Post-sphaleron baryogenesis and $n$–$\overline{n}$ oscillation in non-SUSY SO(10) GUT with gauge coupling unification and proton decay

Abstract

“Post-sphaleron baryogenesis”, a fresh and profound mechanism of baryogenesis accounts for the matter–antimatter asymmetry of our present universe in a framework of Pati–Salam symmetry. We attempt here to embed this mechanism in a non-SUSY SO(10) grand unified theory by reviving a novel symmetry breaking chain with Pati–Salam symmetry as an intermediate symmetry breaking step and as well to address post-sphaleron baryogenesis and neutron–antineutron oscillation in a rational manner. The Pati–Salam symmetry based on the gauge group $\mathrm{SU}(2)_L \times \mathrm{SU}(2)_{R} \times \mathrm{SU}(4)_C$ is realized in our model at $10^{5}$–$10^{6}$ GeV and the mixing time for the neutron–antineutron oscillation process having $\Delta B=2$ is found to be $\tau _{n-\bar{n}} \simeq 10^{8}$–$10^{10}$ s with the model parameters, which is within the reach of forthcoming experiments. Other novel features of the model include low scale right-handed $W^{\pm }_R$, $Z_R$ gauge bosons, explanation for neutrino oscillation data via the gauged inverse (or extended) seesaw mechanism and most importantly TeV scale color sextet scalar particles responsible for an observable $n$–$\bar{n}$ oscillation which may be accessible to LHC. We also look after gauge coupling unification and an estimation of the proton lifetime with and without the addition of color sextet scalars.

1 Introduction

The Standard Model (SM) of particle physics has given us enough reasons to look beyond its framework for dealing with issues like the tiny neutrino masses, matter–antimatter asymmetry of the present universe, dark matter and dark energy, and coupling unification of three fundamental interactions. Among all these, the observed baryon asymmetry of the universe has motivated the scientific community to work upon it since a long time ago. The WMAP satellite data [1, 2], when combined with large scale structure (LSS) data, gives a baryon asymmetry of the universe of $\eta ^{\text{ CMB }} \simeq (6.3\pm 0.3) \times 10^{-10}$, while an independent measurement of the baryon asymmetry carried out by BBN [3] yields $\eta ^{\text{ BBN }} \simeq (3.4 -6.9) \times 10^{-10}$. Two compelling mechanisms, namely leptogenesis [4] and weak scale baryogenesis [5], have been prime tools for explaining the baryon asymmetry of the universe. In leptogenesis the desired lepton asymmetry is created by the lepton number violation as well as out of equilibrium decays of heavy particles, which is subsequently converted into baryon asymmetry by the non-perturbative ($B+L$)-violating sphaleron interactions [6, 7].

Inadequate knowledge of the nature of new physics beyond the standard model leaves us with no choice but to explore all possibilities which may explain the origin of the matter–antimatter asymmetry. Recently a new idea behind baryon asymmetry has been explored named “Post-Sphaleron Baryogenesis (PSB)”, which occurs via the decay of a scalar boson singlet for the standard model having a mass around a few hundreds of GeV and a high dimensional baryon number violating coupling [8–10], where the Yukawa coupling(s) of the scalar(s) act as the source of the CP asymmetry. Apparently, this high dimensional baryon number violating coupling is generated via new physics operative beyond standard model electroweak theory. The mechanism of PSB is based on the idea that the required amount of baryon asymmetry of the universe can be generated below the scale of electroweak phase transition where the sphaleron has decoupled from the Hubble expansion rate. Although the proposal seems interesting it has not yet been incorporated in a realistic grand unified theory. Hence we attempt here to embed the proposal of PSB in a non-SUSY SO(10) GUT with Pati–Salam (PS) symmetry and left–right (LR) symmetry as intermediate symmetry breaking steps.

A detailed study of the literature [11–18] gives an idea of many intriguing features of the SO(10) grand unified theory (including both non-SUSY and SUSY). One of these features is that when a left–right gauge symmetry appears as an intermediate symmetry breaking step in a novel symmetry breaking chain, then the seesaw mechanism can be naturally incorporated into it. In conventional seesaw models associated with thermal leptogenesis the mass scale for heavy RH Majorana neutrino is at $10^{10}$ GeV, which makes it unsuitable for direct detectability at current accelerator experiments like LHC. Therefore, it is necessary to construct a theory having $\mathrm{SU}(2)_{L} \times \mathrm{SU}(2)_{{R}} \times U(1)_{B-L} \times {SU}(3)_C$ and ${SU}(2)_{L} \times \mathrm{SU}(2)_{R} \times \mathrm{SU}(4)_C$ gauge groups as intermediate symmetry breaking steps, which results in low mass right-handed Majorana neutrinos along with $W_{R}$, $Z^\prime$ gauge bosons at the TeV scale. At the same time it should be capable of explaining post-sphaleron baryogenesis elegantly along with other derivable predictions like proton decay and neutron–antineutron oscillation.

We intend to discuss TeV scale post-sphaleron baryogenesis, the neutron–antineutron oscillation having a mixing time close to the experimental limit with the Pati–Salam symmetry or SO(10) GUT as mentioned in a recent work [19] slightly modifying the Higgs content, where nonzero light neutrino masses can be accommodated via the gauged extended inverse seesaw mechanism along with TeV scale $W_{R}$, $Z^\prime$ gauge bosons. As discussed in the work [19] the Dirac neutrino mass matrix is similar to the up-quark mass matrix even with low scale right-handed symmetry breaking. Though the details has been already discussed in the above mentioned work we briefly clarify the point as follows.

In non-SUSY SO(10), the type I seesaw [20–24] contribution to neutrino mass is given by

$$\begin{aligned} m^{I}_\nu = - M_{D} M^{-1}_{R} M^\mathrm{T}_{D}, \end{aligned}$$

where $M_{D}$ is the Dirac neutrino mass matrix, $M_{R}$ is the Majorana neutrino mass matrix for right-handed neutrinos and is related to the right-handed symmetry breaking scale. The Dirac neutrino mass matrix and up-quark mass matrices are similar in a generic SO(10) model that has high scale Pati–Salam symmetry as an intermediate breaking step relating quarks and leptons with each other. Hence, $M_{D} \simeq M_{u}$, which further implies that the $\tau$-neutrino Dirac Yukawa coupling should be equal to the top-quark Yukawa coupling. With $M_{D}\simeq M_{u} \simeq 100$ GeV, the sub-eV scale of the light neutrino consistent with the oscillation data requires the right-handed scale (seesaw scale) to be greater than $10^{13}$ GeV. Such a high seesaw scale makes this idea difficult to probe at any foreseeable laboratory experiments. Hence, as an alternative way, emphasizing its verifiability at LHC, the inverse seesaw mechanism [25–27] has been proposed, with an extra SO(10) fermion singlet $S$ (in addition to the existing fermion content of SO(10)), with a light neutrino mass formula

$$\begin{aligned} m_\nu =\left( \frac{M_{D}}{M}\right) \mu \left( \frac{M_{D}}{M}\right) ^\mathrm{T}, \end{aligned}$$

where $M$ is the $N$–$S$ mixing matrix and $\mu$ is the small lepton number violating mass term for sterile neutrino $S$. The above relation can be recast as

$$\begin{aligned} \left( \frac{m_\nu }{0.1\,\text{ eV }}\right) = \left( \frac{M_{D}}{100\, \text{ GeV }} \right) ^2 \left( \frac{\mu }{\text{ keV }}\right) \left( \frac{M}{10^4\, \text{ GeV }} \right) ^{-2}. \end{aligned}$$

Hence, sub-eV masses for the light neutrinos are consistent with $M_{D} \simeq M_{u}$ (or $Y_{D} \simeq Y_{t}$) which is a generic predictions of high scale Pati–Salam symmetry and compatible with low right-handed symmetry breaking scale ($M_{R}$), since the inverse seesaw formula is independent of $M_{R}$. We have utilized this particular property of low scale right-handed symmetry breaking in studying post-sphaleron baryogenesis and neutron–antineutron oscillation even though a complete discussion on the origin of neutrino masses and mixing via low scale extended inverse seesaw has been omitted.

Here we sketch out the complete work of our paper. In Sect. 2, we briefly discuss non-SUSY SO(10) GUT with a novel symmetry breaking chain, having $\mathcal {G}_{2213}$ and $\mathcal {G}_{224}$ as intermediate symmetry breaking steps. In Sect. 3 we show how gauge coupling unification is achieved in our model. In Sect. 4 we discuss the TeV scale post-sphaleron baryogenesis and embed it within the novel chain of a non-SUSY SO(10) model with self-consistent model parameters. In Sect. 5, we estimate the mixing time for the neutron–antineutron oscillation. In Sect. 6, we present an idea of how low mass scales for the RH Majorana neutrino as well as right-handed gauge bosons $W_R$, $Z^\prime$ are allowed in the model, while explaining light neutrino masses via a gauged extended seesaw mechanism. In Sect. 7 we conclude our work with results and a summary, including a note on the viability of the model at LHC.

