Higgs and Supersymmetry talk

IntroductionCosmology/Astrophysics Implications

MSSMThe Higgs Mass - Evidence for Physics beyond SM

Summarising MSSM Higgs ResultsReferences

Minimal Supersymmetry and Higgs Boson(s)

K. Ahmed

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Outline I

1 IntroductionFundamental ConstituentsThe Couplings UnificationGravity Force and Quantum descriptionSalam’s contribution

2 Cosmology/Astrophysics ImplicationsUnstable Gravitinos as DMExtra Dimensional Theories

3 MSSMHiggs Production at the LHCHiggs DecaysTwo Higgs Doublet Model (2HDM) AnalysisMSSM extension to two Higgs doublets

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Outline II

4 The Higgs Mass - Evidence for Physics beyond SM

5 Summarising MSSM Higgs Results

6 References

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Fundamental ConstituentsThe Couplings UnificationGravity Force and Quantum descriptionSalam’s contribution

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 particlewhich quadratically diverges. The divergent graph arises due to theself 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 II

∆(mH)2S =

λs16π2

[Λ2 −m2

s ln

(Λ2

m2s

)+ . . .

](1)

∆(mH)2F =

λf i2

8π2

[−Λ2 − 3m2

F ln

(Λ2

m2f

)](2)

This divergence is cancelled if one has a corresponding partnercoupled with comparable strength to the scalar Higgs but oppositein 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 thenew physics starts – the hierarchy problem.

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Introduction III

The existence of the matching fermion (spin 12 ) to the scalar Higgs

boson is a requirement of supersymmetry (SUSY) which gives riseto a fermion to every boson and vice versa carrying equal mass inthe exact symmetry limit in order for the cancellation of thedivergence. This is ’naturalness problem’. In other words, in orderto have a ’natural Higgs mass’ SUSY sets an important choice onNew Physics (NP) or physics beyond SM. Further, the Higgs bosonreceives quantum (or loop order) corrections that are limited bythe extent of SUSY breaking (in masses and couplings). A newscale 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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Fundamental Constituents I

Phenomenologically, there are other indicators of SUSY, they are:The fact that in the SM the constituents of matter like quarks andleptons are fermions (spin 1

2 ) [obeying Fermi-Dirac statisticsleading to Pauli Exclusion Principle, i.e., no two identical fermionscan occupy the same state] and bosons carrying force field (spin 1,vector) [obeying Bose statistics, i.e., more than one particlesoccupying the same state] – why this asymmetry?Does nature choose this or is there some underlying subtlesymmetry broken at ordinary energies but may be seen at higherenergies.SUSY affords such symmetrisation between bosons and fermions.In the exact form (unbroken) which is not seen at ordinary energies

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Fundamental Constituents II

(everyday it has the same masses and couplings for both fermionsand bosons).

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The Couplings Unification I

It is found that extrapolation of electromagnetic, weak, and strongcouplings 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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The Couplings Unification II

Figure 2 : Coupling Unification

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The Couplings Unification III

Gauge coupling constants αi = g2i /4π , using Renormalisation

Group (RG) equations, start varying with energy in such a waythat they unify using SUSY at energies of the order of ∼ 1016GeV.In this evolution of various interaction couplings or their inverse tobe precise (in the RG equation), one uses SUSY particles in the1-loop quantum corrections where the coefficients bi of theRenormalisation Group Equation (RGE) assume larger values thantheir SM corresponding coeffiecients. Here bi is defined as

bi = −2πd

dt(α−1

i ), (3)

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The Couplings Unification IV

where t = ln(

qq0

), with q the RGE scale and q0 the SM scale.

Further one uses SU(5) or SO(10) as a grand unified gauge groupand RGE for extrapolation.

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Gravity Force and Quantum description I

Another important requirement for unification theories is the forceof gravity. Theoretically, it is very difficult to develop a quantumtheory of gravity because of divergence problem associated withFeynman diagrams involving interaction with gravity throughgravitons.Superstrings afford a possibility to offset the difficulties ofrenormalisation associated with gravitational field. Supersymmetricgravity theories have been formulated to incorporate grandunification 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 evenin the lightest mass scale, so far. As usual, for NP, physicists waitfor upgradation of accelerator energies.

