PSI – Apr. 10, 2008 Complete two-loop corrections to H → γγ Sandro Uccirati Karlsruhe University In collaboration with C. Sturm, G. Passarino PSI – Apr. 10, 2008 S. Uccirati Page 1
1 � b b W W Z Z PSI – Apr. 10, 2008 � t t Higgs decays in the Standard Model 0.1 � � g g � • H → bb : • 0.01 BR( H ) Dominant process for light Higgs, but huge QCD background. � � 0.001 • H → γγ : • s s � Rare process, but experimentally �� clean. Z � Discovery channel for light Higgs 0.0001 100 130 160 200 300 500 700 1000 M [GeV℄ H • H → WW, ZZ : • Discovery channels for heavy Higgs S. Uccirati Page 2
PSI – Apr. 10, 2008 Lowest order (one-loop) for H → γγ (in SM) • Well-known result • • Ellis-Gaillard-Nanopoulos 1976, Shifman-Vainshtein-Voloshin-Zakharov 1979 • γ real W-mass 0.05 W , Φ complex W-mass H W , Φ 0.04 ) [keV] W , Φ γ 0.03 γ γ → γ (H Γ t 0.02 H t 0.01 t γ 100 110 120 130 140 150 160 170 M [GeV] h S. Uccirati Page 3
PSI – Apr. 10, 2008 Two-loop SM corrections to H → γγ • QCD corrections • • Zheng-Wu ’90, Djouadi-Spira-van der Bij-Zerwas ’91, Dawson-Kauffman ’93, • Melnikov-Yakovlev ’93, Inoue-Najima-Oka-Saito ’94, Steinhauser ’96, Fleischer-Tarasov-Tarasov ’04, Harlander-Kant ’05, Aglietti-Bonciani-Degrassi-Vicini ’06, Passarino-Sturm-U. ’07 • EW corrections • • corrections at O ( G µ m 2 H ) (Korner-Melnikov-Yakovlev ’96) • • corrections at O ( G µ m 2 t ) (Fugel-Kniehl-Steinhauser ’04) • • light-fermion contribution (Aglietti-Bonciani-Degrassi-Vicini ’04) • • top-quark and bosonic contributions for m H < 150 GeV (Degrassi-Maltoni ’05) • • full EW contributions (Passarino-Sturm-U. ’07) • S. Uccirati Page 4
PSI – Apr. 10, 2008 The amplitude of H ( P ) → γ ( p 1 ) + γ ( p 2 ) A µν ( H → γγ ) = g 3 s 2 2 � F D δ µν + � F ( ij ) � p µ θ i p ν j + F ǫ ǫ ( µ, ν, p 1 , p 2 ) . P 16 π 2 i,j =1 Interference with 1-loop ⇓ Bose symmetry Ward identities A µν ( H → γγ ) = g 3 s 2 16 π 2 ( F D δ µν + F P p µ θ 2 p ν 1 ) . Order by order Ward identities: F D + p 1 · p 2 F P = 0 → in pertubation theory Introduce projectors: „ « δ µν − p µ 2 + p µ 1 p ν 2 p ν 1 F D = P µν D A µν , P µν 1 = D n − 2 p 1 · p 2 „ « δ µν − ( n − 1) p µ 2 + p µ 1 p ν 2 p ν 1 1 F P ≡ F (21) = P µν A µν , P µν 1 = − P P P n − 2 p 1 · p 2 p 1 · p 2 S. Uccirati Page 5
PSI – Apr. 10, 2008 The amplitude is computed with the GraphShot package • A FORM code to generate and manipulate the amplitudes in the SM • • A link to FORTRAN libraries for numerical computation • • Authors: G.Passarino, M.Passera, A.Ferroglia, S.Actis, C.Sturm, S.U. • • It is WORK IN PROGRESS (not yet available) • Let’s discover the path to compute Feynman amplitudes ... S. Uccirati Page 6
