Precision measurements constraints on the number of Higgs doublets

# Precision measurements constraints on the number of Higgs doublets

A. E. Cárcamo Hernández    Sergey Kovalenko    Iván Schmidt Universidad Técnica Federico Santa María
and
Centro Científico-Tecnológico de Valparaíso
Casilla 110-V, Valparaíso, Chile
July 17, 2019
###### Abstract

We consider an extension of the Standard Model with an arbitrary number of Higgs doublets (NHDM), and calculate their contribution to the oblique parameters and . We examine the possible limitations on from precision measurements of these parameters. In view of the complexity of the general case of NHDM, we analyze several benchmark scenarios for the Higgs mass spectrum, identifying the lightest CP-even Higgs with the Higgs-like particle recently observed at the LHC with the mass of GeV. The rest of the Higgses are put above the mass scale of GeV, below which the LHC experiments do not detect any Higgs-like signals except for the former famous one. We show that, in a scenario, with all the heavy Higgses degenerate at any scale, there are no limitations on the number of the Higgs doublets. However, upper limits appear for certain not completely degenerate configurations of the heavy Higgses.

## I Introduction

The recent discovery of the 125 GeV scalar particle at the Large Hadron Collider (LHC) :2012gk (); :2012gu () perfectly fills the vacancy of the Higgs boson necessary for the completion of the Standard Model (SM) at the Fermi scale. Surprisingly, the SM with the Higgs boson in this mass range becomes formally self-consistent up to the Planck scale. In the absence of any signal of physics beyond the SM, this fact drastically strengthens the position of this model as the theoretical basis of particle physics.

Although the new observed scalar state has so far all the properties expected of the SM Higgs boson, it is still possible that it could be a light scalar in a multi-Higgs extension of the SM, or a light supersymmetric Higgs boson, or a Higgs boson coming from a strongly interacting dynamics, where the theory becomes nonperturbative above the Fermi scale and the breaking is achieved through some condensate. Now the priority of the LHC experiments will be to measure precisely the couplings of the observed scalar to fermions and gauge bosons, and to establish its quantum numbers in order to identify it with one of these or some other options. On the other hand, searches for new particles beyond the SM are an essential task of the LHC experiments PDG (); Brooijmans:2014eja (); Carena:2013qia (); Bechtle:2013wla (); Heinemeyer:2014uoa (); Dev:2014yca (); Carena:2014nza (); Bhattacharyya:2014oka ().

In this paper, we consider a multi-Higgs extension of the SM, with an arbitrary number of the Higgs electroweak doublets. Our goal is to study possible bounds on the number of Higgs doublets from the precision measurements of the oblique and parameters.

We assume that the Higgs doublets are identical, with hypercharge equal to . Some features such as the relation between the mass and gauge eigenstates in the scalar sector and the relation of the Higgs vacuum expectation values with the symmetry breaking scale GeV presented in the two Higgs doublet model are still fulfilled when the number of Higgs doublets is increased 2N-model ().

The paper is organized as follows. In Sec. II, we briefly describe the theoretical structure of the Higgs doublet model (NHDM). In Sec. III, we compute the one-loop contribution to the and parameters in the NHDM. The bounds on the number of Higgs doublets coming from and parameter constraints at C.L. are computed in Sec. IV. In Sec. V, we summarize our results.

## Ii The Model

We consider an extension of the SM with copies of the complex weak doublet scalar Higgs fields with hypercharge (NHDM). The model scalar potential, invariant with respect to the SM gauge group, is

 V=12N∑i,j=1\boldmath\mathchar2782ijΦ†iΦj+14N∑i,j,k,l=1\boldmath\mathchar277ij,kl(Φ†iΦj)(Φ†kΦl)+N∑i,j,k,l=1\boldmath\mathchar283ij,kl(Φi\boldmath\mathchar2842Φj)(Φk\boldmath\mathchar2842Φl)†. (1)

where is a Pauli matrix in the space and

 \boldmath\mathchar283ij,kl=−% \boldmath\mathchar283ji,kl=−\boldmath\mathchar283ij,lk, (2)

For simplicity, we assume all the parameters in the scalar potential to be real. Then the Hermiticity of the scalar potential (1) implies

