Nonstandard Neutrinos Interactions in a 331 Model with Minimum Higgs Sector
M. Medina1and P. C. de Holanda1
Academic Editor: Hoang Ngoc Long
Received26 Aug 2011
Revised02 Dec 2011
Accepted09 Dec 2011
Published22 Feb 2012
Abstract
We present a detailed analysis of a class of extensions to the SM Gauge chiral symmetry (331 model), where the neutrino electroweak interaction with matter via charged
and neutral current is modified through new gauge bosons of the model. We found the connections
between the nonstandard contributions on 331 model with nonstandard interactions. Through
limits of such interactions in cross-section experiments, we constrained the parameters of the model,
obtaining that the new energy scale of this theory should obey TeV and the new bosons of
the model must have masses greater than 610βGeV.
1. Introduction
Although the standard model (SM) is a good phenomenological theory, describing very well all experimental results, it leaves several unanswered questions that suggest that the SM might be an effective model at low energies, originating from a more fundamental theory. Some of the unexplained aspects in the SM are the existence of three families and lepton flavour violation observed in solar [1β5], atmospheric [6β11], and reactor [12β17] neutrino experiments. These results demonstrate that new physics is required, being interpreted as a sign of physics beyond the SM.
In principle neutrinos new interactions not described by Standard Model can arise in extensions of the SM. We assume that the new physics which induces the nonstandard neutrino interactions (NSIs) [18β29] arises in some models enlarging the symmetry group where the SM is embedded. Models with larger symmetries that may allow us to understand the origin of the families have been proposed [30β34]. In some models, it is also possible to understand the number of families from the cancellation of chiral anomalies, necessary to preserve the renormalizability of the theory [35β37]. This is the case of the or 331 models, which are an immediate extension of the SM [38β46]. There are a great variety of such models, which have generated new expectations and possibilities of solving several problems of the SM.
Our goal is to investigate how NSI with matter can be induced by new physics generated by 331 models. Through the constraints from neutrino elastic scattering experiments on this NSI parameters, we can constrain some values expected for 331 model parameters. We find that the constraints on vacuum expectation values of the model, as well as for the mass of the new bosons, are in full agreement with the limits found in the literature, which makes this class of models a viable theory for a higher energy level.
The paper is organized as follows. In Section 2 we briefly review NSI and present how new interactions can contribute to new matter effects, in addition to the SM electroweak ones. In Section 3 we introduce a specific 331 model and we give the fermion gauge-boson couplings. In Section 4 we calculate the interactions involving neutrinos and how these interactions can be interpreted as new terms beyond SM. Finally, in Section 5 we summarize our main results.
2. Nonstandard Neutrino Interactions
One convenient way to describe neutrino new interactions with matter in the electro-weak (EW) broken phase are the so-called nonstandard neutrino interactions (NSIs), which is a very widespread and convenient way of parameterizing the effects of new physics in neutrino oscillations [18β29]. NSIs with first generation of leptons and quarks for four-fermion operators are contained in the following Lagrangian density [18β22, 24, 25, 28]:
where is the Fermi constant, , and with , and the coefficients encode the deviation from standard interactions between neutrinos of flavor with component -handed of fermions , resulting in a neutrinos of flavor . Then, the neutrino oscillations in the presence of nonstandard matter effects can be described by an effective Hamiltonian, parameterized as
where , is the neutrino energy and with and the electrons and fermions density in the medium, respectively. These parameters can be found in solar [22, 47], atmospheric [20, 48], accelerator [18, 19, 22, 49], and cross-section [18, 19, 21, 50, 51] neutrino data experiment.
We focus on cross-section neutrino experiment, where at low energies the standard differential cross-section for scattering processes has the well-know form:
where is the electron mass, is the incident neutrino energy, and is the electron recoil energy. The quantities and are related to the SM neutral current couplings of the electron and , with . For neutrinos, which take part only in neutral current interactions, we have and while for electron neutrinos, which take part in both charge current (CC) and neutral current (NC) interactions, . In the presence of nonuniversal standard interaction, the cross-section can be written in the same form of (2.3) but with replaced by the effective nonstandard couplings and , leading to the following differential scattering cross-section [19, 21, 50, 51]
3. 331 Model
The success of the standard model (SM) implies that any new theory should contain the symmetry in a low energy limit. Then, it is natural that one possible modification of SM involves extensions of the representation content in matter and Higgs sector, leading to extension of symmetry group to groups with .
In early 90βs, Pisano and Pleitez [38, 39] and Frampton [40] suggested an extension of the symmetry group of electroweak sector to a group , that is, with . The 331 models present some interesting features; for instance, they associate the number of families to internal consistence of the theory, preserving asymptotic freedom.
