CORRECTIONS RADIATIVES EN SUPERSYM´ETRIE ET APPLICATIONS AU CALCUL DE LA DENSIT´E RELIQUE AU-DEL`A DE L’ORDRE DOMINANT

CORRECTIONS RADIATIVES EN SUPERSYM´ ETRIE ET APPLICATIONS AU CALCUL DE LA DENSIT´ E

RELIQUE AU-DEL ` A DE L’ORDRE DOMINANT

Guillaume CHALONS

LAPTH-Universit´e de Savoie 8 Juillet 2010

CHALONS Guillaume THESIS DEFENSE 1/ 50

OUTLINE

1 Going beyond the Standard Model

2 Supersymmetry as a possible solution

3 Need for precise predictions

4 The SloopS code

5 Renormalisation of the Neutralino/Chargino sector

6 Annihilation of a light neutralino

7 Annihilation of a heavy neutralino

OUTLINE

1 Going beyond the Standard Model

2 Supersymmetry as a possible solution

3 Need for precise predictions

4 The SloopS code

5 Renormalisation of the Neutralino/Chargino sector

6 Annihilation of a light neutralino

7 Annihilation of a heavy neutralino

CHALONS Guillaume THESIS DEFENSE 3/ 50

PROBLEMS OF THE STANDARD MODEL

TheStandard Modelseems to be an“incomplete”theory.

Mechanism forgeneratingmass to particles (≡Electroweaksymmetry breaking) yetunknown.

Does not explain theinstabilityof the Higgs mass w.r.t higher orders.

δm²_H⊃ −λ²_f 8π²

„

Λ²−3m²_fln

„ Λ mf

« +...

«

Other masses areprotectedw.r.thigher ordersthanks to asymmetry(chiralfor fermions,gaugefor vector bosons).

No existingsymmetryplaying the same role forscalarbosons.

Cosmology:26%ofmatter energy, only4%identified⇒DARK MATTER SMdoes not explain the“nature”ofDARK MATTER, nocandidatecan explain by itself the present amount ofDM.⇒Need for anewparticle.

DARK MATTERproblem seems to be related to theElectroweaksymmetry breaking

NEED FOR NEW PHYSICS

PROBLEMS OF THE STANDARD MODEL

TheStandard Modelseems to be an“incomplete”theory.

Mechanism forgeneratingmass to particles (≡Electroweaksymmetry breaking) yetunknown.

Does not explain theinstabilityof the Higgs mass w.r.t higher orders.

δm²_H⊃ −λ²_f 8π²

„

Λ²−3m²_fln

„ Λ mf

« +...

«

Other masses areprotectedw.r.thigher ordersthanks to asymmetry(chiralfor fermions,gaugefor vector bosons).

No existingsymmetryplaying the same role forscalarbosons.

Cosmology:26%ofmatter energy, only4%identified⇒DARK MATTER SMdoes not explain the“nature”ofDARK MATTER, nocandidatecan explain by itself the present amount ofDM.⇒Need for anewparticle.

DARK MATTERproblem seems to be related to theElectroweaksymmetry breaking

NEED FOR NEW PHYSICS

CHALONS Guillaume THESIS DEFENSE 4/ 50

PROBLEMS OF THE STANDARD MODEL

TheStandard Modelseems to be an“incomplete”theory.

Mechanism forgeneratingmass to particles (≡Electroweaksymmetry breaking) yetunknown.

Does not explain theinstabilityof the Higgs mass w.r.t higher orders.

δm²_H⊃ −λ²_f 8π²

„

Λ²−3m²_fln

„ Λ mf

« +...

«

Other masses areprotectedw.r.thigher ordersthanks to asymmetry(chiralfor fermions,gaugefor vector bosons).

No existingsymmetryplaying the same role forscalarbosons.

Cosmology:26%ofmatter energy, only4%identified⇒DARK MATTER SMdoes not explain the“nature”ofDARK MATTER, nocandidatecan explain by itself the present amount ofDM.⇒Need for anewparticle.

DARK MATTERproblem seems to be related to theElectroweaksymmetry breaking

OUTLINE

1 Going beyond the Standard Model

2 Supersymmetry as a possible solution

3 Need for precise predictions

4 The SloopS code

5 Renormalisation of the Neutralino/Chargino sector

6 Annihilation of a light neutralino

7 Annihilation of a heavy neutralino

CHALONS Guillaume THESIS DEFENSE 5/ 50

SUPERSYMMETRY AND THE MSSM

Supersymmetry (SUSY) : a solution for physics beyond the SM SymmetrylinkingBosonstoFermions.

Transfer thesymmetryproperties of fermions to scalar bosons tostabilisethe scalarsector.

Not yet observed in nature⇒Brokensymmetry.

MSSM : Minimal Supersymmetric Standard Model =LSUSY +Lsoft. 2 Higgs doublet⇒FiveHiggs bosons :h,H,H^±,A⁰

NEW PARTICLES NEW INTERACTIONS

SUPERSYMMETRY AND THE MSSM

ADVANTAGES Stabilisethe Higgs mass.

If SUSYexact⇐⇒Complete cancellation.

M_SUSY <TeV.

Betterunificationof coupling

“constants“.

R-parity⇒LSP stable Dark Matter candidate: Neutralinoχ˜⁰₁(among other : gravitino, sneutrino,axino· · ·).

