- Email: [email protected]
Future prospects for the determination of the Wilson coefficient C7γ′
Available online at www.sciencedirect.com
Nuclear Physics B (Proc. Suppl.) 234 (2013) 173–176 www.elsevier.com/locate/npbps
Future prospects for the determination of the Wilson coefficient C7γ Andrey Tayduganova,b a Laboratoire
de Physique Th´eorique, CNRS/Universit´e Paris-Sud 11 (UMR 8627) 91405 Orsay, France b Laboratoire de l’Acc´ el´erateur Lin´eaire, Universit´e Paris-Sud 11, CNRS/IN2P3 (UMR 8607) 91405 Orsay, France
Abstract from the direct and the indirect measurements of the photon We discuss the possibilities of assessing a non-zero C7γ (∗) polarization in the exclusive b → sγ decays. We focus on three methods and explore the following three decay modes: B → K ∗ (→ KS π0 )γ, B → K1 (→ Kππ)γ, and B → K ∗ (→ Kπ)+ − . By studying different New Physics scenarios we show that the future measurement of conveniently defined observables in these decays could provide us . with the full determination of C7γ and C7γ Keywords: Rare Decays, Beyond Standard Model, B-Physics 1. Introduction The radiative decay b → sγ has been extensively studied as a probe of the flavour structure of the Standard Model (SM) as well as New Physics (NP), beyond the SM. While the majority of studies has been focused on the prediction of the decay rates of exclusive and inclusive b → sγ decays, relatively few studies of the right-handed currents in these decays have been made. In the SM, the emitted photon is predominantly lefthanded in b, and right-handed in b decays. This is due to the fact that the dominant contribution comes from the () e = 16π chiral-odd dipole operator O7γ 2 mb s L(R) σμν bR(L) . As only left-handed quarks participate in weak interaction, this effective operator induces a helicity flip on one of the external quark lines, which results in a factor mb for bR → sL γL , and a factor m s for bL → sR γR . Hence, the emission of right-handed photons is suppressed by a factor m s /mb . This suppression can be lifted in some NP models where the helicity flip occurs on an internal line, which brings in a factor mNP /mb instead of m s /mb . If the amplitude for b → sγR is of the same order as the SM prediction, or the enhancement of b → sγR goes along with the suppression of b → sγL , the impact on the branching ratio is small since the two helicity amplitudes add incoherently. This implies that there can Email address: [email protected] (Andrey Tayduganov)
0920-5632/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2012.12.008
be a substantial contribution of NP to b → sγ escaping detection when only branching ratios are measured. Therefore, the photon polarization measurement could provide a good test of the SM or at least a useful indication of NP. 2. Various methods for determination of C7γ
The b → sγ process can be described by the effective Hamiltonian1 , 4G F (μb )O7γ (μb ) , Heff = − √ Vtb Vts∗ C7γ (μb )O7γ (μb ) + C7γ 2 (1) ) where the short-distance Wilson coefficient C7γ (C7γ describes the left(right)-handed photon emission ampli /C7γ m s /mb 0.02. tude of b → sγ. In the SM, C7γ Three methods have been proposed for the measurement of the photon polarization or equivalently the ratio /C7γ : C7γ 1 The short-distance QCD effects induce the mixing of O () with 7γ the other operators which we omitted writing in Eq. (1). This effect can be absorbed by defining the so-called “effective” coefficients () eff () . For notational simplicity, whenever C7γ appears in what folC7γ () eff lows, C7γ , evaluated at the scale μb mb, pole = 4.8 GeV, will be understood.
