Exploring dark matter, neutrino mass and R_{K^{(*)},\phi} anomalies in L_{\mu}-L_{\tau} model

Exploring dark matter, neutrino mass and anomalies in model

Shivaramakrishna Singirala krishnas542@gmail.com    Suchismita Sahoo suchismita8792@gmail.com    Rukmani Mohanta rmsp@uohyd.ernet.in School of Physics, University of Hyderabad, Hyderabad-500046, India
Theoretical Physics Division, Physical Research Laboratory, Ahmedabad-380009, India

We investigate Majorana dark matter in a new variant of gauge extension of Standard Model, where the scalar sector is enriched with an inert doublet and a scalar leptoquark. We compute the WIMP-nucleon cross section in leptoquark portal and the relic density mediated by inert doublet components, leptoquark and the new boson. We constrain the parameter space consistent with PLANCK limit on relic density, PICO-60 and LUX bounds on spin-dependent direct detection cross section. Furthermore, we constrain the new couplings from the present experimental data on , , , and mixing, which occur at one-loop level in the presence of and leptoquark. Using the allowed parameter space, we estimate the form factor independent observables and the lepton non-universality parameters and . We also briefly discuss about the neutrino mass generation at one-loop level.

I Introduction

Though the experimental measured values of various physical observables are in excellent agreement with the Standard Model (SM) predictions, there are many open unsolved problems like the matter-antimatter asymmetry, hierarchy problem and the dark matter (DM) content of the universe etc., which make ourselves believe that there is something beyond the SM. In this regard, the study of rare semileptonic decay processes provide an ideal testing ground to critically test the SM and to look for possible extension of it. Although, so far we have not observed any clear indication of new physics (NP) in the sector, there are several physical observables associated with flavor changing neutral current (FCNC) processes which have Aaij et al. (2017, 2014a, 2013a, 2013b, 2014b); Langenbruch (2015) discrepancies. Especially, the observation of anomaly in the angular observables Aaij et al. (2013b) and the decay rate Aaij et al. (2014b) of processes have attracted a lot of attention in recent times. The decay rate of has also deviation compared to its SM prediction Aaij et al. (2013a). Furthermore, the LHCb Collaboration has observed the violation of lepton universality in process in the low region Aaij et al. (2014a)


which has a deviation from the corresponding SM result Bobeth et al. (2007)


In addition, an analogous lepton non-universality (LNU) parameter has also been observed in processes Aaij et al. (2017)


which correspond to the deviation of and from their SM predictions Capdevila et al. (2018)


To resolve the above anomalies, we extend the SM gauge group with a local symmetry. The anomaly free gauge extensions He et al. (1991a, b) are captivating with minimal new particles and parameters, rich in phenomenological perspective. The model is quite simple in structure, suitable to study the phenomenology of DM, neutrino and also the flavor anomalies. It is well explored in dark matter context in literature Patra et al. (2017); Biswas et al. (2016); Kamada et al. (2018); Bauer et al. (2018), in the gauge and scalar portals. The approach of adding color triplet particles to shed light on the flavor sector thereby connecting with dark sector is interesting. Leptoquarks (LQ) are not only advantageous in addressing the flavor anomalies, but also act as a mediator between the visible and dark sector. Few works were already done with this motivation Baek (2018); Mandal (2018); Arcadi et al. (2017); Allahverdi et al. (2018).

Leptoquarks are hypothetical color triplet gauge particles, with either spin-0 (scalar) or spin-1 (vector), which connect the quark and lepton sectors and thus, carry both baryon and lepton numbers simultaneously. They can arise from various extended standard model scenarios Georgi and Glashow (1974); Georgi (1975); Langacker (1981); Fritzsch and Minkowski (1975); Pati and Salam (1974, 1973a, 1973b); Shanker (1982a, b); Schrempp and Schrempp (1985); Gripaios (2010); Kaplan (1991), which treat quarks and leptons on equal footing, such as the grand unified theories (GUTs) Georgi and Glashow (1974); Fritzsch and Minkowski (1975); Langacker (1981); Georgi (1975), color Pati-Salam model Pati and Salam (1974, 1973a, 1973b); Shanker (1982a, b), extended technicolor model Schrempp and Schrempp (1985); Gripaios (2010) and the composite models of quark and lepton Kaplan (1991). In this article, we study a new version of gauge extension of SM with a scalar LQ (SLQ) and an inert doublet, to study the phenomenology of dark matter, neutrino mass generation and compute the flavor observables on a single platform. The SLQ mediates the annihilation channels contributing to relic density and also plays a crucial role in direct searches as well, providing a spin-dependent WIMP-nucleon cross section which is quite sensitive to the recent and ongoing direct detection experiments such as PICO-60 and LUX. The gauge boson of extended symmetry and the SLQ also play an important role in settling the known issues of flavor sector. In this regard, we would like to investigate whether the observed anomalies in the rare leptonic/semileptonic decay processes mediated by transitions, can be explained in the present framework. We analyze the implications of the model on both the DM and flavor sectors, in particular on decay modes. In literature Alok et al. (2017); Bečirević and Sumensari (2017); Hiller and Nisandzic (2017); D’Amico et al. (2017); Bečirević et al. (2016); Bauer and Neubert (2016); Li et al. (2016); Calibbi et al. (2015); Freytsis et al. (2015); Dumont et al. (2016); DorÅ¡ner et al. (2016); de Medeiros Varzielas and Hiller (2015); Dorsner et al. (2011); Davidson et al. (1994); Saha et al. (2010); Mohanta (2014); Sahoo and Mohanta (2016a, b, c, 2015); Kosnik (2012), there were many attempts being made to explain the observed anomalies of rare decays in the scalar leptoquark model.

