Chiral Symmetry in Implicit Regularization: A Review

Abstract

Chiral interactions pose significant challenges for regularization due to the ${\gamma }_5$ Dirac matrix, which is intrinsically four-dimensional. Dimensional regularizations, while widely employed in gauge theories, encounter challenges when treating ${\gamma }_5$ in $d\ne 4$ dimensions, potentially leading to violations of chiral symmetry and the emergence of spurious anomalies. In this work, we examine aspects of Implicit Regularization, a framework formulated to operate in the physical dimension, thereby potentially avoiding ambiguities associated with ${\gamma }_5$. We discuss its implementation and implications for symmetry preservation in chiral gauge theories.

1. Introduction

Precision tests of the Standard Model (SM), consolidated by the discovery of the Higgs boson at the Large Hadron Collider (LHC) [1,2], require increasingly accurate perturbative calculations. These computations often involve dimension-specific objects, such as the matrix ${\gamma }_5$, which is intrinsically four-dimensional. Regularization schemes like dimensional regularization, while powerful for gauge theories, encounter conceptual and technical difficulties when extended to chiral interactions in $d\ne 4$ dimensions. Improper treatment of ${\gamma }_5$ can lead to violations of Ward identities and spurious anomalies, compromising the consistency of theoretical predictions. With future high-luminosity colliders such as the HL-LHC [3] and the proposed FCC [4] aiming to probe rare processes, the reliability of these calculations becomes critical. This motivates the study and refinement of regularization frameworks capable of preserving symmetries while handling dimension-specific structures.

The calculation of decay rates and cross sections requires evaluating radiative corrections to higher orders in a perturbative expansion. The associated multi-loop computations are highly involved, both analytically and computationally. Ultraviolet and infrared divergences typically arise in the integration over loop momenta and phase-space variables, necessitating the use of regularization techniques.

Ultraviolet (UV) divergences in quantum field theory are addressed through regularization and renormalization. A renormalization scheme specifies the prescription for absorbing these divergences into redefined physical parameters, such as masses and coupling constants, at a chosen renormalization scale $\lambda$. While the details of this procedure depend on the scheme, certain signatures of the renormalization group (RG) remain universal. In particular, the first two coefficients of the perturbative $\beta$-function, which governs the running of coupling constants, are scheme-independent. Moreover, in theories like QCD, where the convergence of perturbative expansions is difficult to assess, the RG equations allow identification of scales where observables exhibit minimal sensitivity to higher-order corrections, as demonstrated in Higgs boson production via gluon fusion at $N^3\mathrm{LO}$ [5].

Infrared (IR) divergences arise at the opposite end of the energy spectrum, typically when particles are massless. These singularities occur in two forms: soft, from emissions carrying arbitrarily low energy that escape detection yet accumulate to infinity, and collinear, from partons aligned with a hard particle such that they cannot be experimentally distinguished. According to the Kinoshita–Lee–Nauenberg (KLN) theorem for non-Abelian gauge theories [6,7], IR divergences cancel in physical observables due to the unitarity of the S-matrix. This cancellation requires summing over all degenerate initial and final states within a given energy resolution. In practice, it involves phase-space integrals over coherently combined real-emission and virtual-loop contributions (after UV renormalization) at a fixed perturbative order. Depending on the subtraction method, the cancellation may occur at the integrand level or only after integration.

An effective regularization method must not only remove divergences but also respect the symmetries of the underlying theory, avoiding spurious anomalies and minimizing the need for symmetry-restoring counterterms, which, although legitimate, can be cumbersome in practice. Gauge invariance, unitarity, and causality are fundamental principles of the Standard Model (SM). ’t Hooft and Veltman demonstrated in 1972 that these principles hold to all orders in perturbation theory by introducing dimensional regularization (the HV scheme) to handle ultraviolet divergences [8].

One key property verified in dimensional regularizations such as the HV scheme is momentum-routing invariance (MRI)—the freedom to shift loop momenta in Feynman diagrams—which is deeply connected to ensuring Ward identities be satisfied in the absence of quantum anomalies. In Section 2, we show that MRI also plays a key role in systematizing UV divergences within Implicit Regularization (IReg), a non-dimensional approach that can be applied, for example, to the computation of RG functions [9].

1.1. The ${\gamma }_5$ Problem

In the presence of dimension-specific objects such as the matrix ${\gamma }_5$ or the Levi-Civita tensor, the HV scheme becomes problematic. Not all algebraic properties of ${\gamma }_5$ can be maintained in an arbitrary number of dimensions $d=4-2ϵ$ as $ϵ\to 0$. For example, the anticommutation relation $\{{\gamma }_5,{\gamma }_{\mu }\}=0$ generally fails outside four dimensions, and cyclicity of Dirac traces may be lost. Extensive work has been devoted to defining consistent rules for an extended dimensional algebra involving ${\gamma }_5$ and Lorentz tensors [8,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25].

The loss of ${\gamma }_5$’s anticommuting property in d dimensions breaks gauge invariance and BRST symmetry, which must be reinstated by the use of counterterms (CT). The Breitenlohner–Maison–’t Hooft–Veltman (BMHV) scheme [10] remains the only known all-order perturbative framework to handle ${\gamma }_5$ consistently in dimensional extensions. The Quantum Action Principle (QAP) further strengthens this approach in chiral and supersymmetric (SuSy) theories [26,27,28,29,30,31,32], enabling systematic relations between the dimensionally extended Lagrangian and Ward or Slavnov–Taylor identities order by order, often using the dimensional reduction (DRED) variant (see

Table 1. Regularization schemes and their main properties; please see main text for details. In FDR, FDU, and IReg, all integrations are performed in the physical dimension.

Scheme	Description
CDR	All fields (int./ext.) treated as d-dimensional.
HV	Internal fields d-dim., external strictly 4S.
DRED	All fields quasi-4D ($Q4S$).
FDH	Internal $Q4S$, external 4S.
FDR	local UV/IR subtraction.
FDU	Based on Loop-Tree Duality theorem; local UV/IR subtraction.
IReg	BPHZ-like subtraction.

).

Approaches to simplify the systematization of CT in the BMHV scheme have been forwarded as the “right most” method [18,19], and the chiral covariant handling of CT in a background field method [25]. The former is particularly useful in processes involving open fermionic strings and in the absence of quantum anomalies. The application of the latter is so far restricted to the NLO.

The BMHV scheme introduces additional complexity. Symmetry-restoring counterterms are required not only for physical gauge fields but also for evanescent fields—$ϵ$-scalars—which vanish in the strict four-dimensional limit but are necessary for consistent renormalization. These extra degrees of freedom break gauge invariance and BRST symmetry at intermediate steps and can generate finite contributions to amplitudes, since they appear in series with powers of $ϵ$ comparable to or exceeding those of the original $1/ϵ$ poles [33,34,35,36,37,38]. Handling these contributions requires additional care in the finite structure of amplitudes.

Modern developments include computer-algebra-based algorithms for analytical and numerical evaluation of Feynman integrals. Currently, closed form libraries exist primarily at next-to-leading order (NLO) [39], but automation for high-multiplicity processes remains incomplete. Complexity grows with massive virtual particles and threshold effects. While numerical integration dominates at NLO and beyond, analytical techniques remain essential for systematically removing infrared and ultraviolet divergences.

In this context, regularization frameworks that operate partially or entirely in the physical dimension offer potential advantages, particularly for higher-order calculations and chiral theories. In Table 1, we summarize traditional and emerging schemes according to their dimensionality. In order to better understand the table, we emphasize that, under regularization, momenta, metric, and Dirac matrices are usually treated as objects of an infinite vectorial space. This is usually denoted as $QnS$, where n can be an integer (normally 4) or a real number (usually denoted by d). For instance, in conventional dimensional regularization (CDR) one uses the vectorial space $QdS=4S\oplus Q(-2ϵ)S$. For DRED, on the other hand, one resorts to $Q4S=4S\oplus Q(-2ϵ)S\oplus Q\left(2ϵ\right)S$ for the fields, while amplitude momenta are specified in QdS. We use also the notation $4D$ and $2D$ to specify the physical dimension, independently if $QnS$ is used or not in the regularization procedure.

