# Prospects of discovering new physics in rare charm decays

## Abstract

The LHCb bounds on the branching ratio of the rare decay \(D^0 \rightarrow \mu ^+ \mu ^-\) and the constraints on the branching ratio of \(D^+ \rightarrow \pi ^+ \mu ^+ \mu ^-\) in the nonresonant regions enable us to improve constraints on new physics contributions. Using the effective Lagrangian approach we determine the sizes of the Wilson coefficients allowed by the existing LHCb bounds on rare charm decays. Then we discuss contributions to rare charm meson decay observables in several models of new physics: a model with an additional spin-1 weak triplet, leptoquark models, Two Higgs doublets model of type III, and a \(Z'\) model. Here we complement the discussion by \(D^0 \)–\( {\bar{D}}^0\) oscillations data. Among the considered models, only leptoquarks can significantly modify the Wilson coefficients. Assuming that the differential decay width for \(D^+ \rightarrow \pi ^+ \mu ^+ \mu ^-\) receives a NP contribution, while the differential decay width for \(D^+ \rightarrow \pi ^+ e^+ e^-\) is Standard Model-like, we find that lepton flavor universality can be violated and might be observed at high dilepton invariant mass.

## 1 Introduction

Processes with charmed mesons and top quarks offer an excellent opportunity to search for new physics (NP) in the up-type quark sector. In contrast to B meson physics, which is convenient to search for NP due to good exposure of the short-distance effects, charm quark systems are dominated by large long-distance quantum chromodynamics contributions. Such effects then screen the short-distance contributions of interest. Within the Standard Model (SM) the short-distance physics in rare charm decays is strongly affected by the Glashow–Iliopoulos–Maiani (GIM) mechanism [1]. Namely, box or penguin diagram amplitudes get contributions from down-type quarks which are approximately massless from the weak scale perspective, and this warrants a very effective GIM cancellation. Flavor changing neutral current (FCNC) processes with charm mesons might change charm quantum number for two or one unit (\(|\Delta C| = 2\) or \(|\Delta C|=1\) transitions). The \(|\Delta C| = 2\) transition occurs in \(D^0 \)–\( {\bar{D}}^0\) oscillations and leads to strong constraints on NP from the measured observables as pointed out in [2, 3]. There are two possibilities for NP in the \(|\Delta C| = 2\) transition: the transition might occur at tree level, in which case a new neutral scalar or a vector boson possesses FCNC couplings to *u* and *c* quarks, or at loop level via NP degrees of freedom affecting the box diagrams. The processes with \(|\Delta C|=1\) on the quark level are \(c \rightarrow u \gamma \) and \(c \rightarrow u \ell ^+ \ell ^-\) [4, 5, 6, 7, 8, 9]. Both transitions can be approached in the familiar effective Lagrangian formalism [3]. Additional constraints on NP arise from the down-type quark sector whenever new bosons couple to left-handed quark doublets [10, 11]. Since NP is very constrained by the current experimental results coming from *B* and *K* physics [12] the only chance to observe NP in rare charm decays seems to be when new bosons are coupled to weak singlets. This then allows one to avoid the strong flavor constraints in the down-type quark sector.

^{1}

*K*, or

*B*physics), we comment on the prospects of observing their signals in rare charm decays. To this end, we use the effective Lagrangian encoding the short-distance NP contributions in a most general way. Namely, the experimental results (1) and (3) give us a possibility to constrain NP in \(c\rightarrow u \ell ^+ \ell ^-\) also in a model independent way.

In the case of \(b \rightarrow s \ell ^+ \ell ^-\) transitions, LHCb has recently observed a large departure of the experimentally determined lepton flavor universality (LFU) ratio \(R_K= {\mathrm{BR}(B \rightarrow K \mu ^+ \mu ^-)_{q^2 \in [1,6]\, \mathrm{GeV}^2}}/\mathrm{BR}(B \rightarrow K e^+ e^-)_{q^2 \in [1,6]\, \mathrm{GeV}^2}\) from the expected SM value [15]. This value was found to be \(R_K^\mathrm{LHCb}= 0.745^{+ 0.090}_{-0.074}\pm 0.036\), lower than the SM prediction \(R_K^{SM}= 1.0003 \pm 0.0001\) [16]. This surprising result of LHCb indicates possible violation of LFU in the \(\mu \)–*e* sector. Due to the importance of this result, we investigate whether analogous tests in the \(\mu \)–*e* LFU can be carried out in \(c\rightarrow u \ell ^+ \ell ^-\) processes.

