EFI 22-8 FERMILAB-PUB-22-740-T New tools for dissecting the general 2HDM

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EFI 22-8

FERMILAB-PUB-22-740-T

New tools for dissecting the general 2HDM

Henning Bahl∗1, Marcela Carena†1,2,3, Nina M. Coyle‡1,

Aurora Ireland§1, and Carlos E.M. Wagner¶1,3,4

1Department of Physics and Enrico Fermi Institute, University of Chicago,

5720 South Ellis Avenue, Chicago, IL 60637 USA

2Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, Illinois, 60510, USA

3Kavli Institute for Cosmological Physics, 5640 South Ellis Avenue, University of Chicago,

Chicago, IL 60637

4HEP Division, Argonne National Laboratory, 9700 Cass Ave., Argonne, IL 60439, USA

Two Higgs doublet models (2HDM) provide the low energy eﬀective theory

(EFT) description in many well motivated extensions of the Standard Model.

It is therefore relevant to study their properties, as well as the theoretical

constraints on these models. In this article we concentrate on three relevant

requirements for the validity of the 2HDM framework, namely the perturba-

tive unitarity bounds, the bounded from below constraints, and the vacuum

stability constraints. In this study, we concentrate on the most general renor-

malizable version of the 2HDM — without imposing any parity symmetry,

which may be violated in many UV extensions. We derive novel analyti-

cal expressions that generalize those previously obtained in more restrictive

scenarios to the most general case. We also discuss the phenomenological

implications of these bounds, focusing on CP violation.

∗hbahl@uchicago.edu

†carena@fnal.gov

‡ninac@uchicago.edu

§anireland@uchicago.edu

¶cwagner@uchicago.edu

arXiv:2210.00024v2 [hep-ph] 1 May 2023

Contents

1. Introduction 3

2. The general 2HDM 4

3. Methods for bounding matrix eigenvalues 5

3.1. Frobeniusnorm................................ 5

3.2. Gershgorin disk theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.3. Principalminors ............................... 7

4. Perturbative unitarity 8

4.1. Numericalbound ............................... 8

4.2. A necessary condition for perturbative unitarity . . . . . . . . . . . . . . 11

4.3. Suﬃcient conditions for perturbative unitarity . . . . . . . . . . . . . . . 12

4.4. Numericalcomparison ............................ 14

5. Boundedness from below 15

5.1. Necessary conditions for boundedness from below . . . . . . . . . . . . . 17

5.2. Suﬃcient conditions for boundedness from below . . . . . . . . . . . . . . 18

5.3. Numericalanalysis .............................. 19

6. Vacuum stability 21

6.1. Suﬃcient conditions for stability . . . . . . . . . . . . . . . . . . . . . . . 23

6.1.1. Gershgorin bounds . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6.1.2. Frobeniusbounds........................... 24

6.1.3. Principalminors ........................... 24

6.2. Numericalcomparison ............................ 25

6.3. Vacuum stability in the Higgs basis . . . . . . . . . . . . . . . . . . . . . 26

7. CP Violation in the general 2HDM 28

8. Conclusions 31

A. Higgs basis conversion 33

B. Connection between ζand M2

H±34

1. Introduction

The Standard Model (SM) [1] relies on the introduction of a Higgs doublet, whose

vacuum expectation value breaks the electroweak symmetry [2–4]. This mechanism gen-

erates masses for the weak gauge bosons and charged fermions, as well as potentially

the neutrinos (although there may be other mass sources for the latter). The Stan-

dard Model Higgs sector is the simplest way of implementing the Higgs mechanism for

generating the masses of the known elementary particles. However, it is not the only

possibility, and may be easily extended to the case of more than one Higgs doublet with-

out violating any of the important properties of the SM. Moreover, one of the simplest of

these extensions — two Higgs doublet models (2HDMs) [5] — appears as a low-energy

eﬀective theory of many well motivated extensions of the Standard Model (e.g. those

based on supersymmetry [6–15] or little Higgs [16]).