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A. Tayduganov / Nuclear Physics B (Proc. Suppl.) 234 (2013) 173–176
• An indirect determination of the photon polarization, proposed by Atwood et al. [1], is the measurement of the time-dependent mixing-induced CPasymmetry in the radiative neutral B-mesons decays such as B → K ∗ (→ KS π0 )γ, and Bs → φγ, which is determined by the S parameter, S
2Im[e−iφM C7γ C7γ | 2 |C7γ |2 + |C7γ |
,
(2)
where φ M is the phase in the B − B mixing, which in the SM is φd = 2β 43◦ , and φ s 0, for the Bd and Bs mixing, respectively. Such measurements are expected to be made at future super B factories, reduce the experimental error on the asymmetry parameter S K ∗ γ down to 2%. • A direct determination, proposed by Gronau et al. [2], is based on the study of the angular distribution of the three-body final state, Kππ, coming from the axial vector K1 (1+ )-meson decay, in B → K1 (→ Kππ)γ, and extraction of the polarization parameter λγ , λγ
2 |C7γ | − |C7γ |2 2 |C7γ | + |C7γ |2
.
(3)
In Ref. [3] this method was improved by using a new variable ω, which includes not only the angular dependence but also the dependence on the three-body Dalitz variables which can significantly improve the sensitivity of the measurement of the polarization parameter. Recently measured by the Belle collaboration B(B → K1 (1270)γ) appeared to be comparable to B(B → K ∗ γ), which opened the possibility of measuring the photon polarization in B → K1 γ. • Another indirect way to study the right-handed currents is based on the angular analysis in the semileptonic B → K ∗ (→ Kπ)+ − decay2 . In par2 ticular, two transverse asymmetries, A(2) T (q ) and (im) 2 AT (q ), introduced in Refs. [5, 6], are highly sensitive to the b → sγ process at very low dilepton invariant mass squared q2 = (p+ + p− )2 , 2 lim A(2) T (q ) =
q2 →0
lim
q2 →0
2 A(im) T (q )
=
∗ ] 2Re[C7γ C7γ 2 |C7γ |2 + |C7γ | ∗ ] 2Im[C7γ C7γ 2 |C7γ |2 + |C7γ |
, (4) .
2 In our analysis we do not consider the pollution by the B → K0∗ (→ Kπ)+ − events [4].
These three methods, having their own advantages and disadvantages, can be complementary to each other. Combination of all three of them can in principle put a () coefficients strong constraint on the short-distance C7γ in a model-independent way which then can be used as a constraint in building the NP models. 3. Constraints on C7γ combining various methods of the photon polarization determination by B(B → X s γ) and 3.1. Current constraint on C7γ S K∗ γ In Fig. 1 we show the constraints on C7γ /C7γ available at present and compare them with those that are planned to be obtained from the future measurements. For illustration, we consider two NP scenarios3 : (NP) (NP) ∈ R, C7γ ∈R; • scenario I: C7γ (NP) (NP) = 0, C7γ ∈C; • scenario II: C7γ
In all plots presented in Figs. 1–3, we use the constraint from the inclusive rate. The region outside the gray (dark gray) circle is excluded at 3σ (1σ) level by the current measurement [8], Bexp (B → X s γ) = (3.55 ± 0.24) × 10−4 ,
(5)
which we combined with the SM prediction given in Ref. [9]. In Fig. 1 we show the constraints from already measured B(B → X s γ) and S K ∗ γ . Orange (dark orange) region represents the ±3σ (±1σ) region allowed by the current measurement of S K ∗ γ [8], exp
S K ∗ γ = −0.16 ± 0.22 .
(6)
Performing a χ2 -fit of B(B → X s γ) and S K ∗ γ , we obtain for each considthe 95% and 68% CL regions for C7γ ered NP scenario. One can see from the plots in Fig. 1, that there is still room for NP. Note, however, the ap parent ambiguities in the C7γ − C7γ plane: in scenario I it is fourfold in the C7γ − Cγ plane and twofold in the ] − Im[C7γ ] plane in scenario II, respectively. Re[C7γ Therefore, it is clear that additional observables are re() . quired to pin down the real and imaginary parts of C7γ
3 For
more tested scenarios and details see Ref. [7]
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= 0,
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∈ C (b). The light and dark blue regions
(NP) correspond respectively to the 95% and 68% CL bounds for C7γ ,
obtained from the χ2 -fit of the present measurements of B(B → X s γ) and S K ∗ γ .
