Probing the predictions of an orbifold theory of flavor
Francisco J. de Anda, Newton Nath, José W. F. Valle, Carlos A. Vaquera-Araujo
PProbing the predictions of an orbifold theory of flavor
Francisco J. de Anda, ∗ Newton Nath, † Jos´e W. F. Valle, ‡ and Carlos A. Vaquera-Araujo
4, 5, § Tepatitl´an’s Institute for Theoretical Studies, C.P. 47600, Jalisco, M´exico Instituto de F´ısica, Universidad Nacional Aut´onoma de M´exico, A.P. 20-364, Ciudad de M´exico 01000, M´exico. AHEP Group, Institut de F´ısica Corpuscular – C.S.I.C./Universitat de Val`encia, Parc Cient´ıfic de Paterna.C/ Catedr´atico Jos´e Beltr´an, 2 E-46980 Paterna (Valencia) - SPAIN Consejo Nacional de Ciencia y Tecnolog´ıa, Av. Insurgentes Sur 1582. Colonia Cr´edito Constructor,Del. Benito Ju´arez, C.P. 03940, Ciudad de M´exico, M´exico Departamento de F´ısica, DCI, Campus Le´on, Universidad de Guanajuato,Loma del Bosque 103, Lomas del Campestre C.P. 37150, Le´on, Guanajuato, M´exico
We examine the implications of a recently proposed theory of fermion masses and mixings inwhich an A family symmetry emerges from orbifold compactification. We analyse two variantschemes concerning their predictions for neutrino oscillations, neutrinoless double-beta decay andthe golden quark-lepton unification mass relation. We find that upcoming experiments DUNE as wellas LEGEND and nEXO offer good chances of exploring a substantial region of neutrino parameters. I. INTRODUCTION
The discovery of neutrino oscillations [1, 2] has prompted a great experimental effort toward precision measure-ments [3]. Indeed, the pattern of neutrino mass and mixing parameters is strikingly at odds with the one thatcharacterizes the quark sector, suggesting that it can hardly be expected to happen just by chance. The most popularapproach to bring a rationale to the pattern of neutrino mixing involves the idea that there is some non-Abelianfamily symmetry in nature. In a model-independent way one may assume the existence of some residual CP sym-metry characterizing the neutrino mass matrix, irrespective of the details of the underlying theory [4–6]. A moreambitious approach is, of course, to guess what the family symmetry actually is, and to build explicit flavor models ona case-by-case basis [7–11]. However, pinning down the nature of such symmetry among the plethora of possibilitiesis a formidable task.An interesting theoretical idea has been to imagine the existence of new dimensions in space-time, as a wayto shed light on the possible nature of the family symmetry in four dimensions. In this context, six-dimensionaltheories compactified on a torus have been suggested [12, 13] and a realistic standard model extension has recentlybeen proposed [14] in which fermions are nicely arranged within the framework of an A family symmetry. Thetheory yields very good predictions for fermion masses and mixings, including the “golden” quark-lepton unificationformula [15–19].In this work we focus on the possibility of probing the implications of this theory within the next generation ofneutrino experiments. This includes the long-baseline oscillation experiment DUNE [20, 21] as well as neutrinolessdouble-beta decay (0 νββ for short) searches. In Sec. II we describe the theory framework, identifying two modelsetups, while in Sec. III we determine the potential of upcoming neutrino experiments, such as DUNE and 0 νββ experiments to probe our orbifold compactification predictions. ∗ [email protected] † newton@fisica.unam.mx ‡ valle@ific.uv.es § vaquera@fisica.ugto.mx a r X i v : . [ h e p - ph ] J un II. THEORY FRAMEWORK
Our model features a 6-dimensional version of the Standard Model SU (3) ⊗ SU (2) ⊗ U (1) gauge symmetry, togetherwith 3 right handed neutrinos and supplemented with the orbifold compactification described in our previous paper[14]. The transformation properties of the fields under the gauge and A family symmtetry and their localization onthe orbifold are shown in table I. Field SU (3) SU (2) U (1) A Z Localization L − / ω Brane d c ¯ / ω Brane e c ω Brane Q / ω Brane u c , , ¯ − / (cid:48)(cid:48) , (cid:48) , ω Bulk ν c H u / ω Brane H d − / H ν / ω Brane σ TABLE I. Field content of the model.