PSI – Apr. 10, 2008 1. The Feynman rules • The SM Lagrangian → normal rules for propagators and vertices • • Special rules: • • Higgs vacuum expectation value • H H normal : = 0 special : = 0 • Z-Photon exchange ( g → g (1 + Γ)): • normal : Γ = 0 γ ν = G AZ d ( p 2 ) δ µν + G AZ pp ( p 2 ) p µ p ν , G AZ Z special : d (0) = 0 µ • Renormalization → MS scheme • • Counterterms for couplings, masses, fields, ... • • Finite Feynman amplitudes • S. Uccirati Page 7
PSI – Apr. 10, 2008 2. Generate the amplitude • Group the diagrams into families, paying attention to: • • Permutation of external legs • p 3 p 3 p 3 p 3 p 2 p 1 p 1 p 1 p 1 p 1 p 1 p 2 → p 2 p 2 p 2 p 2 p 3 p 3 • Combinatorial factors (Goldberg strategy) • • Combine the topologies and the Feynman rules • • Introduce projectors • • Compute the trace of Dirac matrices • ⇓ All loop momenta are contracted with other momenta S. Uccirati Page 8
PSI – Apr. 10, 2008 3. Reduction to Basic Integrals • Recursive application of: • • Obvious reduction: • p 2 − m 2 + M 2 2 q.p 1 1 ( q 2 + m 2 ) [( q + p ) 2 + M 2 ] = q 2 + m 2 − ( q + p ) 2 + M 2 − ( q 2 + m 2 ) [( q + p ) 2 + M 2 ] • Mapping on a fixed standard routing for loop momenta: • p 2 p 2 q 2 q 2 + P 0 1 q 2 + p 2 q 2 + p 1 @ q 1 → − q 1 − P q 1 − q 2 q 1 − q 2 − P − P − → B C A q 2 → − q 2 − P q 1 + p 2 q 1 + p 1 q 1 q 1 + P p 1 p 1 • Symmetrization: • p 1 p 2 m 1 m 5 0 1 m 2 m 4 @ q 1 → − q 2 − P m 3 m 3 − P − P − → B C A q 2 → − q 1 − P m 4 m 2 m 5 m 1 p 2 p 1 S. Uccirati Page 9
PSI – Apr. 10, 2008 • We end with integrals up to rank 2: • • 1-loop functions • • 2-loop tadpoles (2 topologies) • T A T B • 2-loop self-energies (4 topologies) • S A S C S E S D • 2-loop vertices (6 topologies) • V E V I V G V M V K V H S. Uccirati Page 10
PSI – Apr. 10, 2008 • Full scalarization of 2-loop self-energies • • Reduction in sub-loops: • d n q 1 q µ Z 2 ] = X q µ 1 1 )[( q 1 − q 2 ) 2 + m 2 2 ( q 2 1 + m 2 1 A new propagator 2 is introduced with spurious mass singularities. q 2 • New tadpoles with dots are generated • • Use integration by parts identities to reduce all tadpoles to: • T B • Full scalarization of 1-loop diagrams • • All 1-loop diagrams with dots are reduced wiht integration by parts • identities S. Uccirati Page 11
PSI – Apr. 10, 2008 Feynman parametrization Consider a general loop integral: Z q µ 1 · · · q µ R D i = ( q + k i ) 2 + m 2 I µ 1 ··· µ R d n q = 1 D N , i N D 1 D 2 · · · D N − S. Uccirati Page 12
PSI – Apr. 10, 2008 Feynman parametrization Z q µ 1 · · · q µ R D i = ( q + k i ) 2 + m 2 I µ 1 ··· µ R d n q = , i N D 1 D 2 D N − D N · · · 1 |{z} |{z} |{z} | {z } x N − (1 − x 1 ) ( x 1 − x 2 ) ( x N − 2 − x N − 1 ) 1 The product of N propagators becomes one propagator to power N Z 1 Z x 1 Z x N − 2 Z dx N − 1 [( q + K ) 2 + M 2 ] − N d n q q µ 1 · · · q µ R I µ 1 ··· µ R = Γ( N ) dx 1 dx 2 · · · N 0 0 0 k µ 1 (1 − x 1 ) + k µ 2 ( x 1 − x 2 ) + . . . + k µ 1 ) + k µ K µ = 1 ( x N − 2 − x N − N x N − 1 N − M 2 ( m 2 1 + k 2 1 )(1 − x 1 ) + ( m 2 2 + k 2 = 2 )( x 1 − x 2 ) + . . . +( m 2 1 + k 2 1 ) + ( m 2 N + k 2 1 − K 2 1 )( x N − 2 − x N − N ) x N − N − N − S. Uccirati Page 13