C7 ΓNPC7SM Γ
1
Figure 1: Current constraints from the combination of the inclusive decay rate and the mixing-induced CP-asymmetry in B → K ∗ (→ (NP) (NP) KS π0 )γ in particular NP scenarios where C7γ ∈ R, C7γ ∈ R
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() in the NP sceFigure 2: Prospect of the future constraints on C7γ
3.2. The expected sensitivity to C7γ in the future measurements In Figs. 2 and 3, we present a future prospect for con() (NP) straining C7γ in the two NP scenarios. The plots are obtained by assuming:
• Improved measurement of the CP-asymmetry parameter S K ∗ γ in B → K ∗ (→ KS π0 )γ. The contour colours correspond to S K ∗ γ allowed by a ±3σ to the present world average (6). Different colours are separated by the size of the current experimental error. That error will be significantly reduced at super B factories. • Potential measurement of the polarization parameter λγ in B → K 1 (1270)γ. The contour colours correspond to λγ ∈ [−1, 1]. The spacing between contours is taken to be σλγ = 0.2, which may be improved by the study of K1 → Kππ decays. That can be made using a detailed experimental study of B → K1 ψ decay. • Potential non-zero measurement of two transverse ∗ im) asymmetries, A(2, ∈ [−1, 1], in B → K + − . T (2) The contours correspond to AT (0) and A(im) T (0) respectively. The interval between the lines represents 20% of uncertainty for each, which, in principle, can be achieved at LHCb. Note that in all these figures we applied the constraint from the measured B(B → X s γ) as allowed by a ±3σ error to the central value (5). In Fig. 2, we present our result for the scenario I. The constraints from S K ∗ γ and A(2) T look very similar in this
(NP) (NP) nario I: C7γ and C7γ are both real. The contour colours in
Fig. (a, b, c) correspond respectively to S K ∗ γ , λγ and A(2) T (0) allowed by a ±3σ error to the central value of Bexp (B → X s γ).
scenario since both of them are proportional to
C7γ C7γ 2 +C 2 C7γ 7γ
() (NP) with C7γ being real numbers. On the other hand, one can see that the shape of the constraint from λγ is quite different. For example, the fourfold ambiguity in the constraints of S K ∗ γ and A(2) T can be reduced to a twofold with the help of the λγ measurement. In addition, one (NP) (NP) observes that the region around the line C7γ = C7γ is quite sensitive to the λγ values, while it is not in the case of S K ∗ γ and A(2) T .
In Fig. 3, we present our result for the scenario II. The constraint from S K ∗ γ is very strong (indeed, assuming that fact that the experimental error will be significantly reduced by super B factories down to 2%, the bound, i.e. the spacing between the adjacent contours will become about 10 times more narrow than those depicted in Fig. 3(a)) but it has an ambiguity along the diagonal. This problem can be partially solved by adding a constraint from λγ which is a circle since λγ is a function /C7γ |2 and therefore is insensitive to the comof |C7γ plex phases. The SM prediction corresponds to the cen(NP) (NP) = C7γ = (0, 0). Near the center tral point C7γ −1, and the sensitivity to C7γ is very low. λγ = λSM γ (NP) (SM) /C7γ | 0.3 (i.e. one is For λγ −0.8 we have |C7γ clearly outside the SM prediction), but inside the circle one cannot distinguish the NP contribution from the SM one.
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• The photon polarization parameter λγ in B → K1 (→ Kππ)γ.
