Observing Left-Right Symmetry in the Cosmic Microwave Background

We consider the possibility of probing left-right symmetric model (LRSM) via cosmic microwave background (CMB). We adopt the minimal LRSM with Higgs doublets, also known as the doublet left-right model (DLRM), where all fermions including the neutrinos acquire masses only via their couplings to the Higgs bidoublet. Due to the Dirac nature of light neutrinos, there exist additional relativistic degrees of freedom which can thermalise in the early universe by virtue of their gauge interactions corresponding to the right sector. We constrain the model from Planck 2018 bound on the effective relativistic degrees of freedom and also estimate the prospects for planned CMB Stage IV experiments to constrain the model further. We find that $W_R$ boson mass below 4.06 TeV can be ruled out from Planck 2018 bound at $2\sigma$ CL in the exact left-right symmetric limit which is equally competitive as the LHC bounds from dijet resonance searches. On the other hand Planck 2018 bound at $1\sigma$ CL can rule out a much larger parameter space out of reach of present direct search experiments, even in the presence of additional relativistic degrees of freedom around the TeV corner. We also study the consequence of these constraints on dark matter in DLRM by considering a right handed real fermion quintuplet to be the dominant dark matter component in the universe.


I. INTRODUCTION
Left-right symmetric models (LRSM) [1][2][3][4][5][6][7][8][9][10][11][12]  Conventionally, the very first proposals and studies of LRSM [1][2][3][4][5] considered a scalar bidoublet for generating fermion masses and also for electroweak symmetry breaking whereas a pair of scalar doublets were introduced for the purpose of left-right symmetry breaking at high energy scale. A very recent detailed study of this model can be found in [22]. On the other hand, the LRSM proposals put forward later [7][8][9][10][11][12] received much more attention due to the possibility of seesaw origin of light neutrino masses through a combination of type I seesaw [23][24][25][26] and type II seesaw [8,[27][28][29][30] or type III seesaw [31]. In the doublet left-right model (DLRM), in its minimal version, there is no such seesaw mechanism as all fermions including neutrinos acquire Dirac masses by virtue of their couplings to the bidoublet scalar. While generating sub-eV neutrino mass in this fashion requires a fine tuning of relevant Yukawa couplings at the level of < 10 −12 , we adopt this minimal scenario to study some of the interesting phenomenological consequences. Radiative generation of light Dirac neutrinos in different left-right symmetric models have also been discussed over last few decades [32][33][34][35][36][37][38][39] which may provide a UV completion of the minimal DLRM we discuss here. Since such UV completions do not drastically change the conclusions we reach in the present work, we stick to the DLRM for the sake of simplicity.
The Dirac nature of light neutrinos in DLRM gives rise to additional relativistic degrees of freedom which can be thermalised in the early universe due to their gauge interactions mediated by right sector gauge bosons. Such additional light degrees of freedom can be probed by precise measurements of the cosmic microwave background (CMB) anisotropies.
