Physics Beyond The Standard Model

Arunansu Sil

Department of Physics

Indian Institute of Technology, Guwahati, India

30th April, 2012 BUE, Cairo University, Egypt

The Standard Model: present status and open problems

The Standard Model of Particle Physics: a mathematically consistent theory which describes strong, weak and

electromagnetic interactions at low energies; highly constrained theory. compatible with (almost) all experimental results.

Why then Beyond Standard Model ?

Theoretical issues: It has 19 free parameters 1. three gauge couplings

2. two parameters in the Higgs sector

3. nine quaks and charged lepton mass

4. three mixing angles and one CP-violating phase in the quark system

5. the QCD parameter 𝜃

Why BSM contd..

In the SM, if we extrapolate the behavior of the forces at higher energies:

the strength of the 3 forces become similar though never exactly the same.

It provides a strong hint toward the long standing goal of unifying the forces of nature.

Perhaps Grand Unification Theory [will be discussed ]

In the SM, quantum corrections apparently make the Higgs boson mass huge. This poses a serious conceptual problem (called hierarchy problem). [will be discussed in the context of Supersymmetry]

Other open problems/less-elegant features:

Non-zero neutrino mass: neutrinos mass can not be accommodated in SM as no right handed neutrino is present in the SM structure: so No Dirac mass.

Neutrino experiments: at least 2 of the 3 three neutrinos are massive ~ 1eV or less.

Open problems of SM contd. solution: new physics can generate dim-5 operator L = yLLHH /M + h.c. M= scale for the new physics i.e. mass of the field that was integrated out. Only three possibilities: introduction of i) a singlet field: right handed neutrino [Type I see-saw] ii) a scalar triplet [Type-II ..] iii) a fermionic triplet [Type III ..]

[This can arise naturally in GUT models]

Charge quantization: can not explain the fractional charge of quarks. [answer from GUT]

No family structure: Why three generations of matter?

Gravity: SM does not include gravity; not even understood whether gravity is to be treated as a gauge theory, since it has not been quantized as the other interactions.

(Extra dimension, String theory)

Questions contd..Cosmic Pie

Dark Energy and Cosmological constant problem: It has been known recently that the expansion of our universe is accelerating and what causes it (Dark Energy) cannot be explained in the SM particle. It requires Vacuum energy density ~ 10-12 eV2

In SM, expected vacuum energy ~ 1048 eV2

almost 60 orders of magnitude difference !

Solution: quintessence or cosmological constant. [Nothing known so far]

Open problem in SM contd.

Almost 25% of our universe is made up with invisible Dark Matter (non baryonic) that cannot be particles within the SM.

Evidence from galactic rotation curve etc. , has to be very weakly interacting particle.

Questions contd..

If there is Supersymmetry, dark matter candidates comes naturally.

There is good evidence that in the very early stage (within first fraction of second after the big bang), the universe went through a stage of extremely rapid expansion, called Inflation. The field responsible for inflation cannot be the SM ones.

[will be discussed]

The idea that universe began from big bang, with a huge burst of energy indicates that it should have evolved into equal parts of matter and antimatter CP symmetry. However we could see mostly particles ! This matter-antimatter asymmetry cannot be explained in the Standard Model.

With the extensions of the SM, we have better understanding of some of

the queries listed above, however those also led to new questions that

could not be formulated earlier.

Grand Unified theories

Standard Model is based on GSM : SU(3)C X SU(2)L X U(1)Y

The SM has 3 gauge interactions described by the corresponding carriers Ga

µ (a = 1 … 8) ; Wiµ (i = 1 … 3) ; and Bµ

Dµ = ∂µ + ig3 ∑a=1,8 Gaµ λa/2 + igw ∑a=1,3Wa

µ σa/2 + igy Bµ Y/2

g3,w,y : coupling for each three interactions (unrelated parameters).

λa: Gell-Mann matrices for SU(3); σa = Pauli matrices for SU(2) and Y: hypercharge operator.

It has also three types of 9 Yukawa matrices

Sin2θw = g2Y /(g2

W + g2Y);

as couplings are dependent on energy scale, µ: Sin2θw (µ) = αY(µ) / [αw(µ)+ αY(µ)]

Grand Unification contd.

The idea of GUT: to reduce all the gauge interactions to one single gauge group with a

single coupling constant.

Also running of the gauge couplings give hint for unification.

Since at low energy, SM is performing extremely well, GUT should break down to

SM group. Breaking of that larger group into GSM by Higgs mechanism.

SM couplings run as they depend on the energy scale, µ [use of RGE]. We want to study

whether we achieve α1 (Mx) = α2 (Mx) = α3 (Mx) = αU

Choice of the gauge group:

1. SM gauge group should then be a subgroup of the grand unified gauge group,

2. all the SM fermions should be included in the GUT matter representations remembering

that list of non-SM fields are minimum with the specific choice.

