PHYSICAL REVIEW D 101, 035001 (2020) - Gravitational wave signals of pseudo-Goldstone dark matter in the Z3 complex singlet model - Inspire HEP
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PHYSICAL REVIEW D 101, 035001 (2020)
Gravitational wave signals of pseudo-Goldstone dark matter
in the Z3 complex singlet model
Kristjan Kannike ,* Kaius Loos,† and Martti Raidal ‡
National Institute of Chemical Physics and Biophysics, Rävala 10, Tallinn 10143, Estonia
(Received 7 August 2019; accepted 14 January 2020; published 3 February 2020)
We study pseudo-Goldstone dark matter in the Z3 complex scalar singlet model. Because the direct
detection spin-independent cross section is suppressed, such dark matter is allowed in a large mass range.
Unlike in the original model stabilized by a parity and due to the cubic coupling of the singlet, the Z3 model
can accommodate first-order phase transitions that give rise to a stochastic gravitational wave signal
potentially observable in future space-based detectors.
DOI: 10.1103/PhysRevD.101.035001
I. INTRODUCTION admits two phases that produce a dark matter candidate.
Firstly, the unbroken Z3 symmetry stabilizes S as a dark
One of the best candidates for the dark matter (DM) is a
matter candidate. Secondly, with the broken Z3 symmetry,
scalar singlet [1,2]. The properties of singlet DM have been
the imaginary part of S, denoted by χ, is still stable due to
studied in detail [3–6] (see [7,8] for recent reviews; see also
the S → S† symmetry of the Lagrangian. The main differ-
Refs. therein). The nonobservation of dark matter by direct
ence between the Z2 and Z3 pseudo-Goldstone DM models
detection experiments [9–11], however, puts severe bounds
is that the potential of the latter contains a cubic S3 term.
on models of DM comprised of weakly interacting massive
There are other possible cubic terms [21–23], but they do
particles (WIMP), pushing the mass of the singlet scalar
not respect the Z3 symmetry.
DM—except around the Higgs resonance—over 1 TeV.
To this date, the Z3 -symmetric complex singlet model
A way to suppress the direct detection cross section is to
has been studied in the unbroken phase. Originally, the
consider as the DM candidate a pseudo-Goldstone with
model was proposed in the context of neutrino physics [24].
derivative couplings to CP-even states [12] (the DM
Detailed analysis of DM phenomenology was carried out in
phenomenology of the imaginary part of the complex
Ref. [25]. Indirect detection of Z3 DM was considered in
scalar singlet was first considered in [3,13], but only in
[26,27]. The cubic coupling can contribute to 3 → 2 scat-
the Higgs resonance region). Because the velocity of DM
tering for Z3 strongly interacting (SIMP) DM [28–31]. The
particles in the Galaxy is small, the prospective signal is
effects of early kinetic decoupling were studied in [32]. The
suppressed by vanishing momentum transfer. This result
Z3 symmetry has been considered as the remnant of a dark
remains practically unaffected when loop corrections to the
Uð1Þ local [33–35] or global [36] symmetry. In the
direct detection cross section are taken into account unbroken phase, the direct detection cross section can also
[14,15]. Despite that, it is possible for pseudo-Goldstone be suppressed if the DM relic density is determined by
DM to show up at the LHC [16–19]. Pseudo-Goldstone semiannihilation processes [37–42]. However, the suppres-
DM with two Higgs doublets was considered in [20]. sion is not as large as for pseudo-Goldstone DM.
