Table Of ContentPhysical parameters and basis transformations in the Two-Higgs-Doublet model
C. C. Nishi∗
Instituto de F´ısica “Gleb Wataghin”
Universidade Estadual de Campinas, Unicamp
13083-970, Campinas, SP, Brasil and
Instituto de F´ısica Teo´rica, UNESP – Sa˜o Paulo State University
Rua Pamplona, 145, 01405-900 – Sa˜o Paulo, Brasil
A direct connection between physical parameters of general Two-Higgs-Doublet Model (2HDM)
potentialsafterelectroweaksymmetrybreaking(EWSB)andtheparametersthatdefinethepoten-
tialsbeforeEWSBisestablished. Thesephysicalparameters,suchasthemassmatrixoftheneutral
Higgsbosons,havewelldefinedtransformationpropertiesunderbasistransformationstransposedto
thefieldsafterEWSB.Therelationsarealsoexplicitlywritteninabasiscovariantform. Violation
of these relations may indicate models beyond 2HDMs. In certain cases the whole potential can
be defined in terms of the physical parameters. The distinction between basis transformations and
reparametrizations is pointed out. Some physical implications are discussed.
I. INTRODUCTION this case, SU(2) doublets with quantum numbers iden-
L
tical to the SM Higgs doublet. The simplest Two-Higgs-
8
DoubletModel(2HDM)hasbeenextensivelystudiedre-
0 The Standard Model (SM) relies on the Higgs mecha-
cently[2,3,4,5,6,7,8,10,11]astheeffectivescalarsec-
0 nism to give masses to all massive gauge bosons and ele-
2 mentary fermions. Such mechanism involves the spon- toroftheMSSMrequirestwoHiggsdoubletsforanomaly
n taneous symmetry breaking (SSB) of the electroweak cancelation[1, 12]. Historically, the addition of one or
more Higgs doublets were considered to implement the
a SU(2)L U(1)Y gauge group to the electromagnetic
J U(1) a⊗sascalarHiggsdoubletacquiresanonzerovac- spontaneous CP violation mechanism (SCPV)[13, 14] as
EM
an alternative source of CP violation.
3 uumexpectationvalue(VEV).Suchschemeimposesuni-
versality constraints on the couplings between fermions Technical difficulties that arise when considering
] and gauge bosons, establishes relations between the NHDMs are twofold: (i) more than one local minimum
h
p masses of the gauge bosons and fixes the couplings be- (orbit), not necessary with the same symmetry breaking
- tween the physical Higgs and the fermions to be propor- pattern, might be present, even at tree level, and (ii) the
p tional to the masses of the latter[1]. reparametrization freedom[11] allowed by the presence
e
The scalar potential, constituted by only one Higgs of the horizontal space formed by the N Higgs-doublets
h
[ doublet,isalsoveryrestrictivesince,fromtheknowledge maymasqueradethenumberofrelevantindependentpa-
of the electroweak (EW) VEV and the physical Higgs rametersandsymmetrypropertiessuchasCPinvariance.
2 mass, the Higgs trilinear and quartic coupling constants Difficulty(i)includesthepossibilityofpotentialswithno
v
are fixed at tree level. However, the physical Higgs bo- remainingU(1)EM symmetryafterEWSB(chargebreak-
0
6 son, which is the only scalar remnant of the EW Higgs ing vacuum)[4, 15, 16] and it forces the stability of the
2 mechanism, was not discovered yet. One then resorts to vacuum to be a relevant issue at tree level[15].
4 indirectmeanstoboundthephysicalHiggsmass,mostof Item (ii) concerns the reparametrization transforma-
. them relying on the higher order perturbative behavior tions induced by basis transformations (or horizontal
2
1 of the SM. Such bounds comes from, e.g., the unitarity transformations[17]) acting on the identical N Higgs-
7 constraintsofthescatteringofgaugebosons,thevalidity doublets. Since all the doublets have the same gauge
0 of the SM up to the Planck scale and the EW vacuum quantum numbers with respect to the SM gauge group,
: stability (see Ref.1 and references therein). Studies con- there is no change in the physical content of the theory
v
i strainingtheHigssmassareveryimportanttoitssearch, if one rotates the fields in such space[18, 19, 21]. Such
X
in particular, in view of the upcoming LHC experiment. possibilitymaymasqueradethenumberofrelevantinde-
r Formodelsextendingtheelectroweaksymmetrybreak- pendent parameters in the theory. More crucially, trans-
a
ing (EWSB) sector of the SM, the relation of the model forming real parameters into complex parameters (for
parameters before EWSB and the physical parameters complex multiplets), CP invariant theories can be dis-
identified after EWSB may not be as minimal as in the guised as CP violating theories. This issue can be solved
SM. Specially for N-Higgs-doublet extensions of the SM in an objective way by noting that CP invariant theories
(NHDMs), the multiplicity of independent parameters remain CP invariant through basis transformations but
may be quite large due to the presence of a horizon- thecorrespondingCPsymmetrytransformationalsoacts
tal space, i.e., the space of identical gauge multiplets, in differentlyindifferentbases. Asapracticalwayofdistin-
guishingtheCPpropertyofatheoryonecanresorttothe
use of CP-odd basis invariants analogous to the Jarlskog
invariant[22] in the quark sector of the SM. In the con-
∗Electronicaddress: ccnishi@ifi.unicamp.br textofNHDMs,numerousofsuchinvariantscanbecon-
2
structed[19] but sufficient conditions for CP invariance lationsbetweenpotentialparametersandthemassmatri-
using a minimum number of invariants could be formu- cesarefound. InSec.III,covariantrelationsbetweenthe
lated only for the potentials of 2HDMs[7, 19, 20, 21, 23] mass matrices and the potential parameters are shown.
and 3HDMs[21]. InSec.IVweshowhowtoachievethetrulyphysicalbasis
Recent advances in the study of 2HDMs include the whereallthemassmatricesarediagonal,pointingoutthe
result that at most two local minima can be present distinction between basis transformations before and af-
whenever there is a discrete set of minima in the or- ter EWSB. Finally, the results and physical implications
bit space[8, 9]. Such result was obtained by using a arediscussedinSec.V.Somepossiblyusefulmaterial,in-
Minkowski structure that emerges naturally in the space cludinganalternativemethodtoensureaboundedbelow
of the fields through suitable change of variables[10, 21]. potential, is presented in the appendices.
