Lecture III
Higgs Bosons in Supersymmetric Models
Outline
• The significance of the TeV-scale—Part 2
• The MSSM Higgs sector at tree-level
• Saving the MSSM Higgs sector—the impact of radiative corrections
• LHC searches for the heavy Higgs states of the MSSM
• Alignment without decoupling in the MSSM?
• Beyond the MSSM Higgs sector
The significance of the TeV scale—Part 2
Despite the tremendous success of the Standard Model (SM) of particle
physics, we know that there are some missing pieces.
• Neutrinos are not massless—perhaps suggestive of a new high energy
(see-saw) scale.
• Dark matter is not accounted for.
• There is no explanation for the baryon asymmetry of the universe.
• There is no explanation for the inflationary period of the very early universe.
• The gravitational interaction is omitted.
Consequently, the SM is an low-energy effective theory. New high energy
scales must exist where more fundamental physics resides.
Indeed, quantum gravitational effects are likely to be relevant only at the
Planck scale,
MPL = (c~/GN)1/2 ≃ 1019 GeV ,
which arises as follows. Consider the gravitational potential energy, GNM2/r,
of a particle of mass M evaluated at its Compton wavelength, r = ~/(Mc).
If this energy is of order the rest mass, Mc2 (i.e., when M ∼ MPL), then
pair production is possible and quantum gravitational effects can no longer
be ignored.
Denote scale at which new physics enters by Λ. Predictions made by the SM
depend on a number of parameters that must be taken as input to the theory.
These parameters are sensitive to ultraviolet physics, and since the physics at
very high energies is not known, one cannot predict their values.
However, one can determine the sensitivity of these parameters to the ultra-
violet scale (which one can take as the Planck scale or some other high energy
scale at which new physics beyond the Standard Model enters).
In the 1930s, it was already appreciated that a critical difference exists
between bosons and fermions. Fermion masses are logarithmically sensitive to
ultraviolet physics. Ultimately, this is due to the chiral symmetry of massless
fermions, which implies that
δmf ∼ mf ln(Λ2/m2
f) .
No such symmetry exists for bosons (in the absence of supersymmetry), and
consequently we expect quadratic sensitivity of the boson squared-mass to
ultraviolet physics, δm2B ∼ Λ2 .
In the SM, m2h = λv2 and m2
W = 14g
2v2 imply that
m2h
m2W
=4λ
g2,
which one would expect to be roughly of O(1). A 125 GeV Higgs boson
satisfies this expectation.
However, the Higgs boson is a consequence of a spontaneously broken scalar
potential,
V (Φ) = −µ2(Φ†Φ) + 12λ(Φ
†Φ)2 ,
where we identify µ2 = 12λv
2 in terms of the vacuum expectation value v of
the Higgs field. The parameter µ2 is quadratically sensitive to Λ. Hence, to
obtain v = 246 GeV in a theory where v ≪ Λ requires a significant fine-tuning
of the ultraviolet parameters of the fundamental theory.
Indeed, the one-loop contribution to the squared-mass parameter µ2 would
be expected to be of order (g2/16π2)Λ2. Setting this quantity to be of order
of v2 (to avoid an unnatural fine-tuning of the tree-level parameter and the
loop contribution) yields
Λ ≃ 4πv/g ∼ O(1 TeV)
A natural theory of electroweak symmetry breaking (EWSB) would seem to
require new physics at the TeV scale to govern the EWSB dynamics.
The Principle of Naturalness
In 1939, Weisskopf announces in the abstract to this paper that “the self-energy of charged particles obeying Bose statistics is found to be quadratically divergent”….
…. and concludes that in theories of elementary bosons, new phenomena must enter at an energy scale of order m/e (e is the relevant coupling)—the first application of naturalness.
Principle of naturalness in modern times
How can we understand the magnitude of the EWSB scale? In the absence of
new physics beyond the SM, its natural value would be the Planck scale (or
perhaps the GUT scale or seesaw scale that controls neutrino masses). The
alternatives are:
• Naturalness is restored by a symmetry principle—supersymmetry—which
ties the bosons to the more well-behaved fermions.
• The Higgs boson is an approximate Goldstone boson—the only other known
mechanism for keeping an elementary scalar light.
• The Higgs boson is a composite scalar, with an inverse length of order the
TeV-scale.
• The naturalness principle does not apply. The Higgs boson is very
unlikely, but unnatural choices for the EWSB parameters arise from other
considerations (landscape?).
Avoiding quadratic sensitivity to Λ with elementary scalars
A lesson from history
The electron self-energy in classical E&M goes like e2/a (a → 0), i.e., it
is linearly divergent. In quantum theory, fluctuations of the electromagnetic
fields (in the “single electron theory”) generate a quadratic divergence. If
these divergences are not canceled, one would expect QED to break down at
an energy of order me/e far below the Planck scale.
The linear and quadratic divergences will cancel exactly if one makes a bold
hypothesis: the existence of the positron (with a mass equal to that of the
electron but of opposite charge).
Weisskopf was the first to demonstrate this cancellation in 1934. . .well,
actually he initially got it wrong, but thanks to Furry, the correct result was
presented in an erratum.
A remarkable result:
�b��a�
e�
�
e�
�
e�
e�
e�
The linear and quadratic divergences of a
quantum theory of elementary fermions are
precisely canceled if one doubles the particle
spectrum—for every fermion, introduce an
anti-fermion partner of the same mass and
opposite charge.
