Virtual Hadronic and Heavy-Fermion O(α 2 ) Corrections to Bhabha Scattering

Transcription

1 Virtual Hadronic and Heavy-Fermion O(α 2 ) Corrections to Bhabha Scattering Janusz Gluza University of Silesia, Katowice in collaboration with S. Actis, M. Czakon and T. Riemann Beijing, China, October 2008

2 Outline 1 Introduction 2 NNLO Fermionic QED Corrections 3 Hadronic Contributions 4 Summary

3 The Luminosity Monitor Luminosity of a collider depends on the machine and the beam: all is complicated (not mentioning errors estimation) dn dt = L(t), N = σ better is to choose a well known process: L = 1 σ Bhabha N Bhabha then we may determine any cross section in (1) dtl(t) = σl (1)

4 Kinematical Regions for Bhabha Two regions where the Bhabha-scattering cross section is large and QED dominated p1 p3 p2 p4 s = (p 1 +p 2 ) 2 = 4E 2 > 4m 2 e, s 10 2 GeV small θ SABS L at LEP, ILC a few degrees t = (p 1 p 3 ) 2 = 4 (E 2 m 2 e ) sin2 θ 2 < 0 s 1-10 GeV large θ LABS L at KLOE θ

5 Experimental Precision on L LABS 10 3 KLOE δl L exp = 0.3% DAΦNE-VEPP-2M 0.1 % SABS 10 4 LEP δl L exp = 0.03% ILC δl L exp = 0.02(1)% Giga-Z Status around 2004, from S. Jadach, hep-ph: Type of correction/error LEPEWWG hep-ph: update Technical precision (0.03%) 0.03% Missing photonic O(α 2 L) 0.10% 0.027% (0.013%) 0.013% Missing photonic O(α 3 L 3 ) 0.015% 0.015% (0.006%) 0.006% Vacuum polarization 0.04% 0.040% 0.025% Light pairs 0.03% 0.010% 0.010% Z-exchange 0.015% 0.015% 0.015% Total 0.11% 0.054% (0.055%) 0.045%

6 NNLO QED Corrections [Bern, Dixon, Ghinculov 00] σ = dφ 2 Re [M 2loop M tree ] Simplification m e = scale problem s, t + simple result ln(x), Li n (x) x = t/s = sin 2 (θ/2) - bad for MCs, m e 0 from Bern,Dixon,Ghinculov, arxiv:hep-ph/ For practical applications Recompute with m e 0 Massive from massless

7 First look, progress during last 2 years Remarkable: photonic, N f = 1, N f = 2 NNLO corrections doubly (triply) cross-checked

8 NNLO photonic and fermionic N f = 1, 2 topologies ½ ¾ ν ξ Î Î Î ¾¹ÐÓÓÔ ÓÜ ÓÖ ÔÖÓ SE loop insertions (without photonic line) are so called fermionic diagrams, rest represents photonic. Closed fermionic loop can be muon, tau, top or light quarks. In general, box B5 is a 4-scale problem: m e, m f, s, t(u).

9 Photonic NNLO massive QED Corrections Photonic corrections (no fermion loop) ( ) ( ) dσ 2 s s = A ln 2 dσ 0 me 2 + B ln me }{{ 2 + }{{} C } non collinear collinear [Arbuzov,Kuraev,Shaikhatdenov 98] [Glover,Tausk,van der Bij 01] A, B [Penin 05] C=C 0 + O(me/s) 2 from massless result Result of A. Penin confirmed by independent calculation [Becher-Melnikov 07]

10 Heavy Fermions with IR Matching excluding O(m 2 e/s) [Penin 05]: photonic corrections follow from a change in the regularization scheme M(m = 0, 1/ǫ ) M(ln(s/m }{{} e) 2, ln(λ 2 /m }{{} e) 2 ) + δm }{{} IR and collinear collinear IR [Mitov-Moch 06]: for 2 n QED/QCD scattering process M(m 0) = j Z 1/2 j M(m = 0) Z j = F D(Q 2, m 0) F D (Q 2, m = 0) [Becher-Melnikov 07]: in QED assuming m 2 e << m 2 f << s, t, u heavy fermions can be included M(m 0) = j Z 1/2 j S(m f ) M(m = 0) }{{} process dependent

