# Lightness of Higgs Boson and Spontaneous CP Violation in Lee Model

###### Abstract

We proposed a mechanism in which the lightness of Higgs boson and the smallness of CP-violation are correlated based on the Lee model, namely the spontaneous CP-violation two-Higgs-doublet-model. In this model, the mass of the lightest Higgs boson as well as the quantities and are in the limit (see text for definitions of and ), namely the CP conservation limit. Here and are the measures for CP-violation effects in scalar and Yukawa sectors respectively. It is a new way to understand why the Higgs boson discovered at the LHC is light. We investigated the important constraints from both high energy LHC data and numerous low energy experiments, especially the measurements of EDMs of electron and neutron as well as the quantities of B-meson and kaon. Confronting all data, we found that this model is still viable. It should be emphasized that there is no standard-model limit for this scenario, thus it is always testable for future experiments. In order to pin down Lee model, it is important to discover the extra neutral and charged Higgs bosons and measure their CP properties and the flavor-changing decays. At the LHC with , this scenario is favored if there is significant suppression in the decay channel or any vector boson fusion (VBF), V+H production channels. On the contrary, it will be disfavored if the signal strengths are standard-model-like more and more. It can be easily excluded at level with several at future colliders, via the accurately measuring the Higgs boson production cross sections.

GBKsong

## I Introduction

How to realize the electro-weak gauge symmetry breaking and CP violation are important topics in the standard model (SM) and beyond the SM (BSM) in particle physics. In order to induce the spontaneous gauge symmetry breaking, the Higgs mechanism was proposed in 1964 higgs (). Meanwhile in the SM the CP violation is put by hand via the complex Yukawa couplings among Higgs field and fermions, namely Kobayashi and Maskawa (KM) mechanism KM (). In 1973, Kobayashi and Maskawa KM () proposed that if there are three generations of fermions, there would be a nontrivial phase which leads to CP violation in the fermion mixing matrix (CKM matrix KM ()cabbibo ()). In a word, one single scalar field plays the two-fold roles. In the SM, only one doublet Higgs field is introduced. After spontaneous symmetry breaking, there exists one physical scalar, the Higgs boson. It is essential to discover and measure the properties of the Higgs boson, in order to test the SM or discover the BSM.

### i.1 Status of experimental measurements on new scalar boson

Experimentally, in July 2012, both CMS disc1 () and ATLAS disc2 () discovered a new boson with the mass around in and final states with the luminosity of about 10. At the LHC, the SM Higgs boson can be produced through the following three processes: (1)gluon-gluon fusion (ggF); (2)vector boson fusion (VBF); (3) associated production with a vector boson (V+H). It can also be produced associated with a pair of top quarks due to the large , but the cross section is suppressed by its phase space and parton distribution function (PDF) of proton. A SM Higgs boson would mainly decay to fermion pairs (, or if heavier than ), massive gauge boson pair (), massless gauge boson pair (), etc. The decay properties for a SM higgs boson are listed in Table 1, for the production and decay properties, see also the reviews pro1 () and pro2 ().

Decay Channel | Branching Ratio () | Relative Uncertainty () |

Total Width |

The updated searches by CMS updc1 (); updc2 (); updc3 (); updc4 () and ATLAS upda1 (); upda2 (); upda3 (); upda4 ()
with the luminosity of about 25 till the end of 2012 ^{1}^{1}1Some new analysis updated
in 2014 are used as well which modify the old results a little bit.
gave the significance and signal strengths (defined as the ratios between observed
and the corresponding SM prediction) for some channels. Because the measurements will be utilized to constrain the new model in this paper, we list the results in
Table 2 for CMS and Table 3 for ATLAS.^{2}^{2}2The VBF events are usually easy to tag with
two jets which have large invariant mass, while sometimes it is difficult to tag a gluon fusion event.

(VBF/V+H) | (ggF) | (combined) | significance | |
---|---|---|---|---|

(VBF/V+H) | (ggF) | (combined) | significance | |
---|---|---|---|---|

The new boson has a combined mass GeV and it is also favored as a particle in spin and parity by the data updc2 (); CPP (); CPP2 () if we assume that there is no CP violation induced by this boson.

The experimental measurements of the new particle are in agreement with the SM predictions within the current accuracy. In the SM the electro-weak fitting results Fit () also favors a light one. It allows a SM Higgs boson lighter than 145GeV at C.L. inferred from the oblique parameters obl () with fixed . However there are still spacious room for the BSM. For example, if we assume that the new particle is a CP-mixing state, the general effective interaction for can be written as updc2 (); CPmix (); CPmix2 ()

(1) |

with . Define

(2) |

where are the partial width for pure CP even (odd) state with . Direct search by CMS gives at C.L. which leads to updc2 (). In a renormalized theory, and which are loop induced are expected to behave as , so they are still not constrained by current LHC data.

