# Probing dark particles indirectly at the CEPC

###### Abstract

When dark matter candidate and its parent particles are nearly degenerate, it would be difficult to probe them at the Large Hadron Collider directly. We propose to explore their quantum loop effects at the CEPC through the golden channel process . We use a renormalizable toy model consisting of a new scalar and a fermion to describe new physics beyond the Standard Model. The new scalar and fermion are general multiplets of the symmetry, and couple to the muon lepton through Yukawa interaction. We calculate their loop contributions to anomalous and couplings which can be applied to many new physics models. The prospects of their effects at the CEPC are also examined assuming a 2‰ accuracy in the cross section measurement.

## I Introduction

One of the major tasks of particle physics is to understand the particle nature of dark matter Ahmed et al. (2010); Bertone et al. (2005); Akerib et al. (2014); Langacker (2009). As the dark matter candidate does not register at the detector and induce a large missing transverse momentum (), one usually searches for the dark matter candidate in the signature of a large together with a bunch of visible particles in the standard model (SM). The method is valid only when there is a large mass gap between the dark matter candidate and its parent particle. However, there could be a scenario in which the dark matter candidate () and its parent particle () are nearly degenerate, e.g. , where denotes the SM particles. The energy of tends to 0 () in the degenerate limit of and . The particle ’s (or their decay products if ’s are not stable) are very soft and cannot register in the detector. It is hard to directly observe or test such new physics signals at the Large Hadron Collider (LHC), and we name it as a “nightmare” scenario.

On the other hand, the new physics particles affect the SM processes through quantum loop corrections, no matter whether they are degenerate or not. Such quantum corrections, if large enough, could be detected at the electron-positron colliders, e.g. the Circular electron-positron collider (CEPC), FCC-ee or International Linear Collider (ILC). In this work we focus on the “nightmare” scenario and explore the potential of measuring the new physics effects in the scattering of at the CEPC with a center of mass energy of 240 GeV. The channel is known as the golden channel which serves as a precision candle owing to its clean background and high detection efficiency Olive et al. (2014). A relative precision of 2‰ on can be reached at ILC Riemann (2001); Baer et al. (2013), and the CEPC Group (2015) is expected to achieve a comparable accuracy.

Dark scalars appear often in various new physics models and have been studied extensively in the literature Jungman et al. (1996); Fox and Poppitz (2009); Cao et al. (2009a); Bertone et al. (2005); Agrawal et al. (2014). Rather than considering a specific complete model, we use a simple toy model to describe the new physics beyond the SM. The toy model consists of a new complex scalar multiplet () and a vector-like fermion (). We demand that the neutral component of serve as the dark matter candidate, while the fermion facilitates the Yukawa coupling of to . In practice we require that be slightly heavier than such that it can decay into and muon lepton pairs ^{1}^{1}1Note that the vector-like fermion , except for a weak gauge singlet, cannot play the role of dark matter candidate as it is constrained severely by the direct detection of the dark matter. However, for the scalar dark matter, it is easy to escape the constraint from LUX data Akerib et al. (2015) if a small mass splitting is generated between the real and imaginary components of the neutral complex scalar..
Our toy model respects the SM gauge symmetry and is renormalizable. Therefore, it can be viewed as a simplified version of a UV-completion model and can be generalized to many new physics models, e.g., the lepto-philic dark matter models Chen and Takahashi (2009); Yin et al. (2009); Fox and Poppitz (2009); Bi et al. (2009); Cao et al. (2009a, 2014).
To ensure the stability of the dark matter candidate, we restrict the mixing of such exotic particles with the SM particles through an exact symmetry, under which the SM fields are all even, whereas the new fields are odd. As a result, the SM particles can only interact with a pair of those exotic particles at a time.

We emphasize that the new physics particles in our toy model can be light, say around , such that the approach of effective field theory Hagiwara et al. (1987); Falkowski and Mimouni (2015); Ellis and You (2015); Wells and Zhang (2016); Bian et al. (2015); Cao et al. (2007, 2011) no longer works, and the full one loop calculation is necessary to address its effects. We use the dimensional regularization to calculate the loop corrections in the on-shell renormalization scheme Denner (1993); Aoki et al. (1982). The analytical results are written in terms of the Passarino-Veltman scalar functions Passarino and Veltman (1979); ’t Hooft and Veltman (1979).

The paper is organized as follows. In Sec. II we first introduce our simplified new physics model with new dark scalar and fermion multiplets. We then calculate the anomalous and couplings in the on-shell renormalization scheme. A simple form of those anomalous couplings are also derived in the approximation of large mass expansion. In Sec. III we evaluate the numerical effects of those anomalous couplings on the cross section of . After taking into account the constraints from dark matter searches at the LHC, we discuss the potential of measuring the loop effects of those dark scalars and fermions through the channel at the CEPC. Finally, we conclude in Sec. IV.

