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

Test of factorization for the long-distance effects from charmonium in

Photo: Granada,, Spain, 2017

B → Kl+l−

NAKAYAMA Katsumasa(Nagoya Univ.) Shoji Hashimoto (KEK)

for JLQCD Collaboration

2018/7/27@Lattice2018,East Lansing,MI,USA

◉ Motivation

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(1): is a penguin-induced FCNC process. (GIM and loop suppressed) (2): Anomaly in experiments: Question: Are the long distance contributions understood? →We calculate the corresponding amplitude by lattice formulation.

B → Kl+l−

B K Oi

J/ψ

l

l

0.1 0.2 0.3 0.4 0.5 0.6 0.7 5 10 15 20 25 dB(B → Kµ+µ−)/dq2(10−7GeV−2) q2(GeV)2 Form factors + CKM + Others Form factors only LHCb14 (B+) LHCb14 (B0) Babar12 CDF11 Belle09

J/ψ ψ(2S)

[D. Du et al. (Fermilab, MILC) 1510.02349]

q2 < m2

J/ψ

3

◇ We focus on the charmonium resonance contribution,

Heff = GF √ 2 2 X

i=1

(V ∗

usVubCiOu i + V ∗ csVcbCiOc i ) − V ∗ tsVtb 10

X

i=3

CiOi !

Oc

1 = (siγµP−cj) (cjγµP−bi)

Oc

2 = (siγµP−ci) (cjγµP−bj)

→ and induce the long-distance contribution,

◉ Charmonium resonance contributions

B

K Oi

J/ψ

l l

Oc

1

Oc

2

which is often estimated by the factorization approximation. How good is that?

◉ . decay amplitudes

4

◇ We’d like to calculate the B decay amplitudes on the lattice.

Formulation is analogous to the decay amplitudes.

[N.H. Christ et al. (RBC, UKQCD) 1507.03094]

◇ The amplitude is calculable from the integration of 4pt-func.

tJ tH tB

tK

Iµ (Ta, Tb, p, k) ' Z tJ+Tb

tJ−Ta

dtH

(0 ⌧ tJ Ta  tJ + Tb ⌧ tK)

Aµ

• q2

= Z d4x hπ(p) |T [Jµ(0)Heff(x)]| K(k)i Aµ

• q2

= Z d4x hK(k) |T [Jµ(0)Heff(x)]| B(p)i K → π K → πll

◇ If we focus on , part is dominant.

5

Z tJ+tb

tJ

dtH

tH tJ Iµ = + tH < tJ

◇ Charm loop would produce contributions like

ω = q m2

J/ψ − q2 ∼ EJ/ψ

Z ∞ dteωte−EJ/ψt = 1 ω − EJ/ψ

Z 0

−∞

dteωteEJ/ψt = 1 ω + EJ/ψ

K B

tH tJ

Z tJ

tJ−ta

dtH

K B

Oi Oi

◉ . decay amplitudes

B → Kl+l−

◇ case

◉ Divergence of the amplitude?

6

[N.H. Christ et al. (RBC, UKQCD) 1507.03094]

K → πl+l−

◇ Some intermediate states have lower than

E

Iµ (Ta, Tb, p, k) = Z ∞ dE ρ(E) 2E hπ(k) |Jµ(0)| E, pi hE, p|Heff(0)|K(p)i EK(p) E ⇣ 1 e[EK(p)−E]Ta⌘

→ Since , they must be subtracted. (e.g. )

K → π, ππ, πππ

Ta → ∞

Iµ = tH tJ

π

K

+(tJ < tH)

+(tJ < tH)

Oi

EK

π

tH tJ

K B

◉ Divergence of the amplitude?

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→ No divergence.

◇ We restrict ourselves in the setup of ◇ case

Iµ (Ta, Tb, p, k) = Z ∞ dE ρS(E) 2E hK(k) |Jµ(0)| E, pi hE, p|Heff(0)|B(p)i EB(p) E ⇣ 1 e[EB(p)−E]Ta⌘

B → Kl+l− Iµ =

+(tJ < tH)

+(tJ < tH)

Oi

mB < mJ/ψ + mK

◉ Amplitude of .

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◇ From the integration of the 4-point correlators, we can

extract the amplitude after taking limit.

