What can we learn from B Kpersonalpages.to.infn.it/~torriell/seminars/files/Mandal.pdf · istituto nazionale di fisica nucleare torino On Phys.Rev. D90 (2014) 096006 and arxiv:1506.04535

i s t i tuto naz i onale d i f i s i c a nucleare tor i no

On Phys.Rev. D90 (2014) 096006 and arxiv:1506.04535

What can we learn from B → K∗`−`+ ?

Rusa MandalThe Institute of Mathematical Sciences

with Rahul Sinha and Diganta Das

July 3, 2015


Outline

1 Introduction

2 Model Independent Framework

Relation among Observables

Hadronic Parameter Extraction

3 New Physics Analysis

4 Summary

Slide 0 — Rusa Mandal, IMSc Phys.Rev. D90 (2014) 096006 & arxiv:1506.04535 (hep-ph)


Introduction

SM amplitude is loop and CKM

suppressed–measurements of severalobservables –valuable probes for indirectsearch of NP.

� Challenge: estimate hadronic uncertainties– long distance QCD effects–form factors and “nonfactorizable“ contributions.

∼ 〈K∗|i∫

d4x e iq·xT{jµem(x),Oi (0)}|B〉

[Khodjamirian et al. ’10]

� Requirement of exact test of SM including all possible effects within it.



Introduction




∼ 〈K∗|i∫






Introduction




∼ 〈K∗|i∫






Introduction




∼ 〈K∗|i∫






Model Independent Framework

� The amplitude : A(B(p)→ K∗(k)`+`−)

=GFα√

2πVtbV

∗ts

[{C9〈K∗|sγµPLb|B〉 −

2C7

q2〈K∗|s iσµνqν(mbPR + msPL)b|B〉

−16π2

q2

∑i={1−6,8}

CiHµi

}¯γµ` + C10〈K∗|sγµPLb|B〉 ¯γµγ5`

]

q=p−k and C7,9,10 are “True” values of Willson coefficients.

� Assuring Lorentz and gauge invariance, hadronic matrix elementparametrization in terms of some ‘unknown’ form factors Xj , Yj and Zj ’s.

〈K∗|sγµPLb|B〉=ε∗ν(X0 q

µqν+X1 (gµν−qµqν

q2)+X2 (kµ−

k.q

q2qµ)qν+iX3 ε

µνρσkρqσ)

〈K∗|i sσµνqνPR,Lb|B〉=ε∗ν(± Y1 (gµν−

qµqν

q2)± Y2 (kµ−

k.q

q2qµ)qν+iY3 ε

µνρσ kρqσ)

Hµi = 〈K∗(ε∗, k)|i∫

d4x e iq·xT{jµem(x),Oi (0)}|B(p)〉

= ε∗ν

(Z1

i (gµν −qµqν

q2) + Z2

i (kµ −k.q

q2qµ)qν + iZ3

iεµνρσ kρqσ

)




� The amplitude : A(B(p)→ K∗(k)`+`−)

=GFα√

2πVtbV

∗ts

[{C9〈K∗|sγµPLb|B〉 −

2C7

q2〈K∗|s iσµνqν(mbPR + msPL)b|B〉

−16π2

q2

∑i={1−6,8}

CiHµi

}¯γµ` + C10〈K∗|sγµPLb|B〉 ¯γµγ5`

]

q=p−k and C7,9,10 are “True” values of Willson coefficients.

� Assuring Lorentz and gauge invariance, hadronic matrix elementparametrization in terms of some ‘unknown’ form factors Xj , Yj and Zj ’s.