## Ii Anomalous couplings of and

We calculate the loop correction to the scattering of from a vector-like fermion and a scalar , where and are the momenta of the electrons and muons. The new fermion and scalar couple to the SM particles through the following interaction:

(1) |

where is the usual covariant derivative with and being the and generators of the field (), respectively, and and being the corresponding coupling strengths. and are the weak eigenstate gauge fields, which are related to the weak bosons by , and , where , with being the weak mixing angle. denotes a general scalar potential. denotes the Yukawa interaction of , and ; depending on the weak isospins of the and fields ( and ), they may couple to either the SM left-handed doublet when , or the right-handed singlet when . Besides, the gauge interaction in the first two terms in Eq. 1 also enters into the loop corrections. We assume no Yukawa interaction of the electron with the new physics fields and , and ignore the electron mass in our calculations. We shall elaborate the anomalous couplings induced by the Yukawa interaction and the purely gauge interaction separately.

The demand that contain an electrically neutral component as the dark matter candidate restricts the value of as follows,

(2) |

In this section we first calculate the anomalous couplings of and for generic and . The analytical results of our simplified model are for arbitrary representations of and , and they can be applied to many new physics models. The requirement of having the dark matter component in is taken into account in our numerical discussion given in Sec. III.

### ii.1 Anomalous couplings induced by the Yukawa interaction

#### ii.1.1 The coupling scenario

When , and couple to the SM left-handed doublet through the following Yukawa interaction,

(3) |

where is the coupling strength and are the Clebsch-Gordan (CG) coefficients to render invariant under the gauge group. The indices label the components of the , and fields, respectively. At one-loop level, the process receives corrections from the diagrams in Fig. 1. Notice that the Yukawa interaction only enters into the self-energy correction of the muon , but does not enter into the self-energy corrections of the weak gauge bosons. Therefore, it does not renormalize the weak sector.

We parameterize the loop corrections to the conserving anomalous couplings of with as following Stange and Willenbrock (1993); Cao et al. (2009b)

(4) |

where is the electrical coupling strength and . Among the four interaction terms, only the vector and axial vertices and are renormalized by the vertex counterterms. The remaining loop-induced Lorentz structures are ultra-violet (UV) finite by themselves, therefore, we decompose Eq. 4 by,

(5) |

where the couplings with subscriptions denote the contributions from the triangle loop corrections, and the terms represent the contributions from the vertex counterterms, as depicted in Fig. 1. They are given by

(6) |

where are the wave function renormalization constants of , and

(7) |

with and being the electroweak quantum numbers of . The renormalization constants ’s are determined from the muon self-energy corrections (see Fig. 1),

(8) |

where are the left/right-handed chirality projectors and is the muon mass. In the on-shell scheme, the finite parts of the counterterms are determined by the requirement that the residue of the fermion propagator at the mass pole is equal to one Denner (1993); Aoki et al. (1982). Therefore, the wave function renormalization constants are fixed by,

(9) |

where denotes taking the real part.

Now we turn to the triangle loop contributions. We first evaluate the and triangle integrals, and derive the and vertices using the defining relations and . Taking the loop diagram in Fig. 1 as an example, upon summing over the loop particle components , it is factorized into a generic one-loop integral, multiplied by

(10) |

where is the third angular momentum operator of the field. The loop is obtained by substituting with in the formula above, yielding simply . The evaluation of the triangle loop diagram in Fig. 1 is similar, giving and as group factors. Note that , we thus have the relation . We also have due to the invariance. Here we choose and as the independent model parameters, and is worked out to be

(11) |

The generic one-loop triangle integrals are evaluated by reducing them fully into the and scalar functions Passarino and Veltman (1979); ’t Hooft and Veltman (1979). After summing the triangle loop contributions with the counterterms according to Eq. 5, we obtain the full results in terms of scalar functions, which are listed in App. C.1. To manifest the cancellation of the UV-divergences, and also to show the decoupling effect explicitly when the loop particles mass is large, we derive those anomalous couplings in the approximation of large mass expansion. See App. B for the approximate expressions of the and scalar functions. The results are given as follows,

(12) |

The and terms are correlated with respect to the electromagnetic current conservation Stange and Willenbrock (1993); Cao et al. (2009b) and appear as

which is the so-called anapole moment term. The anapole moment vanishes at . We also see that the correction to vertex in Eq. 4 vanishes in the Thomson limit, i.e., (and thus ), as consistent with the electrical charge renormalization.

