Calculations for future precision physics at high energies

Transcription

1 Calculations for future precision physics at high energies F. JEGERLEHNER DESY Zeuthen Humboldt-Universität zu Berlin Report on work in Collaboration with M. Kalmykov Seminar, Sep 8, 2004, JINR, Dubna supported by DFG SFB/TR 09

2 Outline of Talk: 1 Perspectives for future precision physics at the ILC 2 Theoretical requirements and challenges 3 State of the art in 2 2 processes 4 The role of α(e) in precision physics 5 Steps towards full two-loop SM calculations 6 Outlook

3 ❶ Perspectives for future precision physics at ILC Giga-Z and Mega-W Precision Observables Most interesting for electroweak precision physics: new scan at Z Z s higher statistics: model independent scan with 5 scan points! plus threshold scan of W W threshold region W pairs (single diagram t channel dominated) mandatory large P e (> 80%) plus either δp e < % or P e +(> 20%) good control of beam-strahlung and energy-spread Remarks: beam-strahlung: Γ Z Γ Z + 60MeV energy-spread: σhad 0 σ0 had 1.8%

4 Most important: a f ρ f v f /a f sin 2 Θ f eff M W sin 2 Θ W Γ had α S (M Z ) Γ b, A b FB ρ b, sin 2 Θ b eff Observable LEP GigaZ σ 0 had = 12π M 2 Z M Z ± ± Γ Z Γ e Γ had Γ 2 Z R l = Γ had Γ l α s (M Z ) ± ± ρ l (0.55 ± 0.10) 10 2 ± N ν ± ±0.004 improved Bhabha required improving selection efficiencies for µ, τ and hadrons by factor 3 assumed

5 most promising: sin 2 Θ l eff via A LR A LR = 1 P σ L σ R σ L + σ R = 2v la l v 2 l + a2 l sin 2 Θ l eff = (1 v e /a l ) 1 4 ratio of total cross sections independent on final state need extremely good control of polarization and others Blondel scheme A LR 10 4 sin 2 Θ l eff ! factor of 13 improvement vs. LEP/SLC Expected improvement factors sin 2 Θ l eff 13 R b 5 A b 15

6 M W threshold scan dominated by one diagram the t channel ν exchange no fragmentation no color recombination etc. More general ILC physics: δm W 6MeV possible big challenge for theory! up to real γ radiation + background e + e virtual particles }{{} Z, W W, γz, ZH }{{} unstable virtual particles }{{} 2f, 2fγ, 4f, 4fγ, }{{} leptons, neutrinos incl. QCD GB production particles??? incl. QCD GB decay quarks, gluons jets of hadrons Monte Carlo event generator

7 ❷ Theoretical requirements and challenges Typical progress required: threshold scan of W W threshold region (single diagram t channel dominated) W pairs MegaW δm W 6MeV possible go from 2 2 LEP 1 to 2 4(6) processes LEP 2/LC (plus γ radiation) at 0.2% level or better new 2 2 scan at Z (A LR,...) Z s GigaZ sin 2 Θ l eff ! higher statistics: model independent scan with 5 scan points! go to next order for 2 2 processes increase precision by factor 10 relative to LEP 1 (plus γ radiation)

8 Note: complexity of calculations grows factorial (a battle against combinatorics) SM in renormalizable gauge without fermion doublet multiplicity 11 types of lines, 57 types of vertices! more loops and/or more legs become a true challenge SUSY extensions even much more complex!

9 In addition: the many mass scale complexity What complicates matters dramatically are the many different mass scales which can be very large and lead to extremely complicated analytic structure of the amplitudes and to dramatic numerical cancellation problems a. a For 1 loop 2 to 2 processes M. Veltman wrote a CDC assembler routine FORMF (which essentially did what FF is doing today in a different way in standard double precision FORTRAN) at 128 digits internal precision which allowed him to guarantee 8 digits final precision in such calculations. Alternatively one has to do the cancellations analytically by appropriate series expansions, usually a tedious task. Examples: W + W -production, ZH-production, t t-production (while f f -production is much easier for the light fermions) Compare QCD : 3 types of lines, 4 types of vertices only mass scales α s Λ QCD, m c, m b, m t except for top production calculations QCD calculations are mostly done in the massless approximation

10 the high precision challenge: GigaZ: 2 2 full 2 loop + leading 3 loop Altogether: new technical challenge signal 3 + background going from a few 100 to several diagrams! Example: # of diagrams at tree level W W production and decay: e + e µ ν µ τ + ν τ 6. e + e e ν e e + ν e 53.