Althoughsomecontroversyremainfromnumericalexam-
ples presenting more than two local minima[24]. Since
the counting of the number of local minima can be a II. PHYSICAL PARAMETERS IN THE PCH
BASIS
very difficult task, the upper bound of the number of
minima is an important result. It was also proved for
2HDMs that (a) charge breaking vacua can not coex- A general 2HDM potential can be divided into its
ist with a neutral vacuum[4, 10] and (b) spontaneously quadratic and quartic parts as
CP violating vacua can not coexist with CP invariant
V =V +V . (1)
vacuum[4, 8, 10]. The result (b) can be extended in a 2 4
weakerversiontoNHDMs: aspontaneouslyCPviolating
The quadratic part is usually written as
extremum always lies above a CP invariant extremum if
the latter exists[16]. The Minkowski structure can be V =Y Φ†Φ , a,b=1,2, (2)
also partially extended to general NHDM potentials[25]. 2 ab a b
Bearing these results in mind, the present article aims where Y is a hermitian matrix and Φ = (φ ,φ ) are
a a1 a2
two goals concerning the 2HDM potential: (i) to extract theHiggsdoubletsforwhichthenotationφ =φ(+) and
a1 a
all the physical parameters identifiable only after EWSB φ =φ(0) isusuallyadoptedwhenthevacuumpreserves
a2 a
and (ii) to study their properties under basis transfor- the electromagnetic symmetry. The quartic part can be
mations. The first goal involves having a more direct conveniently written[21] as
connection between the parameters of the potential and
the physical parameters after EWSB. Since basis trans- V4 = 21Λµνrµrν, µ,ν =0,1,2,3, (3)
formationsareallowedbeforeEWSBandtheyareusually
where Λ is a 4 4 real symmetric matrix while
involvedtoreachthephysicalbasis,itwouldbedesirable
×
tohaveabasiscovariantrelationforthephysicalparam-
r = 1(σ ) Φ†Φ , (4)
eters, which leads to goal (ii). A systematic study of the µ 2 µ ab a b
physical parameters of 2HDMs, including the scalar self-
for µ = 0,1,2,3, are real quadratic combinations of the
interactionsandtheinteractionsofscalarswithfermions doublets. Thematrixσ 1 andσ arethePaulimatri-
0 2 i
and gauge bosons, was carried out in Ref.6. The basis ≡
ces. The quadratic variables r =(r ,r) arefunctionally
µ 0
covariance, however, was not extended to the fields after
free except for the future lightcone constraint[10, 25]
EWSB.
Another aspect of item (i) regards seeking a physical r2 r2 0, r 0. (5)
0− ≥ 0 ≥
parametrization of the 2HDM potential by rewriting the
parametersbeforeEWSBintermsofthephysicalparam- The convention of summation over repeated indices is
eters. As explained in the end of Ref.25, what prevents adopted with Euclidean metric. For example, rµrµ =
the utility of a parametrization depending on physical r02+r2. The Minkowski metric will not be used to avoid
parameters, such as the masses of the physical charged confusion and all indices will be written as lower indices,
and neutral Higgs bosons, is the possibility of the po- differently of Refs.[25] and [10].
tential so defined possess another deeper minimum. The Using the variables rµ, the quadratic part of the po-
existenceofatmosttwolocalminimaalreadyameliorate tential in Eq.(2) can be cast into the form
thesituation. Suchparametrizationalsoexcludesbycon-
V =M r , (6)
struction the potentials without nontrivial minima. For 2 µ µ
the cases we know there is only one local minimum and
where M has four independent components. The num-
henceitisalsotheglobalone,suchparametrizationisun- µ
ber of free parameters contained in Y and M are the
ambiguous. Thequestionthatremainsistoknowifsuch
same and they are indeed related by
parametrization can cover all possible 2HDM potential
containing only one global minimum. Y =M 1σ M =Tr[σ Y]. (7)
The ouline is as follows: in Sec.II we find the mass µ2 µ ↔ µ µ
matrix for neutral scalars in the basis where the mass Wewanttoparametrizethepotentialintermsofphys-
matrixforchargedHiggsbosonsisalreadydiagonal. Re- ical quantities that are defined after EWSB. Confining
3
ourselves to neutral vacua, the first choice of physical where the null eigenvalue corresponds to the charged
parameters will be the masses of physical particles, i.e., Goldstone. We can divide the quadratic part of the po-
onechargedscalarandthreeneutralscalars. Inaddition, tential of Eq.(9) into
more parameters such as the EW vacuum expectation
(cid:12) (cid:12) (cid:12)
value (VEV), the mixing among the neutral scalars and V2(cid:12)SSB =V2(cid:12)charged+V2(cid:12)neutral. (17)
certaincouplingconstantswillbenecessarytocompletely
For the charged fields we have
parametrize the potential that requires 11 essential pa-
ramToeteexrst.ract the physical masses, we need the quadratic V2(cid:12)(cid:12)charged =m2h+h−, (18)
part of the potential after EWSB that is induced by while
Φa →(cid:104)Φa(cid:105)+Φa, (8) V2(cid:12)(cid:12)neutral = 12m2(t21+t22)+ 12Λµνsµsν, (19)
where(cid:104)Φa(cid:105)isthevacuumexpectationvalue(VEV)ofΦa, = 12ti(MN)ijtj, (20)
usuallyac-numberminimumofthepotentialin(1). The
where, using the parametrization of Eq.(14),
extremum equations are shown in appendix A. With the
shift of Eq.(8), the quadratic part of the potential can v v
s = t , s = t . (21)
be written as[25] 0 −2 3 i 2 i
V2(cid:12)(cid:12)SSB =Φ†a(cid:104)M(cid:105)abΦb+ 12Λµνsµsν, (9) Tnehuetr3a×l s3camlaartsrixgivMenNbiysthemassmatrixforthephysical
where
v2
= m2diag(1,1,0)+ Λ˜
sµ = 12(cid:104)Φa(cid:105)†(σµ)abΦb+h.c., (10) MN 4
0 0 Λ
abnodso(cid:104)nMs,(cid:105)iinsctlhuedminagssthsqeuachreadrgmedatrGixolfdosrttohnee.cha(rTgehdeHmigagsss + v42 Λ0 Λ0 Λ −−Λ00212Λ , (22)
01 02 00 03
squared matrix will be denoted simply as “mass matrix” − − −
fromthispointon.) Suchmatrixcanbecalculatedas[25] where Λ˜ = Λ , i,j = 1,2,3. The physical neutral
ij
{ }
scalars will be orthogonal combinations of t , defined by
M =Y + 1σ Λ r , (11) i
(cid:104) (cid:105) 2 µ µν(cid:104) ν(cid:105) thediagonalizationof N inEq.(22). Themassmatrix
M
inthePCHbasiscanbealsofoundinEqs.(24)and
where N
M
(41) of Ref.6 in a different notation.
r = 1(σ ) Φ † Φ . (12) FromEqs.(11),(15)and(22)wecanfindthefollowing
(cid:104) µ(cid:105) 2 µ ab(cid:104) a(cid:105) (cid:104) b(cid:105)
relation between Y and ,
N
In the Physical Charged Higgs (PCH) basis[25], for a M
(cid:18) (cid:19)
neutral vacuum, the VEVs are simply Y = 1 2Y11 (MN)13−i(MN)23 .
