Determining the photon polarization of the b s using the B K 1 ( - - PowerPoint PPT Presentation
Determining the photon polarization of the b s using the B K 1 ( 1270 ) ( K ) decay Andrey Tayduganov 1 , 2 Andrey.Tayduganov@th.u-psud.fr in collaboration with Emi Kou 1 and Alain Le Yaouanc 2 1 Laboratoire de
Andrey Tayduganov1,2
Andrey.Tayduganov@th.u-psud.fr
in collaboration with Emi Kou1 and Alain Le Yaouanc2
1Laboratoire de l’Accélérateur Linéaire (LAL) 2Laboratoire de Physique Théorique (LPT)
Université Paris-Sud 11, France
Osaka University, 1 November 2011
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1
Introduction: the b → sγ process and the photon polarization
2
The B → K1γ decay and polarization measurement Basic idea and formalism Determination of λγ in the DDLR method
3
Strong interaction decays of the K1-mesons Theoretical model Numerical results
4
Sensitivity studies of λγ measurement in the DDLR method Statistical errors Theoretical uncertainties
5
Future prospects of the photon polarization measurement New Physics constraints combining various methods
6
Conclusions and perspectives
2 / 28
1
Introduction: the b → sγ process and the photon polarization
2
The B → K1γ decay and polarization measurement Basic idea and formalism Determination of λγ in the DDLR method
3
Strong interaction decays of the K1-mesons Theoretical model Numerical results
4
Sensitivity studies of λγ measurement in the DDLR method Statistical errors Theoretical uncertainties
5
Future prospects of the photon polarization measurement New Physics constraints combining various methods
6
Conclusions and perspectives
3 / 28
FCNC processes In the SM, Flavour Changing Neutral Current (FCNC) processes are forbidden at tree level. b s γ, Z
uLγµdLW +
µ → u′ L U u† L Ud L
| {z }
VCKM
d ′
LW + µ
d LγµdLZµ → d
′ L U d† L U d L
| {z }
1
d ′
LZµ 4 / 28
FCNC processes In the SM, Flavour Changing Neutral Current (FCNC) processes are forbidden at tree level. b s γ, Z
uLγµdLW +
µ → u′ L U u† L Ud L
| {z }
VCKM
d ′
LW + µ
d LγµdLZµ → d
′ L U d† L U d L
| {z }
1
d ′
LZµ
They can proceed only via loops. b W s u, c, t γ s′
Lσµνqνb′ RAµ X i=u,c,t
V ∗
is Vib F2(mi)
large F2(mt) ⇒ B(B → Xsγ)exp = (3.55 ± 0.24 ± 0.09) × 10−4 ([HFAG(’10)]) B → Xsγ is sensitive to the effects of new physics beyond the SM
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Why are we interested in measuring the photon polarization of b → sγ ?
b W s γ
V − A V − A
M(b → sγ)SM = 4GF √ 2 V ∗
tsVtbF2
e 16π2 sσµνqν“ mb 1 + γ5 2 | {z }
bR→sLγL
+ ms 1 − γ5 2 | {z }
bL→sRγR
” b εµ∗ In the SM, since ms/mb ≃ 0.02 ≪ 1, photons are predominantly left(right)-handed in the B(B)-decays.
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Why are we interested in measuring the photon polarization of b → sγ ?
b W s γ
V − A V − A
bL ˜ g sR γR (δd
RL)23
˜ bL ˜ sR M(b → sγ)SM = 4GF √ 2 V ∗
tsVtbF2
e 16π2 sσµνqν“ mb 1 + γ5 2 | {z }
bR→sLγL
+ ms 1 − γ5 2 | {z }
bL→sRγR
” b εµ∗ In the SM, since ms/mb ≃ 0.02 ≪ 1, photons are predominantly left(right)-handed in the B(B)-decays. NP can induce new Dirac structures and lead to an excess of right(left)-handed photons, without contradicting with the measured B(B → Xsγ). ⇒ The measurement of the photon polarization could provide a test of physics beyond the SM, namely right-handed currents.
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There are 3 methods proposed to measure the ratio MR/ML (≃ 0 in the SM):
1
Method 1: time-dependent CP asymmetry in B0 → K ∗0(→ KSπ0)γ [Atwood et al., Phys.Rev.Lett.79 (’97)] Sf γ = −ξf 2|MLMR| |ML|2 + |MR|2 sin(φM − φL − φR)
2
Method 2: transverse asymmetries in B0 → K ∗0(→ K −π+)ℓ+ℓ− [Kruger&Matias, Phys.Rev.D71 (’05); Becirevic&Schneider, arXiv:1106.3283 (’11)] A(2)
T = −
Re[MRM∗
L]
|MR|2 + |ML|2 , A(im)
T
= Im[MRM∗
L]
|MR|2 + |ML|2
3
Method 3: K1 three-body decay method in B → K1(→ Kππ)γ [Gronau et al., Phys.Rev.Lett.88, Phys.Rev.D66 (’02)] λγ = |MR|2 − |ML|2 |MR|2 + |ML|2
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1
Introduction: the b → sγ process and the photon polarization
2
The B → K1γ decay and polarization measurement Basic idea and formalism Determination of λγ in the DDLR method
3
Strong interaction decays of the K1-mesons Theoretical model Numerical results
4
Sensitivity studies of λγ measurement in the DDLR method Statistical errors Theoretical uncertainties
5
Future prospects of the photon polarization measurement New Physics constraints combining various methods
6
Conclusions and perspectives
7 / 28
How to measure the polarization: basic idea The angular distribution of the three-body decay of Kres in B → Kresγ decay provides a direct determination of the Kres(⇔ γ) polarization [Gronau et al., Phys.Rev.Lett.88 (’02)]. γ z K ∗ π K symmetric B NO helicity information
2 → 3-body
π γ z K1 π K ∗ K B
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How to measure the polarization: basic idea The angular distribution of the three-body decay of Kres in B → Kresγ decay provides a direct determination of the Kres(⇔ γ) polarization [Gronau et al., Phys.Rev.Lett.88 (’02)]. γ z K ∗ π K symmetric B NO helicity information
2 → 3-body
π γ z K1 π K ∗ K B There are two known K1(1+) states, decaying into Kππ final state via K ∗π and ρK modes: K1(1270) and K1(1400). One of the decay channels B → K1γ, namely B+ → K +
1 (1270)γ, is
finally measured (B = (4.3 ± 1.2) × 10−5), while B+ → K +
1 (1400)γ is
suppressed (B < 1.5 × 10−5) [Belle, Phys.Rev.Lett.94 (’05)]. We investigate the feasibility of determining the photon polarization using the B → K1(1270)γ channel.
