Research Article  Open Access
Biswonath Sahoo, Mainak Chakraborty, M. K. Parida, "Neutrino Mass, Coupling Unification, Verifiable Proton Decay, Vacuum Stability, and WIMP Dark Matter in SU(5)", Advances in High Energy Physics, vol. 2018, Article ID 4078657, 21 pages, 2018. https://doi.org/10.1155/2018/4078657
Neutrino Mass, Coupling Unification, Verifiable Proton Decay, Vacuum Stability, and WIMP Dark Matter in SU(5)
Abstract
Nonsupersymmetric minimal SU(5) with Higgs representations and and standard fermions in is well known for its failure in unification of gauge couplings and lack of predicting neutrino masses. Like standard model, it is also affected by the instability of the Higgs scalar potential. We note that extending the Higgs sector by and not only leads to the popular typeII seesaw ansatz for neutrino masses with a lower bound on the triplet mass GeV, but also achieves precision unification of gauge couplings without proliferation of nonstandard light Higgs scalars or fermions near the TeV scale. Consistent with recent LUX2016 lower bound, the model easily accommodates a singlet scalar WIMP dark matter near the TeV scale which resolves the vacuum stability issue even after inclusion of heavy triplet threshold effect. We estimate proton lifetime predictions for including uncertainties due to input parameters and threshold effects due to superheavy Higgs scalars and superheavy gauge bosons. The predicted lifetime is noted to be verifiable at Super Kamiokande and Hyper Kamiokande experiments.
1. Introduction
Standard model (SM) of strong and electroweak interactions has been established by numerous experimental tests, yet evidences on neutrino mass [1â€“5], the phenomena of dark matter [6â€“25], and baryon asymmetry of the universe (BAU) [8, 26â€“29] call for beyond standard model (BSM) physics. It is well known that grand unified theories (GUTs) [30â€“37] are capable of addressing a number of limitations of the SM effectively. There are interesting theories on neutrino mass generation mechanisms [38â€“45] based upon various seesaw mechanisms such as typeI, typeII, typeIII [46â€“62], linear [63, 64], and inverse [65â€“75]. Interesting models for Dirac neutrino mass origin of the neutrino oscillation data have been also proposed [76, 77]. In the absence of experimental evidence of supersymmetry so far, nonsupersymmetric (nonSUSY) GUTs are being extensively exploited by reconciling to the underlying gauge hierarchy problem through finetuning [78, 79]. Higher rank GUTs like SO(10) and can not define a unique symmetry breaking path to the SM gauge theory because of large number of possibilities with one and more intermediate symmetry breakings consistent with electroweak precision data on , and [80â€“82]. On the other hand, the rank4 minimal SU(5) [32] with Higgs representations and defines only one unique symmetry breaking path to the standard modelTypeI seesaw [46â€“52] needs nonstandard heavy righthanded neutrino, linear and inverse seesaw both need nonstandard fermions and scalars, and typeIII seesaw [38â€“45, 58â€“62] needs only nonstandard fermionic extension for their implementation. Out of these popular seesaw mechanisms, typeII seesaw mechanism is the one which needs a heavy nonstandard triplet scalar [52â€“57, 59]. With a second triplet scalar, it is also capable of predicting baryon asymmetry of the universe [57] which is one of the main motivations behind this investigation. This neutrino mass generation mechanism, gauge coupling unification, dark matter, and vacuum stability are the focus of the present work.
Like the minimal SM, with its fermions per generation and the standard Higgs doublet , the minimal SU(5) with Higgs representations and predicts neutrinos to be massless subject to a tiny eV contribution due to nonrenormalizable Planck scale effect which is nearly orders smaller than the requirement of neutrino oscillation data. As the particle spectrum below the GUT symmetry breaking scale is identically equal to the SM spectrum, like SM, the minimal GUT fails to unify gauge couplings [83â€“87]. Also it predicts instability of the Higgs quartic coupling at mass scales GeV [88â€“90] after which the coupling continues to be increasingly negative at least up to the unification scale.
A number of interesting models have been suggested for coupling unification by populating the grand desert and for enhancing proton lifetime predictions [60â€“62, 91â€“98]. In these models a number of fermion or scalar masses below the GUT scale have been utilised to achieve unification. Interesting possibility of typeIII seesaw [60â€“62] with experimentally verifiable dilepton production [99] at LHC has been also investigated.