2 ( ) +i( ) ( )
(cid:18)0(cid:19) (cid:18) 0 (cid:19) MN 13 MN 23 − MN 33 (23)
(cid:104)Φ1(cid:105)= 0 , (cid:104)Φ2(cid:105)= √v2 , (13) Except for Y11, all elements of Y are directly related to
.
N
whilethedoubletsaftertheshift(8)canbeparametrized M
Hence, wecancompletelyparametrizethepotentialin
as
the PCH basis in terms of the set of 12 parameters
(cid:18) h+ (cid:19) (cid:18) G+ (cid:19)
Φ1 = √1 (t1 it2) , Φ2 = √1 ( t3+iG0) , (14) {v,m,Λ00,MN,Λ0}, (24)
2 − 2 −
where Λ = Λ , i = 1,2,3. For fixed values for the
0 0i
where v = 246GeV, is the electroweak VEV, t , i = { }
i set in Eq.(24), we obtain from Eq.(22) the rest of Λ
1,2,3, are normalized neutral scalar fields, h+ is the µν
by
physical charged Higgs and G+ and G0 are the charged
andneutralGoldstonefieldsrespectively. TheGoldstone Λ = 4 [( ) m2δ ], i,j =1,2, (25)
fields G+ and G0 are absorbed by the longitudinal W+ ij v2 MN ij − ij
and Z0 gauge bosons by the Higgs mechanism. For the 4
Λ = Λ + ( ) , i=1,2, (26)
VEV of Eq.(13), we have i3 0i v2 MN i3
4
r = v2(1,0,0, 1). (15) Λ33 = −Λ00+2Λ03+ v2(MN)33. (27)
µ
(cid:104) (cid:105) 4 −
The quadratic parameter Y depends on more parame-
11
In the PCH basis, the mass matrix for the charged
ters other than as
N
scalars can be written M
v2
(cid:104)M(cid:105)=diag(m2,0) , (16) Y11 =m2− 4 (Λ00−Λ03)+ 12(MN)33. (28)
4
There are 12 free parameters. Among these, 11 are III. PHYSICAL PARAMETERS IN AN
essential and can not be eliminated by reparametriza- ARBITRARY BASIS
tion[21]. Nevertheless, one parameter can be removed
by a remaining U(1) reparametrization freedom due to For a general potential (1), the vacuum expectation
value(VEV)willnotbeintheformofEq.(13). Nonethe-
Φ eiθΦ , Φ Φ . (29)
1 1 2 2 less, we can always parametrize a neutral vacuum as
→ →
Since (cid:18) (cid:19) (cid:18) (cid:19)
v 0 v 0
h+ eiθh+ (30) (cid:104)Φ1(cid:105)= √2 cosθ2v , (cid:104)Φ2(cid:105)= √2 eiξsinθ2v . (32)
→
The VEV in Eq.(15) corresponds to θ = π and ξ = π.
represents the electromagnetic U(1) invariance of the v
EM
In the MSSM, the angle θ corresponds to 2β.
potential, the transformation of Eq.(29) amounts ef- v
fectively to a SO(2) rotation in the t ,t fields. No- To explicit the structure of the horizontal space where
1 2
tice that since Φ = 0, the transformation of Eq.(29) basis transformations act, it is more convenient to de-
1
does not affect(cid:104)the(cid:105)VEVs. Choosing appropriately θ in fine[25]
Eq.(29), one can set one of the following parameters to
u (φ ,φ )T , w (φ ,φ )T . (33)
zero: ( N)12,( N)13,( N)23. In particular, choos- ≡ 11 21 ≡ 12 22
M M M
ing ( ) = 0, we obtain a real symmetric Y and we
MN 23 We can rewrite Eq.(32) as
constrain Λ = Λ . For any choice the overall number
23 03
of free independent parameters should be 11. Of course, u =(0,0)T, w = v (cosθv,sinθv eiξ)T. (34)
different choices, such as Λ01 = 0 or Λ02 = 0, could be (cid:104) (cid:105) (cid:104) (cid:105) √2 2 2
alternatively chosen. Obviously, once a choice is made,
one can not set more than one parameter to zero. More generally, we can rewrite
To ensure the vacuum in Eq.(14) to be a local mini-
mum, it is sufficient to pick positive values for m2 (the w = v U e , (35)
v 2
masssquaredofh+)andforthethreeeigenvaluesof (cid:104) (cid:105) √2
N
M
(the masses squared of t ). Such requirements guarantee
i
whereU isaunitarymatrixinSU(2) ande ,a=1,2,
that the second derivative of the potential around the v H a
arethecanonicalvectorsdefinedby(e ) =δ . Interms
extremum is positive semidefinite. a b ab
of r we get
Thereremainsthequestionofboundednessforthepo- µ
tentialdefinedwithphysicalparameters(24). Firstly,we
v2 v2
fihnaviteetioncEhqo.o(s3e)Λw0o0u≥ld0mbaekceauVseatcaqkuinirgern0eg→at∞ivebvuatlu|ers| (cid:104)rµ(cid:105)= 4 (1,cosξsinθv,sinξsinθv,cosθv)= 4 Rµνnν,
4 (36)
if Λ < 0. Moreover, the following statement can be
00 wheren =(1,0,0, 1)andR canberelatedtoU by
proved: µ − µν v
(cid:18) (cid:19)
For a potential V(r) defined as Eq.(1), sat- R (U ) 1Tr[U†σ U σ ]= 1 0 , (37)
isfying Λ + λ > 0 for all λ , i = 1,2,3, µν v ≡ 2 v µ v ν 0 R˜
00 i i
eigenvalues of Λ˜, it is always possible to ob-
tainΛµνrµrν >0,forallrµ satisfyingEq.(5), with R˜ ={Rij}, i,j =1,2,3, being a rotation matrix in
by making the substitution SO(3)H.
We can rewrite the quadratic part of the potential in
Λ0 cΛ0, c>0, (31) Eq.(9) using u and w of Eq.(33), and their respective
→
VEVs in Eq.(34),
with appropriately small c.
