Massive Neutrinos and (Heterotic) String Theory

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1 Massive Neutrinos and (Heterotic) String Theory Introduction Neutrino preliminaries The GUT seesaw A string primer The Z 3 heterotic orbifold (In collaboration with J. Giedt, G. Kane, B. Nelson.)

2 Neutrino mass Nonzero mass may be first break with standard model Enormous theoretical effort: GUT, family symmetries, bottom up Majorana masses may be favored because not forbidden by SM gauge symmetries GUT seesaw (heavy Majorana singlet). Usually ordinary hierarchy. Higgs triplets ( type II seesaw ), often assuming GUT, Left- Right relations

3 Very little work from string constructions, even though probably Planck scale E. Witten, Nucl. Phys. B 268, 79 (1986). (E 6 difficulties.) C. Coriano and A. E. Faraggi, Phys. Lett. B 581, 99 (2004); A. E. Faraggi and M. Thormeier, Nucl. Phys. B 624, 163 (2002). (Heterotic inspired. Extended seesaw with extra dynamical assumptions.) J. R. Ellis, G. K. Leontaris, S. Lola and D. V. Nanopoulos, Eur. Phys. J. C 9, 389 (1999). (Flipped SU(5). May be seesaw, but nonstandard and non-gut-like Majorana, Dirac matrices. Flatness?) L. E. Ibanez, F. Marchesano and R. Rabadan, JHEP 0111, 002 (2001). (Intersecting brane. L conserved.) I. Antoniadis, E. Kiritsis, J. Rizos and T. N. Tomaras, Nucl. Phys. B 660, 81 (2003). (D-brane. L conserved.) J. Giedt, G. Kane, PL, B. Nelson, this work. (Systematic study of heterotic Z 3 orbifolds.)

4 Key ingredients of most bottom up models forbidden in known constructions (heterotic or intersecting brane) (Due to string symmetries or constraints, not simplicity or elegance) Right-handed neutrinos may not be gauge singlets Large representations difficult to achieve (bifundamentals, singlets, or adjoints) GUT Yukawa relations broken String symmetries/constraints severely restrict couplings, e.g., Majorana masses, or simultaneous Dirac and Majorana masses Small Dirac masses from HDO, extended (TeV-scale) seesaw, or triplet seesaw (with inverted hierarchy) may be more likely

5 Models and spectra Weyl fermion Minimal (two-component) fermionic degree of freedom ψ L ψr c by CPT Active Neutrino (a.k.a. ordinary, doublet) in SU(2) doublet with charged lepton normal weak interactions ν L νr c by CPT Sterile Neutrino (a.k.a. singlet, right-handed) SU(2) singlet; no interactions except by mixing, Higgs, or BSM N R NL c by CPT Almost always present: Are they light? Do they mix?

6 Dirac Mass Connects distinct Weyl spinors (usually active to sterile): (m D ν L N R + h.c.) 4 components, L = 0 I = 1 2 Higgs doublet Why small? HDO? LED? Variant: couple active to antiactive, e.g., m D ν el ν c µr L e L µ conserved; I = 1 ν L h N R v = φ m D = hv

7 Majorana Mass Connects Weyl spinor with itself: 1 2 (m T ν L νr c + h.c.) (active); 1 2 (m S N L cn R + h.c.) (sterile) 2 components, L = ±2 Active: I = 1 triplet or seesaw Sterile: I = 0 singlet or bare mass Mixed Masses Majorana and Dirac mass terms ν L ν c R ν L ν L Seesaw for m S m D Ordinary-sterile mixing for m S and m D both small and comparable (or m S m d (pseudo-dirac))

8 3 ν Patterns Solar: LMA (SNO, Kamland) m ev 2, nonmaximal Atmospheric: m 2 Atm ev 2, near-maximal mixing Reactor: U e3 small

9 Mixings: let ν ± 1 2 (ν µ ± ν τ ): ν 3 ν + ν 2 cos θ ν sin θ ν e ν 1 sin θ ν + cos θ ν e Hierarchical pattern Analogous to quarks, charged leptons ββ 0ν rate very small Inverted quasi-degenerate pattern ββ 0ν if Majorana SN1987A energetics (if U e3 0)? May be radiative unstable

