Physics beyond the Standard Model - Herbstschule Maria Laach

**Physics** **beyond** **the** **Standard** **Model**

Basics and Phenomenology of Supersymmetry

The

Problems

Abdelhak DJOUADI ( LPT Orsay)

of **the** SM: GUTs, SUSY, alternatives

SUSY

and **the** MSSM

Higgs and SUSY particle spectrum in **the** MSSM

Higgs

and sparticles at colliders

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.1/36

1. SUSY–GUTs: general features

Low–energy SUSY comes to **the** rescue!!

SUSY-GUTs are **the** most attractive extensions of **the** SM

For each SM particle, **the**re is a SUSY partner with spin–

¡ diff.

(**the**re are also two Higgs doublets, see **the** discussions later).

Some **the**oretical advantages of SUSY **the**ory, are:

It

is **the** largest symmetry that an S–matrix can have

(so, we go fur**the**r in our quest for/use of symmetries in Nature..).

If

SUSY gauged, we get a spin–

¢ ¡gravitino

and spin–2 graviton

(and **the**refore can include gravitational interaction: conceptual SM pb!).

It

is a natural part of Superstrings (**the**ory of everything!?).

If SUSY is realized at low energies (which means £

¤

£¥

¦

§ TeV),

it solves **the** problems discussed in **the** case of non–SUSY GUTs!!!

Let us take **the** example of **the** MSSM, for instance (details later):

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.2/36

1. SUSY–GUTs: gauge coupling unification

The new SUSY particles will contribute to **the** running of ¢¡¤£

(one has to add a contribution of

§

¡

§

§

¥

¦

£ £

£ £

£ £

§

§

we get after calculation

-1

¥

¥ ¦

¡ α i

60

50

40

30

20

10

¦

¡

¡

¥

¦

¡

§©¨

for gauginos)

¥ £

¦

¥

¥ ¦

U(1)

¦

¡

SU(2)

SU(3)

¡

compared to

¦ ¦

MSSM

0

10 2

10 4

10 6

10 8

10 10 10 12 10 14 10 16 10 18

μ (GeV)

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.3/36

¡

¥

§

!

1. SUSY–GUTs: gauge coupling unification

Alternative view: **the** running couplings meet at a single point

obtained from ¡£¢

¤

¥

Note that **the** small discrepancy is due to experimental errors on©

and also to small threshold corrections (new particles) near ;

two loop corrections must be also included for more precision...

Important: for this to work, we need £

Note also that larger is good to prevent proton decay:

In non–SUSY GUT, ¤

In

¦

§

¡

¡

¡ ¡

¨©

§ ¥ ¦

¦

¤

£¥

§

¡

¥

GeV, 10 times smaller...

SUSY–GUTs: additional colored Higgs/higgsino exchange

£

£¥

¤

§

¤

Larger but very close to experimental bound; to be observed soon?

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.4/36

§

¦

¦ years

§

©

¥

¥ ¡

!!!

§

1. SUSY–GUTs: Yukawa coupling unification

One can also unify **the** (3d gen.) Yukawa couplings at ¤

£

£

£

¡

¡

¡

¤

¤

¤

¥

¥

¥

¢

¦

¦

¦

tan β

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¡ ¦

¡

¡

¡

¢

100

10

§

§

§

1

¨©¨

¥

¨¨

¥

¨

¥

M

α s

SUSY

( M ) = 0.118

Z

λ =

b λ τ

m =

t

¥

¥

¥

λ

¡

¦

¥ ¦

100 120 140 160 180 200

¥

b = λ 3.1

= =

t λb

λτ

¡

¥ ¦

¦

¥ ¥

¥ ¥

¡

¡

¦

¡ ¥

¡ ¥

¥ ¥

m b = 4.4

m b = 4.1

mt(mt) (GeV)

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.5/36

λ t = 3.3

¡

¥ ¡

1. SUSY–GUTs: **the** hierarchy problem

In SUSY, we need to add contributions of **the** new particles to

Ex: in **the** case of fermion loop, add **the** contributions of (2) scalars

Again, quadratic divergences for scalars (see later). But assume:

Scalar

Multiplicative

couplings related to fermion couplings:

factors are **the** same:

¦

¤ (nb:

To simplify, **the** scalars have **the** same mass: ¨¡

and add **the** fermionic and scalar contributions; we get:

¥ ¡

©

¤

£

¥¤

¤

¥

¨

¥ ¤

¤

¦ ¡¢ ¥ ¨

£

¥¥¤

£§¦

.

