Lepton Flavor Violation in the Standard Model with general Dimension-6 Operators.

Transcription

1 Lepton Flavor Violation in the Standard Model with general Dimension-6 Operators. Janusz Rosiek based on JHEP 1404 (2014) 167, A. Crivellin, S. Najjari, JR Qui Nhon, 1 Aug 2014 Lepton Flavor Violation in the SM SM extensions parametrization: effective higher dimension operators Physical observables calculation radiative lepton decays l l γ charged lepton EDMs and g 2 anomaly 3-body LFV charged lepton decays l l l l Z 0 ll decays Numerical results and bounds Conclusions J. Rosiek, Lepton Flavor Violation in the SM with general Dimension-6 Operators. 1

2 1. Lepton Flavor Violation in the SM No flavor and CP violation in the lepton sector of SM with massless neutrinos. Neutrino mass discovery shows explicit LFV violation! Charged lepton sector good laboratory to search for New Physics: SM effects negligible, GIM-suppressed by m 2 ν/m 2 W LFV processes theoretically clean, no non-perturbative QCD effects. J. Rosiek, Lepton Flavor Violation in the SM with general Dimension-6 Operators. 2

3 Two possible approaches: 1. Construct specific New Physics model; calculate LFV observables; compare with experimental bounds to constrain model parameters. Time and labor consuming - full calculation need to be done for each model separately. 2. Parametrize New Physics effects in terms of higher dimension operators. Express LFV observables in terms of Wilson coefficients. Model independent - needs to be done just once. Only Wilson coefficients need to be calculated within specific models. J. Rosiek, Lepton Flavor Violation in the SM with general Dimension-6 Operators. 3

4 2. Effective operator approach Advantages - completeness and automatic gauge invariance. Start from full operator basis: L (4) SM L SM = L (4) SM + 1 Λ k C (5) k Q (5) k + 1 Λ 2 k C (6) k Q (6) ( ) 1 k +O Λ 3 - standard renormalizable dim-4 SM Lagrangian.. Full list of dimension 5- an 6-operators: Buchmiller-Wiler 1986 Reduced to minimal set: Grz adkowski, Iskrzyński, Misiak, JR (64 if baryon number not conserved) independent operators. J. Rosiek, Lepton Flavor Violation in the SM with general Dimension-6 Operators. 4

5 LFV related (ϕ - Higgs doublet, B,W - U(1),SU(2) gauge bosons): llll llxϕ llϕ 2 D and llϕ 3 Q ll ( l i l j )( l k l l ) Q ew ( l o σ µν e j )τ I ϕw I µν Q (1) (ϕ i D µ ϕ)( l i l j ) Q ee (ē i e j )(ē k e l ) Q eb ( l i σ µν e j )ϕb µν Q (3) (ϕ i D I µ ϕ)( l i τ I l j ) Q le ( l i l j )(ē k e l ) Q ϕe (ϕ i D µ ϕ)(ē i e j ) llqq Q eϕ3 (ϕ ϕ)( l i e j ϕ) Q (1) lq ( l i l j )( q k q l ) Q ld ( l i l j )( d k d l ) Q lu ( l i l j )(ū k u l ) Q (3) lq ( l i τ I l j )( q k τ I q l ) Q ed (ē i e j )( d k d l ) Q eu (ē i e j )(ū k u l ) Q eq (ē i e j )( q k q l ) Q ledq ( l a i e j)( d k q a l ) Q(1) lequ ( l a i e j)ε ab ( q b k u l) Q (3) lequ ( l a i σ µνe a )ε ab ( q b k σµν u l ) Complete set - other operators give LFV effects suppressed by m 2 ν/m 2 W. 19 (+1 of dim-5) Wilson coefficients - too many, small predictive power. J. Rosiek, Lepton Flavor Violation in the SM with general Dimension-6 Operators. 5

