(Even more?) Simple GUTs come along with longer proton lifetime and hint of normal neutrino mass hierarchy

From hidden simplicities to a conjectured quantum gravity condition on matter content

Here are ideas and speculations from a more than twenty year old article by Giovanni AMELINO-CAMELIA which resonates interestingly - in my ears at least - with grand unification in the Spectral Pati-Salam Model retrospectively (underlining and bold emphasis as well as {...} are mine as usual):

It is probably worth emphasizing, at least for the benefit of the students ... that we have nothing (from the conceptual viewpoint) to assure us that nature should be describable in terms of simple laws. Still, most of us do expect this simplicity, probably extrapolating from the history of physics, which has proceeded through a series of steps of deeper understanding and simplification (such as the description of the baryon spectrum in terms of the quark model). From the point of view of this expected simplicity, the SM is quite unsatisfactory, since it leaves unanswered several questions; in particular,  

(Qa) Particle physics is described by a gauge theory with the peculiar gauge group GSM ≡SU(3)c⊗SU(2)L⊗ U(1)Y .
(Qb) The corresponding three coupling constants, αs, α2, and αY , are free parameters of the model. (Qc) A peculiar bunch of IRREPs (irreducible representations) of SU(3)c⊗SU(2)L hosts the quarks and leptons.
(Qd) The hypercharge assignments to the quark and lepton IRREPs are arbitrary. (Qe) The Yukawa couplings are arbitrary.
(Qf) The entries of the Cabibbo-Kobayashi-Maskawa matrix are arbitrary.
(Qg) Each of the quarks and leptons of the model is present in triplicate copy (family structure).
(Qh) A peculiar (bunch of) IRREP(s) hosts the Higgs particles.
(Qi) The parameters of the Higgs potential, which determine the Higgs mass(es) and all the aspects of symmetry breaking, are arbitrary.
(Qj) The anomaly cancellation is a (apparently accidental) result of the structure of the (arbitrarily selected) matter content of the model.  

GUTs [models with a (grand)unified description of particle interactions] have been investigated primarily because, as illustrated in the discussion of SO(10) GUTs given in the next section, they address/simplify (Qa)-(Qf) and, in some cases, (Qj), and are therefore good candidates for the description of particle physics if the trend of incremental simplifications of this description is to continue. However, it should be noted that not only GUTs bring no improvement in relation to (Qg), but they actually increase the complexity associated to (Qh) and (Qi). Therefore, from the “aesthetic” viewpoint, GUTs have merits and faults (with the merits outnumbering, but not necessarily outweighing, the faults). 

Phenomenological encouragement for the GUT idea comes from the observed low-energy values of αs, α2, and αY , which appear to be arranged just as needed for unification. Indeed, (although the simplest GUT candidate, minimal SU(5), does not pass this test[1]) there are several examples of GUTs which reproduce these data on the coupling constants while being consistent with the ... experimental lower limit on proton decay τ_p→e+π0 ≥ 9×10³² years {in 1996 and τ_p→e+π0 >1.6×10³⁴ years at 90% confidence using Super-Kamiokandedata from April 1996 to March 2015}. One important feature of phenomenologically consistent GUTs is that they must involve at least one extra scale, besides the unification scale MX, at which the RGEs (renormalization group equations) of the SM couplings are modified... 

Perhaps the simplest GUTs meeting the minimum requirement of agreement with the data on the coupling constants and with the experimental limit on proton decay are some SO(10) models, which are reviewed in the next section. They naturally (see next section) predict a two-scale breaking to GSM; in fact, a typical SO(10) breaking chain is given by  

 

Importantly, in SO(10) the hypercharge Y is the combination of two generators belonging to the Cartan, Y = T3R + (B − L)/ 2 , (2) where B − L and T3R belong respectively to the SU(4)PS (the SU(4) containing SU(3)c and U(1)B−L, which was first considered by Pati and Salam) and the SU(2)R subgroups of SO(10). This leads to a high unification point MX if the intermediate symmetry group G′ contains SU(2)R and/or SU(4)PS, since then, between MZ and MR, the Abelian evolution of αY (predicted by SM) is replaced by the non-Abelian one of either component of Y . MX is connected with the masses of the lepto-quarks that can mediate proton decay, and this SO(10) mechanism for a higher unification point turns out to be useful in allowing to meet the condition  