〈K∗|sγµPLb|B〉=ε∗ν(X0 q

µqν+X1 (gµν−qµqν

q2)+X2 (kµ−

k.q

q2qµ)qν+iX3 ε

µνρσkρqσ)

〈K∗|i sσµνqνPR,Lb|B〉=ε∗ν(± Y1 (gµν−

qµqν

q2)± Y2 (kµ−

k.q

q2qµ)qν+iY3 ε

µνρσ kρqσ)

Hµi = 〈K∗(ε∗, k)|i∫

d4x e iq·xT{jµem(x),Oi (0)}|B(p)〉

= ε∗ν

(Z1

i (gµν −qµqν

q2) + Z2

i (kµ −k.q

q2qµ)qν + iZ3

iεµνρσ kρqσ

)




� Factorizable and nonfactorizable contributions upto all orders are absorbed

C9 → C(j)9 = C9 + ∆C

(fac)9 (q2) + ∆C

(j),(non-fac)9 (q2)︸︷︷︸

∼∑

i Ci Z ij /Xj

2(mb+ms)

q2C7 Yj → Yj =

2(mb + ms)

q2C7 Yj + · · ·

� Effective helicity index due to non-factorizable corrections in C9:

C⊥9 ≡ C(3)9 , C‖9 ≡ C

(1)9 , C 0

9 ≡ C(2)9 κ

where κ = 1 +C

(1)9 − C

(2)9

C(2)9

4k.qX1

4k.qX1 + λ(m2B ,m

2K∗ , q

2)X2




� Angular analysis in well-known helicity frame

`−

`+

K

π

B

K∗ θKθ` φ

� The differential distributiond4Γ(B → K∗`+`−)

dq2 d cos θ` d cos θK dφ

=9

32π

[I s1 sin2

θK +I c1 cos2θK +(I s2 sin2

θK +I c2 cos2θK ) cos 2θ`+I3 sin2

θK sin2θ` cos 2φ

+ I4 sin 2θK sin 2θ` cosφ + I5 sin 2θK sin θ` cosφ + I s6 sin2θK cos θ`

+ I7 sin 2θK sin θ` sinφ + I8 sin 2θK sin 2θ` sinφ + I9 sin2θK sin2

θ` sin 2φ].




� Observables are three helicity fractions FL, F⊥, F‖ and six asymmetriesrelated to LHCb observables [R. Aaij et al. ’12,’13]

F⊥ =1

2(1− FL + 2S3), A4 = −

2

πS4, A5 =

3

4S5,

AFB =−ALHCbFB , A7 =

3

4S7, A8 = −

2

πS8, A9 =

3

2πS9.

� Observables in terms of transversity amplitudes with most generalparametric form

AL,Rλ =

(Cλ9 ∓ C10)Fλ − Gλ At

∣∣m`=0

= 0

Fλ≡Fλ(Xj ), Gλ≡Gλ(Yj )





F⊥ =1

2(1− FL + 2S3), A4 = −

2

πS4, A5 =

3

4S5,


3

4S7, A8 = −

2

πS8, A9 =

3

2πS9.


AL,Rλ =


∣∣m`=0

= 0






F⊥ =1

2(1− FL + 2S3), A4 = −

2

πS4, A5 =

3

4S5,


3

4S7, A8 = −

2

πS8, A9 =

3

2πS9.


AL,Rλ =


∣∣m`=0

= 0






F⊥ =1

2(1− FL + 2S3), A4 = −

2

πS4, A5 =

3

4S5,


3

4S7, A8 = −

2

πS8, A9 =

3

2πS9.


AL,Rλ =


∣∣m`=0

= 0

= (∓C10 − rλ)Fλ + iελ.