### i.2 The issue of lightness of new scalar boson in the SM and BSM

BSM is well motivated because SM can’t account for the matter-dominant universe and provide the suitable dark matter candidate. However BSM scale is usually pushed to a much higher value than that of weak interaction, given the great success of the SM. In such circumstance, the GeV scalar boson is unnatural. In other word, the lightness of the new scalar must link to certain mechanism. The issue of the lightness of the new scalar differs in the SM and the BSM. In the SM, we cannot predict the mass of Higgs boson, and the Higgs boson with the mass GeV simply implies that the interactions are in the weak regime. For example the Higgs boson self-coupling

(3) |

Compared with the strong interactions at low energy, the mass of particle (or we call it which plays a similar role as the Higgs boson) appears at a typical scale . Thus we can argue that the new boson with mass 125.7GeV is rather light compared with the strong interaction. As a side remark, the pion mass is light compared with due to the approximate chiral symmetry. This has motivated the idea that new scalar boson may be the pseudo-Nambu-Goldstone boson for certain unknown symmetry breaking.

Theoretically, in some BSM models there exists a light scalar naturally. For example, (1) in the minimal super-symmetric model, the lightest Higgs boson should be lighter than 140GeV including higher-order corrections susy () (at tree level it should be lighter than the mass of boson); (2) in the little higgs model, a Higgs boson which is treated as a pseudo-Nambu-Goldstone boson must be light due to classical global symmetry and it acquires mass through quantum effects only LH (); (3)similarly, anomalous in scale invariance can also generate a light Higgs boson as well scale (); (4) the lightness of Higgs boson can intimately connect with the spontaneous CP violation zhu (). While the first three approaches base on the conjectured symmetry, the last one utilizes the observed approximate CP symmetry. Historically Lee proposed the spontaneous CP violation in 1973 Lee () as an alternative way to induce CP violation. For the fourth approach, Lee’s idea is extended to account for the lightness of the observed Higgs boson.

### i.3 The lightness of new scalar boson and spontaneous CP violation

CP violation was first discovered in neutral K-meson in 1964 CP (). Experimentally people have already measured several kinds of CP violated effects in neutral K- and B-meson, and charged B meson systems PDG (). These CP violation can be successfully accounted for by the CKM matrix, which is usually parameterized as the Wolfenstein formalism wolf ()

(4) |

The Jarlskog invariant PDG ()Jarskog ()

(5) |

measures the CP violation in flavor sector. The smallness of means the smallness of CP-violation in the real world in SM. Another possible explicit CP-violation comes from the term

(6) |

in the QCD lagrangian str ()str2 (). The parameter is strongly constrained by the neutron electric dipole moment (EDM) measurement dncal ()dn (), namely . Why is extremely small is known as the strong CP problem. It is often interesting, necessary and useful to search for other sources of CP violation beyond the KM-mechanism. As a common reason, for example, CP-violation is one of the conditions to produce the matter-antimatter asymmetry in the universe today sakh (), but SM itself cannot provide the first order electro-weak phase transition and large enough CP-violation to get the right asymmetry between matter and anti-matter PDG (); asy (); bar (); cohen ().

In 1973, Lee proposed a 2HDM (Lee model) Lee () in which all parameters in the scalar potential are real but it is possible to leave a nontrivial phase between the vacuum expectation values (VEV) of the two Higgs doublets. CP can be spontaneously broken in this model. Chen et. al. slc () proposed the possibility that the complex vacuum could lead to a correct CKM matrix, which means that we can set all Yukawa couplings real thus the complex vacuum would become the only source of CP-violation. It is also a possible way to solve strong CP problem, for example, in spontaneous CP-violation scenarios, arises only from the determinant of quark mass matrix. Assuming at tree level, the loop corrections can generate naturally small Gino ()Barr (), the so-called “calculable ” str (). Without imposing symmetry GW (), the Yukawa couplings are arbitrary which will generate the flavor changing neutral currents (FCNC) at tree level. FCNC is severely constrained by experiments. Cheng and Sher proposed an ansatz CS () that the flavor changing couplings should be for two fermions with mass and . One of the authors of this paper had proposed a mechanism zhu () to understand the lightness of Higgs boson in the limit. In this paper, we will explore the relation between the smallness of CP-violation and the lightness of Higgs boson in a similar way in Lee model further. Specifically we will study the full phenomenology of the Lee model and to see whether this model is still viable confronting LHC data and numerous low energy measurements.