The scalar sector consists of three Higgs doublets and an extra singlet scalar σ , all transforming as flavor triplets.They are charged under a Z symmetry, so that H d only couples to down-type fermions (charged leptons and downquarks), H u couples only to up-quarks and H ν only couples to neutrinos.The effective Yukawa terms are given by L Y = y N ν c ν c σ + y ν ( LH ν ν c ) + y ν ( LH ν ν c ) + y d ( Qd c H d ) + y d ( Qd c H d ) + y e ( Le c H d ) + y e ( Le c H d ) + y u ( QH u ) (cid:48) u c + y u ( QH u ) (cid:48)(cid:48) u c + y u ( QH u ) u c , (1)where the symbol () , indicates the possible singlet contractions × × → , and × → , (cid:48) , (cid:48)(cid:48) in A . Alldimensionless Yukawa couplings are assumed to be real due to a CP symmetry.The scalar field σ gets a vacuum expected value (VEV) that breaks spontaneously lepton number and the A familysymmetry, giving large Majorana masses to the right handed neutrinos. The corresponding VEV is aligned as (cid:104) σ (cid:105) = v σ ωω , (2)with ω = e πi/ , the cube root of unity.As A is broken at a high mass scale, the Higgs doublets can obtain the most general spontaneous CP violatingalignment, which we parametrize as (cid:104) H u (cid:105) = v u (cid:15) u e iφ u (cid:15) u e iφ u , (cid:104) H ν (cid:105) = v ν e iφ ν (cid:15) ν e iφ ν (cid:15) ν e iφ ν , (cid:104) H d (cid:105) = v d e iφ d (cid:15) d e iφ d (cid:15) d e iφ d . (3)An important prediction of the model comes from the fact that the charged leptons and down-quarks obtain theirmasses from the same H d , so that the A structure implies the golden relation between their masses [15] m τ √ m µ m e = m b √ m s m d , (4)This relation is in good agreement with experiments [22] and is rather robust against renormalization group running.The explicit form of the mass matrices for the matter fields (up to unphysical rephasings) is given as M u = v u y u (cid:15) u y u (cid:15) u y u (cid:15) u y u (cid:15) u ω y u (cid:15) u ω y u (cid:15) u y u ω y u ω y u ,M d = v d y d (cid:15) d e i ( φ d − φ d ) y d (cid:15) d y d (cid:15) d e i ( φ d − φ d ) y d y d (cid:15) d y d ,M e = v d y e (cid:15) d e − i ( φ d − φ d ) y e (cid:15) d y e (cid:15) d e − i ( φ d − φ d ) y e y e (cid:15) d y e ,M RN = y N v σ ω ωω ω ,M Dν = v ν y ν (cid:15) ν e i ( φ ν − φ ν ) y ν (cid:15) ν y ν (cid:15) ν e i ( φ ν − φ ν ) y ν y ν (cid:15) ν y ν ,M Lν = M Dν ( M RN ) − ( M Dν ) T . (5)In what follows we adopt the standard parametrization for the Cabibbo-Kobayashi-Maskawa (CKM) matrix V CKM = c q c q s q c q s q e − iδ q − s q c q − c q s q s q e iδ q c q c q − s q s q s q e iδ q c q s q s q s q − c q s q c q e iδ q − c q s q − s q s q c q e iδ q c q c q , (6)and the symmetrical presentation of the lepton mixing matrix [23, 24], K = c (cid:96) c (cid:96) s (cid:96) c (cid:96) e − iφ s (cid:96) e − iφ − s (cid:96) c (cid:96) e iφ − c (cid:96) s (cid:96) s (cid:96) e − i ( φ − φ ) c (cid:96) c (cid:96) − s (cid:96) s (cid:96) s (cid:96) e − i ( φ + φ − φ ) c (cid:96) s (cid:96) e − iφ s (cid:96) s (cid:96) e i ( φ + φ ) − c (cid:96) s (cid:96) c (cid:96) e iφ − c (cid:96) s (cid:96) e iφ − s (cid:96) s (cid:96) c (cid:96) e − i ( φ − φ ) c (cid:96) c (cid:96) , (7)with c fij ≡ cos θ fij