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Formalism The decay distribution of B → K 1γ → (Kππ)γ is given by the master formula: dΓ ds13ds23dcos θ ∝ 1 4| J |2(1 + cos2 θ) + λγ 1 2Im[ n · ( J × J ∗)]cos θ λγ = |MR|2 − |ML|2 |MR|2 + |ML|2 ≃ −1(+1) in the SM for B(B) respectively ; ;
p2 | p1× p2|
γ π( p1) K( p3) π( p2) ; x y z θ
p1 − C2(s13, s23) p2 ⇔ K1-decay helicity amplitude.
Since the final state interactions break T-parity and n · ( J × J ∗) is T-odd, the amplitude must involve a strong phase, coming from the interference of at least 2 amplitudes.
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New method The decay distribution of B → K 1γ → (Kππ)γ is given by the master formula: dΓ ds13ds23dcos θ ∝ 1 4| J |2(1 + cos2 θ) + λγ 1 2Im[ n · ( J × J ∗)]cos θ Previous method of Gronau et al. In the original proposal by Gronau et al., only the θ-dependence on the polarization was considered (up-down asymmetry):
Aup−down = R 1
0 dcos θ dΓ dcos θ −
R 0
−1 dcos θ dΓ dcos θ
R 1
−1 dcos θ dΓ dcos θ
= 3 4 λγ R ds13ds23Im[ n · ( J × J ∗)] R ds13ds23| J |2 10 / 28
New method The decay distribution of B → K 1γ → (Kππ)γ is given by the master formula: dΓ ds13ds23dcos θ ∝ 1 4| J |2(1 + cos2 θ) + λγ 1 2Im[ n · ( J × J ∗)]cos θ New method: DDLR (Davier, Duflot, Le Diberder, Rougé) In our work, we take into account the Dalitz variable (s13,s23) dependence, which carries the further information of the polarization (it was pointed out in the ALEPH analysis of τ → a1(→ πππ)ν [Davier et al., Phys.Lett.B306 (’93)]). In this method, we use the quantity, called ω: ω(s13, s23, cos θ) ≡ 2Im[ n · ( J × J ∗)]cos θ | J |2(1 + cos2 θ) [Kou,Le Yaouanc&A.T., Phys.Rev.D83 (’11)]
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Basic idea of the DDLR method Our PDF (i.e. the normalized decay width distribution) can be written as W (s13, s23, cos θ) = f (s13, s23, cos θ) +λγg(s13, s23, cos θ) Using the maximum likelihood method, we obtain λγ as a solution of the equation: ∂ ln L ∂λγ =
Nevents
X
i=1
ωi 1 + λγωi = 0
L =
Nevents
Y
i=1
W (si
13, si 23, cos θi)
ωi ≡ g(si
13, si 23, cos θi)/f (si 13, si 23, cos θi)
Notice: resulting solution does not depend on f and g separately but only on their ratio ω.
ω(s13, s23, cos θ) ≡ 2Im[ n · ( J × J ∗)]cos θ | J |2(1 + cos2 θ)
λγ extraction from ω-distribution λγ = ω ω2 σ−2
λγ = −∂2 ln L
∂λγ2 = N „ ω 1 + λγω «2
the events ⇒ no fit is needed !
ωn ≡ 1 Nevents
Nevents
X
i=1
ωn
i 11 / 28
Estimating the J -function Assuming that this process comes from the vector-pseudoscalar meson interme- diate state (K1 → K ∗π, Kρ), J contains Breit-Wigner denominators for the isobars K ∗, ρ and possibly K1 (these BW-factors are the source of the strong phase in J ) 4 form factors for K1 → K ∗π, Kρ (one can express them in terms of S and D partial wave amplitudes) 2 couplings gK∗Kπ, gρππ (they are determined from the decay widths ΓK∗ and Γρ) There have been very important experimental studies of the K1-decays [ACC- MOR, SLAC, B-factories,. . . ]. Although some of the parameters are determined experimentally, there still remain some difficulties in their interpretation and ap- plication.
◮ Therefore in the following, we estimate the K1 → K ∗π, K1 → Kρ form
factors in the framework of the 3P0 quark-pair-creation model.
