# One-Loop Radiative Correction to the Triple Higgs Coupling

in the Higgs Singlet Model

###### Abstract

Though the 125 GeV Higgs boson is consistent with the standard model (SM) prediction until now, the triple Higgs coupling can deviate from the SM value in the physics beyond the SM (BSM). In this paper, the radiative correction to the triple Higgs coupling is calculated in the minimal extension of the SM by adding a real gauge singlet scalar. In this model there are two scalars and and both of them are mixed states of the doublet and singlet. Provided that the mixing angle is set to be zero, namely the SM limit, is the pure left-over of the doublet and its behavior is the same as that of the SM at the tree level. However the loop corrections can alter -related couplings. In this SM limit case, the effect of the singlet may show up in the -related couplings, especially the triple coupling. Our numerical results show that the deviation is sizable. For (see text for the parameter definition), the deviation can be . For , the can reach . The sizable radiative correction is mainly caused by three reasons: the magnitude of the coupling , light mass of the additional scalar and the threshold enhancement. The radiative corrections for the couplings are from the counter-terms, which are the universal correction in this model and always at O(1%). The coupling can be a complementarity to the triple coupling because of the high precision measurement. In the optimal case, the triple coupling is very sensitive to the BSM physics, and this model can be tested at future high luminosity hadron colliders and electron-positron colliders.

## I Introduction

The standard model (SM) has been extensively tested, especially the deviations for the gauge sector are strongly constrained by the electro-weak precision measurements from the Large Electron-Positron Collider (LEP) Agashe:2014kda (), Tevatron and the large hadron collider (LHC). However, the Yukawa sector and the scalar sector are two regimes which are still not well probed. Since the discovery of the Higgs boson at the Large Hadron Collider (LHC) in 2012 Aad:2012tfa (); Chatrchyan:2012xdj (), the most important task is to measure the properties of the scalar accurately. The measurements will help us understand the nature of the electro-weak symmetry breaking mechanism (EWSB) Englert:1964et (); Higgs:1964ia (); Guralnik:1964eu (); Kibble:1967sv (). If there exists new physics beyond the SM (BSM), it is believed that it is related with the Higgs couplings more or less. The Higgs boson is a door to the unknown new world.

Current measurements of the Higgs couplings with gauge bosons tend to be the SM values. At the same time Higgs couplings with the third generation fermions are inferred from the Higgs production processes at the LHC, which are also consistent with those in the SM. Usually for the model construction, the Higgs couplings with fermions and gauge bosons will have the SM limit at the electro-weak scale. However the triple Higgs coupling can deviate from the SM value largely in this limit. Such feature of the triple Higgs coupling has been studied extensively in the two Higgs doublet model (THDM) Kanemura:2004mg (); Osland:2008aw (), inert Higgs doublet model (IHDM) Arhrib:2015hoa (), Higgs triplet model (HTM) Aoki:2012jj () and models with an additional heavy neutrino Baglio:2016ijw ().

Searching for BSM physics is one of the most important goals of high energy physics. The most direct way is to increase the energy of the colliders and see whether there are new heavy resonances, while it is always hard or even impossible to construct the very high energy colliders because of the limitations from the expanses, technologies and so on. However there are other methods to achieve this goal. The new heavy particles will leave footprints at the electro-weak scale through loop effects. We may have indirect signals for the BSM through some physical quantities which are sensitive to the heavy particles.

The minimal extension of the SM in the scalar sector is to add a real gauge singlet. The Higgs singlet model (HSM) has been studied exhaustively in a lot of papers. For example, Ref. Robens:2015gla () studied a model which includes a symmetry spontaneously breaking real Higgs singlet and the author considered the theoretical and phenomenological constraints of this model. Ref. Chen:2014ask () explored the resonant di-Higgs production in the 14TeV hadron collider with an additional intermediate, heavy mass Higgs boson. Ref. Barger:2007im () considered two scenarios: there was (no) mixing between the SM Higgs and the singlet. Then, they analyzed the constraints from electro-weak precision observables, LHC Higgs phenomenology and dark matter phenomenology. Ref. Bojarski:2015kra (); Costa:2015llh () emphasized the heavy-to-light Higgs boson decay at the NLO. Ref. Kanemura:2015fra () focused on the one-loop radiative corrections in the HSM and they performed the numerical calculations for the , , , , , couplings, but not for triple coupling, which is the main topic in this paper.

In the following, we will make a careful analysis of the triple coupling up to one-loop level in this model in the SM limit. There will be an universal deviation from the SM predictions for the , couplings arising from the wave-function renormalization constant . The numerical results show that the universal correction is small. For the triple coupling, there are still couplings (see Appendix A) in this limit. When the mass of the additional scalar is [90,150] GeV and the coupling is order one, the radiative correction to the triple coupling can be or even larger in the vicinity of double Higgs production. It may be measured at future hadron colliders and electron-positron colliders.

This paper is organized as follows. In Sec. II, we give a detailed description of the model including the theoretical constraints on the parameter space and the radiative correction to the triple & couplings. In Sec. III, we present the numerical results. Sec. IV is devoted to the conclusions and discussions. Feynman rules, related Feynman diagrams and calculational details are collected in the Appendix.