In this class of models, the global Uð1Þ symmetry is The discovery of gravitational waves (GWs) by the
explicitly, but only softly, broken into a discrete subgroup LIGO experiment [43,44] opened a new avenue to probe
which is then broken spontaneously. In Ref. [12], the Uð1Þ new physics. First-order phase transitions generate a
group was explicitly broken into Z2 symmetry. We study stochastic GW background [45–47], which may be dis-
the consequences of breaking Uð1Þ into Z3 . The model coverable in future space-based GW interferometers
[48,49]. While the SM Higgs phase transition is a smooth
*
kristjan.kannike@cern.ch crossover [50,51] and does not generate a GW signal, in
† models with an extended scalar sector, the first-order phase
kaius.loos@gmail.com
‡
martti.raidal@cern.ch transition in the early Universe can become testable by
observations. For a recent review on phase transitions and
Published by the American Physical Society under the terms of GWs, see Ref. [52].
the Creative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to GWs from beyond-the-SM physics with a scalar singlet
the author(s) and the published article’s title, journal citation, have been studied in detail. These models admit a two-step
and DOI. Funded by SCOAP3. phase transition that can be of the first order [53–60] and
2470-0010=2020=101(3)=035001(10) 035001-1 Published by the American Physical SocietyKANNIKE, LOOS, and RAIDAL PHYS. REV. D 101, 035001 (2020)
can potentially produce a measurable GW signal [61–72]. phase, the complex singlet S is a DM candidate, while in the
However, in the Z2 pseudo-Goldstone model, all phase broken Z3 phase, it is its imaginary part, the pseudo-
transitions leading to the correct vacuum are of the second Goldstone χ, which can be the DM. We concentrate on the
order [73], yielding no stochastic GW signal and excluding latter case. (In order to avoid domain walls [75–79] due to
the additional potentially powerful experimental test of this the broken Z3 , we can add to the potential a small term that
class of the SM models. The phenomenology of phase explicitly breaks Z3 , such as the linear term in S, but
transitions and GWs of the Z3 complex singlet model was otherwise does not change the dark matter phenomenology.)
studied in detail in Ref. [74] for the case where only the Without a loss of generality, the vacuum expectation
Higgs boson has a VEV in our vacuum. This work, value (VEV) of S can be taken to be real and positive,
however, disregarded the singlet as a DM candidate, because the degenerate vacua, where χ has a nonzero VEV,
because in the unbroken phase, a sizeable GW signal is are related to the real vacuum by Z3 transitions. This choice
incompatible with the correct relic density. corresponds to a negative value of the parameter μ3.
The goal of this paper is to study the nature of phase The stationary point conditions are
transitions and GW signals in the complex scalar singlet
model in which a Uð1Þ symmetry is softly broken into its hð2λH h2 þ 2μ2H þ λSH s2 Þ ¼ 0; ð3Þ
Z3 subgroup. Just like in the Z2 case, the elastic scattering
pffiffiffi
cross section pseudo-Goldstone DM with matter can be sð4λS s2 þ 3 2μ3 s þ 4μ2S þ 2λSH h2 Þ ¼ 0: ð4Þ
small enough for the DM to be well hidden from direct
detection. We find, however, that unlike in the Z2 case, We see that the model admits four types of extrema: the
strong first-order phase transitions can take place because origin as O ≡ ð0; 0Þ is fully symmetric; the vacuum
the potential contains a cubic term. The resulting stochastic H ≡ ðvh ; 0Þ, with Higgs VEV only, spontaneously breaks
GW background is potentially discoverable in future space- the electroweak symmetry; the vacuum S ≡ ð0; vs Þ, with S
based detectors such as LISA and BBO. VEV only, spontaneously breaks Z3 ; our vacuum HS ≡
The paper is organized as follows. We introduce the ðvh ; vs Þ breaks both symmetries.
