Life without SUSY = La vie sans soucis ?

Proton lifetime estimates out of the limbo of theoretical uncertainties?

I continue my wandering in the valley of non-supersymmetric SO(10) models started at this post or if you prefer I offer to contemplate our quantum phenomenological world from the probably most RADICAL* conservative heuristic point of view ;-) 

The continuous efforts to bring into life a new generation of the large neutrino detectors such as DUNE [1] (a 40 kt LAr TPC detector to be built in the Homestake mine in South Dakota), Hyper-K [2] (about 500 kt fiducial water-Cherenkov detector proposed in order to supersede the “smaller” Super-K [3] in Japan) or perhaps even some variant of the liquid-scintillator machine like the european LENA bring back the questions of the possible fundamental instability of the baryonic matter. It is namely the close complementarity of the planned super-rich neutrino physics programme (with a common belief that not only the neutrino mass hierarchy would be determined but also signals of CP violation in the lepton sector may be observed) with the nucleon decay searches that fuels the hope that processes like p → π⁰e⁺ or p → π⁺ν (or perhaps even p → K⁺ν  favoured by low-energy supersymmetry) may finally be seen. To this end, the projected sensitivity of these new facilities should in practically all channels exceed that of the previous generation (dominated by Super-K) by at least one order of magnitude, thus touching the “psychological” proton lifetime boundary of 10³⁵years.  

Unfortunately, the steady progress on the experimental side has hardly been complemented by any significant improvement in the accuracy of the proton lifetime predictions in theory. This, of course, would not be a true concern on day 1 after the proton decay discovery; however, whenever it would come to more delicate (i.e., quantitative) questions beyond the obvious “Is proton absolutely stable?” like, e.g., “Which models can now be ruled-out?” or “What did we really learn about the (presumably) unified dynamics behind these processes?” the theory would have a difficult time to pull any robust answer up the sleeve.  

This has to do, namely, with the enormous (and often irreducible) theoretical uncertainties plaguing virtually all proton lifetime estimates in the current literature [See, e.g., [4, 5] and the references therein] which by far (often by many orders of magnitude) exceed the relatively small – yet fantastic – factor-of-ten “improvement window” the new generation of facilities may open. The typical reason behind this is either a disparity between the amount of the input information available and needed for any potentially accurate calculation – with supersymmetric grand unified theories (SUSY GUTs) as a canonical example – or the lack of a fully consistent treatment at better than just the leading order (LO). While the first issue is more a matter of fashion and, as such, it can be expected to resolve by itself in the future (especially if the LHC sees no hints of SUSY at the TeV scale) the latter is much more difficult to deal with  due to the parametrically higher level of complication such a next-to-leading (NLO) analysis represents... 

There are actually many sources of theoretical uncertainties of very different origins that have to be taken into account when the ambition is to get the total uncertainty at least within the ballpark of the aforementioned experimental “improvement window”. In the framework of the classical GUTs, these are, traditionally, i) the limited accuracy of the existing estimates of the relevant hadronic matrix elements, ii) the limited accuracy of the determination of the masses of the d=6 baryon and lepton number violating (BLNV) operators ... and iii) the insufficiency of the information about their flavour structure available at low energy. On top of that, the proximity of the GUT scale MG, typically in the 10¹⁶GeV ballpark, to the (reduced) Planck scale M_Pl∼10¹⁸GeV should make one worried about the size of the possible gravity-induced effects which, typically, are not under a good control... 

The typical “internal structure” of the gauge-mediated (left) and scalar-mediated (right) d = 6 baryon and lepton number violating operators in the Standard model. The Xµ stands for the vector leptoquarks transforming, usually, as (3, 2, +5/6 ) + h.c. or (3, 2, − 1/6 ) + h.c. under the SM gauge group while ∆ is a generic symbol for scalar mediators transforming typically as (3, 1, + 1/3 ) or (3, 1, + 4/3 ).