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Salam’s contribution I

Abdus Salam’s contribution to Supersymmetry was seminal. Healongwith John Strathdee published an important paper onSupergauge transformations (Nucl. Phys B 76, p.477 (1974)) andlater the concept of Superfields which puts bosons and fermionstogether in the form of Supersymmetric multiplets as Superfields.These superfields are, however defined over extended coordinatecontaining self-commuting (ordinary) space-time coordinate xµ aswell as four non-commuting fermionic Grassmnian variables θµ.Steven Weinberg in his book titled ”The Quantum Theory ofFields, Vol III: Supersymmetry” refers to Salam’s (and Strathdee’s)fundamental contribution to Supersymmetry and underlyingframework of Super Algebra.

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Salam’s contribution II

In the words of Weinberg: ”a great deal of work can be saved byusing a formalism invented by Salam and Strathdee in which thefields in any supermultiplet are assembled into a simple superfield.”(A. Salam & J. Strathdee, Nucl. Phys. B 76, p. 477 (1974))

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Unstable Gravitinos as DMExtra Dimensional Theories

Cosmology/Astrophysics Implications I

SUSY postulates the Lightest Sypersymmetric Particle (LSP)called Neutralino (X ) which is thought to be a neutral particleexisting as a supersposition state of Higgsino (supersymmetric

Higgs bosons, h01, h

02), Zino (supersymmetric Z 0 boson) and

photino (γ, supersymmetric partner of the photon γ). This particleis believed to be comprising over 20% of matter/energy densitycompared to the corresponding critical energy density required toclose the Universe since the Big Bang. Such an invisible particle ofmatter is called Dark Matter (DM). The mass limit for such a DMcandidate is of the order of hundreds of Giga electron volts. Thereare other DM candidates such as axion, CP (strong) violatingparticle. Such particles energies may be accessible to neutrinotelescopes which are designed to detect 100’s of GeV particles.

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Cosmology/Astrophysics Implications II

However, the annihilation rates of neutralinos predicted fromMinimal Supersymmetric SM (MSSM) variants in celestial bodiesare low if contraints from (Wilkinson Microwave Anisotropy Probe)WMAP and (Large Hadron Collider) LHC are taken into account.

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SUSY also predicts through its R-parity violating model a longlived but unstable viable candidate of DM called ’gravitino’. Thisis estimated at a mass of few to a few hundred GeV and may bepresent 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 detectorsbecause its interaction with normal matter falls inversely withfourth power of the Planck constant G−4

planck.

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Involving extra dimension range of the order of 10−3 − 10−15

meters can also provide DM candidates. Extra dimensions can alsobe accomodated or required by Supersymmetry, string theory orM-theory, where they give rise to ’branons’, weakly interacting andmassive fluctuations of the field that represent the 3-D brane onwhich the Standard world lives. A stable and weakly interactingobject, branon makes a good candidate for DM as a usual ’relicbranon’ left over after a freeze out period during the evolution ofthe Universe accumulating gravitationally in the halos of galaxieswhere due to their high energies they annihilate into SM particles.Such particles as products fo annihilation can then be detected bygamma-ray telescopes, surface arrays or neutrino telescopes.

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Higgs Production at the LHCHiggs DecaysTwo Higgs Doublet Model (2HDM) AnalysisMSSM extension to two Higgs doublets

MSSM I

In order to look for physics beyon SM, higher energy data and morelumminosity collisions are awaited from the LHC. One should thenexpect to study Higgs couplings more accurately. One also looksfor higher energy accelerators like ILC, Higgs e+e− factories, etc.The objectives are to look for (additional) CP-even states predictedby MSSM or NMSSM (one having an additional doublet and onecomplex singlet to the normal Higgs doublet invariant underSU(2)

⊗U(1) gauge group).

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MSSM II

In the MSSM one has two Higgs doublets,

H1 =

(H1

1

H21

)=

((φ0

1)∗

−φ−1

)(4)

H2 =

(H1

2

H22

)=

(φ+

2

φ02

). (5)

Symmetry is broken through vacuum expectation values of theHiggs doublets as,

< H1 >=

(v1

0

)< H2 >=

(0v2

). (6)

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MSSM III

Mixing of Higgs states is introduced through the mixing angles αand β,

tanβ =v2

v1, (7)

where v1, v2 > 0 and 0 ≤ β ≤ π2 .