model in Sec. II. Various theoretical and experimental The mass matrix of CP-even scalars in our HS vacuum
constraints are discussed in Sec. III. DM relic density, is given by
phase transitions, and the predictions for the direct detec-
tion and GW signals are treated in Sec. IV. We conclude in
2λH v2 λSH vvs
2
Sec. V. The details on the effective potential and thermal M ¼ : ð5Þ
corrections are relegated to the Appendix. λSH vvs 2λS v2s þ p3 ffiffi μ3 vs
2 2
The mass matrix (5) is diagonalized by an orthogonal
II. Z3 COMPLEX SINGLET MODEL
matrix,
The most general renormalizable scalar potential of the
Higgs doublet H and the complex singlet S, invariant under cos θ − sin θ
O¼ ; ð6Þ
the Z3 transformation H → H, S → ei2π=3 S, is given by sin θ cos θ
V ¼ μ2H jHj2 þ λH jHj4 þ μ2S jSj2 þ λS jSj4 via diagðm21 ; m22 Þ ¼ OT M2 O. The mixing angle θ is
μ given by
þ λSH jSj2 jHj2 þ 3 ðS3 þ S†3 Þ; ð1Þ
2
λSH vvs
where only the cubic μ3 term softly breaks a global Uð1Þ tan 2θ ¼ : ð7Þ
λH v þ λS v2s − 4p3 ffiffi2 μ3 vs
2
symmetry. Notice that the Z3 symmetry precludes any
quartic couplings that would result in a hard breaking of the
The mixing of the CP-even states h and s will yield two
Uð1Þ. The potential Eq. (1)—just like in the original Z2
CP-even mass eigenstates h1 and h2 The mass of the
pseudo-Goldstone DM model—has an additional discrete
pseudoscalar χ is, taking into account the extremum
Z2 symmetry, S → S† .
conditions,
In the unitary gauge, we parametrize the fields as
0 9
v þ s þ iχ m2χ ¼ − pffiffiffi μ3 vs ; ð8Þ
H ¼ vþh ; S¼ s : ð2Þ 2 2
pffiffi 2
2
which is proportional to μ3 as it explicitly breaks the Uð1Þ
The S → S† is then equivalent to χ → −χ, which makes χ symmetry.
stable even as the Z3 symmetry is broken. The model thus We express the potential parameters in terms of physical
admits two different DM candidates: in the unbroken Z3 quantities in the zero-temperature vacuum, that is the
035001-2GRAVITATIONAL WAVE SIGNALS OF PSEUDO-GOLDSTONE … PHYS. REV. D 101, 035001 (2020)
masses m21 and m21 of real scalars, their mixing angle θ, tree-level values. The model is perturbative [86] if
pseudoscalar mass m2χ , and the VEVs vh and vs , jλH j ≤ 23 π, jλS j ≤ π, and jλSH j ≤ 4π.
We require that our HS vacuum be the global one: this
m21 þ m22 þ ðm21 − m22 Þ cos 2θ implies that
λH ¼ ; ð9Þ
4v2h
9m21 m22
m2χ < ; ð18Þ
3ðm21 þ m22 Þ þ 2m2χ þ 3ðm22 − m21 Þ cos 2θ m21 cos θ þ m22 sin2 θ
2
λS ¼ ; ð10Þ
12v2s
which for sin θ ≈ 0 is approximated by mχ ≲ 3m2. This
bound appears to be stronger than the constraint on the
ðm21 − m22 Þ sin 2θ
λSH ¼ ; ð11Þ cubic coupling from unitarity at finite-energy scattering.
