Theorem on vanishing contributions to $\sin^2\theta_W$ and intermediate mass scale in Grand Unified Theories with trinification symmetry

We prove that the values of the electroweak mixing angle $\sin^2\theta_W$ and intermediate mass scale $M_I$ have vanishing contributions due to one-loop, two-loop and gravitational corrections in Grand Unified Theories which accommodate an intermediate trinification symmetry ($G_{333D}$) invoked with spontaneous D-parity mechanism operative at mass scale greater than $M_I$. The proof of theorem is robust and we verify the results numerically using supersymmetric as well as non-supersymmetric version of $E_6$-GUT.


I. INTRODUCTION
The profound discovery of Higgs boson at Large Hadron Collider glorifies the success of the Standard Model (SM) and the importance of spontaneous symmetry breaking. The final step of spontaneous symmetry breaking for all Grand Unified Theories (GUTs) have to proceed through the SM gauge group and to reproduce all SM observables. The theoretical and experimental determination of known SM observables including the electroweak mixing angle sin 2 θ W is more meaningful if the theoretical predictions from GUT models are consistent with these experimentally measured values. Towards achieving this goal, many theoretical GUTs models-like SU (5) [1], SO(10) [2][3][4][5][6][7][8][9][10], E 6 [11][12][13][14][15] etc with or without superysmmetry (SUSY), have been proposed as prospective theories of nature awaiting experimental feasibility. With the development of String Theories [16][17][18] the attention is now more pointing towards re-investigating the Exceptional Group E 6 as a more plausible choice, as it accommodates most of the nice features of the well known GUT groups SU (5), SU (6) and SO(10) through it's embedding. The E 6 GUT, started in the late 1970's(just after SO(10) GUT models) is primarily introduced to explain the hierarchy problem, the mysterious anomaly cancellations of the Standard Model and the quantization of electric charge etc. Many of the pressing issues of SM can be adequately addressed here, with key predictions on achieving gauge unification, stability of proton, strong CP problem, predictions of dark matter candidature, non-zero neutrino masses and mixing etc. Some of the above phenomenological aspects have been discussed for E 6 gauge model via its maximal subgroups SO(10) ⊗ U (1) [19] and SU (3) C ⊗ SU (3) L ⊗ SU (3) R [20][21][22][23][24][25][26][27].
. It has been shown earlier the vanishing contributions [35,[38][39][40] due to one-loop, two-loop, gravitational as well as threshold in case of intermediate Pati-Salam symmetry [2,4,6] with D-parity [41,42]. However, all these corrections survive for the unification mass scale M U as well as GUT coupling constant.
This paper is organised as follows. In Section-II, we propose a general theorem alongwith its proof for vanishing corrections due to one-loop, two-loop and gravitational effect for the intermediate mass scale M I and the electroweak mixing angle sin 2 θ W in a class of GUT models with intermediate G 333D symmetry. We verify the theorem numerically in E 6 GUT model with and without suspersymmetry in the consequent Section-III and IV respectively. The last Section is devoted to the concluding remarks alongwith a comment on threshold corrections.

