Radiative μ and τ leptonic decays at NLO

Transcription

1 Radiative μ and τ leptonic decays at NLO Matteo Fael Universität Bern Massimo Passera INFN Padova Radcor-Loopfest 215 UCLA - June Work in collaboration with Lorenzo Mercolli, arxiv:

2 BaBar s branching ratio measurements B.R. of radiative leptonic decays (! = 1 MeV) e EXP st 52 sy st 1 sy 1 3 BABAR coll., PRD 91 (215) 5113 Babar experimental precision around 3. More precise than CLEO results: T. Bergfeld et al., PRL 84 (2) st 17 sy 1 2 e, st 35 sy 1 3. To be compared with SM branching ratio at NLO of order ln m l m ln m, 1 for l e, 3 for l. 2

3 Radiative μ and τ leptonic decays Very clean, can be predicted with very high precision. TH formulation in terms of Bouchiat-Michel-Kinoshita-Sirlin parameters allows to test couplings beyond the SM V-A; additional BMKS-like param. accessible in radiative decays. SM background for µ and τ flavour violating decays like µ! e! l µ +! e + e e + Precise data on radiative τ decays may allow to determine its anomalous magnetic moment. 3

4 The τ g-2: SM prediction The Standard Model prediction of the tau g-2 is: aτ SM = (2) x 1-8 QED (.5) x 1-8 EW (3.7) x 1-8 HLO (.2) x 1-8 HHO (vac) + 5 (3) x 1-8 HHO (lbl) a τ SM = (5) x 1-8 Eidelman & MP 27 (mτ/mµ) 2 ~ 28: great opportunity to look for New Physics, and a clean NP test too Muon Tau aew/ah 1/45 1/7 aew/ δah if only we could measure it!! 4

5 The τ g-2: experimental bounds The very short lifetime of the tau makes it very difficult to determine aτ measuring its spin precession in a magnetic field. DELPHI s result, from e + e - e + e - τ + τ - total cross-section measurements at LEP 2 (the PDG value): a τ = -.18 (17) PDG 214 With an effective Lagrangian approach, using data on tau lepton production at LEP1, SLC, and LEP2: -.4 < a τ NP <.6 (95% CL) -.7 < a τ NP <.5 (95% CL) Escribano & Massó 1997 Gonzáles-Sprinberg et al 2 Bernabéu et al, propose the measurement of F2(q 2 =Mϒ 2 ) from e + e - τ + τ - production at B factories. NPB 79 (28) 16 5

6 The τ g-2 via τ radiative leptonic decays: a proposal Must employ the polarized differential decay rate. Study the full phase space. d d m M W 2 d W d NLO a d a d d d W W ã µ ` d µ 5 ` ` ` Under investigation for Belle-II analysis. Should at least improve the Delphi bound. Work in progress with S. Eidelman & D. Epifanov 6

7 Differential decay rates 7

8 Radiative μ and τ leptonic decays: definitions ˆn e e l l l e M : mass of the or. m l : final charged lepton mass. n: polarization vector of the initial or. Mass ratio r m l M. Minimum photon energy. l l 8

9 Differential decay rates (dγ) d d m 2 M 2 W d W d virt d d,µ,µ,µ W W W l l l Kinoshita, Sirlin, PRL 2 (1959) 177; Fronsdal, Uberall, PR 133 (1959) 654 Eckstein, Pratt, Ann. Phys. 8 (1959) 297; Kuno, Okada, RMP 73 (21) 151 9

10 Differential decay rates (dγw) m 2 M 2 W m 2 M 2 W d d m 2 M 2 W d W d virt d d,µ,µ,µ W W W l l l M. Fael, L. Mercolli, MP, PRD 88 (213)

11 Differential decay rates (dγv) d d m 2 M 2 W d W d virt d d NLO corrections computed in the effective Fermi Lagrangian.,µ l Fischer Fischeret etal., al., PRD PRD49 49 (1994) 3426; Arbuzov, Scherbakova, PLB PLB (24) (24) 285; 285 also as part of NNLO corrections to decay: T. van Ritbergen, R. G. Stuart, PRL 82 (1999) 488, C. Anastasiou, K. Melnikov, F. Petriello, JHEP 79 (27) 14, F. Caola, A. Czarnecki, Y. Liang, K. Melnikov, R. Szafron, PRD 9 (214) 5. 12/23 11