· · ·

2 4 6 8 10 12 14 16 18

Log₁₀(Q/1 GeV) 0

10 20 30 40 50 60

α⁻¹ α1

−1

α2

−1

α3

−1

CHALONS Guillaume THESIS DEFENSE 7/ 50

SUPERSYMMETRY AND THE MSSM

COMPLICATIONS Not observedyet, neither theHiggsboson...

Lsoft unkown.

Lots offree parameters ('105).

Calculations becomeextremelytedious and involved.

BEYOND LEADING ORDER IN SUSY At LO :mh<mZ butno Higgs found ! LEP Bound onHiggsmass mh>114GeV

At higher orders : Higgs masscan get large corrections. Generically SUSY processes getlargeradiative corrections. Calculations become even morecomplicated...

RADIATIVE CORRECTIONS ARE IMPORTANT

SUPERSYMMETRY AND THE MSSM

COMPLICATIONS Not observedyet, neither theHiggsboson...

Lsoft unkown.

Lots offree parameters ('105).

Calculations becomeextremelytedious and involved.

BEYOND LEADING ORDER IN SUSY At LO :mh<mZ butno Higgs found ! LEP Bound onHiggsmass mh>114GeV

At higher orders : Higgs masscan get large corrections.

Generically SUSY processes getlargeradiative corrections.

Calculations become even morecomplicated...

RADIATIVE CORRECTIONS ARE IMPORTANT

CHALONS Guillaume THESIS DEFENSE 8/ 50

SUPERSYMMETRY AND THE MSSM

COMPLICATIONS Not observedyet, neither theHiggsboson...

Lsoft unkown.

Lots offree parameters ('105).

Calculations becomeextremelytedious and involved.

BEYOND LEADING ORDER IN SUSY At LO :mh<mZ butno Higgs found ! LEP Bound onHiggsmass mh>114GeV

At higher orders : Higgs masscan get large corrections.

Generically SUSY processes getlargeradiative corrections.

Calculations become even morecomplicated...

RADIATIVE CORRECTIONS ARE IMPORTANT

OUTLINE

1 Going beyond the Standard Model

2 Supersymmetry as a possible solution

3 Need for precise predictions

4 The SloopS code

5 Renormalisation of the Neutralino/Chargino sector

6 Annihilation of a light neutralino

7 Annihilation of a heavy neutralino

CHALONS Guillaume THESIS DEFENSE 9/ 50

PRECISION

RELIC DENSITY OF DARK MATTER

WMAP: 0.0975<ΩDMh²<0.1223(10% precision) PLANCK : 2% precision

COSMOLOGY AND PARTICLE PHYSICS

RELIC DENSITY IN THE STANDARD SCENARIO ΩDMh²'3×10⁻²⁷cm³s⁻¹

hσ(χχ→SM)vi

PRECISION

Need for precise theoretical predictions w.r.t experimental measurements. Precision needed at the level ofσ⇒One-loopcalculations (at least). IfSUSYfound⇒Reconstructionof fundamental underlying parameters. Radiative correctionsmust be undercontrolto be able toconstrainthe cosmologicalunderlyingscenario.

CHALONS Guillaume THESIS DEFENSE 11/ 50

COSMOLOGY AND PARTICLE PHYSICS

RELIC DENSITY IN THE STANDARD SCENARIO ΩDMh²'3×10⁻²⁷cm³s⁻¹

hσ(χχ→SM)vi PRECISION

Need for precise theoretical predictions w.r.t experimental measurements.

Precision needed at the level ofσ⇒One-loopcalculations (at least).

IfSUSYfound⇒Reconstructionof fundamental underlying parameters.

Radiative correctionsmust be undercontrolto be able toconstrainthe cosmologicalunderlyingscenario.

RADIATIVE CORRECTIONS-RENORMALISATION

DIVERGENCES

Due to perturbative development in the coupling constant.

T REE LEV EL

q→ ∞ SELF EN ERGIES

fini +CU V

γ, g q→0

∝ln(λIR) V ERT EX DIAGRAMS

BOX DIAGRAMS

CHALONS Guillaume THESIS DEFENSE 12/ 50

RADIATIVE CORRECTIONS-RENORMALISATION

DIVERGENCES

Due to perturbative development in the coupling constant.

T REE LEV EL

q→ ∞ SELF EN ERGIES

fini +CU V

γ, g q→0

∝ln(λIR) V ERT EX DIAGRAMS

BOX DIAGRAMS

RADIATIVE CORRECTIONS-RENORMALISATION

DIVERGENCES

Due to perturbative development in the coupling constant.

T REE LEV EL

q→ ∞ SELF EN ERGIES

fini +CU V

γ, g q→0

∝ln(λIR) V ERT EX DIAGRAMS

BOX DIAGRAMS

REGULARISATION Isolateinfiniteparts in loops

UV: lnΛUV with cut-off,1/UV poles in DR.

IR: lnλIRwith cut-off,1/IRpoles in DR.

CHALONS Guillaume THESIS DEFENSE 12/ 50

RADIATIVE CORRECTIONS-RENORMALISATION

DIVERGENCES

Due to perturbative development in the coupling constant.

T REE LEV EL

q→ ∞ SELF EN ERGIES

fini +CU V

γ, g q→0

∝ln(λIR) V ERT EX DIAGRAMS

BOX DIAGRAMS

COU N T ER−T ERMS

fini +CU V

γ, g

Sof t(Eγ,g< kc) +Hard(Eγ,g> kc)

∝ln(_kc^λ) ∝ln(_k¹_c) REAL EMISSION

REGULARISATION Isolateinfiniteparts in loops

UV: lnΛUV with cut-off,1/UV poles in DR.