In the process, we have introduced a new CPT-symmetry
that associates a fermion with its anti-particle and guarantees
the equality of their masses.
TeV-scale supersymmetry to the rescue
Take the SM and double the particle spectrum. Introduce supersymmetry
(SUSY), which dictates that for every boson, there is a fermion superpartner
of equal mass and vice versa.
Supersymmetry relates the self-energy of the boson to the self-energy of
its fermionic partner. Since the latter is only logarithmically sensitive to
Λ, we conclude that the quadratic sensitivity of the boson squared-mass to
ultraviolet physics must exactly cancel. Naturalness is restored!
Since no superpartners exist that are degenerate in mass with the
corresponding SM particle, SUSY must be a broken symmetry. Conclusion:
The effective scale of SUSY-breaking cannot be much larger
than of order 1 TeV, if SUSY is responsible for the EWSB scale.
The minimal supersymmetric extension of the SM
The minimal supersymmetric extension of the Standard Model (MSSM)
consists of the fields of the two-Higgs-doublet extension of the Standard
Model and the corresponding superpartners. A particle and its superpartner
together form a supermultiplet. The corresponding field content of the
supermultiplets of the MSSM and their gauge quantum numbers are shown
in the following table.
The enlarged Higgs sector of the MSSM constitutes the minimal structure
needed to guarantee the cancellation of anomalies from the introduction
of the higgsino superpartners. Moreover, without a second Higgs doublet,
one cannot generate mass for both “up”-type and “down”-type quarks (and
charged leptons) in a way consistent with a holomorphic superpotential.
To account for supersymmetry (SUSY) breaking, one adds the most general
set of soft-SUSY-breaking terms consistent with the SM gauge symmetry.
Field content of the MSSM
Super- Super- Bosonic Fermionic
multiplets field fields partners SU(3) SU(2) U(1)
gluon/gluino V8 g g 8 1 0
gauge/ V W± , W 0 W± , W 0 1 3 0
gaugino V ′ B B 1 1 0
slepton/ L (νL, e−L) (ν, e−)L 1 2 −1
lepton Ec e+R ecL 1 1 2
squark/ Q (uL, dL) (u, d)L 3 2 1/3
quark Uc u∗R ucL 3 1 −4/3
Dc d∗R dcL 3 1 2/3
Higgs/ Hd (H0d , H
−d ) (H0
d, H−d ) 1 2 −1
higgsino Hu (H+u , H
0u) (H+
u , H0u) 1 2 1
The fields of the MSSM and their SU(3)×SU(2)×U(1) quantum numbers are listed. The electric charge is
given in terms of the third component of the weak isospin T3 and U(1) hypercharge Y by Q = T3 + 12Y .
For simplicity, only one generation of quarks and leptons is exhibited. For each lepton, quark, and Higgs super-
multiplet, there is a corresponding anti-particle multiplet of charge-conjugated fermions and their associated
scalar partners. The L andR subscripts of the squark and slepton fields indicate the helicity of the corresponding
fermionic superpartners.
More on anomaly cancellation
One-loop VVA and AAA triangle diagrams with three external gauge bosons,
with fermions running around the loop, can contribute to a gauge anomaly.
(Here V refers to a γµ vertex and A refers to a γµγ5 vertex.) A theory that
possesses gauge anomalies is inconsistent as a quantum theory.
To cancel the gauge anomalies, we must satisfy certain group theoretical
constraints.
W iW jB triangle ⇐⇒ Tr(T 23Y ) = 0 ,
BBB triangle ⇐⇒ Tr(Y 3) = 0 .
Example: contributions of the fermions to Tr(Y 3)
Tr(Y 3)SM = 3(
127 +
127 − 64
27 +827
)− 1− 1 + 8 = 0 .
Suppose we only add the higgsinos (H+u , H
0u). The resulting anomaly factor
is Tr(Y 3) = Tr(Y 3)SM + 2, leading to a gauge anomaly. This anomaly is
canceled by adding a second higgsino doublet with opposite hypercharge.
A connection between SUSY-breaking and EWSB
Suppose that ΛSSB is the energy scale of a fundamental theory of SUSY-
breaking. The soft-SUSY-breaking parameters of the MSSM are set at ΛSSB,
and these parameters evolve via RG running down to the EWSB scale. This
provides an elegant mechanism of radiatively-generated EWSB.
Hu
Hd
B
lR
W
lL
tR
tL
qR
qL
g
~
~
~
~
~~
~
~
~
m0
2 2
m1/2
µ0+m0
________
√
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
-200
-100
0
100
200
300
400
500
600
700
The Higgs sector of the MSSM
The Higgs sector of the MSSM is a 2HDM, whose Yukawa couplings and Higgs
potential are constrained by SUSY. Instead of employing to hypercharge-one
scalar doublets Φ1,2, it is more convenient to introduce a Y = −1 doublet
Hd ≡ iσ2Φ∗1 and a Y = +1 doublet Hu ≡ Φ2:
Hd =
(H1d
H2d
)=
(Φ0 ∗
1
−Φ−1
), Hu =
(H1u
H2u
)=
(Φ+
2
Φ02
).