12 Differential decay rates (dγγγ) d d m 2 M 2 W d W d virt d d ,µ 2 l 1 1 Extra soft photon emission integrated analytically: d soft d d M 2 IR divs of soft bremsstrahlung and virtual corrections cancel. Double hard bremss integrated numerically ( ). 12

13 Differential decay rates (full dγnlo) d 6 de l d d l G n p l J p K p l p L ml 2 G E l cos le G MW 2 G W G NLO, same for J and K L E l cos le L NLO NLO! Analytically integrated over neutrinos. Visible particle kinematics E l l. G: isotropic function. ˆn J K L: spindependentparts. l l 13

14 Differential decay rates - comparison with previous results d 6 de l d d l G n p l J p K p l p L One-loop corrections to e differential decay rate previously computed in: A. Fischer et al., PRD 49 (1994) 3426 Only the isotropic part G. Files with results unavailable. A. Arbuzov, E. S. Scherbakova, PLB 597 (24) 285 Full spin dependence, but calculation performed in the m l limit. Perfect agreement with our isotropic function G (in that limit). The anisotropic functions J K differ from ours. L has been overlooked. 14

15 Branching Ratios 15

16 Branching ratios at LO B.R. for a minimum photon energy! LO y G 2 F M ln r ln y 6y y 4 3Li 2 y 2 where y 2 M y 1 y r m l M y 3 y ln y y 33y 2 7y 3 y Kinoshita & Sirlin, PRL 2 (1959) 177; Eckstein & Pratt, Ann. Phys. 8 (1959) 297 In LO y only ln r are kept, terms r are neglected. However terms r 2 were not, and cannot, be neglected in the integrand d (be careful also at NLO!). When r is small, LO y gives a good prediction for the branching ratios LO, compared with result integrated numerically.for 1 MeV: LO num LO e e OFF by ~2% 16

17 Inclusive and exclusive branching ratios at NLO The branching ratio of radiative and leptonic decays for a minimum photon energy : d 4 d LO d virt d 5 d! 2 Excl. Incl. Exc :only one of energy, additional second soft photon. Exc! double soft Excl. Inc :at least one of energy. Inc +!! 1 17

18 Branching ratios: τ results B.R. of radiative leptonic decays (! = 1 MeV) e LO Inc NLO n 1 N n 2 N 1 5 Exc NLO n 19 N n 3 N 1 5 Inc th th Exc th th EXP st 52 sy st 1 sy 1 3 n : numerical errors N : uncomputed NNLO corr. ln r ln M Exc Inc NLO BABAR - PRD 91 (215) 5113 th : combined n N : experimental error of lifetime: s e Exc

19 Branching ratios: μ results B.R. of radiative µ leptonic decays (! = 1 MeV) LO Inc NLO n 6 N 1 4 Exc NLO n 7 N 1 4 Inc th 1 2 Exc th 1 2 Crittenden et al. PR 1961 (1961) 1823 J. Adam et al. arxiv: [hep-ex] D. Počanić et al [nucl-ex]. EXP Preliminary new measurements cuts e MEG E e 45 MeV, 4 MeV st 53 sy 1 8 PIBETA 1 MeV, e st 42 sy

20 Conclusions We studied the differential rates and BRs of the radiative decays l l e and e in the SM at NLO in. QED RC were computed taking into account the full mass dependence r m l M, needed for the correct determination of the BRs. We found agreement with an earlier calculation of the isotropic function G in the r limit,whileouranisotropicfunctionsj and K differ. Also, the function L has been previously overlooked. BRs: our results agree with the experimental value for and BaBar s recent measurement of, both for a minimum photon energy 1 MeV. On the contrary, BaBar s recent precise measurement of e, for the same, differs from our prediction by 3 5. e 2

21 Backup Slides 21

22 Differential decay rates (details) The total differential decay for a polarized or lepton in the tau r.f. for y y is d 6 y dx dy d l d G 2 F m x 1 W m m e G x n p l J y n p K yx n p l p L where x 2E l m, y 2E m, c cos l. The polarization vector n n satisfies n 2 1 and n p. The function G x y c y, and similarly for J and K, is given by G x y c y 4 3yz 2 g x y z g NLO x y z y m M W 2 g W x y z 22

23 Inclusive and Exclusive Branching Ratios (2) lim d 4 d LO d virt d soft d 5 d We use a soft threshold to cancel IR. Soft photon radiation d soft computed thanks to the factorization of the amplitude. Integration over the phase space performed numerically. Double hard emission also checked with MadGraph5. 23