Higgs and Supersymmetry talk

Introduction
Cosmology/Astrophysics Implications
MSSM
The Higgs Mass - Evidence for Physics beyond SM
Summarising MSSM Higgs Results
References
Minimal Supersymmetry and Higgs Boson(s)
K. Ahmed
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Introduction
MSSM
References
Outline I
1 Introduction
Fundamental Constituents
The Couplings Uniﬁcation
Gravity Force and Quantum description
Salam’s contribution
2 Cosmology/Astrophysics Implications
Unstable Gravitinos as DM
Extra Dimensional Theories
3 MSSM
Higgs Production at the LHC
Higgs Decays
Two Higgs Doublet Model (2HDM) Analysis
MSSM extension to two Higgs doublets
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Introduction
MSSM
References
Outline II
4 The Higgs Mass - Evidence for Physics beyond SM
5 Summarising MSSM Higgs Results
6 References
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Introduction
MSSM
References
Introduction I
If one looks at the Higgs boson H, its mass cannot be understood.
Quantum oscillations give rise to self mass of the scalar particle
which quadratically diverges. The divergent graph arises due to the
self coupling of the scalar ﬁeld as shown in Figure 1.
S
H H H H
F
F
Figure 1 : Loop diagrams
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Introduction
MSSM
References
Introduction II
∆(mH)2
S =
λs
16π2
Λ2
− m2
s ln
Λ2
m2
s
+ . . . (1)
∆(mH)2
F =
λf i2
8π2
−Λ2
− 3m2
F ln
Λ2
m2
f
(2)
This divergence is cancelled if one has a corresponding partner
coupled with comparable strength to the scalar Higgs but opposite
in sign as in (2), i.e., if λs = 2|λf |2.
This is fine tuning of coupling and the ultraviolet divergence
(quadrative in mass) essentially defines a cut-off mass squared,
that fixes the limit to the Standard Model (SM) beyond which the
new physics starts – the hierarchy problem.
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Introduction
MSSM
References
Introduction III
The existence of the matching fermion (spin 1
2) to the scalar Higgs
boson is a requirement of supersymmetry (SUSY) which gives rise
to a fermion to every boson and vice versa carrying equal mass in
the exact symmetry limit in order for the cancellation of the
divergence. This is ’naturalness problem’. In other words, in order
to have a ’natural Higgs mass’ SUSY sets an important choice on
New Physics (NP) or physics beyond SM. Further, the Higgs boson
receives quantum (or loop order) corrections that are limited by
the extent of SUSY breaking (in masses and couplings). A new
scale then appears in mass, that is a O(TeV).
At this point a need for SUSY (a theory not a female!) arises.
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Introduction
MSSM
References
Fundamental Constituents I
Phenomenologically, there are other indicators of SUSY, they are:
The fact that in the SM the constituents of matter like quarks and
leptons are fermions (spin 1
2) [obeying Fermi-Dirac statistics
leading to Pauli Exclusion Principle, i.e., no two identical fermions
can occupy the same state] and bosons carrying force ﬁeld (spin 1,
vector) [obeying Bose statistics, i.e., more than one particles
occupying the same state] – why this asymmetry?
Does nature choose this or is there some underlying subtle
symmetry broken at ordinary energies but may be seen at higher
energies.
SUSY aﬀords such symmetrisation between bosons and fermions.
In the exact form (unbroken) which is not seen at ordinary energies
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Introduction
MSSM
References
Fundamental Constituents II
(everyday it has the same masses and couplings for both fermions
and bosons).
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Introduction
MSSM
References
The Couplings Uniﬁcation I
It is found that extrapolation of electromagnetic, weak, and strong
couplings with energy do not meet at a point as shown in Figure 2
(i.e., they do not unite or corresponding forces cannot be uniﬁed):
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Introduction
MSSM
References
The Couplings Uniﬁcation II
Figure 2 : Coupling Uniﬁcation
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Introduction
MSSM
References
The Couplings Unification III
Gauge coupling constants αi = g2
i /4π , using Renormalisation
Group (RG) equations, start varying with energy in such a way
that they unify using SUSY at energies of the order of ∼ 1016GeV.
In this evolution of various interaction couplings or their inverse to
be precise (in the RG equation), one uses SUSY particles in the
1-loop quantum corrections where the coefficients bi of the
Renormalisation Group Equation (RGE) assume larger values than
their SM corresponding coeffiecients. Here bi is defined as
bi = −2π
d
dt
(α−1
i ), (3)
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Introduction
MSSM
References
The Couplings Uniﬁcation IV
where t = ln q
q0
, with q the RGE scale and q0 the SM scale.
