On anomalies in effective models with nonlinear symmetry realization
OOn anomalies in effective models with nonlinear symmetryrealization
Andrej Arbuzov ∗ and Boris Latosh † Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna 141980, Russia Dubna State University, Universitetskaya str. 19, Dubna 141982, RussiaAugust 19, 2020
Abstract
Anomalous features of models with nonlinear symmetry realization are addressed. It isshown that such models can have anomalous amplitudes breaking of its original symmetryrealization. An illustrative example of a simple models with a nonlinear conformal symmetryrealization is given. It is argued that the effective action obtained via nonlinear symmetryrealization should be used to obtain an anomaly-induced action which is to drive the lowenergy dynamics.
Conformal symmetry plays a special role in contemporary physics. Firstly, the Standard Modelof particle physics is almost conformally invariant, as it contains a single primary dimensionfulparameter, namely, the Higgs boson mass. Secondly, the conformal group is related to thespacetime symmetry via the Ogievetsky theorem [1]. The theorem states that the infinite-dimensional coordinate transformation algebra is generated via commutation of generators fromthe Lorentz and conformal groups. At the same time, the conformal group cannot be directlyrealized, as the corresponding Noether currents do not exist in Nature. Consequently, at thecoordinate level the conformal symmetry can appear through a non-linear realization [2]. At thelevel of physical fields the conformal group can also be realized in a non-linear way [3, 4, 5, 6, 7].This paper addresses, perhaps, the simplest model with a non-linear realization of the con-formal symmetry at the level of physical fields [3]. The model was presented in paper [8], it isgiven by the following effective action:Γ = (cid:90) d x (cid:34) (cid:40) σ ( α ) σ ( α ) ε E (cid:32) ψε (cid:33)(cid:41) ∂ µ ψ ∂ µ ψ (1)+ 12 E (cid:32) ψε (cid:33) ∂ µ σ ( α ) ∂ µ σ ( α ) − ε E (cid:32) ψε (cid:33) E (cid:32) ψε (cid:33) ∂ µ ψ σ ( α ) ∂ µ σ ( α ) (cid:35) . The action is recovered via symmetry principles [9, 10, 11, 12, 13] and describes Goldstone scalarmodes ψ and σ ( α ) , ( α ) = 0 , , , α ) is not a Lorentz one, but it is associatedwith conformal group generators. In this expression ε is the symmetry breaking energy scale.Functions E and E are defined as E ( x ) def = e x − x , E ( x ) def = e x − x − x . (2)Finally, it should be noted that within such a simple model ψ may be treated as a dilatonassociated with the spontaneously broken conformal symmetry.The main aim of this paper is to study anomalous behavior of this model. Below it is shownthat the model admits anomalous amplitudes which do not respect the non-linear conformal ∗ [email protected] † [email protected] a r X i v : . [ h e p - t h ] A ug ymmetry realization. Consequently, it is argued that the effective action (1) should be substi-tuted by the anomaly-induced effective action which, in turn, drives the low-energy dynamic ofthe theory.Anomalies are crucial for effective theories. Perhaps, the most well-known example of anoma-lous behavior is given by the axial anomaly [14, 15]. Namely, the following diagram revealsanomalous properties: γ µ γ ν γ σ γ (3)The divergent part of this diagram cannot be renormalized, but it is canceled out completelywhen a symmetric diagram with µ ↔ ν is taken into account. On the contrary, the finite partsof such diagrams do not vanish and generate a new physical interaction corresponding to a realeffect.Anomalies in a fundamental theory lead to undesirable effects such as unitarity violation [16],so it is safe to assume that fundamental theories are free from them. For instance, within theStandard Model all axial anomalies are canceled out [17, 18].The same logic cannot be applied for effective models. First and foremost, the notion of ananomaly is traditionally used within renormalizable field theories. Within non-renormalizablemodels new interactions are generated by loop corrections, so the notion of an anomaly shouldbe refined.In this paper we discuss anomalies in the context of effective models. Because of this itis safe to assume that such models, despite being non-renormalizable, have