arXiv:1704.02299v1 [hep-th] 7 Apr 2017
A Sim(2) invariant dimensional regularization J. Alfaro Facultad de F´ısica, Pontificia Universidad Cat´olica de Chile, Casilla 306, Santiago 22, Chile.
[email protected] April 10, 2017 Abstract 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 successful description of Nature, that is being verified at the LHC with a great precision. 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 nµ at most by a scale factor, so ratios of scalar quantities containing the same number of nµ 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 1
[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 in the Electrodynamics sector of the VSRSM. 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 remark 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. Sim(2) invariant regulator The prescription to regularize the infrared divergences that we have discussed in [9], always produces finite results depending on two fixed null vectors n ¯ µ , nµ . Moreover it preserves gauge invariance because it respects the symmetry of the R R loop integral dpf (pµ ) = dpf (pµ + qµ ) for arbitrary qµ . However ML does not respect Sim(2) symmetry. Below we show how to remedy this. We start from the ML result for the integral(equation(5) of [9]) : Z 1 1 Γ(a + b − ω) dp 2 = (−1)a+b i(π)ω (−2)b (¯ n · q)b [p + 2p.q − m2 + iε]a (n · p)b Γ(a)Γ(b) Z 1 1 , ω = d/2(1) dttb−1 2 2 (m + q − 2n · q¯ n · qt − iε)a+b−ω 0 We trade n ¯ µ by qµ . i.e. n ¯ µ = anµ + bqµ . From the conditions: n ¯ .¯ n = 0,¯ n.n = 1 qµ q2 we get n ¯ µ = − 2(n.q)2 nµ + n.q . Therefore,, Z 1 1 = (2) dp 2 [p + 2p.q − m2 + iε]a (n.p)b 2 b Z 1 q Γ(a + b − ω) 1 (−1)a+b i(π)ω (−2)b , ω = d/2 dttb−1 2 2 Γ(a)Γ(b) 2n · q (m + q (1 − t) − iε)a+b−ω 0 Notice that now (2) 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 keep n ¯ fixed, derive (1) with respect to qµ as many times as necessary and qµ q2 then replace n ¯ µ = − 2(n.q) 2 nµ + n.q . We now compute loop corrections to the QED effective action. The Electrodynamics sector of the VSRSM in the Feynman gauge. 1 1 (∂µ Aµ )2 2 −1 ¯ − M ψ − Fµν F µν − L = ψ i 6 D + 6 nm (n · D) 2 4 4 Dµ = ∂µ − ieAµ , Fµν = ∂µ Aν − ∂ν Aµ 2
Photon Self Energy in VSRSM There are two graphs that contribute to the photon self energy. i 6p+M −
m2 6n 2 n·p
1 dp Tr γµ + nµ (6 n) m2 (n.(p + q))−1 (n.p)−1 (3) 2 p2 − M 2 − m2 + iε i (6 p+ 6 q) + M − m2 6n 2 n·(p+q) 1 γν + nν (6 n) m2 (n.(p + q))−1 (n.p)−1 2 (p + q)2 − M 2 − m2 + iε
iΠ1µν = −(−ie)2
Z
and 2
iΠ2µν = (−1)(ie) nµ nν i
Z
dp(n.p)−1 (n.p)−1 [(n.(q + p))−1 + (n.(−q + p))−1 ] (4) m2 6n i 6 p + M − 2 n·p 1 Tr 6 nm2 2 2 p − M 2 − m2 + iε
Thus Πµν = Π1µν + Π2µν We get: iΠµν
nµ qν + nν qµ 2 nµ nν + − ηµν = A(ηµν q − qµ qν ) + B −q (n.q)2 n.q 2
(5)
with Z 1 i 8x(1 − x) dxΓ(2 − ω) ω 2 (4π) 0 (Me − (1 − x)xq 2 − iε)2−ω Z 1 e2 q 2 (1 − x)2 dx B = −m2 i 2 log 1 − 2 4π 0 (1 − x) Me − q 2 (1 − x)x − iε
A = (−ie)2
(6)
