1.Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China 2.Department of Physics, Yantai University, Yantai 264005, China 3.University of Chinese Academy of Sciences, Beijing 100049, China 4.Institute of Particle Physics, Central China Normal University, Wuhan 430079, China 5.Institute of Theoretical Physics, College of Applied Sciences, Beijing University of Technology, Beijing 100124, China Received Date:2018-12-17 Accepted Date:2019-04-11 Available Online:2019-07-01 Abstract:The doubly weak transition $b\to dd{\bar s}$ is highly suppressed in the Standard Model, which makes it a potential channel for exploring new physics signals. We present a study of the exclusive two-body wrong sign weak decay $\smash{\overline B}^0\to K^+\pi^-$ , which belongs to this class, in the perturbative QCD framework. We perform a model independent analysis for various effective dimension-6 operators for which large effects are possible. We further analyze the considered process in the Randall-Sundrum model, including the custodially protected and the bulk-Higgs Randall-Sundrum models. Exploring the experimentally favored parameter space of these models leads to a large and significant enhancement of the decay rate compared to the Standard Model, which might be accessible in future experiments. We propose to search for the wrong sign decay $\smash{\overline B}^0\to K^+\pi^-$ via flavor-tagged time-dependent analyses, which can be performed at LHCb and Belle-II.
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2.$ {{\overline B}^0 \to K^+ \pi^-} $ decay in the Standard ModelThe doubly weak decay $ b \to dd\bar s $ can only occur in SM through the box diagram, which is highly suppressed. The local effective Hamiltonian for the $ b \to dd\bar s $ transition is given by
with functions $ f(x) $ and $ g(x,y) $ given explicitly in [7], where $ x = m_W^2/m_t^2 $, $ y = m_c^2/m_W^2 $. The exclusive $ \smash{\overline B}^0\to K^+\pi^- $ decay, induced by the $ b \to dd\bar s $ transition, can only occur through the annihilation type Feynman diagrams. In the perturbative QCD framework (PQCD), according to the effective Hamiltonian, the four lowest order annihilation Feynman diagrams for $ \smash{\overline B}^0\to K^+\pi^- $ decay are shown in Fig. 1, where ($ a $) and ($ b $) are factorizable diagrams, while ($ c $) and ($ d $) are nonfactorizable. The initial $ b $ and $ \bar d $ quarks annihilate into $ d $ and $ \bar s $ quarks, which then form a pair of light mesons by hadronizing with another pair of $ u\bar u $ produced perturbatively through the one gluon exchange mechanism. Figure1. Feynman diagrams which contribute to the $ \smash{\overline B}^0\to K^+\pi^- $ decay to leading order, ($ a $) and ($ b $) are factorizable diagrams, ($ c $) and ($ d $) are nonfactorizable.
We consider $ \smash{\overline B}^0 $ meson at rest for simplicity. It is convenient to use light-cone coordinates ($ P^+, P^-, { k}_T $), which are defined as:
The $ \smash{\overline B}^0 $ meson momentum $ P_1 $ and the momenta of $ K^+ $ and $ \pi^- $ meson, denoted respectively by $ P_2 $ and $ P_3 $, can be written as
where the light meson masses have been neglected. In PQCD [27-31], the decay amplitude is factorized into soft ($ \Phi $), hard ($ H $) and harder ($ C $) dynamics characterized by different scales
where Tr denotes the trace over Dirac and color indices, and $ b_i $ are conjugate variables to $ { k}_{i T} $ of the valence quarks. The universal wave function $ \Phi_M(x_i,b_i) $ ($ M = B,K,\pi $), describing hadronization of the quark and anti-quark into a meson $ M $, can be determined in other decays. The explicit formulas are given in Appendix A. The factorization scale $ t $ is the largest energy scale in $ H $, as a function of $ x_i $ and $ b_i $. The Wilson coefficient $ C(t) $ results from the radiative corrections at short distances, which includes the harder dynamics at a scale larger than the $ m_B $ scale, and describes the evolution of the local 4-Fermi operators from $ m_W $ down to the $ t $ scale. Using the threshold resummation [32], the large doubly logarithms ($ \ln^2x_i $) are summed, leading to $ S_t(x_i) $ which suppresses the end-point contributions. $ e^{-S(t)} $, called the Sudakov form factor [33], contains resummation of two kinds of logarithms. One of the large logarithms is due to the renormalization of the ultra-violet divergence $ \ln tb $, and the other is a doubly logarithm $ \ln^2b $ arising from the overlap of collinear and soft gluon corrections. Such factors suppress effectively the soft dynamics, making the perturbative calculation of the hard part $ H $ reliable. By inserting the SM operator given in Eq. (1) into the vertices of each Feynman diagram, we calculate the hard part $ H $ to first order in $ \alpha_s $ and obtain the analytic formulas $ F_{a1} $ and $ {\cal M}_{a1} $, which represent the factorizable and nonfactorizable annihilation diagram contributions, respectively. The explicit expressions for $ F_{a1} $ and $ {\cal M}_{a1} $ are given in Appendix B. It is obvious that these annihilation type contributions are suppressed compared to the emission diagrams, which agrees with the helicity suppression argument. Finally, the total amplitude for the considered process in SM is given as
Obviously, this is well below the current experimental measurement abilities, so this channel can turn out to be an ideal probe of NP effects.
