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

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 $SU(2)_L \times SU(2)_{R} \times SU(4)_C$ 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 $\tau_{n-\bar{n}} \simeq 10^{8}-10^{10}\,\mbox{secs}$ with the model parameters which is within the reach of forthcoming experiments. Other novel features of the model includes low scale right-handed $W^{\pm}_R$, $Z_R$ gauge bosons, explanation for neutrino oscillation data via gauged inverse (or extended) seesaw mechanism and most importantly TeV scale color sextet scalar particles responsible for observable $n-\bar{n}$ oscillation which can be accessible to LHC. We also look after gauge coupling unification and estimation of proton life-time with and without the addition of color sextet scalars.


I. INTRODUCTION
The Standard Model (SM) of particle physics has given us enough reasons to look beyond its framework for dealing with issues like tiny neutrino masses, matter-antimatter asymmetry of the present universe, Dark matter and Dark energy, 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. The WMAP satellite data [1,2], when combined with large scale structures (LSS) data, gives the baryon asymmetry of the universe to be η CMB ≃ (6.3 ± 0.3) × 10 −10 while an independent measurement of baryon asymmetry carried out by BBN [3] yields η BBN ≃ (3.4 − 6.9) × 10 −10 . Two compelling mechanisms namely Leptogenesis [4] and Weak scale baryogenesis [5] have been prime tools for explaining baryon asymmetry of the universe. In leptogenesis the desired lepton asymmetry is created by the lepton number violating 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].
An inadequate knowledge about 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 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 under standard model having mass around few hundreds of GeV and a high dimensional baryon number violating coupling [8][9][10], where the Yukawa coupling(s) of the scalar(s) act as the source of 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 A detail study of the literatures [11][12][13][14][15][16][17][18] gives an idea about many intriguing features of the SO(10) grand unified theory (including both non-SUSY and SUSY). One of these features is that when left-right gauge symmetry appears as an intermediate symmetry breaking step in a novel symmetry breaking chain, then 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 SU(2) L × SU (2)  We intend to discuss TeV scale post-sphaleron baryogenesis, neutron-antineutron oscillation having 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 non-zero light neutrino masses can be accommodated via gauged extended inverse seesaw mechanism along with TeV scale W R , Z ′ 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 breifly clarify the point as follows.
In non-SUSY SO(10), the type I seesaw [20] contribution to neutrino mass is given by as an alternative way, emphasizing on its verifiability at LHC, inverse seesaw mechanism [21,22] has been proposed, with an extra SO(10) fermion singlet S (in addition to the existing fermion content of SO(10)), with light neutrino mass formula where M is the N − S mixing matrix and µ is the small lepton number violating mass term for sterile neutrino S. The above relation can be recasted as Hence, sub-eV mass for light neutrinos are consistent with is a generic predictions of high scale Pati-Salam symmetry and compatible with low righthanded symmetry breaking scale (M R ) since inverse seesaw formula is independent of M R .
We have utilised 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 sacle extended inverse seesaw has been omitted.
Here we sketch out the complete work of our paper. In Sec.II, we briefly discuss non-SUSY SO(10) GUT with a novel symmetry breaking chain, having G 2213 and G 224 as intermediate symmetry breaking steps. In Sec.III we show how gauge coupling unification is achieved in our model. In Sec.IV we discuss the TeV scale post-sphaleron baryogenesis and embed it within the novel chain of non-SUSY SO(10) model with the self-consistent model parameters.
In Sec.V, we estimate the mixing time for neutron-antineutron oscillation. In Sec.VI, we present an idea how low mass scales for RH Majorana neutrino as well as right-handed gauge bosons W R , Z ′ are allowed in the model, while explaining light neutrino masses via gauged extended seesaw mechanism. In Sec.VII we conclude our work with results and summary including a note on viability of the model at LHC.

