Real singlet scalar dark matter extension of the Georgi-Machacek model

# Real singlet scalar dark matter extension of the Georgi-Machacek model

Robyn Campbell    Stephen Godfrey    Heather E. Logan    Alexandre Poulin Ottawa-Carleton Institute for Physics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario K1S 5B6, Canada
October 26, 2016
###### Abstract

The Georgi-Machacek model extends the Standard Model Higgs sector with the addition of isospin-triplet scalar fields in such a way as to preserve the custodial symmetry. The presence of higher-isospin scalars contributing to electroweak symmetry breaking offers the interesting possibility that the couplings of the 125 GeV Higgs boson to both gluons and vector boson pairs could be larger than those of the Standard Model Higgs boson. Constraining this possibility using measurements of Higgs production and decay at the CERN Large Hadron Collider is notoriously problematic if a new, non-Standard Model decay mode of the 125 GeV Higgs boson is present. We study an implementation of this scenario in which the Georgi-Machacek model is extended by a real singlet scalar dark matter candidate, and require that the singlet scalar account for all the dark matter in the universe. The combination of the observed dark matter relic density and direct detection constraints exclude singlet scalar masses below about 57 GeV. Higgs measurements are not yet precise enough to be very sensitive to in the remaining allowed kinematic region, so that constraints from Higgs measurements are so far the same as in the GM model without a singlet scalar. We also find that, above the Higgs pole, a substantial region of parameter space yielding the correct dark matter relic density can escape the near-future direct detection experiments DEAP and XENON 1T for dark matter masses as low as 120 GeV and even have a direct detection cross section below the neutrino floor for  GeV. This is in contrast to the singlet scalar dark matter extension of the Standard Model, for which these future experiments are expected to exclude dark matter masses above the Higgs pole up to the multi-TeV range.

## I Introduction

Since the discovery of a Standard Model (SM)-like 125 GeV Higgs boson at the CERN Large Hadron Collider (LHC) Aad:2012tfa (), the determination of the Higgs boson’s couplings to other particles has become a top priority. At the LHC, these couplings are extracted from signal rates in various resonant Higgs production and decay channels, which can be written in the narrow width approximation as

 Rateij=σiΓjΓtot=κ2iσSMiκ2jΓSMj∑kκ2kΓSMk+Γnew. (1)

Here is the Higgs production cross section in production mode , is the Higgs decay partial width into final state , is the total width of the Higgs boson, the corresponding quantities in the SM are denoted with a superscript, and represents the partial width of the Higgs boson into any new, non-SM final states. The coupling modification factors parameterize the deviations of the Higgs couplings from their SM values LHCHiggsCrossSectionWorkingGroup:2012nn ().

The extraction of the Higgs couplings from these LHC rate measurements is plagued by a well-known “flat direction” Zeppenfeld:2000td () that appears when new decay modes are present. For example, one can imagine a scenario in which all the coupling modification factors have a common value and there is a new, unobserved contribution to the Higgs total width, . In this case the Higgs production and decay rates measurable at the LHC are given by

 Rateij=κ4σSMiΓSMjκ2ΓSMtot+Γnew. (2)

All measured Higgs production and decay rates will be equal to their SM values if

 Γnew=κ2(κ2−1)ΓSMtot≥0. (3)

In particular, a simultaneous enhancement of all the Higgs couplings to SM particles can mask, and be masked by, the presence of new decay modes of the Higgs that are not (yet) directly detected at the LHC.111Measuring such an enhancement in the Higgs couplings would be straightforward at a lepton-collider Higgs factory such as the International Linear Collider (ILC), where a direct measurement of the total Higgs production cross section in can be made with no reference to the Higgs decay branching ratios by using the recoil mass method (see, e.g., Ref. Baer:2013cma ()).