11 N f = 1 NNLO massive QED Corrections Fermionic corrections: electrons in SE loops [Bonciani,Ferroglia,Mastrolia,Remiddi,van der Bij 05] electron loops with full m e dependence recalculated in 2007 [Actis, JG, Czakon, Riemann 07] electron loops with full m e dependence and in small electron limit: [Becher-Melnikov 07] electron loops me 2 << s, t

12 Evaluation of the MIs Electron-Loop Case MIs evaluated using the method of differential equations [Kotikov 91] [Remiddi 97] [Caffo,Czyz,Laporta,Remiddi 98] Three-scale problem: s, t, m e x = s s 4m 2 e s + s 4m 2 e y = 4m 2 e t t 4m 2 e t + t MIs: HPLs [Remiddi,Vermaseren 99] and 2dHPLs [Gehrmann,Remiddi 00] Full agreement with the existing result [Bonciani,Ferroglia,Mastrolia,Remiddi,van der Bij 05] [Actis.,Czakon,JG,Riemann 07]

13 N f = 2 NNLO leptonic massive QED Corrections First results: [Becher-Melnikov 07] me 2 << m2 f << s, t, u [Actis, JG, Czakon, Riemann 07] me 2 << mf 2 << s, t, u [Bonciani, Ferroglia, Penin 07] here: me 2 << m2 f, s, t, u Finally, yet another approach (dispersion relations) [Actis, JG, Czakon, Riemann 08] me 2 << mf 2, s, t, u

14 I: Evaluation of the MIs Heavy Fermions Four-scale problem: s, t, m e, m f new heavy-fermion scale Exploit the hierarchy of scales m 2 e << m 2 f << s, t, u Evaluate the MIs neglecting O(m 2 e /m2 f ), O(m2 e /s), O(m2 f /s) Mellin-Barnes method [Smirnov 99] [Tausk 99] efficient ( ) m I 2L 2 z+ǫ dz e Γi (z, ǫ) mf 2 ( ) m 2 k c e k Γj (z, ǫ) m 2 }{{} k f }{{} integral sum MB representations by AMBRE [Kajda, JG, Riemann 07] analytical continuation in ǫ with MB [Czakon 05] sums with XSummer [Moch,Uwer 05]

15 Example: box MIs, L m (x) = ln( m 2 /x) and R = m 2 /M 2 n B[x,y] = m 4ǫ 1 x ǫ Lm(x) + 1 ζ L m(x) + 1 ǫ 2 L2 m(x) + L m(x)l m(y) 2ζ 2 2ζ 3 + 4L m(x) + L 2 m(x) L3 m(x) 4ζ 2 L m(y) + 2L m(x)l m(y) + L m(x)l 2 m(y) 1 6 L3 m(y) 3ζ L2 m(x) L m(x)l m(y) L2 m(y) ln 1 + y x L m(x) L m(y) Li 2 y + Li 3 y o, x x n Bd[x,y] = m 4ǫ 1 h i L m(x)l m(y) + L m(x)l(r) 2ζ 3 + ζ 2 L m(x) + 4ζ 2 L m(y) xy ǫ + + 2L m(x)l 2 m(y) L3 m(y) 2ζ 2 L(R) + 2L m(x)l m(y)l(r) 1 6 L3 (R) 3ζ L2 m(x) L m(x)l m(y) L2 m(y) ln 1 + y x L m(x) L m(y) Li 2 y Li 3 y o. x x

16 Checks on the Computation UV/IR/collinear structure of the result UV divergencies cancel after inserting counterterms IR divergencies cancel after including single-photon emission O(me n ) from IBPIs cancel collinear divergencies ln ( ) s/me 2 Dirac FF agrees with [Burgers 85] [Kniehl,Krawczyk,Kühn,Stuart 88] Expansions of MIs checked with sector decomposition [Binoth,Heinrich 00]

17 Structure of the result SE 2b 2a 2c reducable diagrams 2d 2e irreducable diagrams After combining the 2-loop terms with the loop-by-loop terms and with soft real corrections: dσ NNLO dω + dσnlo γ dω = dσnnlo,e dω + f e Q 2 f dσ NNLO,f2 dω + f e Q 4 f dσ NNLO,f4 dω + f 1,f 2 e Qf 2 1 Qf 2 dσ NNLO,2f 2. dω

