The rare decay H Zγ in perturbative QCD [arxiv: hep-ph/1505.00561] Thomas Gehrmann, Sam Guns & Dominik Kara June 15, 2015 RADCOR 2015 AND LOOPFEST XIV - UNIVERSITY OF CALIFORNIA, LOS ANGELES Z Z H g q H g q γ γ
Contents 1 Introduction 2 The decay in the Standard Model 3 Calculation of the two-loop amplitude Integral basis and differential equations Renormalization 4 Numerical results 5 Conclusions Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 0/20
Motivation: Experiment H γγ loop-mediated sensitive to new physics clean signature 20% relative precision Branching ratios 10 1 1 ττ cc bb gg WW ZZ LHC HIGGS XS WG 2010 H Zγ more background smaller branching ratio spin-dependent particle correlations through Z ll 2 10 3 10 γγ Zγ 100 120 140 160 180 200 M H [GeV] Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 1/20
Motivation: Experiment H γγ loop-mediated sensitive to new physics clean signature 20% relative precision Branching ratios 10 1 1 ττ cc bb gg WW ZZ LHC HIGGS XS WG 2010 H Zγ more background smaller branching ratio spin-dependent particle correlations through Z ll 2 10 3 10 γγ Zγ 100 120 140 160 180 200 M H [GeV] broader spectrum of observables: more info on H couplings Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 1/20
Motivation: Theory H Zγ Analytical LO result available [Cahn, Chanowitz, Fleishon (1979)] [Bergstrom, Huth (1985)] Numerical NLO result in QCD available [Spira, Djouadi, Zerwas (1992)] H + j production Analytical NLO result: independent check NLO result in QCD available for m t [Schmidt (1997)] [Glosser, Schmidt (2002)] [de Florian, Grazzini, Kunszt (1999)] [Ravindran, Smith, van Neerven (2002)] NNLO calculation in QCD well-advanced for m t [Gehrmann, Glover, Jaquier, Koukoutsakis (2012)] [Chen, Gehrmann, Glover, Jaquier (2012)] [Boughezal, Caola, Melnikov, Petriello, Schulze (2013, 2015)] [Boughezal, Focke, Giele, Liu, Petriello (2015)] Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 2/20
Motivation: Theory H Zγ Analytical LO result available [Cahn, Chanowitz, Fleishon (1979)] [Bergstrom, Huth (1985)] Numerical NLO result in QCD available [Spira, Djouadi, Zerwas (1992)] H + j production Analytical NLO result: independent check NLO result in QCD available for m t [Schmidt (1997)] [Glosser, Schmidt (2002)] [de Florian, Grazzini, Kunszt (1999)] [Ravindran, Smith, van Neerven (2002)] NNLO calculation in QCD well-advanced for m t [Gehrmann, Glover, Jaquier, Koukoutsakis (2012)] [Chen, Gehrmann, Glover, Jaquier (2012)] [Boughezal, Caola, Melnikov, Petriello, Schulze (2013, 2015)] [Boughezal, Focke, Giele, Liu, Petriello (2015)] Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 2/20
Motivation: Theory gg gq qq sum 10 2 [Harlander, Neumann (2013)] 1 σ i,j dσi,j dpt [1/GeV] 10 3 10 4 10 5 10 6 Operator O 1 2 O 1O 2 O 1O 3 O 1O 5 SM SM m t 100 300 500 700 100 300 500 700 100 300 500 700 100 300 500 700 p T [GeV] H + j production EFT is likely to break down at high p T High-priority aim: NLO QCD corrections with full m t dependence Two-loop integrals for H Zγ pave the way Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 3/20
