Beyond the SM

Higgs and physics

Beyond the SM

2.- Supersymmetry

Oscar Vives

8th IDPASC School IFIC, 2128/05/2018

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SM Theoretical Problems

•Assignment of matter Quantum Numbers

•Unication of Gauge Couplings

•Origin of Spontaneous Symmetry Breaking

•Accomodate Quantum Gravity

•Large number of Flavour Parameters

SM Observational Problems

•Need of non-baryonic Dark Matter

•CP violation source for Baryogenesis

•Mechanism of Ination

We DO need Physics Beyond the SM !!

SM Theoretical Problems

•Assignment of matter Quantum Numbers

•Unication of Gauge Couplings

•Origin of Spontaneous Symmetry Breaking

•Accomodate Quantum Gravity

•Large number of Flavour Parameters

SM Observational Problems

•Need of non-baryonic Dark Matter

•CP violation source for Baryogenesis

•Mechanism of Ination

We DO need Physics Beyond the SM !!

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SM Theoretical Problems

•Assignment of matter Quantum Numbers

•Unication of Gauge Couplings

•Origin of Spontaneous Symmetry Breaking

•Accomodate Quantum Gravity

•Large number of Flavour Parameters

SM Observational Problems

•Need of non-baryonic Dark Matter

•CP violation source for Baryogenesis

•Mechanism of Ination

We DO need Physics Beyond the SM !!

SM Theoretical Problems

•Assignment of matter Quantum Numbers

•Unication of Gauge Couplings

•Origin of Spontaneous Symmetry Breaking

•Accomodate Quantum Gravity

•Large number of Flavour Parameters

SM Observational Problems

•Need of non-baryonic Dark Matter

•CP violation source for Baryogenesis

•Mechanism of Ination

We DO need Physics Beyond the SM !!

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SM Theoretical Problems

•Assignment of matter Quantum Numbers

•Unication of Gauge Couplings

•Origin of Spontaneous Symmetry Breaking

•Accomodate Quantum Gravity

•Large number of Flavour Parameters

SM Observational Problems

•Need of non-baryonic Dark Matter

•CP violation source for Baryogenesis

•Mechanism of Ination

We DO need Physics Beyond the SM !!

Supersymmetry

• Only possible extension of symmetry beyond Lie Symmetries (ColemanMandula Theorem).

• Correct Unication of Gauge couplings at M_GUT,

GUT assignment of Quantum numbers (anomaly cancellation).

• Solution of the Hierarchy Problem, strong motivation for low-energy SUSY.

• Natural Mechanism of Electroweak Symmetry Breaking, Radiative Symmetry Breaking.

• SUSY is a necessary ingredient in String Theory.

Local Supersymmetry ⇔ Supergravity.

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ColemanMandula Theorem

In the 60's attempts to combine internal and Lorentz symmetries ...

The only conserved quantities that transform as tensors under Lorentz transformations in a theory with non-zero scattering amplitudes in 4D are the generators of the Poincare group and Lorentz invariant quantum numbers (scalar charges).

2×2 spinless particle scattering, bosonic conserved charge,Σ_µν, h1|Σ_µν|1i=αpµ¹p¹ν+βgµν

So, in the scattering process,

pµ¹p¹ν+pµ²p²ν =p³µpν³+pµ⁴pν⁴ & pµ¹+pµ² =p³µ+pµ⁴

Not possible in a theory with non-zero scattering.

However: ColemanMandula theorem does not forbid conserved spinor charges, Qα (transforming like fermions under Lorentz)

Qα|Bosoni=|Fermioni Qα|Fermioni=|Bosoni [Qα,H] =0 [{Qα,Q¯β˙},H] =0

Supersymmetry Algebra {Qα,Q¯β˙}=2σ^µ

αβ˙Pµ {Qα,Qβ}=0 {Q¯α˙,Q¯β˙}=0 Supersymmetry relates particles of dierent spin with equal quantum numbers and identical masses.