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At the LHC, the SM Higgs boson is produced through fourdifferent channels:Gluon gluon fusion channel: gg → hXVector Boson Fusion (VBF) channel: qq′ → hjjXHiggs boson strahlung channel: qq → hVXHiggs boson and top quark pairassociated production channel: q(gg)→ httX

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Higgs Decays I

(S. Heinemeyer et al. LHC Higgs Section Working GroupCollaboration); arXiv:1307.1347 [hep-ph]The Higgs decay rate into a pair of fermion is given at tree level by

Γ(H → f f ) = NeGFmH

4π√

2m2

f , (8)

where Ne = 3(1) for decays into quaks (leptons). Since the treelevel couplings to other particles are propotional to their masses(squared in the cases of massive vector bosons), the dominantHiggs decays are into the heaviest particles that are kinematicallyaccessible, such as, bb, c c and τ+τ−. However, only τ+τ− decaymode, i.e., H →τ+τ− has recently been observed unambiguously

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Higgs Decays II

(ATLAS and CMS collaborations files).Further

Γ(H →WW ∗) =GFm

3H

8π√

2F (r), (9)

where F (r ≡ mW /mH is a kinematic factor) has been observed.

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Two Higgs Doublet Model (2HDM) Analysis I

Now that Higgs Boson has been discovered, a question ariseswhether it is the Higgs Boson of the SM, or whether there aremore?Two Higgs Doublet model and Supersymmetry offer a possibility ofmore Higgs bosons. We now turn our attention to this possibility.Let φ1 and φ2 be two doublet complex scalar fields with weakhypercharge Y = 1, and belonging to symmetry group SU(2)L.

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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 − v21 )2 + λ2(φ†2φ2 − v2

2 )2

+ λ3

((φ†1φ1 − v2

1 ) + (φ†2φ2 − v22 ))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). Aboveequation gives the most general scalar doublet potential subject todiscrete symmetry φ1 → −φ1 which is only softly violated

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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 thepotential is manifestly,

< φ1 >=

(0v1

)< 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 tovanish 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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Two Higgs Doublet Model (2HDM) Analysis IV

Next let tanβ = v2/v1 (Ratio of expectation values of φ2 to thatof φ1) be an important parameter associated with the 2HDM.Next one removes the Goldstone Boson and determines the Higgsstates by rotating:

G± = φ±1 cosβ + φ±2 sinβ, (13)

and Higgs states taken as orthogonal to Goldstone Bosons,

H± = −φ±1 sinβ + φ±2 cosβ (14)

with mass m2H± = λ4(v2

1 + v22 ). Due to CP invariance assumed

before, the imaginary parts and the real parts of the neutral scalar

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Two Higgs Doublet Model (2HDM) Analysis V

fields decouple. In the imaginary (CP-odd) sector, the neutralGoldstone boson is,

G 0 =√

2(Im φ01 cosβ + Im φ0

2 sinβ) (15)

and the orthogonal neutral physical state is,

A0 =√

2(−Im φ01 sinβ + Im φ0

2 cosβ) (16)

with mass m2A0 = λ6(v2

1 + v22 ).

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MSSM extension to two Higgs doublets I

Supersymmetry requires (for minimal case) 2 Higgs doublets; oneto give masses to charge

(+23

)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(

= v2v1

). Simulations have

been done to see that the renormalisation by the top quarkcoupling is important for one of the Higgs multiplet, and may drivem2

Hunegative at the electroweak scale resulting in the electroweak

symmetry breaking and thus may explain negative sign in thequartic term in the effective SM potential. For a heavy top quarkmass, it is then possible for the electroweak scale to be generatedaround 100 GeV if mt ∼ 100 GeV. For this reason SUSY theoristsactually suggested heavy momentum for the top quark, before its

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MSSM extension to two Higgs doublets II

discovery!Now 2 complex Higgs complex Higgs doublets of the MSSM haveeight degrees of freedom, of which 3 are used by the HiggsMechanism for electroweak symmetry breaking to give mass to theW± boson and Z 0, leaving 5 physical Higgs bosons states of these2 (h, H) are neutral Higgs that are CP-even (scalar), one A isneutral CP-odd (pseudoscalar) and 2 are charged, the H±. At treelevel the masses of the scalar Higgs(es) are:

m2h,H =

1

2(m2

A + m2Z ∓

√((m2

A + m2Z )2 − 4m2

Am2Z cos2 β)) (17)

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MSSM extension to two Higgs doublets III

In general their coupling compared to SM couplings are:

ghVV = sin(β − α)gSMHVV , gHVV = cos(β − α)gSM

HVV (18)

ghAZ = cos(β − α)(g ′

) , ghbb+, ghτ+τ− = − sinα

cosβgSMhbb

, gSMhτ+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 2ndlightest discovered.