18 Irreducable terms dσ NNLO,f2 dω = α2 s ( ) 2ω } {σ NNLO,f2 1 + σ NNLO,f2 2 ln s σ NNLO,f2 1 = ( 1 x + x 2 ) 2 { 1 3x 2 3 ( ) s [55 ( m + ln 2 2 me 2 6 ln e mf 2 + [ ( ) ( s m ln 3 me 2 + ln 3 2 e m 2 f ) ] ) + ln ( 1 + 1/x) ]

19 Structure of the Result dσ ( 2 α ) [ ( ) ( ) ] 2 2ω s = A ln + B ln dσ 0 π s me 2 + C ( ) ( ) ( ) C = C 3 ln 3 s + C 2 ln 2 s s + C 1 ln + C 0 m 2 f m 2 f m 2 f C 3 = 1 9 C 2 = ln ( ) 1 x x x = t s = ( ) sin2 θ 2 1 { 324(1 x + x 2 ) x 14385x x x ζ x ζ x 2 ζ2 4158x 3 ζ x 4 ζ ζ3 864xζ x 2 ζ3 864x 3 ζ x 4 ζ3 672 L x + 966xL x 2016x 2 L x + 966x 3 L x 672x 4 L x + 648ζ2L x 2160xζ2 L x 1 [ 108(1 x + x 2 ) x x x x ζ2 432xζ2 162x 2 ζ x 3 ζ2 432x 4 ζ L x 186xL x + 360x 2 L x 186x 3 L x + 120x 4 L x + 72L 2 x 198xL 2 x + 270x2 L 2 x 198x3 L 2 x + 72x4 L 2 x 444L+x + 672xL+x 846x2 L+x C 0 } x 2 ζ2l x 2160x 3 ζ2 L x + 648x 4 ζ2l x 360L 2 x x L 2 x 1458x 2 L 2 x x 3 L 2 x 360x4 L 2 x + 108L3 x 216xL3 x + 252x2 L 3 x 144 x3 L 3 x + 36x4 L 3 x L+x 5259xL+x x 2 L+x 2775x 3 L+x + 672x 4 L+x + 648ζ2L+x 1944xζ2 L+x + 702x 2 ζ2l+x + 864x 3 ζ2l+x 648x 4 ζ2l+x L xl+x 2106x L xl+x x 2 L xl+x 540x 3 L xl+x 135x 2 L 2 xl+x x 3 L 2 xl+x 108x 4 L 2 xl+x 990 L 2 +x xL 2 +x 1512x 2 L 2 +x + 144x3 L 2 +x + 360x4 L 2 +x 324L xl2 +x + 540x L xl2 +x 216x2 L x L 2 +x 108x 3 L xl 2 +x + 108x 4 L xl 2 +x + 36 L 3 +x 279xL 3 +x + 198x 2 L 3 +x + 18x 3 L 3 +x 36x 4 L 3 +x 36 ( 1 x + x 2) [ 20 ( 1 x + x 2) ] + 3x( 1 + 2x)L x 6( 2 + x) L+x Li2(1 x) + 72 ( 1 x + x 2) [ 10 ( 1 x + x 2) + 3 ( 3 2 x + x 2) ] [ L+x Li2(x) L x 324xL x ] ( ) x + 324x 2 L x 108x 3 L x 216L+x + 324xL+x 324x 2 L+x + 108x 3 L+x Li2 x 1 [ + 108x + 324x 2 324x x 4] [ Li3(1 x) x 648x x 3 216x 4] [ Li3(x) x + 324x 2 108x 3] ( ) x } Li3 x x 3 L+x 120x 4 L+x 216L xl+x + 432xL xl+x 378x 2 L xl+x + 108x 3 L xl+x + 36L 2 +x 153xL2 +x + 135x2 L 2 +x + 9x3 L 2 +x 72x4 L 2 +x 144 ( 1 x + x 2) 2 Li2(1 x) ( 1 x + x 2) ] 2 Li2(x) {}}{ C 1 [S.A.,Czakon,Gluza,Riemann 07] [Becher,Melnikov 07]

20 Present situation, virtual NNLO QED

21 Hadronic contributions 1 Actis, Czakon, JG, Riemann, arxiv: v2, Physical Review Letters, 100, 2008, long paper: arxiv: , submitted to PRD 2 Johann H. Kühn, Sandro Uccirati, arxiv: (a) (b) (c)

22 Beyond m 2 f << s KLOE s = 1 GeV < m τ ILC s = 500 GeV m t < t, u 40 0 < θ < region for luminosity Kinematical regions where expansions don t work General method for including hadronic effects

