Oneloop considerations for coexisting vacua in the CP conserving 2HDM
HJE
Oneloop considerations for coexisting vacua in the CP conserving 2HDM
A.L. Cherchiglia 0 1
C.C. Nishi 0 1
Santo Andre 0 1
Brazil 0 1
0 Universidade Federal do ABC
1 Centro de Matematica, Computac~ao e Cognic~ao
The TwoHiggsDoublet model (2HDM) is a simple and viable extension of the Standard Model with a scalar potential complex enough that two minima may coexist. In this work we investigate if the procedure to identify our vacuum as the global minimum by treelevel formulas carries over to the oneloop corrected potential. In the CP conserving case, we identify two distinct types of coexisting minima  the regular ones (moderate tan ) and the nonregular ones (small or large tan )  and conclude that the tree level expectation fails only for the nonregular type of coexisting minima. For the regular type, the sign of m212 already precisely indicates which minima is the global one, even at oneloop.
Beyond Standard Model; Higgs Physics; Spontaneous Symmetry Breaking

1 Introduction
2 Coexisting normal vacua at treelevel
3 E ective potential at oneloop
4 Parametrization and minimization at oneloop
5 Pole masses at oneloop
6 Numerical survey
7 Results
8 Conclusions
A Deviation of a potential minimum
B Finding more than one normal vacua
C Matrix of second derivatives
D Cubic derivatives
E Cubic and quartic couplings
F Selfenergy for the scalars
G Reparametrization symmetry
After the discovery of the Higgs boson of 125 GeV mass in 2012 [1, 2], all the pieces of
(2HDM) in which just another scalar doublet is added. This model features ve physical
Higgs bosons, three neutral and two charged, instead of just one in the SM. It has been
extensively studied in the literature (see e.g. ref. [4] for a review) partly because more
fundamental theories, for instance the MSSM [5{7], require a similar extended scalar
sector. The model also allows the possibility of other sources of CP violation [8, 9], a feature
that gets even richer when more doublet copies are added [10]. Finally, a more complex
scalar sector can generate a strong enough EW
rstorder phase transition [11{27], a
property that is lacking in the SM [28{33] but is necessary to explain the matterantimatter
asymmetry of our universe.
Another feature that a more involved scalar potential encompasses is the possibility
of di erent symmetry breaking patterns and the 2HDM is no exception [34{40].
With
additional Higgs doublets, this complexity increases substantially [41{44]. In general, for
a su ciently complex potential, there may even exist sets of parameters for which many
local minima coexist. In this case, identifying which one is the global minimum might be
a nontrivial task that involves solving a system of polynomial equations. Usually, when
one is sure that only one minimum is present for a given choice of parameters, such a
task is bypassed by trading some of the quadratic parameters in favor of the vevs and
ensuring the extremum is a local minimum. In the 2HDM, that assumption does not hold
in general and for some parameter ranges it is possible that up to two minima coexist
for the same potential [35, 36]. So a worrisome possibility arises: we may be living in a
metastable vacuum with the possibility to tunnel to the global minimum. That situation
was described in ref. [45, 46] as our vacuum being a panic vacuum. One way of testing
that situation is explicitly calculating the depth of the second minimum and comparing it
to the depth of the rst. However, nding the location of the minima explicitly may be
a di cult or, at least, computationally intensive task. Fortunately, in the same work, the
authors developed a method capable of distinguishing if our vacuum is a panic vacuum by
calculating a discriminant that depends only on the position of our vacuum (see ref. [47] for
a more general test). Although many of the possible scenarios with coexisting minima are
not favored by current LHC data [45, 46], we are interested here in studying if the simple
use of this discriminant can be carried over to the oneloop corrected e ective potential.
Already for the inert doublet model [34] it was found that the potential di erence of the
coexisting minima can change sign when oneloop corrections are taken into account [48].
Therefore, the present work aims to verify the validity of such conclusions for a general CP
conserving 2HDM with softly broken Z2. We focus on the case of two coexisting normal
vacua and study the predictive power of treelevel formulas for the depth of the potential
when oneloop corrections are considered.
The outline of the paper is as follows: in section 2 we review the properties of the
2HDM with softly broken Z2 at treelevel focusing on the possibility of two coexisting
minima. Some results can not be found in previous literature. In section 3 we review the
form of the oneloop e ective potential for our case while section 4 explains our procedure
for ensuring that our vacuum has the correct vacuum expectation value. The procedure
to compute and
x the pole mass of the SM Higgs boson to its experimental value is
explained in section 5. The steps we performed to generate the numerical samples are
{ 2 {
listed in section 6 and the resulting analysis is shown in section 7. Finally, the conclusions
can be found in section 8.
2
Coexisting normal vacua at treelevel
The general 2HDM potential at treelevel is
We will be considering the real softly broken Z2 symmetric case where m212 and 5 are real
real and we employ the parametrization
We will also focus on CP conserving vacua where the vacuum expectation values are
These vevs can be further parametrized by modulus and angle as
h 1i = p
1
2
0
v1
!
;
h 2i = p
1
2
0
v2
!
:
(v1; v2) = v (c ; s ) ;
where v = 246 GeV for our vacuum and we use the shorthands c
cos , s
sin . We
call this type of vacuum a normal vacuum and we denote our vacuum by NV [45, 46] with
v1 > 0 and v2 > 0.
By ensuring the existence of one normal vacuum (our vacuum), a scalar potential with
xed parameters cannot simultaneously have another minimum of a di erent type, namely
a charge breaking vacuum or a spontaneously CP breaking vacuum [37, 38]. Just another
coexisting normal vacuum NV0 with vevs (v10; v20) may exist and this is the only case where
two minima can coexist in the 2HDM potential at tree level [35, 36]: only two minima with
the same residual symmetry may coexist. When the coexisting minima exist, we de ne the
potential di erence as
V
VNV0
VNV ;
so that
V > 0 indicates that our vacuum is the global minimum. We use this convention
for the oneloop potential as well.
To describe the situation of two coexisting normal vacua in more detail, we can write
the extremum equations for nonzero v1 and v2:
1
1
1
1
2 1v12 +
We employ the usual shorthand 345
symmetric limit. The complete solutions of eq. (2.5) for m212 = 0 can be easily found:
there are two degenerate extrema that spontaneously break Z2  ZB+ and ZB
 and
two extrema that preserve Z2  ZP1 and ZP2; the latter are often denoted as inert or
inertlike vacuum (see ref. [48] and references therein). Only one of the pairs ZP1;2 or ZB
may coexist as minima.1 They are characterized by (v1; v2) of the form
=
1
1
2
As the
m212v1v2 term is continuously turned on, the Z2 symmetry is soft but explicitly
broken with a negative (positive) contribution in the rst (fourth) quadrant when m212 > 0.
