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Clockwork Mechanism for Neutrino Masses and Mixing: A Theory of Mass Hierarchy without Small Parameters

This theory explores a clockwork mechanism where fast rotation produces strong hierarchy in neutrino masses and mixing. The theory introduces additional RH neutrinos or a hidden sector to generate a resemblance to generation due to an extra dimension. The interaction with dark matter and light scalar fields further influences the neutrino mass and mixing.

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Clockwork Mechanism for Neutrino Masses and Mixing: A Theory of Mass Hierarchy without Small Parameters

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  1. Part 2 Part 1 Theory of n - masses and mixing A. Yu. Smirnov Max-Planck Institute fuerKernphysik, Heidelberg, Germany Pontecorvo School 2019, Sinaia, Romania, September 2, 3, 2019

  2. Ibarra et all 1711.02070 [hep-ph] Clockwork mechanism fast rotation slow rotation Strong hierarchy (small quantities) without small parameters yL0 G. Giudice, et al It can be considered as special case neutrino mass with multiple RH neutrinos, or neutrino via hidden sector Resembles generation due to extra dimension in deconstruction mode Y H nL one massless state mostly here yRn-1 yRn yR2 yR1 yLn-1 Mixing of massless state in yRn is suppressed by factor qn , q > 1 yR0 yL2 yL1 yL0 1 qn mn = Y<H> Yukawa coupling with massless state Yq-n

  3. How it works nL yL0 yL1 yL2 ... yLn-1 yR0 yR1 yR2 ... yRn-1yRn YH m m’ m m’ m m’ m’ m m’ gear q = m’/m yL0 yL1 yL2 ... yLn-1 S Mixing of massless state in yRn yR0 yR1 yR2 ... yRn 0 0 0 ... -q 1 0 0 ... 0 -q 1 0 ... .... 1 0 ... 0 ... -q qn qn-1 qn-2 ... 1 x 1/N mn = Y<H>/N Suppression factor Massless state ( qnyR0+ qn-1 yR1 + qn-2 yR2 + ... + yRn)/N • q2 - 1 • qnq2 -q-2n Normalization: N2 = S0....n q2j

  4. Medium generated mass Due to interactions with new light scalar fields Strongly restricted Interactions with usual matter (electrons, quarks) due to exchange by very light scalar mn Interactions with scalar field sourced by DM particles Interaction with “Fuzzy” dark matter

  5. Effective mass due to interactions with dark matter H. Davoudiasl, et al 1803.0001 [hep-ph] L = - gXf X X - gn f nLnR + h.c. f - very light scalar field producing long (astronomical) range forces X – Dark matter particle (fermion of GeV mass scale) source of the scalar field mn = gn f From equation of motion for f neglecting neutrino contribution to generation of f - gXnX mf2 mf = 10-20- 10-26eV is mass of scalar f = nX = <X X> is the number density of X gXgnrX mf2mX rX- energy density of DM mn = gX = gn = 10- 19  mn = 0.1 eV Mass depends on local density of DM and different in different parts of the Galaxy and outside

  6. Interactions with fuzzy dark matter Berlin, 1608.01307 [hep-ph] Ultra-light scalar DM, huge density r – as a classical field, solution 2 r (x) mf f (t, x ) ~ cos (mft) Mass states oscillate gff ninj + … Coupling to neutrinos gives contribution to neutrino mass and modifies mixing dm (t) = gf f (t) Dqm (t) = gf f (t) /Dmij Neutrinos propagating in this field will experience time variations of mixing in time with frequency given by mf Period ~ month, bounds from solar neutrinos, lab. experiments New observable effects (and not just renormalization of SM Yukawa and VEV ) if the field has - spatial dependence - different sign for neutrinos and antineutrinos

  7. Soft couplings and small VEV’s Small neutrino masses from gravitational θ-term G. Dvali and L. Funcke, Phys.Rev. D93 (2016) no.11, 113002 arXiv:1602.03191 [hep-ph] No bb0n decay due to large q2 the vertex does not exist ? Neutrino mass generation through the condensate (crossed blue circles) via non-perturbative interaction (green circle). bb0n decay - unique process where neutrinos are highly virtual Certain generic features independent on specific scenario can be considered on phenomenological level

