Common origin of baryon asymmetry and proton decay

# Common origin of baryon asymmetry and proton decay

## Abstract

A successful baryogenesis theory requires a baryon-minus-lepton number violation if it works before the electroweak phase transition. The baryon-minus-lepton number violation could also exist in some proton decay modes. We propose a model to show that the cosmological baryon asymmetry and the proton decay could have a common origin. Specifically, we introduce an isotriplet and two isosinglet leptoquark scalars as well as two isotriplet Higgs scalars to the canonical seesaw model. The decays of the Higgs triplets can generate a desired baryon-minus-lepton asymmetry in the leptoquarks. After the Higgs triplets pick up their seesaw-suppressed vacuum expectation values, the leptoquarks with TeV-scale masses can mediate a testable proton decay.

###### pacs:
98.80.Cq, 14.20.Dh, 14.80.Sv, 23.40.-s

## I Introduction

There is an global anomaly thooft1976 () violating the baryon and lepton numbers by an equal amount in the standard model (SM). At the finite temperatures , the anomalous process becomes strong due to an instanton-like solution, the so-called sphalerons krs1985 (). During the sphaleron epoch, neither the baryon asymmetry nor the lepton asymmetry can survive if the baryon and lepton asymmetries are equal. However, the sphaleron processes will not affect any primordial asymmetry and will convert the asymmetry to a baryon asymmetry and a lepton asymmetry fy1986 (). So, a successful baryogenesis mechanism working above the weak scale should require a number violation which is a pure baryon number violation, a pure lepton number violation or a combined baryon and lepton violation.

The baryon and/or lepton number violation can lead to other interesting phenomena. For example, we can obtain a Majorana neutrino mass term by a lepton number violation of two units, a neutron-antineutron oscillation by a baryon number violation of two units, as well as a two-body proton decay by a baryon number violation of one unit and a lepton number violation of one unit.

In the simplest grand unified theories (GUTs), a baryon asymmetry and an equal lepton asymmetry can be simultaneously produced at the GUT scale through some baryon and lepton number violating interactions, which are also responsible for generating a conserving proton decay. In this GUT baryogenesis scenario, the baryon and lepton asymmetries will be both wiped out by the sphaleron processes.

In this paper we shall show it is possible to realize the baryogenesis and the proton decay by same interactions. For this purpose, we shall extend the SM by an isotriplet and two isosinglet leptoquark scalars, two isotriplet Higgs scalars as well as three right-handed neutrinos. The Majorana masses of the right-handed neutrinos will softly break the lepton number while the trilinear scalar couplings involving the Higgs triplets will softly break both of the baryon and lepton numbers. The number violating processes involving the right-handed neutrinos will be assumed to decouple before the out-of-equilibrium decays of the Higgs triplets. So, the asymmetry from the decays of the Higgs triplets into the leptoquarks can explain the baryon asymmetry in the universe. After the Higgs triplets pick up their seesaw-suppressed vacuum expectation values (VEVs), the leptoquarks can mediate a violating decay pss1983 () of the proton into two antineutrinos and one positron or antimuon. If the leptoquarks are at the TeV scale, the proton decay will be close to the experimental limit.

## Ii The model

For simplicity, we do not write down the full lagrangian. Instead, we only give the terms as below,

 L ⊃ −yik¯lLiϕNRk−12MNk¯NcRkNRk−fΩij¯lcLiiτ2ΩqLj (1) −fδaijδa¯lcLiiτ2qLj−hδaijδa¯ecRiuRj−μaϕTiτ2Σaϕ −κaδ1δ2Tr(Σ†aΩ)+H.c. = −yik(¯νLiϕ0+¯eLiϕ−)NRk−12MNk¯NcRkNRk −fΩij[ω−2/3¯νcLiuLj−1√2ω+1/3(¯νcLidLj+¯ecLiuLj) −ω+4/3¯ecLidLj]−fδaijδa(¯νcLidLj−¯ecLiuLj) −hδaijδa¯ecRiuRj−μa(σ0aϕ0ϕ0+√2σ+aϕ0ϕ− −σ++aϕ−ϕ−)−κaδ+1/31δ+1/32[(σ0a)∗ω−2/3 +(σ+a)∗ω+1/3+(σ++a)∗ω+4/3]+H.c.,

with the SM quarks, leptons and Higgs scalar:

 qLi(3,2,+16) = [uLidLi], uRi(3,1,+23), lLi(1,2,−12) = [νLieLi], eRi(1,1,−1), ϕ(1,2,−12) = [ϕ0ϕ−], (2)

the gauge-singlet right-handed neutrinos:

 NRi(1,1,0), (3)

the -singlet and triplet leptoquark scalars:

 δa(3,1,13)=δ+1/3a, Ω(3,3,13)=⎡⎢⎣1√2ω+1/3  ω+4/3ω−2/3  −1√2ω+1/3⎤⎥⎦, (4)

and the -triplet Higgs scalars:

 Σa(1,3,1) = ⎡⎢⎣1√2σ+a  σ++aσ0a  −1√2σ+  a⎤⎥⎦. (5)

We assign the baryon and lepton numbers as below,

 (B,L)=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩(13,0) for qL and~{}other~{}SM~{}quarks,(0,+1) for NR, lL and~{}other~{}SM~{}leptons,(−13,−1) for Ω and δ,(−1,−3) for Σ,(0,0) for ϕ.

