# Trace and Axial Anomalies on Equal Footing

Authors: Renata Jora
Journal: Advances in High Energy Physics
 (2020)

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## Abstract

We discussed that for some particular nonsupersymmetric theories, a generalized symmetry that includes both the scale and axial transformations and leads to a single current may contain also a pseudoscalar term. The method, inspired by the superconformal anomalies, has important application for low-energy effective models where it allows the introduction of a single complex glueball field with a scalar and a pseudoscalar component on the same footing with the complex meson nonet fields made of quarks. Both axial and trace anomalies are satisfied in accordance to the meson structure and the QCD requirements.


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## Body

## 1. Introduction

One of the most important tools in dealing with physical systems and especially quantum field theoretical models is the incorporation of symmetries, global or gauged or exact or approximate. We know that a standard Lagrangian must be real and thus Hermitian and Lorentz invariant. In some instances, the Lagrangian might be also scale invariant, at least at tree level, thus leading to new class of theories whose properties were explored in detail in the literature. The scale invariance is usually broken at the quantum level [1–4] by the renormalization group equations and at tree level by explicit noninvariant terms in the Lagrangian. One usually introduces the symmetric energy momentum tensor θνμ which may be derived with some complications or not from the canonical energy momentum tensor Tνμ. Then, the amount of breaking of the scale invariance is measured by
(1)∂μDμ=θμμ,where Dμ=θνμxν is the dilatation current and ∂μθνμ=0.The quantum anomalies of a theoretical particle physics model are crucial for describing its properties, and more so for the supersymmetric QCD and the QCD Lagrangians, where they are an important tool for constructing low-energy effective models of hadrons.The effective approach based on the exact realization of axial and scale anomalies for supersymmetric QCD was analyzed by Seiberg [5] and Seiberg and Witten [6] leading to remarkable results regarding the phase structure of these theories. Applications of these methods for QCD were then implemented in [7, 8].It is known that in supersymmetric gauge theories, at least in the holomorphic picture, the axial and trace anomalies belong to the same supermultiplet; and thus, the pseudoscalar and scalar glueballs can be regarded on the same footing. A unified description of both these anomalies for supersymmetric gauge theories was presented in [9]. For example, for the supersymmetric Yang-Mills (Fμνa is the gauge field tensor),
(2)θμμ=3NcF+F∗=−3Ncg232π2FaμνFμνa,∂μJμ5=2iNcF−F∗=Ncg232π2FaμνF~μνa.Here,Jμ5=λ¯σ¯μλa is the axial anomaly current and λ is the gluino field. Moreover the main ingredient of the effective theory is S, a composite chiral superfield with the structure
(3)Sy=Φy+2θΨy+θ2Fy,where
(4)Φ≈λ2,Ψ≈σμνλaFμν,aF≈−12FaμνFμνa−i4εμνρσFaμνFaρσ.Inspired by supersymmetric QCD, we will present here an approach for QCD where the pseudoscalar and scalar glueballs are treated as components of a single complex glueball field leaving for future work any possible direct connection with supersymmetry and the subtleties that might arise from it.In Section2, we discuss from a new point of view the scale symmetry and the construction of the symmetric energy momentum tensor for real and complex scalar field theories and QED. In Section 3, we will slightly modify the scale transformation such that to incorporate also the axial transformation for QED [10–12] and the U1 transformation for the complex scalar field theory. Thus, we will add to the symmetric energy momentum tensor presumably a “pseudoscalar” tensor τνμ such that the associated full Noether current becomes
(5)Jμ=θνμxν+τνμxν,with
(6)∂μθνμ=0,∂μτνμ=0.Finally, the conservation of currents for the extended transformation will be(7)∂μJμ=θμμ+τμμ.In Section4, we will show how this approach makes total sense for a low-energy QCD linear sigma model with two chiral nonets, the model is discussed in detail in [13–20]. Section 5 is dedicated to conclusions.


## 2. Scale Transformation Revisited

We will analyze in some detail the scale transformation for some simple theories with amendments that our findings can be generalized easily to more intricate models.


