Adler–Bardeen theorem and manifest anomaly cancellation to all orders in gauge theories
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Abstract
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 we 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.
1 Introduction
The Adler–Bardeen theorem [1, 2] is a crucial property of quantum field theory, and one of the few tools to derive exact results. In the literature various statements go under the name of “Adler–Bardeen theorem”. They apply to different situations. The original statement by Adler and Bardeen says that (I) the Adler–Bell–Jackiw axial anomaly [3, 4] is one-loop exact. The second statement, which is the one we are going to study here, says that (II) (there exists a subtraction scheme where) gauge anomalies vanish to all orders, if they vanish at one loop. Statement II is important to justify the cancellation of gauge anomalies to all orders in the standard model. A third statement concerns the one-loop exactness of anomalies associated with external fields.
Statement I is expressed by a well-known operator identity for the divergence of the axial current. By means of a diagrammatic analysis, Adler and Bardeen were able to provide the subtraction scheme where that identity is manifestly one-loop exact in QED [1]. They emphasized that higher-order corrections vanish, unless they contain the one-loop triangle diagram as a subdiagram. Thus stated, statement I intuitively implies statement II. However, the original proof of Adler and Bardeen applies only to QED.
Other approaches to the problem have appeared, since the paper by Adler and Bardeen, in Abelian and non-Abelian gauge theories. For a review, see for example [2]. Statement I can be proved using arguments based on the properties of the renormalization group [5, 6, 7], regularization independent algebraic techniques [8], or an algebraic/geometric derivation [9] based on the Wess–Zumino consistency conditions [10] and the quantization of the Wess–Zumino–Witten action. Statement II can also be proved using renormalization-group (RG) arguments, with the dimensional regularization [11] or regularization-independent approaches [12].
More recently, statement II was proved by the author of this paper in standard model extensions with high-energy Lorentz violation [13], which are renormalizable by “weighted power counting” [14]. The approach of [13] is closer to the original approach by Adler and Bardeen, in the sense that it does not make use of RG arguments, algebraic methods or geometric shortcuts, it naturally provides the subtraction scheme where the all-order cancellation is manifest, and it is basically a diagrammatic analysis, although instead of dealing directly with diagrams, it uses the Batalin–Vilkovisky formalism [15, 16, 17] to manage relations among diagrams in a compact and efficient way.
In the present paper we prove statement II in the most general perturbatively unitary, renormalizable gauge theories coupled to matter, and elaborate further along the guidelines of Ref. [13]. We upgrade the approach of [13] in a number of directions, emphasize properties that were not apparent at that time, and expand the arguments that were presented concisely. We also gain a certain clarity by dropping the Lorentz violation. A side purpose of this investigation is to develop new techniques and tools to prove all-order theorems in quantum field theory with a smaller effort.
Our results make progress in several directions. To our knowledge, if we exclude Ref. [13] and this paper, statement II has been proved beyond QED only making use of arguments based on the renormalization group. However, RG arguments do not provide the subtraction scheme where the all-order cancellation is manifest, and they are not sufficiently general. For example, they are powerless when the beta functions identically vanish, so they exclude conformal field theories and finite theories, where, however, the Adler–Bardeen theorem does hold. Actually, RG techniques fail even when the first coefficients of the beta functions vanish [11, 12]. Our approach does not suffer from these limitations. Another reason to avoid shortcuts is that in the past the Adler–Bardeen theorem caused some confusion in the literature, therefore new proofs, and even more generalizations, should be as transparent as possible. In this paper we pay attention to all details.
The all-order cancellation of gauge anomalies is a property that depends on the scheme, but the existence of a good scheme is not evident. Knowing the scheme where the cancellation is manifest is very convenient from the practical point of view, because it saves the effort of subtracting ad hoc finite local counterterms at each step of the perturbative expansion. For example, using the dimensional regularization and the minimal subtraction scheme the cancellation of two-loop and higher-order corrections to gauge anomalies in the standard model is not manifest, and finite local counterterms must be subtracted every time.
To find the right subtraction scheme we need to define a clever regularization technique. It turns out that using the Batalin–Vilkovisky formalism and combining the dimensional regularization with the gauge invariant higher-derivative regularization, the subtraction scheme where the Adler–Bardeen theorem is manifest emerges quite naturally [13].
It is well known that, in general, gauge invariant higher-derivative regularizations do not regularize completely, because some one-loop diagrams can remain divergent. From our viewpoint, this is not a weakness, because it allows us to separate the sources of potential anomalies from everything else. We just have to use a second regulator, the dimensional one, to deal with the few surviving divergent diagrams.
The regularization we are going to use introduces two cutoffs: \(\varepsilon =4-D\), where \(D\) is the continued complex dimension, and an energy scale \(\Lambda \) for the higher-derivative regularizing terms. The regularized action must be gauge invariant in \(D=4\), to ensure that the higher-derivative regulator has the minimum impact on gauge anomalies. The physical limit is defined letting \(\varepsilon \) tend to \(0\) and \(\Lambda \) to \(\infty \). When we have two or more cutoffs, physical quantities do not depend on the order in which we remove them. More precisely, exchanging the order of the limits \(\varepsilon \rightarrow 0\) and \(\Lambda \rightarrow \infty \) is equivalent to change the subtraction scheme. That kind of scheme change is, however, crucial for our arguments.
