Three-loop field renormalization for scalar field theory with Lorentz violation

Applying the counterterm method in minimal subtraction scheme we calculate the three-loop quantum correction to field anomalous dimension in a Lorentz-violating O($N$) self-interacting scalar field theory. We compute the Feynman diagrams using dimensional regularization and $\epsilon$-expansion techniques. As this approximation corresponds to a three-loop term, to our knowledge this is the first time in literature in which such a loop level is attained for a LV theory.


I. INTRODUCTION
In high energy physics the main aspects of many physical effects involving particles and fields such as pair annihilation, Compton effect, positronium lifetime, Bremsstrahlung can be understood by lowest-order perturbative calculations [1,2], although higher-level computations give more precise knowledge about these effects. On the other hand the many-body behavior of some physical systems is satisfactorily described only if higher-order approximations are used for studying them. As an example, both three-level and one-loop quantum corrections for the renormalization group outcome for the correlation function critical exponent η, related to field anomalous dimension, which characterizes a second order phase transition in ferromagnetic systems are null [3,4]. Thus the nonvanishing leading quantum contribution to this critical exponent lies just at two-loop order. As ferromagnetic systems present large thermal fluctuations near to critical point, any higher-loop correction, albeit small, is highly relevant for an accurate determination of the numerical value of a critical exponent. For these systems, the critical exponents up to a five-loop level approximation were evaluated [5,6].
All physical phenomena above are described by theories satisfying certain symmetry principles, one of them is Lorentz invariance. However some of these phenomena and many others are been studied in the limit in which this symmetry is violated. These theories were proposed as natural extensions of their Lorentz-invariant (LI) counterparts [7][8][9][10][11][12][13][14][15][16][17][18][19]. More specifically, in a recent paper [20], the β function and field anomalous dimension γ were calculated up to two-loop approximation for a Lorentz-violating (LV) O(N) scalar field theory. This theory may have many applications in the standard model LV Higgs sector.
The mass in this theory was renormalized up to the same loop level [21]. While the β function and mass were computed up to next-to-leading order, only the leading quantum corrections to field anomalous dimension were obtained. The aim of this Letter is to calculate the γ function up to next-to-leading approximation.
We begin this Letter discussing the bare theory for the O(N) scalar field theory with Lorentz violation and its three-loop diagrammatic expansion for two-point function necessary in this work in the Sec. II. In the Sec. III we will discuss the evaluation of the three-loop level renormalization constant for field renormalization and the respective loop-order Wilson function γ. We will finalize the Letter in Sec. IV with our conclusions.

A. Bare theory
The unrenormalized Euclidean Lagrangian density for the massive self-interacting O(N) LV scalar field theory is given by [20] This Lagrangian density is invariant under rotations in an O(N) internal symmetry field space. The field is a N-component vector field and the last term in Eq. (1) represents its The quantities φ B , m B and λ B are the bare field, mass and coupling constant, respectively. The LV second term above breaks the Lorentz symmetry through the dimensionless symmetric constant coefficients K µν (the components of K µν are chosen such that this two-component mathematical object does not transform as a second order tensor under Lorentz transformations) which are the same for all N components of the vector field. This tensor is responsible for a slight symmetry violation when |K µν | ≪ 1. We can also see that the unrenormalized inverse free propagator in momentum space of the theory is given by q 2 +K µν q µ q ν +m 2 B and thus we have a modified version of a conventional scalar field theory. Another modification comes from the emergence of the factor present in the results for the β and γ functions where δ {µν δ ρσ} ≡ δ µν δ ρσ + δ µρ δ νσ + δ µσ δ νρ .
The factor in Eq. (2) has a similar form in Minkowski space-time [20]. These two forms are connected by a Wick rotation when we have δ µν → η µν where η µν is the Minkowski metric tensor. As it is known [6], the bare two-point vertex function Γ B has two divergent terms: one proportional to external momentum P 2 and another to bare mass m 2 B . In the process of field renormalization for a scalar field theory, it is needed to renormalize just the former. The latter can be used to mass renormalization purposes. Our task is to analyze the three-loop level field renormalization term for this function. This will be the subject of next section.

