中科院理论物理所：Hidden Analytic Relations for Two-Loop Higgs Amplitudes in QCD（讲稿，主讲：靳庆军）

Hidden Analytic Relations for Two- Loop Higgs Amplitudes in QCD 靳庆军（中科院理论物理所） EFT Amplitude Workshop @USTC September 6-9,2019 Based on work in collaboration with Gang Yang 1804.04653,1904.07260

Hidden Analytic Relations for Two￾Loop Higgs Amplitudes in QCD 靳庆军（中科院理论物理所） EFT & Amplitude Workshop @USTC September 6-9, 2019 Based on work in collaboration with Gang Yang 1804.04653, 1904.07260

Content ●Motivations ●Computations ●The hidden relation 。Summary and outlook 2

Content • Motivations • Computations • The hidden relation • Summary and outlook 2

Scattering Amplitudes in Standard Model(SM) Model→ Scattering AmplitudeExperiment Data Higgs production in LHC and CEPC LHC Perfect agreement with SM. Future CEPC:higher accuracy,less background noise.Requires even high loop computation of scattering amplitudes. 3

Scattering Amplitudes in Standard Model (SM) Higgs production in LHC and CEPC Model Scattering Amplitude Experiment Data LHC : Perfect agreement with SM. Future CEPC : higher accuracy, less background noise. Requires even high loop computation of scattering amplitudes. 3

New Methods in Scattering Amplitudes ● New methods developed during the computation of scattering amplitudes in supersymmetric field thoeries:spinor helicity formalism,on shell unitarity cut .. N=4 SYM integrand was computed 12 184 to 5 loops.[Bern et al 2012] N=8 supergravity is free of UV (335) 370 404 divergence to at least 5 loops.[Bern et al 2018] Apply these new methods to the computation of loop amplitude in Standard Model

New Methods in Scattering Amplitudes • New methods developed during the computation of scattering amplitudes in supersymmetric field thoeries: spinor helicity formalism, on shell unitarity cut … • N=4 SYM integrand was computed to 5 loops. [Bern et al 2012] • N=8 supergravity is free of UV divergence to at least 5 loops. [Bern et al 2018] • Apply these new methods to the computation of loop amplitude in Standard Model. 4

Higgs to 3 parton amplitude and the Maximal Transcendental Principle Study the properties of Higgs boson is one of the major tasks of LHC and future CEPC. In LHC,Higgs can be produced by the fusion of gluons and quarks.Higgs to 3 parton amplitudes are very important to understand the LHC data. The maximally transcendental principle:the maximal transcendental part of N=4 SYM and QCD amplitudes are the same. N=4 SYM 2 QCD An "easy"N=4 computation gives information for the“difficult'”QCD amplitude! 5

Higgs to 3 parton amplitude and the Maximal Transcendental Principle 5 The maximally transcendental principle: the maximal transcendental part of N=4 SYM and QCD amplitudes are the same. Study the properties of Higgs boson is one of the major tasks of LHC and future CEPC. In LHC, Higgs can be produced by the fusion of gluons and quarks. Higgs to 3 parton amplitudes are very important to understand the LHC data. An “easy” N=4 computation gives information for the “difficult” QCD amplitude!

Content ●Motivations ●Computations ●The hidden relation ●Summary and outlook 6

Content • Motivations • Computations • The hidden relation • Summary and outlook 6

Higgs effective field theory(HEFT) Integrate out the top quark HTr(F2) HTr(F3)... When the transverse momentum of Higgs particle is less than the top quark mass,the HEFT [Wilczek 1977]is good approximation. -aw品2ca+o(】 Oo=HTr(F2) O1=HT(F”FPF。“)O2=HT(DoF DPF), O3=HTr(DP FouD FH),O=HTr(FDPDe FM). .The leading (dimension 4)contribution of Higgs to 3 parton amplitude was computed in [Gehmann,Jaquier,Glover,Koukoutsakis 2011]. The contributions of dimension 6 operators.[Q.Yang 2018]and [Q.Yang 2019]. 7

