Non-Abelian aether-like term in four dimensions
A. J. G. Carvalho, D. R. Granado, J. R. Nascimento, A. Yu. Petrov
NNon-Abelian aether-like term in four dimensions
A. J. G. Carvalho, D. R. Granado, J. R. Nascimento, and A. Yu. Petrov Departamento de Física, Universidade Federal da ParaíbaCaixa Postal 5008, 58051-970, João Pessoa, Paraíba, Brazil ∗ Duy Tân University, Institute of Research and Development,P809, 3 Quang Trung, Hải Châu, Đà Nẵng, Vietnam † The non-Abelian aether-like Lorentz-breaking term, involving triple and quar-tic self-coupling vertices, is generated from the non-Abelian generalization of theLorentz-breaking extended QED including only a minimal spinor-vector interaction.This term is shown explicitly to be finite and non-ambiguous.
I. INTRODUCTION
The Lorentz symmetry breaking opens broad possibilities for constructing extensions ofknown field theory models. The first steps in this direction were presented in the paradig-matic papers by Kostelecky and Colladay [1, 2]. In these papers a list of possible Lorentz-breaking extensions of the standard model has been presented for the first time. Further,many terms of this list were shown to arise as one-loop perturbative corrections. In the caseof scalar, gauge and gravitational fields such terms arise from the corresponding fermionicdeterminants (for a review of various situations where such arising occurs, see [3]).However, absolute majority of these results describe Lorentz-breaking contributions ofsecond order in fields – for example, Carroll-Field-Jackiw (CFJ) term, aether terms forscalar and gauge fields, higher-derivative contributions for gauge fields. At the same time, itis well known that many phenomenological interesting results are obtained in essentially non-Abelian gauge theories (the most interesting application of these theories consists certainlyin studies of QCD and confinement) whose Lagrangian involves terms up to fourth order infields.The first example of a non-Abelian Lorentz-breaking term is the non-Abelian CFJ term[4, 5] which breaks the Lorentz symmetry but preserves the gauge symmetry. In [6], the ∗ jroberto, [email protected], [email protected] † [email protected] a r X i v : . [ h e p - t h ] A ug authors presented the appropriate scheme for the path integral quantization of Yang-Mills-CFJ system. In order to remove properly the gauge copies, one must restrict the path integralto a subspace of independent dynamical fields called Gribov region. In [7], it was shown forthe first time in the YM system that, in order to deal with the gauge copies, such restrictionis mandatory and as a consequence implies the modification of the gluon propagator. Inthe low-energy limit of the theory, the gluon propagator exhibits the propagation of non-physical particles, which means that we are no longer able to describe the propagation ofthe real degrees of freedom of the theory in its low-energy limit. This feature is interpretedas the description of the confinement problem of gluons, which is an inherent problem ofnon-Abelian gauge theories. In [6], a new non-Abelian Lorentz-breaking model, that is,the non-Abelian aether-like model was proposed, a simplified scheme for the generation ofthe non-Abelian aether-like term was presented and the proper path integral quantizationof the Yang-Mills-aether-like system was treated. By means of the Gribov description ofthe gluon confinement problem mentioned above, in [6, 8], it was verified whether the non-Abelian Lorentz symmetry breaking terms can influence the theory in the low-energy limitin a manner implying that the confinement problem may not occur for certain values of thecoupling constants presented in the theory. In both papers it was shown that due to thesmall value of the Lorentz-breaking parameters ( ≈ − GeV), these terms does not affectthe confinement regimes of the theory. Therefore, in a full analogy with those papers, it isnatural to expect that the term generated by us here can yield only very tiny modificationswithin the confinement scenario.Thus the natural problem consists of studying the various issues related to non-AbelianLorentz-breaking terms. Some examples of such terms were recently listed in [9]. However,up to now, there are only very few studies of such terms – as it was mentioned before, the non-Abelian extension of the CFJ term was considered in [4, 5] and some non-perturbative effectsfor Lorentz-breaking extensions of Yang-Mills theory were discussed in [10]. Therefore, itis natural to look for more new results in this direction. The most natural problem fromthe perturbative generation viewpoint could be the generation of the non-Abelian aetherterm using only minimal couplings. As it is known, this generation makes the result inthe Abelian case to be superficially finite and non-ambiguous [11], avoiding the problem ofregularization dependence. Namely, this scheme of calculations will be generalized to thenon-Abelian case, i.e., we will obtain CPT-even three and four-point functions of the gaugefield.This paper is organized as follows. In the section 2, we give basic definitions; in thesection 3, we perform the one-loop calculations; finally, in the section 4 we present ourresults. In the Appendix, the relevant momentum integrals are given.