Outline of the calculation Qgraf Generate Feynman diagrams for process H(q) Z(p 1 )γ(p 2 ) Z Z Z Z H W H q H g q H g q γ γ γ γ Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 4/20
Outline of the calculation Qgraf Generate Feynman diagrams for process H(q) Z(p 1 )γ(p 2 ) Z Z Z Z H W H q H g q H g q γ γ γ γ Form Project relevant Feynman diagrams onto tensor structure with projector M = A ɛ 1,µ (p 1, λ 1 ) ɛ 2,ν (p 2, λ 2 ) Pµν P 2 P µν = p µ 2 pν 1 (p 1 p 2 )g µν Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 4/20
Outline of the calculation Qgraf Generate Feynman diagrams for process H(q) Z(p 1 )γ(p 2 ) Z Z Z Z H W H q H g q H g q γ γ γ γ Form Project relevant Feynman diagrams onto tensor structure with projector M = A ɛ 1,µ (p 1, λ 1 ) ɛ 2,ν (p 2, λ 2 ) Pµν P 2 P µν = p µ 2 pν 1 (p 1 p 2 )g µν Reduze Reduce Feynman Integrals to set of Master Integrals (MIs) Integration-by-parts identities (IBPs) Laporta algorithm Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 4/20
Outline of the calculation Qgraf Generate Feynman diagrams for process H(q) Z(p 1 )γ(p 2 ) Z Z Z Z H W H q H g q H g q γ γ γ γ Form Project relevant Feynman diagrams onto tensor structure with projector M = A ɛ 1,µ (p 1, λ 1 ) ɛ 2,ν (p 2, λ 2 ) Pµν P 2 P µν = p µ 2 pν 1 (p 1 p 2 )g µν Reduze Reduce Feynman Integrals to set of Master Integrals (MIs) Integration-by-parts identities (IBPs) Laporta algorithm Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 4/20
The amplitude can be further decomposed A = c W A W + q c q A q with c W = cos θ w sin θ w, Born-level contribution: ( ) 2 Q q I 3 q 2 Q q sin 2 θ w c q = N c sin θ w cos θ w NLO QCD corrections: A q (m H, m Z, m q, α s, µ) = A (1) q A (1) = c W A (1) W + c ta (1) t + c b A (1) b (m H, m Z, m q )+ α s(µ) π A(2) q (m H, m Z, m q, µ) Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 5/20
The amplitude can be further decomposed A = c W A W + q c q A q with c W = cos θ w sin θ w, Born-level contribution: ( ) 2 Q q I 3 q 2 Q q sin 2 θ w c q = N c sin θ w cos θ w NLO QCD corrections: A q (m H, m Z, m q, α s, µ) = A (1) q A (1) = c W A (1) W + c ta (1) t + c b A (1) b (m H, m Z, m q )+ α s(µ) π A(2) q (m H, m Z, m q, µ) Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 5/20
The decay width is obtained from the amplitude as Parametrization π G F α 2 Γ = 4 ( ) A 2 2 mh 3 m 2 H mz 2 We are left with computation of MIs Use Landau-type variables to absorb natural roots mh 2 = mq 2 (1 h) 2 h 1 4 m2 q m 2 H h + 1 h 1,, mz 2 = mq 2 (1 z) 2 z 1 4 m2 q m 2 Z z + 1 z 1 Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 6/20
The decay width is obtained from the amplitude as Parametrization π G F α 2 Γ = 4 ( ) A 2 2 mh 3 m 2 H mz 2 We are left with computation of MIs Use Landau-type variables to absorb natural roots mh 2 = mq 2 (1 h) 2 h 1 4 m2 q m 2 H h + 1 h 1,, mz 2 = mq 2 (1 z) 2 z 1 4 m2 q m 2 Z z + 1 z 1 Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 6/20
