Renormalizable Gauge Research Articles

Gauge-invariant quantum fields

Gauge-invariant quantum fields are constructed in an Abelian power-counting renormalizable gauge theory with both scalar, vector and fermionic matter content. This extends previous results already obtained for the gauge-invariant description of the Higgs mode via a propagating gauge-invariant field. The renormalization of the model is studied in the Algebraic Renormalization approach. The decomposition of Slavnov–Taylor identities into separately invariant sectors is analyzed. We also comment on some non-renormalizable extensions of the model whose 1-PI Green’s functions are the flows of certain differential equations of the homogeneous Euler type, exactly resumming the dependence on a certain set of dim. 6 and dim. 8 derivative operators. The latter are identified uniquely by the condition that they span the mass and kinetic terms in the gauge-invariant dynamical fields. The construction can be extended to non-Abelian gauge groups.

Precise perturbative predictions from fixed-order calculations

The intrinsic conformality (iCF) is a general property of the renormalizable gauge theory, which ensures the scale invariance of a fixed-order series at each perturbative order. Following the idea of iCF, we suggest a novel single-scale setting approach under the principle of maximum conformality (PMC) with the purpose of removing the conventional renormalization scheme and scale ambiguities. We call this newly suggested single-scale procedure the PMC∞-s approach, in which an overall effective α s , and hence an overall effective scale is achieved by identifying the {β 0} terms at each order. Its resultant conformal series is scale-invariant and satisfies all renormalization group requirements. The PMC∞-s approach is applicable to any perturbatively calculable observable, and its resultant perturbative series provides an accurate basis for estimating the contribution from the unknown higher-order (UHO) terms. Using the Higgs decay into two gluons up to five-loop quantum chromondynamics (QCD) corrections as an example, we show how the PMC∞-s works, and we obtain and . Here the errors are squared averages of those mentioned in the body of the text. The approximation approach (PAA) and the Bayesian approach (BA) have been adopted to estimate the contributions from the UHO terms. We also demonstrate that the PMC∞-s approach is equivalent to our previously suggested single-scale setting approach (PMCs), which also follows from the PMC but treats the {β i } terms from a different point of view. Thus, proper use of the renormalization group equation can provide a solid way to solve the scale-setting problem.

Weak charges quantization in SU(3)c ⊗ SU(n)L ⊗ U(1)Y gauge models

After proving, in a previous paper, that the electric charge quantization occurs as a natural consequence in renormalizable gauge models, we take here a step further within the same paradigm in order to obtain the precise weak charges quantization. To this end a viable boson mass spectrum is obtained first, once a proper parametrization in the Higgs sector is taken into consideration. Hence, by diagonalizing the neutral bosons mass matrix, the quantized neutral weak charge operators are obtained. The Standard Model phenomenology is not affected at all, as its scale (v SM = 246 GeV) is decoupled from the higher scale (V ∼ 10 TeV) specific to our generalized electro-weak unification.

Lorentz invariance of basis tensor gauge theory

Basis tensor gauge theory (BTGT) is a vierbein analog reformulation of ordinary gauge theories in which the vierbein field describes the Wilson line. After a brief review of the BTGT, we clarify the Lorentz group representation properties associated with the variables used for its quantization. In particular, we show that starting from an SO(1,3) representation satisfying the Lorentz-invariant U(1,3) matrix constraints, BTGT introduces a Lorentz frame choice to pick the Abelian group manifold generated by the Cartan subalgebra of U(1,3) for the convenience of quantization even though the theory is frame independent. This freedom to choose a frame can be viewed as an additional symmetry of BTGT that was not emphasized before. We then show how an [Formula: see text] permutation symmetry and a parity symmetry of frame fields natural in BTGT can be used to construct renormalizable gauge theories that introduce frame-dependent fields but remain frame independent perturbatively without any explicit reference to the usual gauge field.

Techni-Pati-Salam composite Higgs model

Composite Higgs models can be extended to the Planck scale by means of the partially unified partial compositeness (PUPC) framework. We present in detail the Techni-Pati-Salam model, based on a renormalizable gauge theory $SU(8)_{PS}\times SU(2)_L\times SU(2)_R$. We demonstrate that masses and mixings for all generations of standard model fermions can be obtained via partial compositeness at low energy, with four-fermion operators mediated by either heavy gauge bosons or scalars. The strong dynamics is predicted to be that of a confining $Sp(4)_{\rm HC}$ gauge group, with hyper-fermions in the fundamental and two-index anti-symmetric representations, with fixed multiplicities. This motivates for Lattice studies of the Infra-Red near-conformal walking phase, with results that may validate or rule out the model. This is the first complete and realistic attempt at providing an Ultra-Violet completion for composite Higgs models with top partial compositeness. In the baryon-number conserving vacuum, the theory also predicts a Dark Matter candidate, with mass in the few TeV range, protected by semi-integer baryon number.