From what has been said one may conclude that it is virtually impossible to provide a theoretically robust NLO prediction for the proton lifetime in the classical GUT framework as the proximity of the Planck scale and the size of its effects on the vital ingredients of these calculations makes the theoretical uncertainties way larger than the desired order-of-magnitude ballpark. Nevertheless, there is a very special setting in which the leading order gravity effects may remain under control to such a degree that it is at least worth trying to perform an NLO analysis within.

Indeed, the renormalizable non-supersymmetric SO(10) model in which the GUT-scale gauge symmetry is broken by the 45-dimensional adjoint scalar representation and the Yukawa couplings are governed by the SO(10) vector(s) and 5-index antisymmetric tensor(s) (i.e., 10 and 126), see, e.g., [9, 25], may overcome the main issues discussed above. Due to the antisymmetry of 45 the leading order gravity smearing effects in the gauge matching are absent and, hence, the masses of the BLNV mediators may be, in principle, determined to a sufficient accuracy. At the same time, the symmetric nature of all the Yukawa couplings at play may justify the use of the formulae (7) and, thus, overcome also the issue with the limited amount of the flavour information available at the low scale. 

Interestingly enough, this very special and beautiful model has been ignored for more than two decades due to the peculiar tachyonic instabilities revealed in its scalar sector back in 1980’s [26–28]; only recently it has been shown [10] that this is a mere artefact of the tree-level approximations used therein and that the model makes perfect sense as a truly quantum theory. Since then, it has been a subject of several dedicated analyses [6, 7, 29] which will hopefully culminate into a complete and robust NLO prediction in a not-so-distant future.

Theoretical Uncertainties in Proton Lifetime Estimates
Helena Kolešová, Michal Malinský, Timon Mede(Submitted on 19 Oct 2015)

A handful of testable Wimps?

One of the most promising class of candidates for DM is the so-called weakly-interacting massive particle (WIMP). These are electrically neutral and colorless particles which have masses of O(10^(2-3)) GeV and couple to SM particles via weak-scale interactions. Their thermal relic abundance can explain the current energy density of DM. Such particles are predicted in many new-physics models; for example, the lightest neutralino in the supersymmetric (SUSY) SM is a well-known candidate for WIMP DM [2].  

For a WIMP to be DM, it should be stable or have a sufficiently long lifetime compared to the age of the Universe. To assure that, it is usually assumed that there is a symmetry which stabilizes the DM particle. For instance, in the minimal SUSY SM (MSSM), R-parity makes the lightest SUSY particle stable and thus a candidate for DM in the Universe [2]. Similarly, Kaluza-Klein parity in universal extra dimensional models [3] and T-parity in the Littlest Higgs model [4] yield stable particles, which can also be promising DM candidates. The ultraviolet (UV) origin of such a symmetry is, however, often obscure; thus it would be quite interesting if a theory which offers a DM candidate and simultaneously explains its stability can be realized as a UV completion rather than introducing the additional symmetry by hand.  

In fact, grand unified theories (GUTs) can provide such a framework. Suppose that the rank of a GUT gauge group is larger than four. In this case, the GUT symmetry contains extra symmetries beyond the SM gauge symmetry. These extra symmetries should be spontaneously broken at a high-energy scale by a vacuum expectation value (VEV) of a Higgs field. Then, if we choose the proper representation for the Higgs field, there remains discrete symmetries, which can be used for DM stabilization [5–10]. The discrete charge of each representation is uniquely determined, and thus we can systematically identify possible DM candidates for each symmetry 