23 Dispersion Relations Obtain fermionic corrections inserting Π R in µν γ g µν q 2 + i δ g µα q 2 + i δ ( q 2 g αβ q α q β) Π R (q 2 ) Represent Π R through a dispersion integral g βν q 2 + i δ Π R (q 2 ) = q2 π 4M 2 dz z Im Π(z) q 2 z + i δ Leptons, top (perturbative) α Im Π f (z) = F ǫ π π θ z 4 mf 2 3! ǫ me 2 mf 2 Qf 2 C f n βf (z) h io 3 βf 2 (z) 2 Light quarks (non pert.) Unitarity ImΠ had (z) = α 3 R had(z) R had (z) = σ({e+ e γ hadrons}; z) (4πα 2 )/(3z)

24 Effectively we must calculate following MIs p1 m p3 γ M t p2 m p4 M2 M8 M M M1 M6 M7(dotγ) M M4 M3 M5 M again expanded from MB representations

25 Box Master, M 8 d D k [ ] [ ] (k 2 z) (k + p 3 ) 2 me 2 (k + p 3 p 1 ) 2 (k + p 3 p 1 p 2 ) 2 me 2 1 { 1 [ ( ) ( = ln m2 e + ln z ) ( ln z )] 2 ζ 2 s (t z) ǫ t s t ( )[ + ln m2 e 1 ( ) ( t 2 ln m2 e + ln z ) ( + ln z ) ( 2 ln 1 z )] t s t t 3 2 ln2( z ) ( + ln z ) ( ln z ) t s t ( 2 ln 1 z )[ ( ln z ) ( ln z ) ] ( Li z )} + O(m 2 t s t s e).

26 IR finite results with boxes The 4 direct and 4 crossed fermionic 2-loop box diagrams have to be combined with other diagrams for an IR-finite contribution: Real Radiation 1loop 1loop 2loop Tree + + assembling all diagrams, the terms in ln(s/m 2 e) drop out and the total contribution of fermionic box diagrams is free of collinear divergencies a sensible, infrared safe cross-section contains the complete sum of all the single IR-divergent diagrams, or no one of them

27 Numbers from Bonciani, Ferroglia, Penin, 08 Hadronic leptonic, m π m f, plus change of kernels agreement (in all digits for µ) and for m τ = 1.7 GeV [but not GeV] :-))

28 Hadronic contributions 100 R had (s) 10 1 R had, Burkhardt R had, update s [GeV 2 ]

29 Hadronic contributions

30 Hadronic contributions in contrast to PRL paper, narrow resonances are implemented replacing the rapidly varying cross section ratio with the parametrization R res (z) = 9π α 2 M resγ e+ e res δ(z Mres). 2 For the numerical evaluation of the contribution due to the narrow resonances, we use the values listed in the Burkhardt s routine

31 Hadronic contributions resonance M res [GeV] Γ e+ e res [kev] ω(782) φ(1020) J/ψ(1S) ψ(2s) ψ(3770) ψ(4040) ψ(4160) ψ(4415) Υ(1S) Υ(2S) Υ(3S) Υ(4S) Υ(10860) Υ(11020) Table: taken directly from Burkhardt routines.

32 Agreement with KU, , 2008 for IPAR=0 θ [ ] s [GeV] θ = 20 1 θ = θ = 3 M Z θ = vertices [µ+τ+hadr.] vertices [e] soft pairs e + e rest: e µ τ hadr Table: dispersion-based approach (first line) and the one the analytical expansion (second line), neglecting O(mf 2 /x), where x = s, t, u. When mf 2 > x, the entry is suppressed. soft pairs: Arbuzov, Kuraev, Merenkov, Trentadue, 1995

33 Agreement with KU, , 2008 for IPAR=0 s [GeV] 1 10 MZ 500 vertices [µ+τ+hadr.] vertices [e] soft pairs e + e rest: e µ τ hadr Table: Numerical values for the differential cross section in nanobarns at a scattering angle θ = 90, in units of 10 4.

34 PRD paper 6 s = 1 GeV * dσ 2 /dσ photonic muon electron total non-photonic hadronic I and II θ

35 PRD paper * dσ 2 /dσ EW had I had II s 1/2 = M Z photonic muon electron total non-photonic hadronic I and II θ

36 Summary NNLO QED Bhabha corrections have been extensively studied by different groups and they reach the level of several permille in the regions needed for measuring the luminosity the results have to be compared and interfaced with the available MC generators