# Exact Lorentz-violating all-loop ultraviolet divergences in scalar field theories

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## Abstract

In this work we evaluate analytically the ultraviolet divergences of Lorentz-violating massive O(*N*) \(\lambda \phi ^{4}\) scalar field theories, which are exact in the Lorentz-violating mechanism, firstly explicitly at next-to-leading order and latter at any loop level through an induction procedure based on a theorem following from the exact approach, for computing the corresponding critical exponents. For attaining that goal, we employ three different and independent field-theoretic renormalization group methods. The results found for the critical exponents show that they are identical in the three distinct methods and equal to their Lorentz-invariant counterparts. Furthermore, we show that the results obtained here, based on the single concept of loop order of the referred terms of the corresponding \(\beta \)-function and anomalous dimensions, reduce to the ones obtained through the earlier non-exact approach based on a joint redefinition of the field and coupling constant of the theory, in the appropriate limit.

## 1 Introduction

Lorentz symmetry is one of the most fundamental symmetries of nature and the possibility of its violation was a theme of intense investigation in the last years, usually as a finite perturbative expansion at some Lorentz-violating (LV) parameters and loop number, both in high energy [1, 2, 3, 4, 5, 6, 7, 8] as well as in low energy [9, 10, 11] physics. In the latter realm, the critical exponents were computed, at least at first order in the Lorentz-violating (LV) parameters \(K_{\mu \nu }\) and any loop level for LV scalar field theories [9, 10, 11]. For this purpose, this evaluation was possible by means of the application of a non-exact approach based on a joint redefinition of the field and coupling constant of the theory. In this work, we present an exact approach, which naturally takes into account the effect of the LV parameters exactly and furthermore for all loop orders. Moreover, we will show that the referred exact approach gives expressions for the \(\beta \)-function as well as for the corresponding fixed point and anomalous dimensions, besides critical exponents, and that these expressions reduce to the ones obtained in the earlier non-exact approach in the appropriate limit.

In this work, we compute analytically the critical exponents for massive O(*N*) \(\lambda \phi ^{4}\) scalar field theories with Lorentz violation. This computation is exact in the LV mechanism. For this purpose, we apply three distinct field-theoretic renormalization group methods and they involve the same theory renormalized at different renormalization schemes. In this field-theoretic formulation, if the critical exponents present the same values when obtained through the three methods, this means that they are universal quantities and we have the confirmation of the universality hypothesis. These universal quantities characterize the critical behavior of distinct systems as a fluid and a ferromagnet. When the critical behavior of two or more distinct systems is characterized by the same critical exponents, we say that they belong to the same universality class. The universality class inspected here is the O(*N*) one, which encompasses the particular models: Ising (\(N=1\)), XY (\(N=2\)), Heisenberg (\(N=3\)), self-avoiding random walk (\(N=0\)) and spherical (\(N \rightarrow \infty \)) for short-range interactions [12]. The critical exponents depend on the dimension *d* of the system, *N* and symmetry of some *N*-component order parameter (magnetization for magnetic systems), and on whether the interactions present are of short- or long-range type. Much work probing the dependence of the critical exponents on the obvious parameters as *d* [13, 14] and *N* [15, 16, 17] was published. Just a few publications addressed the symmetry of the order parameter [18, 19]. The aim of this work is to probe the exact effect of the LV mechanism on the values for the critical exponents.

This paper is organized as follows: In next three sections, we compute analytically and explicitly the next-to-leading loop order quantum corrections to the critical exponents for LV O(*N*) self-interacting \(\lambda \phi ^{4}\) scalar field taking into account the LV mechanism exactly, by applying three distinct field-theoretic renormalization group methods. In Sect. 5 we generalize the results for all loop levels. At the end, we present our conclusions.

## 2 Exact Lorentz-violating next-to-leading order critical exponents in the Callan–Symanzik method

*N*) scalar field theory whose bare Lagrangian density in Euclidean spacetime is given by [6, 7, 8]

*u*as \(\lambda = u m^{\epsilon }\), where

*m*, at the loop level considered, is used as an arbitrary momentum scale, thus we can consider the momenta as dimensionless quantities. The same relation between the corresponding bare quantities \(\lambda _{B}\) and \(u_{0}\) can be also defined as \(\lambda _{B} = u_{0}m^{\epsilon }\). We renormalize these correlation functions multiplicatively

*N*) symmetry factors, we have \(\propto \) , \(\propto \) , \(\propto \) \(\propto \) . Finally, the only diagrams to be evaluated are the ones. Thus we can write the 1PI vertex parts as