#### ii.1.2 The coupling scenario

Now we consider the case that and couple to the SM right-handed singlet through the following Yukawa interaction,

(13) |

where , with . The loop-induced anomalous couplings therefrom are similar to the coupling scenario, because they come from the same sort of diagrams in Fig. 1. Now , since vertex does not conserve the quantum number. We have by the gauge symmetry. Choosing as the independent quantum number, we present the full result of the anomalous couplings in terms of scalar functions in App. C.2. In the approximation of large mass expansion, they become

(14) |

Note that the remarks following Eq. II.1.1 also apply to the results above.

### ii.2 Anomalous couplings induced by the purely gauge interaction

The gauge interactions enter into the loop corrections of the channel propagators, as shown in Fig. 2. For convenience, we collect Fig. 2 through Fig. 2 and also parametrize the parts apart from the initial state matrix element as the anomalous couplings,

(15) |

where . As in Eq. 5, we decompose the couplings into the loop and counterterm parts,

(16) |

where the couplings with subscriptions denote the contributions from the two-point loop corrections. The counterterm parts of the anomalous couplings are given as,

(17) |

where , and the terms in the brackets come from the vector-vector counterterms depicted in Fig. 2, while the terms are from the vertex counterterms shown in Fig. 2. Writing , they are given as follows:

(18) |

with

(19) |

where , , and are the renormalization constants of wave function, electrical charge, weak mixing angle and the -boson mass, respectively. Since and carry the same electroweak quantum numbers, the initial state counterterms in Fig. 2 equal those in Fig. 2, and can be written as,

(20) |

which are UV finite by themselves.

To renormalize the weak sector parameters, in the on-shell mass scheme we fix the mass and wave function renormalization constants by requiring that the renormalized parameters of the theory actually be equal to the physical parameters, i.e., the renormalized mass parameters be equal to the real parts of the poles of the corresponding propagators, and the residues of the propagators of the renormalized fields be equal to one. We further renormalize the electrical charge by equating it with the -coupling for on-shell external particles in the Thomson limit. In the on-shell scheme the weak mixing angle is a derived quantity. We follow Sirlin’s definition Sirlin (1980) to define it as using the renormalized gauge boson masses. To the one-loop order we obtain

(21) |

Now we evaluate the loop diagrams in Fig. 2 through 2. Upon summing over the loop particle components, they are factorized into the corresponding generic self-energy integrals, multiplied by the group factors and , where

(22) |

are the Casimir invariants in representation and of the scalar and the fermion , and and are their dimensions. As before, the generic self-energy integrals are reduced to one-loop scalar functions. We present the full result of the anomalous couplings in terms of scalar functions in App. C.3. In the approximation of large mass expansion, they become

(23) |

and . Notice that the purely counterterm corrections to vertex exactly, since the electrical charge is renormalized to the coupling strength at zero momentum transfer.

## Iii Numerical results

We choose our observable to be the deviation from the SM tree-level cross section
,
where stands for the SM tree-level cross sections ^{2}^{2}2The SM corrections to have been calculated in Ref. Passarino and Veltman (1979); Bardin et al. (1997); Hahn et al. (2003). and is the sum of the SM cross section and the new physics one-loop virtual corrections. Therefore, is the cross section of the interference between the SM and the new physics virtual corrections. Note that this is both theoretically consistent, as the corrections to the cross sections are complete to this order in the perturbation series, and also numerically robust because the new physics one-loop amplitude squared is negligible compared to the interference contribution.
Ignoring the electron mass, the correction is given below in terms of the anomalous couplings,

(24) |

The terms in the first round brackets come from the SM amplitudes, while those from the second round brackets come from the new physics loop corrections. and label the chirality of the initial state electrons (positrons). The index , running through {}, labels the SM vector and axial-vector couplings of the final state pair with . Note that . The index , running through {}, labels the new physics loop-induced contributions, with

(25) |

Note that in the formula above, include both the Yukawa and the gauge corrections to the matrix elements; see Eqs. 4 and 15. are from the counterterm corrections to the initial state matrix elements; see Eqs. 20. The functions are given by

where , , and are the usual Mandelstam variables.

Now we are ready to discuss our numerical results. The SM input parameters are chosen as follows Mohr et al. (2015):

while the weak mixing angle is fixed by . The loop corrections are calculated with the help of LoopTools package Hahn and Perez-Victoria (1999); van Oldenborgh (1991). We choose the independent model parameters to be the Yukawa coupling strength , the loop particle mass , and the quantum numbers of the field .