Super-weak force and neutrino masses

We consider an anomaly free extension of the standard model gauge group GSM by an abelian group to GSM x U (1)Z . The condition of anomaly cancellation is known to fix the Z-charges of the particles, but two. We fix one remaining charge by allowing for all possible Yukawa interactions of the known left handed neutrinos and new right-handed ones that obtain their masses through interaction with a new scalar field with spontaneously broken vacuum. We discuss some of the possible consequences of the model. Assuming that the new interaction is responsible for the observed differences between the standard model prediction for the anomalous magnetic moment of the muon or anti-muon and their measured values, we predict the size of the vacuum expectation value of the new scalar field.


Introduction
The remarkable experimental success of the standard model of elementary particle interaction [1] leaves very little room for the explanation of the observed deviations from it. This success story has culminated in the discovery of the Higgs particle [2,3], which could not have happened without the immense theoretical input to the design of the accelerator and the experiments. With this discovery also a new era of particle physics has arrived as there is no established model that can guide us to new discoveries. Therefore, theories that might incorporate the existing deviations from the standard model are desirable.
The most outstanding experimental observations that cannot be explained by the standard model are the (i) abundance of dark matter in the universe; (ii) non-vanishing neutrino masses; (iii) leptogenesis; * (iv) accelerating expansion of the universe, signalling the existence of dark energy [4]. † In addition to (i)-(iv), (v) inflation in the early universe is also considered a fairly established fact, although there is no direct proof for it. All these facts have to be explained by such an extension of the standard model that respects (a) the high precision confirmation of the standard model at collider experiments (b) and the lack of finding new particles beyond the Higgs boson by the LHC experiments [5,6]. There is one more feature of the standard model, the metastability of vacuum [7,8] that does not necessarily require new physics, but if new physics exists, it should not worsen the stability, but possibly push the vacuum to the stability region.
In addition to the experimental success of the standard model, it is also highly efficient being based on the concepts of local gauge invariance and spontaneous symmetry breaking [9,10]. The only exception of economical description is the relatively large number of Yukawa couplings of the fermions needed to explain their masses. The generation of the fermion masses however, is also highly efficient in the sense that it uses the same spontaneous symmetry breaking of the scalar field to which all other particles owe their masses. In this spirit, it is reasonable to expect that the non-vanishing masses of the neutrinos should be explained by Yukawa couplings, too. Also, the choice of the gauge groups and number of family replications look arbitrary and presently these are determined by phenomenology only.
Clearly, the neutrino masses must play a fundamental role in the possible extensions of the standard model. As the gauge and mass eigenstates of the neutrinos differ, they must feel a second force to the gauge interaction. The second force can be a Yukawa coupling to a scalar. Such explanation of neutrino masses in general requires the assumption of the existence of right-handed neutrinos and perhaps a new scalar field.
In the spirit of economy and level of arbitrariness explained above, in this article we propose an extension of the zoo of particles in the standard model with three right-handed neutrinosand the gauge symmetry of the standard model Lagrangian Such extensions have already been considered in the literature extensively ‡ . In particular, it was shown that the charge assignment of the matter fields is constrained by the requirement of anomaly cancellations up to two free charges [14]. To define the model completely, one has to take a specific choice for these remaining free charges. In this article we propose that the mechanism for the generation of neutrino masses fixes the values of the U (1) Z charges up to an overall scale that can be embedded in the U (1) Z coupling.
The difference between our proposal and existing studies is two-fold. Our primary goal is not the prediction of new observable phenomena at collider experiments, but first focus only on the unexplained phenomena (i-iv), with respecting the observations (a) and (b). As the deviations from the standard model are related to the intensity frontier of particle physics, we assume that the new U (1) Z interaction is secluded from the standard model. We also propose the model in a region of the parameter space that has received little attention before.
2 Definition of the model

