Quantum aspects of antisymmetric tensor field with spontaneous Lorentz violation
Sandeep Aashish Sukanta Panda







[]sandeepa16@iiserb.ac.in



[]sukanta@iiserb.ac.in



Department of Physics, Indian Institute of Science Education and Research, Bhopal 462066, India


























We study the quantization of a simple model of antisymmetric tensor field with spontaneous Lorentz violation in curved spacetime. We evaluate the 1-loop corrections at first order of metric perturbation, using a general covariant effective action approach. We revisit the issue of quantum equivalence, and find that it holds for non-Lorentz-violating modes but breaks down for Lorentz violating modes.









Quantum aspects of antisymmetric tensor field with spontaneous Lorentz violation
    Sukanta Panda
    December 30, 2023
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§ INTRODUCTION

The quest for quantizing gravity is ultimately related to understanding physics at the planck scale, candidates for which include string theory and loop quantum gravity. A difficulty that the development of such theories faces, is our inability to probe high energy scales, owing to the limitations of current particle physics experiments <cit.>. This has led to significant efforts towards finding low energy signatures using effective field theory tools that could be relevant in current and near future experiments in both particle physics and early universe cosmology. Phenomenologically, this amounts to detecting planck supressed variations to standard model and general relativity while maintaining observer independence, termed as standard model extension (SME) <cit.>. 

There is substantial evidence of SME effects from string theory and quantum gravity, according to which certain mechanisms could lead to violation of Lorentz symmetry <cit.>, which is a fundamental symmetry in general relativity that relates all physical local Lorentz frames. In principle, Lorentz violation can be introduced in a theory either explicitly, in which case the Lagrange density is not Lorentz invariant, or spontaneously, so that the Lagrange density is Lorentz invariant but the physics can still display Lorentz violation <cit.>. However, theories with explicit Lorentz violation have been found to be problematic due to their incompatibility with Bianchi identities in Riemann geometry <cit.>, and are therefore not favourable for studies involving gravity.

Another consequence of string theory, at low energies, is the appearance of antisymmetric tensor field along with a symmetric tensor (metric) and a dilaton (scalar field) as a result of compactification of higher dimensions <cit.>. Until recently, antisymmetric tensor had not recieved serious consideration in studies of early universe cosmology, in particular inflation, due to some generic instability issues <cit.>, but some recent studies have shown that presence of antisymmetric tensor field is likely to play a role during inflation era <cit.>. Hence, as a natural extension, an interesting exercise is to consider Lorentz violation in conjunction with antisymmetric tensor (see, for example Ref. <cit.>).

Altschul  in Ref. <cit.> explored in detail spontaneous Lorentz violation with antisymmetric tensor fields, and found the presence of distinctive physical features with phenomenological implications for tests of Lorentz violation, even with relatively simple antisymmetric field models with a gauge invariant kinetic term. 

Our interest in the present work is to take first steps to extend the classical analysis in Ref. <cit.> to quantum regime. We focus on the formal aspects of quantization of antisymmetric tensor field with spontaneous Lorentz violation, and primarily restrict ourselves to dealing with two issues. First, we set up the framework to evaluate the one-loop effective action using covariant effective action approach <cit.>. For simplicity, we consider an action with only quadratic order terms, but in a nearly flat spacetime (Minkowski metric η_μν plus a classical perturbation κ h_μν). This yields one-loop corrections at O(κħ), involving terms upto first order in h_μν. Second, we check the quantum equivalence of the quadratic action considered in the first part with a classically equivalent vector theory, at 1-loop level. The issue of quantum equivalence in curved spacetime is interesting because a free massive antisymmetric tensor theory (no Lorentz violation) is known to be equivalent to a massive vector theory at classical and quantum level due to topological properties of zeta functions <cit.> but, such properties do not hold when Lorentz symmetry is spontaneously broken <cit.>. In fact, it was demonstrated by Seifert in Ref. <cit.> that interaction of vector and tensor theories with gravity are different when topologically nontrivial monopole-like solutions of the spontaneous symmetry breaking equations exist. The method presented here is quite general in terms of its applicability to models with higher order terms in fields. 

In Sec. <ref>, we briefly review spontaneous Lorentz violation in antisymmetric tensor and introduce the classical action considered in this work. The notations used here are largely inspired by Ref. <cit.>. We discuss the covariant effective action technique and its application to derive 1-loop corrections in Sec. <ref>. We also calculate the various propagators required to solve the 1-loop integrals. In Sec. <ref>, we consider the classically equivalent vector theory and calculate 1-loop corrections to compare with the results of Sec. <ref>, to check the quantum equivalence. 






§ SPONTANEOUS LORENTZ VIOLATION AND CLASSICAL ACTION

Spontaneous symmetry breaking occurs when the equations of motion obey a symmetry but the solutions do not, and is effected via fixing a preferred value of vacuum (ground state) solutions. In general relativity, physically equivalent coordinate (or observer) frames are related via general coordinate transformations and local Lorentz transformations. In a given observer frame, fixing the vacuum expectation value (vev) of a tensor or vector field leads to spontaneous breaking of Lorentz symmetry, since all couplings with vev have preferred directions in spacetime <cit.>. 

Spontaneous Lorentz violation in tensor field Lagrangians can be introduced through a potential term that drives a nonzero vacuum value of tensor field. For an antisymmetric 2-tensor field B_μν, we assume,

    ⟨ B_μν⟩ = b_μν.