₁

02

∼

q (quark)

˜q (squark)

11 2

∼

g (gluon) g˜ (gluino)

Chiral supermultiplet Gauge supermultiplet

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Hierarchy Problem

δme =2α_em

π melog Λ

me δm²_H(f) =−2Nf|λ_f|²

16π²[Λ²− 2m²_fln Λ mf] with me →0, chiral symmetry No symmetry protects m_H² . . .

Quadratic div.!!

Λ =10¹⁹ GeV

δme = 0.24 me δm_H² ' 10³⁰ GeV

Supersymmetry

⇒

δm_H²(˜f)=−2Nf˜

λ˜f

16π²[Λ²− 2m²_˜f ln Λ

m˜f +. . .]

δm_H²(f) +δm²_H(˜f)=0 if N_f =N˜f,|λ_f|²=−λ_˜f and m_f =m˜f

⇒ Supersymmetry + Chiral symmetrysolve hierarchy problem.

But ... no scalars degenerate with the SM fermions, SUSY broken!! Soft Supersymmetry breaking

•Preserve cancellation of Quadratic divergencies requires dimensionless couplings still supersymmetric: |λ_f|² =−λ_˜f

•SUSY only broken in couplings with possitive mass dimension: Soft Breaking m_˜f²=m_f²+δ²

δm²_H(f) +δm_H²(˜f)'2N_f|λ_f|² 16π²δ²ln Λ

m^˜_f +. . . to solve hierarchy problem δ <∼1 TeV

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δm_H²(˜f)=−2Nf˜

λ˜f

16π²[Λ²− 2m²_˜f ln Λ

m˜f +. . .]

δm_H²(f) +δm²_H(˜f)=0 if N_f =N˜f,|λ_f|²=−λ_˜f and m_f =m˜f

⇒ Supersymmetry + Chiral symmetrysolve hierarchy problem.

But ... no scalars degenerate with the SM fermions, SUSY broken!!

Soft Supersymmetry breaking

•Preserve cancellation of Quadratic divergencies requires dimensionless couplings still supersymmetric: |λ_f|² =−λ_˜f

•SUSY only broken in couplings with possitive mass dimension: Soft Breaking m_˜f²=m_f²+δ²

δm²_H(f) +δm_H²(˜f)'2N_f|λ_f|² 16π²δ²ln Λ

m^˜_f +. . . to solve hierarchy problem δ <∼1 TeV

δm_H²(˜f)=−2Nf˜

λ˜f

16π²[Λ²− 2m²_˜f ln Λ

m˜f +. . .]

δm_H²(f) +δm²_H(˜f)=0 if N_f =N˜f,|λ_f|²=−λ_˜f and m_f =m˜f

⇒ Supersymmetry + Chiral symmetrysolve hierarchy problem.

But ... no scalars degenerate with the SM fermions, SUSY broken!!

Soft Supersymmetry breaking

•Preserve cancellation of Quadratic divergencies requires dimensionless couplings still supersymmetric: |λ_f|² =−λ_˜f

•SUSYonly broken in couplings with possitive mass dimension:

Soft Breaking m_˜f²=m_f²+δ²

δm²_H(f) +δm_H²(˜f)'2N_f|λ_f|² 16π²δ²ln Λ

m^˜_f +. . . to solve hierarchy problem δ <∼1 TeV

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GUT and coupling unication

Grand Unication

•Simple gauge group unifying SU(3)_C, SU(2)_L and U(1)_Y

•All matter multiplets in 1 generation unied in a single (two) representation of the gauge group:

SU(5) 5¯=























 d^c d^c d^c e⁻ ν_e

























L

10=

























0 u^c u^c u d 0 u^c u d 0 u d 0 e^c 0

























L

S0(10) 16=10+ ¯5+1

⇒Explains the assignment of quantum numbers in the SM

RGE evolution of gauge couplings Running the couplings to high energies they come close at M_GUT '10¹⁶GeV

SM

0 10 20 30 40 50 60

10² 10⁴ 10⁶ 10⁸ 10¹⁰ 10¹² 10¹⁴ 10¹⁶ 10¹⁸

αi

-1

U(1)

SU(2)

SU(3)

µ (GeV)

Standard Model

Using SU(3) and SU(2)_L to predict sinθ_W with correct unication :

sinθ_W^th = 0.214±0.004 sinθ^exp_W = 0.23149±0.00017

MSSM Much better agreement:

sinθ^th_W ' 0.232

sinθ_W^exp = 0.23149±0.00017 iif (M_SUSY '1 TeV).

