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On the effective potential, Horava-Lifshitz-like theories and finite temperature

We calculate the one-loop effective potential at finite temperature for the Horava-Lifshitz-like QED and Yukawa-like theories for arbitrary values of the critical exponent and the space-time dimension. Additional remarks on the zero temperature situation are also presented.

of their renormalization. Within this context, the HL versions of the gauge field theories [3], scalar field theories [4], four-fermion theory [5] and CP N −1 model [6] were considered.
Another important result in this context is the generalization of the Ward identities for the HL-like theories [7].
Another line of investigations on HL-like theories concerns the study of the effective potential. In the works [8][9][10][11] the one-loop effective potential for scalar field theories with different forms of self-couplings and arbitrary values of the critical exponent z, for scalar QED and for the Yukawa models with z = 2 and z = 3 have been obtained. However, a remaining problem was the calculation of the (one-loop) effective potential for the same models with an arbitrary value of the critical exponent. The analysis of these models at zero temperature has been carried out in [12], and its extension for the non-zero temperature case is considered in this paper.
We begin our study of scalar quantum electrodynamics at finite temperature for generic z and d space dimensions by considering in the section 2 that the effective space-time dimension d + z is odd. As we shall demonstrate, at these values of d and z no self-interaction of the scalar field is necessary to achieve the consistency of the model. However, as discussed in the section 3, for d + z even, a divergence occurs demanding the inclusion of a self-interaction of the scalar field. In the section 4 we analyze an HL version of the Yukawa model and show that for odd z the model is in general nonrenormalizable with only one exception happpening if the needed counterterm is proportional to φ 4 . A summary and further comments of our results are presented in section 5.

II. SCALAR QUANTUM ELECTRODYNAMICS WITH d + z ODD
The Lagrangian of the scalar QED with an arbitrary z looks like is a gauge covariant derivative and we assume φ to be massless, for simplicity. By the same reason, we choose the critical exponents for the scalar and vector fields to be the same.
Adding the gauge fixing term [12] the propagators acquire the simple forms For the one-loop calculation of the effective potential, the only relevant vertices are where Φ is a constant background scalar field generated by the shift φ → φ + Φ.
From now on, except where explicitly indicated, the propagators (as well as all momenta) will be taken in the Euclidean space, and everywhere k 2 ≡ k 2 = k i k i , with i running from 1 to d. As we will show shortly, similarly to what happens in the relativistic QED, the perturbative consistency of the model may require the addition of a self-interaction term for the scalar field. For the time being, we discard such possibility so that, as it has been shown in [12] and will be discussed in more details in the next section, the effective potential turns out to be determined by a single integral where M 2 = 2e 2 ΦΦ * and U a , U b and U c are the contributions coming from loops containing only < A 0 A 0 > or < A i A j > propagators and graphs with both the gauge and scalar field propagators, respectively.
In the finite temperature case, following the Matsubara prescription [13], observing that all propagators are bosonic ones, we must change k 0 → 2πnT , where T is the temperature, and n is an integer number. The integral over k 0 is then replaced by the sum: Using the known summation formula [14]: where the additive constant does not depend on E and will be omitted from now on, we have The first term identically reproduces the zero temperature result from [12]. Therefore (throughout this paper we adopt dimensional regularization with minimal subtraction), Notice that U 0 , the first term in the above expression, is finite for d + z odd while, for d + z even, it diverges and requires a subtraction which may be carried out by adding a corresponding counterterm. Therefore, in principle, for the case d + z = 2n, one should introduce into the theory an additional vertex λ(ΦΦ * ) n , to achieve multiplicative renomalizability; the presence of this new self-interaction vertex generates new Feynman diagrams making the evaluation of the one-loop effective potential much more complicated. We will defer the discussion of this situation to the next section and here we will restrict ourselves to the analysis of the case with d + z = 2n + 1. By making the change of variables k z T =k (withk dimensionless), we obtain Thus, for large T , the leading contributions are where and where Li ν (x) denotes the polylogarithm function of order ν. We see that for z = 1 this expression reproduces the temperature dependence found in [14], but, for generic z > 1, the effective potential grows more slowly with the temperature.

III. SCALAR QUANTUM ELECTRODYNAMICS WITH d + z EVEN
As it was pointed out before, for the consistency of the model when d + z = 2n, it is necessary the inclusion of a self-interaction term for the scalar field, so that the Lagrangian then becomes The gauge fixing term and propagators for the gauge field are the same as in the previous section. The propagator for the scalar field will be fixed shortly.
First, we can find the contribution to the effective potential coming from the gauge propagators only. The corresponding Feynman diagrams are depicted in Fig. 1.

