decays in supersymmetry with -parity violation
In the past few years, several hints of lepton flavour universality (LFU) violation have emerged in the and data. Quite recently, the Belle Collaboration has reported the first measurement of the longitudinal polarization fraction in the decay. Motivated by this intriguing result, together with the recent measurements of and polarization, we study decays in the Supersymmetry (SUSY) with -parity violation (RPV). We consider , , and modes and focus on the branching ratios, the LFU ratios, the forward-backward asymmetries, polarizations of daughter hadrons and lepton. It is found that the RPV SUSY can explain the anomalies at level, after taking into account various flavour constraints. In the allowed parameter space, the differential branching fractions and LFU ratios are largely enhanced by the SUSY effects, especially in the large dilepton invariant mass region. In addition, a lower bound is obtained. These observables could provide testable signatures at the High-Luminosity LHC and SuperKEKB, and correlate with direct searches for SUSY.
In recent years, several interesting anomalies emerge in experimental data of semi-leptonic -meson decays. For the ratios with , the latest averages of the measurements by BaBar [1, 2], Belle [3, 4, 5, 6] and LHCb Collaboration [7, 8, 9] give 
Compared to the branching fractions themselves, these ratios have the virtue that, apart from significant reduction of the experimental systematic uncertainties, the CKM matrix element cancels out and the sensitivity to transition form factors becomes much weaker. The SM predictions read 
which are obtained from the arithmetic averages of the most recent calculations by several groups [11, 12, 13, 14]. The SM predictions for and are below the experimental measurements by and , respectively. Taking into account the measurement correlation of between and , the combined experimental results show about deviation from the SM predictions . For the decay, which is mediated by the same quark-level process as , the recent measured ratio at the LHCb  lies within about above the SM prediction . In addition, the LHCb measurements of the ratios , for  and for , are found to be about and lower than the SM expectation, [19, 20], respectively. These measurements, referred to as the , and anomalies, may provide hints of Lepton Flavour University (LFU) violation and have motivated numerous studies of New Physics (NP) both in the Effective Field Theory (EFT) approach [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34] and in specific NP models [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60]. We refer to refs. [61, 62] for recent reviews.
which is consistent with the SM prediction  at . Previously, the Belle Collaboration also performed measurements on polarization in the decay, which gives the result [5, 6]. Angular distributions can provide valuable information about the spin structure of the interaction in the decays, and are good observables to test various NP explanations [66, 67, 68, 69, 70]. Measurements of the angular distributions are expected to be significantly improved in the future. For example, Belle II with data can measure with a precision of .
In this work, motivated by these recent experimental progresses, we study the anomalies in the Supersymmetry (SUSY) with -parity violation (RPV). In this scenario, the down-type squarks interact with quarks and leptons via the RPV couplings. Therefore, they contribute to transition at the tree level and could explain the current anomalies [72, 73, 74]. Besides , we will also study , , and decay. All of them are transition at the quark level, and the latter two decays have not been measured yet. Using the latest experimental data of various low-energy flavour processes, we will derive constraints on the RPV couplings. Then, predictions in the RPV SUSY are made for the five decays, focusing on the distributions of the branching fractions, the LFU ratios and various angular observables. We have also taken into account the recent developments on the form factors [16, 75, 11, 14, 76]. Implications for future searches at the High-Luminosity LHC (HL-LHC) and SuperKEKB are briefly discussed.
This paper is organized as follows: In section 2, we briefly review the SUSY with RPV interactions. In section 3, we recapitulate the theoretical formulae for the various flavour processes, and discuss the SUSY effects. In section 4, detailed numerical results and discussions are presented. We conclude in section 5. The relevant form factors are recapitulated in appendix A.
2 Supersymmetry with -parity violation
where and denote the doublet lepton and quark superfields, respectively. and () are the singlet lepton and quark superfields, respectively. , and indicate generation indices. In order to ensure the proton stability, we assume the couplings are zero. In semi-leptonic meson decays, contribution from the term is through the exchange of sleptons and much more suppressed than the one from the term, which is through the exchange of right-handed down-type squarks . Therefore, we only consider the term in this work. For the SUSY scenario with the term, studies on the anomalies with slepton exchanges can be found in ref. [79, 80].
The interaction with couplings can be expanded in terms of fermions and sfermions as 
where denotes the CKM matrix element. Here, all the SM fermions , and are in their mass eigenstate. Since we neglect the tiny neutrino masses, the PMNS matrix is not needed for the lepton sector. For the sfermions, we assume that they are in the mass eigenstate. We refer to ref.  for more details about the choice of basis. Finally, we adopt the assumption in ref.  that only the third family is effectively supersymmetrized. This case is equivalent to that the first two generations are decoupled from the low-energy spectrum as in ref. [81, 82]. For the studies including the first two generation sfermions, we refer to ref. , where both the and anomalies are discussed.
