Unifying darko-lepto-genesis with scalar triplet inflation

Unifying darko-lepto-genesis with scalar triplet inflation

Chiara Arina chiara.arina@physik.rwth-aachen.de Jinn-Ouk Gong jinn-ouk.gong@cern.ch Narendra Sahu nsahu@iith.ac.in Institut für Theoretische Teilchenphysik und Kosmologie, RWTH Aachen, 52056 Aachen, Germany Theory Division, CERN, CH-1211 Genève 23, Switzerland Department of Physics, IIT Hyderabad, Yeddumailaram 502 205, Andhra Pradesh, India

We present a scalar triplet extension of the standard model to unify the origin of inflation with neutrino mass, asymmetric dark matter and leptogenesis. In presence of non-minimal couplings to gravity the scalar triplet, mixed with the standard model Higgs, plays the role of inflaton in the early Universe, while its decay to SM Higgs, lepton and dark matter simultaneously generate an asymmetry in the visible and dark matter sectors. On the other hand, in the low energy effective theory the induced vacuum expectation value of the triplet gives sub-eV Majorana masses to active neutrinos. We investigate the model parameter space leading to successful inflation as well as the observed dark matter to baryon abundance. Assuming the standard model like Higgs mass to be at 125-126 GeV, we found that the mass scale of the scalar triplet to be GeV and its trilinear coupling to doublet Higgs is so that it not only evades the possibility of having a metastable vacuum in the standard model, but also lead to a rich phenomenological consequences as stated above. Moreover, we found that the scalar triplet inflation strongly constrains the quartic couplings, while allowing for a wide range of Yukawa couplings which generate the CP asymmetries in the visible and dark matter sectors.

Cosmology of theories beyond the SM, dark matter theory, leptogenesis, inflation, baryon asymmetry, particle physics - cosmology connection.
journal: Nuclear Physics B



1 Introduction

A widely accepted theory of the early Universe supposes that there has been a period of cosmic inflation Guth:1980zm; Linde:1981mu; Albrecht:1982wi which not only explains the drawbacks of standard cosmology, but also provides seed for the temperature anisotropy in the cosmic microwave background Lyth:1998xn; Mukhanov:2005sc; Martin:2007bw; Lyth:2009zz; Mazumdar:2010sa. Finding a particle physics model for the inflaton is a non-trivial task however. In the standard model (SM) of particle physics, the only scalar field is the doublet Higgs , whose quartic coupling is not a free parameter once its mass is fixed. Hence a model of chaotic inflation is not possible within the framework of SM. However, by adding one more coupling between the Higgs and gravity Spokoiny:1984bd; Accetta:1985du; Salopek:1988qh, the potential could be made flat enough for producing approximately 60 -folds of inflation. Indeed there is a plateau for value of the field , where is the reduced Planck mass. The phenomenological inflationary constraints are met when matches the amplitude of density perturbations. For instance with a quartic coupling of the non-minimal coupling to gravity is bounded to be , and hence inflation takes place at the unitarity scale GeV Burgess:2009ea; Barbon:2009ya; Hertzberg:2010dc; Lerner:2010mq. This is the so-called Higgs inflation Bezrukov:2007ep; Bezrukov:2008ut; Bezrukov:2010jz. However, the indication of SM like Higgs at 125-126 GeV ATLAS; CMS lead to a metastable vacuum EliasMiro:2011aa; ArkaniHamed:2008ym at around GeV, which is much below the unitarity scale. The current uncertainties in the experimental measurements although allow one to extend the vacuum instability up to Planck scale, but it can only be resolved at future experiments. One of the possibilities to evade this issue is to widen the scalar field content of the SM. Extension of Higgs inflation by means of a scalar singlet or the inert doublet have been discussed in Lerner:2009xg; Lebedev:2011aq; Gong:2012ri; Lebedev:2012zw; EliasMiro:2012ay.