Higgs effective field theory (HEFT) • When the transverse momentum of Higgs particle is less than the top quark mass, the HEFT [Wilczek 1977] is good approximation. • The leading (dimension 4) contribution of Higgs to 3 parton amplitude was computed in [Gehrmann, Jaquier, Glover, Koukoutsakis 2011]. • The contributions of dimension 6 operators. [QJ, Yang 2018] and [QJ, Yang 2019]. Le↵ = Cˆ0O0 + 1 m2 t X 4 i=1 CˆiOi + O ✓ 1 m4 t ◆ O0= HTr(F2) O1= HTr(F ⌫ µ F ⇢ ⌫ F µ ⇢ ), O2 = HTr(D⇢Fµ⌫D⇢F µ⌫), O3= HTr(D⇢F⇢µD￾F ￾µ), O4 = HTr(Fµ⇢D⇢D￾F ￾µ). ⇒ ⇒ HTr(F2) HTr(F3)··· Integrate out the top quark 7

Feynman Diagram vs Unitarity Cut 拉英庄其燕连 。5。 。。,是。 46。 正在立以立议议 4A A出 议议X A A a A AA T A A A Feynman diagrams:a broken vase Imposing cut condition +P2 P3 loop amplitude two on shell tree amplitudes 8

Feynman Diagram vs Unitarity Cut Imposing cut condition loop amplitude two on shell tree amplitudes 8 A A A A F F F F T1 G1 N1 A A A A y y y y T1 C1 N2 A A A A y y y y T1 C2 N3 A A A A U U U U T1 G2 N4 A A A A c c c c T1 C1 N5 A A A A c c c c T1 C2 N6 A A A A V V V V T1 G3 N7 A A A A A A A A T1 C1 N8 A A A A F F F F T2 G1 N9 A A A A y y y y T2 C1 N10 A A A A y y y y T2 C2 N11 A A A A U U U U T2 G2 N12 A A A A c c c c T2 C1 N13 A A A A c c c c T2 C2 N14 A A A A V V V V T2 G3 N15 A A A A A A A A T2 C1 N16 A A A A F F F F T3 G1 N17 A A A A y y y T3 C1yN18 A A A A y y y T3 C2yN19 A A A A U U U T3 G2UN20 A A A A c c c T3 C1c N21 A A A A c c c T3 C2c N22 A A A A V V V T3 G3VN23 A A A A A A A T3 C1AN24 A A Æ A A Feynman diagrams: a broken vase

D-dimensional unitarity cut ● 4-d spinor helicity formalism fails to capture the rational term in non-SUSY theories. A=-2 3+0@ D-dimensional unitarity uses the D-dimensional tree amplitude. A(e1,e2,3,e4)=-(1·e4)(e2·e3）+(e1·3)(e2·e4)-(e1·e2)(3·e4) 2te((-((e +22 more terms The polarization vector summing rule: ∑“e="-grp+gp helicities 9·pi

D-dimensional unitarity cut • 4-d spinor helicity formalism fails to capture the rational term in non-SUSY theories. • D-dimensional unitarity uses the D-dimensional tree amplitude. • The polarization vector summing rule: 9 A￾￾￾￾ = (2 ￾ 2✏)µ4 = 4 3 + O(✏) A(✏1, ✏2, ✏3, ✏4) = ￾ (✏1 · ✏4)(✏2 · ✏3)+(✏1 · ✏3)(✏2 · ✏4) ￾ (✏1 · ✏2)(✏3 · ✏4)) ￾ 2 s h 2t(✏1 · ✏2)(✏3 · ✏4)+(✏1 · p3)(✏2 · p1)(✏3 · ✏4) ￾ (✏1 · p2)(✏2 · p3)(✏3 · ✏4) i + 22 more terms X helicities " µ i "⌫ i = ⌘µ⌫ ￾ qµp⌫ i + q⌫pµ i q · pi

New Strategy of IBP Loop integrand can be reduced using integration by parts(IBP)relations. OOx米xQ王×D -火○ X 工 上 The IBP with cut strategy enhanced the efficiency by a factor of 10. Combine different cuts Cut integrand Reconstruction of integrand Complete integrand IBP with cut IBP Combine different cuts Coefficients c Amplitude 10

New Strategy of IBP 10 Loop integrand can be reduced using integration by parts (IBP) relations. The IBP with cut strategy enhanced the efficiency by a factor of 10