Electroweak standard model with very special relativity

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(1)PHYSICAL REVIEW D 91, 105007 (2015). Electroweak standard model with very special relativity Jorge Alfaro,* Pablo González,† and Ricardo Ávila‡ Facultad de Física, Pontificia Universidad Católica de Chile,Casilla 306, Santiago 22, Chile (Received 26 March 2014; published 6 May 2015; corrected 11 June 2015) The very special relativity electroweak Standard Model (VSR EW SM) is a theory with SUð2ÞL × Uð1ÞR symmetry, with the same number of leptons and gauge fields as in the usual Weinberg-Salam model. No new particles are introduced. The model is renormalizable and unitarity is preserved. However, photons obtain mass and the massive bosons obtain different masses for different polarizations. Besides, neutrino masses are generated. A VSR-invariant term will produce neutrino oscillations and new processes are allowed. In particular, we compute the rate of the decays μ → e þ γ. All these processes, which are forbidden in the electroweak Standard Model, put stringent bounds on the parameters of our model and measure the violation of Lorentz invariance. We investigate the canonical quantization of this nonlocal model. Second quantization is carried out, and we obtain a well-defined particle content. Additionally, we do a counting of the degrees of freedom associated with the gauge bosons involved in this work, after spontaneous symmetry breaking has been realized. Violations of Lorentz invariance have been predicted by several theories of quantum gravity [J. Alfaro, H. Morales-Tecotl, and L. F. Urrutia, Phys. Rev. Lett. 84, 2318 (2000); Phys. Rev. D 65, 103509 (2002)]. It is a remarkable possibility that the low-energy effects of Lorentz violation induced by quantum gravity could be contained in the nonlocal terms of the VSR EW SM. DOI: 10.1103/PhysRevD.91.105007. PACS numbers: 12.60.-i, 11.30.Cp, 12.15.-y, 13.15.+g. I. INTRODUCTION The SUð2ÞL × Uð1ÞR gauge theory of weak and electromagnetic interactions known as the electroweak Standard Model (EW SM) or Weinberg-Salam model is one of the most successful theories of elementary particle physics. It allows one to describe in detail an enormous amount of experimental data. Moreover, precision tests at the LHC have verified both the particle content, the gauge couplings, and the mechanism of spontaneous symmetry breaking (SSB) of the SM. The discovery of the Higgs particle with SM properties at the LHC has completed the picture, leaving a very restrictive range of parameters to be explained by new physics [1]. Therefore, any modification to the SM structure must be very subtle. Nonetheless, SM as it is cannot be the ultimate theory of nature. It does not incorporate the observed fact that neutrinos have mass and it does not incorporate gravity [2]. The main problem of the Weinberg-Salam model is the observation of neutrino flavor oscillations, which imply that neutrinos are massive.1 The SM does not provide an explanation for this fact, since neutrinos are massless in it. If Lorentz symmetry is exact, additional massive particles must be postulated as in the popular seesaw mechanism [4]. These remarks suggest that the subtle modification to the *. jalfaro@uc.cl pgonzalez@ing.uchile.cl Present address: Departamento de Física, FCFM, Universidad de Chile. Blanco Encalada 2008, Santiago, Chile. ‡ raavila@uc.cl 1 Although, it was expected since the 1980s that the neutrinos have mass [3]. †. 1550-7998=2015=91(10)=105007(15). SM must be focused on the generation of neutrino mass and neutrino oscillations, preserving both the symmetries and the particles of the SM. One possibility for including neutrino masses is to introduce a breaking of Lorentz symmetry, through constant background fields causing deviations of Lorentz symmetry [5], but in such proposals the dispersion relation for light is modified. A more conservative alternative would be to keep the essential features of special relativity, like the constancy of the velocity of light, but leave aside rotation symmetry with a subgroup of Lorentz. Such subgroups were identified and used to built what is called very special relativity (VSR) [6]. One of its main features is that the inclusion of P, T, or CP symmetries enlarges VSR to the full Lorentz group. The most interesting of these subgroups are SIM(2) and HOM(2). These subgroups do not have invariant tensor fields besides the ones that are invariant under the whole Lorentz group, and therefore the dispersion relations, time delay, and all classical tests of SR are valid too. However, a nonlocal term is necessary to formulate VSR [7]. VSR admits the generation of a neutrino mass without lepton number violation and without sterile neutrinos. Following this line of thought, in this paper we study a modification of the electroweak Standard Model using VSR as the symmetry of nature. One advantage of this model is that we do not have to include more particles than are currently known. The VSR EW SM is a simple theory with SUð2ÞL × Uð1ÞR symmetry, with the same number of leptons and gauge fields as in the usual electroweak model. However, now it is possible to introduce new mass terms that violate Lorentz invariance. These terms are nonlocal and relevant at low. 105007-1. © 2015 American Physical Society.

(2) JORGE ALFARO, PABLO GONZÁLEZ, AND RICARDO ÁVILA. energies and are able to describe in a straightforward manner the observed neutrino oscillations. The gauge theory formalism necessary to implement the VSR EW SM has been recently developed in Ref. [8]. The model is renormalizable and unitarity is preserved. In the VSR EW SM new processes are allowed, which are consistent with the available data. Neutrino oscillations have the same form as in a Lorentz-invariant theory. We also compute the decay rate of μ− > e þ γ. All of these processes, which are forbidden in the SM, put stringent bounds on the parameters of our model and measure the violation of Lorentz invariance. Violations of Lorentz invariance have been predicted by several theories of quantum gravity [9]. It is an enticing possibility that the low-energy effects of Lorentz violation induced by quantum gravity are embodied in the nonlocal terms of the VSR EW SM. We have organized the paper as follows. In Sec. II, we review the formulation of Yang-Mills fields in VSR. In Sec. III, we define the VSR EW SM gauge bosons, using the formalism of Sec. II. Besides, the VSR Weinberg-Salam model is defined using its various components: gauge, leptons, scalars, and interactions. Section IV contains the description of spontaneous symmetry breaking. The masses of gauge fields and leptons are computed. Section V contains the dispersion relations for the electron (muon, tau) for the general case mL ≠ mR . In Sec. VI we study a novel oscillation: the electron spin oscillation due to VSR. Bounds on some VSR parameters are proposed. In Sec. VII we write the lepton–gauge boson interactions and study the terms which are responsible for neutrino oscillations in the model. In Sec. VIII, we compute the decay rate for the process X− > Y þ γ (flavor changing). Compared with the best experimental bounds available today, we get more restrictions on the VSR parameters. Finally, Sec. IX contains the canonical quantization of the (nonlocal) model. We present our conclusions in Sec. X. Additionally, in Appendix A we study the gauge bosons’ equation of motion and count the degrees of freedom for both the massive and massless cases. Appendix B contains the solutions of the VSR Dirac equation for mL ≠ mR, whereas in Appendix C we obtain the solution of the VSR Dirac equation for the particular but phenomenologically important case mL ¼ mR ¼ m. Next, we use the results of Ref. [8] to build the VSR EW SM based on the SUð2ÞL × Uð1ÞR group and the SM particle representations.. We define the covariant derivative by i Dμ ϕ ¼ ∂ μ ϕ − iAμ ϕ þ m2 nμ ððn · ∂Þ−2 ðn · AÞÞϕ; 2. ð2Þ. where m is a constant with dimension of mass. It measures the departure from Lorentz invariance, since in the term containing it in Eq. (2) a fixed null vector nμ appears. This nonlocal term is invariant under SIM(2) and HOM(2) because the transformations of these subgroups of the Lorentz group change nμ at most by a multiplicative constant factor, which is canceled out by the change of nμ in the denominator. Now, we impose as usual that δðDμ ϕÞ ¼ iΛDμ ϕ:. ð3Þ. Then the gauge transformation for the gauge field is im2 n ½Λ; ððn · ∂Þ−2 ðn · AÞÞ 2 μ m2 im2 þ nμ ððn · ∂Þ−1 ΛÞ − n ððn · ∂Þ−2 ðn · ½A; ΛÞÞ: 2 2 μ ð4Þ. δΛ Aμ ¼ ∂ μ Λ − i½Aμ ; Λ −. We have also checked the closure of the algebra: ½δΛ1 ; δΛ2 Aμ ¼ −iδ½Λ1 ;Λ2 Aμ :. ð5Þ. The commutator of two covariant derivatives defines Fμν , the Aμ field strength, ½Dμ ; Dν ϕ ¼ −iFμν ϕ;. ð6Þ. so we get Fμν ¼ Aν;μ − Aμ;ν − i½Aμ ; Aν −. m2 n ððn · ∂Þ−2 ðn · A;μ ÞÞ 2 ν. m2 n ððn · ∂Þ−2 ðn · A;ν ÞÞ 2 μ im2 þ ½ððn · ∂Þ−2 ðn · AÞÞ; ðnμ Aν − nν Aμ Þ: 2 þ. ð7Þ. It is Hermitian if Aμ is Hermitian: ½D0μ ; D0ν ϕ0 ¼ U½Dμ ; Dν ϕ ¼ Uð−iFμν Þϕ ¼ ð−iF0μν ÞUϕ ð8Þ. with∶ U ¼ eiΛ : We then find that. II. NON-ABELIAN GAUGE FIELDS In this section we review the results of Ref. [8]. We consider a scalar field transforming under a non-Abelian gauge transformation with infinitesimal parameter Λ: δϕ ¼ iΛϕ:. PHYSICAL REVIEW D 91, 105007 (2015). ð1Þ. F0μν ¼ UFμν U−1 :. ð9Þ. It is not difficult to see that a redefinition given by Aμ → Aμ − 12 m2 nμ ððn · ∂Þ−2 ðn · AÞÞ eliminates the modification by the m factor. This means that a modification in the. 105007-2.