II. THE NON-ABELIAN AETHER TERM
Now, let us start with discussion of the non-Abelian aether term. The original aetherterm [12] is known to have the following form L aether = u µ u ν F µλ F νλ , (1)where u µ is a some constant vector, and F µν is the usual stress tensor of the electromagneticfield. Actually, this term is nothing more as the general CPT-even term L even = κ µνλρ F µν F λρ , (2)proposed in [1, 2], for a special form of the constant tensor κ µνλρ (for some issues related tothis CPT-even term, including its impacts for the plane wave solutions of modified Maxwellequations, see also f.e. [13]). It is clear that the non-Abelian analogue of the term (1) canbe written down straightforwardly – it is sufficient to replace the Abelian stress tensor byits non-Abelian analogue [6], i.e. L aether,Y M = u µ u ν tr( F µλ F νλ ) , (3)where F µν = F aµν T a is the non-Abelian, Lie-algebra valued stress tensor.In [14] the scheme for generation of the Abelian aether term was proposed, and thisterm was obtained as a one-loop quantum correction in a theory which involves a magneticcoupling of the fermion to an electromagnetic field. The generalization of the scheme used in[14], to the non-Abelian case is straightforward. In order to, one can start with the followingclassical action: S ψ = Z d x N X i,j =1 ¯ ψ i ( iδ ij ∂/ − g (cid:15) µνλρ F aµν b λ γ ρ ( T a ) ij − mδ ij ) ψ j . (4)where T a are the generators of the corresponding gauge group. The non-Abelian aetherterm was generated for this theory in [6], where it was shown to be finite and ambiguous,similarly to its Abelian analogue, cf. [14].At the same time, the generation of the triple and quartic terms in the Yang-Mills actionusing only minimal couplings is a nontrivial problem. This is the aim we pursue in thispaper. We start with the following action of the spinor coupled to the non-Abelian gaugefield: S = Z d x ¯ ψ i (cid:16) i∂/δ ij − eA/ a ( T a ) ij − mδ ij − b/γ δ ij (cid:17) ψ j . (5)Unlike the nonminimal coupling considered in [6], this action involves the minimal couplingonly, with the corresponding coupling constant being dimensionless. As a result, this theoryis all-loop renormalizable.Within our study, our aim consists in computing the one-loop effective action presentedby the following fermionic determinantΓ (1) = i Tr ln( i∂/ − eA/ a T a − m − b/γ ) , (6)where A/ a = γ µ A aµ , since our vector field is Lie-algebra valued. Expanding the fermionicdeterminant up to the fourth order in external fields, we find (here A/ = A/ a T a ) − i Γ (1) = − e A/ i∂/ − m − b/γ A/ i∂/ − m − b/γ ++ e A/ i∂/ − m − b/γ A/ i∂/ − m − b/γ A/ i∂/ − m − b/γ −− e A/ i∂/ − m − b/γ A/ i∂/ − m − b/γ A/ i∂/ − m − b/γ A/ i∂/ − m − b/γ . (7)To proceed with this calculation, one can use the exact propagator of the spinor whose formin the momentum space is given by [15]: S ( k ) = 1 k/ − m − b/γ = k + b − m + 2( b · k + mb/ ) γ ( k + b − m ) − b · k ) − m b ] ( k/ + m + b/γ ) . (8)However, within our purposes it is more convenient to use the usual propagator of ψ , thatis, < ¯ ψ ( − p ) ψ ( p ) > = p/ − m , since we consider the contributions only up to the second orderin b µ . We note that since the aether term which will be obtained from a minimal couplingis non-ambiguous, the results obtained with use either of the modified propagator or thesimple one will be the same.The result for the second order in A µ can be obtained through the sum of contributionsfor the three diagrams, with two insertions of /bγ , carrying out the expansion of the followingexpression: Γ (1)2 = ie /A i /∂ − m − /bγ /A i /∂ − m − /bγ . (9)In this case, the total contribution is given byΓ (1)2 = e (cid:20) Z d p (2 π ) γ µ ( /p + m ) γ ν ( /p + /k + m ) /bγ ( /p + /k + m ) /bγ ( /p + /k + m )]( p − m ) + Z d p (2 π ) tr[ γ µ ( /p + m ) /bγ ( /p + m ) γ ν ( /p + /k + m ) /bγ ( /p + /k + m )]( p − m ) (10)+ Z d p (2 π ) tr[ γ µ ( /p + m ) /bγ ( /p + m ) /bγ ( /p + m ) γ ν ( /p + /k + m )]( p − m ) (cid:21) A aµ ( − k ) A bν ( k )tr( T a T b ) . The result represents itself as a direct generalization of the Abelian contribution (see f.e.