II. THEOREM AND ITS PROOF
Here, D stands for D-parity known as discrete left-right symmetry. We mainly use E 6 ≡ G GUT in proving the theorem numerically, whereas G GUT can be any GUT model which can accommodate trinification gauge symmetry as highest intermediate symmetry. Now to evaluate the loop-effects, we follow the standard procedure by using the renormalization group equations (RGEs) [43]. The evolution of gauge coupling constant g i (µ) of an intermediate gauge symmetry G i occurring in G GUT → G I → G SM is given by, After simplification, the known analytic formula derived for one loop as well as two loop RGEs for inverse coupling constant valid from µ to the intermediate scale M (M can be of any scale > µ where new theory appears) as, Here, α i = g 2 i /(4π), where g i being the coupling constant for the i th gauge group.
is the one (two)-loop beta coefficients in the mass range µ − M .
The RGEs for evolution of gauge coupling constant in between the mass scale M Z and M I are given by with i = 1Y, 2L, 3C. However, there is an additional gravitational corrections from M I and M U scale emerging from non-renormalizable higher dimensional operators. The corresponding RGEs for evolution of gauge coupling constants are, where, i = 3C, 3L, 3R. The second and third term in the RHS of both eqns. (4) and (5) represent the one-loop effects and two-loop effects respectively while the fourth term of equation (5) is for gravitational contributions.
is the one-loop beta coefficients derived for particle spectrum within mass range M Z −M I (M I −M U ). In the above mass range,the two-loop contributions are respectively, as given by is the two-loop beta coefficients derived for particle spectrum within mass range (5) is to modify the GUT scale boundary condition at µ = M U as, Here ǫ i is the parameter which induces the gravitational correction.
Thus the gravitational contributions ∆ NRO can be put in simplified form, where α G is the GUT-coupling constant. Here it may be noted that the gravitational contribution is due to the non-renormalizable term [44] given by where Φ Φ Φ is the Higgs field responsible for breaking of the GUT symmetry G to G 333D at the mass scale M U . The gravitational correction parameter ǫ i depends on η, VEV of φ and M G . Here it is noteworthy to mention that the non-renormalizable term in the Lagrangian may arise as effects of quantum gravity [32,44] or as a result of compactification of extra dimensions [31]. Thus in eq.(10), the scale M G ≃ M P l ≃ 10 19 GeV, if it is due to quantum gravity effect. However, if it emerges as a result of compactification of extra dimensions then M G ≤ M P l [31].
Analytical proof for sin 2 θ W :-Using the standard procedure, the analytic formula for sin 2 θ W due to contributions arising from one-loop, two-loop and gravitational correction is given by The first, second and third squared-bracketed terms in (11) are due to one-loop, two-loop and gravitational effects repspectively. Now in order to prove the theorem, we need to focus on each term separately.
• The one-loop contributions to the electroweak mixing angle sin 2 θ W is given by Here the parameters A I , B I , A U and B U (as can be inferred from appendix) contains the one-loop effect at µ > M I . However in the first term A U cancelled out in both numerator as well as in denominator which leads to 3/8 only. Second term completely vanishes as as has been shown in appendix), here we may note that B I has only one-loop effect at µ < M I . Thus the modified expression for sin 2 θ W is given by which shows that sin 2 θ W has vanishing contributions from one-loop effects emerging form µ > M I .
• The two-loop contributions to the electroweak mixing angle is given by Now using the similar analysis as in case of oneloop, we can see that the above expression reduces to . This is quite natural in all GUT models with conserved D-parity. Thus sin 2 θ W has vanishing contributions from two-loop effects emerging form µ > M I .
• The important gravitational contribution to sin 2 θ W is as follows, Here we may note that for B U = 0, the second term vanishes identically. In the first term A U cancelled out from both numerator and denominator. Again ) vanishes due to D-parity conservation. As a result of this, there is no effect of gravitational corrections at all to the electroweak mixing angle sin 2 θ W i.e, sin 2 θ W N RO = 0.

Analytical proof for intermediate mass scale M I :-
The analytic formula for intermediate mass scale M I due to one-loop, two-loop and gravitational corrections is read as, Now in order to prove the theorem for intermediate mass scale, we study the expression term by term.
• One loop contribution to intermediate symmetry breaking scale read as, Using the same logic as has been discussed in case of sin 2 θ W , for B U = 0, the above expression reduces to Here the parameter D W D W D W contains experimentally measured values like sin 2 θ W and α em . B I has been already shown to be independent of one-loop effects at µ > M I , hence the proof of the theorem is natural to show the intermediate mass scale M I has vanishing contributions due to one-loop effect emerging from mass scale µ > M I .
• Similarly the two loop effect on M I is exhibited in the 2nd term of eq.(12) given as Here for B U = 0 the expression reduces to It has already been demonstrated that K Θ Θ ΘΘ ′ Θ ′ Θ ′ and B I are independent of two-loop contributions emerging from mass scale µ > M I .
• The gravitational correction for M I as noted in the 3rd term of eq.(12) is as follows, Here again for B U = 0 and E 0 = 0, we see that Thus the gravitational corrections has no effect on the intermediate mass scale.
The proof of the theorem is independent of the choice of particle content and thus, the proof is robust. It can be generalized to both supersymmetric as well as non-supersymmetric version of Grand Unified Theories that accommodates intermediate trinification symmetry We shall now show the stability of electroweak mixing angle sin 2 θ W and intermediate mass scale M I in specific E 6 GUT models with or without supersymmetry by numerical analysis.

III. PREDICTIONS IN SUSY E6 GUT
We consider first the supersymmetric version of E 6 GUT with intermediate trinification symmetry as, (13) At first stage, SUSY-E 6 is broken down to the trinification gauge symmetry G 333D at unification scale M U by assigning non-zero vacuum expectation value (VEV) in the singlet direction (1, 1, 1) ⊂ 650 H which transforms evenly under D-parity. This singlet scalar preserves the symmetry between left and right handed Higgs field. The next stage of symmetry breaking from G 333D to G SM is done by non-zero VEV of (1, 3, 3) 27 ⊕ (1, 3, 3) 27 . The last stage of symmetry breaking G SM to G 31 is done by the weak doublets (1, 2, 1) 27 ⊕ (1, 2, −1) 27 at M Z scale reproducing all known SM fermion masses. Here the supersymmetry is broken at the M Z scale.  In the present case, the gravitational contribution will arise from the non-zero VEV of the Higgs φ (1, 1, 1 , −1, · · · , −1 9 , −1, · · · , −1 MG , which obviously conserves Dparity. Now using the above values of one-loop beta coefficients, we estimated numerically the values of M I , M U , α −1 G and sin 2 θ W for different values of gravitational corrections in terms of ǫ (we skipped the two loop effects for simplicity) as presented in Table.I.