The origin of the notation originates from the Higgs Yukawa Lagrangian:
LYukawa = −hiju (uiRujLH2u − uiRd
jLH
1u)− hijd (d
iRd
jLH
1d − d iRu
jLH
2d) + h.c. .
Note that the neutral Higgs field H2u couples exclusively to up-type quarks
and the neutral Higgs field H1d couples exclusively to down-type quarks. That
is, the Higgs sector of the MSSM is a Type-II 2HDM.
The Higgs potential of the MSSM is:
V =(m2
1 + |µ|2)Hi∗d H
id +
(m2
2 + |µ|2)Hi∗u H
iu −m2
12
(ǫijHi
dHju + h.c.
)
+18
(g2 + g′ 2
) [Hi∗d H
id −Hj∗
u Hju
]2+ 1
2g2|Hi∗
d Hiu|2 ,
where ǫ12 = −ǫ21 = 1 and ǫ11 = ǫ22 = 0, and the sum over repeated indices
is implicit. Above, µ is a supersymmetric Higgsino mass parameter and m21,
m22, m
212 are soft-SUSY-breaking masses. The quartic Higgs couplings are
related to the SU(2) and U(1)Y gauge couplings as a consequence of SUSY.
In the Higgs basis, we can identity the coefficients of the quartic terms of the
scalar potential,
Z1 = Z2 =14(g
2 + g′ 2) cos2 2β , Z3 = Z5 +14(g
2 − g′ 2) , Z4 = Z5 − 12g
2,
Z5 =14(g
2 + g′ 2) sin2 2β , Z7 = −Z6 =14(g
2 + g′ 2) sin 2β cos 2β .
Here, we have used the phase freedom to define the Higgs basis field H2 such
that Z5, Z6 and Z7 are real.
Minimizing the Higgs potential, the neutral components of the Higgs fields
acquire vevs:
〈Hd〉 =1√2
(vd
0
), 〈Hu〉 =
1√2
(0
vu
),
where v2 ≡ v2d + v2u = 4m2W/g
2 ≃ (246 GeV)2 and∗
tan β ≡ vuvd, 0 ≤ β ≤ 1
2π .
At the scalar potential minimum, the squared-mass parameters m21, m
22 and
m212 can be re-expressed in terms of the two Higgs vevs, vu and vd (or
equivalently in terms of mZ and tanβ) and the CP-odd Higgs mass mA,
sin 2β =2m2
12
m21 +m2
2 + 2|µ|2 =2m2
12
m2A
,
12m
2Z = −|µ|2 + m2
1 −m22 tan
2 β
tan2 β − 1.
∗The Higgs fields can be rephased such that the vevs are real and positive. That is, the tree-level MSSM
Higgs sector conserves CP, which implies that the neutral Higgs mass eigenstates are states of definite CP.
At this stage, we can already see the tension with naturalness, if the SUSY
parameters are significantly larger than the scale of electroweak symmetry
breaking. In this case, m2Z will be the difference of two large numbers,
12m
2Z = −|µ|2 + m2
1 −m22 tan
2 β
tan2 β − 1,
requiring some fine-tuning of the SUSY parameters in order to produce the
correct Z boson mass. In the literature, this tension is referred to as the little
hierarchy problem.
In the above equation, µ, m21 and m2
2 are parameters defined at the
electroweak scale. The question of fine-tuning should really be addressed
to the fundamental SUSY-breaking parameters at some high energy scale,
which ultimately determine the low-energy parameters appearing the above
expression.
Higgs bosons of the MSSM
The five physical Higgs particles consist of a charged Higgs pair
H± = H±d sinβ +H±
u cosβ ,
one CP-odd scalar
A0 =√2(ImH0
d sinβ + ImH0u cosβ
),
and two CP-even scalars
h0 = −(√2ReH0
d − vd) sinα+ (√2ReH0
u − vu) cosα ,
H0 = (√2ReH0
d − vd) cosα+ (√2ReH0
u − vu) sinα ,
where we have now labeled the Higgs fields according to their electric charge.
The angle α arises when the CP-even Higgs squared-mass matrix (in the
H0d—H0
u basis) is diagonalized to obtain the physical CP-even Higgs states.
Tree-level MSSM Higgs masses
The charged Higgs mass is given by
m2H± = m2
A +m2W ,
and the CP-even Higgs bosons h0 and H0 are eigenstates of the squared-mass
matrix
M20 =
(m2A sin2 β +m2
Z cos2 β −(m2A +m2
Z) sinβ cosβ
−(m2A +m2
Z) sinβ cosβ m2A cos2 β +m2
Z sin2 β
).
The eigenvalues of M20 are the squared-masses of the two CP-even Higgs
scalars
m2H,h = 1
2
(m2A +m2
Z ±√(m2
A +m2Z)
2 − 4m2Zm
2A cos2 2β
),
and α is the angle that diagonalizes the CP-even Higgs squared-mass matrix.†
†Note the contrast with the SM where the Higgs mass is a free parameter, m2h = 1
2λv2. In the MSSM, all
Higgs self-coupling parameters of the MSSM are related to the squares of the electroweak gauge couplings.
Aside: the decoupling limit of the MSSM
In the limit of mA ≫ mZ, the expressions for the Higgs masses and mixing
angle simplify and one finds
m2h ≃ m2
Z cos2 2β ,
m2H ≃ m2
A +m2Z sin2 2β ,
m2H± = m2
A +m2W ,
cos2(β − α) ≃ m4Z sin2 4β
4m4A
.