The outline of this article is as follows. In Sect. 2 we describe effective Lagrangian of \(|\Delta C|=1\) transition and determine bounds on the Wilson coefficients coming from the experimental limits on \(\mathrm{BR}(D^+ \rightarrow \pi ^+ \mu ^+ \mu ^-)\) and \(\mathrm{BR}(D^0 \rightarrow \mu ^+ \mu ^-)\). Section 3 contains an analysis in the context of specific theoretical models of new physics, contributing to the \(c \rightarrow u\ell ^+ \ell ^-\) and related processes. Section 4 discusses lepton flavor universality violation. Finally, we summarize the results and present conclusions in Sect. 5.

## 2 Observables and model independent constraints

### 2.1 Effective Hamiltonian for \(c \rightarrow u \ell ^+ \ell ^-\)

### 2.2 \(D^+ \rightarrow \pi ^+ \mu ^+ \mu ^-\)

*a*determined by measurements of \(D\rightarrow \pi \ell \nu \) decay spectra. We make an average of four experimental fits to the shape parameters, by taking as input the CLEO-c tagged [20] and untagged analysis [21], BES III [22], and Babar [23] results, all compiled by the HFAG [24]. The fitted shape parameters are \(m_\mathrm{pole} = 1.90(8)\,\mathrm{GeV}\) and \(a= 0.28(14)\). For the normalization of the form factor we rely on the lattice result \(f_+(0) =0.67(3)\) calculated by the HPQCD Collaboration [25]. The shape parameter \(b=1.27(17)\) has also been extracted in lattice simulations [26]. For the tensor current form factor we rely on the fit of lattice data to BK shape as in [26]:

*B*and \(\ell ^-\) in the rest frame of lepton pair whereas

*V*,

*A*,

*S*,

*P*,

*T*, and \(T_5\) are \(q^2\)-dependent functions expressed in terms of hadronic form factors and Wilson coefficients,

\(1\sigma \) ranges and \(90\,\%\) CL upper bounds on resonant branching ratios and amplitude parameters [31]

| \(\rho \) | \(\omega \) | \(\phi \) | \(\eta \) |
---|---|---|---|---|

\(\mathrm{BR}(D^+ \rightarrow \pi ^+ X (\rightarrow \mu ^+ \mu ^-)) [10^{-8}]\) | 3.7 (7) | \(<3\).1 | 160 (10) | 2.0 (3) |

\(|a_X|\) | 1.21 (12) | \(<\)\(0.26\) | 0. 94 (3) | 0.27 (2) |

The magnitude of unknown parameters \(a_X\) (\(X=\rho ,\omega ,\phi ,\eta \)), can be fitted to the measured resonant branching ratios, given in Table 1 [31]. The corresponding values of \(|a_X|\) are given in the second row in Table 1. We treat the relative phases as free parameters. Alternatively, for the relative phases and magnitudes of \(a_X\) one can use flavor structure arguments [18]. In the left-hand panel in Fig. 1 we present the long-distance contributions to the differential branching ratio for \(D^+ \rightarrow \pi ^+ \mu ^+ \mu ^-\) as a function of dilepton invariant mass for a representative set of parameters \(|a_X|\) from the \(1\sigma \) region (Table 1) and random phases of \(a_X\). On the right-hand panel in Fig. 1 we also indicate the interpretation of experimental upper bounds (3) in the case where the total amplitude would be constant, namely in the case where all angular coefficient functions \(a_\ell \), \(b_\ell \), \(c_\ell \) would be independent of \(q^2\). We also estimate the saturation of these bounds by the total resonant decay branching ratio and find for the low- and high-\(q^2\) bin contributions to be smaller than \(7.3\times 10^{-9}\) and \(5.3\times 10^{-9}\), respectively. On the other hand, the short-distance contribution to the total branching ratio of the SM due to the quoted value of \(C_7\) is of the order \(10^{-12}\) and thus negligible.