Our goal in this paper is to study an explicit benchmark model in which this scenario could be realized. We focus on models with extended Higgs sectors. Our first requirement is a model in which the Higgs couplings to and bosons and to fermions can be enhanced relative to those in the SM. To achieve in an extended Higgs model, we need scalars in isospin representations larger than doublets that carry non-negligible vacuum expectation values (vevs). Only a few such models exist that preserve the parameter at tree level: the Georgi-Machacek (GM) model with isospin triplets Georgi:1985nv (); Chanowitz:1985ug (), generalizations of the GM model to higher isospin Galison:1983qg (); Robinett:1985ec (); Logan:1999if (); Chang:2012gn (); Logan:2015xpa (), and an extension of the Higgs sector by an isospin septet with appropriately-chosen hypercharge Hisano:2013sn (); Kanemura:2013mc (); Alvarado:2014jva (). In this paper we choose the GM model as the simplest extension suitable for our purposes. Its phenomenology has been extensively studied Gunion:1989ci (); Gunion:1990dt (); HHG (); Haber:1999zh (); Aoki:2007ah (); Godfrey:2010qb (); Low:2010jp (); Logan:2010en (); Falkowski:2012vh (); Chang:2012gn (); Chiang:2012cn (); Chiang:2013rua (); Kanemura:2013mc (); Englert:2013zpa (); Killick:2013mya (); Englert:2013wga (); Efrati:2014uta (); Hartling:2014zca (); Chiang:2014hia (); Chiang:2014bia (); Godunov:2014waa (); Hartling:2014aga (); Chiang:2015kka (); Godunov:2015lea (). It has also been incorporated into the scalar sectors of little Higgs Chang:2003un (); Chang:2003zn () and supersymmetric Cort:2013foa (); Garcia-Pepin:2014yfa () models, and an extension with an additional isospin doublet Hedri:2013wea () has been considered.

Our second requirement is a new decay mode for the 125 GeV Higgs boson. A particularly attractive prospect is to link Higgs physics to the mystery of dark matter in the universe (for a recent pedagogical review see Ref. Gelmini:2015zpa ()) by allowing the Higgs to decay into pairs of dark matter particles. To this end we extend the GM model through the addition of a real isospin-singlet scalar field , upon which we impose a symmetry . We will require that accounts for the observed dark matter relic abundance in the universe via the standard thermal freeze-out mechanism. Real singlet scalar extensions of the SM Veltman:1989vw (); Silveira:1985rk (); McDonald:1993ex (); Burgess:2000yq (); McDonald:2001vt (); Barger:2007im (); Goudelis:2009zz (); Gonderinger:2009jp (); He:2009yd (); Profumo:2010kp (); Yaguna:2011qn (); Drozd:2011aa (); Djouadi:2011aa (); Kadastik:2011aa (); Djouadi:2012zc (); Cheung:2012xb (); Damgaard:2013kva (); Cline:2013gha (); Baek:2014jga (); Feng:2014vea (); Campbell:2015fra () and of two-Higgs-doublet models He:2008qm (); Grzadkowski:2009iz (); Logan:2010nw (); Boucenna:2011hy (); He:2011gc (); Bai:2012nv (); He:2013suk (); Cai:2013zga (); Wang:2014elb (); Chen:2013jvg (); Drozd:2014yla (); Wang:2014elb (); Campbell:2015fra () have been extensively studied in the literature. These models tend to be tightly constrained by the combination of relic density, dark matter direct-detection limits, and limits on the indirect detection of dark matter annihilation byproducts from nearby dwarf galaxies.

We will find that the situation is rather similar in the singlet scalar dark matter extension of the GM model. The two strongest constraints are the requirement of the correct dark matter relic abundance from thermal freeze-out Kolb:1990vq () and the direct detection cross section limit from the LUX experiment Akerib:2016vxi (). These constraints restrict the allowed range of singlet scalar masses to lie just below the 125 GeV Higgs pole for resonant annihilation (57–62 GeV) or above the boson mass. The constraint from 125 GeV Higgs boson invisible decays is currently weaker than that from direct detection. Constraints coming from Higgs properties and signals also significantly constrain this model. They do however allow for some interesting deviations from the Standard Model that the GM model without the singlet does not allow.