 \boldmath\mathchar277ij,kl=\boldmath\mathchar277ji,lk,\boldmath\mathchar283ij,kl=% \boldmath\mathchar283ji,lk,\boldmath\mathchar278ij=\boldmath\mathchar278ji, (3)

The minimum of the scalar potential is parametrized by N vacuum expectation values

 ⟨Φl⟩=(0vl√2),l=1,2,⋯,N. (4)

We decompose the Higgs fields around this minimum as

 Φl=⎛⎝\boldmath\mathchar286+l1√2(vl+\boldmath\mathchar282l+i% \boldmath\mathchar273l)⎞⎠=⎛⎜⎝1√2(\boldmath\mathchar289l+i\boldmath\mathchar280l)1√2(vl+\boldmath\mathchar282l+i% \boldmath\mathchar273l)⎞⎟⎠ (5)

where

 ⟨\boldmath\mathchar282l⟩=⟨\boldmath\mathchar273l⟩=⟨\boldmath\mathchar289l⟩=⟨\boldmath\mathchar280l⟩=0,l=1,2,⋯,N. (6)

Then the covariant derivative acting on the Higgs doublets takes the form

 D\boldmath\mathchar278Φl = ∂\boldmath\mathchar278Φl−i2gWa\boldmath\mathchar278\boldmath\mathchar284aΦl−i2g′YlB\boldmath\mathchar278Φl (7) = +i2√2⎛⎜⎝2∂\boldmath\mathchar278\boldmath\mathchar280−(gW3\boldmath\mathchar278+g′YlB\boldmath\mathchar278)% \boldmath\mathchar289l−[gW1\boldmath\mathchar278(vl+\boldmath\mathchar282l)+gW2\boldmath\mathchar278\boldmath\mathchar273l]2∂\boldmath\mathchar278\boldmath\mathchar273l−gW1%\boldmath$\mathchar278$\boldmath\mathchar289l+gW2\boldmath\mathchar278\boldmath\mathchar280l−(−gW3\boldmath\mathchar278+g′YlB\boldmath\mathchar278)(vl+\boldmath\mathchar282l)⎞⎟⎠,

where the are the ordinary Pauli matrices and .
The NHDM scalar-gauge boson interactions are given by

 N∑l=1(D\boldmath\mathchar278Φl)(D\boldmath\mathchar278Φl)† = (8) +18N∑l=1{2∂\boldmath\mathchar278% \boldmath\mathchar282l+[gW1\boldmath\mathchar278\boldmath\mathchar280l+gW2\boldmath\mathchar278\boldmath\mathchar289l−(gW3\boldmath\mathchar278−g′YlB\boldmath\mathchar278)]\boldmath\mathchar273l}2 +18N∑l=1{2∂\boldmath\mathchar278% \boldmath\mathchar273l−gW1%\boldmath$\mathchar278$\boldmath\mathchar289l+gW2\boldmath\mathchar278% \boldmath\mathchar280l−(−gW3\boldmath\mathchar278+g′YlB\boldmath\mathchar278)(vl+% \boldmath\mathchar282l)}2.

The connection between the interaction and mass scalar eigenstates is explained in what follows. The charged scalar fields of Eq. (5) are linear combinations of the charged Goldstone bosons and the charged physical scalars. The imaginary parts of the neutral component of the scalar doublets of Eq. (5) are linear combinations of the neutral Goldstone bosons and of the CP-odd neutral scalar fields. The real parts of the neutral component of the scalar doublets of Eq. (5) are linear combinations of the CP-odd neutral scalar fields. Within this framework we consider a scenario where the interaction and mass eigenstates are related in the way analogous to the two Higgs doublet model (2HDM) 2N-model ()

 \boldmath\mathchar282l=N∑j=1RljH0j,\boldmath\mathchar273l=Ql1\boldmath\mathchar2810+N∑j=2QljA0j−1,l=1,2,⋯,N. (9)
 (10)

where:

 vl=vQl1,l=1,2,⋯,N,v2=N∑i=1v2l,N∑l=1RliRlj=\boldmath\mathchar270ij,N∑l=1QliQlj=\boldmath\mathchar270ij. (11)

