Semi-leptonic form factors for B s → K ℓν and B s → D s ℓν Oliver Witzel (RBC-UKQCD collaborations) Lattice 2018, East Lansing, MI, July 27, 2018
RBC- and UKQCD collaborations BNL/RBRC Columbia U U Edinburgh U Southampton Yasumichi Aoki (KEK) Ziyuan Bai Peter Boyle Jonathan Flynn Mattia Bruno Norman Christ Guido Cossu Vera G¨ ulpers Taku Izubuchi Duo Guo Luigi Del Debbio James Harrison Yong-Chull Jang Christopher Kelly Tadeusz Janowski Andreas J¨ uttner Chulwoo Jung Bob Mawhinney Richard Kenway James Richings Christoph Lehner Masaaki Tomii Julia Kettle Chris Sachrajda Meifeng Lin Jiqun Tu Fionn O’haigan Aaron Meyer Bigeng Wang Brian Pendleton Stony Brook University Hiroshi Ohki Tianle Wang Antonin Portelli Jun-Sik Yoo Shigemi Ohta (KEK) Evan Wickenden Tobias Tsang Sergey Syritsyn (RBRC) Amarjit Soni Yidi Zhao Azusa Yamaguchi U Connecticut U Colorado Boulder KEK U Liverpool Tom Blum Oliver Witzel Julien Frison Nicolas Garron Dan Hoying (BNL) MIT Peking U York U (Toronto) Luchang Jin (RBRC) Cheng Tu David Murphy Xu Feng Renwick Hudspith
RBC- and UKQCD collaborations BNL/RBRC Columbia U U Edinburgh U Southampton Yasumichi Aoki (KEK) Ziyuan Bai Peter Boyle Jonathan Flynn Mattia Bruno Norman Christ Guido Cossu Vera G¨ ulpers Taku Izubuchi Duo Guo Luigi Del Debbio James Harrison Yong-Chull Jang Christopher Kelly Tadeusz Janowski Andreas J¨ uttner Chulwoo Jung Bob Mawhinney Richard Kenway James Richings Christoph Lehner Masaaki Tomii Julia Kettle Chris Sachrajda Meifeng Lin Jiqun Tu Fionn O’haigan Aaron Meyer Bigeng Wang Brian Pendleton Stony Brook University Hiroshi Ohki Tianle Wang Antonin Portelli Jun-Sik Yoo Shigemi Ohta (KEK) Evan Wickenden Tobias Tsang Sergey Syritsyn (RBRC) Amarjit Soni Yidi Zhao Azusa Yamaguchi U Connecticut U Colorado Boulder KEK U Liverpool Tom Blum Oliver Witzel Julien Frison Nicolas Garron Dan Hoying (BNL) MIT Peking U York U (Toronto) Luchang Jin (RBRC) Cheng Tu David Murphy Xu Feng Renwick Hudspith
introduction
introduction Bs → K ℓν Bs → Ds ℓν conclusion Why B s meson decays? 1.5 excluded at CL > 0.95 excluded area has CL > 0.95 ◮ Alternative, tree-level determination of | V cb | γ 1.0 ∆ m & ∆ m and | V ub | from B s → D s ℓν and B s → K ℓν s d sin 2 β 0.5 → Commonly used B → πℓν and B → D ( ∗ ) ℓν ∆ m d ε α K β → Long standing 2 − 3 σ discrepancy between γ η 0.0 α V ub V Λ b ub SL α exclusive ( B → πℓν ) and inclusive ( B → X u ℓν ) V ub τ ν -0.5 → B → τν has larger error ε γ K -1.0 CKM sol. w/ cos 2 β < 0 f i t t e r → Alternative, exclusive (Λ b → p ℓν ) determination (excl. at CL > 0.95) ICHEP 16 [Detmold, Lehner, Meinel, PRD92 (2015) 034503] -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 ρ [ http://ckmfitter.in2p3.fr ] 3 / 18