10 Degenerate patterns Motivated by CHDM (no longer needed) Strong cancellations needed for ββ 0ν if Majorana May be radiative unstable

11 4 ν Patterns LSND: m 2 LSND 1 ev 2 Z lineshape: 2.986(7) active ν s lighter than M Z /2 fourth sterile ν S patterns patterns Pure (ν µ ν s ) excluded for atmospheric by SuperK, MACRO Pure (ν e ν s ) excluded for solar by SNO, SuperK More general admixtures possible, but very poor global fits Additional sterile (e.g., 3 + 2) fit better but may have cosmological difficulties

12 The minimal seesaw Active (sterile) neutrinos ν L (N R ) (3 flavors each) L = 1 2 ( νl N L) ( c m T m D m T D m S ) ( ) ν c R N R + hc m T = m T T = triplet Majorana mass matrix (Higgs triplet) m D = Dirac mass matrix (Higgs doublet) m S = m T S = singlet Majorana mass matrix (Higgs singlet)

13 Ordinary (type I) seesaw: m T = 0 and (eigenvalues) m S m D : m eff ν = m Dm 1 S mt D with U P MNS = U e U ν

14 Implementation in GUTs Elegant mechanism for small Majorana masses Leptogenesis Expect small mixings in simplest versions (can evade by lopsided e/d, Majorana textures, etc.) Most models require large Higgs representations, e.g., 126 of SO(10) (alternative: higher dimensional operators) Large Majorana often forbidden, e.g., by extra U(1) s LSND: active-sterile difficult in simple versions

15 Neutrinos in string constructions Key ingredients of most GUT/bottom up models forbidden or different in known constructions (heterotic or intersecting brane) Bifundamentals, singlets, or adjoints; not large representations L may be conserved, or extra U(1) charge for N R String constraints may forbid couplings allowed by 4d symmetries Superpotential terms leading to Majorana masses, or diagonal (same family or same flavor) Majorana usually absent GUT Yukawa relations broken Non-zero superpotential terms may be equal (gauge couplings) Hierarchies from HDO (heterotic), intersection triangles (intersecting brane)

16 Dirac masses Can achieve small Dirac masses (neutrino or other) by higher dimensional operators or by large intersection areas L ν ( S M P l ) p LN c L H 2, S M P l m D ( S M P l ) p H 2 Large p S close to M P l (e.g., anomalous U(1) A ) Small p intermediate scale M P l Similar HDO may give light steriles and ordinary/sterile mixing

17 Majorana masses Can one generate large effective m S from S q+1 S q+1 W ν c ij N i N j (m S ) ij c ij, M q P l consistent with D and F flatness? M q P l Can one have such terms simultaneously with Dirac couplings, consistent with flatness and other constraints? Are bottom-up model assumptions for relations to quark, charged lepton masses maintained?

18 Seesaws in String Constructions No completely realistic constructions Existing constructions usually focus on quark sector Neutrino masses rarely considered, and then as afterthought No construction has yielded GUT-like seesaw Analyze Z 3 heterotic orbifolds (semi-realistic 3- family models) in detail, focussing on neutrino sector (Joel Giedt, G. Kane, PL, Brent Nelson) Large number of possible vacua: Is the minimal seesaw generic? Is some subclass of vacua favored? Any clue about hierarchies, mixings, etc?

19 A String Primer Two classes of quasi-realistic: intersecting D-brane, heterotic Intersecting D-brane Closed strings (gravitons) and open strings ending on D-branes D6-branes: fill ordinary space and 3 of the 6 extra dimensions Stringy implementation of brane world ideas

20 Gauge interactions from strings beginning/ending on stack of parallel branes (one for each group factor) Chiral matter: strings at intersection of branes, e.g., SU(N) SU(M) bifundamental (N, M) a Gauge bosons in adj. U(1) W U(1) b W + Chiral matter in (N, M)