One important difference compared to the singlet scalar extension of the SM is the prospect for future dark matter direct detection experiments to probe the model at heavier singlet masses. While an absence of signal at the planned XENON 1T experiment would exclude singlet scalar masses up to 4.5 TeV in the singlet scalar extension of the SM Cline:2013gha (), in the singlet scalar extension of the GM model a large swath of parameter space with singlet scalar masses as light as 125 GeV remains beyond the reach of XENON 1T. In fact, there is some allowed parameter space with singlet scalar masses near the 125 GeV Higgs pole for resonant annihilation (60–62 GeV) and some with singlet scalar masses above about 150 GeV which have a direct detection cross section that lies below the neutrino floor. This is mainly due to the contribution of the additional scalars in the GM model to the production of the correct relic density, while not contributing strongly to the direct detection cross section.

This paper is organized as follows. In Sec. II we begin with a description of the singlet scalar extension of the GM model. In Sec. III we extend the theoretical constraints on the GM model to include the singlet scalar extension. In Sec. IV we describe the details of the thermal freezeout and imposing the relic abundance constraints on the model parameters while in Sec. V we describe the numerical scan procedure used to map out the allowed parameter space. In Sec. VI we briefly summarize the direct and indirect search constraints on the additional scalars in the GM model. In Sec. VII we compute the dark matter relic abundance and direct and indirect detection cross sections and display the impact of the observational constraints on the allowed parameter space. In Sec. VIII we consider the constraints from the 125 GeV Higgs boson invisible decays and signal strengths in visible channels. Finally in Sec. IX we summarize our conclusions. Feynman rules for couplings involving the singlet scalar are collected in an appendix.

## Ii The Georgi-Machacek model extended by a real singlet scalar

The scalar sector of the GM model Georgi:1985nv (); Chanowitz:1985ug () consists of the usual complex doublet with hypercharge222We use . , a real triplet with , and a complex triplet with . The doublet is responsible for the fermion masses as in the SM. In order to preserve the custodial SU(2) symmetry and avoid large tree-level contributions to the electroweak parameter, the scalar potential is constructed to preserve a global SU(2)SU(2) symmetry, which breaks down to the diagonal subgroup (known as the custodial SU(2) symmetry) upon electroweak symmetry breaking. To make the global SU(2)SU(2) symmetry explicit, we write the doublet in the form of a bidoublet and combine the triplets to form a bitriplet :

 Φ=(ϕ0∗ϕ+−ϕ+∗ϕ0),X=⎛⎜⎝χ0∗ξ+χ++−χ+∗ξ0χ+χ++∗−ξ+∗χ0⎞⎟⎠. (4)

The vacuum expectation values (vevs) are defined by and , where is the unit matrix and the Fermi constant constrains

 v2ϕ+8v2χ≡v2=1√2GF≈(246 GeV)2. (5)

The most general gauge-invariant scalar potential involving these fields and the real singlet , while conserving the global SU(2)SU(2) and the symmetry , is given by

 V(Φ,X) = μ222Tr(Φ†Φ)+μ232Tr(X†X)+λ1[Tr(Φ†Φ)]2+λ2Tr(Φ†Φ)Tr(X†X) (6) +λ3Tr(X†XX†X)+λ4[%Tr(X†X)]2−λ5Tr(Φ†τaΦτb)Tr(X†taXtb) −M1Tr(Φ†τaΦτb)(UXU†)ab−M2Tr(X†taXtb)(UXU†)ab +μ2S2S2+λaS2Tr(Φ†Φ)+λbS2Tr(X†X)+λSS4.

The first three lines of this potential are identical to that given, e.g., in Ref. Hartling:2014zca ().333A translation table to other parameterizations of the GM model scalar potential has been given in the appendix of Ref. Hartling:2014zca (). The last line contains the new terms involving the singlet scalar . Here the SU(2) generators for the doublet representation are with being the Pauli matrices, the generators for the triplet representation are

 t1 = 1√2⎛⎜⎝010101010⎞⎟⎠,t2=1√2⎛⎜⎝0−i0i0−i0i0⎞⎟⎠, t3 = ⎛⎜⎝10000000−1⎞⎟⎠, (7)

and the matrix , which rotates into the Cartesian basis, is given by Aoki:2007ah ()

 U=⎛⎜ ⎜ ⎜⎝−1√201√2−i√20−i√2010⎞⎟ ⎟ ⎟⎠. (8)

We will work in the vacuum in which does not get a vev, so that the symmetry remains unbroken and is stable. The presence of then has no effect on the mass spectrum or potential-minimization conditions of the GM sector of the model, which can be taken from Ref. Hartling:2014zca (). We summarize the physical spectrum here.