2 scalars).

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.6/36

¨¦

¥ ¤ ¡¢ ¤

¨

¨¥

¨¦

¨¦ ¨¤

.

¥¢¡

1. SUSY–GUTs: **the** hierarchy problem

The quadratic divergences have disappeared in **the** sum!!

(**the** same job can be done for (s)contributions of W,Z,H etc..).

Logarithmic divergence still **the**re, but contribution small.

No divergences at all if in addition ¨¦

Symmetry fermions–scalars no

“Supersymmetry” no divergences at all: ¡

Note that if ¦

In summary:

**the**

§

¨¤ (exact SUSY)!

divergence in

is protected!

fine tunning problem is back!!!

– gauge symmetry protects **the** gauge boson masses

– chiral symmetry protects **the** fermion masses

– SUSY is needed to protect **the** scalar masses.

We need SUSY at low energies, ¦

(also to solve **the** gauge unification and Dark Matter problems..).

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.7/36

¦¢

§

¡

!

¡

In

1. SUSY–GUTs: Dark Matter

minimal SUSY models, **the**re is a symmetry called R–parity which

makes that **the** lightest SUSY particle is absolutely stable.

Experimental

In

¦

¦

§

–50

constraints show that this LSP must be quite heavy,

GeV, and thus is non–relativistic (cold dark matter).

most areas of **the** SUSY parameter space, this LSP is electrically

neutral and it interacts very weakly.

For

some values of its mass and couplings, this LSP can have a

relic abundance which is within **the** range given by WMAP data.

The LSP is an ideal candidate for Dark Matter!

Again, for this to work, **the** particle must be lighter than

At least 3 good reasons to believe in SUSY!

in addition to SUSY being part of superstrings and link with gravity!

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.8/36

!

2. Supersymmetry: Basics

Here, we give only basic facts needed later in phenomenology discussion

For details on **the**oretical issues, see **the** lectures of Sven Heinemeyer.

SUSY: a symmetry relating scalars/vector bosons and fermions.

SUSY generators

must

¨ is

©¡

¢

£

¤¢¥

©¦¢

transform

§

¢¥

fermions into bosons and vice–versa:

be an anti–commuting (complicated) object.

also a distinct symmetry generator:

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¢

£

¤¢¥

©¦¢

§

¢¥

Such **the**ories are highly restricted [e.g., no go **the**orem, see SH]

and in a 4-dimension **the**ory with chiral fermions [as in **the** SM]:

¨ carry spin–

¡ with

§ ©¦¢ ¢¥

§ ¨©¦¢ ¢

¥

©¡

©¡

¢

£ ¤¢

¢

£

¤¢¥

L,R helicities and **the**y should obey....

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.9/36

¥

2. Supersymmetry: Basics

.... The SUSY algebra: (which schematically is given by)

©£ :

¥

¢

£

¦

¨¡

¤

¥

generator of space–time transformations.

generators of internal (gauge) symmetries.

SUSY:

¢

£

¢

£

¨¦

¤

¥§

unique extension of **the** Poincaré group of space–time

symmetry to include a four–dimensional Quantum Field Theory...

Single–particle states of **the** **the**ory are in irreducible representations

of **the** SUSY algebra above, which are called supermultiplets.

Fermions and bosons of same supermultiplet are superpartners.

They must have **the** same mass and gauge quantum numbers.

Three types of supermultiplets are needed...

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.10/36

¡

¤

¨

¨¡

¨

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¤

¥§

¨

¤

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2. Supersymmetry: Supermultiplets and Superpartners

Chiral

(or “scalar”) supermultiplet (

with

– 1 two–component Weyl fermion with spin

– 2 real spin–0 scalar = 1 complex scalar ( £§¦

Gauge

(or “vector”) supermultiplet (

¨ £and

¡

¢

¡( £¥¤

– 1 two–component Weyl gaugino–fermion with spin

– 1 real spin–1 masseless gauge vector boson ( £¦

Gravitational supermultiplet:

– 1 two–component Weyl gravitino–fermion with spin

– 1 real spin–2 masseless graviton ( £¦

Ex:

with

)

£©

)

):

and S):

¡ ( £§¤

being 2–component Weyl LH/RH fermions

Each state has a complex spin–0 superpartner noted

One can define

and

¨

so that one has:

two LH chiral supermultiplets for **the** electron:

¢

¢

¢ ¡ ( £§¤

)

)

and

The same for all o**the**r leptons and quarks (except for massless ).