3.Model independent analysis of the $ {{\overline B}^0\to K^+\pi^-} $ decayIn this section we present a model independent analysis of the NP contributions to the exclusive decay $ \smash{\overline B}^0\to K^+\pi^- $. We start with the most general local effective Hamiltonian with all possible dimension-6 operators [3]
The $ \widetilde O_j $ operators represent the chirality flipped operators which can be obtained from $ O_j $ by $ L\leftrightarrow R $ exchange. In SM, only $ {\cal{O}}_1 $ is present. The new physics beyond SM can change the Wilson coefficient of operator $ O_1 $, and it can also provide non zero Wilson coefficients for the other operators. These Wilson coefficients are not free parameters, as all of them also contribute to $ K^0-\smash{\overline K}^0 $ and $ B^0-\smash{\overline B}^0 $ mixing. The experimentally well measured mixing parameters give stringent constraints on the Wilson coefficients. This is the main reason why in many new physics models one can not get large branching ratios for the $ b\to dd \bar s $ or $ b\to ss \bar d $ decays, see for example [34]. We first assume that NP only gives a contribution to the local operator $ {\cal{O}}_1 $ similar to SM, which is true for a large class of NP models including the two-Higgs doublet model (2HDM) with small tan $ \beta $, or the constrained minimal supersymmetry standard model (MSSM) [13]. The NP Hamiltonian for the $ b \to dd\bar s $ transition in this case is the same as for SM in Eq. (1), but with a new Wilson coefficient $ C_{1}^{dd\bar s} $. The decay width of $ \smash{\overline B}^0\to K^+\pi^- $ will also have the same formula as in the SM case. The corresponding Hamiltonians for $ K^0-\smash{\overline K}^0 $ and $ B^0-\smash{\overline B}^0 $ mixing are
where the coefficients in general have the form $ C_{1}^{dd\bar s}\sim \sqrt{C_{1}^{K} C_{1}^{B_d}} $. Since the SM results for $ K^0\text{-}\smash{\overline K}^0 $ and $ B^0\text{-}\smash{\overline B}^0 $ mixing are in good agreement with the experimental data, there is not much room left for the new physics contributions. Next, if the NP contributions come from the non-standard model chiralities, and considering each non-standard operator in $ {\cal{O}}_{j} $ individually, the decay amplitude is given by
where $ j = 2,3,4, 5 $. The explicit expressions for $ F_{a2,a3,a4,a5} $ and $ {\cal M}_{a2,a3,a4,a5} $ are given in Appendix B. Similarly, for $ \widetilde{\cal{O}}_{1-5} $, we have
We define the ratio $ R $ of the branching ratio of the wrong sign decay to the corresponding SM branching ratio of the right sign decay (induced by the $ b \to su\bar u $),
We consider the corresponding LO PQCD prediction for the right sign decay, whose amplitude is given by Eq. (22) of [28], while the numerical predictions of the factorizable and nonfactorizable amplitudes are listed in Table 1 of [28]. With a direct experimental measurement of the ratio $ R $, one can give a constraint for each individual Wilson coefficient of the new physics operator in Eq. (11). As an example, by considering each non-standard model NP operator, we give in Table 1 the upper bound on the corresponding Wilson coefficient by assuming that the current experimental precision can limit the ratio $ R $ to less than $ 0.001 $.
parameter
allowed range/ (GeV?2)
$ \widetilde C_1 $
$<1.1\times 10^{-7} $
$ C_2 $
$<6.3\times 10^{-9} $
$ \widetilde C_2 $
$<6.8\times 10^{-9} $
$ C_3 $
$<5.1\times 10^{-8} $
$ \widetilde C_3 $
$<5.3\times 10^{-8} $
$ C_4 $
$<4.9\times 10^{-9} $
$ \widetilde C_4 $
$<4.2\times 10^{-9} $
$ C_5 $
$<1.6\times 10^{-6} $
$ \widetilde C_5 $
$<7.3\times 10^{-7} $
Table1.Upper bounds on the Wilson coefficients of the non-standard new physics operators obtained by assuming an experimental precision of $ R<0.001 $.