⇒

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RGE evolution of gauge couplings Running the couplings to high energies they come close at M_GUT '10¹⁶GeV

SM

0 10 20 30 40 50 60

10² 10⁴ 10⁶ 10⁸ 10¹⁰ 10¹² 10¹⁴ 10¹⁶ 10¹⁸

αi

-1

U(1)

SU(2)

SU(3)

µ (GeV)

Standard Model

Using SU(3) and SU(2)_L to predict sinθ_W with correct unication :

sinθ_W^th = 0.214±0.004 sinθ^exp_W = 0.23149±0.00017

MSSM Much better agreement:

sinθ^th_W ' 0.232

sinθ_W^exp = 0.23149±0.00017 iif (M_SUSY '1 TeV).

⇒

Radiative Symmetry Breaking

•In MSSM many scalars but (typically) only Higgs gets a vev

•All soft masses possitive O(M_W) at M_GUT

•µH1H2 ∈W , µ∼ O(M_W) (SUSY µproblem)

•Approx. RGE evolution from M_GUT to M_W:

16π²dm²_H2

dt = 6Y_t²(m²_H2+m²_Q3+m²_U3)−6g₂²M₂²−6 5g₁²M₁², 16π²dm²_H1

dt = −6g₂²M₂²−6 5g₁²M₁², 16π²dm_Q²₃

dt = 2Y_t²(m²_H2+m²_Q3+m²_U3)−32 3g₃²M₃²− 6g₂²M₂²− 2

15g₁²M₁²,

⇒

m²_H2 pushed down by Yt

and no SU(3) coupling

EW symmetry breaking occurs naturally as a radiative eect

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Gravity and strings ...

Global bosonic sym. −−−→^Local Gauge Theory Global SUSY −−−→^Local Supergravity

•SUSY transformation parametersξ^α deppend on space-time position. Anticonmutator of 2 SUSY transformations is a translation

{Qα,Q¯β˙}=2σ^µ

αβ˙Pµ

⇓

Local SUSY implies local coordinate tranformations: Gravity

•Superstring can unify gravity with gauge interactions.

Supersymmetry necessary ingredient of consistent String Theory

Non-interacting WessZumino

Weyl fermion + complex boson Ref: Martin, hep-ph/9709356 S =

Z

d⁴xL = Z

d⁴x iψ¯¯σ^µ∂_µψ + ∂^µφ^∗∂_µφ SUSY transformation: scalar↔ fermion, withξ Weyl spinor ([ξ] =−1/2) transf. parameter

δφ=√

2ξ^αψα≡√

2ξψ δφ^∗=√

2ξ¯α˙ψ¯^α˙ ≡√ 2ξ¯ψ¯ δL_scal = +√

2ξ ∂^µψ ∂µφ^∗+√

2ξ ∂¯ ^µψ ∂¯ µφ for a fermion, comparing δL_scal and L_ferm must be:

δψ_α=i√

2(σ^µξ)¯_α∂_µφ δψ¯_α˙ =−i√

2(ξσ^µ)_α˙∂_µφ^∗ δLfer =√

2(ξ σ^µ¯σ^ν∂_νψ)∂_µφ^∗−√

2( ¯ψ ¯σ^νσ^µξ)∂¯ _ν∂_µφ

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So, we arrive at δLfermion =−√

