Limiting Majoron self-interactions from Gravitational waves experiments

TL;DR

This paper explores how gravitational wave experiments can test Majoron models, especially those involving first order phase transitions that produce detectable gravitational waves, and discusses collider detection prospects.

Contribution

It introduces the 'violent Majoron model' scenario where Majoron self-interactions cause phase transitions detectable via gravitational waves, with implications for low-scale B-L symmetry breaking.

Findings

Sub-electroweak scale phase transition can produce observable gravitational waves.

Next-generation detectors like eLISA, DECIGO, and BBO can detect these signals.

Potential collider signatures at CEPC through Higgs portal interactions.

Abstract

We show how Majoron models may be tested/limited in gravitational waves experiments. In particular, the Majoron self-interaction potential may induce a first order phase transition, producing gravitational waves from bubble collisions. We dubbed such a new scenario {\it violent Majoron model}, because it would be associated to a violent phase transition in the early Universe. Sphaleron constraints can be avoided if the global $U(1)_{B-L}$ is broken at scales lower than the electroweak scale, provided that the B-L spontaneously breaking scale is lower than $10\, {\rm TeV}$ in order to satisfy the cosmological mass density bound. The possibility of a sub-electroweak phase transition is practically unconstrained by cosmological bounds and it may be detected within the sensitivity of next generation of gravitational waves experiments: eLISA, DECIGO and BBO. We also comment on the possible detection in CEPC collider, where Majorons's production can be observed from Higgs' portals in missing transverse energy channels.

Figures

Equations

$${\mathcal{L}}_M=fH\bar{L}{\nu }_R+h\sigma {\bar{\nu }}_R{\nu }_R^c+h.c.+V(\sigma ,H),$$

$$V(\sigma ,H)=V_0(\sigma ,H)+V_1(\sigma )+V_2(h,\sigma )$$

$$V_0(\sigma ,H)=$$

$$V_1(\sigma )=\frac{{\lambda }_1}{\Lambda }{\sigma }^5+\frac{{\lambda }_2}{\Lambda }{\sigma }^{\ast }{\sigma }^4+\frac{{\lambda }_3}{\Lambda }({\sigma }^{\ast }{)}^2{\sigma }^3+h.c.$$

$$V_2(H,\sigma )=$$

$$\sigma =\frac{1}{\sqrt{2}}\left(v_{BL}+\rho +i\chi \right).$$

$$M=2h{v}_{BL}$$

$$m=\frac{1}{\sqrt{2}}fv,$$

$$N={\nu }_R+{\nu }_R^c+\frac{m}{M}\left({\nu }_L+{\nu }_L^c\right),$$

$$\nu ={\nu }_L+{\nu }_L^c-\frac{m}{M}\left({\nu }_R^c+{\nu }_R\right),$$

$$m_N\simeq M,m_{\nu }\simeq \frac{m^2}{M},$$

$$f_{collision}\simeq 3.5\times 10^{-4}\left(\frac{\beta }{H_{\ast }}\right)\left(\frac{\bar{T}}{10\mathrm{GeV}}\right){\left(\frac{g_{\ast }(\bar{T})}{10}\right)}^{1/6}\mathrm{mHz},$$

$${\Omega }_{collision}({\nu }_{collision})\simeq$$

$$k{\mathcal{E}}^2{\left(\frac{\bar{H}}{\beta }\right)}^2{\left(\frac{\alpha }{1+\alpha }\right)}^2\left(\frac{V_B^3}{0.24+V_B^3}\right)\left(\frac{10}{g_{\ast }(\bar{T})}\right).$$

$$\mathcal{E}(\bar{T})={\left[T\frac{d{V}_{eff}}{dT}-V_{eff}(T)\right]}_{T=\bar{T}},$$

$$\alpha =\frac{\mathcal{E}(\bar{T})}{{\rho }_{rad}(\bar{T})},{\rho }_{rad}=\frac{{\pi }^2}{30}{g}_{\ast }(T)T^4.$$

$$\beta =-{\left[\frac{d{S}_E}{dt}\right]}_{t=\bar{t}}\simeq {\left[\frac{1}{\Gamma }\frac{d\Gamma }{dt}\right]}_{t=\bar{t}},$$