It is possible to attain a special observer frame in a local Lorentz frame in Riemann spacetime or everywhere in Minkowski spacetime, in which b_μν takes a simple block-diagonal form <cit.>,

    b_μν = 
    (
    0     -a     0     0
    
    a     0     0     0
    
    0     0     0     b
    
    0     0     -b     0
    ),

provided at least one of the quantities X_1=-2(a^2-b^2) and X_2=4ab is nonzero, where a and b are real numbers. Moreover, the analysis of monopole solutions of antisymmetric tensor in Ref. <cit.> showed that for a spherically symmetric nontrivial solution of the equation of motion of B_μν that asymptotically approaches vev, the potential of the form considered below (Eq. (<ref>)) requires putting a=0. As will be seen later on, this choice of a also ensures positivity of certain determinants appearing in loop integral calculations. We thus assume a=0 in the present analysis, although most of the calculations presented here are independent of the structure of b_μν. For later convenience, we also choose b_μνb^μν=1. 

We consider a simple model of a rank-2 antisymmetric tensor field, B_μν, with a spontaneous Lorentz violation inducing potential <cit.>,  

    V(B) = 1/16α^2 (B_μνB^μν - b_μνb^μν)^2.

Again, for the purpose of present analysis, we would like to consider only quadratic order terms in B_μν. To this end, we consider small fluctuations of B_μν about a background value b_μν <cit.>, 

    B_μν = b_μν + B̃_μν.
 
and neglect quartic and cubic terms in fluctuations B̃_μν assuming || B̃_μν|| ≪ ||b_μν||. The resulting potential is,

    V(B) ≈14α^2(b_μνB̃^μν)^2.

Although it may seem at this point that a quadratic Lagrangian might not lead to any significant physical result upon quantization, and it is actually true in case of a flat spacetime, nontrivial physical contributions appear in the 1-loop effective action in curved spacetime as demonstrated in the next section. For notational convenience, we do not explicitly write the tilde symbol for field fluctuations, and assume its use throughout. We thus work with the Lagrangian,

    ℒ = -1/12H_μνλH^μνλ - 14α^2(b_μνB^μν)^2.

The first term in Eq. (<ref>) is the gauge invariant kinetic term, 

    H_μνλ≡∇_μB_νλ + ∇_λB_μν + ∇_νB_λμ,

obeying the symmetry: B^μν⟶ B^μν + ∇_μξ_ν - ∇_νξ_μ for a gauge parameter ξ_μ. The gauge invariance of kinetic term in an otherwise non-gauge invariant Lagrangian (<ref>) gives rise to redundancy problems in the energy spectrum <cit.>, and cannot be removed via usual quantization method. A consistent method to treat this redundancy is given by the Stückelberg procedure <cit.>. According to this procedure, a strongly coupled field called the Stückelberg field is introduced in the symmetry breaking potential term such that the gauge symmetry is restored in a given Lagrangian. The original theory is still recovered in a special gauge (where Stückelberg field is put to zero), however, the advantage is that the redundant degrees of freedom are now encompassed in the Stückelberg field, and can be dealt with using well known quantization frameworks like the Faddeev-Popov method. For a detailed account of this procedure applied to massless and massive antisymmetric tensors, interested reader is referred to Refs. <cit.> respectively, and to Ref. <cit.> for a more recent analysis in the context of covariant effective action.  

The above procedure is applied to (<ref>) via the introduction of a strongly coupled vector field C_μ:

    ℒ = -1/12H_μνλH^μνλ - 1/4α^2[b_μν(B^μν + 1/αF^μν[C])]^2,

so that the Lagrangian (<ref>) becomes gauge invariant (here, F_μν≡∂_μC_ν-∂_νC_μ), and reduces to original Lagrangian (<ref>) in the gauge C_μ=0. The new Lagrangian is invariant under two sets of transformations: (i) gauge transformation of B_μν and shift of field C_μ, 

    B^μν   ⟶    B^μν + ∇_μξ_ν - ∇_νξ_μ, 
    
    C_μ   ⟶    C_μ - αξ_μ,

and, (ii) under the gauge transformation of Stückelberg field C_μ

    C_μ   ⟶    C_μ + ∇_μΛ , 
    
    B^μν   ⟶    B^μν ,

where, ξ_μ and Λ are the corresponding gauge parameters. 
In addition to the above symmetries of fields, there exists a set of transformation of gauge parameters Λ and ξ_μ that leaves the fields B_μν and C_μ invariant,

    ξ_μ   ⟶   ξ_μ + ∇_μψ , 
    Λ   ⟶   Λ + αψ ,

which means that the gauge generators are linearly dependent <cit.>. The gauge fixing procedure for this Lagrangian is covered explicitly in Ref. <cit.>, and to save space we quote directly the final result for the total gauge fixed Lagrangian,

    ℒ_GF = -112H_μνλH^μνλ - 1/4α^2(b_μνB^μν)^2 - 14(b_μνF^μν)^2 - 1/2(b_μνb_ρσ∇^μB^ρσ)^2
     - 1/2α^2C_νC^ν - 1/2(∇_μΦ)^2 - 1/2(∇^μC_μ)^2 - 1/2α^2Φ^2.

The presence of a new scalar field Φ is a direct consequence of gauge-fixing of Stückelberg field, and explicitly displays a scalar degree of freedom that remains hidden in the original Lagrangian (<ref>) with broken gauge symmetry.



