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D 98, 025018 (2018)

Supercurrent anomaly and gauge invariance in the N = 1 supersymmetric Yang-Mills theory

† ‡ Y. R. Batista,1,* Brigitte Hiller,2, Adriano Cherchiglia,3, and Marcos Sampaio3,§ 1UFMG—Universidade Federal de Minas Gerais, ICEx, Dep. de Física, Av. Antônio Carlos, 6627, Belo Horizonte, MG, Brasil, CEP 31270-901 2CFisUC, Department of Physics, University of Coimbra, 3004-516 Coimbra, Portugal 3UFABC—Universidade Federal do ABC, CCNH—Centro de Ciências Naturais e Humanas, Campus Santo Andr´e, Avenida dos Estados, 5001, Santo Andr´e—SP—Brasil, CEP 09210-580

(Received 29 May 2018; published 26 July 2018)

We analyze calculational issues related to the quantum breaking of supercurrent conservation in a supersymmetric non-Abelian Yang-Mills theory. For the sake of simplicity, we take a zero mass gauge multiplet interacting with a massless Majorana spin-1=2 field in the adjoint representation of SUð2Þ. We shed light on a long-standing controversy regarding the perturbative evaluation of the supercurrent anomaly in connection with gauge and superconformal in different frameworks. We find that only superconformal symmetry is unambiguously broken using an invariant four dimensional and compare with the triangle AVV anomaly. Subtleties related to momentum routing invariance in the loops of diagrams and Clifford algebra evaluation inside divergent integrals are also discussed in connection with finite and undetermined quantities in Feynman amplitudes.

DOI: 10.1103/PhysRevD.98.025018

I. INTRODUCTION regularization method that violates , for instance. Anomaly-mediated symmetry breaking is an important The existence of anomalies may be established in a mechanism in field and theory [1]. Its range of regularization independent way. The Adler-Bardeen applications run from phenomenological, such as the anomaly, for instance, can be shown to be determined calculation of the decay rate for neutral pions into two by the topological term TrGG˜ algebraically characterized [2], the computation of quantum numbers in the as a nonvariation under gauge and BRS symmetry. In [8] Skyrme model of hadrons [3] and mechanisms for baryo- was proved that the coefficient of the anomaly is deter- genesis in the [4], to theoretical, namely the mined by convergent one-loop integrals. Moreover, in [9] it study of dualities in , the computation of was shown that, with local coupling, supersymmetric Yang- anomalous couplings in the effective theory of D-, Mills theories have an anomalous breaking of supersym- and the analysis of entropy [5,6]. metry at one-loop order. In the particular case of supersymmetry breaking, it is Perturbative evaluation of a quantum symmetry breaking neither straightforward nor conclusive that supersymmetry is therefore intimately related to regularization issues. A is a symmetry of the full quantum theory in general. specious anomaly stemming from noninvariant regulariza- However, as discussed in [7], there have been claims about tions appears when finite and regularization dependent supersymmetry anomalies which turned out erroneous terms are erroneously incorporated into an amplitude. A because of the difficulty to distinguish between a genuine symmetry preserving regularization is of considerable and a spurious anomaly. The latter is an apparent violation computational utility. Evidently if a model is known before- of a supersymmetric Ward identity due to use of a hand to be anomaly free, the question of whether there exists or not an invariant scheme is irrelevant, should the impo- *[email protected] † sition of Ward identities order by order in perturbation [email protected] ‡ [email protected] theory not to be considered a nuisance. In this case either, §[email protected] one employs an invariant scheme and performs renormal- ization using invariant counterterms or uses noninvariant Published by the American Physical Society under the terms of counterterms to compensate the symmetry breaking. For the the Creative Commons Attribution 4.0 International license. latter strategy to work, a precise knowledge of the symmetry Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, content of the model must be known which often requires and DOI. Funded by SCOAP3. nonperturbative [10] information. In fact, the absence of

2470-0010=2018=98(2)=025018(16) 025018-1 Published by the American Physical Society BATISTA, HILLER, CHERCHIGLIA, and SAMPAIO PHYS. REV. D 98, 025018 (2018) anomalies may be proven without recourse to any they transform among themselves under constant super- regularization by using algebraic properties of the Ward symmetry transformations. As the axial-vector current identities (algebraic ) at least for some has an anomaly, one is compelled to conclude that the particular cases [7,11–14]. supercurrent conservation could be anomalous as well [53].1 Although dimension regularization [15] is tailor-made for The quantum breaking of such symmetry constraints gauge theories, it is less suited to dimensional sensitive translates into the violation of one amid three Ward quantum field theoretical models such as supersymmetric, identities which hold at classical level. As we shall discuss, topological and chiral gauge theories. Naive dimensional different calculational frameworks placed the anomaly in reduction (DRed) can be shown mathematically inconsistent one of the Ward identities. An important result by Abbott, [16]. For instance, Lorentz algebra contractions can lead to Grisaru and Schnitzer [53,56] shows, however, that one equations such as 0 ¼ nðn − 1Þðn − 2Þðn − 3Þðn − 4Þ, valid cannot derive the quantum breaking of the supercurrent only for integer spacetime dimension n and thus it is conservation from the axial-vector current anomaly. incompatible with analytical continuation proposed by Moreover in [57], within a four-dimensional approach dimensional methods. In [17], a consistent version of called preregularization, it was shown that the supercurrent DRed was developed which forbids the use of Fierz anomaly is connected to the inability to reconcile ambi- identities, implying that supersymmetry will also not be guities (in the form of specific momentum routings in the respected in general. Nevertheless, in the same reference it is Feynman diagrams) in a way to preserve simultaneously shown how to identify the breaking of supersymmetry by gauge and supersymmetry. means of the quantum principle, allowing DRed to be The purpose of this work is to shed light on the apparent made operational for particular models to a specific loop clash in the perturbative calculation of the quantum sym- order [18]. In the same vein, there were severe difficulties to metry breaking of the supercurrent conservation in N ¼ 1 renormalize supersymmetric theories in a regularization super YangMills theory in connection with gauge invariance independent way: in the Wess-Zumino gauge useful in and the Rarita-Schwinger constraint (the latter also known as practical calculations, the usual way of treating global superconformal, spin-3=2 or supercurrent trace constraint) by Ward identities was shown to fail [17,19]. [53]. Subtleties related to Dirac algebra and symmetric Anomaly mediated supersymmetry breaking is an impor- integration within divergent amplitudes, parametrization of tant mechanism in addition to Planck scale mediated and arbitrary (finite) regularization dependent terms and gauge mediated scenarios [20]. The former is related to momentum routing invariance in a framework which oper- superconformal anomaly and started back in early 1970s ates in the physical dimension such as IReg will be clarified. with developments in rigid supersymmetry models [21]. Moreover a comparison with results of the supercurrent The study of supersymmetry breaking involves the com- anomaly performed in other regularization frameworks and putation of Ward identities connecting Greens functions of a comparison with the Adler-Bardeen-Bell-Jackiw (ABBJ) the supercurrent to other matrix elements. For those Ward triangle anomaly will be presented. identities to hold, no anomalies of the supercurrent should This contribution is organized as follows. In Sec. II,we exist. Although a manisfest gauge and supersymmetry present some technical tools and apply them to ABBJ invariant regularization is still to be constructed some anomaly. The Feynman rules of SU(2) super Yang-Mills regularization frameworks that operate in the physical Lagrangian with massless Majorana spinors in the presence S dimension do exist [22–51]. In particular, implicit regu- of an external current μ as well as the one-loop correction S → ψ larization (IReg), developed by Batisttel and collaborators, to the process μ þ Aμ compose most of Sec. III.It [23–44] systematically identifies, to arbitrary loop order, also contains the corresponding Ward identities respected at regularization dependent terms as surface terms (resulting classical level and subject to quantum breaking. The results from differences between loop integrals free of external of the Ward identities within different schemes with focus momenta with the same superficial degree of divergence) in IReg follow in Secs. IVand V. Secs. VI and VII contain a without recourse to an explicit regulator. For a comparison general discussion and conclusions regarding subtleties with other similar emergent schemes, see [45]. IReg has appearing in perturbative evaluation of anomalies. All been shown to be adequate to connect momentum routing technical details are left to Appendices. invariance in a diagram, gauge invariance and surface terms in the corresponding Feynman amplitude and therefore it II. IREG AND THE ABBJ TRIANGLE ANOMALY will be used as a tool in this contribution. IReg [23–44] is a regularization framework applicable to We revisit an old controversy regarding the diagrammatic arbitrary loop order proposed as an alternative to dimen- evaluation of the supercurrent anomaly that started with de sional schemes. It operates in the physical dimension of the Witt and Freedman [52]. For concreteness we study N ¼ 1 super Yang-Mills SUð2Þ theory in four spacetime dimen- 1 For a unified discussion about chiral and conformal anoma- sions. In this model, the supercurrent, the axial current and lies, see [54]. Also, see [55] for a study about path integral the stress-energy tensor belongs to the same multiplet, i.e., derivation of anomalies.