It is paramount to restore a thermal bath at the end of inflation to generate visible and dark matter (DM) observed today. At present a number of evidences suggests the existence of DM, which constitutes one quarter of the total energy budget of the Universe Bertone:2004pz; Komatsu:2010fb. However, hitherto a definite mechanism that gives rise to the observed relic abundance of DM is unknown. Usually it is assumed that the DM particle is in thermal equilibrium in the early Universe and freeze-out below its mass scale Kolb:1990vq. However, an alternative scenario to the freeze-out mechanism is that the relic abundance of DM can be accounted by an asymmetric component rather than by the symmetric one Nussinov:1985xr; Barr:1990ca; Dodelson:1991iv; Kaplan:1991ah; Kuzmin:1996he; Fujii:2002aj; Oaknin:2003uv; Hooper:2004dc; Kitano:2004sv; Cosme:2005sb; Farrar:2005zd; Roszkowski:2006kw; McDonald:2006if; Kohri:2009yn; An:2009vq; Kaplan:2009ag; Shelton:2010ta; Davoudiasl:2010am; Haba:2010bm; Gu:2010ft; Blennow:2010qp; McDonald:2010rn; Hall:2010jx; Dutta:2010va; Falkowski:2011xh; Chun:2011cc; Cui:2011ab; Arina:2011cu; Barr:2011cz; Petraki:2011mv; Iminniyaz:2011yp; Graesser:2011wi; Buckley:2011kk; Kouvaris:2011gb; Cirelli:2011ac; vonHarling:2012yn; Davoudiasl:2012uw; Tulin:2012re; Blennow:2012de. Since none of the particles in the SM can be a candidate of DM, one needs to explore physics beyond SM to have a particle physics candidate for DM. Apart from DM, the non-zero neutrino masses as confirmed by the oscillation data are required to be explained in a beyond SM framework. Recall that neutrinos are exactly massless within SM because of the conservation of lepton number up to all orders in perturbation theory.

Besides DM and neutrino mass, an explanation for the observed matter-antimatter asymmetry required for the big bang nucleosynthesis is still missing within the framework of SM. If the reheating temperature is less than electroweak (EW) scale then it is difficult to generate both DM and the observed baryon asymmetry Kohri:2009ka. On the other hand, if the reheating temperature is larger than EW scale, several mechanisms are available which can give rise to required baryon asymmetry, while leaving a large temperature window for creating DM species observed today. In the past years a lot of effort have been made to unify the mechanism giving rise to the asymmetry both in the DM and baryonic sectors Nussinov:1985xr; Barr:1990ca; Dodelson:1991iv; Kaplan:1991ah; Kuzmin:1996he; Fujii:2002aj; Oaknin:2003uv; Hooper:2004dc; Kitano:2004sv; Cosme:2005sb; Farrar:2005zd; Roszkowski:2006kw; McDonald:2006if; Kohri:2009yn; An:2009vq; Shelton:2010ta; Davoudiasl:2010am; Haba:2010bm; Gu:2010ft; Blennow:2010qp; McDonald:2010rn; Hall:2010jx; Dutta:2010va; Falkowski:2011xh; Chun:2011cc; Cui:2011ab; Arina:2011cu; Barr:2011cz; Petraki:2011mv; vonHarling:2012yn; Davoudiasl:2012uw; Walker:2012ka; Heckman:2011sw; MarchRussell:2012hi; Frandsen:2011kt; Belyaev:2010kp. An attempt to unify DM and baryon asymmetry via leptogenesis route has also been proposed by two of the authors in Arina:2011cu, where SM is extended by introducing a scalar triplet and a fermionic doublet dark matter candidate, stable by means of a remnant flavour symmetry. The triplet is taken to be at high scale such that its out-of-equilibrium decay can produce asymmetric DM as well as visible matter through leptogenesis mechanism Ma:1998dx; Hambye:2000ui. Moreover, in the low energy effective theory the induced vacuum expectation value (vev) of the scalar triplet could give rise sub-eV Majorana masses to the active neutrinos. Thus a triple unification of asymmetric DM, baryon asymmetry and neutrino masses in a minimal extension of the SM is achieved.