In this work, we discuss the concrete realization of this scenario in non-SUSY SO(10) GUT models. It is widely known that SO(10) GUTs [11–13] have a lot of attractive features. Firstly, all of the SM quarks and leptons, as well as right-handed neutrinos, can be embedded into 16 representations of SO(10). Secondly, the anomaly cancellation in the SM is naturally explained since SO(10) is free from anomalies. Thirdly, one obtains improved gauge coupling unification [13–20] and improved fermion mass ratios [13, 21] if partial unification is achieved at an intermediate mass scale. In addition, since right-handed neutrinos have masses of the order of the intermediate scale, small neutrino masses can be explained via the seesaw mechanism [22] if the intermediate scale is sufficiently high. SO(10) includes an additional U(1) symmetry, which is assumed to be broken at the intermediate scale. If the Higgs field that breaks this additional U(1) symmetry belongs to a 126 dimensional representation, then a discrete ℤ₂ symmetry is preserved at low energies {(equivalent to matter parity P_M=(−1)^3(B-L))}. One also finds that as long as we focus on relatively small representations (≤ 210), the 126 Higgs field leaving a ℤ₂ symmetry is the only possibility for a discrete symmetry [23, 24]. We focus on this case in the following discussion.  

DM candidates appearing in such models can be classified into two types; one class of DM particles have effectively weak-scale interactions with the SM particles so that they are thermalized in the early universe, while the other class contains SM singlets which are never brought into thermal equilibrium. In the latter case, DM particles are produced out of equilibrium via the thermal scattering involving heavy (intermediate scale) particle exchange processes {see figure below from arxiv.org/abs/1510.03509}. This type of DM is called Non-Equilibrium Thermal DM (NETDM) [25], whose realization in SO(10) GUTs was thoroughly discussed in Ref. [24]. NETDM is necessarily fermionic as scalar DM would naturally couple to the SM Higgs bosons. Depending on the choice of the intermediate-scale gauge group, candidates for NETDM may originate from several different SO(10) representations such as 45, 54, 126 or 210. Although the NETDM candidate itself does not affect the running of the gauge couplings from the weak scale to the intermediate scale, part of the original SO(10) multiplet has a mass at the intermediate scale and does affect the running up to the GUT scale. Demanding gauge coupling unification with a GUT scale above 10¹⁵ GeV leaves us with a limited set of potential NETDM candidates... 

The non equilibrium thermal mechanism for dark matter production 

Stable SO(10) scalar (fermion) DM candidates must be odd (even) under the ℤ₂ symmetry. Therefore fermions must originate in either a 10, 45, 54, 120, 126, 210 or 210' representation, while scalars are restricted to either a 16 or 144 of SO(10). These multiplets must be split and we gave explicit examples of fine-tuning mechanisms in order to retain a 1 TeV WIMP candidate which may be a SU(2)L, singlet, doublet, triplet, or quartet with or without hypercharge. Fermions which are SU(2)L singlets with no hypercharge are not good WIMP candidates but are NETDM candidates and these were considered elsewhere [24]. Our criteria for a viable dark matter model required: gauge coupling unification at a sufficiently high scale to ensure proton stability compatible with experiment; a unification scale greater than the intermediate scale; and elastic cross sections compatible with direct detection experiments. The latter criterion often requires additional Higgs representations to split the degeneracy of the fermionic intermediate scale representations if DM is hypercharged. 

Despite the potential very long list of candidates (when one combines the possible different SO(10) representations and intermediate gauge groups), we found only a handful of models which satisfied all constraints. Among the scalar candidates, the Y=0 singlet and Y=1/2 doublet (often referred to as an inert Higgs doublet [50]) are possible candidates for SU(4)_C⊗SU(2)_L⊗SU(2)_R and SU(3)_C⊗SU(2)_L⊗SU(2)_R⊗U(1)_B-L (with or without a left-right symmetry) intermediate gauge groups. These originate from either the 16 or 144 of SO(10). The latter group (without the left-right symmetry) is also consistent with a state originating from the 144 being a triplet under SU(2)_R. To avoid immediate exclusion from direct detection experiments, a mass splitting of order 100 keV implies that the intermediate scale must be larger than about 3×10^-6 M_GUT for a nominal 1 TeV hyper-charged scalar DM particle. Some of these models imply proton lifetimes short enough to be testable in on-going and future proton decay experiments. 