2 The model

In this section we shall discuss the one-loop gauge coupling unification and estimate the proton life time including short distance enhancement factor to the $d=6$ proton decay operator by reviving the symmetry breaking chain [19]

$$\begin{aligned}&\mathrm{SO}(10) \mathop {\longrightarrow }\limits ^{M_U}\mathrm{SU}(2)_L \times \mathrm{SU}(2)_{R} \times \mathrm{SU}(4)_C D \nonumber \\&\quad \times [\mathcal {G}_{224D},\,(g_{2L} = g_{2R})]\nonumber \\&\quad \mathop {\longrightarrow }\limits ^{M_P} \mathrm{SU}(2)_L \times \mathrm{SU}(2)_{R} \times \mathrm{SU}(4)_C \quad [\mathcal {G}_{224}, (g_{2L} \ne g_{2R})]\nonumber \\&\quad \mathop {\longrightarrow }\limits ^{M_C} \mathrm{SU}(2)_L \times \mathrm{SU}(2)_{R} \times U(1)_{B-L} \times \mathrm{SU}(3)_C \quad [\mathcal {G}_{2213}]\nonumber \\&\quad \mathop {\longrightarrow }\limits ^{M_\Omega } \mathrm{SU}(2)_L \times U(1)_{R} \times U(1)_{B-L} \times \mathrm{SU}(3)_C \quad [\mathcal {G}_{2113}]\nonumber \\&\quad \mathop {\longrightarrow }\limits ^{M_{B-L}} \mathrm{SU}(2)_L \times U(1)_{Y} \times \mathrm{SU}(3)_C \quad [\mathcal {G}_\mathrm{SM} \equiv \mathcal {G}_\mathrm{213}] \nonumber \\&\quad \mathop {\longrightarrow }\limits ^{M_Z} U(1)_\mathrm{em}\times \mathrm{SU}(3)_C \quad [\mathcal {G}_\mathrm{13}]. \end{aligned}$$

(1)

The chain breaks in a sequence, where SO(10) first breaks down to $\mathcal {G}_{224D}, (g_{2L} = g_{2R})$ after the Higgs representation $\langle (1,1,1) \rangle \subset \{54\}_H$ is given a VEV, then the spontaneous breakdown of D-parity occurs in $\mathcal {G}_{224D}, (g_{2L} = g_{2R}) \rightarrow \mathcal {G}_{224}, (g_{2L} \ne g_{2R})$ with the assignment of a VEV to the D-parity odd component $\langle (1,1,1) \rangle$ contained in the Higgs representation $\{210\}_H$. The decomposition of $\{210\}_H$ under $\mathcal {G}_{224}$ is

$$\begin{aligned} \{210\}_H&= (1,1,1) \oplus (2,2,20) \oplus (3,1,15) \oplus (1,3,15) \nonumber \\&\oplus (2,2,6) \oplus (1,1,15). \end{aligned}$$

(2)

Spontaneous D-parity mechanism is aptly utilized here, since the theory allows low mass scale for right-handed Higgs fields around $\mathcal {O}$(TeV), while keeping all its left-handed components at D-parity breaking scale. Now assigning a VEV to the neutral component $\langle (1,1,15)\rangle \subset \{210\}_H$, the Pati–Salam symmetry ($\mathcal {G}_{224}$) breaks down to the left–right symmetry ($\mathcal {G}_{2213}$). The next step of symmetry breaking $\mathcal {G}_{2213} \rightarrow \mathcal {G}_{2113}$ occurs via the VEV $\langle (1,3,0,1)\rangle \subset \{210\}_H$. The right-handed gauge boson $W_R$ acquires a mass in the range of a few TeV and contributes sub-dominantly to neutrinoless double beta decay.

The most desirable symmetry breaking step $\mathcal {G}_{2113} \rightarrow \mathcal {G}_{213}$ is achieved by the $\{126\}_H$ of SO(10) though we have added another Higgs representation $\{16\}_H$ for realization of the gauged inverse seesaw mechanism operative at the TeV scale. The decomposition of the Higgs $\{126\}_H$ under $\mathcal {G}_{224}$ is

$$\begin{aligned} \{126\}_H&= (3,1,10) \oplus (1,3,\overline{10}) \oplus (2,2,15) \oplus (1,1,6).\nonumber \\ \end{aligned}$$

(3)

As we have pointed earlier, due to the D-parity mechanism, the right-handed triplet Higgs field $\Delta _R (1,3,-2,1)$ contained in $(3,1,10)$ gets its mass at the TeV scale, while its left-handed partner $\Delta _L (3,1,-2,1)$ has its mass at D-parity breaking scale $M_P$. As a result of this symmetry breaking, the neutral component of the right-handed gauge boson $Z^\prime$ gets its mass around $\mathcal {O}$(TeV) with the experimental bound $M_{W_R} \ge 2.5$ TeV [30, 31]. The final stage of symmetry breaking $\mathcal {G}_{2113} \rightarrow \mathcal {G}_{213}$ is carried out by giving a VEV to the neutral component of SM Higgs doublet $\langle \phi ^0(2,1/2,1) \rangle$ contained in the bidoublet $\Phi \subset \{10\}_H$.

We shall now check whether SO(10) having TeV scale post-sphaleron baryogenesis, neutron–antineutron oscillation and the gauged inverse seesaw mechanism is consistent with gauge coupling unification. It is found that the coupling constants unify at ($10^{17}$–$10^{18.5}$) GeV with the Higgs fields $\{10\}_H+\{10\}_H^\prime + \{16\}_H + \{126\}_H +\{210\}_H$. Some good reasons behind taking these Higgs fields are as follows. First, the TeV scale post-sphaleron baryogenesis and neutron–antineutron oscillation can be well explained with these parameters, while predicting a $W_R$ gauge boson in the TeV range. Second, it allows $B-L$ breaking ($M_{B-L}$) at the TeV scale resulting in a $Z^\prime$ mass $\ge 1.6$ TeV; moreover, it explains tiny masses for light neutrinos consistent with neutrino oscillation data via the TeV scale gauged inverse seesaw mechanism and LFV decays with branching ratios accessible to ongoing search experiments.

3 Gauge coupling unification and proton decay

3.1 One-loop renormalization group equations (RGEs) for gauge coupling evolution

For simplicity, we consider only the one-loop renormalization group equations (RGEs) for gauge coupling evolution, which can be written as

(4)

where $t=\ln (\mu )$, $\alpha _{i}=g^2_{i}/(4 \pi )$ is the fine structure constant, and $\varvec{a_i}$ is for the one-loop beta coefficients derived for the corresponding $i$th gauge group for which coupling evolution has to be determined. Using the input parameters, the electroweak mixing angle $\sin ^2\theta _W(M_Z)=0.2312$, the electromagnetic coupling constant $\alpha (M_Z) = 127.9$ and the strong coupling constant $\alpha _{S}(M_Z) = 0.1187$ taken from PDG [3, 28, 29], the values of the three coupling constants at the electroweak scale $M_Z=91.187$ GeV can be calculated precisely to be

$$\begin{aligned} \left( \begin{array}{cc} \ \alpha _{2L}(M_Z) \\ \ \alpha _{1Y}(M_Z) \\ \ \alpha _{3C} (M_Z) \end{array}\right) = \left( \begin{array}{cc} \ 0.033493^{+0.000042}_{-0.000038} \\ \ 0.016829 \pm 0.000017 \\ \ 0.118 \pm 0.003 \end{array}\right) , \end{aligned}$$

(5)

where $\{\alpha _{2L}(M_Z), \alpha _{1Y}(M_Z), \alpha _{3C}(M_Z) \}$ denote the fine structure constants for the SM gauge group $\mathcal {G}_{213}=\mathrm{SU}(2)_L \times U(1)_Y \times \mathrm{SU}(3)_C$.

