Neutrino mass, leptogenesis and FIMP dark matter in a $$\mathrm{U}(1)_{B-L}$$ U ( 1 ) B - L model Academic research paper on "Physical sciences"

Eur. Phys. J. C (2017) 77:875 https://doi.org/10.1140/epjc/s10052-017-5436-y

The European Physical journal C

CrossMark

Regular Article - Theoretical Physics

Neutrino mass, leptogenesis and FIMP dark matter in a U(1)B-L model

Anirban Biswas1'2'3, Sandhya Choubey1'2'3^, Sarif Khan12'c

1 Harish-Chandra Research Institute' Chhatnag Road' Jhunsi> Allahabad 211 019' India

2 Homi Bhabha National Institute' Training School Complex' Anushaktinagar Mumbai 400094' India

3 Department of Theoretical Physics' School of Engineering Sciences' KTH Royal Institute of Technology' AlbaNova University Center. 106 91 Stockholm' Sweden

online: 15 December 2017

Received: 16 October 2017 / Accepted: 29 November 2017 / Published © The Author(s) 2017. This article is an open access publication

Abstract The Standard Model (SM) is inadequate to explain the origin of tiny neutrino masses, the dark matter (DM) relic abundance and the baryon asymmetry of the Universe. In this work, to address all three puzzles, we extend the SM by a local U(1)#_l gauge symmetry, three right-handed (RH) neutrinos for the cancellation of gauge anomalies and two complex scalars having non-zero U(1)b-l charges. All the newly added particles become massive after the breaking of the U(1)b-l symmetry by the vacuum expectation value (VEV) of one of the scalar fields $H. The other scalar field, $DM, which does not have any VEV, becomes automatically stable and can be a viable DM candidate. Neutrino masses are generated using the Type-I seesaw mechanism, while the required lepton asymmetry to reproduce the observed baryon asymmetry can be attained from the CP violating out of equilibrium decays of the RH neutrinos in TeV scale. More importantly within this framework, we study in detail the production of DM via the freeze-in mechanism considering all possible annihilation and decay processes. Finally, we find a situation when DM is dominantly produced from the annihilation of the RH neutrinos, which are at the same time also responsible for neutrino mass generation and leptogenesis.

1 Introduction

The presence of non-zero neutrino mass and mixing has been confirmed by observing neutrino oscillations [1,2] among its different flavours. Neutrino experiments have measured the three intergenerational mixing angles (012, 023, 013) and

a e-mail: anirbanbiswas@hri.res.in b e-mail: sandhya@hri.res.in c e-mail: sarifkhan@hri.res.in

the two mass square differences (Am2\ and Am32)1 with an unprecedented accuracy [3-10]. Neutrinos are massless in the Standard Model (SM) of particle physics because in the SM there is no right-handed (RH) counterpart of the left-handed (LH) neutrinos. To generate tiny neutrino masses and their intergenerational mixing angles, as suggested by different experiments, we have to think of some new interactions and/or new particles beyond the Standard Model (BSM). Moreover, there are still some unsolved problems in the neutrino sector. For example, we do not know the exact octant of the atmospheric mixing angle 023 i.e. whether it lies in the lower octant (023 < 45°) or in the higher octant (023 > 45°), the exact sign of Am22, which is related to the mass hierarchy between m2 and m3 (for the normal hierarchy (NH) Am22 > 0, while for the inverted hierarchy (IH) Am22 < 0) and as regards the Dirac CP phase 5, responsible for the CP violation in the leptonic sector. Recently, T2K and Nov A experiments have reported their preliminary result which predicts that the value of Dirac CP phase is around 5CP ~ 270° [11]. Besides these, we do not know whether the neutrinos are Dirac or Majorana fermions. Observation of neutrino-less double p decay [12-17] will confirm the Majorana nature of neutrinos and might also provide important information as regards the Majorana phases, which could be the other source of CP violation in the leptonic sector, if the SM neutrinos are Majorana fermions.

Besides these unsolved problems in the neutrino sector, another well-known puzzle in recent times is the presence of dark matter (DM) in the Universe. Many pieces of indirect evidence suggest the existence of DM. Among the most compelling pieces of evidence of DM are the observed flatness of rotation curves of spiral galaxies [18], gravitational lensing [19], the observed spatial offset between DM and visible mat-

1 A 2 2 2 1 A 2 2 m2+m2

1 Am2 = m2 — m2 and Am2m = m2--

ij i j aim 3 2

ter in the collision of two galaxy clusters (e.g. the Bullet cluster [20], the Abell cluster [21,22]) etc. The latter also imposes an upper bound on the ratio between self interaction and mass of DM particles, which is ^ iS 1 barn/GeV [23]. Moreover, satellite borne experiments like WMAP [24] and Planck [25] have made a precise measurement of the amount of DM present in the Universe from the cosmic microwave background (CMB) anisotropy and the current measured value of this parameter lies in the range 0.1172 < ^DMh2 < 0.1226 at 67% C.L. [25].

Despite the compelling observational evidence for DM due to its gravitational interactions, our knowledge about its particle nature is very limited. The only thing we know about the DM is that it is very weakly interacting and elec-tromagnetically blind. The SM of particle physics does not have any fundamental particle which can play the role of a cold dark matter (CDM), consequently a BSM scenario containing new fundamental stable particle(s) is required. There are earth based ongoing DM direct detection experiments, namely Xenon-1T [26], LUX [27], CDMS [28,29] amongst others, which have been trying to detect the weakly interacting massive particle (WIMP) [30-32] type DM by measuring recoil energies of the detector nuclei scattered by the WIMPs. However, no convincing DM signal has been found yet and hence the MDM-aSI plane for a WIMP type DM is now getting severely constrained. Therefore, invoking particle DM models outside the WIMP paradigm seems to be pertinent at this stage [33]. In the present work we study one of the possible alternatives of WIMP, namely, the feebly interacting massive particle (FIMP) [34-43]. A major difference between the WIMP and FIMP scenarios is that, while in the former case the DM particle is in thermal equilibrium with the plasma in the early Universe and freezes-out when the Hubble expansion rate becomes larger than its annihilation cross section, in the FIMP case the DM is never in thermal equilibrium with the cosmic soup. This is mainly ensured by its extremely weak couplings to other particles in the thermal bath. Therefore, the number density of the FIMP is negligible in the early Universe and increases when the FIMP is subsequently produced by the decays and annihilations of other particles to which it is coupled (very feebly). This process is generally known as freeze-in [34].

In addition to the above two unsolved problems, another long standing enigma is the presence of more baryons over anti-baryons in the Universe, which is known as the baryon asymmetry or the matter-antimatter asymmetry in the Universe. The baryon asymmetry observed in the Universe is

expressed by a quantity YB =-, where nB = nB - ng

is the excess in the number density for baryon over antibaryon while nY and s are the photon number density and the entropy density of the Universe, respectively. At the present epoch, nB = (5.8 - -6.6) x 10-10 at 95% C.L. [44] while at

T ~ 2.73 K, the photon density nY = 410.7 cm-3 [44] and the entropy density s = 2891.2 cm-3 [44] (in natural unit with Boltzmann constant KB = 1). Therefore, the observed baryon asymmetry at the present Universe is YB = (8.249.38) x 10-10, which, although small, is sufficient to produce the ~ 5% energy density (visible matter) of the Universe. To generate the baryon asymmetry in the Universe from a matter-antimatter symmetric state, one has to satisfy three necessary conditions, known as the Sakharov conditions [45-48]. These are: i) baryon number (B) violation, ii) C and CP violation and iii) departure from thermal equilibrium. Since the baryon number (B) is an accidental symmetry of the SM (i.e. all SM interactions are B conserving) and also the observed CP violation in quark sector is too small to generate the requited baryon asymmetry, like the previous cases, here also one has to look for some additional BSM interactions which by satisfying the Sakharov conditions can generate the observed baryon asymmetry in an initially matter-antimatter symmetric Universe.

In this work, we will try to address all of the three above mentioned issues. The non-observation of any BSM signal at LHC implies the concreteness of the SM. However, to address all the three problems, we need to extend the particles list and/or gauge group of SM because as already mentioned, the SM is unable to explain either of them. In our model, we extend the SM gauge group SU(3)c x SU(2)l x U(1)y by a local U(1)B-L gauge group. The B - L extension of the SM [49-52] has been studied earlier in the context of DM phenomenology [53-63] and baryogenesis in the early Universe in Refs. [64-66]. Since we have imposed a local U(1) symmetry, consequently an extra gauge boson (ZBL) will arise. To cancel the anomaly due to this extra gauge boson we need to introduce three RH neutrinos (Ni, i = 1, 2, 3) to make the model anomaly free. Apart from the three RH neutrinos, we also introduce two SM gauge singlet scalars namely and $DM, both of them are charged under the proposed U(1)B-L gauge group. The U(1)B-L symmetry is spontaneously broken when the scalar field $H takes a non-zero vacuum expectation value (VEV) and thereby generates the masses for the three RH neutrinos as well as the extra neutral gauge boson ZBL, whose mass terms are forbidden initially due to the U(1)B-L invariance of the Lagrangian. The other scalar $DM does not acquire any VEV and by choosing appropriate B — L charge $DM becomes naturally stable and therefore, can serve as a viable DM candidate. As mention above, anomaly cancellation requires the introduction of three RH neutrinos in the present model. Therefore we can easily generate the neutrino masses by the Type-I seesaw mechanism after B - L symmetry is broken. Diagonalising the light neutrino mass matrix (mv, for details see Sect. 3.1), we determine the allowed parameter space by satisfying the 3a bounds on the mass square differences (Am\2, Am2tm), the mixing angles (&i2, 9\3 , 023) [67] and the cosmological

bound on the sum of three light neutrinos masses [25]. We also determine the effective mass mpp, which is relevant for neutrino-less double beta decay and compare it with the current bound on mpp from the GERDA phase I experiment [13].

Next, we explain the possible origin of the baryon asymmetry at the present epoch from an initially matter-antimatter symmetric Universe via leptogenesis. We first generate the lepton asymmetry (or B — L asymmetry, YB—L ) from the out of equilibrium, CP violating decays of the RH neutrinos. The lepton asymmetry thus produced has been converted into the baryon asymmetry by the (B + L) violating sphaleron processes, which are effective before and during electroweak phase transition [68-70]. When the sphaleron processes are in thermal equilibrium (1012 GeV ^ T < 102 GeV, T being the temperature of the Universe), the conversion rate is given by [71]

Yb = — 8Nf + 4YB.L . B 22Nf + 13 N$h B L '

where Nf = 3 and N$h = 1, are the number of fermionic generations and number of Higgs doublet in the model, respectively.