0.5
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ReC7 ΓNPC7SM Γ
() in the NP sceFigure 3: Prospect of the future constraints on C7γ (NP) (NP) nario II: C7γ is purely SM-like, i.e. C7γ = 0. The contour colours
in Fig. (a, b, c, d) correspond respectively to S K ∗ γ , λγ , A(2) T (0) and exp (B → A(im) (0) allowed by a ±3σ error to the central value of B T X s γ). 2 The combined measurement of A(2) T (q ) and 2 A(im) T (q ) can, in principle, constraint both Re[C 7γ /C 7γ ] and Im[C7γ /C7γ ] independently on S K ∗ γ and λγ . In contrast to λγ , it is also sensitive to the SM prediction. A pleasant feature of Figs. 2,3 is that the shapes of the resulting plots are quite different in NP scenarios. The four constraints will overlap in scenarios compatiim) and we will be ble with measured S K ∗ γ , λλ and A(2, T () and their phases. In incompatible able to extract C7γ scenarios, the four constraints will not overlap. /C7γ Once again, we stress that we can determine C7γ
from S K ∗ γ only in combination with the B − B mixing phase, φ M . In this paper we assume that NP does not bring any significant contribution to the B − B mixing box diagrams and use the currently measured value, sin 2β = 0.673 ± 0.023 [10]. 4. Conclusions We have studied the prospects for determining the from the future meaWilson coefficients C7γ and C7γ probes surements at LHCb and super B factories. C7γ the right-handed structure of the New Physics models which enter the b → sγ processes. In order to deter mine C7γ , we have used four observables: • The mixing-induced CP-asymmetry parameter S K ∗ γ in B → K ∗ (→ KS π0 )γ.
In principle, these four observables can unambiguously constrain the New Physics contribution to C7γ and C7γ , even when these Wilson coefficients are complex numbers. () 0 and We studied different NP scenarios of C7γ presented the current constraints provided by B(B → X s γ) and S KS π0 γ . Those constraints are still either loose and/or ambiguous. We then showed that the future mea(im) will not only surements of S K ∗ γ , λγ , A(2) T and AT () restrain the allowed range of values for C7γ , but also solve or partially solve the ambiguities in the complex ) plane. (C7γ , C7γ We should emphasize that each of the above quantities has its own advantages and disadvantages depending on the NP scenario. In the scenario I, we found that the bounds coming from S K ∗ γ and from A(2) T are similar. To disentangle the discrete fourfold ambiguity arising from these two constraints, the measurement of λγ could help and reduce this ambiguity to twofold. In the scenario II, λγ plays an important role: although S K ∗ γ bound will be extremely constraining at super B factories, the resulting diagonal ambiguity could be at least partly solved by a constraint provided by (im) are very important the measured λγ . A(2) T and AT since their combination can, in principle, constrain both ] and Im[C7γ ] independently on S K ∗ γ and λγ . Re[C7γ References [1] D. Atwood, M. Gronau and A. Soni, Phys. Rev. Lett. 79 (1997) 185 [hep-ph/9704272]. [2] M. Gronau, Y. Grossman, D. Pirjol and A. Ryd, Phys. Rev. Lett. 88 (2002) 051802 [hep-ph/0107254]. [3] E. Kou, A. Le Yaouanc and A. Tayduganov, Phys. Rev. D 83 (2011) 094007 [arXiv:1011.6593 [hep-ph]]. [4] D. Becirevic and A. Tayduganov, arXiv:1207.4004 [hep-ph]. [5] F. Kruger and J. Matias, Phys. Rev. D 71 (2005) 094009 [hepph/0502060]. [6] D. Becirevic and E. Schneider, Nucl. Phys. B 854 (2012) 321 [arXiv:1106.3283 [hep-ph]]. [7] D. Becirevic, E. Kou, A. Le Yaouanc and A. Tayduganov, JHEP 1208 (2012) 090 [arXiv:1206.1502 [hep-ph]]. [8] D. Asner et al. [Heavy Flavor Averaging Group Collaboration], arXiv:1010.1589 [hep-ex]. [9] A. L. Kagan and M. Neubert, Eur. Phys. J. C 7 (1999) 5 [hepph/9805303]. [10] K. Nakamura et al. [Particle Data Group Collaboration], J. Phys. G G 37 (2010) 075021.