In this article, we realize primordial inflation in the presence of non-minimal coupling to gravity in a scalar triplet () extension of the SM and study the consequent low energy phenomenology. An early attempt of triplet inflation has been discussed in Chen:2010uc within the framework of chaotic inflation, where the quartic coupling of the triplet is supposed to be negligibly small (less than ) and the dominant term in the scalar potential is the triplet mass, around GeV. In presence of the non-minimal coupling of the scalar triplet to gravity the mass scale of the triplet can be much below than GeV without fine tuning the quartic coupling. We take the mass scale of triplet to be around GeV such that it not only give neutrino masses, dark matter abundance and baryon asymmetry, but also evade the possiblity of having a metastable vacuum in the SM EliasMiro:2011aa; ArkaniHamed:2008ym. In presence of non-minimal couplings and to gravity the scalar triplet, together with the SM Higgs field, behaves as inflaton. From this multi-field inflationary scenario a single field model can be retrieved as we demonstrate below. We show that once the heavy mode is settled down at the minimum, the scalar potential is positive definite only if the mass term and the lepton number violating term () are negligible. However, the inflaton can be an admixture of both triplet and SM Higgs moduli or a pure state. We demonstrate in detail how these three cases give rise to different constraints on the model parameter space. Subsequently, we explain how the decay of scalar triplet Arina:2011cu can generate an asymmetric dark matter and visible matter observed today.

The article is organized as follows. In section 2 we briefly underline the main features of the model, which has been introduced in Arina:2011cu and point out new constraints in the parameter space. We then describe the inflationary picture in section 3, where we work out the slow-roll predictions for single field inflation after having discussions regarding the numerical and analytical estimates of all the terms in the scalar potential. The generation of the asymmetries in the dark and visible sectors are discussed in section 4. The ensuing section 5 details the renormalization group (RG) equations accounting for the additional field content with respect to the SM ones. Our results are presented in section 6 and we conclude in section 7. We recall in A the main Boltzmann equations for the production of the asymmetries in both baryonic and DM sectors.

2 Scalar Triplet as the Origin of Inflation and Darko-Lepto-genesis

We extend SM by introducing a scalar triplet , where the quantum numbers in the parenthesis are the charge under the gauge group . Since the hypercharge of is 2, it can have bilinear coupling to the Higgs doublet . As a result the scalar potential involving and can be given as follows:


where the index stands for the Jordan frame, as will be explained in the next section 3. The representation of the scalar triplet is


In the fermion sector we introduce a vector-like doublet with hypercharge  Arina:2011cu. As a result the bilinear couplings of to the lepton doublets , and are given as follows:


where . The covariant derivative is defined as


where represents the Pauli spin matrices. For the hypercharge coupling we have used the grand unified theory (GUT) charge normalisation: .

From (1) and (3) we notice that:

  1. The bilinear coupling of to the Higgs and lepton doublets jointly violate lepton number by two units. Moreover, the couplings are complex and hence can accommodate a net CP violation. As a result the out-of-equilibrium decay of to and in the early Universe can give rise to the observed matter-antimatter asymmetry via leptogenesis route Ma:1998dx; Hambye:2000ui.

  2. The Lagrangian is invariant under a remnant symmetry, with being odd while all the other fields even. This ensures the stability of , the neutral component of , which can be a candidate of dark matter. Hereafter is the inert fermion doublet DM Arina:2011cu. Since the bilinear coupling of to is in general complex, it can accommodate a net CP violation. Therefore, the out-of-equilibrium decay of in the early Universe can generate an asymmetry in DM sector in a similar way the lepton asymmetry is generated via the decay and .

    In the effective theory the bilinear coupling of to and generates a dimension-five operator suppressed by the mass scale of . This is an equivalent type-II seesaw for Majorana mass of DM. Below EW phase transition this operator generates small Majorana mass for as given by


    where is the vev of the SM Higgs. Since is a vector-like Dirac fermion, it can be expressed as a sum of two Majorana fermions, i.e. . Therefore, in a flavour basis , the mass matrix of DM is given by


    Diagonalising the above mass matrix we get two mass eigenstates and with masses and . The mass splitting between the two states is required to be keV in order to explain the high precision annual modulation signal at DAMA Bernabei:2010mq; TuckerSmith:2001hy; Arina:2009um; Arina:2011cu while the null result at Xenon100 Aprile:2011ts. This implies a lower bound on to be


    where we have assumed as required by the parameter of SM.

  3. In the effective low energy theory the bilinear coupling of to lepton and Higgs doublets also generate a dimension-five operator , suppressed by the mass scale of , for neutrino masses. When acquires a vev, this operator then induces sub-eV Majorana masses to active neutrinos given by:


    For , we can easily obtain sub-eV masses of active neutrinos for a wide range of values of the couplings and . For example, taking and to be order unity we need GeV to get sub-eV neutrino masses. For lighter one can get neutrino masses in the ball park of oscillation data by taking smaller values of , yet maintaining vev of to be less than GeV. An advantage for smaller values of is that we can easily explain the required ratio:


    Thus for , we expect .