suitable underlyingfundamental theories. The nonlinear symmetry realization allows one to account for a non-trivialdynamics existing within the fundamental theory. Therefore all divergences appearing within aneffective theory are expected to be regularized with the fundamental theory.In that context it is reasonable to study operators generated at the loop level within effectivemodels. Some of these operators do not respect the original nonlinear realization of the symmetryand they should be understand as footprints of further radiative symmetry breaking occurring inthe fundamental model. Thus, in the context of effective models we use the notion of an anomalyto refer to the presence of such operators. Let us highlight one more time, that the existence ofsuch operators do not introduces new pathologies. Effective models themselves have a limitedapplicability domain, so they are free from the corresponding pathologies, while a fundamentaltheory can always be safely assumed to be free from pathologies.The best example of an effective model anomaly is given by the σ -model for pions [19]. Themodel admits anomalous diagrams which are essential for the observed π → γγ decay [14, 19].In full analogy with the model to be addressed, the pion σ -model is recovered via nonlinearrealization of the chiral symmetry [13]. Its anomalous amplitudes do not respect the non-linearsymmetry realization and thereby violate the original symmetry of the effective action.Such considerations involve three features of anomalies. Firstly, anomalous contributionsdescribe real physical effects which can be probed empirically. Secondly, anomalies break theoriginal model symmetry that exists in the effective action at the classical level. Consequently,the effective action recovered via symmetry principles should be used to obtain an anomaly-induced effective action which, in turn, drives the low energy phenomenology. Most importantly,the anomaly-induced effective action may not respect the original model symmetry. Finally,these conclusions can be universally applied to all models with non-linear symmetry realizationsincluding the model addressed in this paper.The rest of the paper is organized as follows. Firstly, we show that the one-loop three-pointfunction generated by the effective action (1) at the first order in ε − is anomalous off-shelland vanishes on-shell. Secondly, we show that the one-loop four-point function generated bythe effective action at the first order in ε − is anomalous on-shell, but these divergences can beeliminated completely via Ward identities. Finally, we show that a similar argument does nothold for one-loop four-point functions evaluated at the second order in ε − by the effective action.Consequently, we are going to show that the model admits anomalous features and to recoverits anomaly-induced effective action up to ε − order. The paper concludes with a discussion ofresults and their role within gravity models with non-linear symmetry realization.2 Anomalous contributions
A more detailed comment on the studied model (1) is due. As it was noted above, the modelis recovered via symmetry reasoning. Models with nonlinearly realized symmetries, includingnonlinear sigma models, were actively studied in a large number of papers. An explicit methodgenerating a Lagrangian with a given nonlinear realization was found in a series of papers [13,11, 12].The method is implemented as follows. One starts with a big ground G and its subgroup H . As it is well shown in [13, 11, 12] there is an explicit way to construct an action of thebig group G both on the small group H and on a corresponding quotient G / H . This generatestwo realizations of G ; first one represents G on the small group H , while the second one definesa representation of G on G / H . The master formula for such nonlinear realizations is given inmultiple papers [13, 11, 12, 9, 10, 8], so it will not be presented here for the sake of briefness.Most importantly, in papers [11, 12] it was explicitly shown that there is a way to recovercovariant derivatives of fields subjected to such a nonlinear realization. One can recover theirexplicit form knowing only on the big group G and the small group H . Therefore an explicitform of the Lagrangian in a model with a given nonlinear symmetry realization can be obtainedwithout a reference to an explicit form of the transformation laws (see our previous paper [8]with a more detailed discussion).In paper [8] this algorithm was implemented for the conformal group. We have shown that themodel (1) is generated via a nonlinear symmetry realization on subgroup generated by conformaldistortions. To be exact, one has to construct a quotient of the conformal group C (1 ,
