Predictive Theory of Neutrino Masses

PREDICTIVE THEORY OF NEUTRINO MASSES

Olakanmi F. Akinto *,†,‡ and Farida Tahir *,#

*Department of Physics, COMSATS University Islamabad, Pakistan

†Department of Physics, National Mathematical Centre, Sheda-Kwali Abuja, Nigeria

In our recent paper [1] we formulated a predictive theory of neutrino masses by considering the interaction between the infrared sector of the effective theory of quantum gravity and the standard model fields. This allowed us to calculate, for the first time in the history of neutrino physics, the absolute scale of neutrino masses. From this theoretical framework, we obtained quantum-gravitational couplings/effective Majorana dimensionless couplings from the spherically symmetric vacuum solutions arising from the Bose-Einstein statistical modification to gravitation. In the present paper, we show that the same solutions can be obtained directly from the quantum interpretation of gravitational radiation arising from the thermodynamic modification to gravitation. Within this theoretical scheme, we show that the single-field inflationary model, GUTs, dark energy and matter-independent gravitational field of vacuum are all connected to the neutrino mass model.

Keywords: Neutrino masses, GUTs, quantum gravity

I. INTRODUCTION trajectories across the universe. This property makes neutrinos the most eloquent footprint of Neutrinos have been around, literally, since the a possible quantum-classical nature of gravity, beginning of time. In the sweltering moments representing a unique probe for investigating following the Big-Bang epoch, neutrinos were physics at all distance scales [1-2]. among the first sub-atomic particles to emerge from this primordial fireball. Neutrinos are However, the question of how neutrino masses transients, interacting only through the weak arise and their absolute values have not been force and gravity thereby tracing long, lonely answered conclusively: In the standard model

(SM) of particle physics, neutrinos are predicted and 푚3 is the mass of the third distinct neutrino as massless and electrically neutral (sub-atomic) mass-eigenstate, labeled 풱3. Nufit Collaboration fermions, but the experimental results of all the and other world neutrino oscillation data have neutrino-oscillation experiments have shown, also measured, with very good precision (but convincingly, that neutrinos do, indeed, have a with calculative assumption that the minimum vanishingly small mass obtained from squared- neutrino mass is zero ( 푖. 푒. , 푚1 = 0)), the two

2 2 2 mass differences ∆푚푗푖 = 푚푗 − 푚푖 [3]. Precision independent neutrino squared-mass differences

2 +0.21 −5 2 2 measurements of the flux of solar neutrinos as ∆푚21 = (7.42−0.20) × 10 푒푉 and ∆푚31 =

+0.026 −3 2 revealed that fewer electron-type neutrinos (2.517−0.028) × 10 푒푉 for normal mass arrived at the Earth than predicted (about one- ordering (NMO). In this case the mass-sum is

2 2 third of the predicted value). This problem was given as ∑ 푚풱 = √∆푚21 + √∆푚31 ≃ definitely resolved by the SNO experiment in the 0.0588푒푉[4]. 2000s. Eventually solar neutrino data imply that Unfortunately, neutrino oscillations are not electron-neutrinos are linear superpositions of sensitive to the absolute neutrino masses 푚푖 and at least two neutrino mass-eigenstates. In this 푚1 = 0 (for NMO) is an artefact of calculative case the difference between the neutrino assumption used in neutrino-physics community squared-masses is of order ∆푚2 = 푚2 − 21 2 which is in strong tension with the Nature’s 푚2 ~ 10−4푒푉2. Where 푚 and 푚 are the 1 1 2 modus operandi as we have shown in [1]. Hence nonzero masses (푖. 푒. , 푚 > 푚 ≠ 0) of two 2 1 further information about the absolute scale of different neutrino mass-eigenstates, labelled 풱 1 neutrino masses must be investigated from non- and 풱 . Similarly, the precision measurements of 2 oscillation approaches, using beta decay, the flux of atmospheric neutrinos also revealed neutrinoless double beta decay, or cosmological that about one-third of the muon-type neutrinos observations [5]. Presently, there are two known survived passage through the Earth than natural explanations of the smallness of neutrino predicted. The solution to this atmospheric masses with respect to those of the other SM neutrino problem was the realization that muon- elementary particles: (i) the see-saw mechanism neutrinos are also linear superpositions of at [6] and (ii) the mechanism of effective least two neutrino mass-eigenstates, generation of a Majorana/Weinberg mass term where 푚 ≫ 푚 ≠ 0. Here the squared-mass 3 1 by physics beyond the SM (BSM). In both cases difference is of order ∆푚2 = 푚2 − 푚2 ~ 10−3푒푉2 31 3 1 massive neutrinos are Majorana particles,