B−L model based on Q4 symmetry for fermion spectrum with normal neutrino mass ordering

We propose a renormalizable gauge [Formula: see text] extension of the Standard Model (SM) based on [Formula: see text] symmetry and an auxiliary [Formula: see text] symmetry which can explain the observed quark and lepton masses and mixing angles associated to normal neutrino mass ordering through type-I seesaw mechanism. The relation between the atmospheric mixing angle [Formula: see text] and the effective parameters in neutrino sector is analyzed. Two Majorana phases are predicted to be [Formula: see text] and [Formula: see text] and the model also predicts the effective neutrino mass parameters of [Formula: see text], [Formula: see text] which is well consistent with the planning of future experiments. In the quark sector, the model is predictive since it has ten effective parameters that allow to successfully reproduce the experimental values of the experimental values of the ten physical observables of the quark sector.

General renormalization group equations for dimensionful couplings in the MS¯ scheme

Recently, Poole & Thomsen have presented the Renormalization Group Equations (RGEs) in the $\overline{\mathrm{MS}}$ scheme for dimensionless parameters of a general renormalizable gauge theory in a new formalism based on the Local Renormalization Group. In this Letter, we apply the dummy field method to the expressions in this formalism in order to derive the RGEs for the dimensionful couplings which so far have been missing in this approach. The complete set of RGEs has been implemented in a new version of the public code PyR@TE (arXiv:2007.12700) dedicated to the automatic computation of the RGEs for any general renormalizable (non-supersymmetric) gauge theory.

Infinite-order scale-setting using the principle of maximum conformality: A remarkably efficient method for eliminating renormalization scale ambiguities for perturbative QCD

We identify a property of renormalizable SU(N)/U(1) gauge theories, the intrinsic Conformality ($iCF$), which underlies the scale invariance of physical observables and leads to a remarkably efficient method to solve the conventional renormalization scale ambiguity at every order in pQCD: the PMC$_\infty$. This new method reflects the underlying conformal properties displayed by pQCD at NNLO, eliminates the scheme dependence of pQCD predictions and is consistent with the general properties of the PMC (Principle of Maximum Conformality). We introduce a new method to identify conformal and $\beta$-terms which can be applied either to numerical or to theoretical calculations. We illustrate the PMC$_\infty$ for the thrust and C-parameter distributions in $e^+ e^-$ annihilation and then we show how to apply this new method to general observables in QCD. We point out how the implementation of the PMC$_\infty$ can significantly improve the precision of pQCD predictions; its implementation in multi-loop analysis also simplifies the calculation of higher orders corrections in a general renormalizable gauge theory.

First order formulation of the Yang–Mills theory in a background field

The background gauge renormalization of the first order formulation of the Yang–Mills theory is studied by using the BRST identities. Together with the background symmetry, these identities allow for an iterative proof of renormalizability to all orders in perturbation theory. However, due to the fact that certain improper diagrams which violate the BRST symmetry should be removed, the renormalizability must be deduced indirectly. The recursive method involves rescalings and mixings of the fields, which lead to a renormalized effective action for the background field theory.

The Protophobic \(X\)-boson Coupled to Quantum Electrodynamics

The possible origin of an $X$-boson having a mass value around $17 \, \mbox{MeV}$ had motivated us to investigate its interaction with leptons of QED. This new hypothetical particle can possibly be a candidate to describe the so called fifth interaction, in a new physics scenario beyond the Standard Model. The model of the $X$-boson interacting with QED is based on a $U(1) \times U'(1)$ symmetry, where the group $U'(1)$ is attached to the $X$-boson, with a kinetic mixing with the photon. The Higgs sector was revisited to generate the mass for the new boson. Thus, the mass of $17 \, \mbox{MeV}$ fixes a vacuum expected value scale. Thereby, we could estimate the mass of the hidden Higgs field through both the VEV-scale and the Higgs' couplings. A model of QFT was constructed in a renormalizable gauge, and we analyzed its perturbative structure. After that, the radiative correction of the $X$-boson propagator has been calculated at one loop approximation to yield the Yukawa potential correction. The form factors associated with the QED-vertex correction were calculated to confirm electron's anomalous magnetic moment together with the computation of the interaction magnitude. The muon case was discussed. Furthermore, we have introduced a renormalization group scheme to explore the current $X$-boson mass and its coupling constant with the leptons of the Standard Model.