(3) ELECTROWEAK STANDARD MODEL WITH VERY SPECIAL …. ordinary covariant derivative given by Eq. (2) does not affect the observables. Thus, we will use m ¼ 0 from now on. However, VSR allows us to define a new invariant mass term for gauge fields using a new field strength: m2A 1 1 α α ~ ðn Fμα Þ − nμ ðn Fνα Þ : nν Fμν ¼ Fμν þ 2 ðn · DÞ2 ðn · DÞ2. m2 ~ μ ϕ ¼ Dμ ϕ þ 1 ϕ nμ ϕ; D 2n · D. ð11Þ. ~ μ , we can where m2ϕ is a new VSR parameter. Using D introduce different VSR masses for the various matter fields in a covariant manner. III. ELECTROWEAK STANDARD MODEL WITH VERY SPECIAL RELATIVITY In the electroweak model, we have a symmetry given by SUð2ÞL × Uð1ÞR , so a generic field ψ will transform like δψ ¼ iðΛ þ ΘÞψ;. ð12Þ. where Λ and Θ are transformation parameters under SU(2) and U(1), respectively. To define the covariant derivative, we must impose δðDμ ψÞ ¼ iðΛ þ ΘÞDμ ψ:. τi i 0Y ½Dμ ; Dν ψ ¼ −i g Fμν þ g Bμν ψ; 2 2. ð19Þ. Fiμν ¼ ∂ μ Aiν − ∂ ν Aiμ þ gεi jk Ajμ Akν ;. ð20Þ. Bμν ¼ ∂ μ Bν − ∂ ν Bμ :. ð21Þ. such that. ð10Þ We will develop the effect of this element in the next section. Finally, we define the wiggle covariant derivative of the field ϕ by. PHYSICAL REVIEW D 91, 105007 (2015). But, using Eq. (10), we can define 2 ~Fiμν ¼ Fiμν þ mA nν 1 ðnα Fiμα Þ − nμ 1 ðnα Fiνα Þ ; 2 ðn · DÞ2 ðn · DÞ2 ð22Þ 2 ~Bμν ¼ Bμν þ mB nν 1 ðnα Bμα Þ − nμ 1 ðnα Bνα Þ : 2 ðn · ∂Þ2 ðn · ∂Þ2 ð23Þ Now, we have all the elements to build the WeinbergSalam model in VSR. For this we need the gauge fields Bμ and Aiμ , three families of leptons, and a scalar field to implement the Higgs mechanism. Then, we have the following Lagrangians. I) Gauge Lagrangian: The gauge Lagrangian contains two kinds of gauge fields, Bμ and Aiμ . To write the Lagrangian, we use the modified field strengths given by Eqs. (22) and (23). Then, 1 1 ~ ~ μν Lgauge ¼ − F~ iμν F~ μν i − Bμν B : 4 4. ð13Þ. ð24Þ. We can prove that We saw in the last section that for VSR the covariant derivative is not modified. Then, 0. Dμ ψ ¼ ∂ μ ψ − ig Bμ ψ − igAμ ψ; where Aμ ¼. τi 2. Aiμ. and Bμ ¼. Y 2 Bμ .. ð14Þ. Besides, we have. 1 δAμ ¼ ∂ μ Λ − i½Aμ ; Λ; g 1 δBμ ¼ 0 ∂ μ Θ; g. ð25Þ. B~ μν B~ μν ¼ Bμν Bμν þ 2m2B ðnα Bμα Þðn · ∂Þ−2 ðnβ Bμβ Þ:. ð26Þ. Therefore, the Lagrangian is now ð15Þ ð16Þ. 1 m2A α i ðn Fμα Þðn · DÞ−2 ðnβ Fμβ Lgauge ¼ − Fiμν Fμν i − i Þ 4 2 1 m2 − Bμν Bμν − B ðnα Bμα Þðn · ∂Þ−2 ðnβ Bμβ Þ: ð27Þ 4 2. ð17Þ. From this Lagrangian we can see that the equation of motion of Bμ is. ð18Þ. ∂ ν Bμν − m2B nμ ðn · ∂Þ−2 ∂ α ðnβ Bαβ Þ þ m2B ðn · ∂Þ−1 ðnβ Bμβ Þ ¼ 0:. or, by taking the Lie algebra components, 1 δAiμ ¼ ∂ μ ϵi þ εi jk Ajμ ϵk ; g 1 δBμ ¼ 0 ∂ μ ϵ0 ; g. μν μβ i 2 α i −2 F~ iμν F~ μν i ¼ Fμν Fi þ 2mA ðn Fμα Þðn · DÞ ðnβ Fi Þ;. where εi jk is the Levi-Civita symbol and we used Λ ¼ τ2i ϵi and Θ ¼ Y2 ϵ0 . The ordinary field strength in VSR for both gauge fields is given by. ð28Þ Now, if we contract this equation with nμ , we obtain ∂ ν ðnμ Bμν Þ ¼ 0, so. 105007-3.

