High scale flavor alignment in twoHiggs doublet models and its phenomenology
JHE
High scale avor alignment in twoHiggs doublet
Stefania Gori 0 1 3
Howard E. Haber 0 1 2
Edward Santos 0 1 2
0 1156 High Street, Santa Cruz, CA 95064 , U.S.A
1 345 Clifton Court , Cincinnati, OH 45221 , U.S.A
2 Santa Cruz Institute for Particle Physics, University of California , USA
3 Department of Physics, University of Cincinnati
The most general twoHiggs doublet model (2HDM) includes potentially large sources of avor changing neutral currents (FCNCs) that must be suppressed in order to achieve a phenomenologically viable model. The avor alignment ansatz postulates that all Yukawa coupling matrices are diagonal when expressed in the basis of masseigenstate fermion elds, in which case treelevel Higgsmediated FCNCs are eliminated. In this work, we explore models with the avor alignment condition imposed at a very high energy scale, which results in the generation of Higgsmediated FCNCs via renormalization group running from the high energy scale to the electroweak scale. Using the current experimental bounds on avor changing observables, constraints are derived on the aligned 2HDM parameter space. In the favored parameter region, we analyze the implications for Higgs boson phenomenology.
Beyond Standard Model; Heavy Quark Physics; Higgs Physics

HJEP06(217)
1 Introduction 2 The avoraligned 2HDM
at the Large Hadron Collider (LHC) [1{3], attention now turns to elucidating the dynamics
of electroweak symmetry breaking. Many critical question still remain unanswered. What
is the origin of the electroweak scale, and what mechanism ensures its stability? In light
of the existence of multiple generations of fermions, are there also multiple copies of the
scalar multiplets, implying the existence of additional Higgs scalars? If yes, how are the
Higgsfermion Yukawa interactions compatible with the apparent Minimal Flavor Violation
(MFV), which is responsible for suppressed
avor changing neutral currents (FCNCs)?
Motivations for extending the Higgs sector beyond its minimal form have appeared
often in the literature. For example, the minimal supersymmetric extension of the Standard
{ 1 {
Model, which is invoked to explain the stability of the electroweak symmetry breaking
scale with respect to very high mass scales (such as the grand uni cation or Planck scales),
requires a second Higgs doublet [4{7] to avoid anomalies due to the Higgsino partners of the
Higgs bosons. More complicated scalar sectors may also be required for a realistic model of
baryogenesis [8]. Finally, the metastability of the SM Higgs vacuum [9{11] can be rendered
stable up to the Planck scale in models of extended Higgs sectors [12{19]. Even in the
absence of a speci c model of new physics beyond the Standard Model, an enlarged scalar
sector can provide a rich phenomenology that can be probed by experimental searches now
underway at the LHC.
One of the simplest extensions of the SM Higgs sector is the twoHiggs doublet model
(2HDM).1 In its most general form, the 2HDM is incompatible with experimental data
due to the existence of unsuppressed treelevel Higgsmediated FCNCs, in contrast to the
SM where treelevel Higgsmediated FCNCs are absent. To see why this is so, consider
the Higgsfermion Yukawa interactions expressed in terms of interaction eigenstate fermion
elds. Due to the nonzero vacuum expectation value (vev) of the neutral Higgs
fermion mass matrices are generated. Rede ning the left and righthanded fermion
by separate unitary transformations, the fermion mass matrices are diagonalized. In the
SM, this diagonalization procedure also diagonalizes the neutral Higgsfermion couplings,
and consequently no treelevel Higgsmediated FCNCs are present. In contrast, in a generic
2HDM, the diagonalization of the fermion mass matrices implies the diagonalization of
one linear combination of Higgsfermion Yukawa coupling matrices. As a result, treelevel
Higgsmediated FCNCs remain in the 2HDM Lagrangian when expressed in terms of
masseigenstate fermion elds. If it were possible in the 2HDM to realize avordiagonal neutral
Higgs couplings at treelevel (thereby eliminating all treelevel Higgsmediated FCNCs),
then all FCNC processes arising in the model would be generated at the looplevel, with
magnitudes more easily in agreement with experimental constraints.2
eld,
elds
A natural mechanism for eliminating the treelevel Higgsmediated FCNCs was
proposed by Glashow and Weinberg [22] and by Paschos [23] [GWP]. One can implement
the GWP mechanism in the 2HDM by introducing a Z2 symmetry to eliminate half of
the Higgsfermion Yukawa coupling terms. In this case, the fermion mass matrices and
the nonzero Higgsfermion Yukawa coupling matrices (which are consistent with the Z2
symmetry) are simultaneously diagonalized. Indeed, there are a number of inequivalent
implementations of the GWP mechanism, resulting in the socalled Types I [24, 25], and
II [25, 26], and Types X and Y [27, 28] versions of the 2HDM.3
1For a review with a comprehensive list of references, see ref. [20].
2Even in models with avordiagonal neutral Higgs couplings, oneloop processes mediated by the charged
Higgs boson can generate signi cant FCNC e ects involving third generation quarks. Such models, in order
to be consistent with experimental data, will produce constraints in the [mH
, tan ] plane. The most
avor alignment ansatz proposed in ref. [30] asserts a proportionality between the two sets
of Yukawa matrices. If this avoralignment condition is implemented at the electroweak
scale, then the diagonalization of the fermion mass matrices simultaneously yields
avordiagonal neutral Higgs couplings. Moreover, this avoraligned 2HDM (henceforth denoted
as the A2HDM) preserves the relative hierarchy in the quark mass matrices, and provides
additional sources of CPviolation in the Yukawa Lagrangian via the introduction of three
complex alignment parameters. Unfortunately, apart from the special cases enumerated
in ref. [31], there are no symmetries within the 2HDM that guarantee the stability of the
avor alignment ansatz with respect to radiative corrections. As such, avor alignment
at the electroweak scale must be generically regarded as an unnatural netuning of the
Higgsfermion Yukawa matrix parameters. Indeed, the Types I, II, X and Y 2HDMs are
the unique special cases of avor alignment that are radiatively stable after imposing the
observed fermion masses and mixing [32].
In this paper, we consider the possibility that avor alignment arises from New Physics
beyond the 2HDM. Without a speci c ultraviolet completion in mind, we shall assert that
avor alignment is imposed at some high energy scale,
, perhaps as large as a grand
uni cation scale or the Planck scale, where new dynamics can emerge (e.g., see ref. [33]
for a viable model).
Once we impose the
avor alignment ansatz at the scale
, the
e ective
eld theory below this scale corresponds to a 2HDM with both Higgs doublets
coupling to up type and down type quarks and leptons.4 We then employ renormalization
group (RG) evolution to determine the structure of the 2HDM Yukawa couplings at the
electroweak scale. For a generic avor alignment ansatz at the scale , avor alignment in
the Higgsfermion Yukawa couplings at the electroweak scale is violated, thereby generating
Higgsmediated FCNCs. However, these FCNCs will be of Minimal Flavor Violation [35]
type and therefore may be small enough to be consistent with experimental constraints,
depending on the choice of the initial alignment parameters at the scale .
We therefore examine the phenomenology of Higgsmediated FCNCs that arise from
the assumption of avor alignment at some high energy scale, , that, for the purpose
of our analyses, is
xed to be the Planck scale (MP).
We note that similar work was
performed in [36], where meson mixing and B decays were used to constrain the A2HDM
parameter space with avor alignment at the Planck scale. Numerical results were obtained
analytically in [36], using the leading logarithmic approximation. The results of this paper
are rst obtained in the leading log approximation, and then numerically by evolving the
full oneloop renormalization group equations (RGEs) down from the Planck scale to the
electroweak scale. In our work, we discuss the validity of the leading log approximation
and examine additional FCNC processes at high energy (top and Higgs decays) and at low
energy (B meson decays) to place bounds on the A2HDM parameters.
4In practice, one should also append to the 2HDM some mechanism for generating neutrino masses.
An example of incorporating the e ects of neutrino masses and mixing in the context of a 2HDM with
avor changing neutral Higgs couplings can be found in ref. [34]. In this paper, we shall simply put all
avor structure. In the formalism presented in section 2, we
initially allow for the most general form of the Higgs scalar potential and the Yukawa
coupling matrices. In particular, new sources of CPviolation beyond the SM can arise due
to unremovable complex phases in both the scalar potential parameters and the Yukawa
couplings. For simplicity, we subsequently choose to analyze the case of a CPconserving
Higgs scalar potential and vacuum, in which case the neutral masseigenstates consist of
2HDM, in which the Yukawa coupling matrices are diagonal in the basis of quark and
lepton masseigenstates. However, alignment is not stable under renormalization group
running. Following the framework for
avor discussed above, we impose the alignment
condition at the Planck scale and then evaluate the Yukawa coupling matrices of the Higgs
basis at the electroweak scale as determined by renormalization group running, subject to
the observed quark and lepton masses and the CKM mixing matrix. The renormalization
group running is performed numerically and checked in the leading log approximation,
where simple analytic expressions can be obtained. In this context, a comparison with
general Minimal Flavor Violating 2HDMs is performed.
In section 3, we discuss the implications of highscale avor alignment for high energy
processes. We focus on avorchanging decays of the top quark and on the phenomenology
of the heavy neutral CPeven and CPodd Higgs bosons. In section 4, we discuss the
implications of highscale avor alignment for low energy processes. Here we consider constraints
arising from neutral meson mixing observables and from Bs ! `+` , which receive
contributions at treelevel from neutral Higgs exchange, and from the charged Higgs mediated
decay. By comparing theoretical predictions to experimental data, one can already
probe certain regions of the A2HDM parameter space. Additional parameter regions will be
probed by future searches for heavy Higgs bosons and measurements of Bphysics
observables. Conclusions of this work are presented in section 5. Finally, in appendix A we review
the derivation of the Yukawa sector of our model in the fermion masseigenstate basis, and
in appendix B we exhibit the oneloop matrix Yukawa coupling RGEs used in this analysis.
The
Theoretical framework for the 2HDM
Consider a generic 2HDM consisting of two complex, hyperchargeone scalar doublets, 1
2
. The most general renormalizable scalar potential that is invariant under local
SU(2) U(1) gauge transformations can be written as
V = m121 y1 1+m222 y2 2 [m122 y1 2+h:c:]+
+ 4( y1 2)( y2 1)+
1
2 5( y1 2)2+
potential is achieved when the neutral components of the two scalar doublet
quire nonzero vacuum expectation vales, h 10i = v1=p2 and h 20i = v2=p2, where the
elds
ac(potentially complex) vevs satisfy
v
2
is then spontaneously broken, leaving an unbroken U(1)EM gauge group.
In the most general 2HDM, the
elds
1 and
2 are indistinguishable. Thus, it is
HJEP06(217)
always possible to de ne two orthonormal linear combinations of the two doublet
elds
without modifying any prediction of the model. Performing such a rede nition of elds
leads to a new scalar potential with the same form as eq. (2.1) but with modi ed coe cients.
This implies that the coe cients that parameterize the scalar potential in eq. (2.1) are not
directly physical [37].
To obtain a scalar potential that is more closely related to physical observables, one can
introduce the socalled Higgs basis in which the rede ned doublet
elds (denoted below
by H1 and H2) have the property that H1 has a nonzero vev whereas H2 has a zero
vev [37, 38]. In particular, we de ne the new Higgs doublet elds:
H1 =
H1+!
H10
v
i H2 (which does not alter the fact that hH20i = 0). In the Higgs
basis, the scalar potential is given by [37, 38]:
V = Y1H1yH1+Y2H2yH2+[Y3H1yH2+h:c:]+ 2 Z1(H1yH1)2+ 12 Z2(H2yH2)2+Z3(H1yH1)(H2yH2)
1
+Z4(H1yH2)(H2yH1)+
1
2 Z5(H1yH2)2+ Z6(H1yH1)+Z7(H2yH2) H1yH2+h:c: ;
where Y1, Y2 and Z1; : : : ; Z4 are real and uniquely de ned, whereas Y3, Z5, Z6 and Z7 are
potentially complex and transform under the rephasing of H2 ! ei H2 as
[Y3; Z6; Z7] ! e i [Y3; Z6; Z7] and
Z5 ! e 2i Z5 ;
since V must be independent of . After minimizing the scalar potential,
Y1 =
1
2 Z1v2 ;
Y3 =
1
2 Z6v2 :
This leaves 11 free parameters: 1 vev, 8 real parameters, Y2, Z1;2;3;4, jZ5;6;7j, and two
relative phases.
