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 Page 1
HP2.3rd – Firenze, 14-17 September 2010 Standard Model hadronic Higgs production channels g q q g q t V t t H V V H H t V t t g q q g q t H SM Higgs production SM Higgs production 10 5 LHC TeV II 10 3 σ [ fb ] gg → h gg → h σ [ fb ] 10 4 qq → Wh qq → qqh 10 2 10 3 qq → qqh qq → Wh bb → h bb → h 10 qq → Zh gg,qq → tth 10 2 qb → qth gg,qq → tth qq → Zh TeV4LHC Higgs working group TeV4LHC Higgs working group 1 100 200 300 400 500 100 120 140 160 180 200 m h [ GeV ] m h [ GeV ] Hahn,Heinemeyer,Maltoni,Weiglein,Willenbrock [hep-ph/0607308] Gluon-fusion � largest cross section S. Uccirati Page 2
1 � b b W W Z Z HP2.3rd – Firenze, 14-17 September 2010 � t t Higgs decays in the Standard Model 0.1 � � g g b � • H → bb : H • ¯ b 0.01 BR( H ) For light Higgs, huge background W, Z � � • H → WW, ZZ : H • 0.001 s s � W, Z For heavy Higgs �� γ Z � t 0.0001 • H → γγ : H • t 100 130 160 200 300 500 700 1000 M [GeV℄ H t γ γ γ W , Φ , Φ H H W W , Φ , Φ W γ γ For Light Higgs: rare, but clean S. Uccirati Page 3
HP2.3rd – Firenze, 14-17 September 2010 Problems with gauge invariance: H ( P ) → γ ( p 1 ) + γ ( p 2 ) A µν = g 3 s 2 16 π 2 ( F D δ µν + F P p µ θ 2 p ν Amplitude → 1 ) . Ward Identity: F D + p 1 · p 2 F P = 0 1+ G F M 2 � � 2 π 2 Re Σ (1) M 2 H , 0 = M 2 HH ( M 2 √ W Renormalization (Ren) → H ) H 2 F D = F (1) ⊗ (1 + Ren) + F (2) F P = F (1) ⊗ (1 + Ren) + F (2) D D P P • 2-loop level • F (2) + p 1 · p 2 F (2) ( F (1) + p 1 · p 2 F (1) + P ) ⊗ Ren � = 0 D P D � �� � � �� � γ γ Φ Φ H H Φ γ γ Φ Φ � � H W, Φ No “Re” label ∼ M H , 0 → Re S. Uccirati Page 4
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 Page 5
HP2.3rd – Firenze, 14-17 September 2010 Higgs self-energie Σ H ( s, M 2 H , 0 ) = H s H − M 2 H , 0 + Σ H ( s H , M 2 Complex pole: H , 0 ) = 0 • gauge invariant definition • s H = µ 2 • M H , 0 real by construction H − i µ H γ H • � Dyson-resummed Higgs propagator × + × + × + ∆ H ( s ) = . . . × × × ( s − s H ) − 1 � � − 1 Π H ( s ) = Σ H ( s ) − Σ H ( s H ) = 1 + Π H ( s ) , s − s H S. Uccirati Page 6
HP2.3rd – Firenze, 14-17 September 2010 Amplitude for gg → γγ : g γ g γ V V + non-resonant ggH Hγγ ∆ H g γ g γ Signal Background In general S-matrix for i → f : = V i ( s ) ∆ H ( s ) V f ( s ) + B nr S fi � � � � 1 Z − 1 / 2 Z − 1 / 2 = ( s ) V i ( s ) ( s ) V f ( s ) + B nr , H H s − s H Z H = 1 + Π H B nr = non-resonant background Expand the square brackets around s = s H S. Uccirati Page 7
HP2.3rd – Firenze, 14-17 September 2010 S fi = S ( i → H c ) S ( H c → f ) + non resonant terms . s − s H where S ( i → H c ) = Z − 1 / 2 Production : ( s H ) V i ( s H ) H S ( H c → f ) = Z − 1 / 2 Decay : ( s H ) V f ( s H ) H • gauge invariant order per order in perturbation theory • • Diagrams and renormalization evaluated at the complex pole • Σ H ( s ) − Σ H ( s H ) = 1 + ∂ Σ H Z H ( s H ) = 1 + lim ∂s ( s H ) s − s H s → s H • Universal and well-defined parametrization of experimantal data • ⇒ Definition of a gauge-invariant decay width: � Γ( H c → f ) = (2 π ) 4 � � � � 2 d Φ f ( P H , { p f } ) � S ( H c → f ) 2 µ H spins S. Uccirati Page 8