2vs vh If the mass of x ≡ χ or h2 is less than mh =2, then the
Higgs invisible decay width into this particle is given by
1 1
μ2H ¼ − ðm21 þ m22 Þ þ ðm2 − m21 Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4 4vh 2
g m2
× ðvh cos 2θ þ vs sin 2θÞ; ð12Þ Γh→xx ¼ hxx 1 − 4 2x ; ð19Þ
8π mh
1 1 1
μ2S ¼ − ðm21 þ m22 Þ þ m2χ þ ðm2 − m22 Þ with
4 6 4vs 1
× ðvs cos 2θ − vh sin 2θÞ; ð13Þ m2h þ m2χ
gh1χχ ¼ ; ð20Þ
pffiffiffi vS
2 2 m2χ
μ3 ¼ − : ð14Þ 1 1 2
9 vs gh1h2 h2 ¼ 2
m þ m2 ðv cos θ þ vS sin θÞ
vvS 2 h
Both the Higgs doublet and the singlet will get a VEV, with 1 2
the Higgs VEV given by vh ¼ v ¼ 246.22 GeV. We þ vmχ cos θ sin 2θ: ð21Þ
6
identify h1 with the SM Higgs boson with mass
m1 ¼ 125.09 GeV [80]. The invisible Higgs branching ratio is then given by
III. THEORETICAL AND EXPERIMENTAL Γh→χχ þ Γh→h2 h2
BRinv ¼ ; ð22Þ
CONSTRAINTS Γh1 →SM þ Γh→χχ þ Γh→h2 h2
We impose various theoretical and experimental con- which is constrained to be below about 0.24 at 95%
straints on the parameter space of the model. confidence level [87,88] by direct measurements and below
First of all, the potential (1) is bounded from below if about 0.17 by statistical fits of all Higgs couplings [89,90]
pffiffiffiffiffiffiffiffiffiffi (decays of the resulting h2 into the SM may be visible [16],
λH > 0; λS > 0; λSH þ 2 λH λS > 0: ð15Þ but these points are already excluded by direct detection in
any case). In extended Higgs models [91], the mixing
Secondly, we require the couplings to be unitary and phenomenology can be more complicated.
perturbative. The unitarity constraints in the s → ∞ limit The mixing angle between h and s is constrained from
are given by the measurements of the Higgs couplings at the LHC to
j sin θj ≤ 0.5 for m2 ≲ mh and j sin θj ≤ 0.37 for m2 ≳ mh ;
jλH j ≤ 4π; jλS j ≤ 4π; jλSH j ≤ 8π; ð16Þ for the latter case, we have an ever stronger constraint from
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the mass of the W boson down to j sin θj ≤ 0.2 for larger
j3λH þ 2λS 9λ2H − 12λH λS þ 4λ2S þ 2λ2SH j ≤ 8π; ð17Þ values of m2 [92].
Last, but not least, we require that the relic density of χ
be equal to the value ΩDM h2 ¼ 0.120 0.001 from recent
where the last condition, in the λSH ¼ 0 limit, yields jλH j ≤ Planck data [93].
4
3π and jλS j ≤ 2π. We also calculate unitarity constraints at
finite energy with the help of the latest version [81] of the
IV. DIRECT DETECTION AND
SARAH package [82–85]. Scattering at finite energy allows
GRAVITATIONAL WAVE SIGNALS
us to set a bound on the cubic coupling μ3 .
To ensure the validity of perturbation theory, loop For the scan of the parameter space, we choose mχ , m2 ,
corrections to couplings should be smaller than their and sin θ as the free parameters, while vs is used to fit the DM
035001-3KANNIKE, LOOS, and RAIDAL PHYS. REV. D 101, 035001 (2020)
relic density. We generate the free parameter values in the μ2H ðTÞ ¼ μ2H þ cH T 2 ; μ2S ðTÞ ¼ μ2H þ cS T 2 ; ð26Þ
following ranges: mχ ∈½25;1000GeV, m2 ∈½25;4000GeV,
and sin θ ∈ ½−0.5; 0.5. We use the micrOMEGAs package [94] with the coefficients,
to fit the DM relic density and CosmoTransitions [95] for the 1
calculation of phase transitions and the Euclidean action to cH ¼ ð9g2 þ 3g02 þ 12y2t þ 24λH þ 4λSH Þ; ð27Þ
48
determine tunneling rates. The stochastic gravitational wave
signals were calculated using the formulas of Ref. [96] for the 1
cS ¼ ð2λS þ λSH Þ: ð28Þ
nonrunaway case. At the temperature T at which gravita- 6
tional waves are being produced, the spectrum is a function of
For numerical calculations of gravitational wave signals,
the parameters,
the exact expression of the thermal potential is used.