It is noted that the down-type squarks and the scalar leptoquark (LQ) discussed in ref.  have similar interaction with the SM fermions. However, in the most general case, the LQ can couple to the right-handed singlets, which is forbidden in the RPV SUSY. Such right-handed couplings are important to explain the anomaly in the LQ scenario . These couplings can also affect semi-leptonic decays. In particular, their contributions to the decays are found to be small after considering other flavour constraints .
In this section, we will introduce the theoretical framework of the relevant flavour processes and discuss the RPV SUSY effects in these processes.
With the RPV SUSY contributions, the effective Hamiltonian responsible for transitions is given by 
where tree-level sbottom exchange gives
with the Higgs vev . It is noted that this Wilson coefficient is at the matching scale . However, since the corresponding current is conserved, we can obtain the low-energy Wilson coefficient without considering the Renormalization Group Evolution (RGE) effects, i.e., .
where , and , and , and . For each particle in the above decay, its momentum and helicity are denoted as and , respectively. In particular, the helicity of pseudoscalar meson is zero, e.g., . After summation of the helicity of parent hadron , differential decay width for this process can be written as [101, 67]
where , , and . The angle denotes the angle between the three-momentum of and that of in the - center-of-mass frame. With the differential decay width, we can derive the following observables:
The decay width and branching ratio
where is the total width of the hadron .
The LFU ratio
where in the denominator denotes the average of the different decay widths of the electronic and muonic modes.
The lepton forward-backward asymmetry
The polarization fractions
Explicit expressions of the helicity amplitudes and all the above observables can be found in ref.  for decays, and ref.  for decay. The expressions for and are analogical to the ones for and , respectively. Since these angular observables are ratios of decay widths, they are largely free of hadronic uncertainties, and thus provide excellent tests of lepton flavour universality. It is noted that the RPV SUSY effects generate operator with the same chirality structure as in the SM, as shown in eq. (5). It’s straightforward to derive the following relation in all the decays
for , and . Here, vanishing contributions to the electronic and muonic channels are assumed.
The hadronic transition form factors are important inputs to calculate the observables introduced above. In recent years, notable progresses have been achieved in this field [75, 97, 11, 12, 13, 14, 76, 103, 104, 105, 87, 106, 107, 108, 109, 110]. For transitions, it has already been emphasized that the Caprini-Lellouch-Neubert (CLN) parameterization  does not account for uncertainties in the values of the subleading Isgur-Wise functions at zero recoil obtained with QCD sum rules [112, 113, 114], where the number of parameters is minimal . In this work, we don’t use such simplified parameterization but adopt the conservative approach in ref. [11, 14], which is based on the Boyd-Grinstein-Lebed (BGL) parameterization . In addition, we use the transition form factors obtained in the covariant light-front approach . For the transition form factor, we adopt the recent Lattice QCD results in ref. [75, 76]. Explicit expressions of all the form factors used in our work are recapitulated in appendix A.
For transitions, we consider , and decays. Similar to eq. (13), we have
It is noted that the SUSY contributions to both and transitions depend on the same set of parameters, , , and . Therefore, the ratios are related to the decay.
3.2 Other processes
The Flavour-Changing Neutral Current (FCNC) decays and are induced by the and transitions, respectively. In the SM, they are forbidden at the tree level and highly suppressed at the one-loop level due to the GIM mechanism. In the RPV SUSY, the sbottoms can contribute to these decays at the tree level, which result in strong constraints on the RPV couplings. Similar to the transitions, the RPV interactions do not generate new operators beyond the ones presented in the SM. Therefore, we have [74, 73]
where the gauge-invariant function arises from the box and -penguin diagrams in the SM .
The leptonic and couplings are also important to probe the RPV SUSY effects [26, 117]. In particular, and couplings involving left-handed leptons can receive contributions from the loop diagrams mediated by top quark and sbottom. These effects modify the leptonic and couplings as 
where and with the weak mixing angle. The loop functions and have been calculated in refs. [26, 117, 74] and are given by and . Experimental measurements on the couplings have been performed at the LEP and SLD . Their combined results give . The coupling can be extracted from decay data. The measured decay fractions compared to the decay fractions give . Both the leptonic and couplings are measured at few permille level. Therefore, they will put strong bounds on the RPV coupling .
The RPV interactions can also affect -meson decays, e.g., , -meson decays, e.g., , and lepton decays, e.g., . However, as discussed in ref. , their constraints are weaker than the ones from the processes discussed above. In addition, bound from the lifetime [119, 120] is not relevant, since the RPV SUSY contributions to are not chirally enhanced compared to the SM.
Another interesting anomalies arise in the recent LHCb measurements of , which show about deviation from the SM prediction [17, 18] and are refered to as anomalies. The anomalies imply hints of LFU violation in transition. In the RPV SUSY, the left-handed stop can affect this process at the tree level, and the right-handed sbottom can contribute at the one-loop level. However, as discussed in ref. , once all the other flavour constraints are taken into account, no parameter space in the RPV SUSY can explain the current anomaly.