Consider first the limit \(\Lambda \rightarrow \infty \) followed by \(\varepsilon \rightarrow 0\). When \(D\ne 4\) the limit \(\Lambda \rightarrow \infty \) is regular in every diagram and gives back the dimensionally regularized theory: no \(\Lambda \) divergences appear, but just poles in \(\varepsilon \). In this framework there are no known subtraction schemes where the Adler–Bardeen theorem holds manifestly.
Now, consider the limit \(\varepsilon \rightarrow 0\) followed by \(\Lambda \rightarrow \infty \). At fixed \(\Lambda \) we have a higher-derivative theory. If properly organized, that theory is superrenormalizable and contains just a few (one-loop) divergent diagrams, which are poles in \( \varepsilon \) and may be removed by redefining some parameters. At a second stage, we study the limit \(\Lambda \rightarrow \infty \), where \(\Lambda \) divergences appear and are removed by redefining parameters and making canonical transformations. We call the regularization technique defined this way dimensional/higher-derivative (DHD) regularization.
Intuitively, if gauge anomalies are trivial at one loop, there should be no further problems at higher orders, because the higher-derivative regularization is manifestly gauge invariant. Thus, we expect that the DHD regularization provides the framework where the Adler–Bardeen theorem is manifest. However, it is not entirely obvious that the two regularization techniques can be merged to achieve the goal we want. Among the other things, \(\varepsilon \) evanescent terms are around all the time and the \( \mathcal {O}(1/\Lambda ^{n})\) regularizing terms can simplify power-like \( \Lambda \) divergences, causing troubles. Nevertheless, with some effort and a nontrivial amount of work we can prove that all difficulties can be properly dealt with.
Summarizing, the statement we prove in this paper is
Theorem
In renormalizable perturbatively unitary gauge theories coupled to matter, there exists a subtraction scheme where gauge anomalies manifestly cancel to all orders, if they are trivial at one loop.
Once we have this result, we know that no matter what scheme we use, it is always possible to find ad hoc finite local counterterms that ensure the cancellation of gauge anomalies to all orders. Then we are free to use the more common minimal subtraction scheme and the pure dimensional regularization technique.
The paper is organized as follows. In Sects. 2–7 we prove the theorem in non-Abelian Yang–Mills theory coupled to left-handed chiral fermions. This model is sufficiently general to illustrate the key points of the proof, as well as the main arguments and tools, but relatively simple to free the derivation from unnecessary complications. At the end of the paper, in Sect. 8, we show how to include the missing fields, namely right-handed fermions, scalars, and photons, and cover the most general perturbatively unitary renormalizable gauge theory coupled to matter. Section 9 contains our conclusions. In Appendix A we recall the calculation of gauge anomalies in chiral theories. In Appendix B we recall the proof of a useful formula.
The proof for Yang–Mills theory coupled to chiral fermions is organized as follows. In Sects. 2 and 3 we formulate the dimensional and DHD regularization techniques. In Sects. 4–6 we prove the Adler–Bardeen theorem in the higher-derivative theory, studying the limit \(\varepsilon \rightarrow 0\) at \(\Lambda \) fixed. Precisely, in Sect. 4 we work out the renormalization, in Sect. 5 we study the one-loop anomalies and in Sect. 6 we prove the anomaly cancellation to all orders. In Sect. 7 we take the limit \(\Lambda \rightarrow \infty \) and conclude the proof of the Adler–Bardeen theorem for the final theory.
2 Dimensional regularization of chiral Yang–Mills theory
We first prove the Adler–Bardeen theorem in detail in four-dimensional non-Abelian Yang–Mills theory coupled to left-handed chiral fermions. This model offers a sufficiently general arena to illustrate the key arguments and tools of our approach. At the same time, we make some clever choices to prepare the generalization (discussed in Sect. 8) to the most general perturbatively unitary gauge theories coupled to matter. To begin with, in this section we dimensionally regularize chiral gauge theories and point out a number of facts and properties that are normally not emphasized, but are rather important for the arguments of this paper.
To keep the presentation simple we make some simplifying assumptions that do not restrict the validity of our arguments. Specifically, we do not include right-handed fermions and scalar fields, and assume that the groups \(G_{i}\) are non-Abelian, so there is no renormalization mixing among gauge fields, even when more copies of the same simple group are present. In Sect. 8 we explain how to relax these assumptions and cover the most general Abelian and non-Abelian perturbatively unitary renormalizable gauge theories coupled to matter.
