Minimal Lepton Flavor Violation

Transcription

1 Minimal Lepton Flavor Violation Benjamin Grinstein INT, Seattle, October 28, 2008 Vincenzo Cirigliano, Gino Isidori, Mark Wise NPB728, 121(2005) NPB752, 18 (2006) NPB763, 35 (2007)

2 Motivation, rationale, MFV New physics is required to explain the fine tuning puzzle. It must involve new dynamics that become relevant at an energy scale Λ. This new dynamics likely involves new fields, new particles and new interactions. The new dynamics scale Λ must not be much higher than the electroweak scale (the higher the scale the more severe the fine tuning). Quarks and leptons contribute to quadratic divergence in higgs mass divergences depend on quark/lepton masses new dynamics must have flavor dependence New flavor dependent dynamics at a scale ΛF not far above the electroweak scale is a disaster: either ΛF GeV, or fine tune coefficients of dangerous operators (those giving large flavor changing neutral currents ) Unless: avoid large FCNC automatically Minimal Flavor Violation 2

3 MFV is NOT the only possibility e.g., NMFV and generally theories with quark mass suppression But MFV is fairly minimal good if you want to estimate the minimal effect of this new physics in flavor changing processes more predictive, patterns Let s gain some understanding by example. Consider K L πνν 3

4 In the SM: H eff = 4G F 2 l=e,µ,τ C l s L γ µ d L ν l Lγ µ ν l L + h.c. C l = αx( m t M W ) 2π sin 2 θ W V tsv td CKM factor 1 loop factor, X 1 V CKM V CKM = V ud V us V ub V cd V cs V cb V td V ts V tb λ2 λ Aλ 3 ( ρ i η) λ(1 + ia 2 λ 4 η) λ2 Aλ 2 Aλ 3 (1 ρ i η) Aλ 2 (1 + iλ 2 η) + O(λ 6 ). 1 λ 0.22 V ts V td λ 5 4

5 New physics H eff = 1 Λ 2 F l=e,µ,τ C l new s L γ µ d L ν l Lγ µ ν l L + h.c. H eff = 4G F 2 Assume sensitivity to fractional deviation r from SM rate, with l=e,µ,τ C l new 1 C l s L γ µ d L ν l Lγ µ ν l L + h.c. 1 + r 1 + (M W /Λ) 2 A 2 λ 5 /(16π 2 ) 2 For example, r = 4% gives sensitivity to ΛF 10 6 GeV But if new physics has same CKM factors in, then C l new 1 + r 1 + (M W /Λ) 2 1/(16π 2 ) 2 And now r = 4% gives sensitivity to ΛF GeV 5

6 Minimal Flavor Violation (MFV) Premise: Unique source of flavor braking Quark sector in SM, in absence of masses has large flavor (global) symmetry: G F = SU(3) 3 U(1) 2 In SM, symmetry is only broken by Yukawa interactions, parametrized by couplings λ U and λ D L Yuk = H q L λ U u R + H q L λ D d R MFV: all breaking of G F must transform as these When going to mass eigenstate basis, all mixing is parametrized by CKM and GIM is automatic Approach: via effective field theory: at low energies only SM fields 6

7 How does this work? Consider K L πν ν Recall, G F breaking from: L Yuk = H q L λ U u R + H q L λ D d R Implications of G F? use spurion method: q L V L q L u R V u u R d R V d d R λ U V L λ U V u λ D V L λ D V d Effective lagrangian L eff = 1 Λ 2 Ci O i among the operators have, for example In mass basis As needed it includes the factor O = q L (λ U λ U )γ µq L ν L γ µ ν L ( ) VqsV m 2 q qd v 2 s L γ µ d L ν L γ µ ν L q=u,c,t V tsv td m 2 t /v 2 A 2 λ