IV. PREDICTIONS IN NON-SUSY E6 GUT
The non-SUSY version of E 6 GUT with intermediate trinification symmetry is given by We   Table II. We see that the intermediate mass scale M I as well as sin 2 θ W are not affected by the gravitational correction. Here the unification mass scale M U and the GUT gauge coupling constant α G changes with different values of ǫ. In the present model the minimal particle content of E 6 is not enough for proper gauge coupling unification [15] at ǫ = 0, which comes out to be the scale beyond Planck energy. However, if we introduce (1,8,8) ⊂ 650 H , we achieve the unification at 10 14 GeV. In order to achieve for a long proton lifetime the unifcation mass scale has to be enhanced with finite value of ǫ as well as with threshold effects [45]. in a class of grand unified theories which accommodates trinification symmetry invoked with spontaneous D-parity mechanism. We have established the robustness of the proof by considering supersymmetric as well as non-supersymmetric version of E 6 GUT for demonstration purpose. This proof can be generalized to show that there is no effect of GUT-threshold correction on these parameters arising from the mass scale µ > M I . In order to prove this, the SO(10) GUT has to be replaced by E 6 GUT and the intermediate Pati-Salam symmetry G 224D has to be replaced by trinification symmetry (G 333D ). The detailed proof is beyond the scope of this paper which is planned for a separate work.
We, finally, conclude that the origin behind these vanishing contributions to sin 2 θ W and intermediate mass scale M I are primarily because of: (a) Grand Unified Theories like E 6 GUT that accommodates trinification symmetry as an intermediate breaking symmetry. (b) Due to presence of discrete left-right symmetry (D-parity) and the implications of spontaneous D-parity breaking mechanism [41,42] thereby resulted simplified relations for the proof, (c) Due to key matching condition between gauge couplings, α −1 It can also be explained in the GUT models like SU (9) and SO(18) with the trinification symmetry at the intermediate breaking symmetry [46].
With reference to the cosmological issues, it is noteworthy to mention that the present model doesn't have the Domain-Wall problem as the intermediate mass scale M I is high [34] with 10 13 GeV for Non-Susy and 10 16 GeV for SUSY cases(as shown in TableI, II) [47]. As far as proton decay is concerned, the model with high M U allows a stable proton.

Acknowledgments
Chandini Dash is grateful to the Department of Science and Technology, Govt.of India for INSPIRE Fellowship/2015/IF150787 in support of her research work. She is also thankful to Dr.Sudhanwa Patra and IIT Bhilai for the kind hospitality where part of this work is completed.
Appendix A: Formalism for RGEs of gauge couplings in GUTs with trinification symmetry.
We already proved the theorem showing the remarkable property of E 6 GUT or all possible grand unified theories with G 333D trinification intermediate symmetry on vanishing contributions to the electroweak mixing angle sin 2 θ W and intermediate symmetry breaking scale M I due to one-loop, two-loop and gravitational corrections (even can be true for GUT-threshold corrections). The attempts have also been made to show that GUT threshold corrections arising out of super heavy masses or higher dimensional operators identically vanish on sin 2 θ W or the G 224D breaking scale within SO(10) models with G 224D Pati-Salam intermediate symmetry [2,4,6] and we rather carried out our analysis in E 6 GUT with G 333D intermediate trinification symmetry. We aim to derive all the necessary analytic formulas which have been used in the text for proof of the theorem. The simple symmetry breaking chain considered here with intermediate trinification symmetry is given by It is also worth to mention here that b ′ We used two more relations derived based on RGEs for the gauge coupling constants as, Now using equations (3), (4) and (5) where the relevant parameters used in deriving all these key analytic formulas are as follows, Θ ′ 3R (A12) E1 E1 E1 = (8 ǫ3C ǫ3C ǫ3C − 4 ǫ3L ǫ3L ǫ3L − 4 ǫ3R ǫ3R ǫ3R) (A13) E0 E0 E0 = (4 ǫ3L ǫ3L ǫ3L − 4 ǫ3R ǫ3R ǫ3R) (A14) These parameters are characterstics of one loop, two loop and gravitational corrections to the electroweak mixing angle sin 2 θ W and the intermediate mass scale while we focus on two of the parameters B I and B U , in most of times, in the analysis.
, the factor B U vanishes exactly and B I is independent of one-loop effects operative at mass scale µ > M I . Similarly, K Θ Θ ΘΘ ′ Θ ′ Θ ′ can be shown to be independent of two-loop effects emerging from mass scale µ > M I . However, it is found that the unification scale and the GUT gauge coupling constant are fully dependent on these corrections.