$$S_E(T)\simeq \frac{S_3(T)}{T},\Gamma ={\Gamma }_0(T)\exp [-S_E(T)],$$

$${\Gamma }_0(T)\sim T^4,S_3\equiv \int d^{3}r\left({\partial }_i{\sigma }^{†}{\partial }_i\sigma +V_{eff}(\sigma ,T)\right).$$

$$V_{eff}(s,T)\simeq C{T}^2({\sigma }^{†}\sigma )+V(\sigma ,H),$$

$$C=\frac{1}{4}\left(\frac{m_{\sigma }^2}{v_{LB}^2}+{\lambda }_{sH}+h^2-24K_{BL}\right),$$

$$K_{BL}=({\lambda }_2+{\lambda }_3)\frac{v_{BL}}{\Lambda }+{\beta }_2\frac{v_{BL}}{\Lambda }.$$

$$\frac{1}{4}\left[\frac{m_{\sigma }^2}{v_{LB}^2}+\frac{1}{4}{\lambda }_{sH}+h^2\right]\simeq 1$$

$$\Lambda \simeq 400\mathrm{GeV}÷4\mathrm{TeV},$$

$${\mu }_F=\frac{{\sigma }^{NP}(pp\to H)}{{\sigma }^{SM}(pp\to H)}\frac{B{R}^{NP}(H\to F)}{B{R}^{SM}(H\to F)},$$

$$\mathrm{Br}(H\to invisible)=\frac{{\Gamma }_{inv}}{{\Gamma }_{inv}+{\Gamma }_{SM}}<0.51(95\%C.L.),$$

$$C_{H\chi \chi }\le 0.6.$$

$$C_{H\chi \chi }\simeq C_{H\chi \chi }^0=\frac{{\beta }_2{v}_{BL}}{\Lambda }<0.6,$$

$$e^{+}{e}^{-}\to ZH\to Z\bar{b}b,$$

$${\sigma }_{hZ\to b\bar{b}Z}={\sigma }_{hZ}^{SM}\times R_{hZ}\times BR(h\to b\bar{b}),$$

Full text

Lepton/Baryon number violation, Majorons, gravitational waves, colliders

I Introduction

The idea that the neutrino can be identified with its anti-particle, the antineutrino ($\nu=\nu^{c}$), was a wonderful intuition by Ettore Majorana in ’37 Majorana37 . A Majorana mass term for the neutrino must imply a violation of the Lepton number of two units, i.e. $\Delta L=2$. Nowadays, the only realistic test of Majorana’s hypothesis is the neutrinoless double beta decay ($0\nu 2\beta$). The Majorana mass term may be originated by a spontaneous symmetry breaking of a global $U(1)_{L}$ or $U(1)_{B-L}$ extending the Standard Model. This leads to the possibility that a new pseudo Nambu-Goldstone boson, dubbed Majoron, can be coupled with neutrinos and be emitted in the $0\nu\beta\beta$-process Chikashige:1980ui ; Gelmini:1980re ; Schechter:1981cv . Majorons have phenomenological implications not only in $0\nu\beta\beta$ experiments, but they can be limited by astrophysical stellar cooling processes and cosmological bounds. In particular, the spontaneous symmetry breaking scale of $U(1)_{L}$ or $U(1)_{B-L}$ is highly constrained to be higher than the electroweak scale 22a ; 22b ; 24a ; 24b ; Cline:1993ht . On the other hand, it was argued that such a VEV scale cannot be higher than $10\,{\rm TeV}$ for a cosmologically consistent Majoron model Akhmedov:1992hi .

Despite these considerations, the Majoron particle remains very elusive, with not so many direct detection channels being predicted. Nonetheless, the possibility of testing first order phase transitions (F.O.P.T) in the early Universe seems to be more promising after the recent discovery of gravitational waves (GW) at LIGO Caprini:2015zlo ; Kudoh:2005as . In particular, next generations of interferometers like eLISA and U-DECIGO will be also fundamentally important to test gravitational signal produced by Coleman bubbles from FOPT. The production of GW from bubble collisions was first suggested in Refs. Witten:1984rs ; Turner:1990rc ; Hogan:1986qda ; Kosowsky:1991ua ; Kamionkowski:1993fg .