In the general 2HDM, the physical charged Higgs boson is the charged component of
the Higgsbasis doublet H2, and its mass is given by
m2H
= Y2 +
1
2 Z3v2 :
{ 5 {
(2.4)
(2.5)
(2.6)
(2.7)
The three physical neutral Higgs boson masseigenstates are determined by diagonalizing
3 real symmetric squaredmass matrix that is de ned in the Higgs basis [38, 39]
M
2 = v2 BB Re Z6
0
Z1
Im Z6
Re Z6
12 (Z345 + Y2=v2)
s13
s12c23
c12c23
c12s13s23
s12s13s23
c13s23
c12s13c23 + s12s23
s12s13c23
c12s23CC BB
where Z345
Z3 + Z4 + Re Z5.
M2. The diagonalization matrix is a 3
angles: 12, 13 and 23. Following ref. [39],
To identify the neutral Higgs masseigenstates, we diagonalize the squaredmass matrix
3 real orthogonal matrix that depends on three
In light of the freedom to de ne the masseigenstate Higgs elds up to an overall sign, the
invariant mixing angles 12, 13 and
can be determined modulo . By convention, we choose
The physical neutral Higgs states (h1;2;3) are then given by:
1
2
1
2
where the hi are the masseigenstate neutral Higgs elds, cij
Under the rephasing H2 ! e
12 ; 13 are invariant, and
jZ6jei 6 6= 0,5 it is convenient to de ne the invariant mixing angle,
hk = p
qk1
H10
+ qk2H20ei 23 + h:c: ;
where the qk1 and qk2 are invariant combinations of 12 and 13, which are exhibited in
It is convenient to de ne the physical charged Higgs states by
H
e i 23 H2 ;
so that all the Higgs masseigenstate
elds (h1, h2, h3 and H ) are invariant under
H2 ! ei H2.
5If Z6 = 0, then one can always rephase the Higgs basis eld H2 such that Z5 is real. In this basis,
the neutral Higgs boson squaredmass matrix, M2, is diagonal, and the identi cation of the neutral Higgs
boson masseigenstates is trivial.
23
{ 6 {
s12c13
s13
are quite complicated, there are numerous relations among them which take on rather
simple forms. The following results are noteworthy [39, 40]:
one starts out initially with a Lagrangian expressed in terms of the scalar doublet elds i
(i = 1; 2) and interactioneigenstate quark and lepton elds. After electroweak symmetry
breaking, one can reexpress the scalar doublet elds in terms of the Higgs basis elds H1
and H2. At the same time, one can identify the 3
3 quark and lepton mass matrices.
By rede ning the left and righthanded quark and lepton
elds appropriately, the quark
and lepton mass matrices are transformed into diagonal form, where the diagonal elements
are real and nonnegative. The resulting Higgsfermion Yukawa Lagrangian is given by in
eq. (A.16) and is repeated here for the convenience of the reader [40],
3 Yukawa coupling matrices. Note that FR;L
PR;LF , where F = U ,
D, N and E, and PR;L
5) are the right and lefthanded projection operators,
respectively. At this stage, the neutrinos are exactly massless, so we are free to de ne the
physical lefthanded neutrino elds, NL, such that their charged current interactions are
generationdiagonal.6
By setting H10 = v=p2 and H20 = 0, one can relate
U , D, and
E to the diagonal
(uptype and downtype) quark and charged lepton mass matrices MU , MD, and ME ,
6To incorporate the neutrino masses, one can employ a seesaw mechanism [41{45] and introduce three
righthanded neutrino elds along with an explicit SU(2)
U(1) conserving mass term. See footnote 4.
{ 7 {
MU = p
ME = p
v
v
2
2
rephasing H2 ! e
LY must be independent of .
U = diag(mu ; mc ; mt) ;
E y = diag(me ; m ; m ) :
However, the complex matrices F (F = U; D; E) are unconstrained. Moreover, under the
i H2, the Yukawa matrix acquires an overall phase, F
! e
i
F , since
To obtain the physical Yukawa couplings of the Higgs boson, one must relate the
Higgs basis scalar elds to the Higgs masseigenstate elds. Using eqs. (2.13) and (2.14),
the Higgsfermion Yukawa couplings are given by,
couplings in terms of the following three 3
3 hermitian matrices that are invariant with
respect to the rephasing of H2,
F
R
F
I
v
p
where the MF are the diagonal fermion mass matrices [cf. eq. (2.21)] and the Yukawa
coupling matrices are introduced in eq. (2.20). Then, the Yukawa couplings take the
following form:
LY =
MU1=2 qk11 + Re(qk2) UR + i 5 I
U
+ Im(qk2) IU
U
i 5 R
MU1=2U hk
1
v
+
+
+
3
U X
k=1
3
D X
k=1
3
E X
k=1
1
v
1
v
p
v
2
M D1=2 qk11 + Re(qk2) R
D
i 5 I
D
+ Im(qk2) ID + i 5 R
D
ME1=2 qk11 + Re(qk2) R
E
i 5 I
E + Im(qk2) IE + i 5 R
E
3 identity matrix. The appearance of unconstrained hermitian 3
R;I in eq. (2.25) indicates the presence of potential avorchanging
neutral Higgsquark and lepton interactions. If the o diagonal elements of F
pressed, they will generate treelevel Higgsmediated FCNCs that are incompatible with
R;I are
unsupthe strong suppression of FCNCs observed in nature.
with the predictions of the Standard Model. In this paper, we shall identify h1 as the
SMlike Higgs boson. In light of the expression for the h1 coupling to a pair of vector
HJEP06(217)
bosons V V = W +W
or ZZ,
gh1V V
ghSMV V
1. Thus, in the limit of a SMlike Higgs boson, eqs. (2.16)
In addition, eq. (2.19) implies that one additional small quantity characterizes the limit of
a SMlike Higgs boson,
(2.26)
(2.27)
(2.28)
(2.30)
j Im(Z5e 2i 23 )j '
2(m22
m21)s12s13
v2
'
Im(Z62e 2i 23 )v2
m23
m21
1 :
(2.29)
Moreover, in the limit of a SMlike Higgs boson, eq. (2.18) yields
m22
m32 ' Re(Z5e 2i 23 )v2 :
As a consequence of eqs. (2.27) and (2.28), the limit of a SMlike Higgs boson7 can be
achieved if either jZ6j
1 and/or if m2, m3
v. The latter corresponds to the wellknown
decoupling limit of the 2HDM [39, 46, 53].8 In this paper, we will focus on the decoupling
regime of the 2HDM to ensure that h1 is su ciently SMlike, in light of the current LHC
Higgs data [3].
2.3
Neutral scalars of de nite CP
In the exact SMHiggs boson limit, the couplings of h1 are precisely those of the SM
Higgs boson. In this case, we can identify h1 as a CPeven scalar. In general, the heavier
neutral Higgs bosons, h2 and h3 can be mixed CP states. The limit in which h2 and h3
7In the literature, this is often referred to as the alignment limit [46{52]. We do not use this nomenclature
here in order to avoid confusion with
avor alignment, which is the focus of this paper.
8Note that eq. (2.30) implies that in the decoupling limit, m2
v implies that m3
v and vice versa.
{ 9 {
are approximate eigenstates of CP is noteworthy. This limit is achieved assuming that
js13j
js12j. That is,
In the decoupling limit, the ratio of squaredmass di erences in eq. (2.31) is of O(1).
Moreover, unitarity and perturbativity constraints suggest that Re(Z6e i 23 ) cannot be
signi cantly larger than O(1). Hence, it follows that
j Im(Z6e i 23 )j
1 :
In light of eq. (2.10), we can rephase H2 ! ei H2 such that 23 = 0 (mod
),
i.e. c23 =
1. Eqs. (2.29) and (2.32) then yield j Im Z5j ; j Im Z6j
1 : For simplicity
in the subsequent analysis, we henceforth assume that a real Higgs basis exists in which
Z5 and Z6 are simultaneously real. In this case, the scalar Higgs potential and the Higgs
vacuum are CPinvariant, and the squaredmass matrix of the neutral Higgs bosons given
in eq. (2.8) simpli es,
M
2 = BBZ6v2
0
Z1v2
0
Z6v2
0
Y2 + 12 (Z3 + Z4 + Z5)v2
0
0
Y2 + 12 (Z3 + Z4
Z5)v2
1
CC ;
A
boson A = p
2 Im H20 with squared mass,
where 6
sgn Z6, in the real Higgs basis [cf. eqs. (2.11) and (2.12)]. To maintain the
reality of the Higgs basis, the only remaining freedom in de ning the Higgs basis elds is
the overall sign of the eld H2. In particular, under H2 !
H2, we see that Z5 is invariant
whereas Z6 (and Z7) and c23 change sign. We immediately identify the CPodd Higgs
1
2
m2A = Y2 + (Z3 + Z4
Z5)v2 :
Note that the real Higgs masseigenstate eld, A, is de ned up to an overall sign change,
which corresponds to the freedom to rede ne H2 !
H2. In contrast, the charged Higgs
eld H
H2 !
de ned (as a matter of convenience) by eq. (2.14) is invariant with respect to
H2. Indeed, by using eq. (2.34), we can now write H
= 6H2 . In light of
eqs. (2.7) and (2.35),
The upper 2 2 matrix block given in eq. (2.33) is the CPeven Higgs squaredmass matrix,
m2H
= m2A
(Z4
Z5)v2 :
1
2
M2H =
Z1v2
Z6v2
Z6v2
m2A + Z5v2
!
;
(2.31)
(2.32)
and
yields
h = h1 ;
H =
This means that the signs of the elds H and A and the sign of c
rede nition of the Higgs basis eld H2 !
H2.
Note that 0
s
1 in the convention speci ed in eq. (2.12). Moreover, eq. (2.16)
and it therefore follows that 0
s12 ; c12
corresponds to mH
mh and jc
j
(2.19) yield
the SMlike Higgs boson and H as the heavier CPeven Higgs boson. Finally, eqs. (2.15){
0. The decoupling limit
1 [cf. eq. (2.27)], in which case we can identify h as
s
c
=
Z6v2
m2H
1 and c
m2
h
;
Z6
Z1v2 = m2h s2
Z6v2 = (m2h
Z5v2 = m2H s
2
+ m2H c2
m2H )s
c
+ m2h c2
;
;
m2A :
where we have used eq. (2.35) to eliminate Y2. To diagonalize M2H , we de ne the CPeven
masseigenstates, h and H (with mh
mH ) by
H!
h
=
c
s
s
c
!
p
p
mass matrix when expressed relative to the original basis of scalar elds, f 1 ; 2g, which
is assumed here to be a real basis.9 Since the real Higgs masseigenstate elds H and h are
de ned up to an overall sign change, it follows that
is determined modulo . To make
contact with the notation of eq. (2.9), we note that c13 = 1 and c23 = 6 [cf. eq. (2.34)].
Assuming that h1 is the lighter of the two neutral CPeven Higgs bosons, then eq. (2.38)
implies the following identi cations:
In particular, m2h ' Z1v2 in the limit of a SMlike Higgs boson h. Applying eq. (2.40) to
9Given the assumption [indicated above eq. (2.33)] that the scalar Higgs potential and the Higgs vacuum
are CPinvariant, it follows that there must exist a real basis of scalar
elds in which all scalar potential
parameters and the vacuum expectation values of the two neutral Higgs elds, h i0i
p
vi=
2 (for i = 1, 2),
are simultaneously real [54].
(2.38)
(2.39)
(2.40)
(2.41)
(2.42)
(2.43)
(2.44)
(2.45)
(2.46)
(2.47)
1
v
+
+
+
1
v
1
v
p
v
2
X
U KM D1=2( RD
Inserting these results into the general form of the Yukawa couplings given in eq. (2.25), we
obtain the following Higgsfermion couplings in the case of a CPconserving Higgs scalar
potential and vacuum,
LY =
F s
MF + 6c
MF1=2 FR + i"F 5 IF MF1=2 F h
(2.48)
(2.49)
(2.50)
(2.51)
(2.52)
(2.53)
where we have introduced the notation,
in terms of the Higgs Yukawa coupling matrices F simplify,
Moreover, by employing eq. (2.34) in eqs. (2.23) and (2.24), the expressions for FR and IF
"F =
(+1
1
for F = U ;
for F = D; E :
6MF1=2 FRMF1=2 =
i 6MF1=2 IF MF1=2 =
v
v
p
p
The structure of the neutral Higgs couplings given in eq. (2.48) is easily ascertained.