Recent 2018 data from the CMB measurement by the Planck satellite [40] suggests that the effective degrees of freedom for neutrinos as N eff = 2.99 +0.34 −0.33 (1) at 2σ or 95% CL including baryon acoustic oscillation (BAO) data. At 1σ CL it becomes more stringent to N eff = 2.99 ± 0.17. Both these bounds are consistent with the standard model (SM) prediction N SM eff = 3.045 [41][42][43]. Upcoming CMB Stage IV (CMB-S4) experiments are expected to put much more stringent bounds than Planck due to their potential of probing all the way down to ∆N eff = N eff − N SM eff = 0.06 [44]. We use the existing constraints and put strong limits on the scale of left-right symmetry or equivalently the right sector gauge bosons W R , Z R . For comparison, we also check the corresponding bounds for left-right asymmetric scenario by considering different SU (2) R gauge couplings. Interestingly, we find that the bounds on W R , Z R mass from Planck 2018 bound on ∆N eff at 2σ CL are equally competitive as the latest LHC bounds [16,17,20] and much stronger that the corresponding bounds from flavour physics [45]. On the other hand the same Planck 2018 bound at 1σ CL can rule out a much larger mass window for W R , Z R out of reach of present collider experiments. In fact CMB-S4 will be able to probe a much larger region of W R , Z R masses out of existing collider's reach and hence can probe or rule out the minimal model. Since there have been a few recent studies on gauged B − L model with light Dirac neutrinos [46][47][48][49]and corresponding constraints due to Planck 2018 bound on ∆N eff , we also reproduce the corresponding parameter space in gauged B − L model and compare with the one obtained in DLRM. We point out the important difference due to the restricted range of DLRM gauge couplings g R , g BL unlike that in gauged B −L model. We also show the impact of these constraints on dark matter (DM) parameter space in DLRM by considering a right handed fermion quintuplet to be the dominant component of DM which can thermalise by virtue of its interactions with SM mediated by right sector gauge bosons. We calculate the parameter space allowed from observed DM relic and find the leftover parameter space after applying the ∆N eff bound. Finally, we comment on the more stringent Planck 2018 1σ bound (1, 1, 2, −1)  This paper is organised as follows. In section II, we discuss the doublet left-right symmetric model followed by discussion of additional relativistic degrees of freedom due to light Dirac neutrinos in section III. In section IV we briefly discuss dark matter in DLRM particularly focusing on fermion quintuplet DM followed by results and discussion in section V.
We finally conclude in section VI.

II. THE DLRM
We briefly discuss the doublet left-right symmetric model in this section. The fermion and scalar content of the model are given in table I and II respectively. The relevant Yukawa Lagrangian giving masses to the three generations of leptons is given by, where the indices i, j = 1, 2, 3 represent the family indices for the three generations of fermions, Φ = τ 2 φ * τ 2 and τ 2 is Pauli matrix. The gauge structure of the model prevents any renormalisable Yukawa couplings involving the scalar doublets χ L,R . The scalar potential V scalar is given by [22] For details of the minimisation of the scalar potential and resulting symmetry breaking, please refer to [22]. In the symmetry breaking pattern, the neutral component of the Higgs doublet χ R acquires a vacuum expectation value (VEV) to break the gauge symmetry of the DLRM into that of the SM and then to the U (1) of electromagnetism by the VEV of the neutral components of Higgs bidoublet Φ: The VEVs of the neutral components of the Higgs fields can be denoted as where the VEV's k 1 , k 2 satisfy the VEV of the SM namely, v SM = k 2 1 + k 2 2 ≈ 246 GeV. The spontaneous breaking of DLRM gauge symmetry down to U (1) em results in two charged massive vector bosons W L , W R , two neutral massive bosons Z L , Z R and a massless photon as expected. The details of the mass spectrum of gauge bosons are shown in appendix A.
Light Dirac neutrino mass and charged lepton mass are given by where the family indices are suppressed. Without any loss of generality, we make use of rotation in the SU (2) L × SU (2) R space so that only one of the neutral components of the Higgs bidoublet acquires a large vacuum expectation value, k 1 ≈ v SM and k 2 ≈ 0. Under these assumptions, the Dirac neutrino mass matrix is while the charged lepton mass matrix is Therefore, tiny sub-eV Dirac neutrino mass arises due to smallness of Yukawa coupling h while charged lepton masses are generated by corresponding Yukawa coupling h. The details of fermion-gauge boson couplings are shown in appendix B. The details of the scalar mass spectrum is not derived here as we do not need them for our analysis and we refer to [22] for details of the same.