2 leading to the possibility of observing Equation 2 is in agreement with that in [7].The 푑푠 neutrinoless double beta decay [3]. authors derived this formula (푝(푘) = 휇 푑푥휇 푚 푔 ) by considering the propagation of After decades of intense experimental and 푘 휇휈 neutrinos in a gravitational field of a non- theoretical research, we still do not know the rotating spherically symmetric object, which is mechanism that leads to nonzero neutrino described by the Schwarzschild metric. Although masses 푚푖. The purpose of [1] and this review is they arrived at the correct result, their derivation to discuss our formulation of neutrino mass is inconclusive because the values of their mass model constructed from the gravity-embedded term (푚 ) and gravitational field (푔 ) were not Weinberg operator. This idea is based on the fact 푘 휇휈 clearly defined. Here our theoretical intervention that neutrinos only participate in weak and is two-fold: (i) to show that the derivation in [7] gravitational interactions. Hence one could is compatible with the Nature and (ii) to solve Eq. construct a simple theory of neutrino masses by (2) completely within the contextual scheme of considering the interaction between the infrared GUTs and general radiative solutions of the exact sector of the effective theory of quantum gravity Einstein field equations. and the SM fields. Effectively, these masses are generated by the gravity-embedded 5- II. GRAND UNIFIED MODELS, SINGLE-FIELD dimensional Weinberg operator [1]: INFLATIONARY MODEL AND THEORY OF

2 NEUTRINO MASSES 풱퐸푤 푀휇휈 = 푔휇휈 . [1] 훬퐺푈푇 GUTs are one of the most interesting high- Where 푀휇휈 = 푚휇훿휇휈 (휇, 휈 = 0,1,2,3 표푟 1,2,3,4), 0 →푖푓 휇≠휈 energy completions of the SM because they and 훿휇휈 = {1→ 푖푓 휇=휈 [3]. Since 휇 = 휈 for both provide a rich and powerful group-theoretic Majorana neutrino masses 푀휇휈 and gravity, Eq. framework able to solve many BSM problems, (1) reduces to such as hot Big-Bang cosmology. 2 풱퐸푤 푚휇 = 푔휇휈 . [2] 훬퐺푈푇 The standard model of hot Big-Bang cosmology

Where 풱퐸푤(= 246퐺푒푉) is the electroweak relies on the assumption that: as the time (t) vacuum expectation value of Higgs field and approaches zero, the temperature T approaches

훬퐺푈푇 is the spontaneous symmetry-breaking infinity. That is, as t → 0, T → ∞ (where Energy mass-scale of Grand Unified Theories (GUTs). (퐸) = 푀푎푠푠(푀) = 푇푒푚푝푒푟푎푡푢푟푒 (푇), in

natural units 푘퐵 = ℏ = 푐 = 1). Since the

3 behaviour of the universe during the first Where 풢푆푀 = 푆푈(3)퐶 × 푆푈(2)퐿 × 푈(1)푌

−44 fraction of a second 푡 ≲ 10 푠 ( 푎푛푑 푇 ≳ 1.22 × and 풢퐸푀 = 푆푈(3)퐶 × 푈(1)푄. In the type-A 331 1019퐺푒푉) after the Big-Bang can only be a matter model, which is the simplest non-

훬퐺푈푇 for conjecture, it is plausible to construct the supersymmetric model of the type 풢331 → 풢푆푀 , theory of hot Big-Bang cosmology at some the symmetry breaking occurs at the minimum

17 16 temperature 푇0 = 10 퐺푒푉 based on GUTs and energy scale 훬퐺푈푇 = 1.63 × 10 퐺푒푉 for 푀푈 = special solutions of the exact Einstein field 1017퐺푒푉 [9]. (It is worth noting here that 331 equations. At this temperature, the description model can admit any simple groups like of the universe is taken as a set of initial 퐸6, 푆푈(5), 푆푈(7), 푆푂(10), etc. since it is an conditions [8]. This characteristic temperature intermediate gauge group between the SM and

17 (i.e., 푇0 = 10 퐺푒푉) is the unification the scale of unification of the three non- temperature-scale of grand unified models [9]. gravitational interactions). This value