3. GSM consists of total 12 generators, four of them be simultaneously diagonal (Rank 4).

So the minimal choice for the GUT group should be based on a Rank 4 simple group.

Finally SU(5) being Rank 4, enables us to consider only SM fields.

Starting with SU(5)

Each family (one generation) fits nicely into the SU(5) multiplets:

1. In the SM, there are 15 left-handed two-component fermion fields in each family (generation): so it fits nicely in the SU(5) multiplet chosen.

2. We identify the first three of the SU(5) indices as the colour indices and the remaining two as SU(2)L indices.

SU(5) contd.Dµ = ∂µ + ig5 ∑a= 1, 24 Aa

µ λa /2

Gauge bosons: SU(5) adjoint representation has dimension 52 – 1 = 24 and

Its SU(3) X SU(2) decomposition:

24 = (8,1) + (1,3) + (1, 1) + (3, 2) + (3*, 2)

12 extra gauge fields: X and Y bosons

SU(3) gluons W B-field

<H> <Φ>

Spontaneous symmetry breaking: SU(5) ----> GSM ----> SU(3) X U(1)

Minimal choice for Higgs sector: two scalar representations 24H and 5H .

identified with H and Φ = T (since 24 is an adjoint reps, it does not break rank) Ф

__

Yukawa sector: LY = 5F Y5 10F 5*H + 1/8 ε5 10F Y10 10F 5H + h.c. SU(3) Color Triplet

Running and unification

RGE at 1-loop: αi-1 (µ2) = αi

-1 (µ1) – bi /4π ln(µ22 / µ2

1), µ1 > µ2

bi coefficients are group theoretically derived.

Low energy results: α-1 (MZ) ≈ 128 (α = αw Sin2θw )

αs (MZ) = 0.1184(7)

Sin2θw (MZ) = 0.23116(13)

By evaluating how the couplings evolve to high energy scale, one obtains:

Some features of SU(5)

Conclusion: with the bottom-up approach, we see that the SM does not unify exactly within minimal

SU(5). The same observation can be found in Top-bottom approach: impose unification at some scale and run the couplings down to SM scale.

With α-1 (MZ) ≈ 128 and αs (MZ) = 0.1184(7), ΛGUT ~ 6.7 x 1014 GeV with α5

-1 = 41.5

But, Sin2θw (MZ) turns out to be 0.208, so this fails.

Inclusion of Supersymmetry will improve it:

Nice features: Unification of all the interactions within a simple group partial unification of the matter with the two multiplets explanation of electric charge quantization. [eigenvalues of the generators of a simple non-abelian group are

discrete; quarks carry 1/3 unit of the lepton charge as they have 3 colours]

Problems: Lepto-quarks, di-quarks and proton decay: X and Y bosons can mediate proton decay.

u e+

lc q

X X

q qc d uc

B = -1/3 B = 2/3 u u 𝜋0

Doublet triplet splitting problem Neutrino mass can not be accommodated

Supersymmetry Supersymmetry (SUSY): a symmetry between bosons and fermions.

Need for Supersymmetry: (the problem of quadratic divergencies in SM)

two-point function receives a contribution from the fermion loop:

considering q1 = q2 = 0 and k1 = -k2 = k say, with λf Hff

πfHH

(0) = -

fermion-antifermion contribution to the = -

self-energy of the higgs boson in the SM

N(f) = 3 for top quarks = -

1st term: quadratically divergent ! Term is proportional to the cut-off of the theory, Λ.

For Λ ~ MPl, then, MH2 = MH, tree

2 + cg2/4π2 MPl2

f

ff

ff

mk

ii

mk

iitr

kdN

)()()2(2 4

4

222

22

4

42

)()2( f

fff

mk

mkkdN

222

2

224

42

)(

21

)2()(2

f

f

f

f mk

m

mk

kdfN

Solution of quadratic divergence Therefore extreme fine tuning is needed (need to cancel in all order)

Solution through SUSY:The problem is solved by additional contributions to πf

ΦΦ.

Contribution from the 2nd diagram:

Therefore quadratic divergences can be

made to cancel, if required also for the scalar potential to be bounded from

below

We can achieve complete cancellation between fermion and bosonic contributions, if in addition we

have

RL ff

ff

HH mkmk

kdfN

~~

~

22224

4~ 11

)2(')()0(

ff

RL fNfNfN2

~~

'

)()()(

ffmm ~

SUSY formalism If Q is the generator of the symmetry which turns a bosonic into a fermionic state

Q must carry half-integer spin (fermionic and hence anticommutating algebra).

