Electroweak Sudakov logarithms

Transcription

1 Electroweak Sudakov logarithms Stefano Pozzorini MPI Munich LoopFest V, June , SLAC

2 (1 Gauge-bosons pair production with high p T at the LHC 1-loop effects [Accomando, Denner, Hollik, Meier, Kaiser] (2 Single gauge-boson production with high p T at the LHC 1- and 2-loop effects [Kühn, Kulesza, P., Schulze] (3 4-fermion neutral current processes complete 2-loop logarithmic corrections (N 3 LL [Jantzen, Kühn, Penin, Smirnov] (4 Progress towards 2-loop predictions for general processes in NLL approximation [Denner, Jantzen, P.]

3 Introduction High-energy colliders (ILC, LHC will explore the TeV energy scale investigate the mechanism of electroweak symmetry breaking search for new physics Important implications for loop corrections within the Standard Model collider energy characteristic scale of EW corrections (s MW 2 enhancement of EW corrections due to large Sudakov logarithms ln 2 ( s 25 at s 1 TeV

4 Originate from vertex and box diagrams involving virtual weak bosons e ν e W W Z e γ, Z W ± Z W ± α ln 2 ( s e + W + Z e + W Z General form of 1 loop EW corrections for s MW 2 [ ( ( ] s s α C 2 ln 2 + C 1 ln 1 + C 0 + O } {{ } LL } {{ } NLL ( M 2 W s Typical size of logs for 2 2 processes at s 1TeV: effects of O(10% ( δσ1 α ( σ 0 πs 2 log 2 s δσ1 W MW 2 26% + 3α σ 0 πs 2 log s W MW 2 +16% LL NLL

5 The log(s/m 2 terms represent mass singularities and originate from γ, Z, W γ, Z, W diagrams with virtual gauge bosons (γ, Z, W ± coupling to on-shell external legs soft and collinear regions Factorization and universality at one loop M 1 = ( 1 + α 4π legs k δ 1 EW(k } {{ } soft,coll. M 0 Born external-leg factors (universal LL and NLL for arbitrary processes (e, ν, u, d, t, b, γ, Z, W ±, H, g External-leg factors depend only on external-leg quantum numbers δew 1 (k = 1 2 Cew (k log 2 s M 2 + a (kiā(l log r kl s 1 2 Q2 (k [ 2 log s m 2 k l k log M2 λ2 log2 M2 m 2 k a=γ,z,w ± I 2 log M2 λ2 log M2 m 2 k log s M 2 +γew (k log s M 2 ] + l kq(kq(llog r kl s log M2 λ 2 Denner and P. (2001

6 1. One-loop EW corrections in gauge-boson pair production at the LHC Motivation p W ±, Z, γ test YM interactions of gauge bosons, search for anomalous gauge couplings p W ±,Z, γ study region of high invariant mass ŝ = (p V1 +p V2 2 and large scattering angles high P T (V large EW logarithmic corrections in this region

7 Existing calculations for pp V 1 V 2 all processes apart from pp γγ computed process decay corrections WZ, Wγ no NLL Accomando, Denner, P. (2002 Zγ no exact Hollik, Meier (2004 WW, WZ, ZZ yes NLL Accomando, Denner, Kaiser (2005 Zγ, Wγ yes exact+nll Accomando, Denner, Meier (2005 exact O(α results and/or NLL approximation depending on the process most recent calculations include decay of weak bosons

8 pp WW, WZ, ZZ at the LHC Detailed description of final state p Z, W l, ν l l, ν l leptonic decays l, l = e, µ hard photon bremsstrahlung p Z, W l, ν l l, ν l Electroweak corrections double-pole approximation (DPA high-energy region ŝ, ˆt, û NLL approximation Accomando, Denner, Kaiser (2005

9 p V l, ν l l, ν l p V l, ν l l, ν l Double-pole approximation M qq V V lν l l l fact = λ,λ M qq V λ V λ M V λ lν l M V λ l l (p 2 V M2 V + im V Γ V (p 2 V M 2 V + im V Γ V gauge-invariant factorization and separation of scales on-shell production }{{} ŝ M 2 log( ŝ M 2 on-shell decay }{{} M 2 m 2 l λ2 log( λ2 M 2, log( m2 l M 2