§ 1-LOOP EFFECTIVE ACTION





The quantization of theories such as (<ref>) is tricky, because of the symmetries in gauge parameters, as in Eq. (<ref>). Such symmetries lead to a degeneracy in the ghost determinant appearing in the Faddeev-Popov procedure <cit.>, and require special treatment for quantization <cit.>. We follow a general quantization procedure developed in Ref. <cit.> based on DeWitt-Vilkovisky's approach <cit.> that yields covariant and background independent resutls, to deal with the additional symmetries of gauge parameters and derive the 1-loop effective action. 

For a quadratic action not involving quantization of metric the expression for 1-loop effective action in the DeWitt-Vilkovisky's field space notation, about a set of background fields φ̅^i, is given by <cit.>,

    Γ_1[φ̅] = -ln Q_αβ[φ̅] + lnQ̌_a b + 12ln(S^GF_,ij[φ̅]]).

where S^GF is the gauge-fixed action. Let us briefly explain the various (field-space) notations in Eq. (<ref>) (see <cit.> for a detailed introduction). The index i in field space corresponds to all the tensor indices and spacetime dependence of fields in the coordinate space. For example, fields (B_μν(x), C_μ(x),Φ(x)) are denoted by components of φ^i (i = 1,2,3) in field space, where φ^1↔ B_μν(x), φ^2↔ C_μ(x), and φ^3↔Φ(x). The rest of the constructions in field space (tensors, scalar products, connections, field space metric, etc.) are similar to that in a coordinate space. The background fields in this notation, φ̅^i, too carry all the indices of their respective counterparts including coordinate dependence. The object S_,ij represents a derivative in field space, define by,

    S_,ij[φ̅] = (δ^2δφ^jδφ^i S[φ])_φ = φ̅.

Let, ϵ̌^a parametrize the symmetry of gauge parameters, as in Eq. (<ref>), and χ̌^a be the corresponding fixing condition for gauge parameters ϵ^α (can be read off of Eqs. (<ref>) and (<ref>) ), then <cit.>

    Q^α_β = (δδϵ^βχ^α[φ,ϵ,χ̌])_ϵ = 0,

where χ^α is the gauge fixing condition for fields φ^i. Q^α_β is the ghost determinant factor. 

In the present case, corresponding to the symmetries (<ref>), (<ref>) and (<ref>), there are two gauge conditions χ_ξ_ν and χ_Λ, along with a condition χ̌_ψ on the parameters, that lead to three operators Q^ξ_μ_ξ_ν, Q^Λ_Λ and Q̌_ψ^ψ respectively. The results are displayed in TABLE <ref>.


Using these results in Eq. (<ref>), we get

    Γ_1 = -ln Q^ξ_μ_ξ_ν + 12ln(S^GF_,ij[φ̅]]).

S^GF is of course quadratic in fields, and the value of Γ_1 in operator form turns out to be, 

    Γ_1 = iħ/2[ln(_2^μν_  ρσ - α^2b^μνb_ρσ) - ln(_1^μ_ ν -α^2δ^μ_ν) + ln( - α^2)],

where,

    _2^μν_  ρσB^ρσ   ≡   ∇_α∇^αB^μν + ∇_α∇^μB^να + ∇_α∇^νB^αμ + 2b^μνb_ρσb^ασb_βγ∇^ρ∇_αB^βγ, 
    _1^μ_ νC^ν   ≡    2b^νμb_ρσ∇_ν∇^ρC^σ + ∇^μ∇_νC^ν;

and  is the de'Alembertian operator.
In flat spacetime, no physically interesting inferences can be extracted from the above expression. However, in curved spacetime, the operators in Eq. (<ref>) are coupled to the metric g_μν. So, addressing certain issues, like that of quantum equivalence, then becomes nontrivial. Unfortunately, effective action cannot be calculated exactly in such cases <cit.>, and the best way forward is to perform a perturbative study. Therefore, we will consider a nearly flat spacetime instead of a general curved one, so that,

    g_μν(x) = η_μν + κ h_μν(x).

η_μν is the Minkowski metric and h_μν is a perturbation, while κ = 1/M_p(M_p is Planck mass) parametrizes the scale of perturbation.

We can rewrite Γ_1 in integral form by introducing ghost fields c_μ and c̅_μ, 

    Γ_1 = -ln∫ [dη][dc_μ] [dc̅_μ] e^-S_GH,

where, 

    S_GH = η^iS^GF_,ijη^j + c̅^μQ^ξ_μ_ξ_νc^ν,

and η^i are the quantum fluctuations (δ B_μν(x), δ C_μ(x),δΦ(x)). Now, we use Eq. (<ref>) in Eq. (<ref>) and rearrange terms in orders of h_μν:

    S_GH = S_0 + S_1 + O(h_μνh_αβ),

where the subscripts denote the power of h_μν. Substituting Eq. (<ref>) in Eq. (<ref>), and treating S_1 as a perturbation, the integrand can be Taylor expanded to write,

    Γ_1 = -ln(1 + ⟨ S_1⟩ + O(h_μνh_αβ)),

where we have used the normalization for path integral of S_0. The logarithm can be further expanded to yield, upto first order in h_μν,

    Γ_1 = -⟨ S_1⟩ .