If IF 6= 0, then the neutral Higgs
Moreover, the two sign choices, 6 =
elds will exhibit CPviolating Yukawa couplings.10
1 are physically indistinguishable, since the sign of
Z6 can always be ipped by rede ning the Higgs basis eld H2 !
H2. Under this eld
rede nition, F , c
, H and A also ip sign, in which case LY is unchanged.
For completeness, we brie y consider the case where h1 is the heavier of the two neutral
CPeven Higgs bosons. In this case, eq. (2.38) implies the following identi cations,
and
h = 6h2 ;
H = h1 ;
This means that the signs of the elds h and A and the sign of s
rede nition of the Higgs basis eld H2 !
H2. Note that eqs. (2.41){(2.44) are still valid.
10Likewise, if Im Z7 6= 0 in a basis where Z5 and Z6 are real, then the neutral Higgs elds will also possess
CPviolating trilinear and quadralinear scalar couplings.
Invoking the convention given by eq. (2.12) now implies that 0
c
1 and Z6s
Moreover in light of eq. (2.26), if js
is achieved in the limit of jZ6j
j
1 then H is SMlike and m2H ' Z1v2, which
1. No decoupling limit is possible in this case since
mh < mH = 125 GeV. Using eq. (2.53), one can check that eqs. (2.45){(2.47) are modi ed
by taking s
! c
and c
!
s
. As a result, eq. (2.48) remains unchanged.
So far, the parameters and have no separate signi cance. Only the combination, is meaningful.
Moreover the matrices
FR and
IF are generic complex matrices,
which implies the existence of treelevel Higgsmediated
avor changing neutral currents,
as well as new sources of CP violation. However, experimental data suggest that such
Higgsmediated FCNCs must be highly suppressed. One can eliminate these FCNCs by
Higgs potential given in eq. (2.1), which sets 6 = 7 = 0 and gives physical signi cance to
the 1 2 basis choice. This in turn promotes the CPeven Higgs mixing angle
in the real
1 2 basis and tan
v2=v1 to physical parameters of the model.11 The Z2 symmetry
can be extended to the Higgsfermion interactions in four inequivalent ways. In the notation
of the Higgsfermion Yukawa couplings given in eq. (2.48), the FR;I are given by12
1. TypeI: for F = U; D; E, FR = 6 cot 1 and IF = 0.
2. TypeII: UR = 6 cot 1 and IU = 0. For F = D; E, FR =
6 tan 1 and IF = 0.
3. TypeX: ER =
4. TypeY: RD =
6 tan 1 and IE = 0. For F = U; D, FR = 6 cot 1 and IF = 0.
6 tan 1 and ID = 0. For F = U; E, FR = 6 cot 1 and IF = 0.
Inserting these values for the FR and IF into eq. (2.48), the resulting neutral Higgsfermion
Yukawa couplings are avor diagonal as advertised.
From a purely phenomenological point of view, one can simply avoid treelevel
Higgsmediated FCNCs by declaring that the FR and IF are diagonal matrices. In the simplest
generalization of the Type I, II, X and Y Yukawa interactions, one asserts that both the FR
and the IF are proportional to the identity matrix (where the constants of proportionality
can depend on F ). This is called the avoraligned 2HDM, which we shall discuss in the
next subsection.
2.4
The
proportional. When written in terms of fermion masseigenstates, F = p
The avoraligned 2HDM posits that the Yukawa matrices F and F [cf. eq. (2.20)] are
2MF =v is
diagonal. Thus in the A2HDM, the F are likewise diagonal, which implies that treelevel
11Since the existence of a real Higgs basis implies no spontaneous nor explicit CPviolation in the
scalar sector, there exists a
1 2 basis in which the i of eq. (2.1), v1 and v2 (and hence tan ) are
simultaneously real.
12As de ned here, the parameter tan
ips sign under the rede nition of the Higgs basis eld H2 !
H2,
in contrast to the more common convention where tan
is positive (by rede ning H2 !
H2 if necessary).
With this latter de nition, the two cases of 6 =
1 [or equivalently the two cases of sgn(s
c
) =
represent nonequivalent points of the TypeI, II, X or Y 2HDM parameter space. However, we do not
adopt this latter convention in the present work.
Higgsmediated FCNCs are absent. We de ne the alignment parameters aF via,
where the (potentially) complex numbers aF are invariant under the rephasing of the Higgs
basis eld H2 ! ei H2. It follows from eqs. (2.23) and (2.24) that
FR = (Re aF )1 ;
IF = (Im aF )1 :
Inserting the above results into eq. (2.22), the Yukawa couplings take the following form:
LY =
MU qk1 + qk2aU PR + qk2aU PL U hk
MD qk1 + qk2aD PR + qk2aDPL Dhk
ME qk1 + qk2aE PR + qk2aEPL Ehk
1
v
+
+
+
1
v
+
+
+
3
U X
k=1
3
D X
k=1
3
E X
k=1
1
v
1
v
p
v
2
X
This form simpli es further if the neutral Higgs masseigenstates are also states of de nite
CP. In this case, the corresponding Yukawa couplings are given by
LY =
F MF s
+ 6c
Re aF + i F Im aF
5
F h
(2.57)
F MF c
6s
Re aF + i F Im aF
5
F H
F MF
6 Im aF
i F Re aF
5
F A
U a
D KMDPR
a
U MU KPL DH+ + a
E N MEPREH+ + h:c: :
As noted above eq. (2.41), it is convenient to choose a convention in which s
then follows from eq. (2.41) that 6c
=
j
c
j. That is, the neutral Higgs couplings
exhibited in eq. (2.57) do not depend on the sign of c
(which can be ipped by rede ning
the overall sign of the Higgs basis eld H2). Note that in this convention, the signs of the
alignment parameters aF are physical.
The TypeI, II, X and Y Yukawa couplings are special cases of the A2HDM Yukawa
couplings. Since the a
F (F = U; D; E) are independent complex numbers, there is no
preferred basis for the scalar elds outside of the Higgs basis. Thus, a priori, there is no
separate meaning to the parameters
and
in eq. (2.57). Nevertheless, in the special case
of a CPconserving neutral Higgslepton interaction governed by eq. (2.57) with Im aE = 0,
it is convenient to introduce the real parameter tan
via
(2.54)
(2.55)
(2.56)
(2.58)
corresponding to a TypeII or TypeX Yukawa couplings of the charged leptons to the
neutral Higgs bosons. The theoretical interpretation of tan
de ned by eq. (2.58) is as
follows. It is always possible to choose a
1 2 basis with the property that one of the two
Higgslepton Yukawa coupling matrices vanishes. Namely, in the notation of eq. (A.1), we
have 2E;0 = 0, which means that only 1 couples to leptons. In the case of a CPconserving
scalar Higgs potential and Higgs vacuum, we can take the 1 2 basis to be a real basis and
identify tan
or TypeX models, 2
E;0 = 0 does not correspond to a discrete Z2 symmetry of the generic
vi= 2 (for i = 1, 2). However, in contrast to TypeII
p
A2HDM Lagrangian, since we do not require any of the Higgsquark Yukawa coupling
matrices and the scalar potential parameters 6 and 7 to vanish in the same
1 2 basis.
Note that the sign of aE in eq. (2.58) is physical since both 6 and tan
ip sign under
the Higgs basis eld H2 !
H2. In contrast to the standard conventions employed in
the 2HDM with TypeI, II, X or Y Yukawa couplings where tan
is de ned to be positive
[cf. footnote 12], we shall not adopt such a convention here. In practice, we will rewrite
eq. (2.58) as,
where E =
space.
a
E = Ej tan j
;
(2.59)
1 correspond to physically nonequivalent points of the A2HDM parameter
One theoretical liability of the A2HDM is that for generic choices of the alignment
parameters aU and aD, the
avoralignment conditions in the quark sector speci ed in
eq. (2.54) are not stable under the evolution governed by the Yukawa coupling
renormalization group equations. Indeed, as shown in ref. [32], eq. (2.54) is stable under
renormalization group running if and only if the parameters aU and aD satisfy the conditions of
the Type I, II, X or Y 2HDMs speci ed at the end of section 2.3. In the leptonic sector,
since we ignore neutrino masses, the Higgslepton Yukawa couplings are avordiagonal at
all scales. We therefore assume that13
F ( ) = a
F F ( ) ;
for F = U; D;
(2.60)
at some very high energy scale
(such as the grand uni cation (GUT) scale or the Planck
scale). That is, we assume that the alignment conditions are set by some a priori unknown
physics at or above the energy scale . We take the complex alignment parameters aF to
be boundary conditions for the RGEs of the Yukawa coupling matrices, and then determine
the lowenergy values of the Yukawa coupling matrices by numerically solving the RGEs.
To ensure that the resulting lowenergy theory is consistent with a SMlike Higgs boson
observed at the LHC, we shall take mh = 125 GeV, and assume that the masses of H, A
and H
are all of order
H
400 GeV. In this approximate decoupling regime, jc
j is
small enough such that the properties of h are within about 20% of the SM Higgs boson,
as required by the LHC Higgs data [3]. We employ the 2HDM RGEs given in appendix B
from
down to
H , and then match onto the RGEs of the Standard Model to generate
the Higgsfermion Yukawa couplings at the electroweak scale, which we take to be mt or
mZ . Note that the values of Q( H ) = p
the known quark masses via Standard Model RG running.
2MQ( H )=v (for Q = U , D) are determined from
13Under the assumption of a real Higgs basis, 6 = ei 23 is xed via eq. (2.58). This factor, which appears
in eq. (2.54), can then be absorbed into the de nition of aF .
Then, ME ( H ) is determined by the diagonal lepton mass matrix via Standard Model RG
running.
Higgsmediated FCNCs from high scale alignment
To explore the Higgsmediated FCNCs that can be generated in the A2HDM at the
electroweak scale, we establish
avoralignment at some high energy scale, , as for example
at the GUT or Planck scale, and run the oneloop RGEs from the high scale to the
electroweak scale. Thus, we impose the following boundary conditions for the running of the
oneloop 2HDM Yukawa couplings,
HJEP06(217)
where the MQ (Q = U , D) are the diagonal quark matrices, and
H is the scale of the
heavier doublet, taken to be relatively large to guarantee that we are su ciently in the
decoupling limit. For the lepton sector, the corresponding boundary conditions are [cf. eq. (2.59)],
Q
( H ) = p
2MQ( H )=v;
Q
( );
E
E
( H ) = p
2ME ( H )=v;
( H ) = E j tan j
E
( H ):
As noted above for the lepton case (F = E), if E ( ) is proportional to E ( ), then
is proportional to
at all energy scales. Thus, we identify the leptonic alignment parameter
at low energies by tan . More precisely [cf. eqs. (2.21) and (2.59)],
E
( H ) = p
2 E j tan jME ( H )=v :
where aQ represents the aligned part (in general, di erent from a0Q), and
Q the
corresponding degree of misalignment at the high scale.
Satisfying the two boundary conditions for the quark sector [eqs. (2.62) and (2.63)] is not
trivial, since they are imposed at opposite ends of the RG running. For example, to set
avoralignment at the high energy scale, we must know the values of Q( ). This involves
running up
Q( H ) to the high scale, but since the oneloop RGEs are strongly coupled
to the Q matrices, we must supply values for Q( H ) to begin the running.