025018-2 SUPERCURRENT ANOMALY AND GAUGE INVARIANCE IN … PHYS. REV. D 98, 025018 (2018) underlying avoiding some draw- etc., 2w being the degree of divergence of the integrals backs of dimensional methods, for instance the mismatch (hereafter we identify the subscripts 0,1,2 with log, lin, between fermionic and bosonic degrees of freedom that quad). The curly brackets above stand for symmetrization leads to supersymmetry breaking and ambiguities in the γ5 in the Lorentz indices. It is straightforward to show that STs matrix and Levi-Civita tensor algebra due to dimensional are integrals of total derivatives, continuation on the spacetime dimension. Moreover, spu- Z ∂ μ rious evaluation of regularization dependent parameters υ μν k 2wg ¼ 2 2 2−w ; ð5Þ [58], which is usual in some nondimensional frameworks, k ∂kν ðk − m Þ are avoided. IReg operates in momentum space using as μν αβ μα νβ μβ να main strategy the isolation of basic divergent loop integrals ðξ2w − v2wÞðg g þ g g þ g g Þ (BDIs) of a given superficial degree of divergence that Z μ α β ∂ 2ð2 − wÞk k k characterizes the UV divergent behavior of Feynman ¼ : ð6Þ ∂ 2 − 2 3−w amplitudes. The latter is freed from external momentum k kν ðk m Þ dependence by judiciously applying an algebraic identity at The differences between integrals with the same degree of the integrand level: divergence (3) and (4) are regularization dependent and 1 1 ð−1Þðp2 −2p·kÞ should be fixed by symmetry constraints or phenomeno- 2 2 ¼ 2 2 þ 2 2 2 2 : ð1Þ logy [58]. As shown for instance in [29], such STs are ðk−pÞ −m ðk −m Þ ðk −m Þ½ðk−pÞ −m intimately connected with momentum routing invariance It resembles in some aspects others four-dimensional (MRI) in the loops of a Feynman diagram. By consistently programs such as differential renormalization [22,33] setting STs to zero order by order in perturbation theory and the FDR scheme [45,49] in which the intrinsic enforces both MRI and gauge invariance [25,29], allowing divergent pieces are called “vacua”. Infrared divergences us to conjecture that STs are at the root of some symmetry can be regulated either by a fictitious mass at breakings in Feynman diagram calculations. For instance in level or by infrared basic integrals in coordinate space [36]. [29,37–39] it was shown that constrained IReg (i.e., Tensorial basic divergent integrals in turn may be expressed systematically setting STs to vanish) is also a necessary as scalar ones plus surface terms (ST). STs encode most of condition for supersymmetry invariance. Similar results the regularization dependent pieces of explicit regulariza- using different theories were obtained for non-Abelian tions. These features are especially useful for dimensional gauge theories [30,31,38]. More recently, IReg was shown specific quantum field theories. Moreover the IReg scheme to be useful in dealing with γ5 algebra issues in Feynman can be generalized to arbitrary loop order complying with amplitudes [44,60]. the BPHZ renormalization program [59]. After subtraction Finally, a mass dimensional parameter in BDIs can be of subdivergences following the Bogoliubov’s recursion extracted to define a minimum and mass independent formula (devised for subtracting nested and overlapping subtraction scheme via a regularization independent iden- divergences) it is still possible to define new BDIs and tity which, at one loop order, reads surface terms characterizing the divergent behavior at 2 arbitrary loop order. Here, for the sake of brevity, we shall 2 λ2 − m Ilogðm Þ¼Ilogð Þ b ln 2 ; ð7Þ describe only the one-loop structure of IReg. λ To establish our notation, we write one-loop logarithmi- cally basic divergent integrals as2 where Z μ μ μ …μ k 1 …k 2n i 1 2n 2 ≡ ≡ I0 ðm Þ 2 2 2þn ; ð2Þ b 2 ð8Þ k ðk − m Þ ð4πÞ with similar definitions for linearly and quadratically and λ plays the role of a scale. divergent objects. One-loop STs are defined by Derivatives of BDIs with respect to λ yield a constant or another BDI. They are absorbed into renormalization ϒμν μν 2 − 2 2 − μν 2 ≡ υ μν 2w ¼ g I2wðm Þ ð wÞI2wðm Þ 2wg ; ð3Þ constants without explicit evaluation allowing the compu- tation of the usual renormalization group functions. Ξμναβ fμν αβg 2 − 4 3 − 2 − μναβ 2 2w ¼ g g I2wðm Þ ð wÞð wÞI2w ðm Þ μν αβ μα νβ μβ να ≡ ξ2wðg g þ g g þ g g Þ; ð4Þ A. ABBJ anomaly within IREG As an illustration which will serve to interpret the results R R 2 ≡ Λ 4 2π 4 Λ of the supercurrent anomaly, let us present a classical k d k=ð Þ , being a four-dimensional implicit regulator (say, a cutoff) just to justify algebraic operations within example, namely the Adler-Bardeen-Bell-Jackiw (ABBJ) the integrands. triangle anomaly. Calculational details have been presented