23.1.NP with conserved charge
-->
3.1.NP with conserved charge
We start with an NP scenario that involves the exchange of NP fields carrying a conserved charge. A particular example of a scenario of this type is the $ R$-parity violating MSSM [13]. In this class, one can start with NP Lagrangian of a generic form [14]
$ \begin{split} {\cal{L}}_{\rm{flavor}} =& g_{b\to d}(\bar d \Gamma b)X+g_{d\to b}(\bar b \Gamma d)X+g_{s\to d}(\bar d \Gamma s)X\\&+g_{d\to s}(\bar s \Gamma d)X+\rm{h.c.}, \end{split} $
(19)
assuming that field $ X $ with mass $ M_X $ carries a conserved quantum number broken only by the above terms. Flavor changing operators are then obtained after integrating out field $ X $. In this example, one can consider the hierarchies in NP couplings such that the mixing constraints can be trivially satisfied, leaving the $ b \to dd\bar s $ transitions unbounded. To illustrate this point, we consider four scenarios. In scenario 1 (S1) and scenario 2 (S2), we consider that NP matches onto the local operators $ {\cal{O}}_1 $ and $ \cal{\widetilde{O}} $1, respectively, while in scenario 3 (S3) and scenario 4 (S4), NP matches onto the linear combination of local operators $ {\cal{O}}_4 $, $ \cal{\widetilde{O}} $4 and $ {\cal{O}}_5 $, $ \cal{\widetilde{O}} $5, respectively. As S2, S3 and S4 involve local operators with non-standard chiralities, it will be convenient to define the normalized matrix elements of these non-standard operators with respect to the SM operator $ {\cal{O}}_1 $.
where $ j = 2,3,4,5 $. Similarly for $ \cal{\widetilde{O}} $1-5, where we denote the ratio as $ {\widetilde r}_j $. Starting with S1 and S2, the NP Hamiltonians are given by $ [{\cal{H}}_{\rm{S1}}^{dd\bar s}] = C_1^{dd\bar s}\eta_{\rm{QCD}}{\cal{O}}_1 $, $[{\cal{H}}_{\rm{S2}}^{dd\bar s}] = $$ {\widetilde C}_1^{dd\bar s}\eta_{\rm{QCD}}\cal{\widetilde {O}} $1, where $ C_1^{dd\bar s}\!\! =\!\! {\widetilde C}_1^{dd\bar s}\!\! =\!\!\displaystyle\frac{1}{M_{X}^2}\big(g_{b\to d}g_{d\to s}^{\ast}\!+\! g_{s\to d}g_{d\to b}^{\ast}\big) $. As for the RG running of the coefficients $ C_1^{dd\bar s} $ and $ {\widetilde C}_1^{dd\bar s} $ from the weak scale $ m_W $ to $ m_b $, we suppose that there is no new particle within these scales, and that the evolution of both coefficients is the same as in QCD. The evolution factor to NLO is given by $ \eta_{\rm{QCD}}(m_b) = 0.87 $. For S3 and S4, the NP Hamiltonians and the corresponding Wilson coefficients can be written as $[{\cal{H}}_{\rm{NP}}^{dd\bar s}] = f_{j\rm{QCD}}\Big({C_j^{dd\bar s}{\cal{O}}_j}+$$ {\widetilde C}_j^{dd\bar s}{\cal{\widetilde {O}}}_j\Big) $ and $ C_j^{dd\bar s} = $$ 1/{M_{X}^2}(g_{b\to d}g_{d\to s}^{\ast}) $, ${\widetilde C}_j^{dd\bar s} = {1}/{M_{X}^2} $$ (g_{s\to d}g_{d\to b}^{\ast}) $, with $ j = 4, 5 $ for S3 and S4, respectively. Similarly to the previous case, we factor out NLO QCD corrections for the Wilson coefficients in S3 and S4. However, the situation is different here as the operators $ {\cal{O}}_4(\cal{\widetilde {O}}) $4 and $ {\cal{O}}_5(\cal{\widetilde {O}}) $5 mix under renormalization such that the RG evolution operator is a $ 2\times2 $ matrix, so that each Wilson coefficient gets a small mixing induced contribution at NLO, which we ignore. Therefore, keeping only the dominant contributions we obtain $ f_{4\rm{QCD}} = 2 $, while $ f_{5\rm{QCD}} = 0.9 $. Considering the corresponding Hamiltonians in all four scenarios, we get the bounds for $ K^0-\smash{\overline K}^0 $ and $ B^0-\smash{\overline B}^0 $ mixing
where $ j = 1 $ for S1 and S2, while S3 and S4 correspond to $ j = 4, 5 $, respectively. Considering the relations $ |C_{j}^{v}| = 1/(\Lambda_j^v)^2 $, $ |{\widetilde C}_{1}^{v}| = 1/(\Lambda_1^v)^2 $, where $ v = K, B_d $, for $ K^0\text{-}\smash{\overline K}^0 $ and $ B^0\text{-}\smash{\overline B}^0 $ mixing, respectively, the lower bounds on the NP scales $ \Lambda_j^K $ and $ \Lambda_j^{B_d} $ [35] are listed in Table 2, with Im$ (C_j^K) $ additionally constrained from $ \epsilon_K $.
