Higgs Pseudo-Observables Sandro Uccirati KIT In collaboration with - - PowerPoint PPT Presentation

HP2.3rd – Firenze, 14-17 September 2010

Higgs Pseudo-Observables

Sandro Uccirati

KIT In collaboration with G. Passarino, C. Sturm

HP2.3rd – Firenze, 14-17 September 2010

• S. Uccirati

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HP2.3rd – Firenze, 14-17 September 2010

Standard Model hadronic Higgs production channels

g g g g H H H H V V V V t t t t t q q q q q q t t

1 10 10 2 10 3 100 120 140 160 180 200

qq → Wh qq → Zh gg → h bb → h gg,qq → tth qq → qqh

mh [GeV] σ [fb]

SM Higgs production TeV II

TeV4LHC Higgs working group

10 2 10 3 10 4 10 5 100 200 300 400 500

qq → Wh qq → Zh gg → h bb → h qb → qth gg,qq → tth qq → qqh

mh [GeV] σ [fb]

SM Higgs production LHC

TeV4LHC Higgs working group

Hahn,Heinemeyer,Maltoni,Weiglein,Willenbrock [hep-ph/0607308]

Gluon-fusion largest cross section

• S. Uccirati

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HP2.3rd – Firenze, 14-17 September 2010

Higgs decays in the Standard Model

• t
• t

• s
• s
• b
• b

• H → bb:

H b ¯ b

For light Higgs, huge background

• H → WW, ZZ:

H W, Z W, Z

For heavy Higgs

• H → γγ:

H γ γ t t t H γ γ W , Φ W , Φ W , Φ H γ γ W , Φ

For Light Higgs: rare, but clean

• S. Uccirati

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HP2.3rd – Firenze, 14-17 September 2010

Problems with gauge invariance: H(P) → γ(p1) + γ(p2)

Amplitude → Aµν = g3 s2

θ

16 π2 (FD δµν + FP pµ

2 pν 1).

Ward Identity: FD + p1 · p2 FP = 0 Renormalization (Ren) → M 2

H,0 = M 2 H

• 1+ GFM 2

W

2 √ 2 π2 Re Σ(1)

HH(M 2 H)

• FD = F (1)

D

⊗ (1 + Ren) + F (2)

D

FP = F (1)

P

⊗ (1 + Ren) + F (2)

P

• 2-loop level

F (2)

D

+ p1 · p2 F (2)

P

• +

(F (1)

D

+ p1· p2 F (1)

P ) ⊗ Ren

• =

H γ γ Φ Φ Φ H γ γ Φ Φ

No “Re” label ∼MH,0 → Re

• H

W, Φ

• S. Uccirati

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HP2.3rd – Firenze, 14-17 September 2010

• Unstable particles can not be

asymptotic states

• Higgs production and decay are

not well defined ⇓

complete process

pp → γγ + X which consists of

Signal

• pp → (gg → H → γγ) + X
• +

Background

How to extract a pseudo-observable to be termed Higgs partial decay width into two photons which does not violate first principles?

• S. Uccirati

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HP2.3rd – Firenze, 14-17 September 2010

Higgs self-energie ΣH(s, M 2

H,0) =

H

Complex pole: sH − M 2

H,0 + ΣH(sH, M 2 H,0) = 0

• gauge invariant definition
• MH,0 real by construction
• sH = µ2

H − i µH γH

Dyson-resummed Higgs propagator ∆H(s) =

× × + × × + × × +

. . . = (s − sH)−1 1 + ΠH(s) −1 , ΠH(s) = ΣH(s) − ΣH(sH) s − sH

• S. Uccirati

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HP2.3rd – Firenze, 14-17 September 2010

Amplitude for gg → γγ: g g ∆H γ γ V

ggH

V

Hγγ

Signal +

g g γ γ non-resonant

Background

In general S-matrix for i → f: Sfi = Vi(s) ∆H(s)Vf(s) + Bnr =

• Z−1/2

H

(s)Vi(s)

• 1

s − sH

• Z−1/2

H

(s)Vf(s)

• + Bnr,

ZH = 1 + ΠH Bnr = non-resonant background Expand the square brackets around s = sH

• S. Uccirati

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HP2.3rd – Firenze, 14-17 September 2010

Sfi = S(i→Hc) S(Hc →f) s − sH + non resonant terms.

where Production : S(i → Hc) = Z−1/2

H

(sH) Vi(sH) Decay : S(Hc → f) = Z−1/2

H

(sH) V

f(sH)

• gauge invariant order per order in perturbation theory
• Diagrams and renormalization evaluated at the complex pole

ZH(sH) = 1 + lim

s→sH

ΣH(s) − ΣH(sH) s − sH = 1 + ∂ΣH ∂s (sH)

• Universal and well-defined parametrization of experimantal data

⇒ Definition of a gauge-invariant decay width:

Γ(Hc →f) = (2π)4 2 µH

• dΦ

f(PH, {pf})

• spins
• S(Hc → f)
• 2
• S. Uccirati

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HP2.3rd – Firenze, 14-17 September 2010

Analytical continuation

• We have diagrams with complex external squared momenta
• We must understand how is defined the physical Riemann sheet