3.2 Higgs content for the model and corresponding one-loop beta coefficients $\varvec{a_i}$

The Higgs contents for the model used in different ranges of the mass scales under respective gauge symmetries ($\mathcal {G}_I$) with a particular symmetry breaking chain as considered in a recent work [19] where the prime interest was to keep the $W_R$, $Z_R$ gauge bosons at the TeV scale are as follows:

$$\begin{aligned}&{\mathrm {(i)}\, \mu =\mathbf{M}_\mathbf{Z} - \mathbf{M}_{\mathbf{B-L}} }: G=\mathrm{SM} = G_{213},\nonumber \\&\qquad \text{ Higgs: }\,\Phi (2,1/2,1); \nonumber \\&{\mathrm {(ii)}\, \mu =\mathbf{M}_{\mathbf{B-L}} - \mathbf{M}_\mathbf{\Omega }}: G= G_{2113}, \nonumber \\&\qquad \text{ Higgs: }\, \Phi _1 (2,1/2,0,1) \oplus \Phi _2 (2,-1/2,0,1) \nonumber \\&\qquad \qquad \quad \,\,\, \oplus \chi _R (1,1/2,-1,1) \oplus \Delta _R (1,1,-2,1); \nonumber \\&{\mathrm {(iii)}\, \mu =\mathbf{M}_\mathbf{\Omega } - \mathbf{M}_\mathbf{C}}: G= G_{2213}, \nonumber \\&\qquad \text{ Higgs: }\,\Phi _1 (2,2,0,1) \oplus \Phi _2 (2,2,0,1) \oplus \chi _R (1,2,-1,1)\nonumber \\&\qquad \qquad \quad \,\,\, \oplus \Delta _R (1,3,-2,1)\oplus \Omega _R(1,3,0,1), \end{aligned}$$

(6)

$$\begin{aligned}&{\mathrm {(iv)}\, \mu =\mathbf{M}_\mathbf{C} - \mathbf{M}_{\mathbf{\xi }}}: G= G_{224}, \nonumber \\&\qquad \text{ Higgs: }\,\Phi _1(2,2,1)_{10}\oplus \Phi _2(2,2,1)_{10^\prime } \oplus \Delta _R(1,3,\overline{10})_{126}\nonumber \\&\qquad \qquad \quad \,\,\, \oplus \chi _R(1,2,\overline{4})_{16} \nonumber \\&\qquad \qquad \quad \,\,\,\oplus \Omega _R(1,3,15)_{210} \oplus \Sigma (1,1,15)_{210},\nonumber \\&{\mathrm {(v)}\, \mu =\mathbf{M}_\mathbf{\xi } - \mathbf{M}_{\mathbf{P}}}: G= G^\prime _{224}, \nonumber \\&\qquad \text{ Higgs: }\,\Phi _1(2,2,1)_{10}\oplus \Phi _2(2,2,1)_{10^\prime } \oplus \Delta _R(1,3,\overline{10})_{126}\nonumber \\&\qquad \qquad \quad \,\,\, \oplus \chi _R(1,2,\overline{4})_{16} \oplus \Omega _R(1,3,15)_{210} \nonumber \\&\qquad \qquad \quad \,\,\, \oplus \Sigma (1,1,15)_{210} \oplus \xi (2,2,15)_{126},\nonumber \\&{\mathrm {(vi)}\, \mu =\mathbf{M}_\mathbf{P} - \mathbf{M}_\mathbf{U}}: G= G_{224D}, \nonumber \\&\qquad \text{ Higgs: }\,\Phi _1(2,2,1)_{10}\oplus \Phi _2(2,2,1)_{10^\prime } \oplus \Delta _L(3,1,10)_{126}\nonumber \\&\qquad \qquad \quad \,\,\, \oplus \Delta _R(1,3,\overline{10})_{126} \oplus \chi _L(2,1,4)_{16} \!\oplus \! \chi _R(1,2,\overline{4})_{16} \nonumber \\&\qquad \qquad \quad \,\,\, \oplus \Omega _L(3,1,15)_{210} \oplus \Omega _R(1,3,15)_{210}\nonumber \\&\qquad \qquad \quad \,\,\, \oplus \Sigma (1,1,15)_{210} \oplus \xi (2,2,15)_{126} \oplus \sigma (1,1,1)_{210}.\nonumber \\ \end{aligned}$$

(7)

Here we find two categories of Higgs spectrum: Model I having Higgs spectrum as given in Eqs. (6) and (7) excluding the bitriplet Higgs scalar which estimates a proton life time that is far from the reach of search experiments and Model II having the same Higgs spectrum, including the bitriplet Higgs scalar $(3,3,1) \subset \mathcal {G}_{224}$ from the mass scale $M_C$ onwards, which estimates a proton life time very close to the experimental limit. Thus Model II serves our purpose.

The one-loop beta coefficients are found to be the same for both models at the mass scale ranges $M_Z$–$M_{B-L}$, $M_{B-L}$–$M_\Omega$, and $M_\Omega$–$M_{C}$ i.e.,

$$\begin{aligned}&{\mathrm {(i)}\, \mu =\mathbf{M}_\mathbf{Z} - \mathbf{M}_{\mathbf{B-L}} }: G=\mathrm{SM} = G_{2_L 1_Y 3_C},\nonumber \\&\qquad \varvec{a_i}=(-19/6,\, 41/10,\, -7),\nonumber \\&{\mathrm {(ii)}\, \mu =\mathbf{M}_{\mathbf{B-L}} - \mathbf{M}_\mathbf{\Omega }}: G= G_{2_L 1_{R} 1_{(B-L)} 3_C},\nonumber \\&\qquad \varvec{a_i}=(-3,\, 19/4,\, 37/8,\, -7), \nonumber \\&{\mathrm {(iii)}\, \mu =\mathbf{M}_\mathbf{\Omega } - \mathbf{M}_\mathbf{C}}: G= G_{2_L 2_{R} 1_{(B-L)} 3_C},\nonumber \\&\qquad \varvec{a_i}=(-8/3,\, -2/3,\, 23/4,\, -7), \end{aligned}$$

(8)

whereas they differ at the Pati–Salam scale $M_C$ to the unification scale $M_U$ as shown in Table 1.

Table 1 One-loop beta coefficients for different gauge coupling evolutions, without bitriplet Higgs scalar in Model I and with a bitriplet Higgs scalar (3, 3, 1) under the Pati–Salam group $\mathrm{SU}(2)_L \times \mathrm{SU}(2)_R \times \mathrm{SU}(4)_C$ in Model II

The gauge coupling unification for this work is shown in Fig. 1 with the allowed mass scales desirable for our model predictions,

$$\begin{aligned}&M_{B-L}\!=\! 4-7~\mathrm{TeV},\, M_{\Omega } \!=\! 10~\mathrm{TeV}, M_C \!= \!10^{5}-10^{6}\, \text{ GeV },\nonumber \\&M_P\simeq 10^{15.65}~\mathrm{GeV\,\, and\,\,} M_\mathrm{G}\simeq 10^{18.65}~\mathrm{GeV}. \end{aligned}$$

(9)

Fig. 1

Gauge coupling evolution plot having TeV scale $W_R$, $Z_R$ bosons where $M_U=2.65\times 10^{15.8}$ GeV

3.3 Estimation of proton life time for $p \rightarrow \pi ^0\, e^+$

The decay rate for the gauge boson mediated proton decay in the channel $p \rightarrow \pi ^0\, e^+$ including strong and electroweak renormalization effects on the $\mathrm{d}=6$ operator starting from the GUT scale to the proton mass (i.e., 1 GeV) [32, 33] turns out to be

$$\begin{aligned} \Gamma ( p\rightarrow \pi ^0 e^+)&= \frac{\pi }{4}\, A^2_L\, \frac{|\overline{\alpha }_H|^2}{f^2_\pi } \frac{m_p\, \alpha ^2_U}{M^4_U} (1 + \mathcal {F} + \mathcal {D})^2 \mathcal {R}.\nonumber \\ \end{aligned}$$

(10)

In Eq. (10), $A_L=1.25$ is the renormalization factor from the electroweak scale to the proton mass, $\mathcal {D}=0.81$, $\mathcal {F}=0.44$, $\overline{\alpha }_H=-0.011\, \, \text{ GeV }^3$, and $f_\pi =139\, \, \text{ MeV }$, which have been extracted as phenomenological parameters by the chiral perturbation theory and lattice gauge theory. Also $m_p=938.3~\text{ MeV }$ is the proton mass, and $\alpha _U \equiv \alpha _G$ is the gauge fine structure constant derived at the GUT scale. It is worth to note here that the renormalization factor $\mathcal {R}=[(A_{SR}^2+A_{SL}^2)(1+ |{V_{ud}}|^2)^2]$ for SO(10), $V_{ud}=0.974=$ with $A_{SL}(A_{SR})$ being the short-distance renormalization factor in the left (right) sectors, and $V_{ud}$ is the $(1,1)$ element of $V_\mathrm{CKM}$ for quark mixings.

After re-expressing $\alpha _H = \overline{\alpha }_H (1 + \mathcal {F} + \mathcal {D} ) = 0.012~\text{ GeV }^3$, and $\mathcal {A}_R\simeq \mathcal {A}_{L} \mathcal {A}_{SL} \simeq \mathcal {A}_{L} \mathcal {A}_{SR}$, the proton life time can be expressed as

$$\begin{aligned} \tau _p = \Gamma ^{-1}( p\rightarrow \pi ^0 e^+)&= \frac{4}{\pi }\,\frac{f^2_\pi }{m_p}\frac{M^4_U}{\alpha ^2_U} \frac{1}{\alpha ^2_H \mathcal {A}^2_{R}} \frac{1}{\mathcal {F}_q}, \end{aligned}$$

(11)

where $\mathcal {F}_q \simeq 7.6$.