Revisiting RGEs for general gauge theories

We revisit the renormalisation group equations (RGE) for general renormalisable gauge theories at one- and two-loop accuracy. We identify and correct various mistakes in the literature for the β-functions of the dimensionful Lagrangian parameters (the fermion mass, the bilinear and trilinear scalar couplings) as well as the dimensionless quartic scalar couplings. There are two sources for these discrepancies. Firstly, the known expressions for the scalar couplings assume a diagonal wave-function renormalisation which is not appropriate for models with mixing in the scalar sector. Secondly, the dimensionful parameters have been derived in the literature using a dummy field method which we critically re-examine, obtaining revised expressions for the β-function of the fermion mass. We perform an independent cross-check using well-tested supersymmetric RGEs which confirms our results. The numerical impact of the changes in the β-function for the fermion mass terms is illustrated using a toy model with a heavy vector-like fermion pair coupled to a scalar gauge singlet. Unsurprisingly, the correction to the running of the fermion mass becomes sizeable for large Yukawa couplings of the order of O(1). Furthermore, we demonstrate the importance of the correction to the β-functions of the scalar quartic couplings using a general type-III Two-Higgs-Doublet-Model. All the corrected expressions have been implemented in updated versions of the Mathematica package SARAH and the Python package PyR@TE.

The universal character of Zwanziger's horizon function in Euclidean Yang–Mills theories

In light of the recently established BRST invariant formulation of the Gribov–Zwanziger theory, we show that Zwanziger's horizon function displays a universal character. More precisely, the correlation functions of local BRST invariant operators evaluated with the Yang–Mills action supplemented with a BRST invariant version of the Zwanziger's horizon function and quantized in an arbitrary class of covariant, color invariant and renormalizable gauges which reduce to the Landau gauge when all gauge parameters are set to zero, have a unique, gauge parameters independent result, corresponding to that of the Landau gauge when the restriction to the Gribov region Ω in the latter gauge is imposed. As such, thanks to the BRST invariance, the cut-off at the Gribov region Ω acquires a gauge independent meaning in the class of the physical correlators.

Background gauge renormalization and BRST identities

We show how the BRST identities can be used to control the renormalization of the background gauge in QCD, in spite of the fact that certain one-particle reducible graphs have to be omitted in general. We obtain the all order renormalized effective action for the background field theory.

H → γγ, gauge invariance, and the hierarchy problem

The calculation of [Formula: see text] displays interesting behavior which depends on the regulator used in the integration over loop momenta. If calculated using a gauge-invariant regulator, such as dimensional regularization, the calculation yields a unique, finite, gauge-invariant result. If four-dimensional symmetric regulation is used without finite subtractions, additional pieces occur which spoil QED gauge invariance. In both cases, a finite result is obtained, but the particular finite result depends on the regulator utilized in the calculation. While gauge-invariant regulators such as dimensional regularization are normally used, four-dimensional symmetric integration is also physically motivated. Also, the gauge-invariance-violating terms that arise using four-dimensional symmetric integration are of the same form for the fermionic, scalar, and the SM [Formula: see text] loop calculated in renormalizable gauge. This presents an interesting possibility. Inspired by anomaly cancellation, we ask if it is possible that these gauge-invariance-violating terms may cancel in certain models when contributions from all diagrams are included. Here, we calculate the regulator-dependent contributions to [Formula: see text] arising from generic fermion and scalar loops, as well as the Standard Model [Formula: see text] loop contribution, which we evaluate in renormalizable gauge for general [Formula: see text]. We find that a cancellation between such terms is possible, and derive the cancellation condition. Additionally, we find that this cancellation condition ensures QED gauge invariance without finite subtractions for any regulator used, not just for four-dimensional symmetric integration. We additionally relate the regulator-dependent terms in [Formula: see text] to the behavior of quadratically-divergent Higgs tadpole diagrams under shifts of internal loop momentum. Thus, the cancellation condition for the gauge-invariance-violating terms in [Formula: see text] implies a relation between the quadratic divergences in Higgs tadpole diagrams; this has consequences for hypothesized solutions to the hierarchy problem. Lastly, we find that the MSSM obeys our cancellation condition.