025018-3 BATISTA, HILLER, CHERCHIGLIA, and SAMPAIO PHYS. REV. D 98, 025018 (2018) elsewhere [25,29]. Since its discovery [2], the ABBJ which all γ5 should be moved to the rightmost position of anomaly has been calculated within several approaches the amplitude before its spacetime dimensionality is [57,61–68]. An overview on the various regularization altered. Another proposal in four dimensions was discussed schemes applied in the diagrammatic computation of this in [67]. Moreover, in [60], it can be found a thorough anomaly can be found in [6]. There are also derivations discussion on calculations involving Clifford algebra obtained by the path integral measure transformation [69] within Feynman amplitudes evaluated in different schemes. and differential [70,71]. The usual view on the Although the evaluation of the ABBJ anomaly has been diagrammatic derivation of is that it occurs extensively discussed in the literature, we briefly recall due to the momentum routing breaking in the internal lines some aspects of its calculation within IReg with the of Feynman diagrams. Accordingly, the momenta of the purpose of comparing with the supercurrent anomaly. internal lines must assume specific values so that a certain The key features are firstly the parametrization of regu- Ward identity is preserved [72]. This feature was formal- larization dependent (and undetermined) quantities as sur- ized in [57] within a scheme called Preregularization. face terms and secondly MRI in the loops of a Feynman It is not uncommon that an anomaly can apparently be amplitude while working in the physical dimension where removed and reappear in another guise. This is indeed the the Clifford algebra is defined. MRI is known to be fulfilled case with the ABBJ anomaly, which is a property of the in cases where gauge symmetry is not broken at all orders fermion triangle with two vector and one axialvector in perturbation theory upon the use of dimensional regu- vertices. The anomaly may affect either the axial or the larization [15]. vector current, depending on how the theory is regularized. It is legitimate to expect that even in the presence of an However, it is not up to the regularization scheme which anomaly, vector current gauge invariance continues to identity is to be preserved. Ideally the calculational frame- evidence MRI which is built up from energy-momentum work should democratically display the anomaly which, if conservation at a diagram vertices. Indeed that is what we not spurious, contains important physical implications. verify by applying a minimal prescription based on the Besides the triangle anomaly, other examples of inter- symmetrization of the trace over the γ matrices involving changeable anomalies are gravitational anomalies for γ5. This prescription does not make use of the property fermions in two-dimensional spacetime [35]. The super- fγ5; γμg¼0, since the vanishing of such anticommutator current anomaly was thought to have the same property but inside divergent integrals is the origin of ambiguities [25] as we shall discuss in this contribution, at least within a even when applied in the physical dimension [60,74]. Thus, diagrammatic evaluation, this is not the case. for the traces involving four and six Dirac matrices and one Standard dimensional regularization [15] is the most γ5, we use appropriate invariant method for vector gauge invariance. μ β ν ρ 5 μβνρ However, as we have mentioned above, some inconsisten- Tr½γ γ γ γ γ ¼4iϵ and ð9Þ cies may appear regarding the manipulation of dimension- − γ i μ ν α β γ δ 5 specific objects such as the 5 matrix and the Levi-Civita Tr½γ γ γ γ γ γ γ tensor. To circumvent this problem, some rules had to be 4 added to the method, postulating how the dimensional ¼ −gαβϵγδμν þ gαγϵβδμν − gαδϵβγμν − gαμϵβγδν þ gανϵβγδμ continuation of such objects should be performed [73]. βγ αδμν βδ αγμν βμ αγδν βν αγδμ γδ αβμν − g ϵ þ g ϵ þ g ϵ − g ϵ − g ϵ Dimensional methods also break supersymmetry and some ad hoc rules must be incorporated as well. To this end, − gγμϵαβδν þ gγνϵαβδμ þ gδμϵαβγν − gδνϵαβγμ − gμνϵαβγδ; some supersymmetric Ward identities of the underlying ð10Þ model have to be validated to a certain loop order via, say, the quantum action principle [17]. Alternatively, a strategy which can be obtained replacing γ5 by its definition in four that imposes Ward identities to restore broken symmetries i μναβ spacetime dimensions, γ5 ¼ ϵ γμγνγαγβ. A similar order by order in perturbation theory could be employed. 4! approach as encoded in Eq. (10) was used in [60,67]. However, besides being not practical from the calculational Notice that if we had used the following identity to reduce viewpoint, it can also be misleading when there exists a the number of Dirac matrices, genuine anomaly. In the particular case of the AVV anomaly, the triangle μ β ν μβ ν νβ μ μν β μβνρ γ γ γ ¼ g γ þ g γ − g γ − iϵ γργ5; ð11Þ graph contains an axial vertex and care must be exercised with divergent amplitudes involving the dimension-specific γ γργ γ then both Eq. (9) and the anticommuting property 5 5 ¼ object 5 matrix and its Clifford algebra. That is because −γρ would lead to identities regarding the γ5 algebra are not always satisfied under divergent integrals, even in the physical dimension of Tr½γμγβγνγξγαγλγ5¼4iðgβμϵνξαλ þ gβνϵμξαλ − gμνϵβξαλ the model [25,60]. A gauge-invariant prescription for the γ5 λα μβνξ ξλ μβνα ξα μβνλ algebra was proposed in [62] called “rightmost ordering” in − g ϵ þ g ϵ − g ϵ Þ: ð12Þ

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q q

k k2 k k3

5 5 T . . k k1 k k1 p q p q

k k k k3 2 p p

FIG. 1. Triangle diagrams which contribute to the ABBJ anomaly. We label the internal lines with arbitrary momentum routings.

μνα ανβλ However, it is completely arbitrary which three γ matrices pμT ¼ −4iυ0ðα−β −1Þϵ pβqλ; should be taken in order to apply Eq. (11). A different μνα αμβλ q T 4iυ0 α−β −1 ϵ pβq ; choice would give Eq. (12) with Lorentz indexes permuted. ν ¼ ð Þ λ μνβλ Such arbitrariness turns out to be relevant in connection μνα μνβλ ϵ p q T 8iυ0 α−β −1 ϵ pβq − pβq : with symmetry breakings. Thus, the anticommutation ð þ Þα ¼ ð Þ λ 2π2 λ ð17Þ property fγμ; γ5g¼0 should be avoided inside a divergent integral. Moreover, it has been shown that this operation The number υ0ðα − β − 1Þ is arbitrary since υ0 is a (finite) can assign an a priori nonvanishing value to an arbitrary difference between two logarithmic divergences and α and surface term [25]. β are real numbers that we are free to choose as long as The amplitude of the Feynman diagrams in Fig. 1 is Eq. (14) representing energy-momentum conservation given by hold. We can parametrize this arbitrariness in a single 4 υ α − β − 1 ≡ 1 1 Z parameter a by redefining i 0ð Þ 4π2 ð þ aÞ. i i Then the Ward identities become Tμνα ¼ −i Tr γμ γν k =k þ =k1 − m =k þ =k2 − m 1 i μνα − 1 ϵανβλ γ γ μ ↔ ν ↔ pμT ¼ 2 ð þ aÞ pβqλ; × α 5 − þð ;p qÞ: ð13Þ 4π =k þ =k3 m 1 μνα 1 ϵαμβλ qνT ¼ 2 ð þ aÞ pβqλ; where the arbitrary routings kis obey the following relations 4π due to energy-momentum conservation at each vertex μνα 1 μνβλ p q T aϵ pβq : ð þ Þα ¼ 2π2 λ ð18Þ k2 − k3 ¼ p þ q; Notice that the anomaly is democratically displayed in the k1 − k3 ¼ p; Ward identities (18). For vector gauge invariance to be k2 − k1 ¼ q: ð14Þ preserved, one chooses a ¼ −1 and thus the axial identity 1 μνβλ is violated by a quantity equal to − 2 ϵ pβqλ. On the Equation (14) allow us to parametrize the routing k as 2π i other hand, chiral symmetry is maintained at the quantum level for a ¼ 0, and the vector identities are violated. The k1 ¼ αp þðβ − 1Þq; choice a ¼ −1 sets STs to zero [25,29]. In these references k2 ¼ αp þ βq; it was proved that setting STs to zero assures momentum routing invariance and consequently gauge invariance in k3 ¼ðα − 1Þp þðβ − 1Þq; ð15Þ Abelian gauge theories to arbitrary loop order. In other where α and β are arbitrary real numbers which express the words MRI in Feynman diagrams is a necessary and freedom to choose the routing of internal lines. Similar sufficient condition for gauge invariance in Abelian gauge equations for the other diagram are obtained by inter- theories. Although no general proof has been constructed, changing p ↔ q. After taking the trace with the help of the same appears to hold for non-Abelian gauge invariance Eq. (10), we apply the IReg scheme to obtain in Feynman diagram calculations [31,38]. In our result, it was crucial to take the symmetrized version of traces 3 β ˜ involving γ5 matrix as displayed in (10) rather than (12). Tμνα ¼ 4iυ0ðα − β − 1Þϵμναβðq − pÞ þ Tμνα; ð16Þ

3 where υ0 is a surface term (generally explicitly evaluated in In [34] the vector current Ward identities were satisfied when ˜ other regularization schemes) as defined in (5) and Tμνα is the surface term assumed a nonvanishing value that cancelled a finite term in order to preserve gauge symmetry. This was a the finite part of the amplitude sketched in Appendix D. byproduct of identity (12) which was used in that calculation. The vector and axial Ward identities for the massless Therefore, it was thought that the anomaly was due to the theory read: breaking of the momentum routing invariance.

