Supersymmetric contribution to B to rho K and B to pi K^* decays in SCET

We analyze the supersymmetric contributions to the direct CP asymmetries of the decays $B \to \pi K^*$ and $B\to\rho K$ within Soft Collinear Effective Theory. We extend the Standard Model analysis of these asymmetries to include the next leading order QCD corrections. We find that, even with QCD correction, the Standard Model predictions can not accommodate the direct CP asymmetries in these decay modes. Using Mass Insertion Approach, we show that non-minimal flavor SUSY contributions mediated by gluino exchange can enhance the CP asymmetries significantly and thus can accommodate the experimental results.


INTRODUCTION
In the standard model (SM), Charge conjugation Parity (CP) violation and flavour transition arise from the complex Yukawa couplings in the Cabibbo Kobayashi Maskawa (CKM) matrix. The effect of this phase has been first observed in kaon system and confirmed in B decays. However, the expected CP asymmetries in some decay channels for B meson are in contradiction with the experimental measurements carried by Babar and Belle B-factories and proton antiproton collider as Tevatron, with its experiments CDF and D0. The largest discrepancy has been observed in the decay B → Kπ where the world averages for the CP asymmetries of B 0 → K ± π ∓ and B ± → K ± π 0 are given by [1]: A CP (B ± → K ± π 0 ) = 0.050 ± 0.025. (2) which implies that In the SM and using QCD factorization approach, the results of the above two asymmetries read [2]: A CP (B ± → K ± π 0 ) = 7.1 +1.7+2.0+0.8+9.0 −1.8−2.0−0.6−9.7 % (4) A CP (B 0 → K ± π ∓ ) = 4.5 +1.1+2.2+0.5+8.7 −1.1−2.5−0.6−9.5 % , where the first error corresponds to uncertainties on the CKM parameters and the other three errors correspond to variation of various hadronic parameters. These results imply that ∆A QCD CP = 0.025±0.015, which differs from the experimental value by 3.5σ and thus motivate exploring new physics beyond SM.

arXiv:1101.2710v2 [hep-ph] 23 Oct 2011
The decay modes B → πK * and B → ρK are generated at the quark level in the same way as B → Kπ and hence it is interesting to explore hints of New Physics (NP) in these decays. These decay modes are studied within SM in framework of QCDF [2], PQCD [3][4][5][6] and Soft Collinear Effective Theory (SCET) [7]. A detailed comparison between the results for the branching ratios and CP asymmetries in these different factorizations methods can be found in Ref. ([7]). The comparison showed that PQCD results for most B → πK * and B → ρK channels are much larger than SCET results. On the other hand the QCDF results are small and comparable with SCET results but with a relative minus sign. Moreover, in SCET, the direct CP asymmetries of B − → π −K * 0 and B − → ρ −K 0 are zero while the CP asymmetries in other channels are small. Recently, in Ref. ([8]) fits to B → πK * and B → ρK decays are performed where data can be accommodated within the standard model due principally to the large experimental uncertainties, particularly in the CP-violating asymmetries.
One of the four large experiments operating at the Large Hadron Collider (LHC) is LHCb. The main task of the LHCb is to measure precisely the CP asymmetries in B meson decays. These measurements are so important to test the different mechanisms proposed by many models beyond SM to explain the matter-antimatter asymmetry. This test can be regarded as an indirect search for physics beyond SM.
Supersymmetry (SUSY) is one of the most interesting candidates for physics beyond the standard model as it naturally solves the hierarchy problem. In addition, SUSY has new sources for CP violation which can account for the baryon number asymmetry and affect other CP violating observables in the B and K decays. The effects of these phases on the CP asymmetries in semi-leptonic τ decays has been studied in Refs.( [9][10][11]).
In this paper, we analyze the SUSY contributions to the CP asymmetries of the B → πK * and B → ρK decays in the framework of SCET [12][13][14][15]. SCET is an effective field theory describing the dynamics of highly energetic particles moving close to the light-cone interacting with a background field of soft quanta [16]. It provides a systematic and rigorous way to deal with the decays of the heavy hadrons that involve different energy scales. The scaling of fields and momenta in SCET depends on a small parameter λ. Generally λ is defined as the ratio of the smallest and the largest energy scales in the given process. Then, the SCET Lagrangian and effective Hamiltonian are expanded in terms of λ that help to reduce the complexity of the calculations. In addition, the factorization formula provided by SCET is perturbative to all powers in α s expansion.
