On the B → K ∗ `` at low recoil
Ivan Nišandžić,
based on arXiv:1606.00775 with S. Braß and G. Hiller
Technische Universität Dortmund
[email protected]
Brda 2016 - Selected topics in flavour and collider physics
20. October 2016
1/20
Motivation
b → s`` - FCNC processes, loop induced and suppressed in the
SM, potentially sensitive to possible BSM effects.
Better precision enables us to probe higher energy scales, but
also requires better understanding of the long distance dynamics
within the SM (e.g. form factors, charm effects).
B → K ∗ µµ - rich angular information - large potential to
diagnose the NP and long distance dynamics within SM
2/20
Two regions in q 2
Low q 2 region - QCD factorization,
high q 2 region - OPE (this talk).
The limitations of both tools would be in the need for the revisit
in the (near) future
3/20
B → K ∗ `` at high q 2
Above J/ψ and ψ(2S) resonances, the complicated intermediate
charm states (presumably dominated by the wide charm
resonances J P C = 1−− ) show up as wiggles in q 2 distributions.
These effects have been anticipated from the 1970s.
Theoretically controlled approach: Operator Product Expansion
2
(OPE) in 1/Q2 , Q2 ∼ (qmax
, m2b ) of the non-local operator
products
Z
1
µ
Ai ∝ 2 d4 xeikx T {Oi (0)j µ (x)}
(1)
q
in terms of local operators [Grinstein, Pirjol (2004); Beylich,
Buchalla, Feldmann (2011)]:
X
Aµ =
Ci (q 2 )Qµi .
(2)
i
4/20
Leading power corrections from the orders αs /mb , m4c /Q4 - only
of order of few percents. Up to this precision charm contributions
factorize, are universal and are absorbed into effective Wilson
coefficients [Grinstein, Pirjol, (2004); Bobeth, Hiller, van Dyk, (2010)]
eff
C7
eff
C9
=
+
+
−
1
4
20
80
αs
(7) 2
2
C1 − 6C2 A(q ) − C8 F8 (q ) ,
4π
8
4
64
2
C9 + h(q , 0)
C1 + 2C2 + 11C3 − C4 + 104C5 −
C6
2
3
3
3
8 m2
c 4
C1 + C2 + 6C3 + 60C5
3 q2 3
αs
(9) 2
2
2 2
2 C1 B(q ) + 4C(q ) − 3C2 2B(q ) − C(q ) − C8 F8 (q )
4π
1
4
64
4
16
16
2
2
h(q , mb ) 7C3 + C4 + 76C5 +
C6 +
C3 +
C5 +
C6 .
2
3
3
3
3
9
= C7 −
3
1
C3 −
9
C4 −
3
C5 −
9
C6 +
(3)
OPE expected to provide good description of charm effects
within binned observables.
Quark-hadron duality violations enter at some level, but their
magnitude is not known within the OPE itself.
5/20
Plan
Compare the predictions of the OPE with experimental data for
angular observables and branching fractions at low recoil.
Use the Krüger-Sehgal (KS) data driven model of local q 2
distributions, to estimate the uncertainties (limitations of the
OPE) for a chosen binning, independently of the underlying
electro-weak model (SM or BSM)
6/20
Using OPE - Angular Observables
The ”improved” Isgur-Wise relations and OPE lead to universal
short-distance couplings [Bobeth, Hiller, van Dyk (2010); Hambrock, Hiller,
(2012); Hambrock, Hiller, Schacht, Zwicky (2013)]
L,R
L,R
AL,R
f⊥ , AL,R
fk,0 where
⊥ = +i C
k,0 = −i C
C L,R = C9eff ∓ C10 + κ
2m̂b eff
C
ŝ 7
(4)
Then the observables FL , S3 , S4 turn out independent of C L,R
and only depend on form factors (within OPE, IW and no RH
quark currents assumption). [Hambrock, Hiller, (2012); Hambrock, Hiller,
Schacht, Zwicky, (2013)]. If universal, also the wiggles cancel in
FL , S3,4 for infinitesimal bin size, their appearance signals the
non-universal effects.