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.11/36

.

.

)

)

All

¢¡£

2. Supersymmetry: Interactions

fields involved have **the** canonical kinetic energies

¤

¥§¦

¨

£

© ¦

with D **the** covariant derivative. [Note that

£

£

The

£

¤

¤

¨

£

© ¦

¦

¨

£

¦ £

¡

£

have

¥

£

¡

£

4(2) comps.]

interactions are specified by SUSY and gauge invariance:

£

¥

£

£

¥

¤

¤

¡ ¥

¦

All interactions are given by **the** gauge coupling constants ¡ £

© ¦

¦

¦ £

(fundamental prediction of SUSY: same in gauge and Yukawa int.)

At

this stage, a very simple and minimal **the**ory:

Everything is completely specified and no adjustable parameter!

£

© ¦

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.12/36

© ¦

¥

¥ £

¦

¦ ¥

£

¨

2. Supersymmetry: Superpotential

Only freedom: choice of Superpotential W (SUSY and gauge inv!).

It gives **the** scalar potential and Yukawa interactions (fer.–scal.).

– function of **the** superfields ¦ only (not

– Analytic function: no derivative interaction.

– Renormalizability: only terms of dimension 2 and 3.

¡

To obtain **the** interactions explicitly: take

The SUSY tree–level scalar potential is

F–terms

¦

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¢

¢¤£¥

¡

from W through derivatives wrt all scalars

©

¥

¦ ¦

¥ ¡

¦

¨

¨ ¦

¤

¢¤£¥

¢¤£§

¢

§

¦ !).

© £¥

©

¦ ¥.

¡

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.

¤

D–terms

¡ ¥

¦

¦

¦

¦

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©

¦©

¥

¤

corresponding to **the** U(1),SU(2),SU(3) gauge groups:

¨

¦

¨

¦

¡ ¥

¡

¦

©

¦

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.13/36

¦

© ¦

¨

¨© ¦

© ¦

¥

2. Supersymmetry: SUSY–breaking

SUSY cannot be an exact symmetry since no scalars exist with **the**

same mass as known fermions (smultiplets) must be broken.

Spontaneous SUSY breaking?

Means that **the** Lagrangian is invariant under (global) SUSY but

**the** ground state

© is not:

©

and

¨©

Recall: Hamiltonian is related to **the** SUSY charges:

So that one has: ¡

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©

¡

©

©

©

¡

In fact, **the** vacuum energy should be positive:

¥¤

©

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does

¥¤

©¨

requires

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¤

§¤ or

D–term breaking: leads to CCB minima

not work in **the** MSSM!

¤

§¤ or F–term breaking): needs a linear, ©¦

a singlet sfield under

£ ;

not in **the** MSSM!

©

£ ¢

§¦ ,

.

¢¤£

¨©

.

¨¡

term in W

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.14/36

¦

©£

2. Supersymmetry: explicit breaking

Solution: SUSY–breaking occurs in a hidden sector of particles with

no (or very tiny) couplings to **the** visible sector of **the** MSSM.

If mediating interaction is flavor–blind, universal breaking terms.

Examples: gravity (mSUGRA), gauge (GMSB) mediation ...

Many breaking schemes but none is fully satisfactory at **the** moment:

We

Explicit

breaking by hand (also with several possibilities...).

need SUSY breaking at low energy to solve **the** problems:

– Quadratic divergences in **the** Higgs sector.

– Unification of **the** coupling constants of

– Dark Matter problem (existence of a massive stable particle), etc.

In

**the** breaking, we still need to preserve: gauge invariance,

renormalizability, and no quadratic divergence (soft SUSY–breaking).

“Low energy SUSY” effective

¡

¢

¤£

¥

**the**ory at low energy.

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.15/36

¡

¦

¨§

¥

¡

©

¥

.