8 Minimal Lepton Flavor Violation What motivation for MLFV? Aping quark sector GUT s If leptons acquire Dirac masses (like quark sector) then copy from above. Uninteresting: flavor violation in charge lepton sector proportional to tiny neutrino masses Alternative: Majorana masses. (MLFV) Appealing: rationale for small neutrino masses: see-saw mechanism Different from quark sector. So... What are the prediction (for charged lepton ΔF 0 processes) from MLFV in see-saw models? 8

9 Note: LN vs LF Distinguish Lepton Number (LN) violating interactions from Lepton Flavor (LF) violating interactions LN is a U(1) symmetry, assigning unit charge to all leptons (like baryon number for quarks) Majorana mass breaks LN LF is an SU(3) symmetry, mixing different flavors It commutes with U(1) LN, ie, preserves the LN charge 9

10 We want to consider LFV at a low scale (few TeV?), while for see-saw want LNV at an intermediate scale Λ LF Λ LN M planck Two approaches. Field content below and gauge). Then: Λ LF scale is three families of L i and e Ri (plus H Minimal: majorana mass is from non-renormalizable LN breaking interaction Extended: include very heavy ν Ri insofar as it dictates MFV coupling, but then integrate out 10

11 MLFV: Minimal Field Content Assumptions: 1. The breaking of the U(1) LN is independent from the breaking of lepton flavor G LF, with large Λ LN (associated with see-saw) 2. There are only two irreducible sources of G LF breaking, λ e and gν, defined by L Sym.Br. = λ ij e ē i R(H L j L ) 1 2Λ LN g ij ν ( L ci L τ 2 H)(H T τ 2 L j L ) + h.c. (Note: one can also study 2 higgs model case. Note here) Ex: SUSY Triplet Model, A. Rossi, PRD66(2002)

12 MLFV: Extended Field Content Recall, now we include RH neutrinos, flavor group has additional SU(3) νr factor Assumptions: 1. The right handed neutrino mass is flavor neutral, ie, it breaks SU(3) νr to O(3) νr. Denote = M ν δ ij (this makes it still minimal ) M ij ν 2. The right handed neutrino mass is the only source of LN breaking and M ν Λ LFV 3. Remaining LF-symmetry broken only by λ e and λ ν defined by L Sym.Br. = λ ij e ē i R(H L j L ) + iλij ν ν i R(H T τ 2 L j L ) + h.c. Note: after integrating out ν R we have the same lagrangian as for minimal field content, with gν = (λ ν ) T λ ν The distinction is then in what operators are allowed Ex: SUSY with RH degenerate N s, J. Hisano et al, Phys. Rev. D 53, (1996) 12

13 Implementation of MLFV Want to add all possible terms to the lagrangian consistent with assumptions (and usual stuff: Lorentz invariance, gauge symmetry, locality,...) Need characterization of terms that are allowed λ ij e ē i R(H L j L ) 1 g ij 2Λ LN As before, use spurion method. In minimal field content case: ν ( L ci L τ 2 H)(H T τ 2 L j L ) L L V L L L e R V R e R and define λ e V R λ e V L g νg ν Extended field content case g ν V L g νv L with transformation V L V L λ ij e ē i R(H L j L ) + iλij ν ν i R(H T τ 2 L j L ) L L V L L L e R V R e R ν R O ν ν R and define λ e V R λ e V L = λ νλ ν λ ν O ν λ ν V L V L V L δ = λ T ν λ ν δ V L δ V L Also with just like gν for minimal content 13