7/20
Testing the SM (OPE)
One can then test the SM with dB/dq 2 and the observables S5 , AF B :
S5 =
√
ρ2 (q 2 )f0 f⊥
3 2
,
2
2
2 ρ1 (q ) f02 + f⊥
+ fk2
AF B =
ρ1
3ρ2 (q 2 )fk f⊥
,
2
f02 + f⊥
+ fk2
(q 2 )
(5)
where
2
2mb mB eff 1
2
(|C R |2 + |C L |2 ) = C9eff + κ
C
7
+ |C10 | ,
2
q2
1
2mb mB eff ∗
ρ2 (q 2 ) ≡ (|C R |2 − |C L |2 ) = Re C9eff + κ
C
C10 .
7
4
q2
ρ1 (q 2 ) ≡
(6)
8/20
FL , S3 , S4 : OPE + (C 0 9,10 = 0) - comparison with Experiment LHCb,
1512.04442
0.6
0.5
0.00
0.5
-0.05
0.4
0.3
S4
-0.10
S3
FL
0.4
-0.15
0.3
0.2
-0.20
0.2
0.1
-0.25
0.0
15
-0.30
15
16
17
18
19
16
17
18
0.1
15
19
16
q2 @GeV2 D
q2 @GeV2 D
17
18
19
q2 @GeV2 D
0.6
0.5
0.00
0.5
-0.05
0.4
S4
-0.10
0.3
S3
FL
0.4
-0.15
0.3
0.2
-0.20
0.2
0.1
0.0
15
-0.25
16
17
q2 @GeV2 D
18
19
-0.30
15
16
17
q2 @GeV2 D
18
19
0.1
15
16
17
18
19
q2 @GeV2 D
9/20
-0.2
-0.2
-0.3
-0.3
-0.4
-0.4
-0.5
-0.5
16
17
18
-0.6
15
19
16
17
18
0.0
0.0
-0.1
-0.1
-0.2
-0.2
-0.3
-0.4
4
2
0
15
19
16
17
18
19
q2@GeV2D
-0.3
-0.4
-0.5
-0.6
15
6
q2 @GeV2 D
q2 @GeV2 D
AFB
-0.6
15
8
dBdq2 ´108 @GeV-2 D
0.0
-0.1
AFB
0.0
-0.1
S5
S5
S5 , AF B , Br: OPE+SM comparison with Experiment LHCb, 1512.04442, LHCb, 1606.04731
-0.5
16
17
q2 @GeV2 D
18
19
-0.6
15
16
17
18
19
q2 @GeV2 D
Only S4 , S5 show some tension with the OPE in the highest
1GeV2 bin.
10/20
Krueger-Sehgal parametrisation
Krüger-Sehgal (KS) model [Kruger, Sehgal, (1996)] - data from
e+ e− → hadrons is used to obtain charm vacuum polarization,
which is then plugged in the B → K ∗ and corrected with fudge
factors ηc .
Extraction of the charm vacuum polarization hc (q 2 ) from the
e+ e− → hi data [BES Collaboration, (2007)]
R(s) =
σ (e
σ
+ −
e →hi ) (s)
(e+ e− →µ+ µ− )
(s)
.
(7)
11/20
Krueger-Sehgal parametrisation
Using the optical theorem and the dispersion relation:
Im[A(e+ e− → h → e+ e− )] = sσ(e+ e− → h)(s),
π
Im[hc (s)] = Rc (s),
3
Z ∞
s − s0
ds0
Re[hc (s)] = Re[hc (s0 )] +
P
Im[hc (s0 )].
0
0
π
t0 (s − s)(s − s0 )
(8)
12/20
Krüeger-Sehgal parametrisation ctd.
3.0
2.0
1.5
2.0
Re@hc Hq2 LD
Im@hc Hq2 LD
2.5
1.5
1.0
0.5
0.0
14
1.0
0.5
0.0
-0.5
15
16
17
18
19
q2 @GeV2 D
-1.0
14
15
16
17
18
19
q2 @GeV2 D
C9eff (q 2 ) = C9 + 3a2 ηc hc (q 2 ) + . . .