The dimensional schemes CDR and HV are closely related, as are four dimensional helicity (FDH) regularization and DRED, since internal vector fields share the same space–time dimension and in the absence of external vector fields, these schemes become indistinguishable. Differences arise in the properties of the Clifford algebra and in polarization sums, which depend on the associated metric tensors [40]. While treating all objects in d dimensions simplifies the algebra, quasi-four-dimensional ($Q4S$) and strictly four-dimensional ($4S$) spaces can be advantageous in specific contexts. For example, DRED avoids supersymmetry-restoring counterterms up to two loops (and has been verified at three loops for relevant cases) [26,41,42]. Deviations from the anticommutator relation $\{{\gamma }_5,{\gamma }_{\mu }\}=0$ are controlled by evanescent terms (see Section 3), and FDH facilitates the use of spinor-helicity methods [43,44,45], as does FDF (four-dimensional formulation) [46]. These benefits, however, come at the cost of handling $ϵ$-scalar degrees of freedom, which increase computational complexity.

1.2. Regularizing Infinities Beyond NLO Order

Physical observables must remain independent of the chosen regularization method. Therefore, some efforts have been devoted to establishing transition rules between schemes, enabling the use of the advantages of a scheme in intermediate steps. This can be achieved by analyzing the scheme dependence of $\beta$-functions and anomalous dimensions [47]. Currently, these transition rules are known at NNLO for all four-dimensional schemes in non-chiral gauge theories with massless fermions, and for selected cases involving massive fermions. They must be complemented by rules connecting different renormalization schemes.

In QCD, a factorization theorem is also required, and its implementation must not be obstructed by regularization. Factorization separates dynamics at different energy scales, simplifying the treatment of perturbative and non-perturbative contributions to cross sections. Infrared (IR) singularities—soft and collinear poles—can then be isolated and related to parton anomalous dimensions, following a RG-type evolution in the IR sector. NLO transition rules were addressed in [40], where DRED was shown to satisfy the factorization theorem for generic hadron–hadron scattering processes. At NNLO, soft-collinear effective theory (SCET), combined with heavy-quark effective theory, has been instrumental in establishing factorization between soft and hard degrees of freedom in the presence of massive quarks [48].

1.3. Non-Dimensional Schemes

Among non-dimensional schemes, the four-dimensional unsubtracted (FDU) method [49,50] relies on the loop-tree duality (LTD) theorem [51], which connects loop integrals to dual representations closely related to real-emission phase space. In its latest formulation, FDU provides a gauge-invariant, causal, and unitary framework that enables local cancellation of IR and UV divergences at the integrand level to all orders in perturbation theory. It also proposes a local functional form for initial-state collinear splitting functions, offering a fully four-dimensional description for differential observables at colliders [52,53,54,55,56]. The non-dimensional schemes four-dimensional regularization FDR [57] and IReg [58] share a key feature: the complete separation, at the integrand level, of the UV-divergent content of an amplitude from its finite part. Divergent integrals depend only on internal loop momenta and reduce to tensor or scalar structures—called “vacua” in FDR and basic divergent integrals (BDIs) in IReg—without involving physical observables like masses or external momenta. This ensures that the finite part of an amplitude is UV-regularization independent, while divergences are handled through renormalization. The two methods differ in how renormalization is implemented.

In FDR, renormalization is achieved by systematically discarding vacua and avoiding any distinction between UV and IR scales in virtual quantities [59,60,61,62,63,64,65]. Bare parameters are fixed in terms of physical quantities before combining virtual and real contributions, which introduces a physical scale and simultaneously cancels IR singularities. This approach has been applied up to two-loop order for selected processes [47,61].

IReg, on the other hand, is constructed to satisfy the Bogoliubov—Parasiuk—Hepp—Zimmermann (BPHZ) theorem [66,67,68,69,70], ensuring locality and unitarity at all loop orders [71]. BDIs inherently carry a renormalization scale, and IReg distinguishes between UV and IR scales, which is crucial for factorization [72]. This separation facilitates both renormalization and the implementation of the KLN theorem within a BPHZ framework by isolating IR and UV singularities analytically before numerical evaluation of finite integrals.

To date, IReg is the only non-dimensional approach applied to dimension-sensitive processes, including quantum anomalies, Lorentz violation, and supersymmetric theories. Surprisingly, even in the physical dimension, not all ${\gamma }_5$ operations are allowed within divergent integrals, as spurious anomalies and symmetry breakings can occur. As discussed in Section 3, this stems from using $\{{\gamma }_5,{\gamma }_{\mu }\}=0$ in Dirac algebra, which can rearrange $\gamma$-matrix order and lead to symmetric integration—a known source of symmetry violations. To address this, an enhanced IReg was developed using a dimensionally extended space $Q4S=4S\oplus X$, consistent with the Quantum Action Principle (QAP) and providing a reliable algebraic framework [73]. Although this introduces a formal dimensional extension, all integrals remain evaluated in the physical dimension, offering practical advantages.

Finally, bridging dimensional and non-dimensional approaches at intermediate steps has been shown feasible at NLO [74], but generally fails at higher orders [47,72].

This paper is structured as follows. In Section 2, we present the formal construction of Implicit Regularization (IReg), emphasizing gauge invariance symmetry and its implementation within the BPHZ framework. We also discuss how momentum-routing invariance and locality are preserved at all loop orders. Section 3 addresses the subtleties introduced by dimension-specific objects such as ${\gamma }_5$ and the Levi-Civita tensor. We analyze why naive algebraic manipulations fail even in strictly four-dimensional schemes and introduce an enhanced IReg formulation based on a quasi-four-dimensional extension ($Q4S$) that satisfies the Quantum Action Principle (QAP). This approach guarantees the validity of Ward and Slavnov–Taylor identities in chiral and supersymmetric theories without compromising the physical dimension of loop integrals. Finally, in Section 4, we present our concluding remarks, summarizing the interplay between gauge symmetries, regularization schemes, and chiral properties, and outline the prospects and limitations of relating dimensional and non-dimensional schemes beyond next-to-leading order.

2. Gauge Symmetries and Implicit Regularization

Implicit regularization is a scheme that operates essentially in the physical dimension of the underlying quantum field theory [58,75,76]. The core idea is the observation that UV and IR divergences can be cast as basic divergent integrals (BDIs)—that depend only on a subtraction energy scale—in the momentum (UV) and in the configuration (IR) spaces [77,78]. For instance in order to isolate the divergent content of an amplitude in terms of the loop momentum $k_l$, one applies recursively relation Equation (1),

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{{(k_l-p_i)}^2-{\mu }^2}}={\displaystyle \frac{1}{(k_l^2-{\mu }^2)}}-{\displaystyle \frac{(p_i^2-2p_i·k_l)}{(k_l^2-{\mu }^2)\left[{(k_l-p_i)}^2-{\mu }^2\right]}},\end{array}$$

(1)

to all propagators that depend on masses and/or external physical momenta, assuming an implicit regulator to perform such operations in the integrand under the integration sign, say a cut-off. Such algebraic manipulations can potentially violate gauge invariance. However, this invariance can be restored by imposing consistency conditions derived from momentum-routing invariance (MRI) in Feynman diagram loops. These conditions allow us to isolate terms that could break the symmetry and, as a byproduct, provide a way to handle anomalies in a manner that is regularization-scheme-independent. In this sense, the separation of UV divergences is achieved at integrand level, rather than by explicit integration, as occurs in dimensional methods.

Another key idea of the method is to be general enough to not comply with MRI automatically [79]. This is achieved in the course of tensor decomposition of BDIs, which can be all written in terms of scalar BDIs plus arbitrary terms, denoted surface terms. The latter must be null if MRI is to be preserved [9,58,75].

In the context of applications, IReg was employed to obtain NLO corrections for $h\to \gamma \gamma$ [80], $e^{-}{e}^{+}\to {\gamma }^{*}\to q\overline{q}\left(g\right)$ [74], $H\to gg$ [81], $Z\to q\overline{q}$, and $S\to q\overline{q}$ [82], where S is a scalar or pseudo-scalar. In those references, it is shown how to parametrize not only UV divergences in the context of the IReg method, but also IR divergences. To achieve this goal, massless propagators are transformed into massive ones, where the introduced mass plays the role of IR regulator. Non-trivial checks are under active research related to NNLO processes, with promising results [47]. In the present manuscript, we focus on the IR safe process for simplicity, since our main goal is to review the treatment of chiral theories in the context of IReg.