  8. (3 + 1) scheme ne ns nt nm Interpretation Strong perturbation of 3n pattern: n4 mabind~ m4Ua4 Ub4 ~ Dm322 Effect of possible sterile neutrinos can be neglected if Dm241 mass mabind<< ½ Dm212 ~ 3 10-3eV n3 Dm231 n2 |Ua4|2< 10-3 (1 eV/m4) Dm221 n1

  9. Large flavor mixing from steriles ne nm nt meemem met … mmmmmt … … mtt meS mmS mtS no contribution from S to bb0n decay, but S do contribute to oscillations Mass matrix nS mSS … … … mn = ma + mind eV scale seesaw 0.2 0.4 0.4 … 2.8 2.0 … … 3.0 2.0 2.0 4.5 … 2.0 4.5 … … 10.0 mSS 1 eV ma = mind = 10-2eV 10-2eV produce dominant mt-block with small determinant Enhance lepton mixing meSmmSmtSmay have certain symmetry Generate TBM mixing

  10. III. Lepton mixing and flavor symmetries

  11. TBM: deviations and implications Tri-bimaximal mixing P. F. Harrison, D. H. Perkins, W. G. Scott 0.15 2/3 1/3 0 - 1/6 1/3 - 1/2 - 1/6 1/3 1/2 Utbm = Utbm = U23(p/4) U12 0.78 sin2q12= 1/3 0.30 - 0.31 0.62 Not accidental Accidental, numerology, useful for bookkeeping Lowest order approximation which corresponds to weakly broken (flavor) symmetry of the Lagrangian Accidental symmetry (still useful) with some other physics and structures associated There is no relation of mixing with masses (mass ratios) flavons other new particles

  12. The TBM- mass matrix Mixing from diagonalization of mass matrix in the flavor basis mTBM = UTBMmdiag UTBMT mdiag = diag (| m1|, |m2|ei2a, |m3|e i2b) a b b … ½(a + b + c) ½(a + b - c) … … ½(a + b + c) mTBM = c = m3 a = (2m1 + m2)/3, b = (m2 – m1)/3, The matrix has S2 permutation symmetry nm<->nt Mixing is determined by relations between the matrix elements: m12= m13 m22= m33 m11+ m12 = m22 + m23 Eigenvalues –by absolute values of elements

  13. Deriving the TBM-symmetry C.S. Lam Invariance of mass matrices in the flavor basis: Neutrinos ViTmTBM Vi = mTBM Vi = S, Ui • 0 0 • 0 0 1 • 0 1 0 -1 2 2 … -1 2 .. … -1 1 3 nm -nt -permutation S = U = charged leptons diagonal due to symmetry 1 0 0 … w2 0 … … w T = w = exp(-2ip/3) T+ (me+m e)T = me+m e S, T, U –elements of S4

  14. S - symmetry 4 Order 24, permutation of 4 elements Symmetry of cube Generators: S, T S4 = T3 = (ST2)2 = 1 Presentation: Irreducible representations: 1 , 1’, 2, 3, 3’ 3 x 3 = 3’ x 3’ = 1 + 2 + 3 + 3’ Products and invariants New flavor structure 3 x 3’ = 1’ + 2 + 3 + 3’ 1’ x 1’ = 1 2 x 3 = 2 x 3’ = 3 + 3’ 1’ x 2 = 2 2 x 2 = 1 + 1’ + 2

  15. Sn Residual symmetries approach E. Ma, C. S. Lam .... Mixing appears as a result of different ways of the flavor symmetry breaking in the neutrino and charged lepton (Yukawa) sectors. T’ S4 A4 Gf T7 Residual symmetries of the mass matrices Gl Gn Zm Z2 x Z2 or Z2 Generic symmetries which do not depend on values of masses Ml Mn Symmetry transformations in mass bases T U, Sn CP-transformations can be added SnMnSnT = Mn Discrete finite groups Flavons to break symmetries Flavons

  16. Intrinsic symmetries Realized for arbitrary values of neutrino and charged lepton mass can not be broken, always exist In the mass basis for charged leptons for Majorana neutrinos ml = diag( me, mm, mt ) m = diag( m1, m2, m3) Gn Gl ife ifm ift T = diag(e , e , e ) S1 = diag (1, -1, -1) S2 = diag (- 1, 1, -1) fa= 2pka/m Tm = I Zm Si2 = I Z2x Z2 The Klein group S fa= 0 can use subgroup Intrinsic symmetries as residual symmetries