The naïve \(D\)-dimensional continuation of the action (2.1) is not well regularized, because chiral fermions do not have good propagators. To overcome this difficulty, we proceed as follows. As usual, we split the \( D \)-dimensional spacetime manifold \(\mathbb {R}^{D}\) into the product \( \mathbb {R}^{4}\times \mathbb {R}^{-\varepsilon }\) of ordinary four-dimensional spacetime \(\mathbb {R}^{4}\) times a residual \((-\varepsilon ) \)-dimensional evanescent space \(\mathbb {R}^{-\varepsilon }\). Spacetime indices \(\mu ,\nu ,\ldots \) of vectors and tensors are split into bar indices \(\bar{\mu },\bar{\nu },\ldots \), which take the values 0,1,2,3, and formal hat indices \(\hat{\mu },\hat{\nu },\ldots \), which denote the \(\mathbb {R }^{-\varepsilon }\) components. For example, the momenta \(p^{\mu }\) are split into pairs \(p^{\bar{\mu }}\), \(p^{\hat{\mu }}\), or equivalently \(\bar{p}^{\mu }\), \(\hat{p}^{\mu }\). The flat-space metric \(\eta _{\mu \nu }= \) diag\( (1,-1,\ldots ,-1)\) is split into \(\eta _{\bar{\mu }\bar{\nu }}= \) diag\( (1,-1,-1,-1)\) and \(\eta _{\hat{\mu }\hat{\nu }}=-\delta _{\hat{\mu }\hat{\nu }}\). When we contract evanescent components we use the metric \(\eta _{\hat{\mu } \hat{\nu }}\), so for example \(\hat{p}^{2}=p^{\hat{\mu }}\eta _{\hat{\mu }\hat{ \nu }}p^{\hat{\nu }}\). We assume that the continued \(\gamma \) matrices \(\gamma ^{\mu }\) satisfy the continued Dirac algebra \(\{\gamma ^{\mu },\gamma ^{\nu }\}=2\eta ^{\mu \nu }\). We define \(\gamma _{5}=\imath \gamma ^{0}\gamma ^{1}\gamma ^{2}\gamma ^{3}\), \(P_{L}=(1-\gamma _{5})/2\), \(P_{R}=(1+\gamma _{5})/2\) and the charge-conjugation matrix \(C=-\imath \gamma ^{0}\gamma ^{2}\) in the usual fashion. Full \(\mathrm{SO}(1,D-1)\) invariance is lost in most expressions, replaced by \(\mathrm{SO}(1,3)\times \mathrm{SO}(-\varepsilon )\) invariance.
The action (2.1) gives the fermion propagator Open image in new window , which involves only the four-dimensional components \(\bar{p}^{\mu }\) of momenta. Therefore, it does not fall off in all directions of integration for \(p\rightarrow \infty \). Applying the rules of the dimensional regularization, fermion loops integrate to zero. To provide fermions with correct propagators we introduce right-handed \(\psi _{L}^{I}\) -partners \(\psi _{R}^{I}\) that decouple in four dimensions and are inert under every gauge transformations. We include \(\psi _{R}\) and \(\bar{\psi } _{R} \) into the set of fields \(\Phi \). It is not necessary to introduce sources \(K \) for them.
First, observe that the counterterms are \(B\), \(K_{B}\) and \(K_{\bar{C}}\) independent. Indeed, the source \(K_{B}\) appears nowhere in \(S_{r0}\), while \( K_{\bar{C}}\) appears only in \(-\int BK_{\bar{C}}\). Moreover, the gauge-fixing conditions are linear in the fields, and the \(B\)-dependent terms of \( S_{r0}\) are at most quadratic in \(\Phi \). Therefore no nontrivial one-particle irreducible diagrams can have external \(B\) legs.
Second, the action \(S_{r0}\) does not depend on the antighosts \(\bar{C} ^{a_{i}}\) and the sources \(K^{\mu a_{i}}\) separately, but only through the combinations \(K^{\mu a_{i}}+\partial _{\mu }\bar{C}^{a_{i}}\). The \(\Gamma \) functional must share the same property. Indeed, an antighost external leg actually carries the structure \(\partial _{\mu }\bar{C}^{a_{i}}\), since all vertices containing antighosts do so. Given a diagram with \(K^{\mu a_{i}}\) or \(\partial _{\mu }\bar{C}^{a_{i}}\) on external legs, we can construct almost identical diagrams by just replacing one or more legs \(K^{\mu a_{i}}\) with \(\partial _{\mu }\bar{C}^{a_{i}}\), or vice versa.
Third, power counting and ghost-number conservation ensure that the counterterms are linear in the sources \(K\). Using square brackets to denote dimensions in units of mass, we have \([K^{\mu a}]=[K_{C}^{a}]=2\), and \([K_{\psi }]=3/2\). These sources have negative ghost numbers. Therefore, the dimension of a term that is more than linear in \(K\) and has vanishing ghost number necessarily exceeds 4.
2.1 Structure of the dependence on the overall gauge coupling
The \(g\) structures (2.12) and (2.13) are preserved by the antiparentheses: if the functionals \(X(\Phi ,K,g)\) and \(Y(\Phi ,K,g)\) satisfy (2.12), or (2.13), then the functional \((X,Y)\) satisfies (2.12), or (2.13), respectively.