and s fij ≡ sin θ fij where f = q, (cid:96) . The advantage of using the symmetrical parametrization forthe lepton mixing matrix resides in the transparent role of the Majorana phases in the effective mass parametercharacterizing the amplitude for neutrinoless double beta decay (cid:104) m ββ (cid:105) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) j =1 K ej m j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12) c (cid:96) c (cid:96) m + s (cid:96) c (cid:96) m e iφ + s (cid:96) m e iφ (cid:12)(cid:12) , (8)while keeping a rephasing-invariant expression for the Dirac phase δ (cid:96) = φ − φ − φ (9)which affects neutrino oscillation probabilities. A. Model Set-up I (MI)
Following [14], we may further assume that the Higgs’s VEVs preserve conventional (trivial) CP symmetry, andtherefore they are real. Together with the reality of the Yukawa couplings, this implies that the only source of CPviolation is the phase ω . This leads to a very strong predictivity.The model is specified by 15 parameters ( y ν , v ν , y e,d , v d , y u , , v u , (cid:15) u,ν,d , ) that describe 22 low-energy flavor observ-ables: ( m u,c,t,d,s,b,e,µ,τ , m ν , , , θ q , , , δ q , θ l , , , φ , , ) including the neutrino Majorana phases. One extraparameter ( y N v σ ) defines the masses of the 3 right handed neutrinos. Parameter Value y e v d / GeV 1 . y e v d / (10 − GeV) − . y d v d / (10 − GeV) − . y d v d / GeV 2 . y ν v ν / (cid:112) Y N v σ meV × − . y ν v ν / ( (cid:112) Y N v σ meV × − ) 1 . y u v u / (10 − GeV) 6 . y u v u / (10 GeV) 1 . y u v u / GeV − . (cid:15) u / − − . (cid:15) u / − . (cid:15) d / − − . (cid:15) d / − . (cid:15) ν . (cid:15) ν / − − .
23 Observable Data Model best fitCentral value 1 σ range θ (cid:96) / ◦ → . θ (cid:96) / ◦ → . θ (cid:96) / ◦ → . δ (cid:96) / ◦
237 210 →
275 268 m e / MeV 0.489 0.489 → . m µ / GeV 0.102 0.102 → . m τ / GeV 1.745 1.743 → . m / (10 − eV ) 7.55 7.39 → . m / (10 − eV ) 2.50 2.47 → . m / meV 4 . m / meV 9 . m / meV 50 . φ / ◦ φ / ◦ φ / ◦ θ q / ◦ → . θ q / ◦ → . θ q / ◦ → . δ q / ◦ → . m u / MeV 1.28 0.76 → . m c / GeV 0.626 0.607 → . m t / GeV 171.6 170 →
173 171 . m d / MeV 2.74 2.57 → . m s / MeV 54 51 →
57 51 m b / GeV 2.85 2.83 → . χ . TABLE II. Global best-fit of flavor observables within MI . Here CP violation is generated by a fixed phase ω . One can perform a global fit to the flavor observables by defining the chi-square function χ = (cid:88) ( µ exp − µ model ) /σ exp , (10)where the sum runs through the 19 measured physical parameters (note that the overall neutrino mass scale andthe two Majorana phases are currently undetermined). We make use of the MPT package [25] to obtain the flavorobservables from the mass matrices in Eq. (5). Then we scan the 15 free parameters and find the values thatminimize the χ function. Neutrino oscillation parameters are taken from the global fit in Ref. [3], while the rest ofthe observables are taken from the PDG [22]. For consistency of the fit, all quark and charged lepton masses areevolved to the same common scale, which we choose to be M Z . The running of CKM and neutrino mixing parametersis negligible [26, 27]. The results are shown in table II. One sees from the fit that χ = 12 .