The other shortcoming of minimal nonSUSY SU(5) is its inability to predict dark matter which appears to belong to two distinct categories: (i) the weakly interacting massive particle (WIMP) dark matter of bounded mass TeV and (ii) the decaying dark matter which has been suggested to be a possible source of PeV energy IceCube neutrinos.
In this work we implement a novel mechanism for coupling unification and neutrino masses together. When SU(5) is extended by the addition of its Higgs representations and , it achieves two objectives: (i) neutrino mass and mixing generation through typeII seesaw mechanism and (ii) precision gauge coupling unification with experimentally accessible proton lifetime.
But this does not cure the vacuum instability problem persisting in the model as well as the need for WIMP dark matter prediction. Out of these two, as we note in this work, when the dark matter prediction is successfully inducted into the model, the other problem on vacuum stability is automatically resolved.
In contrast to the popular belief on low proton lifetime prediction of the minimal SU(5) [35], we estimate new precise and enhanced predictions of this model including threshold effects [100â€“108] of heavy particles near the GUT scale. Predicted lifetimes are found to be within the accessible ranges of Superkamiokande and Hyperkamiokande experimental search programmes [109].
This paper is organised in the following manner. In Section 2 we discuss neutrino mass generation mechanism in extended SU(5). Section 3 deals with the problem of gauge coupling unification. In Section 4 we make proton lifetime prediction including possible uncertainties. Embedding WIMP scalar DM in SU(5) is discussed in Section 5 with a brief outline on the current experimental status. Resolution of vacuum stability issue is explained in Section 6. We summarise and conclude in Section 7. Renormalization group equations for gauge and Higgs quartic couplings are discussed in the Appendix.
2. Neutrino Mass Through TypeII Seesaw in SU(5)
As noted in Section 1, in contrast to many possible alternative symmetry breaking paths to SM from nonSUSY SO(10) and [80â€“82], SU(5) predicts only one symmetry breaking path which enhances its verifiable predictive capability. Fifteen SM fermions are placed in two different SU(5) representations:Lack of RH in these representations gives vanishing Dirac neutrino mass and vanishing Majorana neutrino mass at renormalizable level. Planck scale induced small Majorana masses can be generated through nonrenormalizable interaction
leading to eV which is too low to explain neutrino oscillation data. Mechanism of Dirac neutrino mass generation has been discussed [76, 77] matching the neutrino oscillation data. Using extensions of the minimal GUT typeIII seesaw origin of neutrino mass has been discussed where the nonstandard fermionic triplet mediates the seesaw. This model can be experimentally tested by the production of likesign dilepton signals at LHC.
TypeII seesaw mechanism for neutrino mass [53â€“57] does not need any nonstandard fermion but needs only the nonstandard lefthanded Higgs scalar triplet with which directly couples with the a dilepton pair. It also directly couples to standard Higgs doublet . As such the standard Higgs VEV can be transmitted as a small induced VEV generating Majorana mass term for the light neutrinos. As this is contained in the symmetric SU(5) scalar representation , the scalar sector of the minimal GUT needs to include in addition to and .
The Yukawa Lagrangiancombined with the relevant part of the Higgs potentialgives rise to the typeII seesaw contribution. In our notation ( generation index) and which are the lepton and scalar doublet of . Here , ( are the Pauli spin matrices) and, similarly, the scalar triplet in the adjoint representation of is expressed as . The Majorana type Yukawa coupling is a matrix in flavor space and is the charge conjugation matrix. Thenwhere different components are given byA diagrammatic representation for typeII seesaw generation of neutrino mass is shown in Figure 1. From the Feynman diagram shown in Figure 1 the induced VEV of the scalar triplet is leading to the typeII seesaw formulaIt is necessary to explain the origin of the breaking scale that occurs in (4), (5), and (8) as well as the Feynman diagram of Figure 1. SU(5) invariance permits the triplet coupling leading to SM invariant coupling . Therefore, in one approach, may be treated as explicitly lepton number violating parameter. Alternatively, it is also possible to attribute a spontaneous lepton number violating origin to this parameter. Since the SM model gauge theory has to remain unbroken down to the electroweak scale, the lepton number violating scale can be generated by the VEV of a Higgs scalar that transforms as a singlet under SM. Such a singlet carrying occurs in the Higgs representation [36, 110]. The part of the SU(5) invariant potential that generates this scale isleading to . The symmetric origin of becomes more transparent if one treats SU(5) as the remnant of or higher rank GUTs like SO(10) and . If unification constraint as discussed below is ignored, the order of magnitude of can be anywhere in the range . But as we will find in the subsequent sections, gauge coupling unification in the present SU(5) framework imposes the lower bound GeV.