(9)
[Chetyrkin, Misiak, Münz (1996)], Bobeth, Misiak, Urban, (2000),
1
a2 =
3
4
C1 + C2 + 6C3 + 60C5
3
= 0.2 at NNLO at the b-mass scale.
(10)
Introduce ”fudge function” ηc ≡ ηc (Kj∗ , q 2 ), j =⊥, k, 0, that corrects for
effects beyond NFA (|ηc = 1| and universal). We take ηc (j) ∈ R, more
generally ηc (j) ∈ C.
13/20
Krüeger-Sehgal parametrisation ctd.
B → Kµ+ µ− wiggles observed in the high q 2 measured by LHCb
[LHCb, 1307.7595]. [Lyon, Zwicky, (2014)] : require large fudge factor
∼ −2.5; one can also probe ψi dependent correction factors.
We expect the correction factor(s) for B → K ∗ ψi to differ wrt
the B → Kψi , e.g.
|ηJ/ψ(ψ(2S)) K ∗ | ' 1, while |ηJ/ψ(ψ(2S)) K | ' 1.5.
(11)
We take in our analysis η 0 s constant but polarization dependent.
With more experimental information, we could be more general.
2 ) [Hiller, Zwicky,
Kinematic relations at the endpoint (q 2 = qmax
(2013)],
√
Ak = − 2A0 , A⊥ = 0,
(12)
and our ansatz C9eff,i (q 2 ) = C9 + 3a2 ηi hc (q 2 ) + . . . (i = 0, k, ⊥),
imply:
η0 = ηk , η⊥ → not constrained
(13)
2 ) = 0.
since f⊥ (qmax
14/20
Krüeger-Sehgal parametrisation ctd.
The observables S7,8,9 provide null tests for the hadronic
universality (vanish in the OPE), (see also Bobeth, Hiller, van Dyk
(2012)), for example:
2mb mB eff
C7 )a2 Im[hc (q 2 )(η0 − η⊥ )] .
S8 ∼ f0 f⊥ (C˜9eff + κ
q2
(14)
We perform the simultaneous fit for η0,⊥ , C9 , C10
4
1 GeV2 bins
2 GeV2 bins
4 GeV2 bins
Η¦
2
0
-2
-4
-3 -2 -1 0 1 2 3
Η°=Η0
15/20
OPE vs Resonance model for 3 different binnings1
2.0
Bin width: 4 GeV 2
2.0
Bin width: 2 GeV 2
Bin width: 1 GeV 2
1.5
1.5
1.0
1.0
1.0
0.5
0.5
0.5
0.0
∆C10
1.5
∆C10
∆C10
2.0
0.0
0.0
-0.5
-0.5
-0.5
-1.0
-1.0
-1.0
-1.5
-1.5-1.0-0.5 0.0 0.5 1.0 1.5 2.0
∆C9
-1.5
-1.5-1.0-0.5 0.0 0.5 1.0 1.5 2.0
∆C9
-1.5
-1.5-1.0-0.5 0.0 0.5 1.0 1.5 2.0
∆C9
2
2
2
Bin width: 2 GeV 2
Bin width: 4 GeV 2
Bin width: 1 GeV 2
1
0
0
0
C'10
C'10
1
C'10
1
-1
-1
-1
-2
-2
-2
-3
-2 -1 0
1 2
C'9
3
4
5
-3
-2 -1 0
1 2
C'9
3
4
5
-3
-2 -1 0
1 2
C'9
3
4
5
1 2
χ /dof = (1.3, 0.8, 1.3) for OPE and (1.0, 0.6, 1.2) for KS
16/20
OPE vs Resonance model for 3 different binnings ctd.
Differences between shaded areas (OPE) and dashed lines (KS) can
serve as a binning dependent systematic uncertainty for OPE.