Finally, in order to address the DM issue, we consider the singlet scalar $DM as a DM candidate. Since the couplings of this scalar to the rest of the particles of the model are free parameters, they could take any value. Depending on the value of these couplings, we could consider $DM as a WIMP or a FIMP. Detailed study on the WIMP type scalar DM in the present U(1)b—l framework has been done in Refs. [59,60,72]. In most of the earlier works, it has been shown that the WIMP relic density is mainly satisfied around the resonance regions of the mediator particles. Moreover, the WIMP parameter space has now become severely constrained due to non-observation of any "real" signal in various direct detection experiments. Thus, as discussed earlier, in this situation the study of scalar DM other than WIMP is worthwhile. Therefore in this work, we consider the scalar field $DM as a FIMP candidate which, depending on its mass, is dominantly produced from the decays of heavy bosonic particles such as hx, h2, Zbl and from the annihilations of bosonic as well as fermionic degrees of freedom present in the model (e.g. Nt, ZBL, hi etc.). In particular, in Ref. [43], we have also studied a SM singlet scalar as the FIMP type DM candidate in a LM — L T gauge extension of the SM. In that work, we have considered the extra gauge boson mass in MeV range to explain the muon (g — 2) anomaly. Consequently, the production of O(GeV) DM from the decay of Z^T is forbidden. Additionally, in that model due to the considered Ln — L T flavour symmetry the neutrino mass matrices (both light and heavy neutrinos) have a particular shape. On the other hand, in the present work, we extensively study the FIMP DM production mechanism from all possible decays

and annihilations other particles present in the model. Moreover, we have found that depending on our DM mass, a sharp correlation exists between the three puzzles of astroparticle physics, namely neutrino mass generation, leptogenesis and DM. Furthermore, in Ref. [40], one of us, along with collaborators, has studied the freeze-in DM production mechanism in the framework of U(1)B—L extension of the SM. However, in that article one has considered an MeV range RH neutrino as the FIMP DM candidate. Thus, in the context of DM phenomenology the current work is vastly different from Ref. [40].

In the non-thermal scenario, most of the production of the FIMP from the decay of a heavy particle occurs when T ~ M, where M is the mass of the decaying mother particle, which is generally assumed to be in thermal equilibrium. Therefore, the non-thermality condition of the FIMP

[73], which in turn imposes a

demands that

severe upper bound on the coupling strengths of the FIMP. Thus the non-thermality condition requires an extremely small coupling of $DM with the thermal bath < 10—10) and, hence, FIMP DM can easily evade all the existing bounds from DM direct detection experiments [26-28].

The rest of the paper has been arranged in the following manner. In Sect. 2 we discuss the model in detail. In Sect. 3 we present the main results of the paper. In particular, we discuss the neutrino phenomenology in Sect. 3.1, baryogenesis via leptogenesis in Sect. 3.2 and non-thermal FIMP DM $DM production in Sect. 3.3. Finally in Sect. 4 we end with our conclusions.

2 Model

The gauged U(1)B—L extension of the SM is one of the most extensively studied BSM models so far. In this model, the gauge sector of the SM is enhanced by imposing a local U(1)B—L symmetry to the SM Lagrangian, where B and L represent the respective baryon and lepton number of a particle. Therefore, the complete gauged group is SU(3)c x SU(2)l x U(1)y x U(1)b— l. Since the U(1)b—l extension of the SM is not an anomaly free theory, we need to introduce some chiral fermions to cancel the anomaly. In order to achieve this, we consider three extra RH neutrinos to make the proposed B - L extension anomaly free. Besides the SM particles and three RH neutrinos, we introduce two SM gauge singlet scalars , $DM in the theory with suitable B — L charges. One of the scalar fields, namely , breaks the proposed U(1)B—L symmetry spontaneously by acquiring a non-zero VEV uBL and thereby generates masses to all the BSM particles. We choose the B - L charge of $DM in such a way that the Lagrangian of our model before the U(1)B—L symmetry breaking does not contain any inter-

action term involving odd powers of $DM. When $H gets a non-zero VEV, this U(1)B-L symmetry breaks spontaneously into a remnant Z2 symmetry under which only $DM becomes odd. The Z2 invariance of the Lagrangian will be preserved as long as the parameters of the Lagrangian are such that the scalar field $DM does not get any VEV. Under this condition, the scalar field $DM becomes absolutely stable and, in principle, can serve as a viable DM candidate. The respective SU(2)L, U(1)Y and U(1)B -L charges of all the particles in the present model are listed in Table 1.

The complete Lagrangian for the model is as follows:

L = Lsm + Ldm + ()h.D'*h) - 4 Fblßv Fbl'v

+- Ni y' D' Ni - V (*h Ah )

- Z yNi*hNcn - J2 y'iL*hNj + h.c., (2) i, j = 1

with *h = ia2*'*h. The term LSM and LDM represent the SM and dark sector Lagrangian, respectively. The dark sector Lagrangian LDM containing all possible gauge invariant interaction terms of the scalar field *DM has the following form:

ldm = (D'*dm)f(D'*dm) - ßDM(*DM*dm)

t 2 t t "^DM (*DM*DM) - kDh (*DM*DM)(*h*h)

-kDH (*DM*DM)(*H*H) ,

where the interactions of $DM with $h and $H are proportional to the couplings and XDH, respectively. The fourth term in Eq. (2) represents the kinetic term for the additional gauge boson ZfL in terms of field strength tensor fBLfv of the U(1)b_l gauge group. The covariant derivatives involving in the kinetic energy terms of the BSM scalars and fermions, $H, $DM and Ni (Eq. (2)), can be expressed in a generic form

Dßf = (d' + i ^bl Öbl (t) Zblm) t,

After the symmetry breaking, the SM Higgs doublet $h and the BSM scalar $H take the following form:

+ Hbl\

V2 ) '

where v = 246 GeV is the VEV of *h, which breaks the SM gauge symmetry into a residual U(1)EM symmetry. The remaining terms in Eq. (2) are the Yukawa interaction terms for the LH and RH neutrinos. As mentioned in the beginning of this section, when the extra scalar field *H gets a non-zero VEV vBL, theproposedU(1)B-L gauge symmetry breaks spontaneously. As a result, the Majorana mass terms for the RH neutrinos, proportional to the Yukawa couplings yNi, are generated. In general, for three generations of RH neutrinos, we will have a 3 x 3 Majorana mass matrix Mr with all off-diagonal terms present. However, in the present scenario for calculational simplicity we choose a basis for the Ni fields with respect to which Mr is diagonal. The diagonal elements, representing the masses of Ni s, are given by

MNi = ~f= VBL •

Like the three RH neutrinos, the extra neutral gauge boson also becomes massive through Eq. (4) when $H picks up a VEV. The mass term ZBL is given by

Mzbl = 2 gBL vbl ■

When both $h and $H obtain their respective VEVs, there will be a mass mixing between the states H and HBL. The mass matrix with respect to the basis H and HBL looks as follows:

■M scalar

2kh V2 khH VBL V

khH VBL V 2k h V

Rotating the basis states H and HBL by a suitable angle a, we can make the above mass matrix diagonal. The new basis states (hi and h2), with respect to which the mass matrix M2scalar becomes diagonal, are some linear combinations of earlier basis states H and HBL. The new basis states, now representing the two physical states, are defined as

where f = $DM, $H, Ni and QBL(f) represents the B - L charge of the corresponding field (listed in Table 1). The quantity V ($h, $H) in Eq. (2) contains the self interaction terms of $H and $h as well as the mutual interaction term between the two scalar fields. The expression of V ($h, $H) is given by

V (*h, *h ) = ß h *h * h

ßh*h*h

kH (*H *h )2

t 2 t t +kh (*h *h ) + khH(*h *h )(*H * H )•

h 1 = H cos a + HBL sin a , h2 = -H sin a + HBL cos a .

where we denote by h1 the SM-like Higgs boson, while h2 is playing the role of a BSM scalar field. The mixing angle between H and HBL can be expressed in terms of the parameters of the Lagrangian (cf. Eq. (2)):

tan 2a =

khH vbl V kh V2 - kH vbl

Table 1 Charges of all particles t^^i. t^i. oi^i.

Gauge Baryon fields Lepton fields Scalar fields

under various symmetry groups ——-:-:-:-:--—--:-:-:-- -

Group Q'L = (u'L, d'L)T u'R dR L'L = (v'L, e'L)T eR NR $DM

SU(2)L 2 112 112 11

U(1)Y 1/6 2/3 -1/3 -1/2 -1 0 1/2 0 0

U(1)b—l 1/3 1/3 1/3 -1 -1 -10 2 ubl

Besides the two physical scalar fields h1 and h2, as mentioned earlier, there is another scalar field ($DM) in the present model, which can play the role of a DM candidate. The masses of these three physical scalar fields h1, h2 and $DM are given by

Ml, = kh v2 + k h vBl

(khv2 — kH vBL)2 + (khH v UBL)2

M2 = kh V2 + kH vS

(kh V2 — kH vBL)2 + (khH V VBL)2 ,

22 MDM = ^DM +

kDh v2 + kDHvBL

where Mx 2 denotes the mass of the corresponding scalar field x.

In this work, we choose Mh2, MDM, nBL, MNi, MZbl , gBL, a, kDh, kDH and kDM as our independent set of parameters. The other parameters in theLagrangian, namely kh, kH, khH, ^ and , can be expressed in terms of these variables as follows [72H

be bounded from below when the following inequalities are satisfied simultaneously [72]:

t4h < 0 ¡4H < 0 >0.

<Ph ' '<PH kh > 0, kH > 0, kDM > 0,

khH > — 2Vkh kh ,

kDh > — 2Vkh kDM,

kDH > —2VkH kDM:

khH + 2/kh k^y kDh + 2y/kh kDMy kDH + 2/kH kDM

+2 y/kh kH kDM + khH\l kDM + kDh\fkH + kDHy/kh > 0. (14)

Besides the lower limits of ks as described by the above inequalities, there are also upper limits on the Yukawa and quartic couplings arising from the perturbativity condition, which demands that the Yukawa and scalar quartic couplings have to be less than V4 n (y < V4 n) and 4 n (k < 4 n), respectively [74].

3 Results

Mh21 + Ml + (Mh22 - Ml) cos 2a

4 vbl '

Ml + Ml + (Ml - Mh22) c°s 2a

4 v2 '

(Ml - m2 ) cos a sin a

v vbl '

(M2 + m22)v + (M2 - Ml)(v cos2a + vbl sin2a)

2 _ -(Mh1- Mh2

-(M2 + Ml)vbl + (Ml - Mh22)(vblcos2a - vsin2a)

„2 _ M2 kDhv2 &DHvbl

FDM = MDM — '

where vBL is defined in terms of MZbl and gBL in Eq. (8).

As we already know, in the present scenario two of the three scalar fields, namely and , obtain VEVs. On the other hand, the remaining scalar field $DM does not have any VEV, which ensures its stability by preserving its Z2 odd parity. Therefore, the ground state of the system is (<$h}, ($H}, ($DM>) = (v, vBL, 0). Now, such a ground state (vacuum) will

2 Throughout the paper we have kept the mass (Mh1) of the SM-like Higgs boson h1 fixed at 125.5 GeV.

3.1 Neutrino masses and mixing

As mentioned earlier, the cancellation of both axial vector anomaly [75,76] and gravitational gauge anomaly [77,78], in U (1)B—L extended SM, requires the presence of extra chi-ral fermions. Hence, in the present model to cancel these anomalies we introduce three RH neutrinos (Nt, i = 13). The Majorana masses for the RH neutrinos are generated only after spontaneous breaking of the proposed B — L symmetry by the VEV of . Also, in the present scenario, as stated earlier, we are working in a basis where the Majorana mass matrix for the three RH neutrinos are diagonal, i.e. Mr = diag (MN1, MNl, MN3). The expression for the mass of the ith RH neutrino (MNi) is given in Eq. (7). On the other hand, the Dirac mass terms involving both left chi-ral and right chiral neutrinos originate when the electroweak symmetry is spontaneously broken by the VEV of the SM Higgs doublet , giving rise to a 3 x 3 complex matrix Md. In general, one can take all the elements of matrix Md as complex but for calculational simplicity and keeping in mind that only three physical phases (one Dirac phase and two Majorana phases) exist for three light neutrinos (Majorana

type), we consider only three complex elements in the lower triangle part of the Dirac mass matrix Md. However, the results we shall present later in this section will not change significantly if we consider all the elements of Md to be complex. The Dirac mass matrix Md we assume has the following structure:

yee ye» yeT \

Md = ( y»e + iy»e y»» y^t I , (15)

, yte + i yxe ytM + i yr». yrr j

where yj = —jv (i, j = e, », t) and the Yukawa coupling V2

yij has been defined in Eq. (2).