Here −e is the electron electric charge, m the electron neutrino mass and Me is the electron mass.qµ is the virtual photon momentum. We first notice that q µ Πµν = 0 as required by U (1) gauge invariance of the photon field. 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 Again there are two graphs that contribute. But one of them vanishes in Feynman gauge. Thus: i 6 p+ 6 q + M − m2 6n Z 2 n·(p+q) 1 1 1 (7) −iΣ(q) = (−ie)2 dp γµ + m2 6 nnµ 2 2 2 n.(p + q) n.q (p + q) − M − m2 + iε 1 iηµν 1 1 γν + m2 6 nnν − 2 2 n.(p + q) n.q p + iε We get: − iΣ(q) = C
6n +D 6q+E n.q 3
(8)
with: C = (−ie)2 m2 [ −ω
2i(4π)
1
Z
dx
0
D = −2(−ie)2 (ω − 1)i(4π)−ω
Z
0
1
dx(1 − x)−1 ln 1 +
dx
0
Z
q 2 (1 − x) (Me2 − q 2 − iε)
+
Γ(2 − ω) ], [µ2 x − x(1 − x)q 2 + (M 2 + m2 )(1 − x) − iε]2−ω
1
Z
E = (−ie)2 2ωM i(4π)−ω
i 16π 2
[µ2 x
− x(1 −
x)q 2
Γ(2 − ω)x , + (M 2 + m2 )(1 − x) − iε]2−ω
1
dx
0
[µ2 x
− x(1 −
x)q 2
Γ(2 − ω) (9) + (M 2 + m2 )(1 − x) − iε]2−ω
Electron-Electron-Photon Proper vertex Γµ (p + q, p)[11] The vertex correction comes from three graphs. Thus: 6 k ′ + M − m2 6n ′ 2 n·k 1 i (−ie)2 γν + nν (6 n) m2 (n.(p + q))−1 (n.(k + q))−1 δΓµ (p + q, p) = dk (k − p)2 2 k ′2 − M 2 − m2 + iε 6 k + M − m2 6n 2 n·k 1 1 2 −1 −1 2 −1 −1 γν + nν (6 n) m (n.k) (n.p) γµ + nµ (6 n) m (n.(k + q)) (n.k) 2 k 2 − M 2 − m2 + iε 2 ˜ µ (p + q, p) (10) +Γ Z
The new graphs are: Z 1 i µ ˜ 6 nm2 nµ (n.p)−1 [(n.(p + q))−1 + (n.(k − q))−1 ] (−ie)2 2 Γ (p + q, p) = dk 2 2 2 (k − p − q) k − M − m + iε Z 1 i (−ie)2 6 nm2 nµ (n.(p + q))−1 [(n.(k + q))−1 + (n.p)−1 ] 2 + dk (k − p)2 k − M 2 − m2 + iε Pole contribution: 1 P Σ(q) = −(−ie) 16π 2 2
P Γµ (p + q, p) = −(−ie)2
6n 2m − 6 q + 4M n.q 2
1 1 16π 2 2 − ω
γµ + 2m2 6 n
1 2−ω
nµ n.pn.(p + q)
The divergent piece satisfies the Ward identity: qµ P Γµ (p + q, p) = P Σ(p) − P Σ(p + q) = −(−ie)2
1 1 16π 2 2 − ω
6 q − 2m2 6 n
1 1 − n.p + n.q n.p
The on-shell proper vertex can be written as follows1 : n σµν o u ¯(p + q) G2 [−iσµν qν 6 n] + G3 6 nQµ + F3 6 nσµν qν 6 n + γ˜µ F1 + F2 i qν u(p) 2M 1 To get this result we must add the contribution of the self energies of photon and electron to this process.
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where: γ˜µ = γµ +
m2 6 nnµ nµ , Qµ = qµ − q 2 n.q 2 n.p(n.p + n.q)
F1 , F2 .F3 , G2 , G3 are forms factors(Lorentz scalar combinations of nµ , pµ , qµ ). Under the Sim(2) scaling nµ → λnµ , F1 , F2 are invariants,F3 → λ−2 F3 ,G3 → λ−1 G3 ,G2 → λ−1 G2 . In the Non-Relativistic(NR) limit we get, keeping terms that are at most linear in qµ : NR limit
Form factor
2Me ϕ↑s ϕs A0 3m2 ↑ i ′ ˆ j qk A0 4M 2 iεijk ϕs σ ϕs n ↑ k iεijk qj ϕs σ ϕs′ Ai −2in0M εijk ϕ↑s σ k ϕs′ qj Ai 2 ↑ ˆ .~σ ϕs′ qj Ai −iεijk nk m M ϕs n i(2M εijk nk ϕ↑s σ j ϕs′ + 2Me ini ϕ↑s ϕs′ )A0 qi 2Me n0 ϕ↑s ϕs′ Qµ Aµ (−4Me εijk nk ϕ↑s ~n.~σ ϕs′ + 4Me n20 εijk ϕ↑s σ k ϕs′ )qj Ai 4Me n0 εijk nj ϕ↑s σ k ϕs′ A0 qi iεijk ϕ↑s σ k ϕs′ Ai qj m2 −i 2M ˆ j ϕ↑s σ k ϕs A0 qj 2 εijk n
F1 (0) F1 (0) F1 (0) G2 (0) G2 (0) G2 (0) G3 (0) F3 (0) F3 (0) F2 (0) F2 (0)