2ξ ∂^µψ ∂µφ^∗−√

2ξ ∂¯ ^µψ ∂¯ µφ+

√2 ∂^µξ¯ψ ∂¯ _µφ+ξψ∂_µφ^∗+ (ξ σ^νσ¯^µψ)∂_µφ^∗

⇒δS = Z

d⁴x(δL_scalar+δL_fermion) =0 Theory is invariant only if the algebra closes, i.e. if the conmutator of two supersymmetric transformations is a symmetry of the theory scalar

(δ_ξ2δ_ξ1−δ_ξ1δ_ξ2)φ=2i ξ₁σ^µξ¯₂−ξ₂σ^µξ¯₁

∂_µφ

But, for fermion, only on-shell,σ¯^µ∂µψ=0, it closes. O-shell we must introduce auxiliary eld, F with Laux =F^∗F and dimensions mass². It does not propagate, the eqs. motion are F =F^∗ =0

we modify the transformation properties δφ=√

2ξψ δφ^∗=√ 2ξ¯ψ¯ δψ_α=i√

2(σ^µξ)¯_α∂_µφ + √ 2 ξ_αF δψ¯α˙ =−i√

2(ξσ^µ)α˙∂µφ^∗ + √ 2ξ¯α˙F^∗ δF=i√

2ξ¯¯σ^µ∂µψ δF^∗=−i√

2∂µψ¯¯σ^µξ Now theory is Supersymmetry invariant even o-shell:

L_WZ =iψ¯¯σ^µ∂µψ + ∂^µφ^∗∂µφ+F^∗F (δ_ξ2δ_ξ1−δ_ξ1δ_ξ2)X =2i ξ₁σ^µξ¯₂−ξ₂σ^µξ¯₁

∂_µX

for X =φ, ψ,F Supersymmetry Algebra ^{{Qα,Q¯β˙}=2σ}α^µβ˙Pµ [Pµ,Qα] =0

{Qα,Qβ}=0 {Q¯α˙,Q¯β˙}=0 Chiral Supermultiplet=ψ, φ,F with four fermionic d.o.f. (2

complex components of ψ) and four bosonic d.o.f. ( 2 complex scalars, φand F )

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we modify the transformation properties δφ=√

2ξψ δφ^∗=√ 2ξ¯ψ¯ δψ_α=i√

2(σ^µξ)¯_α∂_µφ + √ 2 ξ_αF δψ¯α˙ =−i√

2(ξσ^µ)α˙∂µφ^∗ + √ 2ξ¯α˙F^∗ δF=i√

2ξ¯¯σ^µ∂µψ δF^∗=−i√

2∂µψ¯¯σ^µξ Now theory is Supersymmetry invariant even o-shell:

L_WZ =iψ¯¯σ^µ∂µψ + ∂^µφ^∗∂µφ+F^∗F (δ_ξ2δ_ξ1−δ_ξ1δ_ξ2)X =2i ξ₁σ^µξ¯₂−ξ₂σ^µξ¯₁

∂_µX for X =φ, ψ,F Supersymmetry Algebra ^{{Qα,Q¯β˙}=2σµ}αβ˙Pµ [Pµ,Qα] =0

{Qα,Qβ}=0 {Q¯α˙,Q¯β^˙}=0 Chiral Supermultiplet=ψ, φ,F with four fermionic d.o.f. (2

complex components of ψ) and four bosonic d.o.f. ( 2 complex scalars, φand F )

Matter Interactions

General set of renormalizable matter SUSY interactions is given by, Lint=−1

2W^ijψ_iψ_j+WⁱFi+c.c.

with W^ij, Wⁱ functions of bosonic elds [W^ij] =M and

[Wⁱ] =M² ([L] =4, W^ij and Wⁱ cannot be functions ofψand F ) -Lint must be SUSY invariant by itself

⇒

Analitic function of the (complex) scalar elds (not ofφ^∗), at most cubic inφand dimensions of mass³

W = 1

2M^ijφ_iφ_j+1

6λ^ijkφ_iφ_jφ_k, Wⁱ= δW

δφ_i W^ij= δW δφ_iδφ_j

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Fi,F^i∗ eliminated using equations of motion: Fi =−W_i^∗ F^i∗ =−Wⁱ,i.e. functions of scalar elds, no derivatives. Then Lagrangian,