Fermion sector
We consider the usual three fermion families of the standard model extended with one right-handed Dirac neutrino in each family. § We introduce the notation for the quark fields ψ q and for the lepton fields ψ l . In Eq. (2.1) L and R denote the left and right-handed projections of the same field, ¶ ¶ The Weyl spinors of ν L and ν R can be embedded into different Dirac spinors, leading to Majorana neutrinos, without essential changes in the model. However, the negative results of the experiments searching for neutrinoless double β-decay make the Majorana nature of neutrinos increasingly unlikely. Table 1: Assignments for the representations (for SU (N )) and charges (for U (1)) of fermion and scalar fields of the complete model. The right-handed Dirac neutrinos ν R are sterile under the G SM group. The sixth column gives a particular realization of the U (1) Z charges, motivated below, and the last one is added for later convenience.
The Dirac Lagrangian summed over the family replications, is invariant under local G = G SM ⊗ U (1) Z gauge transformations, provided the five gauge fields introduced in the covariant derivative transform as The gauge invariant kinetic term for these vector fields is The field strength T · W µν transforms covariantly under G transformations, T · W µν G −→ U (x) T · W µν U † (x), but B µν and Z µν are invariant, therefore a kinetic mixing term of the U (1) fields is also allowed by gauge invariance: We can get rid of this mixing term by redefining the U (1) fields using the transformation In terms of the redefined fields, the covariant derivative becomes where g Y = g Y tan θ Z and g Z = g Z / cos θ Z . Thus the effect of the kinetic mixing is to change the couplings of the matter fields to the vector field Z µ . Note that we cannot immediately combine the coupling factor (g Z z j − g Y y j ) into a single product of a coupling and a charge. We shall discuss this issue further below.
Gauge symmetry forbids mass terms for gauge bosons. Fermion masses must also be absent because mψψ = mψ L ψ R + mψ R ψ L , but the ψ L , ψ R fields transform differently under G. Thus, the G-invariant Lagrangian describes massless fields in contradiction to observation.

Scalar sector
To solve the puzzle of missing masses we proceed similarly as in the standard model, but in addition to the usual Brout-Englert-Higgs (BEH) field φ that is an SU (2) L -doublet we also introduce another complex scalar χ that transforms as a singlet under G SM transformations. The gauge invariant Lagrangian of the scalar fields is where the covariant derivative for the scalar s (s = φ, χ) is and the potential energy 13) in addition to the usual quartic terms, introduces a coupling term −λ|φ| 2 |χ| 2 of the scalar fields in the Lagrangian. For the doublet |φ| denotes the length |φ + | 2 + |φ 0 | 2 . The value of the additive constant V 0 is irrelevant for particle dynamics, but may be relevant for inflationary scenarios, hence we allow for its nonvanishing value. In order that this potential energy be bounded from below, we have to require the positivity of the self-couplings, λ φ , λ χ > 0. The eigenvalues of the coupling matrix are while the corresponding un-normalized eigenvectors are As λ + > 0 and λ − < 0, in the physical region the potential can be unbounded from below only if u (−) points into the first quadrant, which may occur only when λ < 0. In this case, to ensure that the potential is bounded from below, one also has to require that the coupling matrix be positive definite, which translates into the condition With these conditions satisfied, we can find the minimum of the potential energy at field (2.17) Table 2: Possible signs of the couplings in the scalar potential V (φ, χ) in order to have non-vanishing real VEVs. .
Using the VEVs, we can express the quadratic couplings as simultaneously, which can be satisfied if at most one of the quadratic couplings is smaller than zero. We summarize the possible cases for the signs of the couplings in Table 2.
After spontaneous symmetry breaking of G → SU (3) c ⊗ U (1) Q * * we use the following convenient parametrization for the scalar fields: and We can use the gauge invariance of the model to choose the unitary gauge when and and the vector fields are transformed according to Eq. (2.5). With this gauge choice, the scalar kinetic term contains quadratic terms of the gauge fields from which one can identify mass parameters of the massive standard model gauge bosons proportional to the vacuum expectation value v of the BEH field and also that of a massive vector boson Z µ proportional to w.
We can diagonalize the mass matrix (quadratic terms) of the two real scalars (h and s ) by the rotation where for the scalar mixing angle θ S ∈ (− π 4 , π 4 ) we find The masses of the mass eigenstates h and H are