### 2.1. Scalar Field Theory

We start with a Lagrangian for the scalar field theory(8)L=12∂μΦ∂μΦ−VΦ.Under the scale transformation, the scalar field transforms as(9)Φx⟶exp−σΦxexp−σ,whereas the action changes to
(10)δS=∫δd4xLΦx,x+∫d4x∂L∂xμδxμ+∫d4x∂μ∂L∂∂μΦxδΦx.The conserved current can be determined to be(11)Jμ=∂L∂∂μΦxΦx+∂L∂∂μΦx∂ρΦx−Lηρμxρ.The quantity(12)∂L∂∂μΦx∂ρΦx−Lηρμ=Tρμis the canonical energy momentum tensor. There are different procedures for deriving the symmetric energy momentum tensor θρμ out from the canonical energy momentum tensor in Eq. (12). Here, we will present a very simple way applicable to any theory in general. First, we observe that we need to add an extra term to it that can stem only from the first term in expression (11). We write
(13)∂L∂∂μΦΦ=a∂μΦΦ∂νxν−a∂ρΦΦηνμ∂ρxν=3a∂μΦΦ.Since Eq. (13) is an identity, we immediately determine a=1/3. Then, one can write
(14)∂L∂∂μΦΦ=13∂μΦΦ∂νxν−13∂ρΦΦηνμ∂ρxν=13∂ν∂μΦΦxν−13∂ρ∂ρΦΦηνμxν−13∂ν∂μΦΦxν+13∂ρ∂ρΦΦηνμxν.We drop the total derivatives from the current to get(15)∂L∂∂μΦΦ=−16∂ν∂μΦ2+16∂2Φ2ηνμxν.Then, the symmetric energy momentum tensor is just(16)θνμ=Tνμ+−16∂ν∂μΦ2+16∂2Φ2ηνμ,and this result is consistent with the standard formula for the symmetric energy momentum tensor in the literature.Next, we consider a complex scalar field theory with the Lagrangian(17)L=∂μΦ∗∂μΦ−VΦ,Φ∗.We consider an extension of the scale transformation in whichx′=expσx, and the fields transform as
(18)Φx⟶Φ′x′=exp−σ1+iΦexp−σx,Φ∗x⟶Φ′∗x′=exp−σ1−iΦ∗exp−σx.Note that the above transformation is just the scale transformation associated with aU1 transformation with the same infinitesimal parameter σ. Then, it can be easily determined that
(19)Jμ=∂L∂∂μΦΦ+∂L∂∂μΦ∗Φ∗+∂L∂∂μΦ∂νΦ+∂L∂∂μΦ∗∂νΦ∗−Lηνμxν+i∂L∂∂μΦΦ−i∂L∂∂μΦ∗Φ∗.Finally, one can apply the procedure in Eq. (16) to obtain
(20)Jμ=θνμxν−Kμ,where
(21)Kμ=−i∂μΦ∗Φ+i∂μΦΦ∗.Our aim is to generalize the trace anomaly such that to contain also a contribution from the term in Eq. (21). Specifically, we want to find τνμ such that ∂μτνμ=0 and
(22)Kμ=−τνμxν.We find using the procedure in Eq. (14)
(23)τνμ=−i3∂ν∂μΦ∗Φ+i3∂ρ∂ρΦ∗Φηνμ+h.c.It can be easily checked that(24)Kμ=−τνμxν,∂μτνμ=0.Finally, one can compute the trace of the tensor in Eq. (24) as
(25)−∂μKμ=τμμ=−i∂V∂ΦΦ−∂V∂Φ∗Φ∗,where we applied the equation of motion.


### 2.2. QED

Now, we shall state the results forτνμ briefly for QED [10–12] (for a detailed review, see [21]). Consider the transformation of the fields
(26)ΨLx⟶exp−σ32+iΨLexp−σx′,ΨRx⟶exp−σ3/2−iΨRexp−σx′,together with the corresponding transformation for the conjugate fields. Since the scale transformation contribution in QED is well known, we will consider only the extra contribution coming from the axial transformation included in Eq. (26). Thus, the conserved current has the form
(27)JLμ′=JLμ−i∂L∂∂μΨLΨL,JRμ′=JRμ+i∂L∂∂μΨRΨR.Here,JLμ and JRμ are the standard scale transformation currents. We denote
(28)Kμ=i∂L∂∂μΨLΨL−i∂L∂∂μΨRΨR=Ψ¯γμγ5Ψ,where we applied the equation of motion. Thus, we obtained the standard axial current. Next, we want to determine τνμ such that
(29)Kμ=−τνμxν,∂μτνμ=0.For that, there is only one possible term(30)τνμ=13∂νΨ¯γμγ5Ψ−13∂ρΨ¯γργ5Ψηνμ,where it is clear that the second equation in Eq. (29) is satisfied if one applies the equation of motion in the absence of quark masses. Thus, the full current will be
(31)Jμ′=θνμ+τνμxν,and we were able to write a total current that encapsulates both scale and axial transformations. Finally, the conservation law is
(32)∂μJ′μ=θμμ+τμμand includes both a scalar and a pseudoscalar part.


