Test of factorization for the long-distance effects B Kl + l from - PowerPoint PPT Presentation

Test of factorization for the long-distance effects B → Kl + l − from charmonium in NAKAYAMA Katsumasa(Nagoya Univ.) Shoji Hashimoto (KEK) for JLQCD Collaboration 2018/7/27@Lattice2018,East Lansing,MI,USA Photo: Granada,, Spain, 2017

2 ◉ Motivation B → Kl + l − (1): is a penguin-induced FCNC process. dB ( B → Kµ + µ − ) /dq 2 (10 − 7 GeV − 2 ) 0.7 (GIM and loop suppressed) Form factors + CKM + Others Form factors only 0.6 LHCb14 ( B + ) LHCb14 ( B 0 ) 0.5 Babar12 CDF11 0.4 Belle09 0.3 (2): Anomaly in experiments: 0.2 q 2 < m 2 0.1 J/ ψ J/ ψ ψ (2 S ) 0 0 5 10 15 20 25 q 2 (GeV) 2 [D. Du et al. (Fermilab, MILC) 1510.02349] Question: Are the long distance l contributions understood? J/ ψ → We calculate the corresponding l O i B K amplitude by lattice formulation.

3 ◉ Charmonium resonance contributions ◇ We focus on the charmonium resonance contribution, 2 ! 10 H e ff = G F X X us V ub C i O u cs V cb C i O c ( V ∗ i + V ∗ i ) − V ∗ ts V tb C i O i √ 2 i =1 i =3 l J/ ψ O c 1 = ( s i γ µ P − c j ) ( c j γ µ P − b i ) O c 2 = ( s i γ µ P − c i ) ( c j γ µ P − b j ) l O i B K → and induce the long-distance contribution, O c O c 1 2 which is often estimated by the factorization approximation. How good is that?

4 ◉ . decay amplitudes K → π ◇ We’d like to calculate the B decay amplitudes on the lattice. Formulation is analogous to the decay amplitudes. K → π ll [N.H. Christ et al. (RBC, UKQCD) 1507.03094] Z q 2 � d 4 x h π ( p ) | T [ J µ (0) H e ff ( x )] | K ( k ) i � A µ = Z q 2 � d 4 x h K ( k ) | T [ J µ (0) H e ff ( x )] | B ( p ) i � A µ = ◇ The amplitude is calculable from the integration of 4pt-func. Z t J + T b I µ ( T a , T b , p , k ) ' d t H t J − T a t B t H t J t K (0 ⌧ t J � T a  t J + T b ⌧ t K )

5 ◉ . decay amplitudes B → Kl + l − J/ ψ J/ ψ I µ = + O i O i B K B K t H t J t J t H Z t J + t b Z t J dt H dt H t J t J − t a ◇ Charm loop would produce contributions like Z 0 Z ∞ 1 1 dte ω t e − E J/ ψ t = dte ω t e E J/ ψ t = ω − E J/ ψ ω + E J/ ψ 0 −∞ q ◇ If we focus on , part is dominant. t H < t J m 2 J/ ψ − q 2 ∼ E J/ ψ ω =

6 ◉ Divergence of the amplitude? [N.H. Christ et al. (RBC, UKQCD) 1507.03094] K → π l + l − ◇ case π +( t J < t H ) I µ = O i K π t H t J Z ∞ d E ρ ( E ) h π ( k ) | J µ (0) | E, p i h E, p | H e ff (0) | K ( p ) i ⇣ 1 � e [ E K ( p ) − E ] T a ⌘ I µ ( T a , T b , p , k ) = � 2 E E K ( p ) � E 0 +( t J < t H ) ◇ Some intermediate states have lower than E K E → Since , they must be subtracted. T a → ∞ (e.g. ) K → π , ππ , πππ

7 ◉ Divergence of the amplitude? B → Kl + l − ◇ case J/ ψ +( t J < t H ) I µ = O i B K t H t J Z ∞ d E ρ S ( E ) h K ( k ) | J µ (0) | E, p i h E, p | H e ff (0) | B ( p ) i ⇣ 1 � e [ E B ( p ) − E ] T a ⌘ I µ ( T a , T b , p , k ) = � 2 E E B ( p ) � E 0 +( t J < t H ) ◇ We restrict ourselves in the setup of m B < m J/ ψ + m K → No divergence.