The calculation of Γ_1 thus amounts to evaluating ⟨ S_1⟩, which is a collection of two-point correlation functions of fields. These correlations are just the flat spacetime propagators of fields and can be derived from S_0 using projection operator method. We obtained the expansions of S_GH using xAct packages <cit.> for Mathematica, results of which are presented below:

    S_0    =   ∫ d^4x (- 12α^2 δ C_μδ C^μ -  12α^2 δϕ^2 -  14α^2 (δ B^μν b_μν)^2 
           -  12 (b_α^ν b_βγ b_μν b_ρσδ B^βγ^,αδ B^ρσ^,μ) -  12 (δ C_μ^,μ)^2 
           -  14(b_μν (δ C^ν^,μ -  δ C^μ^,ν))^2 -  12δϕ_,μδϕ^,μ
           + δc̅^μ (- α^2 δ c_μ + δ c^ν_,ν_,μ + 2 b_νμ b_ρσδ c^σ^,ρ^,ν) 
           + 112 (- δ B_νρ_,μ -  δ B_ρμ_,ν -  δ B_μν_,ρ) (δ B^νρ^,μ + δ B^ρμ^,ν + δ B^μν^,ρ) )


    S_1    =   ∫ d^4x ( 12α^2 δ C^μδ C^ν h_μν -  α^2 δ c^μδc̅^ν h_μν -  14α^2 δϕ^2 h^μ_μ
           -  14α^2 δ C_μδ C^μ h^ν_ν -  12α^2 δ c^μδc̅_μ h^ν_ν + 12α^2 δ B^μνδ B^ρσ h_ρ a b^a_σ n_μν
           + 12α^2 δ B^μνδ B^ρσ h_σ a n_μν n_ρ^a -  18α^2 δ B^μνδ B^ρσ h^a_a n_μν n_ρσ + δc̅^μ b^ν_μ b^ρσ h_νρ_,aδ c_σ^,a
           - 2 δc̅^μ h_ν a b^ν_μ b^ρσδ c_σ_,ρ^,a + h_ρ d b^bc b^μν b^ρ_ν b^σ aδ B_bc^,dδ B_σ a_,μ + 12 h_ρ a b^μν b^ρσδ C_σ^,aδ C_ν_,μ
           + 12δc̅^μ h^ρ_ρδ c^ν_,ν_,μ -  14 h^ν_νδϕ_,μδϕ^,μ + 12 h_σ a b^μν b^ρσδ C_ρ^,aδ C_μ_,ν
           -  12 h_ρ a b^μν b^ρσδ C_σ^,aδ C_μ_,ν -  14 h^ρ_ρδ C^μ_,μδ C^ν_,ν -  12δ C^μ h^ρ_ρ_,μδ C^ν_,ν
           -  12δ C^μ b^νρ b^σ a h_μσ_,aδ C_ρ_,ν + δc̅^μ b^ν_μ b^ρσ h_ρσ_,aδ c^a_,ν -  δc̅^μ b^ν_μ b^ρσδ c_σ^,a h_ρ a_,ν
           + 12δc̅^μδ c^ν_,μ h^ρ_ρ_,ν - 2 δc̅^μ h_ρ a b^ν_μ b^ρσδ c_σ^,a_,ν + 12δ c^μδc̅^ν h^ρ_ρ_,μ_,ν
           + δc̅^μ h^a_a b^ν_μ b^ρσδ c_σ_,ρ_,ν + 12 h_μνδϕ^,μδϕ^,ν -  14 h^d_d b^bc b^μν b^ρ_ν b^σ aδ B_σ a_,μδ B_bc_,ρ
           + h_cd b^bc b^μν b^ρ_ν b^σ aδ B_σ a_,μδ B_b^d_,ρ + h_bd b^bc b^μν b^ρ_ν b^σ aδ B_σ a_,μδ B^d_c_,ρ + 12 h_σ d b^ab b^cd b^μν b^ρσδ B_ab_,cδ B_μν_,ρ
           + 12 h_σ a b^μν b^ρσδ C_ν_,μδ C^a_,ρ + 12δ C^μ b^νρ b^σ a h_μσ_,aδ C_ν_,ρ -  12 h_ν a b^μν b^ρσδ C_μ^,aδ C_σ_,ρ
           -  18 h^a_a b^μν b^ρσδ C_ν_,μδ C_σ_,ρ + 14 h^a_a b^μν b^ρσδ C_μ_,νδ C_σ_,ρ + δc̅^μ b^ν_μ b^ρσ h_νσ_,aδ c^a_,ρ
           + δc̅^μ b^ν_μ b^ρσ h_σ a_,νδ c^a_,ρ + δ C^μδ C^ν_,ν h_μ^ρ_,ρ + 14δ B^μνδ B^ρσ_,ν h_μσ_,ρ
           -  δc̅^μ b^ν_μ b^ρσδ c_σ^,a h_ν a_,ρ + δc̅^μ b^ν_μ b^ρσδ c^a_,ν h_σ a_,ρ -  δ c^μδc̅^ν b^ρ_ν b^σ a h_μσ_,a_,ρ
           + δ c^μδc̅^ν b^ρ_ν b^σ a h_σ a_,μ_,ρ + δ c^μδc̅^ν b^ρ_ν b^σ a h_μ a_,σ_,ρ -  18 g_μσ g_ν a g_ρ b h^c_cδ B^σ a^,bδ B^μν^,ρ
           -  14 g_μ a g_ν b g_ρσ h^c_cδ B^σ a^,bδ B^μν^,ρ + 14 g_ν a g_ρ b h_μσδ B^σ a^,bδ B^μν^,ρ + 14 g_μσ g_ρ b h_ν aδ B^σ a^,bδ B^μν^,ρ
           + 12 g_μ a g_ρσ h_ν bδ B^σ a^,bδ B^μν^,ρ + 12 g_μ b g_ρ a h_νσδ B^σ a^,bδ B^μν^,ρ + 14 g_μσ g_ν a h_ρ bδ B^σ a^,bδ B^μν^,ρ
           + 12 g_μ a g_ν b h_ρσδ B^σ a^,bδ B^μν^,ρ + h_νρδ C^μ_,μδ C^ν^,ρ + 12δ B^μν b^b_a b^cd b^ρ_ν b^σ a h_μρ_,bδ B_cd_,σ
           + 12δ B^μν b^b_a b^cd n_μ^ρ b^σ a h_νρ_,bδ B_cd_,σ -  12δ B^μν b^b_a b^cd b^ρ_ν b^σ a h_ρ b_,μδ B_cd_,σ -  12δ B^μν b^b_a b^cd n_μ^ρ b^σ a h_ρ b_,νδ B_cd_,σ
           + 12δ B^μν b^b_a b^cd b^ρ_ν b^σ a h_μ b_,ρδ B_cd_,σ + 12δ B^μν b^b_a b^cd n_μ^ρ b^σ a h_ν b_,ρδ B_cd_,σ -  12 h_ρ a b^μν b^ρσδ C_ν_,μδ C^a_,σ
           -  12 h_ν a b^μν b^ρσδ C^a_,μδ C_ρ_,σ + 12 h_μ a b^μν b^ρσδ C^a_,νδ C_ρ_,σ -  18 h^a_a b^μν b^ρσδ C_μ_,νδ C_ρ_,σ
           + 12δ C^μ b^νρ b^σ aδ C_ρ_,ν h_μ a_,σ -  12δ C^μ b^νρ b^σ aδ C_ν_,ρ h_μ a_,σ -  δc̅^μ b^ν_μ b^ρσδ c^a_,ρ h_ν a_,σ
           + 14δ B^μνδ B^ρσ_,μ h_νρ_,σ -  δc̅^μ b^ν_μ b^ρσδ c^a_,ν h_ρ a_,σ + 14δ B^μν h_νρ_,σδ B_μ^ρ^,σ
           + 14δ B^μν h_μσ_,ρδ B_ν^ρ^,σ + 14δ B^μν h_νσ_,ρδ B^ρ_μ^,σ + 14δ B^μν h_μρ_,σδ B^ρ_ν^,σ)