## 2.1. Scalar Field Theory

We start with a Lagrangian for the scalar field theory(8)L=12∂μΦ∂μΦ−VΦ.Under the scale transformation, the scalar field transforms as(9)Φx⟶exp−σΦxexp−σ,whereas the action changes to
(10)δS=∫δd4xLΦx,x+∫d4x∂L∂xμδxμ+∫d4x∂μ∂L∂∂μΦxδΦx.The conserved current can be determined to be(11)Jμ=∂L∂∂μΦxΦx+∂L∂∂μΦx∂ρΦx−Lηρμxρ.The quantity(12)∂L∂∂μΦx∂ρΦx−Lηρμ=Tρμis the canonical energy momentum tensor. There are different procedures for deriving the symmetric energy momentum tensor θρμ out from the canonical energy momentum tensor in Eq. (12). Here, we will present a very simple way applicable to any theory in general. First, we observe that we need to add an extra term to it that can stem only from the first term in expression (11). We write
(13)∂L∂∂μΦΦ=a∂μΦΦ∂νxν−a∂ρΦΦηνμ∂ρxν=3a∂μΦΦ.Since Eq. (13) is an identity, we immediately determine a=1/3. Then, one can write
(14)∂L∂∂μΦΦ=13∂μΦΦ∂νxν−13∂ρΦΦηνμ∂ρxν=13∂ν∂μΦΦxν−13∂ρ∂ρΦΦηνμxν−13∂ν∂μΦΦxν+13∂ρ∂ρΦΦηνμxν.We drop the total derivatives from the current to get(15)∂L∂∂μΦΦ=−16∂ν∂μΦ2+16∂2Φ2ηνμxν.Then, the symmetric energy momentum tensor is just(16)θνμ=Tνμ+−16∂ν∂μΦ2+16∂2Φ2ηνμ,and this result is consistent with the standard formula for the symmetric energy momentum tensor in the literature.Next, we consider a complex scalar field theory with the Lagrangian(17)L=∂μΦ∗∂μΦ−VΦ,Φ∗.We consider an extension of the scale transformation in whichx′=expσx, and the fields transform as
(18)Φx⟶Φ′x′=exp−σ1+iΦexp−σx,Φ∗x⟶Φ′∗x′=exp−σ1−iΦ∗exp−σx.Note that the above transformation is just the scale transformation associated with aU1 transformation with the same infinitesimal parameter σ. Then, it can be easily determined that
(19)Jμ=∂L∂∂μΦΦ+∂L∂∂μΦ∗Φ∗+∂L∂∂μΦ∂νΦ+∂L∂∂μΦ∗∂νΦ∗−Lηνμxν+i∂L∂∂μΦΦ−i∂L∂∂μΦ∗Φ∗.Finally, one can apply the procedure in Eq. (16) to obtain
(20)Jμ=θνμxν−Kμ,where
(21)Kμ=−i∂μΦ∗Φ+i∂μΦΦ∗.Our aim is to generalize the trace anomaly such that to contain also a contribution from the term in Eq. (21). Specifically, we want to find τνμ such that ∂μτνμ=0 and
(22)Kμ=−τνμxν.We find using the procedure in Eq. (14)
(23)τνμ=−i3∂ν∂μΦ∗Φ+i3∂ρ∂ρΦ∗Φηνμ+h.c.It can be easily checked that(24)Kμ=−τνμxν,∂μτνμ=0.Finally, one can compute the trace of the tensor in Eq. (24) as
(25)−∂μKμ=τμμ=−i∂V∂ΦΦ−∂V∂Φ∗Φ∗,where we applied the equation of motion.


## 2.2. QED

Now, we shall state the results forτνμ briefly for QED [10–12] (for a detailed review, see [21]). Consider the transformation of the fields
(26)ΨLx⟶exp−σ32+iΨLexp−σx′,ΨRx⟶exp−σ3/2−iΨRexp−σx′,together with the corresponding transformation for the conjugate fields. Since the scale transformation contribution in QED is well known, we will consider only the extra contribution coming from the axial transformation included in Eq. (26). Thus, the conserved current has the form
(27)JLμ′=JLμ−i∂L∂∂μΨLΨL,JRμ′=JRμ+i∂L∂∂μΨRΨR.Here,JLμ and JRμ are the standard scale transformation currents. We denote
(28)Kμ=i∂L∂∂μΨLΨL−i∂L∂∂μΨRΨR=Ψ¯γμγ5Ψ,where we applied the equation of motion. Thus, we obtained the standard axial current. Next, we want to determine τνμ such that
(29)Kμ=−τνμxν,∂μτνμ=0.For that, there is only one possible term(30)τνμ=13∂νΨ¯γμγ5Ψ−13∂ρΨ¯γργ5Ψηνμ,where it is clear that the second equation in Eq. (29) is satisfied if one applies the equation of motion in the absence of quark masses. Thus, the full current will be
(31)Jμ′=θνμ+τνμxν,and we were able to write a total current that encapsulates both scale and axial transformations. Finally, the conservation law is
(32)∂μJ′μ=θμμ+τμμand includes both a scalar and a pseudoscalar part.