Short-distance enhancement factor $\mathbf {\mathcal {A}_{SL}}$ extrapolated from GUT scale to $\mathbf {1}$ GeV: For estimating the proton decay rate in the channel $p \rightarrow e^+\pi ^0$ having a dimension-6 operator, one needs to extrapolate the operator from the GUT-scale physics to low energy physics at the scale of $m_p=1\, \text{ GeV }$ [34–36]. With the particular symmetry breaking chain allowed in the non-SUSY SO(10) model (following Ref. [36]), the whole energy range can be separated into the following parts:

• I. from non-SUSY S0(10) GUT scale, $M_U$, to the Pati–Salam symmetry with D-parity ($\mathcal {G}_{224D}$, $g_{2L}=g_{2R}$) invariance scale, $M_P$,

• II. from $M_P$ to the Pati–Salam symmetry without D-parity ($\mathcal {G}_{224}$, $g_{2L} \ne g_{2R}$) scale $M_C$,

• III. from $M_C$ to the $\mathrm{SU}(4)_C$ breaking scale, $M_\Omega$, where we have left–right symmetric model (LRSM) $\mathcal {G}_{2213}$,

• IV. from the left–right symmetry breaking scale ($M_\Omega$) to the $\mathcal {G}_{2113}$ scale ($M_{B-L}$) ,

• V. from the $\mathcal {G}_{2113}$ scale ($M_{B-L}$) to the standard model $\mathcal {G}_{213}$,

• VI. from the standard model to $1\, \text{ GeV }$.

As discussed in Refs. [34–36], the enhancement factor below SM for the $LLLL$ operator is

$$\begin{aligned} \mathcal {A}^{\prime }_{L} = \left[ \frac{\alpha _s (1~\text{ GeV })}{\alpha _s (m_t)}\right] ^{-\frac{4}{2 \cdot (-11+\frac{2}{3}\,n_f)}}, \end{aligned}$$

where $n_f$ denotes the number of quark flavors at the particular energy scale of our interest. Neglecting the effect due to $\alpha _{2L}$ and $\alpha _{Y}$ since their contributions are suppressed as compared to the strong coupling effect $\alpha _s$, this enhancement factor can be expressed in a more explicit manner as

$$\begin{aligned} \mathcal {A}^{\prime }_{L}=\left[ \frac{\alpha _s (1~\text{ GeV })}{\alpha _s (m_c)}\right] ^{2/9} \left[ \frac{\alpha _s (m_c)}{\alpha _s (m_b)}\right] ^{6/25} \left[ \frac{\alpha _s (m_b)}{\alpha _s (m_t)}\right] ^{6/23}.\quad \end{aligned}$$

(12)

Since the model considered here is a non-supersymmetric version of the SO(10) GUT, all other enhancement factors can be written in the same way as

(13)

with $\gamma _i$ (${\varvec{a_i}}$) as the anomalous dimension (one-loop beta coefficients) for the corresponding gauge group $i=\mathrm{SU}(2)_{L}, U(1)_{Y},\, \mathrm{SU}(3)_{C}$. Similarly, one can write the enhancement factor valid for $\mathcal {G}_{2113}$, $\mathcal {G}_{2213}$, $\mathcal {G}_{224}$, and $\mathcal {G}_{224D}$ as

$$\begin{aligned}&\mathcal {A}^{2113}_{SL}=\left[ \frac{\alpha _i (M^0_R)}{\alpha _i (M^+_R)}\right] ^{\frac{- \gamma _i}{2\, {\varvec{a_i}}}}, \nonumber \\&\quad \text{ with }\, i=\mathrm{SU}(2)_{L},\quad U(1)_R,\, U(1)_{B-L},\, \mathrm{SU}(3)_{C}, \nonumber \\&\mathcal {A}^{2213}_{SL}=\left[ \frac{\alpha _i (M^+_R)}{\alpha _i (M_C)}\right] ^{\frac{- \gamma _i}{2\, {\varvec{a_i}}}}, \nonumber \\&\quad \text{ with }\, i=\mathrm{SU}(2)_{L},\quad \mathrm{SU}(2)_R,\, U(1)_{B-L},\, \mathrm{SU}(3)_{C}, \nonumber \\&\mathcal {A}^{224}_{SL}=\left[ \frac{\alpha _i (M_C)}{\alpha _i (M_P)}\right] ^{\frac{- \gamma _i}{2\, {\varvec{a_i}}}}, \nonumber \\&\quad \text{ with }\, i=\mathrm{SU}(2)_{L},\quad \mathrm{SU}(2)_R,\, \mathrm{SU}(4)_{C}, \nonumber \\&\mathcal {A}^{224D}_{SL}=\left[ \frac{\alpha _i (M_P)}{\alpha _i (M_U)}\right] ^{\frac{- \gamma _i}{2\, {\varvec{a_i}}}}, \nonumber \\&\quad \text{ with }\, i=\mathrm{SU}(2)_{L},\quad \mathrm{SU}(2)_R,\, \mathrm{SU}(4)_{C}\, \text{ with } \text{ D-parity }. \nonumber \end{aligned}$$

Hence, the complete short-distance enhancement renormalization factor for this $d=6$ proton decay operator is found to be

$$\begin{aligned} \mathcal {A}_{SL}=\mathcal {A}^\mathrm{SM}_{SL} \cdot \mathcal {A}^{2113}_{SL} \cdot \mathcal {A}^{2213}_{SL} \cdot \mathcal {A}^{224}_{SL} \cdot \mathcal {A}^{224D}_{SL}. \end{aligned}$$

(14)

We have precisely followed the prescription given in Ref. [34, 35] for the derivation of anomalous dimension for the effective $d=6 (LLLL)$ proton decay operator. With a choice of TeV scale particle spectrum used in our model, the unification scale is found to be $M_U=2.65\times 10^{18.5}$ GeV for Model I and $M_U=10^{15.8}$ GeV for Model II. We have estimated the factor $\mathcal {A}_{R}=\mathcal {A}_L \cdot \mathcal {A}_{SL}$ approximately to be $4.36$ with the value of the long distance renormalization factor $A_L=1.25$, which is the same for both models.

With these input parameters, the model under consideration predicts the proton life time to be

$$\begin{aligned} \tau (p \rightarrow e^+ \pi ^0) = 2.6 \times 10^{34}\, \text{ years }, \end{aligned}$$

which is closer to the latest Super-Kamiokande experimental bound [37, 38]

$$\begin{aligned} \tau (p \rightarrow e^+ \pi ^0) \big |_{SK, 2011} > 8.2 \times 10^{33}\, \text{ years }\, \end{aligned}$$

(15)

and aptly supports planned experiments that can reach a bound [39]

$$\begin{aligned}&\tau (p \rightarrow e^+ \pi ^0) |_{HK, 2025} > 9.0 \times 10^{34}\, \text{ years },\nonumber \\&\tau (p \rightarrow e^+ \pi ^0) |_{HK, 2040} > 2.0 \times 10^{35}\, \text{ years }. \end{aligned}$$

(16)

4 TeV scale post-sphaleron baryogenesis

4.1 Basic interaction terms

As already discussed in Sect. 3, the Pati–Salam symmetry survives till few 100 TeV scale playing an important role in the explanation of baryogenesis mechanism and neutron–antineutron oscillation. We need to know all the basic interactions using quarks and diquarks under the high scale Pati–Salam symmetry as well as under low scale SM-like interactions around the TeV scale in order to explain the above phenomena successfully. For that, we take a look at the decomposition of the Pati–Salam Higgs representation $\Delta _R(1,3, \overline{10})$ under the left–right symmetry group $\mathrm{SU}(2)_L \times \mathrm{SU}(2)_R \times U(1)_{B-L} \times \mathrm{SU}(3)_C$ and the SM gauge group $\mathrm{SU}(2)_L \times U(1)_{Y} \times \mathrm{SU}(3)_C$,

$$\begin{aligned}&\Delta (1, 3, \overline{10}) = \{ \Delta _{\ell \ell } (1, 3, -2, 1) \oplus \Delta _{q \ell } (1, 3, -2/3, 3^*)\nonumber \\&\quad \oplus \Delta _{qq} (1, 3, 2/3, 6^*) \quad \text{ under }\, \,\mathcal {G}_{2_L 2_R 1_{B-L} 3_C}, \end{aligned}$$

(17)

$$\begin{aligned}&\quad \Delta (1, 3, \overline{10}) \supset \Delta _{\nu \nu } (1,0,1) \oplus \Delta _{\nu e} (1,1,1) \oplus \Delta _{ee} (1,2,1) \nonumber \\&\qquad \qquad \qquad \qquad \,\,\, \oplus \Delta _{u \nu } (1,-2/3,3^*) \oplus \Delta _{d e} (1,1/3,3^*) \nonumber \\&\qquad \qquad \qquad \qquad \,\,\, \oplus \Delta _{u e} (1,1/3,3^*) \oplus \Delta _{d \nu } (1,1/3,3^*) \nonumber \\&\qquad \qquad \qquad \qquad \,\,\, \oplus \Delta _{u u} (1, -4/3,6^*) \oplus \Delta _{u d} (1,-1/3,6^*) \nonumber \\&\qquad \qquad \qquad \qquad \,\,\, \oplus \Delta _{d d} (1,2/3,6^*) \quad \text{ under }\quad \mathcal {G}_{2_L 1_Y 3_C}, \end{aligned}$$

(18)

where the electric charge is expressed in terms of the generators of the SM group and left–right symmetric group by

$$\begin{aligned} Q=T_{3L}+T_{3R}+{\frac{B-L}{2}}=T_{3L}+Y. \end{aligned}$$

(19)