ρvac We present in Fig. 1, the results of our scans for the direct
α¼ ; ð23Þ
ρrad detection signal. In the left panel, we present, for illus-
tration, the behavior of spin-independent direct detection
where ρrad ¼ g π 2 T 4 =30 and g is the number of relativistic cross section σ SI as a function of DM mass mχ for fixed
degrees of freedom (d.o.f.) in the plasma at temperature T , values of m2 ¼ 300 GeV and sin θ ¼ 0.05. Red lines show
and the current limit from the XENON1T experiment [10] and the
future bound from the projected XENONnT experiment [97].
β dS While the usual Z2 scalar singlet DM is excluded below
¼ T ; ð24Þ
H dT T OðTeVÞ masses, except for the Higgs resonance region, the
suppression of the pseudo-Goldstone cross section in this
where S is the Euclidean action of a critical bubble.
model allows for lighter DM candidates. The two pro-
The micrOMEGAs is also used to compute predictions for
nounced dips in the cross section are given by the Higgs
direct detection signals, to which we add the tiny loop
resonance at mh =2 and the h2 resonance at m2 =2.
correction. Unlike in the Z2 pseudo-Goldstone DM model,
The direct detection signal varies over a wide range and
the tree-level direct detection DM amplitude contains a
is proportional to μ23 . Because the amplitude is proportional
term that does not vanish at zero momentum transfer, so
to sin θ cos θ, larger mixing angles also produce larger
Add ðt ≈ 0Þ ∝ λSH μ3 : ð25Þ direct detection signal. The right panel in Fig. 1 presents the
results of the whole scan, coded in grey-scale by the mixing
For a large part of the parameter space, however, this term is angle j sin θj; empty points are excluded by the constraint
small enough, which allows one to explain the negative on the Higgs invisible branching ratio. The points shown
experimental results from DM direct detection experiments satisfy all other constraints with the exception of the
for a wide range of pseudo-Goldstone DM masses. collider constraints [92] on j sin θj (as seen, most such
In the high temperature approximation (A8), the mass points are already excluded by direct detection). Note that
terms acquire thermal corrections, the BRinv can exclude points with mχ > mh =2, because the
FIG. 1. Left panel: Spin-independent direct detection cross section σ SI as a function of the DM mass mχ for fixed m2 ¼ 300 GeV and
sin θ ¼ 0.05. The bound from the XENON1T experiment is shown in red and the predicted sensitivity of the XENONnT experiment in
dashed red. Right panel: Scatter plot of the results in ðmχ ; σ SI Þ plane in the grey-scale representing the dependence on j sin θj. The upper
bound in σ SI is given by the requirement that the ðvh ; vs Þ minimum be the global one. In unfilled points, the Higgs invisible width
surpasses the experimental limit.
035001-4GRAVITATIONAL WAVE SIGNALS OF PSEUDO-GOLDSTONE … PHYS. REV. D 101, 035001 (2020)
FIG. 2. Left panel: Scatter plot for the peak power of GWs vs the frequency together with the sensitivity curves of the LISA and BBO
experiments. Light red points are forbidden and darker green points are allowed by the results of the XENON1T direct detection searches;
both sets of points pass all other constraints, including the relic density. Right panel: The dependence of the peak power of GWs on the
value of cubic coupling for the points that pass all constraints including direct detection. The color code shows the nature of the phase
transition that gives the strongest signal: O → S in orange, S → S in dark blue, S → HS in cyan, HS → HS in yellow, H → HS in
purple, O → H in red.
Higgs boson can decay invisibly also to two h2 if it is light
enough. Early kinetic decoupling [98,99] may additionally
enhance BRinv several times [32], but, in practice, this
would not change the parameter space of GW signals. In
particular, all our points with a potentially measurable GW
signal have mχ , m2 > mh =2. The upper bound on σ SI arises
from the requirement the HS vacuum be global, which
bound is stronger than that from unitarity that also con-
strains the points from above.