New experimental prospectives in GW experiments have motivated a revival of these ideas in the context of new extensions of the Standard Model Schwaller:2015tja ; Huang:2016odd ; Artymowski:2016tme ; Dev:2016feu ; Katz:2016adq ; Huang:2017laj ; Baldes:2017rcu ; Chao:2017vrq ; Addazi:2016fbj ; Ghorbani:2017jls ; Tsumura:2017knk ; Huang:2017rzf . In other words, the GW data may be used to test new models of particle physics beyond the standard model.

With this paper we suggest to test the Majoron self-interactions and the details of the spontaneous symmetry breaking mechanism of the Lepton symmetry from GW interferometers. This is still a completely open possibility, related to a first order phase transition (F.O.P.T) of the Lepton symmetry in the early Universe. In particular, we show how next generation of interferometers like (e)LISA, U-DECIGO and BBO can test the spontaneous symmetry breaking scale $V_{BL}=10\,{\rm GeV}\div 10\,{\rm TeV}$. We dub such a Majoron particle associated to the F.O.P.T. a violent Majoron. We will discuss how the current limits from LHC in missing transverse energy channels do not exclude the possibility of a violent Majoron. We emphasize that in the framework of the violent Majoron model, the CEPC collider will be able to test an energy scale overlapping the one associated to the eLISA sensitivity for GW signals from F.O.P.T.

II The model

We consider the extension of the Standard Model described by the gauge groups $SU_{c}(3)\times SU(2)_{L}\times U(1)_{Y}\times U(1)_{B-L}$, in which the lepton and baryon numbers are promoted to a $U(1)_{B-L}$ global symmetry111The implication of the Majoron in neutron-antineutron transitions were recently discussed in Refs. Berezhiani:2015afa ; Addazi:2015pia ; Addazi:2015ata ; Addazi:2016rgo . . We introduce a complex scalar field coupled to neutrinos and to the Higgs boson that spontaneously breaks the $U(1)_{L}$ symmetry:

[TABLE]

where $h,f$ are Yukawa matrices of the model, and

[TABLE]

the potential, with

[TABLE]

and higher order terms

[TABLE]

and

[TABLE]

In principle, the scales of new physics entering non-perturbative operators may be different at each others. For convention, we parametrize their differences in the couplings $\lambda_{i},\beta_{i}$.

When $\sigma$ gets a VEV, it may be decomposed in a real and a complex field:

[TABLE]

After the global $U(1)_{L}$ symmetry breaking, the RH neutrino acquires a Majorana mass term

[TABLE]

and a Dirac mass for the LH neutrino

[TABLE]

where $|\langle H\rangle|=v$ and $v/\sqrt{2}=174\,\rm GeV$.

The seesaw relations are obtained for $M>\!\!>m$, namely

[TABLE]

[TABLE]

i.e.

[TABLE]

$m_{N}$ standing for the mass of the right-handed neutrino $N$. The Majoron corresponds to the pseudo-scalar field $\chi$, which is the Nambu-Goldstone boson of the spontaneously broken $U(1)$ symmetry, while the real scalar $\rho$ gets a mass $m_{\rho}\sim O(1)\,v_{s}$ once the self-coupling is assumed to be $O(1)$.

In the Majoron model higher order terms, like the one entering $V_{1}$, are desired in order to induce a mass contribution that would not be allowed at perturbative level. For example, these may be induced either by gravitational effects Akhmedov:1992hi or by exotic instantons (see e.g. Ref. Addazi:2014ila ) at lower scales than the Planck scale222However, related theoretical aspects within the context of string theory are not completely understood. For instance a global $U(1)_{L}$ might be thought as a local $U(1)_{L}$ from flavor branes, since in string theory no exact global $U(1)$ symmetry can arise. This possibility seems highly bounded by the weak gravity conjecture, pointed out recently in Ref. Addazi:2016ksv ..