3. The MSSM: gauge group

As discussed earlier, in SUSY **the**ories:

– For each SM particle, **the**re is a SUSY partner with spin

– SUSY must be broken at a low scale £

¤

£¥

§

¡difference.

To solve **the** unfication, hierachy and dark matter problems of **the** SM.

The MSSM is **the** most economic low energy SUSY extension of SM.

The MSSM is based on **the** following assumptions:

Minimal

gauge group:

£

The SM spin–1 gauge bosons [

gaugino partners

Superfields

¥

¥

¥¦

¥

§

¡¡

©¢

¤£

¡

¡ §

¦

©¢

¦ are

8 1 0

1 3 0

1 1 0

¡

¢

£

¥

¦ and ¡

¡

¦

§

in vector superfields.

¢

§

¢

§ ]

¥

¡

©

and **the**ir spin–

Particle content

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.16/36

£

§

£

£ ¦

§¦¦

¡

Minimal

3. The MSSM: particle content

particle content:

– Three fermion generations [as in SM no

partners, **the** sfermions

– No chiral anomalies ( ¤

SUSY invariant way (no conjugate

two chiral superfields with

Superfield

¡

¡£¢

¡©¡

¡ ¦¡

¡¡

¡

¡

¡

...] and **the**ir spin–0 SUSY

, combined in chiral supermultiplets.

¤

field

)

and fermion mass generation in a

§ and

for u–quarks), we need:

§ .

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¤£

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(

Particle content

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.17/36

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©

¤¦¥

§ ¥

¥

¥

(

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§

), (

¨

¨

,

,

,

, (

¨ ¥

¤ ¨

¨ §

¨ ¤¦¥

¨ § ¥

¨ ¥

¨

¡

¡

)

)

)

R–parity conservation:

3. The MSSM: R–parity

To eliminate terms violating B and L numbers (and proton decay):

Discrete and multiplicative symmetry called R-parity or

Then

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¦

The consequences of

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§¢¢

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§

¨

¨

¡

¡¢¢

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¥

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§

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conservation are very important:

– SUSY particles always produced in pairs.

– SUSY particles decay into an odd number of SUSY particles.

– The lightest SUSY particle (LSP) is absolutely stable.

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.18/36

¥

¢¡ :

3. The MSSM: Superpotential

At this stage, we have a globally supersymmetric Lagrangian.

Everything

No

Only

is specified by SUSY and gauge invariance.

additional parameter compared to SM.

freedom, **the** choice of **the** Superpotential.

The most general Superpotential compatible with SUSY, gauge

invariance, renormalizability and R–parity conservation is:

©

¢

¢

¡£¢

¤¥

¦§¨

©

©

denote **the** Yukawa couplings among **the** three generations

(and which simply a generalisation of **the** SM Yukawa interaction).

supersymmetric

Higgs–higgsino parameter with dimension of mass

(it is thus a supersymmetric parameter, see later....).

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.19/36

©

Soft

3. The MSSM: soft–SUSY breaking

SUSY breaking:

To explicitely break Supersymmetry without reintroducing **the**

quadratic divergences (**the** so–called soft SUSY–breaking), we add by

hand a collection of soft terms (of dimension two and three):

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¦

¡¨

¢

¡¦

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¢

¦

£

¨

¨ ¤ ¥

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§

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¢

¨

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¥

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§

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¢

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¢

¨

¥

§

¨

¨

¢

¨

¨

A ra**the**r complicated and problematic potential indeed!

Too

Leads

many parameters and thus not very predictive.

generically to a problematic phenomenology.

¨

¨

§

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.20/36

¨

¨

¨

§

3. The MSSM: **the** parameters

In **the** most general case (mixing and phases): 105 free parameters!

complex gaugino masses §¡

§

¢

¢

£

£

£

¢ hermitian mass matrices ¥¤

¢ complex

§ matrix

: 45

trilinear coupling matrices

: 6

for **the** bilinear B coupling : 4

Higgs masses squared,

©¨

©¨

: 2

¦ : 54

111–6 (due to constraints from symmetries and Higgs sector)=105.

For “generic” sets of **the**se parameters, leads to severe problems:

large

flavor changing neutral currents [FCNC]

unacceptable

color

an

amount of additional CP–violation

and/or charge breaking minima

incorrect value of **the** Z boson mass, etc......