Fλ≡Fλ(Xj ), Gλ≡Gλ(Yj ), rλ≡Re(Gλ)

Fλ− Re(Cλ9 ), ελ≡ Im(Cλ9 )Fλ − Im(Gλ)




� 9 observables in terms of 10 parameters–

FLΓf = 2F02(r0

2 + C102) + 2ε0

2

F‖Γf = 2F‖2(r‖2 + C 2

10

)+ 2ε‖

2

F⊥Γf = 2F⊥2(r⊥2 + C 210

)+ 2ε⊥

2

√2πA4Γf = 4F0F‖

(r0r‖+ C 2

10

)+4ε0ε‖

√2A5Γf = 3F0F⊥C10

(r0 + r⊥

)AFBΓf = 3F‖F⊥C10

(r‖ + r⊥

)√

2A7Γf = 3C10

(F0ε‖ − F‖ε0

)πA8Γf = 2

√2(F0r0ε⊥ − F⊥r⊥ε0

)πA9Γf = 3

(F⊥r⊥ε‖ − F‖r‖ε⊥

)

� Allows to eliminate almost all the hadronic parameters except for one–P1 which is unaltered by higher order QCD corrections (in αs) andnonfactorizable effects– reliable theoretical input [Beneke/Feldmann ’00]

P1 =F⊥F‖

= −

√λ(m2

B ,m2K∗ , q

2)

2EK∗mB

in leading 1/mB expansion




� 9 observables in terms of 10 parameters–

FLΓf = 2F02(r0

2 + C102) + 2ε0

2

F‖Γf = 2F‖2(r‖2 + C 2

10

)+ 2ε‖

2

F⊥Γf = 2F⊥2(r⊥2 + C 210

)+ 2ε⊥

2

√2πA4Γf = 4F0F‖

(r0r‖+ C 2

10

)+4ε0ε‖

√2A5Γf = 3F0F⊥C10

(r0 + r⊥

)AFBΓf = 3F‖F⊥C10

(r‖ + r⊥

)√

2A7Γf = 3C10

(F0ε‖ − F‖ε0

)πA8Γf = 2

√2(F0r0ε⊥ − F⊥r⊥ε0

)πA9Γf = 3

(F⊥r⊥ε‖ − F‖r‖ε⊥

)� Allows to eliminate almost all the hadronic parameters except for one–

P1 which is unaltered by higher order QCD corrections (in αs) andnonfactorizable effects– reliable theoretical input [Beneke/Feldmann ’00]

P1 =F⊥F‖

= −

√λ(m2

B ,m2K∗ , q

2)

2EK∗mB

in leading 1/mB expansion




A4 =2√

2ε‖ε0

πΓf+

8A5AFB

9π(F⊥ −

2ε2⊥

Γf

)

+√

2

√(FL −

2ε20

Γf

)(F⊥ −

2ε2⊥

Γf

)−

8

9A2

5

√(F‖ −

2ε2‖

Γf

)(F⊥ −

2ε2⊥

Γf

)−

4

9A2

FB

π(F⊥ −

2ε2⊥

Γf

)

� ελ/√

Γf have iterative solutions proportional to A7, A8, A9 and onehadronic parameter P1.

� Asymmetries A7,8,9 ∼ 0 in 3fb−1 LHCb data indicates small complexcontributions to the amplitude. [LHCb-CONF-2015-002]

� Converged solutions of ελ/√

Γf ∼ 0 for each set of observables.




A4 =2√

2ε‖ε0

πΓf+

8A5AFB

9π(F⊥ −

2ε2⊥

Γf

)

+√

2

√(FL −

2ε20

Γf

)(F⊥ −

2ε2⊥

Γf

)−

8

9A2

5

√(F‖ −

2ε2‖

Γf

)(F⊥ −

2ε2⊥

Γf

)−

4

9A2

FB

π(F⊥ −

2ε2⊥

Γf

)

� ελ/√

Γf have iterative solutions proportional to A7, A8, A9 and onehadronic parameter P1.

� Asymmetries A7,8,9 ∼ 0 in 3fb−1 LHCb data indicates small complexcontributions to the amplitude. [LHCb-CONF-2015-002]

� Converged solutions of ελ/√

Γf ∼ 0 for each set of observables.