III Gravitational Waves signal from Majorons

The spontaneous symmetry breaking of the $U(1)_{B-L}$ can be catalyzed by a first order phase transition (F.O.P.T). This induces the generations of Coleman’s bubbles expanding at high velocity, which generate a stochastic cosmological background of gravitational radiation. Gravitational waves are generated by three main processes: i) bubble-bubble collisions; ii) turbulence induced by the bubble’s expansion in the plasma; iii) sound waves induced by the Bubble’s running in the plasma. The peak frequency of the GW signal produced by bubble collision has a frequency

[TABLE]

in which $\beta$ is related to the size of the bubble wall and is expressed in (16), $\bar{T}$ is the temperature at the F.O.P.T., $g_{*}(\bar{T})$ label degrees of freedom involved and the GW intensity is estimated as follows

[TABLE]

[TABLE]

The coefficient $k$ introduced above has a numerical value $k\simeq 2.4\times 10^{-6}$, while

[TABLE]

[TABLE]

In (15) $\rho_{rad}$ stands for the radiation energy density, while $\bar{T}\simeq v_{BL}$ is the first order phase transition temperature, defined by

[TABLE]

in which

[TABLE]

[TABLE]

$V_{B}$ represents the velocity of the bubble. The various values of $V_{B}$ will determine the amount of corrections from turbulence and sonic waves discussed later.

The effective potential is the model dependent part of the Eq.(12). In particular the effective potential get thermal corrections which can be treated in the same approximation performed in Ref. Delaunay:2007wb :

[TABLE]

where

[TABLE]

with

[TABLE]

The case of $K_{BL}\simeq 4\times 10^{-2}$ corresponds to the one testable by eLISA, U-DECIGO and BBO. From Eq. (19), assuming $\lambda_{2,3},\beta_{2}=1$, $v_{BL}\simeq 1\div 100\,{\rm GeV}$ and

[TABLE]

this corresponds to a scale of

[TABLE]

while the GW signal is among $10^{-5}\div 10^{-3}\,{\rm Hz}$. The effect of the Higgs as a dynamical particle are suppressed as $O(v_{BL}^{2}/v^{2})$, which are totally negligible for $v_{BL}\simeq 1\div 10\,{\rm Gev}$ compared to $O(1)$ uncertanties from from Bubble collisions and expansion details.

Let us remark that in principle other contributions from turbulence and sound waves may affect the estimate of the new physics scale by a factor $O(1)$, since at least they will affect the power spectrum density of GW by a factor $O(10)$. This may lower the scale of the new physics by a factor $3$.

Let us compere these order of magnitude semi-analytical estimations with numerical simulations. In Fig.1, we show numerical plots in a realistic set of parameters, using the same model independent spectrum parameterization of Ref.Caprini:2015zlo . We also consider the contribution of turbulence and shock waves as in Ref.Caprini:2015zlo . The results are in good $O(1)$ agreement with the estimations inferred above.

III.1 LHC constrains

From LHC important constrains on the Higgs decay into invisible channels are set. Let us define

[TABLE]

where $F=\gamma\gamma,WW,ZZ,\tau,\tau$ label final states. In Tab/Fig. 1 we show the limits from various channels on Higgs decays. Comparing Eq.(20) with limits from LHC in Fig. 2, we can set a bound on the $C_{H\chi\chi}$-parameter in the Higgs decay rate.

In particular, the model independent limit placed on the invisible decays branching ratio is (see e.g. Chatrchyan:2014tja )

[TABLE]

which corresponds to the bound

[TABLE]

This amounts to find a maximal bound on the hierarchy between the $v_{\sigma}$ and $\Lambda$ scale that corresponds to the value

[TABLE]

where $C_{H\chi\chi}^{0}$ is the leading order contribution to the $C_{H\chi\chi}$, which is originated from the operator in Eq.(5) parametrized by $\beta_{2}$. Such a constrain is easily compatible with GW signals in eLISA and cosmological bounds. For example, fixing $v_{BL}=100\,{\rm GeV}$ and $\beta_{2}=1$, $\Lambda>166\,{\rm GeV}$ is enough to avoid LHC constraints, while for eLISA $\Lambda\simeq 4\,{\rm TeV}$ is large enough to generate a detectable GW signal – see the previous section on gravitational waves discussed above.

III.2 electron-positron colliders and invisible Higgs decays

In CEPC, the Higgs decay rate will be probed with a factor $O(5\div 10)$ of sensitivity higher than LHC.