Now, with respect to the Majorana basis (vOL (NaR

and ((vaL)c NaR)T one can write down the Majorana mass matrix for both left and right chiral neutrinos using Md and Mr matrices in the following way:

m = i 0 T MA .

\Mvt Mr)

Since MD and MR are both 3 x 3 matrices (for three generations of neutrinos), the resultant matrix M will be of order 6 x 6 and it is a complex symmetric matrix which reflects its Majorana nature. Therefore, after diagonalisation of the matrix M, we get three light and three heavy neutrinos, all of which are Majorana fermions. If we use the block diagonal-isation technique, we can write the light and heavy neutrino mass matrices in the leading order as

mv - -MD Mr 1Mdt mN - Mr .

Here MR is a diagonal matrix and the expression of all the elements of mv in terms of the elements of Md and Mr matrices are given in Appendix A. After diagonalising the mv matrix we get three light neutrino masses (mi, i = 1, 2, 3), three mixing angles (012,013 and 023) and one Dirac CP phase 5.

We use the Jarlskog invariant JCP [79] to determine the Dirac CP phase 5, which is defined as

JCP = - sin 2012 sin 2023 sin 2013 cos 013 sin 5. (19)

Moreover, the quantity JCP is related to the elements of the Hermitian matrix h = mvm\ in the following way:

Im (h13h23h31)

Am21 Am22 Am31

where in the numerator Im(X) represents the imaginary part

of X, while in the denominator Am2. = m2 — mj. Once we

ij 1 j

determine the quantity JCP (from Eq. (20)) and the intergen-erational mixing angles of neutrinos, one can easily determine the Dirac CP phase using Eq. (19).

In the present scenario we have 12 independent parameters coming from the Dirac mass matrix. The RH neutrino mass matrix, in principle, should bring about three additional parameters. However, as we will discuss in detail in Sect. 3.2, two of the RH neutrino masses are taken to be nearly degenerate. In particular, the condition of resonant leptogenesis requires that MN2 — MN1 = r1/2, where r1 is the tree level decay width of N1 and is seen to be ~ 10—11 GeV. Therefore, for all practical purposes we have MN1 ~ MN2, and the RH neutrino mass matrix only brings about two independent parameters, MN1 and MN3. Thus, we have 14 independent parameters, which we vary in the following ranges:

1 TeV <

< 3 TeV,

< 15 TeV.

1 < ^ïlfi x 1010 <

x 108 < 1000 (i. j = e. n, T. i = j = e).

< ^^ x 108 <

1 < ^ y^ x 109 < 1000.

(i = T, j = e, IX) .

We try to find the allowed parameter space which satisfies the following constraints on three mixing angles (6j) and two mass square differences (Am2j), JCP obtained from neutrino oscillation data and the cosmological bound on the sum of three light neutrino masses. These experimental/observational results are listed below.

• Measured values of three mixing angles in 3a range [67]: 30° < 0i2 < 36.51°, 37.99° < 023 < 51.71° and 7.82° < 0i3 < 9.02°.

• Allowed values of two mass squared differences in 3a range [67]:

6.93 < -21 eV2 < 7.97 and 2.37 < -31 eV2 <

10-5 10-3 2.63 in 3a range.

• The above-mentioned values of the neutrino oscillation parameters also put an upper bound on the absolute value of Jcp from Eq. (19), which is | Jcpi< 0.039.

• Cosmological upper bound on the sum of three light neutrino masses i.e. Y,i mi < 0.23 eV at 2a C.L. [25].

While it is possible to obtain both normal hierarchy (NH) (m 1 < m 2 < m 3) and inverted hierarchy (IH) (m 3 < m 1 < m2) in this scenario, we show our results only for NH for brevity. Similar results can be obtained for IH.

In the left panel of Fig. 1, we show the variation of the JCP parameter (as defined in Eq. (19)) with the Dirac CP phase 5. From this plot one can easily notice that there are two allowed ranges of the Dirac CP phase, 0° < 5 < 90° and 270° < 5 < 360°, respectively, which can reproduce the neutrino oscillation parameters in the 3a range. Since the Jarlskog invariant JCP is proportional to sin 5 (Eq. 19), we get both positive and

Fig. 1 Left panel: Variation of Jcp with 5. Right panel: Variation of neutrino-less double p decay parameter mpp with m 1

negative values of JCP, symmetrically placed in the first and fourth quadrants. However, the absolute values of JCP always lie below 0.039. Also, here we want to mention that, from the recent results of the T2K [80] experiment, the values of 5 lying in the fourth quadrant are favourable compared to those in the first quadrant. In the right panel of Fig. 1, we show the variation of the neutrino-less double p decay parameter, mpp, with the mass of lightest neutrino, m1. mpp is an important quantity for the study of neutrino-less double p decay as the cross section of this process is proportional

to mpp =

Ei=1(^PMNS)2ei mi

= (mv)ee (see Appendix

B for details), where (mv)ee (Eq. (A1)) is the (1,1) element of the light neutrino mass matrix mv. The nature of this plot is very similar to the usual plot in the mpp-m 1 plane for the normal hierarchical scenario [81]. In the same plot, we also show the current bound on mpp from the KamLand-Zen experiment [17].

3.2 Baryogenesis via resonant leptogenesis

As we have three RH neutrinos in the present model, in this section we study the lepton asymmetry generated from the CP violating out of equilibrium decays of these heavy neutrinos at the early stage of the Universe. The B — L asymmetry thus produced is converted into the baryon asymmetry through sphaleron transitions which violate the B + L quantum number, while conserving the B - L charge. The sphaleron processes are active between temperatures of ~ 1012 GeV and ~ 102 GeV in the early Universe. At high temperatures the sphalerons are in thermal equilibrium and subsequently they freeze out at around T ~ 100-200 GeV [82,83], just before electroweak symmetry breaking (EWSB). To produce suf-

ficient lepton asymmetry, which would eventually be converted into the observed baryon asymmetry, one requires RH neutrinos with masses £ 108 — 109 GeV [82,84]. This is the well-known scenario of the "normal" or "canonical" leptoge-nesis. However, detection of these very massive RH neutrinos is beyond the reach of LHC and other future colliders. Here we consider the RH neutrinos to be in the TeV mass range to allow for their detection at collider experiments. It has been shown that with RH neutrinos in the TeV mass-scale range, it is possible to generate adequate lepton asymmetry by considering the two lightest RH neutrinos, N1 and N2, to be almost degenerate. More specifically, we demand that MNl — MN1 ~ r1/2, where r13 is the total decay width of the lightest RH neutrino, N1. This scenario is known as resonant leptogenesis [83,85-87].

Figure 2 shows the tree level as well as one loop decay diagrams of the lightest RH neutrino, N1. These diagrams are applicable for all the three RH neutrinos. Here L represents the SM lepton, which can either be a charged lepton or a left chiral neutrino, depending on the nature of the scalar field (charged4 or neutral) associated in the vertex, while Nj denotes the remaining two RH neutrinos, N2 and N3, for the case of N1 decay. In order to produce baryon asymmetry in the Universe we need both C and CP violating interactions, which is one of the three necessary conditions (see the Sakharov conditions [45] given in Sect. 1) for baryogenesis. Lepton asymmetry generated from the out of equilibrium

3 The typical value of r is ~ 10 11 GeV (see Fig.3), while MNt ~ O(TeV). Hence we take Mn1 = Mn2 throughout this work.

4 Since these processes occurred before EWSB, we have both charged

and neutral scalars in the SM.

Fig. 2 Feynman diagrams for the decay of lightest RH neutrino N1

*j h Nj k

Fig. 3 Left panel: Variation of the CP asymmetry parameter S1 with out of equilibrium condition of N1. All the points in both plots satisfy the mass of N1. Right panel: Variation of total decay width of N1 with the neutrino oscillation data in the 3a range Mn1 . A black solid line represents the upper bound of 1 coming from

decay of the RH neutrinos is determined by the CP asymmetry parameter (e*), which is given by (for details see Appendix C)

fi2 - —

Im [(MvMv%]

£1 - -

2 (MdMdf)n (MvMv^22 ' 1 1 Im [(Mv Mv %]

T2 + T2 (MvMvT)n (MvMvT)22 2 1 T2

r2 , r2 £2 • 1 1 + i 2

In the left panel of Fig. 3, we show the variation of the CP asymmetry parameter, e1, generated from the decay of the RH neutrino N1, with the mass of N1. Here we see that, for the considered ranges of MN1 (1000 GeV < Mn1 < 10,000 GeV) and other relevant Yukawa couplings (see Eq. (21)), the CP asymmetry parameter e 1 can be as large as ~ 10—2, which is significantly large compared with e1 in the "normal" leptogenesis case (e1 ~ 10—8 for MN1 ~ 1010 GeV) [82]. In the right panel of Fig. 3, we plot the variation of total decay width of N1 with MN1 . From this plot, one can easily notice that in the present scenario, T1 lies between ~ 10—12 and 10—9 GeV for the entire considered

range of MN1 . All the points in the two panels satisfy the neutrino oscillations data in the 3a range, while the black solid line in the right panel provides the upper bound on r1, obtained from the out of equilibrium conditions for N1 i.e. r1 < 3 H (MN1) [82] where H is the Hubble parameter at T = Mn1 .

Next, we calculate the B — L asymmetry generated from the decays as well as the pair annihilations of the RH neutrinos N1 and N2. In order to calculate the net B — L asymmetry produced from the interactions of N1 and N2 at a temperature of the the Universe of T ~ 150 GeV (freeze-out temperature of sphaleron), we have to solve a set of three coupled Boltz-mann equations. The relevant Boltzmann equations [82,83] for calculating YNi and YB—L are given by

dFN1 dz

Mpi z Jg*(z)

1.66 mN1

1 gs(z)

2 n 2 Mpi MN1 Jg*(z) 45

<r1> (^N1 - FNq)

x «a v>N1, Zbl + <a v>N1,(, fl^) (>N1 - (YN1 f)

Fig. 4 Feynman diagrams for the annihilations of RH neutrinos

dYN2 dz

Mpi zVgÂz)

1.66 M2Ni ga(z)

2 n 2 M pi MniJ g*(z) 45

<r2> (Yn2 - YN2)

X (<<V>N2, Zbl + <<v>n2,t,hj (yn2 - (YN2)2) :

dYB-L dz

'■Vg*(z)

1.66 m2,

Ye-lYnL

2 YLeq

+ ej(YNj - YN.i^j <rj>

where Yx = — denotes the comoving number density of s

X, with nx being the actual number density and z = —. The Planck mass is denoted by Mpl. The quantity g+(z) is a function of gp and gs, the effective degrees of freedom related to the energy and entropy densities of the Universe, respectively, and it obeys the following expression [30]:

y/g*(z) =

jgp(z)

1 dln gs(z)\ 3 dlnz /

Before EWSB, the variation of gs (z) with respect to z is negligible compared to the first term within the brackets and hence

one can use Vg+(z) —

gs(z) jgp(z)

. The equilibrium comoving

number density of X (X = Ni, L ), obeying the Maxwell-Boltzmann distribution, is given by [30]

YXX1 (z) =

45 gx / Mx z

M MMxr z)

mnJ ■

where gX and MX are the internal degrees of freedom and

mass of X, respectively, while gs (MNL) is the effective

degrees of freedom related to the entropy density of the Uni-

verse at temperature T =

-.K2( MMN1 z) i

is the modified

Bessel function of order 2. The relevant Feynman diagrams, including both decay and annihilation of Ni, are shown in Figs. 2 and 4. The expression of thermal averaged decay

width <ri}, which is related to the total decay width r of Ni , is given as

<V > = V

K2 ( ^z)

The thermally average annihilation cross sections (av}Ni, Zbl and (av}Ni, Zbl, appearing in the Boltzmann equations (Eqs. (25) and (26)) for the processes shown in Fig. 4, can be defined in a generic form,

<< v> N,, x =

16 mn, mn1 g2Ni K^ ^z)

' 4 M 2

< N,, x K1

Vs ds.

where aNi, x is related to the actual annihilation cross section aNi, x by the following relation:

, x = 2 gNi (s - 4 Mm) <Ni,

where gNi = 2 is for the internal degrees of freedom of the RH neutrino Ni. The expression of aNi, Zbl and aNi, t, Hbl for the present model is given in Ref. [83].