III. ∆N eff IN DLRM
Effective number of relativistic degrees of freedom is defined as where ρ rad = ρ γ + ρ ν is the net radiation content of the universe. As mentioned earlier, the SM prediction is N SM eff = 3.045 [41][42][43] which is also consistent with the constraint from precision measurement of Z boson decay width at LEP N ν = 2.984 ± 0.008 [14]. Any deviation of N eff from N SM eff will therefore indicate the presence of additional relativistic species thermalised in the early universe. While these additional relativistic degrees of freedom can not fully thermalise with the SM bath through interactions mediated by Z boson due to strong LEP bound, they can thermalise via additional interactions or mediating particles not yet observed in direct search experiments. The right handed neutrinos in DLRM provides such an example. They can thermalise with the SM bath in the early universe due to the interactions mediated by right sector gauge bosons, as depicted by the Feynman diagrams shown in figure 1. We consider negligible mixing between left and right sector gauge bosons and hence ignore the contributions coming from processes likē Additionally, the scalar mediated interactions are negligible due to tiny Dirac Yukawa couplings.
To estimate the contribution in ∆N eff we need to check the decoupling temperature of the right handed neutrinos. The decoupling occurs when the expansion rate of the universe becomes more than the interaction rate. Hence, the decoupling temperature can be calculated from the following equality where Γ(T ) is the interaction rate and H(T ) is the expansion rate of the universe. The interaction rate can be written as where the number density n ν R for a relativistic neutrino can be written as and the annihilation cross sections of right handed neutrinos are given in Appendix C.
The expansion rate of the universe can be written as where g ν R is the internal degrees of freedom for right-handed neutrinos. Thus, the contribution of ν R to effective relativistic degrees of freedom can be estimated as where N ν R represents the number of relativistic right-handed neutrinos, g * (T ) corresponds to the relativistic degrees of freedom at temperature T, g * s (T ) corresponds to the relativistic entropy degrees of freedom at temperature T 1 and T dec ν R , T dec ν L are the decoupling temperatures for ν R and ν L respectively. Thus, depending upon the decoupling temperature of ν R and hence g * (T dec ν R ), the additional contribution to ∆N eff can be kept within experimental upper limits. Lower the strength of ν R interaction with SM bath or higher the mediator mass of ν R -SM interactions, larger will be g * (T dec ν R ) and hence smaller will be ∆N eff . Similar analysis for U (1) B−L extension of the SM can be found in [46][47][48][49] whereas some estimates in the context of radiative Dirac neutrino mass in LRSM were made in [35,37]. gauge couplings g R . 1 We use g * and g * s interchangeably, which is true in SM at high temperatures.

IV. DARK MATTER IN DLRM
The data from Planck experiment which restricts the effective relativistic degrees of freedom in our universe also reveal that more than 26% of present universe's energy density is composed of a non-luminous and non-baryonic form of matter, known as dark matter. Apart from recent cosmology based experiments like Planck, there have been several astrophysical evidences for many decades suggesting the presence of DM [50][51][52]. In terms of abundance is conventionally reported as [40]: Ω DM h 2 = 0.120 ± 0.001 at 68% CL. Given that none of the SM particles can be a viable DM candidate, several BSM proposals have been put forward among which the weakly interacting massive particle (WIMP) paradigm is the most popular one. In this framework, a DM particle having masses and interactions similar to those around the electroweak scale gives rise to the observed relic after thermal freeze-out, a remarkable coincidence often referred to as the WIMP Miracle [53].
The minimal DLRM discussed above does not have a stable DM candidate. One can however, minimally extend the model by including additional scalar or fermionic multiplets in the spirit of minimal dark matter scenario [54][55][56]. In these models, the dark matter candidate is stabilised either by a Z 2 = (−1) B−L subgroup of the U (1) B−L gauge symmetry or due to an accidental symmetry at the renormalisable level due to the absence of any renormalisable operator leading to dark matter decay. Such minimal dark matter scenario in LRSM has been studied recently by the authors of [57,58]. Some more recent works on DM in LRSM can be found in [35][36][37][59][60][61][62][63][64][65][66].  [57,58] are no longer stable in DLRM due to the presence of renormalisable interactions with lighter fields. We therefore consider the option of larger fermion multiplet as DM, and the minimal scenario is to consider a real fermion quintuplet of B − L charge 0.