The opposite is true for negative m212. The e ect of adding the m212 term is di erent for
the two types of coexisting minima which we denote by ZB
and ZP1;2 from their m212 ! 0
limit. We also denote the ZB
minima as regular and ZP1;2 as nonregular simply because
it is much more probable to generate models with the former pair than the latter for generic
values of tan
and other parameters.
The two degenerate spontaneously breaking minima ZB : (v1; v2) deviate to ZB+ :
(v1; v2) and ZB : (v10; v20), respectively, and the degenerate potential depth,
. We note that simultaneous sign ips of both
v1; v2 is a gauge symmetry and do not count as a degeneracy. Hence we adopt the
convention that v1 > 0 while v2 can attain both signs so that we only analyze the rst and fourth
quadrant in the (v1; v2) plane.
V0(v1; v2) =
also deviates di erently lifting the degeneracy. In rst approximation in small m212 and in
the deviation of the vevs, the potential depths change respectively by the amount V
m212v1v2, so that the depth di erence of the two minima is
VZB
VZB
VZB+ = V
V+
2m122v1v2 :
1Note that one of ZP1;2 may not be a minimum whereas ZB are always degenerate.
2As long as the solutions for vi2 give positive solutions.
{ 4 {
ZB+ :
where we adopt the convention that all v1; v2; v~1; v~2 are positive. The speci c values of the
vevs are given by
HJEP1(207)6
for the Z2 breaking extrema2 and
(2.6)
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)
where
by
v2
1
2
with m2Hi = mi2i + 12 345v~j2, (ij) = (2; 1) or (1; 2), is the second derivative in the vi direction
around ZPj . When m212 > 0, the deviations vi are positive and the two minima enter the
rst quadrant. Otherwise they move to the fourth quadrant. The potential depth then
changes from the Z2 limit
m411 ;
indeed it gets deeper as m212 increases from zero while the nonstandard vacuum ZB
respectively. The depth di erence of the coexisting ZP1;2 is
VZP = V0jZP1
V0jZP2
m411
where we show the depth di erence calculated exactly against m212, both
normalized by the appropriate power of v = 246 GeV (NV). Only potentials with two
minima are selected and the free parameters are taken as
ftan ; ; m2H+ ; m2A; m2H0 ; m2h; m122g ;
with mh = 125 GeV, 1
tan
50, fmH+ ; mA; mH g ranging from 90 GeV to 1 TeV
(mH > mh),
0:1
cos(
20;000 GeV2
m212
6000 GeV2 and
is constrained near alignment,
)
0:1. Simple bounded from below and perturbativity constraints are
also imposed [4]. The blue points end around m212=v2
0:07 because the nonstandard
minimum gets pushed up until the point where it disappears. In contrast, the right panel
shows the normalized depth di erence with respect to the ratio v0=v of the values for NV0
and NV. We can see for the blue points that the vacuum that lies deeper has a larger
vacuum expectation value. The method we employed to calculate the location of NV0 is
described in appendix B.
{ 5 {
to the standard vev (NV).
We con rm the approximation (2.11): for ZB
the sign of m212 discriminates between
our vacuum being the global minimum (m212 > 0) or just a local metastable vacuum
(m212 < 0). This behavior, however, does not apply for the points (red) that deviate from
the inertlike vacua, ZP1;2, where the nonstandard vacuum may lie deeper despite m212 being
positive. For the generic values of t as used above the density of nonregular coexisting
minima is very low so that only a handful of coexisting nonregular minima is obtained
jointly with the regular points. To generate a su cient number of nonregular points we
further produced another sample (most of the red points) by restricting 20
t
50 and
positive m212.
To accurately distinguish among the di erent cases, ref. [45, 46] constructed a very
useful discriminant D that ensures that our vacuum is always the global minimum if D is
positive.3 Since that discriminant was derived assuming that v1; v2 are both positive, we
cannot apply it to NV0 when v20 < 0. So we rederived the discriminant allowing the vevs
to be negative with the result
D =
1
v4 m122(m121
k2m222)s2 (t2
k2) ;
(2.18)
where k
( 1= 2)1=4 and we have normalized to obtain a dimensionless quantity. This
discriminant is useful because it can be obtained by using only the angle
calculated in
one vacuum and cases with only one minimum are automatically taken into account. The
discriminating power of D is shown in gure 2 where the depth di erence is plotted against
D calculated using NV. Obviously we could have calculated the discriminant for NV0,
obtaining a D0 with sign opposite to D. That implies that the quantity that depends on
the vevs, s2 (t2
k2), must have opposite signs when calculated for NV and NV0.
Our main goal here is to analyze if the discriminant power of m212 and D carries over
to the oneloop e ective potential.
3For D = 0 but m122 6= 0 the discriminant is inconclusive.
{ 6 {
inant (2.18).
3
E ective potential at oneloop
We can now consider the e ective potential
with the oneloop contribution
V = V0 + V1l ;
V1l =
1
The masses Mk2('i) correspond to the scalar elddependent eigenvalues of the treelevel
mass matrices of all particles of the theory while
is the renormalization scale. We are
already assuming a renormalization scheme with minimal subtraction (MS) and, for the
gauge sector, the Landau gauge and dimensional reduction (DRED), following the scheme
of ref. [49, 50]. The parameters contained in V0 are thus the renormalized parameters.
The integer coe cients jckj count all the degrees of freedom for each particle k including
color, charge and spin, while the sign of ck is determined by its boson/fermion character:
positive for bosons and negative for fermions. For example, for the top quark we have
ct =
3
2
2 corresponding to its 3 colors, 2 particle/antiparticle and 2 spin degrees
of freedom. We should note that the e ective potential is generically a gauge dependent
quantity but its value at an extremum is not [51].
As we will focus on normal vacua, we can consider that the e ective potential depends
only on the two real values '1; '2 in the real neutral directions:4
1 = p
1
2
0 !
'1
;
2 = p
1
2
0 !
'2
:
(3.3)
4For a generic eld dependence modulo gauge freedom, we would need two more real directions.
{ 7 {
We reserve the symbols v1; v2 in eq. (2.2) to values at a minimum. So the elddependent
gauge boson masses retain the same functional form as in the SM with v2 = v12 + v2:
2
M W2 ('i) =
4
1 g2('21 + '22) ;
1
4
MZ2 ('i) =
where yt;b are the Yukawa couplings of the third family quarks normalized to the SM
values and the enhancement factor 1=hs i should be considered as the
xed value at the
NV minimum at oneloop. We emphasize that information with the bracket h i. For the
type II, we should replace the Mb dependence on '2 and hs i by '1 and hc i respectively.
We will see that, as usual, the top correction dominates the fermion loops and the di erence
between type I or type II is negligible for the oneloop corrections except for excessively
large tan
which we do not consider. It is also justi ed that we only consider the e ects
of the top and bottom quarks; see gure 3 and comments in the text.