This paper is organized as follows. In Sec. II, we briefly review the decay amplitude for B → M 1 M 2 within SCET framework. Accordingly, we analyze the CP asymmetries and branching ratios for B → πK * and B → ρK within SM in Sec. III. In Sec. IV, we discuss the SUSY contributions to the CP asymmetries of the B → πK * and B → ρK decays. We give our conclusion in Sec. V.
The amplitude of B → M 1 M 2 where M 1 and M 2 are light mesons in SCET can be written as follows Here A LO B→M1M2 denotes the leading order amplitude in the expansion 1/m b , A χ B→M1M2 denotes the chirally enhanced penguin amplitude, A ann B→M1M2 denotes the annihilation amplitude and A c.c B→M1M2 denotes the long distance charm penguin contributions. In the following we give a brief account for each amplitude.
A. Leading order amplitude At leading power in (1/m b ) expansion, the full QCD effective weak Hamiltonian of the ∆ B = 1 decays is matched into the corresponding weak Hamiltonian in SCET I by integrating out the hard scale m b .
Then, the SCET I weak Hamiltonian is matched into the weak Hamiltonian SCET II by integrating out the hard collinear modes with p 2 ∼ Λm b and the amplitude of the ∆ B = 1 decays at leading order in α s expansion can be obtained via [17]: At leading order in α s expansion, the parameters ζ B ( are given by [19] c (f ) 1 N C 9,10 + C 10,9 + ∆c , and b (f ) where ω 2 = m b u and ω 3 = −m bū . u andū = 1 − u are momentum fractions for the quark and antiquark where µ M is the chiral scale parameter generate the so called Chirally enhanced penguins amplitude A χ B→M1M2 [18]. µ M for kaons and pions can be of order (2GeV ) and therefore chirally enhanced terms can compete with the order α s (µ h )(Λ/m b ) terms. The chirally enhanced amplitude for B → M 1 M 2 decays is given by [18] The factors µ M are generated by pseudoscalars and so they vanish for vector mesons [18]. The pseudoscalar light cone amplitude φ M pp (u) is defined as [24,25] φ M pp are commonly expressed in terms of the first few terms in the Gegenbauer series As before, following the same procedure for treating ζ BM J (z) we take ζ BM χ (z) as [18] ζ BM The hard kernels R K , R π , R J K , R J π , R χ K and R χ π can be expressed in terms of Clebsch-Gordan coefficients for the different final states as [18] Summation over q = u, d, s is implicit and c χ i and b χ i are expressed in terms of the short-distance Wilson coefficients as [18] The ∆c χ i and ∆b χ i terms denote perturbative corrections that can be found in Ref. ([18]).

C. Annihilation amplitudes
Annihilation amplitudes A ann B→M1M2 have been studied in PQCD and QCD factorization in Refs.( [26][27][28][29]). Within SCET, the annihilation contribution becomes factorizable and real at lead- [30]. In our numerical calculation, we do not include the contributions from penguin annihilation as their size is small and contains large uncertainty compared to the other contributions [18,24].

D.
Long distance charm penguin amplitude The long distance charm penguin amplitude A c.c B→M1M2 is given as follows where δ cc is the strong phase of the charm penguin. The modulus and the phase of the charm penguin are In this section, we analyze the SM contribution to the CP asymmetries and the branching ratios for B → πK * and B → ρK decays. We follow ref. [18] and work in the next leading order of α s expansion.  The decay modes B → πK * and B → ρK are generated at the quark level via b → s transition and thus we can decompose their amplitudes A according to the unitarity of the CKM matrix as Here    [1]. The first uncertainty in the predictions is due to the uncertainties in SCET parameters while the second uncertainty is due to the uncertainties in the CKM matrix elements.
Turning now to the second case where A tree u = 0, to a good approximation we can write which shows also that the long distance charm penguin gives the dominant contribution to the amplitude, as A tree u will be suppressed by a factor |λ s u | ∼ 0.02|λ s c | in comparison to A cc c . Thus in all cases the long distance charm penguin gives the dominant contribution and as a consequence the amplitude in each decay mode will be of the same order of the long distance charming penguin amplitude.
For decay modes which do not receive contribution from charm penguin one expects very small branching ratios. Hence non-perturbative charming penguin plays crucial rule in the branching ratios using SCET.
The branching ratios of the decay modes B → πK * and B → ρK are given in Tables I where the first uncertainty in the predictions is due to the uncertainties in SCET parameters while the second uncertainty is due to the uncertainties in the CKM matrix elements. As can be seen from that Table, within SM, the branching ratios are in agreements with their corresponding experimental values in most of the decay modes.