Real transversity amplitude limit: ελ=0

√4(FL+F‖+

√2πA4

)F⊥−

16

9

(AFB +

√2A5

)2=

√4F‖F⊥−

16

9A2

FB +

√4FLF⊥−

32

9A2

5

� Solving for any observable:

A4 =8A5AFB

9πF⊥+

8√

98FLF⊥−A2

5

√94F‖F⊥−A2

FB

9πF⊥[Das/Sinha ’12]

A5 =πA4AFB

2F‖±

√( 9

4F‖F⊥−A2

FB)(2F‖FL−π2A24)

2F‖

AFB =πA4A5

FL±

√( 9

8FLF⊥−A2

5)(2F‖FL−π2A24)

FL

� Local tension in LHCb data in observable P′5 which is related to A5:

P′5 =4

3

A5√FL(1− FL)

.



Real transversity amplitude limit: ελ=0

√4(FL+F‖+

√2πA4

)F⊥−

16

9

(AFB +

√2A5

)2=

√4F‖F⊥−

16

9A2

FB +

√4FLF⊥−

32

9A2

5

� Solving for any observable:

A4 =8A5AFB

9πF⊥+

8√

98FLF⊥−A2

5

√94F‖F⊥−A2

FB

9πF⊥[Das/Sinha ’12]

A5 =πA4AFB

2F‖±

√( 9

4F‖F⊥−A2

FB)(2F‖FL−π2A24)

2F‖

AFB =πA4A5

FL±

√( 9

8FLF⊥−A2

5)(2F‖FL−π2A24)

FL

� Local tension in LHCb data in observable P′5 which is related to A5:

P′5 =4

3

A5√FL(1− FL)

.



Numerical Analysis

LHCbRelation

0 5 10 15 20-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

q2

A4

A4 deviates by ∼ 2σ only at 0.1 ≤ q2 ≤ 0.98 GeV2



Numerical Analysis

LHCbRelation

0 5 10 15 20-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

q2

AFB

AFB deviates by ∼ 2σ in 11.0 ≤ q2 ≤ 12.5 GeV2 and 15 ≤ q2 ≤ 17 GeV2



Numerical Analysis

]4c/2 [GeV2q0 5 10 15

5'P

-1

-0.5

0

0.5

1

preliminaryLHCb

SM from DHMV

I LHCb: 2.9σ discrepancies

in each bin [4, 6] GeV2

and [6, 8] GeV2.



Numerical Analysis

]4c/2 [GeV2q0 5 10 15

5'P

-1

-0.5

0

0.5

1

preliminaryLHCb

SM from DHMV



and [6, 8] GeV2.

LHCbRelation

0 5 10 15 20-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

q2

P5′

I Relation: Complete

agreement for entire q2

region.



Numerical Analysis

]4c/2 [GeV2q0 5 10 15

5'P

-1

-0.5

0

0.5

1

preliminaryLHCb

SM from DHMV



and [6, 8] GeV2.

LHCbRelation

0 5 10 15 20-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

q2

P5′

I Relation: Complete

agreement for entire q2

region.



Hadronic parameter extraction

� Most general SM amplitude in real limit:

AL,Rλ = (∓C10 − rλ)Fλ

Notation : P1 =F⊥F‖

, P2 =F⊥F0

, ζ =F2⊥C

210

Γf, u2

λ =2F2⊥r

2λ

Γf.

� Five independent parameters in five equations of observables

F⊥ = u⊥2 + 2ζ

FLP22 = u0

2 + 2ζ

A2FB =

9ζ

2P12

(u‖ ± u⊥

)2

A25 =

9ζ

4P22

(u0 ± u⊥

)2

A4 =

√2

πP1P2

(2ζ ± u0u‖

)

Solution for experimentallyobserved data

I ζ : F⊥ solved using C10 value andmeasured branching fraction Γf

I P1 : F‖ solved using obtained F⊥

I P2 : F0 solved using obtained F⊥







, P2 =F⊥F0

, ζ =F2⊥C

210

Γf, u2

λ =2F2⊥r

2λ

Γf.