(푖. 푒. , 훬 = 1.63 × 1016퐺푒푉) is consistent with Within the Big-Bang cosmology it is assumed 퐺푈푇 the value predicted (푖. 푒. , 훬 ≈ 2.0 × 1016퐺푒푉 ) that the universe expands and cools. During this 퐺푈푇 by many minimal supersymmetric GUTs and cooling epoch, the universe passes through some string-derived 퐺(224) [10-12]. critical temperatures corresponding to characteristic energy scales. These transitions We now have a hierarchy of two vacuum are connected with symmetry breakings and the expectation values 풱퐸푤 and 훬퐺푈푇 regulated by inflationary expansion of the early universe [8]. Majorana neutrino mass (see Eqs 1 and 2):

A general picture thus emerges of a grand unified 2 풱퐸푤 〈푚〉푒푓푓 ≡ = 3.7126푚푒푉. [4] model which begins with a simple gauge group 훬퐺푈푇

풢퐺푈푇 and a valid symmetry at the highest (This value is to be compared with the effective

17 〈 〉 energy 푀푈 = 10 퐺푒푉. As the energy is Majorana neutrino mass 1.2푚푒푉 ≲ 푚푒푒 ≲ considerably lowered (due to inflation), the 4푚푒푉 given within the range of 5-sigma in [13]). theory undergoes a two-step hierarchy of Since GUTs play an important role in the spontaneous symmetry breaking into successive inflationary dynamics of the early universe, their subgroups: imprints could be found in the cosmic microwave

훬퐺푈푇 풱퐸푤 background (CMB) observations by the Planck 풢퐺푈푇 → 풢푆푀 → 풢퐸푀 [3] satellite [14]. Hence for the cosmic inflation to

4 be compatible with the symmetry-breaking of light elements are consistent with the scheme given in Eq. (3), the vacuum energy existence of a cosmic neutrino background

0 (훬푖푛푓) responsible for the inflationary expansion (C풱B) with thermodynamic temperature 푇휈 . of the universe must be equal to 훬 : For 퐺푈푇 The thermodynamic temperatures of CMB and single-field inflationary model, the observed C풱B are related by [3, 17] amplitude of CMB anisotropies implies that [14] 1/3 0 11 0 푇훾 = ( ) 푇휈 . [7] 푟 1/4 4 훬 = 2.2 × 1016퐺푒푉 ( ) . [5] 푖푛푓 0.2 0 −4 Where 푇휈 ~2 × 10 푒푉. As the universe expands Where 푟 is the tensor-to-scalar ratio. Its peak and the relic neutrino background cools, the value has been measured from the Planck behaviour of neutrinos changes from that of temperature measurement, BICEP2 and keck ultrarelativistic relics (radiation) to that of instruments to be 푟 = 0.06 [15]. This same upper nonrelativistic species (matter) as long as value had been previously obtained when the 0 ∑ 푚풱 ≫ 푇휈 [4]. This radiation-matter transition spectral index 푛푠 = 0.98 in [16]. With this value is governed by 퐸 = 푇 = 푀 in natural units (푘퐵 = of 푟(= 0.06), Eq. (5) reduces to ℏ = 푐 = 1). This transition should leave an

16 훬퐺푈푇 = 훬푖푛푓 = 1.63 × 10 퐺푒푉 [6] imprint in the large - scale structure of the universe in such a way that precision Hence Eq. (3) can be regarded as the particle- measurements could provide nontrivial physics mechanism responsible for the cosmic information on the neutrino masses. In this case inflation. the relic neutrino background is best described A. Cosmology and Gravitational as a homogeneous mixture of the neutrino mass- couplings eigenstates 풱푖, where cosmic surveys are A very different way to get information on the expected to provide information on the absolute scale of neutrino masses is from the individual neutrino masses 푚푖. By combing 푀 = study of the CMB radiation spectrum (with 푇 with the spherically symmetric vacuum

0 thermodynamic temperature 푇훾 ) as well as the solutions of Einstein field equations in their study of the large scale structure in the universe. isotropic form, and using Majorana conditions

퐶 In the standard model of cosmology, neutrinos (풱퐿 = 풱퐿 , Ѳ = 푛휋 & 푠푖푛휃 = 0, 푎푛푑 푟 = √3) [1], are predicted to be relics of the hot Big-Bang. we get Here, the measurements of the relic abundance

푇 2 (1− ) 푚1 = 0.6675푚푒푉 , 푔 = 2√3 , [8] 00 푇 2 (1+ ) 2√3 푚2 = 14.4439푚푒푉, [14] 푇 4 푔 = (1 + ) , [9] 11 2√3 푚3 = 43.3316푚푒푉,

푇 4 푚 = 0푚푒푉. 푔 = 3 × (1 + ) , [10] 4 22 2√3 And the mass-sum for the three active 푔33 = 0. [11] neutrinos is given as