Two consequences are immediately apparent. First, mA ≃ mH ≃ mH±, up
to corrections of O(m2Z/mA). Second, cos(β − α) = 0 up to corrections
of O(m2Z/m
2A). This is the decoupling limit, since at energy scales below
approximately common mass of the heavy Higgs bosons H± H0, A0, the
effective Higgs theory is precisely that of the SM.
In particular, we will see that in the limit of cos(β − α) → 0, all the h0
couplings to SM particles approach their SM limits.
Tree-level MSSM Higgs couplings
1. Higgs couplings to gauge boson pairs (V =W or Z)
gh0V V = gVmV sin(β − α) , gH0V V = gVmV cos(β − α) ,
where gV ≡ 2mV /v. There are no tree-level couplings of A0 or H± to V V .
2. Higgs couplings to a single gauge boson
The couplings of V to two neutral Higgs bosons (which must have opposite
CP-quantum numbers) is denoted by gφA0Z(pφ − p0A), where φ = h0 or H0
and the momenta pφ and p0A point into the vertex, and
gh0A0Z =g cos(β − α)
2 cos θW, gH0A0Z =
−g sin(β − α)
2 cos θW.
3. Summary of Higgs boson–vector boson couplings
The properties of the three-point and four-point Higgs boson-vector boson
couplings are conveniently summarized by listing the couplings that are
proportional to either sin(β − α) or cos(β − α) or are angle-independent. As
a reminder, cos(β − α) → 0 in the decoupling limit.
cos(β − α) sin(β − α) angle-independent
H0W+W− h0W+W− —
H0ZZ h0ZZ —
ZA0h0 ZA0H0 ZH+H− , γH+H−
W±H∓h0 W±H∓H0 W±H∓A0
ZW±H∓h0 ZW±H∓H0 ZW±H∓A0
γW±H∓h0 γW±H∓H0 γW±H∓A0
— — V V φφ , V V A0A0 , V V H+H−
where φ = h0 or H0 and V V =W+W−, ZZ, Zγ or γγ.
4. Higgs-fermion couplings
Since the neutral Higgs couplings to fermions are flavor-diagonal, we list only
the Higgs coupling to 3rd generation fermions. The couplings of the neutral
Higgs bosons to ff relative to the SM value, gmf/2mW , are given by
h0bb (or h0τ+τ−) : − sinα
cosβ= sin(β − α)− tanβ cos(β − α) ,
h0tt :cosα
sinβ= sin(β − α) + cotβ cos(β − α) ,
H0bb (or H0τ+τ−) :cosα
cosβ= cos(β − α) + tanβ sin(β − α) ,
H0tt :sinα
sinβ= cos(β − α)− cotβ sin(β − α) ,
A0bb (or A0τ+τ−) : γ5 tan β ,
A0tt : γ5 cot β ,
where the γ5 indicates a pseudoscalar coupling. Note that the h0ff couplings
approach their SM values in the decoupling limit, where cos(β − α) → 0.
Similarly, the charged Higgs boson couplings to fermion pairs, with all particles
pointing into the vertex, are given by‡
gH−tb =g√2mW
[mt cot β PR +mb tanβ PL
],
gH−τ+ν =g√2mW
[mτ tanβ PL
].
Especially noteworthy is the possible tanβ-enhancement of certain Higgs-
fermion couplings. The general expectation in MSSM models is that tan β
lies in a range:
1 <∼ tan β <∼mt
mb.
Near the upper limit of tanβ, we have roughly identical values for the top
and bottom Yukawa couplings, ht ∼ hb, since
hb =
√2mb
vd=
√2mb
v cosβ, ht =
√2mt
vu=
√2mt
v sinβ.
‡Including the full flavor structure, the CKM matrix appears in the charged Higgs couplings in the standard
way for a charged-current interaction.
Saving the MSSM Higgs sector—the impact of radiative corrections
The tree-level Higgs mass result leads to a startling prediction,
mh ≤ mZ| cos 2β| ≤ mZ .
This is clearly in conflict with the observed Higgs mass of 125 GeV. However,
the above inequality receives quantum corrections. The Higgs mass can
be shifted due to loops of particles and their superpartners (an incomplete
cancellation, which would have been exact if supersymmetry were unbroken):
h0 h0 h0 h0t t1,2
m2h<∼ m2
Z +3g2m4
t
8π2m2W
[ln
(M2S
m2t
)+X2t
M2S
(1− X2
t
12M2S
)],
where Xt ≡ At−µ cot β governs stop mixing and M2S is the average squared-
mass of the top-squarks t1 and t2 (which are the mass-eigenstate combinations
of the interaction eigenstates, tL and tR).
A quick and dirty derivation of the logarithmic correction to the Higgs mass
Consider the one-loop effective potential for a gauge theory coupled to matter,
Veff(Φ) = Vscalar(Φ) + V (1)(Φ) .