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1
Introduction: the b → sγ process and the photon polarization
2
The B → K1γ decay and polarization measurement Basic idea and formalism Determination of λγ in the DDLR method
3
Strong interaction decays of the K1-mesons Theoretical model Numerical results
4
Sensitivity studies of λγ measurement in the DDLR method Statistical errors Theoretical uncertainties
5
Future prospects of the photon polarization measurement New Physics constraints combining various methods
6
Conclusions and perspectives
13 / 28
3P0 Qark-Pair-Creation Model (QPCM)
In order to compute J , we use QPCM [Le Yaouanc et al., Phys.Rev.D8 (’73), Phys.Rev.D9 (’74)] to describe the intermediate K1 → K ∗π, ρK decays.
1
QPCM is one of the simplest and most successful quark models which has a good predictive power.
2
The model has just one(!) universal phenomenological parameter- the quark pair-creation constant γ.
3
It is very good especially to compute the P-wave particles (and in this sense, better than the flux-tube-breaking model, for some case).
Basic idea Instead of being created from quark lines, q¯ q is created from anywhere within the hadronic matter and has the quantum numbers of the vacuum ⇒ q¯ q-pair must be in a 3P0 state, SU(3) singlet and
q1( k1) ¯ q2( k2) γ ¯ q( k3) q( k4) A B C
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QPCM predictions for the K1-decays Mass eigenstates K1(1270) and K1(1400) are considered as mixtures of 13P1(K1A) and 11P1(K1B) states [Suzuki, Phys.Rev.D47 (’93)]: |K1(1270) = |K1Asin θK1 + |K1Bcos θK1 |K1(1400) = |K1Acos θK1 − |K1Bsin θK1
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QPCM predictions for the K1-decays Mass eigenstates K1(1270) and K1(1400) are considered as mixtures of 13P1(K1A) and 11P1(K1B) states [Suzuki, Phys.Rev.D47 (’93)]: |K1(1270) = |K1Asin θK1 + |K1Bcos θK1 |K1(1400) = |K1Acos θK1 − |K1Bsin θK1 Partial wave amplitudes AS(K1(1270) → K ∗π/ρK) = SK∗/ρ( √ 2sin θK1 ∓ cos θK1) AD(K1(1270) → K ∗π/ρK) = DK∗/ρ(−sin θK1 ∓ √ 2cos θK1) AS(K1(1400) → K ∗π/ρK) = SK∗/ρ( √ 2cos θK1 ± sin θK1) AD(K1(1400) → K ∗π/ρK) = DK∗/ρ(−cos θK1 ± √ 2sin θK1)
SV = γ r 3 2 2IV
1 − IV
18 , DV = γ r 3 2 IV
1 + IV
18 IV
m=0,±1 = 1
8 Z d3 kYm
1 (
kP − k)ψ(P) ( k)ψ(V ) (− k)ψ−m(K1)
1
( kP + k)
The model has parameters fixed by the previous studies [Le Yaouanc et al., Phys.Rev.D8 (’73); Godfrey&Isgur, Phys.Rev.D32 (’85)]; we follow the 1st ref.
⇒ In this model, J can be computed in terms of γ and θK1.
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Comparison to data To understand the K1 → K ∗π, ρK decays, first one has to explain the
K1(1270) : BKρ/BK∗π = 4.16 ± 1.56, B(K∗π)D/B(K∗π)S = 0.54 ± 0.15 K1(1400) : BKρ/BK∗π = 0.01 ± 0.01, B(K∗π)D/B(K∗π)S = 0.04 ± 0.01 K1-width issue: when the mass of the resonance at the peak is close to a decay threshold, different definitions of the resonance width are no longer equivalent.
We found that ΓPDG
K1(1270) = ΓFWHH K1(1270) = (90 ± 8) MeV/c2 is
less by a factor ∼ 2 than Γpeak
K1(1270).
For the model computations and extraction of {γ, θK1} use the K1 partial decay widths, calculated using the K-matrix formalism.
κπ-issue: PDG assigns B(K1(1270) → K ∗
0 (1430)π) = (28 ± 4)%.
Using QPCM, we predict that B(K1(1270)→K ∗
0 (1430)π)
B(K1(1270)→K ∗(892)π) < 0.01%.
Belle found B(K1(1270) → K ∗
0 (1430)π) ≃ 2%
[Belle,Phys.Rev.Lett.94(’10)]. However, it’s not obvious that “κ” of ACCMOR is K ∗
0 (1430).
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Model parameters fit χ2
min = 0.61,
γ ≃ 4.0 ± 0.5, θK1 ≃ (59 ± 10)◦
20 40 60 80 0.0 0.2 0.4 0.6 0.8 1.0 ΘK1 deg 1CL 1 2 3 4 5 6 7 0.0 0.2 0.4 0.6 0.8 1.0 Γ 1CL
4 6 8 10 20 40 60 80 Γ ΘK1 deg
b1ΩΠD b1ΩΠS K11400KΡS K11400KΠS K11270KΠS The D-waves and the dominant channel K1(1270) → Kρ are not taken into account in the fit due to the dangerous threshold and width effects. 17 / 28
Kρ/K ∗π relative phase issue Im[ n · ( J × J ∗)] and consequently λγ are very sensitive to the relative phase between K ∗π and Kρ: δρ ≡ arg » AS(K1 → Kρ) × AP(ρ → ππ) AS(K1 → K ∗π) × AP(K ∗ → Kπ) –
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Kρ/K ∗π relative phase issue Im[ n · ( J × J ∗)] and consequently λγ are very sensitive to the relative phase between K ∗π and Kρ: δρ ≡ arg » AS(K1 → Kρ) × AP(ρ → ππ) AS(K1 → K ∗π) × AP(K ∗ → Kπ) – The quark model predicts δQPCM
ρ
= 0. Belle data
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Kρ/K ∗π relative phase issue Im[ n · ( J × J ∗)] and consequently λγ are very sensitive to the relative phase between K ∗π and Kρ: δρ ≡ arg » AS(K1 → Kρ) × AP(ρ → ππ) AS(K1 → K ∗π) × AP(K ∗ → Kπ) – The quark model predicts δQPCM
ρ
= 0. Belle data prediction with δρ = 0 prediction with δρ = π We confirm the sign, predicted by QPCM. Belle and ACCMOR found a small “off-set” phase δρ ∼ 30◦ ⇒ it is assumed as a source of systematic uncertainty to λγ.