With no a priori knowledge of which values of Q( H ) lead to
avoralignment at
the high scale, we begin the iterative process by assuming avoralignment at
lowscale alignment parameter a0Q,
This avoralignment will be broken during RGE evolution to the high scale, and a
procedure is needed to reestablish
avoralignment at the high scale. To accomplish this, we
decompose Q( ) into parts that are aligned and misaligned with
Q( ), respectively,
Q
( H ) = a0Q Q
( H ):
Q
To minimize the misaligned part of Q( ), we implement the cost function,
Q
3
X
i;j=1
3
X
i;j=1
j iQj j2 =
Q
j ij ( )
a
Q Q
ij ( )j2;
which, once minimized, provides the optimal value of the complex parameter aQ for
avoralignment at the high scale,
(2.68)
(2.69)
(2.70)
a
Q
Pi3;j=1 iQj ( ) iQj ( )
Pi3;j=1 iQj ( ) iQj ( )
:
Q
( );
We subsequently impose avoralignment at the high scale using this optimized alignment
HJEP06(217)
parameter,
D below
and evolve the oneloop RGEs back down to
H . In principle, further running of U and
H can regenerate o diagonal terms. However, these e ects are extremely small
and can be ignored in practice. At
H , we use (2.62) to match the boundary conditions
for the 2HDM and SM. At this point, the matrices
longer diagonal, so we must rediagonalize
respectively transforming U and
D (at the scale
U and
D at the scale
H are no
U and
D in analogy with eq. (A.12) [while
H ) in analogy with eq. (A.13)]. We can
now evolve U and
D down to the electroweak scale to check the accuracy of the resulting
quark masses. If any of the quark masses di er from their experimental values by more
than 3%, we reestablish the correct quark masses at the electroweak scale,14 run back up
to
H , and then rerun this procedure repeatedly until the two boundary conditions are
satis ed. The result is avoralignment between Q( ) and Q( ), and a set of Q matrices
at the electroweak scale that provide a source of FCNCs.
In our iterative procedure, we demand that all scaledependent Yukawa couplings
remain
below
nite from the electroweak scale to the Planck scale (i.e., Landau poles are absent
= MP). This restricts the range of the possible seed values, a0Q, used in eq. (2.66)
to initialize the iteration. Consequently, the alignment parameters aU and aD cannot be
too large in absolute value. Constraints on the alignment parameters due to Landau pole
considerations during oneloop RG running have been given in ref. [59]. In our analysis,
the allowed values of aU and aD consistent with the absence of Landau poles at all scales
below
are exhibited in gure 1.15 Assuming
H = 400 GeV, these considerations lead
to bounds on the alignment parameters evaluated at the Planck scale,
= MP,
jaU j . 0:8 and jaDj . 80 ;
(2.71)
which are consistent with the results previously obtained in ref. [59].
14Starting the RG evolution at mZ, we use a ve avor scheme to run up to mt and a six avor scheme
above mt. Running quark mass masses at mZ and mt are obtained from the RunDec Mathematica software
package [55, 56], based on quark masses provided in ref. [57]. We
x the initial value of the top Yukawa
coupling yt(mt) = 0:94, corresponding to an MS top quark mass of mt(mt) = 163:64 GeV [58]. For
simplicity, the e ects of the lepton masses are ignored, as these contribute very little to the running.
15If a Landau pole in one of the Yukawa coupling matrices arises at the scale
, then both the
corresponding
( ) diverge, whereas their ratio, aQ, remains nite.
= MP are exhibited. The blue points occupy the region of the A2HDM parameter space where
the prediction for all entries of the
Q matrices lie within a factor of 3 from the results obtained
with the full running. The red points occupy the region where the leading log approximation yields
results quite di erent from the full RG running.
Leading logarithm approximation
In the limit of small alignment parameters, it is possible to obtain approximate analytic
solutions to the oneloop RGEs provided in appendix B. One can express the Q matrices
at the low scale as
U
D
( H ) ' a
( H ) ' a
U U
D D
( H ) +
( H ) +
1
1
16 2 log
U );
D);
;
:
(2.72)
(2.73)
(2.74)
(2.75)
where D D; D U ; D D; D U are the functions de ned in eqs. (B.10){(B.14) and
U ( H ) and
D( H ) are proportional to the diagonal quark mass matrices, MU and
MD respectively, at the scale
H , according to eq. (2.21). Working to one loop order
and neglecting higher order terms, it is consistent to set F = a
F F = a
F p
2MF =v (for
F = U; D; E) in the corresponding functions,16
U
D
( H )ij ' a
( H )ij ' a
U
D
ij
ij
p2(MU )jj +
p2(MD)ii +
v
v
4(M2U )2jvj3 log
p
4(M2D2)ivi3 log
p
H
H
k
k
(aE
aU ) 1 + aU (aE)
ij Tr(ME2 )
(aE
aD) 1 + aD(aE)
ij Tr(ME2 )
+(aD
aU ) 1 + aU (aD)
3 ij Tr(M D2)
2 X(M D2)kkKikKjk
+(aU
aD) 1 + aD(aU )
3 ij Tr(MU2 )
2 X(MU2 )kkKkiKkj
16The misalignment contributions exhibited in eqs. (2.74) and (2.75) were computed for the rst time in
ref. [60].
It follows that there is a large hierarchy among the several o diagonal terms of the
matrices,
D( H )ij
D( H )ji
U ( H )ij
U ( H )ji
(MD)ii
(MD)jj
(MU )jj
(MU )ii
1; for i < j ;
1; for i < j:
eq. (2.20) whereas
D is daggered.
The inequality given in eq. (2.76) was previously noted in ref. [36], and provides the
justi cation for ignoring iDj relative to jDi, for i < j.17 This hierarchy of Yukawa couplings
is reversed for iUj . This reversal can be traced back to the fact that U is undaggered in
It is noteworthy that the leading log results for the o diagonal terms of the Q
matrices obtained in eqs. (2.74) and (2.75) and the corresponding full numerical calculation are
typically within a factor of a few. Even for small alignment parameters, there can be some
small discrepancies between the two approaches that can be traced back to the higher order
terms that were neglected in eqs. (2.74) and (2.75). These higher order terms are not
negligible due to the running performed between the electroweak scale and the high energy scale
. The leading log approximation describes less and less accurately the numerical results at
larger and larger alignment parameters. This is shown in gure 1, where the blue points
correspond to the parameter regime in which the leading log approach leads to results within
a factor of 3 of the results obtained numerically for all the elements of the U and
D
matrices. In contrast, the red points correspond to the parameter regime in which the leading
log approximation leads to results quite di erent from what is obtained by the full running.
2.7
A particular type of Minimal Flavor Violation
In the quark sector of the A2HDM, only the two Yukawa coupling matrices
U and
D
break the SU(3)Q
SU(3)U
SU(3)D global avor symmetry of the electroweak Lagrangian
involving quarks. For this reason, our model can be thought in terms of a speci c realization
of a Minimal Flavor Violating (MFV) 2HDM [29]. In particular, in a general 2HDM with
MFV one can write the Yukawa Lagrangian as
LY;MFV = QLYuURH1y + QLYdyDRH1 + QLAuURH2y + QLAydDRH2 + h:c:;
(2.78)
with H1; H2 the two Higgs doublets in the Higgs basis as de ned in section 2 and QL; UR; DR
avor eigenstate quarks. In general, Au; Ad can be expressed by the in nite sum [35]
(2.76)
(2.77)
Au =
Ad =
X
n1;n2;n3
X
n1;n2;n3
n1n2n3 (YdYdy)n1 (YuYuy)n2 (YdYdy)n3 Yu;
u
d
n1n2n3 (YdYdy)n1 (YuYuy)n2 (YdYdy)n3 Yd;
(2.79)
(2.80)
relations U = p
2 u and D = p
2 yd.
with generic O(1) complex coe cients un;id. In order to determine the coe cients un;id in
the A2HDM, we rotate to the quark masseigenstate basis: Yu !
U ; Au !
U ; Yd !
D,
17To make contact with the Higgs basis Yukawa couplings u and d employed by ref. [36], we note the
as computed in our model in the leading logarithmic approximation. The dark purple region is
favored by the measurement of Bs mixing, the purple region by Bd mixing, and the dark pink
(pink) region by the phase (mass di erence) of the Kaon mixing system. D meson mixing does
not give any interesting bound on the parameter space and it is not shown in the
gure. Right
panel: the corresponding bounds from Bs mixing obtained by scanning the parameter space and
using the full RG running. The yellow, red, and green points correspond to a Wilson coe cient of
< 1=3; [1=3; 1]; > 1 relative to the value that yields the present bound from Bs mixing.
Kaon mixing system, and the pink region by the KK mass di erence. D mixing does
not give any interesting bound on the parameter space and is therefore omitted in the
gure. Bs mixing leads to the most stringent bound and it constrains aD to be smaller
than
4:7 at sizable values of aU . Additionally, the bound from the measurement of CP
violation in Kaon mixing (dark pink) is signi cantly more stringent than the bound from
the mass di erence of the Kaon system (in pink). This is due to the fact that the real
and imaginary parts of the Wilson coe cient of the Kaon system have a similar magnitude
(under the assumption that aU and aD are real). In particular, the ratio of the imaginary
and real parts of the Wilson coe cient is directly related to the phase of the CKM matrix:
Im(C4K )=Re(C4K ) = Im((K32)2K321)=Re((K32)2K321). In contrast, the SM Wilson coe cient
has an imaginary part that is much smaller than the real part. Small di erences between
the constraints from CP violation and the mass di erence also exist in the Bs and Bd
systems. In
gure 6, we only show the most constraining bound in each system, i.e. the
mass di erence in Bs mixing and the phase in Bd mixing.
The right panel of gure 6 shows the corresponding results for the Bs mixing system
obtained by scanning the parameter space and using the full RGrunning. The points
in yellow have a Wilson coe cient smaller than 1/3 the present bound on the Wilson
coe cient; in red we present the points with a Wilson coe cient smaller than the present
bound, and
nally in green we present the points that have been already probed by the
measurement of the Bs mixing observables. In the limit of sizable aU & 0:7, we do not nd
points with aD & 4, in rough agreement with the leading log result.
4.2
The Bmeson rare decays Bs;d ! `+` receive contributions from the exchange of the
Higgs bosons H, A and h at treelevel. This is in contrast to the numerical analysis of
ref. [96], where the avor misalignment at the electroweak scale is set to zero. The neutral
Higgs exchange contributions to the leptonic decay amplitude are proportional to m` and
hence are largest in the case of Bs;d !
+
. However, it is more di cult to tag the
decay to jets and leptons at the LHC and Bfactory detectors, as compared to muons. For
this reason, the present LHCb bounds [97], BR(Bs(d) !
are relatively weak as compared to the SM prediction [98],
+
) . 3
At sizable values of tan , the main contributions to Bs;d !
H and A exchange, as they are enhanced by the second power of tan . Furthermore, in
the cos(
) = 0 limit, the light Higgs (h) contribution vanishes at treelevel. For this
reason, we shall focus henceforth on the heavy Higgs contributions that are given by [99],
BR(Bs;d !
BR(Bs;d !
+
+
where BR(Bs;d !
)
)SM
+
' jSs;dj2 + jPs;dj2
1 + ys;d
Re(Ps2;d)
Re(Ss2;d)
jSs;dj2 + jPs;dj2
1
1 + ys;d
; (4.6)
)SM is the SM prediction for the branching ratio extracted from an
untagged rate. In particular, ys = (6:1
0:7)% and yd
0 have to be taken into account
when comparing experimental and theoretical results, and
0:19)
+
(4.5)
are typically due to
The Ci are the Wilson coe cients corresponding to the Lagrangian
with operators
Ss;d
Ps;d
mBs;d (CsS;d
Cs0S;d) s
mBs;d (CsP;d
Cs0P;d)
(Cs1;0d
+
1
C1S0Ms;d
C1S0Ms;d
4m2
m2Bs;d
;
C1S0Ms;d
C100 s;d) :
2m
2m
i
Os(0)S =
Os(0)P =
Ls =
X(CiOi + Ci0Oi0) + h:c: ;
mBs
mBs
mb (sPR(L)b)(``);
mb (sPR(L)b)(` 5`);
O1(00)s = (s
PL(R)b)(`
5`);
and the corresponding ones for the Bd system. In the limit of cos(
coe cients arising from heavy neutral Higgs exchange are given by
) = 0, the Wilson
CsP =
Cs0P =
mb
mBs p3D2 m
(4.7)
(4.8)
(4.9)
(4.10)
(4.11)
(4.12)
(4.13)
(4.14)
butions to the O100 operators.
( )
and the analogous results for the Bd system. There are no treelevel New Physics
contrisin(
sin(
If cos(
) is nonvanishing, then the scalar Wilson coe cients CsS and C0S given
s
in eqs. (4.13) and (4.14) due to H exchange should be changed accordingly, tan
) tan
+ cos(
) and
D
D sin(
contributions arise due to h exchange; the corresponding contributions are obtained from
CsS and Cs0S given in eqs. (4.13) and (4.14) by making the following replacements, tan
)
cos(
) tan , D
D cos(
) and mH ! mh.