  17. Symmetry group condition D. Hernandez, A.S. 1204.0445 If intrinsic symmetries are residual symmetries of the unique symmetry group (follow from breaking of unique group)  bounds on elements of mixing matrix Inversely, Si and T are elements of covering group. By definition product of these elements (taken in the same basis) also belongs to the finite discrete group: ( UPMNSSi UPMNS+ T)p = I p -integer For each i the equation gives two relations between mixing parameters Two such equations for i = 1,2 fix the mixing matrix completely  TBM Z2x Z2 TBM In general, allows to fix mixing matrix up to several possibilities

  18. Example Gn= Z2 Two relations D. Hernandez, A Y S. 1304.7738 [hep-ph] for column of the mixing matrix: |Ubi|2 = |Ugi|2 1 + a 4 sin2 (pk/m) |Uai|2 = S4 a = Tr(UPMNSSi UPMNS+ T) k, m, p integers which determine symmetry group d = +/-1030 Schemes with non-zero 1-3 mixing can be obtained TBM1 TBM2 TBM

  19. A_4 symmetry E. Ma G Branco, H P Nilles Symmetry group of even permutations of 4 elements A4 Symmetry of tetrahedron Generators: S, T Presentation of the group: no U = Amt S2 = 1 T3 = 1 (ST)3 = 1 Irreducible representations: 3, 1, 1’, 1’’ Products and invariants 3x3 = 3 + 3 + 1 + 1’ + 1’’ 1’ x 1’’ ~ 1

  20. A_4 symmetry breaking A4 ``accidental’’ symmetry due to particular selection of flavon representations and configuration of VEV’s Amt T S Ml diagonal Mn TBM-type In turn, this split originates from different flavor assignments of the RH components of Nc and lc and different higgs multiplets

  21. An A4model G. Altarelli D. Meloni symmetry Yukawa sectors A4 Z4 Charged lepton Neutrinos 1 i -1 -i 3 L L 1 n k i i hd x’ hu fT i 1’ Nc i 1 i lc -1 at multiplets 1, i, -1 1’’ fS M < fT> = v S (0, 1, 0) x 1 k = 0, … n = 1, … 1 < fS> = v S (1, 1, 1) Flavon sector U(1)R Particular selection of representations x’ x fS fT 0 2 x0 fS0 fT0 Vacuum alignment Driving fields -1 1 1 GUT-scale or higher?

  22. Generic problem m = F(Y, v) Mechanism of mass generation VEV’s Yukawa couplings VEV alignment • different contributions • high order corrections follow from independent sectors: Scalar potential Yukawa sector TBM tune by additional symmetries All these components should be correlated ``Natural’’ – consequence of symmetry? one step constructions do not work

  23. More problems No connection to masses Mass hierarchy from additional symmetries U(1) Froggatt- Nielsen mechanism n f L power n is determined by U(1) charges of the corresponding operators Discrete symmetries – restricted possibilities to explain mass spectrum (degenerate, partially degenerate spectra) Missing representations problem

  24. Relating to mass degeneracy Symmetry which left mass matrices invariant for specific mass spectra: m1 = m2 , m3 D. Hernandez, A.S. Partially degenerate spectrum Gn = SO(2) x Z2 Transformation matrix Sn = O2 m1 m2 Relation: sin2 2q23 = +/- sin d = cosk = = 1 Majorana phase +/- p/2 maximal 1-2 mixing is undefined Small corrections to mass matrix lead to 1-2 mass splitting and 1-2 mixing

  25. CP-violation, phase Certain values of d follow from the flavor symmetries without additional assumptions Specific transformations which can be responsible (control) value of the phase Non-trivial CP phase : from generalized CP transformations when also flavor is changed Usually maximal CP violation is predicted

  26. Sn D Flavor and CP symmetry Gfx HCP T’ A4 S4 Flavons D(96) D(48) D(27) GnxHnCP GlxHlCP Residual symmetries of the mass matrices Mn R. Mohapatra, C C Nishi, F. Feruglio, C. Hagedorn, R. Ziegler, E Ma, G-J. Ding, S.F. King, C. Luhn, A. J. Stuart, Y-L. Zhou, Ml  no CP violation d = 0 Gn = Z2 x Z2  usually d = 0, +/- p/2 , p C. Hagedorn, et al Gn = Z2 dmay depend on free parameter and take any value Y. Farzan, A.S.