11 Other examples: tree unitary gauge, loops linear R ξ gauge process # diagrams loop level e + e 4f [4fγ] 6 to 144 [14 to 1008] tree (1) e + e µ ν µ u d [e + e d d] 1907 [8522] 1 loop e + e e + e e + e loop V V (V = W, Z) 50 1 loop W W 4084 (792 1pi) 2 loop Z Z 2348 (616 1pi 2 loop } (2) proliferation in number of final state channels: 81 essentially different channels in e + e 4f a a e + e u dµ ν µ, u dτ ν τ, c sµ ν µ, c sτ ν τ, t bµ ν µ, t bτ ν τ, c sdū, t bdū, t bs c, ν τ τ + µ ν µ, u ddū, c ss c, t bb t, ν µ µ + µ ν µ, ν τ τ + τ ν τ, ν e e + e ν e, u de ν e, c se ν e, t be ν e, ν µ µ + e ν e, ν τ τ + e ν e, µ + µ τ + τ, µ + µ ν τ ν τ, τ + τ ν µ ν µ, ν τ ν τ ν µ ν µ, µ + µ e + e, τ + τ e + e, µ + µ ν e ν e, τ + τ ν e ν e, ν µ ν µ ν e ν e, ν µ ν µ e + e, µ + µ µ + µ, τ + τ τ + τ, ν µ ν µ ν µ ν µ, e + e e + e, ν e ν e ν e ν e, ūuµ + µ, ūuτ + τ, ūu ν µ ν µ, ccµ + µ, ccτ + τ, cc ν µ ν µ, ttµ + µ, ttτ + τ, tt ν µ ν µ, ddµ + µ, ddτ + τ, dd ν µ ν µ, ssµ + µ, ssτ + τ, ss ν µ ν µ, bbµ + µ, bbτ + τ, bb ν µ ν µ, ūue + e, cce + e, tte + e, dde + e, sse + e, bbe + e, ūu ν e ν e, cc ν e ν e, tt ν e ν e, dd ν e ν e, ss ν e ν e, bb ν e ν e, ūu ss, ūu bb, cc dd, cc bb, tt dd, tt ss, dd ss, dd bb, ūu cc, ūu tt, ūuūu, cc cc, dd dd, ss ss, bb bb. (1) F.J, K. Kołodziej (2) F.J, M. Kalmykov, O. Veretin

12 ❸ State of the art in electroweak 2 2 processes Complete 2 2 at 1 loop calculations have been performed long time ago. (Veltman, Passarino 79, Sirlin, Marciano, 79,...) Today largely automatized, standard procedures (Böhm et al. [Feynarts/Feyncalc, FF/Looptools], Tanaka et al. [GRACE/BASES], Bardin et al. [SANC], Passarino et al. ). At the 2 loop level a substantial set of leading corrections have been calculated (heavy top, heavy Higgs, QCD corrections, QED-corrections). For no channel of 2f 2f a complete 2 loop calculation exists! This, in spite of the fact, that such a calculation would have been very important for the interpretation of the Higgs mass bound from LEP/SLC. Most SM calculations concern ρ, r, Γ(Z b b), etc. Typically: leading effects, more recently complete for µ- decay rate (static limit) O((G µ m 2 t ) 2 ) (van der Bij, Hoogeveen, 87, Consoli et al., 89) O(G µ m 2 t α s ) (Fleischer et al. 93, Avdeev et al. 94, Chetyrkin et al. 95)

13 O(G 2 µm 2 t M 2 Z ) (Degrassi, Gambino, Vicini, 96) O(α 2 ) in α and for µ decay (van Ritbergen, Stuart, 99) 2 loop bosonic corrections to Z and W self energies { pole mass vs. MS mass } (F.J, Kalmykov, Veretin 01) 2 loop fermion contributions to µ decay ( r) (Freitas, Hollik, Walter, Weiglein, 02) 2 loop bosonic contributions to µ decay ( r) (Awramik&Czakon, Onishchenko&Veretin, 02) 2 loop fermionic corrections to Z and W self energies { pole mass vs. MS mass } (F.J, Kalmykov, Veretin 02) complete 2 loop to r M W (Awramik, Czakon, Freitas, Weiglein 03)