The golden channel will be the process

[TABLE]

with a cross section

[TABLE]

where $R$ is a suppression factor related to the coupling of the Higgs boson to $Z$. According to Ref. TLEP , the number of Higgs boson produced at Center of Mass energy $\sqrt{s}=240\,{\rm GeV}$ from the $e^{+}e^{-}\rightarrow HZ$ channel will be $2\times 10^{6}$, higher then the other channels. The total cross section of the process can be measured with $0.4\%$ of precision, while $\sigma_{HZ}\times Br(h\rightarrow b\bar{b})$ can reach precision $0.2\%$ for $\sqrt{s}=240\,{\rm GeV}$ TLEP , while it can reach one order of magnitude more then LHC in the $\sqrt{s}=350\,\rm GeV$ (see Fig. 12 of TLEP ). In Fig. 3 we consider the constraints on Eq. (21) overimposed to current LHC constrains, cosmological bounds and future eLISA sensitivity regions.

Let us conclude this section with an important remark. We would like to stress again that the Fig.2 was displayed fixing $\beta_{2}$ to be one. However, fixing different values of the $\beta_{2}$ parameter will change the region plot shown. Assuming smaller values of $\beta_{2}$, the constrain on the $\Lambda$ from colliders are relaxed. Let us also note that the case $\beta_{2}<<1$ is still possible. This case will correspond to suppress the invisible channels in colliders, rendering the Gravitational interferometers bounds stronger then colliders one: the gravitational power spectrum depends on a large set of initial parameters $\lambda_{1,2,3},\beta_{1,2,3}$. With $\beta_{2}<<\lambda_{1,2,3},\beta_{1}$, the GW signal is still unsuppressed. It is worth to mention that for $\beta_{2}\simeq 0$, Invisible channels can be generated from radiative corrections, but, of course, with a strong suppression factor.

III.3 Cosmological limits

III.3.1 Sphaleron bounds

Relating the Majoron model to the pre-sphaleron leptogenesis, stringent constrains are provided from washing-out processes of $(B+L)$-violating sphaleronic interactions in the Standard Model.

First of all, we remind that the bound on the cosmological neutrino mass is

[TABLE]

where $T_{BL}$ is the temperature at which the $L$ (or $B-L$) asymmetry is generated. The cosmological neutrino mass bound sets in turn a bound on $T_{BL}$ from Eq. (24) that reads

[TABLE]

The neutrino mass bound used in all estimations of these papers consider the cosmological and neutrinoless double beta decay bounds (see Ref. PDG ). From sphalerons constraints, one can get out the following bound on $v_{s}$, i.e.

[TABLE]

where

[TABLE]

which reduces to $Y\simeq 1/27$ 22a ; 22b , is compatible with Big Bang Nucleosynthesis constraints 24a ; 24b , and $U_{hl}$ represents the neutrino mixing matrix. Eq. (26) can be related to a bound on the Yukawa matrices:

[TABLE]

Eq. (28) provides a very strong bound on $h$-couplings that reads

[TABLE]

This bound is so strong to be valid even for tiny gravitationally induced (Planck scale suppressed) effects, or when the mixing $U_{hl}$ are very small. Finally, let us comment on possible way-out to this bound. The sphaleron bound can be relaxed, allowing $v_{BL}>v_{Higgs}$, in electroweak baryogenesis scenarios and post-sphaleron baryogengesis scenarios. In this case, also gravitational waves signal from the baryogengesis scenarios should be observable – as recently discussed in Ref. Huang:2016odd .

III.3.2 Cosmological density bound

Cosmological density constraints can be set distinguishing the two cases:

[TABLE]

In the case (A), the Majoron mass is dominated by the $\beta_{1}$-term and casts

[TABLE]

In the case (B) the mass of the Majoron reads

[TABLE]

with

[TABLE]

where $M$ is the RH neutrino mass while $v$ is the Higgs VEV. The cosmological density constraint on the Majorons reads

[TABLE]

for $n_{\chi}\simeq n_{\gamma}$ and for $v=v_{BL}$ and $\lambda,\beta<10^{-2}$. On the other hand, for $\chi$ decoupling sufficiently rapidly, it should be possible that $n_{\chi}<\!\!<n_{\gamma}$. Consequently the constraints on $\chi$ can be weaker than the above.