If we regulate the divergence of the loop correction by a momentum cutoff Λ,
then
V (1)(Φ) =Λ2
32π2StrM2
i (Φ) +1
64π2Str
{M4i (Φ)
[ln
(M2i (Φ)
Λ2
)− 1
2
]},
where M2i (Φ) are the relevant squared-mass matrices for spin 0, 1
2 and 1, in
which the scalar vacuum expectation values are replaced by the corresponding
scalar fields, and the supertrace is defined as
StrM2 ≡∑
J
(−1)J(2J + 1)CJ TrM2J ,
where the sum is taken over J = 0, 12 and 1, and CJ is a counting factor.
In particular, CJ = 1 for Majorana spinors and real bosons, CJ = 2 for
complex bosons, etc. In softly-broken SUSY theories, StrM2i (Φ) = 0.
We focus on the contributions of the top quark and its scalar superpartners,
tL and tR (“stops”). For simplicity, we ignore stop mixing, in which case,
m2t = m2
tL= m2
tR≃M2
S + 12h
2tv
2u , mt =
1√2htvu .
Hence,
V (1)(vd, vu) =3
64π2·2·2
{(M2
S + 12h
2tv
2u)
2
[ln
(M2S + 1
2h2tv
2u
Λ2
)− 1
2
]
−h4tv
4u
4
[ln
(12h
2tv
2u
Λ
)− 1
2
]},
where we have included a color factor of 3, a factor of 2 for complex bosons
and a second factor of 2 to account for both tL and tR.
Exercise: Repeat the above computation including the effects of stop mixing.
Including the tree-level piece, we can write
V (vd, vu) =12(vd vu)
(m2
1 −m212
−m212 m2
2
)(vd
vu
)
+1
32(g2 + g′ 2)(v2d − v2u)
2 + V (1)(vd, vu) .
Requiring that vu is a minimum of the scalar potential, we set ∂V/∂vu = 0,
which yields
m22 = m2
12
vdvu
− 18(g
2 + g′ 2)(v2d − v2u)2 − 1
vu
∂V (1)
∂vu.
Using the minimum condition to eliminate m22, we obtain,
∂2V
∂v2u= m2
12
vdvu
+ 14(g
2 + g′ 2)v2u −1
vu
∂V (1)
∂vu+∂2V (1)
∂v2u.
We now insert the expression previously derived for V (1) into the above result.
The end result is:
∂2V (1)
∂v2u− 1
vu
∂V (1)
∂vu=
3
8π2h4tv
2u ln
(m2t
m2t
).
In particular, notice that the factors of Λ have canceled out!
We therefore see that the effect of the top quark and top squark loops is to
modify the element M222 of the CP-even Higgs squared-mass matrix,
δM222 =
3g2m4t
8π2m2W sin2 β
ln
(m2t
m2t
),
after writing ht =√2mt/vu and vu = (2mW/g) sinβ. Diagonalizing the CP-
even Higgs squared-mass matrix in the limit of mA ≫ mZ and incorporating
the radiative correction obtained above, we obtain the radiatively-corrected
upper bound,
m2h ≤ m2
Z cos2 2β +3g2m4
t
8π2m2W
ln
(m2t
m2t
),
which reproduces the logarithmic correction that was quoted earlier.
The state-of-the-art computation includes the full one-loop result, all the
significant two-loop contributions, some of the leading three-loop terms,
and renormalization-group improvements. The final conclusion is that
mh <∼ 130 GeV [assuming that the top-squark mass is no heavier than a
few TeV], which is compatible with the observed Higgs boson mass.
Maximal mixing corresponds to choosing the MSSM Higgs parameters in such a way thatmh is maximized (for a fixed tan β). This occurs for Xt/MS ∼ 2. As tan β varies, mh
reaches is maximal value, (mh)max ≃ 130 GeV, for tan β ≫ 1 and mA ≫ mZ.
Radiatively-corrected Higgs couplings
Although radiatively-corrections to couplings tend to be at the few-percent
level, there is some potential for significant effects:
• large radiative corrections due to a tanβ-enhancement (for tanβ ≫ 1)
• CP-violating effects induced by complex SUSY-breaking parameters that
enter in loops
In the MSSM, the tree-level Higgs–quark Yukawa Lagrangian is
supersymmetry-conserving and is given by Type-II structure,
Ltreeyuk = −ǫijhbHi
dψjQψD + ǫijhtH
iuψ
jQψU + h.c.
Two other possible dimension-four gauge-invariant non-holomorphic Higgs-
quark interactions terms, the so-called wrong-Higgs interactions,
Hk∗u ψDψ
kQ and Hk∗
d ψUψkQ ,
are not supersymmetric, and hence are absent from the tree-level Yukawa
Lagrangian.
Nevertheless, the wrong-Higgs interactions can be generated in the effective
low-energy theory below the scale of SUSY-breaking. In particular, one-loop
radiative corrections, in which supersymmetric particles (squarks, higgsinos
and gauginos) propagate inside the loop can generate the wrong-Higgs
interactions.
Hi∗u
Qi∗Qi
D∗D
×gaψiQ ψD
(a)
Hi∗u
UU∗
QiQi∗
×ψHuψHdψiQ ψD
(b)
One-loop diagrams contributing to the wrong-Higgs Yukawa effective operators. In (a), the cross (×) corresponds to a factor of the
gluino mass M3. In (b), the cross corresponds to a factor of the higgsino Majorana mass parameter µ. Field labels correspond to
annihilation of the corresponding particle at each vertex of the triangle.