L_chiral = L_WZ +L_int = iψ¯¯σ^µ∂_µψ + ∂^µφ^∗∂_µφ+

− 1

2W^ij ψ_iψ_j − 1

2W^ij∗ ψ¯_iψ¯_j − WⁱW_i^∗ The scalar potential V(φ, φ^∗) is given by

V(φ, φ^∗) =WⁱW_i^∗ =FⁱF_i^∗ =M_ji^∗M^ikφ^j∗φ_k+ 1

2M^ikλ^∗_jnkφ_iφ^j∗φ^n∗ +1

2M_ik^∗λ^jnkφ^i∗φ_jφ_n+ 1

4λ^ijnλ^∗_klnφ_iφ_jφ^k∗φ^l∗ Superpotential: general SUSY

invariant matter interactions

Example: top Yukawa

W = ht QLHt_R^c Gives rise to the Lagrangian, L_int = − 1

2W^ij ψ_iψ_j − 1

2W^ij∗ ψ¯_iψ¯_j − WⁱW_i^∗

=−1 2 h_th

H Q_Lt_R^c + ˜Q_L ˜ht_R^c + ˜t_R^c Q_l˜hi +c.c.

−h²_t

|H˜t_R^c|²+|HQ˜L|²+|Q˜L˜t_R^c|² and the Feynman rules

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Vector Superelds

Gauge multilplet: massless vector boson A^aµ + Weyl gauginoλ^a + auxiliary eld D^a

L_gauge =−1

4F_µν^a F^µνa+i¯λ^aσ¯^µDµλ^a+1 2D^aD^a Fµν^a =∂_µA^aν−∂_νA^aµ+gf^abcA^bµA^cν Dµλ^a =∂_µλ^a−gf^abcA^bµλ^c Lgauge is already Supersymmetric with the transformations (Wess-Zumino gauge)

δA^aµ = iξ¯σ¯µλ^a − iλ¯^aσ¯µξ δλ^a_α =−1

2 (σ^µ¯σ^νξ)_αFµν^a − iξ_αD^a δD^a = −ξ¯σ¯^µDµλ^a + Dµ¯λ^aσ¯^µξ

Gauge Interactions

Replacing∂_µ→Dµ=∂_µ+igT^aA^a_µ obtain Gauge invariant Lagrangian, but not SUSY invariant, we need gaugino and D interactions,

φ^∗T^aψ λ^a λ¯^a ψ¯T^aφ φ^∗T^aφD^a

Then the full Lagrangian is Supersymmetric replacing also in the SUSY transformations derivatives by covariant derivatives, L=Lgauge + Lchiral+ i√

2 g

φ^∗T^aψ λ^a + ¯λ^a ψ¯T^aφ

+ g φ^∗T^aφD^a where W must also be Gauge invariant by itself. Moreover D^a can be eliminated using eqs of motion,

D^a = −g φ^∗T^aφ

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The gauge invariant SUSY Lagrangian,

L = iψ¯_Lσ¯^µDµψ_L + D^µφ^∗_LDµφ_L+ (L→R^c) + 1

4F^µνa F^µνa

−i λ¯^a¯σ^µDµλ^a + i √ 2 g

φ^∗_iT^aψ_i λ^a + ¯λ^aψ¯_iT^aφ_i

− 1

2W^ij ψ_iψ_j − 1

2W^ij∗ ψ¯_iψ¯_j − WⁱW_i^∗ −1 2 g²(P

iφ^∗_iT^aφ_i)² therefore the gauge interactions are (U(1)),

SUSY Lagrangian

• All interactions determined by gauge quantum numbers and Superpotential

•Write all kinetic terms with covariant derivativesplus gaugino interactions and Dterms

• W must be a gauge invariant analitic function of the scalar elds

•Matter interactions determined by WⁱW_i^∗ and W^ijψ_iψ_j

⇒

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Minimal Supersymmetric Standard Model

• Must include all Standard Model particles and interactions

• Supersymmetric partners and SUSY SM interactions

• Supersymmetry must be softly broken

LH SM fermion ↔ Scalar partner Chiral supermultiplets Q,u_R^c,d_R^c,L,e_R^c Q˜,˜u_R^c,˜d_R^c,˜L,˜e_R^c

2 Higgs ↔ fermionic part.

H₁, H₂ H˜₁, H˜₂ gauge bosons ↔ fermionic part.