## 3. Application to a Generalized Linear Sigma Model

In this section, we will show in detail how one can introduce in a consistent way from the point of view presented here the scalar and pseudoscalar glueballs in a linear sigma model which is a low-energy description of QCD. The presence and properties of the scalar and pseudoscalar glueballs in the QCD spectrum were the subjects of relevant works in the literature. In [22, 23], the trace anomaly in low-energy QCD and some features of the scalar glueball were discussed in different frameworks. Furthermore, in [24–26], the axial anomaly and spectrum and decays of the pseudoscalar glueball were analyzed in low-energy QCD.As an application of our method, we consider a generalized linear sigma model [13–19] with two chiral meson nonets, one with a quark antiquark structure and the other one with a four quark composition. We denote this nonets by M and M′, where M=S+iΦ, M′=S′+iΦ′, and S and S′ represent the scalar states and Φ and Φ′ the pseudoscalar ones. The fields M and M′ transform in the same way under chiral SU3 transformations
(33)M⟶ULMUR†,M′⟶ULM′UR†but transform differently under U1A transformation
(34)M⟶e2iνM,M′⟶e−4iνM′.In order for the model to be consistent, it is useful to introduce terms that mock up the axial and scale anomalies which read(35)∂μJμ5=g216π2NFF~F=G,θμμ=∂μDμ=−βg2gFF=H.Note that in this section we use a different convention for the metric tensor with respect to Section2. Here, F is the SU3C field tensor, F~ is its dual, NF is the number of flavors, βg is the beta function for the coupling constant, Jμ5 is the axial current, and Dμ is the dilatation current. The field H may be associated in low-energy QCD with the scalar glueball whereas G with the pseudoscalar one. The Lagrangian has the form
(36)L=−12Tr∂μM∂μM†−12Tr∂μM′∂μM′†−V0M,M′−VSB,where V0M,M′ is a function made from SU3L×SU3R (but not necessarily U1A) invariants formed out of M and M′. The leading choice of terms corresponding to eight or fewer underlying quarks plus antiquark lines at each effective vertex reads [17]
(37)V0=−c2TrMM†+c4aTrMM†MM†+d2TrM′M′†+e3aεabcεdefMdaMebM′fc+H.c.+c3γ1lndetMdetM†+1−γ1lnTrMM′†TrM′M†2.All the terms except the last two (which mock up the axial anomaly) have been chosen to also possess theU1A invariance.For a scalar and pseudoscalar Lagrangian of the type in Eq. (36) but also contains scalar and pseudoscalar glueballs, the trace anomaly reads [20]
(38)θμμ=M∂V∂M+∂V∂M†M†+4H∂V∂H+4G∂V∂G−4V,where we applied the equation of motion in the results of Section 2.From Eq. (23), we obtained in the previous section in terms of Φ
(39)τμμ=−i3∂μ∂μΦ∗Φ+i3∂ρ∂ρΦ∗Φημμ+h.c.=−i∂V∂ΦΦ−∂V∂Φ∗Φ∗.Adjusted for the generalized linear sigma model, the axial anomaly contribution in the regular case for a single chiral nonetM is (note that for this case, the kinetic term has negative sign)
(40)τμμ=iM∂V∂M−M†∂V∂M†.Now, we shall extend the scale transformation to include the axial one as discussed in Section2. For that, we denote Q=H+iG and consider the generalized transformation
(41)M⟶exp−1+iσMxexp−σ,M†⟶exp−1−iσM†xexp−σ,M′⟶exp−1−2iσM′xexp−σ,M′†⟶exp−1+2iσM′†xexp−σ,Q⟶exp−4σQxexp−σ,Q∗⟶exp−4σQ∗xexp−σ.The transformation in Eq. (41) makes sense as an expansion in the infinitesimal parameter σ.Then, the conservation of current should be of the form(42)θμμ+τμμ=H−G,where we used Eq. (35). Moreover, we fixed the transformation law for the field M′ to correspond to the correct axial anomaly. A detailed discussion of the most general structure of the possible terms that mock up the scale and trace anomalies is given in [20]. Here, we will present a first-order term that leads to the correct conservation law
(43)V=V1+V1†,where
(44)V1=Qλ1lnQΛ4+λ2lndetMΛ3+λ3lnTrMM′†Λ3,V1†=Q∗lnλ1Q∗Λ4+λ2lndetM†Λ3+λ3lnTrM′M†Λ3,V=Qλ1lnQΛ4+λ2lndetMΛ3+λ3lnTrMM′†Λ3+Q∗lnλ1Q∗Λ4+λ2lndetM†Λ3+λ3lnTrM′M†Λ3.For the transformation introduced in Eq. (41), the trace of the scale and axial currents read
(45)θμμ=M∂V∂M+M†∂V∂M†+M′∂V∂M′+M′†∂V∂M′†+4Q∂V∂Q+4Q∗∂V∂Q∗−4V,τμμ=iM∂V∂M−M†∂V∂M†−2M′∂V∂M′+2M′†∂V∂M′†.We list below the contributions of various terms(46)M∂V1∂M=Q3λ2+λ3,∂V1†∂M†M†=Q∗3λ2+λ3,M′∂V1†/∂M′=Q∗λ3,∂V1∂M′†M′†=Qλ3,4Q∂V1∂Q=4Qλ1lnQΛ4+λ2lndetMΛ2+λ3lnTrMM′†Λ2+4Qλ1,4Q∗∂V1†∂Q∗=4Q∗λ1lnQ∗Λ4+λ2lndetM†Λ2+λ3lnTrM′M†Λ2+4Q∗λ1.The final results for the combined trace and axial anomalies are(47)θμμ=H+iG3λ2+2λ3+4λ1+H−iG3λ2+3λ3+4λ1=H6λ2+4λ3+8λ1,τμμ=iH+iG3λ2+3λ3−iH−iG3λ2+3λ3=−G6λ2+6λ3.Then, a sufficient and necessary condition for Eq. (42) to be fulfilled is
(48)6λ2+4λ3+8λ1=1,6λ2+6λ3=1.As a consistency check, it is important to notice that the trace and axial anomalies would be equally satisfied by the potential in Eq. (44) also when the standard transformation for axial and trace anomalies would be applied.