Finally, we briefly comment the direct searches for the sbottoms at the LHC. Using data corresponding to at 13 TeV, the CMS collaboration has performed search for heavy scalar leptoquarks in channel. The results can be directly reinterpreted in the context of pair produced sbottoms decaying into top quark and lepton pairs via the RPV coupling . Then, the mass of the sbottom is excluded up to at 95% CL .
4 Numerical results and discussions
In this section, we proceed to present our numerical analysis for the RPV SUSY scenario introduced in section 2. We will derive constraints on the RPV couplings and study their effects to various processes.
The most relevant input parameters used in our numerical analysis are presented in table 1. Using the theoretical framework described in section 3, the SM predictions for the , , , and decays are given in table 2. In order to obtain the theoretical uncertainties, we vary each input parameter within its range and add each individual uncertainty in quadrature. For the uncertainties induced by form factors, we will also include the correlations among the fit parameters. In particular, for the decay, we follow the treatment of ref.  to obtain the statistical and systematic uncertainties induced by the form factors. From table 2, we can see that the experimental data on the ratios , and deviate from the SM predictions by , and , respectively.
In the RPV SUSY scenario introduced in section 2, the relevant parameters to explain the anomalies are and . In section 3, we know only the three products of the RPV couplings, , appear in the various flavour processes. In the following analysis, we will assume these products are real and derive bounds on them. We impose the experimental constraints in the same way as in refs. [124, 125]; i.e., for each point in the parameter space, if the difference between the corresponding theoretical prediction and experimental data is less than error bar, which is evaluated by adding the theoretical and experimental errors in quadrature, this point is regarded as allowed at level. From section 3, it is known that the RPV couplings always appear in the form of in all the decays. Therefore, we can take without loss of generality, which is equivalent to absorb into . Furthermore, the choice of is compatible with the direct searches for the sbottoms at CMS . In the SUSY contributions to the couplings and in eq. (16), additional dependence arises in the loop functions and , respectively. As can be seen in the next subsection, our numerical results show such dependence is weak and the choice of does not lose much generality.
As shown in table 2, the current experimental upper bounds on the branching ratio of and are one order above their SM values. However, since the SUSY contributes to these decays at the tree level, the RPV couplings are strongly constrained as
at level. For the leptonic and couplings, the current measurements on and have achieved to the precision of few permille. We find that the latter can give stronger constraint, which reads
or , at level. It is noted that this upper bound prevents the coupling from developing a Landau pole below the GUT scale .
As discussed in section 3, the RPV interactions affect transitions via the three products . After considering the above individual constraints at level, parameter space to explain the current measurements on , , and are shown in figure 1 for . We can see that the decays and other flavour observables put very stringent constraints on the RPV couplings. The combined constraints are slightly stronger than the individual ones in eq. (17) and (18). It is also noted that, after taking into account the bounds from and , the decays are very sensitive to the product . As a result, the current anomalies give a lower bound on . Finally, the combined bounds in figure 1 read numerically,
As can be seen, a weak lower bound on is also obtained. In addition, although the constraints from the polarization fraction are much stronger than the ones from the polarization fraction , this observable can’t provide further constraints on the RPV couplings. Due to the previous discussions, we show the combined upper bound on as a function of in figure 0(d). It can be seen that the upper limit of just changes around 20% by varying from to . Therefore, one can approximately obtain the allowed parameter space for from figure 0(a)-0(c) by timing a factor of .
In the parameter space allowed by all the constraints at level, correlations among several observables are obtained, which are shown in figure 2. In these figures, the SUSY predictions are central values without theoretical uncertainties. From figure 1(a), we can see that the central value of and are strongly correlated, as expected in eq. (13). It is noted that the SUSY effects can only enhance the central value of by about 8%, so that approach to, but still lie outside, the range of the HFLAV averages. Therefore, future refined measurements will provide a crucial test on the RPV SUSY explaination of anomalies. At Belle II, precisions of measurements are expected to be about 2-4%  with a luminosity of . From figure 1(b), it can be seen that both and deviate from their SM predictions. The lower bound for the latter is , which is due to the lower bound on obtained in the last section. Compared to the SM prediction , such significant enhancement makes this decay an important probe of the RPV SUSY effects. In the future, Belle II with data can measure its branching ratio with a precision of 11% . Another interesting correlation arises between and . As shown in figure 1(f), the RPV SUSY effects always enhance and suppress simultaneously. When approaches to the SM value , the branching ratio of maximally deviates from its SM prediction. In figure 1(d) and 1(e), we show the correlations involving decay. It can be seen that the SUSY prediction on almost lies in the SM range. Since the future Belle II sensitivity at is comparable to the current theoretical uncertainties , much more precise theoretical predictions are required in the future to probe the SUSY effects.
Using the allowed parameter space at level derived in the last subsection, we make predictions on the five decays, , , , and decays. In table 2, the SM and SUSY predictions on the various observables in these decays are presented. The SUSY predictions have included the uncertainties induced by the form factors and CKM matrix elements. At present, there are no available measurements on the and decays. From table