FIG. 1: Contributions involving gauge propagators only.
There are two types of such contributions -the first of them, U a , is given by the sum of loops of < A 0 A 0 > propagators, and -the second one, U b , of loops of < A i A j > propagators.
They are completely analogous, up to an overall factor (U b carries the factor d), and they together contribute as (cf. [11,12]) Repeating the calculations of the Section II, one can show that the results for U a + U b at zero and finite temperature, reproduce the expressions (9,10,11) with the only difference that the overall factor d is replaced by d + 1.
Now, let us obtain the background-dependent effective propagators of the scalar fields.
These propagators will be represented by bold straight lines.
Besides, we also must use the background-dependent propagators of the gauge field, Actually, we will use not these propagators themselves but the objects derived from them: whose Fourier transforms in the Euclidean space are The presence of the new propagators will lead to a contribution to the effective action emerged from the "crossed" sector (those graphs involving both gauge and matter propagators shown at Fig. 3) given by . . . where So, we can write The integral over momenta, as well as the discretization of the zero component of the momentum in order to implement finite temperature, is straightforward but the result is highly cumbersome. Nevertheless it can be performed in the following way.
To simplify this expression, we introduce µ ± = µ ± √ MM, so that µ + = λn(2n − 1)(ΦΦ * ) n−1 , and µ − = λn(ΦΦ * ) n−1 and define q 2 = k 2 0 + k 2z .Using that MΦΦ +MΦ * Φ * = 2λn(n − 1)(ΦΦ * ) n we get After some algebraic transformations, we can rewrite this expression as By cancelling the term with ln(q 2 + µ + ), we arrive at It follows from (15) that U a = 1 2 d d kdk 0 (2π) d+1 ln(q 2 + M 2 k 2z−2 ). Thus, we can cancel some additional terms and get To close the calculations, let us obtain the contribution to the effective potential generated by the self-coupling of the scalar fields. It follows from (17) that in this case we have a new contribution to the effective action To calculate this determinant it is convenient to perform the Fourier transform, which after a Wick rotation to the Euclidean space yields Up to an irrelevant additive constant, the evaluation of this expression gives Using (26), we can write the complete one-loop effective potential: The term proportional to d (that is, U b ) is given by the expressions (9-11). The last term, that is, those originated from U d involving ln(q 2 + µ + ), yields the zero temperature result U (1) plus the finite temperature contribution U (1) At high temperatures we obtain where A is given by (12), and It remains to analyse the first term from (30) which looks like Unfortunately, this integral cannot be done in a closed form. We present the results only for two particular cases.
(i) When the contribution of the gauge coupling dominates, we can choose λ ≃ 0. Then one has µ − = 0 and the integral I is just an irrelevant constant, independent of the classical fields. The complete contribution to the effective potential in this case comes from the terms (9)(10)(11), while the term proportional to λ is essential only on the tree level.
(ii) When the contribution of the scalar self-coupling dominates, we can choose g ≃ 0 in this term. In this case, one has M = 0, so, I ≃ 1 2 d d kdk 0 (2π) d+1 ln(q 2 + µ − ), which, similarly to expressions (31) and (33), yields at zero temperature and at high temperature. Here the effective potential is reduced to the sum of the expressions (31) and (36) at zero temperature, and of (33) and (37) at the high temperature.
We close this section with a discussion of the renormalizability of the model. First, one reminds that we introduced a self-coupling of the scalar field since, at λ = 0, the one-loop effective potential diverges if d + z = 2n with n integer (see (9)) so, the counterterms (ΦΦ * ) n is needed. At the same time, the new vertex (ΦΦ * ) n generates new contributions as in (31), which diverge if 1 + d z = 2ñ, withñ an integer. These contributions are proportional to (ΦΦ * )ñ (n−1) , therefore, to achieve multiplicative renormalizability, one must, in principle, introduce a new vertex (ΦΦ * )ñ (n−1) , which, again modifies the classical action. The only exceptional situation, when this modification is not necessary, is the caseñ = n n−1 . For n andñ integer, the only solution is n =ñ = 2. Therefore, we conclude that only the vertex (ΦΦ * ) corresponds to the renormalizable interaction, with d = 3 and z = 1, that is, just the usual scalar QED. We note, however, that in the cases, where either d + z is odd (that is, the case considered in the previous section), or d z is not an odd number (i.e. either even, or fractionary one), this problem simply will not arise, since, for d+z odd, there is no divergent contributions to the one-loop effective potential. There are, of course, additional restrictions on d and z arising from the fact that, in the renormalizable theories, dimensions of couplings must be non-negative, i.e. z − d + 2 ≥ 0 (for the coupling e) and d + z − n(d − z) ≥ 0 (for the coupling λ). However, these restrictions play a role only at higher loop orders.
To study the one-loop effective potential, it is enough to treat the scalar field as purely external, and to consider the spinor field to be massless since a nontrivial mass implies only in a redefinition of the Φ field. In this case, the loop expansion ends at the one-loop contribution. However, if we assume that Φ is also dynamical (which, in particular, is necessary to proceed renormalization if the contribution to the effective potential diverges), with the same critical exponent z as the ψ, its free Lagrangian is the same as in the theory (1).
Notice that the mass dimension of h is (3z − d)/2, and the theory is (super)renormalizable for z ≥ d/3 -in particular, it is renormalizable in the usual case (z = 1) Yukawa model in (3 + 1)-dimensional space.
The one-loop effective potential corresponding to the Lagrangian (38), looks like for the HL Yukawa models there are no one-loop divergences. This, of course, does not preclude the existence of divergences in higher orders which must be removed by an adequate renormalization scheme.
The models have the same high temperature limit proportional to T 1+d/z as they should but different next to leading behaviors unless d = z. Thus for z > 1 the efffective potential decays with the temperature more slowly than in the usual case [14].