In these models, the SM doublets are part of triplets. In quark sector three new quarks are included to build the triplets, while in lepton sector we can use the right-handed neutrino to such role [38, 40]. Another option is to invoke three new heavy leptons, charged or not, depending on the choice of charge operator [41, 42]. In SM the electric charge operator is constructed as a combination of diagonal generators of . Then, it is natural to assume that this operator in is defined in the same way. The most general charge operator in is a linear superposition of diagonal generators of symmetry groups, given by
where the group generator is defined as with , , being the Gell-Mann matrices for , where the normalization chosen is and is the identity matrix, and and are two parameters to be determined. Then the charge operator in (3.1) acts on the representations 3 and 3* of having the following form:
where we have two free parameters to obtain the charge of fermions, and ( can be determined by anomalies cancellation). However, is necessary to obtain doublets of isospins correctly incorporated in the model [41, 42, 45]. Then we can vary to create different models in 331 context, being a signature that differentiates such models. For , we have the original 331 model [38, 39].
To have local gauge invariance, we have the following covariant derivative: and a total of 17 mediator bosons: one field associated with , eight fields associated with , and another eight fields associated with , written in the following form:
where
Therefore, charge operator in (3.2) applied over (3.4) leads to and . Then the mediator bosons will have integer electric charge only if . A detailed analysis shows that if and are associated with the fundamental representation 3, then and will be associated with antisymmetric representation .
3.1. The Representation Content
There are many representations for the matter content [46], for instance, [38]. But we note that if we accommodate the doublets of in the superior components of triplets and antitriplets of , and if we forbid exotic charges for the new fermions, we obtain from (3.2) the constrain (assuming ). Since a negative value of can be associated to the antitriplet, we obtain that is a necessary and sufficient condition to exclude exotic electric charges in fermion and boson sector [41].
The fields left- and right-handed components transform under as triplets and singlets, respectively. Therefore the theory is chiral and can present anomalies of Alder-Bell-Jackiw [52, 53]. In a non-abelian theory, in the fermionic representation , the divergent anomaly is given by
where are the matrix representations for each group generator acting on the basis with helicity left or right. Therefore, to eliminate the pure anomaly , we should have that . We use the fact that has two fundamental representations, 3 and , then its generators should be associated to and , respectively, that is,
but we know that the matrix representations for each group generator satisfies that [54]. So, we can see that for the anomalies to be canceled, the number of fields that transform as triplets (first term in equation above) and antitriplets under has to be the same; that is, two triplets quark families Γ 3 (color) one antitriplet quark family Γ 3 (color) + 3 antitriplet lepton families. This implies that two families of quarks should transform differently than the third family, as will be discussed in next paragraph.
Usually the third quark family is chosen to transform in a different way than the first two families. But we will assume that the first family transform differently, to address the fact that while and . To state this in a clearer way, we recall that in SM the doublets are , with . We can see that the first component of leptons doublets and first quark family is lighter than the second component. But for the second and third quark families, the opposite occurs. Then we use this idea to justify that first quark family transform as leptons.
3.2. Minimal 331 Model on Scalar Sector
Among the different possibilities of 331 models, we will present a detailed study on a minimal model on scalar sector without exotic electric charges for quarks and with three new leptons without charged [41] , where the fermions present the following transformation structure under :
where . We note that the leptons multiplets consist of three fields , the corresponding neutrinos , and new neutral leptons . We can also see that the multiplet associated with the first quark family consists of down and up quarks and a new quark with the same electric charge of quark up (named ), while the multiplet associated with second (third) family consists of SM quarks of second (third) family and a new quark with the same electric charge of down quark (named ()). The numbers on parenthesis refer to the transformation properties under , and , respectively. With this choice, the anomalies are cancelled in a nontrivial way [55], and asymptotic freedom is guaranteed [56β59].
3.2.1. Scalar Sector and the Yukawa Couplings
The scalar fields have to be coupled to fermions by the Yukawa terms, invariants under . In lepton sector, these couplings can be written as
and writing only three terms in quarks sector, for example,
As usual in these class of models, we impose colorless Higgs (i.e., selecting only the multiplets that transform as singlets under ). We note that we need only three Higgs multiplets, , and , to couple the different fermionic fields and generate mass through spontaneous symmetry breaking. In (3.9) and (3.10) we note that quantum numbers of triplets and are the same, which leads us to consider models with two or three Higgs triplets. We will adopt the first option, two Higgs triplets, due to the simpler scalar sector in comparison with the scenario with three triplets [41β44].
