Top mass effects in gluon fusion processes

Top mass effects in gluon fusion processes Thomas Rauh AEC, University of Bern Ramona Gröber, Andreas Maier, TR arxiv:1709.07799 JHEP 1803 (2018) 020

Outline 2 Gluon fusion processes Validity of approximations Padé approximants Expansion around the top threshold Results Conclusions & outlook

Gluon fusion processes 3 Single Higgs production H Dominant Higgs production process σ(pp H+X) [pb] 2 10 M(H)= 125 GeV 10 1 1 10 pp H (N3LO QCD + NLO EW) pp qqh (NNLO QCD + NLO EW) pp WH (NNLO QCD + NLO EW) pp ZH (NNLO QCD + NLO EW) pp bbh (NNLO QCD in 5FS, NLO QCD in 4FS) pp tth (NLO QCD + NLO EW) pp th (NLO QCD, t-ch + s-ch) LHC HIGGS XS WG 2016 2 10 6 7 8 9 10 11 12 13 14 15 s [TeV]

Gluon fusion processes 3 Single Higgs production H Dominant Higgs production process Convergence is very slow 50 40 30 σ [pb] 20 10 LHC13TeV PDF4LHC15.0 PP->H+X LO NNLO NLO N3LO 30 50 70 90 110 130 150 170 190 210 230 250 μ [GeV] [Mistlberger 2018]

Gluon fusion processes 4 Higgs plus jet production g H Transverse momentum dependence Convergence is very slow dσ/dp TH (pb/gev) 0.12 0.1 0.08 0.06 0.04 p TH LO NLO NNLO ratio 3.5 3 2.5 2 1.5 1 p TH NLO/LO NNLO/NLO NNLO/LO 0.02 0.5 0 0 50 100 150 200 p TH (GeV) 0 0 100 200 300 400 500 600 p TH (GeV) [Chen, Gehrmann, Glover, Jaquier 2014]

Gluon fusion processes 5 Higgs plus Z production Z H Formally NNLO, but large Convergence is very slow sh=gev ff (exact) H;LO ff (exp 0) H;NLO ff (re scale) H;NLO ff (diff) H;NLO ff [2=2] H;NLO 7 11.2 23.9 24.9 26.1 26.5 8 16.0 35.2 35.4 37.2 38.8 13 52.4 129 113 121 140 14 61.8 155 133 142 168 NLO scale variation +21% 21% +20% 20% +14% 17% +13% 16% [Hasselhuhn, Luthe, Steinhauser 2016]

Gluon fusion processes 6 Higgs pair production H Measurement of trilinear Higgs coupling Convergence is very slow H ff H [fb] K (N)NLO X NNLO [%] LO 22.7 LO+NLO j 0 36.4 1.60 LO+NLO j 0+NNLO j 0 39.7 1.75 0 LO+NLO j 0+NNLO j 1 38.7 1.70 2:5 LO+NLO j 0+NNLO j 2 40.5 1.78 +2:0 [Grigo, Hoff, Steinhauser 2015]

Gluon fusion processes 7 Z pair production Z Z Higgs width from interference with gg H ZZ [Caola, Melnikov 2013] Convergence is very slow [Campbell, Czakon, Ellis, Kirchner 2016]

Gluon fusion processes at NLO 8 H g H [Spira, Djouadi, Graudenz, Zerwas 1995;... ] [Jones, Kerner, Luisoni 2018] Z H H H [Borowka et al. 2016] Z Z

Gluon fusion processes at NLO 9 Large mass expansion (LME) m t ŝ H g H [Dawson 1991; Djouadi, Spira, Zerwas 1991] [Harlander, Neumann, Ozeren, Wiesemann 2012;... ] Z H H H [Hasselhuhn, Luthe, Steinhauser 2016] [Grigo, Hoff, Steinhauser 2015; Degrassi, Giardino, Gröber 2016] Z Z [Melnikov, Dowling 2015; Campbell, Ellis, Czakon, Kirchner 2016; Caola et al. 2016]

Validity of the large mass expansion 10 Higgs pair production at LO Fails near the top threshold M HH 2m t 350 GeV 0.12 0.1 full LME dσlo/dmhh [fb/gev] 0.08 0.06 0.04 0.02 0 300 400 500 600 700 800 900 M HH [GeV]