If the superpartners are heavy, then one can derive an effective field theory
description of the Higgs-quark Yukawa couplings below the scale of SUSY-
breaking (MSUSY), where one has integrated out the heavy SUSY particles
propagating in the loops.
The resulting effective Lagrangian is:
Leffyuk = −ǫij(hb + δhb)ψbH
idψ
jQ +∆hbψbH
k∗u ψkQ
+ǫij(ht + δht)ψtHiuψ
jQ +∆htψtH
k∗d ψkQ + h.c.
In the limit of MSUSY ≫ mZ,
∆hb = hb
[2αs3π
µM3I(Mb1,Mb2
,Mg) +h2t
16π2µAtI(Mt1
,Mt2, µ)
],
where, M3 is the Majorana gluino mass, µ is the supersymmetric Higgs-mass
parameter, and b1,2 and t1,2 are the mass-eigenstate bottom squarks and top
squarks, respectively. The loop integral is given by
I(a, b, c) = a2b2 ln(a2/b2) + b2c2 ln(b2/c2) + c2a2 ln(c2/a2)
(a2 − b2)(b2 − c2)(a2 − c2).
Note that I(a, b, c) ∼ 1/max(a2, b2, c2) in the limit where at least one of the
arguments of I(a, b, c) is large. Hence, the one-loop contributions to ∆hb do
not decouple when M3 ∼ µ ∼ At ∼Mb ∼Mt ∼MSUSY ≫ mZ.
Phenomenological consequences of the wrong-Higgs Yukawas
The effects of the wrong-Higgs couplings are tan β-enhanced modifications of
some physical observables. To see this, rewrite the Higgs fields in terms of
the physical mass-eigenstates (and the Goldstone bosons):
H1d =
1√2(v cosβ +H0 cosα− h0 sinα+ iA0 sin β − iG0 cosβ) ,
H2u =
1√2(v sin β +H0 sinα+ h0 cosα+ iA0 cosβ + iG0 sinβ) ,
H2d = H− sinβ −G− cosβ ,
H1u = H+ cosβ +G+ sinβ .
For simplicity, we neglect below possible CP-violating effects due to complex
couplings. Then, the b-quark mass is:
mb =hbv√2cosβ
(1 +
δhbhb
+∆hbtan β
hb
)≡ hbv√
2cosβ(1 + ∆b) ,
which defines the quantity ∆b.
In the limit of large tanβ the term proportional to δhb can be neglected, in
which case,∆b ≃ (∆hb/hb)tan β .
Thus, ∆b is tanβ–enhanced if tanβ ≫ 1. As previously noted, ∆b survives
in the limit of large MSUSY; this effect does not decouple.
The effective Yukawa Lagrangian (neglecting CP-violating effects) yields,
Lint = −∑
q=t,b,τ
[gh0qqh
0qq + gH0qqH0qq − igA0qqA
0qγ5q]+[bgH−tbtH
− + h.c.].
The one-loop corrections can generate measurable shifts in the decay rate for
h0 → bb:
gh◦bb = −mb
v
sinα
cosβ
[1 +
1
1 +∆b
(δhbhb
−∆b
)(1 + cotα cotβ)
].
At large tanβ ∼ 20—50, ∆b can be as large as 0.5 in magnitude and of either
sign, leading to a significant enhancement or suppression of the Higgs decay
rate to bb.
Non-decoupling effects in h0 → bb: a closer look
The origin of the non-decoupling effects can be understood by noting that
below the scale MSUSY, the effective low-energy Higgs theory is a completely
general 2HDM. Thus, it is not surprising that the wrong-Higgs couplings do
not decouple in the limit of MSUSY → ∞.
However, suppose that mA ∼ O(MSUSY). Then, the low-energy effective
Higgs theory is a one-Higgs doublet model, and thus gh0bb must approach its
SM value. Indeed in this limit,
cos(β − α) =m2Z sin 4β
2m2A
+O(m4Z
m4A
),
1 + cotα cotβ = −2m2Z
m2A
cos 2β +O(m4Z
m4A
).
Thus the previously non-decoupling SUSY radiative corrections do decouple
as expected.
LHC searches for the heavy Higgs states of the MSSM
• At present, the LHC search channel with the greatest reach looks for
gg → (H or A) → τ+τ− (at moderate values of tan β) ,
gg → bb(H or A) → bbτ+τ− (at large values of tan β) .
No signal above background has been observed. This provides model-
dependent limits on the MSSM parameter space which rules out regions of
the mA–tan β plane.
• The precision Higgs data suggests that the mixing of the observed Higgs
boson with a non-SM-like Higgs eigenstate is small. This points to the
decoupling regime which provides a lower bound on the value of mA
(except in the parameter region of alignment without decoupling).
[GeV]Am
200 300 400 500 600 700 800 900 1000
βta
n
0
1
2
3
4
5
6
7
8
9
10
PreliminaryATLAS
1 Ldt = 4.64.8 fb∫= 7 TeV, s
1 Ldt = 20.3 fb∫= 8 TeV, s
b, bττ, ZZ*, WW*, γγ →Combined h
]dκ, uκ, VκSimplified MSSM [
Exp. 95% CL Obs. 95% CL
Regions of the (mA, tan β) plane excluded in a simplified MSSM model via fits tothe measured rates of Higgs boson production and decays. The likelihood contourscorresponding approximately to 95% CL, are indicated for the data and expectationassuming the SM Higgs sector. The light shaded and hashed regions indicate the observedand expected exclusions, respectively. Taken from ATLAS-CONF-2014-010.