Since we want to constrain the right sector gauge bosons from cosmology bound on ∆N eff , we particularly focus on right handed fermion quintuplet DM whose relic abundance depends upon the strength of its annihilation through right sector gauge bosons.
In the pure left-right symmetric setup, one has to introduce a pair of left and right handed fermion quintuplets (having same mass) which can be written in component form as Since we are discussing a general scenario with g L = g R , we consider the left fermion quintuplet to be very heavy and decoupled from the low energy phenomenology. Even in the pure left-right symmetric limit g L = g R , one can make the left quintuplet decouple from the low energy phenomenology by introducing a parity odd scalar singlet whose non-zero VEV at a very high scale splits the right and left fermion masses. Such proposals where the left-right discrete symmetry or parity gets broken spontaneously before SU (2) R × U (1) B−L gauge symmetry were put forward long ago in [67][68][69]. While all the components of fermion multiplet have same tree level masses, at radiative level, there arises a mass splitting between charged (with Q) and neutral components given by [57,58], where s M = sin θ M ≡ tan θ W g L g R , s W = sin θ W , r X = M X /M and L where c Q = (2 + Q + 1)(2 − Q) and Q is the electromagnetic charge of the quintuplet component.

V. RESULTS AND DISCUSSION
Using the recipe discussed in previous section, we first calculate the decoupling temperature of right handed neutrinos from the thermal bath for different values of W R , Z R mass and gauge coupling g R . The variation of decoupling temperature with W R mass for different values of g R is shown in figure 2. Although both W R and Z R masses play role in right handed neutrino interactions with the thermal bath, we show the variation of decoupling temperature as well as other physical quantities only in terms of W R mass. This is due to the fact that Z R mass typically depends upon W R mass and is heavier than it, similar to Z and W masses of the SM. Also, we are not restricting ourselves to pure left-right symmetric limit g R = g L and considering different values of g R as well. Decoupling temperature rises for lower values of gauge coupling as well as higher values of W R mass as seen from figure 2 which is expected as the corresponding rate of interactions decreases.
We then show the contribution to ∆N eff in figure 3 as functions of decoupling temperature as well as W R mass. Along with the Planck 2018 bound mentioned earlier, we also show the CMB-S4 sensitivity [71] as well as the Planck 2018 1σ limit while the latter is same as SPT-3G sensitivity [72]. Clearly, Planck 2018 bound at 2σ CL itself rules out W R mass Unlike in [46][47][48][49] where similar constraints on U (1) B−L gauge boson was obtained, the crucial difference in DLRM is that here one can not tune the gauge couplings for a particular value of gauge boson mass in order to suppress the contribution to ∆N eff . This is because the gauge couplings of SU (2) R and U (1) B−L are not arbitrary but related to the gauge coupling of U (1) Y (at the scale of left-right symmetry breaking) as Since g Y is known, one can not change g R , g BL arbitrarily within their perturbative limits 2 .