The proof is shown in appendix B V2(cid:12)(cid:12)SSB =u†(cid:104)M(cid:105)u+w†(cid:104)M(cid:105)w+ 21Λµνsµsν, (38)
The only problem that could make such physical
where
parametrization not viable is the possibility that the po-
tential defined for a given set (24) possess another mini- s = 1 w †σ w+h.c. (39)
mum that lies deeper than the one defined in (13). This µ 2(cid:104) (cid:105) µ
possibilityisrealandnumericalexamplescanbequickly The first term of Eq.(38) corresponds to the mass term
deviced. The problem is not so severe because at most of the charged scalars, one physical and one Goldstone,
twodistinctlocalminimaarepossibleforboundedbelow inanarbitrarybasis. Therespectivemassmatrixis M ,
(cid:104) (cid:105)
potentials containing two Higgs doublets[8]. Although, which is defined by Eq.(11).
in Ref.24, some numerical examples of 2HDM potentials Therelationbetween M inanarbitrarybasisandits
(cid:104) (cid:105)
withmorethantwominimawereapparentlydevised[24]. diagonal form(16) in the PCH basis is given by
On the other hand, this parametrization excludes pon-
tentials without nontrivial minima by construction. U† M U =diag(m2,0). (40)
v(cid:104) (cid:105) v
5
We can then reach the PCH basis by the substitutions where ˆr is the unit vector in the direction of r . The
(cid:104) (cid:105) (cid:104) (cid:105)
relation (48) is proved in appendix C.
Φa =(Uv)abΦ(cid:48)b, (cid:104)Φa(cid:105)=(Uv)ab(cid:104)Φ(cid:48)b(cid:105), (41) To write the second and third terms of Eq.(38) in
terms of t it is necessary to find the parametrization of
i
or equivalently w in terms of t and G0. The desired covariant relation
i
is
w =U w(cid:48), u=U u(cid:48), (42)
v v
1
w = (iG01 +t σ ) wˆ , (49)
with the same substitutions valid for their respective √2 2 i i (cid:104) (cid:105)
VEVs.
Since the basis for which Y,Λ, r is defined is com- where wˆ = w / w . One can confirm that Eq.(49)
pletely arbitrary, the covaria{nce of(cid:104) (cid:105)M} is valid between satisfies(cid:104) E(cid:105)q.(4(cid:104)7)(cid:105)u|s(cid:104)ing(cid:105)|Eq.(10). The covariance can be
(cid:104) (cid:105)
anybasisandnotonlywithrespecttothePCHbasis. (A also checked,
detailed account of the basis covariance of M is given
(cid:104) (cid:105) 1
in appendix D.) Indeed, we can write[25] w(cid:48) =Uw = (iG01 +t(cid:48)σ ) wˆ(cid:48) , (50)
√2 2 i i (cid:104) (cid:105)
M =m2(cid:2)1 wˆ wˆ †(cid:3), (43)
2
(cid:104) (cid:105) −(cid:104) (cid:105)(cid:104) (cid:105) where
where wˆ = w / w .
(cid:104) (cid:105) (cid:104) (cid:105) |(cid:104) (cid:105)| t(cid:48) =R t , (51)
Wecantrytoextendthebasiscovarianceforthemass i ij j
matrix for the physical neutral scalars. We keep the no-
and R is related to U by a relation similar to Eq.(37).
tation to denote such mass matrix. Obviously, the ij
MN ThedependenceonG0 isfixedbyimposingthatEq.(49)
second and third term of Eq.(38) is covariant by basis
reduces to Eq.(14) in the PCH basis. Notice Eq.(49)
transformations for w, such as the transformation (42)
differsfromEq.(46)ofRef.6asthefieldst transformas
with arbitrary U . The question, however, is if we can i
v vectors under SO(3) .
find appropriate fields t , with suitable transformation H
i We can thus rewrite the second term of Eq.(38) in
properties, that renders covariant by some basis
transformation, keeping EMq.(N20) form invariant. We ob- terms of ti using Eqs.(43) and (49) as
vtioinustlyhewPanCtHtobraesciso.verEq.(22)forMN andEq.(21)for w† M w = m2[t2 ( ˆr t)2]. (52)
i
(cid:104) (cid:105) 2 − (cid:104) (cid:105)·
TheimmediateextensionofEq.(14)todefinet inany
i
basis is flawed because a basis transformation over w in The third term of Eq.(38) can be also easily rewritten
Eq.(14) would mix ti with the neutral Goldstone G0. In in terms of ti by using Eq.(47). The sum of the second
other words, with w(cid:48) in the PCH basis given by and the third terms of Eq.(38) defines the mass matrix
for the physical neutral scalars by
1 (cid:18) t(cid:48) it(cid:48) (cid:19)
w(cid:48) = √2 −t1(cid:48)3−+iG20 , (44) w†(cid:104)M(cid:105)w+ 21Λµνsµsν = 12(MN)ijtitj, (53)
we can not define giving the basis covariant relation
(cid:18) (cid:19)
w = √12 −tt13−+iitG20 (45) MN = m2(cid:104)13− (cid:104)r(cid:105)r(cid:104)r(cid:105)2T(cid:105)+|(cid:104)r(cid:105)|(cid:104)Λ˜ +Λ00(cid:104)r(cid:105)r(cid:104)r(cid:105)2T(cid:105)
|(cid:104) (cid:105)| |(cid:104) (cid:105)|
+ r Λ T+Λ r T, (54)
because, in general, (cid:104) (cid:105) 0 0(cid:104) (cid:105)
w =Uvw(cid:48), (46) where |(cid:104)r(cid:105)| = v2/4. Equation (54) is the generalization
(cid:54) of Eq.(22) for an arbitrary basis.
such as for Uv =eiθσ1/2. Equation (54) illustrates two points: (i) the whole po-
The solution is to promote Eq.(21) to define the real tential after EWSB can be completely defined in terms
fields t as of the set
i
2 m2,Λ00,Λ0, r , N , (55)
t s , i=1,2,3. (47) { (cid:104) (cid:105) M }
i i
≡ v
in an arbitrary basis, since Λ˜ can be written in terms
ThedefinitionofEq.(47)ensuresthattiwouldtransform of the set. Moreover, the objects in the set have the
as vectors under SO(3)H when SU(2)H transformations sametransformationpropertiesunderthereparametriza-
are applied to w and (cid:104)w(cid:105) in the definition (39) of si. tion group SU(2)H as the set {M0,Λ00,Λ0,M,Λ˜} that
The s0 component depends on ti by the basis invariant defines the potential before EWSB[21]: two scalars[26],
relation two vectors and one rank-2 tensor. (ii) We can define a
v Physical Neutral Higgs (PNH) basis, in contrast to the
s0 = 2(cid:104)ˆr(cid:105)·t, (48) PCH basis, being the basis where MN is diagonal. In
6
general, this basis will coincide neither with the PCH IV. PHYSICAL BASIS (P-BASIS)
basis nor with the basis with diagonal Λ˜ (the canonical
CP basis in Ref.[21]). The Physical basis (P-basis) should be defined as the
Therelations(23)and(28)canbewritteninthebasis basis where all the fields possess definite masses. From
covariant form Sec.II, we conclude that the mass matrix for physical
neutral scalars ( ) in the PCH basis will be not di-
Y = 21−[m21σ2i−(M(cid:104)rN0(cid:105))Λij0(cid:104)0rˆ−j(cid:105)Λ. 0·(cid:104)r(cid:105)](cid:0)12−σi(cid:104)rˆi(cid:105)(cid:1) (56) aisgodniaagloinnalg(ePneNrHaMl.bNaFsriso)mwoSuecld.ImII,ixththeebpahsyissiwcahlecrheaMrgeNd
Higgs h+ with the charged Goldstone G+. Thus, nei-
We made use of the relation ther the PCH basis nor the PNH basis coincide with the
Physical basis.