  4. In the presence of the non-minimal couplings of and to gravity, the scalar potential (1) can give rise to inflation in the early Universe Bezrukov:2007ep; Bezrukov:2008ut; Bezrukov:2010jz. The scale of inflation at which the power spectrum is normalized (see later section) is GeV, which is much below the Planck scale. At the end of inflation, the Universe becomes radiation dominated, during which the interactions of as given in (3) generate asymmetries in visible and DM sectors.

3 Scalar Triplet – Higgs Inflation

3.1 Action in the Einstein frame

The model for the scalar fields has been defined in the previous section. The scope of this section is to work out the action for inflation. The physical fields are defined in the Jordan frame denoted by an index . We introduce for both scalar components non-minimal couplings to the Ricci scalar . Hence the action in the Jordan frame is:


with the reduced Planck mass set to unity, i.e. .

In the Jordan frame the couplings make the gravitational interactions non-standard. It is therefore convenient to perform a conformal transformation into the Einstein frame, for which we put no index, to retrieve the standard form of the Einstein equations as far as gravity concern, but at the expense of having non-standard kinetic terms for the scalar fields. A conformal transformation preserves the causal structure of space-time in both frames and is given by a smooth and strictly positive function of the fields:


Note that both frame are equivalent for small field values. The metric and the potential transform as:


The doublet and triplet scalar fields are defined in the unitary gauge as following:


where and account for the two degrees of freedom of the triplet neutral component, defined as .

Now taking the large field limit and redefining fields as:


(10) reads:


Note that the kinetic part is highly non-trivial for all fields , and . However the potential, with the field redefinition, takes the form:


3.2 Scalar potential analysis

During inflation the mass eigenvalue of is very large as compared to the Hubble parameter Gong:2012ri. Therefore, is minimized at and we find the effective theory for the light inflatons. The action then becomes:


with . Note that the stabilization of demands important constraints on the couplings, which will be discussed in the following section. For a finite value of , with


we can further approximate the kinetic sector as


where .

For the potential, as can be seen from (3.1), it consists of three contributions – quartic, quadratic and the -terms. Since the latter two are exponentially suppressed, one may be tempted to drop them from the beginning for simplicity. However we must check explicitly if quartic term is really dominant, only after then we can make any simplification. First let us compare the quartic term with the quadratic mass term:


Here we first assume the quartic term is dominant, which normalizes the combination from the amplitude of the power spectrum (see later section). We will justify this assumption a posteriori. Then, with the typical value of during inflation, say , we have so that the ratio becomes


It is not difficult to set this ratio negligibly small with large enough and not too large and : for ( GeV), this ratio becomes which can be easily made small, and even easier if we let smaller than . For the triplet term with we can proceed similarly, and obtain


which looks more stringent than and there indeed is a tension: with large enough and and not too large this ratio may be close to 1 and we should not neglect . However there is another constraint that the potential be positive everywhere. For simplicity, let us neglect which can be made easily negligible, then the potential is


which should be positive definite. This gives


We can easily note that is a mildly increasing function of with the values 1.12364 at and 7.63495 at . Thus, to guarantee the positivity of the potential until the end of inflation where provided that is dominant, we should demand


which in turn gives, combined with (26),


That is, the positivity of the potential demands that the quartic term be dominant, with the fraction of the triplet term contribution at most percent. Further, returning back to (25), using (29) we find


Thus, for , remains indeed negligible compared with unless is very large. However too large will pull down the unitarity scale further, greatly harming the validity of the effective theory: if , may compete with up to percent, but the unitarity scale may well be saturated near and the low energy approximation cannot be trusted. So not too large guarantees negligible contribution of . All these a posteriori justify our assumption at the beginning that the potential is dominated by the quartic term so that .

This estimate gives us the idea that the contributions of to the observable quantities are not significant. To check this, we first compute numerically the change in the number of -folds as follows. We compute from the moment , when the scale of our interest exits the horizon, to , the end of inflation, with a given set of initial conditions of and . Then we repeats with slightly different initial conditions to find and . Then, we find according to the change in the initial field values. In table 1 we show and for several values of , and (note that the amplitude of the power spectrum fixes for a given ). Single field analytic estimate (52) gives (54.0031) for (5.5).