Further one uses SU(5) or SO(10) as a grand uniﬁed gauge group
and RGE for extrapolation.
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Introduction
MSSM
References
Gravity Force and Quantum description I
Another important requirement for unification theories is the force
of gravity. Theoretically, it is very difficult to develop a quantum
theory of gravity because of divergence problem associated with
Feynman diagrams involving interaction with gravity through
gravitons.
Superstrings afford a possibility to offset the difficulties of
renormalisation associated with gravitational field. Supersymmetric
gravity theories have been formulated to incorporate grand
unification of forces including gravity as SUSY GUTS.
Supersymmetry is used as a precursor in most of these theories.
However, there is no experimental evidence of SUSY particles even
in the lightest mass scale, so far. As usual, for NP, physicists wait
for upgradation of accelerator energies.
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Introduction
MSSM
References
Salam’s contribution I
Abdus Salam’s contribution to Supersymmetry was seminal. He
alongwith John Strathdee published an important paper on
Supergauge transformations (Nucl. Phys B 76, p.477 (1974)) and
later the concept of Superfields which puts bosons and fermions
together in the form of Supersymmetric multiplets as Superfields.
These superfields are, however defined over extended coordinate
containing self-commuting (ordinary) space-time coordinate xµ as
well as four non-commuting fermionic Grassmnian variables θµ.
Steven Weinberg in his book titled ”The Quantum Theory of
Fields, Vol III: Supersymmetry” refers to Salam’s (and Strathdee’s)
fundamental contribution to Supersymmetry and underlying
framework of Super Algebra.
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Introduction
MSSM
References
Salam’s contribution II
In the words of Weinberg: ”a great deal of work can be saved by
using a formalism invented by Salam and Strathdee in which the
ﬁelds in any supermultiplet are assembled into a simple superﬁeld.”
(A. Salam & J. Strathdee, Nucl. Phys. B 76, p. 477 (1974))
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Introduction
MSSM
References
Cosmology/Astrophysics Implications I
SUSY postulates the Lightest Sypersymmetric Particle (LSP)
called Neutralino ( ˜X) which is thought to be a neutral particle
existing as a supersposition state of Higgsino (supersymmetric
Higgs bosons, ˜h0
1, ˜h0
2), Zino (supersymmetric Z0 boson) and
photino (˜γ, supersymmetric partner of the photon γ). This particle
is believed to be comprising over 20% of matter/energy density
compared to the corresponding critical energy density required to
close the Universe since the Big Bang. Such an invisible particle of
matter is called Dark Matter (DM). The mass limit for such a DM
candidate is of the order of hundreds of Giga electron volts. There
are other DM candidates such as axion, CP (strong) violating
particle. Such particles energies may be accessible to neutrino
telescopes which are designed to detect 100’s of GeV particles.
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Introduction
MSSM
References
Cosmology/Astrophysics Implications II
However, the annihilation rates of neutralinos predicted from
Minimal Supersymmetric SM (MSSM) variants in celestial bodies
are low if contraints from (Wilkinson Microwave Anisotropy Probe)
WMAP and (Large Hadron Collider) LHC are taken into account.
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Introduction
MSSM
References
SUSY also predicts through its R-parity violating model a long
lived but unstable viable candidate of DM called ’gravitino’. This
is estimated at a mass of few to a few hundred GeV and may be
present in the halos of galaxies as a component of DM.
Gravitinos decay could be seen in neutrino telescopes. However,
gravitino DM cannot be detected directly in normal detectors
because its interaction with normal matter falls inversely with
fourth power of the Planck constant G−4
planck.
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Introduction
MSSM
References
Involving extra dimension range of the order of 10−3 − 10−15
meters can also provide DM candidates. Extra dimensions can also
be accomodated or required by Supersymmetry, string theory or
M-theory, where they give rise to ’branons’, weakly interacting and
massive ﬂuctuations of the ﬁeld that represent the 3-D brane on
which the Standard world lives. A stable and weakly interacting
object, branon makes a good candidate for DM as a usual ’relic
branon’ left over after a freeze out period during the evolution of
the Universe accumulating gravitationally in the halos of galaxies
where due to their high energies they annihilate into SM particles.
Such particles as products fo annihilation can then be detected by
gamma-ray telescopes, surface arrays or neutrino telescopes.
