SCET approach to power-enhanced QED correction in B s → µ + µ − (M. Beneke, C. Bobeth, R. Szafron, Phys. Rev. Lett. 120, 011801) Robert Szafron Technische Universit¨ at M¨ unchen SCET Workshop 19-22 March 2018 Amsterdam Robert Szafron 1/18
Outline ◮ Motivation for precision flavor physics ◮ Power-enhanced correction ◮ SCET approach ◮ Numerical results ◮ Conclusions Robert Szafron 1/18
Why do we need to know the QED corrections in flavour physics? m 2 b / m 2 ◮ Large logarithmic ln � � enhancements can mimic ℓ lepton-flavor universality violation ◮ Soft photon approximation employed so far is not suitable for virtual photons that can resolve the B-meson. ◮ Factorization theorems do not yet exist for QED ◮ Expected precision of measurements may require the inclusion of QED corrections or at least a proof that no effects above 1% exist Our starting point B s → µ + µ − Robert Szafron 1/18
B s → µ + µ − In the SM the process is ◮ loop suppressed (FCNC) Robert Szafron 2/18
B s → µ + µ − In the SM the process is ◮ loop suppressed (FCNC) ◮ helicity suppressed (scalar meson decaying into energetic muons, vector interaction), A ∼ m µ Robert Szafron 2/18
B s → µ + µ − In the SM the process is ◮ loop suppressed (FCNC) ◮ helicity suppressed (scalar meson decaying into energetic muons, vector interaction), A ∼ m µ ◮ purely leptonic final state allows for a precise SM prediction, QCD contained in the meson decay constant f B s Highly suppressed in SM and can be computed with a good precision! Robert Szafron 2/18
Theory Status Weak EFT (at the scale m b ) matching coefficients ◮ 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, 101801, 2014] Robert Szafron 3/18
Theory Status Weak EFT (at the scale m b ) matching coefficients ◮ 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, 101801, 2014] ◮ QED corrections below the m b scale not included Robert Szafron 3/18
Theory Status Weak EFT (at the scale m b ) matching coefficients ◮ 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, 101801, 2014] ◮ QED corrections below the m b scale not included Real radiation - only ultra-soft photons are important ◮ ISR is small [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] Robert Szafron 3/18
QED corrections in QCD bound-states The final state has no strong interaction – QCD is contained in the decay constant q (0) γ µ γ 5 b (0) | ¯ 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 e iqx 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 Robert Szafron 4/18
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 Robert Szafron 5/18
Diagrams in the Weak EFT Power-enhanced ¯ ¯ ¯ ¯ b ℓ b ℓ b ℓ b ℓ s ℓ s ℓ s ℓ s ℓ Not power-enhanced ¯ ¯ ¯ ¯ b ℓ b ℓ b ℓ b ℓ s ℓ s ℓ s ℓ s ℓ ¯ ¯ ¯ b ℓ b ℓ b ℓ s ℓ s ℓ s ℓ e, µ, τ Robert Szafron 6/18
The correction at the 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 Robert Szafron 7/18
The correction at the 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 Robert Szafron 7/18
The correction at the 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 Robert Szafron 7/18
The correction at the 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 Robert Szafron 7/18
The correction at the 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 ◮ Double logarithmic enhancement due to endpoint singularity Robert Szafron 7/18
Helicity suppression Can the helicity suppression be relaxed? ¯ ¯ ℓ ℓ b b ℓγ µ γ 5 ℓ → m ℓ ¯ Λ QCD ¯ m ℓ ¯ ¯ ℓ c γ 5 ℓ ¯ ℓγ µ γ ν ℓ → ℓ c γ 5 ℓ ¯ B s B s c c m b q ¯ q ¯ ℓ ℓ Without QED: � � m ℓ u ( p ℓ ) = u c ( p ℓ ) + O E ℓ For m ℓ → 0 the amplitude has to vanish 1 Annihilation and helicity flip take place at the same point r � m b Robert Szafron 8/18
Helicity suppression Can the helicity suppression be relaxed? ¯ ¯ ℓ ℓ b b ℓγ µ γ 5 ℓ → m ℓ ¯ Λ QCD ¯ m ℓ ¯ ¯ ℓ c γ 5 ℓ ¯ ℓγ µ γ ν ℓ → ℓ c γ 5 ℓ ¯ B s B s c c m b q ¯ q ¯ ℓ ℓ With QED: 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 Robert Szafron 8/18
Operators at the hard scale α em q γ µ P L b )(¯ Q 9 = 4 π (¯ ℓγ µ ℓ ) α em q γ µ P L b �� ¯ � � = ¯ ℓγ µ γ 5 ℓ Q 10 4 π e q σ µν P R b � � Q 7 = 16 π 2 m b ¯ F µν Four-quark operators can be treated as a modification of C 9 and C 7 9 ( q 2 ) C eff C eff 7 Robert Szafron 9/18
C 9 contribution Typical SCET I → SCET II problem but with m ℓ � = 0 � � � C i ( µ b ) O i → du H i ( u ) J i ( u ) , i i Weak operators are matched on B-type SCET I currents � �� � χ C ( n + s ) γ µ ℓ C ( n + t ) γ ⊥ J 9 ( s , t ) = ⊥ P L h ν (0) µ ℓ C (0) � dr J X ( u ) = n + p ℓ 2 π exp [ − iun + p ℓ r ] J X (0 , r ) H 9 ( u ) = C eff 9 ( u ) + O ( α em ) Robert Szafron 10/18
SCET I diagrams ◮ We need L (1) ξ q to convert the hc-quark into a soft quark ◮ For pure hc-interaction, mass is power suppressed ⊥ ∼ λ , L (1) m ℓ ∼ λ 2 , p hc hc , m ◮ For pure c-interaction, mass is included in the leading power Lagrangian, m ℓ ∼ λ 2 ∼ p c ⊥ , L (0) c , m Robert Szafron 11/18
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