Renormalization in Nonminimal Lorentz-Violating Field Theory

TL;DR

This paper explores the renormalization process in a nonminimal Lorentz-violating scalar-fermion model with dimension-five operators, showing some divergences are reduced and the scalar mass is modified.

Contribution

It is the first to analyze renormalization in a nonminimal Lorentz-violating model with specific higher-dimension operators, providing explicit radiative correction calculations.

Findings

Some divergences are improved and become finite.

The scalar pole mass is modified, affecting asymptotic states.

Radiative corrections are explicitly computed for the scalar self-energy.

Abstract

We provide the first step towards renormalization in a nonminimal Lorentz-violating model consisting of normal scalars and modified fermions with mass dimension five operators. We compute the radiative corrections corresponding to the scalar self-energy, and we show that some divergencies are improved and in the scalar sector they are finite. The pole mass of the scalar two-point function is found and shown to lead to modifications of asymptotic states.

Equations

$$\mathcal{L}=\frac{1}{2}{\partial }_{\mu }\varphi {\partial }^{\mu }\varphi -\frac{1}{2}{M}^2{\varphi }^2+\bar{\psi }\left(i\partial to0.0pt\text{hss}/-m-\bar{\alpha }mnto0.0pt\text{hss}/\right)\psi +g_2\bar{\psi }nto0.0pt\text{hss}/(n\cdot \partial {)}^2\psi +g\bar{\psi }\varphi \psi ,$$

$$i\Pi (p)$$

$$Q_{\mu }(k)$$

$$R_{\mu }(p+k)$$

$$\tilde{\Pi }(p)$$

$$q_0$$

$$q_1$$

$$q_n$$

$$({\Gamma }_{\varphi ,R}^{(2)}{)}^{-1}=p^2-M^2+{\Pi }_R(p),$$

$${\Pi }_R(p)=p^2{A}_{\varphi }+m^2{B}_{\varphi }+(n\cdot p{)}^2{C}_{\varphi },$$

$$i{A}_{\varphi }$$

$$i{B}_{\varphi }$$

$$i{C}_{\varphi }$$

$$({\Gamma }_{\varphi ,R}^{(2)}{)}^{-1}{|}_{{\bar{P}}_{\varphi }^2=0}=0.$$

$$0$$

$$M_{\mathrm{ph}}^2=\frac{(M^2-B_{\varphi }{m}^2)}{(1+A_{\varphi })},\bar{y}=\frac{C_{\varphi }}{1+A_{\varphi }},$$

$$\frac{d({\Gamma }_{\varphi ,R}^{(2)}{)}^{-1}}{d{\bar{P}}_{\varphi }^2}{|}_{{\bar{P}}_{\varphi }^2=0}=Z_{\varphi }^{-1},$$

Full text

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1 Introduction

The Standard-Model Extension (SME) can be regarded as an effective generalized framework to accommodate possible effects of suppressed CPT and Lorentz violation. The SME comprises two different sectors, the minimal sector with operators of mass dimension lower or equal to four [1], and the nonminimal sector with higher mass dimensional operators [2]. Several bounds have been given using ultrahigh sensitivity experiments which have allowed to test various predictions of Lorentz breakdown in the standard model of particles and gravity [3].

A key feature that distinguishes nonminimal models of Lorentz violation from minimal ones is the indefinite metric that arises due to the higher time derivative terms. The indefinite metric, as well known, introduces a pseudo unitarity relation for the $S$ matrix, which can imply the loss of conservation of probability. Recently it has been shown that by adopting a Lee-Wick prescription [4], it is possible to have a consistent unitary theory [5].

The renormalization of quantum field theories incorporating Lorentz violation can be modified due to radiatively induced operators having different structures not initially present in the original Lagrangian [6]. As a general statement, one can say that one-loop Lorentz-violating corrections may have a more dramatic effect on asymptotic states that they do have in the typical case. The modified asymptotic space has been mainly shown in computations for the mass pole of two-point functions. In this work we study the renormalization with particular focus on finite radiative corrections associated to monminimal terms and the effect of modifying the renormalization conditions due to Lorentz violation.

2 The Yukawa-like model

Consider the Lagrangian density [7]

[TABLE]

where $n^{\mu}$ is a preferred four-vector, $g_{2}$ is a constant with dimension of $(\text{m}ass)^{-1}$ and $g$ is a typical Yukawa coupling. As a first example of quantum corrections in our model, we study the contribution with two external scalar legs depicted at Fig. 1.

It is represented by the integral

[TABLE]

where we define

[TABLE]

We can write the correction to the scalar propagator up to second-order in $p$ as

[TABLE]

where

[TABLE]

The renormalized scalar two-point function is given by

[TABLE]

with

[TABLE]

where $A_{\phi}$, $B_{\phi}$ and $C_{\phi}$ are constants that can be deduced from the expressions (2), (7), (8) being

[TABLE]

We consider the ansatz $\bar{P}^{2}_{\phi}=p^{2}-{M}^{2}_{{\rm ph}}+\bar{y}(n\cdot p)^{2}$ in terms of the two unknown constants $M_{{\rm ph}}$ and $\bar{y}$, Both constants can be determined using the renormalization condition

[TABLE]

Hence, from (9) replacing the value of $p^{2}$ and using the condition (12), we arrive at the equation

[TABLE]

Due to the independence of each term, we find the two constants

[TABLE]

and in consequence also the scalar pole mass $\bar{P}_{\phi}^{2}$ which dictates how asymptotic states propagate. Substituting the above expressions in Eq. (9) and using the normalization condition

[TABLE]

we identify the finite wave function renormalization constant $Z_{\phi}^{-1}=1+A_{\phi}$.

3 Conclusions

For nonminimal Lorentz-violating models, one should, in general, expect an indefinite metric leading to a nontrivial structure of poles and a pseudo unitary relation for the $S$ matrix. Also, the nonstandard structure of radiative corrections in general leads to a modified asymptotic space. For the model we have focus on we have considered the prescription for locating the poles to be dictated by unitarity conservation requirements. We have found that some radiative corrections are improved, and in fact they are finite in the scalar sector. The pole extraction for the Yukawa-like model has been successfully provided in the scalar sector.

Acknowledgments

The work by A. Yu. P. has been supported by the CNPq project No. 303783/2015-0. CMR acknowledges support by FONDECYT grant 1191553.