Differential equations 1 Choose MI 2 Compute derivative of integrand with respect to internal mass and external invariants 3 Use IBPs to relate resulting integrals to original MI [Kotikov (1991)] [Remiddi (1997)] [Gehrmann, Remiddi (2000)] Full system takes form of total differential N d I(h, z) = R k (ɛ) d log(d k ) I(h, z) k=1 d 1 = z d 5 = h + 1 d 9 = h 2 hz h + 1 d 2 = z + 1 d 6 = h 1 d 10 = h 2 z hz h + z d 3 = z 1 d 7 = h z d 11 = z 2 hz z + 1 d 4 = h d 8 = hz 1 d 12 = z 2 h hz z + h... d N Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 7/20
Differential equations 1 Choose MI 2 Compute derivative of integrand with respect to internal mass and external invariants 3 Use IBPs to relate resulting integrals to original MI [Kotikov (1991)] [Remiddi (1997)] [Gehrmann, Remiddi (2000)] Full system takes form of total differential N d I(h, z) = R k (ɛ) d log(d k ) I(h, z) k=1 d 1 = z d 5 = h + 1 d 9 = h 2 hz h + 1 d 2 = z + 1 d 6 = h 1 d 10 = h 2 z hz h + z d 3 = z 1 d 7 = h z d 11 = z 2 hz z + 1 d 4 = h d 8 = hz 1 d 12 = z 2 h hz z + h... d N Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 7/20
Differential equations 1 Choose MI 2 Compute derivative of integrand with respect to internal mass and external invariants 3 Use IBPs to relate resulting integrals to original MI [Kotikov (1991)] [Remiddi (1997)] [Gehrmann, Remiddi (2000)] Full system takes form of total differential N ( ) d I(h, z) = R k ɛ d log(d k ) I(h, z) k=1 d 1 = z d 5 = h + 1 d 9 = h 2 hz h + 1 d 2 = z + 1 d 6 = h 1 d 10 = h 2 z hz h + z d 3 = z 1 d 7 = h z d 11 = z 2 hz z + 1 d 4 = h d 8 = hz 1 d 12 = z 2 h hz z + h... d N Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 7/20
Differential equations Canonical form [[Henn (2013)]] 12 d M(h, z) = ɛ R k d log(d k ) M(h, z) k=1 reduced number of polynomials d k can be integrated in terms of GHPLs: x 1 G (w 1,..., w n ; x) dt G (w 2,..., w n ; t) 0 t w 1 ( ) G 0n ; x logn x n! leads to linear combinations of GHPLs of homogeneous weight Change basis from Laporta integrals I to canonical integrals M Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 8/20
Differential equations Canonical form [[Henn (2013)]] 12 d M(h, z) = ɛ R k d log(d k ) M(h, z) k=1 reduced number of polynomials d k can be integrated in terms of GHPLs: x 1 G (w 1,..., w n ; x) dt G (w 2,..., w n ; t) 0 t w 1 ( ) G 0n ; x logn x n! leads to linear combinations of GHPLs of homogeneous weight Change basis from Laporta integrals I to canonical integrals M Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 8/20
Integral basis 1 Start with Laporta basis: [Gehrmann, von Manteuffel, Tancredi, Weihs (2014)] Topology under consideration must have at most linear dependence on ɛ be triangular in D = 4 dimensions I 1 I 2 I 3 I 4 I 5 I 6 I 7 I 8 I 9 I 10 I 11 I 12 Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 9/20
Integral basis 1 Start with Laporta basis: [Gehrmann, von Manteuffel, Tancredi, Weihs (2014)] I 13 I 14 I 15 I 16 I 17 I 18 I 19 I 20 I 21 I 22 I 23 I 24 I 25 I 26 I 27 I 28 Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 9/20