Validity of the large mass expansion 11 Higgs pair production at NLO Use a rescaling to improve the prediction from the large-m t limit dff rescaled LME NLO =dx = dfflme NLO =dx dfflo LME dffexact =dx LO =dx : dσ/d mhh [fb/gev] 0.20 0.15 0.10 0.05 LO B-i. NLO HEFT NLO FTapprox LO basic HEFT NLO basic HEFT NLO dσ/d pt,h [fb/gev] 0.25 0.20 0.15 0.10 0.05 LO B-i. NLO HEFT NLO FTapprox LO basic HEFT NLO basic HEFT NLO K factor 0.00 2.0 1.5 1.0 0.5 300 400 500 600 700 800 900 1000 mhh [GeV] K factor 0.00 2.0 1.5 1.0 0.5 0 100 200 300 400 500 pt,h [GeV] [Borowka, Greiner, Heinrich, Jones, Kerner, Schlenk, Zirke 2016]

FTapprox 12 Top mass dependence in real radiation Improvements from including the full top mass dependence in the real part (1 loop) at NLO [Maltoni, Vryonidou, Zaro 2014] and the double real part (1 loop) at NNLO [Grazzini et al. 2018] 0.20 s = 14 TeV s = 14 TeV 0.20 0.15 dσ/dmhh (fb/gev) 0.10 NNLOB-proj NNLONLO-i NNLOFTapprox NLO dσ/dp T,h1 (fb/gev) 0.15 0.10 NNLOB-proj NNLONLO-i NNLOFTapprox NLO 0.05 0.05 0.00 1.4 0.00 1.8 1.3 1.6 ratio to NLO 1.2 1.1 1.0 0.9 0.8 300 400 500 600 700 800 ratio to NLO 1.4 1.2 1.0 0.8 0.6 0 100 200 300 400 500 Mhh (GeV) pt,h1 (GeV)

Padé Approximation 13 [Broadhurst, Fleischer, Tarasov 1993; Fleischer, Tarasov 1994;... ] a; b; H = ytŝ 2mt s 2ı abt F A 1 F (z) Im(z) 1 F z = ŝ 4m 2 t LME 1 1 Re(z) 1

Padé Approximation 14 [Broadhurst, Fleischer, Tarasov 1993; Fleischer, Tarasov 1994;... ] Im(!) F a; b; H = ytŝ 2mt s 2ı abt F A 1 F (z) z = ŝ 4m 2 t z = -1 1 LME 1 z = 1 Re(!) Conformal mapping: z = 4! (1+!) 2-1

Padé Approximation 14 [Broadhurst, Fleischer, Tarasov 1993; Fleischer, Tarasov 1994;... ] Im(!) F a; b; H = ytŝ 2mt s 2ı abt F A 1 F (z) z = ŝ 4m 2 t z = -1 1 LME 1 z = 1 Re(!) Conformal mapping: z = 4! (1+!) 2-1 Padé approximation: [n=m](!) = Ansatz: Fix a i ; b i from LME F (z) = [n=m]`!(z) 1 + a R z from F P n i=0 a i! i 1+ P m i=1 b i! i ; a R [0:1; 10] z 0

Padé Approximation for gg H 15 F 1l 3.0 2.5 LME 2.0 Re 1.5 Im 1.0 0.5 0.0 0.0 0.5 1.0 2.0 5.0 20 z = m 2 H /(4m 2 t ) F 2l 10 5 0 Re LME -5 0.0 0.5 1.0 2.0 5.0 20 z = m H 2 /(4m t 2 ) Im 20 Padé approximants [1/3], [2/2], [3/1] from LME up to 1 m 8 t. Exclude approximants with poles for Re(z) [0; 8]; Im(z) [ 1; 1]. At NLO some instabilities emerge above the top threshold.