F⊥ = u⊥2 + 2ζ

FLP22 = u0

2 + 2ζ

A2FB =

9ζ

2P12

(u‖ ± u⊥

)2

A25 =

9ζ

4P22

(u0 ± u⊥

)2

A4 =

√2

πP1P2

(2ζ ± u0u‖

)











, P2 =F⊥F0

, ζ =F2⊥C

210

Γf, u2

λ =2F2⊥r

2λ

Γf.


F⊥ = u⊥2 + 2ζ

FLP22 = u0

2 + 2ζ

A2FB =

9ζ

2P12

(u‖ ± u⊥

)2

A25 =

9ζ

4P22

(u0 ± u⊥

)2

A4 =

√2

πP1P2

(2ζ ± u0u‖

)








� Fλ related to well-known form-factors V and A1,2

F⊥ =N√

2√λ(m2

B ,m2K∗ , q

2)V (q2)

mB + mK∗

F‖ =− N√

2(mB + mK∗ )A1(q2)

F0 =−N√q2

8mBmK∗A12(q2)

where A12 is a linear combination of A1 and A2.

� At low recoil energy of K∗ only three independent form factors describethe decay in HQET framework. [Grinstein/Prijol ’04]

V (q2)

T1(q2)=

A1(q2)

T2(q2)=

A2(q2)

T3(q2)

m2B

q2

� Relation between form factors neglecting nonfactorizable effects:

u⊥ = u‖ = u0




� Fλ related to well-known form-factors V and A1,2

F⊥ =N√

2√λ(m2

B ,m2K∗ , q

2)V (q2)

mB + mK∗

F‖ =− N√

2(mB + mK∗ )A1(q2)

F0 =−N√q2

8mBmK∗A12(q2)

where A12 is a linear combination of A1 and A2.

� At low recoil energy of K∗ only three independent form factors describethe decay in HQET framework. [Grinstein/Prijol ’04]

V (q2)

T1(q2)=

A1(q2)

T2(q2)=

A2(q2)

T3(q2)

m2B

q2

� Relation between form factors neglecting nonfactorizable effects:

u⊥ = u‖ = u0



NP Analysis

� Fit to experimental measurements and comparison with LCSR [Bharucha

et.el ’15] and Lattice [Horgan et.el ’15] results –

χ2 =

(F ex⊥−F⊥

∆F ex⊥

)2

+

(F exL −FL

∆F exL

)2

+

(Aex

4 −A4

∆Aex4

)2

+

(A2

FBex−A2

FB

2AexFB∆Aex

FB

)2

+

(A2

5ex−A2

5

2Aex5 ∆Aex

5

)2



NP Analysis

� Significant disagreements in high q2 region–

11.0 ≤ q2 ≤ 12.5 GeV2: P1 deviates by 5.1σ, P2 by 4.8σ and ζ by 2.8σ

15 ≤ q2 ≤ 17 GeV2: P1 deviates by 6.4σ, P2 by 6.2σ and ζ by 2.1σ



NP Analysis

q2 range in GeV2 V (q2) A1(q2) A12(q2)

11.0 ≤ q2 ≤ 12.5 0.172± 0.006 0.539± 0.027 0.462± 0.0285.65σ 2.43σ 2.82σ

15.0 ≤ q2 ≤ 17.0 0.713± 0.004 0.638± 0.026 0.505± 0.0166.25σ 3.36σ 4.64σ

17.0 ≤ q2 ≤ 19.0 1.936± 0.007 0.678± 0.025 0.498± 0.0144.28σ 3.82σ 4.64σ

� chi-square function

χ2 =

(PSM − PFit

)2

δP2SM + δP2

Fit

� Reasonable agreements with LCSR results in low q2 region.

� Large Discrepancies with Lattice results in q2≥11GeV2.

� u⊥ = u‖ = u0 does not hold for best fit values in large q2 region.