[11]) and looks likeΓ (1)2 = − e π m b µ F µνa b λ F bλν tr( T a T b ) = − κe π m b µ F µνa b λ F aλν , (11)where F µνa = ∂ µ A νa − ∂ ν A µa is the Abelian part of the stress tensor, and the generators arenormalized through the relation tr( T a T b ) = κδ ab , with κ = 0 is a some real number definingnormalization of the generators. This result can be represented asΓ (1)2 = − κe π m Π λραβ ∂ λ A aρ ∂ α A aβ , (12)where Π λραβ = η ρβ b λ b α − η ρα b λ b β − η λβ b ρ b α + η λα b ρ b β , (13)so that Π λραβ ∂ λ ∂ α is a transversal operator. Straightforward analysis of corresponding Feyn-man diagrams allows to show that just this operator arises when third-order and fourth-ordercontributions are obtained. We note that these, higher-order contributions are essentiallynon-Abelian, being absent in the U (1) case. III. THIRD-ORDER CONTRIBUTION
Our starting point is the three-point contribution to the effective action of the gauge field A aµ given by Γ (1)3 = ie /A i /∂ − m − /bγ /A i /∂ − m − /bγ /A i /∂ − m − /bγ . (14)Here, we use the usual propagator of the spinor field.Throughout our computation we consider the terms up to the second order in b µ (propor-tional to b µ b λ , but we disregard all terms proportional to b since they yield only Lorentz-invariant contributions). The terms of the second order in b µ are given by Fig. 1. FIG. 1. Contributions of the third order in the external fields, with two insertions of /bγ . We note that the algebraic factors accompanying our quantum corrections are the sameas in the usual Lorentz-invariant Yang-Mills theory since b/γ insertion is proportional to theKronecker symbol δ ij in the isotopic space (for a general discussion of quantum aspects ofthe Yang-Mills theory, see the classic papers [16]). These factors will yield first order instructure constants f abc for the three-point function, and the second order for the four-pointfunction.Carrying different contractions in a manner similar to calculations of higher-point func-tions presented in [16, 17], we find that, explicitly, the contribution (a) looks likeΓ (1)3 ,a = e Z d p (2 π ) tr " γ µ /p − m /bγ /p − m γ ν /p − /k − m /bγ /p − /k − m γ ρ ×× /p − /k − /k − m A aµ ( k ) A bν ( k ) A cρ ( k )tr( T a [ T b , T c ]) , (15)where k + k + k = 0.We expand the propagators up to the first order in external momenta k and k , andrewrite this contribution in the form:Γ (1)3 ,a = e Z d p (2 π ) (cid:20) [ γ µ ( /p + m ) /bγ ( /p + m ) γ ν ( /p + m ) /k ( /p + m ) /bγ ( /p + m ) γ ρ ( /p + m )]( p − m ) + [ γ µ ( /p + m ) /bγ ( /p + m ) γ ν ( /p + m ) /bγ ( /p + m ) /k ( /p + m ) γ ρ ( /p + m )]( p − m ) + [ γ µ ( /p + m ) /bγ ( /p + m ) γ ν ( /p + m ) /bγ ( /p + m ) γ ρ ( /p + m )( /k + /k )( /p + m )]( p − m ) (cid:21) ×× A aµ ( k ) A bν ( k ) A cρ ( k )tr( T a [ T b , T c [) . (16)The contributions (a), (b) and (c) yield equal results by symmetry reasons.On the other hand, for (d), (e) and (f) contributions, we proceed in a similar way. The(d) contribution looks likeΓ (1)3 ,d = e Z d p (2 π ) " γ µ /p − m γ ν /p − /k − m /bγ /p − /k − m /bγ /p − /k − m γ ρ ×× /p − /k − /k − m A aµ ( k ) A bν ( k ) A cρ ( k )tr( T a [ T b , T c ]) , (17)After expansion in external momenta and keeping only the first-order terms in this expan-sion, one findsΓ ( d )3 = e (cid:20) Z d p (2 π ) γ µ ( /p + m ) γ ν ( /p + m ) /k ( /p + m ) /bγ ( /p + m ) /bγ ( /p + m ) γ ρ ( /p + m )( p − m ) + Z