Two-loop QCD Corrections to the Heavy Quark Form Factors p.1

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1 Two-loop QCD Corrections to the Heavy Quark Form Factors Thomas Gehrmann Universität Zürich UNIVERSITAS TURICENSIS MDCCC XXXIII Snowmass Linear Collider Worksho005 Two-loop QCD Corrections to the Heavy Quark Form Factors p.1

2 Introduction Top quark pair production at ILC Differential distributions in threshold region understood at NNLO talks of A. Hoang, T. Teubner Continuum description only at NLO, insufficient for precision studies Observables: dσ t t /dp T, A t F B,... Anomalous couplings of top quarks at ILC Observables: g 2 from e + e t tγ, weak axial vector charges from angular distributions,... talk of U. Baur require precise understanding of higher order effects on Standard Model couplings Two-loop QCD Corrections to the Heavy Quark Form Factors p.2

3 Heavy Quark Form Factors Vector and axial vector form factors γ, Z 0 = ( i) v Q F 1 (s, m 2 )γ µ + v Q 1 2m F 2(s, m 2 )iσ µν ( + ) ν «+a Q G 1 (s, m 2 )γ µ 1 γ 5 + a Q 2m G 2(s, m 2 )γ 5 ( + ) µ Scalar and pseudoscalar form factors H, A = i m v ˆSQ F S (s, m 2 ) + ip Q F P (s, m 2 )γ 5 Two-loop QCD Corrections to the Heavy Quark Form Factors p.3

4 Heavy Quark Form Factors Two-loop QCD corrections (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) Two-loop QCD Corrections to the Heavy Quark Form Factors p.4

5 Two-loop QCD Form Factors Method of calculation W. Bernreuther, R. Bonciani, R. Heinesch, T. Leineweber, P. Mastrolia, E. Remiddi, TG project form factors from Feynman diagrams reduce loop integrals to master integrals integration-by-parts equations (K. Chetyrkin, F. Tkachov) Lorentz invariance equations (E. Remiddi, TG) Laporta algorithm (S. Laporta) evaluate master integrals using differential equations R. Bonciani, P. Mastrolia, E. Remiddi Renormalisation: hybrid scheme heavy quark mass and wave function: on-shell Zm OS, Z2 OS QCD coupling constant: MS Zg MS, Z3 MS Slavnov-Taylor identity: Z 1F = Z MS g Z OS 2 q Z MS 3 Two-loop QCD Corrections to the Heavy Quark Form Factors p.5

6 Two-loop QCD Form Factors Anomaly contributions (a) (b) consider four mass combinations: massive/massless for inside/outside quark use d-dimensional γ 5 (S. Larin) all mass combinations finite after renormalisation all mass combinations fulfil anomalous ward identities p µ Λ R Q,µ = 2mΛR Q i α s 4π T R F R Q recover axial contribution to Z 0 hadrons: G 1 (s, m, 0) G 1 (s, 0, 0) B. Kniehl, J. Kühn Two-loop QCD Corrections to the Heavy Quark Form Factors p.6

7 Two-loop QCD Form Factors Results form factors expressed as function of y = s s 4m 2 s + s 4m 2 in terms of rational factors and harmonic polylogarithms up to weight 4 E. Remiddi, J. Vermaseren form factors expanded at threshold in β = 4m 2 /s and for asymptotic energy in r = s/m 2 results checked using Ward identities partial results in literature A. Hoang, J. Kühn, T. Teubner; A. Czarnecki, K. Melnikov; M. Beneke, A. Signer, V.A. Smirnov Two-loop QCD Corrections to the Heavy Quark Form Factors p.7

8 Two-loop QCD Form Factors Threshold and high energy expansion consider finite anomalous form factors C F T R Re G 1 (s, m, m) C F T R Re G 2 (s, m, m) s / 2m exact asymptotic (r -1 ) asymptotic (r -2 ) threshold (β 0 ) threshold (β 2 ) s / 2m exact asymptotic (r -1 ) asymptotic (r -2 ) threshold (β 0 ) threshold (β 2 ) threshold expansion requires at least β 2 -terms; limited range of applicability asymptotic energy expansion converges better; larger range of applicability Two-loop QCD Corrections to the Heavy Quark Form Factors p.8

9 Static Form Factors Form factors at zero momentum transfer ( talk of U. Baur) F 1 (s = 0) = 1 (current conservation) v γ,z Q F 2(s = 0) = (g 2) γ,z (anomalous magnetic moment) a Z Q G 1(s = 0) = a Z,eff. Q (effective axial vector charge) Affect search for anomalous couplings top (µ = m t ) bottom (µ = m b ) bottom (µ = m Z ) (g 2) γ,(1l) Q (g 2) γ,(2l) Q (g 2) Z,(1l) Q (g 2) Z,(2l) Q (g 2) γ Q (g 2) Z Q G 1,Q Experimental sensitivity s = 500 GeV: (g 2) γ,z t : ± (G 1,t 1) : ± s = 800 GeV: (g 2) γ,z t : ± (G 1,t 1) : ± Two-loop QCD Corrections to the Heavy Quark Form Factors p.9

10 Forward-backward Asymmetry Forward-backward asymmetry of heavy quarks PSfrag replacements s m t = GeV m t = 178 GeV m t = GeV is very sensitive on the Higgs mass displays 2.3σ deviation for b-quarks at LEP1 at NNLO, receives contributions from 2, 3 and 4 parton final states two-parton final state infrared finite on its own 0.4 two-parton contribution to 0.3 A t t F B = σ A σ S PSfrag replacements A t t FB (1) A t t FB(α s ) A t t FB (α2 s ) s Two-loop QCD Corrections to the Heavy Quark Form Factors p.10

11 Summary and Conclusions computed two-loop QCD corrections to heavy quark form factors γ, Z 0, H, A Q Q crucial ingredient to dσ t t and A t t continuum F B in the tested convergence of threshold and large energy expansion sizable two-loop effects on determination of anomalous couplings Two-loop QCD Corrections to the Heavy Quark Form Factors p.11