Threshold expansion 16 F 1l F 2l z 1 2ı(1 z) 3=2 + 13ı 3 (1 z)5=2 + O `(1 z) 7=2 ; z 1 C F ı 2 (1 z)ln(1 z) + ı ˆ3ı 2 (C F C A) 40C F + 4C A (1 z) 3=2 12 +C F 2ı 2 3 (1 z)2 ln(1 z) + O `(1 z) 5=2 ; ln(1 z) incompatible with Padé approximation, extended ansatz [n=m]`!(z) F = +s(z) 1 + a R z ln(1 z) absorbed into subtraction function s s analytic for z 0, constructed from,

Padé Approximation with threshold expansion 17 3.0 2.5 2.0 Re LME 3.0 2.5 2.0 Re LME F 1l 1.5 1.0 Im F 1l 1.5 1.0 Im 0.5 0.0 0.5 0.0 Thr 0.0 0.5 1.0 2.0 5.0 20 z = m H 2 /(4m t 2 ) threshold excl. (1 z) i log 0 0.0 0.5 1.0 2.0 5.0 20 z = m 2 H /(4m 2 t ) 10 5 Re LME Im 10 8 6 Re LME F 2l F 2l 4 Im 0 2-5 0.0 0.5 1.0 2.0 5.0 20 0 Thr -2 0.0 0.5 1.0 2.0 5.0 20 z = m H 2 /(4m t 2 ) z = m H 2 /(4m t 2 )

Padé approximation for vacuum polarisation 18 Compare numerical result for vacuum polarization at 3 loops to Padé approximants with n + m = 60 approx. exact 5 4 3 2 1 0 1 2 3 4 10 5 10 10 10 15 Re ( Π (2) ) n 0 l Im ( Π (2) ) n 0 l approx. exact 0.2 0.1 0 0.1 0.2 0.3 10 5 10 10 10 15 0 0.2 0.5 10 20 1 2 5 z [Maier, Marquard 2017] Re ( Π (2) ) n 1 l Im ( Π (2) ) n 1 l 0 0.2 0.5 1 2 5 z [Chetyrkin, Kühn, Steinhauser 1995; Hoang, Mateu, Zebarjad 2008; Kiyo, Maier, Maierhöfer, Marquard 2009]

Padé approximation for 2 2 processes 19 H H = y 2 t s 2ı abt F z ˆA 1 F 1 + A 2 F 2 Three ratios: z = ŝ ; r 4mt 2 H = m2 Hŝ ; r pt = p2 Ṱ s Given a phase space point (z; r H ; r pt ): 1 Compute threshold and large mass expansion for given r H ; r pt 2 Construct 100 approximants in z 3 Evaluate approximants for given z (physical m t )

Padé Approximation for gg( H ) ZZ 20 [Campbell, Ellis, Czakon, Kirchner 2016] 2 Re Z Z Z Z H! exact LME up to 1 mt 12 [3/3] [n/m] with n; m {2; 3} LME up to 1 mt 12 [3/3] [n/m] with n; m {2; 3}

Threshold expansion 21 Near threshold tops can only be on-shell when they are non-relativistic: p 0 t m t m t (1 z), ~p m t 1 z Hierarchy: m t (1 z) m t 1 z mt External gluons set directions n; n for collinear modes

Threshold expansion 21 Near threshold tops can only be on-shell when they are non-relativistic: p 0 t m t m t (1 z), ~p m t 1 z Hierarchy: m t (1 z) m t 1 z mt External gluons set directions n; n for collinear modes p 0 n p mt hard mt n-collinear hard mt 1 - z soft mt 1 - z mt(1-z) ultrasoft potential mt(1-z) ultrasoft n-collinear mt(1-z) mt 1 - z mt Potential Non-Relativistic QCD p mt(1-z) mt 1 - z mt n p Soft-Collinear Effective Theory [Pineda, Soto 97; Beneke et al. 98; Brambilla et al. 99] [Bauer et al. 00-01; Beneke et al. 02]

Threshold expansion 21 Near threshold tops can only be on-shell when they are non-relativistic: p 0 t m t m t (1 z), ~p m t 1 z Hierarchy: m t (1 z) m t 1 z mt External gluons set directions n; n for collinear modes p 0 n p mt hard mt n-collinear hard mt 1 - z soft mt 1 - z mt(1-z) ultrasoft potential mt(1-z) ultrasoft n-collinear mt(1-z) mt 1 - z mt Potential Non-Relativistic QCD p mt(1-z) mt 1 - z mt n p Soft-Collinear Effective Theory [Pineda, Soto 97; Beneke et al. 98; Brambilla et al. 99] [Bauer et al. 00-01; Beneke et al. 02] Unstable-Particle Effective Theory [Beneke, Chapovsky, Signer, Zanderighi 03, 04]

Threshold expansion 22 Master formula Integrating out the hard scale gives the following structure X Z D z 1 ia gg F = d 4 x F T k;l C (k) gg t t C(l) t t F +C gg F F io gg F (0) gg EFT : h i E io (l) t t F (x)io(k) gg t t (0) gg EFT O (k) gg t t O (l) t t HH + O (k) gg HH Resonant contribution: Propagation of non-relativistic top pair. Contains non-analytic terms 1 z and ln(1 z). Non-resonant contribution: Contribution from hard top loop (local operator in EFT). Analytical in (1 z).