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¯ αβ a In summary, we have seen that within a four-dimensional Sμϵ ¼ −iðψ¯ aγμσ ϵÞFαβ ð23aÞ regularization scheme such as IReg and taking into account some subtleties related to Clifford algebra inside divergent or equivalently, by the Majorana condition for ϵ and ψðxÞ integrals (symmetrization of the trace), arbitrary momen- tum routing amounts to gauge invariance. Such a routing αβ a ϵ¯Sμ ¼ −iðϵσ¯ γμψ aÞFαβ: ð23bÞ arbitrariness in a Feynman graph is always accompanied by a surface term, which is set to zero on gauge invariance Likewise, defining global supersymmetric transforma- grounds. Indeed even in the case when the axial Ward tions as in (22) but substituting γμ with γμγ5 and σμν with identity is chosen to be verified in the ABBJ anomaly, the σμνγ5 leads to a conserved Noether current, which is just momentum routing may be absorbed in the choice of the (23) with γμ → γμγ5. The result is not affected by which arbitrary ST. Thus it seems plausible to disregard particular definition of the transformation one uses as a γ5 factor can momentum routings in the discussion of Ward identities, be absorbed into the transformation properties of the even in anomalous cases, in favor of intrinsic arbitrary fields [75].4 parameters hidden in perturbation theory in the form of In order to study the on mass-shell anomalies of the STs. The latter should be left as an adjustable parameter if supercurrent we consider the process not fixed on symmetry principles of the underlying model [25,58]. Sμ → ψ þ Aμ ð24Þ

III. SUPERCURRENT ANOMALY IN SUSY for on-shell and fermions (therefore, no need to NON-ABELIAN GAUGE THEORY consider mass and wave function renormalization at the The N ¼1 supersymmetric and gauge invariant Yang-Mills one-loop level). The Feynman rules for the bubble and SU(2) Lagrangian in the Wess-Zumino gauge reads [21] triangle graphs that contribute to the process (24) required to evaluate quantum breakings of the supercurrent Ward 1 1 a aμν i a μ a a a identities are displayed in Fig. 2 in the Feynman gauge. In L − FμνF ψ¯ γ Dμψ C C ¼ 4 þ 2 ð Þ þ 2 [52] it was shown that matrix elements of this current 1 between physical states are gauge invariant and conserved: η∂μ abη ∂ μ 2 þ a Dμ b þ 2ξ ½ μAa ; μ ∂μ S 0: ≡ L L L hphysj jphysi¼ ð25Þ inv þ þ gauge; ð19Þ The external physical state is a fermion with momentum p where and color index a, and a gauge field with momentum q and ε a a a b c color index b, namely ju; p; aij ;q;bi. The transition Fμν ¼ ∂μAν − ∂νAμ þ gϵ AμAν; ð20Þ abc amplitude to vacuum state reads 1 2 3 ψ a a ¼ ; ; are color indices, is a massless Majorana μ ¯ ab ν phys S phys 0 Sμ u; p; a ε;q;b Sμν ε q u p : spinor (transforming in the adjoint representation), super- h j j i¼h j j ij i¼ bð Þ að Þ c a symmetric partner of the gauge fields Aν and C is an ð26Þ auxiliary field which we will drop out later as it does not couple to any field in the computation of the supercurrent. For on-shell bosons and fermions all factors of p= in ab 2 2 The covariant derivative is defined as Sμν adjacent to uaðpÞ vanish as well as factors of p , q ab qν Sμν a a b c or in : ðDμψÞ ¼ ∂μψ þ gϵabcAμψ : ð21Þ εν 0 0 2 2 0 L ðqÞqν ¼ ; pu= ðpÞ¼ ;p¼ q ¼ ; ð27Þ The action correspondent to inv is invariant under supersymmetry transformations, in which p= should be moved to the rightmost position to a a apply (27). δAν ¼ iϵγ¯ μψ ; ð22aÞ The Ward identities which express supersymmetry, a μν a a gauge and superconformal (spin-3=2 constraint) invariance δψ ¼ σ ϵFμν þ ϵC ; ð22bÞ of the supercurrent read a μ a δC ¼ ϵγ¯ Dμψ ; ð22cÞ 4Indeed should one employ the supercurrent definition with γ where σμν 1 γμ; γν and ϵ is a constant spinor. the 5-matrix into the Feynman rules, we see that applying, for ¼ 4 ½ instance, the rightmost positioning (gauge invariant) prescription The Noether current associated to the invariance of (19) for the γ5 matrix [62] leads to the same results for the quantum under (22) is corrections modulo a γ5 factor.

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ab FIG. 2. Feynman rules: (a) Gauge and fermion and interaction vertices. (b) Supercurrent Sμν vertices.

μ μ ab ab ab ab ab ab ab p − q Sμν 0; Sμν ≡ Σμν Σμν Σμν Σμν Σμν ; ð Þ ¼ ð28Þ ð Þ1−loop ¼ A þ B þ C þ D ð32Þ

ν ab q Sμν ¼ 0; ð29Þ in which μ ab γ Sμν ¼ 0; ð30Þ Z where qν is the momentum of the external gauge field. They 4 2δab α ab ρ g NA Σμν σρνγμγαγ ; are readily satisfied at tree level A ¼ 2 2 k ðk þ sAÞ ðk þ p þ sAÞ Z 2 ab βνη ab tree β ab ηβ 2g δ N ðSμν Þ ¼ −2iσβνγμq δ ; ð31Þ Σab −σ γ B μνB ¼ μ 2 2 ; k ðk þ sBÞ ðk þ q þ sBÞ Z 4 2δab αζ but an anomaly occurs at quantum level. Many authors ab g NCν Σμν ¼ −σαζγμ ; have calculated this anomaly and there has been some C ðk þ s Þ2ðk − p þ s Þ2ðk − q þ s Þ2 Z k C C C controversy about whether the anomaly is in the divergence 4 2δab ωη ab g NDν or in the trace of the supercurrent. We display their results Σμν ¼ σωηγμ ; D 2 − 2 − 2 in Table I. k ðk þ sDÞ ðk p þ sDÞ ðk q þ sDÞ We proceed with the calculation of the supercurrent at ð33Þ one-loop order in a fully four-dimensional setting using IReg. Feynman rules displayed in Fig. 2 yield, for the diagrams depicted in Fig. 3, the following superficially linearly divergent amplitude: with

TABLE I. Supercurrent anomaly in SYM N ¼ 1 in the Wess-Zumino gauge within different regularization schemes. Notice that the three Ward identities cannot be satisfied simultaneously at quantum level just as the AVV triangle.

Framework Gauge invariance Supercurrent conservation Spin constraint (Rarita Schwinger constraint) Rosenberg method [53] ✓✗ ✓ Preregularization [57] ✓ (✗) ✗ (✓) ✓ Cutoff [57] ✓ (✗) ✗ (✓) ✓ Analytical regularization [76] ✓✗ ✓ Point-splitting [77,78] ✓✗(✓) ✓ (✗) Dimensional reduction [75,79] ✓✗(✓) ✓ (✗) Dimensional regularization [78,80] ✓✓ ✗

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FIG. 3. Bubble and triangle graphs contributing to the process Sμ → ψ þ Aμ.

α α NA ¼ðk þ p þ sAÞ ; conditions at the beginning), a precise cancellation of spurious infrared regulators will take place among the βνη − 2 − − NB ¼ gβνðk þ q þ sBÞη þ gνηð k þ q sBÞβ various terms. At this stage, on-shell conditions (27) can