(4) JORGE ALFARO, PABLO GONZÁLEZ, AND RICARDO ÁVILA. ∂ ν Bμν þm2B ðn·∂Þ−1 ðnβ Bμβ Þ ¼ 0; → ∂ 2 Bμ −∂ μ ∂ ν Bν þm2B Bμ −m2B ðn·∂Þ−1 ∂ μ ðn·BÞ ¼ 0; ð29Þ. PHYSICAL REVIEW D 91, 105007 (2015). hypercharge Y ¼ −1 and the singlets have Y ¼ −2. So, using Eq. (14), we have ig0 τi i Dμ La ¼ ∂ μ þ Bμ − ig Aμ La ; 2 2. and ∂ ν ðnμ Bμν Þ ¼ 0; 2. ν. → ∂ ðn · BÞ − ðn · ∂Þ∂ ν B ¼ 0:. Dμ Ra ¼ ð∂ μ þ ig0 Bμ ÞRa : ð30Þ. On the other hand, we need to fix the gauge freedom. We can use the Lorentz gauge plus an additional restriction from VSR, which is given by ∂ μ Bμ ¼ 0;. ð31Þ. nμ Bμ ¼ 0:. ð32Þ. We will see that m2L is the mass matrix of neutrinos, which generates the oscillation between the different families. III) Scalar Lagrangian: The scalar Lagrangian contains a complex doublet scalar field ϕ¼. If we use this in Eqs. (29) and (30), we obtain ∂ 2 Bμ þ m2B Bμ ¼ 0:. . ν0aL e0aL. ϕþ ; ϕ0. ðϕÞ. ~ μ ϕÞ† ðD ~ μ ϕÞ − Vðϕ† ϕÞ Lscalar ¼ ðD ðϕÞ. ð33Þ. From this equation we can see that Bμ has mass mB . We can obtain a similar result from the free equation of motion of Aiμ , where the mass is mA . Therefore, the Lagrangian (27) describes massive gauge fields, but we will see that it preserves two degrees of freedom (see Appendix A). In Sec. IVA, we will study the free dynamic using Eq. (27) after spontaneous symmetry breaking is realized. II) Leptonic Lagrangian: The leptonic Lagrangian contains three SU(2) doublets La ¼. ð35Þ. ¼ ðDμ ϕÞ† ðDμ ϕÞ − m2ϕ ϕ† ϕ − Vðϕ† ϕÞ; with Vðϕ† ϕÞ ¼ μ2 ϕ† ϕ þ λðϕ† ϕÞ2 :. ð36Þ. We notice that the term proportional to m2ϕ can be absorbed by redefining μ2 þ m2ϕ → μ2 . Therefore, our scalar Lagrangian is reduced to. ;. Lscalar ¼ ðDμ ϕÞ† ðDμ ϕÞ − μ2 ðϕ† ϕÞ − λðϕ† ϕÞ2 :. ð37Þ. The hypercharge of ϕ is Y ¼ 1, so. where ν0aL ¼ 12 ð1 − γ 5 Þν0a and e0aL ¼ 12 ð1 − γ 5 Þe0a , and three SU(2) singlets Ra ¼ e0aR ¼ 12 ð1 þ γ 5 Þe0n . As is usual, we make the assumption that there is no right-handed neutrino. The index a represents the different families and the index 0 shows that the fermionic fields are the physical fields before breaking the symmetry of the vacuum. The Lagrangian is ba μ ~ ðRÞ ba R ~ ðLÞ Llepton ¼ L̄b iγ μ ½D μ La þ R̄b iγ ½D μ a 1 2 ba μ ba α −1 La ¼ L̄b iγ δ Dμ þ ½mL nμ ðn Dα Þ 2 1 þ R̄b iγ μ δba Dμ þ ½m2R ba nμ ðnα Dα Þ−1 Ra 2 i ¼ iL̄a DLa þ L̄b n½m2L ba ðnα Dα Þ−1 La þ iR̄a DRa 2 i þ R̄b n½m2R ba ðnα Dα Þ−1 Ra ; ð34Þ 2. where m2L and m2R are Hermitian matrices in family indices (ba), and they could depend on γ 5 . The doublets have. Dμ ϕ ¼. ig0 τ ∂ μ − Bμ − ig i Aiμ ϕ: 2 2. ð38Þ. From Eq. (38), we will obtain the masses of each field after spontaneous symmetry breaking. For the moment, the dynamics of the Higgs is not important, so we will focus on the gauge and lepton fields. IV) Interaction Lagrangian: The interaction Lagrangian is Lint ¼ −½Γba L̄b ϕRa − ½Γ† ba R̄b ϕ† La ;. ð39Þ. where Γ is a matrix associated with the Yukawa interaction. Therefore, the final Lagrangian is given by L ¼ Lgauge þ Llepton þ Lscalar þ Lint :. ð40Þ. Now, we can proceed to break the symmetry using the Higgs mechanism.. 105007-4.

(5) ELECTROWEAK STANDARD MODEL WITH VERY SPECIAL …. IV. SPONTANEOUS SYMMETRY BREAKING To break the symmetry, we need to find the vacuum of the system. For this we search for a solution to ∂V ∂ϕ ¼ 0. From Eq. (36) it can be noted that this occurs for hϕi ¼ 0 or 1 0 hϕi ¼ pffiffiffi ; 2 v qffiffiffiffiffiffi 2 where v ¼ −μλ . It is useful to work in the unitary gauge. In this gauge, the Goldstone bosons are removed from the Lagrangian by a gauge transformation. After the gauge transformation, we can use ϕ¼. Thus, the operator Q ¼ T 3L þ Y2 is still a symmetry of the vacuum. This means that the gauge field associated to Q will keep its original mass before SSB, the VSR mass mG . Please see Appendix A Eq. (A9). A. Gauge fields After spontaneous symmetry breaking, we have contributions to the quadratic term in the gauge fields from Eqs. (27) and (37). Then, 1 i μν m2A α i μβ −2 Lgauge free ¼ − Fμν Fi − ðn Fμα Þðn · DÞ ðnβ Fi Þ 4 2 ð2Þ 1 m2 − Bμν Bμν − B ðnα Bμα Þðn · ∂Þ−2 ðnβ Bμβ Þ 4 2 v2 þ 0 1 ðg0 Bμ þ gτi Aiμ Þ† 8 0 j 0 × ðg Bμ þ gτj Aμ Þ ; 1. 0 ; vþH pffiffi 2. where H is the Higgs boson. From the nonzero value for the vacuum, we have a new quadratic term in the fields, so they will obtain an additional contribution to the mass. To compute the mass, we will study the free part in the Lagrangian for each field. Notice that 0 0 Y 3 TL þ ¼ ¼ Qhϕi: 2 v 0. ð41Þ. PHYSICAL REVIEW D 91, 105007 (2015). where the Pauli matrices are 0 1 0 −i ; τ2 ¼ ; τ1 ¼ 1 0 i 0. τ3 ¼. 1 0 0 −1. ;. ð42Þ. ð43Þ. and the subindex (2) means that we keep up to quadratic terms in the field. Evaluating Eqs. (20), (21), and (43) in Eq. (42), we obtain. 1 Lgauge free ¼ Aiμ ðð∂ 2 þ m2A Þδμν − ð∂ μ þ m2A ðn · ∂Þ−1 nμ Þð∂ ν þ m2A ðn · ∂Þ−1 nν Þ þ m2A nμ nν ðn · ∂Þ−2 ð∂ 2 þ m2A ÞÞAνi 2 1 þ Bμ ðð∂ 2 þ m2B Þδμν − ð∂ μ þ m2B ðn · ∂Þ−1 nμ Þð∂ ν þ m2B ðn · ∂Þ−1 nν Þ þ m2B nμ nν ðn · ∂Þ−2 ð∂ 2 þ m2B ÞÞBν 2 v2 g2 μ 1 v2 ðA1 Aμ þ Aμ2 A2μ Þ þ ðg0 Bμ − gA3μ Þðg0 Bμ − gAμ3 Þ: þ 8 8. ð44Þ. Now, to diagonalize this Lagrangian, we study the case when mA ¼ mB ≡ mG and 1 − A1μ ¼ pffiffiffi ðW þ μ þ W μ Þ; 2. i − A2μ ¼ pffiffiffi ðW þ μ − W μ Þ; 2. g0 Aμ − gZμ A3μ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; g2 þ g02. gAμ þ g0 Zμ Bμ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : g2 þ g02. ð45Þ. Then, Lgauge free. v 2 g2 μ 2 2 ∂ þ mG þ ¼ δν − ð∂ μ þ m2G ðn · ∂Þ−1 nμ Þð∂ ν þ m2G ðn · ∂Þ−1 nν Þþm2G nμ nν ðn · ∂Þ−2 ð∂ 2 þ m2G ÞÞW νþ 4 1 v2 ðg2 þ g02 Þ μ δν − ð∂ μ þ m2G ðn · ∂Þ−1 nμ Þð∂ ν þ m2G ðn · ∂Þ−1 nν Þ ∂ 2 þ m2G þ þ Zμ 2 4 1 þm2G nμ nν ðn · ∂Þ−2 ð∂ 2 þ m2G ÞÞZν þ Aμ ðð∂ 2 þ m2G Þδμν − ð∂ μ þ m2G ðn · ∂Þ−1 nμ Þð∂ ν þ m2G ðn · ∂Þ−1 nν Þ 2 2 μ −2 2 2 ν þmG n nν ðn · ∂Þ ð∂ þ mG ÞÞA : ð46Þ W −μ. 105007-5.