The physical fields can be organized by their transformation properties under the custodial SU(2) symmetry into a custodial fiveplet, a custodial triplet, and three custodial singlets, one of which is . The custodial-fiveplet and -triplet states are given by

 H++5=χ++,H+5=(χ+−ξ+)√2, H05=√23ξ0−√13χ0,r, H03=−sHϕ0,i+cHχ0,i, (9)

and their complex conjugates, where the vevs are parameterized by

 cH≡cosθH=vϕv,sH≡sinθH=2√2vχv, (10)

and we have decomposed the neutral fields into real and imaginary parts according to

 ϕ0→vϕ√2+ϕ0,r+iϕ0,i√2,χ0→vχ+χ0,r+iχ0,i√2, ξ0→vχ+ξ0. (11)

The masses within each custodial multiplet are degenerate at tree level and can be written (after eliminating and in favor of the vevs) as444Note that the ratio is finite in the limit , (12) which follows from the minimization condition  Hartling:2014zca ().

 m25 = M14vχv2ϕ+12M2vχ+32λ5v2ϕ+8λ3v2χ, (13) m23 = M14vχ(v2ϕ+8v2χ)+λ52(v2ϕ+8v2χ)=(M14vχ+λ52)v2.

The gauge singlet remains a mass eigenstate, with physical mass-squared given by

 m2S=μ2S+2λav2ϕ+6λbv2χ, (14)

which we require to be positive to avoid breaking the symmetry.

The other two custodial SU(2)–singlet mass eigenstates are given by

 h = cosαϕ0,r−sinαH0′1, H = sinαϕ0,r+cosαH0′1, (15)

where

 H0′1=√13ξ0+√23χ0,r. (16)

The mixing angle and masses are given by

 sin2α=2M212m2H−m2h,cos2α=M222−M211m2H−m2h, (17) m2h,H=12[M211+M222∓√(M211−M222)2+4(M212)2],

where we choose , and

 M211 = 8λ1v2ϕ, M212 = √32vϕ[−M1+4(2λ2−λ5)vχ], M222 = M1v2ϕ4vχ−6M2vχ+8(λ3+3λ4)v2χ. (18)

## Iii Theoretical Constraints on Lagrangian Parameters

The singlet scalar dark matter extension of the GM model has 13 free parameters, two of which can be fixed by and the 125 GeV Higgs mass. Before scanning over the remaining parameters, we first study the relevant theoretical and experimental constraints. The theoretical constraints come from (1) perturbative unitarity imposed on scalar scattering amplitudes, (2) the requirement that the scalar potential be bounded from below, and (3) that the custodial SU(2)-preserving minimum is the true global minimum of the potential.

### iii.1 Perturbative unitarity of 2→2 scattering amplitudes

The scalar couplings in Eq. 6 can be bounded by perturbative unitarity of the 2 2 scalar field scattering amplitudes. These constraints were studied in the original GM model in Refs. Aoki:2007ah (); Hartling:2014zca (); here we extend them to include the real singlet scalar.

The partial wave amplitudes are related to the matrix element of the process by:

 M=16π∑J(2J+1)aJPJ(cosθ), (19)

where is the (orbital) angular momentum and are the Legendre polynomials. Perturbative unitarity requires that the zeroth partial wave amplitude, , satisfy or . Because the 2 2 scalar field scattering amplitudes are real at tree level, we adopt the second, more stringent, constraint. We will use this to constrain the magnitudes of the scalar quartic couplings .

We work in the high energy limit, in which the only tree-level diagrams that contribute to scalar scattering are those involving the four-point scalar couplings since all diagrams involving scalar propagators are suppressed by the square of the collision energy. Thus the dimensionful couplings , , , and are not constrained directly by perturbative unitarity. In the high energy limit we can ignore electroweak symmetry breaking and include the Goldstone bosons as physical fields (this is equivalent to including scattering processes involving longitudinally polarized and bosons). We neglect scattering processes involving transversely polarized gauge bosons or fermions.