14 ρ in limit of zero mass gauge bosons O(αα s ), O(α 2 ) and leading 3 loop (Faisst, Kühn, Seidensticker, Veretin 03) top-quark propagator { on-shell vs. MS scheme, RG in broken phase } O(α 2 s) (F.J., Tarasov, Veretin 98) and O(αα s ) (F.J, Kalmykov 03) full 2 loop fermion contributions to sin 2 Θ lept eff M H bound (Awramik, Czakon, Freitas, Weiglein, 04) Results: Full set of SM 2 loop counterterms: α, masses, wave function renormalization CT s M W prediction in terms of α, G µ and M Z complete 2 loop up to O(m 2 µ/mw 2 ) sin 2 Θ lept eff prediction in terms of α, G µ and M Z full 2 loop fermionic corrections Still waiting for complete 2 loop 2f 2f (e + e -annihilation or scattering)

15 First complete two loop calculation of 1 3: Fermi constant G µ in terms of α M Z and M W (low energy expansion excellent approximation): (Awramik&Czakon 02, Onishchenko&Veretin 02) (Awramik, Czakon, Freitas & Weiglein 03) m H (GeV) r (α) r (αα s) r (αα2 s ) r (α2 ) ferm r(α2 ) bos r (G2 F α sm 4 t ) r (G3 F m6 t ) Table 1: The numerical values ( 10 4 ) of the different contributions to r specified in the table are given for different values of m H and M W = GeV (the W and Z masses have been transformed so as to correspond to the real part of the complex pole).

16 First full two loop fermionic calculation of sin 2 θ lept eff : Effective weak mixing parameter sin 2 θ lept eff in terms of α M Z and M W : (Awramik, Czakon, Freitas & Weiglein 04) Table 2: Difference to previous approximate result including terms of O(α 2 m 2 t ( ) ( ) M H sin 2 θ lept eff sin 2 θ lept eff ZFITTER GeV [?] ) from (Degrassi et al. 97). Implemented in the most recent version of ZFITTER, version 6.40.

17 Most accurate electroweak precision predictions Calculate sin 2 Θ i from α, G µ and M Z : 2 Gµ M 2 Z sin2 Θ i cos 2 Θ i = πα 1 1 r i Sirlin 1980 Various definitions of sin 2 Θ i, which coincide at tree level: From weak gauge boson masses, from electroweak gauge couplings and from the neutral current coupling of the charged fermions: sin 2 Θ W = 1 M W 2 MZ 2 sin 2 Θ g = e 2 /g 2 sin 2 Θ f = 1 4 Q f ( 1 v ) f a f, f ν r i = r i (α, G µ, M Z, m H, m f t, m t ) = α f i (sin 2 Θ i ) ρ + r i rem universal predictions for M W, A LR, A f F B, Γ f, δ α δm W, δ sin 2 Θ i,

18 ❹ The role of α em (s) in precision physics Uncertainties of hadronic contributions to effective α are a problem for electroweak precision physics: α, G µ, M Z most precise input parameters partially non-perturbative relationship precision predictions sin 2 Θ f, v f, a f, M W, Γ Z, Γ W, α(m Z ), G µ, M Z best effective input parameters for VB physics (Z,W) etc. δα α δg µ G µ δm Z M Z δα(m Z ) α(m Z ) (present) δα(m Z ) α(m Z ) (ILC requirement) LEP/SLD: sin 2 Θ eff = (1 g V l /g Al )/4 = ± δ α(m Z ) = δ sin 2 Θ eff = affects Higgs mass bounds!!! For perturbative QCD contributions very crucial: precise QCD parameters α s, m c, m b, m t Lattice-QCD

19 Evaluation of α(m Z ) Non-perturbative hadronic contributions α (5) had (s) can be evaluated in terms of σ(e + e hadrons) data via dispersion integral: α (5) had (s) = αs 3π where ( P E 2 cut 4m 2 π + P E 2 cut γ (s ) s (s s) ds Rdata γ (s ) s (s s) ds RpQCD R γ (s) σ(0) (e + e γ hadrons) 4πα 2 3s ) R Compilation: Davier, Eidelman et al. 02 Theory = pqcd: Groshny et al. 91, Chetyrkin et al ω Φ e + e J/ψ 1S ψ 2S hadrons ψ 3770 QCD exclusive data s (GeV) ϒ 1S ϒ 2S 3S 4S BES γγ2 ϒ Crystal B. PLUTO γ had Π γ(q 2 ) γ γ σtot had (q 2 ) had 2 R 3 2 e + e hadrons PLUTO MD1 QCD LENA JADE 1 Crystal B. MARK J ϒ F. Jegerlehner Seminar, JINR, Dubna Sep 8, 2004 s (GeV)

