On the radiative corrections in the Horava-Lifshitz z=2 QED

We calculate one-loop contributions to the two and three point spinor-vector functions in z=2 Horava-Lifshitz QED. This allows us to obtain the anomalous magnetic moment.

tion of the effective potential in different HL-like scalar theories, and in Yukawa-like theory and QED, including the finite temperature case [8,9] (for a review on HL-like field theory models, see also [10]). Furthermore, in our previous paper [11], we provided an analysis of Lorentz symmetry restoration and the existence of anomalies in z = 2 spinor and scalar QED. A natural continuation of this study could consist in obtaining the HL-like analogue of the anomalous magnetic moment. This is the problem addressed in the present paper.
The structure of this work is as follows. In the section 2, we describe the classical action, propagators and vertices of z = 2 QED. In the section 3 we calculate the contributions to the three-point vector-spinor function and to the quadratic part of the fermionic Lagrangian.
Based on these results, the one-loop renormalization of the model is analyzed. A Summary is devoted to the discussion of the results.

II. CLASSICAL ACTION, PROPAGATORS AND VERTICES
We consider the Horava-Lifshitz-like (HL-like) spinor QED. For the sake of concreteness, we restrict ourselves to the case z = 2, as in [11]. In this case, the Lagrangian describing the model we are interested is where D 0,i = ∂ 0,i − ieA 0,i is a gauge covariant derivative, with the corresponding gauge transformations being ψ → e ieξ ψ,ψ →ψe −ieξ , and A 0,i → A 0,i + ∂ 0,i ξ. Our metric is (+, −, −, −), and ∆ denotes the d-dimensional Laplacian. The parameters a and b were introduced to keep track of the contributions associated to the higher derivative terms.
The free propagator of the fermionic field is where ω = a k 2 + m 2 , and P ± = 1±γ 0 2 are orthogonal projectors. To find the propagator for the vector field, we choose to work in the Feynman gauge by adding to (1) the gauge fixing Lagrangian [9], yielding to the free propagators the forms The interaction vertices are where In the momentum space they look like with all momenta chosen to be entering the corresponding vertices.
The mass dimensions for the case d = 3 on which we concentrate in this paper are: 1/2 for e, 3/2 for A 0 , 1/2 for A i and 3/2 for ψ (as usual for HL theories, we have 1 for ∂ i and z = 2 for ∂ 0 ). Thus, we can conclude that unlike the five-dimensional theory considered in [6], our theory is super-renormalizable. Indeed, the degree of superficial divergence for a generic graph in d space dimensions can be shown to be where E 0 , E i , E ψ are the numbers of external A 0 , A i , ψ legs respectively, V 1,2,3,4 are the numbers of corresponding vertices. Thus, QED with z = 2 is super-renormalizable if d < 4, renormalizable if d = 4 and non-renormalizable if d > 4. Various interesting aspects as renormalization and infrared properties of the theory with d = 4 were treated in [6]. Motivated by the physical relevance of the usual (z = 1) QED in d = 3 space dimensions, here we will consider its extension to z = 2. In particular, we will focus on the anomalous magnetic moment of the fermionic field. Needless to say, our interest in this issue comes from the fact that the computation of the electron magnetic moment in relativistic QED is one of the hallmarks in the development of field theory. In our case, the calculation is more complex because the additional derivative couplings lead us to proceed a complete analysis of the one loop corrections of all trilinear vertexes. However, as our model is super-renormalizable these corrections are finite.
From the tree approximation to the vertexψ(p 2 )σ ij ψ(p 1 )F ij (p), where F ij is an external magnetic field, we may extract the normal magnetic moment as being 2eaψ Sψ, where S is the spin operator.