Under these conditions, only the zeroth partial wave amplitude contributes to , so that the constraint corresponds to . This condition must be applied to each of the eigenvalues of the coupled-channel scattering matrix including each possible combination of two scalar fields in the initial and final states. Because the scalar potential is invariant under , the scattering processes preserve electric charge and hypercharge and can be conveniently classified by the total electric charge and hypercharge of the incoming and outgoing states. We include a symmetry factor of for each pair of identical particles in the initial and final states.

The basis states for are,

 χ++∗χ++, χ+∗χ+, ξ+∗ξ+, ϕ+∗ϕ+, ξ0ξ0√2, χ0∗χ0, ϕ0∗ϕ0, S2√2, Sξ0. (20)

Scattering amplitudes involving these states yield eight distinct eigenvalues of ,

 x±2 =4λ1−2λ3+4λ4±√(4λ1+2λ3−4λ4)2+4λ25, y1 =16λ3+8λ4, y2 =4λ3+8λ4, zb =4λb, z1,2,3 =Roots(P(z)), (21)

where , , and are the roots of the polynomial,

 P(z)=det⎛⎜⎝24λ1−z12λ24λa12λ228λ3+44λ4−z6λb4λa6λb12λS−z⎞⎟⎠. (22)

We have followed the notation of Refs. Aoki:2007ah (); Hartling:2014zca () where possible. Note that the pair of eigenvalues of Refs. Aoki:2007ah (); Hartling:2014zca () is recovered by taking in .

The basis states for and are,

 ϕ+ξ+∗, ϕ0ξ0, χ+ϕ+∗, χ0ϕ0∗, Sϕ0. (23)

Scattering amplitudes involving these states yield four additional distinct eigenvalues of ,

 y3 =4λ2−λ5, y4 =4λ2+2λ5, y5 =4λ2+4λ5, za =4λa. (24)

Scattering amplitudes involving basis states with other values of and only repeat eigenvalues that have already been found. Note that by adding the real singlet scalar we have replaced the two eigenvalues of Refs. Aoki:2007ah (); Hartling:2014zca () with five new eigenvalues . We obtain the unitarity bounds by requiring that the absolute value of each of the eigenvalues in Eqs. (21) and (24) be less than .

The three unitarity constraints can be made more algebraically tractable by replacing them with three equivalent conditions as follows. First, since Eq. (22) is linear in , we can solve the equation for as a function of the root ,

 λS(z)=16⎛⎝z2+2λ2a(7λ3+11λ4−18z)+9λ2b(3λ1−18z)−18λ2λaλb2(7λ3+11λ4−18z)(3λ1−18z)−9λ22⎞⎠. (25)

This function has two poles, across which changes sign. There are thus three values of that yield the same value of , corresponding to the three roots of the polynomial . We now require that all three of these roots satisfy . For this to be possible, the two poles in must also lie at values between and . The positions of these two poles are given by , where

 x±1 = 12λ1+14λ3+22λ4 (26) ±√(12λ1−14λ3−22λ4)2+144λ22.

Therefore we require , reproducing two of the unitarity constraints from the original GM model Aoki:2007ah (); Hartling:2014zca (). The third condition restricts to lie in the range for which the three roots of all lie within ,

 λminS<λS<λmaxS, (27)

where and from Eq. (25).

To summarize, we will require that the following constraints from perturbative unitarity be satisfied:

 8π> ∣∣12λ1+14λ3+22λ4±√(12λ1−14λ3−22λ4)2+144λ22∣∣=|x±1|, 8π> ∣∣4λ1−2λ3+4λ4±√(4λ1+2λ3−4λ4)2+4λ25∣∣=|x±2|, 8π> |16λ3+8λ4|=|y1|, 8π> |4λ3+8λ4|=|y2|, 8π> |4λ2−λ5|=|y3|, 8π> |4λ2+2λ5|=|y4|, 8π> |4λ2+4λ5|=|y5|, 8π> |4λa|=|za|, 8π> |4λb|=|zb|, λS< 16(4π+2λ2a(7λ3+11λ4−π)+9λ2b(3λ1−π)−18λ2λaλb2(7λ3+11λ4−π)(3λ1−π)−9λ22), λS> 16(−4π+2λ2a(7λ3+11λ4+π)+9λ2b(3λ1+π)−18λ2λaλb2(7λ3+11λ4+π)(3λ1+π)−9λ22). (28)