The symmetry of the underlying QFT theory is one feature that regularization methods should comply with. In the context of non-chiral abelian theories (for instance, QED), it is well-known that momentum-routing invariance is directly connected to the conservation of the abelian gauge symmetry. For IReg, it was shown in [9] that non-null surface terms are in the root of the breaking of momentum-routing invariance at arbitrary loop order, which directly causes the breaking of abelian gauge symmetry. Thus, one can define a constrained version of IReg, where surface terms are automatically set to zero. For chiral abelian theories, the same reasoning still applies for vectorial Ward identities (related to non-chiral vertexes) [83,84]. For non-abelian gauge theories, the proof of gauge invariance is more involved. In the context of dimensional regularization methods, it is more directly performed by employing the quantum action principle (QAP) [85]. It is thus possible to show that QCD is preserved to arbitrary loop order by dimensional method. On the other hand, using this framework, one can show that none of the dimensional methods do preserve supersymmetry (SUSY) to arbitrary loop order [26]. For IReg, a proof showing that the method complies with the QAP is still lacking, although pertinent pre-requisites such as numerator/denominator consistency and invariance under shifts of the integration momenta were shown to be fulfilled [73]. Thus, we have at present working examples in the context of QCD [72], Electroweak Standard Model [86], and SUSY [77,84,87] at two-loop order.

The present status of IReg can be summarized as follows:

(i)

Complying with BPHZ for abelian gauge and chiral theories at all orders in perturbation theory.

(ii)

In non-abelian theories there are non-trivial examples worked out up to and including two-loop order.

(iii)

The QAP has not been proven explicitly; however, the formal requirements for its successful implementation underly the IReg procedure through a normal form (discussed in the next sections).

(iv)

Work in progress contemplates higher order non-abelian processes and the formal proof of the QAP within IReg.

Working Example: One Loop Implementation

In the remainder of this section, we present examples illustrating the rules of IReg in the context of chiral theories. For simplicity, we restrict our discussion to one-loop amplitudes; a pedagogical introduction to two-loop applications can be found in [88]. Our first case focuses on two-point functions at one loop, working within a two-dimensional Euclidean space. In this setting, spinorial QED serves as the simplest theory to embed, where we concentrate on the polarization tensor (the photon two-point function). After a straightforward application of the Feynman rules, we obtain

$$\begin{array}{cc}\hfill {\mathrm{\Pi }}_{\mu \nu }\left(p\right)& =-\int {\displaystyle \frac{d^{2}k}{4{\pi }^2}}\mathrm{tr}\left({\gamma }_{\mu }{\displaystyle \frac{1}{\cancel{k}-\cancel{p}}}{\gamma }_{\nu }{\displaystyle \frac{1}{\cancel{k}}}\right)=-\mathrm{tr}\left({\gamma }_{\mu }{\gamma }_{\alpha }{\gamma }_{\nu }{\gamma }_{\beta }\right)B_{\alpha \beta }.\hfill \end{array}$$

(2)

where

$$\begin{array}{c}\hfill B_{\alpha \beta }=\int {\displaystyle \frac{d^{2}k}{4{\pi }^2}}{\displaystyle \frac{k_{\beta }{(k-p)}_{\alpha }}{k^2{(k-p)}^2}}\end{array}$$

(3)

The integral $B_{\alpha \beta }$ is ultraviolet (UV) divergent, so regularization is required. Note that the denominator depends not only on the loop momentum but also on the physical (external) momentum. This dependence prevents us from isolating a UV-divergent quantity that is free of physical variables. As discussed earlier in this section, this is the central idea behind IReg. However, a naive Taylor expansion around $p=0$ cannot be applied, as it would also introduce infrared (IR) divergences. Explicitly,

$$\begin{array}{c}\hfill B_{\alpha \beta }\sim \int {\displaystyle \frac{d^{2}k}{4{\pi }^2}}{\displaystyle \frac{k_{\beta }{k}_{\alpha }}{k^4}}+\mathcal{O}\left(p\right)\end{array}$$

(4)

To circumvent this problem, we first introduce a fictitious mass regulator in all propagators, for instance,

$$\begin{array}{c}\hfill B_{\alpha \beta }=\underset{{\mu }^2\to 0}{lim}\int {\displaystyle \frac{d^{2}k}{4{\pi }^2}}{\displaystyle \frac{k_{\beta }{(k-p)}_{\alpha }}{(k^2-{\mu }^2)[{(k-p)}^2-{\mu }^2]}}\end{array}$$

(5)

In this case, the Taylor expansion would behave better as long as the limit on the mass is not enforced. In IReg, we go a step further: instead of resorting to any approximation, we re-sum the UV-finite part. For our particular example, we obtain

$$\begin{array}{c}\hfill B_{\alpha \beta }=\underset{{\mu }^2\to 0}{lim}\int {\displaystyle \frac{d^{2}k}{4{\pi }^2}}\left[{\displaystyle \frac{k_{\beta }{k}_{\alpha }}{{(k^2-{\mu }^2)}^2}}-{\displaystyle \frac{k_{\beta }{k}_{\alpha }(p^2-2k.p)}{{(k^2-{\mu }^2)}^2[{(k-p)}^2-{\mu }^2]}}-{\displaystyle \frac{k_{\beta }{p}_{\alpha }}{(k^2-{\mu }^2)[{(k-p)}^2-{\mu }^2]}}\right]\end{array}$$

(6)

The first term is the so-called BDI, which is logarithmic divergent in this case. We define

$$\begin{array}{c}\hfill I_{log}{\left({\lambda }^2\right)}_{{\alpha }_1\dots {\alpha }_{2n}}=\int {\displaystyle \frac{d^{2}k}{4{\pi }^2}}{\displaystyle \frac{k_{{\alpha }_1}\dots k_{{\alpha }_{2n}}}{{(k^2-{\lambda }^2)}^{n+1}}}\end{array}$$

(7)

In this approach, the BDI can be tensors of any rank. It is possible, however, to have only scalar BDI. To achieve this goal, one can resort to identities as below

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial k_{\beta }}}{\displaystyle \frac{k_{\alpha }}{(k^2-{\lambda }^2)}}={\displaystyle \frac{{\delta }_{\alpha \beta }}{(k^2-{\lambda }^2)}}-{\displaystyle \frac{2k_{\alpha }{k}_{\beta }}{{(k^2-{\lambda }^2)}^2}}\end{array}$$

(8)

By integrating both sides, it is possible to express all BDIs in terms of $I_{log}\left({\lambda }^2\right)$ only. The terms with the total derivative are denoted surface terms, and they must be set to zero in order to comply with momentum-routing invariance which automatically leads to gauge invariance. Finally, the evaluation of the finite part is standard. Notice, however, that both the finite part and the BDI are dependent on the fictitious mass $\mu$. Since the integral was IR-safe from the beginning, the limit ${\mu }^2\to 0$ must be well-behaved. It can be shown that

$$\begin{array}{c}\hfill I_{log}\left({\lambda }^2\right)=I_{log}\left({\mu }^2\right)-log{\displaystyle \frac{{\mu }^2}{{\lambda }^2}}\end{array}$$

(9)

allowing us to isolate a proper UV-divergence, since $\lambda \ne 0$. The final result is

$$\begin{array}{c}\hfill B_{\alpha \beta }\left(p\right)=\int {\displaystyle \frac{d^{2}k}{4{\pi }^2}}{\displaystyle \frac{{(k-p)}_{\alpha }{k}_{\beta }}{k^2{(k-p)}^2}}={\displaystyle \frac{1}{4\pi }}\left\{{\displaystyle \frac{{\delta }_{\alpha \beta }}{2}}\left(I_{log}\left({\lambda }^2\right)-log{\displaystyle \frac{p^2}{{\lambda }^2}}\right)+\left({\delta }_{\alpha \beta }-{\displaystyle \frac{p_{\alpha }{p}_{\beta }}{p^2}}\right)\right\}.\end{array}$$

(10)

Returning to the polarization tensor, we still need to plug in the trace

$$\begin{array}{c}\hfill \mathrm{tr}\left({\gamma }_{\mu }{\gamma }_{\rho }{\gamma }_{\nu }{\gamma }_{\sigma }\right)=2\left({\delta }_{\mu \rho }{\delta }_{\nu \sigma }-{\delta }_{\mu \nu }{\delta }_{\rho \sigma }+{\delta }_{\mu \sigma }{\delta }_{\nu \rho }\right)\end{array}$$

(11)

which gives the end result

$$\begin{array}{c}\hfill {\mathrm{\Pi }}_{\mu \nu }\left(p\right)=-{\displaystyle \frac{1}{4\pi }}\left({\delta }_{\mu \nu }-2{\displaystyle \frac{p_{\mu }{p}_{\nu }}{p^2}}\right)\end{array}$$