18 / 28
1
Introduction: the b → sγ process and the photon polarization
2
The B → K1γ decay and polarization measurement Basic idea and formalism Determination of λγ in the DDLR method
3
Strong interaction decays of the K1-mesons Theoretical model Numerical results
4
Sensitivity studies of λγ measurement in the DDLR method Statistical errors Theoretical uncertainties
5
Future prospects of the photon polarization measurement New Physics constraints combining various methods
6
Conclusions and perspectives
19 / 28
Results: Monte Carlo simulation We estimate the sensitivity of future experiments to λγ using “ideal” (i.e. de- tector effects and background are not taken into account) MC simulation.
γ
from B → K1(1270)γ Nevents 103 104 (I) B+ → K +π−π+γ ±0.18 ±0.06 (II) B+ → K 0π+π0γ ±0.12 ±0.04 (III) B0 → K 0π+π−γ ±0.18 ±0.06 (IV) B0 → K +π−π0γ ±0.12 ±0.04 For 10k events the error on λγ is 10%. [Kou,Le Yaouanc&A.T., Phys.Rev.D83 (’11)]
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Results: Monte Carlo simulation We estimate the sensitivity of future experiments to λγ using “ideal” (i.e. de- tector effects and background are not taken into account) MC simulation.
γ
from B → K1(1270)γ Nevents 103 104 (I) B+ → K +π−π+γ ±0.18 ±0.06 (II) B+ → K 0π+π0γ ±0.12 ±0.04 (III) B0 → K 0π+π−γ ±0.18 ±0.06 (IV) B0 → K +π−π0γ ±0.12 ±0.04 For 10k events the error on λγ is 10%. [Kou,Le Yaouanc&A.T., Phys.Rev.D83 (’11)] The use of the Dalitz plot information improves the sensitivity by a factor 2 compared to the pure angular cos θ-fit (or Aup−down).
AupdownI, III DDLRI, III DDLRII, IV 200 500 1000 2000 5000 0.0 0.2 0.4 0.6 0.8 1.0 Nevents ΣΛΓstat.
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K1 mixing angle
1
Generate fake MC “data” using λth
γ = 0.5, θth K1 = 59◦, δth ρ = 0, 30◦. 2
Use the “data” and calculate λobs
γ
using DDLR method.
3
In step 2 in order to evaluate theoretical uncertainties we vary θmodel
K1
randomly according to Gaussian distribution with mean value θth
K1 = 59◦
and standard deviation σθK1 = 10◦: λobs
γ
= ωmodel ω2model
21 / 28
K1 mixing angle
1
Generate fake MC “data” using λth
γ = 0.5, θth K1 = 59◦, δth ρ = 0, 30◦. 2
Use the “data” and calculate λobs
γ
using DDLR method.
3
In step 2 in order to evaluate theoretical uncertainties we vary θmodel
K1
randomly according to Gaussian distribution with mean value θth
K1 = 59◦
and standard deviation σθK1 = 10◦: λobs
γ
= ωmodel ω2model
γ
λ
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.02 0.04 0.06 0.08 0.1
δth
ρ = 0
λobs
γ
∈ [0.42; 0.60] at 68%CL
21 / 28
K1 mixing angle
1
Generate fake MC “data” using λth
γ = 0.5, θth K1 = 59◦, δth ρ = 0, 30◦. 2
Use the “data” and calculate λobs
γ
using DDLR method.
3
In step 2 in order to evaluate theoretical uncertainties we vary θmodel
K1
randomly according to Gaussian distribution with mean value θth
K1 = 59◦
and standard deviation σθK1 = 10◦: λobs
γ
= ωmodel ω2model
γ
λ
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.02 0.04 0.06 0.08 0.1
δth
ρ = 0
λobs
γ
∈ [0.42; 0.60] at 68%CL
γ
λ
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.01 0.02 0.03 0.04 0.05 0.06 0.07
δth
ρ = 30◦
λobs
γ
∈ [0.40; 0.64] at 68%CL
21 / 28
1
Introduction: the b → sγ process and the photon polarization
2
The B → K1γ decay and polarization measurement Basic idea and formalism Determination of λγ in the DDLR method
3
Strong interaction decays of the K1-mesons Theoretical model Numerical results
4
Sensitivity studies of λγ measurement in the DDLR method Statistical errors Theoretical uncertainties
5
Future prospects of the photon polarization measurement New Physics constraints combining various methods
6
Conclusions and perspectives
22 / 28
b W s γ
V − A V − A
bL ˜ g sR γR (δd
RL)23
˜ bL ˜ sR M(b → sγ) = 4GF √ 2 V ∗
tsVtb
e 16π2 mbsσµνqν“ C eff
7γ
1 + γ5 2 | {z }
bR→sLγL
+ C ′ eff
7γ
1 − γ5 2 | {z }
bL→sRγR
” b εµ∗ Br measurement of the inclusive and exclusive b → sγ processes (B(B → Xsγ)exp = (3.55 ± 0.24 ± 0.09) × 10−4 [HFAG(’10)]) is not a direct polarization determination: B ∝ |C eff SM
7γ
+ C eff NP
7γ
|2 + |C ′ eff NP
7γ
|2 In the SM, C (′) eff
7γ
are real. Some NP models have extra sources of CP-violation ⇒ C (′) eff
7γ
can be complex.