For the scalar contribution we need to calculate the eigenvalues of the matrix of second
derivatives of V0 for generic values of 'i. These mass matrices are shown in appendix C
and their eigenvalues correspond to MS2('i) of the 8 scalars S 2 fG ; H ; G0; A0; H0; h0g.
Due to charge and CP conservation, the mass matrices are still separated into three sectors:
two charged scalars and its antiparticles, two CP odd scalars and two CP even scalars. We
emphasize that e.g. MG2 0 ('i) is nonvanishing at 'i away from any treelevel minimum. It
is the second derivative of the whole e ective potential at oneloop that will vanish in the
directions of the Goldstone modes.
4
Parametrization and minimization at oneloop
We are interested in surveying the cases where the e ective potential at oneloop (3.1)
continues to have two local minima, one of which should be our vacuum with v = pv12 + v22 =
246 GeV. The vevs v1; v2 no longer satisfy the treelevel minimization relations in (2.5)
but should now minimize the whole e ective potential V0 + V1l.
We need a convenient
parametrization to ensure that one minimum has the appropriate value of v.
To parametrize V0, we will use as input the usual 8 quantities
fv1; v2; ; m2H+ ; m2A; m2H0 ; m2h; m122g ;
(4.1)
where vi satisfy the minimization equations (2.5). It is clear that these quantities de ne
V0 unambiguously by
xing the 8 parameters fm211; m222; m212; 1
; 2
; 3
; 4; 5g; see e.g.
ref. [45]. When we add the oneloop contribution, it is clear that the true minimum will
{ 8 {
We can see that mi2i ! 0 and V1l ! 0 in the limit where we turn o all couplings of the
scalars to other particles including selfcouplings. For small couplings, it is also expected
that the physical masses are close to the masses fm2H+ ; m2A; m2H0 ; m2hg used as input. It is
possible to use a di erent renormalization scheme where all the masses and mixing angles
at tree level are maintained at oneloop [52]. Our scheme, however, avoids the need to
deal with infrared divergences coming from the vanishing Goldstone masses [22{27, 53, 54].
This problem is more severe at higher loop orders [55].
Now the minimization equations at oneloop can be separated into a treelevel part
which leads to the treelevel equations written in (2.5), and a oneloop part that de nes
mi2i by
= 0 ;
i = 1; 2 ;
mi2i =
i = 1; 2 :
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
be shifted by a small amount from the position (v1; v2) at treelevel. Instead of correcting
for that shift, we add the nite counterterms
to the potential and adjust the values of mi2i so that v1; v2 continue to be a minimum at
oneloop.5 This means that the oneloop e ective potential (3.1) is now rewritten as
V =
We can separate the derivative of V1l into its contribution from scalars (S), vector bosons
(V ) and fermions (F ):
1
h3syti22 mt2 ln
mt2
2
1 + g2m2W ln
1 +
h3sybi22 mb2 ln
m2
b
2
m2W
2
1
;
1
;
for each i = 1; 2. The dimensionless coe cients
iS are given in appendix D and we
use lowercase letters for mW;Z;t;b because they correspond to the actual values in the SM
when we use our vacuum NV. For charged particles the contribution of the antiparticle
can be taken into account by doubling the contribution of the particle. The fermion part
corresponds to the type I model. For the type II model, we must replace hs i ! h
c i in
the couplings to the b quark. We can see in gure 3 that the contributions from scalars are
5This procedure is equivalent to trading m121; m222 in favor of v1; v2 [48].
{ 9 {
large for m211 and m222 while the top contribution is also large and negative for m222. The
contribution from the bottom quark is negligible for tan
50 and there is no appreciable
di erence between the type I or type II model. Thus for de niteness we consider the type I
model. We also note that the scalar masses and coe cients depend on m211; m222 and (4.5)
must be solved selfconsistently.
The only remaining task is to write MS2(vi) in terms of the input parameters (4.1). We
note that MS2(vi) should be computed from the second derivatives of V0 + V . But the part
coming from V0 at 'i = vi corresponds to the usual masses at treelevel because (v1; v2)
still corresponds to a minimum of V0. Therefore, these matrices will have the generic form
HJEP1(207)6
Speci cally, the mass matrices for the di erent sectors read
M(vi) = M
tree +
M :
s
2
s c
s
2
s c
s c !
c
2
+
s c !
c
2
+
m211
m211
m222
!
m222
!
;
;
Mchar(vi) = m2H+
Modd(vi) = m2A
Meven(vi) =
(4.7)
(4.8)
(4.9)
(4.10)
(4.11)
(4.12)
(4.13)
(4.14)
2 5 are the masses squared for the charged
Higgs and the pseudoscalar at treelevel; see e.g. [4]. Clearly the
rst and second
matrices contain each a vanishing eigenvalue corresponding to the charged (G ) and
neutral (G0) Goldstone bosons in the limit where
(p2 Im 0 p
1
; 2 Im
02) and (p2 Re 01; p
from the parametrization i = ( i+; i0)T.
2 Re 02) in eqs. (4.8), (4.9) and (4.10), respectively,
mi2i ! 0.
We use the basis ( 1+
; 2+),
Diagonalization of the treelevel part of Meven(vi) de nes the angle
through
where
Using the same notation we can nd the eigenvalues shifted by
mi2i as
U y MterveeenU
= diag(m2H0 ; m2h0 ) ;
U
=
c
s
s !
c
:
U y+ Mchar(vi)U + = diag(MG2 + ; M H2+ ) ;
U y0 Modd(vi)U 0 = diag(MG2 0 ; MA20 ) ;
U y0 Meven(vi)U 0 = diag(M H20 ; Mh20 ) ;
where the angles +; 0; 0 are shifted from ; ;
by a small amount due to mi2i:
+ =
+
+ ;
0 =
+
0 ;
0 =
+
:
The explicit forms for
+;0 and
can be seen in appendix D.
The previous section showed a way of ensuring that one of the minima of the e ective
potential at oneloop corresponded to our vacuum with v = 246 GeV and that tan
= v2=v1
could be used as input at oneloop. The following task to describe a realistic 2HDM at
oneloop is to ensure that the SM higgs boson mass corresponds to the experimentally
measured value [56]:
It is clear that at oneloop we cannot use this value for the treelevel parameter mh in (4.1).
Instead, we must check that the pole mass corresponds to (5.1).6
We follow ref. [50] to calculate the pole masses m^S of all the scalars S including the
SM higgs boson by computing the selfenergies of the theory at oneloop. We focus in this
section on the CP even sector which will give rise to the pole masses of h and H restricted
to the case m^H > m^ h. The selfenergy for the other sectors are given in appendix F.