2.2 Properties of the dimensional regularization of chiral theories
Now we recall a few properties of the dimensional regularization of chiral theories, which are important for the rest of our analysis. It is well known that divergences are just poles in \(\varepsilon \). Instead, the terms that disappear when \(D\rightarrow 4\), called “evanescences”, may be of two types: formal or analytic. Analytically evanescent terms, briefly denoted by “aev”, are those that factorize at least one \( \varepsilon \), such as \(\varepsilon F_{\mu \nu }F^{\mu \nu }\), \(\varepsilon \bar{\psi }_{L}\imath \,/{D}\psi _{L}\), etc. Formally evanescent terms, briefly denoted by “fev”, are those that formally disappear when \( D\rightarrow 4\), but do not factorize powers of \(\varepsilon \). They are built with the tensor \(\delta _{\hat{\mu }\hat{\nu }}\) and the evanescent components \(\hat{x}\), \(\hat{p}\), \(\hat{\partial }\), \(\hat{\gamma }\), \(\hat{A}\) of coordinates, momenta, derivatives, gamma matrices and gauge fields. Examples are Open image in new window , \( (\partial _{\hat{\mu }}A_{\nu }^{a})(\partial ^{\hat{\mu }}A^{\nu a})\), etc.
Feynman diagrams may generate “divergent evanescences”, briefly denoted by “divev”. They are made of products between poles and formal evanescences, such as \((\partial _{\hat{\mu }}A_{\nu }^{a})(\partial ^{\hat{\mu }}A^{\nu a})/\varepsilon \). The theorem of locality of counterterms demands that we renormalize divergent evanescences away, together with ordinary divergences (see below). However, this makes sense only if we can define divergent evanescences unambiguously, which could be problematic due to the observations made above. For example, if we multiply both sides of formula (2.14) by \(1/\varepsilon \) we get a relation of the type “divev \(=\) finite \(+\) divev”.
Ultimately, the problem does not arise in the theories we are considering here, for the following reasons. Both the classical action and the counterterms are local functionals, equal to integrals of local functions of dimension 4. In the paper we also show that the first nonvanishing contributions to the anomaly functional (2.11) are local, equal to integrals of local functions of dimension 5. A fermion bilinear \(\bar{\psi }_{1}\gamma ^{\rho _{1}\cdots \rho _{k}}\psi _{2}\) has dimension 3, so power counting implies that the classical action, as well as counterterms and local contributions to anomalies, cannot contain products of two or more fermion bilinears. Therefore, they are not affected by the ambiguities discussed above. Those ambiguities can only occur in the convergent sector of the theory, where they are harmless, since both analytic and formal evanescences must eventually disappear.
Thanks to the properties just mentioned, it is meaningful to require that the action \(S_{r0}\), as well as its extensions constructed in the rest of this paper, do not contain analytically evanescent terms. More precisely, the coefficients of every Lagrangian terms should be equal to their four-dimensional limits. This request is important to avoid unwanted simplifications between \(\varepsilon \) factors and \(\varepsilon \) poles, when divergent parts are extracted from bilinear expressions such as \( (\Gamma ,\Gamma )\). It can be considered part of the definition of the minimal subtraction scheme. For the same reason, we must be sure that the antiparentheses do not generate extra factors of \(\varepsilon \), or poles in \(\varepsilon \), which is proved below.
Finite nonevanescent contributions will be called “nev”. We need a convention to define these quantities precisely, otherwise they can mix with evanescent terms. For example, we need to state whether \(\bar{C}\partial ^{2}C\), or \(\bar{C}\bar{\partial }^{2}C\), or a combination such as \((1+\alpha \varepsilon )\bar{C}\bar{\partial }^{2}C+\beta \bar{C}\hat{\partial }^{2}C\), where \(\alpha \) and \(\beta \) are constants, is taken to be nonevanescent. The convention we choose is that nonevanescent terms are maximally symmetric with respect to the \(D\)-dimensional Lorentz group. For the arguments of this paper we just need to focus on local functionals contributing to counterterms and anomalies. In the case of counterterms the nonevanescent terms are those appearing in the action \(S_{r0}\), which are \(\mathrm{SO}(D)\) -invariant when chiral fermions are switched off. In the case of anomalies the nonevanescent terms are \(\mathrm{SO}(D)\)-invariant unless they contain the tensor \(\varepsilon ^{\mu \nu \rho \sigma }\) or chiral fermions.
2.3 Evanescent extension of the classical action
2.4 Structure of correlation functions
Now we analyze the evaluation of correlation functions. We use the same notation for a function and its Fourier transform, since no confusion is expected to arise.
The analytic expansion around \(\varepsilon =0\) of (2.25) or (2.27) is defined by expanding the scalars \(G^{i}(k)\) in powers of \( \varepsilon \) without affecting the evanescent components of external momenta. The analytic limit is the order zero of the analytic expansion, once the poles in \(\varepsilon \) have been subtracted away. The formal limit \(\varepsilon \rightarrow 0\) is the limit where the evanescent components of gauge fields, external momenta and fermion bilinears are dropped. The limit \(\varepsilon \rightarrow 0\) is the analytic limit followed by the formal limit.