↓ i0+ Feynman prescription Example:

s m

= ∆ − 1 dx ln χ, χ = −s x (1−x) + m2 − i0+

• Complex mass:

m2 → µ2 − iµγ

• Imχ does not change sign
• Complex s:

s → M 2 − iMΓ

• Imχ changes sign

→ Problem General rule: lim

γ,Γ→0 Ampl (s, m) = Ampl (M 2, µ)

If Reχ < 0 and Imχ > 0 (second quadrant): lim

γ,Γ

H→0 Im[ln χ] = π

= Feynman prescription for real masses (µ2 → µ2 − i0) = −π

• S. Uccirati

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HP2.3rd – Firenze, 14-17 September 2010

• If Reχ < 0 and Imχ > 0 (second quadrant), we have to change the

definition of the log. Analytical continuation on the second Riemann sheet: ln(z) → ln−(z) = ln(z) − 2 i π θ(−Rez) θ(Imz)

• second quadrant

⇔

move the cut on the positive imaginary axis

• This changes the computation of loop functions (analytical continuation for

Lin, HPLs, etc.)

• Change of the integration contour in integral representations:
• The integration contour (x ∈ [0, 1]) never crosses the cut of ln χ

(negative real axis), but . . .

• . . . it can cross the cut of ln− χ (positive imaginary axis) → Problem

In the example this happens for M 2 ≥ 4 µ2 & µΓ − Mγ ≥ 0

• S. Uccirati

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HP2.3rd – Firenze, 14-17 September 2010

General strategy in parametric space:

• Diagrams → integrals of polynomial (quadratic in some variables)

to negative/non-integer power .

• S. Uccirati

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HP2.3rd – Firenze, 14-17 September 2010

General strategy in parametric space:

• Diagrams → integrals of polynomial (quadratic in some variables)

to negative/non-integer power

• Pick up one variable x (quadratic):

χ = a x2 + b x + c Reχ = 0, Imχ = 0 → Hyperbolas .

Re x Im x 1 | Reχ Imχ Reχ Imχ

contour

• S. Uccirati

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HP2.3rd – Firenze, 14-17 September 2010

General strategy in parametric space:

• Diagrams → integrals of polynomial (quadratic in some variables)

to negative/non-integer power

• Pick up one variable x (quadratic):

χ = a x2 + b x + c Reχ = 0, Imχ = 0 → Hyperbolas Find the cuts ⇔ study intersections

• f hyperbolas.

.

Re x Im x 1 | Reχ Imχ>0 Reχ Imχ>0

cut contour

• S. Uccirati

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HP2.3rd – Firenze, 14-17 September 2010

General strategy in parametric space:

• Diagrams → integrals of polynomial (quadratic in some variables)

to negative/non-integer power

• Pick up one variable x (quadratic):

χ = a x2 + b x + c Reχ = 0, Imχ = 0 → Hyperbolas Find the cuts ⇔ study intersections

• f hyperbolas.
• Deform just the contour of x, for general

values of the others

• Deformation for the general case can be

easily automatized (numerically)

Re x Im x 1 Reχ Imχ>0 Reχ Imχ>0

cut contour

• S. Uccirati

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HP2.3rd – Firenze, 14-17 September 2010

Numerical effects: Notation

RMRP → Real Masses and Real Momenta. The usual on-shell scheme where all masses and all Mandelstam in- variants are real. CMRP → Complex Masses and Real Momenta. The complex mass scheme (Denner-Dittmaier-Roth-Wieders [hep-ph/0505042]) with complex internal W and Z poles (extend- able to top complex pole) but with real, external, on-shell Higgs and with the standard LSZ wave-function renormalization. CMCP → Complex Masses and Complex Momenta. The (complete) complex mass scheme with complex internal W and Z poles and complex, external, Higgs where the LSZ procedure is carried out at the Higgs complex pole (on the second Riemann sheet).

• S. Uccirati

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HP2.3rd – Firenze, 14-17 September 2010

Numerical effects: H → gg

5 10 15 20 25 30 35 120 200 300 400 500 Γ(H→gg)[MeV] µH [GeV]

RMRP CMCP (Γt=0) CMCP (Γt=13.1GeV)

• S. Uccirati

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HP2.3rd – Firenze, 14-17 September 2010

[GeV]

H

µ 300 320 340 360 380 400 420 440 460 480 500 H) [pb] → (pp σ 0.02 0.04 0.06 0.08 0.1 0.12 0.14

= 3 TeV s

Numerical effects: pp → H

CMRP CMCP .

• S. Uccirati

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HP2.3rd – Firenze, 14-17 September 2010

Numerical effects: H → γγ

20 40 60 80 100 120 140 150 200 250 300 350 400 Γ(H→γγ)[KeV] µH [GeV] RMRP CMRP CMCP

• S. Uccirati

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HP2.3rd – Firenze, 14-17 September 2010

Numerical effects: H → ¯ bb

• 7
• 6
• 5
• 4
• 3
• 2
• 1

150 200 250 300 350 400 RMRP CMRP CMCP

• S. Uccirati

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HP2.3rd – Firenze, 14-17 September 2010

Summary

• Proposal for a gauge invariant parametrization of experimental

distributions for Higgs physics

• Gauge invariant definition of production cross section and decay width
• Numerical effects: negligible below t¯

t threshold, but sizable for large MH

• Computational recipe:

Analytical continuation and contour distortion for diagrams with complex Mandelstan invariants

• S. Uccirati

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