23 / 28
B(B → Xsγ)exp = (3.55 ± 0.24 ± 0.09) × 10−4 [HFAG(’10)]
BrBXsΓ
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 ReC7 Γ
effC7 Γ eff
ImC7 Γ
effC7 Γ eff 24 / 28
3 different methods give different constraints:
1
ACP(B → KSπ0γ): Sexp
KSπ0γ = −0.15 ± 0.2
[HFAG(’10)] σ(SKSπ0γ)SuperB ≈ 0.02 at 75 ab−1
0.75 0.55 0.35 0.15 0.05 0.05 0.25 0.25 0.45 0.45
ACPB0KSΠ0Γ
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 ReC7 Γ
effC7 Γ eff
ImC7 Γ
effC7 Γ eff
24 / 28
3 different methods give different constraints:
1
ACP(B → KSπ0γ): Sexp
KSπ0γ = −0.15 ± 0.2
[HFAG(’10)] σ(SKSπ0γ)SuperB ≈ 0.02 at 75 ab−1
2
λγ potential measurement from ω-distribution in B → K1(1270)γ: σ(λγ)th ∼ 0.2
0.75 0.55 0.35 0.15 0.05 0.05 0.25 0.25 0.45 0.45
ACPB0KSΠ0Γ
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 ReC7 Γ
effC7 Γ eff
ImC7 Γ
effC7 Γ eff
0.8 0.6 0.4 0.2
ΛΓB K11270 Γ
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 ReC7 Γ
effC7 Γ eff
ImC7 Γ
effC7 Γ eff
24 / 28
3 different methods give different constraints:
1
ACP(B → KSπ0γ): Sexp
KSπ0γ = −0.15 ± 0.2
[HFAG(’10)] σ(SKSπ0γ)SuperB ≈ 0.02 at 75 ab−1
2
λγ potential measurement from ω-distribution in B → K1(1270)γ: σ(λγ)th ∼ 0.2
3
A(2)
T and A(im) T
potential measurement from the angular analysis of B0 → K ∗0(→ K −π+)ℓ+ℓ−: σ(A(2)
T )LHCb ≈ 0.2 at 2 fb−1
0.75 0.55 0.35 0.15 0.05 0.05 0.25 0.25 0.45 0.45
ACPB0KSΠ0Γ
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 ReC7 Γ
effC7 Γ eff
ImC7 Γ
effC7 Γ eff
0.8 0.6 0.4 0.2
ΛΓB K11270 Γ
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 ReC7 Γ
effC7 Γ eff
ImC7 Γ
effC7 Γ eff
0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8
AT
2B0Kll
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 ReC7 Γ
effC7 Γ eff
ImC7 Γ
effC7 Γ eff
0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8
AT
imB0Kll
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 ReC7 Γ
effC7 Γ eff
ImC7 Γ
effC7 Γ eff
24 / 28
C eff
7γ
is assumed to be purely SM-like (i.e. C eff (NP)
7γ
= 0) B(B → Xsγ)exp = (3.55 ± 0.24 ± 0.09) × 10−4 [HFAG(’10)] Sexp
KSπ0γ = −0.15 ± 0.2
[HFAG(’10)]
BrBXsΓexp ACP
expB0KSΠ0Γ
SM
0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 ReC7 Γ
effC7 Γ eff
ImC7 Γ
effC7 Γ eff 25 / 28
C eff
7γ
is assumed to be purely SM-like (i.e. C eff (NP)
7γ
= 0) B(B → Xsγ)exp = (3.55 ± 0.24 ± 0.09) × 10−4 [HFAG(’10)] Sexp
KSπ0γ = −0.15 ± 0.2
[HFAG(’10)] λSM
γ
≃ −1 ± 0.2 A(2)SM
T
≃ A(im)SM
T
≃ 0 ± 0.2
BrBXsΓexp ACP
expB0KSΠ0Γ
SM
0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 ReC7 Γ
effC7 Γ eff
ImC7 Γ
effC7 Γ eff 25 / 28
1
Introduction: the b → sγ process and the photon polarization
2
The B → K1γ decay and polarization measurement Basic idea and formalism Determination of λγ in the DDLR method
3
Strong interaction decays of the K1-mesons Theoretical model Numerical results
4
Sensitivity studies of λγ measurement in the DDLR method Statistical errors Theoretical uncertainties
5
Future prospects of the photon polarization measurement New Physics constraints combining various methods
6
Conclusions and perspectives
26 / 28
1
We study the B → K1γ decay to determine the photon polarization in the b → sγ process in order to search the effects of New Physics beyond the SM.
27 / 28
1
We study the B → K1γ decay to determine the photon polarization in the b → sγ process in order to search the effects of New Physics beyond the SM.
2
We propose a new quantity ω(s13, s23, cos θ), which contains all the information on polarization in each event and allows to reduce the error
27 / 28
1
We study the B → K1γ decay to determine the photon polarization in the b → sγ process in order to search the effects of New Physics beyond the SM.
2
We propose a new quantity ω(s13, s23, cos θ), which contains all the information on polarization in each event and allows to reduce the error
3
We obtain the statistical accuracy < 10% for the SM-prediction for λγ for 10k events of the B → K1(1270)γ decay.