21 Family replication from multiple intersections on compactified geometry Yukawa interactions exp( A ijk ) hierarchies Existing models: conserved L; no diagonal (Majorana) triangles (1, 1) (1,3) (1, 1) (1,2) (1,0) (1,1) T 2 T 2 T 2 H Q 3 L U 3 SU(3) Q 2 L U2 Q 1 L SU(3) SU(2) U U(1) 1 SU(3)

22 The Z 3 Heterotic Orbifold }{{} E 8 }{{} E 8 SU(3) SU(2) U(1) 5 hidden closed strings Three families automatic Tremendous symmetry, stringy selection rules restricted couplings Chiral fermions, N = 1 supersymmetry orbifold Anomalous U(1) A ; F and D flatness; vacuum, restabilization

23 The Z 3 Orbifold Compactify on three two-torii T 2

24 n = 0 strings closed on R 2 ; winding states n = 1, 2 closed on T 2 (not R 2 ) Z 2 orbifold: identify x and x two fixed points; twisted states closed on orbifold (n = 0) e 2 (n = 1) e 1 e 2 x x x x (n = 2) e 1 x x

25

26 Z 3 orbifold: identify 2π/3 rotations three fixed points

27 Anomalous U(1) A ; F and D flatness; vacuum restabilization One linear combination of the U(1) 5 may be anomalous Green-Schwarz mechanism cancels anomaly in 4d U(1) SU(N) SU(N) U(1) + = 0 B 2 SU(N) SU(N) Fayet-Iliopoulous term added to the D- term of U(1) A ξ FI = g2 STR 192π 2 Tr QA M 2 PL

28 Supersymmetry is restored when certain scalar fields acquire VEV s such that D- and F flatness conditions are satisfied: D A i Q (A) i S i 2 + ξ FI = 0 D a i Q (a) i S i 2 = 0 F i W S i = 0; W = 0 VEVs S i reduce gauge symmetries, give masses (restabilization) Other S i VEVs can acquire intermediate scale masses by radiative breaking

29 Search for Minimal Seesaw Look for structure in Z 3 heterotic orbifold: ( W eff = (ν i, N i ) 0 (m D ) ij (m D ) ji (m M ) ij ) ( νj N j ) Require simultaneous Majorana and Dirac couplings, and appropriate hypercharge Don t insist on realistic quark sector Majorana mass from S 1 S n 2 NN/M n 3 PL Dirac mass from S 1 S d 3 NLH u/m d 3 PL Only 5 embeddings into E 8 E 8, 4 realistic hidden sector groups 175 models in 20 patterns with same ξ FI (Giedt)

30 Pattern No. G hid r FI Species SO(10) U(1) 3 No U(1) A SO(10) U(1) SU(5) SU(2) U(1) SU(5) SU(2) U(1) SU(5) SU(2) U(1) SU(5) SU(2) U(1) SU(5) SU(2) U(1) SU(5) SU(2) U(1) SU(4) SU(2) 2 U(1) SU(4) SU(2) 2 U(1) SU(4) SU(2) 2 U(1) SU(4) SU(2) 2 U(1) SU(3) SU(2) 2 U(1) SU(3) SU(2) 2 U(1) SU(3) SU(2) 2 U(1) SU(3) SU(2) 2 U(1) SU(3) SU(2) 2 U(1) SU(3) SU(2) 2 U(1) SU(3) SU(2) 2 U(1) SU(3) SU(2) 2 U(1)

31 Classified superpotential terms of degree 9 Large number (O(50)) fields in each, half are SM singlets None are singlets under all U(1) s Huge number of terms, but small wrt number of fields due to symmetries/selection rules r FI = ξ FI /M PL S i /M PL

32 Pattern

33 Require F and D flatness Examined 3 models from each pattern (conjecture: all models in pattern equivalent) Studied subset of flat directions with 1d D flatness and minimal F -flatness (more general directions very complicated) Huge number of D-flat directions, reduced drastically by F -flatness

34 Pattern w/o w/3 w/

35 For each surviving direction, looked for candidate Majorana mass terms S 1 S n 2 NN, where the S i 0 for that direction Only two patterns out of 20 (2.6 and 1.1) have candidate Majorana mass terms Must still check: Is there a surviving hypercharge Y with Y N = 0? Are there candidate Dirac couplings S 1 S d 3 NLH u at low enough order? Do L, H, and quark candidates have correct Y?