3) on asubgroup generate by conformal distortion generators D , K ( µ ) . Here D is a uniform conformaldistortion operator, K ( µ ) are operators corresponding to conformal distortions orthogonal to thecorresponding coordinate lines, and ( µ ) = 0 · · · ε − expansion:Γ = (cid:90) d x (cid:34)
12 ( ∂ψ ) + 12 ( ∂σ ) + 1 ε ψ (cid:34) ∂ µ σ ( α ) ∂ µ σ ( α ) + 12 σ ( α ) (cid:3) σ ( α ) (cid:35) + 14 ε (cid:40) σ ( ∂ψ ) + 76 ψ ( ∂σ ) − ψ∂ µ ψ · σ∂ µ σ (cid:41) + O ( ε − ) (cid:35) . (4)It should be noted that the expansion in terms of the inverse symmetry breaking energy ε − matches the expansion in N -particle interaction functions. In other words, the term describing N -particle interaction belongs to ε − N -order of the expansion. Because of this the effectiveaction (1) is not a momentum expansion.Up to the order ε − tree-level diagrams of the effective action are ψ ψ = ik , σ ( α ) σ ( β ) = ik δ ( α )( β ) , (5) k p, ( α ) q, ( β )= − i ε (cid:34) p · q + 12 k (cid:35) δ ( α )( β ) ,p p q , ( α ) q , ( β )= − i ε (cid:34) p · p + 73 q · q −
56 ( p + p ) · ( q + q ) (cid:35) δ ( α )( β ) . ψ evaluated by the effective action at the first order in ε − reads lp q = π ε (cid:104) p q l C ( l, p, , , − l ( l + p + q − p · q ) B ( l, , − p ( l + p + q − l · q ) B ( p, , − q ( l + p + q − l · p ) B ( q, , (cid:105) . (6)Here B and C are Passarino-Veltman integrals [20] defined in A. The expression has a non-vanishing divergent part proportional to the following operators: (cid:3) ψ ( ∂ψ ) , ψ ( (cid:3) ψ ) . (7)In full agreement with the logic presented above, the effective action (1) lacks these operatorsas they do not respect the nonlinear symmetry realization. Therefore the diagram dynamicallybreaks the original effective action symmetry. Another point that should be noted is the factthat these operators differ in dimension compared with the three-particle interaction operatorsin the effective action (1). Consequently, they can be viewed as second order terms in (cid:3) /ε expansion, i.e., as higher order terms in the momentum expansion.However, three-points functions alone can hardly be considered as a sufficient evidence ofanomalous behavior, as they all vanish on-shell due to kinematics. Because of this the corre-spondent anomalous contributions cannot be probed empirically. To proceed with the search foranomalies we are going to evaluate one-loop four-point functions given by the effective action atthe first order in ε − .Four-point functions can have divergent parts which do not vanish on-shell. That can bedirectly shown via dimensional considerations. Namely, a one-loop four-particle function for ψ can generate operator ε − N ( ∂ψ ) ψ N − that does not vanish on-shell. In the particular case of afour-point function for ψ such a diagram reads (in terms of the Mandelstam variables ): p p q q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) on-shell = iπ ε d − s ( s d + 2 t ) B ( p + p , ,
0) + t ( t d + 2 s ) B ( p − p , , . (8)The complete one-loop four-point ψ function matrix element on-shell contains the followingdivergent part:+ + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) on-shell = − iπ ε ( s + t + u ) (cid:34) d − (cid:35) . (9)Its divergent part is proportional to the following operators:( ∂ψ ) , ψ∂ µ ∂ ν ψ ∂ µ ψ ∂ ν ψ . (10)They do not vanish on-shell, thus such an anomalous contribution can be probed empirically.Nonetheless, these divergences can be eliminated via Ward identities. In full analogy withgauge theories, the model admits a symmetry which establishes certain relations between ampli-tudes, no matter the fact that the symmetry is realized in a nonlinear way.To obtain Ward identities it is required to use the field transformation law. It is given by thefollowing expression [8]:exp (cid:34) i u ( µ )( ν ) L ( µ )( ν ) + iθD + iθ ( µ ) K ( µ ) (cid:35) exp (cid:104) iφD + iσ ( α ) K ( α ) (cid:105) = exp (cid:104) iφ (cid:48) D + iσ (cid:48) ( α ) K ( α ) (cid:105) exp (cid:34) i u (cid:48) ( µ )( ν ) L ( µ )( ν ) (cid:35) . (11) s = ( p + p ) , t = ( p + q ) , u = ( p + q ) . u ( µ )( ν ) , θ ( µ ) , and θ are the transformation parameters, while φ = ε − ψ is a dimensionlessfield variable. The transformation law reads φ (cid:48) = φ + θ ,σ µ + θ µ + 12 ( σ ν u νµ + φθ µ − σ µ θ ) + · · · = σ (cid:48) µ − σ (cid:48) ν u νµ + · · · . (12)In the expression for σ we neglected terms containing higher orders of the transformation pa-rameters.In such a way the field ψ on its own admits a shift symmetry. The Ward identities aregenerated by the infinitesimal action of the transformations on the Feynman integral. At the ε − -level Ward identities for ψ read (cid:90) D [ ψ, σ ( α ) ] (cid:34) ε (cid:40) ( ∂σ ) + 12 σ (cid:3) σ (cid:41)(cid:35) e i Γ = 0 . (13)This identity shows that all diagrams with at least one external ψ line connected to the three-point function vanish due to the symmetry. Consequently, all diagrams discussed before vanishand anomalous contributions are eliminated.The same argument does not hold for the four-particle function, as the Ward identities at the ε − level read0 = (cid:90) D [ ψ, σ ( α ) ] (cid:34) ∂ µ (cid:40) ∂ µ ψ σ − ψ σ∂ µ σ (cid:41) + 75 ψ ( ∂σ ) − ∂ψ σ∂σ (cid:35) e i Γ . (14)Because of the gradient term the identity cannot exclude the corresponding interaction from thetheory.Consequently, one should search for one-loop anomalous amplitudes generated by the effectiveaction (1) at the second level in ε − . Such amplitudes exist and they are given by the following: p p q q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) on-shell = i π
64 1 ε s B ( p + p , , . (15)+ + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) on-shell = − i π
32 1 ε ( s + t + u ) (cid:34) d − (cid:35) . (16)Divergences of these diagrams appear on-shell and can be probed empirically. At the same timethey cannot be eliminated via Wards identities, so they lead to real effects. Finally, in fullagreement with the symmetry reasoning operator ( ∂ψ ) which is generated by the divergent partof the diagrams is missing in the original effective action, as it does not respect the non-linearsymmetry realization.It is required to make a comment on the renormalizability of the model. The fact thatone-loop amplitudes generate operators missing in the original effective action (4) shows thatthe model cannot be renormalized by the standard technique. On the other hand it can only beconsidered as an effective model applicable in the low energy regime. The complete model which,indeed, should be renormalizable and free from anomalies must explicitly contain a mechanismof spontaneously breaking down the conformal symmetry and generating the energy scale ε .The nonlinear symmetry realization technique used to generate the effective action (1) can onlyrecover the dynamics of low energy modes. It is safe to assume that the divergent part of thesediagrams can be renormalized when treated in the complete theory. It also should be highlightedthat aforementioned diagrams alongside diagrams to be discussed further generate divergent localparts and finite non-local parts. Their role is discussed in the last section of the paper.Finally, it is possible to recover the anomaly-induced effective action. To do this it is requiredto evaluate the rest one-loop four-particle diagrams:5 p q q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) on-shell = − π ε s (cid:34) d − (cid:35) , (17) p p q q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) on-shell = + + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) on-shell = − i π ε ( s + t + u ) (cid:34) − d + finite part (cid:35) . (18)These expressions allow one to restore the anomaly-induced action:Γ anomaly = (cid:90) d x (cid:34) − ψ (cid:3) ψ − σ (cid:3) σ + 14 ε (cid:40) σ ( ∂ψ ) + 76 ψ ( ∂σ ) −