12−(cid:104)wˆ(cid:105)(cid:104)wˆ(cid:105)† = 21(cid:2)12−σi(cid:104)rˆi(cid:105)(cid:3). (57) To achieve the P-basis, we need independent basis
transformations on the upper (u) and lower components
One can check Eq.(56) reduces to Eqs.(23) and (28) in (w) of the doublets, i.e.,
the PCH basis. It is also important to remark that Y in
Eq.(56)isindependentontheparticularVEV.Foradif- u Uuu, w Uww, (66)
→ →
ferentminimumofthesamepotential(orextremumifwe
do not require positive definite and m2), ,m2 where Uu,Uw are different transformations in SU(2).
N N
M M The transformations in Eq.(66) are legitimate basis
and r differ in such a way that Y is the same. In addi-
(cid:104) (cid:105) transformations only after EWSB since they still pre-
tion, we can write the (ij)=(33) component of Eq.(23)
serve the EM symmetry, for a neutral vacuum, but do
in the following basis covariant form
not preserve the SU(2) U(1) gauge structure of the
L Y
⊗
wˆ †Y wˆ = 1 rˆ ( ) rˆ . (58) doublets,exceptforUu =Uw. Thegeneralgroupofbasis
(cid:104) (cid:105) (cid:104) (cid:105) −2(cid:104) i(cid:105) MN ij(cid:104) j(cid:105) transformations generated by Eq.(66) is SU(2) SU(2)
⊗
instead of SU(2) valid before EWSB.
To obtain the interaction terms[25] H
The P-basis can be achieved either from the PCH ba-
(cid:12) sis or from the PNH basis. The latter choice is more
V3(cid:12) = Λµνsµrν, (59)
SSB convenient. Let us choose the PNH basis for which
V4(cid:12)(cid:12)SSB = 21Λµνrµrν, (60) =diag(m2,m2,m2). (67)
MN 1 2 3
in terms of the real fields t , we should calculate r in
i µ
Eq.(4) using Eq.(47). Splitting The VEV w will be in the general form of Eq.(35),
(cid:104) (cid:105)
different from the PCH basis. The respective r would
µ
(cid:104) (cid:105)
r =x +y , (61) be parametrized in the form of Eq.(36).
µ µ µ
In the PNH basis, the fields t and G0, contained in
i
where w, already have definite masses. The components of u,
however, are combinations of the physical charged fields
x = 1u†σ u, (62) h+ and G+. The relation between u and the physical
µ 2 µ
fields is given by
y = 1w†σ w, (63)
µ 2 µ
u=U u(cid:48), (68)
v
we obtain the following covariant relations for y,
where u(cid:48) refers to u in the PCH basis,
y = 1[(G0)2+t2], (64)
0 4 u(cid:48) =(h+,G+)T. (69)
y = 1[(G0)2 t2] rˆ + 1( ˆr t)t + 1G0(t ˆr ) .(65)
i 4 − (cid:104) i(cid:105) 2 (cid:104) (cid:105)· i 2 ×(cid:104) (cid:105) i
In a basis invariant form, we know the component of u
Since each component of u = (u ,u )T is a combination parallel to w is the charged Goldstone,
1 2
(cid:104) (cid:105)
of the physical charged Higgs h+ and the charged Gold-
stoneG+, thereisnoneedtowritethemintermsofreal wˆ †u=G+. (70)
(cid:104) (cid:105)
fields. The expression (62) can be kept as the covariant
The orthogonal direction contains the physical h+.
relation.
Obviously, the quadratic part of the potential after
A last observation about Eq.(65) concerns the trans-
EWSB will be
formationpropertiesofG0underrefections,i.e.,CPsym-
mteermtryo.fTEoq.k(e6e5p)tthoebterathnesfsoarmmeataisonthperporpeecretdieinsgoftetrhmeslawset V2(cid:12)(cid:12)SSB =m2h+h−+ 21m2it2i . (71)
conclude that G0 should be a pseudoscalar (scalar under Theremainingtasktocompletelydefinethepotentialaf-
SO(3) and changing sign under reflection or CP) and ter EWSB is to write the interaction terms in Eqs.(59)
H
consequentlyCP-oddirrespectiveoftheCPpropertiesof and (60) in terms of t ,G0,h+,G+ . The sole depen-
i
{ }
the potential. (See appendix D of Ref.6.) dence of those interaction terms on u comes from x in
µ
7
Eq.(62). Thecomponentx isbasisindependentandcan potentialwithanontrivialvacuum. Amongtheelements
0
be readily written of the set, it is clear that the masses are physical observ-
ables. Ontheotherhand,theconnectionofthecoupling
x0 = 21(G−G++h−h+). (72) constants and mixing matrices appearing in the interac-
tion terms with physical observables is not direct. For
The spatial components can be written
example, devising scattering observables to extract the
x =(R˜ ) x(cid:48) , (73) three parameters composing Λ0, present in V3 and V4,
i v ij j does not seem a straightfoward task. The form of V
3
where (R˜ ) =Tr[U†σ U σ ] and in terms of the physical fields, given in appendix E, re-
v ij v i v j inforce such difficulty. The explicit form of V and V
3 4
x(cid:48) = 1(h−G++h+G−), (74) in the Physical basis can be also found in Eqs.(57)–(60)
1 2
of Ref.6, although the dependence on the mass matrix
x(cid:48) = −i(h−G+ h+G−), (75)
2 2 − of the neutral scalars are not explicitly shown. An at-
x(cid:48) = 1(G−G+ h−h+). (76) tempttoextracttheobservableparametersinthe2HDM,
3 2 −
aiming to identify the presence of discrete symmetries
The variables sµ can be written in terms of ti using through measurements, was made in Ref.27. Neverthe-
Eqs.(47) and (48) while the variables yµ are defined in less,separatingthesetofEq.(55)andfindingtherelation
Eqs.(64) and (65). of other parameters with the set is important to estab-
lishthenumberofindependentparameterspossible. The
violation of any relation between parameters would indi-
V. DISCUSSIONS
cate a model with a scalar sector distinct of the 2HDM.