The scalar selfcouplings can be extracted from V0. Given a set of real scalars Si that
interact through the quartic vertex
igijkl and cubic vertex
igijk, the selfenergy
ij for
Si{Sj coming from scalars in the loop with incoming momentum p2 = s is given by [49, 50]
16 2 iSj (s) =
1 X gijkkA(mk) +
1 X gijklB(mk; ml; s) ;
2
assuming we are in the basis with diagonalized masses (quadratic part of V0 + V ). The A
and B functions are the PassarinoVeltman functions [59] which, in the notation of ref. [48],
read
6Another possibility is to use a more physical renormalization condition to ensure this [52]. The
renormalization of the entire theory is discussed in ref. [57, 58].
2
k
Z 1
0
A(m)
m2 log
B(m1; m2; s)
dt log
2 kl
1 ;
m2
2
t m21 + (1
t) m22
2
t(1
t)s
:
(5.1)
(5.2)
(5.3)
(5.4)
(5.5)
The A function represents the oneloop graph with one quartic vertex (tadpole) and the
B function represents the oneloop graph with two vertices and two internal lines. Hence
the factors 1=2 are the symmetry factors that appear in front of the Feynman diagrams for
identical elds. These functions appear after renormalizing these diagrams using the MS
prescription. Simpli cations for vanishing s can be found in ref. [48]. The contributions
coming from gauge bosons and fermions in the loop are also shown in appendix F and all
the necessary cubic and quartic couplings of the theory are explicitly shown in appendix E.
For example, the SM higgs selfenergy is given by
where gh2h2 = gh4 and ghh2 = gh3 are the quartic and cubic selfcouplings for h and all MS
refer to MS(vi). However, the mixing to the other CP even scalar H in
Hh cannot be
neglected.
Due to charge and CP conservation, the selfenergy for the di erent scalars decouple
into separate pieces for the CP even, CP odd and charged sectors. The selfenergy for the
CP even sector is given by the matrix
even(s) =
HH (s)
hH (s)
Hh(s)!
hh(s)
:
The oneloop pole squared masses m^2H ; m^ 2h for the CP even scalars H; h are the solutions
where we take only the real part of the selfenergy because we are not interested in the
decay widths. The m^2h corresponds to the solution that continually approaches Mh2 in the
limit where we turn o the interactions. A similar consideration applies to all pole squared
masses.
6
Numerical survey
We describe here the procedure we used to survey the models. For de niteness, we adopted
the relevant xed parameters of the SM to be
g = 0:6483 ;
g0 = 0:3587 ;
v = 246:954 GeV ;
yt = 0:93697 ;
yb = 0:023937 :
(5.6)
(5.7)
(6.1)
The rst four values were taken from refs. [55, 60] as the running parameters of the SM at
the top pole mass
= 173:34 GeV and the bottom Yukawa was adapted from its mass value
mb = 4:18 GeV [56]. Among these parameters, only the top Yukawa appreciably a ects
the oneloop e ective potential together with the quartic scalar selfcouplings. We use a
xed renormalization scale
= 300 GeV for all calculations and note that the running of yt
from the top mass scale only amounts to a small di erence. The running for the rest of the
parameters are even less relevant. We remark that any choice of the renormalization scale
is allowed, since the di erence in depth of the potential at two extrema is a renormalization
scaleindependent quantity [61]. However, from a practical point of view, it is desirable
that the logarithms in the e ective potential do not become too \large" so as to lead to
numerical instabilities [62]. We have checked our calculation for some di erent values of
the renormalization scale and the value of
= 300 GeV proved to be a stable choice.
Among the input parameters in (4.1), we xed the standard vev v as above and took
the rest of the parameters randomly in the range shown in the rst row of table 1, restricted
to mH > mh. After checking for simple perturbativity and bounded from below conditions
at treelevel,7 we picked only the points where the shifted masses squared MS2(vi) were
7Since we work with a xed renormalization scale where the oneloop corrections are not large, we expect
the tree level relations to be valid to a good approximation [64{66]. For bounded from below conditions,
this is a conservative choice as oneloop corrections may enlarge the possible parameter space [67].
Sample
t
cos(
) mH+ (GeV) mA (GeV) mH (GeV) mh (GeV)
G
NR
1
20
50
50
0:1
0:1
90
1000
90
1000
90
1000
0
150
6000
6000
dash denotes the same range as in the previous row.
positive and the solutions for the shift
mi2i in (4.5) were real. Then we further selected
only the points where the pole mass m^h calculated as in section 5 fell in the experimental
range of (5.1).8 In this way, we generated the sample G with 294437 points among which
4525 had two minima at oneloop, 17 of which were nonregular.
To
nd the second
minimum, we explicitly minimized the real part of the e ective potential (3.1) starting
from the nonstandard minimum at treelevel and then retained only the points where the
value of the potential at that minimum was real [63]. Given the small number of
nonregular points, we generated another sample denoted as NR focusing only on nonregular
points by imposing large t as in the second row of table 1; other ranges were kept the
same, except for m212 which was chosen positive. Sample NR thus contained 185905 points
among which 1563 had two minima at oneloop. Hereafter, unless explicitly speci ed, we
will consider only the joint sample of G and NR. However, we would like to emphasize
that, even though samples G and NR have a similar number of points, the parameter space
scanned by sample NR represents only a small portion of the parameter space probed by
sample G. This justi es our de nition of nonregular points.
After the selections described above, the input parameters mH+ ; mA; mH get roughly
con ned to the range [90; 500] and the nonstandard higgsses acquire pole masses in a
similar range with slightly smaller maximal value for m^H+ . In contrast, the distribution
for t is homogeneous in the range of table 1 for the whole sample but, as we select only
the points with two minima at oneloop, it gets separated into two ranges, 1
t . 3:8 for
the regular minima, and 24 . t
50 for the nonregular ones.
To check our numerically implemented formulas, we performed the following
consistency checks:
1. Vanishing of the pole masses for the Goldstone bosons G ; G0 for zero external
momentum for the three cases where we successively add the oneloop scalar, vector
boson and fermion contributions.
2. Equality of the pole masses for the CP even higgsses H; h for zero external
momentum and the eigenvalues of the explicitly calculated second derivative matrix of the
e ective potential in the real neutral directions (3.3) in all three cases of successive
addition of the oneloop scalar, vector boson and fermion contributions.
8The exact adopted procedure di ers slightly with respect to the range of mh: instead of postselecting
only the values of mh for which the pole mass coincided with the experimental range, we randomly selected
the input mh in the range [0; 200] GeV and then later varied only this value searching for the correct
pole mass m^h. If a solution were found, we kept that point. This procedure resulted in the approximately
homogeneous distribution of mh in the range shown in table 1. This modi cation speeded up the generation
of points.
HJEP1(207)6
m222 (green points) while the fermions contribute negatively.