Turning now to the SM predictions for the CP asymmetries which are presented in Table II where, as before, where the first uncertainty in the predictions is due to the uncertainties in SCET parameters while the second uncertainty is due to the uncertainties in the CKM matrix elements . Clearly from the Table, the SM predictions for the CP asymmetries of B + → π 0 K * + has different sign in comparison with the experimental measurement and the predicted CP asymmetries in many of the decay modes are in agreement with the experimental measurements due to the large errors in these measurements. Moreover, we see from the Table that, the predicted CP asymmetry ofB → π 0K * 0 and B + → ρ 0 K + disagree with the experimental results within 1σ. error of the experimental data. This can be attributed to the lack of the weak CP violating phases as SM Wilson coefficients are real and the only source of the weak phase is the phase of the CKM matrix.
Note, SCET provides large strong phases and thus with new sources of weak CP violation one would expect enhancement in these asymmetries. In the next section we consider the case of SUSY models with non universal A terms where new sources of weak CP phases exist.
∆ IJ LL = ∆ JI LL and ∆ IJ RR = ∆ JI RR , but no such relation holds for ∆ LR . It is often to set (m 2 wherem is the average sfermion mass. The Flavour Changing structure of the A − B sfermion propagator is exhibited by its non-diagonality and it can be expanded as where a, b = (1, 2, 3) are flavor indices and I is the unit matrix. It is convenient to define a dimensionless quantity (δ f AB ) ab ≡ (∆ f AB ) ab /m 2 . As long as (∆ f AB ) ab is smaller thanm 2 we can consider only the first order term in (δ f AB ) ab of the sfermion propagator expansion. The parameters (δ f AB ) ab can be constrained through vacuum stability argument [33], experimental measurements concerning FCNC and CP violating phenomena [34]. Recent studies about other possible constraints can be found in Refs.( [35][36][37]). At next leading order in α s expansion, the dominant SUSY contributions to our decay modes are originated from diagrams mediated by the exchange of gluino and chargino. The complete expressions for the gluino and chargino contributions to the Wilson coefficients can be found in Refs. [34,[38][39][40].
After including SUSY contributions to the mentioned decays and keeping the dominant terms we find The mass insertions (δ u RL ) 32 and (δ u LR ) 32 are not constrained by b → sγ and so we can set them as (δ u RL ) 32 = (δ u LR ) 32 = e iδu where δ u is the phase that can vary from −π to π. It should be noted that in order to have a well defined Mass Insertion Approximation scheme, it is necessary to have |(δ f AB ) ab | < 1 but here in order to maximize the SUSY CP-violating contributions we take it of order one. Applying b → sγ constraints leads to the following parametrization [41] In the following we present our results for the CP asymmetries. In our analysis we consider two scenarios, the first one with a single mass insertion where we keep only one mass insertion per time and take the other mass insertions to be zero and the second scenario with two mass insertions will be considered only in the cases when one single mass insertion is not sufficient to accommodate the experimental measurement. After setting the different mass insertions as mentioned above, we see from Eq.(24) that, the terms that contain the mass insertions (δ u RL ) 32 and (δ u LR ) 32 will be small in comparison with the other terms and thus we expect that their contributions to the asymmetries will be small.
These terms are obtained from diagrams mediated by the chargino exchange and thus we see that gluino contributions give the dominant contributions as known in the literature.
We start our analysis of the direct CP asymmetries by considering the first scenario in which we take only one mass insertion corresponding to the gluino mediation and set the others to be zero. The left diagram corresponds to ACP (B + → π +K * 0 ) while the right diagram corresponds to ACP (B + → π 0 K * + ). In both diagrams we take only one mass insertion per time and vary the phase of from −π to π. The horizontal lines in both diagrams represent the experimental measurement to 1σ. The left diagram corresponds to ACP (B 0 → π 0K * 0 ) while the right diagram corresponds to ACP (B 0 → π − K * + ). In both diagrams we take only one mass insertion per time and vary the phase of from −π to π. The horizontal lines in both diagrams represent the experimental measurement to 1σ.