Cosmological constraints that can be derived from Majorons heavily rely on the assumption that the Majoron is out of equilibrium, and on the decay channels that are allowed by the particular model instantiations.

If we consider massive Majorons and stable LH neutrinos, limits on the Yukawa coupling $h$ can be derived from the see-saw relation and the cosmological constraints on the neutrinos mass density, i.e.

[TABLE]

where $v$ is the Higgs expectation value. LH electrons and RH neutrinos are in thermal equilibrium via the interactions

[TABLE]

with an interaction rate

[TABLE]

Thermal equilibrium, as realized above, happens for

[TABLE]

For $T<M$, RH neutrinos go out of thermal equilibrium, disappearing from the thermal bath. At this stage, the relevant interaction of the scalar complex field $\sigma$ is with LH neutrinos, with a coupling of the order of

[TABLE]

In the case of a spontaneous symmetry breaking scale of $v_{BL}\simeq 1\div 100\,{\rm GeV}$ suggested above while discussing GW signals, the limit on the Majoron coupling becomes very stringent and reads

[TABLE]

Let us remark that, assuming $M>10\,{\rm GeV}$ os so, $\sigma$ goes out of equilibrium for a temperature of about $T\simeq M$. As a consequence, Majoron density in the present Universe is

[TABLE]

in which $g_{*}(T)$ represents the effective number of light particle species at a temperature $T$; $T_{RH}$ and $T_{0}$ are the decoupling temperature for the RH neutrinos and the presente temperature of the Universe respectively.

Further constrains on the $v_{BL}$ from Majoron decays must be considered. The Majoron decay into two neutrinos

[TABLE]

has a decay time

[TABLE]

Eq. (43) constrains the Majoron relic density as follows (see Ref. Akhmedov:1992hi ):

[TABLE]

where $\tau_{U}$ is the age of the Universe. Eq. (44) leads to

[TABLE]

where $\delta\rho/\rho\leq 10^{-4}$ are the initial density fluctuations.

On the other hand relativistic decay products of $\chi$ must be redshifted enough to maintain a matter dominated Universe, i.e. to avoid constrains on dark radiation:

[TABLE]

where

[TABLE]

and

[TABLE]

This leads to the following bound

[TABLE]

leading to

[TABLE]

Such a bound can be generalized for higher order operators in the complex scalar sector, leading to

[TABLE]

Nonetheless, bounds on $v_{BL}$ provided by higher $n$-order contributions are less stringent.

III.3.3 Dark Matter

The Majoron particle can provide a new candidate of Dark Matter Akhmedov:1992hi ; DM . For sub-electroweak phase transition considered, the Majoron mass is parametrized by Eq.(31). For a $v_{BL}\simeq 10\,{\rm GeV}$ phase transition, the Majoron is naturally Kev-ish, while the overproduction problem is avoided. This means that the Majoron can compose Warm Dark Matter. The Majoron Dark Matter paradigm can be tested from colliders missing energy channels and from gravitational waves experiments. This certainly enforces the naturalness and phenomenological health of our proposal.

IV Conclusions and remarks

We have shown how gravitational waves experiments may provide useful informations on the Majoron self-interaction potential. In particular, the possibility of a first order phase transition at a scale of about $1\div 100\,{\rm GeV}$ is still unbounded by any cosmological limits, such as non-perturbative electroweak effects — sphalerons — and cosmological density abundance. The main message of this paper is that such a scale overlaps the sensitivity of future gravitational waves’ interferometers, like eLISA, U-DECIGO and BBO. In fact, a scale of about $1\div 100\,{\rm GeV}$ falls into the range of frequencies around $10^{-5}\div 10^{-3}\,{\rm Hz}$. However, an observation of a stochastic gravitational waves signal in eLISA should imply a new physics scale of UV completion for the violent Majoron model of about $3\,{\rm TeV}$ or so. This means that the production of Majorons in colliders may provide a complementary test-bed for this model. For instance, the detection of Majorons in missing energy channels can be tested in the future collider CPEC from Higgs invisible decays.

Acknowledgements.

AM wishes to acknowledge support by the Shanghai Municipality, through the grant No. KBH1512299, and by Fudan University, through the grant No. JJH1512105.