(12)

It is clear that it is not transverse, implying that the Ward identity ($p_{\mu }{\mathrm{\Pi }}_{\mu \nu }=0)$ is not satisfied. Therefore, only setting surface to zero is not sufficient to guarantee a gauge-invariant result. This issue is dealt with using the definition of a normal form of the method. The normal form in IReg is obtained after all Lorentz contractions and Dirac algebra simplifications are performed. In our particular example, the Dirac trace must be evaluated before the regularization is applied. After performing the trace, we find

$$\begin{array}{c}\hfill {\mathrm{\Pi }}_{\mu \nu }\left(p\right)=-\left[4B_{\mu \nu }\left(p\right)-2{\delta }_{\mu \nu }{B}_{\alpha \alpha }\left(p\right)\right],\end{array}$$

(13)

where

$$\begin{array}{c}\hfill B_{\alpha \alpha }=\int {\displaystyle \frac{d^{2}k}{4{\pi }^2}}{\displaystyle \frac{k^2-k.p}{k^2{(k-p)}^2}}=\underset{{\mu }^2\to 0}{lim}\int {\displaystyle \frac{d^{2}k}{4{\pi }^2}}{\displaystyle \frac{1}{{(k-p)}^2-{\mu }^2}}-\int {\displaystyle \frac{d^{2}k}{4{\pi }^2}}{\displaystyle \frac{k.p}{(k^2-{\mu }^2)[{(k-p)}^2-{\mu }^2]}}\end{array}$$

(14)

Notice that we have simplified the numerator against the denominator, related to terms containing $k^2$. This is a must in IReg, since symmetric integration is not allowed. It will be clear soon that $B_{\alpha \alpha }\ne {\delta }_{\alpha \beta }{B}_{\alpha \beta }$ in IReg. In Equation (14) notice that only the first term is UV-divergent. It be can readily written as a BDI (since surface terms are null, it is possible to perform shifts in the integrals), and the final result is

$$\begin{array}{c}\hfill B_{\alpha \alpha }={\displaystyle \frac{1}{4\pi }}\left(I_{log}\left({\lambda }^2\right)-log{\displaystyle \frac{p^2}{{\lambda }^2}}\right).\end{array}$$

(15)

Notice, however, that

$$\begin{array}{c}\hfill {\delta }_{\alpha \beta }{B}_{\alpha \beta }={\displaystyle \frac{1}{4\pi }}\left(I_{log}\left({\lambda }^2\right)-log{\displaystyle \frac{p^2}{{\lambda }^2}}\right)+{\displaystyle \frac{1}{4\pi }}.\end{array}$$

(16)

Returning to Equation (13), we obtain

$$\begin{array}{c}\hfill {\mathrm{\Pi }}_{\mu \nu }\left(p\right)={\displaystyle \frac{1}{\pi }}\left({\displaystyle \frac{p_{\mu }{p}_{\nu }}{p^2}}-{\delta }_{\mu \nu }\right),\end{array}$$

(17)

which is transverse and respects the Ward identity.

The main point is that defining a normal form in IReg is essential to obtain gauge-invariant results. At two loops, this becomes even more critical, since one must handle sub-divergences in addition to the overall UV divergence of the Feynman amplitude. Further discussions on this topic can be found in [72,88,89]. In this work, however, we restrict ourselves to one-loop amplitudes, where the normal form is achieved after performing Lorentz contractions and Dirac algebra, canceling whenever possible terms containing $k^2$ in the numerator against those in the denominator.

Since IReg is, in principle, defined in the physical dimension, it was expected that identities valid only at finite dimension could be employed without ambiguity. Particularly relevant are identities involving the ${\gamma }_5$ matrix, such as $\{{\gamma }_{\mu },{\gamma }_5\}=0$. This expectation, however, is too naive [73]. In the next section, we discuss how to handle ${\gamma }_5$ consistently within the context of IReg.

3. Chiral Symmetry

As mentioned in the previous section, the treatment of ${\gamma }_5$ requires particular care not only in dimensional schemes but also when using schemes that operate in the physical dimension such as IReg [73,90]. Before presenting the details, let us recall the core difficulties in the context of dimensional regularization (DReg).

In four dimensions, ${\gamma }_5$ is defined by

$${\gamma }_5\equiv {\displaystyle \frac{i}{4!}}{ϵ}_{\mu \nu \rho \sigma }{\gamma }^{\mu }{\gamma }^{\nu }{\gamma }^{\rho }{\gamma }^{\sigma },\{{\gamma }_5,{\gamma }_{\mu }\}=0,{\gamma }_5^2=1.$$

(18)

In DReg [8], however, the Dirac algebra is analytically continued to $d=4-2ϵ$ dimensions, while the Levi-Civita tensor in the definition above is kept intrinsically four-dimensional. This mismatch leads to a fundamental obstruction: one cannot simultaneously preserve (i) complete anticommutation $\{{\gamma }_5,{\gamma }_{\mu }\}=0$ for all $\mu$, (ii) cyclicity of the Dirac trace, and (iii) the correct realization of axial Ward identities, as is the case for the Adler–Bell–Jackiw anomaly [91,92]. This tension already appears in the one-loop AVV triangle: imposing full anticommutation together with naive trace cyclicity causes the anomaly to vanish, contradicting the known physical result.

Consequently, any consistent use of ${\gamma }_5$ in DReg requires adopting a specific prescription that sacrifices either strict anticommutation or strict trace cyclicity. The corresponding violation must be compensated for by the addition of symmetry-preserving finite renormalizations to ensure the model predictability.

A well established scheme is the Breitenlohner–Maison–’t Hooft–Veltman (BMHV) prescription [85], in which the Dirac space is split into a direct sum of a strictly four-dimensional physical subspace and an evanescent $(d-4)$-dimensional complement $\mathcal{V}={\overline{V}}_4\oplus {\widehat{V}}_{ϵ}$. Dirac matrices are decomposed as

$${\gamma }_{\mu }={\overline{\gamma }}_{\mu }+{\widehat{\mathrm{\gamma }}}_{\mu },\{{\overline{\gamma }}_{\mu },{\overline{\gamma }}_{\nu }\}=2{\overline{g}}_{\mu \nu },\{{\widehat{\gamma }}_{\mu },{\widehat{\gamma }}_{\nu }\}=2{\widehat{g}}_{\mu \nu },\{{\overline{\gamma }}_{\mu },{\widehat{\gamma }}_{\nu }\}=0,$$

(19)

with $\overline{g}·\widehat{g}=0$ and $\overline{g}+\widehat{g}=g$. The matrix ${\gamma }_5$ is kept purely four-dimensional, defined by ${\gamma }_5=i{\overline{\gamma }}^0{\overline{\gamma }}^1{\overline{\gamma }}^2{\overline{\gamma }}^3$, so that

$$\{{\gamma }_5,{\overline{\gamma }}_{\mu }\}=0$$

(20)

as well as

$$\begin{array}{c}\hfill \{{\gamma }_5,{\gamma }_{\mu }\}=\{{\gamma }_5,{\overline{\gamma }}_{\mu }\}+\{{\gamma }_5,{\widehat{\gamma }}_{\mu }\}=0+2{\widehat{\gamma }}_{\mu }{\gamma }_5=2{\widehat{\gamma }}_{\mu }{\gamma }_5\end{array}$$

(21)

In this construction the ${\gamma }_5$ matrix explicitly anti-commutes with Dirac matrices in four dimensions. Also, the spaces are orthogonal to each other ${\overline{V}}_4\perp {\widehat{V}}_{ϵ}$, meaning that ${\widehat{\gamma }}_{\mu }{\gamma }_5={\gamma }_5{\widehat{\gamma }}_{\mu }$. In other words, $[{\gamma }_5,{\widehat{\gamma }}_{\mu }]=0$. Although this setup is consistent to all loop orders, the underlying gauge symmetries are broken, and have to be restored by finite counterterms; see [93] for a recent review.

Naively imposing $\{{\gamma }_5,{\gamma }_{\mu }\}=0$ in d dimensions (the so called NDR approach) simplifies algebra but is inconsistent beyond very special cases: already at one loop it yields a vanishing AVV anomaly. A compromise often used in practice is to define ${\gamma }_5$ by the $d=4$ Levi–Civita tensor and keep $\{{\gamma }_5,{\gamma }_{\mu }\}=0$ only inside fully symmetrized traces; however, this must be justified a posteriori by checking Ward identities and, when necessary, by adding finite counterterms.