10 Example: electroweak NLL corrections to q q WZ Longitudinal polarization M 0 GBET = W + φ + χ δm 1 M 0 = α 4π { log 2 ŝ [ ] C ew q + C ew L Φ + log ŝ + log ˆt ŝ s2 W c 2 Y ql log ˆt 3C ew q + 4C ew W û L Φ 3 2s 2 W [ m 2 t 2 ( s 2 log û W ŝ ]} b (1 2 Transverse polarization M 0 = W + T W + + Z T W + W + T T + δm 1 M 0 = α 8π log2 + α 4π log ŝ MW 2 } + 3C ew q L { ŝ 1 s 2 W [ { [ 2C ew q + C ew L W 1 + log ˆt ŝ + log û ŝ + c 2 Wcos ˆθ c 2 W cos ˆθ s 2 W Y q L ]} c 2 W c 2 W cos ˆθ s 2 W Y log ˆt q û L ] Z T Z T derived from Denner and P. (2001

11 Ñ Ü È È Ñ Ü p max T (l distribution for pp WZ eν eµ + µ at the LHC e-05 Ì Ðµ» Î Born cross section EW cross section 1e Cuts for LHC detectors p T (l, p miss T > 20 GeV, η l < 3 recombination of soft and collinear photons (E γ < 2GeV, R lγ < 0.1 Cuts to select gauge-bosons resonances M(l l M Z < 20 GeV M T (lν l M W < 20 GeV Cuts to select high-energy region p T (Z > 300 GeV Large negative corrections increase with p T 30% at p max T (l 800 GeV! Ì Ðµ Î

12 Ñ Ü È È Ñ Ü p max T (l distribution for pp ZZ e+ e µ + µ at the LHC e-05 Ì Ðµ» Î Born cross section EW cross section 1e Cuts for LHC detectors p T (l, p miss T > 20 GeV, η l < 3 recombination of soft and collinear photons (E γ < 2GeV, R lγ < 0.1 Cuts to select gauge-bosons resonances M(l l M Z < 20 GeV Cuts to select high-energy region M inv (l ll l > 500 GeV, y(zz < 3 Large negative corrections p T dependence similar as for WZ size larger than for WZ 50% at p max T (l 800 GeV! Ì Ðµ Î

13 Ñ Ü È È Ñ Ü p max T (l distribution for pp WW e+ ν e µ ν µ at the LHC Ì Ðµ» Î Born cross section EW cross section Cuts for LHC detectors p T (l > 20GeV, p miss T > 25 GeV, η l < 3 recombination of soft and collinear photons (E γ < 2GeV, R lγ < 0.1 Cuts to select gauge-bosons resonances W reconstruction not possible Cuts to select high-energy region M inv (l l > 500 GeV, y(l l < 3 Large negative corrections behaviour and size similar as for WZ 35% at p max T (l 800 GeV! Ì Ðµ Î

14 Electroweak corrections ( EW vs statistical error ( stat = 1/ 2Lσ 0 pp WZ lν l l l pp ZZ l ll l pp WW l ν l ν l l M cut inv (leptons[gev] EW [%] stat [%] EW [%] stat [%] EW [%] stat [%] estimate based on L = 100fb 1 and final states with l, l = e or µ EW and stat grow with M inv (leptons EW stat up to M inv 1 TeV

15 Ñ Ü È Electroweak corrections vs anomalous couplings Accomando, Kaiser (2005 L = g WWV [g V 1 (W µν Wµ V ν W µ V νw µν + κ V W µ W νv µν + λ ] MW 2 W ρµ Wµ ν V νρ Limits from LEP2 and Tevatron: g Z , κγ 0.061, 0.07 λ Ì Ðµ» Î p T (Z > 250 GeV Born NLO 2a 2b 3a 3b 4a 4b Scenario λ g Z 1 κ γ 2a/2b 0 ± a/3b 0 0 ±0.04 4a/4b ± e pp WZ eν e µ + µ : large effects at high p T Ñ Ü Ì È Ðµ Î anomalous couplings spoil gauge cancellations and (in general enhance σ electroweak corrections reduce σ and increase the sensitivity to anomalous couplings