Since the fields $\Delta _{\nu \nu } \text{(S) }$, $\Delta _{uu}$, $\Delta _{ud}$, $\Delta _{ud}$ and quark fields are mainly responsible for the nonzero baryon asymmetry and neutron–antineutron oscillation, we need to know the exact interactions among them. The desirable interaction Lagrangian for diquark Higgs scalars with the SM quarks at the TeV scale which will yield an observable neutron–antineutron oscillation and the post-sphaleron baryogenesis is

$$\begin{aligned}&\mathcal {L} \supset \frac{f_{ij}}{2}\, \Delta _{d d} d_i d_j \!+\! \frac{h_{ij}}{2}\, \Delta _{u u} u_i u_j \!+\! \frac{g_{ij}}{2\sqrt{2}} \Delta _{ud} (u_i d_j \!+\! d_i u_j) \nonumber \\&\qquad + \frac{\lambda }{2} \Delta _{\nu \nu } \Delta _{dd} \Delta _{ud} \Delta _{ud} + \frac{\lambda ^\prime }{2} \Delta _{\nu \nu } \Delta _{uu} \Delta _{dd} \Delta _{dd} + \text{ h.c. } \nonumber \\&\quad \subset F\, (\psi ^\mathrm{T}_{R a}\, C^{-1}\, \tau _2\, \varvec{\tau } \cdot \Delta ^\dagger _{ab}\, \psi _{R b} + \text{ L }\leftrightarrow \text{ R }) \nonumber \\&\qquad +\,\text{ h.c. }\,\, \text{ under } \ \mathcal {G}_{224}, \end{aligned}$$

(20)

where $F$, $f,h,g$ are the Majorana couplings and $\tau$ is the generator for the $\mathrm{SU}(2)$ group.

Within the SO(10) framework, the Yukawa couplings obey the boundary condition $f_{ij} = h_{ij} = g_{ij}$ in the $\mathrm{SU}(2)_L \times \mathrm{SU}(2)_R \times \mathrm{SU}(4)_C \times D$ limit and the same holds true for quartic Higgs couplings $\lambda =\lambda ^\prime$ as well. All fermions are right-handed (when the chiral projection on the operator is suppressed) and a fermion field under the high scale Pati–Salam symmetry $\mathcal {G}_{224}$ transforms as

$$\begin{aligned} \psi _{L,R}=\left( \begin{array}{cccc} u_1 &{} u_2 &{} u_3 &{} \nu \\ d_1 &{} d_2 &{} d_3 &{} e \end{array} \right) _{L,R}. \end{aligned}$$

(21)

The diquark Higgs scalars transforming under the SM gauge group $\mathrm{SU}(2)_L \times U(1)_Y \times \mathrm{SU}(3)_C$ are denoted by the quantum numbers

$$\begin{aligned}&\Delta _{\nu \nu } (1, 0, 1),\, \Delta _{u^c u^c} (1, -4/3, 6^*),\, \Delta _{d^c d^c} (1, 2/3, 6^*),\,\,\nonumber \\&\text{ and }\,\, \Delta _{u^c d^c} (1, -1/3, 6^*). \end{aligned}$$

(22)

It is clear from Eq. (20) that the Higgs field $\Delta _{\nu \nu } (1, 0, 1) \subset \Delta _R(1, 3, -2, 1) \subset (1, 3, \overline{10})$ is a neutral complex field. The breaking of $\mathcal {G}_{2113} \rightarrow \mathcal {G}_{213}$ is achieved by assigning a VEV to its neutral component $\Delta _{\nu \nu } \subset \Delta _{R} (1,0,-2,1)$. Its real component acquires a VEV in the ground state which can be represented as $\Delta _{\nu \nu } = v_{B-L} + \frac{1}{\sqrt{2}} (S_r + i \rho )$, while the field $\rho$ gets absorbed by the gauge boson corresponding to the gauge group $U(1)_{B-L}$. Therefore, the remaining real scalar field $S_r$ is indeed the physical Higgs particle which serves our purpose of explaining post-sphaleron baryogenesis and the neutron–antineutron oscillation.

4.2 General expression for CP asymmetry

Without loss of generality, if we consider the particle and antiparticle decay modes of $S_r$ ($S_r$ being its own antiparticle) i.e., $S_r \rightarrow u^c d^c u^c d^c d^c d^c$, which gives a change of baryon number $\Delta B_{(S_r \rightarrow 6 q^c)} = +2$ and $S_r \rightarrow \overline{u^c} \overline{d^c} \overline{u^c} \overline{d}^c \overline{d}^c \overline{d}^c$, which gives $\Delta B_{(S_r \rightarrow 6 \overline{q^c})} = -2$, then the $CP$-asymmetry in baryon number produced by these decays can be quantified as

$$\begin{aligned} \varepsilon _{CP}&= \frac{\Delta B_{(S_r \rightarrow 6 q^c)}\, \Gamma (S_r \rightarrow 6 q^c)}{\Gamma _\mathrm{tot}} \!+\!\frac{\Delta B_{(S_r \rightarrow 6 \overline{q^c})}\, \Gamma (S_r \!\rightarrow \! 6 \overline{q^c})}{\Gamma _\mathrm{tot}} ,\nonumber \\&= \frac{(+2)\, \Gamma (S_r \rightarrow 6 q^c) +(-2)\, \Gamma (S_r \rightarrow 6 \overline{q^c})}{\Gamma _\mathrm{tot}} = 2\, \frac{\Gamma -\bar{\Gamma }}{\Gamma _\mathrm{tot}}, \nonumber \\ \end{aligned}$$

(23)

where $\Gamma _\mathrm{tot} = \Gamma +\bar{\Gamma }$ is the total decay rate with $\Gamma \equiv \Gamma (S_r \rightarrow 6 q^c)$ and $\bar{\Gamma }\equiv \Gamma (S_r \rightarrow 6 \overline{q^c})$. It is evident from Eq. (23) that we need divergent partial decay rates for particle and antiparticle decays in order to produce the correct amount of baryon asymmetry and hence we should derive the general conditions under which $\Gamma$ and $\bar{\Gamma }$ can be different. It is worth to mention here that the other decay modes of $S_r$ have been ignored for simplicity by adjusting the corresponding couplings involved in the respective decay modes.

In generic situations where the theory is CPT-conserving, there can never be a difference between $\Gamma$ and $\bar{\Gamma }$ if one considers only the tree-level process depicted in Fig. 2, since $\Gamma =\bar{\Gamma }$ at tree level. It is found that the nonzero contribution to $\varepsilon _{CP}$ comes from the interference between the tree-level graph (shown in Fig. 2) and the one-loop corrections (shown in Fig. 3).

Fig. 2

Feynman diagram representing the decay of $S\rightarrow 6 q$ at tree level in order to explain post-sphaleron baryogenesis operative at the TeV scale. Since $S$ is a real scalar field, the decay mode $S \rightarrow 6 \overline{q}$ is possible by reversing the arrow direction of the quark field

Fig. 3

Feynman graphs of the one-loop vertex correction for $\Gamma (S_r \rightarrow 6 q^c)$

4.3 Constraints on post-sphaleron baryogenesis

Here we illustrate how post-sphaleron baryogenesis is slightly different from any other standard baryogenesis process. For post-sphaleron baryogenesis to be successful in explaining the required matter–antimatter asymmetry of our universe, few extra conditions must be satisfied by the model parameters along with the Sakharov conditions that say that particle interaction must (i) violate baryon number, $B$, (ii) violate $C$ and $CP$, and (iii) be out of thermal equilibrium. Firstly, the $S_r$ Higgs scalar should be lighter than the other members contained in the Pati–Salam multiplet $(1,3, \overline{10})$ i.e., the diquark Higgs scalars $\Delta _{qq}$ so that the baryon number conserving decays involving on-shell $\Delta _{qq}$ are kinematically forbidden. Secondly, the out of equilibrium baryon number violating decays should occur after the electroweak phase transition so that it will not be affected by the sphaleron processes which is proactive at $>$TeV scale. We point out that Ref. [9] neatly elaborates the mechanism of post-sphaleron baryogenesis.