Here GeV is the conventional electroweak symmetry breaking scale. The fields () and () are the CP-even and CP-odd neutral Higgs bosons, respectively. Similarly to the gauge bosons which are defined in terms of and , the charged Higgs and Goldstone bosons are related to the component fields in (10) as

 H±j=H1j∓iH2j√2,    \boldmath\mathchar281±=\boldmath\mathchar2811∓i\boldmath\mathchar2812√2 (12)

Thus we assumed the following:

1. The rotation matrix , which relates the neutral Goldstone boson and the CP odd neutral Higgses with the interaction eigenstate scalars () in Eq. (9), is the same as the one that relates the components of the charged Goldstone bosons and Higgses with the corresponding interaction eigenstates , () in Eqs. (10), (12).

2. The vacuum expectation values of Higgs fields () are related to the common symmetry breaking scale GeV through the first relation in Eq. (11).

Both assumptions are generalizations of the corresponding relations of the 2HDM 2N-model (). In the case of NHDM, these relations are not true everywhere in the parametric space but only in a certain part of it. Adopting the above assumptions, we limit ourselves to a region in the parametric space of the NHDM, which is motivated (hinted) by the 2HDM.

## Iii One-loop contribution to the T and S parameters.

In this section we calculate one-loop contributions to the oblique parameters and defined as Peskin:1991sw (); Peskin:1991sw2 (); epsilon-approach (); epsilon-approach2 (); Barbieri:2004 (); Barbieri-book ():

 T=Π33(q2)−Π11(q2)\boldmath\mathchar267EM(MZ)M2W∣∣∣q2=0,           S=2sin2\boldmath\mathchar274W\boldmath\mathchar267EM(MZ)dΠ30(q2)dq2∣∣∣q2=0. (13)

Here , , and are the vacuum polarization amplitudes with , and external gauge bosons, respectively, where is their momentum. Let us note that, in the aforementioned definitions of the oblique and parameters, it is assumed that the new physics is not light compared to and .

### iii.1 T parameter

The interaction Lagrangian, relevant for the computation of one-loop contributions to the parameter in Eq. (13), is

 L(T)int = gg′v2\boldmath\mathchar2811W1\boldmath\mathchar278B\boldmath\mathchar278+gg′v2N∑i=1Pi1H0iW3\boldmath\mathchar278B\boldmath\mathchar278+g2(\boldmath\mathchar2810∂\boldmath\mathchar278\boldmath\mathchar2811−\boldmath\mathchar2811∂\boldmath\mathchar278% \boldmath\mathchar2810)W1\boldmath\mathchar278+g2N−1∑i=1(A0i∂\boldmath\mathchar278H1i−H1i∂\boldmath\mathchar278A0i)W1\boldmath\mathchar278 (14) +g2N∑i=1Pi1(% \boldmath\mathchar2812∂\boldmath\mathchar278H0i−H0i∂\boldmath\mathchar278\boldmath\mathchar2812)W1\boldmath\mathchar278+g2N∑i=1N−1∑j=1Pi,j+1(H2j∂\boldmath\mathchar278H0i−H0i∂\boldmath\mathchar278H2j)W1\boldmath\mathchar278 +g2(\boldmath\mathchar2812∂\boldmath\mathchar278\boldmath\mathchar2811−% \boldmath\mathchar2811∂\boldmath\mathchar278\boldmath\mathchar2812)W3\boldmath\mathchar278+g2N−1∑i=1(H2i∂\boldmath\mathchar278H1i−H1i∂\boldmath\mathchar278H2i)W3\boldmath\mathchar278+g2N∑i=1Pi1(H0i∂\boldmath\mathchar278\boldmath\mathchar2810−% \boldmath\mathchar2810∂\boldmath\mathchar278H0i)W3\boldmath\mathchar278 +g2N∑i=1N−1∑j=1Pi,j+1(H0i∂\boldmath\mathchar278A0j−A0j∂\boldmath\mathchar278H0i)W3\boldmath\mathchar278.

where is given by

 Pij=N∑l=1RliQlj. (15)

By definition it satisfies the inequality

 0≤Pij≤1 (16)

As seen from Eq. (14), the parameter (13) at one-loop level receives contributions from the diagrams shown in Fig. 1.