In Fig. 2, we present the predicted stochastic GW signal
together with the predicted sensitivity curves of the future
LISA and the BBO satellite experiments. We used bubble wall
velocity vw ¼ 0.5 and the fraction κ turb ¼ 0.05κ v in a
turbulent motion that is converted into gravitational waves
following [96], where the fraction κ v of vacuum energy
converted into gravitational waves is given in [100]. To avoid
cluttering the plot, we only show the peak power of each GW
spectrum. In the left panel, light red points are excluded by the
XENON1T direct detection constraints, while the darker green
points are still allowed by direct detection; all points not
satisfying the other constraints have been excluded. In the
right panel, we depict the dependence of the GW signal on
the cubic coupling μ3 . The color code shows the nature of the
phase transition that yields the strongest signal in a sequence
of phase transitions. The largest signal is produced by the
O → S transitions, enhanced by a sizable cubic coupling.
Because mχ ∝ μ3 , there is a similar dependence on mχ =m2 .
The strongest signals can be produced at roughly mχ ≈ 2.3m2.
In the left panel of Fig. 3, we show the correlation FIG. 3. Left panel: Scatter plot for the peak power of GWs vs
direct detection cross section. The points pass all other con-
between the gravitational wave signal and the direct
straints. Right panel: Signal-to-noise ratios of the data points for
detection cross section. the BBO experiment. Light red points are forbidden and darker
We have also calculated the signal-to-noise ratio for the green points are allowed by the results of the XENON1T direct
BBO experiment, shown in the right panel of Fig. 3 with the detection searches; both sets of points pass all other constraints.
same color code as Fig. 1. The horizontal thick line shows The horizontal thick line shows SNR ¼ 1.
035001-5KANNIKE, LOOS, and RAIDAL PHYS. REV. D 101, 035001 (2020)
ACKNOWLEDGMENTS
We would like to thank Matti Heikinheimo and Christian
Gross for useful comments. This work was supported by
the Estonian Research Council Grant No. PRG434, the
Grant No. IUT23-6 of the Estonian Ministry of Education
and Research, and by the European Union through the
ERDF Centre of Excellence program Project No. TK133.
APPENDIX: THERMAL EFFECTIVE POTENTIAL
At the one-loop level, the quantum corrections to the
scalar potential in the MS renormalization scheme are given
by
X 1
FIG. 4. β=H vs α. Light red points are forbidden and darker 4 m2i
green points are allowed by the results of the XENON1T direct ΔV ¼ ni mi ln 2 − ci ; ðA1Þ
i
64π 2 μ
detection searches.
where ni are the d.o.f. of the ith field, mi are field-
SNR ¼ 1; the actual detection threshold is expected to be dependent masses, and the constants ci ¼ 32 for scalars
within an order of magnitude of it [96]. and fermions and ci ¼ 56 for vector bosons. The masses and
Because the direct detection cross section is proportional d.o.f. ni of the fields are given in Table I. The field-
to μ23 , the points on the left panel of Fig. 2 with the highest dependent masses of h and s are given by the eigenvalues
GW signal are excluded. Note, however, that the direct m21;2 of the mass matrix of the CP-even eigenstates with
detection signal can still be small even for a sizeable GW elements given by
signal, because the Higgs portal λSH is on the order of (0.01).
This is in contrast with λSH ∼ 1 in Ref. [74] with the 1
ðm2R Þ11 ¼ μ2H þ 3h2 λH þ λSH s2 ; ðA2Þ
assumption that the Z3 symmetry is unbroken in our vacuum. 2
Figure 4 shows the range of β=H vs α with the same 2
ðmR Þ12 ¼ λSH hs; ðA3Þ
color code as Fig. 1. While Ref. [101] showed that for α
close to unity the gravitational wave signal is suppressed, 3 pffiffiffi 1
ðm2R Þ22 ¼ μ2S þ 3λS s2 þ 2μ3 s þ h2 λSH : ðA4Þ
our values of α are quite smaller than one. 2 2
We neglect the contributions of the Goldstone bosons G0
V. CONCLUSIONS
and G as unimportant numerically. To calculate the
We have studied the Z3 complex scalar singlet DM effective potential in the case of negative field-dependent
model, where only the cubic coupling of the singlet masses, we substitute ln m2i → ln jm2i j, which is equivalent
explicitly breaks a global Uð1Þ symmetry. The model to analytical continuation [102]. We set the renormalization
has two phases with stable DM. In the phase where the scale to μ ¼ M t .