MX ≥ 3.2×10¹⁵ GeV , (3)  

which is necessary... for agreement with the present experimental limit on proton decay. Although (relatively) simple GUTs, such as SO(10), can work, they are affected by the hierarchy problem, and this causes many to prefer SUSY (supersymmetric) GUTs... In this paper (but not necessarily elsewhere) I take the position that for the GUTs (whether they are SUSY or not) the aesthetic advantages and the consistency with the observed low-energy values of the gauge coupling constants outweigh the damage done in regard to (Qh) and (Qi). This motivates me to look for possible ways to associate hidden simplicities to the apparently complicated GUT structures affecting (Qh) and (Qi) (and (Qg)); the reader is warned of the fact that the resulting discussion is accordingly quite speculative.

From ... properties of the smallest IRREPs of SO(10) one can easily see that the typical pattern of SSB of SO(10) to GSM has two steps; indeed, with the exception of the singlet in the 144, the little group of all the above mentioned GSM singlets is larger than GSM. Actually, either for phenomenological or for technical reasons, some of the above mentioned GSM singlets, cannot be used for the first SSB step...The previous considerations lead to four scenarios, in which the first steps of breaking are: 

(Ia) SO(10) → SU(3)c⊗SU(2)L⊗SU(2)R⊗U(1)B−L×D 

(Ib) SO(10) → SU(4)PS⊗SU(2)L⊗SU(2)R 

(Ic) SO(10) → SU(3)c⊗SU(2)L⊗SU(2)R⊗U(1)B−L 

(II) SO(10) → SU(4)PS⊗SU(2)L⊗SU(2)R×D,  

The type-I SO(10) models require that an appropriate vector3 in the space of GSM singlets of the 210 acquires a v.e.v. (vacuum expectation value) at the GUT scale. Analogously the type-II model requires that the GSM singlet of the 54 acquires a v.e.v. at the GUT scale. An appealing[10] possibility for the completion of the models of type-Ia,b,c and type-II is the one of realizing the second SSB step, at a scale MR, with the GSM singlet of the 126⊕126 representation, and the third SSB step with a combination of the SU(3)c⊗U(1)e.m.-invariant vectors of two 10’s, in such a way to avoid the unwanted relation mt = mb ... Through the see-saw mechanism, the scale MR is related to the masses of the (almost) left-handed neutrinos... 

The type-Ia,b,c and type-II models have been studied... and they have been found to be consistent with the unification of couplings and the experimental bound on the proton lifetime, although in the case of the type-II model the consistence with the experimental bound on the proton lifetime is only marginal... Within the see-saw mechanism, one also finds that these models predict masses for the (almost) left-handed neutrinos in an interesting range.  

This phenomenology depends however on the values of the parameters of the Higgs potential, which are free inputs of the model. Most importantly, as mentioned above, the parameters of the Higgs potential must be chosen so that the desired SSB pattern is realized. Although this does not involve a particular fine tuning[10, 11, 12, 13], it does introduce an element of undesirable arbitrariness in the models. Similarly, the “matter ingredients” of the models (e.g., in the type-I models, 16⊕16⊕16 for the fermions plus 10⊕10⊕126⊕126⊕210 for the Higgs bosons) is selected with the only constraint of reproducing observation (i.e. the matter content is not constrained by any requirement of internal consistency of the models)... 

One way to render a GUT more predictive would be the discovery of a dynamical mechanism (quasi) fixing the values of the parameters of the Higgs potential at the GUT scale MX. In this section I discuss one such mechanism which might be available when looking at the GUT as an effective low-energy description of a more fundamental theory (possibly including gravity).  