(6) JORGE ALFARO, PABLO GONZÁLEZ, AND RICARDO ÁVILA. PHYSICAL REVIEW D 91, 105007 (2015). From this Lagrangian, we can find the free equations of motion. They are v 2 g2 2 2 2 −1 2 −1 W ∂ þ mG þ μ − ð∂ μ þ mG ðn · ∂Þ nμ Þðð∂ · W Þ þ mG ðn · ∂Þ ðn · W ÞÞ 4 þ m2G nμ ðn · ∂Þ−2 ð∂ 2 þ m2G Þðn · W Þ ¼ 0;. ð47Þ. v2 ðg2 þ g02 Þ 2 2 ∂ þ mG þ Zμ − ð∂ μ þ m2G ðn · ∂Þ−1 nμ Þðð∂ · ZÞ þ m2G ðn · ∂Þ−1 ðn · ZÞÞ 4 þ m2G nμ ðn · ∂Þ−2 ð∂ 2 þ m2G Þðn · ZÞ ¼ 0;. ð48Þ. ð∂ 2 þ m2G ÞAμ − ð∂ μ þ m2G ðn · ∂Þ−1 nμ Þðð∂ · AÞ þ m2G ðn · ∂Þ−1 ðn · AÞÞ þ m2G nμ ðn · ∂Þ−2 ð∂ 2 þ m2G Þðn · AÞ ¼ 0:. Notice that all of the gauge field equations are have the form. ð49Þ. polarizations. It is certainly a prediction that should be investigated in appropriate experiments, for example at the LHC.. ð∂ 2 þ M2 ÞV μ − ð∂ μ þ m2G ðn · ∂Þ−1 nμ Þ B. Lepton fields. × ðð∂ · VÞ þ m2G ðn · ∂Þ−1 ðn · VÞÞ þ m2G nμ ðn · ∂Þ−2 ð∂ 2 þ m2G Þðn · VÞ ¼ 0;. ð50Þ. where M is a mass term. Using the result from Appendix A, we obtain the following results. (i) W degrees of freedom. The mass is μ has qthree ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi v2 g2. 2 MW ¼ 4 þ mG for polarizations perpendicular vg to nμ , and MW ¼ 2 for the longitudinal polarization. (ii) q Zμffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi has three degrees of freedom. The mass is MZ ¼ ffi v2 ðg2 þg02 Þ 4. þ m2G for polarizations perpendicular pffiffiffiffiffiffiffiffiffiffi v g2 þg02 to nμ , and M Z ¼ for the longitudinal 2 polarization. (iii) Aμ has two degrees of freedom and the mass is only MA ¼ mG. One thing that we must consider is the fact that the photon gains mass in VSR. Of course, this mass must be very tiny. Some widely accepted bounds for the photon mass are as follows. (i) The most accepted bound uses a magnetohydrodynamics argument concerning the properties of the Solar wind at Pluto’s orbit, giving mG ≤ 10−18 eV [10]. (ii) Measures of the galactic magnetic field are only possible if the photon mass is zero; this has given a constraint of mG ≤ 3 × 10−27 eV [11]. On the other hand, the W μ and Zμ bosons will exhibit different propagations for perpendicular and longitudinal polarizations (just like birefringence), but the difference is extremely small since it depends on the photon mass mG . The bounds of the photon mass give us a great idea of the similarity of the masses of W μ and Zμ for different. In order to see what happens to the leptons, we look at the diagonal (in flavor) part of Lint . In particular, we will study the electron family in more detail. Lint is now Gv Lint ¼ − peffiffiffi ðēR eL þ ēL eR Þ þ higher-order terms: 2. ð51Þ. To determine the mass eigenstates we look at the equations of motion provided by the quadratic piece of the Lagrangian. Introducing ψ¼. eR ; eL. we get 1 2 Ge v −1 i ∂ þ nm̄ ðn · ∂Þ − pffiffiffi ψ ¼ 0; 2 2 5. ð52Þ 5. 1þγ where m̄2 ¼ m2R PR þ m2L PL , PL ¼ 1−γ 2 , and PR ¼ 2 . For the neutrino, we get. m2L −1 i ∂þ νL ¼ 0: nðn · ∂Þ 2. ð53Þ. That is, the neutrino mass is mν ¼ mL .2 The only pole that the neutrino propagator has is at p2 ¼ m2L . Please see Appendix D.. 105007-6. 2.

(7) ELECTROWEAK STANDARD MODEL WITH VERY SPECIAL …. V. DISPERSION RELATIONS FOR mL ≠ mR In this section, we write the solution of the VSR Dirac equation for the electron for the case mL ≠ mR . The particle solution is 1 2 Ge v −1 p − nm̄ ðn · pÞ − pffiffiffi us ¼ 0; 2 2. smaller than the contribution to the electron mass given by SSB, in a perturbative approach it makes sense to use the spinors that solve the Dirac equation with mL ¼ mR ¼ 0 to describe the initial and final states of the electron. So, the probability of measuring spin down will be proportional to. ð54Þ. Pð↑ → ↓Þ ¼ 2R2 ð1 − cos ððE↑ − E↓ ÞtÞÞ ðE↑ − E↓ Þt 2 2 ¼ 4R sin 2. where u1 ¼. 1 Gv p þ peffiffiffi nU 1 ; 2ðn · pÞ 2. p2 ¼ m2R þ. G2e v2 2. with. and p0 > 0;. p2. ¼. m2L. G2 v2 þ e 2. for a certain constant R, and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2L þ M2 − m2R þ M2 1 m2L 1 m2R −1− ≃M 1þ 2 M2 2 M2. ð55Þ. 1 Ge v p ffiffi ffi u2 ¼ pþ nU 2 ; 2ðn · pÞ 2. E↑ − E↓ ¼. with. and p0 > 0:. ðm2L − m2R Þ ; 2M 2π jm2L − m2R j ¼ ; T 4M 8πM : →T¼ 2 jmL − m2R j ¼. ð56Þ. The antiparticle solution is 1 2 Ge v −1 p ffiffi ffi vs ¼ 0; p − nm̄ ðn · pÞ þ 2 2. ð57Þ. where 1 Ge v p ffiffi ffi p− nU 1 ; v1 ¼ 2ðn · pÞ 2 p2 ¼ m2R þ. G2e v2 2. PHYSICAL REVIEW D 91, 105007 (2015). with. and p0 > 0;. 1 Ge v v2 ¼ p − pffiffiffi nU 2 ; with 2ðn · pÞ 2 2 2 G v p2 ¼ m2L þ e and p0 > 0: 2. ð58Þ. ð60Þ. To put some bound on this effect, we can imagine that the anisotropy of VSR has a cosmological origin, perhaps a primordial magnetic field. Such fields B have been bounded by 10−17 G<B<10−9 G [12]. Assuming that these primordial magnetic fields induce the electron spin flip, we get the estimation jm2L − m2R j ¼ gμB B ≲ 10−17 eV 2M → jm2L − m2R j ≲ 10−11 eV2 :. jE↑ − E↓ j ¼. ð59Þ. U 1 and U 2 are constant spinors (see Appendix B). Therefore, if mL ≠ mR , the electron, muon, and tau are actually composed of two different particles with slightly different masses. Due to this, in the next section we explore a novel electron oscillation and put some plausible bounds on jmL − mR j. VI. ELECTRON-SPIN PRECESSION Consider an electron at rest with spin up in the z direction. Since mL and mR are expected to be much. ð61Þ. This bound is very strong. This means that mL ¼ mR is probably an excellent approximation in almost every case. However, the possibility of an electron-spin precession must not be ignored. VII. LEPTON–GAUGE BOSON INTERACTIONS We now consider three lepton families eb ; νb ; b ¼ 1…3. Keeping up to first-order terms in the gauge fields (because higher-order terms are strongly suppressed by the smallness of the gauge coupling and the mass terms introduced by VSR), the interaction terms in Fourier space are. 105007-7.