A WORD ABOUT INFRARED DIVERGENCIES

γ, g q→0

∝ln(λIR)

V ERT EX DIAGRAMS _{γ, g}

Sof t(Eγ,g< kc) +Hard(Eγ,g> kc)

∝ln(_kc^λ) ∝ln(_k¹

c) REAL EMISSION

Originate from

,→ Masslessgauge bosons (γ,g) coupling toon-shellexternal legs.

,→ Softandcollinearregions of integration over boson momenta (appear as double logln²(λIR)or1/²_IR).

Addingreal emissionremove unphysical dependency in the cut-offλIRor1/²_IR. Integration over3-particles phase spacecan becomplicated.

Usually for DM calculation2→2processes are enough, but if real corrections' vertex corrections,2→3processes should also be included.

If c.m energy√

sMV, EW bosons behave like aphoton⇒Mass singularities insoftandcollinearlogs∝ln²(s/M_W²)

CHALONS Guillaume THESIS DEFENSE 13/ 50

A WORD ABOUT INFRARED DIVERGENCIES

V=W, Z s≫MV²

∝ln²(s/MV²) V ERT EX DIAGRAMS

V=W, Z

s≫MV²

∝ln²(s/MV²) REAL EMISSION

Originate from

,→ Masslessgauge bosons (γ,g) coupling toon-shellexternal legs.

,→ Softandcollinearregions of integration over boson momenta (appear as double logln²(λIR)or1/²_IR).

Addingreal emissionremove unphysical dependency in the cut-offλIRor1/²_IR. Integration over3-particles phase spacecan becomplicated.

Usually for DM calculation2→2processes are enough, but if real corrections' vertex corrections,2→3processes should also be included.

If c.m energy√

sMV, EW bosons behave like aphoton⇒Mass singularities insoftandcollinearlogs∝ln²(s/M_W²)

A WORD ABOUT LOOP INTEGRALS

Looptensorintegralsreducedto a basis ofscalarintegrals[Passarino-Veltman (1979)]

Reduction method rely on a kinematical ingredient : TheGram Determinant.

For 2→2 processes,Gram determinantvanishes when relative velocityv→0 In this case reduction methodinefficient⇒differentapproach

Segmentationhas been used to study theanalyticalandnumericalbehaviour for v→0[Boudjema-Semenov-Temes (2005)].

1 D₀D₁D₂D₃ =

„ 1

D₀D₁D₂−α 1

D₀D₂D₃−β 1

D₀D₁D₃+ (α+β−1) 1 D₁D₂D₃

«

× 1

A+ 2`·(s3−αs1−βs2)

A = (s²₃−M₃²)−α(s²₁−M₁²)−β(s₂²−M₂²)−(α+β−1)M₀². D_i = (`+s_i)²−M²_i, s_i=

i

X

j=1

p_j

Relevant mostly forindirect detection:χχ→W⁺W⁻, γγ· · · in our galaxy (v'10⁻³c).

CHALONS Guillaume THESIS DEFENSE 14/ 50

A WORD ABOUT LOOP INTEGRALS

Looptensorintegralsreducedto a basis ofscalarintegrals[Passarino-Veltman (1979)]

Reduction method rely on a kinematical ingredient : TheGram Determinant.

For 2→2 processes,Gram determinantvanishes when relative velocityv→0 In this case reduction methodinefficient⇒differentapproach

Segmentationhas been used to study theanalyticalandnumericalbehaviour for v→0[Boudjema-Semenov-Temes (2005)].

1 D0D1D2D3

=

„ 1

D0D1D2−α 1

D0D2D3−β 1 D0D1D3

+ (α+β−1) 1 D1D2D3

«

× 1

A+ 2`·(s₃−αs₁−βs₂)

A = (s²₃−M²₃)−α(s²₁−M²₁)−β(s₂²−M₂²)−(α+β−1)M₀². D_i = (`+s_i)²−M²_i,s_i=

i

X

j=1

p_j

Relevant mostly forindirect detection:χχ→W⁺W⁻, γγ· · · in our galaxy

SINGULARITIES IN LOOPS USING SEGMENTATION

Singularitiesarise inscalartriangleC0and boxD0loop integrals whenβ→0.

Results innumerical instabilities.

f, M

f , M¯ γ f , M¯ f, M

C0 β→0

−−−→ − π² Q²β β=v/2 =p

1−4M²/Q²

v

0 0.02 0.04 0.06 0.08 0.1

-12000 -10000 -8000 -6000 -4000 -2000 0

a -19.74 ± 1.347e-07

b 33.99 ± 7.08e-06 a -19.74 ± 1.347e-07

b 33.99 ± 7.08e-06 a/v+b

C₀×Q²

a=−2π2=−19.7392...

If two heavy massesMand one internal mass very smallmM

CHALONS Guillaume THESIS DEFENSE 15/ 50

SINGULARITIES IN LOOPS USING SEGMENTATION

Singularitiesarise inscalartriangleC0and boxD0loop integrals whenβ→0.

Results innumerical instabilities.

f, M

f , M¯ γ f , M¯ f, M

C0 β→0

−−−→ − π² Q²β β=v/2 =p

1−4M²/Q²

v

0 0.02 0.04 0.06 0.08 0.1

-12000 -10000 -8000 -6000 -4000 -2000 0

a -19.74 ± 1.347e-07

b 33.99 ± 7.08e-06 a -19.74 ± 1.347e-07

b 33.99 ± 7.08e-06 a/v+b

C₀×Q²

a=−2π2=−19.7392...