For the reasons explained above, the analytic and formal limits may be ambiguous in the convergent sector of the theory, but they are unambiguous in the divergent sector. More importantly, the limit \(\varepsilon \rightarrow 0\) is always unambiguous. Since the tensors \(G^{\mu _{1}\cdots \mu _{p}}\) are regular when any evanescent components \(\hat{k}\) of the external momenta \(k\) are set to zero, the formal limits of (2.25) and (2.27) are well defined.
When we use the expressions “\(\mathcal {O}(\varepsilon )\)” or “ev” we mean any quantity that vanishes in the limit \(\varepsilon \rightarrow 0\). Clearly, ev \(=\) aev \(+\) fev.
2.5 Locality of counterterms
Now we comment on the locality of counterterms. The forms of the regularized propagators ensure that a sufficient number of derivatives with respect to physical \(\bar{k}\) and/or evanescent \(\hat{k}\) components of external momenta \(k\) kills the overall divergences of Feynman diagrams. If we subtract the divergent evanescences, together with the ordinary divergences, up to some order \(n\), then both ordinary divergences and divergent evanescences of order \(n+1\) are polynomial in \(\bar{k}\) and \(\hat{k}\). The \(S_{r0}\) -extension \(S_{r}=S_{r0}+S_{\text {ev}}\) of (2.15) allows us to subtract all of them in a way that is efficient for the proof of the Adler–Bardeen theorem.
On the other hand, it is safe to subtract the divergent evanescences order by order, together with nonevanescent divergences. In this paper we adopt this prescription.
2.6 Properties of the antiparentheses
- (i)
the antiparentheses \((X_{\text {conv}},Y_{\text {conv}})\) of convergent functionals \(X_{\text {conv}}\) and \(Y_{\text {conv}}\) are convergent;
- (ii)
the antiparentheses \((X_{\text {conv}},Y_{\text {ev}})\) of convergent functionals \(X_{\text {conv}}\) and evanescent functionals \(Y_{\text {ev}}\) are evanescent;
- (iii)
the antiparentheses \((X,Y)\) do not generate either poles in \(\varepsilon \) or factors of \(\varepsilon \) if \(X\), \(Y\) and \((X,Y)\) do not involve products of two or more fermion bilinears.
- (iii\(^{\prime }\))
- the antiparentheses \((X_{\text {A}},Y_{\text {B}})\) of functionals \(X_{\text {A}}\) and \(Y_{\text {B}}\) with the properties specified by their subscripts A and B , satisfy the identitiesas long as \(X_{\text {A}}\), \(Y_{\text {B}}\) and \((X_{\text {A}},Y_{\text {B}})\) do not involve products of two or more fermion bilinears.$$\begin{aligned}&(X_{\text {fev}},Y_{\text {nev/fev}}) \!=\!\text {fev},\qquad (X_{\text {divev} },Y_{\text {nev/fev/divev}})\!=\!\text {divev},\nonumber \\&(X_{\text {ev}},Y_{\text {fev} })\!=\!\text {ev}, \quad \left. (X_{\text {nev}},Y_{\text {div}})\right| _{\text {div}} \!=\!(X_{\text {nev} },Y_{\text {div}}),\nonumber \\&\left. (X_{\text {nev}},Y_{\text {nev}})\right| _{ \text {nev}}=(X_{\text {nev}},Y_{\text {nev}}),\nonumber \\&\left. (X_{\text {nev}},Y_{\text {nevdiv}})\right| _{\text {nev div}} =(X_{\text {nev}},Y_{\text {nevdiv}}), \end{aligned}$$(2.32)
It remains to study the relation between \(L_{ij(X,Y)}\) and \(L_{iX}\), \(L_{jY}\). The antiparentheses can produce index contractions by means of the paired functional derivatives \(\delta /\delta A_{\mu }\)–\(\delta /\delta K^{\mu }\) and \(\delta /\delta \psi \)–\(\delta /\delta K_{\psi }\). Clearly, no such operations can generate poles in \(\varepsilon \). This observation is sufficient to prove statements (i) and (ii).
Statement (iii) also says that the antiparentheses cannot convert formal \( \varepsilon \) evanescences into analytic ones. It applies, for example, to local functionals \(X\) and \(Y\) that are equal to the integrals of functions of dimensions \(n_{X},n_{Y}\) \(\leqslant 5\), such that \(n_{X}+n_{Y}\) \( \leqslant 8\), because then \(X\), \(Y\) and \((X,Y)\) cannot contain products of two or more fermion bilinears. In the paper we will apply statement (iii) to the divergent contributions to \(\Gamma \) and the first nonvanishing contributions to the anomaly functional \(\mathcal {A}\) of (3.10).