Threshold expansion 22 Master formula Integrating out the hard scale gives the following structure X Z D z 1 ia gg F = d 4 x F T k;l C (k) gg t t C(l) t t F +C gg F F io gg F (0) gg EFT : h i E io (l) t t F (x)io(k) gg t t (0) gg EFT O (k) gg t t O (l) t t HH + O (k) gg HH The experience with the triangle form factor (single Higgs production) shows that knowledge of non-analytic terms is sufficient for a good reconstruction. Therefore, we only consider the resonant part.

Threshold expansion 23 Resonant matrix element Ladder exchanges of potential gluons yield Coulomb singularities ( s= 1 z) k =... Can be resummed into a non-relativistic Green function 8 X < 1 nrlo; A resonant A 0 (z = 1) 1 z 2l+1 k=0 k s 1 z : s; 1 z 2s ; s 1 z; (1 z) nrnlo; nrnnlo;

Threshold expansion 23 Resonant matrix element Ladder exchanges of potential gluons yield Coulomb singularities ( s= 1 z) k =... Can be resummed into a non-relativistic Green function 8 X < 1 nrlo; A resonant A 0 (z = 1) 1 z 2l+1 k=0 k s 1 z : s; 1 z 2s ; s 1 z; (1 z) nrnlo; nrnnlo; Collinear nrnlo corrections are scaleless.

Threshold expansion 23 Resonant matrix element Ladder exchanges of potential gluons yield Coulomb singularities ( s= 1 z) k =... Can be resummed into a non-relativistic Green function 8 X < 1 nrlo; A resonant A 0 (z = 1) 1 z 2l+1 k=0 k s 1 z : s; 1 z 2s ; s 1 z; (1 z) nrnlo; nrnnlo; Collinear nrnlo corrections are scaleless. Virtual ultrasoft gluon exchange between collinear gluons is scaleless. Ultrasoft ttg interaction is power-suppressed after decoupling transformation (nrnnnlo). [Beneke et al. 2010]

Threshold expansion 24 Resonant matrix element: ingredients 2l+3 1 z 2l+2 1 z 2l+1 1 z 2l 1 z 2l 1 1 z LO NLO NNLO nrlo nrnlo nrnnlo α 0 s α 1 s α 2 s nrlo: C (k) gg t t ; C(l) t t HH at tree level P-wave Green function G P (1 z) at nrlo [Beneke, Piclum, TR 2013] nrnlo: C (k) gg t t ; C(l) t t HH at one loop P-wave Green function G P (1 z) at nrnlo [Beneke, Piclum, TR 2013] O( 0s ; s) at nrnnlo: C (k) gg t t ; C(l) t t HH for (1 z) power-suppressed operators O( 0s ; s) of P-wave Green function G P (1 z) at nrnnlo

Padé approximation for gg HH 25 LO results dσlo/dmhh [fb/gev] 0.12 0.1 0.08 0.06 0.04 0.02 full [n/m] w/o THR [n/n ± 1, 3] 0 300 400 500 600 700 800 900 M HH [GeV]

Padé approximation for gg HH 26 NLO results compared to [Borowka et al. 2016] 0.00080 0.00070 p T = 100 GeV 0.00060 0.00050 Vfin 0.00040 0.00030 0.00020 0.00010 full reweighted HEFT [n/m] w/o thres [n/n ± 0, 2] 350 400 450 500 550 600 650 700 M HH [GeV] V f in 10 4 M HH [GeV] p T [GeV] HEFT [n=m] [n=n ± 0; 2] full 336.85 37.75 0:912 0:996 ± 0:004 0:990 ± 0:001 0:996 ± 0:000 350.04 118.65 1:589 1:933 ± 0:012 1:937 ± 0:010 1:939 ± 0:061 411.36 163.21 4:894 4:326 ± 0:183 4:527 ± 0:069 4:510 ± 0:124 454.69 126.69 6:240 5:300 ± 0:192 5:114 ± 0:051 5:086 ± 0:060 586.96 219.87 7:797 4:935 ± 0:583 5:361 ± 0:281 4:943 ± 0:057 663.51 94.55 8:551 5:104 ± 1:010 4:096 ± 0:401 4:120 ± 0:018