*l*such insertions. As is well known, an extra composite field insertion is responsible for one additional power of the propagator in the corresponding 1PI vertex part. We can then work in the ultraviolet limit,

*i.e.*, in the limit where the external momenta \(P_{i}/m \rightarrow \infty \). After taking this limit, the rhs can be neglected in comparison with the lhs, order by order in perturbation theory. This is, in essence, the content of Weinberg’s theorem [22]. Therefore the 1PI vertex parts satisfy the renormalization group equation, thus permitting us to apply the theory of scaling for these functions and evaluate the \(\beta \)-function and anomalous dimensions as well as the corresponding critical exponents. The LV coefficients can now be considered exactly by noting that \(q^{2} + K_{\mu \nu }q^{\mu }q^{\nu } \equiv (\delta _{\mu \nu } + K_{\mu \nu })q^{\mu }q^{\nu }\) = \(q^{t}(\mathbb {I}\) + \(\mathbb {K})q\), where

*q*is a

*d*-dimensional vector whose representation is a column matrix and \(q^{t}\) is a row matrix and \(\mathbb {I}\) and \(\mathbb {K}\) are matrix representations of the identity and \(K_{\mu \nu }\), respectively. Thus setting \(q^{\prime } = \sqrt{\mathbb {I} + \mathbb {K}}q\), the LV mechanism is shown explicitly through two contributions. The first of them is displayed through the volume elements of

*d*-dimensional integrals \(d^{d}q^{\prime } = \sqrt{\mathrm{det}(\mathbb {I} + \mathbb {K})}d^{d}q\); thus \(d^{d}q = d^{d}q^{\prime }/\sqrt{\mathrm{det}(\mathbb {I} + \mathbb {K})}\). This LV full or exact contribution \(\varvec{\Pi } = 1/\sqrt{\mathrm{det}(\mathbb {I} + \mathbb {K})}\) reduces to its perturbative counterpart \(\varPi \simeq \varPi ^{(0)} + \varPi ^{(1)} + \varPi ^{(2)}\) for small violations of the Lorentz symmetry, where \(\varPi ^{(i)}\) is the LV contribution of order

*i*in \(K_{\mu \nu }\) [6, 7, 8, 9, 10, 11]. The other LV modification of the theory is that involving the external momenta. It can be seen in the momentum-dependent

*d*-dimensional integrals when evaluated in dimensional regularization in \(d = 4 - \epsilon \) that

*d*-dimensional sphere. Its finite value in four-dimensional spacetime is \(\hat{S}_{4}=2/(4\pi )^{2}\). This definition is convenient as to each loop integration we have a factor of \(\hat{S}_{4}\) at four dimensions, thus avoiding the appearance of Euler–Mascheroni constants in the middle of the calculations [20]. Now setting \(q^{\prime } \rightarrow P^{\prime }\) and \(q \rightarrow P\), \(P^{\prime 2} = P^{2} + K_{\mu \nu }P^{\mu }P^{\nu }\). As is well known, from all diagrams displayed above, we need to compute only four of them [20]. They are shown in Appendix A. When we absorb \(\hat{S}\) in a redefinition of the coupling constant and use the Feynman diagrams for computing the \(\beta \)-function and anomalous dimensions by writing the Laurent expansion

*u*. This term is fundamental for making possible expansions in quantum field theory and is essential in the renormalization group and \(\epsilon \)-expansion techniques developed by Wilson, specially with applications to critical phenomena [23, 24, 25] in \(d < 4\). Its second one-loop term is of second order in

*u*, but it has acquired only a linear power of \(\varvec{\Pi }\). The last one, although being of third order in

*u*, must be of second order in \(\varvec{\Pi }\), since it is of two-loop order. Similar arguments can be utilized to the other terms of the anomalous dimensions of Eqs. (36) and (37) as well. Thus, the exact approach permit us to see that each loop term is accompanied of a power of the LV full \(\varvec{\Pi }\) factor as it is shown by the general theorem displayed in last section. This procedure is valid at all intermediate steps of the program. Another interesting point is that, in this method, the \(\beta \)-function and anomalous dimensions depend on the LV coefficients at its exact form only through the LV \(\varvec{\Pi }\) factor and on the symmetry point employed. We need to compute the nontrivial solution of the \(\beta \)-function. The trivial one leads to the mean field or Landau critical exponents and can be obtained mathematically by a factorization procedure resulting in the factorization of a single power of

*u*in the equation for the \(\beta \)-function. This procedure results in the nontrivial fixed point given by

## 3 Exact Lorentz-violating next-to-leading order critical exponents in the Unconventional minimal subtraction scheme

## 4 Exact Lorentz-violating next-to-leading order critical exponents in the BPHZ method

*i*-th loop order renormalization constants for the field, renormalized coupling constant and composite field, respectively. They are given by where

*S*is the symmetry factor for the corresponding diagram and so on when some

*N*-component field is considered. By using the diagrams in Appendix C, we see that the \(\beta \)-function and anomalous dimensions are given by

## 5 Exact Lorentz-violating all-loop order critical exponents

For computing the critical exponents for all loop levels, we can employ any of the methods aforementioned since the critical exponents, being universal quantities, must be the same if evaluated at any renormalization scheme. For this purpose, we will employ the BPHZ method which is the most general one. Before that, we need to assert the following theorem.