Vector supermultiplets Bµ,Wµⁱ,Gµ^a B˜,W˜ⁱ,g˜^a

MSSM Superpotential

Includes the Yukawa interactions of the SM

W =Y_d^ijQ_iH₁d_Rj^c +Y_e^ijL_iH₁e_Rj^c +Y_u^ijQ_iH₂u_Rj^c +µH₁H₂

•Need of two Higgs doublets Y_Qi +Y_dR^c = 1

6 +1 3 = 1

2 ⇒ Y_Hd =−1 2 Y_Qi +Yu^c_R = 1

6−2 3 =−1

2 ⇒ Y_Hu = 1 2

in the SM H_u =H and H_d =H^∗. However, a Superpotential containing H^∗ would be non-supersymmetric.

A second doublet also required by anomaly cancellation.

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R Parity Other Gauge invariant terms can appear in W W∆L=1 =λ^ijkLiLje_Rk^c +λ^0ijkLiQjd_Rk^c +ⁱLiH2

W∆B=1 =λ^00ijk u_Ri^c d_Rj^c d_Rk^c

violate baryon or lepton number by 1 unit. If bothλ⁰ andλ⁰⁰6=0

⇒ rapid proton decay!!

p⁺→e⁺π⁰⇒λ^{0 ∗}₁₁₂·λ⁰⁰₁₁₂ ≤2×10⁻²⁷

New discrete symmetry, R-parity, forbids these terms R_P = (−1)^3B+L+2S

SM particles and Higgs bosons RP = +1,superpartners RP =−1

R Parity Other Gauge invariant terms can appear in W W∆L=1 =λ^ijkLiLje_Rk^c +λ^0ijkLiQjd_Rk^c +ⁱLiH2

W∆B=1 =λ^00ijk u_Ri^c d_Rj^c d_Rk^c

violate baryon or lepton number by 1 unit. If bothλ⁰ andλ⁰⁰6=0

⇒ rapid proton decay!!

p⁺→e⁺π⁰⇒λ^{0 ∗}₁₁₂·λ⁰⁰₁₁₂ ≤2×10⁻²⁷

New discrete symmetry, R-parity, forbids these terms R_P = (−1)^3B+L+2S

SM particles and Higgs bosons RP = +1,superpartners RP =−1

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RP conserved in the MSSM

• W∆L=1 and W∆B=1 absent in the MSSM.

•Lightest Supersymmetric Particle (LSP) stable (dark matter).

• Any sparticle decays into nal state with odd number of LSP.

• In colliders, Supersymmetric particles only produced in pairs.

It is also possible an R_P violating MSSM with W∆L=1 or W∆B=1

(not both) and stable proton.

•L or B is violated

•LSP not stable anymore (not dark matter candidate).

•Single production of SUSY particles possible.

RP conserved in the MSSM

• W∆L=1 and W∆B=1 absent in the MSSM.

•Lightest Supersymmetric Particle (LSP) stable (dark matter).

• Any sparticle decays into nal state with odd number of LSP.

• In colliders, Supersymmetric particles only produced in pairs.

It is also possible an R_P violating MSSM with W∆L=1 or W∆B=1

(not both) and stable proton.

• L or B is violated

• LSP not stable anymore (not dark matter candidate).

• Single production of SUSY particles possible.

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Soft SUSY Breaking

•SUSY must be broken, m˜e6=m_e, mg˜ 6=0.