After substituting the mass insertions given in eq.(25) in eq.(24) we find that the first and third terms in the amplitudes B + → π +K * 0 and B + → π 0 K * + will be approximately equal and both of them will be smaller than the second term. As a consequence, one predicts that the asymmetries generated by the mass insertions (δ d LL ) 23 and (δ d RL ) 23 will be equal and in the same time these asymmetries will be smaller than the case of using (δ d LR ) 23 which can be seen from Fig.1. In that Figure,we plot the CP asymmetries, A CP (B + → π +K * 0 ) and A CP (B + → π 0 K * + ) versus the phase of the (δ d AB ) 32 where A and B denote the chirality i.e. L and R. for 3 different mass insertions. The horizontal lines in both diagrams represent the experimental measurements to 1σ. As can be seen from Figure1 left, for all gluino mass insertions, the value of the CP asymmetry A CP (B + → π +K * 0 ) is enhanced to accommodate the experimental measurement of the asymmetry within 1σ for many values of the phase of the mass insertions. On the other hand, Figure 1right shows that the CP asymmetry A CP (B + → π 0 K * + ) is enhanced to accommodate the experimental measurement within 1σ for all values of the phase of the mass insertions. The point we stress here is that SUSY Wilson coefficients provide source of large weak phases, which are needed for accommodation of CP asymmetries.
In Fig.2 we plot the two asymmetries, A CP (B 0 → π 0K * 0 ) and A CP (B 0 → π − K * + ) versus the phase of the (δ d AB ) 32 as before. As can be seen from Fig.2 left, A CP (B 0 → π 0K * 0 ) lies within 1σ range of its experimental value for many values of the phase of the mass insertion (δ d LR ) 23 only. The reason for that is as before,(see eq.(24)) the two mass insertions (δ d LL ) 23 and (δ d RL ) 23 will give equal contributions to the CP asymmetries which will be smaller than the case of using (δ d LR ) 23 . On the other hand, Fig.2 right, we see that A CP (B 0 → π − K * + ) can be accommodated within 1σ for many values of the phase of the three gluino mass insertions.
Finally we discuss the CP asymmetries of the decay modes B + → ρ + K 0 and B + → ρ 0 K + . After substituting the mass insertions given in eq.(25) in eq.(24), we find that the first and third terms in the amplitudes B + → ρ + K 0 and B + → ρ 0 K + will be no longer equal as previous cases and thus we expect their contributions to the asymmetries will be different which can be seen from Fig.(3) where, as before, we plot A CP (B + → ρ + K 0 ) and A CP (B + → ρ 0 K + ) versus the phase of the (δ d AB ) 23 . In Fig.(3) we do not show the horizontal lines representing the 1σ range of the experimental measurement as the three In both diagrams we assume that the two mass insertion have equal phases and we vary the phase from −π to π. The horizontal lines in both diagrams represent the experimental measurements to 1σ.
curves of the A CP (B + → ρ + K 0 ) corresponding to the three gluino mass insertions totally lie in this 1σ range for all values of the phase of the mass insertions. On the other hand, Fig.3 right, we see that A CP (B + → ρ 0 K + ) can not be accommodated within 1σ for any value of the phase of all gluino mass insertions. This motivates us to consider the second scenario with two mass insertions.
In Fig.4, we plot the CP asymmetry, A CP (B + → ρ 0 K + ) versus the phase of the mass insertion for 2 different mass insertions. The left diagram correspond to gluino contributions where we keep the two mass insertions (δ d LR ) 23 and (δ d RL ) 23 and set the other mass insertions to zero. The right diagram correspond to both gluino and chargino contributions where we keep the two mass insertions (δ d LR ) 23 and (δ u RL ) 32 and set the other mass insertions to zero. In both diagrams we assume that the two mass insertion have equal phases and we vary the phase from −π to π. As before, the horizontal lines in both diagrams represent the experimental measurement to 1σ. As can be seen from Fig.4 left, two gluino mass insertions can not accommodate the experimental measurement for any value of the phase of the mass insertion. On the other hand from Fig.4 right, two mass insertions one corresponding to chargino contribution and the other corresponding to gluino contribution can not accommodate the experimental measurements. We find that in order to accommodate the CP symmetry in this case the Wilson coefficient Cg 9 should be increased at least by a factor −6π/α without violating any constraints on the SUSY parameter space.
We show the corresponding diagram in Fig.5.  23 and set the other mass insertions to zero. We assume that the two mass insertion have equal phases and we vary the phase from −π to π. The horizontal lines in the diagram represent the experimental measurements to 1σ.

V. CONCLUSION
Within Soft Collinear Effective Theory, we extend the Standard Model analysis of the B → πK * and B → ρK asymmetries to include the next leading order QCD corrections. We find that, even with QCD correction, the Standard Model predictions can not accommodate the direct CP asymmetries in these decay modes.
We have analyzed the SUSY contributions to the direct CP asymmetries of the decay modes B → ρK and B → πK * using the Mass Insertion Approximation. Contrarily to SM, our results show that these direct CP asymmetries can be significantly enhanced by the SUSY contributions mediated by gluino exchange and thus accommodate the experimental results.