, , , , 42.0850 35.9478
the same as above but 64.3191 54.2294
the same as above but 42.0884 35.9508
, , , , 42.1880 36.0828
the same as above but 64.5161 54.4816
the same as above but 42.1880 36.0828
, , , , 42.1785 36.0711
the same as above but 64.4986 54.4600
the same as above but 42.1785 36.0711
Table 1: Numerical estimates of the contributions of the fields and to the total number of -folds . For the first case the bound (29) is saturated, while the rest examples satisfy it trivially.

3.3 Constraints on the scalar potential

From now on, as discussed in the previous section, we only consider single field case where term does not contribute and only drives inflation. However as mentioned at the beginning, we assume that is stabilized already. For this to happen, we need to study in detail this stabilization which gives constraints on the couplings. These constraints do affect low energy phenomenology by incorporating RG equations, even which does not participate in the inflationary dynamics but whose RG equation does include quartic couplings.

We first have to ensure that the potential, quartic terms alone, is positive definite everywhere. This is necessary because we may not have to ensure . Indeed, we must have , and


for positive potential.

Coming back to the definition of the potential in terms of and , as the mass eigenvalue for is very large compared to (see Appendix B of Gong:2012ri and Achucarro:2010jv; Achucarro:2010da; Achucarro:2012sm) we assume is stabilized at throughout the whole process of our interest. The different minima in which the heavy field quickly sets in, are found minimizing the potential part independent of :


The minima are listed below together with the corresponding minimum energy and constraints for vacuum stability. At and inflation is driven by pure Higgs () or pure triplet (). At the finite minimum, inflation is driven by an admixture of both fields.

  1. : Then, becomes a constant, i.e. vacuum energy, of the value:


    We demand that and . Then, we must satisfy the conditions


    Note that the first condition is also equivalent to demanding that the numerator of (33), which is essentially a quadratic equation of , is always positive, i.e. the equation has no solution of .

  2. : In this case so this corresponds to pure Higgs inflation, i.e. the Higgs moduli alone drives inflation. becomes a constant, i.e. vacuum energy, of the value


    In this case gives

  3. : In this case so this corresponds to pure triplet inflation (in this case the triplet moduli alone drives inflation) with:


    In this case gives


Notice that because of (39) and (43) for pure Higgs and triplet inflation is preferred.

3.4 Slow-roll analysis for single field inflation

Figure 1: Left: Two slow-roll parameters (47) and (48) are shown as a function of the inflaton field. The magenta solid line denotes while the black dotted line stands for , as labeled. Right: The inflationary potential (46) is depicted as a function of the inflaton, solid gray line. In both panels the light red region indicates where the slow roll condition break down.

Provided that the quartic potential alone is dominant over quadratic or triplet contributions to the potential, we may estimate the inflationary predictions using the so-called formalism Starobinsky:1986fxa; Sasaki:1995aw; Sasaki:1998ug; Gong:2002cx; Lyth:2005fi. Essentially, the formalism tells us that the perturbation in the number of -folds, which is the same in both frames Gong:2011qe, is equivalent to the curvature perturbation on super-horizon scales. Then the slow-roll approximation, described by the parameters and is working well111Note that this approximation is equivalent at first order to the slow-roll predictions obtained with the Hubble flow parameters , as described in Martin:2006rs..

Before going into the detail of slow-roll inflation let us make comments about the reheating. Inflation not only consists of the slow-roll period but also a reheating phase since it permits to link inflation with the subsequent radiation dominated era. This phase is connected to the potential part close to the minimum and takes place during a few -folds. The reheating phase is poorly known and technically difficult to model properly. To take into account uncertainties on this post inflationary phase we use the reheating parameter described in Martin:2010kz as


having supposed the simplest model of a scalar field coupled to radiation and that the effective fluid (inflaton plus radiation) with energy density and pressure is conserved and stands for the mean equation of state parameter during reheating. In addition is defined as the total number of -folds during reheating


being the number of -folds at which reheating is completed and the radiation dominated period begins while is the total number of -folds during inflation. We assume instantaneous reheating, namely at the end of inflation the Universe enters straightaway in the radiation dominated era with equation of state . This is equivalent to consider the reheating parameter equal to 1 or in (44). This can be understood physically because the pre-/reheating stage can not be distinguished from the radiation dominated era and therefore can not affect the inflationary predictions.