27 / 28
1
We study the B → K1γ decay to determine the photon polarization in the b → sγ process in order to search the effects of New Physics beyond the SM.
2
We propose a new quantity ω(s13, s23, cos θ), which contains all the information on polarization in each event and allows to reduce the error
3
We obtain the statistical accuracy < 10% for the SM-prediction for λγ for 10k events of the B → K1(1270)γ decay.
4
Perspective: the right-handed currents will be very strictly constrained by the future experiments, LHCb and SuperB. It was demonstrated that combining the three methods, we will be able to constrain C ′ eff
7γ /C eff 7γ
quite precisely.
27 / 28
28 / 28
Theoretical uncertainties: c-quark loop and soft gluon contribution An additional helicity enhancement can be ob- tained by considering (at parton level) a three- particle final state: b → sγg. The dominant contribution comes from the four-quark opera- tor O2 = (sLγµcL)(cLγµbL) b s c ¯ c γ g O2 The soft (| kg|2 ≪ 4m2
c) gluon contribution is estimated to be [Ball&Zwicky,
Phys.Lett.B642(’06)]: M(B → K
∗γR)
M(B → K
∗γL)
≃ ms mb −C2 C7 L − ˜ L 36m2
cmbT B→K∗ 1
(0) ≃ ms mb × (0.8 ± 0.2) Soft gluon correction enhances the leading term up to 20%. Additional hadronic corrections are expected to be smaller.
30 / 28
Basic idea of the DDLR method Our probability density function (i.e. the normalized decay width distribution) can be written as W (s13, s23, cos θ) = f (s13, s23, cos θ) + λγg(s13, s23, cos θ) = f (1 + λγω) Then, the log-likelihood function for a sample of N measurements is: ln L = ln
N
Y
i=1
W (si
13, si 23, cos θi) = N
X
i=1
ln(1 + λγωi) +other terms independent of λγ Using the maximum likelihood method, we obtain λγ as a solution of the following equation: ∂ ln L ∂λγ =
N
X
i=1
ωi 1 + λγωi = N ω 1 + λγω = 0 Since W depends on λγ, one can reduce a multi-dimensional fit to a
31 / 28
Simple solution for λγ Our PDF can be rewritten as W ′(ω) = ϕ(ω)(1 + λγω) where ϕ(ω) is an unknown function which is very hard to determine analytically. The moments can be defined as ωn = Z 1
−1
ωnW ′(ω)dω In our case, ϕ(ω) turns out to be an even function of ω. Therefore ˙ ω2n−1¸ = λγ R 1
−1 ω2nϕ(ω)dω
˙ ω2n¸ = R 1
−1 ω2nϕ(ω)dω
⇒ λγ = ˙ ω2n−1¸ ω2n
32 / 28
ω-distribution
ω
0.2 0.4 0.6 0.8 1
Number of events
200 400 600 800 1000 1200 1400 1600 1800 2000
= +1
γ
λ = -1
γ
λ
Example of MC ω-distribution for 10k of B+ → (K +π−π+)K1(1270)γ events with purely right-handed (red) and left-handed (blue) photons.
33 / 28
Using the method of minimal χ2 χ2(λγ) = X
i
` Ndata
i
− Nth
i (θK1, δρ, λγ)
´2 σ2
Ndata
i
→ min we obtain the estimator λobs
γ .
If θK1 or δρ is changed according to prediction of another model, the minimization of χ2 will give another value for λobs
γ . 34 / 28
PDG: B(K1(1270) → K ∗
0 (1430)π) = (28 ± 4)%
[Belle,Phys.Rev.Lett.94(’10)]: B(K1(1270) → K ∗
0 (1430)π) ≃ 2%
Our QPCM prediction: B(K1(1270) → K ∗
0 (1430)π)
B(K1(1270) → K ∗(892)π) < 0.01%
K11270KΠ K11270K0
1430Π
0.4 0.6 0.8 1.0 1.2 1 2 3 4 5 6 7 sKΠ GeV2c4 Amplitude 2
35 / 28
The K-matrix is defined from the S-matrix (S = 1 + 2iρ
1 2 Tρ 1 2 ) as
K −1 ≡ T −1 + iρ, K = K † Kij = X
a′
fa′ifa′j ma′ − m The partial and total K-matrix widths can be defined as Γa′i(m) = 2f 2
a′iρii(m)
Γa′(m) = X
i
Γa′i(m) where ρij(m) = 2ki(m)δij/m is the two-body phase space factor.