4. This indicates a relativelygood global fit, with some tension in the description of quark CP violation, as seen from the table. The origin for thisis traced to the absence of a free parameter describing CP violation, as discussed above.
B. Model Set-up II (MII)
We can now relax the assumption that all the Higgs VEVs are real, allowing them to be general complex numbers.However, we keep the assumption that the Yukawa couplings are real. This reinstates 2 physical phases φ ν,d − φ ν,d ,now increasing to the number of free parameters to 17, for a total of 22 flavor observables. Parameter Value y e v d / (10 − GeV) − . y e v d / GeV 1 . y d v d / (10 − GeV) − . y d v d / GeV 2 . y ν v ν / (cid:112) Y N v σ meV − . y ν v ν / (cid:112) Y N v σ meV 7 . y u v u / (10 − GeV) 6 . y u v u / (10 GeV) 1 . y u v u / GeV 7 . (cid:15) u / − − . (cid:15) u / − − . (cid:15) d / − . (cid:15) d / − . (cid:15) ν / − − . (cid:15) ν / − . φ d − φ d ) /π − . φ ν − φ ν ) /π .
093 Observable Data Model best fitCentral value 1 σ range θ (cid:96) / ◦ → . θ (cid:96) / ◦ → . θ (cid:96) / ◦ → . δ (cid:96) / ◦
237 210 →
275 198 . m e / MeV 0.489 0.489 → . m µ / GeV 0.102 0.102 → . m τ / GeV 1.745 1.743 → . m / (10 − eV ) 7.55 7.39 → . m / (10 − eV ) 2.50 2.47 → . m / meV 24 . m / meV 25 . m / meV 55 . φ / ◦ . φ / ◦ . φ / ◦ . θ q / ◦ → . θ q / ◦ → . θ q / ◦ → . δ q / ◦ → . m u / MeV 1.28 0.76 → . m c / GeV 0.626 0.607 → . m t / GeV 171.6 170 →
173 171 . m d / MeV 2.74 2.35 → . m s / MeV 54 51 →
57 54 m b / GeV 2.85 2.76 → . χ . TABLE III. Global best-fit of flavor observables within MI . Here there are two free CP violation phases. In this general setup, we loose predictivity for the physical CP violating phases δ l,q , leading to a drastic improvementof the global fit, achieving a minimum χ = 1 . δ q .As mentioned above, a characteristic feature of our schemes is the golden quark-lepton mass relation given in Eq. 4.We now turn to study this prediction as obtained from our global fits of flavor observables within Models I and II. InFig. 1 we use the golden quark-lepton mass relation in MI and MII to make predictions for the down and strangequark masses. Here the cyan bands stand for the 1, 2 and 3 σ regions compatible with the exact golden relation m τ / √ m µ m e = m b / √ m s m d at the M Z scale, and the yellow contours are the 1, 2 and 3 σ regions for the quark massparameters measured at the same scale. To better appreciate the predictive power of our framework, we have variedrandomly the parameters of MII around the best fit point in Table III and we have determined the shape of theparameter region consistent at 3 σ with all the 19 measured parameters of the model. This region is shown in purplein Fig. 1. The corresponding contour for MI is not shown as it is very similar, given the fact that the golden relationis not very sensitive to the improvement of the CP violating phases in MII , compared to MI . However, one can seethat the best fit point for scheme MII , indicated by the black cross, is now compatible at 1 σ with the exact goldenformula.
40 45 50 55 60 65 702.02.53.03.54.0 � � / ��� � � / � � � ×× FIG. 1. Prediction for the down-quark and strange-quark masses at the M Z scale. The cyan contours represent the 1, 2 and3 σ allowed regions from the golden relation m τ / √ m µ m e = m b / √ m s m d . The yellow contours show the 1, 2 and 3 σ ranges ofthe measured quark masses at the M Z scale [27]. The blue region is the allowed parameter space consistent at 3 σ with theglobal flavour fit in Table III. The red (black) cross indicates the location of the best fit point for MI ( MII ). III. PROBING NEUTRINO PREDICTIONS
In this section we present a close-up of the neutrino predictions of our orbifold compactification schemes, examiningalso the capability of future experiments to test them.