operators
$K^0-\smash{\overline K}^0$
$B^0-\smash{\overline B}^0$
parameter
lower limit/TeV
parameter
lower limit/TeV
${\cal{O}}_1$
$\Lambda_1^K$
$1\times 10^3$
$\Lambda_1^{B_d}$
210
${\cal{O}}_2$
$\Lambda_2^K$
$7.3\times 10^3$
$\Lambda_2^{B_d}$
$1.2\times 10^3$
${\cal{O}}_3$
$\Lambda_3^K$
$4.1\times 10^3$
$\Lambda_3^{B_d}$
600
${\cal{O}}_4$
$\Lambda_4^K$
$17\times 10^3$
$\Lambda_4^{B_d}$
$2.2\times 10^3$
${\cal{O}}_5$
$\Lambda_5^K$
$10\times 10^3$
$\Lambda_5^{B_d}$
$1.3\times 10^3$
Table2.Lower bounds on the NP scales coming from $K^0-\smash{\overline K}^0$ and $B^0-\smash{\overline B}^0$ mixing [35].
To keep the $ b \to dd\bar s $ transition unbounded and large, we take as case-I $ g_{d\to s} = g_{b\to d} = 1 $, where the bounds in (21) can be trivially satisfied, if for instance $ g_{s\to d} = g_{d\to b} = 0 $. As mixing bounds do not constrain $ M_{X} $ in this case, the ratio $ R_X $ can be as large as possible. For example, after assuming that the NP scale $ M_{X} $ lies around the TeV scale, the obtained ratio $ R_X $ in each scenario is listed under case-I in Table 3. One can see that the resulting ratio $ R_X $, after satisfying the mixing constraints, is very large in each scenario compared to the SM result ($ R_{\rm{SM}} $). Therefore, the measurement of the ratio $ R_X $, with an upper limit for the $ \smash{\overline B}^0\to K^+\pi^- $ branching ratio if no signal is found, will provide the most stringent constraint on the operators in this example of NP.
scenarios
$ R_X $
$ R_{\rm{SM}} $
$ M_X $/TeV
case-I
$ M_X $/TeV
case-II
S1
1.0
0.085
10
$ 8.5\times 10^{-6} $
$ 6.8\times 10^{-15} $
S2
0.074
$ 7.3\times 10^{-6} $
S3
55
$ 0.005 $
S4
0.002
$ 1.9\times 10^{-7} $
Table3.Ratio of the branching fraction for the wrong sign decay to the branching fraction for the right sign decay, after satisfying the mixing constraints, in the case of NP involving conserved charge $ (R_X) $. $ \rm{S}1-\rm{S}4 $ represent NP scenarios corresponding to the presence of different NP operators (see text for details).