SUSY Algebra:

Weyl spinor index:

left handed: α = 1, 2 right handed:

Mass dim. Q = ½

Dim: ½ 1 3/2

Superfield formalism:

4D spacetime superspace

fermionic coordinate (dim: -1/2)

SUSY contd.. Chiral superfield: SM fermions are chiral (their left and right-handed components transfer

differently under SU(2) X U(1))

Transformation law:

dim: 1 1 -1/2 3/2 -1 2spin: 0 0 1/2 0

Wess-Zumino model

Kahler potential:

Superpotential:

Canonical Kinetic term:

Mass, interaction terms

F has no kinetic terms; auxiliary field.

= 0

(F-flat condition)

where Fi are solutions of equation of motion:

Scalar potential: >0 (positive definite)

SUSY vacuum: Vmin = 0

Vector superfield:

MSSM Gauge invariant superpotential:

Yukawa couplings µ-term

B/L number violating terms:

Proton decay:

R-parity: forbids

Under R‐parity Even Odd Neutral LSP: Candidate for Dark matter

SM fields SUSY partners

Particle content

SUSY breaking SUSY must be broken (but SOFTLY)! Otherwise

1. F-term should be nonzero

2. Needs to be softly broken to maintain

the cancellation of quadratic divergences.

SUSY is broken in a Hidden Sector

gravity/gauge interaction etc.

Spurions:

Effective interaction between

X and MSSM fields:

Hidden SectorFx

MSSM

SUSY breaking contd..

Sfermion mass:

A-term:

Gaugino mass:

For gravity mediation, M = MPl

(In case of gauge mediation, this can be replaced be the messenger mass)

Soft SUSY breaking terms in MSSM

mass2 correction to Higgs doublets is negative!

leads to radiatiove EW symmetry breaing.

SUSY Breaking model building

Most common approach to mediate the supersymmetry breaking to the MSSM sector: 1. Gravity mediation; F1/2 ~ 1012 GeV2. Gauge mediation; low energy susy breaking.

Here we will focus upon the supersymmetry breaking sector(based on Metastable SUSY introduced breaking by ISS)

Conventional picture of Dynamical Supersymmetry Breaking (DSB) The idea is to use an asymptotically free gauge theory like QCD, so that

the smallness of the SUSY breaking scale can be explained dynamically…

Msusy~ MCutoff exp(c/g2(MCutoff)) V Kahler potential is hard to

control! Metastable SUSY breaking

overcomes this….

Vacuum Energy VDSB > 0 Ф

SQCD

Gauge group Flavour

SU(NC) Nf

SQCD Duality (electric)

Local SU(Nc); Nf flavours Qi, Q’j

Coupling becomes ~ O(1) at the scale Λ

Global : SU(Nf) x SU(Nf) x U(1)B x U(1)R

Q : Nf 1 Q’ : 1 Nf

Superpotential: Introduce quark mass (ISS approach) [Intriligator, Seiberg, Shih, 2007]

WUV = Tr m Q Q’ ; m « Λ

Breaks Chiral symmetry.

R(Q) = R(Q’) =1

Duality contd..

1. Local SU(Nf - Nc) ;

Nf flavours qi, q’j + singlet N2f “mesons” Φij = Q’i Qj / Λ

2. Coupling gm ~ O(1) , i.e. becomes strong at Λ

It would be IR free for Nc + 1 < Nf ≤ 3 Nc /2 β function: b0 = 2Nf – 3Nc < 0 3. The running of the gauge coupling of g SU(Nf – Nc):

4. Same global group. g SU(Nc)

SU(Nf -Nc)

Λ E

fc NNEg E

e23

)(

82

2

WWUVUV = Tr m Q Q’ = Tr m Q Q’

Duality continued…(magnetic)

4. Superpotential:

WIR = Tr (q’Φq) (Yukawa int.)

+ Tr (mΦΛ) (quark mass) + P Λ3 [det Φ/ΛNf]1/P ; P = Nf – Nc (Non Perturbative )

RIR (Φ) = 2 , RIR (q) = - RIR(q’)

5. The Kahler potential for the IR free fields is smooth near the origin and can be taken to be canonical,

222 |'||||| qqTrK

ISS bottom-up

Minimal example with Nf = 7 and Nc = 2

From Rank condition

So Supersymmetry is broken by rank condition with Vacuum energy

V = Nc µ4

Φ = 0 0 , q= q+ , q’ = q’+ , q’+q = µ2 IP 0 Φ+ 0 0

The pseduo moduli Φ+ is lifted by 1-loop effective potential. The local Non-SUSY Vacuum is therefore:

0' 2*

bac

bcaa

b

qqW

F ab

Magnetic

SU(7) X U(1)B X U(1)R global

SUSY Breaking Vacuum Φ = 0 0 , q= q’T = µ2.Ip 0 0 0 .INc

The non-perturbative superpotential gives negligible contribution to the effective potential around this vacuum and can be ignored here.