53 ( ψ∂ψ )( σ∂σ ) (cid:41) + 1 ε (cid:40) c ( ∂ψ ) + c , ( ∂ψ ) ( ∂σ ) + c , ( ∂ψ ) ( σ (cid:3) σ ) + c , ( ∂σ ) ( ψ (cid:3) ψ )+ c , ( ∂σ ) + c , ( ∂σ ) ( σ (cid:3) σ ) (cid:41)(cid:35) . (19) Results are analogous to the well-known case of the anomaly in the σ -model [14, 19]. In the lowenergy regime the original symmetry of the model is spontaneously broken. The effective action,nonetheless, respects the nonlinear realization, because of which some interactions do not appearin the model. Anomalous amplitudes have divergent parts proportional to operators missing inthe effective action. This feature is understood as a direct indication of a dynamical breaking ofthe effective action symmetry.In full analogy with the given arguments one can conclude that an effective gravity actionrecovered via a nonlinear symmetry realization at the classical level should not be consideredsufficient for a description of real low-energy gravitational processes. It should be used further torestore the corresponding anomaly-induced effective action which, in turn, should be implementedfor calculations of physical observables. The studied model shows that an effective action andthe corresponding anomaly-induced effective action can have not only different forms, but alsodifferent symmetries.It should be noted that a similar anomalous behavior takes place in completely differentgravity models. Namely, it appears also in Galileon models [21, 22]. In the flat spacetimeGalileon models describe a scalar field with the so-called generalized Galilean symmetry. Becauseof the symmetry the theory admits second-order field equations and it is free from ghost states.However, at the one-loop level the theory generates higher derivative terms that break the originalmodel symmetry [23, 24]. Models of that class can be considered effective as the generalizedGalilean symmetry can be induced by auxiliary dimensions [21, 22]. Similar phenomenology ispreserved even in a curved spacetime. Within generalized Galileons (Horndeski models) one-loopeffects generate both new interactions [25] and higher-derivative terms [26].These considerations highlight the special role of symmetries in effective models. Quantumanomalies can hide the original symmetry of the model thereby complicating its empirical veri-fication. Consequently, even if the conformal symmetry (or any other symmetry) is realized innature in a nonlinear way, it still may not be probed easily due to anomalies.Perhaps, the simplest reliable way to verify a theory with such anomalous features is givenby non-local contributions of anomalous amplitudes. Such amplitudes generate finite non-localcontributions alongside the divergent parts. For instance, within the model studied in this paperthe one-loop four-point ψ function generates the following non-local operator:6 p q q → − i π ε p · p q · q ln (cid:32) − ( p + p ) µ (cid:33) . (20)This operator has a finite coupling and it is to affect real physical processes. Therefore theexact value of the coupling can be recovered from empirical data. These couplings are completelyfixed by the original effective action (1) and they are not affected by the anomaly. It should alsobe noted that the role of non-local operators generated at the one-loop level is actively studiedwithin effective gravity [27, 28, 29, 30].Concluding, one can argue that anomalies of effective models with a nonlinear symmetryrealization should be subjected to a more detailed consideration. The model studied it thispaper alongside the other mentioned results shows that anomalies have non-trivial effects on thelow-energy phenomenology. A Expressions for diagrams
The following definitions of Passarino-Veltman integrals are used [20, 31]: B ( p, m , m ) def = (2 πµ ) − d iπ (cid:90) d d k (cid:2) ( k − m ) (( k − p ) − m ) (cid:3) − = − d − , (21) C ( p, p − q, q, m , m , m ) def = (2 πµ ) − d iπ (cid:90) d d k (cid:2) ( k − m ) (( k − p ) − m ) (( k − q ) − m ) (cid:3) − . One-loop on-shell four-particle diagrams used in the paper are given by the following expres-sions: p p q q = i π ε s B ( p + p , , , p p q q = i π ε t B ( p + q , , , (22) p p q q = i π ε u B ( p + q , , , p p q q = i π ε s B ( p + p , , ,p p q q = i π ε t B ( p + q , , , p p q q = i π ε u B ( p + q , , . References [1] V. I. Ogievetsky. Infinite-dimensional algebra of general covariance group as the closure offinite-dimensional algebras of conformal and linear groups.
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