Further, we assume a higher NP scale at $ 10 $ TeV. The resulting PQCD prediction for the ratio $ R_X $ in corresponding scenarios is listed in Table 3 as case-II. In scenario S3, the predicted ratio, after satisfying the mixing constraints, is large enough to provide stringent constraint for the coefficient of the operators $ {\cal{O}}_4 $ or $ \cal{\widetilde{O}} $4. On the other hand, the obtained ratio $ R_X $ in (S1, S2) and S4 give nine and eight orders of magnitude increase, respectively, compared to the SM result, although it is very difficult to reach such a precision experimentally. As case-III, we consider the situation where accidentally one of the $ g_i $ couplings is very small. For example, if we take $ g_{s\to d} = 0 $ and all the other $ g_i = 1 $, then $ K^0-\smash{\overline K}^0 $ mixing bounds in (21) are trivially satisfied, while $ B^0-\smash{\overline B}^0 $ mixing yields a lower bound on $ M_{X} $. In this case, the bound on $ M_{X} $ is in each scenario more stringent compared to the previous cases. Therefore, the predictions for the ratio $ R $ obtained for scenarios S1, S2, S3 and S4 are of the order of $ 10^{-11} $, $ 10^{-12} $ and $ 10^{-16} $, respectively. However, compared to $ R_{\rm{SM}} $, as given in Table 3, S1 and S2 give four and three orders of magnitude enhancement, respectively. 23.2.NP scenarios with MFV and NMFV -->
3.2.NP scenarios with MFV and NMFV
In MFV [36], the coefficients can be considered as $ C_j^v = \displaystyle\frac{F_j^v}{[{(\Lambda_{\rm{MFV}})}_{_j}^{v}]^2} $, where $ v = dd\bar s, K, B_d $ for the $ b \to dd\bar s $ transition, and $ K^0-\smash{\overline K}^0 $ and $ B^0-\smash{\overline B}^0 $ mixing, respectively. First, we consider the class of MFV models in which $ \Delta F = 2 $ processes are strongly dominated by the same single operator with the $ (V-A)\times (V-A) $ structure [36], as in the low energy effective Hamiltonian of SM. All the other non-standard operators ($ {\cal{O}}_{2-5} $ and $ \widetilde{\cal{O}}_{1-5} $), if generated, are suppressed and do not contribute significantly. For instance, in the MFV effective theory [36], the non-standard dimension-6 FCNC operators for $ \Delta F = 2 $ processes involving external down-type quarks can be constructed with the bilinear FCNC structure of the form $ \bar D_R \lambda_d \lambda_{\rm{FC}} Q_L $, where $ \lambda_{\rm{FC}} $ represents the effective coupling for all FCNC processes with external down-type quarks. Therefore, the constructed operators will be suppressed by the Yukawa couplings ($ \lambda_{d_i} $) corresponding to the external masses. Example of this class of models are 2HDM II and MSSM, provided that $ \tan \beta $ (the ratio between the two vacuum expectation values of the two-Higgs doublet) is not too large. Therefore, in the MFV case with small/moderately large $ \tan \beta $, we have $ F_1^v = F_{\rm{SM}}^v $ and $ F_{j\neq1}^v = 0 $. $ F_{\rm{SM}}^v $ are the appropriate CKM matrix elements, such as $ F_{\rm{SM}}^{dd\bar s} = V_{tb}V_{td}^*V_{ts}V_{td}^* $, $ F_{\rm{SM}}^{K} = (V_{ts}V_{td}^*)^2 $ and $ F_{\rm{SM}}^{B_d} = (V_{tb}V_{td}^*)^2 $. In this case, the UTfit collaboration has given, from a global fit, a model independent lower bound on the MFV scale $ {(\Lambda_{\rm{MFV}})}_{_1} $ at 95% probability, such that $ {(\Lambda_{\rm{MFV}})}_{_1}>5.5\;(5.1) $ TeV for small (moderately large) $ \tan \beta $ [35]. We define the suppression scales that include the hierarchy of the NP induced flavor changing couplings, such that
With all parameters known, one observes that the suppression scale $ \Lambda_1^{dd\bar s} $ is the geometric average of the NP scales in $ K^0-\smash{\overline K}^0 $ and $ B^0-\smash{\overline B}^0 $ mixing, $ \Lambda_1^{dd\bar s} = \sqrt {\Lambda_1^K \Lambda_1^{B_d}} $. The resulting order of magnitude prediction for the ratio $ R({\cal{O}}_1) $ is $ 10^{-16} $ for small $ \tan \beta $ , while it is of the same order as the SM prediction for moderately large $ \tan \beta $. Next, we consider the class of MFV models, often called generalized MFV models (GMFV), in which the non-standard operators ($ {\cal{O}}_{2-5} $ and $ \widetilde{\cal{O}}_{1-5} $) are allowed to contribute significantly to the $ \Delta F = 2 $ processes. The above mentioned models, 2HDM II and MSSM, transform to GMFV models for very large $ \tan \beta $ [37]. Generally, in models with a two-Higgs doublet and very large $ \tan \beta $, the most prominent contributions from the non-standard operators to the $ \Delta F = 2 $ processes with external down-type quarks, come from the $ {\cal{O}}_{4} $ operator, which originates in the MFV effective theory as $ (\bar D_R \lambda_d \lambda_{\rm{FC}} Q_L) $$(\bar Q_L \lambda_{\rm{FC}} \lambda_d D_R) $ [36]. Although still suppressed by the Yukawa couplings ($ \lambda_{d_i} $), the corresponding $ {\cal{O}}_{4} $ operators in the case of $ B_q^0-\smash{\overline B}_q^0 $ mixing ($ q = d, s $), give non-negligible contributions for very large $ \tan \beta $, so that [36]
In this case, the UTfit collaboration obtained the lower bound on $ {(\Lambda_{\rm{MFV}})}_{_4} $, which represents the NP scale corresponding to the mass of non-standard Higgs bosons [35]
In order to give a prediction for the ratios $ R({\cal{O}}_4) $ and $ R(\cal{\widetilde{O}} $4), it is required to opt for a specific model to obtain the $ a_{0,1,2} $ factors, which represent the $ \tan \beta $-enhanced loop factors of $ \cal{O} $(1). However, irrespective of the particular values of $ a_{0,1,2} $ and $ \tan \beta $, the prediction of $ R({\cal{O}}_4) $ and $ R(\cal{\widetilde{O}} $4) will be smaller than $ R_{\rm{SM}} $ because of the suppressed Yukawa couplings in $ C_4^{dd\bar s} $ and $ {\widetilde C}_4^{dd\bar s} $. In NMFV [38], we have $ |F_j^v| = F_{\rm{SM}}^v $ with an arbitrary phase, so that the strict correlation between the Wilson coefficients is lost, and we have approximately $ \Lambda_j^{dd\bar s}\sim \sqrt {\Lambda_j^K \Lambda_j^{B_d}} $. Considering each operator $ {\cal{O}}_j $ separately, one can obtain the approximate value of the suppression scale $ \Lambda_j^{dd\bar s} $ by using the bounds from Table 2. The corresponding PQCD predictions for the ratio $ R({\cal{O}}_j) $, with $ j = 1,2,3,4,5 $, are of $ {\cal{O}}(10^{-12}) $, $ {\cal{O}}(10^{-13}) $, $ {\cal{O}}(10^{-14}) $, $ {\cal{O}}(10^{-14}) $ and $ {\cal{O}}(10^{-17}) $, respectively. Although the PQCD amplitudes for the $ {\cal{\widetilde{O}}}_{j} $ operators are slightly different, the predictions of the order of magnitude for the ratios $ R({\cal{\widetilde{O}}}_{j}) $ remain the same. The model-independent results of the UTfit group [35] suggested that the scale of heavy particles mediating tree-level FCNC in NMFV models must be heavier than $ \sim 60 $ TeV. An application of these results to the RS-type models [39] showed that the measured value of $ \epsilon_K $ implies that the mass scale of the lightest KK gluon must lie above $ \sim 21 $ TeV, if the hierarchy of the fermion masses and weak mixings is solely due to the geometry, and the 5D Yukawa couplings are anarchic and of $ \cal{O} $(1). A follow up study of the $ {\rm{RS}}_c $ model [40], while confirming the results of [39], pointed out that there exist regions in parameter space, without much fine-tuning of the 5D Yukawa couplings, which satisfy all $ \Delta F = 2 $ and EW precision constraints for the masses of the lightest KK gauge bosons, $ M_{\rm{KK}}\simeq3 $ TeV. Therefore, in a specific NMFV model the bounds, including the accidental cancellations of the contributions of different operators, can be weaker. We study the $ \smash{\overline B}^0\to K^+\pi^- $ decay in the RS-type models below. Moreover, while considering any specific NP model one must consider additional bounds, apart from those from $ K^0-\smash{\overline K}^0 $ and $ B^0-\smash{\overline B}^0 $ mixing. Finally, we assume that the NP scale is at the mass scale probed by $ K^0-\smash{\overline K}^0 $ mixing, such that the corresponding $ \Lambda_j^K $ are given in Table 2, and all flavor violating couplings are of $ {\cal{O}}(1) $. The resulting ratio $ R({\cal{O}}_1) $ is one order of magnitude larger than the SM prediction, while for all other cases, it is of the same order of magnitude as the SM value, or even smaller. Similarly, for the $ {\cal{\widetilde{O}}}_{j} $ operators, the order of magnitude of the predictions for $ R({\cal{\widetilde{O}}}_{j}) $ remains the same as for $ R({\cal{O}}_j) $.