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Introduction
MSSM
References
Higgs Decays
MSSM I
In order to look for physics beyon SM, higher energy data and more
lumminosity collisions are awaited from the LHC. One should then
expect to study Higgs couplings more accurately. One also looks
for higher energy accelerators like ILC, Higgs e+e− factories, etc.
The objectives are to look for (additional) CP-even states predicted
by MSSM or NMSSM (one having an additional doublet and one
complex singlet to the normal Higgs doublet invariant under
SU(2) U(1) gauge group).
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Introduction
MSSM
References
Higgs Decays
MSSM II
In the MSSM one has two Higgs doublets,
H1 =
H1
1
H2
1
=
(φ0
1)∗
−φ−
1
(4)
H2 =
H1
2
H2
2
=
φ+
2
φ0
2
. (5)
Symmetry is broken through vacuum expectation values of the
Higgs doublets as,
< H1 >=
v1
0
< H2 >=
0
v2
. (6)
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Introduction
MSSM
References
Higgs Decays
MSSM III
Mixing of Higgs states is introduced through the mixing angles α
and β,
tan β =
v2
v1
, (7)
where v1, v2 > 0 and 0 ≤ β ≤ π
2 .
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Introduction
MSSM
References
Higgs Decays
At the LHC, the SM Higgs boson is produced through four
diﬀerent channels:
Gluon gluon fusion channel: gg → hX
Vector Boson Fusion (VBF) channel: qq → hjjX
Higgs boson strahlung channel: q¯q → hVX
Higgs boson and top quark pair
associated production channel: ¯q(gg) → ht¯tX
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Introduction
MSSM
References
Higgs Decays
Higgs Decays I
(S. Heinemeyer et al. LHC Higgs Section Working Group
Collaboration); arXiv:1307.1347 [hep-ph]
The Higgs decay rate into a pair of fermion is given at tree level by
Γ(H → ¯f f ) = Ne
GF mH
4π
√
2
m2
f , (8)
where Ne = 3(1) for decays into quaks (leptons). Since the tree
level couplings to other particles are propotional to their masses
(squared in the cases of massive vector bosons), the dominant
Higgs decays are into the heaviest particles that are kinematically
accessible, such as, ¯bb, ¯c¯c and τ+τ−. However, only τ+τ− decay
mode, i.e., H →τ+τ− has recently been observed unambiguously
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Introduction
MSSM
References
Higgs Decays
Higgs Decays II
(ATLAS and CMS collaborations ﬁles).
Further
Γ(H → WW ∗
) =
GF m3
H
8π
√
2
F(r), (9)
where F(r ≡ mW /mH is a kinematic factor) has been observed.
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Introduction
MSSM
References
Higgs Decays
Two Higgs Doublet Model (2HDM) Analysis I
Now that Higgs Boson has been discovered, a question arises
whether it is the Higgs Boson of the SM, or whether there are
more?
Two Higgs Doublet model and Supersymmetry oﬀer a possibility of
more Higgs bosons. We now turn our attention to this possibility.
Let φ1 and φ2 be two doublet complex scalar ﬁelds with weak
hypercharge Y = 1, and belonging to symmetry group SU(2)L.
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Introduction
MSSM
References
Higgs Decays
Two Higgs Doublet Model (2HDM) Analysis II
The Higgs potential which breaks (spontaneously)
SU(2)L U(1)Y down to U(1)EM is,
V (φ1, φ2) = λ1(φ†
1φ1 − v2
1 )2
+ λ2(φ†
2φ2 − v2
2 )2
+ λ3 (φ†
1φ1 − v2
1 ) + (φ†
2φ2 − v2
2 )
2
+ λ4[(φ†
1φ1)(φ†
2φ2) − (φ†
1φ2)(φ†
2φ1)]
+ λ5[Re(φ†
1φ2) − v1v2 cos ξ]2
+ λ6[Im(φ†
1φ2) − v1v2 sin ξ]2
(10)
where the λi are real parameters (Hermiticity requirement). Above
equation gives the most general scalar doublet potential subject to
discrete symmetry φ1 → −φ1 which is only softly violated
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Introduction
MSSM
References
Higgs Decays
Two Higgs Doublet Model (2HDM) Analysis III
(by dim. 2 terms, viz: whose coeﬃcient is λ4).
Assuming that all λi are non-negative, then the minimum of the
potential is manifestly,
< φ1 >=
0
v1
< φ2 >=
0
v2eiξ , (11)
which breaks SU(2)L U(1)Y down to U(1)EM, as desired.