14 Then write all operators of dimension 5, 6,... consistent with assumptions. For µ eγ, µ + N e + N, need two lepton field ops: O (1) LL = L L γ µ L L H id µ H O (2) LL = L L γ µ τ a L L H τ a id µ H O (3) LL = L L γ µ L L O (4d) LL O (4u) LL Ops with LL QL γ µ Q L = L L γ µ L L dr γ µ d R = L L γ µ L L ū R γ µ u R O (5) LL = L L γ µ τ a L L QL γ µ τ a Q L Ops with RL O (1) RL = g H ē R σ µν λ e L L B µν O (2) RL = gh ē R σ µν τ a λ e L L W a µν O (3) RL = (D µh) ē R λ e D µ L L O (4) RL = ē Rλ e L L QL λ D d R O (5) RL = ē Rσ µν λ e L L QL σ µν λ D d R O (6) RL = ē Rλ e L L ū R λ U iτ 2 Q L O (7) RL = ē Rσ µν λ e L L ū R σ µν λ U iτ 2 Q L We have neglected Δ 2 We have neglected, hence no RR operators (λ e ) 2 14

15 For µ eeē need, in addition, four lepton operators O (1) 4L = L L γ µ L L LL γ µ L L O (2) 4L = L L γ µ τ a L L LL γ µ τ a L L O (3) 4L = L L γ µ L L ē R γ µ e R O (4) 4L = δ njδ mi L i Lγ µ L j L L m L γ µ L n L O (5) 4L = δ njδ mi L i Lγ µ τ a L j L L m L γ µ τ a L n L 15

16 Up to dimension 6 operators, the new interactions are L eff = 1 Λ 2 LFV 5 i=1 ( ) c (i) LL O(i) LL + c(i) 4L O(i) 4L + 1 Λ 2 LFV 2 j=1 c (j) RL O(j) RL + h.c. with coefficients naively c 1 We can now study the phenomenology of MLFV with minimal field content. Useful to look at parameters first 16

17 Use G LF symmetry to rotate to the mass eigenstate basis (v = Higgs vev) λ e = m l v g ν = Λ LN v 2 = 1 v diag(m e, m µ, m τ ) U m ν U = Λ LN v 2 U diag(m ν1, m ν2, m ν3 )U U is the PMNS matrix. It is determined from neutrino mixing: U ce iα 1/2 se iα 2/2 s 13 e iδ se iα1/2 / 2 ce iα2/2 / 2 1/ 2 se iα1/2 / 2 ce iα2/2 / 2 1/ 2 Here c cos θ sol s sin θ sol θ sol 32.5 s13 is poorly known, s13 < oops! sorry: two different δ

18 Hence, amplitudes are given in terms of Λ LN or M ν and Λ LFV (actually only ratio Λ LN /Λ LFV or M ν /Λ LFV ) Coefficients, c, of order 1 Low energy measured (or measurable) masses and mixing angles In particular, the following two combinations appear in the operators: = { Λ 2 LN v 4 Um 2 νu minimal field content M ν v 2 Um ν U extended field content, CP limit δ = δ T = { ΛLN v 2 U m ν U minimal field content M ν v 2 U m ν U extended field content = λ νλ ν Note: recall in extended case, which is not directly related to mass. It is in CP limit, which we assume for simplicity (and further minimality). 18

19 MLFV: Phenomenology

20 μ eγ, μ-to-e conversion and their relatives I: minimal field content ( ) 4 B li l j (γ) = ΛLN R li lj(γ)(s 13, δ; c (i) ) Λ LFV - since Δ U(m ν ) 2 U, only differences of m 2 enter; these are measured! - s13 and δ unknown PMNS parameters (scan on δ) - choose c (i) of order one for the estimate - ratio of scales can be large: R Au µ e R µ eγ R Al µ e s 13 perturbative gν Λ LN (1 ev/m ν ) GeV so Λ LFV 1 TeV Λ LN /Λ LFV

21 Predictive: l l γ patterns are independent of unknown input parameters (scales cancel in ratios, in this case c (i) s cancel too, and all other parameters are from long distance) B µ eγ B τ µγ δ = δ = 0 B µ eγ B τ eγ δ = π/ δ = π δ = π s 13 s 13 21