4.4 Out of equilibrium condition

For effectively creating the baryon asymmetry of the universe via post-sphaleron baryogenesis, the decays of $\Gamma (S_r \rightarrow 6 q^c)$ should satisfy the out of equilibrium condition, which is described by $\Gamma _{S_r} \lesssim H(T)$ where $\Gamma = \Gamma (S_r \rightarrow 6 q^c)=\frac{36}{(2 \pi )^9} \frac{(\mathbf Tr [f^\dagger f])^3\, \lambda ^2 M^{13}_S}{6 M^{12}_\Delta }$ is the total decay width and $H \simeq 1.66 \sqrt{g^*_s}\, \frac{T^2}{M_\text {Pl}}$ is the Hubble parameter with the reduced Planck mass $M_\mathrm{Pl } \simeq 1.2 \times 10^{18}\,\mathrm GeV$ and $g^*_s$ is the number of relativistic degrees of freedom. In order to satisfy the out of equilibrium condition, we should have

$$\begin{aligned}&\Gamma _{S_r} \simeq H |_{(T=T_d)} \nonumber \\&\quad \Rightarrow \text{ T }_d =\left[ \frac{36\,\lambda ^2\,\left( \mathbf Tr [f^\dagger f] \right) ^3\,M_\mathrm{Pl } M^{13}_S}{ (2 \pi )^9\, 1.66\, g^{1/2}_{*} (6 M_\Delta )^{12} }\right] ^{1/2} \simeq 6.1 \nonumber \\&\qquad \qquad \qquad \times \left( \frac{M^{13}_S}{M^{12}_{\Delta }}\right) ^{1/2} \,\text{ GeV }^{1/2}. \end{aligned}$$

(24)

To illustrate the mechanism of post-sphaleron baryogenesis, we require extra fields $\Delta _{uu}$, $\Delta _{ud}$, and $\Delta _{dd}$ as color sextets and $\mathrm{SU}(2)_L$ singlet scalar bosons that couple to the right-handed quarks contained in the Pati–Salam multiplet $(1,3,\overline{10})$. For the set of model parameters $M_S=500$ GeV, $M_\Delta \simeq 1{,}000$ GeV, the decoupling temperature is found to be 2 GeV, which is well below the EW scale where the sphaleron has been decoupled. Hence, it is inferred from the above equation that the decay of $S$ goes out of equilibrium around $T\simeq M_S$. Below this temperature ($T < M_S$), the decay rate falls very rapidly as the temperature lowers down.

4.5 Estimation of net baryon asymmetry

Now we concentrate on estimating the CP asymmetry coming from the interference term between the tree level and the one-loop level diagrams for the decay of $S_r$, which is shown in Figs. 2 and 3, respectively. For a discussion of baryon number violation in the loop diagram and the necessary derivation of the interference diagram, interested readers may go through reference [9]. In the present work, we only check whether or not the representative set of model parameters provide the correct number for the required baryon asymmetry of the universe. Hence, without going deep into the derivation, we simply here write the calculated CP asymmetry for post-sphaleron baryogenesis via decay of $S_r$ with baryon number violating interactions:

$$\begin{aligned}&\varepsilon _\mathrm{wave} \simeq \frac{g^2}{64 \pi \text{ Tr }(f^\dagger f)} f_{j \alpha } V^*_{j \beta } f_{i \alpha } \delta _{i3} \frac{m_t m_j}{m^2_t-m^2_j} \nonumber \\&\qquad \qquad \,\,\times \sqrt{\left( 1-\frac{m^2_W}{m^2_t} +\frac{m^2_\beta }{m^2_t}\right) ^2- 4 \frac{m^2_\beta }{m^2_t}} \nonumber \\&\qquad \qquad \,\, \times \bigg [2 \left( 1-\frac{m^2_W}{m^2_t} +\frac{m^2_\beta }{m^2_t}\right) + \left( 1+\frac{m^2_\beta }{m^2_t}\right) \nonumber \\&\qquad \qquad \,\,\times \left( \frac{m^2_t}{m^2_W} +\frac{m^2_\beta }{m^2_t}-1\right) - 4 \frac{m^2_\beta }{m^2_W}\bigg ], \end{aligned}$$

(25)

$$\begin{aligned}&\varepsilon _\mathrm{vertex} \simeq \frac{g^2}{32 \pi \text{ Tr }(f^\dagger f)} f_{i \beta } V^*_{i \beta } f_{i \alpha } \delta _{i3} \frac{m_j m_\beta }{m^2_W} \nonumber \\&\qquad \qquad \,\,\times \left[ 1+\frac{9 m^2_W}{M^2_S}\text{ ln }\left( 1+\frac{M^2_S}{3 m^2_W} \right) \right] , \end{aligned}$$

(26)

$$\begin{aligned}&\varepsilon _\mathrm{CP} =\varepsilon _\mathrm{wave} + \varepsilon _\mathrm{vertex}. \end{aligned}$$

(27)

Here the expression in Eq. (25) represents the CP asymmetry coming from interference between the tree and one-loop self energy diagram while the expression in Eq. (26) represents the CP asymmetry due to interference of the tree and one-loop vertex diagram (see Ref. [9] for details). In the above expression, $V$ is the well-known CKM matrix in the quark sector, $i,j$ correspond to the up-quark indices $u,c,t$, while $\alpha ,\beta$ represent down-quark indices $d,s,b$. A sum over repeated indices (Einstein convention) is assumed here. The $\delta _{i3}$ is due to the fact that the CP asymmetry is nonzero only when we have a top quark in the final state (since only the CKM elements involving the third generation have a large imaginary part).

As mentioned earlier, the mechanism of post-sphaleron baryogenesis provides a natural explanation for the observed baryon asymmetry of our universe i.e., $\eta _B\simeq 10^{-10}$. Using $m_c=1.27$ GeV, $m_b=4.25$ GeV, $m_t=172$ GeV, the CKM mixing elements $V_\mathrm{CKM}$, and the Yukawa couplings relevant for color scalar particles in their allowed range, the CP asymmetry via the decay of $S_r$ through loop diagrams with the exchanges of $W^{\pm }$ bosons is estimated to be $10^{-8}$. A further dilution of the baryon asymmetry arises from the fact that $T_d \ll M_S$, since the decay of $S_r$ releases entropy into the universe. As a result the final baryon asymmetry, taking into account the dilution factor, becomes

$$\begin{aligned} \eta _B = \varepsilon _\mathrm{CP} \times \left( \frac{T_d}{M_S} \right) , \end{aligned}$$

(28)

where $T_d$ is the decoupling temperature of the color scalar and $M_S$ is the mass of the scalar. The condition $T_d/M_S \ge 10^{-2}$ otherwise leads to suppressed baryon asymmetry, which finally results in a baryon asymmetry in the range of $10^{-10}$. The scatter plot between the final baryon asymmetry including dilution factor ($\eta _B$) with this phase ($\delta _{i3}$) is shown in Fig. 4.

Fig. 4

Estimation of final baryon asymmetry in terms of CP asymmetry with overall phase $\delta$ contained in the CKM mixing matrix

5 Observable neutron–antineutron oscillation with TeV scale diquark Higgs scalars

5.1 Feynman amplitudes for neutron–antineutron oscillation

We consider the contributions arising only from the RH diquark Higgs fields having masses at the TeV scale while ignoring the contributions from LH diquark Higgs fields, since they have masses around the $\text{ eV }$ range. The Feynman diagrams contributing to the neutron–antineutron oscillation are shown in Fig. 5 (loop diagram), Fig. 6a and b. Our prime goal is to estimate the mixing time for this loop diagram, clarifying why we have suppressed other contributions within our model parameters.

Fig. 5

Loop contributions to neutron–antineutron oscillation in the post-sphaleron baryogenesis operative at the TeV scale

Fig. 6

Feynman diagrams contributing to neutron–antineutron oscillation. The figure in the left-panel involves two $\Delta _{u^c d^c}$ and one $\Delta _{d^c d^c}$ bosons whereas the figure in the right panel involves two $\Delta _{d^c d^c}$ and one $\Delta _{u^c u^c}$ bosons. The structure of the theory is such that these tree-level contributions are suppressed in the present work

There are two types of contributions to the $n$–$\overline{n}$ oscillation in the right-handed sector at loop level, (i) one involving one $u^c u^c$-type and two $d^c d^c$-type, (ii) another one involving one $d^c d^c$-type and two $u^c d^c$-type $\Delta$-bosons. The Feynman amplitude for the second type of contribution where one needs to change the two $b^c$ quarks to two $d^c$ quarks from the already generated effective operator $u^c d^c b^c u^c d^c b^c$ via a second order weak interactions (given in Fig. 5) can be written

$$\begin{aligned} \mathcal {A}^\mathrm{1-loop}_{n-\overline{n}}&\simeq \frac{(f_{ud})_{11} (f_{ud})_{13} (f_{dd})_{13}\, \lambda v_{B-L}}{M^4_{u^c d^c} M^2_{d^c d^c}}\, \frac{g^4\, V^2_{td}\, m^2_b\,m^2_t}{(16 \pi ^2)^2 M^4_{W_L}}\,\nonumber \\&\times \text{ log }\left( \frac{m^2_b}{M^2_{W_L}}\right) . \end{aligned}$$

(29)

The Feynman amplitude for tree-level processes shown in Fig. 6a and b (which are suppressed with the choice of our model parameters) can be written

$$\begin{aligned} \mathcal {A}^\mathrm{tree}_{n-\overline{n}}&= \mathcal {A}^\mathrm{(a)}_{n-\overline{n}}+ \mathcal {A}^\mathrm{(b)}_{n-\overline{n}} \nonumber \\&\simeq \frac{(f_{dd})_{11} (f_{ud})^2_{11}\, \lambda \, v_{B-L}}{M^4_{u^c d^c} M^2_{d^c d^c}} + \frac{(f_{uu})_{11} (f_{dd})^2_{11}\, \lambda \, v_{B-L}}{M^4_{d^c d^c} M^2_{u^c u^c}}.\nonumber \\ \end{aligned}$$

(30)

5.2 Prediction for neutron–antineutron mixing time $\tau _{n-\overline{n}}$

Before estimating the $n$–$\overline{n}$ oscillation mixing time one should carefully fix the input parameters in order to satisfy flavor changing neutral current (FCNC) constraints and to give correct amount of baryon asymmetry of the universe. For example, using a diquark sextet Higgs scalar mass around TeV scale, the corresponding Yukawa coupling $(f_{dd})_{11}\simeq 0.001$–$0.1$ along with the other allowed range of model parameters contradicts the FCNC constraints and hampers post-sphaleron baryogenesis even though it predicts a neutron–antineutron oscillation time (as shown in Fig. 6) within the experimental search limits. So this means that one has to choose the Majorana Yukawa coupling $f$ accordingly. Now we briefly discuss how this choice of $f$ can be achieved within the framework of SO(10) (elaborated in Ref. [19]).