(8) JORGE ALFARO, PABLO GONZÁLEZ, AND RICARDO ÁVILA. Lint lept;gauge. ð1 − γ 5 Þm̄2L þ ð1 þ γ 5 Þm̄2R ; m̄2 ¼ 2. νM L ¼ V l νL ;. ð64Þ. ð63Þ. and m̄2R. are 3 × 3 nondiagonal Hermitian matrices. To zeroth order in m̄2 , from Eq. (62) we can see that eL , eR , and νL are the flavor states, and eL and eR are the mass states, but νL is not a mass eigenstate because its leading nonzero mass is m̄. 0. ð62Þ. Then, for neutrinos the relation between both states is. where k and q are the momentum of the gauge and lepton 5 fields, respectively, we have used uR;L ¼ 1γ 2 , and. where m̄2L. PHYSICAL REVIEW D 91, 105007 (2015). gg n 2 ba ba μ −1 −1 μ ¼ − pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ēb δ γ þ ½m̄ ðn · ðk þ qÞÞ ðn · qÞ n ea Aμ 2 g2 þ g02 g02 n − pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ēb δba γ μ þ ½m̄2 ba ðn · ðk þ qÞÞ−1 ðn · qÞ−1 nμ ea Zμ 2 g2 þ g02 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 2 ba g2 þ g02 ba μ −1 −1 μ þ ēbL δ γ þ ½m̄L ðn · ðk þ qÞÞ ðn · qÞ n eaL Zμ 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 02 n 2 ba g þg ba μ −1 −1 μ − ν̄bL δ γ þ ½m̄L ðn · ðk þ qÞÞ ðn · qÞ n νaL Zμ 2 2 g n 2 ba ba μ −1 −1 μ p ffiffi ffi ν̄bL δ γ þ ½m̄L ðn · ðk þ qÞÞ ðn · qÞ n eaL W þ − μ 2 2 g n − pffiffiffi ēbL δba γ μ þ ½m̄2L ba ðn · ðk þ qÞÞ−1 ðn · qÞ−1 nμ νaL W −μ ; 2 2 0. where νM L is the mass state and V l is a unitary transformation, such that V l m̄2L V †l is a diagonal matrix. Because the mass and flavor states of the neutrinos are not the same, we will have an oscillation between different states, where V l is the mixing matrix, which corresponds to the Pontecorvo-Maki-Nakagawa-Sakata matrix:. c13 c12. e−iδ s13. c13 s12. 1. B V †l ¼ @ −s23 s13 c12 eiδ − c23 s12. −s23 s13 s12 eiδ þ c23 c12. C s23 c13 A;. −c23 s13 c12 e þ s23 s12. −c23 s13 s12 e − s23 c12. c23 c13. iδ. with cij ¼ cosðθij Þ and sij ¼ sinðθij Þ. Therefore, a VSR nondiagonal mass matrix term is a natural form to describe neutrino oscillations. On the other hand, if we include the terms with VSR mass in Eq. (62), we can see that we need a nonlocal unitary transformation to diagonalize the interactions and obtain the flavor states. Actually, this means that we will have oscillations in all of the leptons. However, introducing this unitary transformation is complicated and unnatural, so we will think of these terms as new, very small interactions of the order of VSR parameters. These interactions will generate transitions from one family to another. This is the subject of the next section.. iδ. where e is the electron charge. So, we have the process given by Fig. 1, X → Y þ γ. If X and Y are leptons with mX > mY , the corresponding decay rate is given by. dΓ ¼ 4παðj½m̄2L ðXYÞ j2 þ j½m̄2R ðXYÞ j2 Þ ×. One of the flavor-changing interactions is e − ēm ½m̄2 mn nðn · ðk þ qÞÞ−1 ðn · AÞðn · qÞ−1 en ; 2. ðni ϵpi ðkÞÞðnj ϵpj ðkÞÞ ðnl pl Þðnm qm Þ. ð2πÞ4 δð4Þ ðq − p − kÞ d3 k d3 p ; 2mX ð2πÞ3 2Eγ ðkÞ ð2πÞ3 2EY ðpÞ 2. VIII. X → Y þ γ. ð65Þ. e 1 ≃ 137 is the fine-structure constant, Eγ ðkÞ ¼ where α ¼ 4π pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 k, EY ðpÞ ¼ p þ m2Y , ϵpi is the photon polarization, and ni is the space component of the null vector nμ. We can choose j~nj ¼ 1. Because of n~ , this decay has a privileged direction, given by the polarization. We will study the unpolarized case, so we must sum over p and use. 105007-8.

(9) ELECTROWEAK STANDARD MODEL WITH VERY SPECIAL …. PHYSICAL REVIEW D 91, 105007 (2015). So, we can compute the branching ratio given by BðX; YÞ ¼ ¼. ΓðX → Y þ γÞ ΓðX → Y þ ν̄Y þ νX Þ 48π 3 αðj½m̄2L ðXYÞ j2 þ j½m̄2R ðXYÞ j2 Þ. m8X G2F 4 mP 10 3 ¼ 47.05 × 10 π α mX 2 ½m̄L ðXYÞ 2 ½m̄2R ðXYÞ 2 : × þ m2X m2X . FIG. 1. X decay to Y plus photon, where ðX; YÞ ¼ ðe; τ; μÞ and mX > mY .. ð68Þ. Using Eq. (65), we can see that s23 s13 ðc212 −s212 Þ þs12 c23 c12 cosðδÞ δm2 j½m̄2L ðeμÞ j2 ¼c213 2 2 þs23 s13 Δm2 þc223 c213 s212 c212 sin2 ðδÞðδm2 Þ2 ; ð69Þ m2 þm2. where δm2 ¼ m22 − m21 and Δm2 ¼ m23 − 1 2 2 , where mi are the neutrino masses corresponding to different families. In this model, the mixing angles have the usual values, namely [13] s212 ¼ 0.307; 0.0241ðNHÞ; 2 s13 ¼ 0.0244ðIHÞ; 0.386ðNHÞ; s223 ¼ 0.392ðIHÞ; 1.08πðNHÞ; δ¼ 1.09πðIHÞ;. FIG. 2. X decay to Y plus neutrino-antineutrino, where ðX; YÞ ¼ ðe; τ; μÞ and mX > mY .. X ki kj : ϵpi ðkÞϵpj ðkÞ ¼ δij − ~ jkj p. Now, if we evaluate Γ in the X particle rest frame, such ~ with mX ≫ mY , and we use n~ ¼ ẑ, we that q ¼ ðmX ; 0Þ obtain ΓðX → Y þ γÞ ≃. αðj½m̄2L ðXYÞ j2 þ j½m̄2R ðXYÞ j2 Þ 4m3X. where (NH) is the normal hierarchy and (IH) is the inverted hierarchy. So, if we evaluate this in Eq. (69), we obtain . :. j½m̄2L ðeμÞ j. ð66Þ. On the other hand, we have the known process X → Y þ ν̄Y þ νX (see Fig. 2), where the decay rate is G2 m5 ΓðX → Y þ ν̄Y þ νX Þ ≃ F X3 ; 192π. δm2 ¼ 7.54 × 10−5 ½eV2 ; 2.43 × 10−3 ½eV2 ðNHÞ; 2 Δm ¼ 2.42 × 10−3 ½eV2 ðIHÞ;. ð67Þ. where GF ¼ 1.01 × 10−5 m−2 P and mP is the proton mass.. ¼. 2.07 × 10−4 ½eV2 ðNHÞ; 2.10 × 10−4 ½eV2 ðIHÞ;. ð70Þ. and the branching ratio with Eq. (70) is B¼. 4.43 × 10−25 ½eV2 ðNHÞ; 4.56 × 10−25 ½eV2 ðIHÞ:. The best upper limit on the branching ratio is Bðμ; eÞ < 5.7 × 10−13 [14], so the predicted branching ratio is much smaller than the current experimental bound.. 105007-9.