3 DHD regularization
3.1 The DHD limit
The basic idea behind the DHD regularization is to “first send \(\varepsilon \) to zero, then \(\Lambda \) to infinity”. However, we must formulate the rules of such limits more precisely, since certain caveats demand attention. We distinguish the higher-derivative theory from the final theory. The higher-derivative theory is the one defined by the classical action \(S_{\Lambda }\) (or \(\tilde{S}_{\Lambda }\), if we use the tilde parametrization), where the scale \(\Lambda \) is kept fixed and treated like any other parameter, instead of a cutoff. It is super-renormalizable and regularized by the dimensional technique. Its divergences, which are poles in \(\varepsilon \), are subtracted in the next section using the minimal subtraction scheme. The final theory is obtained by taking the limit \(\Lambda \rightarrow \infty \) on the renormalized higher-derivative theory, after subtracting the \(\Lambda \) divergences that emerge in that limit.
- (i)First, consider analytic evanescences in \(\varepsilon \) multiplied by products \(\Lambda ^{k}\ln ^{k^{\prime }}\Lambda \), such as \(\varepsilon \Lambda ^{2}\ln \Lambda \). Since we first send \(\varepsilon \) to zero, these quantities are not true divergences and must be neglected. In any case, they cannot be subtracted away, because the theorem of locality of counterterms does not apply to them. Consider for example the integralwhere for the purposes of our present discussion the mass \(m\) can also play the role of an external momentum. Expanding the right-hand side in powers of \(\varepsilon \) we find that the \(\mathcal {O}(\varepsilon ^{0})\) terms, which are equal to$$\begin{aligned}&\int \frac{\mathrm {d}^{D}p}{(2\pi )^{D}}\frac{\Lambda ^{4}}{ (p^{2}+m^{2})(\Lambda ^{4}+(p^{2})^{2})}\\&\quad =\frac{\Lambda ^{4-\varepsilon }m^{2} \left[ \cos \left( \frac{\pi \varepsilon }{4}\right) +\frac{\Lambda ^{2}}{ m^{2}}\sin \left( \frac{\pi \varepsilon }{4}\right) -\frac{\Lambda ^{\varepsilon }}{m^{\varepsilon }}\right] }{2^{D}\pi ^{(D-2)/2}\Gamma (\frac{ D}{2})(\Lambda ^{4}+m^{4})\sin \left( \frac{\pi \varepsilon }{2}\right) }, \end{aligned}$$have a \(\Lambda \)-divergent part that is polynomial in \(m\), as expected, while the \(\mathcal {O}(\varepsilon ^{1})\) terms have a \(\Lambda \)-divergent part that contains expressions such as$$\begin{aligned} \frac{1}{32\pi ^{2}}\left( \pi \Lambda ^{2}-2m^{2}\ln \frac{\Lambda ^{2}}{ m^{2}}\right) +\mathcal {O}\left( \frac{m}{\Lambda }\right) , \end{aligned}$$which are not polynomial in \(m\).$$\begin{aligned} \quad \Lambda ^{2}\ln \frac{\Lambda ^{2}}{m^{2}},\quad m^{2}\ln ^{2}\frac{\Lambda ^{2}}{m^{2}}, \end{aligned}$$
- (ii)
Next, consider formal evanescences times \(\Lambda ^{k}\ln ^{k^{\prime }}\Lambda \), such as \((\ln \Lambda )\partial _{\mu }A_{\hat{ \nu }}\partial ^{\mu }A^{\hat{\nu }}\). These can (actually, must, for the reasons explained in Sect. 2.5) be subtracted away (as long as their coefficients are calculated at \(\varepsilon =0\)), because the form of regularized propagators ensures that counterterms are polynomial in both physical and evanescent components of external momenta and fields.
- (iii)
Formally evanescent expressions multiplied by products \(\Lambda ^{k}\ln ^{k^{\prime }}\Lambda \) and factors of \(\varepsilon \) are just like case (i) and should not be subtracted away.
- (iv)
For completeness, we point out a fourth type of \(\varepsilon \) -evanescent \(\Lambda \) divergences, that is to say, nonlocal contributions of type (ii), which can appear as artifacts of inconvenient manipulations. Precisely, because of the ambiguities encoded in (2.14) some quantities of type (i) can be converted into nonlocal divergences of type (ii). These conversions should just be avoided. To this purpose, it is sufficient to note that the structure (2.20) of diagrams and the expansion of the integrals \(G^{\mu _{1}\cdots \mu _{p}}\) only generate \( \varepsilon \)-evanescent \(\Lambda \) divergences of types (i), (ii) and (iii). In the event that “aev \(\rightarrow \) fev conversions” of type (2.14) are accidentally applied, nonlocal divergences of type (ii) can just be ignored, because they cannot mix with the local terms belonging to the power-counting renormalizable sector and they are resummable into contributions of type (i).
We can thus define the procedure with which we renormalize the final theory and define the physical quantities. We call it the DHD limit. We still organize the contributions to \(\Gamma \) and \(\mathcal {A}\) in the form (2.20). Referring to (2.25) and (2.27), the DHD limit is made of the analytic limit \(\varepsilon \rightarrow 0\), followed by the limit \(\Lambda \rightarrow \infty \), followed by the formal limit \( \varepsilon \rightarrow 0\). We also have the DHD expansion, that is to say, the analytic expansion around \(\varepsilon =0\) followed by the expansion around \(\Lambda =\infty \).