In Eq. (1), the Majorana masses of the right-handed neutrinos break the lepton number, the trilinear couplings of the Higgs scalars break both the baryon number and the lepton number, while other terms conserve the baryon and lepton numbers. Clearly, the Majorana masses and the trilinear scalar couplings also break the number. Note that the baryon and lepton numbers are only allowed to softly break. We hence have forbidden the Yukawa couplings of the Higgs triplets to the leptons and the Yukawa couplings of the leptoquarks to the quarks.

The Higgs doublet will develop a VEV:

 ⟨ϕ0⟩≃174GeV, (7)

to spontaneously break the electroweak symmetry. The Higgs triplets then can pick up their seesaw-suppressed VEVs:

 ⟨σ0a⟩≃−μ∗a⟨ϕ0⟩2M2Σa≪⟨ϕ0⟩, (8)

like the Higgs triplets in the type-II seesaw model mw1980 ().

## Iii Neutrino masses

Although the Yukawa couplings of the Higgs triplets to the lepton doublets are absent from Eq. (1), they can appear at one-loop order as shown in Fig. 1. We calculate the effective Yukawa couplings to be

 L ⊃ −12(ga)ij¯lcLiiτ2ΣalLj+H.c. (9) = −12(ga)ij(σ0a¯νcLiνLj+√2σ+a¯νcLieLj−σ++a¯ecLieLj) +H.c..

with

 (ga)ij ≃ 14π2μa⎛⎝−M2ΣaM2Nk+iπ⎞⎠y∗ik1MNky∗jk (10) ≃ i4πμa(mν)ij⟨ϕ0⟩2  for  M2ΣaM2Nk≪1.

Here the canonical type-I seesaw minkowski1977 () formula have been adopted,

 L⊃−12mν¯νcLνL+H.c.  with  mν=−y∗⟨ϕ0⟩2MNy†. (11)

The effective Yukawa couplings (9) will also contribute to the neutrino masses through the type-II seesaw mechanism,

 (δmν)ij=(ga)ij∑a⟨σ0a⟩≃−i4π∑a|μa|2M2Σa(mν)ij. (12)

Clearly, the type-I seesaw could dominate over the type-II seesaw,

 δmν≪mν  for  |μa|2≪M2Σa. (13)

The neutrino mass matrix can be diagonalized by

 mν=Udiag{m1,m2,m3}UT, (14)

where are the mass eigenvalues while is the mixing matrix with three mixing angles, one Dirac CP phase and two Majorana CP phases. The neutrino oscillation experiments have given some information on the neutrino masses and mixing such as stv2011 ()

 m22−m21 = (7.09−8.19)×10−5eV2; (15a) m23−m21 = {−(2.08−2.64)×10−3eV2,(2.18−2.73)×10−3eV2. (15b)

Furthermore, the cosmological observations komatsu2011 () have put an upper bound on the sum of the neutrino mass eigenvalues,

 ∑imi < 0.58eV. (16)

## Iv Baryogenesis

We assume that the lepton number violating processes involving the right-handed neutrinos will decouple before the decays of the Higgs triplets and then the final asymmetry should be generated by the decays of the Higgs triplets. Below the seesaw scale , the scattering processes should have the rate fy1990 ():

 ΓA=1π3∑m2i⟨ϕ0⟩4T3. (17)

By requiring

 ΓA

where the Hubble constant is given by

 H(T)=(8π3g∗90)12T2MPl, (19)

with being the Planck mass and being the relativistic degrees of freedom (the SM fields plus an isotriplet and two isosinglet leptoquark scalars), the lepton number violating processes will decouple when the temperature falls down to

 T≃2×1013GeV(2.5×10−3eV2∑im2i). (20)

In the following demonstration, we hence shall consider the mass spectrum as below,

 MΣ1,2<1013GeV

As shown in Fig. 2, the Higgs triplets have the following decay modes:

 {Σ→δ1 δ2 Ω,ϕ∗ϕ∗,lcLlcL;Σ∗→δ∗1 δ∗2 Ω∗,ϕ ϕ,lLlL. (22)

Therefore, the decays of the Higgs triplets can produce a asymmetry in the leptoquarks and the leptons if CP is not conserved. To quantify the asymmetry, we can define a CP asymmetry in the decays of the Higgs triplets ,

 εa=εδ1δ2Ωa+εlLa, (23)

with

 εδ1δ2Ωa = 2ΓΣa→δ1δ2Ω−ΓΣ∗a→δ∗1δ∗2Ω∗Γa, (24a) εlLa = 2ΓΣa→lcLlcL−ΓΣ∗a→lLlLΓa. (24b)