Fermion masses
We already discussed that explicit mass terms of fermions would break SU (2) L ⊗ U (1) Y invariance. However, we can introduce gauge-invariant fermion-scalar Yukawa interactions where h.c. means hermitian conjugate terms and the parameters c D , c U , c are called Yukawa couplings that are matrices in family indices and summation over the families is understood implicitly. The dot product abbreviates products of SU (2) components: andL ≡ ν ,¯ . The Z-charge of the BEH field is constrained by U (1) Z invariance of the Yukawa terms to z φ = Z 2 − Z 1 , which works simultaneously for all three terms.
After spontaneous symmetry breaking and fixing the unitary gauge, this Yukawa Lagrangian becomes We see that there are mass terms with mass matrices The neutrino oscillation experiments suggest non-vanishing neutrino masses and the weak and mass eigenstates of the left-handed neutrinos do not coincide. In principle, the charge assignment of our model allows for the following gauge invariant Yukawa terms of dimension four for the neutrinos for arbitrary values of Z 1 and Z 2 if the superscript c denotes the charge conjugate of the field, ν c = −iγ 2 ν * and the Z-charge of the right-handed neutrinos and the new scalar satisfy the relation z χ = −2z ν R . There are two natural choices to fix the Z-charges: (i) the leftand right-handed neutrinos have the same charge, or (ii) those have opposite charges. In the first case we have which is solved by Z 1 = Z 2 and it leads to the charge assignment of the U (1) B−L extension of the standard model, studied in detail (see for instance, [15] and references therein). In the second case which is solved by Z 1 = Z 2 /7. As the overall scale of the Z-charges depends only on the value of the gauge coupling g Z , we set Z 2 freely. For instance, choosing Z 2 = 7/6 implies Z 1 = 1/6 and the Z-charge of the BEH scalar is while that of the new scalar is While we cannot exclude the infinitely many cases when the magnitudes of Z-charges of the left-and right-handed neutrinos differ, we find natural to assume that Eq. (2.31) is valid. The corresponding Z-charges are given explicitly in the sixth column of Table 1.
After the spontaneous symmetry breaking of the vacuum of the scalar fields Eq. (2.29) leads to the following mass terms for the neutrinos: The off-diagonal elements represent interaction terms that look formally like Dirac mass terms, − i,j ν i,L (m D ) ij ν j,R + h.c. After spontaneous symmetry breaking the quantum numbers of the particles ν c i,L and ν i,R are identical, hence they can mix. Thus the propagating states will be a mixture of the left-and right-handed neutrinos. Those states can be obtained by the diagonalization of the full matrix M (0, 0), for which a possible parametrization is given for instance in Ref. [16].
In order to understand the structure of the matrix M (0, 0) better, we first diagonalize the matrices m D and M M separately by a unitary transformation and an orthogonal one. Defining we can rewrite the neutrino Yukawa Lagrangian as Assuming the hierarchy m i M j , we can integrate out the right-handed (heavy) neutrinos and obtain an effective higher dimensional operator with Majorana mass terms for the left-handed neutrinos Thus, if m i M i , then the mixing between the light and heavy neutrinos will be very small, the ν i,L can be considered as the mass eigenstates that are mixtures of the left-handed weak eigenstates, and whose masses can be small naturally as suggested by phenomenological observations.
As we can only observe neutrinos through their charged current interactions, it is more natural to use the flavour eigenstates than the mass eigenstates. In the flavour basis, the couplings of the leptons to the W boson are diagonal:

Re-parametrization into right-handed and mixed couplings
Having set the Z-charges of the matter fields, we can re-parametrize the couplings to Z using the new coupling Then the covariant derivative in Eq. (2.9) becomes where r j = z j − y j and its values are given explicitly in the last column of Table 1. Thus, if a U (1) Z extension of G SM is free of gauge and gravity anomalies, then it is equivalent to a U (1) R extension with tree-level mixed coupling g ZY [17], related to the kinetic mixing parameter θ Z by Eq. (2.43).
Particle phenomenology of the standard model suggests that the interaction of the fermions through the Z vector boson must be suppressed significantly. The origin of such a suppression can be either a small coupling to Z or the large mass of Z . Usual studies in the literature focus on the latter case. Here we explore the former possibility.
The complete Lagrangian is the sum of the pieces given in Eqs.

Mixing in the neutral gauge sector
The neutral gauge fields of the standard model and the Z mix, which leads to mass eigenstates A µ , Z µ and T µ . The mixing is described by a 3 × 3 mixing matrix as For the Weinberg mixing angle θ W we have the usual value sin θ W = g Y / g 2 L + g 2 Y . We introduce the notion of reduced coupling defined by γ i = g i /g L , i.e. γ L = 1. Then we have and for the mixing angle θ T of the Z -boson we find is the usual ratio of the scalar vacuum expectation values. For small values of the new couplings γ ZY and γ Z , implying small κ, we have The charged current interactions remain the same as in the standard model. The neutral current Lagrangian can be written in the form where the first term is the usual Lagrangian of QED, the second one is a neutral current coupled to the Z 0 boson, and the third one is the neutral current coupled to the T boson, while the new neutral current has the same dependence on fermion dynamics with different coupling strength: To define the perturbation theory of this model explicitly, we present the Feynman rules in Appendix A.