### iii.2 Requirement that the scalar potential be bounded from below

We next examine the constraints on the scalar couplings imposed by requiring that the scalar potential be bounded from below. The constraints that must be satisfied at tree level for the scalar potential to be bounded from below can be determined by considering only the terms that are quartic in the fields, because these terms dominate at large field values. Following the approach of Ref. Arhrib:2011 (), we parametrize the potential using the following definitions,

 r =√Tr(Φ†Φ)+Tr(X†X)+S2, r2cos2γsin2β =Tr(Φ†Φ), r2sin2γsin2β =Tr(X†X), r2cos2β =S2, ζ =Tr(X†XX†X)(Tr(X†X))2, ω =Tr(Φ†τaΦτb)Tr(X†taXtb)Tr(Φ†Φ)Tr(X†X). (29)

Making these substitutions, we can write the quartic part of the potential as

 V4=r4(1+tan2γ)2(1+tan2β)2xTAy, (30)

where

 x=⎛⎜⎝1tan2βtan4β⎞⎟⎠,y=⎛⎜⎝1tan2γtan4γ⎞⎟⎠, (31)

and

 (32)

The first fraction in Eq. (30) is always positive, and grows with the overall field excursion . The term in Eq. (30) can be positive or negative; we require it to be positive to ensure that the potential is bounded from below. This term can be expressed as a bi-quadratic in with coefficients being other bi-quadradics in . A bi-quadratic of the form will be positive for all values of if the following conditions are satisfied:

 a>0,c>0,andb+2√ac>0. (33)

In our case this leads to the following constraints on the elements of the matrix in Eq. (32):

 0

where it should be understood that repeated indices are summed over, and is a unit vector with a in the th component and zeros everywhere else. The last two conditions do not provide any new information as they are always satisfied when the others are, but we list them for completeness.

The ranges of the parameters and are given, as in the original GM model Hartling:2014zca (), by

 ζ∈[13,1],ω∈[−14,12]. (35)

For a given value of , we can write , where Hartling:2014zca ()

 ω±(ζ)=16(1−B)±√23[(1−B)(12+B)]12, (36)

with

 B≡√32(ζ−13)∈[0,1]. (37)

Therefore, we can write our constraints as follows:

 λ1 >0, λ4 >{−13λ3 for λ3≥0,−λ3 for λ3<0, λ2 >⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩12λ5−2√λ1(13λ3+λ4)for λ5≥0,λ3≥0,ω+(ζ)λ5−2√λ1(ζλ3+λ4)for λ5≥0,λ3<0,ω−(ζ)λ5−2√λ1(ζλ3+λ4) for λ5<0, λa >−2√λ1λS, λb >⎧⎪ ⎪⎨⎪ ⎪⎩−2√(13λ3+λ4)λS for λ3≥0,−2√(λ3+λ4)λS for λ3<0, λS >0. (38)

The first three of these constraints are identical to those in the original GM model, while the last three are new.

We note that the full parameter space of the quartic scalar couplings as allowed by perturbative unitarity and the requirement that the scalar potential be bounded from below can be covered by scanning over the following ranges. For the couplings , the ranges are the same as in the original GM model Hartling:2014zca (),

 λ1∈(0,π3),λ2∈(−2π3,2π3),λ3∈(−π2,3π5), λ4∈(−π5,π2),λ5∈(−8π3,8π3). (39)

For the new couplings , , and in the singlet scalar dark matter extension of the GM model, the ranges are,555The upper limits of these ranges come from the unitarity constraints in Eq. (28). The upper limit on comes directly from . The upper limit on comes from the upper and lower bounds on : for large enough these two bounds meet each other, and the least stringent bound on comes from taking all other quartic couplings equal to zero in these expressions. The upper limit on comes directly from the expression in Eq. (28), which is least stringent when all other quartic couplings are set to zero. The lower limit on comes from an interplay of the bounded-from-below constraint in Eq. (38) and the upper bound on from Eq. (28) when and . The lower limit on comes from an interplay of the constraint in Eq. (38) and the bound on and from in Eq. (28). The least stringent limit occurs when . The lower limit on comes trivially from Eq. (38).