(10) JORGE ALFARO, PABLO GONZÁLEZ, AND RICARDO ÁVILA. and the equation of motion is given by. IX. CANONICAL QUANTIZATION Since VSR introduces nonlocal terms, it is quite important to show how canonical quantization works in this case. Moreover, we have to check that it gives the same results as the path-integral quantization. Since nonlocality means that the equations of motion are integral-differential equations, we have to fix a point of view with regards to the quantization of such theories. Our perspective is to accept the results of the path-integral quantization, which is a Lagrangian quantization and is better defined in this case. However, to understand the particle content of the model, we must have a canonical formulation in terms of creation and annihilation operators. With this in mind, we will develop a canonical second quantization, such that it agrees with the path-integral quantization, although we will have to introduce anticommutation rules that are noncanonical. We believe that this is due to the presence of second-class constraints, and thus it is the Dirac bracket that defines the anticommutation relations instead of the Poisson bracket. Aside from this subtle point (which needs to be clarified in the future), we obtain a canonical second quantization which permits a particle interpretation of the model. The propagator defined in the canonical formulation coincides with the pathintegral result. The creation operators describes particles of definite momentum and charge. We start from a local leptonic Lagrangian [15], where the canonical formalism is well defined, L ¼ iψ̄∂ψ − Mψ̄ψ þ iχ̄ðn · ∂Þϕ þ iϕ̄ðn · ∂Þχ im im im im þ χ̄nψ − ψ̄nχ þ ϕ̄ψ − ψ̄ϕ; 2 2 2 2. m2 −1 i ∂þ − M ψ ¼ 0: nðn · ∂Þ 2. Pψ ¼ iψ † ; Pχ ¼ in0 ϕ̄; Pϕ ¼ in0 χ̄; and the local Hamiltonian is given by m m 0 i HL ¼ −Pψ γ γ ∂ i ψ þ iMψ − nχ − ϕ 2 2 Pϕ i Pχ m m þ n ∂ i ϕ − nψ þ ni ∂ i χ − ψ : 2 2 n0 n0. ð75Þ. ð76Þ. Using the canonical commutation relations fψðxÞi ; Pψ ðx0 Þj geqt ¼ iδð~x − x~0 Þδij ; fϕðxÞi ; Pϕ ðx0 Þj geqt ¼ iδð~x − x~0 Þδij ; fχðxÞi ; Pχ ðx0 Þj geqt ¼ iδð~x − x~0 Þδij. ð77Þ. and Eq. (76), we reproduce the Lagrangian equations of motion (72). A. Second quantization. ð71Þ. The most general solution of Eq. (74) is X ~Þ þ ψ̂ −ðsÞ ðt; x~ÞÞ; ψðt; x~Þ → ψ̂ðt; x~Þ ¼ ðψ̂ þ ðsÞ ðt; x. ψ̂ −ðsÞ ðt; x~Þ ð72Þ. from which we can deduce ϕ¼−. ð78Þ. s. ~Þ ψ̂ þ ðsÞ ðt; x. i∂ψ − Mψ −. m ðn · ∂Þ−1 nψ; 2 m χ ¼ − ðn · ∂Þ−1 ψ; 2 m ϕ̄ ¼ − ðn · ∂Þ−1 ψ̄n; 2 m χ̄ ¼ − ðn · ∂Þ−1 ψ̄; 2. ð74Þ. Now, the canonically conjugated variables are. where ψ is the lepton field, ϕ and χ are auxiliary fields, and nμ ¼ ðn0 ; ni Þ, so that ∣~n∣ ¼ ∣n0 ∣ and nμ ¼ ðn0 ; −ni Þ. The Lagrangian equations of motion are im im nχ − ϕ ¼ 0; 2 2 m ðn · ∂Þϕ þ nψ ¼ 0; 2 m ðn · ∂Þχ þ ψ ¼ 0; 2. PHYSICAL REVIEW D 91, 105007 (2015). Z ¼ Z ¼. d3 k 1 ~x ~ ðsÞ ðkÞe ~ −iEðkÞtþik·~ pffiffiffiffiffiffiffiffiffiffiffiffiffi âs ðkÞu ; 3 ð2πÞ 2EðkÞ d3 k 1 ~x ~ ðsÞ ðkÞe ~ iEðkÞt−ik·~ pffiffiffiffiffiffiffiffiffiffiffiffiffi b̂†s ðkÞv ; 3 ð2πÞ 2EðkÞ. ð79Þ. where â† and b̂† are the creation operators for particles ~ and vðsÞ ðkÞ ~ are and antiparticles, respectively, and uðsÞ ðkÞ solutions of (see Appendix C) m2 −1 k− nðn · kÞ − M uðsÞ ¼ 0; 2 m2 k− nðn · kÞ−1 þ M vðsÞ ¼ 0: 2. ð73Þ. 105007-10. Using the equations of motion, we get.

(11) ELECTROWEAK STANDARD MODEL WITH VERY SPECIAL …. Z. HL ¼ i Z. PHYSICAL REVIEW D 91, 105007 (2015). m 0 m 0 m 0 m 0 3 † 0 i i i d x −ψ γ γ ∂ i ψ þ iMψ − nγ χ − γ ϕ þ χ̄ n ∂ i ϕ − γ nψ þϕ̄ n ∂ i χ − γ ψ 2 2 2 2. d3 xðψ † γ 0 ðγ 0 ∂ 0 ψÞ þ χ̄ðn0 ∂ 0 ϕÞ þ ϕ̄ðn0 ∂ 0 χÞÞ Z n0 m2 3 † −1 † 0 −1 ¼ i d x ψ ∂0ψ þ ððn · ∂Þ ψ Þγ nððn · ∂Þ ∂ 0 ψÞ : 2. ¼i. From Eq. (80), we see that ψ and ψ † are no longer canonically conjugated variables as in Eq. (77). Instead, they satisfy the anticommutation relations given by Eq. (88). Finally, if we use the properties given by Eq. (C3), we obtain X Z d3 k ~ âs ðkÞ ~ − b̂s ðkÞ ~ b̂†s ðkÞÞ ~ HL ¼ EðkÞðâ†s ðkÞ ð2πÞ3 s X Z d3 k ~ âs ðkÞ ~ þ b̂†s ðkÞ ~ b̂s ðkÞÞ: ~ EðkÞðâ†s ðkÞ ð81Þ ¼ 3 ð2πÞ s Besides, we can prove that the electric charge is Z n m2 Q ¼ d3 x ψ † ψ þ 0 ððn · ∂Þ−1 ψ † Þγ 0 nððn · ∂Þ−1 ψÞ : 2. where we used ∂ μ ¼ ð∂ 0 ; −∂ i Þ. Then, following the same procedure that we used to calculate Eqs. (81) and (83), we obtain. P ¼ i. ¼. Now, the energy-momentum tensor is im2 −1 nðn · ∂Þ − M ψ T μν ¼ iψ̄γ ν ∂ μ ψ − ημν ψ̄ i∂ þ 2 þ. im2 ððn · ∂Þ−1 ψ̄Þnnν ∂ μ ððn · ∂Þ−1 ψÞ; 2. ð84Þ. where γ μ ¼ ðγ 0 ; −γ i Þ. Then, the momentum operator is. s. d3 k i † ~ ~ − b̂s ðkÞ ~ b̂†s ðkÞÞ ~ k ðâs ðkÞâs ðkÞ ð2πÞ3. s. d3 k i † ~ ~ þ b̂†s ðkÞ ~ b̂s ðkÞÞ: ~ k ðâs ðkÞâs ðkÞ ð2πÞ3. XZ. ð86Þ. Assuming the standard anticommutation relations for the creation and annihilation operators, fas ðpÞ; ar ðqÞ↑ g ¼ ð2πÞ3 δð3Þ ðp − qÞ; fbs ðpÞ; br ðqÞ↑ g ¼ ð2πÞ3 δð3Þ ðp − qÞ;. ð87Þ. we compute h0jψ a ðxÞψ̄ b ðyÞj0i Z d3 p 1 X usa ðpÞūsb ðpÞe−ip·ðx−yÞ ¼ ð2πÞ3 2Ep s Z d3 p 1 m2 n −ip·ðx−yÞ ¼ pþM− e 2 ðn · pÞ ab ð2πÞ3 2Ep Z m2 n d3 p 1 −ip·ðx−yÞ ¼ i∂ x þ M þ i e 2 ðn · ∂ x Þ ab ð2πÞ3 2Ep and. Z. d3 xT i0 Z im2 3 0 ¼ d x −iψ̄γ ∂ i ψ − ððn · ∂Þψ̄Þnn0 ∂ i ððn · ∂ÞψÞ 2 Z ¼ −i d3 x ψ † ∂ i ψ. Pi ¼. n0 m2 −1 † 0 −1 þ ððn · ∂Þ ψ Þγ nððn · ∂Þ ∂ i ψÞ ; 2. XZ. B. Propagator. ð82Þ Notice that in Eq. (82) the same combination of ψ and ψ † appears as in Eq. (80). This must be so, because Q will generate a U(1) gauge transformation of the field upon using the anticommutation relations. We can use Eq. (78) to obtain X Z d3 k † ~ âs ðkÞ ~ þ b̂s ðkÞ ~ b̂†s ðkÞÞ ~ ðâs ðkÞ Q¼ 3 ð2πÞ s X Z d3 k ~ âs ðkÞ ~ − b̂†s ðkÞ ~ b̂s ðkÞÞ: ~ ðâ†s ðkÞ ð83Þ ¼ 3 ð2πÞ s. ð80Þ. ð85Þ. h0jψ̄ b ðyÞψ a ðxÞj0i Z d3 p 1 X v̄sb ðpÞvsa ðpÞeip·ðx−yÞ ¼ ð2πÞ3 2Ep s Z d3 p 1 m2 n ip·ðx−yÞ ¼ p−M− e 2 ðn · pÞ ab ð2πÞ3 2Ep Z m2 n d3 p 1 ip·ðx−yÞ ¼ − i∂ x þ M þ i e : 2 ðn · ∂ x Þ ab ð2πÞ3 2Ep. 105007-11.