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The Higgs Mass - Evidence for Physics beyond SM? I

(J. Ellis, arXiv:1312.5672 [hep-ph]) CMS and ATLAS results ofHiggs mass are quite consistent and a naive global average (for theHiggs mass) is

mH = (125.6± 0.4)GeV (20)

And this average is quite consistent with the electroweak databased on one-loop level SM collaboration to ∆X 2 ∼ 1.5 level.However, when effective Higgs potential is considered then thereare problems. When self renormalisation effects are taken intoaccount for the Higgs field coming from Higgs self-coupling andHtt coupling, one can write the Higgs self-coupling as:

λQ =λ(v)

1− 34π2λ(v) ln Q2

v2

+ . . . , (21)

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The Higgs Mass - Evidence for Physics beyond SM? II

where Q is some renormalisation scale above the electroweak scalev . And due to Htt coupling; i.e., when

λ(Q) = λ(v)

[1− 3

4π2λ(v) ln

Q2

v2

]−1

= λ(v)− 3m4t

4π2v4ln

Q2

v2+ . . . , (22)

Where in the above equation, non-leading terms with RGE solutionhave been ignored.

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One notes that renormalisation of the Higgs self coupling in theRGE solution for large Q values tends toincrease λ(Q) in (18) leading to a landau singullarity (Landau Polearises for largeQ2 values relative to v2 as: Q2 = v2 exp(4π2/3λ(v)) ). While in(19) it decreases the Higgs self coupling λ(Q) with increasingQ-values. At some point when Q is sufficiently large relative to v(electroweak scale), λ(Q) is driven to negative values. This wouldset 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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[G. Degrassi, et al. JHEP 1208, (2012) 098]The measured value of mH plus mt ' 173 GeV would drive thequartic self-coupling λ to negative values for some energy scale∼ 1010 to 1014 GeV, if no physics beyond SM intervenes at lowerenergy scale as shown:

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Figure 3 : Higgs mass Mh in GeV

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Figure 4 : Higgs pole mass Mh in GeV

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The instability of the vacuum having large negative value for largeQ−value of the order of 1010 to 1014 GeV (approaching Planckscale/Planckian era) is hard to reconcile with the present value ofcosmological 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 themass of the Higgs boson can be extended and defined to higherloops graphs also, in the same self-consistent way. Also as we sawthat existence of the Higgs mass as found alongwith top quarkmass found also empirically provides through electroweak vacuumstability 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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The implications of the mass requirement of mH and mt areplotted in Figures 3 and 4. The result in (23) is based onNNLO-SM calculation by Giuseppe Degrassi et al,(CERN-PH-TH/2012 134 RM3-TH/12-9) and says that forvacuum stability for Q values from 1013 − 1014 GeV, the mass ofMH > (129.4± 1.8) GeV.

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It predicts, Higgs mass

m2h,H =

1

2(m2

A + m2Z ∓

√(m2

A + m2Z )2 − 4m2

Am2Z cos2 2β) (24)

β = tan− 1(v2v1

)Couplings,

ghVV = sin(β − α)gSMHVV

gHVV = cos(β − α)gSMHVV

ghAZ = cos(β − α)g ′

2 cos θW

ghbb, ghτ+.τ− = − sinα

cosβgSMhbb

, gSMhτ+.τ− , (25)

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where α, β are two mixing angles for the 2 complex doublet as in2HDM.For mA >> mW , as seen mH ∼ mA ∼ m±H are very similar. But

formA small compared to mZ such thatm2

A

m2Z' 0, then mA may be a

Higgs lighter than the one discovered at mh ' 125GeV.

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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 GroupCollaboration (arXiv:1307.1347 [hep-ph])

J. Ellis, Higgs Physics (arXiv:1312.5672 [hep-ph])

G. Degrassi, et al., Higgs mass and vacuum stability in theStandard Model at NNLO, JHEP 1208, (2012) 098(arXiv:1205.6497 [hep-ph])

P. Binetruy, Supersymmetry: Theory, Experiment andCosmology, Oxford University Press (2006)

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Higgs and Supersymmetry talk