A. Neutrino oscillations at DUNE
We start by quantifying the capability of the DUNE experiment to test the oscillation predictions resulting from the A family symmetry, as realized from a six-dimensional spacetime after orbifold compactification. Before presentingdetails about the simulated results, we first give a brief technical overview of the simulation details of DUNE, theproposed next generation superbeam neutrino oscillation experiment at Fermilab, USA [20, 21]. The collaborationplans to use neutrinos from the Main Injector (NuMI) at Fermilab as a neutrino source. In this experiment, the firstdetector will record particle interactions near the beam source, at Fermilab. On the other hand, the neutrinos fromFermilab will travel a distance of 1300 km before reaching the far detector situated at the underground laboratory of“Sanford Underground Research Facility (SURF)” in Lead, South Dakota. The proposed far detector will use four10 kton volume of liquid argon time-projection chambers (LArTPC). The expected neutrino flux corresponding to 1.07MW beam power gives 1 . × protons on target (POT) per year for an 80 GeV proton beam energy. We followthe same procedure as given in [28] for performing our numerical analysis of DUNE. The GLoBES package [29, 30]along with the auxiliary files as mentioned in [21] has been utilized for the simulation. We adopt 3.5 years runningtime in both neutrino and antineutrino modes, with a 40 kton total detector volume. In the numerical analysis, wealso take into account both the appearance and disappearance channels of neutrinos and antineutrinos. In addition,both the signal and background normalization uncertainties for the appearance as well as disappearance channelshave been taken into account in our analysis, as mentioned in the DUNE CDR [21]. θ l δ l / π * MI σ σ σ FIG. 2.
DUNE sensitivity region in the (sin θ l , δ l ) plane. The ‘star-mark’ represents the latest neutrino oscillation best-fit [3], whilethe ‘black-dot’ is the predicted best-fit, as given in Table II. Given that normal mass ordering (i.e., m < m < m ) of neutrinos is currently preferred over the inverted one(i.e., m < m < m ) at more than 3 σ [3], we focus on the first scenario throughout this work.In what follows, we examine the sensitivity regions of DUNE in the (sin θ l , δ l ) for different seed points. Theseare shown at 1 σ (dark-orange), 2 σ (orange), and 3 σ (lighter-orange) confidence level, respectively. θ l δ l / π * σ σ σ θ l δ l / π * σ σ σ FIG. 3.
DUNE (sin θ , δ l ) sensitivity regions in models MI (left) and MII (right), assuming the corresponding best-fit points obtainedin setup MI (left-panel) and MII (right-panel), respectively, as indicated by the ‘star-marks’.
In Fig. 2, we show the expected 1, 2 and 3 σ DUNE sensitivity regions in the (sin θ l , δ l ) plane. Here we assumemodel setup MI and take the latest neutrino oscillation best-fit point from [3] as benchmark, as indicated by theblack ‘star-mark’. The corresponding MI theory predictions are indicated by the brown 3 σ confidence level region,and its best-fit point, as given in Table II, is shown by the black ‘dot’. One sees that DUNE will be able to rule outpredicted correlation between sin θ l , and δ l for MI at 1 σ C. L. In contrast, we note that the predicted region inmodel
MII covers the full DUNE sensitivity contours, so we do not show this plot in Fig. 2. In other words, if thecurrent best fit value of the oscillation parameters remains, DUNE will not be able to rule out the predictions for
MII even at 1 σ C. L.We now change our seed points, adopting as benchmarks the (sin θ l , δ l ) best-fit points predicted in each of themodels described above. The resulting DUNE sensitivity regions are given in Fig. 3.One sees from the left-panel that, if the MI predicted value of (sin θ l , δ l ) is the true beanchmark value, thenDUNE (after 3.5 running time in both neutrino and antineutrino modes) can rule out maximal value of θ l i.e.,sin θ l = π/ σ confidence level. On the other hand, by adopting the MII predicted best-fit as the true seedvalue, we notice from the right-panel that DUNE can rule out maximal value of sin θ l at 1 σ , whereas it can ruleout δ l = 3 π/ σ confidence level. B. Neutrinoless double-beta decay