We need more constrained MSSMs

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.21/36

4. Constrained MSSMs: pMSSM

A phenomenologically viable MSSM is defined by assuming:

all

Mass

soft SUSY–breaking parameters are real (no new CP viol).

1st/2d

and trilinear cpls. for sfermions diagonal (no FCNC)

sfermion generation universality (no pb. with Kaons)

Phenomenological MSSM (pMSSM) with 22 free parameters:

£ ¢

¢©¨

¢¨

¢¨

¤

¡ :

**the** ratio of **the** vevs of **the** two–Higgs doublet fields.

¢¨

£¦¥ ¢

: **the** Higgs mass parameters squared.

:

:

§ :

¢¨

¢¨

**the** bino, wino and gluino mass parameters.

¢¨

¢¨

¢¨

¢¨

: 1st/2d generation sfermion mass para.

: third generation sfermion mass para.

**the** third generation trilinear couplings.

**the** first/second generation trilinear couplings.

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.22/36

In fact:

You can trade ¢

(in fact:

4. Constrained MSSMs: pMSSM

and

¡ can

£

¤

£¦¥ ¢

with more ”physical” and

be determined from ESWB, see later).

in general not relevant for phenomenology.

(enter only in ”light” flavor physics:

If

£¢

¤ §

, neutron edm, ....).

you focus on a given sector (Higgs, gauginos, sfermions):

only few parameters to deal with and model indep. analyses....

( ¢¨

You

¥ phenomenologically

more viable model than general MSSM

can also use common soft–SUSY breaking terms in many cases

¢¨

¢¨

; ¢¨

¢¨

¢¨

;

; etc..)

and one ends with an even more restrictive set of parameters,

¥ much

more predictive model than general MSSM

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.23/36

¨©

§ ¦

.

4. Constrained MSSMs: mSUGRA

Almost all problems of MSSM solved at once if soft SUSY–breaking

parameters obey a set of universal boundary conditions at ¡

Underlying assumption: SUSY–breaking occurs in a hidden sector

communicating with visible sector through gravitational interactions.

¥ Universal

Besides

¢ ¢

¢

§ unification

soft terms emerge if interactions are “flavor–blind”:

which fix **the** scale

Unification of gaugino, scalar masses and trili. couplings at

Universal gaugino masses: §

Universal scalar masses:

Universal trilinear couplings:

Also: B and

§

§

from

¥

¥

¨

requiring of EWSB and minimization of

¡

¡

¨

©¨

©¨

¡

¨

¡

¡

§

§

¢

§

§

¨

©

§

¥¤£

©

¥ ¤

¦ §

¡¦

¦

¢ .

GeV:

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.24/36

¥

¡

¢

4. Constrained MSSMs: mSUGRA

Only 4.5 param:

¡

All soft breaking parameters at ¢ are obtained through RGEs.

With ¡

¢

§

¥ £

mass (GeV)

800

700

600

500

400

300

200

100

0

¥ ¤

10 2

¦ §

§

¨

GeV and £¡

M 3

mbR ,Q ~ ~

L

m~ tR

m 1

M 2

Radiative EWSB occurs since

m~ τL

Evolution of sparticle masses

m~ τR

10 4

¥ EWSB more natural in MSSM (

M 1

m 2

¨

10 6

Q (GeV)

¦

¦

¥

¥

£

¤

©

§

©

¦¥¨§

¥¨©

10 10 10 17

at scale

¡

/

:

___

\ m +

¨

m

m

1/2

0

2 2

0 μ

from RGEs) than in SM!

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.25/36

4. Constrained MSSMs: GMSB

In GMSB, SSB transmitted to MSSM fields via SM gauge interactions.

Hidden sector for SUSY–break. contains messengers fields, ¢¡

quark/lepton-like pairs coupled to a gauge singlet chiral superfield

The spotential is

¤©§

¦£¨§

¥

¥

¦£

¤© with

¦£ havings vevs. and

SSB are generated by (1or2) loop corrections at scale

with

¦

§

§

§

¥

¥

(generated at two–loops).

¨

§

¢

¢

,

§

¡

m and s label messengers

and scalars; f/g are one/two loop functions; N/C are Dynkin/Casimirs..