These relations should be constrained by experimental
Equation(58) relates the depth of the potential in the data and studies of the bounds on the mass of the phys-
minimum with the mass matrix of the physical neutral ical charged Higgs[28, 29] or of the decay width of the
scalars. We can obtain bounds on the depth of the po- physical Higgs bosons[2, 30] already exist in the litera-
tential from the relations ture. Ofcourse,higherordereffects,suchastheexchange
of quarks, would modify these tree level relations. The
V((cid:104)r(cid:105))= 12V2((cid:104)r(cid:105))=−V4((cid:104)r(cid:105)), (77) numberofminimamaybealsomodifiedwhenhigheror-
der contributions are taken into account. The existence
where r represents an extremum while V and V re-
(cid:104) (cid:105) 2 4 of at most two minima, for example, may not be true
fer respectively to the quadratic and quartic part of the
beyond tree level[31].
potentialbeforeEWSB,definedinEqs.(2)and(3),eval-
Anotheraspectoftheidentificationofphysicalparam-
uatedintheextremum. ThefirstequalityofEq.(77)can
eters concerns the remaining reparametrization freedom
be written using Eq.(58) as
such as the one in Eq.(29). That rephasing transfor-
V( r )= 1 r rˆ T( ) rˆ . (78) mation freedom is particularly important when count-
(cid:104) (cid:105) −2(cid:104) 0(cid:105)(cid:104) i(cid:105) MN ij(cid:104) j(cid:105) ing the number of parameters of the mixing matrix R˜
v
From the relation above, we can deduce the following in Eq.(73). Since R˜v appears in the couplings involv-
bounds for the depth of a minimum r , ing the physical charged fields it may seem that it is
(cid:104) (cid:105) a physical rotation matrix, needing three angles for its
1 r m2 V( r ) 1 r m2, (79) parametrization. However, only two angles are physi-
− 2(cid:104) 0(cid:105) 3 ≤ (cid:104) (cid:105) ≤−2(cid:104) 0(cid:105) 1
cal. The reason is that the reminiscent reparametriza-
where m23 and m21 are respectively the greatest and the tion freedom induced by Eq.(29) can remove one angle.
least eigenvalue of N. We can conclude that a mini- Suchreparametrizationfreedomisequivalenttorotations
M
mum will be deeper if the respective masses for the neu- around r . Anexplicitparametrizationusingtwoangles
(cid:104) (cid:105)
tral scalars and the value of v are greater. is available in Eqs.(E8) and (E9) of appendix E.
A different physical bound can be extracted from con- The case of CP conserving potentials includes the
dition (B11) necessary for bounded below potentials and MSSM2HDMpotential(seeRef.2)andcanbeeasilyan-
ferTo(mΛ˜t+heΛpo1si)teive>de0finwieteanrersisveofaMt N. FromEq.(54)and aYly).zeIdnabdydsietitotnin,gifΛth2ier=ei0snfoorSiP(cid:54)=CV2,awnedhMav2e=r0 (=or0r.eIanl
⊥ 00 ⊥ (cid:104) 2(cid:105)
thiscase, fromEq.(54), weseetheneutralscalart does
2
m2 m2 < v2Λ , (80) not mix with other scalars and corresponds to a CP-odd
− 3 4 00 field with mass
where e⊥ is any unit vector orthogonal to (cid:104)ˆr(cid:105). The last m2 =m2+ v2Λ . (81)
inequality means the mass of the charged Higgs can not 2 4 22
bearbitrarilylargecomparedtothemassesoftheneutral
scalars. Relation (81) is equivalent to a known relation encoun-
In Sec.III we have found the set of Eq.(55) could be tered in the MSSM [see Eq.(10) of Ref.12], where the
chosenasthephysicalparametersthatdefinethe2HDM pseudoscalar t is usually called A and v2Λ = m2 .
2 4 22 − W
8
The remaining neutral scalars t ,t are CP-even and The corresponding basis transformation group will be
1 3
their mass matrix can be also read from Eq.(54). SO(N2 1) SU(N), the first factor acting on the neu-
− ⊗
Itisimportanttostressthattheoriginalbasistransfor- tral scalars and the second on the charged scalars in-
mations valid before EWSB forming the SU(2) group dependently. The enlargement of the basis transforma-
H
could be explicitly transposed to the fields after EWSB. tion group after EWSB compared to the basis transfor-
Although the possibility of transposition could be fore- mation group before EWSB is greater in NHDMs, with
seen, various properties of the transformations after N >2, thaninthetwo-Higgsdoubletcase(N =2). But
EWSB could not be anticipated. For example, the basis the difference is not just quantitative. For the 2HDM
transformations after EWSB mix the physical charged potential, the basis transformation group after EWSB,
Higgs h+ with the charged Goldstone while the neutral SU(2) SU(2), is just the double of the basis trans-
⊗
fields ti mix among them [through the same SU(2)H formation group SU(2)H before EWSB, which can be
∼
SO(3) ]withoutmixingwiththeneutralGoldstonethat understood as the original basis transformation acting
H
transforms as a scalar of SU(2) . independently on the upper u and lower w components
H
From the discussions of Sec.IV, we can see there is of the doublets, as described in Eq.(66). For N >2, the
an important distinction between basis transformation factor SO(N2 1) necessary to diagonalize N and,
− M
and reparametrization. The transformations of Eq.(66) consequently, necessary to reach the Physical basis, can
constitute legitimate basis transformations that preserve not be thought as the original reparametrization group
the gauge structure after EWSB but they do not config- SU(N)H acting independently on the lower components
ure as reparametrization transformations. On the one w of the doublets.
hand, only the original SU(2) basis transformations
H
that preserves the SU(2) gauge structure configure as Finally, we can say that a nontrivial horizontal struc-
L
reparametrization transformations. On the other hand, ture in the scalar sector of a theory enriches the lat-
the maximal semisimple group of transformations which ter significantly, opening the possibility of different phe-
mix four real fields, t and G0, is SO(4). In addition, nomenology such as different symmetry breaking pat-
i
if we do not impose the kinetic part to be invariant, terns. At the same time, the theory becomes less predic-
thereparametrizationgroupSU(2) canbeextendedto tive as much more free parameters are available. Never-
H
SL(2,c)[10]. theless,usefulphysicalinformationcanbeextractedfrom
thehorizontalstructurebyclassifyingthetransformation
As an terminological issue, the term basis transforma-
properties of the parameters appearing in the potential.
tions (or horizontal transformations) should be accom-
These properties constrain the relations between param-
panied by the gauge structure that they preserve to be
etersbeforeandafterSSB,relating, forinstance, vectors
precise. For example, for the 2HDM treated here, it is
of the horizontal group with vectors. In the 2HDM po-
important to specify if the basis transformations act be-
tential analyzed here, we could relate, for example, the
fore [SU(2) U(1) ] or after EWSB [U(1) ].