The fermion candidates were even more restrictive. Models with Y=0 must come from a SU(2)_L triplet (singlets are not WIMPs). In this case only one model was found using the SU(4)_C⊗SU(2)_L⊗SU(2)_R intermediate gauge group and requiring additional Higgses ... at the intermediate scale. Models with Y=1/2 doublets were found for SU(4)_C⊗SU(2)_L⊗U(1)_R with a singlet fermion required for mixing, and SU(4)_C⊗SU(2)_L⊗SU(2)_R with a triplet fermion for mixing. In both cases, additional Higgses ... are required at the intermediate scale...

Weakly-Interacting Massive Particles in Non-supersymmetric SO(10) Grand Unified Models Natsumi Nagata, Keith A. Olive, Jiaming Zheng

(Submitted on 2 Sep 2015)

One route from GUT scale to the SM with a break at Pati-Salam?

The discovery of the Higgs boson at the Large Hadron Collider (LHC) at CERN is a major milestone of the successes of the standard model (SM) of particle physics. Indeed, with all the quarks and leptons and force carriers of the SM now detected and the source of spontaneous symmetry breaking identified there is a well-deserved sense of satisfaction. Nonetheless, there is a widely shared expectation that there is new physics which may be around the corner and within striking range of the LHC. The shortcomings of the standard model are well-known. There is no candidate for dark matter in the SM. The neutrino is massless in the model but experiments indicate otherwise. At the same time the utter smallness of this mass is itself a mystery. Neither is there any explanation of the matter-antimatter asymmetry seen in the Universe. Besides, the lightness of the Higgs boson remains an enigma if there is no physics between the electroweak and Planck scales. 

Of the several alternatives of beyond the standard model extensions, the one on which we focus in this work is the left-right symmetric (LRS) model [1, 2] and its embedding within a grand unified theory (GUT). Here parity is a symmetry of the theory which is spontaneously broken resulting in the observed left-handed weak interactions. The left-right symmetric model is based on the gauge group SU(2)_L⊗SU(2)_R⊗U(1)_B-L and has a natural embedding in the SU(4)_C⊗SU(2)_L⊗SU(2)_R Pati-Salam model [3] which unifies quarks and leptons in an SU(4)_C symmetry. The Pati-Salam symmetry is a subgroup of SO(10) [4, 5]. These extensions of the standard model provide avenues for the amelioration of several of its shortcomings... 

In a left-right symmetric model emerging from a grand unified theory, such as SO(10), one has a discrete symmetry SU(2)_L↔SU(2)_R – referred to as D-parity [15] – which sets g_L=g_R. Both D-parity and SU(2)_R are broken during the descent of the GUT to the standard model, the first making the coupling constants unequal and the second resulting in a massive W_R. The possibility that the energy scale of breaking of D-parity is different from that of SU(2)_R breaking is admissible and well-examined. The difference between these scales and the particle content of the theory controls the extent to which g_L≠g_R. 

In this work we consider the different options of {non-supersymmetric} SO(10) symmetry breaking. It is shown that a light W_R  goes hand-in-hand with the breaking of D-parity at a high scale, immediately excluding the possibility of g_L=g_R. Breaking of D-parity above the scale of inflation, in fact, is usually considered a good feature for getting rid of unwanted toplogical defects such as domain walls [16, 17]. The other symmetries that are broken in the passage to the standard model are the SU(4)_C and SU(2)_R of the Pati-Salam (PS) model. The stepwise breaking of these symmetries and the order of their energy scales have many variants. There are also a variety of options for the scalar multiplets which are used to trigger the spontaneous symmetry breaking at the different stages. We take a minimalist position of (a) not including any scalar fields beyond the ones that are essential for symmetry breaking, and also (b) impose the Extended Survival Hypothesis (ESH) corresponding to minimal fine-tuning to keep no light extra scalars. With these twin requirements we find that only a single symmetry-breaking route – the one in which the order of symmetry breaking is first D-parity, then SU(4)_C, and finally SU(2)_R – can accommodate a light MW_R ... 