If two heavy massesMand one internal mass very smallmM

SINGULARITIES IN LOOPS USING SEGMENTATION

Singularitiesarise inscalartriangleC0and boxD0loop integrals whenβ→0.

Results innumerical instabilities.

f, M

f , M¯ γ f , M¯ f, M

C0 β→0

−−−→ − π² Q²β β=v/2 =p

1−4M²/Q²

v

0 0.02 0.04 0.06 0.08 0.1

-12000 -10000 -8000 -6000 -4000 -2000 0

a -19.74 ± 1.347e-07

b 33.99 ± 7.08e-06 a -19.74 ± 1.347e-07

b 33.99 ± 7.08e-06 a/v+b

C₀×Q²

a=−2π2=−19.7392...

If two heavy massesMand one internal mass very smallmM

CHALONS Guillaume THESIS DEFENSE 15/ 50

SINGULARITIES IN LOOPS USING SEGMENTATION

Singularitiesarise inscalartriangleC0and boxD0loop integrals whenβ→0.

Results innumerical instabilities.

f, M

f , M¯ V, m f , M¯ f, M

C0 β→0

−−−→ − π m×M β=v/2 =p

1−4M²/Q²

v

0 0.02 0.04 0.06 0.08 0.1

-12000 -10000 -8000 -6000 -4000 -2000 0

a -19.74 ± 1.347e-07

b 33.99 ± 7.08e-06 a -19.74 ± 1.347e-07

b 33.99 ± 7.08e-06 a/v+b

C₀×Q²

a=−2π2=−19.7392...

If two heavy massesMand one internal mass very smallmM

FROM TREE TO LOOPS : NEED FOR AUTOMATION

At tree-level we have for ˜χ⁰₁χ˜⁰₁→WW 7 diagrams.

Some efficienttree-levelcodes already exist forrelic densitycalculations : DarkSUSY[Bergstr¨omet al.(2004)]

micrOMEGAs[B´elanger, Boudjema, Pukhov, Semenov (2002)]

· · ·

At one-loop we have'7000diagrams

Then for anaccurateandreliablerelic density prediction atone-looporder we need :

→ A coherentrenormalisation schemeand a choice ofinput parameters.

→ To generatecounter-terms, for SUSYgigantictask.

→ To compute ahugeamount of loop diagrams.

→ Loop Integrals library to handleGram determinantwhenv→0.

→ To deal withIRandcollinear divergencies→include bremsstrahlung.

→ To evaluatemany processesenteringhσvi.

NEED FOR AUTOMATION

CHALONS Guillaume THESIS DEFENSE 16/ 50

FROM TREE TO LOOPS : NEED FOR AUTOMATION

At tree-level we have for ˜χ⁰₁χ˜⁰₁→WW 7 diagrams.

Some efficienttree-levelcodes already exist forrelic densitycalculations : DarkSUSY[Bergstr¨omet al.(2004)]

micrOMEGAs[B´elanger, Boudjema, Pukhov, Semenov (2002)]

· · ·

At one-loop we have'7000diagrams

Then for anaccurateandreliablerelic density prediction atone-looporder we need :

→ A coherentrenormalisation schemeand a choice ofinput parameters.

→ To generatecounter-terms, for SUSYgigantictask.

→ To compute ahugeamount of loop diagrams.

→ Loop Integrals library to handleGram determinantwhenv→0.

→ To deal withIRandcollinear divergencies→include bremsstrahlung.

→ To evaluatemany processesenteringhσvi.

NEED FOR AUTOMATION

FROM TREE TO LOOPS : NEED FOR AUTOMATION

At tree-level we have for ˜χ⁰₁χ˜⁰₁→WW 7 diagrams.

Some efficienttree-levelcodes already exist forrelic densitycalculations : DarkSUSY[Bergstr¨omet al.(2004)]

micrOMEGAs[B´elanger, Boudjema, Pukhov, Semenov (2002)]

· · ·

At one-loop we have'7000diagrams

Then for anaccurateandreliablerelic density prediction atone-looporder we need :

→ A coherentrenormalisation schemeand a choice ofinput parameters.

→ To generatecounter-terms, for SUSYgigantictask.

→ To compute ahugeamount of loop diagrams.

→ Loop Integrals library to handleGram determinantwhenv→0.

→ To deal withIRandcollinear divergencies→include bremsstrahlung.

→ To evaluatemany processesenteringhσvi.

NEED FOR AUTOMATION

CHALONS Guillaume THESIS DEFENSE 16/ 50

FROM TREE TO LOOPS : NEED FOR AUTOMATION

At tree-level we have for ˜χ⁰₁χ˜⁰₁→WW 7 diagrams.

Some efficienttree-levelcodes already exist forrelic densitycalculations : DarkSUSY[Bergstr¨omet al.(2004)]

micrOMEGAs[B´elanger, Boudjema, Pukhov, Semenov (2002)]

· · ·

At one-loop we have'7000diagrams

Then for anaccurateandreliablerelic density prediction atone-looporder we need :

→ A coherentrenormalisation schemeand a choice ofinput parameters.

→ To generatecounter-terms, for SUSYgigantictask.

→ To compute ahugeamount of loop diagrams.

→ Loop Integrals library to handleGram determinantwhenv→0.