16 Electroweak corrections vs anomalous couplings Ý Ðµ Accomando, Kaiser (2005 L = g WWV [g V 1 (W µν Wµ V ν W µ V νw µν + κ V W µ W νv µν + λ ] MW 2 W ρµ Wµ ν V νρ Limits from LEP2 and Tevatron: g Z , κγ 0.061, 0.07 λ p T (Z > 250 GeV Born NLO 2a 2b 3a 3b 4a 4b Scenario λ g Z 1 κ γ 2a/2b 0 ± a/3b 0 0 ±0.04 4a/4b ± pp WZ lν l l l : large effects at small y(zl Ý Ðµ (in general dip from radiation zero filled by anomalous couplings dip enhanced by electroweak logarithmic corrections

17 One-loop predictions for pp WW, ZZ, WZ based on NLL approximation how precise is the NLL approximation?

18 pp Wγ, Zγ at the LHC Detailed description of final state l, ν l leptonic decays l = e, µ p Z, W l, ν l hard photon bremsstrahlung Electroweak corrections p γ leading-pole approximation (LPA exact O(α calculation high-energy region ŝ, ˆt, û NLL approximation Accomando, Denner and Meier (2005

19 p T (γ distribution for pp Wγ lν l γ at the LHC ¹¾ ¹½ ¹ ½¼ ½ Æ ±µ ¹½¼ Ô Ì» ε ½¼ ½ ½¼ ¾ ¹¾¼ ¹ ¼ ¼ Born cross section ew. corr.. cross section ¼ ½¼¼ ¼ ¾¼¼ ¼¼ ¼¼ ¾¼¼ ¼¼ ¼¼ Ô Ì Îµ Ô ¼¼ ¼¼ ¼¼ ¼¼ ½¼ ½¼ ¼¼ Cuts for LHC detectors p T (l > 20 GeV, p miss T > 50 GeV, η l,γ < 2.5, R γl > 0.7 recombination of soft and collinear photons (E γ < 2GeV, R lγ < 0.1 Cut to select W resonance M T (lν l M W < 20 GeV Large negative corrections increase with p T EW stat up to p T (γ 700 GeV 23% at p T (γ 700 GeV 1%-level agreement with virtual NLL approximation [Accomando, Denner, P. (2002]

20 Virtual corrections to pp Wγ: exact corrections vs NLL approximation p γ,c T (GeV NLL const ln (ˆt/ŝ ln 2 (ˆt/ŝ % 0.26% 1.78% 3.95% % 0.58% 1.97% 3.50% % 0.74% 1.91% 3.05% % 0.87% 1.79% 2.62% Complete high-energy approximation C 2 ln 2 ( ŝ + C 1 ln ( ŝ } {{ } NLL: correct description of large effects at high energy + B 0 + B 1 ln ( ˆt ŝ ( ˆt + B 2 ln 2 + O ŝ } {{ } non-enhanced terms: several percent effect (-5.5%! Percent-level precision requires exact O(α corrections or at least complete high-energy approximation ( M 2 W ŝ }{{} suppressed ( 0.5%

21 2. pp Z+jet and pp γ +jet at the LHC pp Z + jet pp γ + jet Precision measurement σ(gq Zq = O(αα S : large! 10 0 clean signature (Z leptons dσ dp [ pb/gev ] T L gluon-pdf at 1% Explore TeV energy scale σ(p T > 1TeV 25 fb events/year 10 6 requires theoretical precision at percent p T [GeV] level!

22 pp Z + jet (pp γ + jet similar Z,W ± Z,W ± Z,W ± Z,W ± Partonic reactions q q Zg, gq Zq, g q Z q crossing symmetry dominant contribution from initialstate gluons W ± Z,W ± Z,W ± Z,W ± Z,W ± W W ± W W W ± Weak corrections (virtual Z,W exact 1-loop calculation for finite p T compact 1-loop approximation for large p T dominant 2-loop effects at large p T Kühn, Kulesza, P., Schulze (2005