6 1. Only dim-5 term (Weinberg operator) Q νν = ε ab ε cd ϕ a ϕ c (l b i )T Cl d j does not contribute directly to LFV processes in the charged lepton sector. 2. Q eϕ3 = (ϕ ϕ)( l i e j ϕ) gives off-diagonal corrections to fermion masses. Rediagonalization LFV effect in charged lepton sector negligible. 3. Contributions of most of the 2-lepton-2 quark operators vanish or suppressed by m l /M W powers. 4. Goldstone/Higgs couplings to leptons - always suppressed by m l /M W powers we neglect all physical Higgs diagrams. 9 operators remain: 2: (llϕx)-type operators - modify γll vertex 3: (ll)(ϕdϕ)-type operators - modify Zll,Wl v vertices 3: 4-lepton contact couplings 1: 2-lepton 2-quark coupling. J. Rosiek, Lepton Flavor Violation in the SM with general Dimension-6 Operators. 6

7 3. Effective lepton-photon coupling Related observables: radiative lepton decays (µ eγ), EDMs and AMMs of charged leptons. The general form of the flavor violating photon-lepton vertex:: V fi µ llγ = i [ γ µ fi (F Λ 2 VL P L +F fi VR P R)+(F fi SL P L +F fi SR P R)q µ + (F fi TL iσµν P L +F fi ] TR iσµν P R )q ν Most important: tensor F TL,F TR. Tree level LFV contribution exist: q l i l f i ( e δ fi +iσ µν [ C fi γl P L +C fi γr P ] ) R qν C γr fi = C γl fi = v 2 Λ 2 ( cw C fi eb s WC fi ) ew J. Rosiek, Lepton Flavor Violation in the SM with general Dimension-6 Operators. 7

8 lσ µν lf µν coupling non-renormalizable - generated radiatively, Wilson coefficients C eb,c ew inherently contain loop suppression factors. Other operators can be generated at tree level, like contact 4-lepton coupling via contraction of the heavy boson propagator. It make sense to add 1-loop terms from other operators to tree-level C eb,c ew terms! 1-loop calculation relatively complicated - quadruple- and quintuplevertices appear with new Dirac structures and momentum dependence. J. Rosiek, Lepton Flavor Violation in the SM with general Dimension-6 Operators. 8

9 l i l j l j l j l j W Z 0 G 0 ν j l f l i l f l i a) b) c) W l f l i W G G W G ν j ν j ν j l f l i l f l i d) e) f) G l f l i G ν j l f l i γ G µ G ν j l f l i l f g) h) i) W G G W G G l i l f l i l f l i j) k) l) l f l i l j (d j,u j ) l f l i m) n) l j l f + reducible diagrams J. Rosiek, Lepton Flavor Violation in the SM with general Dimension-6 Operators. 9

10 Non-trivial internal test gauge invariance, requires for i j: F VL = F VR = 0 for p 2 i = m2 l i, p 2 f = m2 l f, q 2 = 0. Calculations performed in two independent approaches: 1. Passarino-Veltman reduction of tensor loop integrals in the fixed Feynman gauge (R ξ with ξ = 1) 2. Asymptotic expansion of loop integrals in the general R ξ gauge. Both approaches agree and confirm explicit gauge invariance. J. Rosiek, Lepton Flavor Violation in the SM with general Dimension-6 Operators. 10

11 Final 1-loop results for tensor form-factors compact and simple: ZWG fi FTL ZWG fi FTR = 4e [C (1)fi = 4e [C (1)fi m f (1+s 2 ) W (C (3)fi 3(4π) 2 m i (1+s 2 ) W (C (3)fi 3(4π) 2 m f +C fi ϕe m i)( 3 2 s2 W ) ] m i +C fi ϕem f)( 3 2 s2 W ) ] 4l fi FTL = 2e 3 (4π) 2 j=1 Cfjji le m j 4l fi FTR = 2e 3 (4π) 2 j=1 Cjifj le m j ql fi FTL = 16e ( ) 3 3(4π) 2 j=1 C(3)fijj lequ m uj log m2 u j µ 2 ql fi FTR = 16e ( ) 3 3(4π) 2 j=1 C(3)fijj lequ m uj log m2 u j µ 2 Q (3) lequ = ( l a i σ µνe a )ε ab ( q b k σµν u l ) can only be loop generated - its (infinite) 1-loop contribution double-loop suppressed can be neglected. J. Rosiek, Lepton Flavor Violation in the SM with general Dimension-6 Operators. 11