þ gβηð2k þ q þ 2sBÞν; be judiciously applied to simplify the final result. We emphasize that, without a proper regularization scheme αζ − γ γα ζ − ζ ζ NCν ¼ð=k =q þ =sCÞ νð=k þ =sCÞ ðk p þ sCÞ; for infrared divergences, the naive application of on-shell ωη − − γ − − ω conditions can lead to spurious cancellations (see NDν ¼ð =k þ p= =sDÞ βð k þ q sDÞ βη β η Appendix B)asin[53,57,76]. × ½ð−2k þ q − 2sDÞνg þðk − 2q − sDÞ δν ηδβ þðk þ q þ sDÞ ν: ð34Þ IV. LORENTZ DECOMPOSITION OF THE ONE-LOOP CORRECTION TO THE αβ 1 α β We used σ ≡ 4 ½γ ; γ and, for SUð2Þ, C2ðGÞ¼2. SUPERCURRENT Moreover allows the following decomposition Σab α α α for μν si ≡ mip þ niq ; ð35Þ ab Σμν ¼ =qγνγμB0 þ gμν=qB1 þ qμγνB2 þ pνγμB3 … ði ¼ A; ;DÞ are arbitrary internal momentum routings. γ In evaluating the integrals displayed in (33),itis þ pμ νB4 þ pνqμ=qB5 þ pμpν=qB6; ð36Þ important to bear in mind that, for any regularization that in which on-shell conditions (27) were taken into account operates in the physical dimension, the order in which the (and thus the coefficients of γνγμp=, gμνp=, qνγμ, qμqν=q and Clifford algebra is performed (before or after integration in internal momenta) yields different results.5 In order to avoid qμqνp= are zero). ambiguous symmetric integrations in the physical space- The coefficients B0, B1, B2, B3, and B4 possess regu- time dimension (and define a consistent framework with a larization dependent STs (and an in unique relation to the quantum action principle)[60,81] the B0) which is renormalizable and plays no role in the Ward algebra should be executed before integration. We illustrate identities. On the other hand B5 and B6 are finite and well this feature in Appendix A. determined (regularization independent). Notice that since In the amplitudes (33), we refrain to apply on-shell the supercurrent does not couple to any field in the conditions (27) at the level of the integrals which avoids the Lagrangian, the anomaly does not spoil the renormaliz- introduction of a proper regularization scheme to treat ability of the theory. The Ward identities (30) in terms of infrared divergences. Therefore, only a fictitious mass μ is the Lorentz decomposition of the one-loop correction to the added in the propagators to regularize spurious infrared supercurrent (36) read divergences that appear when ultraviolet divergences are ν ab μ2 μ → 0 q Σμν ¼ðB1 þ B2 þ p · qB5Þqμ=q þ p · qγμB3 isolated in terms of a BDI, namely Ilogð Þ. The limit is taken in the end of the calculation. The regularization þðB4 þ p · qB6Þpμ=q; independent relation (7) introduces a renormalization group μ μ ab ðp − q ÞΣμν ¼½B1 − B3 þðB5 − B6Þp · qpν=q scale λ ≠ 0 in the BDI’s in terms of which renormalization constants are defined. As the sum of amplitudes in Eq. (32) þðB2 − B4Þp · qγν; contributing to the one-loop correction of the supercurrent μ ab γ Σμν ¼ðB2 − B1Þ=qγν þ 2ð2B3 þ B4 þ p · qB6Þpν: is infrared finite (since we refrain to apply on-shell ð37Þ 5The FDR scheme [49], although defined in the physical dimension, avoids this issue by appending a new set of rules to In Table II, we display the values assumed by the the method which ultimately results in the introduction of “extra” coefficients B1, …, B4 in terms of the coefficient B5.As integrals, absent in IReg, for instance. we shall show in the next section B6 ¼ −B5.

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TABLE II. Values assumed by B1; …;B4 coefficients in terms of the coefficient B5 (momentum routing and surface term independent) for an anomaly in each of the Ward identities. We have used the result B5 ¼ −B6.

W. I. Anomaly B1 B2 B3 B4 Gauge ✓ Susy ✓ −2B5ðp · qÞ B5ðp · qÞ 0 B5ðp · qÞ μ ab Spin-3=2 γ Σμν ¼ 3B5ðp · qÞ=qγν Gauge ✓ − μΣab 3 − γ B5 B5 Susy ðp qÞ μν ¼ 2 B5ðp · qÞ½pν=q ðp · qÞ ν − 2 ðp · qÞ − 2 ðp · qÞ 0 B5ðp · qÞ Spin-3=2 ✓ ν ab 2 Gauge q Σμν ¼ −B5ðp · qÞqμ=q þ B5ðp · qÞ γμ − 2B5ðp · qÞpμ=q Susy ✓ −B5ðp · qÞ −B5ðp · qÞ B5ðp · qÞ −B5ðp · qÞ Spin-3=2 ✓

ab V. ON-SHELL EXPRESSION OF Σμν identities involving the supercurrent just as in the ABBJ anomaly. The B coefficients involve regularization depen- In Appendix C, we exhibit details of the computation of dent terms which somewhat explain the controversial the amplitudes which lead to the one-loop correction to the ab results using different frameworks exposed in Table I. supercurrent Σμν . Here, we show the complete expression Under the light of the ABBJ anomaly example discussed in order to discuss the relation between gauge invariance, earlier, one can easily verify that STs in (38) are always surface terms and MRI in Feynman diagrams. This is accompanied by arbitrary momentum routings and thus accomplished in a fully four-dimensional framework in setting STs to zero amounts to implement MRI and which, as discussed earlier, we regularize the amplitudes consequently vector gauge invariance, namely taking heed of subtleties involving the noncommuting character of operations like Clifford algebra contractions ν ab and on-shell/massless limits under the integration sign of a q Σμν ¼ 0; μ ab divergent amplitude. Thus, for the coefficients appearing in ðp − qÞ Σμν ¼ 0; Eq. (36), we get (see Appendix C) 3 μ ab i 2 ab γ Σμν g δ =qγν; ¼ 4π2 ð39Þ B0 2 ¼ −6I ðλ Þþð−2n þ 2n þ 2n − 13Þν0 g2δab log C D A showing that the supersymmetry Noether current remains 2 − 2 2 −3 − π2 6 p − 9 p · q conserved and the spin-3=2 constraint presents an anomaly. þ 3 b þ ln λ2 ln λ2 − 2 2 q p · q VI. A DIGRESSION ON THE RESULTS OF þ 12 ln þ 6Li2 1 − λ2 p2 THE LITERATURE 2 6 1 − p · q Although the value of the anomaly agrees in the different þ Li2 2 ; q frameworks, there is no consensus regarding which Ward B1 identity is anomalous at one-loop level. In [53], the same ¼ 4½ðn − n − 2n þ 3Þν0 þðn − n Þξ0 − 2b; g2δab D A C C D model was discussed, and an evaluation of the anomaly in the divergence of the supercurrent was presented. Their B2 ¼ 4½ðn − n − 2n − 3Þν0 þðn − n Þξ0 þ b; strategy was based in imposing both gauge invariance and 2δab D A C C D g the spin constraint which maintains the spin-3=2 character B3 2 5 2 − 4 2 ν 2 − ξ of the supercurrent. The latter was obtained at the cost of 2 ab ¼ ½ð mD þ mA mC þ Þ 0 þ ðmC mDÞ 0; αβ g δ factoring out the matrix product σ γμ throughout the B4 calculation in the amplitudes, ¼ 4½ð−2m − m − 2Þν0 þðm − m Þξ0 þ b; g2δab A C C D Σab σαβγ Σab δabσαβγ γ − γ B5 4b B6 μν ¼ μ αβν ¼ μ½A0ð βgαν αgβνÞ=q ¼ ¼ − : ð38Þ 2δab 2δab g ðp · qÞ g þ A1ðgβνpα − gανpβÞþA2ðpαγβ − pβγαÞγν γ − γ γ − γ where Li2 is the dilogarithm function and we made explicit þ A3ðqβ α qα βÞ ν þ A4ðpαqβ pβqαÞ ν=q the routing dependence as defined in (35). þ A5ðqαγβ − qβγαÞpν=q þ A6ðpβγα − pαγβÞpν=q Table II shows apparently the possibility of a “demo- − − cratic” display of the anomaly among the three Ward þ A7ðpβqα pαqβÞpν þ A8ðgανqβ gβνqαÞ; ð40Þ