Here

 Γa = ΓΣa=ΓΣa→δ1δ2Ω+ΓΣa→ϕ∗ϕ∗+ΓΣa→lcLlcL (25) ≡ ΓΣ∗a=ΓΣ∗a→δ∗1δ∗2Ω∗+ΓΣ∗a→ϕϕ+ΓΣ∗a→lLlL

is the decay width. We can calculate the tree-level decay width:

 Γa = 18π(164π2|κa|2+|μa|2M2Σa+164π2|μa|2∑im2i⟨ϕ0⟩4) (26) ×MΣa,

and the one-loop CP asymmetries:

 εδ1δ2Ωa = 12πIm(κ∗aκbμaμ∗b)|κa|21M2Σa−M2Σb (27a) ×(1+164π2M2Σa∑im2i⟨ϕ0⟩4) ×164π2|κa|2164π2|κa|2+|μa|2M2Σa+164π2|μa|2∑km2k⟨ϕ0⟩4, εlLa = −1128π3Im(κ∗aκbμ∗aμb)|μa|2M2ΣaM2Σa−M2Σb (27b) ×164π2|μa|2∑im2i⟨ϕ0⟩4164π2|κa|2+|μa|2M2Σa+164π2|μa|2∑km2k⟨ϕ0⟩4.

When the Higgs triplets go out of equilibrium, their CP violating decays can generate a asymmetry in the leptoquarks and as well as the leptons . For example, we consider the weak washout region, where the out-of-equilibrium condition can be described by the following quantity,

 Ka=ΓaH∣∣T=MΣa≲1. (28)

The induced asymmetry then can approximate to kt1990 ()

 nB−Ls∼3×εag∗forKa≲1, (29)

where the factor means the three components of the decaying Higgs triplets. If is much lighter than , the final asymmetry should come from the decays of . Alternatively, if and have a small mass split, both of them will significantly contribute to the final asymmetry. In this case, the CP asymmetry could be resonantly enhanced fps1995 ().

Since the leptoquarks decays into the leptons and the quarks, their asymmetries can be transferred to a baryon asymmetry and a lepton asymmetry through the sphaleron processes. The final baryon asymmetry in the universe should be krs1985 ()

 nBs=2879nB−Ls. (30)

## V Proton decay

Due to the VEVs of the Higgs triplets , the component of the leptoquark triplet will have a trilinear coupling with the leptoquark singlets , i.e.

 L ⊃ −ρδ+1/31δ+1/32ω−2/3+H.c.%   with (31) ρ=κ1⟨σ0∗1⟩+κ2⟨σ0∗2⟩.

By integrating out the leptoquark scalars, we can obtain a low-scale effective Lagrangian as below,

 Leff = ∑a≠bρ∗fΩijfδamnhδbklm2Ωm2δam2δb(¯νcLiuLj)(¯νcLmdLn)(¯ecRkuRl) (32) +H.c..

The dominant proton decay thus should be

 p → e+R(μ+R)+νcLi+νcLj, (33)

as shown in Fig. 3. Clearly, the proton decay violates the number by two units. We can roughly estimate the proton decay width by

 Γp→e+νν = ∑ijΓp→e+R+νcLi+νcLj ∝ ∑a≠b∑ij|fΩi1|2|fδaj1|2|hδb11|2|ρ|2m4Ωm4δam4δb, Γp→μ+νν = ∑ijΓp→e+R+νcLi+νcLj (34) ∝ ∑a≠b∑ij|fΩi1|2|fδaj1|2|hδb21|2|ρ|2m4Ωm4δam4δb.

## Vi Parameter choice

We now give an example of the parameter choice to show that our model can simultaneously generate a desired baryon asymmetry and an testable proton decay. We take

 MΣ1=0.2MΣ2=1012GeV, |μ1|=0.2|μ2|=3×109GeV, |κ1|=|κ2|=0.1, sin(κ∗1κ2μ∗1μ2|κ1κ2μ1μ2|)=−0.32, (35)

to derive the out-of-equilibrium quantity

 K1≃0.62, (36)

and the CP asymmetry

 ε1≃εΩ1≃1.2×10−8≫εlL1. (37)

The final baryon asymmetry then would be

 ηB=7.04×nBs≃7.04×3×2879ε1g∗≃6.57×10−10, (38)

which is consistent with the cosmological observations komatsu2011 ().

From the above parameter choice, we can also read the trilinear coupling among the leptoquarks,

 |ρ|≃1.1eV. (39)

Such a tiny parameter means the proton decay (V) can naturally have a life time close to the experimental limits nakamura2010 () and even if the leptoquarks are at the TeV scale.

## Vii Summary

In this paper, we have shown that the baryon asymmetry and the proton decay can have a common origin. In our model, the decays of the heavy Higgs triplets can produce a asymmetry in the TeV-scale leptoquarks. Due to the sphalerons, we eventually can obtain a baryon asymmetry to explain the baryon asymmetry in the universe. Benefited from the seesaw-suppressed VEVs of the Higgs triplets, the leptoquarks can have a trilinear coupling to mediate an observable proton decay.

Acknowledgement: PHG is supported by the Alexander von Humboldt Foundation. US thanks R. Cowsik for arranging his visit as the Clark Way Harrison visiting professor.

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