Now taking CP-conserving state which requires the phase ξ to
vanish and λ5 = λ6, then the last two terms can be combined as,
|φ1†φ2 − v1v2eiξ
|2
→ |φ1†φ2 − v1v2|2
(ξ → 0) (12)
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Introduction
MSSM
References
Higgs Decays
Two Higgs Doublet Model (2HDM) Analysis IV
Next let tan β = v2/v1 (Ratio of expectation values of φ2 to that
of φ1) be an important parameter associated with the 2HDM.
Next one removes the Goldstone Boson and determines the Higgs
states by rotating:
G±
= φ±
1 cos β + φ±
2 sin β, (13)
and Higgs states taken as orthogonal to Goldstone Bosons,
H±
= −φ±
1 sin β + φ±
2 cos β (14)
with mass m2
H± = λ4(v2
1 + v2
2 ). Due to CP invariance assumed
before, the imaginary parts and the real parts of the neutral scalar
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Introduction
MSSM
References
Higgs Decays
Two Higgs Doublet Model (2HDM) Analysis V
ﬁelds decouple. In the imaginary (CP-odd) sector, the neutral
Goldstone boson is,
G0
=
√
2(Im φ0
1 cos β + Im φ0
2 sin β) (15)
and the orthogonal neutral physical state is,
A0
=
√
2(−Im φ0
1 sin β + Im φ0
2 cos β) (16)
with mass m2
A0 = λ6(v2
1 + v2
2 ).
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Introduction
MSSM
References
Higgs Decays
MSSM extension to two Higgs doublets I
Supersymmetry requires (for minimal case) 2 Higgs doublets; one
to give masses to charge +2
3 quarks, Hu and the other to charge
−1
3 quarks and charged leptons, Hd . The ratio of their vacuum
expectation values are denoted as β = v2
v1
. Simulations have
been done to see that the renormalisation by the top quark
coupling is important for one of the Higgs multiplet, and may drive
m2
Hu
negative at the electroweak scale resulting in the electroweak
symmetry breaking and thus may explain negative sign in the
quartic term in the eﬀective SM potential. For a heavy top quark
mass, it is then possible for the electroweak scale to be generated
around 100 GeV if mt ∼ 100 GeV. For this reason SUSY theorists
actually suggested heavy momentum for the top quark, before its
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Introduction
MSSM
References
Higgs Decays
MSSM extension to two Higgs doublets II
discovery!
Now 2 complex Higgs complex Higgs doublets of the MSSM have
eight degrees of freedom, of which 3 are used by the Higgs
Mechanism for electroweak symmetry breaking to give mass to the
W ± boson and Z0, leaving 5 physical Higgs bosons states of these
2 (h, H) are neutral Higgs that are CP-even (scalar), one A is
neutral CP-odd (pseudoscalar) and 2 are charged, the H±. At tree
level the masses of the scalar Higgs(es) are:
m2
h,H =
1
2
(m2
A + m2
Z ((m2
A + m2
Z )2
− 4m2
Am2
Z cos2
β)) (17)
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Introduction
MSSM
References
Higgs Decays
MSSM extension to two Higgs doublets III
In general their coupling compared to SM couplings are:
ghVV = sin(β − α)gSM
HVV , gHVV = cos(β − α)gSM
HVV (18)
ghAZ = cos(β − α)(
g
) , gh¯bb+
, ghτ+τ− = −
sin α
cos β
gSM
h¯bb
, gSM
hτ+τ− .
(19)
If mA >> mW , then from (17) ma ∼ mH ∼ mH± . However if mA
is small and mH ∼ 125 GeV, then mA is smal then mH is 2nd
lightest discovered.
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Introduction
MSSM
References
The Higgs Mass - Evidence for Physics beyond SM? I
(J. Ellis, arXiv:1312.5672 [hep-ph]) CMS and ATLAS results of
Higgs mass are quite consistent and a naive global average (for the
Higgs mass) is
mH = (125.6 ± 0.4)GeV (20)
And this average is quite consistent with the electroweak data
based on one-loop level SM collaboration to ∆X2 ∼ 1.5 level.
However, when effective Higgs potential is considered then there
are problems. When self renormalisation effects are taken into
account for the Higgs field coming from Higgs self-coupling and
Ht¯t coupling, one can write the Higgs self-coupling as:
λQ =
λ(v)
1 − 3
4π2 λ(v) ln Q2
v2
+ . . . , (21)
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Introduction
MSSM
References
The Higgs Mass - Evidence for Physics beyond SM? II
where Q is some renormalisation scale above the electroweak scale
v. And due to Ht¯t coupling; i.e., when
λ(Q) = λ(v) 1 −
3
4π2
λ(v) ln
Q2
v2
−1
= λ(v) −
3m4
t
4π2v4
ln
Q2
v2
+ . . . , (22)
Where in the above equation, non-leading terms with RGE solution
have been ignored.