We now turn to the predictions for neutrinoless double beta decay in Models MI and MII and confront themwith experimental sensitivities [31–38]. This is shown in Fig. 4. The values for the effective mass (cid:104) m ββ (cid:105) consistentat 3 σ with the measured flavor observables (mainly neutrino oscillation parameters) obtained from the global fit arerepresented by the green contour for the case of the “constrained” model MI , and by the blue one for MII . Thetheory-predicted regions are obtained by allowing the free parameters to vary randomly from the best fit point whilesimultaneously complying at 3 σ with all the measured observables of the global fit. One sees that the predicted regionfor MII becomes wider, while the region for MI remains quite small. This is due to the effect of the variation of theavailable free phases in MII φ ν,d − φ ν,d , which are directly related to the Majorana phases. In contrast, in MI the - - - - - - - - � � / �� < � ββ > / � � NO KamLAND - Zen ( Xe ) SNO + Phase IILEGENDnEXO D i s f av o r e db y C o s m o l og y FIG. 4. Effective Majorana neutrino mass parameter (cid:104) m ββ (cid:105) as a function of the lightest active neutrino mass m . Here green(blue) contour represents the predicted (cid:104) m ββ (cid:105) parameter space consistent at 3 σ with the global flavor fit for model setup MI(MII), and the best-fit value is shown by the red (black) dot. The current KamLAND-Zen limit is shown by the light-yellowband, and the projected sensitivities for future experiments are indicated in dashed horizontal lines, see text for details. only available CP violating phase is fixed, leading to sharply predicted 0 νββ decay amplitude which can not deviatemuch from its best fit value.Interestingly enough, predictivity is not destroyed by the inclusion of those extra phases, and MII still has upperand lower bounds for both the effective mass (cid:104) m ββ (cid:105) and the lightest neutrino mass parameter. As a visual guide forthe experimental searches of 0 νββ , in Fig. 4 the horizontal yellow band indicates the current experimental limits fromKamland-Zen (61 −
165 meV) [31], while the dashed lines correspond to the most optimistic sensitivities projected forSNO + Phase II (19 −
46 meV) [35], LEGEND (10 . − . . − . νββ experimentsLEGEND and nEXO. Finally, the vertical gray band represents the current sensitivity of cosmological data from thePlanck collaboration [39]. IV. CONCLUSION
We have investigated the implications of a recently proposed theory of fermion masses and mixings based on an A family symmetry that arises from the compactification of a 6-dimensional orbifold. We have analysed two variantionsof the idea, a “constrained” one, in which CP violation is strictly predicted, and another where CP phases are freeto vary. We have quantified the predictions of these schemes for neutrino oscillations, neutrinoless double-beta decayand the golden quark-lepton mass formula. We have found that the projected long baseline experiment DUNE canprobe the model predictions concerning the maximality of the atmospheric mixing or the value of the CP phase ina meaningful way. Likewise, the next generation of neutrinoless double-beta decay experiments, especially LEGENDand nEXO, could probe our model MII in a substantial region of parameters.
ACKNOWLEDGMENTS
Work is supported by the Spanish grants SEV-2014-0398 and FPA2017-85216-P (AEI/FEDER, UE), PROME-TEO/2018/165 (Generalitat Valenciana) and the Spanish Red Consolider MultiDark FPA2017-90566-REDC. CAV-Ais supported by the Mexican C´atedras CONACYT project 749 and SNI 58928. NN is supported by the postdoc-toral fellowship program DGAPA-UNAM, CONACYT CB-2017-2018/A1-S-13051 (M´exico) and DGAPA-PAPIITIN107118. [1] T. Kajita, “Nobel Lecture: Discovery of atmospheric neutrino oscillations,”
Rev.Mod.Phys. (2016) 030501.[2] A. B. McDonald, “Nobel Lecture: The Sudbury Neutrino Observatory: Observation of flavor change for solar neutrinos,” Rev.Mod.Phys. (2016) 030502.[3] P. de Salas et al. , “Status of neutrino oscillations 2018: 3 σ hint for normal mass ordering and improved CP sensitivity,” Phys.Lett.