→ To deal withIRandcollinear divergencies→include bremsstrahlung.

→ To evaluatemany processesenteringhσvi.

NEED FOR AUTOMATION

FROM TREE TO LOOPS : NEED FOR AUTOMATION

At tree-level we have for ˜χ⁰₁χ˜⁰₁→WW 7 diagrams.

Some efficienttree-levelcodes already exist forrelic densitycalculations : DarkSUSY[Bergstr¨omet al.(2004)]

micrOMEGAs[B´elanger, Boudjema, Pukhov, Semenov (2002)]

· · ·

At one-loop we have'7000diagrams

Then for anaccurateandreliablerelic density prediction atone-looporder we need :

→ A coherentrenormalisation schemeand a choice ofinput parameters.

→ To generatecounter-terms, for SUSYgigantictask.

→ To compute ahugeamount of loop diagrams.

→ Loop Integrals library to handleGram determinantwhenv→0.

→ To deal withIRandcollinear divergencies→include bremsstrahlung.

→ To evaluatemany processesenteringhσvi.

NEED FOR AUTOMATION

CHALONS Guillaume THESIS DEFENSE 16/ 50

FROM TREE TO LOOPS : NEED FOR AUTOMATION

At tree-level we have for ˜χ⁰₁χ˜⁰₁→WW 7 diagrams.

Some efficienttree-levelcodes already exist forrelic densitycalculations : DarkSUSY[Bergstr¨omet al.(2004)]

micrOMEGAs[B´elanger, Boudjema, Pukhov, Semenov (2002)]

· · ·

At one-loop we have'7000diagrams

Then for anaccurateandreliablerelic density prediction atone-looporder we need :

→ A coherentrenormalisation schemeand a choice ofinput parameters.

→ To generatecounter-terms, for SUSYgigantictask.

→ To compute ahugeamount of loop diagrams.

→ Loop Integrals library to handleGram determinantwhenv→0.

→ To deal withIRandcollinear divergencies→include bremsstrahlung.

→ To evaluatemany processesenteringhσvi.

NEED FOR AUTOMATION

FROM TREE TO LOOPS : NEED FOR AUTOMATION

At tree-level we have for ˜χ⁰₁χ˜⁰₁→WW 7 diagrams.

Some efficienttree-levelcodes already exist forrelic densitycalculations : DarkSUSY[Bergstr¨omet al.(2004)]

micrOMEGAs[B´elanger, Boudjema, Pukhov, Semenov (2002)]

· · ·

At one-loop we have'7000diagrams

Then for anaccurateandreliablerelic density prediction atone-looporder we need :

→ A coherentrenormalisation schemeand a choice ofinput parameters.

→ To generatecounter-terms, for SUSYgigantictask.

→ To compute ahugeamount of loop diagrams.

→ Loop Integrals library to handleGram determinantwhenv→0.

→ To deal withIRandcollinear divergencies→include bremsstrahlung.

→ To evaluatemany processesenteringhσvi.

NEED FOR AUTOMATION

CHALONS Guillaume THESIS DEFENSE 16/ 50

FROM TREE TO LOOPS : NEED FOR AUTOMATION

At tree-level we have for ˜χ⁰₁χ˜⁰₁→WW 7 diagrams.

Some efficienttree-levelcodes already exist forrelic densitycalculations : DarkSUSY[Bergstr¨omet al.(2004)]

micrOMEGAs[B´elanger, Boudjema, Pukhov, Semenov (2002)]

· · ·

At one-loop we have'7000diagrams

Then for anaccurateandreliablerelic density prediction atone-looporder we need :

→ A coherentrenormalisation schemeand a choice ofinput parameters.

→ To generatecounter-terms, for SUSYgigantictask.

→ To compute ahugeamount of loop diagrams.

→ Loop Integrals library to handleGram determinantwhenv→0.

→ To deal withIRandcollinear divergencies→include bremsstrahlung.

→ To evaluatemany processesenteringhσvi.

NEED FOR AUTOMATION

FROM TREE TO LOOPS : NEED FOR AUTOMATION

At tree-level we have for ˜χ⁰₁χ˜⁰₁→WW 7 diagrams.

Some efficienttree-levelcodes already exist forrelic densitycalculations : DarkSUSY[Bergstr¨omet al.(2004)]

micrOMEGAs[B´elanger, Boudjema, Pukhov, Semenov (2002)]

· · ·

At one-loop we have'7000diagrams

Then for anaccurateandreliablerelic density prediction atone-looporder we need :

→ A coherentrenormalisation schemeand a choice ofinput parameters.

→ To generatecounter-terms, for SUSYgigantictask.

→ To compute ahugeamount of loop diagrams.

→ Loop Integrals library to handleGram determinantwhenv→0.

→ To deal withIRandcollinear divergencies→include bremsstrahlung.

→ To evaluatemany processesenteringhσvi.

NEED FOR AUTOMATION

CHALONS Guillaume THESIS DEFENSE 16/ 50

FROM TREE TO LOOPS : NEED FOR AUTOMATION

At tree-level we have for ˜χ⁰₁χ˜⁰₁→WW 7 diagrams.

Some efficienttree-levelcodes already exist forrelic densitycalculations : DarkSUSY[Bergstr¨omet al.(2004)]

micrOMEGAs[B´elanger, Boudjema, Pukhov, Semenov (2002)]

· · ·

At one-loop we have'7000diagrams

Then for anaccurateandreliablerelic density prediction atone-looporder we need :

→ A coherentrenormalisation schemeand a choice ofinput parameters.