introduction Bs → K ℓν Bs → Ds ℓν conclusion Why B s meson decays? [HFLAV] R(D*) BaBar, PRL109,101802(2012) 0.5 ∆ χ 2 = 1.0 contours Belle, PRD92,072014(2015) ◮ Alternative tests of lepton flavor violations LHCb, PRL115,111803(2015) SM Predictions Belle, PRD94,072007(2016) 0.45 → Determine e.g. R D ( ∗ ) from B s decays R(D)=0.300(8) HPQCD (2015) Belle, PRL118,211801(2017) s LHCb, FPCP2017 R(D)=0.299(11) FNAL/MILC (2015) to compare with R D ( ∗ ) from B decays Average R(D*)=0.252(3) S. Fajfer et al. (2012) 0.4 0.35 σ 4 D ( ∗ ) ≡ BF ( B → D ( ∗ ) τν τ ) R τ/µ σ BF ( B → D ( ∗ ) µν µ ) 0.3 2 0.25 HFLAV FPCP 2017 χ 0.2 P( 2 ) = 71.6% 0.2 0.3 0.4 0.5 0.6 R(D) ◮ Nonperturbative lattice calculation favor B s over B decays (higher precision) ◮ Only the spectator quark differs: R D ( ∗ ) may be a good proxy for R D ( ∗ ) s 3 / 18
introduction Bs → K ℓν Bs → Ds ℓν conclusion Why B s meson decays? [HFLAV] R(D*) R(D*) BaBar, PRL109,101802(2012) BaBar, PRL109,101802(2012) 0.5 0.5 ∆ ∆ χ χ 2 2 = 1.0 contours = 1.0 contours Belle, PRD92,072014(2015) Belle, PRD92,072014(2015) ◮ Alternative tests of lepton flavor violations LHCb, PRL115,111803(2015) LHCb, PRL115,111803(2015) Average of SM predictions SM Predictions Belle, PRD94,072007(2016) Belle, PRD94,072007(2016) 0.45 0.45 → Determine e.g. R D ( ∗ ) from B s decays ± R(D)=0.300(8) HPQCD (2015) R(D) = 0.299 0.003 Belle, PRL118,211801(2017) Belle, PRL118,211801(2017) ± s LHCb, PRL120,171802(2018) LHCb, FPCP2017 R(D)=0.299(11) FNAL/MILC (2015) R(D*) = 0.258 0.005 to compare with R D ( ∗ ) from B decays Average Average R(D*)=0.252(3) S. Fajfer et al. (2012) 0.4 0.4 0.35 0.35 σ σ 4 4 D ( ∗ ) ≡ BF ( B → D ( ∗ ) τν τ ) R τ/µ σ 2 σ BF ( B → D ( ∗ ) µν µ ) 0.3 0.3 2 0.25 0.25 HFLAV HFLAV ◮ HFLAV updated SM prediction, R D ( ∗ ) : Summer 2018 FPCP 2017 χ χ 0.2 0.2 P( P( 2 2 ) = 71.6% ) = 74% averaging [Bigi, Gambino PRD94(2016)094008] 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 [Bernlochner, Ligeti, Papucci, Robinson PRD95(2017)11500] R(D) R(D) [Bigi, Gambino, Schacht JHEP11(2017)061][Jaiswal, Nandi, Patra JHEP12(2017)060] ◮ Nonperturbative lattice calculation favor B s over B decays (higher precision) ◮ Only the spectator quark differs: R D ( ∗ ) may be a good proxy for R D ( ∗ ) s 3 / 18
introduction Bs → K ℓν Bs → Ds ℓν conclusion | V ub | from exclusive semileptonic B s → K ℓν decay ℓ � q 2 = M 2 B s + M 2 K − 2 M B s E K ν W b u B s K s ◮ Conventionally parametrized by (neglecting term ∝ m 2 ℓ f 2 0 ) � 3 / 2 G 2 � K − q 2 � 2 − 4 M 2 2 ×| V ub | 2 d Γ( B s → K ℓν ) � M 2 B s + M 2 B s M 2 ×| f + ( q 2 ) | = F dq 2 192 π 3 M 3 K Bs experiment known nonperturbative input CKM 4 / 18