36 / 28
CP-asymmetry in B0 → K ∗0(→ KSπ0)γ, Bs → φγ ∼ ms ∼ mb B B f γL ∼ mb ∼ ms B B f γR Time-dependent CP-asymmetry in neutral B-mesons results from the interference of mixing and decay [Atwood et al.,Phys.Rev.Lett.79 (’97)]. ACP(t) ≡ Γ(Bq(t) → f γ) − Γ(Bq(t) → f γ) Γ(Bq(t) → f γ) + Γ(Bq(t) → f γ) ≃ Sf γ sin(∆mt) Sf γ = −ξf 2|MLMR| |ML|2 + |MR|2 sin(φM − φL − φR) where φL,R = arg(ML,R) and φM is the Bq − ¯ Bq mixing phase. In the SM b → sγ |MR/ML| ≃ ms/mb, φL = φR ≃ 0 B0 − B φM = 2β ≃ 43◦ Bs − Bs φM ≃ 0 ⇒ 8 > < > : SKSπ0γ ≃ − 2ms
mb sin 2β ≪ 1
Sexp
KSπ0γ
= −0.15 ± 0.2[HFAG(’10)] Sφγ ≃ 0
37 / 28
Transverse asymmetries in B0 → K ∗0(→ K −π+)ℓ+ℓ− Analysis of the angular distributions in B0 → K ∗0(→ K −π+)ℓ+ℓ− in the low ℓ+ℓ− inv.mass region and measurement of the transverse asymmetries [Kruger&Matias,Phys.Rev.D71(’05)]. d 2Γ dq2dφ = 1 2π dΓ dq2 » 1 + 1 2FT(q2) “ A(2)
T (q2) cos 2φ + A(im) T
(q2) sin 2φ ”– A(2)
T (q2) = − 2Re[MRM∗ L]
|MR|2 + |ML|2 A(im)
T
(q2) = 2Im[MRM∗
L]
|MR|2 + |ML|2 In the heavy quark and large EK∗ limit (⇔ q2 → 0) A(2)
T (0) = 2Re[C eff 7γ C ′ eff ∗ 7γ
] |C eff
7γ |2 + |C ′ eff 7γ |2
A(im)
T
(0) = 2Im[C eff
7γ C ′ eff ∗ 7γ
] |C eff
7γ |2 + |C ′ eff 7γ |2
z ℓ− ℓ+ K − π+ B0 K ∗0 φ θK∗ θℓ
38 / 28
Here we compare the precision of x ≡ |C ′ eff
7γ /C eff 7γ | measurement, using the
methods of the ACP measurement in B0 → (KSπ0)K∗γ and λγ = x2−1
1+x2 deter-
mination in B → K 1γ. The error of x determination will be dependent on the measured value of λγ(⇔ x): σx = (1 + x2)2 4x σλγ For some values of x, considerably different from the SM (i.e. 0), one can obtain a better sensitivity, compared to the ACP-method.
ΣΛΓ0.05 ΣΛΓ0.10 ΣΛΓ0.15 ΣΛΓ0.20 SM
0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.05 0.10 0.15 0.20 0.25 C7 Γ
effC7 Γ eff
Σ C7 Γ
effC7 Γ eff
For instance, one can see from the Fig. that if we measure λγ < 0.8(⇔ x > 0.3) with the error σλγ ≈ 0.1, we can have a smaller error on x, compared to the estimated σx ≃ 0.1 from potential measurement of ACP at LHCb [LHCB-ROADMAP4-001].
39 / 28
Extra sources of flavour violation in SUSY In the SM, the quark masses come from Yukawa couplings: LSM
Yukawa = υuRyuuL + υd RyddL + h.c.
In SUSY, the squark mass can come from any combination of left and right couplings: LMSSM
soft
= ˜ q†
Lm2 Q˜
qL + ˜ u†
Rm2 U˜
uR + ˜ d †
Rm2 D ˜
dR + υ2˜ u†
RAU˜
qL + υ1˜ d †
RAD˜
qL + . . . m2
Q,U,D and AU,D are not diagonal in the quark eigenmass basis ⇒ the
squark propagators can change flavour and chirality: ˜ bL ˜ sL (δd
LL)23
˜ bL ˜ sR (δd
RL)23
˜ bR ˜ sL (δd
LR)23
˜ bR ˜ sR (δd
RR)23 40 / 28
Extra sources of flavour violation in SUSY: Mass Insertion Approximation Quark and squark mass matrices are rotated in the same way to diagonalize the fermion mass matrix (the super-CKM basis). In this basis, the couplings of fermions and sfermions to neutral gauginos are flavour diagonal, leaving the source of flavour violation in the
AB)ij.
The sfermion propagator can then be expanded as ˙ ˜ qAi ˜ q∗
Bj
¸ = i(k2 − m2
˜ q − ∆q AB)−1 ij
≃ iδij k2 − m2
˜ q
+ i(∆q
AB)ij
(k2 − m2
˜ q)2 + . . .
The flavour violation in SUSY can be parametrized in a model independent way by the dimensionless parameters (δq
AB)ij ≡ (∆q AB)ij
m2
˜ q 41 / 28
K 1|¯ sσµνγ5qνb|B = iǫµνρσεν∗
K1pρ Bpσ K12T B→K1 1
(q2) K 1|¯ sσµνqνb|B = T B→K1
2
(q2) ˆ ε∗
K1µ(m2 B − m2 K1) − (ε∗ K1 · pB)(pB + pK1)µ
˜ + T B→K1
3
(q2)(ε∗
K1 · pB)
" qµ − q2 m2
B − m2 K1
(pB + pK1)µ # T B→K1
1
(0) = T B→K1
2
(0) in order to avoid a kinematic singularity at q2 = 0. Since the outgoing photon is on-shell, q2 = 0 and qµε∗µ = 0 ⇒ the matrix element is parametrized with only one form factor T K1
1 (0):
K 1LγL|O7γ|B = K 1RγR|O ′
7γ|B = i e
8π2 mb(m2
B − m2 K1)T B→K1 1