To estimate the uncertainties of the OPE predictions for a given
binning, we suggest the ratios:
R
1 = R
bin
2
ρKS
1 dq
OP E dq 2
bin ρ1
bin [GeV2 ]
1
2
12
Table:
R
,
15 − 19
(0.85,1.16)
(0.82,1.0)
(0.86,1.05)
2 = R
bin
2
ρKS
2 dq
OP E dq 2
bin ρ2
15 − 17
(0.81,1.30
(0.74,1.13)
(0.87,1.05)
R
,
17 − 19
(0.87,1.03)
(0.85,0.91)
(0.84,1.05)
12 = R
bin
2
ρKS
2 dq
OP E dq 2
bin ρ2
15 − 16
(0.76,1.20)
(0.71,1.17)
(0.95,1.06)
16 − 17
(0.84,1.38)
(0.78,1.08)
(0.78,1.05)
R
· Rbin
E dq 2
ρOP
1
bin
2
ρKS
1 dq
17 − 18
(0.84,1.03)
(0.76,0.95)
(0.75,1.05)
. (15)
18 − 19
(0.86,1.05)
(0.84,0.97)
(0.93,1.05)
0
Ratios k for different q 2 -bins and 1σ ranges of parameters η0 , η⊥ and C9,10 . The coefficients C9,10
→ 0.
This suggests that OPE performs better at endpoint bins than at
the lower q 2 bins and the bin of maximal size (15 − 19)GeV2 .
17/20
Conclusions
So far, good performance of the SM+OPE, although large BSM
effects are allowed.
Small binnings and more precise data is going to probe the limits
of the OPE.
B → K ∗ `` provides large(r) number of angular observables to
hopefully disentangle the SD and LD effects.
CP-averages S7,8,9 important for testing the hadronic universality
of charm effects.
Consistency between fits to Wilson coefficients between high q 2 ,
low q 2 and inclusive decays are important to decide the fate of
the B → K ∗ µµ anomaly.
18/20
Backup slides - The notation
Let us review the notation and the conventions.
The effective Hamiltonian:
X
4GF
Hef f = − √ Vts∗ Vtb
Ci (µ)Oi + h.c
2
i
(16)
We use CMM basis O1 − O8 [Chetyrkin, Misiak, Münz (1996)], e.g.
O1c = (s̄L γµ T a cL )(c̄L γµ T a bL ), O2c = (s̄L γµ cL )(c̄L γµ bL ).
(17)
EM and QCD dipole operators:
e
gs
O7 =
(s̄σ µν PR b)Fµν , O8 =
mb (s̄σ µν PR T a b)Gaµν
2
(4π)
(4π)2
The semileptonic operators:
αem
¯ µ `), O10 = αem (s̄γµ PL b)(`γ
¯ µ γ5 `).
O9 =
(s̄γµ PL b)(`γ
4π
4π
0
O9,10
with PL → PR in quark currents
(18)
(19)
19/20
Angular Observables in B → K ∗ ``
Complete information about the decay in full four-fold angular
distributions:
3
d4 Γ
=
J(q 2 , cos θ` , cos θK , φ),
dq 2 d cos θ` d cos θK dφ
8π
d4 Γ̄
3 ¯ 2
J(q , cos θ` , cos θK , φ),
=
2
dq d cos θ` d cos θK dφ
8π
(20)
with
J(q 2 , θ` , θK , φ) = J1s sin2 θK + J1c cos2 θK + (J2s sin2 θK + J2c cos2 θK ) cos 2θ`
+ J3 sin2 θ` sin2 θK cos 2φ + J4 sin 2θ` sin 2θK cos φ
+ J5 sin θ` sin 2θK cos φ + J6 cos θ` sin2 θK
+ J7 sin θ` sin 2θK sin φ + J8 sin 2θ` sin 2θK sin φ
+ J9 sin2 θ` sin2 θK sin 2φ.
The LHCb Collaboration measured the CP-averaged ratios2
Ji + J¯i
Si ≡
.
dΓ/dq 2 + dΓ̄/dq 2
2
LHCb
Note the different convention FL = FLLHCb , S3,5,7,9 = 34 S3,5,7,9
, S4,8 =
LHCb
3 LHCb
= 4 S5
, AF B = −AF B
LHCb
− 43 S4,8
, S5
(21)
20/20