12 Final result for F TL,F TR depend on 6 Wilson coefficients: tree level C eb,c ew one loop C (1),C(3),C ϕe,c le. Applications of effective lepton-photon coupling: radiative lepton decays l i l f γ (µ eγ, τ eγ,µγ): m 3 l i ( F fi 2 + Br [ l i l f γ ] = 16πΛ 4 Γ TR li charged leptons anomalous magnetic moments: a li = 2m l i eλ 2 Re[ FTR ii ] charged leptons electric dipole moments d li = 1 Λ 2 Im[ FTR ii ] F fi TL 2) J. Rosiek, Lepton Flavor Violation in the SM with general Dimension-6 Operators. 12

13 4. Tree-level decays: l l l l and Z 0 l f l+ i 3-body charged lepton decays - various final state compositions: 3 leptons of the same flavor: µ ± e ± e + e, τ ± e ± e + e and τ ± µ ± µ + µ. 3 distinguishable leptons: τ ± e ± µ + µ and τ ± µ ± e + e. 2 lepton of the same flavor and charge and 1 with different flavor and opposite charge: τ ± e µ ± µ ± and τ ± µ e ± e ± (exotic, L = 2!). Tricky phase space integral, photon propagator 1/q 2 1/m 2 l diverges in corners of phase space photon contribution enhanced by logarithmic factor log(m 2 l /m2 l ). Z 0 l f l+ i decays - interesting observation: γll and Zll decays depend on orthogonal combinations ( c W C fi eb s WC fi ) ( ew, sw C fi eb +c WC fi ) ew. J. Rosiek, Lepton Flavor Violation in the SM with general Dimension-6 Operators. 13

14 5. Numerical analysis and bounds on Wilson coefficients Assumption: no fine-tuning or large cancellations. Then: anomalous tree-level γll coupling best constrained by radiative lepton decays Zll coupling and contact 4 lepton couplings best constrained by the three-body charged lepton decays First approximation - constrain γll couplings from limits on Br[l i l f γ], assuming all other Wilson coefficients negligible (C γ = c W C fi eb s WC fi ew ): C 12 2 γ + ( ) 2 C 21 2 Λ Br[µ eγ] γ TeV , 13 C 13 2 γ + ( ) 2 C 31 2 Λ Br[τ eγ] γ TeV , 8 C 23 2 γ + ( ) 2 C 32 2 Λ Br[τ eγ] γ TeV Numbers dividing the branching ratios the current experimental bounds. J. Rosiek, Lepton Flavor Violation in the SM with general Dimension-6 Operators. 14

15 Strong bounds on C γ for Λ 1 TeV: for µ e transitions 10 6 for τ µ,e transitions. Next: assume Cγ fi 0 and use the bounds from the Br(l i l f l f l f ): with C li l f l f l f given by ( ) 2 Λ Br[µ eee] C µeee TeV 1 10, 12 ( ) 2 Λ Br[τ eee] C τeee TeV , 8 ) 2 Br[τ µµµ] , 8 C τµµµ ( Λ 1 TeV C li l f l f l f = C fffi le ( C (1)fi ( C (1)fi +C (3)fi +C (3)fi ϕe ) ) 2 +C fiff le ee 2 +2 C fiff le 0.54 C fi 2 +2 C fiff C fi 2. ϕe J. Rosiek, Lepton Flavor Violation in the SM with general Dimension-6 Operators. 15