3.3. Model with Two Higgs Triplets
For the models with two Higgs triplets, we obtain (note that in this model we assumed e )
Assuming the following choice to the Higgs triplets vacuum expectation value (VEV) [41] and , we associate with the mass of the new fermions, which lead us to assume . We expand the scalar VEVs in the following way:
The real (imaginary) part is usually called CP-even (CP-odd) scalar field. The most general potential can be written as
Demanding that in the displaced potential the linear terms on the field should be absent, we have, in tree-level approximation, the following constraints:
The analysis of such equations shows that they are related to a minimum in scalar potential with the value . Then, replacing (3.12) and (3.14) in (3.13), we can calculate the mass matrix in basis through the relation , obtaining
Since (3.15) has vanishing determinant, we have one Goldstone boson and two massive neutral scalar fields and with masses (note that if , we obtain two Goldstone bosons, and , and a massive scalar field with mass , where ; then imposing leads to and )
where real values for βs produce positive mass to neutral scalar fields only if and , which implies that . A detailed analysis shows that when in (3.13) is expanded around the most general vacuum, given by (3.12) and using constrains in (3.14), we do not obtain pseudoscalar fields . This allows us do identify three more Goldstone bosons, , and . For the mass spectrum in charged scalar sector on basis, the mass matrix will be given by
with two eigenvalues equal to zero, equivalent to four Goldstone bosons and two physical charged scalar fields with large masses given by , which leads to the constrain .
This analysis shows that, after symmetry breaking, the original twelve degrees of freedom in scalar sector leads to eight Goldstone bosons (four electrically neutral and four electrically charged), four physical scalar fields, two neutral (one of which being the SM Higgs scalar), and two charged. Eight Goldstone bosons should be absorbed by eight gauge fields as we will see in next section.
3.3.1. Gauge Sector with Two Higgs Triplets
The gauge bosons interaction with matter in electroweak sector appears with the covariant derivative for a matter field as
where are Gell-Mann matrices of algebra and is the charge of abelian factor of the multiplet in which acts. The matrix contains the gauge bosons with electric charges , defined by the generic charge operator in (3.1). For the matrix will have the following form:
where and nonphysical gauge bosons on nondiagonal entries, , are defined in (3.5) with , and
Then for the 331 model we are considering , we have two neutral gauge bosons, and , and four charged gauge bosons, and . The three physical neutral eigenstates will be a linear combination of , and . After breaking the symmetry with , , and using covariant derivative for the triplets , we obtain the following masses for the charged physical fields:
and the following physical eigenstates:
The neutral sector in approximation for leads to the following masses for the neutral physical fields:
We can see from (3.21) and (3.23) that we have one nonmassive boson, which we associate with the photon, and four massive neutral fields, where the mass of one of them is proportional to while the other three have masses proportional to (new energy scale). Therefore we can associate the field with SM and the fields , , and with three new neutral bosons. We note that (3.23) contains two same of the eigenvalues; thus, the and components have the same mass, and this conclusion contradicts the previous analysis in [41], but this is in agreement with [43, 44]. We also have four massive charged fields, where two of them have masses proportional to . Thus we can associate the fields to the SM fields , while the fields are new bosons. The eigenstates , and can be related to the physical eigenstates , and by
Assuming for , we obtain
where, again, and
We note that all are of order . So, assuming , for a new energy scale of order , all the βs are negligible.
3.3.2. Charged and Neutral Currents
The interaction between gauge bosons and fermions in flavor basis is given by the following Lagrangian density:
where represents any right-handed singlet and any left-handed triplet. We can write , and in lepton sector, we obtain
where
In quark sector we have that for the first family triplet , and for the singlets , and , we have and , respectively. Then we have
For second and third families we know that for the triplets and and , for the singlets , respectively, where . Then we obtain for
4. Neutrinos Interactions with Matter in 331 Model
It is well known that neutrino oscillation phenomenon in a material medium, as the sun, earth, or in a supernova, can be quite different from the oscillation that occurs in vacuum, since the interactions in the medium modify the dispersion relations of the particles traveling through it [60]. From the macroscopic point of view, the modifications of neutrino dispersion relations can be represented in terms of a refractive index or an effective potential. And according to [60, 61], the effective potential can be calculated from the amplitudes of coherent elastic scattering in relativistic limit.
In the present 331 model, the coherent scattering will be induced by neutral currents, NC, mediated by bosons , and and by charged currents, CC, mediated by bosons and . Following [61], we calculate in next sections the neutrino effective potentials in coherent scattering.
4.1. Charged Currents
The first term of (3.30) shows that the interaction of charged leptons with neutrinos occurs only through the gauge bosons ; then, by (3.22) we obtain that the interaction through charged bosons is given by
The amplitude for the neutrino elastic scattering with charged leptons in tree level through CC is given by (note from (4.1) that only left-handed leptons interact with neutrinos, as in SM)
For low energies , the effective Lagrangian is given by
where we used the Fierz transformation [62] to go from (4.2) to (4.3). Replacing (3.21) in (4.3), we obtain
where we used to denote the term that appears from the new charged boson. We can see that for a new energy scale the term that comes from the new boson does not contribute to the process, as expected, since the new charged boson has a mass of the order of the new energy scale of the theory (see (3.21)).