d p (2 π ) γ µ ( /p + m ) γ ν ( /p + m ) /bγ ( /p + m ) /k ( /p + m ) /bγ ( /p + m ) γ ρ ( /p + m )( p − m ) (18)+ Z d p (2 π ) γ µ ( /p + m ) γ ν ( /p + m ) /bγ ( /p + m ) /bγ ( /p + m ) /k ( /p + m ) γ ρ ( /p + m )( p − m ) + Z d p (2 π ) γ µ ( /p + m ) γ ν ( /p + m ) /bγ ( /p + m ) /bγ ( /p + m ) γ ρ ( /p + m )( /k + /k )( /p + m )( p − m ) (cid:21) ×× A aµ ( k ) A bν ( k ) A cρ ( k )tr( T a [ T b , T c ]) . The diagrams (d), (e) and (f) also yield equal results by symmetry reasons. Due to thepresence of the commutators, all contributions turn out to be proportional to the first orderin f abc , just as in the Lorentz-invariant case [16].The complete result for the three-point function is a sum of results for (a) – (f) diagramsgiven by Fig. 1. Through straightforward comparison with contributions to the two-pointfunction, it is shown to be proportional to the same tensor Π λραβ (13) arising within thecalculation of the two-point function, explicitly,Γ (1)3 = κe π m f abc Π λραβ ∂ λ A aρ A bα A cβ , (19)or, as is the same, Γ (1)3 = κe π m b µ f abc F µνa b λ A bλ A cν . (20)We note that the constant factor accompanying this term is appropriate to form the gaugecovariant expression for the non-Abelian gauge invariant action. IV. FOURTH-ORDER CONTRIBUTION
Now, we turn to the fourth-order contribution. It is given byΓ (1)4 = − e /A i /∂ − m − /bγ /A i /∂ − m − /bγ /A i /∂ − m − /bγ /A i /∂ − m − /bγ . (21)The relevant contribution is presented by the Feynman diagram depicted at Fig. 2. FIG. 2. Contribution of the fourth order in the external fields, with two insertions of /bγ . Explicitly, considering the cycle ( I ) − ( III ) − ( V ) − ( V I ), with each contribution issimilar to the ( I ), we have for I contribution:Γ ( I )4 = − e Z d p (2 π ) tr " γ µ ( /p + m ) p − m /bγ ( /p + m ) p − m γ ν ( /p − /k + m )( p − k ) − m /bγ ( /p − /k + m )( p − k ) − m ×× γ ρ ( /p − /k − /k + m )( p − k − k ) − m γ λ ( /p − /k + m )( p − k ) − m A aµ A bν A cλ A dρ tr([ T a , T b ][ T c , T d ]) . (22)Here, the commutators again arise due to various manners of carrying out the contractions,as in [17]. Now, to form the contribution to the Yang-Mills-aether term, only zero order inmomenta must be kept, so,Γ ( I )4 = − e Z d p (2 π ) tr " γ µ ( /p + m ) /bγ ( /p + m ) γ ν ( /p + m ) /bγ ( /p + m ) γ ρ ( /p + m ) γ λ ( /p + m )( p − m ) ×× A aµ A bν A cλ A dρ tr([ T a , T b ][ T c , T d ]) . (23)The diagrams ( II ) , ( IV ) are equal. Explicitly, the contribution ( IV ) isΓ ( IV )4 = − e Z d p (2 π ) tr " γ µ ( /p + m ) p − m γ ν ( /p − /k + m )( p − k ) − m /bγ ( /p − /k + m )( p − k ) − m ×× γ λ ( /p − /k − /k + m )( p − k − k ) − m γ ρ ( /p − /k + m )( p − k ) − m /bγ ( /p − /k + m )( p − k ) − m ×× A aµ A bν A cλ A dρ tr([ T a , T b ][ T c , T d ]) . (24)Keeping again only zero order in momenta, we haveΓ ( IV )4 = − e Z d p (2 π ) tr " γ µ ( /p + m ) γ ν ( /p + m ) /bγ ( /p + m ) γ λ ( /p + m ) γ ρ ( /p + m ) /bγ ( /p + m )( p − m ) ×× A aµ A bν A cλ A dρ tr([ T a , T b ][ T c , T d ]) . (25)The diagrams ( V II ) − ( V III ) − ( IX ) − ( X ) yield the same contributions. Explicitly,the contribution ( X ) isΓ ( X )4 = − e Z d p (2 π ) tr " γ µ ( /p + m ) p − m γ ν ( /p − /k + m )( p − k ) − m γ λ ( /p − /k − /k + m )( p − k − k ) − m ×× γ ρ ( /p − /k + m )( p − k ) − m /bγ ( /p − /k + m )( p − k ) − m /bγ ( /p − /k + m )( p − k ) − m ×× A aµ A bν A cλ A dρ tr([ T a , T b ][ T c , T d ]) . (26)Again, keeping only zero order in momenta, we haveΓ ( X )4 = − e Z d p (2 π ) tr " γ µ ( /p + m ) γ ν ( /p + m ) γ λ ( /p + m ) γ ρ ( /p + m ) /bγ ( /p + m ) /bγ ( /p + m )( p − m ) ×× A aµ A bν A cλ A dρ tr([ T a , T b ][ T c , T d ]) . (27)Calculating all traces and using the integrals listed in the appendix A, we show that