Symmetry breaking routes of SO(10) distinguished by the order of breaking of SU(2)_R, SU(4)_C, and D-parity. The SO(10) scalar multiplets responsible for symmetry breaking at every stage have been indicated. Only the DCR (red solid) route can accommodate the light W_R scenario... 

Scalar fields considered when the ordering of symmetry-breaking scales is M_D>M_C>M_R The submultiplets contributing to the RG evolution at different stages according to the Eextended Survival Hypothesis are shown. D-parity (±) is indicated as a subscript.

... Before turning to SO(10) we briefly remark about Pati-Salam partial unification within this route. Because there are four steps of symmetry breaking this is an underdetermined system. For this work, M_R is restricted to be in the O(TeV) range. The scale MC is taken as the other input in the analysis. At the one-loop level the results can be analytically calculated using the beta-function coefficients in eq. (23). The steps can be identified from eqs. (25) and (26). The latter determines MD once MC is chosen. η is then fixed using eq. (25). For example, for M_C=10⁶  GeV one gets η=0.63 when M_R=5 TeV. Within the Pati-Salam model the upper limit of M_D is set by the Planck mass M_Planck . We find that in such a limit one has M_C=10^17.6 GeV and η=0.87 for M_R=5 TeV...

The observation by the CMS collaboration of a 2.8σ excess in the (2e)(2j) channel around 2.1 TeV can be interpreted as a preliminary indication of the production of a right-handed gauge boson W_R. Within the left-right symmetric model the excess identifies specific values of η=g_R/g_L, r=MN_e/MW_R, and VN_e {N_e is the electronic right-handed neutrino and V parametrizes the mixing between the electron and its right-handed neutrino}. We stress that even with g_L=g_R and VN_e=1 the data can be accommodated by an appropriate choice of r. We explore what the CMS result implies if the left-right symmetric model is embedded in an SO(10) GUT. η ≠1 is a consequence of the breaking of left-right D-parity. We find that a W_R in the few TeV range very tightly restricts the possible routes of descent of the GUT to the standard model. The only sequence of symmetry breaking which is permitted is M_D>M_C>M_R with a D-parity breaking scale ≥ 10¹⁶ GeV. All other orderings of symmetry breaking are excluded. Breaking of left-right discrete parity at such a high scale pushes g_L and g_R  apart and one finds 0.64 ≤ η ≤ 0.78. The unification scale, M_U, has to be as high as ∼10¹⁸ GeV so that it is very unlikely that proton decay will be seen in the ongoing experiments. The SU(4)_C-breaking scale, M_C, can be as low as 10⁶ GeV, which may be probed by rare decays such as K_L → µe and B_d,s → µe or n − n̅ oscillations...

 The ATLAS collaboration has also presented evidence [32] for an enhancement around 2 TeV in the di-boson – ZZ and WZ – channels in their 8 TeV data. Our interpretation of the excess in the (ee)(jj) channel in terms of a W_R by itself fails to provide an explanation of the above. If the W_R production is normalised to the former then it falls an order of magnitude short of the di-boson rates. It has been shown that interpretation of the di-boson observations as well as the (ee)(jj) data is possible if the LRS model is embellished with the addition of some other fermionic states [33, 34].

Implications of the CMS search for W_R on Grand Unification  

Triparno Bandyopadhyay (Calcutta Univ), Biswajoy Brahmachari (Vidyasagar Evening Coll),Amitava Raychaudhuri (Calcutta Univ)
(Submitted on 10 Sep 2015)

//Updated with two figures on December 1 2015.
//Title slighlty edited on 29 December 2016

//* the adjective RADICAL was added on 25 September 2021 (cf https://twitter.com/bardot_cedric/status/1438867394227752962?s=20 )