• The infamous LHC wedge region of large mA and moderate values of
tanβ will be difficult to probe at the LHC. For values of mA > 2mt, the
H, A → tt is a challenging signal. In a limited parameter regime with
2mh < mH < 2mt and moderate tan β, the decay H → hh may provide
some opportunity for discovery.
[GeV]Am
200 300 400 500 600 700 800 900 1000 1100 1200
βta
n
10
20
30
40
50
60
70
80
= 3
00 G
eV
Hm
= 5
00 G
eV
Hm
= 1
000 G
eV
Hm
= 123.5 GeVh
m
= 125 GeVh
m
= 126 GeVh
m
= 126.3 GeVh
m
Observed
Expected
σ1
σ2
ATLAS Run-I (Obs.)
-1=13 TeV, 3.2 fbsPreliminary,ATLAS
= 1 TeVSUSY
scenario, Mmod+
hMSSM m
, 95% CL limitsττ→H/A
95% CL Excluded:
Observed Expectedσ 1± 3 GeV± 125 ≠MSSMhm
Expected Expectedσ 2± 7 + 8 TeV(HIG-14-029)
CMSPreliminary
(13 TeV)-12.3 fb
(GeV)Am
200 400 600 800 1000 1200 1400
βta
n
10
20
30
40
50
60 scenariomod+
hm
The alignment limit without decoupling in the MSSM
Recall that in the Higgs basis, the CP-even neutral Higgs squared-mass matrix
is given by
M2 =
(Z1v
2 Z6v2
Z6v2 m2
A + Z5v2
).
The mixing angle of rotation from the Higgs basis to the mass basis is α− β.
It follows that in both the decoupling limit (mH ≫ mh) and the alignment
limit without decoupling [|Z6| ≪ 1 and m2H − Z1v
2 ∼ O(v2)],§
cos2(β − α) =Z26v
4
(m2H −m2
h)(m2H − Z1v2)
≪ 1 ,
Z1v2 −m2
h =Z26v
4
m2H − Z1v2
≪ O(v2) .
§Note the upper bound m2h ≤ Z1v
2 on the mass of h is saturated in the exact alignment limit.
The MSSM values of Z1 and Z6 (including the leading one-loop corrections):
Z1v2 = m2
Zc22β +
3v2s4βh4t
8π2
[ln
(M2S
m2t
)+X2t
M2S
(1− X2
t
12M2S
)],
Z6v2 = −s2β
{m2Zc2β −
3v2s2βh4t
16π2
[ln
(M2S
m2t
)+Xt(Xt + Yt)
2M2S
− X3t Yt
12M4S
]}.
where M2S ≡ mt1
mt2, Xt ≡ At − µ cotβ and Yt = At + µ tanβ.
As previously noted, m2h ≤ Z1v
2, is consistent withmh ≃ 125 GeV for suitable
choices for mS and Xt. Exact alignment (i.e., Z6 = 0) can now be achieved
due to an accidental cancellation between tree-level and loop contributions,¶
m2Zc2β =
3v2s2βh4t
16π2
[ln
(M2S
m2t
)+Xt(Xt + Yt)
2M2S
− X3t Yt
12M4S
].
That is, Z6 ≃ 0 is possible for a particular choice of tan β.¶See M. Carena, H.E. Haber, I. Low, N.R. Shah and C.E.M. Wagner, Phys. Rev. D 91, 035003 (2015).
200 250 300 350 400 450 500
MA [GeV]
5
10
15
20
25
tan
β
0
5
10
15
20
-2 ln(L)mhalt
scenario (µ=3mQ)
FeynHiggs-2.10.2SusHi-1.4.1HiggsBounds-1.2.0
95% CL excl.
Carena et al.
mhmod+
(µ=200GeV)
200 250 300 350 400 450 500
MA [GeV]
5
10
15
20
25
tan
β
0
5
10
15
20
∆χHS2
mhalt
scenario (µ=3mQ)
FeynHiggs-2.10.2SusHi-1.4.1HiggsSignals-1.3.0
95% CL
Constraints from LHC Higgs searches in the alignment benchmark scenario (with µ = 3MS).
Left panel: distribution of the exclusion likelihood from the CMS φ → τ+τ− search and the
observed 95% CL exclusion line as obtained from HiggsBounds.
Right panel: likelihood distribution, ∆χ2HS obtained from testing the signal rates of the
Higgs boson h against a combination of Higgs rate measurements from the Tevatron and
LHC experiments, obtained with HiggsSignals.
See P. Bechtle, S. Heinemeyer, O. Stal, T. Stefaniak and G. Weiglein, EPJC 75, 421 (2015).
Likelihood analysis: allowed regions in the tanβ–mA plane
Preferred parameter regions in the (MA, tan β) plane (left) and (MA, µAt/M2S) plane
(right), where M2S = mt1
mt2, in a pMSSM-8 scan. Points that do not pass the direct
constraints from Higgs searches from HiggsBounds and from LHC SUSY particle searches
from CheckMATE are shown in gray. Applying a global likelihood analysis to the points
that pass the direct constraints, the color code employed is red for ∆χ2h < 2.3, yellow
for ∆χ2h < 5.99 and blue otherwise. The best fit point is indicated by a black star.