III. ANOMALOUS MAGNETIC MOMENT IN THE SPINOR QED
To proceed with the one-loop analysis we will now study low momentum corrections to the vector-spinor three point function and in particular determine its contribution to the anomalous magnetic moment. That study is based in the Feynman diagrams depicted in the Figs. 1 and 2.
The twelve possible contributions to the effective Lagrangian coming from Fig. 1 are: Here p = −(p 1 + p 2 ) is the momentum entering with the external gauge field. As indicated, the spatial parts of these integrals are dimensionally regularized with the parameter d attaining its physical value, d = 3, at the end of the calculation.
The contributions from Fig. 2

look like
Up to first order in the momenta, we may summarize the result of the calculations of the previous expressions as T i =ψT i ψ, i = 1, 2, . . . 14, with s = a + b, so, T 9 = e 3 ab 2 16s 5/2 mπ (p 2 − p 1 ) l A l , T 10 = e 3 a 2 8bs 3/2 mπ γ 0 A 0 , T 11 = − e 3 a 3 48bs 3/2 mπ σ ij F ij , Notice that, up to the order that we considered, T 4 and T 5 vanish (this is so because, at zeroth order in the momenta, they involve the product σ ij δ ij = 0). Let us then analyze the remaining contributions. First of all, T 1 , T 3 , T 10 involve only A 0 and cannot yield the anomalous magnetic moment. Instead, they produce Also, T 8 , T 9 , T 12 and T 13 give Finally, we found the following contribution for the vertex V 4 : Thus, up to one-loop order, the complete triple interaction term being the sum of V 1 , V 3 and V 4 is modified to Notice that these results are consistent with the gauge invariance of the model. In fact, for z = 2 there are two independent gauge invariant interactions, namely, aψ(iγ i D i ) 2 ψ, that we have considered, and a 1ψ (D i ) 2 ψ. Observe that the inclusion of the last interaction has the effect of changing a to a + a 1 in the vertices V 2 and V 3 and in the quadratic part of the fermionic Lagrangian. However, it has no effect in the vertices V 1 and V 4 . For simplicity, we do not consider the a 1ψ (D i ) 2 ψ vertex for the moment. We note that although we did not calculate explicitly the correction to the vertex V 2 , the gauge invariance requires the same renormalization as in the vertex V 3 and in the Laplacian term occurring in (14).
The renormalization of the model follows by making a reparametrization as follows Therefore, written in terms of the renormalized quantities, the fermionic Lagrangian is where the renormalization constants are chosen so that the two point vertex function satisfies and also the three point vertex function (one A 0 field and a pairψ, ψ) at zero momenta obeys Γ (2,1) = ie. Using these conditions, we get Summarizing, the renormalized Lagrangian (up to one-loop) is given by: −e 2 aψψA i A i (Z a + one−loop corrections) + (higher derivatives).
The anomalous magnetic moment induced by the radiative corrections can be read immediately from this Lagrangian. It must be noted that no correction of the form iψγ k ∂ k ψ or 1 4 F ik F ik was induced, what makes Lorentz symmetry restoration to be a very improbable goal (this is in agreement with the results of [5] for pure boson field models and for the fermionic QED in 1+4 dimensions). However, Lorentz symmetry could possibly emerge at low energies if the model included from the beginning the term iψγ k ∂ k ψ.
After this renormalization procedure the magnetic vertex V 3 becomes This is our final result for the magnetic moment up to one-loop.

IV. SUMMARY
Motivated by the fundamental role played by anomalous magnetic moment in the development of relativistic (z = 1) QED, here we calculated the z = 2 analogous term. Due to the existence of various new vertices, the calculation turns out to be more involved than in the z = 1 QED, what prompted us to proceed a complete one loop analysis of the renormalizarion process. Our computation shows an essential difference with the one in the usual QED -while in the usual QED the one-loop <ψAψ > triangle diagram contributes not only to the magnetic moment but also to the renormalization of the electric charge, in this case this contribution yields no divergences, being finite from the very beginning, and even finite renormalization of the electric charge does not occur at the one-loop order. This is a consequence of the fact that the Ward identity Z = Z e is obeyed (see 18) and that in this model the gauge field is not renormalized, which, in its turn, implies that the Lorentz symmetry restoration cannot occur. Thus, our study may be relevant at very high energies where Lorentz invariance is presumably broken. On the other hand, at low energies the inclusion of the term linear in the spatial derivatives is certainly mandatory.
A natural continuation of this study could consist in obtaining the possible contributions to the anomalous magnetic moment in HL-like QED with higher values of z, as well as in higher spatial dimensions where the theory will by renormalizable rather than superrenormalizable. We plan to carry out this study in a forthcoming paper.