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2.1. TypeII Seesaw Fit to the Neutrino Oscillation Data
2.1.1. Neutrino Mass Matrix from Oscillation Data
The effective light neutrino mass matrix is diagonalised by a unitary matrix (in PMNS parametrisation which is written as ) and yields three mass eigenvalues . The light neutrino mass matrix () can be reconstructed aswhere PMNS matrix is parameterised using the PDG convention [111] aswhere with , is the Dirac CP phase, and are Majorana phases.
Here we present our numerical analysis within and limits of experimental data. As we do not have any experimental information about Majorana phases, they are varied in the whole interval randomly. From the set of randomly generated values we pick only one set of and use them for our numerical estimations. The procedure adopted here can be repeated to derive corresponding solutions for the Majorana coupling matrix for other sets of randomly chosen Majorana phases. Although very recently and limits of Dirac CP phase have been announced [1], we prefer to use only their central value as an example. For our present analysis we choose a single set of from a number of sets derived by random sampling and also a single value of close to the best fit value. For our analysis all possible values of the solar and atmospheric mass squared differences and mixing angles have been taken which lie within the (or ) limit of the oscillation data as determined by recent global analysis [1]. Summary of the global analysis is presented in Table 1. At first we analyze the limits imposed on the neutrino Yukawa couplings by oscillation constraints taking into account both the mass ordering of light neutrinos, normal ordering (NO), and inverted ordering (IO). In this case we use only one fixed value of the lightest neutrino mass eigenvalue and the other two mass eigenvalues are calculated using the experimental values of the mass squared differences. In this case we represent the bounds on the elements of Yukawa matrix in a tabular form. Later we proceed to estimate the bounds on the matrix elements imposed by experimental constraints of oscillation observables. In this analysis instead of fixed lightest neutrino mass eigenvalue, we vary it in the range eV. The other two mass eigenvalues are calculated using ranges of solar and atmospheric mass squared differences. As already explained we use a single set of randomly chosen and the central value of quoted in the Table 1. The variation of matrix elements is expressed in terms of their moduli () and the corresponding phases ) with are shown graphically in Figures 2, 3, and 4 in the NO case. It is clear from the plots that for each single value of there is a band of allowed values of and . This band signifies the allowed range of the corresponding matrix element for that single value of . To represent the bounds in a more transparent manner we produce another set of plots as in Figures 5 and 6 where we show the allowed values of and for a fixed value of . It is to be noted that in this present work graphical representation is done for normally ordered light neutrinos only. Similar kind of exercise can be carried out for inverted mass ordering also.

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2.1.2. Majorana Yukawa Coupling for Bounds of Neutrino Oscillation Data
We now estimate the matrix for the normally ordered (NO) case. For this purpose we take the mass of the lightest neutrino as eV. Then using the ranges of solar and atmospheric mass squared differences for NO case, as mentioned in Table 1, the other two neutrino mass eigenvalues are calculated. Obviously we get a range of values of and . Plugging in these mass eigenvalues along with all possible combinations and the mixing angles within the bound in (11) we obtain large number of sets of matrix. Thus we also get respective bounds on the elements of the matrix (or equivalently on the Yukawa coupling matrix ) corresponding to the oscillation constraints. As mentioned earlier we use single set of randomly chosen Majorana phases while the Dirac CP phase is chosen close to its central value. The effective light neutrino mass matrix and the coupling matrix are connected through the induced VEV which is obtained by assuming the dimensionful coupling where GeV and the electroweak VEV is GeV. With these considerations we estimate the bound on the elements of matrix and present them in Table 2. In the inverted mass ordering the smallest mass eigenvalue is which is set to be equal to eV. The other two eigenvalues are calculated using the limit of the solar and atmospheric mass squared differences. In this case also we are able to put a bound on the modulus and phase of the Yukawa coupling matrix following the same procedure as done in the case of NO. The constrained parameters () for inverted mass ordering are given in Table 3.