B782 (2018) 633–640, arXiv:1708.01186 [hep-ph] .[4] P. Chen et al. , “Generalized µ − τ reflection symmetry and leptonic CP violation,” Phys.Lett.
B753 (2016) 644–652, arXiv:1512.01551 [hep-ph] .[5] P. Chen et al. , “Classifying CP transformations according to their texture zeros: theory and implications,”
Phys.Rev.
D94 (2016) 033002, arXiv:1604.03510 [hep-ph] .[6] P. Chen et al. , “CP symmetries as guiding posts: Revamping tribimaximal mixing. II.,”
Phys. Rev.
D100 no. 5, (2019)053001, arXiv:1905.11997 [hep-ph] .[7] K. Babu, E. Ma, and J. W. F. Valle, “Underlying A(4) symmetry for the neutrino mass matrix and the quark mixingmatrix,”
Phys.Lett.
B552 (2003) 207–213.[8] G. Altarelli and F. Feruglio, “Discrete Flavor Symmetries and Models of Neutrino Mixing,”
Rev.Mod.Phys. (2010)2701–2729, arXiv:1002.0211 [hep-ph] . [9] S. Morisi and J. W. F. Valle, “Neutrino masses and mixing: a flavour symmetry roadmap,” Fortsch.Phys. (2013)466–492, arXiv:1206.6678 [hep-ph] .[10] S. F. King, A. Merle, S. Morisi, Y. Shimizu, and M. Tanimoto, “Neutrino Mass and Mixing: from Theory toExperiment,” New J.Phys. (2014) 045018, arXiv:1402.4271 [hep-ph] .[11] P. Chen et al. , “Warped flavor symmetry predictions for neutrino physics,” JHEP (2016) 007, arXiv:1509.06683[hep-ph] .[12] F. J. de Anda and S. F. King, “An S × SU (5) SUSY GUT of flavour in 6d,” JHEP (2018) 057, arXiv:1803.04978[hep-ph] .[13] F. J. de Anda and S. F. King, “ SU (3) × SO (10) in 6d,” JHEP (2018) 128, arXiv:1807.07078 [hep-ph] .[14] F. J. de Anda, J. W. F. Valle, and C. A. Vaquera-Araujo, “Flavour and CP predictions from orbifold compactification,”
Phys.Lett.
B801 (2020) 135195, arXiv:1910.05605 [hep-ph] .[15] S. Morisi et al. , “Relating quarks and leptons without grand-unification,”
Phys.Rev.
D84 (2011) 036003, arXiv:1104.1633 [hep-ph] .[16] S. King et al. , “Quark-Lepton Mass Relation in a Realistic A Extension of the Standard Model,”
Phys.Lett.
B724 (2013) 68–72, arXiv:1301.7065 [hep-ph] .[17] S. Morisi et al. , “Quark-Lepton Mass Relation and CKM mixing in an A4 Extension of the Minimal SupersymmetricStandard Model,”
Phys.Rev.
D88 (2013) 036001, arXiv:1303.4394 [hep-ph] .[18] C. Bonilla et al. , “Relating quarks and leptons with the T flavour group,” Phys.Lett.
B742 (2015) 99–106, arXiv:1411.4883 [hep-ph] .[19] M. Reig, J. W. Valle, and F. Wilczek, “SO(3) family symmetry and axions,”
Phys.Rev.
D98 (2018) 095008, arXiv:1805.08048 [hep-ph] .[20]
DUNE
Collaboration, R. Acciarri et al. , “Long-Baseline Neutrino Facility (LBNF) and Deep Underground NeutrinoExperiment (DUNE),” arXiv:1512.06148 [physics.ins-det] .[21]
DUNE
Collaboration, T. Alion et al. , “Experiment Simulation Configurations Used in DUNE CDR,” arXiv:1606.09550[physics.ins-det] .[22]
Particle Data Group
Collaboration, M. Tanabashi et al. , “Review of Particle Physics,”
Phys.Rev.
