Enhanced QED correction to B s → µ + µ − (M. Beneke, C. Bobeth, R. Szafron, Phys. Rev. Lett. 120, 011801) Robert Szafron Technische Universit¨ at M¨ unchen 9 January 2018 XXIV Cracow Epiphany Conference 1/22
Outline ◮ Motivation s → µ + µ − in the SM ◮ B 0 ◮ QED corrections in QCD bound states ◮ Results ◮ Conclusions 2/22
The Quest for New Physics How can we discover BSM physics? We need a measurement that shows significant discrepancy with the SM prediction 3/22
The Quest for New Physics How can we discover BSM physics? We need a measurement that shows significant discrepancy with the SM prediction Apart from the direct searches at the LHC, the good candidates are ◮ Precise low-energy measurements (e.g. g − 2, Lamb shift) ◮ Rare process suppressed/forbidden in the SM (e.g. CLFV, FCNC) 3/22
The Quest for New Physics How can we discover BSM physics? We need a measurement that shows significant discrepancy with the SM prediction Apart from the direct searches at the LHC, the good candidates are ◮ Precise low-energy measurements (e.g. g − 2, Lamb shift) ◮ Rare process suppressed/forbidden in the SM (e.g. CLFV, FCNC) The key point is We need a precise SM theoretical prediction! 3/22
Strange B-meson A good candidate for NP searches ◮ A meson composed of b and s quarks b 0 ◮ B 0 s and B s oscillate B s ◮ Among many its decay modes, s ¯ s → µ + µ − is particularly B 0 important for NP searches 4/22
B s → µ + µ − In the SM the process is ◮ loop suppressed (FCNC) SM helicity and loop suppression make it very sensitive to BSM scalar interactions. 5/22
B s → µ + µ − In the SM the process is ◮ loop suppressed (FCNC) ◮ helicity suppressed (scalar meson decaying into energetic muons) SM helicity and loop suppression make it very sensitive to BSM scalar interactions. 5/22
B s → µ + µ − In the SM the process is ◮ loop suppressed (FCNC) ◮ helicity suppressed (scalar meson decaying into energetic muons) ◮ purely leptonic final state allows for a precise SM prediction, QCD contained in the meson decay constant f B s SM helicity and loop suppression make it very sensitive to BSM scalar interactions. 5/22
Experimental status B s → µ + µ − has been observed by LHCb and CMS in 2013 [R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 111, 101805] [S. Chatrchyan et al. (CMS Collaboration), Phys. Rev. Lett. 111, 101804] Joint publication: [CMS Collaboration and LHCb Collaboration, Nature 522, 68–72] B ( B s → µ + µ − ) LHCb + CMS = (2 . 8 +0 . 7 − 0 . 6 ) · 10 − 9 Most recent update B ( B s → µ + µ − ) LHCb = (3 . 0 +0 . 7 − 0 . 6 ) · 10 − 9 [R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 118, 191801] 6/22