The paper is structured as follows. We describe the particle content, relevant Lagrangian and interaction terms, pattern of symmetry breaking in section-II. We derive the mass eigenstates of the new fermions and the scalar spectrum in section-III. We then provide a detailed study of DM phenomenology in prospects of relic density and direct detection observables in section-IV. Mechanism of generating light neutrino mass at one-loop level is illustrated in section-V. Section-VI contains the additional constraint on the new parameters obtained from the existing anomalies of the flavor sector, like , , , and mixing. We then investigate the impact of additional gauge symmetry on the , LNU parameters and optimized observables in section-VII. We summarize our findings in Section-VIII.

Ii New model with a scalar leptoquark

We study the well known anomaly free extension of SM with three neutral fermions , with charges and respectively. A scalar singlet , charged under the new is added to spontaneously break the local gauge symmetry. We also introduce an inert doublet () and a scalar leptoquark with charges and to the scalar content of the model. We impose an additional symmetry under which all the new fermions, and the leptoquark are odd and rest are even. The particle content and their corresponding charges are displayed in Table. 1 .

Table 1: Fields and their charges of the proposed model.

The Lagrangian of the present model can be written as


where the scalar potential is


The gauge symmetry is spontaneously broken to by assigning a VEV to the complex singlet . Then the SM Higgs doublet breaks the SM gauge group to low energy theory by obtaining a VEV . The new neutral gauge boson associated with the extension absorbs the massless pseudoscalar in to become massive. The neutral components of the fields and can be written in terms of real scalars and pseudoscalars as


The inert doublet is denoted by , with . The masses of its charged and neural components are given by


The masses obtained by the colored scalar and the gauge boson are


In the whole discussion of the results, we consider the benchmark values for the masses of the scalar spectrum as TeV.

Iii Mixing in the fermion and scalar sector

The fermion and scalar mass matrices take the form


One can diagonalize the above mass matrices by , where


with and .

We denote the scalar mass eigenstates as and , with is assumed to be observed Higgs at LHC with GeV and GeV. The mixing parameter is taken minimal to stay with LHC limits on Higgs decay width. We indicate and to be the fermion mass eigenstates, with the lightest one () as the probable dark matter in the present work.

Iv Dark matter phenomenology

iv.1 Relic abundance

The model allows the dark matter () to have gauge and scalar mediated annihilation channels. The possible contributing diagrams are provided in Fig. 1 which are mediated by . Majorana DM in portal (upper row in Fig. 1 ) has already been well explored in literature Singirala et al. (2017); Nanda and Borah (2017). Here, we focus on -mediated channels (middle and bottom rows in Fig. 1 ) contributing to DM observables, which we later make connection with radiative neutrino mass as well as flavor observables.

Figure 1: Feynman diagrams contributing to relic density.
Figure 2: Behavior of relic density plotted against DM mass with TeV, shown with varying and (left panel) and (right panel). Black horizontal dotted lines denote the range of PLANCK limit (Aghanim et al., 2018).

The relic abundance of dark matter is computed by


Here the Planck mass, and denotes the total number of effective relativistic degrees of freedom. The function reads as


The thermally averaged annihilation cross section is given by the expression


where , denote the modified Bessel functions and , where is the temperature. The analytical expression for the freeze out parameter is


Here represents the number of degrees of freedom of the dark matter particle .