introduction Bs → K ℓν Bs → Ds ℓν conclusion Nonperturbative input ◮ Parametrizes interactions due to the (nonperturbative) strong force ◮ Use operator product expansion (OPE) to identify short distance contributions ◮ Calculate the flavor changing currents as point-like operators using lattice QCD ⇒ Nonperturbative calculation: lattice QCD → Additional challenge m b = 4 . 18GeV ∼ 1000 × m d m c = 1 . 28GeV ∼ 270 × m d 5 / 18
introduction Bs → K ℓν Bs → Ds ℓν conclusion Set-up ◮ RBC-UKQCD’s 2+1 flavor domain-wall fermion and Iwasaki gauge action ensembles → Three lattice spacings a ∼ 0.11 fm, 0.08 fm, 0.07 fm; one ensemble with physical pions [PRD 78 (2008) 114509][PRD 83 (2011) 074508][PRD 93 (2016) 074505][JHEP 1712 (2017) 008] ◮ Unitary and partially quenched domain-wall up/down quarks [Kaplan PLB 288 (1992) 342] , [Shamir NPB 406 (1993) 90] ◮ Domain-wall strange quarks at/near the physical value ◮ Charm: M¨ obius domain-wall fermions optimized for heavy quarks [Boyle et al. JHEP 1604 (2016) 037] → Simulate 3 or 2 charm-like masses then extrapolate/interpolate ◮ Effective relativistic heavy quark (RHQ) action for bottom quarks [Christ et al. PRD 76 (2007) 074505] , [Lin and Christ PRD 76 (2007) 074506] → Builds upon Fermilab approach [El-Khadra et al. PRD 55 (1997) 3933] → Allows to tune the three parameters ( m 0 a , c P , ζ ) nonperturbatively [PRD 86 (2012) 116003] → Smooth continuum limit; heavy quark treated to all orders in ( m b a ) n 6 / 18
B s → K ℓν
introduction Bs → K ℓν Bs → Ds ℓν conclusion B s → K ℓν form factors ◮ Parametrize the hadronic matrix element for the flavor changing vector current V µ in terms of the form factors f + ( q 2 ) and f 0 ( q 2 ) M 2 Bs − M 2 M 2 Bs − M 2 � q µ � p µ B s + p µ � K | V µ | B s � = f + ( q 2 ) + f 0 ( q 2 ) q µ K − K K q 2 q 2 t V µ t q b u t 0 t sink s l ◮ Calculate 3-point function by → Inserting a quark source for a “light” propagator at t 0 → Allow it to propagate to t sink , turn it into a sequential source for a b quark → Use another “light” quark propagating from t 0 and contract both at t 8 / 18
introduction Bs → K ℓν Bs → Ds ℓν conclusion Determining B s → K ℓν form factors f + and f 0 on the lattice ◮ Updating calculation [PRD 91 (2015) 074510] with new values for a − 1 and RHQ parameters ◮ New analysis directly fitting form factors and accounting for excited state contributions ◮ On the lattice we prefer using the B s -meson rest frame and compute f � ( E K ) = � K | V 0 | B s � / � f ⊥ ( E K ) p i K = � K | V i | B s � / � 2 M B s and 2 M B s ◮ Both are related by √ 2 M Bs f 0 ( q 2 ) = � ( M B s − E K ) f � ( E K ) + ( E 2 K − M 2 � K ) f ⊥ ( E K ) M 2 Bs − M 2 K f + ( q 2 ) = 1 √ � � f � ( E K ) + ( M B s − E K ) f ⊥ ( E K ) 2 M Bs 9 / 18
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