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Introduction
MSSM
References
One notes that renormalisation of the Higgs self coupling in the
RGE solution for large Q values tends to
increase λ(Q) in (18) leading to a landau singullarity (Landau Pole
arises for large
Q2 values relative to v2 as: Q2 = v2 exp(4π2/3λ(v)) ). While in
(19) it decreases the Higgs self coupling λ(Q) with increasing
Q-values. At some point when Q is suﬃciently large relative to v
(electroweak scale), λ(Q) is driven to negative values. This would
set instability in the electroweak vacuum if,
mH < 129.4 + 1.4
mt − 173.1GeV
0.7
− 0.5
αS (mZ ) − 0.1184
0.0007
± 1.0TH ]GeV (23)
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Introduction
MSSM
References
[G. Degrassi, et al. JHEP 1208, (2012) 098]
The measured value of mH plus mt 173 GeV would drive the
quartic self-coupling λ to negative values for some energy scale
∼ 1010 to 1014 GeV, if no physics beyond SM intervenes at lower
energy scale as shown:
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Introduction
MSSM
References
Figure 3 : Higgs mass Mh in GeV
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Introduction
MSSM
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Figure 4 : Higgs pole mass Mh in GeV
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Introduction
MSSM
References
The instability of the vacuum having large negative value for large
Q−value of the order of 1010 to 1014 GeV (approaching Planck
scale/Planckian era) is hard to reconcile with the present value of
cosmological constant related to vacuum energy is nearly zero.
Within SM such a low mass (23) is hard to realise with SUSY.
Once this is done at the one-loop level, then it is shown that the
mass of the Higgs boson can be extended and deﬁned to higher
loops graphs also, in the same self-consistent way. Also as we saw
that existence of the Higgs mass as found alongwith top quark
mass found also empirically provides through electroweak vacuum
stability the requirement that Higgs mass satisfying:
mH < 129.4 + 1.4
mt − 173.1GeV
0.7
− 0.5
αS (mZ ) − 0.1184
0.0007
± 1.0TH ]GeV
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Introduction
MSSM
References
The implications of the mass requirement of mH and mt are
plotted in Figures 3 and 4. The result in (23) is based on
NNLO-SM calculation by Giuseppe Degrassi et al,
(CERN-PH-TH/2012 134 RM3-TH/12-9) and says that for
vacuum stability for Q values from 1013 − 1014 GeV, the mass of
MH > (129.4 ± 1.8) GeV.
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Introduction
MSSM
References
It predicts, Higgs mass
m2
h,H =
1
2
(m2
A + m2
Z (m2
A + m2
Z )2 − 4m2
Am2
Z cos2 2β) (24)
β = tan− 1 v2
v1
Couplings,
ghVV = sin(β − α)gSM
HVV
gHVV = cos(β − α)gS
MHVV
ghAZ = cos(β − α)
g
2 cos θW
gh¯bb, ghτ+.τ− = −
sin α
cos β
gSM
h¯bb
, gSM
hτ+.τ− , (25)
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Introduction
MSSM
References
where α, β are two mixing angles for the 2 complex doublet as in
2HDM.
For mA >> mW , as seen mH ∼ mA ∼ m±
H are very similar. But
formA small compared to mZ such that
m2
A
m2
Z
0, then mA may be a
Higgs lighter than the one discovered at mh 125GeV.
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Introduction
MSSM
References
References
A. Salam & J. Strathdee, Nucl. Phys. B 76, p. 477 (1974)
S. Weinberg, The Quantum Theory of Fields, Vol III:
Supersymmetry, Cambridge University Press (2000)
S. Heinemeyer et al., LHC Higgs Section Working Group
Collaboration (arXiv:1307.1347 [hep-ph])
J. Ellis, Higgs Physics (arXiv:1312.5672 [hep-ph])
G. Degrassi, et al., Higgs mass and vacuum stability in the
Standard Model at NNLO, JHEP 1208, (2012) 098
(arXiv:1205.6497 [hep-ph])
P. Bin´etruy, Supersymmetry: Theory, Experiment and
Cosmology, Oxford University Press (2006)
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