→ To generatecounter-terms, for SUSYgigantictask.

→ To compute ahugeamount of loop diagrams.

→ Loop Integrals library to handleGram determinantwhenv→0.

→ To deal withIRandcollinear divergencies→include bremsstrahlung.

→ To evaluatemany processesenteringhσvi.

OUTLINE

1 Going beyond the Standard Model

2 Supersymmetry as a possible solution

3 Need for precise predictions

4 The SloopS code

5 Renormalisation of the Neutralino/Chargino sector

6 Annihilation of a light neutralino

7 Annihilation of a heavy neutralino

CHALONS Guillaume THESIS DEFENSE 17/ 50

SLOOPS CODE

Evaluation of one-loop diagrams including acompleteandcoherent renormalisation ofeach sectorof the MSSM with anOS scheme.

Modularity between different renormalisation schemes.

Non-lineargauge fixing.

Handles alarge numberof Feynman diagrams.

RENORMALISATION OF THE MSSM SECTORS

FERMION SECTOR

Input parameters as in the Standard Model

GAUGE SECTOR

Input parameters : α(0),MW,MZ thencw=MW/MZ

HIGGS SECTOR

Input parameters : M_A0,t_β=v2/v1 . Several definitions forδt_β: DR :δtβ is a pure divergence

CHALONS Guillaume THESIS DEFENSE 19/ 50

RENORMALISATION OF THE MSSM SECTORS

FERMION SECTOR

Input parameters as in the Standard Model

GAUGE SECTOR

Input parameters : α(0),MW,MZ thencw=MW/MZ

HIGGS SECTOR

Input parameters : M_A0,t_β=v2/v1 . Several definitions forδt_β: DR :δtβ is a pure divergence

RENORMALISATION OF THE MSSM SECTORS

FERMION SECTOR

Input parameters as in the Standard Model

GAUGE SECTOR

Input parameters : α(0),MW,MZ thencw=MW/MZ

HIGGS SECTOR

Input parameters : M_A0,t_β=v2/v1 . Several definitions forδt_β : DR :δt_β is a pure divergence

MH:δtβis defined from the measurement of the massmH

A⁰τ τ:δtβis defined from the decayA⁰→τ⁺τ⁻(vertex∝mτt_β)

CHALONS Guillaume THESIS DEFENSE 19/ 50

RENORMALISATION OF THE MSSM SECTORS

FERMION SECTOR

Input parameters as in the Standard Model

GAUGE SECTOR

Input parameters : α(0),M_W,M_Z thencw=M_W/MZ

HIGGS SECTOR

Input parameters : M_A0,tβ=v2/v1 . Several definitions forδtβ : DR :δtβ is a pure divergence

MH:δtβis defined from the measurement of the massmH

A⁰τ τ:δtβis defined from the decayA⁰→τ⁺τ⁻(vertex∝mτtβ)

RENORMALISATION OF THE MSSM SECTORS

FERMION SECTOR

Input parameters as in the Standard Model

GAUGE SECTOR

Input parameters : α(0),M_W,M_Z thencw=M_W/MZ

HIGGS SECTOR

Input parameters : M_A0,tβ=v2/v1 . Several definitions forδtβ : DR :δtβ is a pure divergence

MH:δtβis defined from the measurement of the massmH

A⁰τ τ:δtβis defined from the decayA⁰→τ⁺τ⁻(vertex∝mτtβ)

SFERMIONS SECTOR

Input parameters : 3 sfermions massesmd˜₁,md˜₂,mu˜₁ and 2 conditions forA_u,d

CHALONS Guillaume THESIS DEFENSE 19/ 50

RENORMALISATION OF THE MSSM SECTORS

FERMION SECTOR

Input parameters as in the Standard Model

GAUGE SECTOR

Input parameters : α(0),M_W,M_Z thencw=M_W/MZ

HIGGS SECTOR

Input parameters : M_A0,t_β=v2/v1 . Several definitions forδtβ : DR :δtβ is a pure divergence

MH:δtβis defined from the measurement of the massmH

A⁰τ τ:δtβis defined from the decayA⁰→τ⁺τ⁻(vertex∝mτtβ)

NEUTRALINOS/CHARGINOS SECTOR

GAUGE FIXING

Linear gauge fixing

LGF = − 1

ξW|∂µW^µ++iξW

g 2vG⁺|²

− 1 2ξZ

(∂µZ^µ+ξZ

g 2cw

vG⁰)²

− 1 2ξA

(∂µA^µ)²

Γ^VV= ⁻ⁱ

q2−M2 V+i

»

gµν+ (ξV−1) ^qµqν

q2−ξVM2 V

–

CHALONS Guillaume THESIS DEFENSE 20/ 50

GAUGE FIXING

Linear gauge fixing

LGF = − 1

ξW|∂µW^µ++iξW

g 2vG⁺|²

− 1 2ξZ

(∂µZ^µ+ξZ

g 2cw

vG⁰)²

− 1 2ξA

(∂µA^µ)²

Γ^VV= ⁻ⁱ

q2−M2 V+i

»

gµν+ (ξ_V−1) ^qµqν

q2−ξVM2 V

–

ξW,Z,A= 1(Feynman gauge)