II. 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] SO(10) The chain breaks in a sequence, where SO(10) first breaks down to G 224D , (g 2L = g 2R ) after the Higgs representation (1, 1, 1) ⊂ {54} H is given a VEV, then the spontaneous breakdown As we have pointed earlier, due to D-parity mechanism, the right-handed triplet Higgs For simplicity, we consider only the one-loop renormalization group equations(RGEs) for gauge coupling evolution which can be written as where, t = ln(µ), α i = g 2 i /(4π) is the fine structure constant, and a i a i a i is the one-loop beta coefficients derived for the the corresponding i th gauge group for which coupling evolution has to be determined. Using the input parameters, electroweak mixing angle where Higgs content for the model and corresponding one-loop beta coefficients a i a i a i The Higgs contents for the model used in different ranges of mass scales under respective gauge symmetries (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 TeV scale are as follows, Here we find two categories of Higgs spectrum; Model-I having Higgs spectrum as given in eqn. ( 6) and eqn. ( 7)  The one-loop beta coefficients are found to be the same for both the models at mass scale whereas, they differ at Pati-Salam scale M C to the Unification scale M U as shown in Table.I.
The gauge coupling unification for this work is shown in Fig. 1 with the allowed mass scales desirable for our model predictions,   The decay rate for the gauge boson mediated proton decay in the channel p → π 0 e + including strong and electroweak renormalization effects on the d = 6 operator starting from the GUT scale to the proton mass (i.e, 1 GeV) [26,27] comes out to be In the eq. (10) short-distance renormalization factor in the left (right) sectors, and V ud is the (1, 1) element of V CKM for quark mixings.
the proton life time can be expressed as where F q ≃ 7.6.
Short distance enhancement factor A SL extrapolated from GUT scale to 1 GeV: For estimating proton decay rate in the channel p → e + π 0 having dimension-6 operator, one needs to extrapolate the operator from the GUT scale physics to the low energy physics at the scale of m p = 1 GeV [28][29][30]. With the particular symmetry breaking chain allowed in the non-SUSY SO(10) model (following the ref. [30]), the whole energy range can be separated into following parts V. from standard model to 1 GeV .
As discussed in refs. [28][29][30], the enhancement factor below SM for the LLLL operator is where, n f denotes the number of quark flavors at the particular energy scale of our interest.
Neglecting the effect due to α 2L and α Y since their contributions are suppressed as compared to the strong coupling effect α s , this enhancement factor can be expressed in a more explicit manner as Since the model considered here is non-supersymmetric version of SO (10) GUT, all other enhancement factors can be written in the same way as Hence, the complete short distance enhancement renormalization factor for this d = 6 proton decay operator is found to be We have earnestly followed the prescription given in ref. [28,29] 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 × 10 18.5 GeV for Model-I and M U = 10 15.8 GeV for Model-II. We have estimated the factor A R = A L · A SL , approximately, to be 4.36 with the value of long distance renormalization factor A L = 1.25 which is the same for both the models.
With these input parameters, the model under consideration predicts the proton life time to be τ (p → e + π 0 ) = 2.6 × 10 34 yrs that is closer to the latest Super-Kamiokande experimental bound [31,32] τ (p → e + π 0 ) SK,2011 > 8.2 × 10 33 yrs , and ably supports planned experiments that can reach a bound [33] τ (p → e + π 0 ) HK, where the electric charge is expressed in terms of the generators of the SM group and leftright symmetric group as, Since the fields ∆ νν (S), ∆ uu , ∆ ud , ∆ ud and quark fields are mainly responsible for non-zero 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 TeV scale which will yield observable neutron-antineutron oscillation and post-sphaleron baryogenesis is where F , f, h, g are the Majorana couplings and τ is the generator for SU(2) group.
Within the SO(10) framework, the Yukawa couplings obey the boundary condition, f ij = h ij = g ij in the SU(2) L × SU(2) R × SU(4) C × D limit and the same holds true for quartic Higgs couplings λ = λ ′ as well. All fermions are right-handed (when chiral projection on the operator is suppressed) and a fermion field under the high scale Pati-Salam symmetry G 224 transforms as, where Γ tot = Γ +Γ is the total decay rate with Γ ≡ Γ(S r → 6q c ) andΓ ≡ Γ(S r → 6q c ). It is evident from eqn (23) that we need divergent partial decay rates for particle and antiparticle decays in order to produce correct amount of baryon asymmetry and hence we should derive the general conditions under which Γ andΓ 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 Γ andΓ if one considers only the tree-level process depicted in Fig. 2 since Γ =Γ at tree level. It is found that the nonzero contribution to ε CP comes from the interference between the tree-level graph (shown in 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 make it a point here that ref. [9] neatly elaborates the mechanism of post-sphaleron baryogenesis.

D. Out of equilibrium condition
For effectively creating the baryon asymmetry of the universe via post-Sphaleron baryogenesis, the decays of Γ(S r → 6q c ) should satisfy the out of equilibrium condition, which M Pl , is the Hubble parameter with the reduced Planck mass M Pl ≃ 1.2 × 10 18 GeV and g * s is the number of relativistic degrees of freedom. In order to satisfy the out of equilibrium condition, we should have To illustrate the mechanism of post-sphaleron baryogenesis, we require extra fields ∆ uu , ∆ ud and ∆ dd as color sextets and SU(2) L singlet scalar bosons that couple to the righthanded quarks contained in the Pati-Salam multiplet (1,3,10). For set of model parameters M S = 500 GeV, M ∆ ≃ 1000 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 ≃ M S . Below this temperature (T < M S ), the decay rate falls very rapidly as the temperature cools down.

E. 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 Fig.2 and Fig.3 respectively. For discussion on baryon number violation in the loop diagram and 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 note here down, the calculated CP-asymmetry for post-sphaleron baryogenesis via decay of S r with baryon number violating interactions.
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 α, β represent to down-quark indices d, s, b.
Sum over repeated indices (Einstein convention) is implicitly assumed here. The δ i3 is due to the fact that the CP asymmetry is non-zero only when we have a top quark in the final state (since only the CKM elements involving third generation have a large imaginary part).
As mentioned earlier,the mechanism of post-sphaleron baryogenesis provides a natural 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 ≥ 10 −2 , otherwise leads to suppressed baryon asymmetry, which finally results a baryon asymmetry in the range of 10 −10 . The scatter plot between the final baryon asymmetry including dilution factor (η B ) with this phase (δ i3 ) is shown in Fig. 4.  Fig. 5) can be written as, And, the Feynman amplitude for tree level processes shown in Fig. 6(a) and Fig. 6(b) (which are suppressed with the choice of our model parameters), can be written as,  Fig. 7 (Fig. 8).
The n −n amplitude can be translated into the n − n oscillation time as,  Table.II.
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 SU(2) L × SU(2) R × 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 as  shown in Fig. 9 with the allowed mass scales desirable for our model predictions,

VI. 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 neutrino via p + p → W ± R → ℓ ± α + N R , from which N R can be further decayed into N R → W * R → ℓ ∓ β = 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 TeV scale. A more pleasant situation is that the model, though nonsupersymmetric, predicts similar branching ratios as in supersymmetric models for LFV processes like µ → eγ, τ → µγ, and τ → eγ. And the predicted branching ratios for these Besides all these points, the model can also predict a number of verifiable new physical quantities like (i) new non-standard contribution to 0ν2β 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, (iv) experimentally verifiable proton decay modes such as p → e + π 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 ≃ 10 TeV.

VII. CONCLUSION
We have closely studied the mechanism of post-sphaleron baryogenesis, that can potentially explain matter-antimatter asymmetry of the present universe, by analyzing the basic