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μ αβ μ because the identity γ σ γμ ¼ 0 ensures γμS ¼ 0. The In order to preserve gauge invariance in analytic regulari- Σab Σab coefficients A0, A1, A2, A3 and A8 that multiply tensors of zation, in (32) either the bosonic ( μνA and μνB) or the shift k Σab Σab first rank, after a in linearly divergent integrals in , fermionic ( μνC and μνD) propagators should be modified. contain surface terms due to the shift in the momentum Using Point-Splitting regularization, in [77] was shown an integral as well as finite unambiguous terms. On the other anomaly for spinor current which, however, could be hand A4, A5, A6 and A7 coefficients that multiply third rank removed by a redefinition of the gauge invariant conserved tensors are finite and unambiguous. Thus the imposition of current keeping the spin-3=2 constraint satisfied. ν ab gauge invariance, q Σαβν ¼ 0, leads to relations among the Hagiwara and collaborators [78], on the other hand, Ai’s in such a way that the supercurrent anomaly is employed dimensional, Pauli-Villars and point-splitting determined only by A4 to A7 coefficients. They applied regularizations to show that there exists an anomaly in the Rosenberg method [75] which assigns values for the the superconformal current but not in the divergence of the surface terms so that Ward identities are respected, viz. supercurrent. They emphasize the importance of picking up gauge invariance and spin-3=2 constraint. The latter con- a definite regularization in studying the anomaly of the αβ μ straint derives from the algebraic identity γμσ γ ¼ 0. supercurrent. Otherwise, due to its specific structure, all αβ The problem in maintaining the structure σ γμ factored the contributions to the one-loop correction would have the αβ out is that it clashes with the property of performing the form σ γμΣαβν which naturally grants privilege to the spin- μ Dirac algebra before integration to avoid spurious sym- 3=2 constraint to be satisfied and consequently ∂ Sμ ≠ 0. metric integration in the physical dimension. While this In Ref. [82], it is argued that it is not a matter of simply approach seems a matter of choice, it is not immaterial as redefining loop momenta that would shift the anomaly from discussed in Appendix A. More importantly, an anomaly of the supersymmetric current into the spin-3=2 constraint as the supercurrent suggests that a similar situation may this would have important implications on the multiplet develop in the context of models (as pointed structure of currents and anomalies. out in [53] itself) which may pose problems due to In [83], the supercurrent and superconformal anomalies renormalizability issues. In a subsequent work [56],an were evaluated for off-shell N ¼ 1 supersymmetric Yang- anomaly killing mechanism was developed at one-loop Mills theory within the Fujikawa method and the heat level by including interactions of the Yang-Mills meson and kernel regularization scheme. They obtain no one-loop Majorana spinors with the noninteracting scalar Wess- supercurrent anomaly and a superconformal anomaly that Zumino multiplet. Evidently we can map the expansion agrees with our calculation. (36) into (40). For instance, the coefficients that define the Finally, dimensional methods were discussed in [75,78– spin-3=2 constraint in (37) read 80].In[80], conventional dimensional regularization was employed, setting the anomaly at the superconformal B1 ¼ B2 ¼ 8ðA3 − A0 − ðp · qÞA4Þ; sector, a result that was later confirmed by [78].In[79], dimensional reduction was used, with the same outcome. 4 2 − 2 B3 ¼ ½A1 þ A2 ð A6 þ A7Þðp · qÞ; As discussed in this reference, for the calculation at hand at B4 ¼ −8ðA1 þ 2A2Þ; least at one-loop order, the two schemes are equivalent. ϵ 8 2 The reason is that -scalars, which need to be introduced in B6 ¼ ð A6 þ A7Þ; ð41Þ the dimensional reduction scheme to avoid a mismatch 3 2 between fermionic and bosonic degrees of freedom, do not which automatically yields zero for the spin- = constraint contribute to the Ward identities studied here. In [75], in (37). instead of embedding the theory in a quasi-four-dimensional In [57] within a fully four-dimensional approach method space (which will enforce the introduction of ϵ-scalars, for called preregularization, STs are explicitly evaluated using instance), the supercurrent was defined in strictly four symmetric integration (Appendix B) but arbitrary momen- μ αβ dimensions. Therefore, the identity γ σ γμ ¼ 0 is satisfied tum routings are chosen in such a way that gauge invariance μ by construction which leads to γμS ¼ 0. Since gauge or the supersymmetry Noether current is conserved at one- 3 2 symmetry still holds in this scheme, the only available loop level while the spin- = constraint is maintained as ∂μS their Lorentz decomposition is similar to [53]. Ward identity to be violated will be μ, which is the result Analytic regularization [76] also preserves gauge invari- found by the authors. ance and the spin-3=2 constraint. On-shell conditions are These results are all summarized in Table I. applied before Feynman parameter integrations. Fermionic and bosonic propagators are modified in this method VII. DISCUSSION AND CONCLUSIONS κ introducing a regularization parameter , It is well known that in dimensional specific models which includes chiral, supersymmetric and topological 1 =k þ p= 1 1 → → : ð42Þ field theories, Feynman diagram calculations cannot be − 2 1þκ − 2 − 2 1þκ =k p= ½ðk þ pÞ ðk qÞ ½ðk qÞ handled in standard dimensional regularization. Despite

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2 some generalizations in dimensional methods to tackle the ν ab ig ab q Σμν ¼ δ ½ðp · qÞγμ − 2pμ=q − qμ=qð44Þ algebra of violating objects such as the γ5-matrix 4π2 constructed in [19], issues related to the possibility of spurious anomalies and bundersome Ward identities to be and supercurrent and spin-3=2 conservation fulfilled is imposed order by order on the Green’s functions make it obtained only if ν0 ¼b=2, ξ0 ¼5b=6 and 5mD þ6mA −2mC ¼ worthwhile seeking for a framework that fully operates in 6, 2nD þ 3nA þ nC ¼ 3. the physical dimension. Yet IReg adequately addresses the Our results in IReg agree with those obtained in dimen- problem of evaluation of undetermined regularization sional regularization and dimensional reduction (in the dependent surface terms related to momentum routing in case in which the theory is embedded in a quasi-four- Feynman diagrams to all loop orders, care must be exercised dimensional space as customary). Although, at one-loop in the formal treatment of Lorentz tensors and γ5 matrices as order, there is no difference between these two-dimensional discussed in [60] even in nondimensional methods [25,44]. schemes for the Ward identities here studied, we believe In the particular instance of the supercurrent anomaly, we that IReg would reproduce the results of dimensional have seen that Clifford algebra does not commute with reduction at higher loop (in this case, it is expected that integration in the loop momenta. Novel schemes that do not the dimensional schemes differ by finite terms, due to the rely on dimensional continuation in the spacetime dimen- inclusion of ϵ-scalar contributions). This fact has recently sion have been proposed and developed. The motivation for been observed in the case of nonsupersymmetric theories this progress has been to broaden the conceptual basis as [45,60]. Finally, it should be noticed that, as reported in [60], well as to enable efficient, automated analytical and numeri- the correlation between IReg and dimensional schemes is cal calculational methods to test beyond the standard model only possible if subtleties involving the Clifford algebra are theories within precision observables. dealt with properly. By doing so one ultimately obtains a Relying on a fully four-dimensional approach in which consistent framework for Feynman diagram calculations in regularization dependent terms are left to be fixed on the physical dimensions, in the sense that all regularization- symmetry grounds, we have calculated the supercurrent dependent terms that appear are directly connected with anomaly of the zero mass Yang-Mills multiplet interacting MRI and gauge symmetry as in [25,29]. with a single massless Majorana spin 1=2 field trans- 2 forming in the adjoint representation of SUð Þ for sim- ACKNOWLEDGMENTS plicity. Momentum routing invariance, an obviously desired property of Feynman diagram calculations, is M. S. acknowledges research Grant No. 303482/2017-6 known to be connected to gauge invariance as discussed 784 from CNPq-Brazil and financial support by FAPESP. in the introduction. We have seen that setting surface terms B. H. acknowledges partial support from the FCT (Portugal) to zero automatically implements momentum routing Project No. UID/FIS/04564/2016. A. C. acknowledges invariance in the one-loop correction to the supercurrent. financial support from CAPES (Coordenação de The quadridivergence of the supercurrent as well as gauge Aperfeiçoamento de Pessoal de Nível Superior), Brazil. symmetry remains conserved at quantum level, whereas the B. H. and A. C. would like to acknowledge networking superconformal invariance translated by the spin-3=2 con- support by the COST Action CA16201. straint is broken as seen in Eq. (39). Unlike the ABBJ anomaly in which the anomaly shifts between the Ward identities as shown in Eq. (18), the APPENDIX A: DISCUSSION ON THE violation of one of the three Ward identities as shown in AMBIGUITIES RELATED TO THE CLIFFORD Table II is not affected by a simple choice of STs, ALGEBRA displacing the anomaly from one Ward identity to another. Consider the following divergent piece of an amplitude This suggests that the “democracy” in the perturbative in four dimensions, calculation of the anomaly as seen in ABBJ anomaly [58] Z μ ν does not occur. For example, for an anomalous supercurrent γ γ k k ≡ γ γ μν 2 μ ν 2 2 3 μ νIlogðm Þ: ðA1Þ conservation, k ðk − m Þ