22 Overall normalization 10-8 B τ µγ exp limit Bµ eγ B µ eγ If s 13 is small, look at tau modes s 13 Here Λ LN /Λ LF V = and c (1) RL c(2) RL = 1 Belle and BaBar have bounds (summer 05) of a few 10-7 for Br(τ l γ) and Br(τ lll) 22

23 μ eγ, μ-to-e conversion and their relatives II: extended field content Replace Λ 2 LN/Λ 2 LFV by vm ν /Λ 2 LFV Now Δ U m ν U so amplitudes depend on overall neutrino mass scale (ie, lightest neutrino mass) m lightest ν = 0 m lightest ν R Au µ e R µ eγ R Al µ e s = 0.2 ev R Au µ e R µ eγ R Al µ e s 13 ( ) 2 B li l j (γ) = vmν Rli lj(γ)(s Λ 2 13, m lightest ν ; c (i) ) LFV perturbative λ ν M ν GeV; with Λ LF V 1 TeV, 23 vm ν Λ 2 LF V 10 9

24 One final note: results depend on hierarchy of neutrino masses, normal (m ν1 m ν2 m ν3 ) vs. inverted (m ν1 m ν2 m ν3 ) NORMAL INVERTED 10-8 B τ µγ 10-8 B τ µγ δ = 0 B µ eγ δ = π B µ eγ δ = π exp limit Bµ eγ δ = 0 exp limit Bµ eγ s 13 s 13 (vm ν )/Λ 2 LFV = c (1) RL c(2) RL = 1 shading: 0 m lightest ν 0.02 ev 24

25 ( ) 4 Γ µ 3e /Γ µ eν ν = [ a a 2 8Re(a 0a ) 4Re(a 0a + )+6I a 0 2] Λ LN Λ aeµ LFV 2 minimal ( ) 2 vm ν Λ 2 beµ LFV 2 extended a + = sin 2 θ w (c (1) LL + c(2) LL ) + c(3) 4L a = (sin 2 θ w 1 2 )(c(1) LL + c(2) LL ) + c(1) 4L + c(2) 4L + 2δ eµδ ee eµ (c (4) 4L + c(5) 4L ) a 0 = 2e 2 (c (1) RL c(2) RL ) 3l Decays: 4L operators beμ aeμ s 13 beμ s 13 Π s 13 25

26 Γ τ eµ µ = Γ τ eν ν v 4 eτ 2 Λ 4 LFV [ a ã 2 4Re[a 0(a + + ã )] + 12Ĩ a 0 2] Γ τ µµē = Γ τ eν ν v 4 2δ eτ δ µµ 2 Λ 4 LFV c (4) L + c(5) L 2 NORMAL HIERARCHY INVERTED HIERARCHY 2deΜ dee aeμ m min ev 2deΜ dee aeμ m min ev d xx is δ xx 26

27 GUTs GUTs connect MFV in quark and lepton sectors Better motivation for MLFV New effects (e.g., LFV even for Dirac neutrino) Includes thoroughly studied models (e.g., SUSY-GUTs) 27

28 MFV-GUTs in a nut-shell three families of left handed fields: ψ i 5 χ i 10 N i 1 (d c R, L L ) (Q L, u c R, e c R) i = 1, 2, 3 In the absence of masses, symmetric under SU(3) 5 SU(3) 10 SU(3) 1 Include symmetry breaking (here with one higgs): λ ij 5 ψt i χ j H 5 + λ ij 10 χt i χ j H 5 1 M (λ 5) ij ψ T i Σχ j H 5 gives bad mass relations for light families λ u λ 10, λ d λ T e λ 5 Σ 24; M large; freedom to fix mass relations λ u λ 10, λ d (λ 5 + ɛλ 5), λ T e ( λ ɛλ 5), ɛ = MGUT /M λ ij 1 N T i ψ j H 5 + M ij R N T i N j neutrino masses (Dirac+Majorana) spurion transformation laws: Q L V 10 Q L u R V10 u R d R V 5 d R L L V 5 L L e R V10 e R 28 λ 10 V 10 λ 10 V 10 λ 5 V 5 λ 5 V 10 λ 5 V 5 λ 5 V 10 λ 1 V 1 λ 1 V 5 M R V 1 M R V 1 connect lepton to quark MFV