 §.§ Propagators

We use the projection operator method <cit.> to invert the operators in S_0 and derive the Green's functions or propagators. In the operator form, S_0 can be recast as

    S_0 =∫ d^4x (  14 B^μν𝒪^B_μν,αβB^αβ + 12 C^μ𝒪^C_μνC^ν + 1/2Φ𝒪^ΦΦ)

where,

    𝒪^B_μν,αβ   =   /2(η_μαη_νβ-η_μβη_να)  + 1/2(∂_μ∂_βη_να-∂_ν∂_βη_μα-∂_μ∂_αη_νβ+∂_ν∂_αη_μβ)
           - (α^2 + 2 (b_ρσ∂^ρ)^2) b_μνb_αβ,
    𝒪^C_μν   =    2 b_σμb_ρν∂^σ∂^ρ + ∂_μ∂_ν - α^2η_μν,
    𝒪^Φ   =    - α^2.

At this point, we would like to point out that a calculation for the propagator of B_μν using projector method was first performed in Ref. <cit.> recently. However, their calculation did not account for the Stückelberg field and as a result our operator (𝒪^B)_μν,αβ is different from the one in Ref. <cit.>, which misses the contribution from gauge-fixing term  2 (b_ρσ∂^ρ)^2 b_μνb_αβ. Fortunately, this term is merely an addition to mass, α^2, and ends up not contributing to the propagator, (𝒪^B)^-1_μν,αβ. So, we end up getting an identical result for the propagator, barring complex infinity terms that can be ignored (see appendix for details of projection operators P^(1),...,P^(6)),

    (𝒪^B)^-1_μν,αβ (x,x')=∫d^4p(2π^n)e^-ip·(x-x')(1/p^2 P^(1)_μν, αβ + b^2/(b_ρσp^σ)^2( P^(4)_μν, αβ+ P^(5)_μν, αβ)),

There are no massive propagating modes in Eq. (31) and only one massless mode propagates, as concluded in Ref. <cit.>. The second pole describes a massless pole propagating in an anisotropic medium, which for our choice of b_μν gives,

    b^2((p^2)^2 + (p^3)^2) = 0.

Contrary to the claim in Ref. <cit.> where these modes were described as non-physical due to a negative sign appearing in energy-momentum relations as a result of a different choice of b_μν, we note that for our choice of b_μν which corresponds to monopole solutions of antisymmetric tensor, energy terms (p^0) disappear altogether. 

For the Stückelberg field C_μ, spontaneous Lorentz violating term appears in the kinetic part (first term in Eq. <ref>), which makes inverting 𝒪^C_μν a little tricky. New projector operators have to be defined apart from the longitudinal and transverse momentum operators, that also have a closed algebra, so that any operator D_μν can be then expanded in terms of these projectors. We define,

    𝒫^(1)_μν = p_μp_νp^2;   𝒫^(2)_μν = η_μν - p_μp_νp^2; 
    𝒫^(3)_μν = 1(b_ρσp^σ)^2 b_σμb_ρνp^σp^ρ .