2 2 3 2 Performing the Dirac algebra and using that =k ¼ k gives μ ab ig ab ðp − qÞ Σμν ¼ δ ðpν=q − ðp · qÞγνÞð43Þ Z 8π2 2 γ γ μν 2 2 m μ νIlogðm Þ¼Ilogðm Þþ 2 2 3 ; ðA2Þ k ðk − m Þ and spin-3=2 constraint and gauge invariance satisfied one must have both ν0 ¼b=2, ξ0 ¼5b=6 and 5mD þ6mA −2mC ¼ where the second term on the rhs is finite and regularization −6, 2nD þ 3nA þ nC ¼ 0. On the other hand, an anomalous independent. On the other hand, using (3) enables us to gauge Ward identity, write

025018-11 BATISTA, HILLER, CHERCHIGLIA, and SAMPAIO PHYS. REV. D 98, 025018 (2018) μν 2 gμν 2 gμν b 4π γμγνI ðm Þ¼γμγν I ðm Þþ υ0 − bγ þ b ln ; ðA9Þ log 4 log 4 ϵ E m2 2 ¼ I ðm Þþυ0; ðA3Þ log that differs by a term −b=2 from (A8). Such a term can spuriously break symmetries in the underlying theory. which is the result performing the Dirac algebra after Thus, should we use nondimensional methods to tackle manipulating the integrand (and eventually performing the divergent Feynman amplitudes of theories that are υ integration). Recall that 0 is a regularization dependent sensitive to dimensional continuation on the spacetime surface term. Such an ambiguity is present in any strictly dimension, care must be exercised with both the Clifford four-dimensional regularization (as long as one refrains to algebra which does not commute with integration and append a set of rules to circumvent this problem as done in symmetric integration [60]. the FDR scheme) and a prescription must be adopted in order to avoid spurious symmetry breaking. As a matter of APPENDIX B: DISCUSSION ON THE fact such an ambiguity is related to symmetric integration AMBIGUITIES RELATED TO ON-SHELL LIMITS which at one-loop level means to set Consider the UV divergent and IR finite integral (con- Z Z 2 0 fμ1μ2 μ2 −1μ2 g 2n sidering that on-shell conditions such as p ¼ are not μ μ μ 2 g ::g n n k 2 1 2 2n immediately applied), k k ::k fðk Þ¼ 2 ! fðk Þ; ðA4Þ k k ð nÞ Z 1 2 2 : ðB1Þ the curly brackets standing for symmetrization on Lorentz k k ðk þ pÞ indices, which is well known to be a forbidden operation for divergent integrals in the physical dimension [81], while We evaluate it by introducing an infrared regulator μ in it is allowed within dimensional regularization. In order to intermediate steps and taking the limit μ → 0. The regu- avoid symmetrical integration, one should perform the larization independent relation (7) introduces the renorm- 2 Dirac algebra (contractions, traces) before manipulating alization scale λ ≠ 0 as below Z Z the amplitude integrand. Indeed, within dimensional regu- 1 1 larization, performing the γ-matrices contraction before ¼ lim k2ðk þ pÞ2 μ→0 ðk2 − μ2Þ½ðk þ pÞ2 − μ2 integration, k k 2 Z λ2 − − p 2 2 ¼ Ilogð Þ b ln 2 þ b: μν before gμν d k λ γ γ 2 γ γ μ νIlogðm Þ ¼ μ ν 2 2 3 DR d ðk − m Þ k However, applying on-shell conditions from the start yields d 4π ϵ ¼ b ð1 − ϵ=2ÞΓðϵÞ; ðA5Þ Z d m2 1 lim μ→0 ðk2 −μ2Þ½ðkþpÞ2 −μ2 k Z Z whereas using that 1 kα ¼ lim −2pα μ→0 2 2 2 2 2 2 2 2 ϵ k ðk −μ Þ k ðk −μ Þ ½ðk −2k·p−m Þ μν 2 b 4π μν Z Z Z Γ ϵ 1 α α Ilogðm Þ ¼ 2 ð Þg ; ðA6Þ p2↓0 1 k þp ðz−1Þ DR 4 m −4 ¼ lim 2 2 2 pα dzz 2 2 3 μ→0 ðk −μ Þ 0 ½k −μ k k gives μ2 ¼I ðλ2Þ−blimln : log μ→0 λ2 ϵ μν after b 4π γ γ 2 Γ ϵ 4 − 2ϵ μ νIlogðm Þ ¼ 2 ð Þð Þ: ðA7Þ DR 4 m Notice that the two results differ not only by logarithmi- cally infrared divergent terms (setting p2 → 0 in the first 2 with b defined as in (8), which is just the same result as case and μ → 0 in the second), but also by a constant 2b. (A5), namely Therefore, to use the latter, a proper infrared regularization scheme is needed, since without it spurious cancellations in 4π the case of the full one-loop supercurrent calculation may b − γ − b b E þ b ln 2 : ðA8Þ take place. ϵ m 2 APPENDIX C: EXPLICIT RESULTS On the other hand, while Eq. (A2) reproduces the results of DR displayed in (A8), performing the Dirac algebra after The coefficients Bi, i ¼ 0; …; 6 in the Lorentz decom- integration as in (A3) yields, in DR position of the one-loop correction to the supercurrent (36)

025018-12 SUPERCURRENT ANOMALY AND GAUGE INVARIANCE IN … PHYS. REV. D 98, 025018 (2018) receive contributions from each of the amplitudes in D 2δab −2 λ2 2 − 4 ν Eq. (33) that represent the diagrams of Fig. 3, namely B0 ¼ g Ilogð Þþ ðnD Þ 0 2bπ2 p2 2ðp · qÞ A B C D 0 … 6 2b − 4b ln − − 7b ln Bi ¼ Bi þ Bi þ Bi þ Bi ;i¼ ; ; : ðC1Þ þ 3 þ λ2 λ2 2 2 Here, we display each contribution separately as well as q ðp · qÞ þ 5b ln − þ 4bLi2 1 − λ2 p2 some useful results of integrals cast in IReg. 2ðp · qÞ A A A 2 ab þ 4bLi2 1 − ; ðC10Þ −2B ¼ B ¼ B ¼ −4g δ n ν0; 2 0 1 2 A q 1 A A 2 ab 2 B3 ¼ − B4 ¼ 2g δ I ðλ Þþð2mA þ 1Þν0 2 log 2 2 D g ab 2 2 B ¼ δ 3I ðλ Þþ9ð3 þ 2n Þν0 p 1 9 log D − b ln − þ 2b ; λ2 2 6 1 − 3 ξ − 3 ðp · qÞ − 16 A A þ ð nDÞ 0 b ln 2 b ; ðC11Þ B5 ¼ B6 ¼ 0: ðC2Þ λ