As seen from the left panel of Fig. 2, the relic density with -channel contribution is featured to meet the PLANCK limit (Aghanim et al., 2018) near the resonance in propagator (), i.e., near . We restrict our discussion to the mass region (in GeV), , and also is considered to be sufficiently large such that its resonance doesn’t meet the PLANCK limit below TeV region of DM mass. Now, in this mass range of DM, the channels mediated by drive the relic density observable, where the gauge coupling controls the -channel contribution, while are relevant in -channel contributions. The relevant parameters in our investigation are . The effect of these parameters on the relic abundance is made transparent in Fig. 2 , where we fixed , in order to explain neutrino mass at one loop level. Left panel shows the variation of relic density with varying gauge parameters and , right panel depicts the behaviour with varying parameter. No significant constraint on , parameters is observed, however relic density has an appreciable footprint on parameter space, which will be discussed in the next section.

iv.2 Direct searches

Moving to direct searches, the WIMP-nucleon cross section is insensitive to direct detection experiments as couples differently to Majorana fermion (axial-vector type) and quarks (vector type) Agrawal et al. (2010); Okada and Orikasa (2012). The -channel scalar () exchange can give spin-independent contribution, but it doesn’t help our purpose of study. In the scalar portal, one can obtain contribution from spin-dependent (SD) interaction mediated by SLQ, of the form


The corresponding cross section is given by Agrawal et al. (2010)


where the angular momentum , GeV for nucleon. The values of quark spin functions are provided in (Agrawal et al., 2010). Now, it is obvious that it can constrain the parameters and . Fig. 3 left panel displays parameter space (green and red regions) remained after imposing PLANCK Aghanim et al. (2018) limit on current relic density. Here, the region shown in green turns out to be excluded by most stringent PICO-60 Amole et al. (2017) limit on SD WIMP-proton cross section, as seen from the right panel.

Figure 3: Left panel depicts the parameter space consistent upto level of PLANCK limit (Aghanim et al., 2018) on relic density. Right panel gives the SD WIMP-proton cross section as a function of DM mass. Dashed lines represent the recent bounds obtained from PICO-60 Amole et al. (2017) and LUX Akerib et al. (2017). Green (red) data points in both the panels represent PLANCK allowed and PICO excluded (PLANCK and PICO allowed).

V Radiative neutrino mass

To generate light neutrino mass at one-loop level, we can write the interaction term using the inert doublet as


The corresponding diagram is shown in Fig. 4 . Assuming is much greater than the expression for the radiatively generated neutrino mass Ma (2006) is given by


Here and the fermion mass eigenstates . With a sample parameter space, and TeV, one can explain neutrino mass () near eV scale. Thus, the light neutrino mass generation can be successfully achieved in the proposed model.

Figure 4: Radiative generation of neutrino mass.

Vi Flavor Phenomenology

The general effective Hamiltonian responsible for the quark level transition is given by Bobeth et al. (2000, 2002)


where is the Fermi constant and denote the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements. The ’s stand for the Wilson coefficients evaluated at the renormalized scale Hou et al. (2014), where the sum over includes the current-current operators and the QCD-penguin operators . The quark level operators mediating leptonic/semileptonic processes are given as


where denotes the fine-structure constant and are the chiral operators. The primed operators are absent in the SM, but can exist in the proposed model.

The previous section has discussed the available new parameter space consistent with the DM observables which are within their respective experimental limits. However, these parameters can be further constrained from the quark and lepton sectors, to be presented in the subsequent sections.

vi.1 mixing

In this subsection, we discuss the constraint on the new parameters from the mass difference between the meson mass eigenstates (), which characterizes the mixing phenomena. In the SM, mixing proceeds to an excellent approximation through the box diagram with internal top quark and boson exchange. The effective Hamiltonian describing the transition is given by Inami and Lim (1981)


where , is the QCD correction factor and is the loop function Inami and Lim (1981) with . Using Eqn. (22), the mass difference in the SM is given as


The SM predicted value of by using the input parameters from Patrignani et al. (2016); Charles et al. (2015) is


and the corresponding experimental value is Patrignani et al. (2016)


Even though the theoretical prediction is in good agreement with the experimental oscillation data, it does not completely rule out the possibility of new physics.

Figure 5: Box diagrams of mixing with leptoquark in the loop.

The box diagrams for mixing in the presence of singlet SLQ and are shown in Fig. 5 . The effective Hamiltonian in the presence NP is given by




with and are the loop functions Baek (2018). Using Eqn. (26), the mass difference of mixing due to the exchange of and is found to be


Including the NP contribution arising due to the SLQ exchange, the total mass difference can be written as


Using Eqns. (24) and (25) in (29), one can put bound on and parameters.

vi.2 process

The rare semileptonic process is mediated via quark level transitions. In the current framework, the transitions can occur via the exchanging one-loop penguin diagrams shown in Fig. 6 .

Figure 6: Penguin diagram of processes, where with leptoquark in the loop.

The matrix elements of the various hadronic currents between the initial meson and meson in the final state are related to the form factors as follows Bobeth et al. (2007); Ball and Zwicky (2005)


where and denote the 4-momenta and mass of the meson and is the momentum transfer. By using Eqn. (30), the transition amplitude of process is given by


where and are the four momenta of charged leptons and is the loop function Baek (2018); Hisano et al. (1996). Now comparing this amplitude (31) with the amplitude obtained from the effective Hamiltonian (20), we obtain a new Wilson coefficient associated with the right-handed semileptonic electroweak penguin operator as


The differential branching ratio of process with respect to is given by