Integral basis 1 Start with Laporta basis: [Gehrmann, von Manteuffel, Tancredi, Weihs (2014)] I 13 I 14 I 15 I 16 I 17 I 18 I 19 I 20 I 21 I 22 I 23 I 24 I 25 I 26 I 27 I 28 Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 9/20
Integral basis di 15 2 Perform transformation to canonical basis by integrating out homogeneous parts in D = 4 dz = h z d 1 d 2 d 3 d 4 d 9 d 10 (2 z (h 2 + 1) h (z + 1) 2 ) { 3 2 h z (h 1)2 ( ( h 2 + 1 ) ( z 2 + 1 ) h (z + 1) 2) I 7 h [ h 4 z ( z 2 + 1 ) h ( h 2 + 1 ) ( z 4 + z 3 + 4z 2 + z + 1 ) 1 z +h ( 2 z 4 + 4z 3 + 2z 2 + 4z + 1 ) + z ( z 2 + 1 )] (2 I 13 + I 14 ) [2 h 6 z ( 2 z 2 + 1 ) 2 h z ( h 4 + 1 ) (z + 1) 2 ( 2 z 2 z + 2 ) +h 2 ( h 2 + 1 ) ( z 6 + 8 z 5 + 17 z 4 + 8 z 3 + 17 z 2 + 8 z + 1 ) + 2 z 2 ( z 2 + 1 )] I 15 } di 15 dh = h z d 1 d 4 d 9 d 10 (2 z (h 2 + 1) h (z + 1) 2 ) { 3 2 z2 (h 1) 3 (h + 1) I 7 + z ( h 2 1 ) ( z ( h 2 + 1 ) h ( z 2 + 1 )) (2 I 13 + I 14 ) h2 1 h [2z 2 ( h 4 + 1 ) 2hz (z + 1) 2 ( h 2 + 1 ) + h 2 ( z 4 + 2z 3 + 6z 2 + 2z + 1 )] I 15 } Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 10/20
Integral basis 2 Perform transformation to canonical basis by integrating out homogeneous parts in D = 4 M 15 = z (h2 + 1) h (z + 1) 2 z (h 2 + 1) h (z + 1) 2 [ 3 (h 1) 2 2 h I 7 (h z)(hz 1) (2 I 13 + I 14 ) hz (z2 1)(h 2 ] + 1 h (z + 1)) I 15 hz Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 10/20
Integral basis 2 Perform transformation to canonical basis by integrating out homogeneous parts in D = 4 [ dm 15 = ɛ (M 2 + 32 ) M 7 + 5 M 13 + M 14 4 M 15 d log(d 1 ) ( ) 3 + 2 M 7 + 2 M 13 + M 14 2 M 15 d log(d 2 ) (M 2 + 32 ) M 7 + M 13 M 14 M 15 d log(d 3 ) (M 2 + 2 M 6 + 52 ) M 7 + M 13 M 14 M 15 d log(d 4 ) + 3 M 7 d log(d 6 ) + (M 2 + M 6 + 2 M 7 + 3 M 13 2 M 15 ) d log(d 7 ) + (M 2 M 6 + M 7 + 3 M 13 2 M 15 ) d log(d 8 ) ( ) ] 3 2 M 7 + 2 M 13 + M 14 M 15 d log(d 9 ) Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 10/20
Integration Integrate differential equation in h or z up to constant C(z) or C(h) Use boundary conditions: h = 1 mh 2 = 0 z = 1 mz 2 = 0 h = z mh 2 = mz 2 h = 1 z m 2 H = m 2 Z Perform transformations with the help of symbol and coproduct: G (w 1 (x),..., w n (x); x) G (a 1,..., a n ; x) G (w 1 (x),..., w n (x); y) G (c 1 (y),..., c n (y); x) Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 11/20
Integration Integrate differential equation in h or z up to constant C(z) or C(h) Use boundary conditions: h = 1 mh 2 = 0 z = 1 mz 2 = 0 h = z mh 2 = mz 2 h = 1 z m 2 H = m 2 Z Perform transformations with the help of symbol and coproduct: G (w 1 (x),..., w n (x); x) G (a 1,..., a n ; x) G (w 1 (x),..., w n (x); y) G (c 1 (y),..., c n (y); x) Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 11/20
Integration Integrate differential equation in h or z up to constant C(z) or C(h) Use boundary conditions: h = 1 mh 2 = 0 z = 1 mz 2 = 0 h = z mh 2 = mz 2 h = 1 z m 2 H = m 2 Z Perform transformations with the help of symbol and coproduct: G (w 1 (x),..., w n (x); x) G (a 1,..., a n ; x) G (w 1 (x),..., w n (x); y) G (c 1 (y),..., c n (y); x) Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 11/20
