that are real in the physical region. We have checked that the interference of tree and two-loop helicity amplitudes, summed over helicities and colours, reproduces the previous results for the finite remainders for interference of tree and two-loop amplitudes given in Refs. [1,2].

Subtraction terms for one-loop amplitudes with one unresolved parton

Fully differential next-to-next-to-leading order calculations require a method to cancel infrared singularities. In a previous publication, I discussed the general setup for the subtraction method at NNLO. In this paper I give all subtraction terms for electron-positron annihilation associated with one-loop amplitudes with one unresolved parton. These subtraction terms are integrated within dimensional regularization over the unresolved one-particle phase space. The results can be used with all variants of dimensional regularization (conventional dimensional regularization, the 't Hooft-Veltman scheme and the four-dimensional scheme).

Two-loop QCD helicity amplitudes for massless quark-massless gauge boson scattering

We present the two-loop helicity amplitudes for the scattering of massless quarks and massless gauge bosons in QCD. We use projector techniques to compute the coefficients of the general tensor describing the two-quark two-boson amplitude in conventional dimensional regularisation and use these coefficients to derive the helicity amplitudes in the 't Hooft-Veltman scheme. The structure of the infrared divergences agrees with that predicted by Catani while expressions for the finite remainders for q qbar -> gluon gluon, q qbar -> gluon gamma, q qbar -> gamma gamma and the processes related by crossing symmetry are given in terms of logarithms and polylogarithms that are real in the physical region. We have checked that the interference of tree and two-loop helicity amplitudes, summed over helicities and colours, reproduces the previous results for the finite remainders for interference of tree and two-loop amplitudes given in Refs. [1,2]. We also find agreement with the two-loop helicity amplitudes for quark-gluon scattering presented in Ref. [3].

Supersymmetric regularization, two-loop QCD amplitudes, and coupling shifts

We present a definition of the four-dimensional helicity (FDH) regularization scheme valid for two or more loops. This scheme was previously defined and utilized at one loop. It amounts to a variation on the standard 't Hooft-Veltman scheme and is designed to be compatible with the use of helicity states for "observed" particles. It is similar to dimensional reduction in that it maintains an equal number of bosonic and fermionic states, as required for preserving supersymmetry. Supersymmetry Ward identities relate different helicity amplitudes in supersymmetric theories. As a check that the FDH scheme preserves supersymmetry, at least through two loops, we explicitly verify a number of these identities for gluon-gluon scattering (gg to gg) in supersymmetric QCD. These results also cross-check recent non-trivial two-loop calculations in ordinary QCD. Finally, we compute the two-loop shift between the FDH coupling and the standard MS-bar coupling, alpha_s. The FDH shift is identical to the one for dimensional reduction. The two-loop coupling shifts are then used to obtain the three-loop QCD beta function in the FDH and dimensional reduction schemes.