The three steps that define the DHD limit are unambiguous in the divergent sector, which does not contain products of more than one fermion bilinears. Instead, the first and third steps are ambiguous in the convergent sector. What is important is that the DHD limit is also unambiguous in the convergent sector.
4 Renormalization of the higher-derivative theory
In this section and the next two we study the higher-derivative regularized theory \(\tilde{S}_{\Lambda }\), keeping \(\Lambda \) fixed and (mostly) using the tilde parametrization. We first work out the renormalization of the theory, then study its one-loop anomalies and finally prove the anomaly cancellation to all orders.
5 One-loop anomalies
In this section we study the one-loop anomalies, and relate those of the final theory, which are trivial by assumption, to those of the higher-derivative theory, which turn out to be trivial as a consequence.
We know how \(\mathcal {\tilde{A}}_{\Lambda \text {nev}}^{(1)}\) depends on \(\tilde{g}\). The other dimensionful parameters of \(\tilde{S} _{\Lambda }\) (such as \(\tilde{\zeta }_{i}\) and \(\tilde{\xi }_{i}\)), as well as the powers of \(\Lambda \) multiplying various terms (such as \(\widetilde{\bar{ \psi }}_{L}^{I}\imath \widetilde{\,/{D}}\tilde{\psi }_{L}^{I}\)), have dimensions greater than 4. They cannot contribute to \(\mathcal {\tilde{A}} _{\Lambda \text {nev}}^{(1)}\), because the local functions \( \mathcal {\tilde{A}}^{a}\) are polynomial in them and have dimension 4. Thus, \( \mathcal {\tilde{A}}_{\Lambda \text {nev}}^{(1)}\) can only depend on \(\tilde{g}\tilde{C}\), \(\tilde{g}\tilde{A}\), \(\tilde{r}_{i}\), \( \lambda ^{\prime }\), \(\xi ^{\prime }\), \(\eta _{1i}\) and \(\eta _{2i}\). Using (5.3), switching to nontilde variables, and recalling that \(\tilde{g} \tilde{A}=gA\), \(\tilde{g}\tilde{C}=gC\), we see that \(\mathcal {A}_{\Lambda \text {nev}}^{(1)}\) is \(\Lambda \) independent. Now we show that actually \(\mathcal {A}_{\Lambda \text {nev}}^{(1)}\) coincides with the one-loop anomaly \(\mathcal {A}_{f\text {nev}}^{(1)}\) of the final theory.
To prove this fact, we need to take \(\Lambda \) to infinity and study the DHD limit at one loop. A more comprehensive study of the DHD limit will be carried out later. The terms that are divergent in this limit are denoted by “Ddiv”, to distinguish them from the divergences considered so far, which strictly speaking were “\(\varepsilon \)div”. Recall that, according to the definition of DHD limit, the \(\Lambda \)-divergent parts cannot contain analytic \(\varepsilon \) evanescences, but can contain formal \(\varepsilon \) evanescences.
One-loop gauge anomalies vanish when the trace appearing in (5.11) vanishes. Typically, the cancellation is possible when the gauge group is a product group and the theory contains various types of fermionic fields in suitable representations, as in the standard model.
The next step is to prove the anomaly cancellation to all orders in the higher-derivative theory. After that, we will have to complete the DHD limit by renormalizing the \(\Lambda \) divergences.
6 Manifest Adler–Bardeen theorem in the higher-derivative theory
This is not the final result we want, though. To get there we still need to take \(\Lambda \) to infinity and complete the DHD limit.
7 Manifest Adler–Bardeen theorem in the final theory
We are finally ready to study anomaly cancellation to all orders in the final theory. In this section we study the \(\Lambda \) dependence and complete the DHD limit, according to the rules of Sect. 3.1. The subtraction of \(\Lambda \) divergences proceeds relatively smoothly, and preserves the master equation to all orders up to terms that vanish in the DHD limit.
- (I)
\(\Gamma _{n}\) has a regular limit for \(\varepsilon \rightarrow 0\) at fixed \(\Lambda \), and
- (II)the local functionalis “truly \(\varepsilon \)-evanescent at fixed \(\Lambda \)”, that is to say a local functional such that \(\langle \mathcal {E}_{n}\rangle \) tends to zero when \( \varepsilon \rightarrow 0\) at fixed \(\Lambda \).$$\begin{aligned} (S_{n},S_{n})\equiv \mathcal {E}_{n} \end{aligned}$$(7.1)
Moreover, since the canonical transformations generated by (7.4) act multiplicatively on fields and sources, the operations \(T_{n}\) act on the \( \Gamma \) functional precisely as they act on the action. Therefore, \(\Gamma _{n+1}=T_{n}\Gamma _{n}\). Since the operations \(T_{n}\) are \(\varepsilon \) -independent, we conclude that \(\Gamma _{n+1}\) is regular when \(\varepsilon \rightarrow 0\) at fixed \(\Lambda \), to all orders in \(\hbar \), which promotes the inductive assumption (I) to \(n+1\) loops.