Now, since usual matter has only leptons from first family, we will restrain our calculations to the neutrino interactions with first family standard model particles. The term in (4.4) can be calculated following [61], where we have the correspondence spin, velocity, and , where is the electronic density. Assuming nonpolarized medium and vanishing average velocity, we obtain that (4.4) can be written as
The modifications on electronic neutrino dispersion relations can be represented by the following effective potential:
Disregarding the term since we are assuming , and remembering that in Section 3.3.1 we associated boson with SM boson , we can easily associate
We note that (4.7) gives limits for the VEV of one of the Higgs triplets. Under assumption , we can write , from which we can see that the maximum value of is achieved when we consider , in which replacing leads to
4.2. Neutral Current
The Lagrangian for neutrino elastic scattering with fermions through NC is given by
For low energies, we have that with and (4.9), and following the same procedure of Section 4.1, we obtain
4.2.1. Leptons Sector
From (3.31) and (3.24), we obtain that for the known neutral leptons
By (4.11), (4.12), and (4.13), we obtain that vertex interactions with neutrinos can be written as
where
We note from (4.14) that neutrinos do not interact electrically, as expected. For charged leptons, from (3.31) and (3.24), we obtain
and therefore
where
and, again, . We note that by (4.20) we can make the association . Then for , (4.15)β(4.17) and (4.21)β(4.23) lead to
Since intermediate neutral bosons in (4.9) do not distinguish between different lepton flavors, the interaction through NC with electron is described by the following effective potential:
where
and index refers to neutrino flavor. We note that the potential through CC comes from interactions of electron neutrinos with left-handed electrons, while the effective potential through NC comes from left- and right-handed electrons.
Considering both NC and CC, we can write the effective potential felt by neutrinos as , where
Comparing with SM expression for such potential:
we can find that
where we adopt in what follow, the convention that denotes SM-like part of the model; thus, the new terms beyond SM can be associated with the parameters s in NSI [63]. So, in the approximation , for , we obtain
We note that on limit , we recover SM. The NSIs are a subleading interaction, as expected. By (4.31) and (4.32), we obtain .
4.2.2. Quarks Sector
For the quarks of the first family, the Lagrangian density in (3.33) describes the interactions with gauge bosons , and ; then, by (3.24) and (3.25) we obtain the following interactions for up quarks:
The couplings quark-quark-boson for the first family are given by
where
We note that (4.34) and (4.35) reflect the fact that quarks interact electrically through photons with coupling constant , as in SM. The effective Lagrangian at low energies for neutrino interaction with quarks up through neutral currents are given by (4.10) with :
where is the up quarks average density.
SM predictions, using result of (4.7), can be written as
where
By comparison, we obtain
Then we can say that , where
Again, we obtain universal NSI, as for the electrons. We note that and in the limit we recover SM.
For down quarks by (3.33) and (3.24), we obtain that
where
Then by (4.10) for , we obtain the following effective Lagrangian for NC:
and the effective potential felt by neutrinos when crossing a medium composed by a density of quarks is , where
Then we can easily see that in SM the NC effective potential for neutrinos in a -quark medium, using result of (4.7), will be given by
Then from (4.47)β(4.50), we obtain
and neglecting terms of order , for , we obtain that , where
Then we obtain . Note that again in limit we recover the SM.
5. Results
In last sections we saw that in 331 model we chose, all NSI parameters are universal and diagonal and will not affect oscillation experiments. However, measurements of cross-section will be sensitive to such parameters, through modifications on [51]. We will now compare our results with those obtained in cross-section measurements. We will assume .
In Table 1 we can see that constrains in lead to ., while the constrains in lead to , and the constrains in ( lead to . If has its maximum value of 174.105 GeV, then βTeV. We note also that by we obtain ; then, for βTeV and βGeV, we obtain 142βGeVβGeV. We therefore cannot predict any hierarchy to the VEVβs and . Based on those results, we obtain the following inferior limits for the new gauge bosons masses:
6. Conclusion
We presented in this work a procedure to show that models with extended gauge symmetries can lead to neutrino nonstandard interactions, respecting the Standard Model Gauge symmetry , without spoiling the available experimental data and reproducing the known phenomenology at low energies. We also have shown that with an assumption about a mass hierarchy for the Higgs triplets VEVβs we could qualitatively address the mass hierarchy problem in standard model. Finally we obtained limits for the triplets VEVβs based on limits for NSI in cross-section experiments.
We believe that the class of model presented here is an interesting theoretical possibility to look for new physics beyond SM. We restrained our work to a simple scenario, but flavor-changing interactions can be naturally introduced in the model, leading to new constraints on NSI.
Acknowledgments
The authors would like to thank Alex Dias and Marcelo Guzzo for valuable discussions. One of them (M. Medina) would like to thank CNPq for financial support.
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