8 ◉ Amplitude of . B → Kl + l − ◇ From the integration of the 4-point correlators, we can T a,b → ∞ extract the amplitude after taking limit. Z ∞ d E ρ S ( E ) h K ( k ) | J µ (0) | E, p i h E, p | H e ff (0) | B ( p ) i ⇣ 1 � e [ E B ( p ) − E ] T a ⌘ I µ ( T a , T b , p , k ) = � 2 E E B ( p ) � E 0 +( t J < t H ) Z q 2 � d 4 x h K ( k ) | T [ J µ (0) H e ff ( x )] | B ( p ) i � A µ = A µ ( q 2 ) = − i T a,b →∞ I µ ( T a , T b , k , p ) lim

9 B → Kl + l − Factorization method for decay

10 ◉ Factorization ◇ Assume that l ong-range gluon exchange can be ignored J/ ψ O i B K J/ ψ h i ∝ h i h i O i B K 1 h P K | J cc (Vol . ) h 0 | J cc ν J cc ν ( c i γ µ P − c i )( s j γ µ P − b j ) | P B i = µ | 0 ih P K | V µ | P B i → We test this assumption by the lattice calculation.

11 ◉ Factorization ◇ Factorizable operator and non-factorizable operator O F O NF Fierz trans. O (1) O c = ( c i γ µ P − c i ) ( s j γ µ P − b j ) 1 = ( s i γ µ P − c j ) ( c j γ µ P − b i ) F O c 2 = ( s i γ µ P − c i ) ( c j γ µ P − b j ) ⇣ ⌘ O (8) c i [ T a ] ij γ µ P − c j ( s k [ T a ] kl γ µ P − b l ) NF = 2 = 1 3 O (1) + 2 O (8) 1 = O (1) O c O c F NF F O (8) ◇ Assume non-factrizable operator could be ignored NF 2 = 1 → We test this assumption . O c 3 O c 1 O l 2 ' � 0 . 7 O l K → ππ case, Lattice. 1 [P.A. Boyle et al. (RBC, UKQCD) 1212.1474]

12 Preliminary result for the test of factorization

13 ◉ Current status L 3 × T ( × L s ) meas am uds a − 1 β am c am b am π aE K am J/ ψ am B 4.17 2.453(4) 32 3 × 64( × 12) 100 0.04 0.44037 0.68808 0.2904(5) 0.3513(16) 1.2809(6) 1.063(11) 4.35 3.610(9) 48 3 × 96( × 8) 90 0.025 0.27287 0.66619 0.1986(3) 0.2378(9) 0.8701(3) 0.9543(19) GeV ' 714 MeV ' 855 MeV ' 3 . 14 GeV ' 3 . 44 GeV Mobius domain-wall fermion with 2+1 flavor. ◆ up and down mass same as strange. m b = { (1 . 25 2 ) m c , (1 . 25 4 ) m c } ◆ “ bottom” mass: k = 2 π ◆ F inite momentum in the final state L (1 , 0 , 0)

14 ◉ 4-point functions Z d 3 x d 3 y e − i q · y D E φ K ( t K , k ) T [ J µ ( t J , y ) H e ff ( t H , x )] φ † Γ (4) µ ( t H , t J , p , k ) = B (0 , p ) 1.0 × 10 -12 1.0 × 10 -14 B K 1.0 × 10 -16 t J t H 1.0 × 10 -18 log(Correlator) J/ ψ K B 1.0 × 10 -20 1.0 × 10 -22 1.0 × 10 -24 B K t H t J 1.0 × 10 -26 1.0 × 10 -28 v1v1 cv1 abs 1.0 × 10 -30 0 5 10 15 20 25 30 35 40 45 t H t/a

15 ◉ 4-point functions 2 O 1 h i h O c 1 i 1.0 = 1.5 h i h i (Fact . ) Factorization 1 0.5 0 0 5 10 15 20 25 t H J/ ψ O 1 B

16 ◉ 4-point functions 1 O 1 /O 2 O c 2 /O c h O c 2 i 1 1/3 0.8 h O c 1 i 0.6 1 Factorization = 3 0.4 0.2 0 -0.2 -0.4 -0.6 0 5 10 15 20 25 t H J/ ψ B

17 ◉ TO DO LIST We have to… (1):determine the lattice renormalization constants. (2):Input more realistic momentum. k ≥ 2 π E B (0) = E J/ ψ ( k ) + E K ( k ) L (2 , 2 , 2) (3):Input or extrapolate to physical quark masses. (4):complete integration and taking limit to extract amplitude.

18 ◉ Summary B → Kl + l − ◇ We study the charmonium contribution to by the lattice calculation. K → π l + l − ◇ is calculable analogously to for B → Kl + l − lighter bottom quark masses. ◇ As a first step, we study the accuracy of the factorization approximation. ◇ Sizable non-factorizable contribution is observed in the long-distance region.

19 ◉ 4 point functions a = 2 . 45 GeV 2 O 1 1.0 1.5 1 Factorization 0.5 0 0 2 4 6 8 10 12 14 16 18 t H

20 ◉ 4 point functions a = 2 . 45 GeV 1 O c 2 /O c O 1 /O 2 1 1/3 0.8 0.6 Factorization 0.4 0.2 0 -0.2 -0.4 -0.6 0 2 4 6 8 10 12 14 16 18 t H