23 Analytic 1-loop results for qq Zg (compact expressions in terms of scalar integrals M1 2 = 8π 2 αα S (N 2 c 1 + α 2π V =Z,W ± λ=r,l (I V I V qλ H A 1 (M2 V ] { ( [ 2 ( I Z q H δC A λ q λ + c W s 3 T 3 q I Z λ q [2H 0 δc N λ q + α λ 2π W 1 s 2 W H N 1 (M2 W ] } Asymptotic expansion: ŝ, ˆt, û H A/N 1 (M V 2 = [ Re g A/N 0 (M V 2 ˆt 2 +û 2 ˆtû [ ( g 0 N (M2 W = 2 g A 0 (M2 V = log2 ( 2 4 D γ E +log 4πµ 2 M 2 Z log 2( ˆṱ 3 u 2 [log2( ˆṱ +log 2( ûŝ ] 20π2 s 9 ŝ M 2 V +3log g N 1 (M2 V = ga 1 (M2 V [log ( ûŝ ( +ga/n 1 (M V 2 ˆt 2 û 2 ˆtû +ga/n 2 (M V 2 ] ( ( ( ]+log 2 ŝ log M W 2 2 ˆt log M W 2 2 ŝ M V 2 log π 3 +2 û + 32 [log2( +log ˆṱs 2( ( ( ûŝ +log ˆṱ +log ûŝ ]+ 7π2 s 3 2 5, ( ˆṱ ]= 1 s 2 [log2( ûŝ log 2( ˆṱ s ] g 2 N (M2 V = ga 2 (M2 V = 2[log2( ˆṱ +log 2( ( ( ûŝ +log ˆṱ +log ûŝ ] 4π 2 s s Very compact expressions NLL: predicted by process-independent formula [Denner and P. (2001] NNLL: contains also π 2,log(ˆt/û,... not growing with energy

24 Dominant two-loop contributions for qq Zg M2 2 = M α 3 α S (N 2 c 1 ˆt 2 +û 2 [ ] I Z q C ew λ q X 1 + c W λ s 3 T 3 q X 2 λ W T3 q λ Yq λ 8s 4 W ˆtû λ=l,r { X IV q λ [ 1 2 I Z q λ ( ( I Z q C ew λ q + c W λ s 3 T 3 q λ W b 1 c 2 W ( Y q λ b 2 s 2 W ] } C qλ + c W s 3 T 3 q b 2 X 3 λ W Kühn, Kulesza, P., Schulze (2005 includes LL and NLL terms ( X 1 = ln 4 ( X 2 = ln 4 ( X 3 = ln 3 ŝ ˆt ŝ M W 2 ( 6 ln 3 ( + ln 4 ŝ û ln 4 ( ŝ derived from process-independent results for two-loop Sudakov corrections Melles (2001; Denner, Melles, P. (2003

25 One-loop corrections to dσ/dp T for pp Z + jet R had NLO/LO R had R had NLL/LO NNLL/LO Large negative corrections increase with p T % at p T 1 TeV (b R had R had NLL/NLO NNLL/NLO agreement with Maina, Moretti, Ross (2004 Quality of high-energy approx. NLO, NLL, NNLL overlap! (c p T [GeV] NLL NLO NLO NNLL NLO NLO very precise (much better than for Wγ MS input: α S (M 2 Z = 0.13, α = 1/128.1, s2 w = PDFs: LO MRST2001, µ F = µ R = p T

26 1- and 2-loop corrections to σ(p T > p cut T for pp Z+jet Size of corrections at p T 1TeV 1-loop: -26% loop: -26% + 4% = -22% Comparison with statistical error NLO/LO-1 NNLO/LO-1 statistical error T [GeV] p cut L = 300fb 1, Z leptons ( σ/σ stat 2% at 1 TeV 2-loop effects not negligible! MS input: α S (M 2 Z = 0.13, α = 1/128.1, s2 w = PDFs: LO MRST2001, µ F = µ R = p T

27 1- and 2-loop corrections to σ(p T > p cut T for pp γ + jet 0.00 Size of corrections at p T 1TeV 1-loop: -12.4% loop: -12.4% + 1.6% = -10.8% NLO/LO-1 NNLO/LO-1 statistical error T [GeV] p cut Comparison with statistical error L = 300fb 1 ( σ/σ stat 1.3% at 1 TeV 2-loop effects stat. error! input: α S (M 2 Z = 0.13, α = 1/137, s2 w = 1 M2 W /M2 Z PDFs: LO MRST2001, µ F = µ R = p T