The fermion contribution to
while the contribution to
m222 (red curve) is practically the same for the type I and II models
m211 (orange curve) applies only to the type II case but vanishes for the
type I case. The contributions from the gauge bosons are shown in the purple dashed line.
only whereas the red points consider all the contribution in the type I or II model.
7
Results
Let us rst quantify the shifts mi2i in (4.5) for the di erent contributions. The contribution
coming from the gauge bosons only depends on the value of v and, for fermions, it depends
on v and . The scalar contribution depends on many parameters coming from the scalar
potential. The dependence of the di erent contributions on tan
are shown in gure 3. We
can see that the dominant contribution for
m211 comes from the scalars whereas
m222 also
has large positive contributions from scalars but they are partly canceled by the negative
contribution from fermions (top). Such a partial cancellation in
m222 can be clearly seen
in gure 4 where we show the contribution only from scalars (green points) and all the
blue (red) points represent (non)regular points.
contributions (red points). The orange curve in
gure 3, which quanti es the bottom
contribution to
m211 in the type II model, shows that it is negligible in the range of tan
we are interested in and our calculation that uses the type I model applies equally well
to the type II case. All the di erent contributions are calculated using eq. (4.6) and the
scalar contribution in particular depends on the
mi2i themselves and these are taken as
the total contributions. The remaining contributions of gauge bosons (purple dashed curve
in
gure 3) are much smaller. The contribution from scalars are calculated using the
whole sample (G + NR) described in section 6 which also includes the points with only one
vacuum.
The deviation of the location of our vacuum when we add V do V0 is illustrated in
gure 5 where we show the ratio of t for V0 + V (ttree+ ) to that of V0 (t ) against the ratio
of the vev for V0+ V (vtree+ ) to that of V0 (v). We only show the points with two coexisting
minima and divide the points between the regular ones (blue) and the nonregular ones
(red). We can see that as mi2i get shifted by
mi2i all points with two minima have their
vevs decreased while t
mostly increases for the regular points and mostly decreases for the
nonregular points. Note that the nonregular points only consider large t
whereas the
regular points only include moderate t roughly up to 3:8. As the location of (v1; v2) for our
vacuum is the same for V0 and the oneloop corrected potential (4.3) we can also interpret
this plot as the modi cation of the vev location of V0 + V compared to V0 + V + V1l. If
we had considered all the points including the points with only our vacuum, the majority
of points would follow the behavior of the regular points in blue.
For the case of coexisting vacua, we can see a clear di erence in behavior between the
regular points and the nonregular ones in
gure 6 where we plot the potential di erence
with respect to m212. The left panel shows these quantities using the treelevel potential
V0 while the right panel is the same plot for the oneloop potential (4.3). We can clearly
see that the potential di erence goes continuously to zero as m212 ! 0 for the blue points
representing the regular minima. Moreover, for positive m212 our vacuum is guaranteed to
be the global minimum in both tree level or oneloop potential. This behavior is opposite
potential (2.1) is considered. This plot should be compared with
gure 1. Right: the full 1loop
e ective potential (4.3) is considered. The green points represent a change in sign of the tree level
prediction for the potential di erence.
(red), scalars (purple), and vectors bosons (green) to the oneloop potential di erence of the right
panel of gure 6. In the left plot only regular points are considered while the right plot shows the
behavior of the nonregular ones.
for negative m212. The reason is that only m212 breaks explicitly the Z2 symmetry of the
theory and it controls the degeneracy breaking of the spontaneously breaking minima. This
behavior is not followed by the nonregular minima that do not have degenerate minima
in the m212 ! 0 limit. Some points (green) in which our vacuum is not the global one at
treelevel even get inverted and become the global minimum as the oneloop corrections
are added. The same conclusion is reached if we had compared the oneloop potential to
V0 + V instead: some cases where our minimum is not the deepest at treelevel becomes
the global minimum at oneloop.
We can have an idea of the di erent oneloop contributions in gure 7 where we separate
the potential di erence in
gure 6 into its di erent contributions. Regarding the regular
points (left plot), it can be seen that the oneloop potential di erence is almost entirely due
ence. The color coding follows gure 6.
to the treelevel contribution (blue) since the contributions from V (yellow), fermions (red)
and scalars (purple) approximately cancel each other while the contribution from gauge
bosons (green) is negligible compared to the others. This behavior justi es our choice
for the renormalization scale. For the nonregular points (right plot), no clear pattern
emerges. We can also see in
gure 8 that there are points where the potential di erence
is raised as well as points where it is lowered by the oneloop corrections for both regular
and nonregular points.
Considering that the discriminant (2.18) test is applicable for all cases where at least
one normal vacuum is known, we can investigate if it is still a good predictor for the
oneloop potential with two coexisting minima. We can adapt the treelevel discriminant to
oneloop in the following three ways:
D1 = eq. (2.18) ;
D2 = eq. (2.18) mi2i!mi2i+ mi2i; v!v(0); ! (0) ;
D3 = eq. (2.18) mi2i!mi2i+ mi2i :
(7.1)
The quantity D1 is the discriminant calculated with V0 as the whole potential while D2
is calculated by using V0 + V in (4.3). In the latter case, the quadratic parameters mi2i
get shifted and the location of the minimum, denoted above as (v(0); (0)) or as tree +
(superscript/subscript) in
gure 5, do not coincide with the ones used as input, denoted
as (v; ). The last adaptation D3 considers the shifts in the quadratic parameters but
keeps the vevs as (v; ). We will test here if any of these discriminants are capable of
distinguishing if our vacuum is the global one at oneloop only using parameters at
treelevel.9
9Strictly speaking, some calculation at oneloop is required for some of these quantities depending on
how the calculation is set up. Using the splitting of the potential in the form (4.3), D1 is the most natural
quantity to use and there is no oneloop calculation required. If V0 +
V is considered as the potential at
treelevel, D2 or D3 are the natural quantities depending on which minimum is taken.
points. The green points mark the cases where D1 predicts the opposite depth di erence.
For the regular coexisting minima, gure 6 shows that m212 is already a good predictor
of the global minimum, but we can test the discriminants in (7.1). The result of this test
is shown in
gure 9 where the potential di erence at oneloop is plotted against the three
discriminants. We can see in the left and middle plots that D1 and D2 correctly predict
the global minimum of the oneloop e ective potential while the right plot shows that D3
fails for more than 10% of the points.
In constrast, the failure of all the discriminants (7.1) for the nonregular points (of
samples G and NR) can be seen in
gure 10 which shows the potential depth di erence
as a function of the discriminants, similarly to the regular points in
gure 9; note that
the horizontal scale is very di erent. The green points in the second and fourth quadrants
mark the cases where the discriminant D1 predicts the opposite behavior at oneloop. We
can clearly see that all discriminants fail for a signi cant portion of points, not necessarily
for the same ones. From the property of D1, we could have seen its wrong prediction in
gure 6 as well.