As discussed, a consistent way to deal with dimension-specific objects (such as ${\gamma }_5$) is to employ the BHMV scheme. In this case, not only Dirac matrices but all objects (external momenta, etc.) are split into a four-dimensional and evanescent part. This is akin to using the vectorial space $QdS=4S\oplus Q(-2ϵ)S$.

At first sight, one might question whether the appearance of an evanescent quantity can pose any difficulty in physical amplitudes, since the final results must ultimately be taken in the physical dimension. The subtlety, however, is that an evanescent term can multiply divergent loop integrals, thereby generating finite contributions once the limit to four dimensions is performed. This mechanism is well-known in dimensional schemes and will reappear in the context of IReg.

To discuss this point in the simplest possible setup, consider the UV-divergent integral in $2D$ space

$$\begin{array}{c}\hfill I_{\mu \nu }=\int d^{2}k{\displaystyle \frac{k_{\mu }{k}_{\nu }}{{(k^2+m^2)}^2}}\end{array}$$

(22)

It is straightforward to evaluate the following trace by employing the definition of ${\gamma }_5$ in two dimensions

$$\begin{array}{cc}\hfill \mathrm{tr}\left(\{{\gamma }_5,{\gamma }_{\mu }\}{\gamma }_{\nu }{\gamma }_{\rho }{\gamma }_{\sigma }\right)& =\mathrm{tr}\left({\gamma }_5{\gamma }_{\mu }{\gamma }_{\nu }{\gamma }_{\rho }{\gamma }_{\sigma }\right)+\mathrm{tr}\left({\gamma }_{\mu }{\gamma }_5{\gamma }_{\nu }{\gamma }_{\rho }{\gamma }_{\sigma }\right)\hfill \\ \hfill & =-4\left(g_{\mu \nu }{ϵ}_{\rho \sigma }-g_{\mu \rho }{ϵ}_{\nu \sigma }+g_{\mu \sigma }{ϵ}_{\nu \rho }\right)\hfill \end{array}$$

(23)

By now considering the case where the trace is multiplied by the integral $I_{\mu \nu }$, one obtains in IReg

$$\begin{array}{c}\hfill \left(\mathrm{tr}\left({\gamma }_5{\gamma }_{\mu }{\gamma }_{\nu }{\gamma }_{\rho }{\gamma }_{\sigma }\right)+\mathrm{tr}\left({\gamma }_{\mu }{\gamma }_5{\gamma }_{\nu }{\gamma }_{\rho }{\gamma }_{\sigma }\right)\right)I^{\mu \sigma }=4\pi {ϵ}_{\rho \nu }\ne 0\end{array}$$

(24)

This result explicitly shows that identities such as $\{{\gamma }_{\mu },{\gamma }_5\}=0$ do not hold in general, even for IReg. The reason boils down to the fact that $I_{\alpha \alpha }\ne {\delta }_{\alpha \beta }{I}_{\alpha \beta }$ in IReg, as we discussed in the last section.

Another interesting example is given by the VA (vector-axial) vertex, which is the two-dimensional analogue of the AVV anomaly in four dimensions. Explicitly,

$$\begin{array}{cc}\hfill {\mathrm{\Pi }}_{\mu \nu }^5\left(p\right)& =-\int {\displaystyle \frac{d^{2}k}{4{\pi }^2}}\mathrm{tr}\left({\gamma }_{\mu }{\displaystyle \frac{1}{\cancel{k}-\cancel{p}}}{\gamma }_{\nu }{\gamma }_5{\displaystyle \frac{1}{\cancel{k}}}\right)=-\mathrm{tr}\left({\gamma }_{\mu }{\gamma }_{\alpha }{\gamma }_{\nu }{\gamma }_5{\gamma }_{\beta }\right)B_{\alpha \beta }.\hfill \end{array}$$

(25)

As discussed in the previous section, the next step is to define a normal form within IReg. We have shown that, to maintain gauge invariance supported by vanishing surface terms, all Dirac and Lorentz algebra manipulations must be completed before applying IReg regularization rules. For clarity, consider the consequences of lifting this condition in the current scenario. For this purpose, we only need the expression for $B_{\alpha \beta }\left(p\right)$ evaluated within IReg, given in Equation (10), together with the corresponding trace. Using the two-dimensional definition of ${\gamma }_5$, one obtains

$$\mathrm{tr}\left({\gamma }_{\mu }{\gamma }_{\alpha }{\gamma }_{\nu }{\gamma }_5{\gamma }_{\beta }\right)=2\left(-{ϵ}_{\beta \nu }{\delta }_{\alpha \mu }+{ϵ}_{\mu \nu }{\delta }_{\alpha \beta }-{ϵ}_{\alpha \nu }{\delta }_{\beta \mu }+{ϵ}_{\beta \alpha }{\delta }_{\mu \nu }-{ϵ}_{\mu \alpha }{\delta }_{\beta \nu }-{ϵ}_{\beta \mu }{\delta }_{\alpha \nu }\right).$$

(26)

Thus, the amplitude is given by

$${\mathrm{\Pi }}_{\mu \nu }^5\left(p\right)=-{\displaystyle \frac{1}{2\pi }}\left({ϵ}_{\mu \nu }-{\displaystyle \frac{2{ϵ}_{\alpha \nu }{p}_{\mu }{p}_{\alpha }}{p^2}}\right)$$

(27)

It is now straightforward to obtain the Ward identities:

$$\begin{gathered} p_{\mu }{\mathrm{\Pi }}_{\mu \nu }^5\left(p\right)={\displaystyle \frac{1}{2\pi }}{ϵ}_{\mu \nu }{p}_{\mu }, \\ {p}_{\nu }{\mathrm{\Pi }}_{\mu \nu }^5\left(p\right)={\displaystyle \frac{1}{2\pi }}{ϵ}_{\nu \mu }{p}_{\nu }, \end{gathered}$$

(28)

In this case, the anomaly is shared equally between the vector and axial Ward identities. Recall that, since IReg preserves (vector) gauge invariance, performing the integral evaluation in IReg before completing the trace manipulation is not appropriate. To address this, we define a normal form in IReg for expressions involving ${\gamma }_5$ by carrying out all Dirac and Lorentz algebraic simplifications beforehand. The remaining challenge is to determine how to evaluate traces that contain the ${\gamma }_5$ matrix.

The naive approach would be to impose the identity $\{{\gamma }_{\mu },{\gamma }_5\}=0$. However, as previously discussed, since $I_{\alpha \alpha }\ne {\delta }_{\alpha \beta }{I}_{\alpha \beta }$, this option is generally not viable. Nevertheless, for illustration, let us examine what happens if we adopt this assumption in the amplitude calculation. In this case, it is more convenient to analyze the Ward identities directly. For the vectorial one,

$$\begin{array}{cc}\hfill p_{\mu }{\mathrm{\Pi }}_{\mu \nu }^5\left(p\right)& =-\int {\displaystyle \frac{d^{2}k}{4{\pi }^2}}\mathrm{tr}\left((\cancel{p}-\cancel{k}+\cancel{k}){\displaystyle \frac{1}{\cancel{k}-\cancel{p}}}{\gamma }_{\nu }{\gamma }_5{\displaystyle \frac{1}{\cancel{k}}}\right)\hfill \\ & \hfill =\int {\displaystyle \frac{d^{2}k}{4{\pi }^2}}\mathrm{tr}\left({\gamma }_{\nu }{\gamma }_5{\displaystyle \frac{1}{\cancel{k}}}\right)-\int {\displaystyle \frac{d^{2}k}{4{\pi }^2}}\mathrm{tr}\left({\displaystyle \frac{1}{\cancel{k}-\cancel{p}}}{\gamma }_{\nu }{\gamma }_5\right)\\ & =0\hfill \end{array}$$

(29)

where we have performed the shift $k\to k+p$ in the second integral of the second line. For the axial Ward identity

$$\begin{array}{cc}\hfill p_{\nu }{\mathrm{\Pi }}_{\mu \nu }^5\left(p\right)& =-\int {\displaystyle \frac{d^{2}k}{4{\pi }^2}}\mathrm{tr}\left({\gamma }_{\mu }{\displaystyle \frac{1}{\cancel{k}-\cancel{p}}}(\cancel{p}-\cancel{k}+\cancel{k}){\gamma }_5{\displaystyle \frac{1}{\cancel{k}}}\right)\hfill \\ & \hfill =\int {\displaystyle \frac{d^{2}k}{4{\pi }^2}}\mathrm{tr}\left({\gamma }_{\mu }{\gamma }_5{\displaystyle \frac{1}{\cancel{k}}}\right)+\int {\displaystyle \frac{d^{2}k}{4{\pi }^2}}\mathrm{tr}\left({\gamma }_{\mu }{\displaystyle \frac{1}{\cancel{k}-\cancel{p}}}{\gamma }_5\right)\\ & =0\hfill \end{array}$$

(30)

where we have used $\{{\gamma }_{\mu },{\gamma }_5\}=0$ in the second integral of the second line, and used the same identity again to obtain the third line. In this case, the vectorial Ward identity is respected, but the anomaly is not reproduced.