PSI – Apr. 10, 2008 Feynman parametrization Z q µ 1 · · · q µ R D i = ( q + k i ) 2 + m 2 I µ 1 ··· µ R d n q = 1 D N , i N D 1 D 2 · · · D N − Z 1 Z x 1 Z x N − 2 Z dx N − 1 [( q + K ) 2 + M 2 ] − N d n q q µ 1 · · · q µ R = Γ( N ) dx 1 dx 2 · · · 0 0 0 k µ 1 (1 − x 1 ) + k µ 2 ( x 1 − x 2 ) + . . . + k µ 1 ) + k µ K µ = 1 ( x N − 2 − x N − N x N − 1 N − M 2 ( m 2 1 + k 2 1 )(1 − x 1 ) + ( m 2 2 + k 2 = 2 )( x 1 − x 2 ) + . . . +( m 2 1 + k 2 1 ) + ( m 2 N + k 2 1 − K 2 1 )( x N − 2 − x N − N ) x N − N − N − Integration in d n q is performed ”Z 1 Z x 1 Z x N − “ N − n 2 n n 1 ( M 2 ) 2 − N 2 Γ I N = i π dx 1 dx 2 · · · dx N − 2 0 0 0 ”Z 1 Z x 1 Z x N − “ N − n 2 n n I µ 1 1 ( − K µ 1 ) ( M 2 ) 2 Γ 2 − N = i π dx 1 dx 2 · · · dx N − N 2 0 0 0 ”Z 1 Z x 1 Z x N − K µ 1 K µ 2 + M 2 δ µ 1 µ 2 » – “ N − n 2 n n I µ 1 µ 2 ( M 2 ) 2 Γ 2 − N = i π dx 1 dx 2 · · · dx N − 1 N 2 2 N − n − 2 0 0 0 I µ 1 µ 2 µ 3 = . . . N S. Uccirati Page 14
PSI – Apr. 10, 2008 4. Analytical cancellations of divergences Extraction of the UV poles • 1-loop diagrams → trivial (Γ( ǫ/ 2)) • • 2-loop diagrams: • • Overall divergency → trivial (Γ( ǫ )) • • Singularities coming from sub-loops → hidden in the integrand • p 2 [1] = q 2 1 + m 2 1 m 5 [2] = ( q 1 − q 2 ) 2 + m 2 Z d n q 1 d n q 2 = 1 2 − P V I = , [3] = q 2 2 + m 2 m 4 m 2 π 4 [1] [2] [3] [4] [5] 3 [4] = ( q 2 + p 1 ) 2 + m 2 m 1 | {z } 4 m 3 x [5] = ( q 2 + P ) 2 + m 2 p 1 5 | {z } y 1 ,y 2 ,y 3 Z 1 Z dS 3 ( y 1 , y 2 , y 3 ) [ x (1 − x )] − ǫ/ 2 (1 − y 1 ) ǫ/ 2 − 1 V − 1 − ǫ = C ǫ dx 0 • The single pole can always be expressed in terms of 1-loop functions. • m 1 p 2 m 5 I = m 2 m 2 − P V + finite part . × 3 3 m 4 m 3 p 1 m 2 S. Uccirati Page 15
PSI – Apr. 10, 2008 Collinear divergencies They come from the coupling of light particles (m) with massless particles q 1 q 1 Single p m p m = ⇒ m divergency m q 1 + p q 1 + p m m m m Double m = ⇒ m ′ m m divergency m ′ m ′ m ′ • Single divergency: Subtraction method • J 1 = µ 4 − n � 1 d n q 1 1 + m 2 )[( q 1 + p ) 2 + m 2 ][( q 1 − q 2 ) 2 + M 2 ] . ( q 2 i π 2 S. Uccirati Page 16
PSI – Apr. 10, 2008 After parametrization Z 1 Z z dy 1 V = [ A − y ( q 2 + p ) 2 ] y + m 2 (1 − y ) , A = ( q 2 + p z ) 2 + M 2 . J 1 = dz V , 0 0 Add and subtract: V − 1 = ( A y + m 2 ) − 1 0 „ 1 Z 1 Z z Z 1 Z z « 1 V − 1 J 1 = dz dy A y + m 2 + dz dy V 0 0 0 0 0 " # Z 1 Z 1 Z 1 Z z − ln m 2 dz 1 dz 1 A ln Az dy A − y ( q 2 + p ) 2 − 1 1 + O ( m 2 ) . = A + + dz s s y A 0 0 0 0 Example: p 2 p 2 M 5 M 5 � 1 ln m 2 M 4 − P − P M 3 = dz + finite part M 4 s 0 m M 3 m (1 z ) p 1 − p 1 zp 1 The coefficients of the log are 1-loop functions S. Uccirati Page 17
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