L⊗ Y EM rank-2 tensor Λ˜, appearing before EWSB, with the mass
Ingeneral,thehorizontalgroupafterSSBwillbelarger
matrix of the neutral scalars , only extractable after
than the horizontal group before SSB. It should be re- MN
EWSB.Moreover,theserelationswerebasisinvariant. It
marked that usually the physical mixing parameters be-
isimportanttonoticethatthetransformationproperties
longtotheadditionalbasistransformationsonlyallowed
of the parameters refer to the horizontal group SU(2)
after SSB. For example, the CKM matrix for quarks H
actingontheHiggsdoubletsbeforeEWSB.Althoughthe
comes from the difference between the rotations on the
horizontalgroupactingonthefieldsafterEWSBcouldbe
fields u ,c ,t and d ,s ,b necessarytodiagonal-
L L L L L L
{ } { } larger, the transformation properties of the parameters
izetherespectivemassmatrices; applyingbasistransfor-
followed essentially from the original horizontal group
mations before EWSB, it is only possible to diagonalize
valid before EWSB. Obviously, a transformation in the
one of the up or down quark Yukawa coupling matrices.
enlarged horizontal group is usually necessary to reach
Asimilarstructureappearsin2HDMsforwhichthemix-
ingamongneutralscalars,thematrixR˜ ,appearsasthe the Physical basis where all fields have definite masses.
v
difference between the PCH basis and the PNH basis.
ForgeneralN-Higgs-doubletmodels(NHDMs),theco-
variant relation for the mass matrix of neutral scalars
can be easily written by generalizing Eqs.(54) and (49)
toN Higgsdoublets. Thecovariantrelationforthemass
matrix of charged scalars M was found in Ref.[25].
(cid:104) (cid:105)
The fields t , however, will transform as a vector of
i
adjSU(N) ,livinginarealvectorspaceofN2 1dimen- Acknowledgments
H
sions. Since,ingeneral,atransformationinSO−(N2 1),
−
a larger group than adjSU(N) , will be required to di-
H
agonalize , the Physical Neutral Higgs (PNH) ba- The author would like to thank Igor Ivanov for usefull
N
M
sis can not be reached by reparametrization but only discussions. This work was supported by Funda¸c˜ao de
by general horizontal transformations valid after EWSB. Amparo `a Pesquisa do Estado de S˜ao Paulo (Fapesp).
9
APPENDIX A: EXTREMUM EQUATIONS Equation (B6) is found by projecting Eq.(B5) to Λ
0
while Eq.(B7) is equivalent to the requirement ˆrˆr = 1.
·
Any neutral extremum rµ of the potential in Eq.(1) The components of ˆr perpendicular to Λ0 can be found
should satisfy the followin(cid:104)g ex(cid:105)tremum equations[25] from Eq.(B5) once ˆrΛ0 is known.
·
For any extremum satisfying Eq.(B5) we find for
M0+Λ00 r0 +Λ0 r =m2, (A1) Eq.(B1) the value
(cid:104) (cid:105) ·(cid:104) (cid:105)
Mi+Λi0(cid:104)r0(cid:105)+Λ˜ij(cid:104)rj(cid:105)=−m2(cid:104)rˆi(cid:105), (A2) V4(cid:12)(cid:12)extremum = 12r2(−λ)(cid:12)(cid:12)extremum. (B9)
wherei=1,2,3andm2 isthemasssquaredofthephys-
WeseealltheLagrangemultipliersλcorrespondingtoan
ical charged Higgs. The minus sign on the righthand
extremumshouldbenegative. Inparticular,thegreatest
sideofEq.(A2)istheonlyreminiscentoftheMinkowski
of them should be negative.
metric adopted in Ref.25.
In the basis for which Λ˜ + Λ 1 = diag(a ,a ,a ),
Theoriginalextremumequationonthedoubletsreads 00 3 1 2 3
a >a >a ,
1 2 3
M w =0, (A3)
(cid:104) (cid:105)(cid:104) (cid:105) (cid:88)3 Λ2
for u =0 and w =0. Equation (A3) means w is an f(x)= 0i . (B10)
eige(cid:104)nv(cid:105)ector of M(cid:104) (cid:105)w(cid:54) ith null eigenvalue. (cid:104) (cid:105) −i=1 ai+x
(cid:104) (cid:105)
A plot of f(x), with a >0, can be seen in Fig.1 jointly
i
APPENDIX B: BOUNDED BELOW CONDITION with the solution of greatest λ. We see there are at least
two extrema corresponding to the least and greatest λ.
The intermediary extrema may not exist depending on
We seek here the necessary conditions for a bounded
the minimum slope of the curves. For example, in Fig.1,
below potential using a method distinct to the ones
for a x a , there is no solution for f(cid:48)(x)=1.
adopted in Refs.10 and 3. We will restrict ourselves to − 2 ≤ ≤− 3
From the schematic view of Fig.1 we see a > 0 is
positive definite V . i
4
necessary to have positive λ and consequently positive
Rewriting V4 for r0 = r we obtain −
| | definiteV ,unlessΛ =0fornonpositivea . Inageneral
4 0i i
V = 1rT(Λ˜ +Λ 1 )r+ rrΛ . (B1) basis, it is necessary that
4 2 00 3 | | · 0
All variables ri will be treated here as c-numbers. We Λ00+eigenvalues(Λ˜)>0, (B11)
seekthedirectionrforwhichthepotentialincreasesmore
slowly. We minimize then unlessΛ0 havenullprojectioninsomeeigenvectordirec-
tion. To assure V is positive definite is necessary and
4
V4(cid:48) =V4+ 12λ(r2−1), (B2) sufficient to have the greatest Lagrange multiplier
constrainingrtobeintheunitsphereusingtheLagrange
maxλ<0. (B12)
multiplier method.
Differentiating, From Eq.(B6), the distance between the greatest x
min
and the greatest λ is ˆrΛ = ˆrΛ .
∂V4(cid:48) = Λ˜ijrj +Λ00ri+Λ0i r +rˆi(rΛ0)+λri,(B3) Finally, if ai > 0 a|n·d o0n|e m−ak·es0Λ0i small enough,
∂r | | · | |
i we can always find λ < 0, proving the assertion pre-
∂∂Vλ4(cid:48) = 12(r2−1). (B4) cfe(dxi)nggetEqc.lo(s3e1r).to Athse|xΛ-0aix|isg.etInsmspalelceira,l,thfreomcufrv(exs) of
≥
f(x) , a =min(a ), we can conclude that
Equation (B3) yields |a1,a2→a3 3 i
ˆr= (cid:2)Λ˜ +(Λ00+λ+ˆrΛ0)13(cid:3)−1Λ0. (B5) 2|Λ0|<a3 (B13)
− ·
is a sufficient condition.