Thus, in **the** GMSB model **the**re are six basic input parameters

¡

plus **the** mass of **the** very light gravitino (which is **the** LSP).

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.26/36

§

¥

¤£ .

¢

4. Constrained MSSMs: AMSB

In AMSB, SUSY breaking occurs also in hidden sector (e.g. extra dims)

and is transmitted to visible sector via (e.g. super–Weyl) anomalies.

Gaugino, scalar masses and trilinear couplings are simply related to

**the** scale dependence of **the** gauge and matter kinetic functions.

In terms of gravitino mass

¨ ,

¡ functions for and

couplings

and anomalous dimensions of chiral superfields, SSB terms are:

¡£¢

§

¤ ¥¤

¨

© ©

¥¤

¡

¡§¦ ¨

© ©¨

¥¤

RG invariant equations valid at any scale (make a predictive model).

(

and

¡ terms

are obtained as usual by requiring EWSB).

However, picture spoiled by tachyonic sleptons ¢

¥ add a non anomalous contribution to soft masses

In minimal AMSB with a universal ¢ ,

©

¨

¡

¨

¡ ¨

¨ ,

¨

¨

¦

**the** inputs are:

©

in general!

¢

to ¢

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.27/36

5. The SUSY spectrum:

The basic Lagrangian, gives **the** currents and corresponding states:

we need to turn **the** current eigenstates into **the** mass eigenstates

Charginos:

Neutralinos:

Sfermions:

Higgs

mixtures of **the** charged higgsinos and gauginos

¨

¨¢¡

¨§

mixtures of **the** neutral higgsinos and gauginos

¨

¨©

¨¥¡

©

¨ ¡

©

¤

© §

£

¤

£¤

mixing between LH and RH sfermions of same flavor

bosons: 2 doublets of complex fields

¥ 3 degrees of freedom to generate §

¥ 5 degrees of freedom left:

¨

¨

£

¨

¨

¨

§¡

¨

¤

©

§

¨

¨

¤

©

¨

§

¤

©

§©

¨

¨

¨

§

¨©

¦ dof

To determine **the** mass and **the** mixing angle of **the** physical states

find relevant mass matrices and diagonalize **the**m.... include RC?

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.28/36

5. The SUSY spectrum: charginos

The general chargino mass matrix, in terms of , and

diagonalized by:

(Pauli § to make **the**¤

¡

¢

£ ¤

§

¡

¥

¡

masses

§

£

§

¢

¥

¡

positive and

Simple analytical formulae for **the** masses ¢

For limiting cases, interpretation much simpler.

£

¤

In **the** opposite limit,

§

¡

§

§

,

¤

¤

**the** roles of¤

¦ §

¡

¡

¡

¢

§¨

§

©

¡

©

©

¡ , is

rotation matrices)

and mixing angles.

¡

¤

¡

§

are reversed.

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.29/36

¢

¦

5. The SUSY spectrum: neutralinos

For neutralinos, **the** 4x4 mass matrix depends on

In **the**

¢

¨

¢

Diagonalized by a single real matrix

For

¦©

£

,¤

©

¨

¡

¡

¦

¦

¨

§ © ¨

is bino

In **the** opposite limit,

¥

¥

¤

¤

§

©

basis,

§

,¤

©

¤

¤

¡

£

,

¨§

¨§

¡

¥

¥

¡

it is given by

wino

£ .

Again for

§

©

¡

¡

¡

¡

and¤

¥

¥

§

§

§

¤

¡

©

§

¡

§¥¤

¡

¥

¥

higgsinos (

**the** roles are again reversed.

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.30/36

¡

§ .

:

.

5. The SUSY spectrum: gluinos and RGEs

Finally, **the** gluino mass is identified with at **the** tree–level

¡£¢

In constrained models with boundary conditions at **the** high energy

scale ¤ , **the** evolution of **the** gaugino masses given by RGEs

¦

© ¥

¥¦¨§

¢

¦§

§

where in all sparticles contribute to **the** evolution from Q to ¤ .

Equations are related to those of **the** gauge couplings

With inputs at scale and common value at ¤

¥

¢

¢

¢

¢

¢

.