29 get old mixing structures (to be included in composite operators), like quarks: QL λ uλ u Q L, dr λ d λ uλ u Q L leptons: LL λ 1 λ 1 L L, ē R λ e λ 1 λ 1 L L but also get interesting new ones, like quarks: QL (λ e λ e) T Q L, d R λ T e (λ e λ e) T Q L, dr (λ e λ 1 λ 1 )T Q L, leptons: LL (λ d λ d )T L L, d R (λ eλ e ) T d R, dr (λ 1 λ 1 )T d R, ē R (λ d λ d λ d )T L L, ē R λ u λ uλ T d L L, ē R λ u λ ue R, ē R (λ d λ d )T e R, going over to quark/lepton mass basis, introduce two new mixing matrices C = V T e R V dl, G = V T e L V dr so get, for example ē R λ u λ ue R ē R λ u λ uλ T d L L ē R λ u λ uλ e L L ē R [ C (q) C ] er ē R [ C (q) λd G ] el ē R [ C (q) C ] λe e L where (q) ij V λ 2 CKM u V CKM = m2 t v 29 2 (V CKM) 3i(V CKM ) 3j + O(m 2 c,u/m 2 t )

30 quick example (probably out of time by now): τ µγ, τ eγ & µ eγ [ ] L eff = v Λ 2 ēr c 1 λ e λ 1 λ 1 + c 2 λ u λ uλ e + c 3 λ u λ uλ T d σ µν e L F µν just like MLFV above Generalizes Barbieri & Hall ( λ 5 = 0, C = G = 1 ) New mixing structures C = V T e R V dl Independent of Mν G = V T e L V dr Hierarchical Large: for Λ=10TeV Br(µ eγ) ( m 2 t v 2 ) λ 2 (m τ /v), (τ µ) λ 3 (m τ /v), (τ e) λ 5 (m µ /v), (µ e) (λ = 0.22) R. Barbieri, L.J. Hall, Phys. Lett. B 338 (1994) 212 R. Barbieri, L.J. Hall, A. Strumia, Nucl. Phys. B 445 (1995)

31 Conclusion (I think the case is made that) Non-tuned theories of flavor with LHC-scale new-physics may reasonably be expected to exhibit charged lepton LFV (lepton flavor violation) at levels that are accessible through experiments like MEG and the next generation, the proposed and mu2e Not water tight, only compelling. What could go wrong? -Accidents (small coefficients c) -No GUTs, Light right handed neutrinos (or purely Dirac) Leptogenesis? Ask Vincenzo 31

32 More slides

33 W W Part of loop graph (W is virtual). For any one intermediate quark amplitude is s u,c,t d Sum over intermediate quarks and expand q For first term use V qd V qsf (m 2 q/m 2 W ) q q M D W F (m 2 q/m 2 W, µ/m W ) V qd V qs = 0 V qd V qs [ F (0) + m2 q M 2 W and for second q F (0) + q u ] V qd V qs = V ud V us m 2 qv qd Vqs = (m 2 q m 2 u)v qd Vqs q u 33 (jump back)

34 Decays of/to hadrons Hopelessly small! Br π 0 µ + e Υ τµ τ πµ

35 We have also explored the effects of deleting a class of operators. For example: assume 4L operators are not present Can we get 3l decays? Yes, through loops Need care in loops of light quarks: chiral lagrangian does the job Result: amplitude is ~0.1 of 4L ops (large logs) Equivalently, these give a ~20% correction to rate Patterns are similar to those from 4L µ Ô (i) LL e π, K π, K q q V i µ V EM ν γ e e 35 V i µ V EM ν