The values of ˆrΛ corresponding to an extremum is
· 0 For r > r we can parametrize r = eχ r, χ >
given by the roots of 0 0
| | | | | |
0. The analysis of the minimization of V for fixed r
4
| |
ˆrΛ0 =f(ˆrΛ0+λ), (B6) is equivalent to the preceding analysis replacing Λ00
· · Λ e2χ and Λ Λ eχ. If Λ /Λ 1, conditio→n
00 0i 0i 0 00
constrained by (B13) is preserved→for χ>0 on|ce it| is val≤id for χ=0. If
Λ /Λ >1 and λ >0, a sufficient condition is
df(x) | 0| 00 i
=1, (B7)
dx Λ 2
0
| | <min(λ ), (B14)
for x=ˆrΛ0+λ. The function f(x) is defined by Λ00 i
·
f(x) Λ T(cid:2)Λ˜ +(Λ +x)1 (cid:3)−1Λ . (B8) where λ are the eigenvalues of Λ˜.
0 00 3 0 i
≡−
10
The equation (40) can thus be written as
f(x)
U†(cid:0)Y + 1σ Λ r (cid:1)U =Y(cid:48)+ 1σ Λ(cid:48) r(cid:48) , (D2)
wherevtherel2atµionµbνe(cid:104)twν(cid:105)eenv Y(cid:48),Λ(cid:48), r2(cid:48) µinµνth(cid:104)eνP(cid:105)CHbasis
{ (cid:104) (cid:105)}
and Y,Λ, r in the original basis is
{ (cid:104) (cid:105)}
x Y(cid:48) = U†YU , (D3)
min v v
x r(cid:48) = RT r , (D4)
λ (cid:104) µ(cid:105) µν(cid:104) ν(cid:105)
Λ(cid:48) = RT Λ R , (D5)
µν µα αβ βν
andR =R (U )isgivenbyEq.(37). Hence,thefirst
µν µν v
term of (38) is form invariant,
a a a x=0 u† M u=u(cid:48)† M(cid:48) u(cid:48). (D6)
− 1 − 2− 3 (cid:104) (cid:105) (cid:104) (cid:105)
` ´
FIG. 1: Plot of typical f(x). The dot lies at x ,f(x )
min min
wherex isthegreatestvaluethatsatisfiesf(cid:48)(x)=1. Equa- APPENDIX E: INTERACTION TERMS
min
tion (B6) defines the value of λ depicted as the intersection
of the line x−xmin+f(xmin) with the x-axis. The interaction terms can be simplified into
vV3(cid:12)(cid:12) = (cid:2)u†Yu+w†Yw(cid:3)t(cid:107)+m2[x0t(cid:107) xt]
2 SSB − − ·
APPENDIX C: PROOF OF EQ.(48) + r Λ t (cid:2)u 2+ 1t2(cid:3)
(cid:104) 0(cid:105) 0⊥· ⊥ | ⊥| 2 ⊥
+(x+y)T t, (E1)
N
M
From the completeness of the σ matrices[32],
µ
21(σµ)ab(σµ)cd =δadδcb, (C1) V4(cid:12)(cid:12)SSB = 21Λ00(r02−r2)+(r0−(cid:104)ˆr(cid:105)·r)Λ0·r
we can calculate, for a neutral vacuum (cid:104)w(cid:105)=(cid:54) 0, + rT2Mr0Nr + 12(cid:0)Λ00− mr02 (cid:1)r2⊥, (E2)
(cid:104) (cid:105) (cid:104) (cid:105)
r s + r s = 1Tr[ w w †σ ]Tr[σ (w w †+h.c.)]
(cid:104) 0(cid:105) 0 (cid:104) i(cid:105) i 4 (cid:104) (cid:105)(cid:104) (cid:105) µ µ (cid:104) (cid:105) where
= 1Tr[(w w †+h.c.) w w †]
2 (cid:104) (cid:105) (cid:104) (cid:105)(cid:104) (cid:105)
= 2(cid:104)r0(cid:105)s0, (C2) u⊥ ≡ u−(cid:104)wˆ(cid:105)(cid:104)wˆ(cid:105)†u, (E3)
t t ˆr , (E4)
where r0 = w 2/2=v2/4. Hence, (cid:107) ≡ ·(cid:104) (cid:105)
(cid:104) (cid:105) |(cid:104) (cid:105)| t t ˆr t , (E5)
⊥ (cid:107)
≡ −(cid:104) (cid:105)
s0 = rˆi si, (C3) u 2 = x x ˆr , (E6)
(cid:104) (cid:105) ⊥ 0
| | − ·(cid:104) (cid:105)
(cid:112) t2 = 2(y y ˆr ), (E7)
since r0 = ri ri . With the same reasoning, one ⊥ 0− ·(cid:104) (cid:105)
(cid:104) (cid:105) (cid:104) (cid:105)(cid:104) (cid:105)
can prove
and Λ is analogous to t .
0⊥ ⊥
One can also explicit the matrices U and R˜(U ) in
s y =s y . (C4) v v
0 0 i i
Eqs.(68) and (73) choosing an explicit parametrization:
APPENDIX D: BASIS COVARIANCE FOR (cid:104)M(cid:105) Uv = U(θv,ξ)iσ2, (E8)
R˜ = R˜(θ ,ξ)R˜(π,0)=R˜(U ) (E9)
v v v
It is important to stress that the definition of the
charged mass matrix M is covariant by basis transfor- where
(cid:104) (cid:105)
mation (41) in the following sense. The definition of the
charged mass matrix in Eq.(11) is valid in any basis, in U(θ,ϕ) ≡ e−i12σ3ϕe−i12σ2θ. (E10)
particular, in the PCH basis, R˜(θ,ϕ) e−iJ3ϕe−iJ2θ. (E11)
≡
diag(m2,0)= M(cid:48) =Y(cid:48)+ 1σ Λ(cid:48) r(cid:48) . (D1) The generators of rotations are defined by i(J ) =(cid:15) .
(cid:104) (cid:105) 2 µ µν(cid:104) ν(cid:105) k ij ijk
[1] M. Gomez-Bock, M. Mondragon, M. Muhlleitner, SymmetryBreakingandHiggsPhysics,”arXiv:0712.2419
M. Spira and P. M. Zerwas, “Concepts of Electroweak [hep-ph].