Integration Results in terms of GHPLs up to weight four G (a 1,..., a n ; h) with a i {0, ±1, z, 1 z, J z, 1 J z, K ± z, L ± z } G (b 1,..., b n ; z) with b i {0, ±1, c, c} c = 1 ( 1 + ) i 3 K ± z = 1 (1 + z ± ) 3 + 2 z + z 2 2 2 z J z = 1 z + z 2 L ± z = 1 (1 + z ± ) 1 + 2 z 3 z 2 2 z verified through differential equation in other variable checked numerically against SecDec [Borowka, Heinrich, Jones, Kerner, Schlenk, Zirke (2015)] Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 12/20
Integration Results in terms of GHPLs up to weight four G (a 1,..., a n ; h) with a i {0, ±1, z, 1 z, J z, 1 J z, K ± z, L ± z } G (b 1,..., b n ; z) with b i {0, ±1, c, c} c = 1 ( 1 + ) i 3 K ± z = 1 (1 + z ± ) 3 + 2 z + z 2 2 2 z J z = 1 z + z 2 L ± z = 1 (1 + z ± ) 1 + 2 z 3 z 2 2 z verified through differential equation in other variable checked numerically against SecDec [Borowka, Heinrich, Jones, Kerner, Schlenk, Zirke (2015)] Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 12/20
Integration Example M 15 = ɛ 2 [ π2 6 G( 1 z, 0; h) + G(z, 0; h) G(0; z)g( 1 ; h) G(0; z)g(z; h) + G(0; z)g(0; h) z 2G(0, 0; h) + G(1, 0; z)] [ + ɛ 3 5ζ 3 + 2G( 1 z, 1 z, 0; h) 2G( 1 2π2, z, 0; h) + z 3 G( 1 z ; h) + 6G( 1, 1, 0; h) z 3G( 1 z, 0, 0; h) + 2G( 1 z, 1, 0; h) + 2G(z, 1 π2, 0; h) 2G(z, z, 0; h) + G(z; h) z 3 6G(z, 1, 0; h) + 5G(z, 0, 0; h) 2G(z, 1, 0; h) G(K z, 1 z, 0; h) + G(K z, z, 0; h) π2 6 G(K z ; h) + 2G(K z, 0, 0; h) G(K + z, 1 z, 0; h) + G(K + z, z, 0; h) π2 6 G(K + z ; h) + 2G(K + π2 z, 0, 0; h) + 3 G( 1; z) + 2G( 1, 0; z)g( 1 ; h) + 2G( 1, 0; z)g(z; h) z 2G( 1, 0; z)g(0; h) + G( 1, 0, 0; z) 2G( 1, 1, 0; z) G(0, 1 z π2, 0; h) G(0; z) 2 Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 13/20
Integration Example + 2G(0; z)g( 1 z, 1 z ; h) + 2G(0; z)g( 1 z, z; h) 2G(0; z)g( 1 z, 0; h) + 2G(0; z)g(z, 1 z ; h) + 2G(0; z)g(z, z; h) 2G(0; z)g(z, 0; h) G(0; z)g(k z, 1 z ; h) G(0; z)g(k z, z; h) + G(0; z)g(k z, 0; h) G(0; z)g(k + z, 1 z ; h) G(0; z)g(k + z, z; h) + G(0; z)g(k + z, 0; h) G(0; z)g(0, 1 ; h) G(0; z)g(0, z; h) + G(0, z, 0; h) + G(0; h)g(1, 0; z) + 12G(0, 1, 0; h) z G(0, 0; z)g( 1 z ; h) G(0, 0; z)g(z; h) G(0, 0; z)g(k z ; h) G(0, 0; z)g(k + z ; h) + 2G(0, 0; z)g(0; h) + G(0, 0; h)g(0; z) G(0, 0, 0; z) 8G(0, 0, 0; h) + 2G(0, 1, 0; z) + 4G(0, 1, 0; h) π2 3 G(1; z) 2G(1, 1, 0; z) 2G(1, 0; z)g( 1 ; h) 2G(1, 0; z)g(z; h) z +G(1, 0; z)g(k z ; h) + G(1, 0; z)g(k + z ; h) + 2G(1, 0, 0; z) 6G(1, 0, 0; h) + G(1, 1, 0; z) ] + O ( ɛ 4) Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 13/20
Renormalization (a) quark mass M q and Yukawa coupling Y q in OS scheme A (2,a) q (m H, m Z, M q ) = A (2) q,bare + Z OS A (1) q + δm OS C (1) q m q Renormalization constants Z OS = α s(µ) π 16 i C F π2 S ɛ 4 δm OS = m q Z OS 3 2ɛ ɛ (1 2ɛ) Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 14/20
Renormalization (a) quark mass M q and Yukawa coupling Y q in OS scheme A (2,a) q (m H, m Z, M q ) = A (2) q,bare + Z OS A (1) q + δm OS C (1) q m q Renormalization constants Z OS = α s(µ) π 16 i C F π2 S ɛ 4 δm OS = m q Z OS 3 2ɛ ɛ (1 2ɛ) Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 14/20