## Ii Model

We introduce a real additional gauge singlet with hyper-charge besides the SM Higgs doublet . Then, we can write the scalar potential as

(1) |

Evidently, the singlet doesn’t have any Yukawa interactions or gauge interactions with the SM fields. The scalar fields in the unitary gauge can be parametrized as

Without loss of generality, we can set to be zero by shifting the , namely the redefinition of the field. After EWSB, the two tadpoles are . are the coefficients of the fields in the Lagrangian. At tree level, . Then . Mass terms of the scalar fields are

(2) | ||||

After diagonalizing the mass matrix, we get the following expressions

(3) | ||||

In the above expressions, we use instead of to avoid the confusion with the parameter in the Lagrangian. are the simplified notations for . From now on, we will choose the parameters as the inputs. According to the definitions of in Eq. 2 and Eq. 3 , can be expressed by the new inputs as

(4) |

### ii.1 Constraints on the parameter space

In the SM limit (), will decouple from the fermions and gauge bosons because of the scaling factor . Thus, it will evade all the present experimental constraints. But the SM Higgs will still couple with by the interacting vertices . And this makes great influence on the triple coupling which is discussed later. All the analyses below will be carried out under the SM limit assumption, namely . When is very large, the scalar potential will become . It must be bounded from below, so we have

(5) |

Another constraint we should consider is the so-called perturbative unitarityLee:1977eg (). S-wave amplitude should satisfy the relation , where is given by . Here, are the Mandelstam variables as usual, and is the scattering amplitude. According to the Goldstone equivalence theorem, massive vector boson is dominated by the longitudinal polarization at high energy. So we need only to consider the two-to-two scattering processes with initial and final states: . Similar analyses have been discussed in many papers Dawson:1988va (); Cynolter:2004cq (). This is a matrix, but it will be reduced into a matrix in the SM limit. A subtlety one may caution is an extra for the same initial and final states, which is often ignored in many papers. After some trivial calculations, we have a matrix

(6) |

Three eigenvalues of the matrix are

Then, we have the constraints from perturbative unitarity

(7) |

In the SM limit, . Together with the bounded constraints in Eq. 5, we get the following parameter space in Fig. 1. The interesting feature is that there are no constraints on .

### ii.2 One-loop radiative correction to the triple couplings in the SM limit

We will calculate the deviation of the triple coupling from the SM value originated from one-loop radiative correction in the SM limit. During the calculations, we adopt the conventions from Ref. Denner:1991kt (). There is no doubt that the loop particles must be the additional scalar . To gauge the deviation from the SM value, we define as

(8) |

In the following, we will present the numerical results for for the chosen model parameters.

Similarly, The deviation of the coupling from the SM value originated from one-loop radiative correction in the SM limit is defined as

(9) |

The analytical expressions can be found in Appendix C. The related Feynman rules and calculations in the SM limit are given in the Appendix.

## Iii Numerical results

In this section, we will do some numerical evaluations of for different model parameters. We set as in the SM. The deviation of is mainly determined by , where one of the Higggs boson with momentum is off-shell. The dominant contribution is from the triangle diagram which is proportional to . We choose the allowed value of , respectively, and study the dependence on using LoopToolsHahn:1998yk (). Behaviours of are shown in Fig. 2, Fig. 3.

It is easy to see that the correction to the triple coupling is sensitive to . If the coupling can reach order one, the deviation can be very large for in the vicinity of double Higgs production (). For , the can be . For , the can almost approach . The sizable radiative correction is mainly caused by three reasons: order one coupling , light mass of the additional scalar and the threshold enhancement. In this case, the triple coupling is very sensitive to BSM physics. Experimentally, The deviation of the triple coupling may be probed through production channel at future hadron colliders Dawson:2013bba (); Yao:2013ika (); Cao:2013cfa (). The model may also be probed through and production channels at future electron-positron colliders Abramowicz:2013tzc (); Asner:2013psa (); Tian:2010np (). At a low energy electron-positron colliders with 240GeV or so and high luminosity, can also be detected indirectly McCullough:2013rea (); Shen:2015pha (); Cao:2014ita ().

Additionally, we consider the comparison between the triple and coupling. Numerical results are shown in Fig. 4:

We can find that the is very small, compared to the triple coupling. Owing to the high precision measurement of coupling, it can be complementary to the triple coupling.

## Iv Conclusions

The radiative correction to the triple coupling is calculated in the minimal extension of the SM by adding a real gauge singlet scalar. In this model there are two scalars and and both of them are mixed states of the doublet and singlet. Provided that the mixing angle is set to be zero, is the pure left-over of the doublet and its behavior is the same as that of the SM at the tree level. However the radiative corrections from the singlet can alter -related couplings which deviate from the SM values. Our numerical results show that the deviation is sizable. For , the can be . For , the can reach . The sizable radiative correction is mainly caused by three reasons: the magnitude of the coupling , light mass of the additional scalar and the threshold enhancement. The radiative correction for the coupling can be a complementarity to the triple coupling because of the high precision measurement. In the optimal case, the triple coupling is very sensitive to BSM physics, and this model can be tested at future high luminosity hadron colliders and electron-positron colliders.

### Acknowledgements

We would like to thank Gang Li, Chen Shen, Chen Zhang and Yang Li for helpful discussions. This work was supported in part by the Natural Science Foundation of China (Grants No. 11135003 and No. 11375014).

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## Appendix

## Appendix A Related Feynman rules

## Appendix B Tadpole and self-energy of the SM Higgs from the additional scalar

Tadpole of the SM Higgs and the renormalization constant :

Self-energy of the SM Higgs and the renormalization constants :

## Appendix C Calculational details for the one-loop radiative correction

One-loop radiative correction for the triple coupling:

Assuming the Higgs bosons with momentum are on shell, while the Higgs boson with momentum is off shell, that is . We can get the following analytical expression for the triple coupling in the SM limit, which is the deviation from the SM prediction.

are the coefficients of the vertex up to one-loop level in the HSM and SM respectively, but is the tree level coefficient in the SM. If there is an imaginary part in , we just extract the real part. Because the imaginary part is not observable at this order due to the interference with tree level amplitude.

Similarly, we get the one-loop radiative correction for the coupling:

The different point for the deviation of the coupling is that it is independent of .