q2 2 B 2 ab 2 2g 2 B0 ¼ −3g δ I ðλ Þ − b ln − þ 2b ; D δab 3 λ 9 −5 2 ν log λ2 B2 ¼ 9 Ilogð Þþ ð þ nDÞ 0 BB ¼ BB ¼ BB ¼ BB ¼ BB ¼ BB ¼ 0: ðC3Þ 2 1 2 3 4 5 6 6 1 − 3 ξ − 3 ðp · qÞ 20 þ ð nDÞ 0 b ln 2 þ b ; ðC12Þ λ 2 C 2 ab 2 ðp·qÞ B g δ −I λ − 2n 5 ν0 b 2b ; 0 ¼ logð Þ ð C þ Þ þ ln λ2 þ 2g2 D δab −6 λ2 45 ν ðC4Þ B3 ¼ 9 Ilogð Þþ mD 0 p2 2 6 1 − 3 ξ − 16 9 − 2g þ ð mDÞ 0 b þ b ln 2 C δab −3 λ2 9 3 − 4 ν λ B1 ¼ 9 Ilogð Þþ ð nCÞ 0 2ðp · qÞ 2 − 3b ln ; ðC13Þ 6 3 − 1 ξ 3 ðp · qÞ − 20 λ2 þ ð nC Þ 0 þ b ln λ2 b ; ðC5Þ 4ig2 2 2 D δab 6 λ2 − 9ν 3 1 − 3 ξ 19 C g ab 2 B4 ¼ Ilogð Þ 0 þ ð mDÞ 0 þ b B ¼ δ −3I ðλ Þ − 9ð1 þ 4n Þν0 9 2 9 log C 2 2 2 − 9 − p 3 ðp · qÞ ðp · qÞ b ln 2 þ b ln 2 ; ðC14Þ 6 3n − 1 ξ0 3b − 2b ; λ λ þ ð C Þ þ ln λ2 ðC6Þ 2 8g2δab b 2g 2 D D C δab −3 λ 9 1 − 4 ν −2B ¼ B6 ¼ − : ðC15Þ B3 ¼ 9 Ilogð Þþ ð mCÞ 0 5 3 ðp · qÞ 2ðp · qÞ 6 3m − 1 ξ 3b − 2b ; þ ð C Þ 0 þ ln λ2 ðC7Þ The following integrals are useful (the on-shell limits (p2 → 0;q2 → 0) have already been judiciously taken): 4g2 Z C δab 3 λ2 − 9 ν 1 λ2 B4 ¼ Ilogð Þ mC 0 λ2 2 − 9 2 2 ¼ Ilogð Þþ b þ b ln 2 ; ðC16Þ k k ðk þ pÞ p 2ðp · qÞ þ 3ð3m − 1Þξ0 − 3b ln þ 8b ; ðC8Þ C λ2 Z α α λ2 k p − λ2 ν − 2 − − 2 2 ¼ 2 Ilogð Þþ 0 b bln 2 ; 8g2δab b k k ðk þ pÞ p BC ¼ −2BC ¼ − : ðC9Þ 5 6 3 ðp · qÞ ðC17Þ

025018-13 BATISTA, HILLER, CHERCHIGLIA, and SAMPAIO PHYS. REV. D 98, 025018 (2018) Z 1 APPENDIX D: CALCULATION OF THE FINITE k2ðk − pÞ2ðk − qÞ2 PART OF THE AVV TRIANGLE k b −4 22 − 4 2 − p · q We perform the computation of the finite part of the ¼ ln ln ln 2 ˜ 2p · q q triangle diagram, Tμνα. Since it does not depend on the 2 routing, we choose k1 ¼ 0, k2 ¼ q e k3 ¼ −p to get: p · q π − ln2 − þ ; ðC18Þ Z q2 3 − γ i γ i γ γ i Z Tμνα ¼ i Tr μ − ν − α 5 − − kα k =k m =k þ =q m =k p= m k2ðk −pÞ2ðk − qÞ2 þðμ ↔ ν;p↔ qÞ k 2 2 β ˜ b p q ¼ −8iυ0ϵμναβðq − pÞ þ Tμνα; ðD1Þ ¼ −pα ln − þ qα ln − ; ðC19Þ 2p · q 2p · q 2p ·q in which Z k2 2p · q λ2 2 − ˜ λ 2 2 2 2 2 ¼ Ilogð Þþ b b ln 2 ; Tμνα ¼ 4ibfϵαμνλq ðp ξ01ðp;qÞ−q ξ10ðp;qÞÞ k k ðk − pÞ ðk − qÞ λ λ 2 λ τ ϵαμνλq 1 2m ξ00 p;q 4ϵανλτp q ξ01 p;q ðC20Þ þ ð þ ð ÞÞ ½ð ð Þ Z −ξ02ðp;qÞÞpμ þξ11ðp;qÞqμðμ ↔ ν;p↔ qÞg: ðD2Þ kαkβ 2 2 2 ξ k ðk − pÞ ðk − qÞ The functions nmðp; qÞ are defined as k αβ 2 Z Z g 2 p · q 1 1−z znym ¼ I ðλ Þ − ν0 þ b 3 − ln ξ 4 log λ2 nmðp; qÞ¼ dz dy ; ðD3Þ 0 0 Qðy; zÞ b p2 q2 þ pαpβ ln − − qαqβ ln − 4p · q 2p · q 2p · q with 2 1− 2 1− −2 − 2 þðpαqβ þ pβqαÞ ; ðC21Þ Qðy;zÞ¼½p yð yÞþq zð zÞ ðp·qÞyz m ðD4Þ obeying the property ξ ðp; qÞ¼ξ ðq; pÞ. Z α β η nm mn k k k Those integrals satisfy the following relations which we k2ðk −pÞ2ðk − qÞ2 have also used in the derivation of Eq. (D2) k 1 λ2 fη αβg 2 2 ¼ p g 3I ðλ Þ − 3ξ0 þ b 8 þ 3ln q ξ11ðp;qÞ −ðp · qÞξ02ðp;qÞ 36 log 2p · q 1 1 1 1 λ2 − 2 2 2 2 2ξ fη αβg 2 ¼ Z0ððp þ qÞ ;m Þþ Z0ðp ;m Þþq 01ðp;qÞ ; þ q g 3I ðλ Þ − 3ξ0 þ b 8 þ 3ln 2 2 2 36 log 2p · q b ðD5Þ − ðpfαpβqηg þ pfαqβqηgÞ 12p ·q 2 p ξ11ðp;qÞ − ðp · qÞξ20ðp;qÞ α β η 2 α β η 2 bp p p − p bq q q − q 1 1 1 þ 6 ln 2 þ 6 ln 2 ; − 2 2 2 2 2ξ p · q p · q p · q p · q ¼ 2 2Z0ððp þ qÞ ;m Þþ2Z0ðq ;m Þþp 10ðp;qÞ ; ðC22Þ ðD6Þ Z 2 α k k 2 q ξ10ðp; qÞ − ðp · qÞξ01ðp; qÞ k2ðk − pÞ2ðk − qÞ2 k 1 ðp þ qÞα 2p · q − 2 2 2 2 2ξ λ2 − ν 2 − ¼ 2 Z0ððp þ qÞ ;m ÞþZ0ðp ;m Þþq 00ðp; qÞ ; ¼ 2 Ilogð Þ 0 þ b ln λ2 ; ðD7Þ ðC23Þ 2 p ξ01ðp; qÞ − ðp · qÞξ10ðp; qÞ where 1 2 2 2 2 2 η αβ η αβ β αη α βη ¼ −Z0ððp þ qÞ ;m ÞþZ0ðq ;m Þþp ξ00ðp; qÞ ; pf g g ¼ p g þ p g þ p g 2 pfαpβqηg ¼ pβpηqα þ pαpηqβ þ pαpβqη: ðD8Þ

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2 2 2 where Z ðp ;m Þ is defined as q ξ20ðp;qÞ−ðp·qÞξ11ðp;qÞ k 1 1 2 1 2 3 2 ¼ − þm ξ00ðp;qÞ þ p ξ01ðp;qÞþ q ξ10ðp;qÞ ; Z 2 2 2 2 1 2 − 2 1 − 2 2 k m p zð zÞ Zkðp ;m Þ¼ dzz ln 2 : ðD11Þ ðD9Þ 0 m

2 p ξ02ðp;qÞ−ðp·qÞξ11ðp;qÞ 1 1 1 3 The derivation of the relations(D5)–(D10) are easily − 2ξ 2ξ 2ξ ¼ 2 2þm 00ðp;qÞ þ2q 10ðp;qÞþ2p 01ðp;qÞ ; performed by integration by parts. A complete review on the evaluation of one-loop n-point functions in a four- ðD10Þ dimensional setup can be found in [84].

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