At last, to make sure that the points for which the discriminant test fails include
phenomenologically realistic points, we have checked the viability of the red/green points in
gure 6 by considering phenomenological constraints implemented in the 2HDMC code [68,
69]. We found that in the case of the type I model around half of the green points are
allowed by experimental constraints such as the S, T, U precision electroweak parameters
and data from colliders implemented in the HiggsBounds and HiggsSignals packages [70{
73], while in the type II model they are all excluded. For the type II model we have also
included constraints coming from Rb measurements [74] as well as B meson decays [75{78]
which prove to be very strong by setting mH+ > 480 GeV independently of the value of
tan . For type I, since the red/green points have tan
> 10, these constraints do not
impose any further restrictions [79, 80]. We also checked that all points still respect simple
bounded from below and perturbativity constraints [4].
8
Conclusions
We have studied the oneloop properties of the real TwoHiggsdoublet model with softly
broken Z2 with respect to the possibility of two coexisting normal vacua.
The softly
broken nature must remain at oneloop and the case of two coexisting normal minima can
be classi ed into two very distinct types depending on their nature in the vanishing m212
limit: regular minima that spontaneously break the symmetry and the nonregular minima
(or minimum) that preserve the symmetry, i.e., they are inertlike in the symmetric limit.
Since in the rst case the two minima are degenerate in the symmetry limit, even at
oneloop, they are connected by the Z2 symmetry and then they should di er only by the sign
of v2. After the inclusion of the m212 term, the sign of v2 continue to be opposite for our
vacuum and the nonstandard one so that the two regular minima are found in the rst
and fourth quadrants in the (v1; v2) plane. In contrast, the nonregular minima deviate
from the inertlike minima and both deviate to the rst quadrant when m212 is positive.
The vacua that spontaneously break Z2 in the m212 ! 0 limit behave rather regularly
and we can distinguish which coexisting minimum is the global one by just examining the
sign of m212: when it is negative our vacuum is only a metastable local one and the opposite
is true if it is positive. For this type of coexisting vacua, the discriminant at tree level (D1;2
in eq. (7.1)) is still a good predictor of the nature of the minimum it is calculated with.
For the nonregular coexisting vacua, m212 is positive in the convention that our vacuum
has both vevs positive and it cannot be used as an indicator. At tree level, the discriminant
of ref. [45, 46] is a very convenient way of testing if our vacuum is the global one because
only the location of our minimum is required. However, at oneloop, this discriminant is
not a precise indicator of which minima is the global one. We have found realistic cases
where our vacuum is not the global minimum at treelevel but it becomes the global one
after the addition of the oneloop corrections. Few cases for the opposite behavior were
also found. As the discriminant e ectively distinguishes the sign of the potential di erence
between the coexisting minima at treelevel, the latter itself is also a not good indicator for
the nonregular minima. This is reminiscent of the exact Z2 symmetric case (inert model)
investigated in ref. [48]. On the other hand, we were unable to
nd a discriminant that
works for both regular and nonregular minima at oneloop. Finding a simple and precise
criterion for global minimum does not seem to be an easy task as that was not achieved
even in the simpler exact Z2 limit. We also emphasize that for our parametrization which
enforces our vacuum to be a minimum from the start, and for the chosen generic ranges of
parameters (sample G), the occurrence of nonregular minima, as suggested by its name,
is much rarer: only 38% correspond to the nonregular cases and, among then, only 0:3%
are coexisting minima.
In summary, the softbreaking term m212 controls the lifting of the degeneracy of the
regular coexisting minima (moderate tan ) even at oneloop and can be used as the sole
indicator of which minimum is the global one. That is not true when two coexisting
nonregular vacua (large or small tan ) exist: the discriminant that is a precise indicator at
treelevel is not reliable at oneloop and explicit calculation of the potential depths must
be carried out.
Acknowledgments
HJEP1(207)6
A.L.C. acknowledges nancial support by Brazilian CAPES (Coordenac~ao de
Aperfeicoamento de Pessoal de N vel Superior) and C.C.N. acknowledges partial support by Brazilian
Fapesp grant 2014/191646 and 2013/220798, and CNPq 308578/20163. The authors
thank Pedro Ferreira for useful discussions while this work was developed.
A
Deviation of a potential minimum
Take a potential V0(') depending on real scalar elds 'i for which we know a minimum
(extremum) 'i = 'i satisfying
We want to quantify how the location of the minimum and the value of the potential deviate
when we add a small perturbation U on the potential as
Since the rst term vanishes due to (A.1), we get the deviation to rst order
where Ui
The deviation in the value of the minimum can be equally expanded around ' as
We assume V0 has no at direction around '.
We rst quantify the deviation of the location of the minimum as
The derivative of the perturbed potential gives
'j + O( 2) :
V = V0 + U :
'i = 'i + 'i :
'j =
Mji1Ui ;
V (' + ') = V0(') + U (')
1
2 UiMij1Uj + O( 3) ;
(A.1)
(A.2)
(A.3)
(A.4)
(A.5)
(A.6)
after using (A.5). Generically the third term contributes to deepen the potential. That is
the dominant contribution when the perturbation vanishes on the unperturbed minimum:
U (') = 0. The latter happens for the m212 term when perturbing the inertlike minima
while for the spontaneously broken Z2 minima the dominant term is the second, linear in
the perturbation.
(B.1)
(B.2)
(B.3)
(B.4)
(B.5)
(B.6)
Finding more than one normal vacua
Let us rewrite the minimization equations in eq. (2.5) as
where
A X = M ;
A =
345
1
345
2
!
;
v2!
2m212=v1v2 depends on the vevs.
end, we can formally write (B.1) as X = A 1M and equate v12v22 = 4m412= 2. We obtain
To nd solutions (v1; v2) of (B.1) for m212 6= 0, we rst need an equation for . For that
2
2 = [m121
2] ;
where 1 =
! 0 and m413245!m12v112v22=4. When m212 6= 0, we can nd the possible values for
1m222 and 2 = 345m222
2
2m121. The m212 ! 0 limit is taken as
(extrema)
from the quartic equation
2 = 2[m121
2] :
trema in this case [35, 36]. We also can see that the root 345 +p
We know there are at most four real solutions coinciding with the maximal number of
ex1 2 from the lefthandside
is always positive due to bounded below conditions. Possible solutions for (B.1) depends
on the sign of
and we allow both signs for v1v2. We can see that for the same , ipping
the sign of m212 is equivalent to ipping the sign of v1v2.