GAUGE FIXING

Non-Lineargauge fixing

LGF = − 1

ξW|(∂µ−ie˜αAµ−igcwβZ˜ µ)W^µ+

+iξW

g

2(v+˜δh⁰+ωH˜ ⁰+iκG˜ ⁰+iρA˜ ⁰)G⁺|²

− 1 2ξZ

(∂µZ^µ+ξZ

g 2cw

(v+˜h⁰+˜γ⁰_H)G⁰)²

− 1 2ξA

(∂µA^µ)²

ξ_W,Z,A= 1(Feynman gauge)

CHALONS Guillaume THESIS DEFENSE 20/ 50

GAUGE FIXING

Non-Lineargauge fixing

LGF = − 1

ξW|(∂µ−ie˜αAµ−igcwβZ˜ µ)W^µ+

+iξW

g

2(v+˜δh⁰+ωH˜ ⁰+iκG˜ ⁰+iρA˜ ⁰)G⁺|²

− 1 2ξZ

(∂µZ^µ+ξZ

g 2cw

(v+˜h⁰+˜γ⁰_H)G⁰)²

− 1 2ξA

(∂µA^µ)²

Wµ⁻

Aν

G⁻ eMW(1 +αg˜ µν)

ξW,Z,A= 1(Feynman gauge)

OUTLINE

1 Going beyond the Standard Model

2 Supersymmetry as a possible solution

3 Need for precise predictions

4 The SloopS code

5 Renormalisation of the Neutralino/Chargino sector

6 Annihilation of a light neutralino

7 Annihilation of a heavy neutralino

CHALONS Guillaume THESIS DEFENSE 21/ 50

NEUTRALINO/CHARGINO SECTOR

SUMMARY AT TREE-LEVEL

The lightest neutralino ˜χ⁰₁can be a good DM candidate if R-parity conserved.

Mass matrices in the (Be,Wf⁰,He₁⁰,He₂⁰) basis

and (Wf^±,He_1,2^±) one

Y= 0 B B B

@

M1 0 −cβsw MZ sβsw MZ

0 M2 cβcw MZ −sβcw MZ

−cβsw MZ cβcw MZ 0 −µ

sβsw MZ −sβcw MZ −µ 0 1 C C C A

| {z }

−→N ( ˜χ0 1,χ˜0

2,χ˜0 3,χ˜0

4)

,

X= M2 √

2sβMW

√2cβMW µ

!

| {z }

−−−→( ˜U,V χ± 1,χ˜±

2)

Diagonalisation + Decomposition⇒6 eigenstates/eigenvalues : 4 neutralinos ˜χ⁰_i and 2 charginos ˜χ^±_i .

,→ χ˜⁰₁=N11Be+N12Wf⁰+N13He₁⁰+N14He₂⁰ with

4

X

j=1

N_1j² = 1

The value of eachN_1jdetermine thenatureofχ˜⁰₁and itscouplingsto other particles.

m_χ˜0 i,m

χ˜^±_i complicated functions ofM1,M2,µ,MW,MZ,sw,t_β. We are left with3 parameters(M1,M2,µ) to be reconstructed/defined.

NEUTRALINO/CHARGINO SECTOR

SUMMARY AT TREE-LEVEL

The lightest neutralino ˜χ⁰₁can be a good DM candidate if R-parity conserved.

Mass matrices in the (Be,Wf⁰,He₁⁰,He₂⁰) basis and (Wf^±,He_1,2^±) one

Y= 0 B B B

@

M1 0 −cβsw MZ sβsw MZ

0 M2 cβcw MZ −sβcw MZ

−cβsw MZ cβcw MZ 0 −µ

sβsw MZ −sβcw MZ −µ 0 1 C C C A

| {z }

−→N ( ˜χ0 1,χ˜0

2,χ˜0 3,χ˜0

4)

,X= M2 √

2sβMW

√2cβMW µ

!

| {z }

−−−→( ˜U,V χ± 1,χ˜±

2)

Diagonalisation + Decomposition⇒6 eigenstates/eigenvalues : 4 neutralinos ˜χ⁰_i and 2 charginos ˜χ^±_i .

,→ χ˜⁰₁=N11Be+N12Wf⁰+N13He₁⁰+N14He₂⁰ with

4

X

j=1

N_1j² = 1

The value of eachN_1jdetermine thenatureofχ˜⁰₁and itscouplingsto other particles.

m_χ˜0 i,m

χ˜^±_i complicated functions ofM1,M2,µ,MW,MZ,sw,t_β. We are left with3 parameters(M1,M2,µ) to be reconstructed/defined.

CHALONS Guillaume THESIS DEFENSE 22/ 50

NEUTRALINO/CHARGINO SECTOR

SUMMARY AT TREE-LEVEL

The lightest neutralino ˜χ⁰₁can be a good DM candidate if R-parity conserved.

Mass matrices in the (Be,Wf⁰,He₁⁰,He₂⁰) basis and (Wf^±,He_1,2^±) one

Y= 0 B B B

@

M1 0 −cβsw MZ sβsw MZ

0 M2 cβcw MZ −sβcw MZ

−cβsw MZ cβcw MZ 0 −µ

sβsw MZ −sβcw MZ −µ 0 1 C C C A

| {z }

−→N ( ˜χ0 1,χ˜0

2,χ˜0 3,χ˜0

4)

,X= M2 √

2sβMW

√2cβMW µ

!

| {z }

−−−→( ˜U,V χ± 1,χ˜±

2)

Diagonalisation + Decomposition⇒6 eigenstates/eigenvalues : 4 neutralinos ˜χ⁰_i and 2 charginos ˜χ^±_i .