•Solve hierarchy problem, broken by terms of possitive mass dimension Soft Supersymmetry Breaking,and Msusy ≤ O(1 TeV).

• Gaugino masses L⁽_soft¹⁾ = 1

2

M₁ B˜B˜ +M₂W˜W˜ +M₃g˜g˜ +h.c.

• Scalar masses

L⁽_soft²⁾ = (m²Q˜)_ijQ˜iQ˜_j^∗+ (m²˜u)_ij˜u^c_Ri˜u^c_Rj^∗+ (m²d˜)_ijd˜_Ri^c d˜_Rj^c∗+ (m²˜L)_ijL˜i˜L^∗_j+ (m²˜e)_ij˜e_Ri^c ˜e_Rj^c∗+ (m_H²₁)H1H₁^∗+ (m_H²₂)H2H₂^∗

• Trilinear couplings and Bterm

L⁽_soft³⁾ = (Y_d^A)^ijQ˜iH1d˜Rj+ (Y_e^A)^ijL˜iH1˜e_Rj^c + (Y_u^A)^ijQ˜iH2˜u^c_Rj+BµH1H2

CMSSM

• Minimal and simple realization of the MSSM.

• Similar sparticle masses in a general MSSM.

•Any generic MSSM must include at least the CMSSM physics.

• Main dierence in FCNC and CP violation observables.

• Representative MSSM example for collider phenomenology.

Minimal number of new SUSY paramenters

m₀² → Universal scalar mass. M₁/2 → Common gaugino mass.

A₀ → Universal trilinear. B → Soft Higgs mass.

µ → Susy Higgs mass. tanβ → Ratio of Higgs vevs.

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Soft breaking parameters dened at∼M_GUT. Evolve them to M_W with Renormalization Group Equations (RGE).

•Gauge couplings and gaugino masses dα⁻_a¹

dt =−b_a

2π dM_a dt = 1

8π²b_ag_a²M_a dMa/g_a²

dt =0 ⇒

⇒M1

α₁ = M2

α₂ = M3

α₃ ba= (33

5 ,1,−3) t =lnQ Q₀

• Yukawa couplings (bτ unication ...) dyt

dt = yt

16π²

6y_t²+y_b²−16

3 g₃²−3g₂²−13 15g₁²

dy_b

dt = y_b 16π²

6y_b²+y_t²+yτ²−16

3 g₃²−3g₂²− 7 15g₁²

dyτ

dt = yτ

16π²

4yτ²+3y_b²−3g₂²−9 5g₁²

•Soft masses 16π²d

dtm²_H2 = 6y_t²(m_H²₂+m²_Q3+m²_U3)−6g₂²M₂²−6 5g₁²M₁² 16π²d

dtm²_H1 = −6g₂²M₂²−6 5g₁²M₁² 16π²d

dtm²_Q3 = 2y_t²(m_H²₂+m²_Q3+m²_U3)−32

3 g₃²M₃²−6g₂²M₂²− 2 15g₁²M₁² 16π²d

dtm_U²₃ = 4y_t²(m_H²₂+m²_Q3+m²_U3)−32

3 g₃²M₃²−32 15g₁²M₁² Full set of RGEs in literature. Fortran codes, ISASUSY, SOFTSUSY, SPHENO . . .

Mass of H₂ pushed down by top and absence of SU(3)coupling

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Radiative Symmetry Breaking

30/35

Collider phenomenology

Sparticle production

Gluino and Squark production (1st gen.) dominates through

Squark & gluino decays

•If two body decays with strong coupling are allowed they always dominate

˜q →q˜g, g˜→qq˜

Otherwise squarks decay to quark chargino or neutralino through electroweak couplings

q˜→qχ⁰_i, q⁰χ⁻_i and gluinos decay through an o-shell squark

g˜→qqχ⁰_i, qq⁰χ⁻_i

32/35

LHC constraints

Production of coloured particles with long decay chains. . .

Search of jets + missing E_T:

34/35

Stop (˜t) production:

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