Renormalization (a) quark mass M q and Yukawa coupling Y q in OS scheme A (2,a) q (m H, m Z, M q ) = A (2) q,bare + Z OS A (1) q + δm OS C (1) q m q Renormalization constants Counterterms Z OS = α s(µ) π 16 i C F π2 S ɛ 4 δm OS = m q Z OS 3 2ɛ ɛ (1 2ɛ) Z Z C (1) q = H q + H q + H q Z γ γ γ Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 14/20
Renormalization (b) quark mass M q in OS, Yukawa coupling y q in MS scheme A (2,b) q (m H, m Z, m q, µ) = A (2,a) q (m H, m Z, m q (µ)) + A (1) q (m H, m Z, m q (µ)) The finite shift in the amplitude is induced by expressing the OS quantities in terms of MS quantities: M q = m q (µ) (1 + ) Y q = y q (µ) (1 + ) ( ) = α s(µ) C F 1 + 3 π 4 log µ 2 m 2 q(µ) Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 15/20
Renormalization (c) quark mass m q and Yukawa coupling y q in MS scheme A (2,c) q (m H, m Z, m q, µ) = A (2,b) q (m H, m Z, m q (µ)) + A(1) q (m H, m Z, M q ) M q is defined through the following replacements in A (1) q : h = h 2 h h 1 h + 1 z = z 2 z z 1 z + 1 Mq=m q(µ) Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 16/20
Numerical results Analytic continuation from Euclidean to Minkowski region Region I: m 2 Z < m2 H < 4 m2 q top quark amplitude Region II: m 2 Z < 4 m2 q < m 2 H not needed Region III: 4 m 2 q < m 2 Z < m2 H bottom quark amplitude NLO decay width Γ (2) in renormalization schemes (a), (b) and (c) [ Γ (2,a) = [ Γ (2,b) = 7.04500 + 0.42617 α s(µ) π 7.06369 + α ( s(µ) π µ=m H = 7.06363 kev < 10 5 [ Γ (2,c) = 7.02908 + α s(µ) π ( µ=m H = 7.05402 kev 3 o / oo ] kev µ=mh = 7.06033 kev 2 o / oo 0.53038 0.76333 log m 2 t (µ) + 0.01224 log µ 2 m 2 b(µ) µ 2 )] kev )] µ 2 0.64310 + 0.10551 log m 2 t (µ) + 0.01446 log µ 2 kev m 2 b(µ) Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 17/20
Numerical results Analytic continuation from Euclidean to Minkowski region Region I: m 2 Z < m2 H < 4 m2 q top quark amplitude Region II: m 2 Z < 4 m2 q < m 2 H not needed Region III: 4 m 2 q < m 2 Z < m2 H bottom quark amplitude NLO decay width Γ (2) in renormalization schemes (a), (b) and (c) [ Γ (2,a) = [ Γ (2,b) = 7.04500 + 0.42617 α s(µ) π 7.06369 + α ( s(µ) π µ=m H = 7.06363 kev < 10 5 [ Γ (2,c) = 7.02908 + α s(µ) π ( µ=m H = 7.05402 kev 3 o / oo ] kev µ=mh = 7.06033 kev 2 o / oo 0.53038 0.76333 log m 2 t (µ) + 0.01224 log µ 2 m 2 b(µ) µ 2 )] kev )] µ 2 0.64310 + 0.10551 log m 2 t (µ) + 0.01446 log µ 2 kev m 2 b(µ) Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 17/20
Numerical results Partial width (a) (b) (c) Γ (1) WW 7.83473859 7.83473859 7.83473859 Γ (1) Wt 0.83278001 0.80390896 0.83655274 Γ (1) Wb 0.02206641 0.01288843 0.00904595 Γ (1) tt 0.02212973 0.02062193 0.02233069 Γ (1) tb 0.00117276 0.00066123 0.00048294 Γ (1) bb 0.00002094 0.00000714 0.00000324 Γ (1) 7.04500291 7.06368591 7.02908279 Γ (2) Wt 0.02203714 0.00078280 0.02457012 Γ (2) Wb 0.00586227 0.00072730 0.00175365 Γ (2) tt 0.00117120 0.00004016 0.00131174 Γ (2) tb 0.00031156 0.00003731 0.00009362 Γ (2) bt 0.00003103 0.00000064 0.00001418 Γ (2) bb 0.00001585 0.00000081 0.00000078 Γ (2) 7.06033332 7.06368591 7.05401616 Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 18/20