16 Bounds on Wilson coefficient of the LFV 4-lepton and the Z 0 -leptonlepton vertices for Λ O(1) TeV: 10 5 for µ e transitions 10 2 for τ µ and τ e transitions. Bounds weaker than for anomalous photon couplings, but these coefficients can be tree level generated, so potentially larger. Last bound from Z 0 l ± f l i : C (1)12 C (1)13 C (1)23 +C (3)12 +C (3)13 +C (3) C 12 ϕe 2 + C 13 ϕe 2 + C 23 ϕe 2 + C 12 Z 2 + C 13 Z 2 + C 23 Z 2 + C 21 Z 2 + C 31 Z 2 + C 32 Z ( ) 2 2 Λ Br [ Z 0 µ ± e ] TeV ( ) 2 2 Λ Br [ Z 0 τ ± e ] TeV ( ) 2 2 Λ Br [ Z 0 τ ± µ ] TeV Less stringent but constrain orthogonal combination of C eb,c ew. J. Rosiek, Lepton Flavor Violation in the SM with general Dimension-6 Operators. 16

17 Correlations of Wilson coefficients. Example: correlation of Z(W)ll-γll couplings. 12 C Φe Μ eee,μ eγ and Z Μe C ew C Φe Τ eee,τ eγ and Z Τe C 13 ew C Φe Τ ΜΜΜ,Τ ΜΓ and Z ΤΜ C 23 ew 10 3 Allowed regions in the C fi ew Cfi ϕe plane for Λ = 10 TeV. Red, yellow: l i l f γ, l i l f l f l f. The contour lines: Br(Z 0 l f l i ). Photon couplings better constrained by l lγ, Z,W couplings by l 3l decays. J. Rosiek, Lepton Flavor Violation in the SM with general Dimension-6 Operators. 17

18 Constraints from Z 0 l ± f l i decays weaker but still useful as they are complementary to bounds from l i l f γ: Τ ΜΓ and Z ΤΜ C eb Allowed regions from Br[Z 0 τµ] (yellow) and Br[τ µγ] (blue) in the plane for Λ = 1 TeV. CeW 23 C23 eb 23 C ew J. Rosiek, Lepton Flavor Violation in the SM with general Dimension-6 Operators. 18

19 Ratios of decay rates independent of New Physics scale Λ. In the photon domination scenario decay rates strictly related: R fi Br(l i 3l f ) Br(l i l f γ) = α f 3π (logm2 m 2 11 i 4 ) Crucial for experiments: no need to test µ eγ and µ 3e separately? C Φ CΓ Br Μ eγ Br Μ eee C Φe C Γ C Φ CΓ R fi ratio in the C fi ϕe/c fi Br Τ eγ Br Τ eee C Φe C Γ γ C (1)fi C Φ CΓ Br Τ ΜΓ Br Τ ΜΜΜ /C fi γ plane. C Φe C Γ J. Rosiek, Lepton Flavor Violation in the SM with general Dimension-6 Operators. 19

20 Lepton number conserving observables. Anomalous magnetic moments: Example: muon g 2 anomaly. a µ = Re [ ] ( ) Cle Cγ 22 1 TeV, Λ To be compared with measurement: a exp µ (2.7±0.8) 10 9 Electric Dipole Moments (normalized to current exp. bounds): d e /d exp e = Im [ ] ( Cle Cγ 11 1 TeV Λ d µ /d exp µ = 36.1 Im [ ] ( ) Cle Cγ 22 1 TeV Λ d τ /d exp τ = 0.69 Im [ ] ( ) 2 Cγ 33 1 TeV Λ ) 2 J. Rosiek, Lepton Flavor Violation in the SM with general Dimension-6 Operators. 20

21 6. Conclusions Several experimentally well constrained LFV processes calculated within the SM extended with all LFV dim-6 operators. Predictions in terms of Wilson coefficients - all relevant contributions included, results automatically gauge-invariant. Approximate numerical formulae based on current exp. bounds - specific NP models can be tested just calculating Wilson coefficients. Typical bounds on LFV Wilson coefficients discussed depending on New Physics scale Λ - usually very strong for Λ = O(1) TeV. Examples of correlations between various Wilson coefficients shown. J. Rosiek, Lepton Flavor Violation in the SM with general Dimension-6 Operators. 21