The SM Wilson coe cient takes the form [100],
HJEP06(217)
!
!
(4.15)
(4.16)
(4.17)
(4.18)
(4.19)
and the predicted branching ratios are given by
C1S0Ms;d =
4:1
as obtained in [98] with the inclusion of O( em) and O( s2) corrections. These values are
in relatively good agreement with the experimental results. The combination of the LHCb
and the CMS measurements at Run I for the Bs and Bd decays to muon pairs are [101]:
BR(Bs !
+
+
) = (2:8+00::76)
) = (3:9+11::64)
10 9;
and Bd !
Note the much larger uncertainty in the latter decay mode.
The ATLAS collaboration has also reported a Run I search for Bs !
+
, which
yielded BR(Bs !
+
) = (0:9+10::18)
10 9 [102], although this measurement is not yet
competitive with eq. (4.18). Very recently, LHCb reported a new measurement for Bs;d !
+
using Run II data [103]. Their result, BR(Bs !
+
) = (2:8
0:6)
10 9, agrees
very well with the LHCb and CMS combination quoted in eq. (4.18). In contrast, the new
LHCb Bd measurement is closer to the SM prediction, BR(Bd !
In the following, we will compare the predictions of the A2HDM with the LHCb and CMS
combination shown in eqs. (4.18) and (4.19). In the coming years, the two branching ratios
will be measured much more accurately by the LHC. In particular, the Bs and Bd branching
fractions will be measured by each experiment with a precision of
13% and
48% at
+
) = (1:6+10::19) 10 10.
RunIII, improving to 11% and 18%, respectively, at the HLLHC [104].
In gure 7, we show the constraints from the measurement of Bs !
(right panel) as functions of aU and aD, with xed tan
+
(left panel)
= 10, E = +1[see
) = 0, and mA = mH = 400 GeV, based on the leading logarithmic
approximation. The pink shaded region denote the parameter space favored by the CMS
and LHCb combined results at the 2 level, namely
BR(Bs !
BR(Bs !
+
+
)
)SM
)
)SM
[0:8; 6:6]:
(4.20)
21Fixing a di erent sign, E = 1, leads to the same results, with the exchange (aU ; aD) ! ( aU ; aD).
+
(right panel) relative to the SM, as a function of aU and aD, with xed tan
) = 0, and mA = mH = 400 GeV. The regions in pink are allowed at the 2
level by the
present measurements. The purple shaded regions are anticipated by the more precise HLLHC
measurements, assuming a measured central value equal to the SM prediction. The gray shaded
regions produce Landau poles in the Yukawa couplings below MP.
The purple shaded region in gure 7 is the parameter space favored at 2
by the HLLHC
measurement, assuming a measured central value equal to the SM prediction. Comparing
the region in pink to the region in purple, one can get a sense of the improvement the
HLLHC can achieve in testing our model. The expected experimental error at the HLLHC
is comparable to the present theory error. For this reason, an additional improvement
can be achieved via a more precise calculation of the SM prediction for the two branching
ratios, with the bene t of more accurate measurements of the CKM elements that will be
obtained at the LHCb and at Belle II in the coming years.
The present measurement of Bs !
our model. The measurement of Bd !
values of jaDj (cf. the white region where jaDj
+
+
constrains sizable values of aU and aD in
also sets an interesting constraint at smaller
3 and the values of jaU j are sizable),
+
)exp=BR(Bd !
+
)SM
3:7. However, the deviation from the SM prediction is
since the central value of the measurement is larger than the SM prediction: BR(Bd !
not yet statistically signi cant, due to the large experimental uncertainty. Nevertheless, a
sizable suppression of the Bd decay mode is presently disfavored. As expected, the contours
for BR(Bs;d !
+
)=BR(Bs;d !
)SM in the two panels of gure 7 are very similar.
This is due to the fact that our model is a particular type of MFV model in the leading
logarithmic approximation [cf. section 2.7]. In particular, MFV models generically predict
)=BR(Bs !
BR(Bd !
)SM=BR(Bs !
+
)SM, with
corrections arising only from ms=mb and md=mb terms. For this reason, it is di cult in
our model to enhance one decay mode, while suppressing the other.
+
)
+
(right panel) relative to the SM, as a function of M (the mass of the heavy scalar and
pseudoscalar) and aD. We x tan
= 10, aU = 0:2, and cos(
) = 0. The pink regions are the
regions allowed at the 2
level by the present measurements. The purple regions are anticipated
by the more precise HLLHC measurements, assuming a measured central value equal to the SM
prediction. The gray shaded regions produce Landau poles in the Yukawa couplings below MP.
+
(left panel) and for Bd !
+
(right panel)
relative to the SM, obtained via scanning the parameter space and using the full RG running, at
xed tan
) = 0, and mA = mH = 400 GeV. The yellow, red, green and blue points
corresponds to branching ratios normalized to the SM prediction < 0:4; [0:4; 1:1]; [1:1; 10]; > 10.
In boldface we denote the range preferred by the LHCb and ATLAS measurement of Bs !
+
to the Bd;s !
4.3
B !
decays
It is also interesting to investigate the bounds as a function of the heavy Higgs boson
masses. In gure 8, we show the same constraints in the (M; aD) plane, where M
mA =
mH , having xed tan
) = 0. Sizable regions of parameter
space are allowed, even for values of M as small as
300 GeV. Finally, in
gure 9, we
show the results obtained through scanning the parameter space and utilizing the full RG
running. These plots are qualitatively similar to the contour plots of gure 7 obtained in
the leading logarithmic approximation, although the heavy Higgs exchange contributions
decay rates computed using the full RG running are somewhat larger
(at large alignment parameters) than the corresponding leading log results.
HJEP06(217)
The leptonic decays B ! ` are interesting probes of the Higgs sector of our model and
particularly of the charged Higgs couplings, since the charged Higgs boson mediates
treelevel New Physics contributions to these decay modes. The
channel is the only decay
mode of this type observed so far. The present experimental world average is [105]22
and is in relatively good agreement with the SM prediction [106]23
(4.21)
(4.22)
(4.23)
(4.24)
(4.25)
(4.26)
where we have de ned the SM Wilson coe cient CSuMb = 4GF Kub= 2 and CRub(L) are the
ub
Wilson coe cients of the OR(L) = (uPR(L)b)( PL ) operators.
p
In particular [107],
CRub(L) =
1
m2H
p
LR(RL) 2m
ub
v
tan ;
with
LR(RL) the two charged Higgs couplings H+uLbR, H+uRbL given by
ub
This leads to the branching ratio,
)
)SM
m2B
mb
p
v tan
2Kubm2H
X
i
Kui 3Di + Kib i1
U
2
:
22Updated results and plots available at: http://www.slac.stanford.edu/xorg/hfag.
23Updated results and plots available at: http://ckm tter.in2p3.fr.
In our model, the New Physics contribution to this decay reads
BR(B !
BR(B !
)
)SM
= 1 +
m2B
mbm
Cub
L
CSuMb
CRub 2
;
uLbR =
X Kui 3Di ;
i
= 1
uRbL =
X Kib iU1 :
i
panel: leading log predictions, where the pink region is favored by the measurement of B !
The purple region is anticipated by future measurement at Belle II, under the assumption that
.
the central value of the measurement is given by the SM prediction. Right panel: result of the
parameter space scan, using the full RG running. Yellow, red, green and blue points correspond to
the ratios < 0:2; [0:79; 1:71]; [1:71; 3]; > 3, respectively. In boldface we denote the range preferred
by the present world average for BR(B !
).
In the leading logarithmic approximation, the most important contributions come from the
second term of the above expression (/ 3Di ), as one can easily deduce from eqs. (2.74)
and (2.75).
In
gure 10, we show our numerical results as obtained using the leading log
approximation (left panel) and the scan of the parameter space using the full RGEs,
having
xed mH
= 400 GeV and tan
= 10.
A very large region of parameter is still
allowed by the measurement of B !
. In particular, in the leading logarithmic
apment parameter, aU . Indeed, in the pink region shown in the left panel of
proximation, every value jaDj . 17 is allowed, irrespective of the value of the other
aligngure 10,
[0:79; 1:71], consistent with the current measurements.
This is no longer the case when we consider the scan based on the full RGrunning. In this
case, a few points at large values of jaU j are excluded by the measurement of BR(B !
(see the blue points in the right panel of the gure). In the left panel of gure 10, we also
)
exhibit the purple shaded region of parameter space that would be favored by the future
Belle II measurement, under the assumption that the central value of the measurement
is given by the SM prediction for this branching ratio [cf. eq. (4.22)]. The allowed region
of parameter space is expected to shrink considerably, thanks to the anticipated accuracy
of the Belle II measurement with a total error of the order of
5% [108], leading to an
allowed range, BR(B !
)=BR(B !
)SM
improvement in the SM prediction of this B meson decay mode.
[0:86; 1:14], where we have assumed no
We have explored the consequences of avoralignment at a very high energy scale on avor
observables in the two Higgs doublet Model (2HDM). Flavor alignment at the electroweak
scale generically requires an unnatural
netuning of the matrix Yukawa couplings. If
avor alignment is instead imposed at a higher energy scale such as the Planck scale,
perhaps enforced by some new dynamics beyond the SM, then the
avor misalignment
at the electroweak scale due to RG running will generate new sources of FCNCs. The
resulting treelevel Higgsmediated FCNCs are somewhat suppressed and relatively mildly
constrained by experimental measurements of avorchanging observables.
We require that the alignment parameters at the high scale remain perturbative. In
particular, no Landau poles are encountered during RG running. These requirements lead
to an upper bound on the values of the alignment parameters at the Planck scale. This
in turn provide an upper bound on the size of FCNCs generated at the electroweak scale.
The avorchanging observables considered in this paper that provide the most sensitive
probe of the avoraligned 2HDM parameter space are meson mixing and rare B decays
such as Bs;d !
and B !
. We also considered constraints from LHC searches
with Bs !
II" scenario.
bb; +
of heavy Higgs bosons (the most important of which are searches for pp ! b(b)H; H !
), and measurements of the couplings of the observed (SMlike) Higgs boson. The
most stringent constraint on the
avoraligned 2HDM parameter space arises from the
measurement of the rare decay Bs !
+
.
We investigated the predictions of the avoraligned 2HDM in the regions of the
parameter space not yet probed by the measurements listed above. The top rare
avor
changing decays, t ! uh, t ! ch, are generated at treelevel. However, once we impose
constraints from Higgs coupling measurements, the predicted branching ratios for these
neutral avor changing top decays are beyond the LHC reach. Furthermore, the model
predicts a novel phenomenology for the heavy Higgs bosons. In particular, the heavy Higgs
bosons can have a sizable branching ratios into a bottom and a strange quark, and the
ratios, BR(H ! tt) : BR(H ! bb) : BR(H !
the predictions of the more common Type I and II 2HDMs. These features are exhibited
+
), can be very di erent, if compared to
in our summary plots in gures 11 and 12.
tan
cos(
In gure 11, we summarize the constraints on the (aU ; aD) parameter space, with xed
= 10 (upper panels) and tan
= 3 (lower panels). In both panels, we x the values
) = 0 and mA = mH = mH
= 400 GeV. The region favored by all avor
constraints is shown in reddishbrown. At sizable values of aD, the most relevant constraint
comes from the measurement of Bs !
+
(tan region). Bs meson mixing also sets an
interesting bound on the parameter space (bluegray region). It o ers some complementary
, as it does not depend on the particular value of tan . Moreover, it will be
able to probe the small region of parameter space with aU > 0 and sizable values of aD
favored by the measurement of Bs !
+
value in agreement with the SM prediction.24 The measurement of B !
imposes only a
in the case of a future measurement with a central
24We use the results in [109] for the future prospects in measuring Bs mixing, corresponding to the \Stage
HJEP06(217)
computed in the leading log approximation. We x cos(
) = 0, mA = mH = mH
= 400 GeV,
tan
= 10 (upper panels), and tan
= 3 (lower panels).
The contours represent the ratio
BR(H ! bb)m2 =[BR(H !
)3mb2], where 1 is the Type I and Type II 2HDM prediction. The
reddishbrown regions are favored by all avor constraints. The green region is favored by the
measurement of B !