Padé approximation for gg ZZ 27 Vector coupling /z (1) B VV 6 4 2 0 x = 0.1 Re Preliminary Im LME to z 6 No THR /z (1) B VV 6 x = 0.1 LME to z 6 THR to z 5/2 4 Re 2 0 Im -2-2 (2) B VV /z no ln (-4 z) terms 0.0 0.5 1.0 2.0 5.0 20 250 200 150 100 50 0-50 -100 x = 0.1 Re Im z 0.0 0.5 1.0 2.0 5.0 20 z LME to z 6 No THR threshold excl. (1 z) i log 0 (2) B VV /z no ln (-4 z) terms 0.0 0.5 1.0 2.0 5.0 20 z 250 200 150 100 50 0-50 -100 x = 0.1 LME to z 6 THR to z 2 Re Im 0.0 0.5 1.0 2.0 5.0 20 z

Padé approximation for gg ZZ 28 Axial-vector coupling 1.0 0.5 x = 0.1 1.0 0.5 x = 0.1 (1) r Z 2 BAA 0.0-0.5-1.0 Re Im (1) r Z 2 BAA 0.0-0.5-1.0 Re Im no ln (-4 z) terms (2) r Z 2 BAA -1.5-2.0 20 10 0-10 -20-30 -40 LME to z 6 No THR 0.0 0.5 1.0 2.0 5.0 20 x = 0.1 Re Im 0.0 0.5 1.0 2.0 5.0 20 z z LME to z 6 No THR threshold excl. (1 z) i log 0 Preliminary no ln (-4 z) terms (2) r Z 2 BAA -1.5-2.0 20 10 0-10 -20-30 -40 LME to z 6 THR to z 5/2 0.0 0.5 1.0 2.0 5.0 20 x = 0.1 Re Im 0.0 0.5 1.0 2.0 5.0 20 z z LME to z 6 THR to z 2

Conclusions 29 Proof of principle: Padé approximations can capture top mass corrections in gluon fusion processes Threshold expansion is necessary to model the peak region Systematic improvement through more expansion terms (LME, threshold expansion, small mass expansion)

Outlook 30 Higher orders in threshold expansion

Outlook 30 Higher orders in threshold expansion Combination with small-mass expansion 0.30 m 4 t dσ LO (gg HH)/dθ θ=π/2 [fb/rad] 0.25 0.20 0.15 0.10 0.05 m 8 t m 16 t m 32 t exact mh 0 mt 0.00 250 500 750 1000 1250 1500 1750 2000 s [GeV] [Davies, Mishima, Steinhauser, Wellmann 2018]

Outlook 30 Higher orders in threshold expansion Combination with small-mass expansion Mass effects in three-loop form factors

Outlook 30 Higher orders in threshold expansion Combination with small-mass expansion Mass effects in three-loop form factors Mass effects in gg( H ) ZZ interference

Outlook 30 Higher orders in threshold expansion Combination with small-mass expansion Mass effects in three-loop form factors Mass effects in gg( H ) ZZ interference Electroweak corrections to Higgs pair production

Backup

Small-p T expansion 32 [Bonciani, Degrassi, Giardino, Gröber 2018] Expand for p 2 T + m2 H ŝ; 4m2 t σ LO partonic [fb] 1 0.8 0.6 0.4 full HEFT O(1/m 8 t ) O(p 0 T + m0 h ) O((p 2 T + m2 h )2 ) Vfin 0.00080 0.00070 0.00060 0.00050 0.00040 0.00030 full HEFT O(p 0 T + m0 h ) O(p 2 T + m2 h ) O((p 2 T + m2 h )2 ) 0.2 0.00020 0.00010 pt = 100 GeV 0 0 100 200 300 400 500 600 700 800 900 s [GeV] 400 500 600 700 800 900 MHH [GeV]