Nuclear Physics B (Proc. Suppl.) 234 (2013) 173–176 www.elsevier.com/locate/npbps
Future prospects for the determination of the Wilson coefficient C7γ Andrey Tayduganova,b a Laboratoire
de Physique Th´eorique, CNRS/Universit´e Paris-Sud 11 (UMR 8627) 91405 Orsay, France b Laboratoire de l’Acc´ el´erateur Lin´eaire, Universit´e Paris-Sud 11, CNRS/IN2P3 (UMR 8607) 91405 Orsay, France
Abstract from the direct and the indirect measurements of the photon We discuss the possibilities of assessing a non-zero C7γ (∗) polarization in the exclusive b → sγ decays. We focus on three methods and explore the following three decay modes: B → K ∗ (→ KS π0 )γ, B → K1 (→ Kππ)γ, and B → K ∗ (→ Kπ)+ − . By studying different New Physics scenarios we show that the future measurement of conveniently defined observables in these decays could provide us . with the full determination of C7γ and C7γ Keywords: Rare Decays, Beyond Standard Model, B-Physics 1. Introduction The radiative decay b → sγ has been extensively studied as a probe of the flavour structure of the Standard Model (SM) as well as New Physics (NP), beyond the SM. While the majority of studies has been focused on the prediction of the decay rates of exclusive and inclusive b → sγ decays, relatively few studies of the right-handed currents in these decays have been made. In the SM, the emitted photon is predominantly lefthanded in b, and right-handed in b decays. This is due to the fact that the dominant contribution comes from the () e = 16π chiral-odd dipole operator O7γ 2 mb s L(R) σμν bR(L) . As only left-handed quarks participate in weak interaction, this effective operator induces a helicity flip on one of the external quark lines, which results in a factor mb for bR → sL γL , and a factor m s for bL → sR γR . Hence, the emission of right-handed photons is suppressed by a factor m s /mb . This suppression can be lifted in some NP models where the helicity flip occurs on an internal line, which brings in a factor mNP /mb instead of m s /mb . If the amplitude for b → sγR is of the same order as the SM prediction, or the enhancement of b → sγR goes along with the suppression of b → sγL , the impact on the branching ratio is small since the two helicity amplitudes add incoherently. This implies that there can Email address: [email protected] (Andrey Tayduganov)
0920-5632/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2012.12.008
be a substantial contribution of NP to b → sγ escaping detection when only branching ratios are measured. Therefore, the photon polarization measurement could provide a good test of the SM or at least a useful indication of NP. 2. Various methods for determination of C7γ
The b → sγ process can be described by the effective Hamiltonian1 , 4G F (μb )O7γ (μb ) , Heff = − √ Vtb Vts∗ C7γ (μb )O7γ (μb ) + C7γ 2 (1) ) where the short-distance Wilson coefficient C7γ (C7γ describes the left(right)-handed photon emission ampli /C7γ m s /mb 0.02. tude of b → sγ. In the SM, C7γ Three methods have been proposed for the measurement of the photon polarization or equivalently the ratio /C7γ : C7γ 1 The short-distance QCD effects induce the mixing of O () with 7γ the other operators which we omitted writing in Eq. (1). This effect can be absorbed by defining the so-called “effective” coefficients () eff () . For notational simplicity, whenever C7γ appears in what folC7γ () eff lows, C7γ , evaluated at the scale μb mb, pole = 4.8 GeV, will be understood.