28 Ratio of the p T distributions for pp γ + jet and pp Z + jet 1.30 p T M Z : strong p T dependence dσ Z dp T dσ γ dp T / LO NLO NNLO due to M Z p T M Z : weak p T dependence ratio determined by γ/z couplings and up/down PDFs 0.80 PDF-uncertainties cancel 0.70 QCD corrections cancel p T [GeV] loop and 2-loop EW corrections p T =1TeV: =0.81 p T =2TeV: =0.95

29 Electroweak Sudakov logarithms beyond one loop Typical size of LL and NLL corrections for 2 2 processes at s 1TeV 1-loop effects ( δσ1 α log 2 s σ 0 LL πs 2 W ( δσ1 + 3α σ 0 πs 2 log s W MW 2 NLL 26% +16% 2-loop effects (back-of-the-envelope estimate ( δσ2 + α2 σ 0 LL 2π 2 s 4 log 4 s W MW 2 3.5% ( δσ2 3α2 σ 0 π 2 s 4 log 3 s 4.1% W NLL How to compute higher-order logarithmic corrections? first approach based on resummation techniques for mass singularities in QED, QCD extended to EW theory non-trivial due to mass gap in the gauge sector M A = 0 M Z M W how to separate soft-collinear singularities due to massless photons?

30 InfraRed Evolution Equation (IREE Dependence of matrix elements on transverse-momentum cut-off µ T M log(µ T = K(µ TM factorization of mass sing. (QCD kernel known for SU(N 2 regimes with exact gauge symmetry µ T M W : kernel insensitive to mass gap as in symmetric SU(2 U(1 theory with M A = M Z = M W µ T M W : weak bosons frozen and kernel as in QED Integration of IREE yields double factorization and exponentiation { MW } { } dµ s T dµ T M(µ 0 = exp K 2 (µ T exp K 1 (µ T µ 0 µ T M W µ T M Born Fadin, Lipatov, Martin, Melles (2000

31 2 loop EW corrections for s General form α 2 [ C 4 ln 4 ( s } {{ } LL + C 3 ln 3 ( s } {{ } NLL + C 2 ln 2 ( s } {{ } NNLL + C 1 ln 1 ( s }{{} N 3 LL + C 0 ] Results based on the IREE approach LL for arbitrary processes [Fadin, Lipatov, Martin, Melles (2000] NLL for arbitrary processes [Melles (2001,2002,2003,2004] NNLL for massless f f f f [Kühn, Moch, Penin, Smirnov (2000,2001,2003] N 3 LL for massless f f f f [Jantzen, Kühn, Penin, Smirnov (2004,2005]

32 3. N 3 LL for massless f f f f Jantzen, Kühn, Penin, Smirnov (2004,2005 Step A: logarithmic corrections within SU(2 Higgs model with M W ± = M Z = M using QCD resummation techniques decomposition of 4-fermion amplitude A = = ig2 s F2 à evolution equation for logarithmic corrections to the form factor F [Mueller(1979, Collins(1980, Sen (1981] F = F ln s = [ s ] dx x γ(α(x + ζ(α(s + ξ(α(m2 W F input for SU(2 Higgs model from explicit 2-loop N 3 LL calculation using expansion by regions [Jantzen, Smirnov (2006] evolution equation for reduced amplitude à [Sen(1983, Botts, Sterman (1987,1989] à = à ln s = χ(α(s à input for matrix of soft anomalous dimension χ derived from existing 2-loop QCD results (Higgs mechanism irrelevant

33 Step B: IREE approach for extension to EW theory including photons (i SU(2 results translated to SU(2 U(1 group with M γ = M (ii photonic singularities for M γ = 0 assumed to factorize according to IREE prescription A SU(2 A SU(2 U(1 Mγ =M A QED Mγ =0 A Mγ QED =M A SU(2 U(1 Mγ =M confirmed by 2-loop form factor calculation for SU(2 U(1 with M γ = 0 and sin 2 (θ W = 0 mixing corrections due to nonabelian γ W interaction expected but only for coefficient of ln 1 term and suppressed by sin 2 (θ W 0.2