These operators satisfy a closed algebra, as shown in TABLE <ref>.

Using these operators, 𝒪^C_μν in momentum space can be written as,

    𝒪^C_μν = - 2(b_ρσp^σ)^2 𝒫^(3)_μν - (p^2 + α^2 ) 𝒫^(1)_μν - α^2 𝒫^(2)_μν.

Assuming that (𝒪^C)^-1_μν in momentum space has the form,

    (𝒪^C)^-1_μν = m_1𝒫^(1)_μν + m_2𝒫^(2)_μν + m_3𝒫^(3)_μν,

we use the identity 𝒪𝒪^-1 = ℐ to obtain,

    (𝒪^C)^-1_μν(x,x') = ∫d^4p(2π^n)e^-ip·(x-x')(-1p^2 + α^2𝒫^(1)_μν - 1α^2𝒫^(2)_μν + 1α^2(b_ρσp^σ)^2(b_ρσp^σ)^2 + α^2/2𝒫^(3)_μν).

Here, a massive scalar mode with pole at α propagates while another anisotropic mode propagates with mass α/√(2). Terms in 𝒫^(2) contain a massless pole and an additive poleless term which does not contribute to correlations and can be ignored. For Φ, the scalar propagator is given by,

    (𝒪^Φ)^-1 (x,x') = ∫d^4p(2π^n)e^-ip·(x-x')1p^2 + α^2.




 §.§ Quantum corrections

Since all terms in ⟨ S_1⟩ are local, they correspond to tadpole diagrams. We solve these integrals in two steps: first, the derivatives of field fluctuations are transformed to momentum space by substituting Eqs. (<ref>), (<ref>), and (<ref>). We also perform by-parts integrals to get rid of derivatives of h_μν, so that in all expressions below, a coefficient h_μν is understood to be present but not explicitly written. The Fourier transformed ⟨ S_1⟩ then has terms of the form,

    ∫ d^4x A(x) ⟨∂^mδ ∂^nδ⟩⟶∫ d^4x d^4p(2π^n) A(x) (-ip)^m (ip)^n⟨δ_pδ_p⟩,

where, tensor indices of A(x) and δ have been omitted for convenience. δ is the quantum field fluctuation, and ⟨δ_pδ_p⟩ represents the propagator(s) in momentum space. 

The second step is to replace ⟨δ_pδ_p⟩ with values of Green's function and evaluate the integrals. We primarily use the results in Ref. <cit.> to evaluate the divergent terms of most of the integrals, except those involving anisotropic term (b_ρσp^σ)^2. There are two types of poleless integrals coming from Eq. (<ref>):

    ∫ d^4x d^4p(2π^n) A(x)p^μ...p^βp^2 ;   ∫ d^4x d^4p(2π^n) A(x)p^μ...p^β(b_ρσp^σ)^2,

with upto four p^μ's in the numerator. The first integral vanishes due to the lack of a physical scale <cit.>. To solve the second integral, we use the approach developed in <cit.>, and find that it also does not have any physical contribution. 

Next, there are broadly three types of integrals with non-zero poles arising from the rest of propagators:

    ∫ d^4x d^4p(2π^n) A(x)p^μ...p^βp^2 + α^2 ;   ∫ d^4x d^4p(2π^n) A(x)p^μ...p^βp^2(p^2 + α^2);   
       ∫ d^4x d^4p(2π^n) A(x)p^μ...p^β(b_ρσp^σ)^2 + α^2/2;

Again, the solutions to first two types of integrals are available in Ref. <cit.>. We solve the third type of integral as follows. Following <cit.>, we write

    ∫ d^4p 1(b_ρσp^σ)^2 + α^2/2 = ∫ d^4p ∫_0^∞ dθexp[-θ((b_ρσp^σ)^2 + α^2/2)].

Integrating over d^4p, followed by writing the integral over θ in terms of Γ function leads to familiar expressions encountered in dimensional regularization, which finally yields the divergent part as (ϵ = n-4),

    divp(∫ d^4p 1(b_ρσp^σ)^2 + α^2/2) = -π^2α^22√((b_μρb^ρν))2ϵ,
 
which is identical to that of a scalar propagator integral except for the √((b_μρb^ρν)) in the denominator. For our choice of b_μν, Eq. (<ref>) with a=0 and b=1/√(2), this term becomes a diagonal matrix,

    b_μρb^ρν = diag(0   0   1/2   1/2),

implying that the determinant is zero. It turns out however, that this determinant appears as a factor in the denominator of the divergent part of effective action, and hence we use a regularization factor ϵ' to write,
 
    b_μρb^ρν = lim_ϵ'→ 0 diag(ϵ'   ϵ'   1/2   1/2).