 λa∈(−2π(3√2−2)7,2π),λb∈(−4π√33,4π3), λS∈(0,2π3). (40)

Within these ranges, the conditions in Eqs. (28) and (38) must still be applied and any points in violation discarded.

### iii.3 Conditions to avoid alternative minima

Finally we check that the scalar potential does not contain any deeper minima that spontaneously break the custodial symmetry or that give the singlet a vev.

The constraints on the parameters required to ensure that the desired electroweak-breaking and custodial SU(2)-preserving minimum is the true global minimum were studied for the original GM model in Ref. Hartling:2014zca (). These continue to apply in the singlet-extension that we study here and we implement them as follows. Using and from Eq. (29) and introducing the additional parameters

 σ = Tr(Φ†τaΦτb)(UXU†)abTr(Φ†Φ)[Tr(X†X)]12, ρ = Tr(X†taXtb)(UXU†)ab[Tr(X†X)]32, x2 = Tr(Φ†Φ), y2 = Tr(X†X), z2 = S2, (41)

the scalar potential can be written as

 V= μ222x2+μ232y2+μ2S2z2+(λ2−λ5ω)x2y2 +λax2z2+λby2z2+λ1x4+(λ3ζ+λ4)y4 +λSz4−M1σx2y−M2ρy3. (42)

The parameters , , and capture the dependence on which component(s) of obtain a vev. The correct custodial SU(2)-preserving vacuum corresponds to , , , and  Hartling:2014zca (). For a given set of Lagrangian parameters, we check that these values yield the lowest value of the potential by using the convenient parameterization Hartling:2014zca ()

 ζ = 12sin4θ+cos4θ, ω = 14sin2θ+1√2sinθcosθ, σ = 12√2sinθ+14cosθ, ρ = 3sin2θcosθ, (43)

and scanning over .

We then check that the potential does not have any deeper minima in which gets a vev. If , and are all positive, then cannot get a vev. We only have to worry about this possibility if one or two of these parameters are negative (all three cannot be negative because we require ). Taking yields two possible extrema,

 z = 0, z2 = −14λS(μ2S+2λax2+2λby2). (44)

We then take and , plug in each of the two solutions for from Eq. (44), solve for the possible values of and in each case, and then plug these back into to obtain the depth of the potential at each extremum. Points are discarded if a minimum with is deeper than the desired one with .

## Iv Thermal Relic Density

We now turn to constraints from the dark matter relic abundance. We assume that the scalar dark matter candidate constitutes all of the dark matter. We will use the observed relic density to fix a combination of and . We show that the direct detection constraints restrict the dark matter mass to be near half the Higgs mass around GeV or above approximately GeV.

### iv.1 Thermal Freezeout

The relic density of through thermal freeze-out in the early universe is determined by the annihilation cross section for . We calculate the thermally averaged cross section as a function of temperature using Gondolo:1990dk (); Edsjo:1997bg ():

 ⟨σ12→34vrel⟩=g1g2T32π4neq1neq2 (45) ×∫∞4m2Sσ12→34[s−4m2S]2√sK1(√sT)ds,

where is the usual Mandelstam variable, is the number of internal degrees of freedom of , is the relative velocity of particles and , and is the modified Bessel function of the second kind of order 1. We use the thermally averaged total annihilation cross section as input for the usual Boltzmann equation Kolb:1990vq (); Gondolo:1990dk (); Edsjo:1997bg ():

 dnSdt+3HnS=−⟨σvrel⟩[n2S−(neqS)2], (46)

where is the Hubble parameter and is the number density of . Here is the equilibrium number density of and is given by Gondolo:1990dk (); Edsjo:1997bg ():

 neqS =gS(2π3)∫e−ES/Td3pi =gSm2ST2π2K2(mS/T) (47) ≈