B. Bare three-loop contribution to two-point function
The single component field (N = 1) three-loop diagrams for the unrenormalized bare two-loop function are [6] As we are not interested in diagrams proportional to m 2 B , which is the case of tadpole diagram for all orders in the tensor K µν [20], we see both topologically and mathematically that the last three diagrams have, at least, a tadpole diagram on their expressions as seen So these diagrams do not contribute to field renormalization.
has no dependence on external momentum and, consequently, will be discarded for γ function calculation purposes. Although it has no tadpole on its expression, we can show explicitly that it is proportional to m 2 B . This is achieved by using the "partial-q" ∂q µ /∂q µ = d [22] identity. Introducing in the diagram above, it can be written as being proportional to m 2 where Finally the first diagram can be cast into another form using once again the "partial-q" identity (8). Thus we obtain where The first term in Eq. (13) is proportional to unrenormalized mass m 2 B and will be not used. We will use the second term which will be shown to be proportional to external momentum P 2 and will give the three-loop contribution to γ function. This will be approached in the next section.

A. Renormalized theory and the three-loop contribution to γ function
In this Letter we apply the counterterm method in minimal subtraction scheme [22] in the renormalizaton process and use a given fixed notation [6]. In this scheme the renormalized theory is attained after some diagrams are added to initial bare perturbative diagrammatic expansion for cancelling infinities where the external momenta are held arbitrary, although the final renormalized theory satisfies these conditions at vanishing external momenta. These new diagrams are called counterterm diagrams and can be seen as generated by a few terms added to original unrenormalized Lagrangian density. As a consequence of this addition, the finite theory (with all bare parameters substituted by their renormalized counterparts) is now composed by both the initial and counterterm Lagrangian density and the divergences of the theory are absorbed by renormalization constants, Z φ for field renormalization. Thus the renormalized theory is attained, whose n-point functions with n ≥ 1 satisfy the Callan-Symanzik equation where Γ (n) ≡ Γ (n) (k 1 , ..., k n ; m, g, µ) and where | B indicates that we have to calculate the parameters in the bare theory. The β and γ m (associated to mass renormalization) functions were obtained earlier [20,21], g is the renormalized dimensionless coupling constant given by g = λµ −ǫ and µ is an arbitrary mass parameter. The three-loop quantum correction to γ function for a self-interacting N-component scalar field is the objective of this work. Thus, all we need is the field renormalization constant up to three-loop. It is given by [6] Z φ (g, ǫ −1 ) = 1 + 1 P 2 The operator K extracts only the divergent terms of diagrams and the renormalization constant Z φ (g, ǫ −1 ) is a Laurent expansion in ǫ and a function of g. The factor S is the symmetry factor for a scalar theory with O(N) symmetry for the respective diagram and so on. We utilize dimensional regularization [22][23][24] to regularize the diagrams in ǫ-expansion where ǫ = 4 − d. In the equation above we have the definitionsŜ d = S d /(2π) d = 2/(4π) d/2 Γ(d/2) [21]. The factor S d = 2π d/2 /Γ(d/2) is the surface area of a unit d-dimensional sphere and has the finite valueŜ 4 = 2/(4π) 2 in four dimensions. In four-dimensional space, each loop integration contributes with a factor S 4 . We use the integral expressed in this way because it is more convenient. It avoids the appearing of Euler-Mascheroni constants in the middle of calculations [25]. As these constants are not present in the renormalized theory, if we would not use the Eq. (19) the referred constants would have to cancel precisely. All diagrams have the renormalized free propagator in their expressions and can be expanded in a Taylor series 1 (q 2 + K µν q µ q ν + m 2 ) n = 1 (q 2 + m 2 ) n 1 − n K µν q µ q ν q 2 + m 2 + n(n + 1) 2! K µν K ρσ q µ q ν q ρ q σ (q 2 + m 2 ) 2 + ... (20) in the K µν small parameters.
The first diagram in the Eq. (18) is the sunset diagram. Its expression is given by and we can write it as a sum of others two integrals. Using one more time the "partial-q" [21], we get where The integral A(P ) is part of a term proportional to m 2 and is commonly used in mass renormalization. The integral B(P ) is proportional to P 2 and was used to renormalize the field and to calculate the γ function up to two-loop [20], namely γ 2−loop (g) = (N + 2)g 2 Π 2 /36(4π) 4 .
The third diagram in Eq. (18) is a counterterm diagram. It can be written as an operation over the sunset diagram, namely where c 1 g is the first order coupling constant counterterm and is proportional to divergent term for "fish" diagram calculated previously [20]. As c 1 g introduces a first order pole in the counterterm diagram, we have to calculate the sunset finite part. Introducing Feynman parameters [25], making the change of integration variables q 1 + P → q ′ 1 and after q ′ 1 → q 1 and after momentum integration for zero order in K, the integral B (0) (P ) assumes the form As the calculation of the finite part of a diagram involves more effort than just its pole term [26], we will make our calculations up to O(K). Adding this contribution to the counterterm diagram we have (see A) where The second diagram present in the Eq. (18) was written as a sum of others two integrals.