Where T is the thermodynamic temperature. 3 ∑푖=1 푚푖 = 푚1 + 푚2 + 푚3 = 0.0584푒푉. [15]

To further analyze Eqs 8-11, it is convenient to This value is to be compared with the minimum put the thermodynamic temperature T in value, ∑ 푚풱 ≃ 0.0588푒푉, obtained from the dimensionless form (by using Eq. (7)) since the neutrino-oscillation experiments [3-4]. Eq. (15) radial distance 푟 is dimensionless. Defining is also consistent with the 0.058푒푉 ≲ ∑ 푚휈 ≲

1 푇0 11 0.062푒푉 obtained within the range of 5-sigma in 푇 → 훾 = ( )3 = 1.4010. [12] 푇0 4 휈 [13]. From Eqs 8-12, the gravitational couplings are B. Atmospheric Neutrino and Solar calculated as Neutrino Squared-Mass Differences

푔00 = 0.1798, (i) The atmospheric neutrino squared-mass 푔11 = 3.8905, [13] difference is calculated (from Eq. (14)) as 푔22 = 11.6715 , 2 2 2 2 −3 2 ∆푚퐴 = ∆푚31 = 푚3 − 푚1 = 1.88 × 10 푒푉 [16] 푔33 = 0. Equation 16 is consistent with the measurement Equation 13, which is the matter-independent of atmospheric neutrinos and antineutrinos in gravitational field of vacuum, is to be compared 2 the MINOS far detector, where ∆푚31 = 1.9 × with the values of 푔휇휈 in [1]. 10−3푒푉2 for 90% single-parameter confidence From Eqs (2) and (13), the individual neutrino intervals [18]. masses are calculated as (ii) The solar neutrino squared-mass difference

is calculated (from Eq. (14)) as

2 2 2 2 −4 2 ∆푚⊙ = ∆푚21 = 푚2 − 푚1 = 2.08 × 10 푒푉 [17]

Equation 17 is consistent with all solar neutrino consistent with the observed value 휌훬 ≃ ((2.37 ±

4 oscillation experiments with KamLAND data 0.05) × 10−3푒푉) [21]. measured as ∆푚2 = ∆푚2 ≳ 2.0 × 10−4푒푉2 at 푆표푙 21 III. CONCLUDING REMARKS 99.73% C.L.[19]. We have explored a predictive neutrino mass Thus the mass-sum of the three active neutrinos model with a gravity-embedded Weinberg can be expressed in terms of the first part of Eq. operator, in which we have generated the (14), Eqs 16 and 17 as neutrino masses through the interaction

∑ 2 2 2 2 푚휈 = 푚1 + √∆푚21 + 푚1 + √∆푚31 + 푚1 = 0.0585푒푉 [18] between the infrared sector of the effective C. Cosmological Constant/Dark Energy theory of quantum gravity and the standard model fields. Embedding gravity into a two-step If truly the vacuum energy owes its existence to spontaneous symmetry breaking mechanism of the asymmetry/broken symmetry of the cosmos, grand unified models, we predict three active then the effective Majorana neutrino mass

neutrinos with nonzero masses 푚1,2,3 ≠ 0 and one regulates the value of vacuum energy as 휖푣푎푐 =

inactive neutrino with zero mass 푚4 = 0. This 〈푚〉푒푓푓 . Eq. (4) is to be compared with 휖푣푎푐 ≈ remarkable yet simple idea owes its unique 3.6푚푒푉 in [20]. Hence the proper vacuum existence to the quantum interpretation of energy density (or dark energy), responsible for gravitational radiation arising: from the Bose- the accelerated expansion of the universe, is Einstein statistical modification to gravitation in given as [1]: [1], and to the spherically symmetric vacuum 4 휌 = 푇푣푎푐 = 푔 [〈푚〉 ] = (2.42 × 10−3푒푉)4 [19] 푣푎푐 00 00 푒푓푓 solutions arising from the thermodynamic This value is to be compared with two different modification to gravitation as established in this theoretical approaches: (i) from the resummed paper. These two different theoretical quantum gravity in the context of asymptotic approaches only point to one conclusion that

−3 4 safety, we have 휌훬 ≃ (2.4 × 10 푒푉) [21]. And (ii) neutrinos produced in a weak interaction from low energy quantum gravity in the context (governed by extended electroweak theory) of QCD Veneziano ghost, we have must propagate in a gravitational field in order