Their partial contributions, assuming the cutoff to be much larger than the masses of the scalar particles, are

 T(\boldmath\mathchar2811B)≃−316% \boldmath\mathchar281cos2\boldmath\mathchar274Wln(Λ2m2W), (17)
 N∑i=1T(H0iB)≃316\boldmath\mathchar281cos2\boldmath\mathchar274WN∑i=1P2i1ln⎛⎜⎝Λ2m2H0i⎞⎟⎠, (18)
 N∑i=1N−1∑j=1T(H0iA0j)≃116% \boldmath\mathchar267EM(MZ)\boldmath\mathchar2812v2N∑i=1N−1∑j=1P2i,j+1F(Λ2,m2H0i,m2A0j), (19)
 N−1∑i=1T(H1iH2i)≃116\boldmath\mathchar267EM(MZ)\boldmath\mathchar2812v2N−1∑i=1G(Λ2,m2H±i), (20)
 N∑i=1N−1∑j=1T(H0iH2j)≃−116% \boldmath\mathchar267EM(MZ)\boldmath\mathchar2812v2N∑i=1N−1∑j=1P2i,j+1F(Λ2,m2H0i,m2H±j), (21)
 N−1∑i=1T(H1iA0i)≃−116\boldmath\mathchar267EM(MZ)\boldmath\mathchar2812v2N−1∑i=1F(Λ2,m2H±i,m2A0i). (22)

The subscripts in denote the internal lines of the diagrams in Fig. 1. The functions and are defined as

 F(Λ2,m21,m22)=Λ2−m41m21−m22ln(Λ2m21)−m42m22−m21ln(Λ2m22), (23)
 G(Λ2,m2)=limm1,m2→mF(Λ2,m1,m2)=\allowbreakΛ2−2m2ln(Λ2m2)+m2 (24)

Collecting all the contributions together, we find the one-loop contribution to the parameter coming from the scalar sector of the NHDM:

 T=∑abTab ≃ −316\boldmath\mathchar281cos2\boldmath\mathchar274WN∑i=1P2i1ln⎛⎜⎝m2H0im2W⎞⎟⎠+116\boldmath\mathchar267EM(MZ)\boldmath\mathchar2812v2N−1∑i=1[m2H±i−h(m2A0i,m2H±i)] (25) +116\boldmath\mathchar267EM(MZ)\boldmath% \mathchar2812v2N∑i=1N−1∑j=1P2i,j+1[h(m2H0i,m2A0j)−h(m2H0i,m2H±j)] = −316\boldmath\mathchar281cos2\boldmath\mathchar274Wln(m2hm2W)+3(1−P211)16\boldmath\mathchar281cos2\boldmath\mathchar274Wln(m2H0m2h) +N−116\boldmath\mathchar267EM(MZ)\boldmath\mathchar2812v2[m2H±i−h(m2A0i,m2H±i)] +116\boldmath\mathchar267EM(MZ)\boldmath% \mathchar2812v2N∑i=2N−1∑j=1P2i,j+1[h(m2H0i,m2A0j)−h(m2H0i,m2H±j)],

where we identified the lightest CP-even Higgs with the LHC Higgs-like particle with the mass GeV.

The function is given by:

 (26)

We can split the parameter as , where is the contribution from the SM, while contain all the contributions involving the heavy scalars:

 TSM=−316\boldmath\mathchar281cos2\boldmath\mathchar274Wln(m2hm2W), (27)
 ΔT ≃ +116\boldmath\mathchar2812v2\boldmath\mathchar267EM(MZ)N∑i=1N−1∑j=1P2i,j+1[h(m2H0i,m2A0j)−h(m2H0i,m2H±j)].