Two Higgs doublet models may diﬀer in the mechanism of generation of fermion

masses. If both Higgs doublets couple to fermions of a given charge, their couplings

will be associated to two diﬀerent, complex sets of Yukawa couplings, which would form

two diﬀerent matrices in ﬂavor space. The fermion mass matrices would be the sum

of these, each multiplied by the corresponding Higgs vacuum expectation value. So

diagonalization of the fermion mass matrices does not lead to the diagonalization of the

fermion Yukawa matrices. Such theories are then associated with large ﬂavor violating

couplings of the Higgs bosons at low energies — a situation which is experimentally

strongly disfavored. Hence, it is usually assumed that each charged fermion species

couples only to one of the two Higgs doublets. In most works related to 2HDM, this

is accomplished by implementing a suitable Z2symmetry. The diﬀerent possible charge

assignments for this Z2symmetry then ﬁx the Higgs–fermion coupling choices and deﬁne

diﬀerent types of 2HDMs.

This Z2symmetry not only ﬁxes the Higgs–fermion couplings but also forbids certain

terms in the Higgs potential that are far less problematic with respect to ﬂavor violation.

As a starting point for an investigation of the phenomenological implications of these

terms, we will in this work discuss the theoretical bounds on the boson sector of the

theory (without any need to specify the nature of the Higgs-fermion couplings). We will

concentrate on the constraints that come from the perturbative unitarity of the theory,

the stability of the physical vacuum, and the requirement that the eﬀective potential is

bounded from below. Existing works [5, 17–29] focus either on the Z2-symmetric case

or only provide a numerical procedure to test these constraints in the general 2HDM

(see Ref. [30] for a recent work on analytic conditions for boundedness-from-below).

We will go beyond current studies by deriving analytic bounds that apply to the most

general, renormalizable realization of 2HDMs. Our conditions will be given in terms

of the mass parameters and dimensionless couplings of the 2HDM tree-level potential.

At the quantum level, however, these parameters are scale dependent; although we will

refrain from doing so here, one can apply these conditions at arbitrarily high energy

scales by using the renormalization group evolution of these parameters.

Our article is organized as follows. In Section 2 we introduce the scalar sector of

the most general 2HDM that deﬁnes the framework for most of the work presented

in this article. Section 3 reviews three theorems from linear algebra which will allow

us to derive analytic bounds in the coming sections. In Section 4, we concentrate on

the requirement of perturbative unitarity. Section 5 presents the bounds coming from

the requirement that the tree-level potential be bounded from below. In Section 6,

we discuss the vacuum stability. Finally, we reserve Section 7 for a brief analysis of

the phenomenological implications (focusing on CP violation) and Section 8 for our

conclusions. A Table listing the most relevant ﬁndings of our work may be found at the

beginning of the Conclusions.

2. The general 2HDM

As emphasized above, we focus on the scalar sector of the theory. In general, gauge

invariance implies that the potential can only include bilinear and quartic terms. Each

of the three bilinear terms has a corresponding mass parameter, of which two (m2

11 and

22) are real while the third, m2

12, is associated with a bilinear mixing of both Higgs

doublets and may be complex.

Regarding the quartic couplings in the scalar potential, the two associated with self

interactions of each of the Higgs ﬁelds, λ1and λ2, must be real and, due to vacuum

stability, positive. There are two couplings associated with Hermitian combinations of

the Higgs ﬁelds, λ3and λ4, which must be real, though not necessarily positive. The

coupling λ5is associated with the square of the gauge invariant bilinear of both Higgs

ﬁelds, and it may therefore be complex. The couplings λ6and λ7are associated with

the product of Hermitian bilinears of each of the Higgs ﬁelds with the gauge invariant

bilinear of the two Higgs ﬁelds, and, as with λ5, they may be complex. The most general

scalar potential for a complex 2HDM is therefore:

V=m2

11Φ†

1Φ1+m2

22Φ†

2Φ2−(m2

12Φ†

1Φ2+h.c.)

+λ1

2(Φ†

1Φ1)2+λ2

2(Φ†

2Φ2)2+λ3(Φ†

1Φ1)(Φ†

2Φ2) + λ4(Φ†

1Φ2)(Φ†

2Φ1)

+λ5

2(Φ†

1Φ2)2+λ6(Φ†

1Φ1)(Φ†

1Φ2) + λ7(Φ†

2Φ2)(Φ†

1Φ2) + h.c.,

(1)

with Φ1,2= (Φ+

1,2,Φ0

1,2)Tbeing complex SU(2) doublets with hypercharge +1.