(12) JORGE ALFARO, PABLO GONZÁLEZ, AND RICARDO ÁVILA. PHYSICAL REVIEW D 91, 105007 (2015). That is, Z SF ðx − yÞ ¼. 2. m n d4 p iðp þ M − 2 n·pÞ −ip·ðx−yÞ ; e ð2πÞ4 p2 − M2 − m2 þ iε. which coincides with the path-integral result. This calculation shows that the relations (87) are correct. It follows that the model describes particles of definite energy-momentum and charge. Finally, after using the equations of motion of the auxiliary fields, we have that the canonical anticommutation relations are Z d3 p 1 X fψ a ðxÞ; ψ b ðyÞ† geqt ¼ ðusa ðpÞu†sb ðpÞe−i~pð~x−~yÞ þ vsa ðpÞv†sb ðpÞei~pð~x−~yÞ Þ ð2πÞ3 2Ep s Z d3 p 1 X ðusa ðpÞu†sb ðpÞ þ vsa ð−pÞv†sb ð−pÞÞe−i~pð~x−~yÞ ¼ ð2πÞ3 2Ep s Z d3 p −i~pð~x−~yÞ m2 0 1− ¼ e nγ : ð88Þ ~ Þ2 Þ ð2πÞ3 2n0 ðE2p − ðn̂ · p ab. Just like we said before, we believe that Eq. (88) came from a Dirac bracket due to the presence of second-class constraints on the model, which shall be clarified in the future. X. CONCLUSIONS AND OPEN PROBLEMS In this paper, we applied the VSR formalism to the electroweak Standard Model. This modification admits the generation of a neutrino mass without lepton number violation and without sterile neutrinos or another types of additional particles. However, a nonlocal term is necessary. So, the VSR EW SM is a simple theory with SUð2ÞL × Uð1ÞR symmetry and with the same number of leptons and gauge fields as in the electroweak Standard Model, but now we have nonlocal mass terms that violate Lorentz invariance. Besides, it is renormalizable and unitary. First, we reviewed the formulation of Yang-Mills fields in VSR, developed in Ref. [8], and then we used this to define the VSR EW SM gauge bosons and find the equations of motion after spontaneous symmetry breaking. With this, we concluded that the number of degrees of freedom are not modified with respect to the usual electroweak SM: W μ and Zμ have three degrees of freedom and Aμ has two. However, the masses are modified: the pffiffiffiffiffiffiffiffiffiffi v g2 þg02 vg and Z are M ¼ and M ¼ , masses of W μ W Z μ 2 2 respectively, for longitudinal polarizationqwith respect to ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v2 ðg2 þg02 Þ v2 g2 2 nμ , and MW ¼ þ m2G 4 4 þ mG and M Z ¼ for perpendicular polarizations with respect to nμ . On the other hand, the photon Aμ has a unique mass MA ¼ mG for the two polarizations. We presented some bounds on mG . In a future work, all of these predictions should be investigated in appropriate experiments, for example at the LHC.. Second, we solved the equations of motion for the leptons. A modified dispersion relation was produced and, in the particular case of neutrinos, they obtain mass without lepton number violation or sterile neutrinos. Besides, we can produce neutrino oscillations. For the electron (muon, tau), we obtained an interesting effect in the case mL ≠ mR , i.e., an electron-spin oscillation. This means that the electrons (muon and tau) are actually composed of two different states with slightly different masses. In fact, we found an extremely strong bound: jm2L − m2R j ≲ 10−11 eV2 . Therefore, mL ¼ mR is an excellent approximation. Third, we analyzed the lepton–gauge boson interactions to study new processes that are forbidden in the usual electroweak model. In particular, we computed the decay rate for X− > Y þ γ, where X and Y are leptons with mX > mY . We obtained a more restrictive condition for the branching ratio compared with the best experimental bounds available today. Finally, we analyzed the canonical quantization of the model. For this, we used auxiliary fields to eliminate the nonlocal terms and obtain a local Hamiltonian. Then, we performed the quantization. To come back to the nonlocal formalism, we used the equations of motion of the auxiliary fields. However, they are integral-differential equations. This produced a noncanonical anticommutation relation for the fermion field. So, we decided to accept the results of the path-integral quantization as the correct point of view with regards to the quantization, and we proved that this produces the correct expressions for the propagator, Hamiltonian, and charge operator in terms of creation and annihilation operators within the canonical second quantization. We believe that this noncanonical anticommutation relation came from a Dirac bracket due to the presence of second-class constraints on the model. This point shall be clarified in a future work.. 105007-12.