Aµ(q2) = −i lim

Ta,b→∞ Iµ(Ta, Tb, k, p)

Iµ (Ta, Tb, p, k) = Z ∞ dE ρS(E) 2E hK(k) |Jµ(0)| E, pi hE, p|Heff(0)|B(p)i EB(p) E ⇣ 1 e[EB(p)−E]Ta⌘

B → Kl+l− Ta,b → ∞ Aµ

• q2

= Z d4x hK(k) |T [Jµ(0)Heff(x)]| B(p)i

+(tJ < tH)

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Factorization method for decay

B → Kl+l−

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◇ Assume that long-range gluon exchange can be ignored

◉ Factorization

h i i h h i ∝

→ We test this assumption by the lattice calculation.

B K J/ψ Oi

hPK|Jcc

ν (ciγµP−ci)(sjγµP−bj)|PBi =

1 (Vol.)h0|Jcc

ν Jcc µ |0ihPK|Vµ|PBi

K B

Oi

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◇ Factorizable operator and non-factorizable operator

◉ Factorization

OF ONF

O(1)

F

= (ciγµP−ci) (sjγµP−bj) O(8)

NF =

⇣ ci [T a]ij γµP−cj ⌘ (sk [T a]kl γµP−bl)

Oc

1 = O(1) F

Oc

2 = 1

3O(1)

F

+ 2O(8)

NF ◇ Assume non-factrizable operator could be ignored

O(8)

NF

→ We test this assumption . Fierz trans.

Oc

2 = 1

3Oc

1

Oc

1 = (siγµP−cj) (cjγµP−bi)

Oc

2 = (siγµP−ci) (cjγµP−bj)

Ol

2 ' 0.7Ol 1

K → ππ case, Lattice.

[P.A. Boyle et al. (RBC, UKQCD) 1212.1474]

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Preliminary result for the test of factorization

13

◉ Current status

◆ up and down mass same as strange. ◆ “bottom” mass: ◆ Finite momentum in the final state

mb = {(1.252)mc, (1.254)mc}

' 714 MeV ' 855 MeV ' 3.14 GeV ' 3.44 GeV

k = 2π L (1, 0, 0)

β a−1 L3 × T(×Ls) meas amuds amc amb amπ aEK amJ/ψ amB 4.17 2.453(4) 323 × 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) 483 × 96(×8) 90 0.025 0.27287 0.66619 0.1986(3) 0.2378(9) 0.8701(3) 0.9543(19)

Mobius domain-wall fermion with 2+1 flavor.

GeV

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◉ 4-point functions

1.0×10-30 1.0×10-28 1.0×10-26 1.0×10-24 1.0×10-22 1.0×10-20 1.0×10-18 1.0×10-16 1.0×10-14 1.0×10-12 5 10 15 20 25 30 35 40 45

log(Correlator) t/a

v1v1 cv1 abs

B K

tH tJ

K B

tH tJ

B K

Γ(4)

µ (tH, tJ, p, k) =

Z d3xd3ye−iq·y D φK (tK, k) T [Jµ (tJ, y) Heff (tH, x)] φ†

B(0, p)

E

J/ψ

tH

15

◉ 4-point functions

Factorization

0.5 1 1.5 2 5 10 15 20 25

tH

O1 1.0

B

O1

J/ψ

hOc

1i

(Fact.)

h i

i

h h i

=

• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 1 5 10 15 20 25

tH

O1/O2 1/3

16

◉ 4-point functions

Factorization =

Oc

2/Oc 1

hOc

2i

hOc

1i

1 3

B

J/ψ

◉ TO DO LIST

17

We have to… (1):determine the lattice renormalization constants. (2):Input more realistic momentum. (3):Input or extrapolate to physical quark masses. (4):complete integration and taking limit to extract amplitude.

EB(0) = EJ/ψ(k) + EK(k) k ≥ 2π L (2, 2, 2)

◉ Summary

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◇ We study the charmonium contribution to

by the lattice calculation.

◇ is calculable analogously to for

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.

B → Kl+l− K → πl+l− B → Kl+l−

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◉ 4 point functions

Factorization

a = 2.45 GeV

0.5 1 1.5 2 2 4 6 8 10 12 14 16 18

tH

O1 1.0

• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 1 2 4 6 8 10 12 14 16 18

tH

O1/O2 1/3

20

◉ 4 point functions

Factorization

Oc

2/Oc 1

a = 2.45 GeV