To calculate the B — L asymmetry at around T ~ 150 GeV, we have to numerically solve the set of three coupled Boltzmann equations (Eqs. (25)-(27)) using Eqs. (28)-(32). However, we can reduce the two flavour analysis (when both N2 and N1 are separately considered) into one flavour case by considering the parameters of the Md matrix in such a way that the decay widths of N1 and N2 are of the same order, i.e. r1 ~ r2. Hence, the CP asymmetry generated from the decays of both N1 and N2 are almost identical (e1 ~ e2; see Eqs. (C9)-(C10)). In this case, the net B — L asymmetry is equal to twice of that is being generated from the CP violating interactions of the lightest RH neutrino N1 [83]. Hence instead of solving three coupled differential equations we now only need to solve Eqs. (25) and (27). The results we have found by numerically solving Eqs. (25) and (27) are plotted in Fig. 5. In this plot, we show the variation of YN1 and YB—L with z for MN1 = 2000 GeV, aBL = 3 x 10—4

Fig. 5 Variation of Yn1 (green dashed line) and Yb-l (blue dashed-dotted line) with z where the other parameters have been kept fixed at

MNl = 2000 GeV, aBL ( = g|r ) = 3 x 10-4, Mzbl = 3000 GeV

Table 2 Baryon asymmetry of the Universe generated for three different values of Mn1 and si

MNi [GeV] «1 YB = S

1600 1800 2000 4.4 x 10-4 8.7121 x 10-11 2.25 x 10-4 8.7533 x 10-11 1.8 x 10-4 8.5969 x 10-11

Table 3 Couplings of FIMP (0dm) with Zbl, h1 and h2

Vertex abc Vertex factor Sabc

0DM 0DM ZBLß 0DM ^DM hi 0DM ^DM h2 gBL "BL (P2 - f1)ß — (kphv cos a + Xdh^bl sin a) (kDhV sin a - XDhvbl cos a)

and Mzbl = 3000 GeV.5 While solving the coupled Boltz-mann equations we consider the following initial conditions: Yn1 (TB) = YNq and Yb-l = 0 with TB being the initial temperature, which we take as 20 TeV. Thereafter, the evolutions of YNl and Yb-l are governed by their respective Boltzmann equations. From Fig. 5, one can notice that initially up to z ~ 1 (T ~ MNl), the comoving number density of YN1 does not change much as a result of the B - L asymmetry produced from the decay, and the annihilation of N1 is also less. However, as the temperature of the Universe drops below the mass of MN1, there is a rapid change in the number density of N1, which changes around six orders of magnitude between z = 1 and z = 20. Consequently, the large change in YN1 significantly enhances the B - L asymmetry YB-L and finally Yb-l saturates to the desired value around ~ 10-10, when there are practically no N1 left to produce any further B — L asymmetry.

The produced B - L asymmetry is converted to net baryon asymmetry of the Universe through the sphaleron transitions while they are in equilibrium with the thermal bath. The quantities Yb-l and YB are related by the following equation [71]:

Yb = -2 X — Yb-l(Tf),

where Tf ~ 150 GeV is the temperature of the Universe up to which the sphaleron process, converting B — L asymmetry to a net B asymmetry, maintains its thermal equilibrium. The extra factor of 2 in the above equation is due to the equal contribution to YB-L arising from the CP violating interactions of N2 as well. Finally, we calculate the net baryon asymmetry YB for three different masses of the RH neutrino

5 The considered value of Mzbl and the corresponding gauge coupling gBL satisfy the upper bounds obtained from LEP [95,96] and more recently from LHC [59].

N1 and CP asymmetry parameter s1. The results are listed in Table 2. In all three cases, the final baryon asymmetry lies within the experimentally observed range for YB, i.e. (8.239-9.375) x 10-11 at 95% C.L. [44].

3.3 FIMP dark matter

In the present section we explore the FIMP scenario for DM in the Universe, by considering the complex scalar field 0DM as a corresponding candidate. As described in Sect. 2, the residual Z2 symmetry of 0DM makes the scalar field absolutely stable over the cosmological time scale and hence can play the role of a DM candidate. Since 0DM has a non-zero B - L charge nBL, DM talks to the SM as well as the BSM particles through the exchange of extra neutral gauge boson ZBL and two Higgs bosons present in the model; one is the SM-like Higgs, h 1, while the other one is the BSM Higgs, h2. The corresponding coupling strengths, in terms of the gauge coupling gBL, B - L charge nBL, mixing angle a and As, are listed in Table 3. As the FIMP never enters into thermal equilibrium, these couplings have to be extremely feeble in order to make the corresponding interactions non-thermal. For the case of the 0DM $DM ZBLM coupling, we will make the B - L charge of 0DM extremely tiny so that this interaction enters into the non-thermal regime. In principle, one can also choose the gauge coupling gBL to be very small; however, in the present case we will keep the values of gBL and MZbl fixed at 0.07 and 3 TeV, respectively, as these values reproduce the observed baryon asymmetry of the Universe (see Sect. 3.2). Also, there is another advantage of choosing tiny nBL: this will make only 0DM out of equilibrium, while keeping ZBL in equilibrium with the thermal bath. Moreover, due to the non-thermal nature, the initial number density of FIMP is assumed to be negligible and as the temperature of

Fig. 6 Feynman diagrams for all the possible production modes of ^dm before EWSB

Hbl, *h, Zbl *

hbl.Zbl y ""

N *DM Zbl

n. HBL, ZBL '

BL. *h. ZBL

S N v *DM

the Universe begins to fall down, they start to be produced dominantly from the decays and annihilation of other heavy particles.

In the present scenario, we consider all the particles except 0dm to be in thermal equilibrium. Before EWSB, all the SM particles are massless.6 In this regime, production of ,DM occurs mainly from the decay and/or annihilation of BSM particles, namely ZBL, HBL, and Ni .Also, before EWSB the annihilation of all four degrees of freedom of the SM Higgs doublet, , can produce ,DM. Feynman diagrams for all the production processes of ,DM before EWSB are shown in Fig. 6.

After EWSB, all the SM particles become massive and consequently, besides the BSM particles, ,DM can now also be produced from the decay and/or annihilation of the SM particles. The corresponding Feynman diagrams are shown in Fig. 7. In generating the vertex factors for the different vertices to compute the Feynman diagrams as listed in Fig. 6 and Fig. 7, we use the LanHEP [88] package.

In order to compute the relic density of a species at the present epoch, one needs to study the evolution of the number density of the corresponding species with respect to the temperature of the Universe. The evolution of the number density of 0DM is governed by the Boltzmann equation containing all possible number changing interactions of ,DM. The Boltzmann equation of ,DM in terms of its comoving

number density Y,DM = , where n and s are the actual number density and entropy density of the Universe, is given by

dYfoM _ 2Mpi Zyg(Z)

6 Although the SM particles acquire thermal masses before EWSB, we neglect these masses, as in this regime this approximation will not affect the DM production processes significantly.

1.66 M2 gs(z)

J2 <TX ^dm^dm y,dm }

X=zbl, ä1, h2 4n2 MpiMh1^gAz) 45 1.66 z2

!>vpp^,DM,dm>(Ypq2 - Y,DM}

+ <ct v

, \(y eqYeq — Y2 ) h1h2^0DM,DM h1 h2 0DM >

where z = -j-, while V g*(z), gs(z) and M pi are the same as those in Eqs. (25)-(27) of Sect. 3.2. In the above equation (Eq. (34)), the first term represents the contribution coming from the decays of ZBL, h1 and h2. The expressions of equilibrium number density, YX^(z) (X is any SM or BSM particle except 0DM), and the thermal averaged decay width, <TX^DM^DM>, can be obtained from Eqs. (29) and (30), respectively, by only replacing MN1 with MX, the mass of the decaying mother particle. As mentioned above, before EWSB, the summation in the first terms is over h2 and ZBL only, as there will be no contribution from the SM Higgs decay, as such a trilinear vertex (h 1^DM^DM) is absent before EWSB and after EWSB there will be contributions to the relic density of 0DM from all the decaying particles. The DM production from the pair annihilations of the SM and BSM particles are described by the second term of the Boltzmann equation. Here, summation over p includes all possible pair annihilation channels, namely W+ WZZ, Zbl Zbl , NiNi, hihi, tt. However before EWSB, pair annihilations of the BSM particles and the SM Higgs doublet, , contribute to the production processes (i.e. p = ZBL, Ni, HBL, ; see Fig. 6). The third

Fig. 7 Production processes of 0DM from both SM as well as BSM particles after EWSB

0DM h ZBL

hi, Zbl ^ VW.

Ni, t iDM

hi, Zbl

hi, hj, ZBL

WZ, Zbl

^n- hi, Zbl

W -,Z,Zbl

hi,hi^ y 0DM

hi, h^' v v 9dm

term, which is present only after EWSB, is another production mode of 0DM from the annihilation of h1 and h2. The expressions of all the relevant cross sections and decay widths for computing the DM number density are given in Appendix E. The most general form of thermally averaged annihilation cross section for two different annihilating particles of mass MA and MB is given by [43]

fi = ys2 + (M2a - M2B)2 - 2 s (M2a + M2B),

f2 = Vs - (Ma - Mb )2 Vs - (Ma + Mb)2 , (aVAB ^4dm4dm )

8 M AMI T K2( K2( f

oAB^0DM0DM . . Tr (\fs X / -=- f 1 f2 K1 I -

(Ma+MB )2 Vs

Finally, the relic density of 0DM is obtained using the following relation between Uh2 and Y^DM(0) [89,90]:

Uh2 = 2.755 x 108 (MGeV^) Y*dm (0), (36)

where Y^DM (0) is the value of the comoving number density at the present epoch, which can be obtained by solving the Boltzmann equation.

The contribution to DM production processes from decays as well as annihilations of various SM and BSM particles

depends on the mass of 0DM. Accordingly, we divide the rest of our DM analysis into four different regions, depending on MDM and the dominant production modes of 0DM.