36 Pattern 2.6 Six directions have Majorana mass terms of form I monomial : (4, 4, 6, 7, 18, 35, 43, 43), Eff. Maj. mass : (4, 5, 5) Numbers refer to a classification of the chiral matter superfields I-monomial lists S i fields with VEVs (of order r FI M PL 0.1M PL ) Underlined fields are the S i, others (N 5 ) are Majorana neutrinos Family indices suppressed

37 However, no Dirac couplings involving N 5 through degree d 6, i.e., none through order S d 3 N 5 LH u Light seesaw masses would be of order m ν (rd 3 FI v u ) 2 r FI M PL r 2d 7 FI 10 5 ev }{{} d>6 < ev Also eight directions of form I monomial : (4, 4, 7, 18, 19, 27, 43, 43), Eff. Maj. mass : (7, 7, 19, 27, 43, 43, 43, 34, 34) However, no Dirac couplings of degree < 9 m ν ev

38 Pattern 1.1 No anomalous U(1) A ; VEVs may still be determined, e.g., by radiative breaking of non-anomalous, typically at intermediate scale Two classes of directions with Majorana masses, but first has no Dirac couplings through (needed) degree 6. Second class promising: I monomial : (3, 3, 8, 21, 22, 29, 46, 72), Eff. Maj. mass : (8, 22, 46, 72, 9, 9) There is also a candidate Dirac mass: N 9 L 36 L 64, where L 36, L 64 are two SU(2) doublets

39 Can define appropriate hypercharge for all fields L 36 = L, L 64 = H u (family indices suppressed) A second set of Majorana and Dirac couplings of higher degree also present (not shown) No realistic quark Yukawas (and no GUT-type relations) Undesired doubling of leptons and Higgs Apparently, we have found an example of a seesaw, even if not fully realistic! We were about to study family structure (scale, hierarchy, mixings)

40 The Fatal Flaw The same direction has degree 3 mass terms coupling N 9 to other fields Ñ: W mix = λs 8 N 9 Ñ 14 + λs 22 N 9 Ñ 27 + λs 72 N 9 Ñ 50 + λs 46 N 9 Ñ 81 with B C A L = (ν L Ñ N) 0 0 A 0 0 B A B C ν LÑ N, Three massless and six supermassive neutrinos! (no additional terms generated to needed order) This could also occur for other apparent seesaws

41 Outlook Neutrino mass likely due to large or Planck scale effects, but little previous work in string context No viable examples of minimal seesaw in huge class of Z 3 orbifold vacua Could consider more general vacua (two independent VEVs, cancellations of F terms) Other types of orbifolds and heterotic constructions? Will also have strong gauge and stringy constraints. (L conserved in existing intersecting brane) Even if a few examples are found, they don t appear generic

42 Consider alternatives seriously Small Dirac masses from high degree terms (very common in constructions) (could also give light sterile ν s and mixing) Extended seesaws, m ν m 2+k D /M 1+k, with k 1 and low (e.g., TeV) scale M Higgs triplet models: non-trivial to embed in strings (higher level), but very predictive (e.g., inverted hierarchy with nearly bi-maximal mixing) (B. Nelson, PL)

43 Extended (TeV) Seesaw? m ν m p+1 /m p S, p > 1 (e.g., m 100 MeV, m S 1 TeV for p = 2) ν L, N R, N R (3 flavors each) L = 1 ( νl N c L 2 N 0 m D m D L) c m T D 0 m SS m T D m T SS 0 νc R N R N R + hc or L = 1 2 ( νl N c N L L) c 0 m D 0 m T D 0 m SS 0 m T SS m S νc R N R N R + hc (Faraggi et al.: may occur in specific heterotic model, with dynamical assumptions.)