34 Relative 2-loop logarithmic corrections to σ(e + e d d ( α 2 [2.79 ln 4 ( s ln 3 ( s ln 2 ( s ln 1 ( s ] 4π sin 2 θ w LL (ln 4 NLL (ln 4 + ln 3 NNLL (ln 4 + ln 3 + ln 2 N 3 LL (ln 4 + ln 3 + ln 2 + ln s 1/2 [GeV] Subleading logarithms increasingly large coefficients alternating signs Logarithmic approximation total 2-loop correction very small + residual theoretical error O(10 3 oscillating, bad convergence + better convergence expected for other processes

35 Open question: 2-loop EW logs for production of heavy particles (Z,W,b,t,H? in contrast to massless fermions heavy particles couple directly to Higgs and Yukawa sectors and the role of symmetry breaking becomes crucial NLL resummation prescriptions [Melles (2001,2002,2003,2004] based on IREE splitting of EW theory in symmetric SU(2 U(1 and QED regime effect of symmetry breaking assumed to be negligible (not proven Confirmed by diagrammatic two loop calculations based on EW Feynman rules LL for arbitrary processes [Beenakker, Werthenbach (200,2002; Denner, Melles, P. (2003] NLL for gluon-fermion form factor [P. (2004] NLL for general processes: work in progress...

36 4. Two-loop electroweak NLLs for arbitrary processes: preliminary results Goal derive 2-loop NLLs for arbitrary electroweak processes (light- and heavy fermions, gauge bosons, Higgs diagrammatic 2-loop calculation based on electroweak Feynman rules Tools for process-independent analysis already developped electroweak collinear Ward identities to isolate and factorize mass singularities within ξ = 1 gauge automatic algorithm for multi-scale 2-loop diagrams in NLL approximation First results for general fermionic processes f 1 f 2 f 3... f n M = f 1 f 3 in the limit (p i + p j 2 MW 2 f 2 f n Denner, Jantzen, P. (in preparation

37 A 1- and 2-loop diagrams that give rise to NLL mass singularities (ξ = 1: virtual gauge bosons (a i = W ±,Z, γ coupling to external fermions a a 1 1 a 2 a 3 a1 a 3 a 2 a 3 B Soft and collinear approximations for fermion-boson vertices lim q µ xp µ [ p p ] = 2ep µ IV (Dirac structure disappears Vµ(q

38 C Reduction to factorizable contributions diagrams with collinear gauge bosons that couple to external and internal lines a 2 a 3 cancel against non-factorizable contributions from diagrams with gauge bosons coupling only to external lines a a 1 1 a 2 a 3 a1 a 3

39 Reduction to factorizable contributions at one-loop level Factorizable (F: external-leg exchange of soft/collinear gauge bosons F = i ( i neglegt collinear momenta in hard matrix element j j Non-factorizable (N: cancelled by diagrams with collinear gauge bosons coupling to internal lines V µ + j i V µ N = 0 Cancellation mechanism collinear vertex prop. to q µ collinear Ward identities for spontaneously broken nonabelian theories V µ + j i N V µ qµ = 0 Denner, P. (2001

40 Analogous collinear Ward identities derived at the two-loop level [Denner, Jantzen, P. (2006] only factorizable contributions remain F i j F i j F a 3 i j F i j F i j F i j F j k i F j a1 a 3 i k F i j k l Structure of single factorizable contribution i a F 2 = M a 0 3 j ( ie 3 g 2 ǫ a 3 Iā2 i Iā1 i Iā3 j }{{} gauge couplings: SU(2 matrix D(M a1, M a2, M a3 ; r ij }{{} 2-loop integral

41 D Evaluation of 2-loop integrals For every topology various configurations different scales 3 Mandelstam invariants: r ij = (p i +p j 2 F W γ,z W F γ,z W W F W W γ,z heavy particles: M W, M Z, M H, m t massless particles: γ, light fermions ( Loop integrals in the high-energy limit L = log C mn ǫ m L n m,n s M W 2 C mn = C mn ( MZ M W, M H M W, NLL approximation with same counting for L and ǫ 1 poles 1 and D = 4 2ǫ m t, r ij M W s 2-loop LLs = ǫ 4, ǫ 3 L 1, ǫ 2 L 2, ǫ 1 L 3, L 4 NLLs = ǫ 3, ǫ 2 L 1, ǫ 1 L 2, L 3