,→ χ˜⁰₁=N11Be+N12Wf⁰+N13He₁⁰+N14He₂⁰ with

4

X

j=1

N_1j² = 1

The value of eachN_1jdetermine thenatureofχ˜⁰₁and itscouplingsto other particles.

m_χ˜0 i,m

χ˜^±_i complicated functions ofM1,M2,µ,MW,MZ,sw,t_β. We are left with3 parameters(M1,M2,µ) to be reconstructed/defined.

NEUTRALINO/CHARGINO SECTOR

SUMMARY AT TREE-LEVEL

The lightest neutralino ˜χ⁰₁can be a good DM candidate if R-parity conserved.

Mass matrices in the (Be,Wf⁰,He₁⁰,He₂⁰) basis and (Wf^±,He_1,2^±) one

Y= 0 B B B

@

M1 0 −cβsw MZ sβsw MZ

0 M2 cβcw MZ −sβcw MZ

−cβsw MZ cβcw MZ 0 −µ

sβsw MZ −sβcw MZ −µ 0 1 C C C A

| {z }

−→N ( ˜χ0 1,χ˜0

2,χ˜0 3,χ˜0

4)

,X= M2 √

2sβMW

√2cβMW µ

!

| {z }

−−−→( ˜U,V χ± 1,χ˜±

2)

Diagonalisation + Decomposition⇒6 eigenstates/eigenvalues : 4 neutralinos ˜χ⁰_i and 2 charginos ˜χ^±_i .

,→ χ˜⁰₁=N11Be+N12Wf⁰+N13He₁⁰+N14He₂⁰ with

4

X

j=1

N_1j² = 1

The value of eachN_1jdetermine thenatureofχ˜⁰₁and itscouplingsto other particles.

m_χ˜0 i,m

χ˜^±_i complicated functions ofM1,M2,µ,M_W,M_Z,sw,t_β.

We are left with3 parameters(M1,M2,µ) to be reconstructed/defined.

CHALONS Guillaume THESIS DEFENSE 22/ 50

NEUTRALINO/CHARGINO SECTOR

SUMMARY AT TREE-LEVEL

The lightest neutralino ˜χ⁰₁can be a good DM candidate if R-parity conserved.

Mass matrices in the (Be,Wf⁰,He₁⁰,He₂⁰) basis and (Wf^±,He_1,2^±) one

Y= 0 B B B

@

M1 0 −cβsw MZ sβsw MZ

0 M2 cβcw MZ −sβcw MZ

−cβsw MZ cβcw MZ 0 −µ

sβsw MZ −sβcw MZ −µ 0 1 C C C A

| {z }

−→N ( ˜χ0 1,χ˜0

2,χ˜0 3,χ˜0

4)

,X= M2 √

2sβMW

√2cβMW µ

!

| {z }

−−−→( ˜U,V χ± 1,χ˜±

2)

Diagonalisation + Decomposition⇒6 eigenstates/eigenvalues : 4 neutralinos ˜χ⁰_i and 2 charginos ˜χ^±_i .

,→ χ˜⁰₁=N11Be+N12Wf⁰+N13He₁⁰+N14He₂⁰ with

4

X

j=1

N_1j² = 1

The value of eachN_1jdetermine thenatureofχ˜⁰₁and itscouplingsto other particles.

INPUT PARAMETERS IN CHARGINO/NEUTRALINO SECTOR

AND AT ONE-LOOP

The 4x4 neutralino CT mass matrix is defined as :

δY = 0 B

@

δM1 0 δY13 δY14

0 δM2 δY23 δY24

δY13 δY23 0 −δµ δY14 δY24 −δµ 0

1 C A

The 2x2 CT chargino mass matrix is :

δX = 0 B B

@

δM2

√2s_βM_W

„

1 2

δM2 W M2

W

+c_β^2δt_t^β

β

«

√2cβMW

„

1 2

δM2 W M2

W −s_β^2δt_t^β

β

«

δµ

1 C C A

At one-loop : 3 counter-terms (δM1,δM2,δµ) →3 renormalisation conditions.

Our choice : ON-SHELL renormalisation conditions obtained from 3physical massesm^TL_χ˜

i =mphys

χ˜i .

Multiplechoicesavailableto choose 3 masses among the 6 (≡20). Advantage: gaugeinvariantdefinition.

CHALONS Guillaume THESIS DEFENSE 23/ 50

INPUT PARAMETERS IN CHARGINO/NEUTRALINO SECTOR

AND AT ONE-LOOP

The 4x4 neutralino CT mass matrix is defined as :

δY = 0 B

@

δM1 0 δY13 δY14

0 δM2 δY23 δY24

δY13 δY23 0 −δµ δY14 δY24 −δµ 0

1 C A

The 2x2 CT chargino mass matrix is :

δX = 0 B B

@

δM2

√2s_βM_W

„

1 2

δM2 W M2

W

+c_β^2δt_t^β

β

«

√2cβMW

„

1 2

δM2 W M2

W −s_β^2δt_t^β

β

«

δµ

1 C C A

At one-loop : 3 counter-terms (δM1,δM2,δµ) →3 renormalisation conditions.

Our choice : ON-SHELL renormalisation conditions obtained from 3physical massesm^TL_χ˜

i =mphys

χ˜i .

Multiplechoicesavailableto choose 3 masses among the 6 (≡20).

Advantage: gaugeinvariantdefinition.