Therefore, we need to evaluate the trace without employing the identity $\{{\gamma }_{\mu },{\gamma }_5\}=0$. This is achieved by using the definition of ${\gamma }_5$ in two dimensions: ${\gamma }_5={ϵ}_{\mu \nu }{\gamma }^{\mu }{\gamma }^{\nu }/2$. In the language of [90], we employ a symmetrization of the trace. We obtain Equation (26), and the correlator is given by

$$\begin{array}{cc}\hfill {\mathrm{\Pi }}_{\mu \nu }^5\left(p\right)& =-2\int {\displaystyle \frac{d^{2}k}{4{\pi }^2}}\left(-{ϵ}_{\beta \nu }{\delta }_{\alpha \mu }+{ϵ}_{\mu \nu }{\delta }_{\alpha \beta }-{ϵ}_{\alpha \nu }{\delta }_{\beta \mu }+{ϵ}_{\beta \alpha }{\delta }_{\mu \nu }-{ϵ}_{\mu \alpha }{\delta }_{\beta \nu }-{ϵ}_{\beta \mu }{\delta }_{\alpha \nu }\right){\displaystyle \frac{{(k-p)}_{\alpha }{k}_{\beta }}{{(k-p)}^2{k}^2}}\hfill \end{array}$$

(31)

$$\begin{array}{l}={\displaystyle \frac{{ϵ}_{\nu \alpha }}{\pi }}\left({\displaystyle \frac{p_{\mu }{p}_{\alpha }}{p^2}}-{\delta }_{\mu \alpha }\right)\hfill \end{array}$$

(32)

It is straightforward to calculate the WI

$$\begin{array}{cc}\hfill p_{\mu }{\mathrm{\Pi }}_{\mu \nu }^5\left(p\right)& =0\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill p_{\nu }{\mathrm{\Pi }}_{\mu \nu }^5\left(p\right)& ={\displaystyle \frac{{ϵ}_{\mu \nu }{p}_{\nu }}{\pi }}\hfill \end{array}$$

(34)

which shows that the anomaly is distributed only on the axial WI.

Even though we could correctly reproduce the anomaly in the axial WI, the symmetrization of the trace is a valid approach only in particular cases. If we have open fermionic lines, or more complex cases where the ${\gamma }_5$ matrix appears in the propagator (not only in the vertex) [94], we need to devise a more general approach. This is achieved by defining IReg in a quasi-space $QnS$. In practice, we define the fermionic algebra in such a way that all Dirac matrices (except ${\gamma }_5$) live in $QnS$, while ${\gamma }_5$ itself is kept strictly in the integer dimension. This separation ensures that the delicate algebraic identities involving ${\gamma }_5$, especially those entering Ward identities and anomaly calculations, are treated consistently.

The price to pay is that the anticommutation relation between ${\gamma }_5$ and the $QnS$ Dirac matrices becomes modified. Instead of the naive $\{{\gamma }_5,{\gamma }_{\mu }\}=0$, one finds

$$\{{\gamma }_5,{\gamma }_{\mu }\}=2{\gamma }_5{\widehat{\gamma }}_{\mu },$$

(35)

where ${\widehat{\gamma }}_{\mu }$ denotes the evanescent component. This relation mirrors the BMHV structure and guarantees that potentially ambiguous terms are confined to the evanescent sector. Once loop integrals are evaluated, the finite remnants arising from ${\widehat{\gamma }}_{\mu }$ contributions can be tracked explicitly and tested against the physical Ward identities.

To recapitulate, let us consider $n=4$: The quasispace $Q4S=4S\oplus X$, where the space $X=Q\left(2ϵ\right)S\oplus Q(-2ϵ)S$ in DRED. However the way momenta are treated in IReg differs from DRED. For example the loop momenta in DRED reside in $4S\oplus Q(-2ϵ)S$ and the extra space $Q\left(2ϵ\right)S$ is the space of $ϵ$ scalar momenta. In IReg the momenta are in $Q4S$. To highlight this distinction X stands for the space of evanescent contributions, fields and momenta, that arise applying the algebra Equations (19)–(21) and needs not be further specified [73]. Examples of how the space X is handled in practical calculations are given below.

This framework has two important consequences. First, it allows one to reproduce the correct anomaly structure without relying on ad hoc symmetrizations of traces. Second, it provides a systematic procedure applicable beyond simple correlators, such as in processes involving open fermionic lines, higher-loop diagrams [95] or when ${\gamma }_5$ appears inside propagators [94]. In these cases, the $QnS$ construction prevents inconsistencies by enforcing a well-defined ordering of algebraic manipulations prior to the application of IReg. The price to pay is the appearance of symmetry-breaking terms that must be removed by finite symmetry restoring counterterms. In the context of IReg, examples at one-loop are found in [73] while for two-loop one can resort to [95]. In the context of DReg, there are recent examples at two [30], three [32], and four loop [29].

In summary, the quasi-space $QnS$ approach provides the missing generality: it extends the consistency of IReg in the presence of ${\gamma }_5$ to arbitrary amplitudes, while maintaining control over anomalies and preserving gauge invariance. This makes $QnS$ the natural analogue, within IReg, of the BMHV treatment of ${\gamma }_5$ in dimensional regularization.

Finally, for illustration, we return to the amplitude we have been discussing. The vectorial Ward identity can be calculated as in Equation (29). We notice by passing that this result can be generalized to arbitrary loop order since the ${\gamma }_5$ acts as a spectator. The proof in the context of IReg can be found in [90]. For the axial WI, we can employ the identity $\{{\gamma }_{\mu },{\gamma }_5\}=2{\gamma }_5\widehat{{\gamma }_{\mu }}$ to obtain

$$\begin{array}{cc}\hfill p_{\nu }{\mathrm{\Pi }}_{\mu \nu }^5\left(p\right)& =-\int {\displaystyle \frac{d^{2}k}{4{\pi }^2}}\mathrm{tr}\left({\gamma }_{\mu }{\displaystyle \frac{1}{\cancel{k}-\cancel{p}}}(\cancel{p}-\cancel{k}+\cancel{k}){\gamma }_5{\displaystyle \frac{1}{\cancel{k}}}\right)\hfill \\ & \hfill =\int {\displaystyle \frac{d^{2}k}{4{\pi }^2}}\mathrm{tr}\left({\gamma }_{\mu }{\gamma }_5{\displaystyle \frac{1}{\cancel{k}}}\right)+\int {\displaystyle \frac{d^{2}k}{4{\pi }^2}}\mathrm{tr}\left({\gamma }_{\mu }{\displaystyle \frac{1}{\cancel{k}-\cancel{p}}}{\gamma }_5\right)-2\int {\displaystyle \frac{d^{2}k}{4{\pi }^2}}\mathrm{tr}\left({\gamma }_{\mu }{\displaystyle \frac{1}{\cancel{k}-\cancel{p}}}{\gamma }_5\widehat{\cancel{k}}{\displaystyle \frac{1}{\cancel{k}}}\right)\\ & =\int {\displaystyle \frac{d^{2}k}{4{\pi }^2}}\mathrm{tr}\left({\gamma }_{\mu }{\gamma }_5{\displaystyle \frac{1}{\cancel{k}}}\right)+\int {\displaystyle \frac{d^{2}k}{4{\pi }^2}}\mathrm{tr}\left({\gamma }_{\mu }{\displaystyle \frac{1}{\cancel{k}-\cancel{p}}}{\gamma }_5\right)-2\int {\displaystyle \frac{d^{2}k}{4{\pi }^2}}\mathrm{tr}\left({\gamma }_{\mu }{\displaystyle \frac{1}{\cancel{k}-\cancel{p}}}{\gamma }_5{\displaystyle \frac{k^2-{\overline{k}}^2}{k^2}}\right)\hfill \\ & =\int {\displaystyle \frac{d^{2}k}{4{\pi }^2}}\mathrm{tr}\left({\gamma }_{\mu }{\gamma }_5{\displaystyle \frac{1}{\cancel{k}}}\right)-\int {\displaystyle \frac{d^{2}k}{4{\pi }^2}}\mathrm{tr}\left({\gamma }_{\mu }{\displaystyle \frac{1}{\cancel{k}-\cancel{p}}}{\gamma }_5\right)+2\int {\displaystyle \frac{d^{2}k}{4{\pi }^2}}\mathrm{tr}\left({\gamma }_{\mu }{\displaystyle \frac{1}{\cancel{k}-\cancel{p}}}{\gamma }_5{\displaystyle \frac{{\overline{k}}^2}{k^2}}\right)\hfill \\ & =0+{\displaystyle \frac{{ϵ}_{\mu \nu }{p}_{\nu }}{\pi }}\hfill \end{array}$$