One way to prevent Higgs-induced ﬂavor violation in the fermion sector is to introduce

aZ2parity symmetry under which each charged fermion species transforms as even or

odd. The Higgs doublets are assigned opposite parities and couple only to those charged

fermions that carry their own parity. In such a scenario, the terms accompanying the

couplings λ6and λ7would violate parity symmetry and hence should vanish. The mass

parameter m2

12 is also odd under the parity symmetry but induces only a soft breaking

of this symmetry, which does not aﬀect the ultraviolet properties of the theory. Thus it

is usually allowed.

There are alternative ways of suppressing ﬂavor violating couplings of the Higgs to

fermions which do not rely on a simple parity symmetry and hence allow for the presence

of λ6and λ7terms. One example is the ﬂavor-aligned 2HDM [31]. Alternatively, one

can impose a parity symmetry in the ultraviolet but allow the eﬀective low energy ﬁeld

theory to be aﬀected by operators generated by the decoupling of a sector where this

symmetry is broken softly by dimensionful couplings which do not respect the parity

symmetry properties. One example of such a theory is the NMSSM in the presence of

heavy singlets, as discussed in Ref. [32]. In this case, the presence of the couplings λ6

and λ7is essential to allow for the alignment of the light Higgs boson with a SM-like

Higgs, leading to a good agreement with precision Higgs physics even in the case of large

Higgs self couplings.

So we see that it is not necessary to restrict to the Z2-symmetric 2HDM with vanishing

λ6and λ7to avoid Higgs induced ﬂavor violation in the fermion sector.1Further, it is

phenomenologically interesting to study the 2HDM in full generality with these terms

present. One consequence would be the possibility of having charge-parity (CP) violation

in the bosonic sector. Indeed, to keep good agreement with Higgs precision data [33,

34], one is normally interested in studying 2HDM in (or close to) the exact alignment

limit — the limit in which one of the neutral scalars carries the full vacuum expectation

value and has SM-like tree-level couplings [35–40]. If one imposes exact alignment in

the Z2-symmetric 2HDM, however, CP is necessarily conserved, as will be explained in

detail in Section 7. In the full 2HDM, on the other hand, one can have CP violation

whilst remaining in exact alignment thanks to the presence of λ6and λ7terms. This

CP violation could manifest in the neutral scalar mass eigenstates as well as bosonic

couplings (see also Ref. [41]), providing many potential experimental signatures. With

this motivation in mind, we keep λ6and λ7non-zero throughout this work.

3. Methods for bounding matrix eigenvalues

In this work, much of the analysis of perturbative unitarity and vacuum stability involves

placing bounds on matrix eigenvalues. In the most general 2HDM, analytic expressions

for these constraints are either very complicated or simply can not be formulated. To

obtain some analytic insight, we derive conditions which are either necessary or suﬃcient.

Their derivation is based on three linear algebra theorems which we brieﬂy review here.

3.1. Frobenius norm

One may derive a bound on the magnitude of the eigenvalues of a matrix using the

matrix norm. The following deﬁnition and theorem are needed:

Theorem: The magnitude of the eigenvalues eiof a square matrix Aare bounded from

above by the matrix norm: |ei| ≤ ||A||.

1Note that non-vanishing λ6,7induces ﬂavor violation via Higgs mixing. This eﬀect is, however, loop

suppressed.

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EFI22-8FERMILAB-PUB-22-740-TNewtoolsfordissectingthegeneral2HDMHenningBahl*1,MarcelaCarena1,2,3,NinaM.Coyle1,AuroraIreland1,andCarlosE.M.Wagner¶1,3,41DepartmentofPhysicsandEnricoFermiInstitute,UniversityofChicago,5720SouthEllisAvenue,Chicago,IL60637USA2FermiNationalAcceleratorLaboratory,P.O.Box50...

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EFI 22-8 FERMILAB-PUB-22-740-T New tools for dissecting the general 2HDM

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