### Theorem 1

Consider a given Feynman diagram in momentum space of any loop order in a theory represented by the Lagrangian density of Eq. (1). Its evaluated expression in dimensional regularization in \(d = 4 - \epsilon \) can be written as a general functional \(\varvec{\Pi }^{L}\mathcal {F}(u,P^{2} + K_{\mu \nu }P^{\mu }P^{\nu },\epsilon ,\mu )\) if its LI counterpart is given by \(\mathcal {F}(u,P^{2},\epsilon ,\mu ,m)\), where *L* is the number of loops of the corresponding diagram.

### Proof

A general Feynman diagram of loop level *L* is a multidimensional integral in *L* distinct and independent momentum integration variables \(q_{1}\), \(q_{2}\), ..., \(q_{L}\), each one with volume element given by \(d^{d}q_{i}\) (\(i = 1, 2,...,L\)). As showed in the last section, the substitution \(q^{\prime } = \sqrt{\mathbb {I} + \mathbb {K}}q\) transforms each volume element: \(d^{d}q^{\prime } = \sqrt{det(\mathbb {I} + \mathbb {K})}d^{d}q\). Thus \(d^{d}q = d^{d}q^{\prime }/\sqrt{det(\mathbb {I} + \mathbb {K})} \equiv \varvec{\Pi }d^{d}q^{\prime }\), \(\varvec{\Pi } = 1/\sqrt{det(\mathbb {I} + \mathbb {K})}\). Then the integration in *L* variables results in a LV overall factor of \(\varvec{\Pi }^{L}\). Now setting \(q^{\prime } \rightarrow P^{\prime }\) in the substitution above, where \(P^{\prime }\) is the transformed external momentum, we have \(P^{\prime 2} = P^{2} + K_{\mu \nu }P^{\mu }P^{\nu }\). So a given Feynman diagram, evaluated in dimensional regularization in \(d = 4 - \epsilon \), assumes the expression \(\varvec{\Pi }^{L}\mathcal {F}(u,P^{2} + K_{\mu \nu }P^{\mu }P^{\nu },\epsilon ,\mu )\), where \(\mathcal {F}\) is associated to the corresponding diagram if the LI Feynman diagram counterpart evaluation results in \(\mathcal {F}(u,P^{2},\epsilon ,\mu )\).

*L*is the number of loops of the corresponding graph. Thus we can write the \(\beta \)-function and anomalous dimensions for all loop levels

## 6 Conclusions

We have evaluated analytically the ultraviolet divergences of Lorentz-violating massive O(*N*) \(\lambda \phi ^{4}\) scalar field theories, which are exact in the Lorentz-violating mechanism, firstly explicitly at next-to-leading order and later at any loop level through an induction procedure based on a theorem following from the exact approach, for computing the corresponding critical exponents. For this purpose, we have employed three different and independent field-theoretic renormalization group methods. We have found equal critical exponents in the three methods and these are furthermore identical to their Lorentz-invariant counterparts. We have also showed that the exact approach, which reduces to the non-exact one in its limited range of applicability, besides being exact, is capable of furnishing the expressions for the all-loop LV radiative quantum corrections to the \(\beta \)-function and anomalous dimensions considering just a single concept, that of the loop number of the corresponding terms of these functions. Furthermore, the present exact approach, when applied to the referred theory, is the first one in the literature to the best of our knowledge. Thus it can inspire the exact solution of problems involving considering the exact effect of LV mechanisms in many physical phenomena ranging from high (standard model extension for example) to low energy physics (corrections to scaling, finite-size scaling, amplitude ratios, critical exponents in geometries subject to different boundary conditions, Lifshitz points etc. [29, 30, 31, 32].

## Notes

### Acknowledgements

With great pleasure the authors thank the kind referee for helpful comments. PRSC and MISJ would like to thank Federal University of Piauí and FAPEAL (Alagoas State Research Foundation), CNPq (Brazilian Funding Agency) for financial support, respectively.

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