(36)

where we performed the shift $k\to k+p$ in some of the integrals as well as used the fact that odd integrals in k are null. Notice also how the X space is dealt with in practice in the context of IReg. The point is to rewrite all objects in the X space in terms of differences of objects in $QnS$ and $nS$, whose rules in IReg are well-defined. In the case of $QnS$, one must always simplify $k^2$ terms in numerator against denominator, in order to avoid symmetric integration. For $nS$, on the other hand, it is possible to write ${\overline{k}}^2={\overline{g}}_{\alpha \beta }{k}^{\alpha }{k}^{\beta }$. The integral with $k^{\alpha }{k}^{\beta }$ is dealt with as usual, and the end result is contracted against ${\overline{g}}_{\alpha \beta }$. After all manipulations, we reproduce the result obtained employing the symmetrization of the trace as expected.

Example in $4D$

The next example is in four dimensional physical space. Let us clarify why the previous $2D$ examples are now helpful and how they are connected to the physically relevant $4D$ situation:

1.

The algebraic properties of ${\gamma }_5$ used later in $4D$ are first illustrated in $2D$ for pedagogical clarity. The normal form of the ${\gamma }_5$ algebra is simpler to demonstrate in $2D$, and the same structural properties extend to $4D$.

2.

The subsequent $4D$ example shows explicitly that the regularization must be carried out in $QnS$ space whenever the fermion propagator contains ${\gamma }_5$. In such cases, operations such as symmetrization of traces no longer commute with the manipulations of ${\gamma }_5$, as demonstrated in Equation (38).

3.

The $2D$ discussion is not used for any actual computation in the $4D$ case; it serves only to build intuition for why dimensional continuation requires the use of $QnS$ space when chiral projectors appear.

4.

The text below clarifies the $4D$ example involving the propagator

$$D_B={\displaystyle \frac{1}{\cancel{k}-\cancel{b}{\gamma }_5}},$$

and explains how naive symmetrization fails in $Q4S$ unless $\{{\gamma }_{\mu },{\gamma }_5\}=0$.

If one uses that the anticommutator $\{{\gamma }_{\mu },{\gamma }_5\}=0$ and the notation $P_R,P_L$ for the usual right and left chiral projectors, one obtains for $D_B$ and its inverse in the massless limit

$$D_B={\displaystyle \frac{1}{\cancel{k}-\cancel{b}{\gamma }^5}},D_B^{-1}=\left((\cancel{k}-\cancel{b})P_R+(\cancel{k}+\cancel{b})P_L\right)$$

(37)

where the four-vector $b_{\mu }$ is constant valued.

In $Q4S$, however, using the Equation (21) leads to

$$D_B{D}_B^{-1}=1-\left({\displaystyle \frac{1}{\cancel{k}-\cancel{b}}}+{\displaystyle \frac{1}{\cancel{k}+\cancel{b}}}\right)(\widehat{\cancel{k}}-\widehat{\cancel{b}}{\gamma }^5),$$

(38)

which reduces to the unit matrix only if the anticommutator is zero. These operations are done prior to evaluating a trace and thus condition the outcome of its symmetrization.

This type of propagator occurs for instance in the “Bumblebee” model of Lorentz violation [96,97], in which Lorentz violation is induced through a Coleman and Weinberg mechanism [98] and $b_{\mu }$ is related to the existence of a non-trivial vacuum expectation value of a timelike axial-vector $B_{\mu }$ background field, $b_{\mu }=(b,0,0,0)\sim <0\left|B_{\mu }\right|0>$. Here we concentrate just on the regularization technicalities and refer to the bibliography for the physical relevance of this type of models.

It is known that in the BMHV approach a finite valued gap equation emerges associated to the polarization vector ${\mathrm{\Pi }}^{\mu }$ given by the fermionic tadpole diagram.

First let us consider the regularization defined in the physical space, such that $\{{\gamma }^{\mu },{\gamma }^5\}=0$ is employed

$$\begin{array}{cc}\hfill {\mathrm{\Pi }}^{\mu }=Tr\left[{\int }_k{\displaystyle \frac{1}{\cancel{k}-\cancel{b}{\gamma }^5}}{\gamma }^{\mu }{\gamma }^5\right]& =Tr\left[{\int }_k\left({\displaystyle \frac{1}{\cancel{k}-\cancel{b}}}{P}_L+{\displaystyle \frac{1}{\cancel{k}+\cancel{b}}}{P}_R\right){\gamma }^{\mu }{\gamma }^5\right]\hfill \\ \hfill & =2{\int }_k{\displaystyle \frac{k^{\mu }-b^{\mu }}{{(k-b)}^2}}-2{\int }_k{\displaystyle \frac{k^{\mu }+b^{\mu }}{{(k+b)}^2}}\hfill \\ \hfill & =2\int {\displaystyle \frac{k^{\mu }[{(k+b)}^2-{(k-b)}^2]-b^{\mu }[{(k+b)}^2+{(k-b)}^2]}{{(k-b)}^2{(k+b)}^2}}.\hfill \end{array}$$

(39)

After Feynman parametrization one obtains

$$\begin{array}{c}\hfill {\mathrm{\Pi }}^{\mu }=Tr\left[{\int }_k{\displaystyle \frac{1}{\cancel{k}-\cancel{b}{\gamma }^5}}{\gamma }^{\mu }{\gamma }^5\right]=-2{\int }_0^{1}dx{\int }_k{\displaystyle \frac{2b^{\mu }(k^2-\mathrm{\Delta })-4k^{\mu }(b·k)}{{(k^2-\mathrm{\Delta })}^2}}\end{array}$$

(40)

where $\mathrm{\Delta }=4b^{2}x(x-1)+{\mu }^2$, and ${\mathrm{\mu }}^2$ is a fictitious mass introduced in the propagators of Equation (39).

As compared to the expression in the framework of dimensional regularization in the BMHV scheme [97]

$$Tr\left[{\int }_k{\displaystyle \frac{1}{\cancel{k}-\cancel{b}{\gamma }^5}}{\gamma }^{\mu }{\gamma }^5\right]=-2{\mu }^{4-D}{\int }_0^{1}dx\int {\displaystyle \frac{d{k}^d}{{\left(2\pi \right)}^d}}{\displaystyle \frac{2b^{\mu }(k^2-M^2-2{\widehat{k}}^2)-4k^{\mu }(b·k)}{{(k^2-M^2)}^2}}$$

(41)

where $M^2=4b^{2}x(x-1)$, one observes that the main difference is the appearance of a term containing ${\widehat{k}}^2$.

On the other hand, after promoting the Lagrangian to $Q4S$, the propagator of the Bumblebee model reads after being rationalized

$${D_B}^{Q4S}={\displaystyle \frac{1}{\cancel{k}-\cancel{b}{\gamma }^5}}={\displaystyle \frac{k^2+{\overline{b}}^2+(2k·\overline{b}+[\widehat{\cancel{k}},\overline{\cancel{b}}]){\gamma }^5}{{(k-\overline{b})}^2{(k+\overline{b})}^2-4{\widehat{k}}^2{\overline{b}}^2}}(\cancel{k}+\overline{\cancel{b}}{\gamma }^5).$$

(42)

Here the $\cancel{k}$, $\cancel{b}$ were extended to $Q4S$, while ${\gamma }_5$ stays in $4S$.