. Bluegray and tan regions are favored by Bs mixing and Bs !
respectively. The gray shaded regions produce Landau poles in the Yukawa couplings below MP.
The left and right panels represent the bounds as they are now and as projected for the coming
+
400 GeV, cos(
) = 0, and tan
= 10.
Blue points correspond to points allowed by the
, but not by the measurement of Bs mixing or Bs !
are allowed by the measurements of B !
and of meson mixing but not by Bs !
. Green points
+
. Red
points are allowed by all constraints. The left and right panels represent the bounds as they are now
+
and as projected for the coming years, as detailed in section 4. In the solid white region, Landau
poles in the Yukawa couplings are produced below MP.
relatively weak constraint on the parameter space (green region). For values of tan
= 10
(or larger), in the region of parameter space favored by present and future avor constraints,
the ratio m2 BR(H ! bb)=3mb2 BR(H !
+
) is smaller than the ratio predicted by Type
I and II 2HDM in most of the Aligned 2HDM parameter space. The parameter space is
somewhat less constrained at lower values of tan , as shown in the lower panels of gure 11.
In
gure 12, we present the corresponding results obtained in the numerical scan
with full RG running, with
xed cos(
) = 0, mA = mH = mH
= 400 GeV, and
tan
= 10. The qualitative features of the leading log approximation continue to hold. In
particular, we again see that Bs !
+
provides the most stringent constraint on the
aligned 2HDM parameter space. Note that in order to emphasize the comparison of the
constraints obtained from the di erent B physics observables in
gures 11 and 12, we do
not include the constraints due to the LHC searches for the heavy Higgs bosons decaying
into fermion pairs in these gures. As shown in
gures 4 and 5 for the heavy Higgs mass
values quoted above, in the region of the Aligned 2HDM parameter space consistent with
no Landau poles below MP, the current LHC limits on H and A production eliminate the
parameter regime with jaDj & 30{40 and jaU j . 0:1.
In considering the phenomenological implications of extended Higgs sectors, the most
conservative approach is to impose only those constraints that are required by the
current experimental data. In most 2HDM studies in the literature, the Yukawa couplings
are assumed to be of Type I, II, X or Y. In this paper, we have argued that the current
experimental data allows for a broader approach in which the Yukawa couplings are
approximately aligned in
avor at the electroweak scale. The resulting phenomenology can
yield some unexpected surprises. We hope that the search strategies of future Higgs studies
at the LHC will be expanded to accommodate the broader phenomenological framework
of the (approximately) avoraligned extended Higgs sector.
Acknowledgments
H.E.H. gratefully acknowledges Paula Tuzon for numerous interactions during her two
month long visit to Santa Cruz in 2010{2011. Her work on the aligned 2HDM provided
inspiration for this work. S.G. thanks Wolfgang Altmannshofer for discussions. H.E.H.
and E.S. are supported in part by the U.S. Department of Energy grant number
DESC0010107. S.G. acknowledges support from the University of Cincinnati. S.G. and H.E.H.
are grateful to the hospitality and the inspiring working atmosphere of both the Kavli
Institute for Theoretical Physics in Santa Barbara, CA, supported in part by the National
Science Foundation under Grant No. NSF PHY1125915, and the Aspen Center for Physics,
supported by the National Science Foundation Grant No. PHY1066293, where some of the
research reported in this work was carried out.
A
Review of the Higgsfermion Yukawa couplings in the Higgs basis
In a general 2HDM, the Higgs fermion interactions are governed by the following interaction
Lagrangian:25
0 0
LY = Q0L ea aU;0 UR0 + QL a( aD;0)yDR0 + EL a( aE;0)y ER0 + h:c: ;
(A.1)
summed over a = a = 1; 2, where
1;2 are the Higgs doublets, ea
i 2 a, Q0L and EL0 are
are vectors in the quark and lepton avor spaces, and aU;0; aD;0; aE;0 are 3
the weak isospin quark and lepton doublets, and UR0 , DR0, ER0 are weak isospin quark and
lepton singlets.26 Here, Q0L, EL0, UR0 , DR0, ER0 denote the interaction basis states, which
3 matrices in
quark and lepton
avor spaces.
Note that a
U;0 appears undaggered in eq. (A.1), whereas the corresponding Yukawa
coupling matrices for downtype fermions (D and E) appear daggered. In this convention,
the transformation of the Yukawa coupling matrices under a scalar eld basis change is
the same for both uptype and downtype fermions. That is, under a change of basis,
a ! Uab b (which implies that ea ! ebUbya), the Yukawa coupling matrices transform as
a ! Uab bF and a
F
F y !
bF yUbya (for F = U , D and E), which re ects the forminvariance
of LY under the basis change.
The neutral Higgs states acquire vacuum expectation values,
(A.2)
(A.3)
where v^av^a = 1 and v = 246 GeV. It is also convenient to de ne
where 12 =
21 = 1 and 11 = 22 = 0.
25We follow the conventions of ref. [39], in which covariance is manifest with respect to U(2)
avor
transformations, a ! Uab b [where U 2 U(2)], by implicitly summing over barred/unbarred index pairs
of the same letter.
26The right and lefthanded fermion elds are de ned as usual: R;L
PR;L , where PR;L
12 (
1
5
).
h 0ai = p ;
vv^a
2
w^b
v^a ab ;
Following refs. [37, 39], we de ne invariant and pseudoinvariant matrix Yukawa
couplings,
where F = U , D or E. Inverting these equations yields
F;0
v^a aF;0 ;
F;0
w^a aF;0 ;
F;0 =
a
F;0v^a + F;0w^a :
Note that under the U(2) transformation, a ! Uab b
,
F;0 is invariant and
F;0
! (det U ) F;0 :
(A.6)
The Higgs elds in the Higgs basis are de ned by H1
v^
a a and H2
w^a a, which
can be inverted to yield
a = H1v^a + H2wa [39]. Rewriting eq. (A.1) in terms of the Higgs
basis elds,
0
LY = Q0L (He1 U;0 + He2 U;0) UR0 + QL (H1 D;0 y + H1 D;0 y) DR0
0
+EL (H1 E;0 y + H1 E;0 y) ER0 + h:c:
The next step is to identify the quark and lepton masseigenstates. This is
accomplished by replacing H1 ! (0 ; v=p2) and performing unitary transformations of the left
and righthanded uptype and downtype fermion multiplets such that the resulting quark
and charged lepton mass matrices are diagonal with nonnegative entries. In more detail,
we de ne:
PLU = VLU PLU 0 ;
PLE = VLEPLE0 ;
PRU = VRU PRU 0 ;
PRE = VRDPRE0 ;
PLD = VLDPLD0 ;
PLN = VLEPLN 0 ;
PRD = VRDPRD0 ;
and the CabibboKobayashiMaskawa (CKM) matrix is de ned as K
for the neutrino elds, we are free to choose V N = VLE since neutrinos are exactly massless
L
in this analysis.27 In particular, the unitary matrices VLF and VRF (for F = U , D and E)
VLU V D y : Note that
L
are chosen such that
v
v
MU = p2 VLU U;0V U y = diag(mu ; mc ; mt) ;
R
MD = p2 VLD D;0 yVRD y = diag(md ; ms ; mb) ;
ME = p2 VLE E;0 yVRE y = diag(me ; m ; m ) :
It is convenient to de ne
U = VLU U;0VRU y ;
U = VLU U;0VRU y ;
D = VRD D;0V D y ;
L
D = VRD D;0V D y ;
L
E = VRD E;0V E y ;
L
E = VRD E;0V E y :
L
27Here we are ignoring the righthanded neutrino sector, which gives mass to neutrinos via the seesaw
mechanism.
(A.4)
(A.5)
(A.7)
(A.8)
(A.9)
(A.10)
(A.11)
(A.12)
(A.13)
which is a physical observable. The matrices
D and
E are independent
pseudoinvariant complex 3
3 matrices. The Higgsfermion interactions given in eq. (A.7) can be
rewritten in terms of the quark and lepton masseigenstates,
LY = U L( U H10 y + U H20 y)UR
+N L( E yH1+ + E yH2+)ER + EL( E yH10 + E yH20)ER + h:c:
(A.16)
B
Renormalization group equations for the Yukawa matrices
We rst write down the renormalization group equations (RGEs) for the Yukawa matrices
16 2 (d=d ) = 16 2(d=dt), the RGEs are given by [32]:
aU;0, a
D;0 and aE;0. De ning D
D a
U;0 =
8gs2 +
4
9 g2 +
17 g0 2
12
D a
D;0 =
8gs2 +
4
9 g2 +
+ 3Tr aU;0( bU;0)y + aD;0( bD;0)y + Tr aE;0( bE;0)y
U;0
b
2( bD;0)y aD;0 U;0 + aU;0( bU;0)y bU;0 + 2 ( bD;0)y bD;0 U;0 +
b a
1
12 bU;0( bU;0)y aU;0 ;
Eq. (A.6) implies that under the U(2) transformation, a ! Uab b
F is invariant and
F
! (det U ) F ;
for F = U , D and E. Indeed, F is invariant since eqs. (A.9){(A.11) imply that
MF = p
2
F ;
U ,
D a
E;0 =
4
9 g2 +
+ bE;0( bE;0)y aE;0 +
12 aE;0( bE;0)y bE;0 :
+ 3Tr ( bD;0)y aD;0 + ( bU;0)y aU;0 + Tr ( bE;0)y aE;0
D;0
b
2 bD;0 aU;0( bU;0)y + bD;0( bD;0)y aD;0 +
2 a
1 D;0 bU;0( bU;0)y +
12 aD;0( bD;0)y bD;0;
E;0 + 3Tr ( bD;0)y aD;0 + ( bU;0)y aU;0 + Tr ( bE;0)y aE;0
a
and 2
F;0 =
The RGEs above are true for any basis choice. Thus, they must also be true in the
Higgs basis in which v^ = (1; 0) and w^ = (0; 1). In this case, we can simply choose 1F;0 =
F;0
F;0 to obtain the RGEs for the
F;0 and
F;0. Alternatively, we can multiply
eqs. (B.1){(B.3) rst by v^a and then by w^a. Expanding ay, which appears on the righthand
sides of eqs. (B.1){(B.3), in terms of y and y using eq. (A.5), we again obtain the RGEs for
the F;0 and F;0. Of course, both methods yield the same result, since the diagonalization
matrices employed in eqs. (A.9){(A.11) are de ned as those that bring the mass matrices
to their diagonal form at the electroweak scale. No scale dependence is assumed in the
diagonalization matrices, and as such they are not a ected by the operators D.
(A.14)
(A.15)
(B.1)
(B.2)
E;0
b
(B.3)
+ U;0( U;0y U;0+ U;0y U;0)+12( D;0y D;0+ D;0y D;0) U;0
+1( U;0 U;0y+ U;0 U;0y) U;0;
+ 3Tr U;0 U;0y+ D;0 D;0y +Tr E;0 E;0y U;0 2 D;0y D;0 U;0+ D;0y D;0 U;0
+ U;0( U;0y U;0+ U;0y U;0)+12( D;0y D;0+ D;0y D;0) U;0
+1( U;0 U;0y+ U;0 U;0y) U;0;
+ 3Tr U;0 U;0y+ D;0 D;0y +Tr E;0 E;0y U;0 2 D;0y D;0 U;0+ D;0y D;0 U;0
+ 3Tr D;0y D;0+ U;0y U;0 +Tr E;0y E;0] D;0 2( D;0 U;0 U;0y
+ D;0 U;0 U;0y)+( D;0 D;0y+ D;0 D;0y) D;0+1 D;0( U;0 U;0y+ U;0 U;0y)
+1 D;0( D;0y D;0+ D;0y D;0);
2
2
2
2
2
2
(B.4)
(B.5)
(B.6)
(B.7)
(B.8)
(B.9)
+( D;0 D;0y+ D;0 D;0y) D;0+1 D;0( U;0 U;0y+ U;0 U;0y)
+1 D;0( D;0y D;0+ D;0y D;0);
+ 3Tr D;0y D;0+ U;0y U;0 +Tr E;0y E;0 E;0
+( E;0 E;0y+ E;0 E;0y) E;0+1 E;0( E;0y E;0+ E;0y E;0);
+ 3Tr D;0y D;0+ U;0y U;0 +Tr E;0y E;0] D;0 2( D;0 U;0 U;0y+ D;0 U;0 U;0y)
+ 3Tr D;0y D;0+ U;0y U;0 +Tr E;0y E;0 E;0
4
4
4
4
D
U =
D
U =
D
D =
D
D =
D
E =
D
E =
8gs2+ 49 g2+
5 g02
9 g2+ 15 g02
4
9 g2+ 15 g02
4
4
2
2
2
2
E
E
U
U
(B.10)
(B.11)
(B.12)
(B.15)
(B.16)
(B.17)
Using eqs. (A.12) and (A.13), we immediately obtain the RGEs for the F and F .