With these inputs in xAct<cit.>, the final result for the divergent part of 1-loop effective after some further manipulations, is obtained as,

    divp(Γ_1)    =   116π^2ϵ( α^4 κ h^a _a   + 1√((b_μρb^ρν))( -  116α^4 κ h_μ b b_a^b b^μ a -  332α^4 κ h^b_b b_aμ b^μ a
           -  112α^4 κ h_μ c b_a^b b_b^ν b^c_ν b^μ a -  5192α^4 κ h_bc b_aμ b^bν b^c_ν b^μ a + 1192α^4 κ h_ν c b_b^c b^bν b_μ a b^μ a
           + 196α^4 κ h_bc b^bν b^c_ν b_μ a b^μ a -  1384α^4 κ h^c_c b_a^ν b_bν b_μ^b b^μ a -  148α^4 κ h_ν c b_a^ν b^c_b b_μ^b b^μ a
           + 5192α^4 κ h_bc b_a^ν b^c_ν b_μ^b b^μ a -  1384α^4 κ h^c_c b_aμ b^bν b^μ a b_ν b + 5384α^4 κ h^c_c b^bν b_μ a b^μ a b_ν b
           + 5192α^4 κ h^c_c b_a^ν b_μ^b b^μ a b_ν b + 1192α^4 κ h_μ c b_a^b b_b^ν b^μ a b_ν^c + 1192α^4 κ h_bc b_aμ b^bν b^μ a b_ν^c
           -  7192α^4 κ h_bc b^bν b_μ a b^μ a b_ν^c -  196α^4 κ h_bc b_a^ν b_μ^b b^μ a b_ν^c -  1384α^4 κ h^c_c b_a^b b_b^ν b^μ a b_νμ
           -  196α^4 κ h_bc b^c_ν b_μ^b b^μ a b^ν_a + 196α^4 κ h_μ c b_a^b b^c_ν b^μ a b^ν_b))

where ϵ = n-4 (as n→ 4) is the divergence parameter from dimensional regularization. Eq. (<ref>) presents the divergent piece of one loop corrections of antisymmetric tensor field theory with spontaneous Lorentz violation at leading order in field fluctuations in a nearly flat spacetime, and is valid for a vacuum value that supports monopole solutions. The one-loop divergence structures in principle lead to corrections to parameters (or couplings) in the classical action through counterterms (for example, in Ref. <cit.>). Studying such corrections is interesting at higher orders in background fields, but lie beyond the scope of present work. Also, it is not easy to compare theories with and without Lorentz violation in the present context, because the simplest Lorentz violating potential contains upto quartic order terms in fields; while without Lorentz violation, the potential(s) that have been studied in the past <cit.> are quadratic in field components.




§ QUANTUM EQUIVALENCE

The classical Lagrangian (<ref>) can be written in an equivalent form where the field B_μν can be eliminated through the introduction of a vector field, so that the resulting Lagrangian describes a classically equivalent vector theory with spontaneous Lorentz violation. In this section, we will check their quantum equivalence at one-loop level. 

Checking classical equivalence of two theories is an interesting theoretical exercise, because it provides insight into the degrees of freedom and dynamical properties of theories that may be described by very different fields, like in 2-form, 1-form or a scalar field theories, and thus may lead to several simplifications in a given theory. This problem naturally extends to the quantum regime, and it is certainly not trivial to prove quantum equivalence of two classically equivalent theories especially in curved spacetime. For instance, it can be shown that a massive 2-form field is quantum equivalent to a massive vector field because of some special topological properties of zeta functions <cit.>. However, it is extremely difficult to perform similar analyses when, for example, the Lorentz symmetry is broken <cit.>. In flat spacetime, establishing quantum equivalence is indeed trivial, because there is no field dependence in Γ_1 (Eq. (<ref>)) and hence effective actions of two theories do not possess any physical distinction.

On the contrary, in curved spacetime, the presence of metric makes things interesting. Only problem is, the effective action cannot be calculated exactly. So, our best bet, in this case, is to do a perturbative study like the one in the previous section. 

Classical equivalence of Eq. (<ref>) was explored in Ref. <cit.>, it was found to be equivalent to,

    ℒ = 12B_μνℱ^μν - 1/2C^μC_μ - 14α^2(b_μνB^μν)^2,

where,

    ℱ_μν = 12ϵ_μνρσF^μν.

C_μ is a vector field and F_μν is as defined before. We choose to continue with the same symbol for vector and Stückelberg field to avoid unnecessary complications. Eq. (<ref>) can be written exclusively in terms of C_μ through the use of projection operators,

    T_||μν = b_ρσT^ρσ b_μν, 
    
    T_⊥μν = T_μν - T_||μν,

for any two-rank tensor T_μν, and subsequently using the equations of motion for B_||μν and B_⊥μν, to obtain,

    α^2ℒ = 14(b̃_μνF^μν)^2 - 1/2α^2 C^μC_μ,

where we have defined b̃_μν = 12ϵ_μνρσb^ρσ. Note that Lorentz violation enters Eq. (<ref>) through the kinetic term, although it is still gauge-symmetric. A similar exercise of applying Stückelberg procedure leads to the gauge fixed action in flat spacetime,

    S̃_0 = ∫ d^4x ( 12 C^μ𝒪^C'_μνC^ν + 1/2Φ𝒪^ΦΦ)

where, 

    𝒪^C'_μν = -12(b̃_σμb̃_ρν∂^σ∂^ρ + b̃_σνb̃_ρμ∂^σ∂^ρ) + 12∂_μ∂_ν - 12α^2η_μν.