Add the N.P. term

This form is exact and uniquely determined by the symmetries of the theory and generated by the SU(Nf -Nc) gaugino condensation.

The Susy Minimum is given by:

WNP = Λs3(det Φ / ΛΛ77

ss ) )1/21/2

Restoration of Supersymmetry

Φ0 = <Φ> = µ[Λ/µ]X = µ/εx , x = (3Nc -2Nf)/Nc

<q> = <q’> = 0, ε = µ/Λ So we find the SUSY vacuua in the magnetic theory, staying below its cut-off; Φ0 < Λ.

V

m QQ’ 1 loop correc.

Wdyn

0 Φ0 Φ

Metastability

Lifetime of metastable vacuum:

Γ = e-s4 , s4 = ΔΦ4 / V+ ≈ ε-4x » 1

By choosing µ « Φ0 < Λ , it is possible to ensure the metastability

of the non-susy minimum.

ISS contd.

Local SU(5), with Nf = 7, Qi (fundamental), Q’i (anti-fundamental)

Superpotential: W = mQQ’ µ2 = Λs m since Φ = QQ’/Λs

Chirality breaking (UV) Susy breaking (IR)Choice of m is bit adhoc! We want to generate it dynamically

through inflation…. Coupling like f(X) QQ’ mQQ’

Addition of the N.P. term in the superpotential allows for SQCD completions.

Electric

GMSB and dISS

SUSY R-symmetry Gaugino mass by R ≠0, 2 fields Messengers qi>2 in 5 of SU(5)GUT

W = qi Φij qj’ - µ2 TrΦ + mq1 q2

------------------------------------------------------------------------------------------------------------------------------

SU(7)flavour SU(2) x SU(5) Ξ SU(5)GUT

IR UV------------------------------------------------------------------------------------------------------------------------------------------

WdISS= mqdet q1q2 WdISS= Be /M2pl

Bm /ΛS3 Ξ Be = det Q SU(5)

For Nc=2, det qi ≈ q2 breaks baryon number it fixes R(q) = 1

pl

s

q Mm

2

3

GMSB contd.

Now <Φ> ≠ 0 , R-symmetry is broken. FΦ ~ mq Λs = µ2; Fq ~ mqµ mgaugino≠0 since <q> ≠ 0 Metastability requires µ ~ mq « Λs

for mgaugino ~ TeV, µ ~ 109 GeV [Abel et. al]

mq = µ2/Λs ~ TeV, Λs ~ TeV

What could be the origin of it?

Perhaps Inflation !

E ~ 100 GeV (corresponds to time t ~ 10-10 sec considering the evolution of the universe)

Perhaps the early universe was the ultimate particle accelerator, and will provide the first glimpse of physics at the scale of Grand Unified Theories (GUTs), or even the Planck scale.

An interesting question is whether SUSY plays a fundamental role too at the early stages of the evolution of the universe and, more specifically, during Inflation !

Inflation: Inflationary Universe theory is an add-on to the standard Big-bang Cosmology: introduced as a resolution to the Horizon and the Flatness problem .

It corresponds to a period of accelerated expansion of the very early universe. a(t) ~ exp(Ht).

Modeling of Inflation

A slowly rolling scalar field:

Go back to Friedmann equation: which gives a(t) ~ exp(Ht)

i.e. exponential expansion of the Universe.

Slow Roll parameters :

0

2.

2

3

8V

G

a

aH

0303 '.

'...

VHVH

2'2

2

V

VmplV

Vmpl

''2

Can SUSY play any role ? Indeed!

Vacuum Energy during Inflation: the level of density and temperature fluctuations observed in the present universe, δH = 1.94×10-5, require

the inflaton potential to be extremely flat, e.g. in Chaotic inflation model, V = c Φ4

c ~ 10-13 to be consistent with observations. Also the inflaton should be of the order of the Planck mass during inflation.

This is where supersymmetry comes to rescue.

For example in Supersymmetric Hybrid Inflation model, no such small parameter

is required.

How to realize inflation in a supersymmetric framework?

Consider the superpotential:

INFLATON WATERFALL FIELD RESPONSIBLE FOR REHEATINGScalar Potential:

For S > M and 0 =א , local minimum with vacuum energy V ~ k2M4 : responsible for Inflation!

End of Inflation: when Sc < M, the mass term of S becomes negative, and the system ends V Inflation up in the supersymmetric minimum at

M and S = 0 = <א>

0 Χ

NfMkSW ,)( 22

....222222 MkSkV

Motivation

On the other hand we know that the world is not globally supersymmetric: In Nature, supersymmetry must be a broken symmetry.

We wanted to study whether there could be a relation between the Inflation

vacuum energy and supersymmetry breaking.

SET UP: 3 fold Universe

SUGRA

INFLATION SUSY Breaking(ISS)

MSSM

GMSBREHEAT

RH

RH