It is found in Ref. [19] that all charged fermion masses and CKM mixing can be fitted well at the GUT scale within the framework of SO(10) with two kinds of structures; I) with a single Higgs representation $126_H$, II) with two Higgs representations $126_H$, $126_H^\prime$. As has been derived, structure I with Yukawa coupling $f_{126_H} =$ diag(0.0236, $-0.38$, 1.5) estimates $n$–$\bar{n}$ oscillation mixing time to be $10^9$ s, which does not serve our purpose. Rather we consider structure II where the dominant contribution to the $n$–$\overline{n}$ oscillation comes from the loop diagram while suppressing the tree-level contribution. This choice of having two Higgs bosons, $126_H$, $126_H^\prime$, not only fits the fermion masses at the GUT scale, but it also allows the RH neutrino Majorana mass and hence the corresponding Yukawa coupling $f_{126_H^\prime }$ as per our requirement. Due to the second Higgs representation ${126}^{\prime }$ with its Yukawa coupling $f^{\prime }$ to fermions we get $v_{\xi ^{\prime }}= 1$–$100$ MeV following the same procedure, provided all other components are at the GUT scale except $\xi ^{\prime }(2,2,15)$, which is at the intermediate scale $M_{\xi ^{\prime }}=10^{13}$–$10^{14}$ GeV. By treating the mass of $\xi (2,2,15) \subset 126$ to remain at its natural GUT-scale value, its induced VEV is negligible and precision unification with large GUT-scale value is unaffected except for phenomenologically inconsequential additional threshold effects. Then defining $F= f^{\prime }v_{\xi ^{\prime }}$ gives exactly the same fit to the GUT-scale fermion masses and mixings but now with the diagonal structure $f_i^{\prime } =(0.0236, -0.38, 1.5)$. But since $<\Delta _R^{\prime }>=0$ and only $\Delta _R \subset {126}_H$ with VEV $v_R$ is used to break $G_{2113}$, the coupling $f$ and hence $M_N$ are allowed to have any $3 \times 3$ form without any restriction. In order to suppress the tree-level contributions to the $n$–$\overline{n}$ oscillation as shown in Fig. 6 which otherwise causes problems in baryon asymmetry, we particularly choose the Majorana coupling $f_{dd}$ as per our requirement, i.e., ${f_{dd}}_{11}\le 10^{-5}$.

Using this particular choice of Yukawa couplings i.e., ${f_{dd}}_{11},\, {f_{dd}}_{22},\le 10^{-5}$, and others in the range of $0.001$–$1.0$, one can calculate the mixing time for neutron–antineutron oscillation as a function of the mass of the color Higgs scalar ($B-L$ breaking scale) as shown in Fig. 7 (Fig. 8).

Fig. 7

Estimation of $\tau _{n-\bar{n}}$ as a function of diquarks mass $M_{\Delta _{ud}}$

Fig. 8

Estimation of $\tau _{n-\bar{n}}$ as a function of $B-L$ breaking scale $v_{B-L}$, while keeping the other model parameters within their allowed range, consistent with the mechanism of post-sphaleron baryogenesis

The $n$–$\bar{n}$ amplitude can be translated into the $n$–$\overline{n}$ oscillation time:

$$\begin{aligned} \tau _{n-\bar{n}}^{-1} = {\delta m}_{n-\bar{n}} = C_{\mathrm{\small QCD}} (\mu _\Delta , 1\,\text{ GeV }) |A_{n-\bar{n}}^\mathrm{1-loop}| \end{aligned}$$

(31)

with $C_{\mathrm{\small QCD}} (\mu _\Delta , 1\,\text{ GeV }) = 0.1 \text{ GeV }^6$ as used in Ref. [9]. The estimated $n$–$\bar{n}$ oscillation time for various choices of model parameters i.e., ${f_{ud}}_{11} \le {10}^{-5}$, $M_S = (100-5{,}000)GeV$, $B-L$ breaking scale from (3–5) TeV and the masses of ${M_\Delta }_{ud/dd}$ between $M_S$ and $V_{B-L}$, $\lambda \simeq 0.01$–$1.0$ is presented in Table 2.

Table 2 Numerical estimation of neutrino–antineutrino oscillation time

5.3 Coupling unification including diquarks at the TeV scale

It is evident that the post-sphaleron baryogenesis and neutron–antineutron oscillation phenomena require the existence of color Higgs scalars, having masses around TeV scale. In this subsection, we intend to examine whether unification of the gauge couplings is still possible after the addition of extra color scalars $\Delta _{ud}$, $\Delta _{dd}$, $\Delta _{uu}$ to the existing particle content as noted in Sect. 3, by studying their respective renormalization group equations. The one-loop beta coefficients derived for the present model along with their gauge symmetry groups, range of mass scales, and spectrum of Higgs scalars necessary for gauge coupling unification to explain the TeV scale post-sphaleron baryogenesis and neutron–antineutron oscillation are given by

$$\begin{aligned}&{\mathrm {(i)}\, \mu =\mathbf{M}_\mathbf{Z} (\mathbf{91.817}~\text{ GeV }) - \mathbf{M}_\mathbf{T} (\mathbf{1}~\text{ TeV })}: \nonumber \\&\qquad \mathcal {G} = \mathcal {G}_{2_L 1_Y 3_C}\equiv \text{ SM }\nonumber \\&\qquad \text{ Higgs: }\, \Phi (2, 1/2, 1)_{10}:\quad \varvec{ a_i}=(-19/6,\, 41/10,\, -7); \end{aligned}$$

(32)

$$\begin{aligned}&{\mathrm {(ii)}\, \mu =\mathbf{M}_\mathbf{T}(\mathbf{1}\, \text{ TeV }) - \mathbf{M}_{\mathbf{B}-\mathbf{L}}(\mathbf{3}\, \text{ TeV }):} \quad \mathcal {G} = \mathcal {G}_{2_L 1_Y 3_C}, \nonumber \\&\qquad \text{ Higgs: }\, \Phi (2,1/2,1)_{10} \oplus S(1, 0, 1)_{126}\subset \Delta _R \nonumber \\&\qquad \qquad \quad \,\,\, \oplus \Delta _{u^c d^c}(1, -1/3, 6^*)_{126} \oplus \Delta _{d^c d^c} (1, 2/3, 6^*)_{126}\, \nonumber \\&\qquad \qquad \quad \,\,\, \oplus \Delta _{u^c u^c} (1, -4/3, 6^*)_{126}: \nonumber \\&\qquad \varvec{a_i} = (-19/6,\, 207/30,\, -27/6),\end{aligned}$$

(33)

$$\begin{aligned}&{\mathrm {(iii)}\, \mu \!=\!\mathbf{M}_{\mathbf{B-L}}(\mathbf{3}\, \text{ TeV }) \!-\! \mathbf{M}_\mathbf{\Omega }(\mathbf{10}\, \text{ TeV }):} \quad \mathcal {G} \!=\! \mathcal {G}_{2_L 1_R 1_{B-L} 3_C},\nonumber \\&\qquad \text{ Higgs: }\,\Phi _1(2, 1/2, 0, 1)_{10} \oplus \Phi _2(2, -1/2, 0, 1)_{10^\prime }\nonumber \\&\qquad \qquad \quad \,\,\, \oplus \Delta _R(1, 1, -1, 1)_{126} \oplus \chi _R(1, 1/2, -1/2, 1)_{16},\, \nonumber \\&\qquad \qquad \quad \,\,\,\oplus \Delta _{u^c d^c}(1, 1, -2/3, 6^*)_{126}\nonumber \\&\qquad \qquad \quad \,\,\, \oplus \Delta _{d^c d^c} (1, 0, -2/3, 6^*)_{126}\nonumber \\&\qquad \qquad \quad \,\,\,\oplus \Delta _{u^c u^c} (1, 0, -2/3, 6^*)_{126}: \nonumber \\&\qquad \varvec{a_i} = (-3,\, 35/4,\, 45/8, -27/6), \end{aligned}$$