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A. Tayduganov / Nuclear Physics B (Proc. Suppl.) 234 (2013) 173–176
• An indirect determination of the photon polarization, proposed by Atwood et al. [1], is the measurement of the time-dependent mixing-induced CPasymmetry in the radiative neutral B-mesons decays such as B → K ∗ (→ KS π0 )γ, and Bs → φγ, which is determined by the S parameter, S
2Im[e−iφM C7γ C7γ | 2 |C7γ |2 + |C7γ |
,
(2)
where φ M is the phase in the B − B mixing, which in the SM is φd = 2β 43◦ , and φ s 0, for the Bd and Bs mixing, respectively. Such measurements are expected to be made at future super B factories, reduce the experimental error on the asymmetry parameter S K ∗ γ down to 2%. • A direct determination, proposed by Gronau et al. [2], is based on the study of the angular distribution of the three-body final state, Kππ, coming from the axial vector K1 (1+ )-meson decay, in B → K1 (→ Kππ)γ, and extraction of the polarization parameter λγ , λγ
2 |C7γ | − |C7γ |2 2 |C7γ | + |C7γ |2
.
(3)
In Ref. [3] this method was improved by using a new variable ω, which includes not only the angular dependence but also the dependence on the three-body Dalitz variables which can significantly improve the sensitivity of the measurement of the polarization parameter. Recently measured by the Belle collaboration B(B → K1 (1270)γ) appeared to be comparable to B(B → K ∗ γ), which opened the possibility of measuring the photon polarization in B → K1 γ. • Another indirect way to study the right-handed currents is based on the angular analysis in the semileptonic B → K ∗ (→ Kπ)+ − decay2 . In par2 ticular, two transverse asymmetries, A(2) T (q ) and (im) 2 AT (q ), introduced in Refs. [5, 6], are highly sensitive to the b → sγ process at very low dilepton invariant mass squared q2 = (p+ + p− )2 , 2 lim A(2) T (q ) =
q2 →0
lim
q2 →0
2 A(im) T (q )
=
∗ ] 2Re[C7γ C7γ 2 |C7γ |2 + |C7γ | ∗ ] 2Im[C7γ C7γ 2 |C7γ |2 + |C7γ |
, (4) .
2 In our analysis we do not consider the pollution by the B → K0∗ (→ Kπ)+ − events [4].
These three methods, having their own advantages and disadvantages, can be complementary to each other. Combination of all three of them can in principle put a () coefficients strong constraint on the short-distance C7γ in a model-independent way which then can be used as a constraint in building the NP models. 3. Constraints on C7γ combining various methods of the photon polarization determination by B(B → X s γ) and 3.1. Current constraint on C7γ S K∗ γ In Fig. 1 we show the constraints on C7γ /C7γ available at present and compare them with those that are planned to be obtained from the future measurements. For illustration, we consider two NP scenarios3 : (NP) (NP) ∈ R, C7γ ∈R; • scenario I: C7γ (NP) (NP) = 0, C7γ ∈C; • scenario II: C7γ
In all plots presented in Figs. 1–3, we use the constraint from the inclusive rate. The region outside the gray (dark gray) circle is excluded at 3σ (1σ) level by the current measurement [8], Bexp (B → X s γ) = (3.55 ± 0.24) × 10−4 ,
(5)
which we combined with the SM prediction given in Ref. [9]. In Fig. 1 we show the constraints from already measured B(B → X s γ) and S K ∗ γ . Orange (dark orange) region represents the ±3σ (±1σ) region allowed by the current measurement of S K ∗ γ [8], exp
S K ∗ γ = −0.16 ± 0.22 .
(6)
Performing a χ2 -fit of B(B → X s γ) and S K ∗ γ , we obtain for each considthe 95% and 68% CL regions for C7γ ered NP scenario. One can see from the plots in Fig. 1, that there is still room for NP. Note, however, the ap parent ambiguities in the C7γ − C7γ plane: in scenario I it is fourfold in the C7γ − Cγ plane and twofold in the ] − Im[C7γ ] plane in scenario II, respectively. Re[C7γ Therefore, it is clear that additional observables are re() . quired to pin down the real and imaginary parts of C7γ
3 For
more tested scenarios and details see Ref. [7]
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= 0,
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∈ C (b). The light and dark blue regions
(NP) correspond respectively to the 95% and 68% CL bounds for C7γ ,
obtained from the χ2 -fit of the present measurements of B(B → X s γ) and S K ∗ γ .