We show the allowed region of these two gauge couplings in figure 5. While we still have a large region within perturbative limits, we have chosen g R to be either equal to g L or smaller while keeping g BL also below order one for our benchmark analysis. For g R > g R the Planck bound becomes even more stringent, as we found in the scan plot shown in g R − M W R plane in To check the impact of these constraints on DM parameter space, we then calculate relic of right fermion quintuplet DM. For DM relic calculation, we first implement the model in SARAH [74] and then feed the model files into micrOMEGAs [75] for relic calculations. We then consider three benchmark combinations of (g R , W R ) while keeping the scale of left-right gauge symmetry breaking v R fixed. The resulting variation of DM relic as a function of DM mass is shown in left panel plot of figure 7. The resonance corresponding to W R , Z R masses are clearly visible in this plot. Unlike in quintuplet DM scenario in triplet LRSM [58], here the two resonances are quite close to each other due to smaller ratio of Z R to W R mass in DLRM. The results shown in left panel plot of figure 7 also agree with that shown in [66]. We then scan the parameter space of W R , Ω 0 R masses and show the region satisfying correct DM relic in right panel plot of figure 7. Multiple allowed values of DM mass for a fixed W R mass are arising due to annihilation and coannihilations of Ω 0 We also apply the corresponding bounds on W R mass from Planck constraints on ∆N eff at 2σ CL as horizontal shaded lines so that the region below the respective lines are disallowed. Clearly, a large part of the DM parameter space specially for g R = g L gets ruled out by ∆N eff bounds. We do not show other existing bounds on W R mass from flavour or LHC data as they are either equally or less strong compared to the bounds derived here. are not arbitrary in DLRM due to their non-trivial connection to g Y of standard model.
We also show the impact of ∆N eff constraints on dark matter parameter space in DLRM.
While DLRM does not have a dark matter candidate on its own, we incorporate the pres-ence of an additional fermion quintuplet DM in the minimal DM spirit. Since such a real fermion quintuplet does not have any renormalisable coupling with other fermions or scalars of DLRM, the relic abundance of DM, the neutral component of right-handed fermion quintuplet, depends crucially on its annihilation and coannihilation mediated by W R , Z R gauge bosons. We constrain the parameter space satisfying correct DM relic by using the respective ∆N eff bounds for different g R . We find available parameter space satisfying correct DM relic even after applying Planck 2018 bound on ∆N eff at 2σ CL.
We also compare our results in view of more stringent Planck 2018 1σ bound N eff = 2. Covariant derivatives of the scalar fields in DLRM can be written as The corresponding kinetic Lagrangian of scalar fields are Considering the scalar vevs as, The charged vector boson mass matrix can be written as whereas the neutral vector boson mass matrix is As expected, the neutral gauge boson mass matrix has one vanishing eigenvalue, corresponding to massless photon. After diagonalisation of the mass matrices we can represent the gauge fields in terms of physical gauge boson states as Also, we can express these couplings as, sinθ W = e g L and cosθ W = e g Y with θ W being the Weinberg angle. In DLRM, Z Lµ and Z Rµ will also mix as the bi-doublet Φ transform nontrivially under both SU (2) L and SU (2) R gauge groups. The mixing can be represented as where the mixing angle can be written as The charged vector boson states are where, we can have the corresponding masssquared terms for the charged physical gauge bosons as, Note that we have taken k 2 = 0 which is equivalent to vanishing tree level mixing angle ζ. One can however generate radiative mixing between charged vector bosons, but that is typically very small < 10 −7 [37]. Note that although we write W 1 , W 2 , Z, Z as physical massive gauge boson states here to show the details, in the main text we continue to use W L , W R , Z L , Z R for better clarity.

Appendix B: Fermion-gauge boson interactions in DLRM
In this section we note down the fermion interactions with massive vector bosons. The kinetic term of leptons in DLRM is given by where, The same kinetic Lagrangian is, in fact, applicable to quarks too if we include gluons in the covariant derivative. We show the interactions with neutral massive vector bosons in table  The interaction of neutrino ν, charged leptons with W 2 (or W R ) is similar to the ones with W L except that g L is replaced by g R : where we have ignored the details of right handed lepton mixing matrix, taking it to be a unit matrix. We also notice a difference in ν R coupling to Z boson from what was reported in [22]. While in our case ν R coupling to Z is suppressed by corresponding Z − Z mixing angle, the authors of [22] found identical couplings of ν L and ν R to Z boson. This will however lead to very large interaction rate of ν R and will make it very difficult to satisfy the stringent bounds on N eff .

Appendix C: Annihilation cross-sections of right handed neutrinos
The annihilation cross sections of ν R mediated by right sector gauge bosons are