(34)

$$\begin{aligned}&{\mathrm {(iv)}\, \mu =\mathbf{M}_\mathbf{\Omega }(10^{4}\, \text{ GeV }) - \mathbf{M}_\mathbf{C}(\mathbf{10}^{\mathbf{5}}-\mathbf{10}^{\mathbf{6}}\, \text{ GeV }):}\nonumber \\&\qquad \mathcal {G} = \mathcal {G}_{2_L 2_R 1_{B-L} 3_C}, \nonumber \\&\qquad \text{ Higgs: }\,\Phi _1(2, 2, 0, 1)_{10} \oplus \Phi _2(2, 2, 0, 1)_{10^\prime } \nonumber \\&\qquad \qquad \quad \,\,\, \oplus \Delta _R(1, 3, -1, 1)_{126}\oplus \chi _R(1, 2, -1/2, 1)_{16},\, \nonumber \\&\qquad \qquad \quad \,\,\, \oplus \Delta _{u^c d^c} (1, 3, -2/3, 6^*)_{126} \nonumber \\&\qquad \qquad \quad \,\,\, \oplus \Delta _{d^c d^c}(1, 3, -2/3, 6^*)_{126} \nonumber \\&\qquad \qquad \quad \,\,\, \oplus \Delta _{u^c u^c} (1, 3, -2/3, 6^*)_{126} \oplus \Omega _R(1, 3, 0, 1)_{210} \nonumber \\&\qquad \varvec{a_i} = (-8/3,\, 4/3,\, 55/4, -2). \end{aligned}$$

(35)

In analogy to the above discussion, we have two scenarios; one without bitriplet and another with bitriplet Higgs scalar (3, 3, 1) under the Pati–Salam group $\mathrm{SU}(2)_L \times \mathrm{SU}(2)_R \times \mathrm{SU}(4)_C$, while its effect has been included from $M_C$ onwards to the unification scale $M_U$. Accordingly, we have estimated the one-loop beta coefficients for these two scenarios:

$$\begin{aligned}&{\mathrm {(v)} \, \mu =\mathbf{M}_\mathbf{C} - \mathbf{M}_\mathbf{\xi }:} \quad \mathcal {G} = \mathcal {G}_{2_L 2_R 4_C}, \nonumber \\&\qquad \text{ Higgs: }\,\Phi _1(2, 2, 1)_{10} \oplus \Phi _2(2, 2, 1)_{10^\prime } \oplus \Delta _R(1, 3, \overline{10})_{126}\nonumber \\&\qquad \qquad \quad \,\,\, \oplus \chi _R(1, 2, \overline{4})_{16} \oplus \Omega _R(1, 3, 15)_{210},\nonumber \\&\qquad \varvec{a_i} = (-8/3,\, 29/3,\, -14/3), \end{aligned}$$

(36)

$$\begin{aligned}&{\mathrm {(vi)} \, \mu =\mathbf{M}_\mathbf{\xi } - \mathbf{M}_\mathbf{P}:} \quad \mathcal {G} = \mathcal {G}_{2_L 2_R 4_C},\nonumber \\&\qquad \text{ Higgs: }\,\Phi _1(2, 2, 1)_{10},\, \Phi _2(2, 2, 1)_{10^\prime },\,\Delta _R(1, 3, \overline{10})_{126},\,\nonumber \\&\qquad \qquad \quad \,\,\, \chi _R(1, 2, \overline{4})_{16}, \Omega _R(1, 3, 15)_{210} + \xi (2, 2, 15)_{126^\prime },\nonumber \\&\qquad \varvec{ a_i} = (7/3,\, 44/3,\, 2/3), \end{aligned}$$

(37)

$$\begin{aligned}&{\mathrm {(vii)}\, \mu =\mathbf{M}_\mathbf{P} - \mathbf{M}_\mathbf{U}:} \quad \mathcal {G} = \mathcal {G}_{2_L 2_R 4_C},\nonumber \\&\qquad \text{ Higgs: }\,\Phi _1(2, 2, 1)_{10},\, \Phi _2(2, 2, 1)_{10^\prime },\, \Delta _R(1, 3, \overline{10})_{126},\,\nonumber \\&\qquad \qquad \quad \,\,\, \Delta _L(3, 1, 10)_{126},\, \chi _R(1, 2, \overline{4})_{16},\, \chi _L(1, 2, 4)_{16},\, \nonumber \\&\qquad \qquad \quad \,\,\, \Omega _R(1, 3, 15)_{210},\, \Omega _L(3, 1, 15)_{210},\, \xi (2, 2, 15)_{126^\prime },\,\nonumber \\&\qquad \qquad \quad \,\,\, \Sigma ^\prime (1,1,15)_{210},\, \nonumber \\&\qquad \varvec{a_i} = (44/3,\, 44/3,\, 6). \end{aligned}$$

(38)

The gauge coupling unification after the addition of extra color sextet scalars particles is shown in Fig. 9 with the allowed mass scales desirable for our model predictions,

$$\begin{aligned}&M_{B-L}= 4-7~\mathrm{TeV},\, M_\Omega \!=\! 10~\mathrm{TeV},\, M_C \!=\! 10^{5}\!-\!10^{6}\, \text{ GeV },\nonumber \\&M_P\simeq 10^{14.65}~\mathrm{GeV\,\, and\,\,} M_\mathrm{U}\simeq 10^{16.25}~\mathrm{GeV}. \end{aligned}$$

(39)

Fig. 9

Coupling unification for the present model where $\Delta _{u^c d^c}$, $\Delta _{d^c d^c}$, and $\Delta _{u^c u^c}$ have been included at the TeV scale keeping in mind that these particle mediate neutron–antineutron oscillation and baryon asymmetry and including $\xi (2,2,15)$ around $10^{12}$ GeV in order to fit the fermions masses at the GUT scale

6 Viability of the model

As already known, the lepton flavor and lepton number violating dilepton signals can be probed from the production of heavy RH Majorana neutrinos via $p+p \rightarrow W^\pm _R \rightarrow \ell ^\pm _\alpha +N_R$, from which $N_R$ may further decay into $N_R \rightarrow W^*_R \rightarrow \ell ^{\mp }_\beta = 2j$. This process, being the main channel for $N_R$ production via on-shell $Z_R$ production and $W_R$ fusion, needs to be verified at LHC and our model suits the purpose, since we have $W_R$, $Z_R$ gauge bosons and scalar diquarks at the TeV scale. A more pleasant situation is that the model, though non-supersymmetric, predicts similar branching ratios as in supersymmetric models for LFV processes like $\mu \rightarrow e\gamma$, $\tau \rightarrow \mu \gamma$, and $\tau \rightarrow e\gamma$. The predicted branching ratios for these LFV decays, being closer to the current experimental search limits can be used to verify the left–right framework in this model. Moreover, the estimated neutron–antineutron oscillation mixing time, gauge coupling unification and proton life time in the model stay in the range of ongoing search experiments.

Besides all these points, the model can also predict a number of verifiable new physical quantities like (i) a new non-standard contribution to $0\nu 2\beta$ rate in the $W_L$–$W_L$ channel, (ii) contributions to branching ratios of lepton flavor violating (LFV) decays, (iii) leptonic CP-violation due to non-unitarity effects, and (iv) experimentally verifiable proton decay modes such as $p\rightarrow e^+\pi ^0$, provided the gauged inverse seesaw mechanism is found to be operative. We find it appropriate to mention here that these physical quantities were also discussed in a recent work [19], but in that model the asymmetric left–right gauge symmetry was incorporated at $\simeq 10$ TeV.

7 Conclusion

We have closely studied the mechanism of post-sphaleron baryogenesis, which can potentially explain the matter–antimatter asymmetry of the present universe, by analyzing the basic interactions using quarks and diquark Higgs scalars under high scale Pati–Salam symmetry and low scale SM-like interactions at the TeV scale. The study estimates the total baryon asymmetry to be $\eta _B \simeq \mathcal{O}(10^{-10})$ and neutron–antineutron oscillation with mixing time to be $\tau _{n-\bar{n}}\simeq \mathcal{O}(10^{-10}$–$10^{-8})$ s, which may be accessible at ongoing search experiments. We have made a humble attempt to embed the framework of PSB in a non-SUSY SO(10) model with Pati–Salam symmetry as a low scale intermediate breaking step where we have shown a strong interlink between post-sphaleron baryogenesis and neutron–antineutron oscillation operative at the TeV scale and we laid out a novel mechanism of inducing the required CP asymmetry via the SM $W_L^{\pm }$ loops.

More essentially, we have embedded the TeV scale LR model within the framework of the SO(10) model, where the predicted mass for light neutrinos matches with the neutrino oscillation data. Our calculations indicate that the TeV scale masses of $W^\pm _R$ and heavy RH neutrinos can also give dominant non-standard contributions to neutrinoless double beta decay, which may sound crucial to the experimentalists. Some more good features of the model are an explanation of the nonzero light neutrino masses via the extended/inverse seesaw mechanism, a new non-standard contribution to neutrinoless double beta decay, and leptonic CP-violation from non-unitary effects.