Masses of the gauge bosons
The photon is massless, while the masses of the massive neutral bosons are and where M W = 1 2 vg L and we assumed M T < M Z . Indeed, in order to have M Z within the experimental uncertainty of the known measured value, we need θ T 0 (precise constraint will be presented elsewhere), which justifies the expansions at κ = 0, where we used Eq. (2.51) and M Z = wg Z . Thus τ can also be written as the ratio of the masses of the two massive neutral gauge bosons, justifying our assumption on the hierarchy of masses. In fact, unless w v, we find M T M Z .

Free parameters
There are five parameters in the scalar sector, λ φ , λ χ , λ, v and w that has to be determined experimentally, while the values of µ φ and µ χ (at tree level) are given in Eq. (2.18). However, it is more convenient to use parameters that can be measured more directly, for instance, Then the other parameters, expressed in terms of the free ones, are w = v tan β, (first indices are to be used if λ φ v 2 < λ χ w 2 , second ones otherwise) (2.65)

Possible consequences
Our hope in devising this model is to explain the established experimental observations listed in the introduction. We envisage the following scenario: • The massive T vector boson is a natural candidate for WIMP dark matter if it is sufficiently stable, i.e. its mass is below the threshold of electron-positron pair production, which requires that the new force is super-weak, τ ∼ 10 −5 . Such a light vector boson is not yet excluded by beam-dump experiments (see for instance [18] and references therein). A new technology to search for electron recoils from the interaction of sub-GeV dark matter particles with electrons in silicon start to become sensitive to dark matter searches of mass as low as about 500 keV [19]. • Majorana neutrino mass terms for the right-handed neutrinos and Yukawa interactions between the left-and right-handed neutrinos and the BEH vacuum are generated by the spontaneous symmetry breaking of the scalar fields as outlined in Sect. 2.3. This scenario provides a possible origin of neutrino oscillations and effective Majorana mass terms for the left-handed neutrinos. • The neutrino Yukawa terms provide a source for the PMNS matrix as shown in in Sect. 2.3, which in turn can produce leptogenesis (and hence baryogenesis). • The vacuum of the χ scalar has a charge z j = −1 (or r j = −1) that may be a source of the current accelerated expansion of the universe. • The second scalar together with the established BEH field can cause hybrid inflation.
In order that the model makes these explanations credible, we have to find answer to the following question: Is there any region of the parameter space of the model that is not excluded by experimental results, both established in standard model phenomenology and elsewhere? Of course, answering this question requires studies well beyond the scope of a single article. Here we shall focus on the constraints over the parameter space that can be obtained from the standard model phenomenology and in particular from the anomalous magnetic moment of the muon.