After the analysis of the previous sections the actual potential is, with ,


Its behavior as a function of the field is shown in the right panel of figure 1. Note that inflation takes place for trans-Planckian values of the field. The shaded region denotes the breakdown of the slow-roll approximation, that we discuss straightaway. We can define the slow-roll parameters in terms of the potential as


Both parameters, as functions of , are shown in the left panel of figure 1, as labeled. Note that both are positive and monotonically increasing functions of the inflaton. The red region indicates the breakdown of slow-roll condition . Then, the number of -folds becomes, using the slow-roll equation ,


To determine the latter, we identify this moment as when . Then we easily find


The total number of -folds after the Hubble length exit for instantaneous reheating is given by , using (44), enough to solve the flatness and horizon problems. We can find by plugging into (49) as


From (49) we can immediately find


The power spectrum is normalized at the pivot scale of WMAP7, : . The power spectrum of scalar perturbation at the pivot scale is given by:


Thus as quoted several times before, is fixed once we fix or vice versa, as


Also, the spectral index is given by


which lies well within the 2 range of WMAP7,  Komatsu:2010fb.

4 Asymmetric DM and Leptogenesis

From (3) we see that there are three different channels available for the decay of scalar triplet : , and . Since the couplings are in general complex, the quasi-equilibrium decay of via these channels produce asymmetries in lepton and DM sectors.

Figure 2: The CP asymmetry for leptogenesis and asymmetric DM generated by the interference of tree and one-loop self energy correction diagrams are shown in the first and second raw respectively.

The CP asymmetry in either sector arises via the interference of tree level with the one-loop self energy diagram as shown in figure 2. From these diagrams we see that to generate net CP asymmetry at least two scalar triplets and are required. As a result the interaction of and is described by a complex mass matrix instead of a single mass term as mentioned in (1). The diagonalisation of the flavour basis spanned by () gives rise to two mass eigenstates with masses and . The complex conjugate of are given by . Unlike the flavour eigenstates and , the mass eigenstates and are not CP eigenstates and hence their decay can give rise to CP asymmetry. Assuming a mass hierarchy in the mass eigenstates of the triplets, the final asymmetry arises by the decay of lightest triplet and . The CP asymmetries are defined as


where the front factor 2 takes into account of two similar particles are produced per decay. From figure 2, the asymmetries are estimated to be:




is the total decay rate of the lightest triplet. In the numerical calculations we will use this total decay rate as: , where and are the branching fractions in the decay channels and respectively. In the following we set eV and therefore the total decay rate depends only on three variables, namely , and .

When fails to compete with the Hubble expansion scale of the Universe, decays away and produces asymmetries in either sectors. As a result the yield factors are given by:


where , is the entropy density and , are the efficiency factors, which take into account the depletion of asymmetries due to the number violating processes involving , and . At a temperature above EW phase transition a part of the lepton asymmetry gets converted to the baryon asymmetry via the sphaleron processes. As a result the baryon asymmetry is:


where is the number of generations and is the number of scalar doublets. From (62) and (63) the DM to baryon ratio is given by:


where is the proton mass. From this equation it is clear that the criteria can be satisfied by adjusting the ratio and , where the efficiency factor:


can be obtained by solving the relevant Boltzmann equations Arina:2011cu; Hambye:2005tk; Chun:2006sp given in A. The ratio of CP asymmetries is


From the above equation we observe that if , then we get


Taking and we get the ratio of CP asymmetries in a broad range: .

5 Renormalisation Group Equations in Scalar Triplet Model

The RG equations of the scalar, gauge and Yukawa couplings in SM have been extensively discussed in the literature, see for example DeSimone:2008ei; Espinosa:2007qp for discussions relative to the cosmological framework. However, in the presence of scalar triplet the RG evolution of these couplings change because of the additional lepton number violating interactions of the scalar triplet with SM Higgs and leptons, as it has been described first in Chao:2006ye and then improved by Schmidt:2007nq; Gogoladze:2008gf, which will be our main references. Moreover, in our case, the triplet couples to the inert fermion doublet dark matter . In the following we list the modification to the standard running for as well as the RG equations for the different couplings pertaining to such as , , , and the non-minimal couplings to gravity and .

Having defined , where is the renormalization scale, the RG equations of the quartic couplings in the scalar potential including the triplet are given by