42 Example of 2-loop diagram in NLL approximation i F a 3 = M0 ie 3 g 2 ǫ a 3 Iā2 i Iā1 i Iā3 j ( s r ij 2ε D(M a1, M a2, M a3 ; r ij j D(M W, M W, M W ; r ij NLL = 1 6 L log ( r ij s L 3 3L 2 ǫ 1 5L 3 D(M W, M W, M Z ; r ij NLL = 1 ( 3 log M 2 Z M W 2 L 3 D(0, M W, M W ; r ij NLL = 1 3 L3 ǫ 1 log ( rij s D(M W, M W, 0; r ij NLL = 1 3 L3 ǫ 1 log ( r ij s + L 2 ǫ L3 L 2 ǫ L4 7 ( 3 log rij s L 2 ǫ L4 7 3 log ( r ij s All loop integrals evaluated and cross checked with 2 independent methods automatic algorithm based on sector decomposition [Denner, P. (2004] expansion by regions + Mellin barnes representation [Smirnov, Jantzen (2006] L 3 + 2L 2 ǫ L3 L 3 6ǫ 3 6Lǫ 2

43 E Complete 2-loop amplitude δm 2 = 1 2 i j,,a 3 [ i F + j i F + j i a F 2 + a 3 j F i j + i F + j F i j ] i j k,,a 3 [ j k i F + F j a1 a 3 i k ] i j k l, F i j k l Summations over topologies, combinations of gauge bosons (W ±,Z, γ and external legs (i, j, k, l = 1,..., n using global gauge invariance and group-theoretical identities

44 Result for fermionic processes f 1 f 2 f 3... f n 2-loop NLL corrections M (2 = ( α 4π 2 [ 1 2 ( F em 2 + F em F sew (F sew 2 + G sew ] M (0 can be expressed in terms of 1-loop NLL corrections, L = log(s/ F em = 1 2 j i I A i IA j F sew = 1 2 j i a=a,z,± { L L3 ǫ 1 4 L4 ǫ 2 + [ ( 2ǫ 2 3ǫ 1 + 2ǫ 1 rij log s Iā i Ia j ] K(ǫ, M W ; r ij [ ( 3 2 log rij s }{{} K(ǫ, M W ; r ij ] (2L + L 2 ǫ + 13 L3 ǫ 2 } F i j and 1-loop β-function coeff. 1-loop functions G sew = 1 2 n i=1 [ ( 2 ] b (1 Yi 1 g2 1 + b ( g2 2 C F,i ǫ [ K(2ǫ, M W, s ( s ǫ K(ǫ, M W, s] µ 2

45 Result for fermionic processes f 1 f 2 f 3... f n Two-loop results consistent with double factorization and exponentiation [( α ] M (0 + M (1 + M (2 exp F 4π em exp [ ( α 4π F sew + ( α 4π 2 Gsew ] M (0 electromagnetic singularities factorize and exponentiate separately remaining part as within a symmetric SU(2 U(1 theory with M γ = M W = M Z in agreement with IREE prescription [Kühn et. al. (2000, Melles (2003]

46 Summary Large EW logs for gauge boson production with high p T at LHC (1 WW, WZ, ZZ, Wγ, Zγ (p T 500 GeV 1-loop logs: 15% to 30% non-log terms sizable for Wγ, Zγ ( 6% unknown for WW, WZ, ZZ Recent progress at 2-loop level (3 f f f f all 2-loop logs from ln 4 to ln 1 included large cancellations between leading and subleading terms complete two-loop smaller than 1% (2 Z + jet and γ + jet (p T 1TeV 1-loop logs: 26% (Z and 12.5% (γ non-log terms small (1% 2-loop NLLs significant: +4% (Z and +1.5% (γ (4 towards NLLs for arbitrary processes diagrammatic with EW Feynman rules all aspects of symmetry breaking collinear Ward identities, automatic algorithm for loop calculations first results for fermionic processes