The tadpole contribution to the gap equation is now expressed as

$$\begin{array}{c}\hfill {{\mathrm{\Pi }}^{\mu }}_{Q4S}=Tr\left[{\int }_k{\displaystyle \frac{1}{\cancel{k}-\cancel{b}{\gamma }^5}}{\gamma }^{\mu }{\gamma }^5\right]=Tr\left[{\int }_k{\displaystyle \frac{k^2+{\overline{b}}^2+(2k·\overline{b}+[\widehat{\cancel{k}},\overline{\cancel{b}}]){\gamma }^5}{{(k-\overline{b})}^2{(k+\overline{b})}^2-4{\widehat{k}}^2{\overline{b}}^2}}(\cancel{k}+\overline{\cancel{b}}{\gamma }^5){\gamma }^{\mu }{\gamma }^5\right]\end{array}$$

(43)

which can be recast as

$${{\mathrm{\Pi }}^{\mu }}_{Q4S}\left(b\right)=-2{\int }_0^{1}dx{\int }_k{\displaystyle \frac{2{\overline{b}}^{\mu }(k^2-\mathrm{\Delta })-4k^{\mu }\left(\overline{b}·k\right)}{{(k^2-\mathrm{\Delta })}^2}}+8{\int }_0^{1}dx{\int }_k{\displaystyle \frac{{\overline{b}}^{\mu }{\widehat{k}}^2}{{(k^2-\mathrm{\Delta })}^2}}.$$

(44)

and compared to Equation (40). The only difference in the first term of Equation (44) as compared to the result written in the physical dimension Equation (40) is that the internal momentum is now defined in $Q4S$. The integrals of Equation (40) have been effected using the rules of IReg, separating BDI from the finite pieces, yielding zero for both [94]. All the steps that lead to these integrals can be repeated in the first therm of Equation (44) which is then also null. One gets finally that Equation (44) reduces to the integral

$${\mathrm{\Pi }}^{\mu }\left(b\right)=8{\int }_0^{1}dx{\int }_k{\displaystyle \frac{{\overline{b}}^{\mu }{\widehat{k}}^2}{{(k^2-\mathrm{\Delta })}^2}}.$$

(45)

In order to evaluate this integral, we use that ${\widehat{k}}^2=k^2-{\overline{k}}^2$. As already discussed in the framework of IReg, symmetric integration is not allowed in divergent integrals

$$\begin{array}{c}\hfill {\int }_k{k}^{2}f\left(k\right)\ne g_{\alpha \beta }{\int }_k{k}^{\alpha }{k}^{\beta }f\left(k\right),\end{array}$$

(46)

implying that Lorentz contraction and regularization do not commute, once the internal momenta are in $Q4S$. On the other hand, for contracted internal momenta in $4S$, we have

$$\begin{array}{c}\hfill {\int }_k{\overline{k}}^{2}f\left(k\right)={\overline{g}}_{\alpha \beta }{\int }_k{k}^{\alpha }{k}^{\beta }f\left(k\right).\end{array}$$

(47)

Thus, the gap equation is given as

$${{\mathrm{\Pi }}^{\mu }}_{Q4S}\left(b\right)=8{\overline{b}}^{\mu }\left[{\int }_0^{1}dx{\int }_k{\displaystyle \frac{k^2}{{(k^2-\mathrm{\Delta })}^2}}-{\overline{g}}_{\alpha \beta }{\int }_0^{1}dx{\int }_k{\displaystyle \frac{k^{\alpha }{k}^{\beta }}{{(k^2-\mathrm{\Delta })}^2}}\right].$$

(48)

which has been evaluated in [94] to result in the finite contribution to the gap equation

$${\mathrm{\Pi }}^{\mu }\left(b\right)=i{\displaystyle \frac{{\overline{b}}^{\mu }}{3{\pi }^2}}{b}^2$$

(49)

and complies with the result obtained from Equation (41) in the BMHV scheme [97].

This simple exercise shows again the X space at work, which is crucial to get a finite contribution for the gap equation. In its absence the null result of Equation (40) in $4S$ is obtained.

We conclude this example by mentioning that although the Lagrangian in $Q4S$ breaks the global chiral symmetry associated to the model, it is possible to show that the symmetry violating terms vanish in the evaluation of the gap equation [94]; therefore, it is not necessary to resort to counterterms in this case.

To summarize the type of operations which are allowed and required in IReg in chiral models, we distinguish the case (1) where all operations can be performed in strict physical dimension from the case (2) where the extension to $Q4S$ algebra is indispensable:

1a—Symmetrization of Dirac traces in closed fermion loops has been shown to be a necessary and sufficient condition to ensure the following:

(i)

Guarantee gauge invariance to all orders in chiral abelian gauge theories. Here all surface terms (ST) are set to zero to implement MRI;

(ii)

Reproduce the Adler–Bell–Jackiw triangle anomaly. In this case STs are kept arbitrary until all integrations are effected and only then fixed according to the Ward Identities of a theory.

1b—The “right most position” approach is applicable in open fermionic strings at NLO. Work in progress at NNLO.

2—In the case that the ${\gamma }_5$ matrix is present in the propagators and not only in the interaction vertices.

In this case, in order to be able to identify spurious from physical anomalies, it is essential to resort to the QAP. In IReg it has not yet been proven that the method complies with it in general, but for specific examples it is possible to identify the CT necessary. This was done in [73,95].

4. Concluding Remarks

In this work we have surveyed the interplay between gauge symmetries, regularization schemes, and chiral properties in quantum field theories, choosing a $2D$ framework to address working examples in a pedagogical manner. The discussion emphasized how different regularization methods (mainly dimensional regularization, and implicit regularization) embody distinct advantages and shortcomings when it comes to preserving gauge invariance and chiral consistency. The subtleties highlighted here illustrate that no single scheme is universally optimal: the appropriate choice often depends on the particular physical problem, symmetry constraints, and renormalization requirements under consideration.

As already mentioned in Section 1.3 of the introduction, bridging among dimensional and non-dimensional schemes is proven to be possible at NLO. At higher orders, it is generally not so. As shown in [95] this is also the case for chiral theories; nevertheless, whenever NLO order terms are present in the higher order evaluation, for instance, in the counterterm structures, the bridging remains a useful tool. It has also been shown at NLO and NNLO for several examples [72,81,82,86,95] that $ϵ$ scalar degrees of freedom that are required in DRED are absent in IReg. It is not possible to infer at present if this property of IReg is general at higher orders in chiral theories.

Crucial to the consistency of IReg, in the sense that possible ambiguities arising in the manner Clifford algebra and corresponding traces are evaluated within UV divergent integrals, is the existence of an underlying normal form. As guiding principles, it is required that UV integrals are invariant under shifts of the integration momenta as well as that numerator–denominator consistency is fulfilled. The first requirement implies that momentum root invariance (MRI) is implemented by construction, since surface terms must be set to zero. As a consequence, abelian gauge invariance is also implemented by construction. The second requirement implies that $k^2\ne g_{\alpha \beta }{k}^{\alpha }{k}^{\beta }$, whether or not in the presence of ${\gamma }_5$, i.e., symmetric integration is not allowed.

In the present work examples, the normal form is achieved by cancelling squared loop momenta in numerator and denominator, after evaluation of the trace and prior to regularization. When ${\gamma }_5$ is present, the normal form requires in addition that $\{{\gamma }_{\mu },{\gamma }_5\}\ne 0$. This can be achieved by formally adopting a $Q4S=4S\oplus X$ extension which defines the BHMV procedure for IReg, thus potentially inheriting the loss of gauge invariance and BRST symmetry associated therewith, which must be restored by appropriate counterterms. At NLO we have been able to show empirically that a symmetrization of the trace or the “right most position” prescription in open fermion strings are sufficient to achieve consistent results whenever the ${\gamma }_5$ matrix is absent from fermion propagators and resides only in the vertices, not needing to extend to $Q4S$. However this turns out to be equivalent as symmetrization and “right most position” comply with the $Q4S$ approach when applicable. Moreover the symmetrization of the trace becomes cumbersome if more than one ${\gamma }_5$ matrix is present, as it is traded by its definition in terms of ${\gamma }_{\mu }$ matrices, four in $4S$ space for each. The $Q4S$ extension allows us to rely on the X space as the determining factor needed to handle consistently the non anti-commuting property of ${\gamma }_5$, and still evaluate all integrations strictly in $4S$. An example in $4D$ is discussed that illustrates these points.

Our analysis of chiral symmetries further underscored the delicate balance between regularization techniques and the preservation of physical principles. In particular, the treatment of anomalies remains a guiding test of any regularization method, as anomalies encapsulate both profound physical phenomena and potential pitfalls in theoretical consistency.