I observe that, besides leading to the possibility of an increased predictivity (in the sense clarified above) of the SSB pattern, viewing GUTs as effective low-energy descriptions of a more fundamental theory, with the associated RG implications, requires a modification of the conventional tests of the naturalness of a GUT. These conventional tests typically assign a “low grade” to GUTs in which a fine tuning of the Higgs parameters is needed for a phenomenological SSB pattern; however, the effective-theory viewpoint on GUTs would require to check whether the phenomenological SSB pattern corresponds to fine tuning of the Higgs parameters at the scale M∗. It is plausible that a scenario requiring no fine tuning of the Higgs parameters at M∗ might correspond via the RG running (for example in presence of an appropriate infra-red fixed point) to a narrow region (apparent fine tuning) of the Higgs parameter space at MX , where the SSB is decided. On the other hand, it is also plausible that a SSB pattern corresponding to a significant portion of the Higgs parameter space actually requires some level of fine tuning at M∗ (for example, the considered portion of Higgs parameter space might be “disfavored” by the RG running). 

Concerning the scale M∗ at which the GUT becomes relevant as an effective low-energy theory, it should be noticed that, while  any scenario with M∗ >MX is plausible, the present (however limited) understanding of physics beyond the GUT scale MX suggests that M∗ could be within a few orders of magnitude of the Planck scale MP . In fact, it is reasonable to expect that beyond the GUT there is a theory incorporating gravity (a quantum gravity), and MP is the scale believed to characterize this more fundamental theory. 

It is also important to realize that the type of RG naturalness that I am requiring for GUTs is really a minimal requirement once the GUT is seen as an effective low-energy description of a more fundamental theory. In order to get a consistent GUT from this viewpoint one would also want that “nothing goes wrong” in going from the scale M∗ to the scale MX . For example, SSB should not occur “prematurely” at a scale µSSB such that MX <µSSB <M∗ , and the running of the masses involved in the RGEs should be taken into account. In relation to this point, it is interesting that the investigation of the RG naturalness of the parameters of the Higgs potential might ultimately help understanding also the emergence of the GUT scale. At present this scale is just a phenomenological  input of a GUT, resulting from the observed (low-energy) values of the GSM coupling constants, but it would be interesting to see it emerging as a scale within the GUT. By studying the RGEs for the parameters of the Higgs potential one might find such a scale; for example, assuming not-too-special initial conditions for the parameters at the scale M∗, one might find that the running of the parameters is such that SSB of the GUT occurs typically in the neighborhood of a certain scale hopefully a phenomenologically reasonable one).  

I also want to stress that one could consider additional consistency/naturalness conditions in order to render the GUT consistent with a working cosmological (early universe evolution) scenario. Such conditions should be properly formulated in the language of finite temperature field theory, and should take into account the fact that (contrary to the expectations often expressed in the literature) the dependence of couplings on the renormalization scale is different from their temperature dependence... 

As illustrated by the review of SO(10) GUTs ... GUTs typically involve a remarkably complex matter content. Most notably, the Higgs sector consists of several carefully selected IRREPs of the GUT group, and, like in the SM, the fermionic sector of leptons and quarks is arbitrary and is plagued by the family triplication. This complexity might well be telling us that the GUT idea needs drastic revisions; however, in this paper I take the point of view that the complexity of the matter content might be only apparent. I therefore want to mention a few appealing scenarios in which this complexity might arise from a fundamental simplicity. For continuity with the line of argument advocated in the previous section, let me start by mentioning the possibility that as an effective low-energy description of a more fundamental theory, the (effective) matter content of the GUT at the scale MX might be fixed by the RG running. It is in fact plausible that some IRREPs tend to get heavy masses via RG running, whereas the masses of other IRREPs (the ones relevant for symmetry breaking and low-energy phenomenology) might tend to be light (i.e. order MX or less). This type of running of masses (or other parameters) might even be responsible for the cancellation of anomalies at low energies; indeed, the RG running is known to be easily driven by symmetries... 