*D*meson, \(f_D= 209(3)\,\mathrm{MeV}\), has been averaged over \(N_f = 2+1\) lattice simulations [32, 33, 34]. In the SM this decay is dominated by the intermediate \(\gamma ^*\gamma ^*\) state that is electromagnetically converted to a \(\mu ^+ \mu ^-\) pair. It was estimated in [9] that \(\mathrm{BR}(D^0 \rightarrow \mu ^+ \mu ^-) \simeq 2.7\cdot 10^{-5}\times \mathrm{BR}(D^0 \rightarrow \gamma \gamma )\), and, together with the upper bound \(\mathrm{BR}(D^0 \rightarrow \gamma \gamma ) < 2.2\times 10^{-6}\) at \(90\,\%\) CL [35], this leads to the limit \(\mathrm{BR}(D^0 \rightarrow \mu ^+ \mu ^-)^\mathrm{SM} \lesssim 10^{-10}\).

## 3 Constraints on the Wilson coefficients

Maximal allowed values of the Wilson coefficient moduli, \(|{\tilde{C}}_i| = |V_{ub} V_{cb}^* C_i|\), calculated in the nonresonant regions of \(D^+ \rightarrow \pi ^+ \mu ^+ \mu ^-\) in the low lepton invariant mass region (\(q^2 \in [0.0625,0.276]\,\mathrm{GeV}^2\)), denoted by \(\mathrm{I}\), in the high invariant mass region (\(q^2 \in [1.56,4.00]\,\mathrm{GeV}^2\)), denoted by \(\mathrm{II}\), and from the upper bound \(\mathrm{BR}(D^0 \rightarrow \mu ^+ \mu ^-) < 7.6 \times 10^{-9}\) [13]. The last row gives the maximal value for the case where \({\tilde{C}}_9= \pm {\tilde{C}}_{10}\). All the quoted bounds have been derived for real \(C_{i}\). The bounds for \({\tilde{C}}_i\) apply also to the chirally flipped coefficients \({\tilde{C}}_j^\prime \)

\(|{\tilde{C}}_i|_\mathrm{max}\) | |||
---|---|---|---|

\(\mathrm{BR}(\pi \mu \mu )_\mathrm{I}\) | \(\mathrm{BR}(\pi \mu \mu )_\mathrm{II}\) | \(\mathrm{BR}(D^0\rightarrow \mu \mu )\) | |

\({\tilde{C}}_7\) | 2.4 | 1.6 | – |

\({\tilde{C}}_9\) | 2.1 | 1.3 | – |

\({\tilde{C}}_{10}\) | 1.4 | 0.92 | 0.63 |

\({\tilde{C}}_S\) | 4.5 | 0.38 | 0.049 |

\({\tilde{C}}_P\) | 3.6 | 0.37 | 0.049 |

\({\tilde{C}}_T\) | 4.1 | 0.76 | – |

\({\tilde{C}}_{T5}\) | 4.4 | 0.74 | – |

\({\tilde{C}}_9= \pm {\tilde{C}}_{10}\) | 1.3 | 0.81 | 0.63 |

The high invariant dilepton mass bin is more restrictive than the low dilepton invariant mass bin. Due to the parity conservation in \(D\rightarrow \pi \) transition the bounds for \({\tilde{C}}_j^\prime \), \( j=7,9,10,S,P\) are the same as for \({\tilde{C}}_j\).

We turn to the discussion of specific models in the next section.