NP Analysis

q2 range in GeV2 V (q2) A1(q2) A12(q2)

11.0 ≤ q2 ≤ 12.5 0.172± 0.006 0.539± 0.027 0.462± 0.0285.65σ 2.43σ 2.82σ

15.0 ≤ q2 ≤ 17.0 0.713± 0.004 0.638± 0.026 0.505± 0.0166.25σ 3.36σ 4.64σ

17.0 ≤ q2 ≤ 19.0 1.936± 0.007 0.678± 0.025 0.498± 0.0144.28σ 3.82σ 4.64σ

� chi-square function

χ2 =

(PSM − PFit

)2

δP2SM + δP2

Fit

� Reasonable agreements with LCSR results in low q2 region.

� Large Discrepancies with Lattice results in q2≥11GeV2.

� u⊥ = u‖ = u0 does not hold for best fit values in large q2 region.



Summary

� Derived relation is parametrically exact in the SM limit and incorporates

I resonance contributions

I electromagnetic correction to hadronic operators at all orders

I all factorizable and nonfactorizable contributions

I not limited by the accuracy of estimation of short and long distanceeffects.

� In real transversity amplitude limit the disagreements in AFB and A4 insome q2 bins indicate the absence of a consistent set of form factors.

� Form factors extracted from experimental data show large discrepancy inlow recoil region of K∗ meson.

� Possible source of error ⇒ finite bin size effect!

� If not ⇒ possibility of unaccounted for some operators: New Physics ?



Summary












Summary












Summary














Back Up



Expressions of the amplitudes:

AL,R⊥ = N

√2λ

12(m2

B ,m2K∗ , q

2)[(C

(3)9 ∓ C10)X3 − Y3

]AL,R‖ = 2

√2N[(C

(1)9 ∓ C10)X1 − ζ Y1

]AL,R

0 =N

2mK∗√

q2

[(C

(2)9 κ∓ C10)

{4k.qX1 + λ(m2

B ,m2K∗ , q

2)X2

}− ζ{

4k.qY1 + λ(m2B ,m

2K∗ , q

2)Y2

}]At = −

N

m∗K

√q2λ1/2(m2

B ,m2K∗ , q

2)C10 X0,

where,

κ = 1 +C

(1)9 − C

(2)9

C(2)9

4k.qX1

4k.qX1 + λ(m2B ,m

2K∗ , q

2)X2,

λ(a, b, c)≡a2 +b2 +c2−2(ab + bc + ac) and N is the normalization constant.



Angular coefficients

I s1 =(2 + β2)

4

[|AL⊥|

2 + |AL‖|

2 + (L→ R)]

+4m2

q2Re(AL

⊥AR⊥∗

+AL‖A

R‖∗

)

I c1 = |AL0 |2 +|AR

0 |2 +4m2

q2

[|At |2 +2Re(AL

0AR0∗

)]

I s2 =β2

4

[|AL⊥|

2 + |AL‖|

2 + (L→ R)]

I c2 = −β2[|AL

0 |2 + (L→ R)]

I3 =β2

2

[|AL⊥|

2 − |AL‖|

2 + (L→ R)]

I4 =β2

√2

[Re(AL

0AL‖∗

) + (L→ R)]

I5 =√

2β[Re(AL

0AL⊥∗

)− (L→ R)]

I s6 = 2β[Re(AL

‖AL⊥∗

)− (L→ R)]

I7 =√

2β[Im(AL

0AL‖∗

)− (L→ R)]

I8 =1√

2β2[Im(AL

0AL⊥∗

) + (L→ R)]

I9 = β2[Im(AL

‖∗AL⊥) + (L→ R)

]Slide 21 — Rusa Mandal, IMSc Phys.Rev. D90 (2014) 096006 & arxiv:1506.04535 (hep-ph)