  27. Do we use correct flavor symmetries? Do we use flavor symmetries correctly? Promising development. Convincing?

  28. Modular symmetries Another realization of symmetry approach inspired by string theory Symmetry related to (orbifold) compactification of extra dimensions and primary realized on the moduli fields which describe geometry of the compactified space. For single modulus field t the modular transformation reads a t + b c t + d tgt = The 2x2 matrices of integer numbers a b c d with determinant ad – bc = 1 Form the group G = SL(2, Z) special linear ... Finite subgroup of G : GN = G / G (N)

  29. Modular group Finite subgroup of G is quotient group of level N: GN = G / G (N) where G (N) - is congruence subgroup of N of G of level N. GN is called the modular group G3 , G4, G5 are isomorphic to A4, S4, A5, correspondingly In SUSY, the chiralsuperfields transform as kI jI (ct + d) r(g)jI kI is the weight of multiplet r(g) is the representation of g element of the group GN Appearance of the weight factor in transformations is new element which leads to new consequences

  30. Modular forms Another key element of formalism fi(t) - holomorphic functions of modulus field t Transformation properties are similar to those of superfields kf fi(t) fi(gt) = (ct + d) r(g)ijfj(t) Form multiplet of GN whose dimension is determined by level N and weight kf For instance for N = 3 and kf = 2, fi form triplet with components Y1(t) = 1 + 12q + 36q2 + ... i2p t Y2(t) = - 6q1/3 (1 + 7q + ... ) q = e Y3(t) = - 18 q2/3 (1 + ... ) Data fit: t = 0.0117 + i 0.995 weak hierarchy Yi = (1, - 0.74, - 0.27)

  31. Invariance invariance requires For terms of potential Y j1j2j3 r1x r2 x r3 x rY = I for product of A4 representations Siki + kY = 0 for weights Additional condition which acts as Froggatt- Nielsen factors Yukawa couplings form multiplets they are fixed by symmetry

  32. Models J Griado and F. Feruglio 1807.01125 [hep-ph] flavon L EcNc Y j A4 3 1, 1’, 1’’ 3 3 3 k 1 -4 -1 2 3 Y lowest order modular form tand jare fixed by fitting on m3 /m2, 12 mixing and 13 mixing Predictions: sin 2 q23 = 0.46 d /p = 1.434 a21/p = 1.7 a31/p = 1.2

  33. Gui-Jun Ding, S.F. King, Xiang-Gan Liu 1907.11714 [hep-ph] Models No flavons Flavor from single modulus field t L EcNc Y A4 3 1, 1’’, 1’ 3 3 k 2 2 0 -2 Higher order modular forms Y(2), Y(4) , Y(6) constructed as products of Y(2) All Yukawas are modular forms tis fixed by fitting 12 mixing and 13 mixing Predictions: sin 2 q23 = 0.58 m1 = 0.0946 eV Cosmological bound? d /p = 1.6 m2 = 0.0950 eV a21/p = 0.15 m3 = 0.1071 eV a31/p = 1.00 Normal ordering mee = 0.095 eV

  34. Dm412 = 1 - 2 eV2 IV. Models of LSND Neutrino Mass

  35. n M. Shaposhnikov, et al MSM Everything below EW scale  small Yukawa couplings • - small neutrino mass • lepton asymmetry • via oscillations • can be produced in • B-decays (BR ~ 10-10) etc, SHiP L R Origin of this scale, Why at EW? BAU 0.1 - few GeV split ~ few kev - radiative decays  3.5 keV line? WDM Decouples from generation of neutrino mass, RHN? 3 - 10 kev Extensions of model G.K Karananas, M. Shaposhnikov Unification: Constrained GUT’s? Unnatural Seesaw –small Dirac Yukawas Flavor structure, mixing?

  36. Bounds on nu-MSM NuSTAR Tests of Sterile-Neutrino Dark Matter: New Galactic Bulge Observations and Combined Impact Brandon M. Roach et al, arXiv:1908.09037 [astro-ph.HE] The impact of Nustar data on the νMSM parameter space. The tentative signal at E≃3.5 keV (Bulbul:2014,Boyarsky:2014) - red point.