(Taken from P. Bechtle, H.E. Haber, S. Heinemeyer, O. Stal, T. Stefaniak, G. Weiglein and
L. Zeune, in preparation.)
The alignment limit of the NMSSM Higgs sector
In the MSSM, a supersymmetric Higgsino mass term µ appears, whose
magnitude is similar to that of a typical SUSY-breaking mass. Although
phenomenology demands it, there is no theoretical reason for these two mass
scales to be connected.
This problem is addressed in the next-to-minimal supersymmetric extension
of the Standard Model (NMSSM). In this model, a new singlet supermultiplet
is added consisting of a complex singlet scalar field S and its fermionic
superpartner. An effective µ-term appears, µ = λvs, where λ governs the
coupling of the singlet scalar to the Higgs doublet fields and vs is the vev of
the singlet scalar field.
The analog of the Higgs basis fields are:
h ≡√2Re H0
1 − v , H ≡√2Re H0
2 , HS ≡√2 (Re S − vs) ,
G ≡√2 Im H0
1 , A ≡√2 Im H0
2 , AS ≡√2 Im S .
CP is not automatically conserved in the NMSSM Higgs sector, but we shall
assume it for simplicity. In this case, the symmetric squared-mass matrix for
the CP-even scalars in the Higgs basis is
M2S =
Z1v2 Z6v
2√2 v
[C1 + (Zs1 + 2Zs5)vs
]
M2
A + Z5v2 v
√2
[C3 + C4 + 2(Zs3 + Zs7 + Zs8)vs
]
−C1
v2
2vs+ 3(C5 + C6)vs + 4(Zs4 + 2Zs9 + 2Zs10)v
2s
,
where M 2A ≡ 2µ(Aλ + κvs)/s2β and µ ≡ λvs. Note that κ governs the
self-coupling of the singlet scalar field. The coefficients of the scalar potential
appear in the squared-mass matrix M2,
V ∋ . . .+ 12Z1(H
†1H1)
2 + . . . +[12Z5(H
†1H2)
2 + Z6(H†1H1)H
†1H2 + h.c.
]+ . . .
+S†S[Zs1H
†1H1 + . . .+ (Zs3H
†1H2 + h.c.) + Zs4S
†S]
+{Zs5H
†1H1S
2 + . . . + Zs7H†1H2S
2 + Zs8H†2H1S
2 + Zs9S†S S2 + Zs10S
4 + h.c.}
+[C1H
†1H1S + . . .+ C3H
†1H2S + C4H
†2H1S + C5(S
†S)S + C6S3 + h.c.
].
Exact alignment occurs when M212 = M2
13 = 0. That is,
Z6 = 0 , C1 + (Zs1 + 2Zs5)vs = 0 .
In the NMSSM, including the leading one-loop radiative corrections,
Z1v2 = (m2
Z − 12λ
2v2)c22β +12λ
2v2 +3v2s4βh
4t
8π2
[ln
(M2S
m2t
)+X2t
M2S
(1− X2
t
12M2S
)],
Z6v2 = −s2β
{(m2
Z − 12λ
2v2)c2β −3v2s2βh
4t
16π2
[ln
(M2S
m2t
)+Xt(Xt + Yt)
2M2S
− X3t Yt
12M4S
]}.
Note that the squared-mass bound of the SM-like Higgs boson is now modified
at tree-level by a positive quantity,
m2h ≤ (Z1v
2)tree = m2Zc
22β +
12λ
2v2s22β .
Regions of parameter space exist in which the term proportional to λ2 provides
a significant contribution to m2h.
In contrast to the MSSM, in the NMSSM one can set Z6 = 0 and obtain
mh = 125 GeV, with only small contributions from the one-loop radiative
corrections. This leads to a preferred choice of NMSSM parameters,
λ ∼ 0.65 , tan β ∼ 2
Β
Λ
Λ = ±
=
±=
=
Β
HL
Λ =
The exact alignment limit also requires that M213 = 0. In the NMSSM,
M2
As22β
4µ2+κs2β2λ
= 1 .
which lead to further correlations of the NMSSM parameter space.
H L
HL
Λ Κ = Λ �
H L Β = =
Near the alignment limit, we have mA ≃ mH ≃MA.
Phenomenological Implications of the NMSSM Higgs sector
[reference: M. Carena, H.E. Haber, I. Low, N.R. Shah and C.E.M. Wagner, Phys. Rev. D 93,
035013 (2016)]
• Numerous H → HH and H → V H channels are kinematically allowed.
• The parameters µ and MA are correlated by the alignment conditions.
• Typical values of |µ| ∼ 100–300 GeV yield chargino/neutralino states that
are approximately unmixed higgsino (with mass ∼ |µ|) and singlino (with
mass ∼ 2|κµ/λ|) states. Thus, we expect significant branching ratios of
H,A→ χ+χ− , χ0i χ
0j , H± → χ±χ0
i .
• Imposing perturbativity up to the Planck scale implies that κ <∼ 12λ.
Consequently, the singlino is the LSP (denoted as χ01). A potential
decay mode of the neutral higgsino is χ02 → χ0
1 + h.