theresult is proportional to the second order in structure constants, as it must be by the gaugesymmetry reasons, and to the operator Π λραβ , and has the following formΓ (1)4 = − κe π m Π λραβ f abm f cdm A aλ A bρ A cα A dβ , (28)0or, as is the same, Γ (1)4 = − κ e π m f abm f cdm b µ A µa A νb b λ A cλ A dν . (29)The sum of the expressions (11), (20) and (29) yields the desired resultΓ (1) = − κ e π m b µ F aµν b λ F λνa , (30)where F µνa = ∂ µ A νa − ∂ ν A µa − ef abc A µb A νc (31)is the non-Abelian stress tensor. Thus, we conclude that we have succeeded to generate thenon-Abelian aether-like term. V. SUMMARY
In this paper, we have performed the generation of non-Abelian aether-like term. Ourstarting point was the theory of Dirac spinor minimally coupled to the non-Abelian gaugefield, where the Lorentz symmetry breaking has been introduced through the usual b/γ term,that is, the same model considered in [5] and representing itself as a straightforward non-Abelian generalization of the model used within studies of a Lorentz-breaking extension ofQED.The main significance of our result consists in the fact that while up to now, mostof the papers devoted to study of perturbative aspects of Lorentz-breaking theories wereconcentrated either on quadratic finite contributions (for a review on finite corrections see[3]) or on renormalization of coupling vertices (see f.e. [18]), here we performed perturbativegeneration of a finite non-Abelian, fourth-order contribution. We note that our result is validfor an arbitrary gauge group. It is the second example of generation of a non-Abelian termcarried out with use of only minimal couplings, after [5], ever realized. Actually, our resultis the next-order contribution to the expansion of the effective action of the non-Abeliangauge field coupled to fermions, after the non-Abelian CFJ term [5]. The advantage of ourapproach in comparison with [6] consists in the fact that unlike the scheme presented in [6],the calculation performed in this paper is carried out on the base of a minimal coupling,being hence explicitly superficially finite and hence ambiguity-free. Thus, we have confirmed1that the non-Abelian Lorentz-breaking terms can arise as quantum corrections in a somefundamental theory, as it follows from the concept of emergent dynamics.As it was discussed in [6], the impact of the aether term should be small in comparisonwith the usual Yang-Mills term since the Lorentz-breaking parameters are very small. It isnatural to expect that there will be only small modifications in qualitative description of theconfinement generated by the aether term. Nevertheless we note that there are many issuesrelated to the non-Abelian aether term within the confinement problem and other contextswhich need to be studied. We expect to perform such studies in our next papers. Acknowledgements.
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Here we list the integrals used to perform our computations: Z d p (2 π ) p − m ) = − i π m ; (A1) Z d p (2 π ) p ( p − m ) = i π m ; (A2) Z d p (2 π ) p µ p ν ( p − m ) = ig µν π m ; (A3) Z d p (2 π ) p ( p − m ) = − i π m ; (A4) Z d p (2 π ) p − m ) = − i π m ; (A5) Z d p (2 π ) p ( p − m ) = − i π m ; (A6) Z d p (2 π ) p µ p ν ( p − m ) = − ig µν π m ; (A7) Z d p (2 π ) p ( p − m ) = i π m ; (A8) Z d p (2 π ) p − m ) = − i π m ; (A9) Z d p (2 π ) p ( p − m ) = i π m (A10) Z d p (2 π ) p µ p ν ( p − m ) = ig µν π m ; (A11) Z d p (2 π ) p ( p − m ) = − i π m ; (A12) Z d p (2 π ) p µ p ν p ρ p σ ( p − m ) = − i π m ( g µν g ρσ + g µρ g νσ + g µσ g νρ ) ; (A13) Z d p (2 π ) p µ p ν p ρ p σ ( p − m ) = i π m ( g µν g ρσ + g µρ g νσ + g µσ g νρ ) ; (A14) Z d p (2 π ) p µ p ν p ρ p σ ( p − m ) = − i π m ( g µν g ρσ + g µρ g νσ + g µσ g νρ ) ..