3.3.1 Mdm <

Mh1 Mh2 Mzb

, the SM and BSM

222 particles decay dominated region

In this case DM is dominantly produced from the decays of all three particles, namely h1, h2 and ZBL. Therefore, in this case the U(1)B-L part of the present model directly enters into DM production. Moreover, in this mass range, 0DM can also be produced from the annihilations of the SM and BSM particles, however, we find that their contributions are not as significant as those from the decays of h1; h2 and ZBL. In the left panel and right panel of Fig. 8, we show the variation of DM relic density with z. In the left panel, we show the dependence of DM relic density with the initial temperature Tin. The initial temperature (Tin) is the temperature up to which we assume that the number density of DM is zero and its production processes start thereafter. We can clearly see from the figure that, as long as the initial temperature is above the mass of the BSM Higgs (Mh2 ~ 500 GeV), the final relic density does not depend on the choice of the initial temperature and reproduces the observed DM relic density of the Universe for the chosen values of the model parameters as written in the caption of Fig. 8. If we reduce the initial temperature from 500 GeV, i.e. for Tin = 251 GeV, the decay contribution of the BSM Higgs, h2, becomes less, since the corresponding number density of h2 for Tin < Mh2 is Boltzmann sup-

/ / / / - / / r 1 - 1 1

1 - 1 1 1 1 ~ 1 ( i j Decay Annihilation ....... n h2 = 0.12

0.01 0.1 1 10 100 1000 z (=Mh/T)

Fig. 8 Left (right) panel: Variation of relic density Q.H2 with z for different initial temperature (contributions to Q,h2 coming from decay and annihilation), where the other parameters are fixed at XDh =

0.01 0.1 1 10 100 1000 z (=Mh/T)

8.75 x 10-13, XDH = 5.88 x 10-14, nBL = 1.33 x 10-10, MDM = 50 GeV, Mzbl = 3000 GeV, №L = 0.07, Mh1 = 125.5 GeV and Mh2 = 500 GeV, a = 10-4

pressed (exponentially suppressed), which is clearly shown by the blue dashed-dotted line. Hence, if we reduce the initial temperature (Tin) further, i.e. Tin < , Mhi ~ 42 GeV, then the number densities of both SM-like Higgs h i and BSM Higgs h2 become Boltzmann suppressed and, hence, a smaller amount of DM production will occur, which is evident from the left panel of Fig.8 (represented by the yellow dashed-dotted line). On the other hand in the right panel of Fig. 8, we show the contributions to the DM relic density coming from decay and annihilation. The magenta dotted horizontal line represents the present day observed DM relic density of the Universe. The green dashed line represents the total decay contribution arising from the decays of both h1, h2 and ZBL, whereas the net annihilation contribution coming from the annihilation of all the SM as well as BSM particles is shown by the blue dashed-dotted line. There is a sudden rise in the annihilation contribution which occurs around the Universe temperature T ~ 154 GeV (i.e. the EWSB temperature). After the EWSB temperature, all the SM particles become massive and hence the sudden rise in the annihilation part because of the appearance of the annihilation channels W + W-, Z Z, h1 h1, h1 h2. The plot clearly implies that the lion share of the contribution comes from the decay of the two Higgses h1, h2 and of ZBL, while for the considered values of the model parameters the annihilation contribution is subdominant. Moreover, in this case we cannot enhance the annihilation contribution by increasing the parameters kDh, kDH and nBL as these changes will result in the over-production of DM from the decays of h1, h2 and Zbl.

In the left panel of Fig. 9, we show how the individual decay contribution from each scalar varies with z. Here we consider the values of the scalar quartic couplings kDh = 8.75 x 10-13 and kdh = 5.88 x 10-14 and the (B - L) charge of 0DM nBL = 1.33 x 10-10. From this plot we can see that before EWSB the SM-like Higgs h1 cannot decay to a pair of 0DM as in this epoch it has no coupling with the latter. In this regime the decay of the BSM Higgs h2 and ZBL contribute, while after EWSB even the SM-like Higgs starts contributing to the DM production and hence we get an increased relic density (right side of EWSB). Its worth mentioning here that while generating the plot in the left panel of Fig. 9, we take the scalar quartic couplings kDh, kDH and B - L charge of 0DM nBL of different strengths such that the contributions of the two scalars (h 1 and h2) and the extra gauge boson to the DM relic density are of equal order. This is because for the case of the BSM Higgs h2 decay the coupling kDH multiplied by the B - L symmetry breaking VEV vBL is relevant, while for the decay of the SM-like Higgs h1, the product of the parameter kDh and the EWSB VEV v is relevant and the contribution from the decay of ZBL, DM charge nBL is relevant. Since in the present case vBL > v, the magnitudes of the two quartic couplings kDh and kDH are of different order (see Table3). On the other hand, in the right panel of Fig. 9, we show the variation of the relic density with z for four different values of the DM mass MDM. From Eq. (36), one can see that the DM relic density is directly proportional to the mass MDM and as a result when the other relevant couplings remain unchanged Uh2 increases with MDM. This feature is clearly visible in the right panel for the cases with MDM = 10 GeV (black solid

£ №

z (=Mh,/T)

Fig. 9 Left panel: variation of decay contributions of the two Higgs bosons to Uh2 separately with z. Right panel: Variation of relic density Uh2 with z for different values of the DM mass Mdm . Other parameters

Mdm = 10 GeV Mdm = 30 GeV M dm = 50 GeV M dm = 75 GeV

n h2 = 0.12

z (=Mh,/T)

valuehavebeenkept fixed at XDh = 8.75x10-13, XDH = 5.88x10-14, nBL = 1.33x10-10, Mdm = 50GeV(fortheleftpanel), Mzbl = 3000 GeV, gBL = 0.07, Mh1 = 125.5 GeV and Mh2 = 500 GeV, a = 10-4

line), MDM = 30 GeV (red dashed line) and 50 GeV (green dashed line), respectively. However, for MDM = 75 GeV (blue dashed-dotted line) Uh2 does not rise equally because for this value of the DM mass the decay of h1 to a pair of 0DM and 0DM becomes kinematically forbidden and hence, there is no equal increment in this case.

In the left panel and right panel of Fig. 10, we show how the relic density varies with z for different values of scalar quartic couplings ADh and ADH, respectively. In each panel, one can easily notice that there exists a kink around the EWSB region. However, in the left panel, the kink occurs for a higher value of ADh while in the right panel, the situation is just opposite. We have already seen in the left panel of Fig. 9 that before EWSB only h2 decay is contributing to the DM relic density and at the EWSB region SM-like Higgs h 1 also starts contributing. A kink will always appear in the relic density curve when contribution of the SM-like Higgs boson h 1 to Uh2 is larger compared to that of the BSM Higgs h2 and extra gauge boson ZBL i.e. r, , ,t > r, , ,t , rv i it . The values of scalar quartic

«2 ^VDMVDM ZBL ^VDMVDM n

couplings ADh and ADH in the left panel of Fig. 9 are such that r, , ,t and T7 , ,t always remain large

h2 ^VDMVDM ZBL ^VDMVDM J b

compared to rh1^DM^DM and hence no kink is observed in the total relic density curve. However, in the present figure (in the left panel of Fig. 10) we do have kinks around the EWSB region, because in the left panel with ADH = 8.316 x 10-14 and nBL = 1.33 x 10-10, r, , ,t > ^^ ,t , r7 i it condition is satisfied only for the case with

ZBL ^VDMVDM J

larger value of ADh = 1.237 x 10-11 (ADh >> ADH) while in the right panel with a fixed value of ADh = 1.237 x 10-12,

the above condition is not maintained because the zBL decay channel dominates.

In the left panel of Fig. 11, we show the allowed region in the coupling plane (ADh-Adh) which reproduces the observed DM relic density (0.1172 < Uh2 < 0.1226). In this figure, we clearly indicate the dominant DM production processes when MDM varies between 10 GeV to 100 GeV i.e. DM production from the decays of h1 , h2 or both or entirely from the annihilations of the SM particles like WZ, h1 etc. The parameters which are related to the ZBL decay (gBL, nBL) have been kept fixed at 0.07 and 1.33 x 10-10, respectively, so at every time an equal amount of ZBL decay contribution remains present. As illustrated in the figure, when the parameter ADh is small compared to the other parameter ADH then among the two scalars it is the BSM Higgs h2 which is mainly contributing to the DM production, while for the opposite case, the production of 0DM becomes h1 dominated and in between the two scalars contribute equally. Apart from that, if the mass of 0DM is greater

than the half of the SM-like Higgs mass (i.e. MDM > —— ), then DM production from h1 decay becomes kinematically forbidden. In this case, however, the production from the decays of h2 and ZBL is still possible. Now, the deficit in DM production can be compensated by the production from self annihilation of the SM particles like h 1, W± and Z ; for this we need to increase the parameter ADh. Moreover, by increasing ADh (decreasing ADH simultaneously) we can arrive at a situation where DM production is entirely dominated by the annihilations of the SM particles and this situation has been indicated by a pink coloured arrow in the left panel of Fig. 11.

10-3 -

10-6 -

Xdh = 8.316 x 10-1

--- Xdh = 1.237 x 10-1

-— XDH = 1.237 x 10-1 -— XDH = 1.237 x 10-1

..... n h2 = 0.12

10-3 -

10-6 - I

XDh = 1.237 x 10-1

---- Xdh = 8.316 x 10-1

----- Xdh = 8.316 x 10-1

----- Xdh = 8.316 x 10-1

.......... n h2 = 0.12

z (=Mh1/T)

100 1000 0.01

1 10 z (=Mh/T)

100 1000

Fig. 10 Left (Right) panel: Variation of relic density Q,h2 with z for three different values of Xdh (Xdh), where the other parameters are fixed at XDH = 5.88 x 10-14 (XDh = 8.75 x 10-13), nBL = 1.33 x 10-10, MDM = 50 GeV, Mzbl = 3000 GeV, gBL = 0.07, Mhl = 125.5 GeV and Mh, = 500 GeV, a = 10-4

Fig. 11 Left (Right) panel: Allowed region in the kDh-kDH (Mh2-a) plane where other parameters are fixed at Mzbl = 3000 GeV, gBL = 0.07, nBL = 1.33 x 10-10, Mh1 = 125.5

On the other hand, in the right panel of Fig. 11 we present the allowed region in the Mh2-a plane which satisfies the relic density bound. From this figure one can see that with the increase of Mh2, the allowed values of mixing angle a decrease. The reason behind this decrement is the vacuum stability conditions as given in Eq. (14). The region satisfying both the relic density bound and the vacuum stability conditions is shown by the green dots, while in the other part of the Mh2-a plane the quantity becomes positive which is undesirable in the context of the present model (see Eq. (14)).