We should mention that there are also cosmological implication for Lee model. In this model, CP is a spontaneously broken discrete symmetry thus it may face the domain wall problem Dom () during the electro-weak phase transition. It is argued that if there is a small initial bias thus one of the vacuum states is favored, the domain walls would disappear soon Dom ()domsol1 (), for example, if there is small explicit CP-violation expli (). In the soft CP breaking model, the electro-weak baryogenesis effects is estimated by Cohen et. al. cohen () at early time, and was estimated again by Shu and Zhang shu () after including LHC data. They found that the observed matter-anti-matter asymmetry can be explained. It is also discussed numerically that an inflation during the symmetry breaking would forbid the domain wall production domsol2 ().

This paper is organized as following. Section II presents the Lee model and the scenario that lightness of Higgs boson and smallness of CP violation are correlated. Section III and IV contain the constraints on Lee model from high energy and low energy data respectively. Section V studies the perspectives for Lee model for future experiments. The last section collects our conclusions and discussions.

## Ii The Lee model: mass spectrum and couplings

We begin with the description of Lee model Lee () assuming that in the whole lagrangian there are no explicit CP-violation terms,
which means all the CP-violation effects come from a complex vacuum ^{3}^{3}3For a review on two Higgs doublet models (2HDM), the interested reader can read Ref. 2HDM (). For the Lee model, the interactions of scalar fields read Lee ()

(7) |

Here

(8) |

are the two higgs doublets. We can get the masses of gauge bosons

(9) |

by setting . Defining as the real(imaginary) part of , we can write a general potential as

(10) | |||||

in which we can always perform a rotation between and to keep the coefficient of term zero in . We can also write the general Yukawa couplings as

(11) |

in which and all Yukawa couplings are real.

Minimizing the higgs potential, and for some parameter choices, we can get a nonzero phase difference between two higgs VEVs, which would induce spontaneous CP violation. We can always perform a gauge transformation to get at least one of the VEVs real like in (8). When , we can express

(12) | |||||

(13) |

is identified as as usual. We also have an equation about

(14) |

which requires . Of course, the couplings must keep the vacuum stable, for the conditions see Appendix A for details.

All the CP-violation effects in the real world are small (see the data in PDG ()) corresponding to the
smallness of the off-diagonal elements in the CKM-matrix which leads to the smallness of the Jarlskog
invariant. As a limit, when ,^{4}^{4}4We write
for short in this paper.
or we may write instead since always holds,
there would be no CP-violation in the scalar sector. The CKM-matrix would be real thus there would be no
CP-violation in flavor sector as well. In this paper we will consider the small limit,
in which all CP-violation effects tends to zero as . We treat the whole world as an
expansion around the point without CP-violation.

The two higgs doublets contain 8 degrees of freedom, 3 of which should be eaten by massive gauge bosons as Goldstones. So there are 5 physical scalars left, 2 of which are charged and 3 of which are neutral. If CP is a good symmetry, there will be 2 CP even and 1 CP odd scalars among the 3 neutral ones. However, when CP is spontaneously breaking, the CP eigenstates will mix with each other thus the neutral scalars have no certain CP charge. We have the Goldstones as

(15) | |||||

(16) |

The charged Higgs boson is the orthogonal state of the charged Goldstone as

(17) |

and its mass square should be

(18) |

While for the neutral part, we write the mass square matrix as in the basis . The symmetric matrix is

(19) |

and its three eigenvalues correspond to the masses of three neutral bosons.

We expand the matrix in series of as

(20) |

to get the approximate analytical behavior of its eigenvalues and eigenstates. Certainly we have

(21) |

which means a zero eigenvalue of thus there must be a light neutral scalar when is small. To the leading order of , for the lightest scalar , we have

(22) | |||||

(23) | |||||

While for the two heavier neutral Higgs, we have

(24) |

in which are the other two eigenvalues of and

(25) |

where . The physical states are

(26) |

For all the details about scalar spectra and its small expansion series, the interested reader can see Appendix B.

From the Yukawa couplings we will get the mass matrixes for fermions as

(27) | |||

(28) |

We can always perform the diagonalization for with matrixes and as

(29) |

And is the CKM matrix.

In this scenario, the couplings for the discovered light Higgs boson should be modified from SM by a factor as

(30) | |||||

where the factors and must be real, but and may be complex. According to (C.1)-(C.4) in Appendix C, to the leading order of , we straightforwardly have

(31) | |||||

(32) | |||||

(33) |

and the coupling including charged higgs should be

(34) |

where

(35) | |||||

(36) | |||||

We choose all the nine free parameters as nine observables in Higgs sector: masses of four scalars and ; vacuum expected values and two mixing angles for neutral bosons. The mixing angles are represented as and .