44 Triplet models Introduce Higgs triplet T = (T ++ T + T 0 ) T with weak hypercharge Y = 1 Majorana masses m T generated from L ν = λ T ij L it L j if T 0 0 Old Gelmini-Roncadelli model: T 0 EW scale with spontaneous L violation Excluded by Z Majoron + scalar (equivalent to N ν = 2) Modern triplet models (type II seesaw) break L explicity by T HH couplings, giving large Majoron mass (Lazarides, Shafi, Wetterich, Mohapatra, Senjanovic, Schechter, Valle, Ma, Hambye, Sarkar, Rossi,...) Often considered in SO(10) or LR context, with both ordinary and triplet mechanisms competing and with related parameters, but can consider independently.

45 General SUSY case W ν = λ T ij L it L j + λ 1 H 1 T H 1 + λ 2 H 2 T H 2 +M T T T + µh 1 H 2 T, T are triplets with Y = ±1, M T GeV. Typically, T 0 λ H /m T m ν ij = λt ij λ 2 v 2 2 M T

46 String constructions Expect λ T ij = 0 for i = j (off-diagonal) mν ii = 0 Also, need multiple Higgs doublets H 1,2 with λ 1,2 off diagonal Partial explanation: SU(2) triplet with Y 0 requires higher level embedding, e.g., of SU(2) SU(2) SU(2) (Have Z 3 constructions with some but not all of the features.) W λ T 1j L 1(2, 1)T (2, 2)L j (1, 2), j = 2, 3 yields m ν = 0 a b a 0 0 b 0 0 Typical string case: a = b

47 HDO (or SU(2) SU(2) SU(2) SU(2)) can give m ν 23 0 For m ν = 0 a b a 0 c b c 0 can take a, b, c real w.l.o.g. by redefinition of fields (not true for general m ν ) Tr m ν = 0 and m ν = m ν m 1 + m 2 + m 3 = 0

48 m 2 Atm ev 2, m ev 2 two solutions For m 2 =0 (a) m i 1, 1 2, 1 (ordinary, with shifted masses) 2 (b) m i 1, 1, 0 (inverted) With m 2 0 (a) m i = 0.054, 0.026, ev ( m i = ev (cosmology)) (b) m i = 0.046, 0.045, ev ( m i = ev (cosmology)) m ν a m ν b 0 a b a 0 0 b 0 0 (a) leads to unrealistic mixing matrix consider (b)

49 A special texture The L e L µ L τ conserving texture m ν 0 a b a 0 0 b 0 0 has been considered phenomenologically by many authors (Zee; Barbieri, Hall, Smith, Strumia, Weiner; King, Singh; Ohlsson; Barbieri, Hambye, Romanino; Lebed, Martin; Babu, Mohapatra; Lavignac, Masina, Savoy; Feruglio, Strumia, Vissani; Altarelli, Feruglio, Masina)

50 m ν 0 a b a 0 0 b 0 0 New aspects Strong string motivation Motivation for special case a = b Most likely perturbation in 23 element from HOT Diagonalization: tan θ Atm = b/a need b = a for maximal tan 2 θ = 1 (maximal) (experiment tan 2 θ = )

51 Majorana mass matrix m ν Inverted hierarchy Bimaximal mixing for U e = I: U ν

52 Perturbations on m ν cannot give both m 2 and π 4 θ θ C 0.23 without fine-tuning between terms, e.g., 1 m m 2 Atm = ɛ π 4 θ 0.23

53 However, U e I with small angles (comparable to CKM) can can give agreement with experiment (Frampton, Petcov, Rodejohann; Romanino; Altarelli, Feruglio, Masina) U e 1 se 12 0 s e yields θ π 4 se 12 2 = U e3 2 (se 12 )2 2 ( ), 90% (exp : < 0.03) m ββ m 2 (cos 2 θ sin 2 θ ) ev

54 Conclusions Neutrino mass likely due to large or Planck scale effects, but little work in string context Specific orbifold string constructions (heterotic, intersecting brane) not consistent with common GUT and bottom up assumptions for m ν No examples of minimal seesaw in large class of heterotic Z 3 orbifold vacua Small Dirac, extended seesaw, Higgs triplet (inverted hierarchy in string context) may be more likely