Numerical results 7.08 7.075 (a) (b) (c) 7.07 Γ (2) in kev 7.065 7.06 0.4 o / oo 7.055 1.3 o / oo 7.05 7.045 µ = m H 50 100 150 200 250 300 µ in GeV µ = 2m H Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 19/20
Conclusions Renormalization OS scheme MS scheme hybrid scheme with OS mass and MS Yukawa coupling Numerical results Corrections in sub-per-cent range and consistent with each other Residual QCD uncertainty: 1.7 o / oo Checks Confirmation of previously available numerical OS result [Spira, Djouadi, Zerwas (1992)] Agreement with independent calculation [Bonciani, Del Duca, Frellesvig, Henn, Moriello, Smirnov (2015)] Master Integrals Two-loop three-point integrals with two different external legs and one internal mass derived analytically using differential equations Important ingredient to two-loop amplitudes of H + j production Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 20/20
Conclusions Renormalization OS scheme MS scheme hybrid scheme with OS mass and MS Yukawa coupling Numerical results Corrections in sub-per-cent range and consistent with each other Residual QCD uncertainty: 1.7 o / oo Checks Confirmation of previously available numerical OS result [Spira, Djouadi, Zerwas (1992)] Agreement with independent calculation [Bonciani, Del Duca, Frellesvig, Henn, Moriello, Smirnov (2015)] Master Integrals Two-loop three-point integrals with two different external legs and one internal mass derived analytically using differential equations Important ingredient to two-loop amplitudes of H + j production Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 20/20
Conclusions Renormalization OS scheme MS scheme hybrid scheme with OS mass and MS Yukawa coupling Numerical results Corrections in sub-per-cent range and consistent with each other Residual QCD uncertainty: 1.7 o / oo Checks Confirmation of previously available numerical OS result [Spira, Djouadi, Zerwas (1992)] Agreement with independent calculation [Bonciani, Del Duca, Frellesvig, Henn, Moriello, Smirnov (2015)] Master Integrals Two-loop three-point integrals with two different external legs and one internal mass derived analytically using differential equations Important ingredient to two-loop amplitudes of H + j production Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 20/20
Conclusions Renormalization OS scheme MS scheme hybrid scheme with OS mass and MS Yukawa coupling Numerical results Corrections in sub-per-cent range and consistent with each other Residual QCD uncertainty: 1.7 o / oo Checks Confirmation of previously available numerical OS result [Spira, Djouadi, Zerwas (1992)] Agreement with independent calculation [Bonciani, Del Duca, Frellesvig, Henn, Moriello, Smirnov (2015)] Master Integrals Two-loop three-point integrals with two different external legs and one internal mass derived analytically using differential equations Important ingredient to two-loop amplitudes of H + j production Thomas Gehrmann, Sam Guns & Dominik Kara The rare decay H Zγ in perturbative QCD June 15, 2015 20/20