The DHD framework defines a subtraction scheme where the cancellation takes place naturally and manifestly. In any other framework, the right scheme must be identified step-by-step, from two loops onwards, by fine-tuning local counterterms.
Some final comments are in order. Because of (4.7) higher-order divergent terms of the form \(\Lambda ^{p}\)ln\(^{k}\Lambda /\varepsilon \) are generated along the way. They appear in \(S_{R}\) and in the partially renormalized actions \(S_{n}\). Our renormalization procedure (which is just made of redefinitions of parameters, fields, and sources) makes them cancel opposite contributions coming from diagrams. Therefore, they do not appear in the \(\Gamma \) functionals \(\Gamma _{R}\) and \(\Gamma _{n}\), which are indeed regular in the limit \(\varepsilon \rightarrow 0\) at \(\Lambda \) fixed.
In several steps of the proof we have used the fact that \(S_{\Lambda }=S_{r}+ \mathcal {O}(1/\Lambda ^{6})\). It is important that the higher-derivative regularized classical action \(S_{\Lambda }\) does not contain terms with fewer inverse powers of \(\Lambda \). Consistently with this, renormalization does not require to turn them on. The operations \(T_{n}\) may contain power-like divergences, which can generate terms with less than six inverse powers of \(\Lambda \) when they act on \(S_{\Lambda }-S_{r}\). Those terms are at least one loop and not divergent, so they do not affect the structure of the classical action \(S_{\Lambda }\).
8 Standard Model and more general theories
In this section we show how to extend the proof of the previous sections to the standard model and the most general perturbatively unitary, power-counting renormalizable theories. We just need to include photons \( V_{\mu }\), scalar fields \(\varphi \), and right-handed fermions \(\chi _{R}\), which were dropped so far for simplicity. Depending on the representations, we can also add Majorana masses to the fermions \(\psi _{L}\).
Let us describe the nontrivial contributions to the one-loop gauge anomalies \(\mathcal {A}_{f\text {nev}}^{(1)}\). We have terms of the Badreen type and terms proportional to \(C^{u}W_{\mu \nu }^{v}W^{z\mu \nu }\). Using differential forms, the terms of the Bardeen type are linear combinations of \(\mathcal {B}_{1}=\int \mathrm {Tr}\left[ dC{\scriptscriptstyle \wedge }A{\scriptscriptstyle \wedge }dA\right] \) and \(\mathcal {B}_{2}=\int \mathrm {Tr}\left[ dC{\scriptscriptstyle \wedge }A{\scriptscriptstyle \wedge }A{ \scriptscriptstyle \wedge }A\right] \), as in (5.11), where now \(C=C^{\hat{a}}T_{f}^{\hat{a}}\), \(A=A_{\mu }^{\hat{a}}T_{f}^{\hat{a} }dx^{\mu }\), \(d=dx^{\mu }\partial _{\mu }\) and \(T_{f}^{\hat{a}}\) are the matrices \(T^{\hat{a}}\) restricted to the fermions. The coefficient of \( \mathcal {B}_{1}\) is the same as in (5.11), apart from the minus sign associated with right-handed fermions. The coefficient of \( \mathcal {B}_{2}\) is uniquely determined by the coefficient of \(\mathcal {B} _{1}\), but it differs from the one of (5.11) any time \(U(1)\) gauge fields and/or ghosts are involved. The terms proportional to \(CW_{\mu \nu }W^{\mu \nu }\) can only appear in (unusual) situations where global \( U(1) \) gauge symmetries are potentially anomalous. One-loop gauge anomalies are trivial when all these terms cancel out, and there exists a local functional \(\chi (gA)\) such that \(\mathcal {A}_{f\text {nev} }^{(1)}=(S_{K},\chi )\).
With the rules of this section gauge anomalies manifestly cancel to all orders in the most general perturbatively unitary, renormalizable gauge theory coupled to matter, as long as they vanish at one loop. We stress again that the proof we have given also works when the theory is conformal or finite, or the first coefficients of its beta functions vanish, where instead RG techniques are powerless.
9 Conclusions
We have reconsidered the Adler–Bardeen theorem, focusing on the cancellation of gauge anomalies to all orders, when they are trivial at one loop. The proof we have worked out is more powerful than the ones appeared so far and makes us understand aspects that the previous derivations were unable to clarify. Key ingredients of our approach are the Batalin–Vilkovisky formalism and a regularization technique that combines the dimensional regularization with the higher-derivative gauge invariant regularization. The most important result is the identification of the subtraction scheme where gauge anomalies manifestly cancel to all orders. We have not used renormalization-group arguments, so our results apply to the most general perturbatively unitary, renormalizable gauge theories coupled to matter, including conformal field theories, finite theories, and theories where the first coefficients of the beta functions vanish.
In view of future generalizations to wider classes of quantum field theories, we have paid attention to a considerable amount of details and delicate steps that emerge along with the proof. We are convinced that the techniques developed here may help us identify the right tools to upgrade the formulation of quantum field theory and simplify the proofs of all-order theorems.
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