## 4. Proposal for a Lagrangian Containing a Complex Glueball Field

In [20], an effective generalized linear sigma model that contained both scalar and pseudoscalar glueballs was introduced. There, H and G were introduced separately as individual states. Here, we will consider a new version of that Lagrangian where the scalar and pseudoscalar glueballs are parts of the same complex field Q=H+iG. This Lagrangian is consistent with the symmetries and axial and scale anomalies of low-energy QCD and has the form
(49)L=−12Tr∂μM∂μM†−12Tr∂μM′∂μM′†−12QQ∗−3/4∂μQ∂μQ∗+f+fA,S+VSB.In Eq. (49), the kinetic term for the glueball field Q is written such that it is scale invariant. The term f can be
(50)f=−c2TrMM†+c4aTrMM†MM†+d2TrM′M′†+e3aεabcεdefMdaMebM′fc+H.c.+g1QQ∗Λ4,where g1 is a dimensionless coupling constant and Λ is the QCD scale.Furthermore, the axial and scale anomaly terms are encapsulated infA,S=V, where a detailed expression of V is given in Eqs. (43) and (44).The symmetry breaking terms can be(51)VSB=TrAM+M†,where A is a 3×3 diagonal constant matrix proportional to the light quark masses.Note that for each term in the full potential, we chose the minimal ones from a multitude of possible terms.


## 5. Conclusions

In this work, we explored some features of the scale transformation in the context of some simple theories in order to illustrate the possibility of incorporating both the scale and axial anomalies in a single larger transformation. Our method is inspired by the supersymmetric theories, where it is known that all anomalies including the trace and axial ones are part of the same supermultiplet structure [27]. In order to express this at the level of currents, we introduce a new tensor τνμ that has the correct properties such that the net anomalous conservation of current is given by the sum θμμ+τνμ. In some theories, τμμ is a pseudoscalar.This approach allowed us to introduce a comprehensive term that mocks up both the trace and axial anomalies in an effective low-energy QCD model. Consequently, the scalarH and pseudoscalar G glueballs that may be present in a generalized linear sigma model with scalar and pseudoscalar meson nonets can be arranged in a singlet complex field H+iG on the same footing with the quark composite mesons. In this direction, we also proposed a low-energy effective Lagrangian based on the derivations and results in this work. This method may have important application in low-energy QCD models and besides that, in any effective model that displays both trace and axial anomalies.



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