All form factors are evaluated at qµ = 0. Here A0 is the electric potential and Ai is the vector potential.ϕs′ 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 F2 (0)−2n0 M G2 (0)−4F3 (0)Me n20 i(See rows 11, 5 and 9 of the list). Introduce the following integrals: Z dk(n.k + n.q)a1 (n.k)a2 ((k − p)2 )a3 (k 2 − Me2 )a4 ((k + q)2 − Me2 )a5 = int(a1, a2, a3, a4, a5) Z dk(n.k + n.q)a1 (n.k)a2 ((k − p)2 )a3 (k 2 − Me2 )a4 ((k + q)2 − Me2 )a5 kµ = int 11(a1, a2, a3, a4, a5)pµ + int 12(a1, a2, a3, a4, a5)qµ + int 13(a1, a2, a3, a4, a5)nµ Z dk(n.k + n.q)a1 (n.k)a2 ((k − p)2 )a3 (k 2 − Me2 )a4 ((k + q)2 − Me2 )a5 kµ kν = int 21(a1, a2, a3, a4, a5)ηµν + int 22(a1, a2, a3, a4, a5)pµpν + int 23(a1, a2, a3, a4, a5)qµqν + int 24(a1, a2, a3, a4, a5)nµnν + int 25(a1, a2, a3, a4, a5)(pµqν + pν qµ )+ int 26(a1, a2, a3, a4, a5)(pµnν + pν nµ ) + int 27(a1, a2, a3, a4, a5)(nµqν + nν qµ )
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We get: F2 (0) − 2n0 M G2 (0) − 4F3 (0)Me n20 i = −4ie M {int 22(0, 0, −1, −1, −1) − int 11(0, 0, −1, −1, −1)}− 2
2
2ie2 m2 {− int(0, 0, −1, −1, −1) − 2 int 22(0, −1, −1, −1, −1)M n0 + 3 int 11(0, 0, −1, −1, −1)} Evaluating the integrals according to the Sim(2) invariant prescription to o(m2 ), we get: F2 − 4F3 Me n20 i − 2G2 n0 M =
α 2π
where α is the fine structure constant. Therefore to this order the QED result holds. Notice that already at tree level, the model predicts the existence of an anisotropic electric moment of the electron , corresponding to the second line of the list and an anisotropic magnetic moment of the electron, corresponding m2 to the fourth row of the list, both of the order of M 2 [14]. e
2
|~ p| =
3e m 3 m2 |(~ s × n ˆ )| ≤ λe 4Me Me2 8 Me2
(11)
where λ = 2.4 × 10−12 m is the Compton wave length of the electron. Using the best bound on the electric dipole moment of the electron[12], |~ p| < 8.7 × 10−29 ecm.,we ˙ get: m2 < 9.7 × 10−19 Me2 For the muon λ = 1.17 × 10−14 m. Using the best bound on the muon electric dipole moment[13],|~ pµ | < 1.8 × 10−19 ecm.,we ˙ get: m2µ < 4 × 10−7 Mµ2 Bounds using the experimental values of the magnetic moments are much weaker. The Sim(2) invariant regularization opens the way to explore the full quantum possibilities of VSR. They should be systematically studied, both in Particle Physics models as well as in Quantum Gravity models. In particular, possible anisotropies in the propagation of particles at cosmological distances could provide a good test of VSRSM as well as the first evidence of Lorentz invariance violation. Acknowledgements The work of J. A. is partially supported by Fondecyt 1150390, Anillo ACT 1417. J.A. wants to thank H. Morales-T´ecotl, L.F. Urrutia, D. Espriu and R. Soldati for useful remarks.
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[10] Bassetto,A.,Dalbosco,M.Lazzizzera,I. D31(1985)2012.
and
Soldati,R.,
Phys.
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[11] In the vertex calculation we used the program J.A.M.Vermaseren,New features of FORM. math-ph/0010025.
Rev
FORM:
[12] The ACME Collaboration (January 2014). ”Order of Magnitude Smaller Limit on the Electric Dipole Moment of the Electron”,Science. 343 (6168): 269272. [13] G. W. Bennett et al. (Muon (g-2) Collaboration),Phys. Rev. D 80, 052008. [14] See also A. Dunn, T. Mehen, Implications of SU (2)L × U (1)R Symmetry for SIM (2) Invariant Neutrino Masses, arXiv:hep-ph/0610202.
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