A $Sim(2)$ invariant dimensional regularization

We introduce a $Sim(2)$ invariant dimensional regularization of loop integrals. Then we compute the one loop quantum corrections to the photon self energy, electron self energy and vertex in the Electrodynamics sector of the Very Special Relativity Standard Model(VSRSM).

The Weinberg-Salam model(SM) is a very successfull description of Nature, that is being verified at the LHC with a great precission. Moreover, until now, neither new particles nor new interactions have been discovered at the LHC [1]. This cannot be the whole story, though. The SM assumes that the neutrino is a massless particle, whereas we know that the neutrino is massive in order to describe the observed neutrino oscillations [2] If we assume that Lorentz's is an exact symmetry of Nature, we have to introduce new particles and interactions in order to give masses to the observed neutrinos through, for instance, the seesaw mechanism [3].
A new possibility to have a massive neutrino arises in Very Special Relativity(VSR) [4]. Instead of the 6 parameter Lorentz group, a 4 parameters subgroup(Sim (2)) is assumed to be the symmetry of Nature. Sim(2) transformations change a fixed null four vector n µ at most by a scale factor, so ratios of scalar quantities containing the same number of n µ in the numerator as in the denominator are Sim(2) invariant, although they are not Lorentz invariant. In this way it is possible to write a VSR mass term for left handed neutrinos [5].
Recently, we have proposed the SM with VSR [6] (VSRSM).It contains the same particles and interactions as the SM, but neutrinos can have a VSR mass without lepton number violation. Since the electron and the electron neutrino form a SU (2) L doublet, the VSR neutrino mass term will modify the QED of the electron.
A main obstacle in exploring the loop corrections in the VSRSM is the nonexistence of a gauge invariant regulator that preserve the Sim(2) symmetry of the model.
In this letter, we define an appropriate regulator, based on the calculation of integrals using the Mandelstam-Leibbrandt(ML) [7] [8]prescription introduced in [9].We want to emphasize that our method directly lead to the ML prescription, the only one compatible with canonical quantum field theory [10].The regulator preserve gauge invariance, a property inherited from the ML prescription, as well as the Sim(2) symmetry.
Then we proceed to compute one loop corrections. We find the divergent and finite part of the vacuum polarization and electron self energy. Moreover we compute the leading correction to the standard QED result for the anomalous magnetic model of the electron.
We want to emphasize that meanwhile no new particles or interactions are discovered at the LHC or elsewhere, we have to consider the VSRSM as a very strong candidate to describe weak and electromagnetic interactions. It contains all the predictions of the SM plus neutrino masses and neutrino oscillations. It is renormalizable(as we show explicitly in this letter) and unitarity of the S metric is preserved. If future experiments validates the predictions of the model, it would be the first evidence of Lorentz Symmetry violation.

Mandelstam-Leibbrandt(ML) prescription from a hidden symmetry
In this section we review the results of [9]. Let us compute the following simple integral: where f is an arbitrary function.dp is the integration measure in d dimensional space and n µ is a fixed null vector((n·n) = 0). This integral is infrared divergent when (n · p) = 0. The ML is: wheren µ is a new null vector with the property (n ·n) = 1.
To compute A µ we must know what f is , provide an specific form of n µ andn µ , and evaluate the residues of all poles of f (p 2 ) (n·p) in the p 0 complex plane, a difficult task for an arbitrary f .
Instead we want to point out the following symmetry: It preserves the definitions of n µ andn µ : 0 = (n · n) → λ 2 (n · n) = 0 0 = (n ·n) → λ −2 (n ·n) = 0 1 = (n ·n) → (n ·n) = 1 We see from (1) that: Now we compute A µ , based on its symmetries. It is a Lorentz vector which scales under (2) as λ −1 . The only Lorentz vectors we have available in this case are n µ andn µ . But (2) forbids n µ . That is: Multiply by n µ to find (A · n) = a. Thus a = dpf (p 2 ). Finally: By the same arguments, we can compute the generic integral: q µ is an external momentum, a Lorentz vector. F is an arbitrary function. The last relation follows from (2), for a certain f we will find in the following.
Taking the partial derivative respect to q µ in both sides of (3), we obtain that We defined u = p · q,x = q 2 ,y = (n · q)(n · q). () ,u means derivative respects to u.
Assuming that the solution and its partial derivatives are finite in the neighborhood of y = 0, it follows from the equation that f (x, 0) = g(x). That is the partial differential equation has a unique regular solution.
Now we apply this result to compute integrals that appear in gauge theory loops: In this case The unique regular solution of (4) is: In the same way we can compute the whole family of loop integrals: Using dimensional regularization, we obtain: The prescription to regularize the infrared divergences that we have reviewed in chapter 1, always produces finite results depending on two fixed null vectors n µ , n µ . Moreover it preserves gauge invariance because it respects the shift symmetry of the loop integral dpf (p µ ) = dpf (p µ + q µ ) for arbitrary q µ . However ML does not respect Sim(2) symmetry of VSRSM. Below we show how to remedy this.
We start from the ML result for the integral(5). We traden µ by q µ . i.e.n µ = an µ + bq µ 1 . From the conditions:n.n = 0,n.n = 1 we getn µ = − q 2 2(n.q) 2 n µ + qµ n.q . Moreover, we see thatn µ satisfies the scaling (2) and is real for any value of q 2 in Minkowsky space. So, all the conditions to apply the procedure reviewed in section 1 are satisfied. Therefore,, Notice that now (6) respects the Sim(2) invariance of the original integral. The same procedure can be applied to other integrals found in [9]. Notice that first we keepn fixed, derive (5) with respect to q µ as many times as necessary and then replacen µ = − q 2 2(n.q) 2 n µ + qµ n.q . The rationale for this prescription derives from the observation that we could compute the integral with whatever power of p µ in the numerator using Cauchy theorem of residues in p 0 complex plane. In this way it doesn't matter whethern µ depends on q µ or not.
Once we have obtained (6), we notice that it provides a unique analytic continuation of the integral from b < 0 to b > 0. Since for b < 0 we do not need an infrared regulator, we can compute the integral using standard dimensional For instance in d = 3 we must havenµ = a q 2 nµ (n.q) 2 + b qµ n.q + cε µνλ nν (n.q) 2 q λ q 2 with a, b, c pure numbers. This fails to be real for q 2 < 0. regularization. By integration by parts in the integral over t,we can check that (6) gives the right answer for b < 0.