C7 ΓNPC7SM Γ
1
Figure 1: Current constraints from the combination of the inclusive decay rate and the mixing-induced CP-asymmetry in B → K ∗ (→ (NP) (NP) KS π0 )γ in particular NP scenarios where C7γ ∈ R, C7γ ∈ R
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() in the NP sceFigure 2: Prospect of the future constraints on C7γ
3.2. The expected sensitivity to C7γ in the future measurements In Figs. 2 and 3, we present a future prospect for con() (NP) straining C7γ in the two NP scenarios. The plots are obtained by assuming:
• Improved measurement of the CP-asymmetry parameter S K ∗ γ in B → K ∗ (→ KS π0 )γ. The contour colours correspond to S K ∗ γ allowed by a ±3σ to the present world average (6). Different colours are separated by the size of the current experimental error. That error will be significantly reduced at super B factories. • Potential measurement of the polarization parameter λγ in B → K 1 (1270)γ. The contour colours correspond to λγ ∈ [−1, 1]. The spacing between contours is taken to be σλγ = 0.2, which may be improved by the study of K1 → Kππ decays. That can be made using a detailed experimental study of B → K1 ψ decay. • Potential non-zero measurement of two transverse ∗ im) asymmetries, A(2, ∈ [−1, 1], in B → K + − . T (2) The contours correspond to AT (0) and A(im) T (0) respectively. The interval between the lines represents 20% of uncertainty for each, which, in principle, can be achieved at LHCb. Note that in all these figures we applied the constraint from the measured B(B → X s γ) as allowed by a ±3σ error to the central value (5). In Fig. 2, we present our result for the scenario I. The constraints from S K ∗ γ and A(2) T look very similar in this
(NP) (NP) nario I: C7γ and C7γ are both real. The contour colours in
Fig. (a, b, c) correspond respectively to S K ∗ γ , λγ and A(2) T (0) allowed by a ±3σ error to the central value of Bexp (B → X s γ).
scenario since both of them are proportional to
C7γ C7γ 2 +C 2 C7γ 7γ
() (NP) with C7γ being real numbers. On the other hand, one can see that the shape of the constraint from λγ is quite different. For example, the fourfold ambiguity in the constraints of S K ∗ γ and A(2) T can be reduced to a twofold with the help of the λγ measurement. In addition, one (NP) (NP) observes that the region around the line C7γ = C7γ is quite sensitive to the λγ values, while it is not in the case of S K ∗ γ and A(2) T .
In Fig. 3, we present our result for the scenario II. The constraint from S K ∗ γ is very strong (indeed, assuming that fact that the experimental error will be significantly reduced by super B factories down to 2%, the bound, i.e. the spacing between the adjacent contours will become about 10 times more narrow than those depicted in Fig. 3(a)) but it has an ambiguity along the diagonal. This problem can be partially solved by adding a constraint from λγ which is a circle since λγ is a function /C7γ |2 and therefore is insensitive to the comof |C7γ plex phases. The SM prediction corresponds to the cen(NP) (NP) = C7γ = (0, 0). Near the center tral point C7γ −1, and the sensitivity to C7γ is very low. λγ = λSM γ (NP) (SM) /C7γ | 0.3 (i.e. one is For λγ −0.8 we have |C7γ clearly outside the SM prediction), but inside the circle one cannot distinguish the NP contribution from the SM one.