Mh2 MZbl

—-, —ZBL, BSM particles decay

2 ™M 2 2 and SM particles annihilation dominated region

Clearly in this mass region, DM production from the decay of the SM-like Higgs h1 is kinematically forbidden and hence DM has been produced from the decays of h2, ZBL only. However, unlike the previous case, here we find significant contribution to DM relic density arising from the self annihilation of the SM particles namely, h1, W±, Z and t. On the other hand, the annihilations of BSM particles like ZBL, h2 and Ni have negligible effect on DM production processes.

z (= T )

Fig. 12 Variation of the DM relic density Q,h2 with z. The other parameter values have been kept fixed at Xdh = 6.364 x 10-12, XDH = 7.637 x 10-14, nBL = 8.80 x 10-11, MDM = 70 GeV, Mzbl = 3000 GeV, gBL = 0.07, Mh1 = 125.5 GeV, Mh2 = 500 GeV, a = 10-5, MNl ^ MN1 = 2000 GeV and MN3 = 2500 GeV

In Fig. 12, we have shown the variation of the DM relic den-

Mh1 Mh2 MZbl

sity with z for ——— < M^DM < ——, —-—. Since now

the decay of the h 1 to $dm$d>m! is kinematically forbidden, hence we can increase the parameter kDh safely and this will not overproduce DM in the Universe. Due to this moderately large value of kDh, the annihilation channels become important. From Fig. 12 it is clearly seen that in this case the annihilation channel h1h1 ^ ^dm$Dm (Green dashed line)

contributes significantly to the DM production. Therefore in the present case, production of DM has been controlled by the decays of h2, ZBL and the self annihilations of the SM particles and thus directly relates to the U(1)B-L sector of this model.

Mh1 Mh2

, BSM particles decay

2 2 ™M 2 and annihilation dominated region

In this regime of the DM mass, the only surviving decay mode is the decay of the B - L gauge boson ZBL to a pair of 0DM. Apart from that, depending on the choice of the mass of 0DM a significant fraction of DM has been produced from the self annihilation of either BSM Higgs h2. In other words, we can say that in this region the production of DM is BSM particles dominated. In the left panel of Fig. 13 we show the relative contribution of dominant production modes of DM to Uh2 for a chosen value of MDM = 450 GeV. From this plot one can easily notice that in the case when MDM < Mh2, the almost entire fraction of DM is produced from the decay of ZBL (green dashed line) and self annihilation of the BSM Higgs h2 (solid turquoise line). This is because, as in this case the production of 0DM from h2 decay is kinematically forbidden, one can increase the parameter XDH so that the annihilation channel h2h2 ^ 0DM^DMt, which is mainly proportional to XDH (due to four point interaction), becomes significant.

On the other hand, in the right panel we consider a situation where almost the entire DM has been produced from the decay of B - L gauge boson. For this purpose, we choose

AT m^ m^

mdm > 2,2

fi h2 = 0.12

Decay + Annihilation Zbl Decay Ni Ni Annihilation ZBLZBL Annihilation h2h2 Annihilation

/ mh,. z (= t )

, , . mh. mh2 mdm > 2,2

fi h2 = 0.12

Zbl Decay

ZBLZBL + N iNi + h2h2 Annihilation

z (= t )

Fig. 13 Left (Right) panel: Variation of the DM relic density Uh2 with z. The other parameter values have been kept fixed at kDh = 2.574 x 10-12 (7.212 x10-14), kDH = 3.035 x 10-11 (8.316 x 10-14),

nBL = 3.4 x 10-11 (6.2 x 10-11), MDM = 450 GeV (600 GeV), Mzbl = 3000 GeV, gBL = 0.07, Mh1 = 125.5 GeV, Mh2 = 500 GeV, a = 10-5, MN2 ^ MN1 = 2000 GeV and MN3 = 2500 GeV

2000 4000 6000 8000 10''

Mz„, [GeV]

Fig. 14 Allowed region in the Mzbl —gBL plane which produces observed DM relic density. Solid lines (black and red) are the upper Limits on the gauge coupling gBL for a particular mass of Zbl obtained from LHC and LEP, respectively. Other relevant parameters used in this plot are 250GeV < MDM < 5000GeV, ADh = 7.212 x 10-14,

Adh = 8.316

Mh2 = 500 GeV, a = 10-5, MN2

MN1 = 2000 GeV and MN3 = 2500 GeV

MDM > Mh2 and a larger value of nBL = 6.2 x 10-11. Similar to the previous case (i.e. MDM < Mh2) here also, the production of 0DM from h2 decay still remains forbidden. However, as the sum of the final state particles masses is larger than that of the initial state, in this case the h2h2 annihilation mode becomes suppressed. Moreover, to make the contribution of the h2 annihilation even more suppressed we reduce the quartic couplings ADh and ADH accordingly. As a result other annihilation channels, e.g. ZBLZBL, NiNi, also become inadequate as these channels are mediated by the exchange of h 1 and h2. Although RH neutrinos can annihilate to 0DM^DMt via ZBL, we cannot increase the contribution of ZBL mediated diagrams because for that one has to further increase the B - L charge of 0DM (nBL), which results in an over production of DM in the Universe from ZBL decay. From the right panel of Fig. 13, one can easily notice that in this situation ZBL decay is the most dominant DM production channel (red dashed line), while the total contributions from the annihilations of h2, ZBL and Ni are negligible. Therefore, for the entire mass range of 0DM,

. Mh1 Mh2 M Zbl ^ ,,

i.e. —, —< MfoM < BL , the DM production processes are always related to the U( 1)B-L sector of the present model by receiving a sizeable contribution from ZBL decay.

In Fig. 14, we show the allowed region (green coloured points) in the MZBL-gBL plane, which reproduces the observed DM relic density. While generating this plot we vary 250 GeV < Mdm < 5000 GeV and 10-11 < nBL < 10-8. In this region, as mentioned above, dominant contributions to the DM relic density arise from ZBL decay and annihilation of the BSM Higgs h2. In this figure, the black

solid line represents the current upper bound [59,91,92] on gBL for a particular mass of ZBL from LHC,7 while the limit [95—97] from LEP8 has been indicated by the red solid line respectively. Therefore, the region below the red and black solid line is allowed by the collider experiments like LHC and LEP. The benchmark value of gBL, Mzbl (= 0.07, 3000 GeV) for which we have computed the baryon asymmetry in the previous section (Sect. 3.2) is highlighted by a blue coloured star. Hence, in this regime the extra gauge boson ZBL immensely takes part in achieving the correct ballpark value of the DM relic density and also at the same time ZBL plays a significant role in obtaining the observed value of the matter—antimatter asymmetry of the Universe.

3.3.4 M,

Mh1 Mh2 M Zbl „cm

—r1, —T2, , , BSM particles

2 2 2 annihilation dominated region

Finally, in this range of the DM mass the entire production of 0DM from the decays of h1, h2 and ZBL becomes kinemat-ically inaccessible. Therefore, in this case all three parameters, namely ADh, ADH and nBL, become free and we can make sufficient increment to these parameters so that either scalar medicated (h1, h2) or gauge boson mediated (ZBL) annihilation processes of Ni, ZBL or both, can be the dominant contributors in DM production.

Similarly, in the left panel and right panel of Fig. 15, we show two different situations where the DM production is dominated by scalar (h1, h2) mediated diagrams and gauge boson ZBL mediated diagrams, respectively. In the left panel, by keeping the nBL value low and adjusting the parameters ADh and ADH one can achieve the correct value DM relic density and on the other hand, in the right panel we keep the values of ADh and ADH sufficiently low and by suitably adjusting the DM charge nBL we achieve the correct value of the DM relic density. Therefore, in this region, a strong correlation exists among the neutrino sector, U(1)B-L sector and DM sector as the entire DM is now being produced from NiNi and ZBLZBL annihilations.

In Fig. 16, we show the allowed parameter space in the MDM—MN1 plane by DM relic density. In order to generate this plot we vary the DM mass in the range 1500 GeV < Mdm < 3000 GeV, the RH neutrino masses 1500 GeV <

7 To get the bound in the Mzbl—gBL plane from the LHC, ATLAS and CMS collaborations consider the Drell—Yan processes (p p ^ Zbl ^ 11, with l = e or /x) and by searching the dilepton resonance they put a lower bound on Mzbl for a particular value of the extra gauge coupling gBL. For updated bounds on the mass of the extra neutral gauge boson at 13 TeV centre of mass energy of LHC, see Refs. [93,94].

8 LEP consider the processes e+ e- ^ f f (f = e) above the Z-pole mass and by measuring its cross section they put lower limit on the ratio between the gauge boson mass and gauge coupling, which is ^BL > 6—7 TeV.

gBL —

, t z (= "

Fig. 15 Left (Right) Panel: Variation of the DM relic density Uh2 with z when dominant contributions are coming from scalar hi (gauge boson Zbl) mediated annihilation channels. The other relevant parameter values have been kept fixed at XDh = 7.017 x 10-12 (7.212 x 10-13),

z (= t)

XDH = 6.307 x 10-11 (8.316 x 10-12), nBL = 1.0 x 10-10 (1.34 x 10-8), MDM = 1600 GeV, Mzbl = 30004 GeV, gBL = 0.07, Mh1 = 125.5 GeV, Mh2 = 500 GeV, a = 10-5, MNl ^ MN, = 2000 GeV and MN3 = 2500 GeV

Fig. 16 Allowed region in the Mdm-Mn1 plane which mimics the observed DM relic density. The blue coloured star represents our benchmark point (MDM = 1600 GeV, MN1 = 2000 GeV)

MNi < 10000 GeV (i = 1, 2), MN1 < MN3 < MNl + 5000 GeV and 10-10 < nBL < 10-8. The other relevant parameters have been kept fixed at kDh = 7.212 x 10-13, kDH = 8.316 x 10-12, MZBL = 3000 GeV, gBL = 0.07, Mh2 = 500 GeV, a = 10-5 As discussed above, in this Mh1 Mh2 Mzbl s

regime (Mdm >

-) 0DM is dominantly

2 2 2 produced from the annihilations of ZBL and RH neutrinos.

From this plot one can observe that in this high DM mass

range to obtain the observed DM relic density, the mass of

the lightest RH neutrino cannot be larger than ~ 6000 GeV.

Analogous to Fig. 14, here also we indicate the benchmark

point for which we compute the baryon asymmetry in Sect. 3.2 by a blue coloured star. Therefore, in this case RH neutrinos are very actively taking part in all three processes we consider in this work, namely DM production processes, tiny neutrino mass generation and the generation of required lep-ton asymmetry to reproduce the observed baryon asymmetry of the Universe.

From the above four regions, which are based on the mass of our FIMP DM, it is evident that in the first region DM production mainly happens from the decay of h1, h2 and ZBL and all annihilations are subdominant. Therefore, in this region only the extra neutral gauge boson (ZBL), BSM Higgs (h2) and SM-like Higgs (h1) are taking part in the DM relic density estimates and there is no significant role of the RH neutrinos. In the second region, SM-like Higgs decay does not contribute to DM production processes, hence one can safely increase the quartic coupling kDh and consequently the h 1 h 1 annihilation contribution increases. Similar to the previous regime, here also RH neutrinos have less importance in determining the DM relic density. In the third region, the only decay mode that is involved in DM production is ZBL ^ 0DM^DMt. Since all other decay modes corresponding to h1 and h2 are kinematically forbidden, we can increase both the quartic couplings kDh and kDH appropriately, which eventually enhances the annihilation contribution from the BSM Higgs significantly. Moreover, due to the increment of quartic couplings in this region, ZBLZBL and NiNi annihilation channels start contributing in the DM production processes. Lastly in region four, due to the high value of the DM mass no decay process contributes to DM relic density and

only the BSM particle annihilation contributes. Therefore, in this region by properly adjusting the extra gauge coupling gBL, one can get a sizeable fraction of DM production from the annihilation of the RH neutrinos. Apart from the masses of the involved particles, the annihilation of RH neutrinos mediated by ZBL depends on the extra (B-L) gauge coupling gBL solely. Thus, depending on the mass range of our FIMP DM, we can say that the different model parameters and the additional BSM particles (e.g. ZBL, N, h2) are fully associated to the DM production processes in the early Universe.