Once some solution
= 0 is found, we can nd the vevs from the relation X = A 1M
After ensuring these expression are positive, we can extract v1; v2 and tan
with the sign
convention where v1 > 0 and
v12 =
v22 =
2(m222
2(m211
( 345
( 345
)2
)2
2)
1)
1 2
1 2
> 0 ;
> 0 :
sign(v2) = sign( m122) :
The derivatives are with respect to the elds
around the values in eq. (2.2).
D
Cubic derivatives
The coe cients iS in (4.6) are de ned as
(quadratic) =
0
1
+
a
2
p
m211
m212
m212! :
m222
a = 1; 2
iS =
SS
where S = fG+; H+; G0; A; H; hg, sec = fchar, odd, eveng refers to the di erent scalar
sectors and Usec diagonalizes Msec. The last subindex SS refers to one of the diagonal entries
following the ordering in (4.13).
Explicit calculation leads to
Matrix of second derivatives
The mass matrices at treelevel for the charged, CPodd and CPeven scalar sectors prior
to imposing the minimization conditions are given respectively by
Mchar('i) = h
Modd('i) = h
Meven('i) = h
i =
i =
The term (quadratic) refers to the quadratic contribution given by
+ (quadratic) ;
(C.1)
+ (quadratic) ;
(C.2)
+ (quadratic) :
(C.3)
1A0 = 1s20 + 345c20
1H0 = 3 1c2 0 + 345s 0 s 0 + 2c 0
1h0 = 3 1s2 0 + 345c 0 c 0
2s 0
v2
v2
v2
v2
v2
v1
v2
v1
;
;
;
;
;
;
(C.4)
(C.5)
(D.1)
(D.2)
where the mixing angles +; 0; 0 were de ned in (4.14) with
tan 2 + =
tan 2 0 =
tan 2
=
m2H+
m2A0
The coe cients 2S can be obtained from 1S for each scalar S above by using the
replace
Cubic and quartic couplings
0;+ !
angles
Given that the cubic and quartic couplings do not depend on the quadratic parameters,
they correspond to the treelevel ones listed, e.g., in ref. [81] if we are in the limit 0 !
,
. If we want the couplings with these corrections, we can adapt the rotation
!
0 or
!
+ or
!
0 for the couplings that do not mix the charged sector
with the CPodd sector because in the latter there is an ambiguity in distinguishing +
from 0. Another ambiguity arises if the couplings are written in terms of (v; ) instead of
(v1; v2) because then we need to distinguish
coming from the vevs and the 0;+ coming
from the diagonalization of the shifted mass matrices.
We adopt the convention that
igABCD is the Feynman rule associated to the vertex
ABCD. This is opposite to e.g. ref. [81]. We also abbreviate e.g. gG0G0G0G0 as gG04. The
essential set of quartic couplings is
2c20 + 345c2 0 ;
2c2+ + 3c2 +)
1
1
1
gG04 = 3 1c40 + 2s40 + 2 345s20c20 ;
gh2G02 = 1s2 0c20 + 2c2 0s20 + 345c20 0
gHhG02 = 2 s2 0(
gG02A2 = 43 s22 0( 1 + 2) +
1
2 345c22 0 ;
= 1s2+c20 + 2s20c2+ + 3c2+ 0
gG02H+H
gG02G+H
= 2 s2 +(
gh2G0A = 2 s2 0(
gG0A3 =
gG0AH+H
=
3
1
1
2 v1( 1
i
2
gG02G+G
Coupling
=
=
More couplings can be obtained from simple replacements; see table 2.
The essential set of cubic couplings is
ghG02 =
= v2 2s2+ c 0 v1 1c2+ s 0 + 3(v2c2+ c 0 v1s2+ s 0 )+
1
2 45s2 + (v1c 0 v2s 0 ) ;
3)s2 + s 0 + 2 v2( 2
1
3)s2 + c 0 +
1
2 45c2 + (v1c 0 v2s 0 ) ;
( 4
5)(v1s 0 v2c 0 ) :
The remaining couplings can be obtained through reparametrization symmetries as shown
Replacement
Coupling Obtainable from
Finally we note that, although the reparametrization symmetry already allows a huge
simpli cation in the computation of the cubic and quartic couplings needed for our
calculation, their use can be errorprone. In our routines we have adopted a di erent
approach, namely we expanded the treelevel scalar potential in terms of physical
elds
S = fh; H; A; G0; G+; H+; G ; H g and performed derivatives to obtain the desired
couplings. For instance,
gh3G0 =
ghH2 =
Si=0
(E.3)
HJEP1(207)6
Some couplings are absent because CP is conserved and A; G0 are CP odd.
(F.1c)
F
Selfenergy for the scalars
We show here the di erent contributions to the selfenergy of scalars due to scalars (S),
fermions (F ) and gauge bosons (V ) in the loop.
We rst list the contributions from scalars in the loop for the di erent sectors. For the
CP odd sector we have
gG0SS0 B(MS; MS0 ; s) + 2jgG0G+H j2B(MG+ ; MH ; s) ;
2
(F.1a)
gA2SS0 B(MS; MS0 ; s) + 2jgAG+H j2B(MG+ ; MH ; s) ;
(F.1b)
16 2 SAA(s) =
gA2S2 A(MS) +
gA2SSA(MS)
16 2 SAG0 (s) = 16 2 S
G0A(s) =
gAG0S2 A(MS) +
gAG0SSA(MS)
1
2
X
S=G0;A;H;h
X
S=G+;H+
gASS0 gG0SS0 B(MS; MS0 ; s) + 2gAG+H gG0G+H B(MG+ ; MH ; s) :
1
2
+
1
2
+
+
X
X
S=G0;A;H;h
S = G0; A
S0=H;h
X
S=G0;A;H;h
X
S = G0; A
S0=H;h
X
S = G0; A
S0=H;h
X
S=G+;H+
X
S=G+;H+
X
S=G+;H+
X
The selfenergy for the charged sector is
G+G+ (s) =
gG+G S2 A(MS) +
gG+G SSA(MS)
H+H+ (s) =
gH+H S2 A(MS) +
gH+H SSA(MS)
In the CP even sector we have
16 2 SHH (s) =
G+H+ (s) = 16 2 S
H+G+ (s) =
gG+H S2 A(MS) +
gG+H SSA(MS)
X
S=G+;H+
gG+SS0 gH+SS0 B(MS; MS0 ; s)
jgG+SS0 j2B(MS; MS0 ; s)
X
S=G+;H+
X
S=G+;H+
jgH+SS0 j2B(MS; MS0 ; s)
1
2
X
S=G0;A;H;h
1
2
+
1
2
+
+
X
X
X
X
S=G0;A;H;h
S = G+; H+
S0=G0;A;H;h
S=G0;A;H;h
S = G+; H+
S0=G0;A;H;h
X
S = G+; H+
S0=H;h
(F.2a)
(F.2b)
(F.2c)
(F.2d)
(F.2e)
(F.2f)
(F.3a)
(F.3b)
(F.3c)
(F.3d)
(F.3e)
(F.3f)
)
(F.4)
(F.5)
the general formula [49, 50]:
SFS0 (s) =
Nc
16 2
( tr YfSf0 (YfS0f0 )y
2
where Nc is the number of colors of the fermion in the loop, the B function was given
in (5.4) while BF F (mf ; mf0 ) is de ned as
BF F (mf ; mf0 ; s)
mf mf0 tr[YfSf0 YfS0f0 ]B(mf ; mf0 ; s) ;
Note that couplings such as gG+G A0 = 0 due to CP conservation.