Anomalous magnetic moment of the muon
There is a long standing deviation between the experimental result and predicted standard model value of the anomalous magnetic moment of the muon [20]. The experiments are performed with electrically positive and negative muons separately. As these are produced in pion decay through charged-current weak interactions, the positive muon which is produced in association with a left-handed neutrino must be also left handed, while the negative muon, which emerges with a right-handed anti-neutrino, is right handed. The electromagnetic interaction between the muon and the magnetic field of the experiment preserves the chirality of the muon. The quoted experimental result is the average of the two measurements and larger than the theoretical prediction of the standard model. Here we assume that the differences between the measured values and the standard model predictions for the anomalous magnetic moments of the left-and right-handed muons are indeed different as those belong to different chirality states and due to the neglected neutral current interactions in Eq. (2.57). Based on this assumption-which will be tested by the increased precision of future experiments-, we estimate the allowed values for the ratio tan β of the vacuum expectation values and that of the mixed coupling γ ZY and the right coupling γ Z , As the new U (1) Z sector may influence the standard model phenomenology only within the current experimental uncertainties, the new gauge coupling must be small. Therefore, the use of first order perturbation theory is justified. At one-loop accuracy, the only new contributions to the anomaly constant a µ = (g µ − 2)/2 emerge due to the modified Zμµ interaction and the new interaction Tμµ, both presented in the Appendix. The only new Feynman graph is a triangle with the exchange of a T boson between the muon legs, which is formally identical to the triangle with the exchange of a Z 0 boson between the muon legs as shown in Fig. 1. Consequently, the computation follows the same steps as in the case of the electroweak corrections [21][22][23][24], so we present only the result for the exchange of a Z 0 , and that for the exchange of a T boson, where h f represents h ± f defined in Eq. (A.1) for the right/left-handed muon. The contribution of the Z 0 boson in the standard model is recovered by setting h f = 0 and θ T = 0. Thus, the complete new contribution to the a µ in this model is given by As mentioned before, the standard model phenomenology requires θ T 0, which justifies the expansion in θ T : numerically.
The latest experimental results [20] suggest Here we assume that these differences are due to the new contribution of Eq. (4.5), to deduce estimates for the values of ρ Z and tan β. For the positive (left-handed) muon h µ + /g Z = −ρ Z /2, while for the negative (right-handed) muon h µ − /g Z = −1/2 − ρ Z . Substituting these into Eq. (4.5) and using Eqs. where we assumed that the experimental uncertainties are not anti-correlated. The value of ρ Z is determined by the ratio of the deviations between the experimental values and standard model predictions through the equation The value of tan β is determined by the absolute size of the deviations such that it increases with decreasing differences.
Using the value of ρ Z and the known value of g Y 0.36, it follows from Eq. (4.1) that the new gauge coupling and the kinetic coupling have to be correlated as g Z 0.16 sin θ Z , which however, does not fix the magnitude of these couplings. Using tan β = 0.54 in Eq. (2.49), we obtain τ 0.88γ Z , so γ Z ∼ 10 −5 if τ ∼ 10 −5 , i.e. in the super-weak case. For the vacuum expectation value of the new scalar field we find w = 133 +103 −32 GeV . (4.10)

Conclusions
In this paper we collected the well established experimental observations that cannot be explained by the standard model of particle interactions. We have then proposed an anomaly free extension by a U (1) Z gauge group, which is the simplest possible model. We also assumed the existence of a new complex scalar field with Z-charge only (i.e. neutral with respect to the standard model interactions) and three right-handed neutrinos. In order to fix the Z-charges of the particle spectrum we assumed that the left-and right-handed neutrinos have opposite Z-charges. Thus such a model predicts the existence of (i) a massive neutral vector boson, (ii) a massive scalar particle and (iii) three massive right-handed neutrinos. The left-handed neutrinos remain massless as in the standard model, but their Yukawa interactions with the BEH field and the right-handed neutrinos provides a field theoretical basis for explaining neutrino oscillations and predict effective Majorana masses for the propagating mass eigenstates.
We have discussed how the new neutral gauge field Z µ mixes with those of the standard model (B µ and W µ 3 ) and argued that the mixing results in a new boson T of a small mass related to the small new gauge coupling and small mixing with the standard model vector fields. We also presented the Feynman rules of the model in unitary gauge and collected the new free parameters.
In order that the predictions of the model be credible, we have to answer whether there is any region of the parameter space of the model that is not excluded by experimental results established in standard model phenomenology or elsewhere. To answer such a question with satisfaction, studies well beyond the scope of a single article are needed, which forecasts an exciting research project. As a first check, we computed the contribution of the new vector boson to the anomalous magnetic moment of the muon and used the discrepancy between the prediction of the standard model and the measured values to constrain the vacuum expectation value of the new scalar. This check predicts the ratio of the new gauge coupling and kinetic mixing parameter which can be tested by measurements resulting in increased precision for the anomalous magnetic moment of the muon.

A Feynman rules
The Feynman rules of the model are obtained from the complete Lagrangian in Eq. (2.45). For studying the UV behaviour of the model, it is convenient to use the Feynman rules before SSB, while for low energy phenomenology the rules after SSB are needed. In this paper we concentrated only on a simple application of the latter that did not require renormalization, so rules in the unitary gauge were sufficient. The propagators of the new fields are related trivially to those of the standard fields. Thus, we present only the vertices, neglecting the rules related to QCD, which are unchanged.

Feynman rules after SSB
We present the rules in unitary gauge.