(37) |

The just stands for the vertex strength ratio comparing with that in SM^{5}^{5}5There is a sum rule
due to spontaneous electro-weak symmetry broken, thus only two of the are free,
and here is just the in (31).. In the scalar sector, for non-degenerate neutral Higgs
bosons, a quantity measures the CP violation effects 2HDM ()K ()^{6}^{6}6If at least two of the neutral
bosons have degenerate mass, we can always perform a rotation among the neutral fields to keep .,
while in Yukawa sector, the Jarlskog invariant Jarskog () measures that. In this scenario,
to the leading order of , we have

(38) |

In order to calculate J, we define matrix as

(39) |

We can always choose a basis in which the diagonal elements of are zero. Thus

(40) |

in which using equations (27) and (28), to the leading order of , we have

(41) | |||||

(42) | |||||

To the leading order of , the determinant

(43) | |||||

where , thus

(44) |

According to the equations (38), (44), and (22), we propose that the lightness of the Higgs boson and the smallness of CP-violation effects could be correlated through small since both the Higgs mass and the quantities and to measure CP-violation effects are proportional to at the small limit.

In the following two sections, we will study whether the Lee model is still viable confronting the current numerous high and low energy measurements. From Eq. (31), it is quite clear that couplings of discovered scalar boson differ from those in the SM, namely Lee model does not have SM limit. Provided that LHC obtained only a small portion of its designed integrated luminosity, there would be spacious room for Lee model. In the long run, LHC and future facilities have the great potential to discover/exclude Lee model. We will discuss this part in section V.

## Iii Constraints from High Energy Phenomena

In this model there are two more neutral bosons and one more charged boson pair comparing with SM, these degree of freedoms may affect on the physics at electro-weak scale, and they could also be constrained by direct searches at the LHC. For the discovered boson, SM predicts the decay branching ratios for a Higgs boson with mass 125.7GeV in Table 1. However in Lee model, the modified couplings will change the total width and branching ratios due to equations (31)-(34), together with the production cross sections modified by (33) for gluon fusion and (31) for vector boson fusion and the associated production with vector bosons. Of course, this model may also affect top physics because the couplings between Higgs boson and top quark are not suppressed and it may also change the flavor changing couplings especially for top quark. Thus it is necessary to discuss the constraints to this model from high energy phenomena.

### iii.1 Constraints on heavy neutral bosons

A heavy Higgs boson may decay to , , , (for neutral bosons heavier than ), or (for light charged Higgs and a neutral boson heavier than ). Based on the searches for the SM Higgs boson using diboson final state HH (), masses and couplings of the other two heavier neutral Higgs bosons should be constrained by the data. For a neutral Higgs boson heavier than 350GeV, the resonance search tt () may also give some constraints.

In this scenario, the totol width of a heavy boson can be expressed as

(45) |

where , , , correspond to massive gauge boson pairs, charged Higgs pair, neutral Higgs pair and top quark pair final states respectively. The partial decay width for a heavy neutral Higgs with mass are

(46) | |||||

(47) | |||||

(48) | |||||

(49) |

in unit of its mass. Here we have the vertices

(50) |

The couplings .

The signal strength is defined as

(51) |

for a production channel. The for different channels. For a heavy Higgs with , is very close to 1; while for , has a minimal value of about when . According to (46) -(49), we can estimate that for both and , .

Thus according to the figures in HH (), we have three types of typical choices for the mass of two heavy neutral higgs particles in Table 4. (Here we write the mass of the lighter boson and the heavier one .)

Case | Allowed (GeV) | Allowed (GeV) |
---|---|---|

I | ||

II | ||

III |

### iii.2 Constraints due to Oblique Parameters

After the discovery of the new boson, there are new electro-weak fit for the standard model Fit (). Choosing and , the oblique parameters obl () are

(52) |

with the correlation coefficient between two quantities; or

(53) |

with fixed , where R is the correlation coefficient between S and T. The basic mathematica code to
draw the S-T ellipse can be found on the webpage STcode ()^{7}^{7}7Assuming Gaussian distribution,
the second should be 6.0 instead of 6.8 in the code. See the 36th chapter (statistics)
of the reviews in PDG PDG (), in its 2014 updated version please see the 38th chapter instead.. The contribution
to S and T parameters due to multi-higgs doublets were calculated in ST () (see the formulae in 2HDM ()).

(54) | |||||

(55) | |||||

where is the rate of the coupling to that in SM ( represents above-mentioned ) and . is the reference point for Higgs Boson, and . The functions read (following the fomulae in 2HDM ())