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() in the NP sceFigure 3: Prospect of the future constraints on C7γ (NP) (NP) nario II: C7γ is purely SM-like, i.e. C7γ = 0. The contour colours
in Fig. (a, b, c, d) correspond respectively to S K ∗ γ , λγ , A(2) T (0) and exp (B → A(im) (0) allowed by a ±3σ error to the central value of B T X s γ). 2 The combined measurement of A(2) T (q ) and 2 A(im) T (q ) can, in principle, constraint both Re[C 7γ /C 7γ ] and Im[C7γ /C7γ ] independently on S K ∗ γ and λγ . In contrast to λγ , it is also sensitive to the SM prediction. A pleasant feature of Figs. 2,3 is that the shapes of the resulting plots are quite different in NP scenarios. The four constraints will overlap in scenarios compatiim) and we will be ble with measured S K ∗ γ , λλ and A(2, T () and their phases. In incompatible able to extract C7γ scenarios, the four constraints will not overlap. /C7γ Once again, we stress that we can determine C7γ
from S K ∗ γ only in combination with the B − B mixing phase, φ M . In this paper we assume that NP does not bring any significant contribution to the B − B mixing box diagrams and use the currently measured value, sin 2β = 0.673 ± 0.023 [10]. 4. Conclusions We have studied the prospects for determining the from the future meaWilson coefficients C7γ and C7γ probes surements at LHCb and super B factories. C7γ the right-handed structure of the New Physics models which enter the b → sγ processes. In order to deter mine C7γ , we have used four observables: • The mixing-induced CP-asymmetry parameter S K ∗ γ in B → K ∗ (→ KS π0 )γ.
In principle, these four observables can unambiguously constrain the New Physics contribution to C7γ and C7γ , even when these Wilson coefficients are complex numbers. () 0 and We studied different NP scenarios of C7γ presented the current constraints provided by B(B → X s γ) and S KS π0 γ . Those constraints are still either loose and/or ambiguous. We then showed that the future mea(im) will not only surements of S K ∗ γ , λγ , A(2) T and AT () restrain the allowed range of values for C7γ , but also solve or partially solve the ambiguities in the complex ) plane. (C7γ , C7γ We should emphasize that each of the above quantities has its own advantages and disadvantages depending on the NP scenario. In the scenario I, we found that the bounds coming from S K ∗ γ and from A(2) T are similar. To disentangle the discrete fourfold ambiguity arising from these two constraints, the measurement of λγ could help and reduce this ambiguity to twofold. In the scenario II, λγ plays an important role: although S K ∗ γ bound will be extremely constraining at super B factories, the resulting diagonal ambiguity could be at least partly solved by a constraint provided by (im) are very important the measured λγ . A(2) T and AT since their combination can, in principle, constrain both ] and Im[C7γ ] independently on S K ∗ γ and λγ . Re[C7γ References [1] D. Atwood, M. Gronau and A. Soni, Phys. Rev. Lett. 79 (1997) 185 [hep-ph/9704272]. [2] M. Gronau, Y. Grossman, D. Pirjol and A. Ryd, Phys. Rev. Lett. 88 (2002) 051802 [hep-ph/0107254]. [3] E. Kou, A. Le Yaouanc and A. Tayduganov, Phys. Rev. D 83 (2011) 094007 [arXiv:1011.6593 [hep-ph]]. [4] D. Becirevic and A. Tayduganov, arXiv:1207.4004 [hep-ph]. [5] F. Kruger and J. Matias, Phys. Rev. D 71 (2005) 094009 [hepph/0502060]. [6] D. Becirevic and E. Schneider, Nucl. Phys. B 854 (2012) 321 [arXiv:1106.3283 [hep-ph]]. [7] D. Becirevic, E. Kou, A. Le Yaouanc and A. Tayduganov, JHEP 1208 (2012) 090 [arXiv:1206.1502 [hep-ph]]. [8] D. Asner et al. [Heavy Flavor Averaging Group Collaboration], arXiv:1010.1589 [hep-ex]. [9] A. L. Kagan and M. Neubert, Eur. Phys. J. C 7 (1999) 5 [hepph/9805303]. [10] K. Nakamura et al. [Particle Data Group Collaboration], J. Phys. G G 37 (2010) 075021.