(13) ELECTROWEAK STANDARD MODEL WITH VERY SPECIAL …. In the present work, we did not include quarks in the formalism. We leave the implementation of this part of the VSR EW SM for a future publication. Meanwhile, there are many interesting applications of the model, including studying the processes that have been observed at the LHC, putting bounds on the parameters of the model, and/or describing new physics beyond the SM in preparation for the precision tests that will be available at the next run of the LHC. ACKNOWLEDGMENTS The work of P. G. has been partially financed by Fondecyt 1110378, Anillo ACT 1102, Anillo ACT 1122, and CONICYT Programa de Postdoctorado FONDECYT No. 3150398. R. A. has been partially financed by Fondecyt 1110378 and Anillo ACT 1102. The work of J. A. is partially supported by Fondecyt 1110378, Fondecyt 1150390, and Anillo ACT 1102. The authors want to thank C. Aulakh for very interesting and enlightening discussions.. PHYSICAL REVIEW D 91, 105007 (2015). but a gauge degree of freedom survives, which is given by V 0μ ¼ V μ þ ∂ μ λ → ∂ 2 λ ¼ 0:. From Eq. (A7) we see that this freedom can be used to fix ðn · VÞ ¼ 0; then, ð∂ 2 þ m2G ÞV μ ¼ 0:. We use a plane-wave solution to count the degrees of freedom, V μ ¼ εμ e−ik·x ; → ðk · εÞ ¼ 0. ð∂ þ M ÞV μ − ð∂ μ þ. m2G ðn. m2G ðn. −1. × ðð∂ · VÞ þ. k2 − m2G ¼ 0;. and ðn · εÞ ¼ 0:. So, the gauge field has mass mG and two independent polarizations (two degrees of freedom). (II) M ≠ mG : In this case, Eq. (A1) reduces to. ð∂ · VÞ ¼ 0;. Let us study Eq. (50), 2. ðA9Þ. ð∂ 2 þ M2 ÞV μ − m2G ðn · ∂Þ−1 ð∂ μ − nμ ðn · ∂Þ−1 ∂ 2 Þðn · VÞ ¼ 0;. APPENDIX A. 2. ðA8Þ. ð∂ 2 þ M 2 − m2G Þðn · VÞ ¼ 0:. −1. · ∂Þ nμ Þ. ðA10Þ. · ∂Þ ðn · VÞÞ. þ m2G nμ ðn · ∂Þ−2 ð∂ 2 þ m2G Þðn · VÞ ¼ 0:. ðA1Þ. If we contract with ∂ μ and nμ , respectively, we obtain ðM2 − m2G Þð∂ · VÞ ¼ 0; ð∂ 2 þ M 2 − m2G Þðn · VÞ − ðn · ∂Þð∂ · VÞ ¼ 0:. ðA2Þ. We can see that Eq. (A10) is not a Proca-like equation and it is not gauge invariant; therefore, this case was not included in Ref. [16]. Using a plane-wave solution, V μ ¼ εμ e−ik·x , we obtain ðk2 − M 2 Þεμ þ m2G ðn · kÞ−1 ðkμ − nμ ðn · kÞ−1 k2 Þðn · εÞ ¼ 0; ðk · εÞ ¼ 0;. From these equations, we have two cases. (I) M ¼ mG : In this case, Eq. (A1) reduces to. 2. ðk − M þ. ð∂ 2 þ m2G ÞV μ − ∂ μ ð∂ · VÞ − m2G ðn · ∂Þ−1 ∂ μ ðn · VÞ ¼ 0;. ðA4Þ. Additionally, we have a gauge invariance given by δV μ ¼ ∂ μ ϵ;. m2G Þðn. · εÞ ¼ 0:. The third equation tells us that. ðA3Þ ∂ 2 ðn · VÞ − ðn · ∂Þð∂ · VÞ ¼ 0:. 2. ðA5Þ. because Eqs. (A3) and (A4) are just like Eqs. (29) and (30), respectively. Thus, we must fix the gauge. Using the Lorentz gauge ð∂ · VÞ ¼ 0, we have that ð∂ 2 þ m2G ÞV μ − m2G ðn · ∂Þ−1 ∂ μ ðn · VÞ ¼ 0;. ðA6Þ. ∂ 2 ðn · VÞ ¼ 0;. ðA7Þ. ðn · εÞ ¼ λδðk2 − M 2 þ m2G Þ; where λ is an arbitrary scalar. So, evaluating this in the other equations, we conclude that the most general solution is εμ ¼ Λμ δðk2 − M 2 Þ þ λðn · kÞ−1 ðkμ − ðM 2 − m2G Þnμ ðn · kÞ−1 Þ × δðk2 − M2 þ m2G Þ;. ðA11Þ. where Λμ is an arbitrary vector such that ðk · ΛÞ ¼ ðn · ΛÞ ¼ 0. In conclusion, we can say that we have three degrees of freedom: λ (one) and Λμ (two). However, the masses change for different polarizations. Respectively, the masses are M2 − m2G and M 2 .. 105007-13.

(14) JORGE ALFARO, PABLO GONZÁLEZ, AND RICARDO ÁVILA. PHYSICAL REVIEW D 91, 105007 (2015). n 0 m2 ¼ 2EðkÞ − δ 0; ðn · kÞ ss n 0 m2 † ~ ~ vðsÞ ðkÞvðs0 Þ ðkÞ ¼ 2EðkÞ − δ 0; ðn · kÞ ss. APPENDIX B. ~ ðs0 Þ ðkÞ ~ u†ðsÞ ðkÞu. When mL ≠ mR , the solutions can be written as states with spin in the n̂ direction. Actually, we can prove that Eqs. (55) and (56) represent spin up and down, respectively. The same result applies for Eqs. (55) and (56). We have that 0 1 1 B n1 þin2 C B n þn C U 1 ¼ N 1 B 0 3 C; @ 1 A. 0 B B U 2 ¼ N2B @. n1 þin2 n0 þn3. n0 m2 ~ ðs0 Þ ð−kÞ ~ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u†ðsÞ ðkÞv δss0 ; 2 2 2 ~ n0 EðkÞ − ð~n · kÞ. n1 −in2 1. − n0 þn3 1 n1 −in2 n0 þn3. C C C A. ðB1Þ. −1. in the Dirac representation, where N 1 and N 2 are normalization parameters.. ~ 0 nuðs0 Þ ðkÞ ~ ¼ 2ðn · kÞδss0 ; u†ðsÞ ðkÞγ ~ 0 nvðs0 Þ ðkÞ ~ ¼ 2ðn · kÞδss0 ; v†ðsÞ ðkÞγ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi † ~ 0 ~ ~ 2 δss0 : uðsÞ ðkÞγ nvðs0 Þ ð−kÞ ¼ 2 n20 EðkÞ2 − ð~n · kÞ. ðC3Þ. APPENDIX C When mR ¼ mL ¼ m, the solutions to the VSR Dirac equation can be written as φs 1 m2 us ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; k − n þ M 2ðn · kÞ m2 0 n0 k0 þ M − 2ðn·kÞ 1 0 and φ2 ¼ ; with φ1 ¼ 0 1 0 1 m2 ; k− n−M vs ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2ðn · kÞ m χs n0 k0 þ M − 2ðn·kÞ 0 −1 with χ 1 ¼ and φ2 ¼ : 1 0. APPENDIX D The only pole that the neutrino propagator has is at p ¼ m2L . In order to see it in a simpler way, we consider the propagator derived from Eq. (53) in two space-time dimensions, 2. 1. γ ¼ σ1 ¼ i γ0 ¼ σ3 ¼. 01. ;. 10 10 ; 0−1. 1 m2L α¼− ; 2 n:p p0 − αn0 iðp1 − αn1 Þ det iðp1 − αn1 Þ − ðp0 − αn0 Þ. These solutions reduce to the standard Dirac solutions in the Pauli-Dirac representation for m ¼ 0 [2]. They satisfy the completeness relations. ¼ −p20 þ 2αp0 n0 − α2 n20 þ p21 − 2αp1 n1 þ α2 n21. 2 X ~ ūs ðkÞ ~ ¼ k − m n þ M; us ðkÞ 2ðn · kÞ s. ðC1Þ. 2 X ~ v̄s ðkÞ ~ ¼k− m n −M vs ðkÞ 2ðn · kÞ s. ðC2Þ. The determinant has been computed for arbitrarily small n:p. We used the property that nμ is a null vector, that is, n2 ¼ 0. Thus, there is only a Lorentz-invariant pole. This result holds in arbitrary space-time dimensions.. [1] CMS Collaboration, Evidence for the direct decay of the 125 GeV Higgs boson to fermions, Nat. Phys. 10, 557 (2014). [2] P. Langacker, The Standard Model and Beyond (CRC Press, Boca Raton, 2009).. [3] D. H. Perkins, Introduction to High Energy Physics (Cambridge University Press, Cambridge, England, 2000). [4] R. Mohapatra, Unification and Supersymmetry: The Frontiers of Quark-Lepton Physics, 3rd ed. (Springer, New York, 2002).. and the orthogonality rules given by. ¼ −p2 þ m2L :. 105007-14.

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