3.3.5 Analytical estimates

So far, we have solved the full Boltzmann equation (Eq.34) for a FIMP 0DM numerically. Apart from this, one can estimate the FIMP relic density (or comoving number density) by using the approximate analytical formula. Let us consider a FIMP (0DM) which is produced from the decay of a particle A i.e., A ^ 0DM 0DMt, where A in the present model can be hi, h2 or ZBL. The contribution of A to the FIMP relic density at the present epoch, considering the effect of both 0dm and <fc>Mt, is given by [34]

^FIMPh2 _ 2.18 x 1027gA MdmFa

gsyJTp

where MA and gA are the mass and internal degrees of freedom of the mother particle A, respectively, while TA is the decay width of the process A ^ 0DM 0DMt. The analytic expressions for TA corresponding to h2, h 1 and ZBL are given in Eqs. (D4), (D7) and (D9) in the appendices. Moreover, gp and gs, as defined earlier, are the degrees of freedom related to the energy and entropy densities of the Universe, respectively. Let us now compare the analytical result with the numerical value which we obtain by solving the Boltzmann equation Eq. (34). For this purpose, let us consider a situation when a significant fraction of our FIMP candidate (0DM) is produced from the decay mode of the BSM Higgs i.e., h2 ^ ^DM^DM-Substituting the values of the model parameters given in the caption of Fig. 9 to Eq. (37), we get the contribution of h2 to the DM relic density, which is

ftFIMPh2 ~ 0.027.

where we consider gp = gs « 100 and gA = 1. This can be compared to the contribution of h2 obtained from the exact numerical estimate shown in the left panel of Fig. 9, which is

ft, , ,t h2 = 0.0276.

«2^<PDM<PDM

Therefore, from the above two estimates it is clearly evident that the analytical result agrees well with the full numerical result. Similarly, for the other decay modes also (i.e. h1, ZBL) one can match the analytical and numerical results.

4 Conclusion

In this work we considered a local U(1)B-L extension of the SM and, to cancel the additional anomalies associated with this gauge symmetry, we introduced three RH neutrinos (Ni, i = 1. 2. 3). Besides the three RH neutrinos, we also introduced two SM gauge singlet scalars, and 0DM. The scalar field , being charged under U(1)B-L, takes a non-zero VEV and breaks the proposed B - L symmetry spontaneously. Moreover, as the scalar field 0DM has also a non-zero B - L charge, one can adjust this charge suitably so that after symmetry breaking we are left with a model with a residual Z2 symmetry and only 0DM behaves as an odd particle under this leftover symmetry. This makes 0DM absolutely stable over the cosmological time scale and hence acts as a DM candidate. After spontaneous breaking of U(1)B-L gauge symmetry, all RH neutrinos and the extra neutral gauge boson ZBL, acquired mass. Due to the presence of the three RH neutrinos in the model, we easily generated Majorana masses for the three light neutrinos by the Type-I seesaw mechanism. This model is also able to explain baryogenesis via leptogenesis, where we generated the lepton asymmetry in the Universe from out of equilibrium, CP violating decays of two degenerate RH neutrinos and converted this lepton asymmetry to the observed baryon asymmetry through the sphaleron transitions.

In explaining the neutrino masses by Type-I seesaw mechanism, we considered a complex Dirac mass matrix Md and a diagonal Majorana mass matrix Mr for the RH neutrinos. In determining the allowed model parameter space, we used the measured values of the neutrino oscillation parameters, namely three mixing angles (012, 013 and 023) and two mass square differences (Am21, Am:;tm) in their current 3a range. In particular, in the current model we could reproduce the whole allowed 3a range of the neutrino oscillation parameters by different combinations of the relevant model parameters. The Dirac CP phase was constrained to lie within two distinct regions. One is the entire first quadrant (0°—90°), while the other one spans the entire fourth quadrant (270°-360°). However, if we considered the T2K result on the Dirac CP phase then the values of 8 lying in the fourth quadrant are more favourable compared to those in the first quadrant. We also computed the magnitudes of the Jarlskog invariant JCP and found that the values of /CP, for the model parameters which satisfy the neutrino oscillation data, always lie below 0.039. Finally, we calculated the values of mpp, the quantity relevant to neutrino-less double j decay, for the allowed model parameter space.

Since we allowed complex Yukawa couplings in the Dirac mass matrix Md, the decays of the RH neutrinos were CP violating. We took the masses of the RH neutrinos in the TeV range and worked in the parameter space where the lightest two RH neutrino states were nearly degenerate, with

their masses separated by their tree level decay width. This scenario led to resonant leptogenesis (or TeV scale lepto-genesis) for the production of observed baryon asymmetry in the Universe from the out of equilibrium decays of the RH neutrinos. We generated the observed baryon asymmetry for three different values of the RH neutrino masses namely MNl = 1600 GeV, 1800 GeV and 2000 GeV, respectively, where the required values of the CP asymmetry parameter (e1) were 4.4 x 10-4, 2.25 x 10-4 and 1.8 x 10-4, respectively. These values of MN1 and e1 were also seen to be allowed by the neutrino oscillation data.

Lastly, we studied the DM phenomenology by considering a FIMP type DM candidate 0DM. We took into account all the production modes of 0DM (both before and after EWSB) arising from the annihilations and decays of the SM as well as BSM particles. We found that depending on the mass of 0DM, the production processes of 0DM can be classified into four distinct categories. These are: (1) SM and BSM particles decay dominated region, (2) BSM particles decay and SM particles annihilation dominated region, (3) BSM particles annihilation and ZBL decay dominated region, and finally (4) BSM particles annihilation dominated region. The first region is characterised by Mh1 Mh2 Mzbl 2

and here DM is mainly pro-

duced from the decays of h1, h2 and ZBL. In the sec-

ond region, DM mass is concentrated between and

Mh2 2 '

< Mdm <

this case, h2, ZBL decays and h 1 h 1, W+ W , ZZ annihilations act as the dominant production modes of FIMP DM.

Mh1 Mh2 Mzbl

In the third region where

< Mdm <

2 2 2 DM has mainly been produced from ZBL decay and from

the annihilation of the BSM Higgs h2 (for MDM < Mh2). Finally, in the last region all three decay modes become

Mh1 Mh2 MZBL kinematically forbidden as MDM > ——, ——, —-— and

hence entire DM is produced from the self annihilations of ZBL and RH neutrinos (Ni). Therefore in all four regions the U(1)B-L gauge boson has played a significant role in DM production, while the effects of the RH neutrinos are important in the last two regions only. We also found that, since for a FIMP candidate 0DM the observed DM relic density (0.1172 < Uh2 < 0.1226) is generated via the freeze-in mechanism; this puts upper bounds on the scalar and gauge portal couplings of 0DM to restrict its over-production i.e. ^Dh < 10-11, Adh ^ 10_10 and «bl < 10_8. Hence, due to such extremely feeble couplings 0DM can easily evade all the constraints coming from any terrestrial DM direct detection experiment.

In conclusion, our spontaneously broken local U(1)B-L extension of the SM with three additional RH and two addi-

tional scalars can explain the three main pieces of evidence for physics beyond the SM, viz., small neutrino masses, matter-antimatter asymmetry of the Universe and DM. Tiny neutrino masses and all mixing angles can be obtained via the Type-I seesaw mechanism where we chose a certain pattern for the real and complex Yukawa couplings. The model gave a definite prediction for the CP violating phase to be measured in the next generation long baseline experiments. The DM candidate is a scalar which is neutral under the SM gauge group and has a non-zero B _ L charge. DM is made stable by virtue of a remnant Z2 symmetry arising after the spontaneous breaking of the U(1)B-L gauge symmetry. This can be achieved by imposing a suitable B _ L charge on 0DM so that the Lagrangian does not contain any odd term of 0DM. This scalar DM can easily be taken as a FIMP candidate which is produced from the decays and annihilations of the SM and BSM particles. Therefore, even if the WIMP type DM is ruled out in the near future from direct detection experiments, the present variant of the U(1)b_l scenario with FIMP DM will still survive. Further, since gBL is of the order of the SM gauge couplings, this model has the potential to be tested in the LHC or in other future collider experiments by detecting B _ L gauge boson ZBL from its SM decay products. Moreover, considering the masses of the RH neutrinos in the TeV scale allows us to simultaneously explain the baryon asymmetry of the Universe from resonant leptogenesis, FIMP DM production via a freeze-in mechanism and neutrino masses and mixing from the TeV scale Type-I seesaw mechanism. Thus, all three phenomena addressed in this article are interconnected.

Acknowledgements SK and AB also acknowledge the HRI cluster computing facility (http://cluster.hri.res.in). The authors would also like to thank the Department of Atomic Energy (DAE) Neutrino Project under the XII plan of Harish-Chandra Research Institute. This project has received funding from the European Union's Horizon 2020 research and innovation programme InvisiblesPlus RISE under the Marie Sklodowska-Curie grant agreement No 690575. This project has received funding from the European Union's Horizon 2020 research and innovation programme Elusives ITN under the Marie Sklodowska-Curie grant agreement No 674896.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Funded by SCOAP3.

Appendix A: Expression for the Majorana mass matrix of light neutrinos

Here we give the expression of all the elements of the light neutrino mass matrix m v in terms of the Yukawa couplings and the RH neutrino masses:

(m v)n (m v) 12 (m v)13

MN1 Mn2 Mn3 yex yxx yex yx2

MN2 yet yrt MN3

Mn3 yee yTe

yee yxe . yee yxe — I

MN1 yex yTx

yee yTe yex y2x -1 I -r^--r

(m v)21 = (m v) 12 (m v)22

y flfl

yx e + y¡e

MN2 MN3 MN1 MN1

2 y e y e

(m v)23 = -

y Xt yTT yxe yTe

y XX yT x + y xe yTe

yT e y x e + y x e yT e + y xx yT X

(m v)31 (m v)32

(m v)13, (m v)23,

(m v)33 = -

y2x + Yre + y2x

MN3 MN1 MN2 MN1 MN2

- i 2[ y2e y2e + y2x y2x

(mv)11 (mv) 12 (m v)13

(mv)21 (mv)22 (mv)23 (mv)31 (mv)32 (mv)33,

(A1) (A2)

Appendix B: Neutrino-less double ft decay parameter

Since the light neutrino mass matrix is Majorana in nature, it is a complex symmetric matrix. A complex symmetric matrix mv can be diagonalised by a unitary matrix UPMNS (defined in Eq. (9)) in the following way:

mdiag = UPMNS m v UPMNS' ^ m v = UpmNS m diag UPMNS.

Now equating the (i, j )th element from both sides of the above equation, we get

(mv)ij = (UPMNS)ik (mdiagW (U]PMNS)k'j•

Since mdia is a diagonal matrix, we can further simplify (m v)ij by using mdiag = mk 8kkr, where mk is the mass of kth light neutrino. Therefore (mv)ij takes the following form:

masses, the intergenerational mixing angles and the phases. Taking i = j = 1, we get the expression of the (1,1) element of mv i.e.