### iii.2 S parameter

The interaction Lagrangian relevant for the computation of the one-loop contribution to the parameter in Eq. (13) is

 L(S)int = g2(\boldmath\mathchar2812∂\boldmath\mathchar278\boldmath\mathchar2811−% \boldmath\mathchar2811∂\boldmath\mathchar278\boldmath\mathchar2812)W3\boldmath\mathchar278+g2N−1∑i=1(H2i∂\boldmath\mathchar278H1i−H1i∂\boldmath\mathchar278H2i)W3\boldmath\mathchar278 (29) +g′2(\boldmath\mathchar2812∂\boldmath\mathchar278\boldmath\mathchar2811−\boldmath\mathchar2811∂\boldmath\mathchar278\boldmath\mathchar2812)B\boldmath\mathchar278+g′2N−1∑i=1(H2i∂\boldmath\mathchar278H1i−H1i∂\boldmath\mathchar278H2i)B\boldmath\mathchar278 +g2N∑i=1Pi1(H0i∂\boldmath\mathchar278\boldmath\mathchar2810−\boldmath\mathchar2810∂\boldmath\mathchar278H0i)W3\boldmath\mathchar278+g2N∑i=1N−1∑j=1Pi,j+1(H0i∂%\boldmath$\mathchar278$A0j−A0j∂%\boldmath$\mathchar278$H0i)W3%\boldmath$\mathchar278$ −g′2N∑i=1Pi1(H0i∂\boldmath\mathchar278\boldmath\mathchar2810−\boldmath\mathchar2810∂\boldmath\mathchar278H0i)B\boldmath\mathchar278−g′2N∑i=1N−1∑j=1Pi,j+1(H0i∂\boldmath\mathchar278A0j−A0j∂\boldmath\mathchar278H0i)B\boldmath\mathchar278.

As follows from this Lagrangian and the definition (13), the parameter at one-loop level receives contributions from the diagrams shown in Fig. 2.

Their partial contributions, assuming the cutoff to be much larger than the masses of the scalar particles, are

 S(\boldmath\mathchar2811\boldmath\mathchar2812)≃112\boldmath\mathchar281\boldmath\mathchar281ln(Λ2m2W), (30)
 N−1∑i=1S(H1iH2i)≃112\boldmath\mathchar281N−1∑i=1ln⎛⎜⎝Λ2m2H±i⎞⎟⎠, (31)
 N∑i=1S(H0i\boldmath\mathchar2810)≃−112\boldmath\mathchar281N∑i=1P2i1ln⎛⎜⎝Λ2m2H0i⎞⎟⎠, (32)
 N∑i=1N−1∑j=1S(H0iA0j) ≃ −112\boldmath\mathchar281N∑i=1N−1∑j=1P2i,j+1(m2A0j−m2H0i)3⎧⎪⎨⎪⎩m6A0j⎡⎢ ⎢⎣ln⎛⎜⎝Λ2m2A0j⎞⎟⎠+56⎤⎥ ⎥⎦−m6H0i⎡⎢⎣ln⎛⎜⎝Λ2m2H0i⎞⎟⎠+56⎤⎥⎦ (33) +3m2H0im2A0j⎡⎢ ⎢⎣m2H0i⎡⎢⎣ln⎛⎜⎝Λ2m2H0i⎞⎟⎠+32⎤⎥⎦−m2A0j⎡⎢ ⎢⎣ln⎛⎜⎝Λ2m2A0j⎞⎟⎠+32⎤⎥ ⎥⎦⎤⎥ ⎥⎦⎫⎪⎬⎪⎭.

As before, the subscripts in denote the internal lines of the diagrams in Fig. 2. Then, the 1-loop Higgs contribution to the parameter in the NHDM is

 S=∑abSab ≃ 112\boldmath\mathchar281⎡⎢⎣N∑i=1P2i1ln⎛⎜⎝m2H0im2W⎞⎟⎠+N∑i=1N−1∑j=1P2i,j+1K(m2H0i,m2A0j,m2H±j)⎤⎥⎦ (34) = 112\boldmath\mathchar281ln(m2hm2W)+112\boldmath\mathchar281⎡⎢⎣N∑i=2P2i1ln⎛⎜⎝m2H0im2h⎞⎟⎠+N∑i=1N−1∑j=1P2i,j+1K(m2H0i,m2A0j,m2H±j)⎤⎥⎦,

where we identified the lightest CP-even Higgs with the LHC Higgs-like particle with the mass GeV. We defined a function

 K(m21,m22,m23) = 1(m22−m2