  37. Mass and mixing from hidden sector Neutrino portal Hidden sector Visible sector SM F flavons S l nL nR S S fermions mD mlD MS S S L-R MD S S GUT mD~ MD = diag mn~ MS MS = non-diagonal mlD origin of UX Basis fixing symmetry Ul ~VCKM MPl m = keV - MeV Gbasis = Z2 x Z2 MS scale

  38. Two sectors UPMNS~ VCKM+ UX New sector related to neutrinos, responsible for large neutrino mixing smallness of neutrino mass Common sector for quarks and leptons. Implies mDn ~ mu ml ~ md May have special symmetries which lead to BM or TBM mixing Q - L unification, GUT CKM physics, hierarchy, of masses and mixings Froggatt-Nielsen (?), relations between masses and mixing

  39. Xun-JieXu , A.S A GUT scheme with Ghidden = S4 and BM mixing SO(10) Visible sector Portal Hidden sector UX UCKM Ghidden= S4 SF 16F 1H 16H 10H basis fixing symmetry Z2 x Z2 Non-trivial charges (0, 1) (1, 0) (1, 1) (0, 1) (1, 0) (1, 1) mass hierarchy no mixing Z2 x Z2 S4 CKM mixing - additional structure Spontaneously broken by flavons MS ~ MPl MD ~ MGUT MS ~ MBM mD~ MD = diag Double seesaw

  40. Embedding D. Hernandez, A.S. 1204.0445 B. Bajc, A.S. To fix UXone can use the residual symmetry approach: Gf non-abelian group (Z2 x Z2 )V (Z2 x Z2 )H UX Embedding leads to general expression (via symmetry group condition) pnij pij |(UX )ij|2 = cos2 p , n -integer Unitarity condition pnij pij similar for the sum over j Si cos2 = 1 This allows to reconstruct the matrix UX up to discrete number of possibilities on of them – BM mixing

  41. Realization for BM mixing explicitly or spontaneously (Z2 x Z2 )V S4 (Z2 x Z2 )H spontaneously Charge assignment Fields 16Fi Si h x f S4 3 3 1 2 3’ other fields are S4 singlets Flavons, singlets of SO(10) VEV alignment < f > ~ (0, 0, 1), < x > ~ (0, 1) UX = UBM MS = MBM dCP = (0.80 – 1.16) p

  42. mD Low scale Left-right symmetric model x P SU(2)L x SU(2)R x U(1)B-L – inverse seesaw mq ~ ml ~ mDn with q-l similarity Fields LL LRcLcRS B – L -1 -1 1 1 0 Higgs bi-doublet nl l nl l F MD ~ MR ~ PeV L R m ~ 10 keV cR cL Higgs doublets inverse seesaw flavor symmetry in m S singlet breaks L-R symmetry m with Majorana mass terms

  43. mD Model with ``left and right’’ singlets x P SU(2)L x SU(2)R x U(1)B-L invariant under global U(1) Higgs bi-doublet Fields LL LR SL SR L 1 1 - 1 - 1 nl l nl l F broken by m-terms L R cR cL keV scale sterile neutrino - DM SL SR mLR m m

  44. Dm412 = 1 - 2 eV2 Conclusions LSND

  45. 3n - paradigm works well, no significant and well established deviations have been found Situation with unknowns is rather uncertain: various preferences are at 2-3 s level and in some cases controversial, hints fragile On theory side, New interesting ideas, models emerge Maybe, neutrino properties from dark sector? However nothing is really established and we are not far from the beginning

  46. Probably correct elements of the theory of neutrino mass and mixing are already among numerous mechanisms, schemes models and the goal is to identify them Still something important can be is missed Feeling is that the theory may not be simple and the progress may not be easy Enormous efforts in determination of matrix elements, cross-sections, systematics, backgrounds... And all this is to measure neutrino parameters Eventually, we measure neutrino parameters to establish the underlying physics. In spite of scepticism searches for true theory of mass and mixing must continue

  47. Connections Any discovery in these fields can have impact on neutrinos LHC observables Higgs physics Neutrinos Dark matter Lepton Axions Flavor Violation Dark energy Anomalies (g-2)m Lepton In B-decays Model dependent, not unique universality

  48. Dm412 = 1 - 2 eV2 Back-up LSND

  49. Tests and problems Normal mass hierarchy, first 2-3 octant dCPl~dCPq Flavons, new fermions, new higgses are at GUT – Planck scale Nothing should be observed at LHC which is responsible for neutrino masses Proton decay New elements related to CKM physics Very strong hierarchy of masses of RH neutrinos: Leptogenesis with second RH neutrino

  50. Dm412 = 1 - 2 eV2 Neutrinos LSND and EW scale

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