A similar calculation of the propagator yields,

    (𝒪^C')^-1_μν(x,x') = ∫d^4p(2π^n)e^-ip·(x-x')(-1p^2 + α^2𝒫^(1)_μν - 1α^2𝒫^(2)_μν - 1α^2(b̃_ρσp^σ)^2-(b̃_ρσp^σ)^2 + α^2/2𝒫̃^(3)_μν),

where 𝒫̃^(3)_μν has the same form as 𝒫^3_μν but with b̃_μν instead of b_μν. Finally, the one-loop effective action is found to be,

    divp(Γ̃_1)    =   116π^2ϵ( α^4 κ h^a _a   + 1√(-(b̃_μρb̃^ρν))(196α^4 κ h_ecb̃_adb̃^adb̃_b^cb̃^be + 132α^4 κ h^b_bb̃^acb̃_ca
           -  116α^4 κ h_abb̃^acb̃_c^b + 1192α^4 κ h_bcb̃_adb̃^adb̃^beb̃^c_e -  196α^4 κ h_acb̃^adb̃_b^eb̃^c_eb̃_d^b
           + 196α^4 κ h_ecb̃_a^bb̃^adb̃_b^cb̃_d^e -  1384α^4 κ h^c_cb̃_a^bb̃^adb̃_beb̃_d^e -  196α^4 κ h_ecb̃_a^bb̃^adb̃^c_bb̃_d^e
           + 5192α^4 κ h_bcb̃_a^bb̃^adb̃^c_eb̃_d^e -  1384α^4 κ h^c_cb̃^adb̃_b^eb̃_d^bb̃_ea + 1384α^4 κ h^c_cb̃_adb̃^adb̃^beb̃_eb
           -  1384α^4 κ h^c_cb̃^adb̃^beb̃_dab̃_eb + 1192α^4 κ h^c_cb̃_a^bb̃^adb̃_d^eb̃_eb -  164α^4 κ h_bcb̃_adb̃^adb̃^beb̃_e^c
           + 1192α^4 κ h_bcb̃^adb̃^beb̃_dab̃_e^c + 1192α^4 κ h_acb̃^adb̃_b^eb̃_d^bb̃_e^c -  148α^4 κ h_bcb̃_a^bb̃^adb̃_d^eb̃_e^c
           -  1192α^4 κ h_dbb̃_a^bb̃^adb̃_ceb̃^ec))

Upon comparing Eqs. (<ref>) and (<ref>), we can immediately notice that the first term is identical, while the rest of terms appearing with b_μν and b̃_μν do not match. The first term arises from the propagator of non-Lorentz violating modes, while all the other terms correspond to contributions from propagator of Lorentz violating modes. Hence, the quantum equivalence holds along non-Lorentz violating modes but not along Lorentz violating modes involving b_μν. This conclusion is validated by the results of Ref. <cit.>, where it was shown that when there are topologically nontrivial monopole-like solutions of the spontaneous symmetry breaking equations, the interaction with gravity of the vector and tensor theories are different.






§ CONCLUSION

Study of spontaneous Lorentz violation with rank-2 antisymmetric tensor is interesting because of the possibility of rich phenomenological signals of SME in future experiments. Since antisymmetric tensor fields are likely to play s significant role in the early universe cosmology, studying their quantum aspect is a natural extension of classical analyses. In a past study <cit.>, it was found that issues like quantum equivalence are difficult to address in a general curved spacetime. In the present work, this problem is overcome by adopting a perturbative approach to evaluating effective action, that is also general enough to be applied to more complicated models including interaction terms.

We quantized a simple action of an antisymmetric tensor field with a nonzero vev driving potential term that introduces spontaneous Lorentz violation, using a covariant effective action approach at one-loop. The one-loop corrections were calculated in a nearly flat spacetime, at O(κħ). We revisited the issue of quantum equivalence, and found that for the non-Lorentz-violating modes (independent of vev b_μν), antisymmetric tensor field is quantum-equivalent to a vector field. However, contributions from the Lorentz violating part of the propagator leads to different terms in effective actions, and as a result, ΔΓ = Γ_1 - Γ̃_1 ≠ 0, i.e. the theories are not quantum equivalent.





 This work was partially funded by DST (Govt. of India), Grant No. SERB/PHY/2017041. 


*



§ PROJECTION OPERATORS FOR B_ΜΝ

The basic projection operators for an antisymmetric tensor are defined as <cit.>,

    [                               P^(1)_μν,αβ                                         =                   1/2(θ_μαθ_νβ-θ_μβθ_να),;                               P^(2)_μν,αβ                                         = 1/4(θ_μαω_νβ-θ_ναω_μβ-θ_μβω_να+θ_νβω_μα), ]

where,

    θ_μν=η_μν-ω_μν,  ω_μν=∂_μ∂_ν/,

are the longitudinal and transverse projection operators along the momentum. To account for the Lorentz violation induced by nonzero vev, four new operators need to be introduced as follows <cit.>:

    P^(3)_μν,αβ   =    P^⊥_μν, αβ,
    
    P^(4)_μν,αβ   =   1/2( ω_μλ P^∥_νλ, αβ - ω_νλ  P^∥_μλ, αβ),
    
    P^(5)_μν,αβ   =   1/2( ω_αλ P^∥_μν, βλ - ω_βλ  P^∥_μν, αλ),
    
    P^(6)_μν,αβ   =   1/4( ω_μα P^∥_νρ, βσ ω^ρσ - ω_να  P^∥_μρ, βσ ω^ρσ. 
          - .ω_μβ  P^∥_νρ, ασ ω^ρσ + ω_νβ  P^∥_μρ, ασ ω^ρσ).

The operators P^(1)_μν,αβ, ⋯ , P^(6)_μν,αβ obey a closed algebra <cit.>. 

The identity element is given by,

    ℐ_μν, αβ = 1/2(η_μαη_νβ-η_μβη_να)=[P^(1)+P^(2)]_μν,αβ.