As we have taken a most general complex symmetric structure of the matrix (or in other words the Yukawa coupling matrix ) without imposing any kinds specific flavor symmetry, it does not have any definite prediction of the Dirac CP violating phase . Any value of in the given range can be accommodated. In this regard few remarks about the present experimental status of the Dirac CP phase are in order. The recent global analysis of oscillation data done in [1] has made it clear that value of the Dirac CP phase is more or less ruled out. In normal mass ordering (NO) is disfavored at more than confidence level whereas for inverted mass ordering (IO) it is more stringent, where is ruled out at more than . The best fit value of in NO and IO is near and , respectively. For the sake of simplicity we work with only the best fit values. We have also estimated the highest and lowest values of the CP violating measure, the Jarlskog invariant for both the mass orderings. For NO: , for IO: when is kept fixed at its best fit value whereas all other observables are varying in their respective ranges.
2.1.3. Majorana Yukawa Coupling for Bounds of Neutrino Oscillation Data
Here we follow exactly the same methodology as the previous case, however the numerical calculations are done with ranges of oscillation data instead of range. Here we are exploring the normally ordered case only. Unlike the previous case the lightest neutrino mass eigenvalue is not kept fixed; it is varied over a range of eV and the corresponding variations of the modulus and phase of Majorana Yukawa couplings are depicted in Figures 2, 3, and 4. The allowed ranges of those quantities for a fixed are also shown in Figures 5 and 6.
3. Gauge Coupling Unification in the Scalar Extended SU(5)
3.1. Lower Bound on the Scalar Triplet Mass
Exercising utmost economy in populating the grand desert, it was noted that the presence of the scalar component at an intermediate mass GeV could achieve gauge coupling unification at GeV [112] but no neutrino oscillation data was available at that time. Using the most recent electroweak precision data [111, 113, 114], in this work we find that this intermediate scalar mass is now reduced by one order, GeV. Similarly the GUT scale is now determined with high precision including all possible theoretical and experimental uncertainties. Noting the result of this work as discussed in Section 2 that typeII seesaw realisation of neutrino mass needs substantially lower than the GUT scale leads to the natural apprehension that the presence at intermediate mass scale would destroy precision unification achieved by . This apprehension is logically founded on the basis that nonvanishing contributions to the and beta functions would misalign the fine structure constants and from the realised unification paths substantially for all mass scales .
We prevent any such deviation from the realisation of precision coupling unification by assuming all the components of to have the identical degenerate mass which is bounded in the following manner:Thus, in order to safeguard precision unification, it is essential that in the present scalar extended SU(5) model (the upper limit is due to our observation that typeII seesaw scale is lower than the GUT scale although, strictly speaking, is possible if typeII seesaw contribution to neutrino mass is ignored).
Thus typeII seesaw realisation of neutrino mass and precision unification in SU(5) needs the additional scalar representations and .
3.2. RG Solutions to Mass Scales
For realistic unification of gauge couplings we use one loop equations [115] supplemented by topquark threshold effects [83] and twoloop corrections [116]
In the range of mass scale we include topquark Yukawa coupling () contribution at the twoloop level with the coefficients in (14) and the RG evolution equation [83]The beta function coefficients in three different mass ranges , , and are
We have used the most recent electroweak precision data [114]
Using RGEs and the combinations and , we have derived analytic formulas for the unification scale and intermediate scale() treating triplet scalar scale () constant as
where and the first term in (20) represent one loop contributions. The terms , , denote the threshold corrections due to unification scale (), intermediate scale (), and GUT fine structure constant ().
Excellent unification of gauge couplings is found for where the number in the exponent is due to GUT scale matching of inverse fine structure constant that is present even if all superheavy masses are exactly at [101, 103â€“105].
3.3. Effects of on Unification
It is well known that when a complete GUT representation is superimposed on an already realised unification pattern in nonSUSY GUTs [117, 118], the GUT scale is unchanged but the inverse fine structure constants change their slopes and deviate from the original paths proportionately so as to increase the unification coupling. As an example in nonSUSY SO(10) [117, 118], at first a precision unification frame has been achieved with the modification of the TeV scale spectrum of the minimal SUSY GUT by taking out the full scalar super partner content of the spinorial super field representation . Then the resulting TeV scale spectrum is [117, 118]which may be recognised to be the same as the corresponding spectrum in the splitSUSY case supplemented by the additional scalar doublet . In (22) â€™s represent nonstandard fermions. Further adjustment of masses of these particles around TeV scale has been noted to achieve degree of precision coupling unification higher than MSSM [117]. After having thus achieved a precision unification, the full is superimposed at the typeII seesaw scale of the nonsupersymmetrised unification framework. Analogous to MSSM, this model [118] predicts a number of fermions as in (22) at the TeV scale which must be verified experimentally at accelerator energies.