4.$ {{\overline B}^0 \to K^+ \pi^-} $ in the custodial RS modelIn the RS model with custodial symmetry, we have a single warped extra dimension with the SM gauge symmetry group enlarged to the gauge group $SU(3)_c\times $$ SU(2)_L\times SU(2)_R\times U(1)_X\times P_{LR} $ [21-23]. In the $ {\rm{RS}}_c $ model, the $ \smash{\overline B}^0\to K^+\pi^- $ decay, occurring through the $ b\to dd\bar s $ transition, is mainly affected by the tree level contributions from the lightest KK excitations of the model, such as the KK gluons $ {\cal{G}}^{(1)} $, KK photon $ A^{(1)} $ and the new heavy EW gauge bosons $ (Z_H, Z^{\prime}) $, while in principle $ Z $ and the Higgs boson should also contribute. Since $ Zb_L{\bar b}_L $ coupling is protected by the discrete $ P_{LR} $ symmetry in order to satisfy the EW precision constraints, the tree-level $ Z $ contributions are negligible. Moreover, in the $ {\rm{RS}}_c $ model, $ \Delta F = 2 $ contributions from the Higgs boson exchanges are of $ {\cal O}(\upsilon^4/{M_{\rm{KK}}}^4) $ [41], which implies that the Higgs FCNCs have limited importance. For the $ \Delta F = 2 $ observables, the Higgs FCNCs provide the most prominent effects on the CP-violating parameter $ \epsilon_K $, but even in this case they are typically smaller compared to the KK-gluons exchange contributions [42]. Therefore, for the considered $ {\rm{RS}}_c $ setup [23], realizing the insignificance of the possible Higgs boson effects in the $ \Delta F = 2 $ processes, we ignore them in our study of the $ \smash{\overline B}^0\to K^+\pi^- $ decay. Furthermore, neglecting the corrections from the EW symmetry breaking, and the small $ SU(2)_R $ breaking effects on the UV brane, we assign a common name to the masses of the first KK gauge bosons
where the KK-scale, $ M_{\rm{KK}}\sim{\cal{O}}(\rm{TeV}) $, sets the mass scale for the low-lying KK excitations. The dominant contribution comes from KK gluons $ ({\cal{G}}^{(1)}) $, while the new heavy EW gauge bosons $ (Z_H, Z^{\prime}) $ can compete with it. The KK photon $ A^{(1)} $ gives a very small contribution. For the $ \smash{\overline B}^0\to K^+\pi^- $ decay, tree level contributions from the lightest KK gluons $ {{\cal{G}}^{(1)}} $, the lightest KK photon $ {A^{(1)}} $ and $ (Z_H, Z^{\prime}) $ lead to the following effective Hamiltonian
where the chosen operator basis is the same as in Eq. (11), and the Wilson coefficients correspond to $ \mu = {\cal{O}}(M_{g^{(1)}}) $. The Wilson coefficients are given by the sum
with $ j = 1,4,5 $ and $ i = VLL, VRR, LR, RL $. We point out that in the $ {{\rm{RS}}_c} $ model, compared to the similar $ K^0-\smash{\overline K}^0 $ and $ B^0-\smash{\overline B}^0 $ mixing processes [40], the $ \smash{\overline B}^0\to K^+\pi^- $ decay receives additional contributions from the $ {\cal{\widetilde{O}}}_{4} $ and $ {\cal{\widetilde{O}}}_{5} $ operators. Using the Fierz transformations, we calculate the contributions to the Wilson coefficients from KK gluons, denoted by $ [C_j^{i}(M_{g^{(1)}})]^{{\cal{G}}^{(1)}} $ in Eq. (27), to be
where $ p_{\rm{UV}} $ parameterizes the influence of brane kinetic terms on the $ SU(3)_c $ coupling. We set $ p_{\rm{UV}}\equiv1 $. Similarly, the contributions coming from the KK photon $ A^{(1)} $ and the new heavy EW gauge bosons $ (Z_H,Z^{\prime}) $, are given by
where the different flavor violating couplings $ \Delta_{L,R}^{db}(V) $ and $ \Delta_{L,R}^{ds}(V) $, with $ V = {\cal{G}}^{(1)} $, $ A^{(1)} $, $ Z^{(1)} $, $ Z_X^{(1)} $, are given in Ref. [40]. These couplings involve overlap integrals with the profiles of the zero mode fermions and shape functions of the KK gauge bosons. Similar to Fig. 1, we calculate the LO diagrams for the $ \smash{\overline B}^0\to K^+\pi^- $ decay in the $ {{\rm{RS}}_c} $ model in the PQCD framework. We adopt $ (F_{a1},F_{a4},F_{a5}) $ and $ ({\cal{M}}_{a1},{\cal{M}}_{a4},{\cal{M}}_{a5}) $ to stand for the contributions of the factorizable and nonfactorizable annihilation diagrams from the $ ({\cal{O}}_1,{\cal{\widetilde{O}}}_{1}), ({\cal{O}}_4,{\cal{\widetilde{O}}}_{4}) $ and $ ({\cal{O}}_5,{\cal{\widetilde{O}}}_{5}) $ operators, respectively. In PQCD, the Wilson coefficients are calculated at the scale $ t $, which is typically of $ {\cal{O}}(1-2) $ GeV. We employ the RG running of the WC's from the scale $ M_{g^{(1)}} $ to $ t $. The relevant NLO QCD factors are given in Ref. [43]. Moreover, one loop QCD and QED anomalous dimensions for the operator basis corresponding to the $ b\to ss\bar d $ and $ b\to dd\bar s $ transitions were derived recently [44]. Finally, the total decay amplitude for the $ \smash{\overline B}^0\to K^+\pi^- $ decay in the $ {{\rm{RS}}_c} $ model is given by
where the obtained expressions for the factorization formulas $ (F_{a1},F_{a4},F_{a5}) $ and $ ({\cal{M}}_{a1},{\cal{M}}_{a4},{\cal{M}}_{a5}) $ are given in Appendix B.