(mv)n = Y^mk №mns)2 k,

which is related to the important parameter mpp of the neutrino-less double j decay [81] by

^ mk №mns)1 k

= l(mv)11l

Appendix C: CP asymmetric parameter calculation for leptogenesis

The amount of lepton asymmetry generated in the out of equilibrium decay of the RH neutrino Ni is parametrised by the CP asymmetry parameter (si), which is defined as

If we consider only the tree level decay process of Ni (first diagram in Fig. 2), there will not be any CP violation. The non-zero CP asymmetry is generated only by the interference between the tree level and the one loop level diagrams. The expression of the CP asymmetry parameter (si) is given by [85,86,98].

Mn, (YL + Sj

' MNj MNj\ 2 1 j =i j j

(MD MD ^

(MdMdt)ii (MdMdt)jj

where Vj and Sj are the contributions coming from the vertex correction and the self energy correction diagrams, respectively (second and third diagrams in Fig. 2). The expressions of Vj and Sj have the following forms [85,86,98]:

Vj = 2

log 1 +

MN. AM2

S■ = -

j (AM2)2 + M2n, T] with

(mv)ij = ^ mk (^PMNs)ik №mns)jk ■

The above equation expresses the elements of the light neutrino mass matrix (Eq. (A1)) in terms of the light neutrino

A MN = MN i - K

and r j denotes the tree level decay width of the RH neutrino Nj (neglecting subdominant one loop corrections), which is given by

MNj 4nv2

(MDMD.

Now, as mentioned in the beginning of this section, the enhancement in the CP asymmetry factor (Eq. (C1)) occurs when two RH neutrinos are almost degenerate i.e. MN2 _

MN1 ~ —. This is known as the resonance condition. In the present scenario, we consider MN3 > MN2 ~ MN1. Therefore, the resonance condition is satisfied only for the two lightest RH neutrinos N2 and N1. Hence we can neglect the contribution of N3 in the CP asymmetry parameter (Eq. (C2)) by considering the summation over only N1 and N2 (i.e. j = 1, 2). Using the resonance condition in Eq. (C3), one

( Mn i \

can easily notice that Sj ~ O I T- I >> 1 ( j = i, 2)

Im \(MvMdt)2j]

Mn, Tj

-- y^ 'J S

' j =Tj=1 MNj Mnj j (MdMdt),, (MdMdt)jj '

where we neglect the quantity Vj, which is, under the present

condition (MN2 - MNl — — ), much smaller compared to Sj. The resonance condition leads to

AMh = MN2 - M2i ,

Tx( T1

= y(2 MNl + -J

- MNl Ti + O (r2). Using Eq. (C7) in Eq. (C3) we get

Si - -

2 MN2 Ti '

Mni t2 + T2

Now, substituting the expressions of Si and S2 in Eq. (C6) and using Im [(MD MD t )2i2] = - Im [(MD MD t)2J,one obtains

Im [(MvMd

e2 - -

£i - -

2 (MdMdf)ii (MdMdf)22 '

Ti T2 Im [(MdMd%]

T2 + T2 (MDMD% (MDMD^22 2 Ti T2

T2 + T262.

Appendix D: Expressions of decay widths of h2, hi and

particles become massive and affect DM production, while before EWSB, those particles have no effect. To take this effect into account we define an extra constant, CASB. In all the equations, the value of the constant CASB = 0 before the EWSB and it is equal to unity i.e. CASB = 1 after the EWSB. Also before the EWSB, there is no mixing between the SM and BSM Higgs bosons, i.e. a = 0. The two vertices which are common to all Higgs mediated diagrams are as follows:

ghi^DM = - (Vkdh cos « + VBLX°h sin ^

gh2^DM = (vkdh sin « - VBL^DH cos a) . Total decay width of h 2:

h2 ^ VV (V = WZ):

gh2 VV =

sin a.

T(h2 ^ VV) =

Casb Mh32 SI2 vv 64 n MV Sv *

V i2MV

where Sv = 2 (i) for the ZZ (W+ W-) final state. h2 ^ hi hi:

T(h2 ^ h i h i) =

h 2 ^ ^DM 0dm

T(h2 ^ 0DM0dm) =

h2 ^ ff:

ghihih2 32 n Mh2

^h2^DM 0DM i6 n Mh2

Mh2 h2

. (D4)

gh2 ff = — sin a,

T(h2 ^ ff) =

Casb nc Mh2 gh2 ff 8n

2 \ 3/2

In the present work, we consider the effect of electroweak symmetry breaking on DM production. After EWSB the SM

nc is the colour charge, for leptons it is i and for quarks it is 3.

The total decay width of the extra Higgs, h 2, in the present case is

Appendix E: Analytical expression of relevant cross sections

r^2 = J2 r(h2 — VV) + r(h2 — hihi)

+ r(h2 — 0DM0DM) + J2 T{h2 — f f)-

Total decay width of h i:

Casb g\ ,t

t(hi — 0dm0dm) =

h10DM0DM

16 n Mhl

The total decay width of the SM-like Higgs boson is

Th1 = cos2 a Tsm + r(h1 — 0DM0DM) , (D8)

where TSM is the total decay width of the SM Higgs boson.

Here we will give the expressions of the cross sections for all relevant processes which take part in the FIMP DM production.

• hihi ^ <DM0dm:

ghihihi = -3 [2 vkh cos3 a + 2 vbl Ah sin3 a

+khH sin a cos a (v sin a + vbl cos a)], ghl hih2 = [6 vAh cos2 a sin a - 6 vbl Ah sin2 a cos a -(2 - 3 sin2 a) vAhH sin a -(i - 3sin2 a)vBL AhH cos a],

^h^dm = ~(kDh C0S2 a + kDH Sin2 a) '

Mh1h1 =

Casb gh'h'h' Zh^DMÏDM (S - Mh1 ) + iMh! Th!

gh1h1 h2 gh20dm0dm

(S - M22 ) + iMh2 Th2 'hlhi0dm0dm ' ah1h1^$DM$DM

s - 4 MDM 2 -JTDT IMh1h112 -

s - 4 Mh21 11

Total decay width of Zbl :

T(Zbl — ff ) = MZBLnc(qfgBL)2

m1bl/n

T(Zbl - v,) = mblg^h - m

m7 „ I 4M2 T(ZBL - NxNx) = MZLgBL(1 -

2 \ 3/2

t(Zbl — 0DM^DM ) =

gBL^BL MZ BL I i 4 mDMX

The total decay width of the extra neutral gauge boson ZBL is

h2h2 — 0DM 0dm:

gh2h2h2 = 3 [2 vkh sin3 a - 2 Vbl^H cos3 a

+khH sin a cos a (v cos a - vbl sin a)], gh2h2h1 =-[6 vkh sin2 a cos a + 6 vbl^h cos2 a sin a -(2 - 3 sin2 a)vBLkhH sin a + (1 - 3 sin2 a)vkhH cos a],

gh2h20dm0dm = - (kDh sin2 ^ +kDH cos2 a) ,

gh2h2 h1 gh10dm0dm (s - M2 ) + iMh1 Th1

Mh2h2 = I Casb

gh2h2h2 gh20dm0dm

(s - M22 ) + iMh2 Th2

dm0dm ' °h2h2 — 0dm0dm

s - 4 mDm

s - 4Mh22

IMh2h2 I2

TZbl = J2 T(ZbL — ff) + T(Z BL — Vx Vx )

+ T(ZBL — NxNx) + T(Zbl — 0DMf0DM) .

h1h2 — 0DM 0DM

gh1h1h2 = [6 vkh cos a sin a

-6 vBL kH sin2 a cos a

— (2 — З sin2 а) vЛhH sin а -(1 - Зsin2 а)vBL Л2Н cos а]

gh2222г = -[6 vЛh sin2 а cos а +6 vB^ H cos а sin а

— (2 — З sin2 а) vBL Л2Н sin а + (1 - З sin2 а^Л2Н cos а],

ghlh2ф¿мФDM = sin а cos ^Dh - ^н);

Mhlh2 = -Casb (g

hlh2ф¿м ФDM

gh2 h2h1 gh2 ф¿мФDM ' (s - Mh22 ) + iMh2 Т22/ g2121 h2 ^^DM

(s - M, ) + iMhl Т21 '

°h1 h2^ф¿MфDM 16n s

s(s - 4 mDm)

^ (s - (Mh, + Mh2)J)(s - (Mh2 - М^ )J)

W + W- ^ фDM ФDM

Ñ7 lMhlh2|n

г WW =

gh2 WW =

2mW cos а

2M^ sin а

Aww = Casb

g21 WW gh^DM

(s - M2 ) + iMhl Т21

gh2 WW^^DM (s - Mh22 ) + iMh2 ^

Mww = 9 i 1 +

(s - 2M^)2S

1 s - 4 м2

WW^фDмфDм l6ns y s - 4M2,

DM IMwwl2 . (E5)

gh2 ZZ ^^¿DM

(s - Mh22 ) + iMh2 Т22/

2 (s - 2M2 )2\

Mzz = - 1 + ---r^ Azz,

zz 9 l + SM4 ¿¿

a.Zz^¿M фDM

tt ^ ФDM ^DM

s - 4 MDM n

-77DT lMzz l2 .

s - 4Mj

gh1tt =--cos а

gh2tt = — sin а , v

gZBL tt = З '

Mtt = Casb

ltt g2 ^¿^DM

(s - M2 ) + iMhl Т2l

gh2tt gh2ф¿мФDM

(s - Mh22 ) + iMh2 ^ 1

З2п s -c

(s - 4Mt2)

/ s - 4MDDM n xj-DM lMtt l2 ,

s - 4 m,

а Zbl t

gBL -BL s - 4MD

«^¿m^m 64ns -c \ s - 4Mt2

s (s - 4mDm) gzbltt

(s - m2bl )2 + т2^ m2

Zbl Zbl

+ а B

»^¿DM^M tí^^^M tí^^^M

Ni Ni ^ фDM ФDM (i = 1, 2, З):

zz ^ фDM ^DM:

г zz =

gh2 ZZ =

2mz cos а

2MZ sin а

Azz = Casb

ghl ZZ gh^¿^DM

(s - M2 ) + iMhl Т21

1 NiNi =

gh2 NiNi =

yNi sin а

yNi cos а

MNi Ni =

Casb ghlNiNi gh^¿MфDм

(s - Mj) + iMhl Т21

gh2NiNi ^j^MФDM

(s - Mj2 ) + iMhj Т2, '

h1h2 j t

NiNi ^0dm0dm

(s-4 M2Nj) 32n s

(s-4mDM^ ,2

(s — 4MN)

|MNiNi |2

NiNi ^0DM0DM

g/Tn/T

192n s

s - 4MDM (s - 4mDm)(s - 4mn,.)

s - 4MN (s - m2 )2 + r2z m

Ni ^ = a h1h2

Zbl Zbl Zbl

NiNi ^0dm0dm NiNi ^0dm0dm NiNi ^dm^dm

zbl zbl ^ 0DM 0dm :

gh1 Zbl Zbl _

gh2 Zbl Zbl

2M2 sin a

2M\ cos a v

gt ° Zbl Zbl0dm0dm

= 2gBLnBL'

lgh1 Zbl Zbl gh10DM 0DM

aZBL ZBL = CASB

(s - M2 ) + lMh1 rh1

gh2 ZblZbl gh20DM0dm

(s - Mh22 ) + i' Mh2 rh2

Zbl Zbl0dm0dm 2 \2 \

(s - 2MZbl)2 mzblzbl = 9 i 1 +-8M- 1 azblZbl,

aZBL Zbl^ 0DM 0dm

s -4 m2

s - 4 MZbl

1 MZbl Zbl |2

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