Z3 is spontaneously broken, the residual CP-like Z2 We add a counterterm potential,
symmetry stabilizes the imaginary part of the complex
singlet S as a pseudo-Goldstone DM candidate. The DM δV ¼ δμ2H jHj2 þ δλH jHj4 þ δμ2S jSj2 þ δλS jSj4
direct detection cross section can be considerably sup- δμ3 3
pressed by the small momentum transfer or the resonance at þ δλSH jSj2 jHj2 þ ðS þ S†3 Þ þ δV 0 ; ðA5Þ
half the mass of the heavy singletlike particle. 2
While we may be unable to discover the pseudo-
Goldstone DM via direct detection, it may be testable by TABLE I. Field-dependent masses and the numbers of d.o.f.
other means. In this model, a stochastic gravitational wave
background can arise from the first-order phase transitions Field i m2i ni
due to the presence of a cubic coupling for the singlet. The h m21 1
strongest signals, which can potentially be observed by the s m22 1
future BBO experiment, are produced in the O → S phase χ μS þ λS s − 3ffiffi2 μ3 s þ 12 λSH h2
2 2 p 1
transition (if present) at mχ ≈ 2.3m2 . The strength of the Z0 1 2 02 2
4 ðg þ g Þh
3
gravitational wave signal is anticorrelated with a small W 1 2 2 6
4g h
mixing angle and is, therefore, greater where the direct t 1 2 −12
2 yt h
detection cross section is smaller (at fixed cubic coupling).
035001-6GRAVITATIONAL WAVE SIGNALS OF PSEUDO-GOLDSTONE … PHYS. REV. D 101, 035001 (2020)
in order to fix the VEVs and the mass matrix in our HS The counterterm potential δV is chosen such as to keep
minimum to their tree-level values. quantum corrections to the masses, to mixing between h
The thermal corrections to the potential are given by and s and to the VEVs zero. The counterterms are given by
T4 X mi 1
VT ¼ 2 ni J ∓ ; ðA6Þ δλH ¼ ð∂ h ΔV − v∂ 2h ΔVÞ; ðA10Þ
2π i T 2v3
1
δλS ¼ ð4∂ s ΔV − vS ∂ 2χ ΔV − 3vS ∂ 2s ΔVÞ; ðA11Þ
where 6v3S
Z ∞ h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 1
J∓ ¼ dyy2 ln 1 ∓ exp − y2 þ x2 ; ðA7Þ δλSH ¼ − ∂ ∂ ΔV; ðA12Þ
vvS h s
0
1
with the − sign applied to bosons and the þ sign to δμ2H ¼ ð−3∂ h ΔV þ vs ∂ h ∂ s ΔV þ v∂ 2h ΔVÞ; ðA13Þ
2v
fermions. In the high-temperature limit m=T ≪ 1, the
1
thermal contributions are given by δμ2S ¼ − ðv ∂ 2 ΔV þ 8∂ s ΔV − 3vS ∂ 2s ΔV
6vS S χ
T2 X − 3v∂ h ∂ s ΔVÞ; ðA14Þ
V T ðTÞ ¼ n m2 : ðA8Þ
24 i i i pffiffiffi
2 2
δμ3 ¼ 2 ðvS ∂ 2χ ΔV − ∂ s ΔVÞ; ðA15Þ
The full thermally corrected effective potential is then 9vS
where we take h ¼ v, s ¼ vS , χ ¼ 0 after taking the
V ð1Þ ¼ V þ ΔV þ δV þ V T : ðA9Þ
derivatives.
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