Adler–Bardeen theorem and manifest anomaly cancellation to all orders in gauge theories

We reconsider the Adler-Bardeen theorem for the cancellation of gauge anomalies to all orders, when they vanish at one loop. Using the Batalin-Vilkovisky formalism and combining the dimensional-regularization technique with the higher-derivative gauge invariant regularization, we prove the theorem in the most general perturbatively unitary renormalizable gauge theories coupled to matter in four dimensions, and identify the subtraction scheme where anomaly cancellation to all orders is manifest, namely no subtractions of finite local counterterms are required from two loops onwards. Our approach is based on an order-by-order analysis of renormalization, and, differently from most derivations existing in the literature, does not make use of arguments based on the properties of the renormalization group. As a consequence, the proof we give also applies to conformal field theories and finite theories.

Multi-loop Integrand Reduction with Computational Algebraic Geometry

We discuss recent progress in multi-loop integrand reduction methods. Motivated by the possibility of an automated construction of multi-loop amplitudes via generalized unitarity cuts we describe a procedure to obtain a general parameterisation of any multi-loop integrand in a renormalizable gauge theory. The method relies on computational algebraic geometry techniques such as Gröbner bases and primary decomposition of ideals. We present some results for two and three loop amplitudes obtained with the help of the MACAULAY2 computer algebra system and the Mathematica package BASISDET.

General Relativity as the Classical Limit of the Renormalizable Gauge Theory of Volume Preserving Diffeomorphisms

The different roles and natures of spacetime appearing in a quantum field theory and in classical physics are analyzed implying that a quantum theory of gravitation is not necessarily a quantum theory of curved spacetime. Developing an alternative approach to quantum gravity starts with the postulate that inertial energy-momentum and gravitational energy-momentum need not be the same for virtual quantum states. Separating their roles naturally leads to the quantum gauge field theory of volume-preserving diffeomorphisms of an inner four-dimensional space. The classical limit of this theory coupled to a quantized scalar field is derived for an on-shell particle where inertial energy-momentum and gravitational energy-momentum coincide. In that process the symmetry under volume-preserving diffeomorphisms disappears and a new symmetry group emerges: the group of coordinate transformations of four-dimensional spacetime and with it General Relativity coupled to a classical relativistic point particle.

Global residues and two-loop hepta-cuts

We examine maximal unitarity in the nonplanar case and derive remarkably compact analytic expressions for coefficients of master integrals with two-loop crossed box topology in massless four-point amplitudes in any gauge theory, thereby providing additional steps towards automated computation of the full amplitude. The coefficients are obtained by assembling residues extracted through integration on linear combinations of higher-dimensional tori encircling global poles of the loop integrand. We recover all salient features of two-loop maximal unitarity, such as the existence of unique projectors for each master integral. Several explicit calculations are provided. We also establish exact equivalence of our results and master integral coefficients recently obtained via integrand-level reduction in any renormalizable gauge theory.

Multifield dynamics of Higgs inflation

Higgs inflation is a simple and elegant model in which early-universe inflation is driven by the Higgs sector of the Standard Model. The Higgs sector can support early-universe inflation if it has a large nonminimal coupling to the Ricci spacetime curvature scalar. At energies relevant to such an inflationary epoch, the Goldstone modes of the Higgs sector remain in the spectrum in renormalizable gauges, and hence their effects should be included in the model's dynamics. We analyze the multifield dynamics of Higgs inflation and find that the multifield effects damp out rapidly after the onset of inflation, because of the gauge symmetry among the scalar fields in this model. Predictions from Higgs inflation for observable quantities, such as the spectral index of the power spectrum of primordial perturbations, therefore revert to their familiar single-field form, in excellent agreement with recent measurements. The methods we develop here may be applied to any multifield model with nonminimal couplings in which the $\mathcal{N}$ fields obey an $SO(\mathcal{N})$ symmetry in field space.

On the gauge dependence of vacuum transitions at finite temperature

Abstract In principle, observables as for example the sphaleron rate or the tunneling rate in a first-order phase transition are gauge-independent. However, in practice a gauge dependence is introduced in explicit perturbative calculations due to the breakdown of the gradient expansion of the effective action in the symmetric phase. We exemplify the situation using the effective potential of the Abelian Higgs model in the general renormalizable gauge. Still, we find that the quantitative dependence on the gauge choice is small for gauges that are consistent with the perturbative expansion.