GeV,

one has for gaugino mass parameters at **the** weak or SUSY scale :

With norm.factor

¡

¦

¡

¦

in , we have

¡

¡

¡

¦

¡

¦

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.31/36

¡

¡

¦

¡

¦ ©

¡

¦©

¡

¦

¡

¡

.

5. The SUSY spectrum: sfermions

Sfermion system described by ¡

¡£¢¥¤§¦

¡¢¥¤§¨

and

with **the** various entries given by

They are diagonalized by

turn **the** current eigenstates

¡

¢

¡

and

©¤ . For 3d generation, mixing

¡¤

¡

¡

¢

¢¤§¦ ¡

¢¤¨ ¡

¤ ¡

¡¤

¤

¤

¤

¡

¤ ¡

¤

3 param.for each species:

¡¤

¤

¡

to be included.

© ¤

¤

rotation

¦

matrices of angle

into **the** mass eigenstates

¤ , which

¡

¡

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.32/36

¡

¡

¡

.

¡

¢

5. The SUSY spectrum: sfermions

Note: mixing very strong in stop sector,

generates mass splitting between

sbottom/stau sectors also for large

In cMSSM with universal ¡

and ¡

¥§¦

¥ ¦ ,

©¨

¡

leading to light

at

masses are simple if Yukawas are small (

¥§

¡

¡

§

¨ §

With inputs at , ¤

¥

¥

¢ ¨

¥

¥§

depend

,

¢

¤ £

¥§¦ ;

.

and

mixing in

**the** RGEs for scalar

on I, Y, color):

¥

¤

¦

¢§

¡

¡

¡

¡

¢

and ¤ ,

one obtains

For 3d generation squarks, Yukawa couplings to be included!!

¡

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.33/36

¡

¡

¢

¡

6. Extensions of MSSM: Rp violation

To avoid fast P decay, we do not need both L and B conservation

In most general W, include

¡

¡

¤

¢ £

¢

¨

¤ ¥

£ ¨

£

¥§¦¤

L=1 or

P decay modes and experimental limits on

However,

no

But,

¥ £

¥ ¤

¤

£ ¨¢ B=1 interactions:

£

¥ ¤

B and

L imply

at least 45 new parameters in **the** general case.

stable LSP and thus no SUSY DM candidate...

rich phenomenology (e.g. s channel sfermion production)

enters in neutrino phenomenology and adresses small masses

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.34/36

©

¨

£

¢

¤

.

6. Extensions of **the** MSSM: CP violation

One can allow for some CP–violating parameters, in particular:

Complex

Complex trilinear

¢

(some

phases rotated away) and©

couplings, in particular

The MSSM Higgs sector stays CP–conserving at **the** tree–level but

complex parameters enter at **the** one–loop level through©

CP

violation is needed for (direct) baryogenesis in MSSM

However,

Complicates

¢ .

and

many new parameters will enter in **the** general case

**the** determination of spectrum but less fine-tunning!

Strongly constrained by data ( ¢¡

¥¤£

No

sign yet of any additionnaly from

¥ and

CP

needs cancelations

in B–factories etc...

One can also allow for flavor non–diagonal interactions, howver:

Parameters

Only

strongly constrained from FCNC, K, B physics...

adds complications/parameters (no **the**ory motivation)...

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.35/36

¢ .

But©

Solution:©

one additional neutralino state:

The©

problem:©

6. Extensions of **the** MSSM: NMSSM

enters EWSB and **the** determination of .

It must be of order SUSY–breaking parameters such as

is a SUSY preserving parameter, comes from

and, a priori, no reason for having©

is related to a vev of an additional field

NMSSM: introduce a gauge singlet superfield

Extended spectrum in NMSSM compared to MSSM:

two additional Higgs particles

less

¨

¢

¢

¨

¢

¡

¡

¡©

¢

¢£

¤ with

©

¥

....

¤

¡

¡ ,

§ ¤ into superpotential

constrained and fine tuned model, richer phenomenology...

Ex: upper bound on h mass is

LEP searches bounds are not valid and h lighter than 100 GeV.

¨

¢

**Maria**–**Laach**, 05–09/09/06 **Physics** **beyond** **the** SM: SUSY – A. Djouadi – p.36/36

¢

¨

¨

¢

¢

¨

¡©

¢

¢

–40 GeV.

¦

.