B s → µ + µ − in the SM ◮ Scales above m b can be integrated-out ◮ Electroweak physics is hidden in the matching coefficients of weak interaction EFT m b ◮ Systematic expansion in m W ◮ Large logs are summed using usual RG evolution [C. Bobeth, P. Gambino, M. Gorbahn, and U. Haisch, 2004; T. Huber, E. Lunghi, M. Misiak, and D. Wyler,2006] 10 L ∆ B =1 = 4 G F � √ C i Q i + h.c. 2 i =1 b W s b s u,c,t u,c,t Z l + l − l + l − 7/22
Theory Status Matching coefficients are known up to ◮ NLO EW [C. Bobeth, M. Gorbahn, E. Stamou, 2014 ] ◮ NNLO QCD [T. Hermann, M. Misiak, M. Steinhauser, 2013] B ( B s → µ + µ − ) TH = (3 . 65 ± 0 . 23) · 10 − 9 [C. Bobeth, M. Gorbahn, T. Hermann, M. Misiak, E. Stamou, M. Steinhauser, Phys.Rev.Lett. 112 (2014) 101801] ◮ QED corrections below the m b scale not included ◮ Estimated size of the correction O ( α em ) ∼ 0 . 3% Real radiation - only ultra-soft photons are important ◮ ISR is small (Low Soft-Photon Theorem) [Y. Aditya, K.Healey, A. Petrov, Phys.Rev. D87 (2013) 074028 ] ◮ FSR - included in the experimental analysis [A. J. Buras, J. Girrbach, D. Guadagnoli, G. Isidori, Eur.Phys.J. C72 (2012) 2172 ] 8/22
Scales in the problem Leptonic decay of B s is a SM multi-scale problem ◮ Electroweak scale m W m 2 W → ∞ ◮ Hard scale m b ◮ Hard-collinear scale Weak EFT � m b Λ QCD ◮ Soft scale Λ QCD m 2 b → ∞ ◮ Collinear scale m µ We take Λ QCD ∼ m µ so the soft SCET I ⊗ HQEFT scale of HQEFT is also a soft scale of SCET I m b Λ QCD → ∞ SCET II ⊕ HQEFT 9/22
QED corrections in QCD bound-states The final state has no strong interaction – QCD is contained in the decay constant q γ µ γ 5 b | ¯ B q ( p ) � = if B q p µ � 0 | ¯ This is no longer true when QED effects are included – non-local time ordered products have to be evaluated � d 4 x T { j QED ( x ) , L ∆ B =1 (0) }| ¯ � 0 | B q � This can be done for QED bound-states but QCD is non-perturbative at low scales 10/22
Helicity suppression Can the helicity suppression be relaxed? ¯ ℓ b ℓγ µ γ 5 ℓ → m ℓ ¯ ¯ ℓ c γ 5 ℓ ¯ B s c m b q ¯ ℓ For m ℓ → 0 the amplitude has to vanish 1 Annihilation and helicity flip take place at the same point r � m b 11/22
Helicity suppression Can the helicity suppression be relaxed? ¯ ℓ b m ℓ ¯ Λ QCD ¯ ℓγ µ γ ν ℓ → ℓ c γ 5 ℓ ¯ B s c ¯ q ℓ √ 1 Annihilation and helicity flip can be separated by r ∼ m b Λ QCD It is still a short distance effect since the size of the meson is 1 r ∼ Λ QCD For m ℓ → 0 the amplitude still vanishes 12/22
SCET approach ¯ b ℓ EFT approach allows to perform systematic expan- Λ QCD sion in m b . Two step matching is required: Ef- h.c. fective weak interaction → ℓ q ¯ SCET I → SCET II γ Modes ◮ Hard collinear, p 2 ∼ Λ QCD m b ◮ Collinear, p 2 ∼ Λ 2 QCD ∼ m 2 µ ◮ Soft p 2 ∼ Λ 2 QCD 13/22