## 4 Impact on specific models

### 4.1 Spin-1 weak triplet

*B*sector anomalies (\(R_{D^{(*)}}\), \(R_K\)) even in the scenario with \(U(2)_q \times U(2)_\ell \) flavor symmetry [11]. The relevant effective Lagrangian that follows from integrating out the vector triplet at tree level reads

*B*meson decays whose experimental value requires \(R_0 = 0.14 \pm 0.04\). The constraint from \(\tau \rightarrow 3 \mu \) implies \(\lambda ^\ell _{\mu \mu } = (0.013 \pm 0.011)(0.15/R_0)\frac{g_q}{g_\ell }\), while the constraint on the \(|\Delta C| = 2\) operators from CP violation in \(D^0 \)–\( {\bar{D}}^0\) mixing results in the inequality \(g_\ell /g_q > 1.26 R_0\). From these ingredients one can estimate the maximum value of \(C_{9}\),

### 4.2 Leptoquarks

There exist several scalar and vector leptoquark (LQ) states which may leave imprint on \(c\rightarrow u \ell ^+ \ell ^-\) transitions [37]. The possible scalar states transform under the SM gauge group as \((3,3,-1/3)\), \((3,1,-1/3)\), and (3, 2, 7 / 6), of which only the latter state conserves baryon and lepton number on the renormalizable level. Thus the mass of the scalar multiplet (3, 2, 7 / 6) can be close to the electroweak scale without destabilizing the proton. In addition, there are four vector LQs which potentially contribute in rare charm decays, and they carry the following quantum numbers: (3, 3, 2 / 3), (3, 1, 5 / 3), (3, 2, 1 / 6), and \((3,2,-5/6)\). Only the first two states have definite baryon and lepton numbers.

Among all scalar LQs we will consider only the baryon number conserving state (3, 2, 7 / 6), which comes with a rich set of couplings that are in general severely constrained by *B* and *K* physics [38]. Then, among the two baryon number conserving vector LQs we will focus on the state in the representation (3, 1, 5 / 3) whose phenomenology is limited to the up-type quarks and charged leptons.

#### 4.2.1 Scalar leptoquark (3, 2, 7 / 6)

*B*,

*K*physics and four fermion operator constraints [40].

#### 4.2.2 Vector leptoquark (3, 1, 5 / 3)

*i*,

*j*. Integrating out \(V^{(5/3)}\) results in the right-handed current operators:

### 4.3 Two Higgs doublet model type III

*h*and

*H*, one pseudoscalar,

*A*, and two charged scalars, \(H^\pm \). In the scenario with MSSM-like scalar potential their masses and mixing angles are related [41],

### 4.4 Flavor specific \(Z^\prime \) extension

## 5 Lepton flavor universality violation

The LFU ratio \(R_\pi ^\mathrm{II}\) at high dilepton invariant mass bin and maximal value of each Wilson coefficient (applies also for the primed coefficients, \({\tilde{C}}_i^\prime \)). It is assumed that NP contributes only to the muonic mode. The SM value of \(R_\pi ^\mathrm{II}\) is given in the first row

\(|{\tilde{C}}_i|_\mathrm{max}\) | \(R_\pi ^\mathrm{II}\) | |
---|---|---|

SM | – | \( 0.999\pm 0.001\) |

\({\tilde{C}}_7\) | 1.6 | \(\sim \)6–100 |

\({\tilde{C}}_9\) | 1.3 | \(\sim \)6–120 |

\({\tilde{C}}_{10}\) | 0.63 | \(\sim \)3–30 |

\({\tilde{C}}_S\) | 0.05 | \(\sim \)1–2 |

\({\tilde{C}}_P\) | 0.05 | \(\sim \)1–2 |

\({\tilde{C}}_T\) | 0.76 | \(\sim \)6–70 |

\({\tilde{C}}_{T5}\) | 0.74 | \(\sim \)6–60 |

\({\tilde{C}}_9= \pm {\tilde{C}}_{10}\) | 0.63 | \(\sim \)3–60 |

\({\tilde{C}}_9' = -{\tilde{C}}_{10}'\big |_{\mathrm{LQ}(3,2,7/6)}\) | 0.34 | \(\sim \)1–20 |