The fermionic corrections to the propagator of a scalar S to a scalar S0 are given by
BF F (mf ; mf0 ; s)
[(s
mf
mf0 )B(mf ; mf0 ; s)
A(mf )
A(mf0 )] :
VS;So0dd(s) =
SS0
V; char(s) =
g
2
g
2
g
2
H
A
G0
H
G
1
1
pyt2 cs 0 1
pyt2 ss 0 1
i pyt2 cs 0 5
bb
pyb2 Cbh1
pyb CH 1
2 b
i pyb2 CbA 5
i pyt2 ss 0 5 i pyb2 CbG 5
ybCbH PR
ybCbG PR
yt cs + PL
yt ss + PL
We denote the vertex of the scalar Sk to the fermions f and f 0 by
contain the 5 matrix while Y refers to the transformation
These Yukawa couplings are listed in table 4 where the coe cients CfS depend on the model
used as in table 5.
Finally, the contributions coming from the gauge bosons in the loop are given by
iYfkf0 and they may
SS0
16 2 4c2w CSeoS00 CSeo0S00 BSV (MS00 ; mZ ; s) +
e+
3
3
e+
+ SS0 4c2w A(mZ ) +
A(mW )
16 2 4c2w CSeoS00 CSeo0S00 BSV (MS00 ; mZ ; s) +
+ SS0 4c2w A(mZ ) +
A(mW )
1 Ce+
;
;
1 Co+
o+
3
2
3
2
16 2 4 SS00 CS0S00 BSV (MS00 ; mW ; s) +
4 SS00 CS0S00 BSV (MS00 ; mW ; s)
+ SS0 s2w BSV (MS; 0; s) + cot22w BSV (MS; mZ ; s)
+ s2wCS+CS+0 s2wm2Z BV V (mZ ; mW ; s) + m2W BV V (0; mW ; s)
o+
1 Co+
2 SS00 CS0S00 BSV (MS00 ; mW ; s)
+ SS0 3s2w cot22w A(mZ ) +
A(mW )
;
3
2
(F.6)
(F.7)
(F.8)
A
G
c 0 0
H
G
c + 0
s + 0
c + 0
H
G
s + 0
c + 0
H
s
0 c
0
C+
S
s
H
c
G
+
+
where there is an implicit summation on S00. The loop functions BSV (mV ; mS; s) and
BV V (mV ; mV 0 ; s) are de ned as
BSV (mS; mV ; s) =
4m2V m2V 0
p
4
4m2V m2V 0 B(0; 0; s) +
B(mS; 0; s) +
B(mV ; 0; s)
(m2V + m2V 0
4m2V m2V 0
m2S
m2V
m2V
A(mV )
4m2V
A(mV 0 ) :
4 m2V 0
B(mV ; mV 0 ; s)
B(0; mV 0 ; s)
s
A(mV ) ;
(F.9)
BV V (mV ; mV 0 ; s) = 2B(mV ; mV 0 ; s) +
The couplings CSxS0 and CSx can be read from the following tables.
G
Reparametrization symmetry
Here we list some useful reparametrization symmetries that allow us to relate di erent
quartic and cubic scalar couplings that are numerous. These relations help us to check
di erent couplings or deduce new ones from a smaller set. Moreover, given simple relations,
they minimize errors and speed up numerical implementation.
The simplest reparametrization is to exchange elds of the same type, one in 1 and
the other in 2, together with a 90 shift in the respective mixing angle:
8H ! h
h !
>: 0 !
H
0 + =2
8G0
>
< 0
A
>
:
! A
!
0
G
0
0 !
0 + =2
Qeven :
Qodd :
Qchar :
ghG02 hG02
! ghG02 j 0! 0 =2
This is what allowed us to relate gHG02 to ghG02 in (E.2a) after 0 !
0
HG02 by the inverse transformation of Qeven. Another
example is gHhAG0 in (E.1k): it is odd by the replacement 0 !
0 + =2 or 0 !
0 + =2.
The reparametrization symmetry above arises by noting that the original elds in
1;2
are left invariant by the transformations and consequently the potential is also invariant.
(F.10)
8 G+
H+
>
<
>
:
!
+ !
! H+
G+
+ + =2
:
(G.1)
=2 since
Q12 :
Qe!o :
1 $
2 ;
m121 $ m222 ;
1 $
8m121 $ m222 ;
1 $
: v1 $ v2
2 ;
>
<
8 H $ h ;
G
0
$ A ;
<
8 0 !
0 !
>
: + !
=2
=2
=2
H!
G0!
A0 ;
G0!
A0
!
H!
8 0 $
v1 !
>
:v2 !
0 ;
0 ;
+ :
0
iv1
iv2
The last type of symmetry we can explore for reparametrization is the original gauge
invariance. Discrete subgroups are the most useful. For example, the reparametrization
arises because of invariance by the gauge transformation
a !
1
i
!
a ;
a = 1; 2 :
For special eld con gurations, we can also de ne an exchange reparametrization
between charged elds and CP even neutral elds as below
If we take the case of CP even elds,
1
2
!
c 0 H
s 0 h
s 0 H + c 0 h
!
;
we see 1;2 are the same after the transformation Qeven. The same is true for the other
pair of scalars and diagonalization angles.
The other reparametrization symmetry we can use is the exchange symmetry of the
original potential (2.1), restricted10 to the case of the CP conserving softly broken Z2,
For the elds with de nite masses at tree level this transformation reads
(G.2)
(G.3)
(G.4)
;
(G.5)
(G.6)
(G.7)
(G.9)
and G+ = G1=p2; H+ = H1=p2 can be chosen by electromagnetic gauge invariance.
Similarly,
Qc$e :
Qc$o :
H!
G1!
H1
G0!
A0
G2!
H2
0 $
+ ;
for G0 = A0 = 0 ;
0 $
+ ;
for H = h = 0 ;
(G.8)
and G+ = iG2=p2; H+ = iH2=p2 can be chosen. These reparametrization symmetries
arise from the gauge symmetry
a !
0 1
1 0
!
a ;
a = 1; 2 :
Note that the vevs should also transform nontrivially and this reparametrization only works
for quartic couplings.
10Some adjustments are needed for the most general potential in (2.1).
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