The angular asymmetries

A4 =

[ ∫ π/2

−π/2

−∫ 3π/2

π/2

]dφ[ ∫ 1

0

−∫ 0

−1

]d cos θK

[ ∫ 1

0

−∫ 0

−1

]d cos θ`

d4(Γ + Γ)

dq2d cos θ`d cos θKdφ∫ 2π

0

dφ

∫ 1

−1

d cos θK

∫ 1

−1

d cos θ`d4(Γ + Γ)

dq2d cos θ`d cos θKdφ

A5 =

[ ∫ π/2

−π/2

−∫ 3π/2

π/2

]dφ[ ∫ 1

0

−∫ 0

−1

]d cos θK

∫ 1

−1

d cos θ`d4(Γ− Γ)


0

dφ

∫ 1

−1

d cos θK

∫ 1

−1



A7 =

[ ∫ π

0

−∫ 2π

π

]dφ[ ∫ 1

0

−∫ 0

−1

]d cos θK

∫ 1

−1



0

dφ

∫ 1

−1

d cos θK

∫ 1

−1



A8 =

[ ∫ π

0

−∫ 2π

π

]dφ[ ∫ 1

0

−∫ 0

−1

]d cos θK

[ ∫ 1

0

−∫ 0

−1

]d cos θ`

d4(Γ− Γ)


0

dφ

∫ 1

−1

d cos θK

∫ 1

−1



A9 =

[ ∫ π/2

0

−∫ π

π/2

+

∫ π

0

−∫ 2π

3π/2

]dφ[ ∫ 1

−1

d cos θK][ ∫ 1

−1

d cos θ`] d4(Γ− Γ)


0

dφ

∫ 1

−1

d cos θK

∫ 1

−1





� The solutions for complex contributions to the amplitudes

ε⊥ =

√2πΓf

(r0−r‖)F⊥

[A9P1

3√

2+

A8P2

4−

A7P1P2r⊥

3πC10

],

ε‖ =

√2πΓf

(r0−r‖)F⊥

[A9r0

3√

2r⊥+

A8P2r‖

4P1r⊥−

A7P2r‖

3πC10

],

ε0 =

√2πΓf

(r0−r‖)F⊥

[A9P1r0

3√

2P2r⊥+

A8r‖

4r⊥−

A7P1r0

3πC10

].

where rλ, P2 and C10 are funtions of observables FL, F⊥, A5 and AFB.

� The terms ελ/Γf are evaluated completely in terms of observables and theform factor ratio P1.



q2 range in GeV2 ε⊥/√

Γf ε‖/√

Γf ε0/√

Γf0.1 ≤ q2 ≤ 0.98 −0.048± 0.116 −0.047± 0.103 0.020± 0.111

1.1 ≤ q2 ≤ 2.5 −0.010± 0.078 −0.010± 0.078 0.078± 0.172

2.5 ≤ q2 ≤ 4.0 −0.009± 0.079 −0.008± 0.080 −0.025± 0.212

4.0 ≤ q2 ≤ 6.0 −0.026± 0.097 0.014± 0.093 0.032± 0.234

6.0 ≤ q2 ≤ 8.0 −0.011± 0.088 −0.046± 0.078 −0.132± 0.129

11.0 ≤ q2 ≤ 12.5 −0.011± 0.050 0.038± 0.074 −0.078± 0.114

15.0 ≤ q2 ≤ 17.0 −0.0003± 0.067 −0.027± 0.071 0.020± 0.072

17.0 ≤ q2 ≤ 19.0 0.006± 0.076 −0.090± 0.090 −0.040± 0.088

Converged ελ/√

Γf mean values with 1σ errors



Lepton Mass Correction

� Seven amplitudes AL,R⊥,‖,0 and At [W. Altmannshofer et al. ’09]

� Including contribution to the observables with index ”o“

Γf → Γof = β2Γf + 3T1

Fλ → F oλ = (β2Fλ + T1)/Γo

f

A5,FB,7 → Ao5,FB,7 = β A5,FB,7/Γo

f

A4,8,9 → Ao4,8,9 = β2 A4,8,9/Γo

f

T1

Γof

=1

3−

4I s2−I c23Γo

f

=1

3−

16

9A10 +

64

27A11

∼ O(m2

q2

)is small and can be extracted

from angular analysis.