(0) The form factors of the mass eigenstates are related to T
B→K1A,B 1
, which can be calculated with LCSR, as following: T B→K1(1270)
1
(0) = T B→K1A
1
(0)sin θK1 + T B→K1B
1
(0)cos θK1 T B→K1(1400)
1
(0) = T B→K1A
1
(0)cos θK1 − T B→K1B
1
(0)sin θK1
42 / 28
The decay amplitude of the axial-vector meson A to some vector (Vij) and pseudoscalar (Pk) mesons can be expressed in the following Lorentz invariant form: M(A → VijPk) = ε(A)
µ T µνε (Vk )∗ ν
, T µν = f g µν + hpµ
Vij pν A
The unknown effective couplings f and h can be related to the partial wave amplitudes AS and AD as f = − „ AS + AD √ 2 « , h = »„ 1 − mVij EVij « AS + „ 1 + 2mVij EVij « AD √ 2 – EVij mA p2
k
where EVij and pk(= − pVij ) are the energy of the vector meson and the momentum of pseudoscalar meson in the A-reference frame. The amplitude of the subsequent decay Vij → PiPj can be parametrized in terms of the effective coupling gVij Pi Pj (which can be determined from the measured partial decay width of Vij): M(Vij → PiPj) = gVij Pi Pj · ε
(Vij ) µ
(pi − pj)µ
43 / 28
Parametrizing the propagation of Vij with the relativistic Breit-Wigner form BWVij (sij) = 1/(sij − m2
Vij − imVij ΓVij ), one can write the total amplitude
M(A → (PiPj)Vij Pk) = ε(A)
µ (f g µν+hpµ Vij pν A)ε (Vij )∗ ν
BWVij (sij)gVij Pi Pj ε
(Vij ) σ
(pi−pj)σ Summing over the Vij-polarizations, one obtains the Lorentz invariant ampli- tude: M(A → (PiPj)Vij Pk) = ε(A)
µ Jµ ijk,
Jµ
ijk = ck(sij)pµ k − ci(sij)pµ i
ck(sij) = gVij Pi Pj " −(f + h(m2
A − pA · pk))
1 + m2
i − m2 j
m2
Vij
! + 2h(pA · pi) # BWVij (sij) ci(sij) = 2gVij Pi Pj f BWVij (sij) If there are several possible channels of the A-decay to the same charged final state P1P2P3, one has to sum over all possible diagrams with different interme- diate vector resonance states: M(A → P1P2P3) = X
Vij
(Ii, I z
i ; Ij, I z j |IVij , I z Vij )(IVij , I z Vij ; Ik, I z k |IA, I z A)
×M(A → (PiPj)Vij Pk) = ε(A)
µ J µ = ε(A) µ (C1(s13, s23)pµ 1 − C2(s13, s23)pµ 2 ) 44 / 28
For the axial meson decay (A = K1A,B) into the ground states of vector (V = K ∗, ρ) and pseudoscalar (P = π, K) mesons, the spacial integrals are given by
IV
m=0,±1 =
Z d3 k1d3 k2d3 k3d3 k4δ( k1 + k2 − kA)δ( k2 + k3 − kV )δ( k4 + k1 − kP)δ( k3 + k4) ×Ym
1 (
k3 − k4)ψ(A)( k1 − k2)ψ(V )( k2 − k3)ψ(P)( k4 − k1) = 1 8 Z d3 kYm
1 (
kP − k)ψ(P) ( k)ψ(V ) (− k)ψ−m(A)
1
( kP + k)
where ψLz
L
are the normalized Fourier transforms of harmonic oscillator meson wave functions:
ψ(i)
0 (
k) = R3/2
i
π3/4 exp −
i
8 ! , ψm(i)
1
( k) = s 2 3 R5/2
i
π1/4 Ym
1 (
k) exp −
i
8 !
Here Ym
1 (
k) = | k|Y m
1 (ˆ
εm k) p 3/4π, Ri is the meson wave function radius and εm are the polarization vectors, defined as ε0 = (0, 0, 1), ε±1 = ∓ 1
√ 2(1, ±i, 0). IV = − 4 √ 3 π5/4 R5/2
A
(RV RP)3/2 (R2
A + R2 V + R2 P)5/2
1 − k2
P
(2R2
A + R2 V + R2 P)(R2 V + R2 P)
4(R2
A + R2 V + R2 P)
! exp " − k2
P
R2
A(R2 V + R2 P)
8(R2
A + R2 V + R2 P)
# IV
1
= 4 √ 3 π5/4 R5/2
A
(RV RP)3/2 (R2
A + R2 V + R2 P)5/2 exp
" − k2
P
R2
A(R2 V + R2 P)
8(R2
A + R2 V + R2 P)
# 45 / 28
The annual yield of the B → (Kππ)K1γ decay can be estimated as following: Nannual = NB × B(B → K1γ) × B(K1 → Kππ) × ǫtot
B(B+ → (K+π−π+)K1(1270)γ) = 4.3 × 10−5 × (0.16 ∗ 4/9 + 0.42 ∗ 1/6) ≃ 0.6 × 10−5, B(B+ → (K0π+π0)K1(1270)γ) = 4.3 × 10−5 × (2 ∗ 0.16 ∗ 2/9 + 0.42 ∗ 1/3) × 1/3 ≃ 0.3 × 10−5
For R LLHCb = 2 fb−1, NB ≃ 8 × 1011, ǫtot ∼ 0.1% (as in B → K ∗γ and Bs → φγ [CERN-LHCB-2007-030]) ⇒ NLHCb
annual(B+ → (K +π−π+)K1(1270)γ) ≈ 5 × 103
NLHCb
annual(B+ → (K 0π+π0)K1(1270)γ) ≈ 2.5 × 103
For R LSuperB = 2 ab−1, NB ≃ 1.6 × 1010, ǫtot ∼ 1% (as in B → K +
1 (1270)γ [Belle (’05)]) ⇒
NSuperB
annual (B+ → (K +π−π+)K1(1270)γ) ≈ 1 × 103
NSuperB
annual (B+ → (K 0π+π0)K1(1270)γ) ≈ 0.5 × 103 46 / 28