# Multi-Higgs doublet models: physical parametrization, sum rules and unitarity bounds

## Abstract

If the scalar sector of the Standard Model is non-minimal, one might expect multiple generations of the hypercharge-1/2 scalar doublet analogous to the generational structure of the fermions. In this work, we examine the structure of a Higgs sector consisting of N Higgs doublets (where N ≥ 2). It is particularly convenient to work in the so-called charged Higgs basis, in which the neutral Higgs vacuum expectation value resides entirely in the first Higgs doublet, and the charged components of remaining N - 1 Higgs doublets are mass-eigenstate fields. We elucidate the interactions of the gauge bosons with the physical Higgs scalars and the Goldstone bosons and show that they are determined by an N × 2N matrix. This matrix depends on (N - 1)(2N - 1) real parameters that are associated with the mixing of the neutral Higgs fields in the charged Higgs basis. Among these parameters, N - 1 are unphysical (and can be removed by rephasing the physical charged Higgs fields), and the remaining 2(N - 1) ^{2} parameters are physical. We also demonstrate a particularly simple form for the cubic interaction and some of the quartic interactions of the Goldstone bosons with the physical Higgs scalars. These results are applied in the derivation of Higgs coupling sum rules and tree-level unitarity bounds that restrict the size of the quartic scalar couplings. In particular, new applications to three Higgs doublet models with an order-4 CP symmetry and with a ${\mathbb{Z}}_{3}$ symmetry, respectively, are presented.

- Authors:

- Univ. of Lisbon (Portugal). Center for Theoretical Particle Physics (CFTP). Dept. of Physics. Inst. Superior Técnico
- Univ. of California, Santa Cruz, CA (United States). Santa Cruz Inst. for Particle Physics

- Publication Date:

- Research Org.:
- Univ. of California, Santa Cruz, CA (United States); Univ. of Lisbon (Portugal)

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25); National Science Foundation (NSF); Foundation for Science and Technology (FCT) (Portugal)

- OSTI Identifier:
- 1501931

- Grant/Contract Number:
- SC0010107; PHY-1066293; CERN/FIS-NUC/0010/2015; UID/FIS/00777/2013

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of High Energy Physics (Online)

- Additional Journal Information:
- Journal Volume: 2017; Journal Issue: 11; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Beyond Standard Model; Higgs physics; CP violation; discrete symmetries

### Citation Formats

```
Bento, Miguel P., Haber, Howard E., Romão, J. C., and Silva, João P..
```*Multi-Higgs doublet models: physical parametrization, sum rules and unitarity bounds*. United States: N. p., 2017.
Web. doi:10.1007/jhep11(2017)095.

```
Bento, Miguel P., Haber, Howard E., Romão, J. C., & Silva, João P..
```*Multi-Higgs doublet models: physical parametrization, sum rules and unitarity bounds*. United States. doi:10.1007/jhep11(2017)095.

```
Bento, Miguel P., Haber, Howard E., Romão, J. C., and Silva, João P.. Wed .
"Multi-Higgs doublet models: physical parametrization, sum rules and unitarity bounds". United States. doi:10.1007/jhep11(2017)095. https://www.osti.gov/servlets/purl/1501931.
```

```
@article{osti_1501931,
```

title = {Multi-Higgs doublet models: physical parametrization, sum rules and unitarity bounds},

author = {Bento, Miguel P. and Haber, Howard E. and Romão, J. C. and Silva, João P.},

abstractNote = {If the scalar sector of the Standard Model is non-minimal, one might expect multiple generations of the hypercharge-1/2 scalar doublet analogous to the generational structure of the fermions. In this work, we examine the structure of a Higgs sector consisting of N Higgs doublets (where N ≥ 2). It is particularly convenient to work in the so-called charged Higgs basis, in which the neutral Higgs vacuum expectation value resides entirely in the first Higgs doublet, and the charged components of remaining N - 1 Higgs doublets are mass-eigenstate fields. We elucidate the interactions of the gauge bosons with the physical Higgs scalars and the Goldstone bosons and show that they are determined by an N × 2N matrix. This matrix depends on (N - 1)(2N - 1) real parameters that are associated with the mixing of the neutral Higgs fields in the charged Higgs basis. Among these parameters, N - 1 are unphysical (and can be removed by rephasing the physical charged Higgs fields), and the remaining 2(N - 1)2 parameters are physical. We also demonstrate a particularly simple form for the cubic interaction and some of the quartic interactions of the Goldstone bosons with the physical Higgs scalars. These results are applied in the derivation of Higgs coupling sum rules and tree-level unitarity bounds that restrict the size of the quartic scalar couplings. In particular, new applications to three Higgs doublet models with an order-4 CP symmetry and with a Z 3 symmetry, respectively, are presented.},

doi = {10.1007/jhep11(2017)095},

journal = {Journal of High Energy Physics (Online)},

issn = {1029-8479},

number = 11,

volume = 2017,

place = {United States},

year = {2017},

month = {11}

}