−3 4 to manifest their oscillation property because 휌훬 ≃ 푐푔푟푎푣 × (3.6 × 10 푒푉) [20], where 푐푔푟푎푣 is a the squared-mass differences ∆푚2 depend only dimensionless gravitational coupling. These 푗푖 on the different values of 푔 . We argue that if three different theoretical approaches are 휇휈

2 the values of 푔휇휈 were to be the same, ∆푚푗푖 physics. The most potent probes of the nature of would vanish completely, and thus no oscillation the neutrino are tests of lepton number would have been observed. Hence neutrino- violation/conservation, especially searches for oscillation phenomenon is a direct consequence neutrinoless double-beta decay. Another of gravity-embedded 5-dimensional Weinberg important prediction of the theoretical scheme operator. If Equation 2 is correct, one can make presented in this paper states that dark a robust prediction that: Neutrinos are Majorana energy/proper vacuum energy density owes its particles and lepton number is not an exact existence to the two-way symmetric breaking symmetry of Nature. Determining the nature of of non-supersymmetric GUT and the matter- the neutrinos (whether Dirac or Majorana independent gravitational field of vacuum. particles) has received a high priority in particle

i [1] Olakanmi F. Akinto and Farida Tahir, “Effective ℒ5 [9] R. A. Diaz, D. Gallego and R. Martinez, Lagrangian and masses of Majorana neutrinos”, Int. J. “Renormalization Group and Grand Unification with of Geom. Methods Mod. Phys. 16 (2021). 331 Models”, Int. J. Mod. Phys. A22 (2007). [2] Giuseppe Gaetano Luciano and Luciano [10] A Salam Memorial Meeting (1997), ed., J. Ellis, et Petruzziello, “Testing gravity with neutrinos: from al., ISBN 98110236190 (P. 98-127). classical to quantum regime”, Int. J. Mod. Phys. D29 [11] J. M. Gipson and K.E. Marshak, “Intermediate (2020). mass scales in the new SO (10) grand unification in the [3] C. Giunti and C.W. Kim, “Fundamentals of one loop approximation”, Phys. Rev. D31 (1985). Neutrino Physics and Astrophysics” (Oxford [12] V.N. Senoguz and Q. Shafi, “Testing University Press, 2007). supersymmetric grand unified models of inflation”, [4] Joseph A. Formaggio, André Luiz C. de Gouvea and Phys. Lett. B567 (2003). R.G. Harmish Robertson, “Direct Measurements of [13] Hiroshi Okada and Yuta Orikasa, “Neutrino mass Neutrino Mass”, Phys. Rep. 914 (2021). model with a modular 푆4 symmetry”, arXiv: [5] Guo-yuan Huang, Werner Rodejohann and Shun 1908.08409v3 [hep-ph] (2021). Zhou, "Effective neutrino masses in KATRIN and [14] Djuna Croon, et al., “GUT Physics in the Era of the future tritium beta-decay experiments”, Phys. Rev. LHC”, Front. Phys. 04 (2019). D101 (2020). [15] P.A.R. Ade, et al., (Keck Array and BICEP2 [6] André de Gouvea, “Neutrino Mass Models”, Annu. Collaboration), “Constraints on Primordial Rev. Nucl. Part. Sci. 66 (2016). Gravitational Waves using Planck WMAP and New [7] N. Fornengo, C. Giunti, C.W. Kim and J. Song, BICEP2/Keck Observations through the 2015 Season”, “Gravitational effects on the neutrino oscillation”, Phys. Rev. Lett. 121 (2018). Phys. Rev. D56 (1997). [16] H.V. Peiris, et al., “First-year Wilkinson [8] Alan H. Guth, “Inflationary universe: A possible Microwave Anisotropy Probe (WMAP) Observations: solution to the horizon and flatness problem”, Phys. Implications for Inflation”, APJS 148 (2003). Rev. D23 (1981). [17] S. Weinberg, “Gravitation and Cosmology: Principles and Applications of General Theory of Relativity” (MIT, John Wiley and Son 1972).

[20] Federico R. Urban and Ariel R. Zhitnitsky, “The [18] P. Adamson, et al., “Measurement of cosmological constant from the QCD Veneziano atmospheric neutrinos and antineutrinos in the ghost”, Phys. Lett. B688 (2010). MINOS Far Detector (MINOS Collaboration)” Phys. [21] B.F.L. Ward, “An Estimate of Λ in Resummed Rev. D86 (2012). Quantum Gravity in the Context of Asymptotic [19] H. Nunokawa, W.J.C. Teves and R. Zukanovich Safety”, Phys. Dark Universe 2 (2013). Funchal, “Determining the oscillation parameters by solar neutrinos and KamLAND”, Phys. Lett. B562 (2003).