SCET approach ¯ b ℓ The quark is only non-local along the direction of the lepton h.c. ℓ q ¯ γ Soft quarks do not contribute to the The quark has a hard- power enhanced correction collinear virtuality – soft glu- ons are power suppressed = − if B q m b � � � 0 | q ( x − ) β h v (0) α � B q ( p ) × 4 � ∞ � 1 + / � � ∂ �� � v d ω e − i ω t n − − n − l γ ν φ + ( ω ) / n + + φ − ( ω ) / γ 5 ⊥ ∂ l ν 2 0 ⊥ αβ 13/22
Operators α em q γ µ P L b )(¯ = 4 π (¯ ℓγ µ ℓ ) Q 9 α em �� ¯ q γ µ P L b � � Q 10 = ¯ ℓγ µ γ 5 ℓ 4 π e q σ µν P R b � � Q 7 = 16 π 2 m b ¯ F µν C eff C eff 7 9 14/22
The correction at amplitude level C 7 C i ¯ γ b ℓ ¯ ¯ b ℓ b ℓ q ′ C 9 , 10 γ q ¯ ℓ q ¯ q ¯ ℓ ℓ γ γ q ¯ ℓ q ¯ q ¯ ℓ ℓ γ ℓγ 5 ℓ + α em m ℓ f B q N C 10 ¯ 4 π Q ℓ Q q m ℓ m B f B q N ¯ i A = ℓ (1 + γ 5 ) ℓ � � ln m b ω u � � 1 � ∞ 9 ( um 2 d ω × 0 du (1 − u ) C eff b ) ω φ B + ( ω ) + ln m 2 0 1 − u ℓ � � + 2 π 2 � − 2 ln m b ω � ∞ ln 2 m b ω − Q ℓ C eff d ω ω φ B + ( ω ) 7 0 m 2 m 2 3 ℓ ℓ 15/22
The correction at amplitude level C 7 C i ¯ γ b ℓ ¯ ¯ b ℓ b ℓ q ′ C 9 , 10 γ q ¯ ℓ q ¯ q ¯ ℓ ℓ γ γ q ¯ ℓ q ¯ q ¯ ℓ ℓ γ ℓγ 5 ℓ + α em m ℓ f B q N C 10 ¯ 4 π Q ℓ Q q m ℓ m B f B q N ¯ i A = ℓ (1 + γ 5 ) ℓ � � ln m b ω u � � 1 � ∞ 9 ( um 2 d ω × 0 du (1 − u ) C eff b ) ω φ B + ( ω ) + ln m 2 0 1 − u ℓ � � + 2 π 2 � − 2 ln m b ω � ∞ ln 2 m b ω − Q ℓ C eff d ω ω φ B + ( ω ) 7 0 m 2 m 2 3 ℓ ℓ ◮ Tree level amplitude 15/22
The correction at amplitude level C 7 C i ¯ γ b ℓ ¯ ¯ b ℓ b ℓ q ′ C 9 , 10 γ q ¯ ℓ q ¯ q ¯ ℓ ℓ γ γ q ¯ ℓ q ¯ q ¯ ℓ ℓ γ ℓγ 5 ℓ + α em m ℓ f B q N C 10 ¯ 4 π Q ℓ Q q m ℓ m B f B q N ¯ i A = ℓ (1 + γ 5 ) ℓ � � ln m b ω u � � 1 � ∞ 9 ( um 2 d ω × 0 du (1 − u ) C eff b ) ω φ B + ( ω ) + ln m 2 0 1 − u ℓ � � + 2 π 2 � − 2 ln m b ω � ∞ ln 2 m b ω − Q ℓ C eff d ω ω φ B + ( ω ) 7 0 m 2 m 2 3 ℓ ℓ ◮ Helicity suppression × power enhancement factor 15/22
The correction at amplitude level C 7 C i ¯ γ b ℓ ¯ ¯ b ℓ b ℓ q ′ C 9 , 10 γ q ¯ ℓ q ¯ q ¯ ℓ ℓ γ γ q ¯ ℓ q ¯ q ¯ ℓ ℓ γ ℓγ 5 ℓ + α em m ℓ f B q N C 10 ¯ 4 π Q ℓ Q q m ℓ m B f B q N ¯ i A = ℓ (1 + γ 5 ) ℓ � � ln m b ω u � � 1 � ∞ 9 ( um 2 d ω × 0 du (1 − u ) C eff b ) ω φ B + ( ω ) + ln m 2 0 1 − u ℓ � � + 2 π 2 � − 2 ln m b ω � ∞ ln 2 m b ω − Q ℓ C eff d ω ω φ B + ( ω ) 7 0 m 2 m 2 3 ℓ ℓ ◮ Convolution from the hard-scale matching 15/22
The correction at amplitude level C 7 C i ¯ γ b ℓ ¯ ¯ b ℓ b ℓ q ′ C 9 , 10 γ q ¯ ℓ q ¯ q ¯ ℓ ℓ γ γ q ¯ ℓ q ¯ q ¯ ℓ ℓ γ ℓγ 5 ℓ + α em m ℓ f B q N C 10 ¯ 4 π Q ℓ Q q m ℓ m B f B q N ¯ i A = ℓ (1 + γ 5 ) ℓ � � ln m b ω u � � 1 � ∞ 9 ( um 2 d ω × 0 du (1 − u ) C eff b ) ω φ B + ( ω ) + ln m 2 0 1 − u ℓ � � + 2 π 2 � − 2 ln m b ω � ∞ ln 2 m b ω − Q ℓ C eff d ω ω φ B + ( ω ) 7 0 m 2 m 2 3 ℓ ℓ ◮ Convolution with the light-cone distribution function 15/22
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