Schematically, we shall write,
D
F =
F ;
D F =
F ;
for F = U , D and E. Explicitly, the corresponding functions at oneloop order are given
by,
U + 3Tr U Uy+ D Dy +Tr E Ey
+ 3Tr U Uy+ D Dy +Tr E Ey
U 2K
Dy DKy U + Dy DKy U
+ U ( Uy U + Uy U )+ 1 K( Dy D+ Dy D)Ky U + 1 ( U Uy+ U Uy) U ;
U + 3Tr U Uy+ D Dy +Tr E Ey
+ 3Tr U Uy+ D Dy +Tr E Ey
U 2K
Dy DKy U + Dy DKy U
+ U ( Uy U + Uy U )+ 1 K( Dy D+ Dy D)Ky U + 1 ( U Uy+ U Uy) U ;
+ 3Tr Dy D+ Uy U +Tr Ey E]
D 2( DKy U Uy+ DKy U Uy)K
+ 3Tr Dy D+ Uy U +Tr Ey E]
D 2( DKy U Uy+ DKy U Uy)K
+( D Dy+ D Dy) D+ 1 DKy( U Uy+ U Uy)K+
E+ 3Tr Dy D+ Uy U +Tr Ey E
D+ 3Tr Dy D+ Uy U +Tr Ey E]
D
(B.13)
D+ 3Tr Dy D+ Uy U +Tr Ey E]
D
(B.14)
2
2
2
2
2
2
For the numerical analysis of the RGEs, it is convenient to de ne
D
DKy ;
D
e
DKy ;
+ 3Tr Dy D+ Uy U +Tr Ey E
+ 3Tr Dy D+ Uy U +Tr Ey E
E+ 3Tr Dy D+ Uy U +Tr Ey E
keeping in mind that the (unitary) CKM matrix K is de ned at the electroweak scale and
thus is not taken to be a running quantity. The RGEs given in eqs. (B.11){(B.16) can now
be rewritten by taking
D
! eD, D
! e
D and K ! 1. The advantage of the RGEs
written in this latter form is that the CKM matrix K no longer appears explicitly in the di
erential equations, and enters only in the initial condition of D at the low scale [cf. eq. (2.62)],
In particular, the high scale boundary condition given by eq. (2.63) also applies to
eD, i.e.,
Open Access.
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
D
p
( H ) =
2MD( H )Ky=v :
eD( ) = a
D D
( ) :
(B.18)
D and
(B.19)
HJEP06(217)
LHC pp collision data at p
[1] ATLAS collaboration, Observation of a new particle in the search for the Standard Model
Higgs boson with the ATLAS detector at the LHC, Phys. Lett. B 716 (2012) 1
[arXiv:1207.7214] [INSPIRE].
[2] CMS collaboration, Observation of a new boson at a mass of 125 GeV with the CMS
experiment at the LHC, Phys. Lett. B 716 (2012) 30 [arXiv:1207.7235] [INSPIRE].
[3] ATLAS and CMS collaborations, Measurements of the Higgs boson production and decay
rates and constraints on its couplings from a combined ATLAS and CMS analysis of the
s = 7 and 8 TeV, JHEP 08 (2016) 045 [arXiv:1606.02266]
[4] P. Fayet, Supergauge invariant extension of the Higgs mechanism and a model for the
electron and its neutrino, Nucl. Phys. B 90 (1975) 104 [INSPIRE].
[5] K. Inoue, A. Kakuto, H. Komatsu and S. Takeshita, Lowenergy parameters and particle
masses in a supersymmetric grand uni ed model, Prog. Theor. Phys. 67 (1982) 1889
[6] R.A. Flores and M. Sher, Higgs masses in the standard, multiHiggs and supersymmetric
models, Annals Phys. 148 (1983) 95 [INSPIRE].
(1986) 1 [Erratum ibid. B 402 (1993) 567] [INSPIRE].
JHEP 11 (2006) 038 [hepph/0605242] [INSPIRE].
[7] J.F. Gunion and H.E. Haber, Higgs bosons in supersymmetric models. 1, Nucl. Phys. B 272
[8] L. Fromme, S.J. Huber and M. Seniuch, Baryogenesis in the twoHiggs doublet model,
[9] F. Bezrukov, M. Yu. Kalmykov, B.A. Kniehl and M. Shaposhnikov, Higgs boson mass and
new physics, JHEP 10 (2012) 140 [arXiv:1205.2893] [INSPIRE].
[10] G. Degrassi et al., Higgs mass and vacuum stability in the Standard Model at NNLO, JHEP
08 (2012) 098 [arXiv:1205.6497] [INSPIRE].
089 [arXiv:1307.3536] [INSPIRE].
[11] D. Buttazzo et al., Investigating the nearcriticality of the Higgs boson, JHEP 12 (2013)
[12] J. EliasMiro, J.R. Espinosa, G.F. Giudice, H.M. Lee and A. Strumia, Stabilization of the
electroweak vacuum by a scalar threshold e ect, JHEP 06 (2012) 031 [arXiv:1203.0237]
[13] O. Lebedev, On stability of the electroweak vacuum and the Higgs portal, Eur. Phys. J. C
72 (2012) 2058 [arXiv:1203.0156] [INSPIRE].
[14] G.M. Pruna and T. Robens, Higgs singlet extension parameter space in the light of the LHC
discovery, Phys. Rev. D 88 (2013) 115012 [arXiv:1303.1150] [INSPIRE].
doublet scenario: a study including LHC data, JHEP 12 (2014) 166 [arXiv:1407.2145]
[17] D. Das and I. Saha, Search for a stable alignment limit in twoHiggsdoublet models, Phys.
Rev. D 91 (2015) 095024 [arXiv:1503.02135] [INSPIRE].
[18] P. Ferreira, H.E. Haber and E. Santos, Preserving the validity of the twoHiggs doublet
model up to the Planck scale, Phys. Rev. D 92 (2015) 033003 [Erratum ibid. D 94 (2016)
059903] [arXiv:1505.04001] [INSPIRE].
[19] D. Chowdhury and O. Eberhardt, Global ts of the twoloop renormalized twoHiggsdoublet
model with soft Z2 breaking, JHEP 11 (2015) 052 [arXiv:1503.08216] [INSPIRE].
[20] G.C. Branco, P.M. Ferreira, L. Lavoura, M.N. Rebelo, M. Sher and J.P. Silva, Theory and
phenomenology of twoHiggsdoublet models, Phys. Rept. 516 (2012) 1 [arXiv:1106.0034]
[21] M. Misiak and M. Steinhauser, Weak radiative decays of the B meson and bounds on MH
in the twoHiggsdoublet model, Eur. Phys. J. C 77 (2017) 201 [arXiv:1702.04571]
[22] S.L. Glashow and S. Weinberg, Natural conservation laws for neutral currents, Phys. Rev.
D 15 (1977) 1958 [INSPIRE].
397 [INSPIRE].
[23] E.A. Paschos, Diagonal neutral currents, Phys. Rev. D 15 (1977) 1966 [INSPIRE].
[24] H.E. Haber, G.L. Kane and T. Sterling, The fermion mass scale and possible e ects of
Higgs bosons on experimental observables, Nucl. Phys. B 161 (1979) 493 [INSPIRE].
[25] L.J. Hall and M.B. Wise, Flavor changing Higgsboson couplings, Nucl. Phys. B 187 (1981)
[26] J.F. Donoghue and L.F. Li, Properties of charged Higgs bosons, Phys. Rev. D 19 (1979) 945
[27] V.D. Barger, J.L. Hewett and R.J.N. Phillips, New constraints on the charged Higgs sector
in two Higgs doublet models, Phys. Rev. D 41 (1990) 3421 [INSPIRE].
[28] M. Aoki, S. Kanemura, K. Tsumura and K. Yagyu, Models of Yukawa interaction in the
two Higgs doublet model and their collider phenomenology, Phys. Rev. D 80 (2009) 015017
[arXiv:0902.4665] [INSPIRE].
conservation vs. minimal avour violation, JHEP 10 (2010) 009 [arXiv:1005.5310]
[30] A. Pich and P. Tuzon, Yukawa alignment in the twoHiggsdoublet model, Phys. Rev. D 80
(2009) 091702 [arXiv:0908.1554] [INSPIRE].
quasialignment with two Higgs doublets and RGE stability, Eur. Phys. J. C 75 (2015) 286
[32] P.M. Ferreira, L. Lavoura and J.P. Silva, Renormalizationgroup constraints on Yukawa
alignment in multiHiggsdoublet models, Phys. Lett. B 688 (2010) 341 [arXiv:1001.2561]
[33] S. Knapen and D.J. Robinson, Disentangling mass and mixing hierarchies, Phys. Rev. Lett.
115 (2015) 161803 [arXiv:1507.00009] [INSPIRE].
[34] F.J. Botella, G.C. Branco, M. Nebot and M.N. Rebelo, Flavour changing Higgs couplings in
a class of two Higgs doublet models, Eur. Phys. J. C 76 (2016) 161 [arXiv:1508.05101]
[35] G. D'Ambrosio, G.F. Giudice, G. Isidori and A. Strumia, Minimal avor violation: an
e ective eld theory approach, Nucl. Phys. B 645 (2002) 155 [hepph/0207036] [INSPIRE].
[36] C.B. Braeuninger, A. Ibarra and C. Simonetto, Radiatively induced avour violation in the
general twoHiggs doublet model with Yukawa alignment, Phys. Lett. B 692 (2010) 189
[arXiv:1005.5706] [INSPIRE].
U.K., (1999) [INSPIRE].
[37] S. Davidson and H.E. Haber, Basisindependent methods for the twoHiggsdoublet model,
Phys. Rev. D 72 (2005) 035004 [Erratum ibid. D 72 (2005) 099902] [hepph/0504050]
[38] G.C. Branco, L. Lavoura and J.P. Silva, CP violation, Oxford University Press, Oxford
[39] H.E. Haber and D. O'Neil, Basisindependent methods for the twoHiggsdoublet model II.
The signi cance of tan , Phys. Rev. D 74 (2006) 015018 [Erratum ibid. D 74 (2006)
059905] [hepph/0602242] [INSPIRE].
[40] H.E. Haber and D. O'Neil, Basisindependent methods for the twoHiggsdoublet model III.
The CPconserving limit, custodial symmetry and the oblique parameters S, T , U , Phys.
Rev. D 83 (2011) 055017 [arXiv:1011.6188] [INSPIRE].
421 [INSPIRE].
! e at a rate of one out of 109 muon decays?, Phys. Lett. B 67 (1977)
1103 [INSPIRE].
Rev. Lett. 44 (1980) 912 [INSPIRE].
[42] M. GellMann, P. Ramond and R. Slansky, Complex spinors and uni ed theories, Conf.
Proc. C 790927 (1979) 315 [arXiv:1306.4669] [INSPIRE].
[43] T. Yanagida, Horizontal symmetry and masses of neutrinos, Prog. Theor. Phys. 64 (1980)
[44] R.N. Mohapatra and G. Senjanovic, Neutrino mass and spontaneous parity violation, Phys.
[45] R.N. Mohapatra and G. Senjanovic, Neutrino masses and mixings in gauge models with
spontaneous parity violation, Phys. Rev. D 23 (1981) 165 [INSPIRE].
[47] N. Craig, J. Galloway and S. Thomas, Searching for signs of the second Higgs doublet,
boson: alignment without decoupling, JHEP 04 (2014) 015 [arXiv:1310.2248] [INSPIRE].