D98 (2018) 030001.[23] J. Schechter and J. W. F. Valle, “Neutrino Masses in SU(2) x U(1) Theories,”
Phys.Rev.
D22 (1980) 2227.[24] W. Rodejohann and J. W. F. Valle, “Symmetrical Parametrizations of the Lepton Mixing Matrix,”
Phys.Rev.
D84 (2011) 073011, arXiv:1108.3484 [hep-ph] .[25] S. Antusch, J. Kersten, M. Lindner, M. Ratz, and M. A. Schmidt, “Running neutrino mass parameters in see-sawscenarios,”
JHEP (2005) 024.[26] Z.-z. Xing, H. Zhang, and S. Zhou, “Updated Values of Running Quark and Lepton Masses,”
Phys.Rev.
D77 (2008)113016, arXiv:0712.1419 [hep-ph] .[27] S. Antusch and V. Maurer, “Running quark and lepton parameters at various scales,”
JHEP (2013) 115, arXiv:1306.6879 [hep-ph] .[28] N. Nath, R. Srivastava, and J. W. F. Valle, “Testing generalized CP symmetries with precision studies at DUNE,”
Phys.Rev.
D99 (2019) 075005, arXiv:1811.07040 [hep-ph] .[29] P. Huber, M. Lindner, and W. Winter, “Simulation of long-baseline neutrino oscillation experiments with GLoBES(General Long Baseline Experiment Simulator),”
Comput.Phys.Commun. (2005) 195.[30] P. Huber, J. Kopp, M. Lindner, M. Rolinec, and W. Winter, “New features in the simulation of neutrino oscillationexperiments with GLoBES 3.0: General Long Baseline Experiment Simulator,”
Comput.Phys.Commun. (2007)432–438.[31]
KamLAND-Zen
Collaboration, A. Gando et al. , “Search for Majorana Neutrinos near the Inverted Mass HierarchyRegion with KamLAND-Zen,”
Phys.Rev.Lett. (2016) 082503, arXiv:1605.02889 [hep-ex] .[32]
CUORE
Collaboration, C. Alduino et al. , “First Results from CUORE: A Search for Lepton Number Violation via0 νββ
Decay of
Te,”
Phys.Rev.Lett. (2018) 132501, arXiv:1710.07988 [nucl-ex] .[33]
EXO
Collaboration, J. Albert et al. , “Search for Neutrinoless Double-Beta Decay with the Upgraded EXO-200Detector,”
Phys.Rev.Lett. (2018) 072701, arXiv:1707.08707 [hep-ex] .[34]
GERDA
Collaboration, M. Agostini et al. , “Improved Limit on Neutrinoless Double- β Decay of Ge from GERDAPhase II,”
Phys.Rev.Lett. (2018) 132503, arXiv:1803.11100 [nucl-ex] .[35]
SNO+
Collaboration, S. Andringa et al. , “Current Status and Future Prospects of the SNO+ Experiment,”
Adv.High Energy Phys. (2016) 6194250, arXiv:1508.05759 [physics.ins-det] .[36]
LEGEND
Collaboration, N. Abgrall et al. , “The Large Enriched Germanium Experiment for Neutrinoless Double BetaDecay (LEGEND),” vol. 1894, p. 020027. 2017. arXiv:1709.01980 [physics.ins-det] .[37] nEXO
Collaboration, J. Albert et al. , “Sensitivity and Discovery Potential of nEXO to Neutrinoless Double BetaDecay,”
Phys.Rev.
C97 (2018) 065503, arXiv:1710.05075 [nucl-ex] .[38] O. Azzolini et al. , “Search for Neutrino-less Double Beta Decay of Zn and Zn with CUPID-0,” arXiv:2003.10840[nucl-ex] .[39]
Planck
Collaboration, N. Aghanim et al. , “Planck 2018 results. VI. Cosmological parameters,” arXiv:1807.06209[astro-ph.CO]arXiv:1807.06209[astro-ph.CO]