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 • ↓ i 0 + Feynman prescription Example: � 1 m χ = − s x (1 − x ) + m 2 − i 0 + s = ∆ − dx ln χ, 0 m 2 → µ 2 − iµγ • Complex mass: Im χ does not change sign • � s → M 2 − iM Γ • Complex s: Im χ changes sign → Problem • � γ, Γ → 0 Ampl ( s, m ) = Ampl ( M 2 , µ ) General rule: lim If Re χ < 0 and Im χ > 0 (second quadrant): Feynman prescription for lim H → 0 Im[ln χ ] = π � = = − π real masses ( µ 2 → µ 2 − i 0) γ, Γ S. Uccirati Page 9
HP2.3rd – Firenze, 14-17 September 2010 • If Re χ < 0 and Im χ > 0 (second quadrant), we have to change the • definition of the log. Analytical continuation on the second Riemann sheet: move the cut on the ln( z ) → ln − ( z ) = ln( z ) − 2 i π θ ( − Re z ) θ (Im z ) ⇔ positive imaginary axis � �� � second quadrant • This changes the computation of loop functions (analytical continuation for • Li n , 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 µ 2 & µ Γ − Mγ ≥ 0 S. Uccirati Page 10
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 Page 11
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 Im x contour • Pick up one variable x (quadratic): • χ = a x 2 + b x + c Im χ Re χ Re χ = 0 , Im χ = 0 → Hyperbolas | 0 1 Re x Re χ Im χ . S. Uccirati Page 12
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 Im x contour • Pick up one variable x (quadratic): • cut χ = a x 2 + b x + c Im χ >0 Re χ Re χ = 0 , Im χ = 0 → Hyperbolas study intersections | Find the cuts ⇔ 0 1 of hyperbolas. Re x Re χ Im χ >0 . S. Uccirati Page 13
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 Im x contour • Pick up one variable x (quadratic): • cut χ = a x 2 + b x + c Im χ >0 Re χ Re χ = 0 , Im χ = 0 → Hyperbolas study intersections Find the cuts ⇔ 0 1 of hyperbolas. Re x • Deform just the contour of x , for general • values of the others Re χ • Deformation for the general case can be • Im χ >0 easily automatized (numerically) S. Uccirati Page 14
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 Page 15
HP2.3rd – Firenze, 14-17 September 2010 Numerical effects: H → gg 35 RMRP CMCP ( Γ t =0) 30 CMCP ( Γ t =13.1GeV) 25 Γ (H → gg)[MeV] 20 15 10 5 0 120 200 300 400 500 µ H [GeV] S. Uccirati Page 16
HP2.3rd – Firenze, 14-17 September 2010 Numerical effects: pp → H 0.14 CMRP CMCP 0.12 0.1 H) [pb] 0.08 → (pp 0.06 σ 0.04 0.02 s = 3 TeV 0 300 320 340 360 380 400 420 440 460 480 500 [GeV] µ . H S. Uccirati Page 17
HP2.3rd – Firenze, 14-17 September 2010 Numerical effects: H → γγ 140 120 100 Γ (H →γγ )[KeV] 80 60 40 RMRP 20 CMRP CMCP 0 150 200 250 300 350 400 µ H [GeV] S. Uccirati Page 18
HP2.3rd – Firenze, 14-17 September 2010 Numerical effects: H → ¯ bb 0 -1 -2 -3 -4 -5 -6 RMRP CMRP CMCP -7 150 200 250 300 350 400 S. Uccirati Page 19
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 M H • • Computational recipe : • Analytical continuation and contour distortion for diagrams with complex Mandelstan invariants S. Uccirati Page 20
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