[50] H.E. Haber, The Higgs data and the decoupling limit, in Proceedings, 1st Toyama
International Workshop on Higgs as a Probe of New Physics 2013 (HPNP2013), Toyama
[51] P.S. Bhupal Dev and A. Pilaftsis, Maximally symmetric two Higgs doublet model with
natural Standard Model alignment, JHEP 12 (2014) 024 [Erratum ibid. 11 (2015) 147]
[52] A. Pilaftsis, Symmetries for Standard Model alignment in multiHiggs doublet models, Phys.
[arXiv:1408.3405] [INSPIRE].
Rev. D 93 (2016) 075012 [arXiv:1602.02017] [INSPIRE].
(1990) 363 [INSPIRE].
[53] H.E. Haber and Y. Nir, Multiscalar models with a highenergy scale, Nucl. Phys. B 335
[54] J.F. Gunion and H.E. Haber, Conditions for CPviolation in the general twoHiggsdoublet
model, Phys. Rev. D 72 (2005) 095002 [hepph/0506227] [INSPIRE].
[55] K.G. Chetyrkin, J.H. Kuhn and M. Steinhauser, RunDec: a Mathematica package for
running and decoupling of the strong coupling and quark masses, Comput. Phys. Commun.
133 (2000) 43 [hepph/0004189] [INSPIRE].
[56] F. Herren and M. Steinhauser, Version 3 of RunDec and CRunDec, arXiv:1703.03751
Chin. Phys. C 40 (2016) 100001 [INSPIRE].
[57] Particle Data Group collaboration, C. Patrignani et al., Review of particle physics,
[58] P. Marquard, A.V. Smirnov, V.A. Smirnov and M. Steinhauser, Quark mass relations to
fourloop order in perturbative QCD, Phys. Rev. Lett. 114 (2015) 142002
[arXiv:1502.01030] [INSPIRE].
[59] J. Bijnens, J. Lu and J. Rathsman, Constraining general two Higgs doublet models by the
evolution of Yukawa couplings, JHEP 05 (2012) 118 [arXiv:1111.5760] [INSPIRE].
[60] M. Jung, A. Pich and P. Tuzon, ChargedHiggs phenomenology in the aligned
twoHiggsdoublet model, JHEP 11 (2010) 003 [arXiv:1006.0470] [INSPIRE].
[61] ATLAS collaboration, Constraints on new phenomena via Higgs boson couplings and
invisible decays with the ATLAS detector, JHEP 11 (2015) 206 [arXiv:1509.00672]
[62] CMS collaboration, Summary results of high mass BSM Higgs searches using CMS runI
data, CMSPASHIG16007, CERN, Geneva Switzerland, (2016).
[63] A. Arhrib, Higgs bosons decay into bottomstrange in two Higgs doublets models, Phys. Lett.
B 612 (2005) 263 [hepph/0409218] [INSPIRE].
[64] G. Abbas, A. Celis, X.Q. Li, J. Lu and A. Pich, Flavourchanging top decays in the aligned
twoHiggsdoublet model, JHEP 06 (2015) 005 [arXiv:1503.06423] [INSPIRE].
[65] A. Greljo, J.F. Kamenik and J. Kopp, Disentangling avor violation in the topHiggs sector
at the LHC, JHEP 07 (2014) 046 [arXiv:1404.1278] [INSPIRE].
[66] A. Arhrib, Top and Higgs avor changing neutral couplings in two Higgs doublets model,
Phys. Rev. D 72 (2005) 075016 [hepph/0510107] [INSPIRE].
[67] G. Eilam, J.L. Hewett and A. Soni, Rare decays of the top quark in the standard and two
Higgs doublet models, Phys. Rev. D 44 (1991) 1473 [INSPIRE].
[68] B. Mele, S. Petrarca and A. Soddu, A new evaluation of the t ! cH decay width in the
Standard Model, Phys. Lett. B 435 (1998) 401 [hepph/9805498] [INSPIRE].
[69] J.A. AguilarSaavedra, Top avorchanging neutral interactions: theoretical expectations
and experimental detection, Acta Phys. Polon. B 35 (2004) 2695 [hepph/0409342]
in pp collisions at p
[arXiv:1509.06047] [INSPIRE].
Higgs decays to bb pairs at p
[70] C. Zhang and F. Maltoni, Topquark decay into Higgs boson and a light quark at
nexttoleading order in QCD, Phys. Rev. D 88 (2013) 054005 [arXiv:1305.7386]
[71] ATLAS collaboration, Search for avourchanging neutral current top quark decays t ! Hq
s = 8 TeV with the ATLAS detector, JHEP 12 (2015) 061
[72] CMS collaboration, Search for the avorchanging neutral current decay t ! qH where the
s = 8 TeV, CMSPASTOP14020, CERN, Geneva
p
[73] CMS collaboration, Search for top quark decays t ! qH with H !
in pp collisions at
s = 8 TeV, CMSPASTOP14019, CERN, Geneva Switzerland, (2014).
[74] S. Gori, Three lectures of avor and CPviolation within and beyond the Standard Model, in
2015 European School of HighEnergy Physics (ESHEP 2015), Bansko Bulgaria, 2{15
September 2015 [arXiv:1610.02629] [INSPIRE].
[75] Top Quark Working Group collaboration, K. Agashe et al., Working group report: top
quark, in Community Summer Study 2013: Snowmass on the Mississippi (CSS2013),
Minneapolis MN U.S.A., 29 July{6 August 2013 [arXiv:1311.2028] [INSPIRE].
[76] M. Selvaggi, Perspectives for top quark physics at highluminosity LHC, PoS(TOP2015)054
[arXiv:1512.04807] [INSPIRE].
arXiv:1607.01831 [INSPIRE].
[77] M.L. Mangano et al., Physics at a 100 TeV pp collider: Standard Model processes,
[78] W. Altmannshofer, S. Gori and G.D. Kribs, A minimal avor violating 2HDM at the LHC,
Phys. Rev. D 86 (2012) 115009 [arXiv:1210.2465] [INSPIRE].
quarks, JHEP 11 (2015) 071 [arXiv:1506.08329] [INSPIRE].
[79] CMS collaboration, Search for neutral MSSM Higgs bosons decaying into a pair of bottom
[80] ATLAS collaboration, Search for minimal supersymmetric Standard Model Higgs bosons
H=A in the
nal state in up to 13:3 fb 1 of pp collisions at p
s = 13 TeV with the
ATLAS detector, ATLASCONF2016085, CERN, Geneva Switzerland, (2016).
13 fb 1 of pp collisions at p
Switzerland, (2016).
pp collisions at p
Geneva Switzerland, (2016).
[81] CMS collaboration, Search for a neutral MSSM Higgs boson decaying into
CMSPASHIG16006, CERN, Geneva Switzerland, (2016).
[82] CMS collaboration, Search for a highmass resonance decaying into a dilepton nal state in
s = 13 TeV, CMSPASEXO16031, CERN, Geneva
protonproton collisions at p
s = 13 TeV with the ATLAS detector,
ATLASCONF2016045, CERN, Geneva Switzerland, (2016).
[83] ATLAS collaboration, Search for new highmass resonances in the dilepton nal state using
[84] ATLAS collaboration, Search for heavy Higgs bosons A=H decaying to a topquark pair in
s = 8 TeV with the ATLAS detector, ATLASCONF2016073, CERN,
[85] M. Carena, S. Gori, A. Juste, A. Menon, C.E.M. Wagner and L.T. Wang, LHC discovery
potential for nonstandard Higgs bosons in the 3b channel, JHEP 07 (2012) 091
[arXiv:1203.1041] [INSPIRE].
[86] BaBar collaboration, J.P. Lees et al., Evidence for an excess of B ! D( )
Phys. Rev. Lett. 109 (2012) 101802 [arXiv:1205.5442] [INSPIRE].
[87] BaBar collaboration, J.P. Lees et al., Measurement of an excess of B ! D( )
and implications for charged Higgs bosons, Phys. Rev. D 88 (2013) 072012
decays,
[arXiv:1303.0571] [INSPIRE].
[88] LHCb collaboration, Measurement of the ratio of branching fractions
B(B0 ! D +
)=B(B0 ! D +
ibid. 115 (2015) 159901] [arXiv:1506.08614] [INSPIRE].
), Phys. Rev. Lett. 115 (2015) 111803 [Addendum
[89] Belle collaboration, M. Huschle et al., Measurement of the branching ratio of
B ! D( )
relative to B ! D( )`
D 92 (2015) 072014 [arXiv:1507.03233] [INSPIRE].
` decays with hadronic tagging at Belle, Phys. Rev.
[90] Belle collaboration, A. Abdesselam et al., Measurement of the branching ratio of
B0 ! D +
relative to B0 ! D +` ` decays with a semileptonic tagging method,
arXiv:1603.06711 [INSPIRE].
[91] A. Abdesselam et al., Measurement of the lepton polarization in the decay B ! D
,
arXiv:1608.06391 [INSPIRE].
(2015) 054018 [arXiv:1506.08896] [INSPIRE].
[92] M. Freytsis, Z. Ligeti and J.T. Ruderman, Flavor models for B ! D( ) , Phys. Rev. D 92
[93] F. Mahmoudi and O. Stal, Flavor constraints on the twoHiggsdoublet model with general
Yukawa couplings, Phys. Rev. D 81 (2010) 035016 [arXiv:0907.1791] [INSPIRE].
[94] Quark Flavor Physics Working Group collaboration, J.N. Butler et al., Working
group report: quark avor physics, arXiv:1311.1076 [INSPIRE].
[95] A. Bevan et al., Standard Model updates and new physics analysis with the unitarity
triangle t, arXiv:1411.7233 [INSPIRE].
[96] X.Q. Li, J. Lu and A. Pich, Bs0;d ! `+`
JHEP 06 (2014) 022 [arXiv:1404.5865] [INSPIRE].
decays in the aligned twoHiggsdoublet model,
[97] L. Martini, Search for new physics in the B meson decays: B(0s) !
, Nuovo Cim. C
39 (2016) 231 [INSPIRE].
Bs;d ! `+` in the Standard Model with reduced theoretical uncertainty, Phys. Rev. Lett.
112 (2014) 101801 [arXiv:1311.0903] [INSPIRE].
(2012) 121 [arXiv:1206.0273] [INSPIRE].
Symmetries and asymmetries of B ! K
JHEP 01 (2009) 019 [arXiv:0811.1214] [INSPIRE].
decays in the Standard Model and beyond,
decay from the
combined analysis of CMS and LHCb data, Nature 522 (2015) 68 [arXiv:1411.4413]
collected during the LHC run 1 with the ATLAS detector, Eur. Phys. J. C 76 (2016) 513
branching fraction and e ective
decays, Phys. Rev. Lett. 118 (2017) 191801
CERNLHCC2015010, CERN, Geneva Switzerland, (2015).
bhadron, chadron and lepton properties as of summer 2014, arXiv:1412.7515 [INSPIRE].
[hepph/0406184] [INSPIRE].
at Belle and Belle II,
PoS(ICHEP2016)131 [arXiv:1701.02288] [INSPIRE].
[arXiv:1309.2293] [INSPIRE].
[15] R. Costa , A.P. Morais , M.O.P. Sampaio and R. Santos , Twoloop stability of a complex singlet extended Standard Model , Phys. Rev. D 92 ( 2015 ) 025024 [arXiv: 1411 .4048] [16] N. Chakrabarty , U.K. Dey and B. Mukhopadhyaya , Highscale validity of a two Higgs [31] F.J. Botella , G.C. Branco , A.M. Coutinho , M.N. Rebelo and J.I. SilvaMarcos, Natural the decoupling limit , Phys. Rev. D 67 ( 2003 ) 075019 [ hep ph/0207010] [INSPIRE].
[48] D.M. Asner et al., ILC Higgs white paper , in Proceedings, Community Summer Study 2013 : Snowmass on the Mississippi (CSS2013), Minneapolis MN U.S.A ., 29 July{6 August 2013 [49] M. Carena , I. Low, N.R. Shah and C.E.M. Wagner , Impersonating the Standard Model Higgs [98] C. Bobeth , M. Gorbahn , T. Hermann, M. Misiak , E. Stamou and M. Steinhauser , [99] W. Altmannshofer and D.M. Straub , Cornering new physics in b ! s transitions , JHEP 08 [100] W. Altmannshofer , P. Ball , A. Bharucha , A.J. Buras , D.M. Straub and M. Wick , [107] A. Crivellin , A. Kokulu and C. Greub , Flavorphenomenology of twoHiggsdoublet models