Another hypothesis which has been gaining some momentum in the literature on low-dimensional Quantum-Gravity toy models is that Quantum Gravity might be quite selective concerning the type of matter “it likes to deal with”, i.e. the requirement of overall consistency of Quantum Gravity might fix the matter content. Results pointing (however faintly) in this direction can be found for example in certain studies of discretized two-dimensional Quantum Gravities[25], and studies of the Dirac quantization of certain two-dimensional Quantum Gravities in the continuum [26 {26'}]... 

Perhaps the only robust concept discussed in this paper is the one concerning the way in which the conventional tests of the naturalness of a GUT need to be modified if the GUT is seen as a low-energy effective description of a more fundamental theory. 

On a more speculative side, I also articulated the hope that the correct GUT (if there is one) could be such that its SSB pattern is essentially predicted (in the sense of the RG naturalness I discussed) rather than being a free input; this would fit well the general trend of increased predictivity at each new stage of our understanding of particle physics. 

I have also looked at the complexity of the matter content of GUTs, and explored the possibility that this might be an apparent complexity, hiding a fundamental simplicity... I have speculated on a few appealing candidates for this simplicity; however, it is reasonable to expect that real progress in this direction will require dramatic new developments. 

Simple GUTs

G. Amelino-Camelia

(Submitted on 9 Oct 1996)

Two challenges for some courageous graduate students in particle physics phenomenology and cosmology:

• Explore the phenomenological consequences of the spectral Pati-Salam model(s) in particular those concerning neutrino masses and mixing* and check its compatibility with current low-energy data and potential falsifiability with future planned experiments. Investigate the possible "Renormalisation Group naturalness" of this simple(r?) GUT(s).  

• Build a cosmological model with a scope from a 10¹² GeV leptogenesis horizon to the current 0.23 eV scale, compatible with current astrophysical data and based only on the matter content of the spectral Pati-Salam model and the phenomenology of mimetic gravity consistent with the spectral action principle. Explore the phenomenological implications in particular concerning the different cosmological backgrounds (gravitational, electromagnetic and neutrino sectors).

* Neutrino may sing "from their GUTs"

From a particle physics phenomenologist perspective, spectral models may appear not very appealing as the matter content does not offer great perspective regarding the discovery of an exotic particle at a man made accelerator or even detector: no sparticle or wimp for instance. Nevertheless the recent article summarized below shows how some minimal grand unified models can help us to stay tuned on the faint neutrino song:

Minimal SO(10) grand unified models provide phenomenological predictions for neutrino mass patterns and mixing. These are the outcome of the interplay of several features, namely: i) the seesaw mechanism; ii) the presence of an intermediate scale where B-L gauge symmetry is broken and the right-handed neutrinos acquire a Majorana mass; iii) a symmetric Dirac neutrino mass matrix whose pattern is close to the up-type quark one. In this framework two natural characteristics emerge. Normal neutrino mass hierarchy is the only allowed, and there is an approximate relation involving both light-neutrino masses and mixing parameters. This differs from what occurring when horizontal flavour symmetries are invoked. In this case, in fact, neutrino mixing or mass relations have been separately obtained in literature. In this paper we discuss an example of such comprehensive mixing-mass relation in a specific realization of SO(10) and, in particular, analyse its impact on the expected neutrinoless double beta decay effective mass parameter hmeei, and on the neutrino mass scale. Remarkably a lower limit for the lightest neutrino mass is obtained (m_lighest  ≥ 7.5×10^-4 eV, at 3 σ level). 

The solid (dashed) lines bound the allowed region in the m_lighest –(m_ee) plane obtained by spanning the 3 σ ranges for the neutrino mixing parameters [9] in case of NH (IH). The dotted (dot-dashed) line is the prediction of eq. (9) on the effective mass, once the NH (IH) best-fit values of the neutrino mixing parameters are adopted [9]. The shaded region represents the 3 σ area obtained according to the neutrino mass-mixing dependent sum rule of eq. (9).

A neutrino mass-mixing sum rule from SO(10) and neutrinoless double beta decay

F. Buccella, M. Chianese, G. Mangano, G. Miele, S. Morisi, P. Santorelli

(Submitted on 2 Jan 2017 (v1), last revised 11 Jan 2017 (this version, v2))

//Last edit February 20, 2017