## 6 Summary and outlook

Motivated by the great improvement of the bounds on rare charm decays by the LHCb experiment we determine the bounds on the effective Wilson coefficients. Existing data implies upper bounds on the effective Wilson coefficients as presented in Table 2. The strongest constraints on \(C_{10}\), \(C_P\), \(C_S\), and \(C_{10}^\prime \), \(C_P^\prime \), \(C_S^\prime \) are obtained from the bound on the branching fraction of \(D^0 \rightarrow \mu ^+ \mu ^-\) decay. The nonresonant differential decay width distribution gives bounds on \(C_i\), \( i=7,9,10, S, P, T, T5\) as well as on the coefficients of the operators of opposite chirality. The constraints are stricter in the high dilepton invariant mass bin than in the low dilepton invariant mass bin, and this statement applies in particular to the contributions of the scalar and pseudoscalar operators. The forward–backward asymmetry is sensitive to the combination of scalar and tensor coefficients at high-\(q^2\).

Next, we have investigated new physics models in which the effective operators may be generated. We have found that the presence of a leptoquark, which is either a scalar and weak doublet, (3, 2, 7 / 6), or has spin-1 and is a weak singlet, (3, 1, 5 / 3), can lead to sizable contributions to the Wilson coefficients \(C_9^\prime \) and \(C_{10}^\prime \). The sensitivity to the LQ scenarios is similar in the high-\(q^2\) bin of \(D^+ \rightarrow \pi ^+ \mu ^+ \mu ^-\) and \(D^0 \rightarrow \mu ^+ \mu ^-\), while \(D^0 \)–\( {\bar{D}}^0\) mixing results in a somewhat stronger constraint. For the Two Higgs doublet model of type III the presence of scalar and pseudoscalar operators enhances the sensitivity in \(D^0 \rightarrow \mu ^+ \mu ^-\) and therefore results in small effects in \(D^+ \rightarrow \pi ^+ \mu ^+ \mu ^-\). We have also discussed a SM extension by a \(Z'\) gauge boson where the tree-level amplitude in \(D^0 \)–\( {\bar{D}}^0\) mixing is a dominant constraint and leaves no possibility of signals in rare charm decays.

Our study indicates the possibility to check whether lepton flavor universality between muonic and electronic channels is valid by means of studying ratios of widths of \(D^+ \rightarrow \pi ^+ \ell ^+ \ell ^-\) at low or high dilepton invariant mass bins, \(R_\pi ^\mathrm{I,II}\). In the SM the two ratios are close to 1, especially in the high-\(q^2\) bin. Assuming the electronic decay is purely SM-like, we find that in the high-\(q^2\) bin the ratio \(R_\pi ^\mathrm{II}\) is in most cases significantly increased with respect to the SM prediction, while there is no clear preference between higher and lower values at low-\(q^2\) bin ratio \(R_\pi ^\mathrm{I}\). In the leptoquark models studied in this paper the ratio may be greatly increased, but a slight decrease cannot be excluded, presently due to the unknown interplay of weak phases with the phases of the resonant spectrum. Chances to observe new physics in rare charm decays are present in models where the connection to the stringent constraints stemming from *B* and *K* flavor physics are hindered. New physics models which fulfill this condition are main candidates to be exposed experimentally by future progress in bounding the rare charm decays \(D \rightarrow \pi \mu ^+ \mu ^-\) and \(D^0 \rightarrow \mu ^+ \mu ^-\), as well as by more precise studies of \(D^0 \)–\( {\bar{D}}^0\) mixing with the potential NP contributions. Alternatively, experimental tests of lepton flavor universality in rare charm decays might point toward the presence of new physics in the charm sector, which can easily be hidden in the case of existing experimental observables.

*Note* While we were finishing this paper another work [48] appeared in which the authors studied rare charm decays.

## Footnotes

- 1.
Note that the high-\(q^2\) bin quoted by the experiment extends beyond the maximal allowed \(q^2_\mathrm{max} = (m_D-m_\pi )^2 = 2.99\,\mathrm{GeV}^2\).

## Notes

### Acknowledgments

We thank Benoit Viaud for constructive comments on the first version of this paper. We thank Martin Bauer for pointing out a typographical error in Eq. (16) in the previous version. We acknowledge support of the Slovenian Research Agency. This research was supported in part by the Munich Institute for Astro- and Particle Physics (MIAPP) of the DFG cluster of excellence “Origin and Structure of the Universe”.

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