The model
The leptonic sector of VSRSM consists of three SU (2) doublets L a = ν 0 where ν 0 aL = 1 2 (1 − γ 5 )ν 0 a and e 0 aL = 1 2 (1 − γ 5 )e 0 a , and three SU (2) singlet R a = e 0 aR = 1 2 (1+γ 5 )e 0 n . We assume that there is no right-handed neutrino. The index a represent the different families and the index 0 say that the fermionic fields are the physical fields before breaking the symmetry of the vacuum.
In this letter we restrict ourselves to the electron family. m is the VSR mass of both electron and neutrino. After spontaneous symmetry breaking(SSB), the electron adquires a mass term M = Gev √ 2 , where G e is the electron Yukawa coupling and v is the VEV of the Higgs. Please see equation (52) of [6]. The neutrino mass is not affected by SSB:M νe = m.
Restricting the VSRSM after SSB to the interactions between photon and electron alone, we get the VSR QED action.ψ is the electron field. A µ is the photon field. We use the Feynman gauge.
We see that the electron mass is M e = √ M 2 + m 2 , where m is the electron neutrino mass.

Feynman rules
To draw the Feynman graphs we used [11] In the following sections, we have

Photon Self Energy in VSRSM
In this section we present the computation of the photon self-energy. In VSRSM it is given by two graphs: Applying the Sim(2) invariant regulator to the addition of the graphs of Figure (2), and after a long calculation, we get: Here −e is the electron electric charge, m the electron neutrino mass and M e is the electron mass.q µ is the virtual photon momentum. We first notice that q µ Π µν = 0 as required by U (1) gauge invariance of the photon field. It is obtained by a straightforward application of the regularized integrals of . Moreover B(q 2 = 0) = 0,therefore the photon remains massless. Also the photon wave function divergence is the same as in QED.

Electron Self Energy in VSRSM
Here we calculate the electron self-energy. Again we have two graphs contributing to the 2-proper vertex. See Figure(  − iΣ(q) = C n n.q + D q + E with: In this subsection we discuss the 3 points proper vertex and verify the Ward-Takahashi identity. This is an important test of the gauge invariance of the regulator. The one loop contribution to Γ µ (p = p+q, p) consists of the addition of 3 graphs (Figure(4)):

Figure 4: One loop contribution to the 3 points proper vertex
As a result of the shift symmetry which is respected by the regulator, dpf (p µ ) = dpf (p µ + q µ ) for arbitrary q µ , we can prove the Ward-Takahashi identity: Here S(p) = i p−M −Σ(p) is the full electron propagator and Γ µ (p + q, p) is the three proper vertex.
Below we explicitly verified that the pole at d = 4 satisfies (11) 2 Pole contribution: The divergent piece satisfies the Ward identity:

Form factors
The on-shell proper vertex can be written as follows: In the Non-Relativistic(NR) limit we get Table 1, keeping terms that are at most linear in q µ .

NR limit
Form factor In the right column we list the form factor. In the left column we have the NR limit of the matrix element accompanying the form factor in (14).All form factors are evaluated at q µ = 0. Here A 0 is the electric potential and A i is the vector potential.ϕ s is a two dimensional constant vector that corresponds to the NR limit of the Dirac spinors.
To show the power of the Sim(2) invariant regularization prescription presented in this letter, we will compute the one loop contribution to the (isotropic)anomalous magnetic moment of the electron. It is given by F 2 (0)−2n 0 M G 2 (0)−4F 3 (0)M e n 2 0 i(See rows 11, 5 and 9 of Table 1).
The Sim(2) invariant regularization opens the way to explore the full quantum possibilities of VSR. They should be systematically studied, in Particle Physics models as well as in Quantum Gravity models.