In contrast, the present model has only the standard Higgs doublet and the WIMP DM scalar singlet near TeV scale as discussed below. Although the TeV scale DM has not been confirmed yet by direct experiments, LUX16 or FemiLATlike experiments may detect it. Moreover, as shown below, the present model ensures vacuum stability through this WIMP dark matter candidate whereas in [118] the vacuum stability and DM issues are yet to be answered. Further, the SM coupling unification scale in [118] being close to the SUSY GUT scale, GeV, predicts proton lifetime nearly times larger than the current experimental limit without threshold effect which is expected to introduce larger uncertainty compared to the present model. It may be more difficult to verify this model by ongoing proton decay experiments. But the present model including such uncertainties is within the experimentally accessible limits. The origin of invariant GUT scale in the presence of in the present model is due to the invariance of the beta function differences which is in this modelThis results in a change in the inverse GUT coupling constant which occurs due to the RG predicted modification
Thus the result of typeII seesaw motivated insertion of into the realised unification framework is to decrease in the inverse GUT fine structure constant while keeping mass scales same as in (21)"which is a effect. It is essential to take this effect into consideration in the topdown approach for consistency with the precision value of the electromagnetic fine structure constant [114]. More important is its visible effect on proton lifetime prediction. It is clear from dependence in (36) of Section 4, (21), and (25) that the this intermediate typeII seesaw scale has a proton lifetime reduction by that further reduces for lower seesaw scales, GeV. But the reduction effect decreases as increases such that the proton lifetime remains unchanged for the limiting value .
In Figure 7 we have shown evolution of inverse fine structure constants of three gauge couplings of SM against mass scales depicting precision unification at GeV.
3.3.1. Implications for Lepton Number and Flavor Violations
It is evident from (13) and (21) that the numerical lower bound on the masses of three members of the triplet in is
Out of these we have discussed in Section 2 how the mediation of gives typeII seesaw contribution to neutrino masses matching with available neutrino oscillation data at  levels for all types of hierarchies: NH, IH, and QD. As a result the HiggsYukwa interaction and induced VEV of the neutral component of the triplet in the present SU(5) model give similar predictions as in the triplet extended SM based analyses [119] or in the leftright symmetric models and SO(10) with large boson mass [45, 120]. Currently a number of experimental investigations are underway to detect the double beta decay process that would establish Majorana nature of neutrino. The most important difference from such SM based phenomenological analyses is that in the present SU(5) model with typeII seesaw all the parameters of the neutrino oscillation data are theoretically predicted by the seesaw mechanism. Even though GeV, it predicts the double beta decay lifetime close to the observable limit of yrs for QD type light neutrino masses eV. On the other hand for NH type of hierarchy the predicted decay rate is much lower with lifetime yrs. Another theoretical contribution to the double beta decay process is due to the mediation of the doubly charged component through the physical process which is negligible because of additional damping of the amplitude caused by the inverse square of its heavy mass GeV. The charged component also mediates a new loop contributions to lepton flavor violating processes such as . Again, because of heavy triplet mass the respective contribution to branching ratio turns out to be much smaller than the corresponding prediction with (supplemented by the oscillation data): [45, 120]. Similarly the tree level mediation of the LFV process by is severely damped out compared to the loop mediated boson contribution.
3.4. Threshold Effects on the GUT Scale
In the single step breaking model discussed in this work, GUT threshold effects due to superheavy degrees of freedom in different SU(5) representations are expected as major sources of uncertainties on unification scale and proton lifetime prediction. We have estimated the threshold uncertainties following the partially degenerate assumption introduced in [121, 122] which states that the superheavy components belonging to the same GUT representation are degenerate with a single mass scale.
The analytic formulas for GUT threshold effects on the unification scale, intermediate scale, and GUT fine structure constant are
In (27) are matching functions due to superheavy scalars (S) and gauge bosons (V) to the three gauge couplings,