5.$ {{\overline B}^0\to K^+\pi^-} $ in the bulk-Higgs RS modelThe bulk-Higgs RS model is based on the 5D gauge group $ SU(3)_c\times SU(2)_V\times U(1)_Y $, where all the fields are allowed to propagate in the 5D space-time including the Higgs field [26]. For the $ \smash{\overline B}^0\to K^+\pi^- $ decay in the bulk-Higgs RS model, we consider contributions from the tree-level exchanges of KK gluons and photons, the $ Z $ boson and the Higgs boson, as well as from their KK excitations and the extended scalar fields $ \phi^{Z(n)} $, which are present in the model. Furthermore, in the bulk-Higgs RS model we consider the summation over the contributions from the whole tower of KK excitations, with the lightest KK gauge boson states having a mass $ M_{g^{(1)}}\approx2.45 $$ M_{\rm {KK}} $. We start with the most general local effective Hamiltonian as given in Eq. (10), containing all possible dimension-6 operators of Eq. (11), and calculate the Wilson coefficients at $ {\cal{O}}(M_{\rm{KK}}) $
where $ Q_d = -1/3 $, $ T_3^d = -1/2 $, and $ N_c = 3 $. $ \beta $ is a parameter of the model related to the Higgs profile. The Higgs and the scalar field $ \phi^Z $ give opposite contributions to the Wilson coefficient $ C_2 $, thus they cancel each other, giving $ C_2 = 0 $. Similarly, $ \widetilde C_2 = 0 $. The expressions for the required mixing matrices $ (\widetilde{\Delta}_{D(d)})_{13}\otimes(\widetilde{\Delta}_{D(d)})_{12} $ and $(\widetilde{\Omega}_{D(d)})_{13}\otimes $$ (\widetilde{\Omega}_{D(d)})_{12} $, in terms of the overlap integrals of boson and fermion profiles, are similar in the bulk-Higgs RS model to the ones given in [34,45], and are obtained by replacement of the flavors in the $ b \to dd\bar s $ transition. The effective Hamiltonian given in Eq. (10) is valid at $ {\cal{O}}(M_{\rm{KK}}) $, which must be evolved to a low-energy scale $ t $ in the PQCD formalism. Hence, for the evolution of the Wilson coefficients we use the formulas for the NLO QCD factors given in Ref. [43]. Similarly to the $ {{\rm{RS}}_c} $ model, the LO PQCD factorization calculation of the total amplitude for the $ \smash{\overline B}^0\to K^+\pi^- $ process yields in the bulk Higgs RS model:
A.Appendix A: Wave functionsThe wave functions describe the hadronization of a quark and anti-quark into a meson $ M $, which is a non-perturbative but universal process. For the incoming $ B $ meson, the wave function is written as
The light-cone distribution amplitudes $ \phi^A_{\pi(K)} $, $ \phi_{\pi(K)}^P $ and $ \phi_{\pi(K)}^T $ have been studied within the QCD sum rules [52-54], and they are expanded in the Gegenbauer polynomials,
where $ C_F = 4/3 $ is the group factor of the $ {SU}(3)_c $ gauge group, and $ r_i = m_{0i}/M_B $ ($ i = K,\pi $). The threshold resummation $ S_t(x) $ is parameterized as [32]
$ S_B $, $ S_K $, and $ S_\pi $ result from resummation of doubly logarithms caused by the overlap of soft collinear gluon corrections and single logarithms due to the renormalization of ultra-violet divergence [33], which are defined as
where $ \beta_0 = (33-2n_f)/3 $ , and $ n_f $ is the number of active flavors. $ \Lambda = 250 {\rm MeV} $ is QCD scale at $ n_f = 4 $. The functions $ h_i $ ($ i = a,b,c,d $) in the decay amplitudes arise from the propagators of the virtual quarks and gluons, which are expressed by