A10 =

∫ 2π

0

dφ

∫ 1

0

d cos θK[ ∫ −1/2

−1

−∫ 1/2

−1/2

+

∫ 1

1/2

]d cos θ`

d4(Γ + Γ)


0

dφ

∫ 1

−1

d cos θK

∫ 1

−1



A11 =

∫ 2π

0

dφ[∫ −1/2

−1

−∫ 1/2

−1/2

+

∫ 1

1/2

]d cos θK

[∫ −1/2

−1

−∫ 1/2

−1/2

+

∫ 1

1/2

]d cos θ`

d4(Γ + Γ)

dq2dcos θ`dcos θKdφ∫ 2π

0

dφ

∫ 1

−1

d cos θK

∫ 1

−1






Ao4 =

2√

2β2ε‖ε0

πΓof

+8β2Ao

5AoFB

9π(F o⊥ −

T⊥Γof

)

+√

2

√(F oL −T0

Γof

)(F o⊥ −

T⊥Γof

)−

8

9β

2Ao5

2

√(F o‖ −T‖Γof

)(F o⊥ −

T⊥Γof

)−

4

9β

2AoFB

2

π(F o⊥ −

T⊥Γof

)

with Tλ = T1 + 2β2ε

2λ ; λ ∈ {0,⊥, ‖}



Hadronic matrix elements

⟨K∗(k) sγµ(1− γ5)b B(p)

⟩=− 2mK∗A0(q2)

ε∗.q

q2qµ −

2i V (q2)

mB + mK∗εµνρσε

∗νpρkσ

− (mB + mK∗ )A1(q2)(ε∗µ −

ε∗.q

q2qµ)

+ A2(q2)ε∗.q

mB + mK∗

[pµ + kµ −

m2B −m2

K∗

q2qµ

]⟨K∗(k) sσµνq

ν(1 + γ5)b B(p)⟩

= 2T1(q2)εµνρσε∗νpρkσ

− iT2(q2)[(m2

B −m2K∗ )ε∗µ − (ε∗.q)(pµ + kµ)

]− iT3(q2)(ε∗.q)

[qµ −

q2

m2B −m2

K∗(pµ + kµ)

]

and A12 =(mB + mK∗ )2(m2

B −m2K∗ − q2)A1(q2)− λ(m2

B ,mK∗ , q2)A2(q2)

16mBm2K∗ (mB + mK∗ )



Case I: q2 =4m2

� leptons carry equalmomenta and recoilagainst K∗

� Isotropic distribution in θ`and φ

� Fλ= 1/3 and A FB,4−9= 0

Ao4 =β→0

√2

π

√F oL−

T1

Γof

√F o‖ −

T1

Γof

=Foλ→1/3

T1/Γof→1/3

0

Case II: q2 =(mB−mK∗ )2[Hiller/Zwicky ’14]

� FL= 1/3 and AFB= 0

� F⊥= 0 and F‖= 2/3

� A5,7,8,9= 0 and A4= 2/3π

A4 =8A5AFB

9πF⊥

+√

2

√FLF⊥ − 8

9A2

5

√F‖F⊥ − 4

9A2

FB

πF⊥

=AFB→0

A5→0

√2√

FLF‖

π=

FL→1/3

F‖→2/3

2

3π


Documents

What can we learn from B Kpersonalpages.to.infn.it/~torriell/seminars/files/Mandal.pdf · istituto nazionale di fisica nucleare torino On Phys.Rev. D90 (2014) 096006 and arxiv:1506.04535