20 Evaluation FJ 2003 update: at M Z = GeV R(s) data up to s = E cut = 5 GeV and for Υ resonances region between 9.6 and 13 GeV perturbative QCD from 5.0 to 9.6 GeV and for the high energy tail above 13 GeV α (5) hadrons (M Z 2 ) = ± ± Adler ± BP 03 α 1 (MZ 2 ) = ± ± Adler ± BP 03

21 Indirect Higgs boson mass measurement m H = GeV e + e τ : δm H 19 GeV Direct lower bound: m H > 114 GeV at 95% CL Indirect upper bound: m H < 260 GeV at 95% CL H W, Z W, Z H W, Z W, Z Final A 0,l fb ± A l (SLD) ± A l (P τ ) ± Q had fb ± Preliminary A 0,b fb ± A 0,c fb ± Average ± m H [GeV] χ 2 /d.o.f.: 9.7 / 5 α (5) had = ± m t = ± 4.3 GeV m t = 174 ± ± 4.3 GeV δm H = +20 GeV new: full 2-loop electroweak fermion contribution to sin 2 Θ lept eff sin 2 θ lept eff δm H = +19 GeV M. Awramik et al (LEP Electroweak Working Group: D. Abbaneo et al. 04)

22 5 Steps towards full two-loop SM calculations Aim: so far little feeling for size of corrections from bosonic sector. Very complex: electroweak SM: 57 vertices, 11 types of lines (fermions as one fermion doublet) multiple factorial growth of complexity; very different mass scales! QED and QCD on electroweak processes: limited number of diagrams relatively small number of diagrams involving top or physical Higgs full gauge boson sector (incl. Higgs- and Fadeev-Popov ghosts) large number of diagrams Steps of technical complications: self energies form factors boxes Complete calculations of observables available so far only for µ decay Awramik&Czakon, Onishchenko&Veretin Full two loop renormalization program: need full set of counter terms. e.g., on-shell renormalization scheme α, M Z and M W as basic parameters (QED like scheme) calculate gauge boson mass counter-terms (equiv. MS vs. pole mass relation) Theoretical issue: About the proper definition of masses of unstable particles (Stuart, 91, Sirlin, 91,..., Kniehl, Sirlin, 98)

23 The pole mass of the weak gauge bosons (at two loops) The mass and width of a massive gauge boson V are defined via the position s P of the pole of the full propagator (=zero of its inverse) s P m 2 V Π V (s P, m 2 V, ) = 0, Π V (p 2, ) transversal part of the one-particle irreducible self-energy (depends on all SM parameters) bare amplitude in terms of bare parameters (m V m V,0, Π V Π V,0 ) renormalized amplitude in terms of renormalized parameters, e.g., MS (no index) Properties of the pole: gauge invariant infrared finite complex in general Defines pole (on shell) mass M and width Γ via s P M 2 imγ.

24 Renormalization The renormalized amplitudes Π V,r (p 2, m 2 V,r, ) = Π(1) V,r (p2, m 2 V,r, ) + Π(2) V,r (p2, m 2 V,r, ) + to two-loops read (indices: 0=bare, r=renormalized) Π (1) V,r (p2, m 2 V,r, ) = 2 3 4Π (1) V,0 (p2, m 2 V,0, ) + (δm 2 V ) (1) (p 2 m 2 V,r) δz (1) 5 V m2 j,0 = m 2 j,r Π (2) V,r (p2, m 2 V,r, ) = + X j (δm 2 j) (1) m 2 j,0 2 e 0 = er! 4Π (2) V,0 (p2, m 2 V,0, ) + Π (1) V,0 (p2, m 2 V,0, ) + (δm 2 V ) (1) δz (1) V 3 Π (1) V,0 + (δe)(1) Π (1) V,0 e + (δm2 V ) (2) (p 2 m 2 V,r) δz (2) 5 V 0 m2 j,0 = m 2 j,r e 0 = er where in the MS scheme order by order the mass-counter-term (δm 2 V ) (j) subtracts the ɛ poles at p 2 = m 2 V,r and the wave-function renormalization counter-term δz (j) V subtracts the ɛ poles remaining when p2 m 2 V,r Strictly speaking the renormalization of the ghost sector (in particular of the gauge parameter) is not discussed here, because its not needed for what follows. In case of the Z the γ Z-mixing is an additional complication (see below).

25 Pole mass and γ Z-mixing In the neutral gauge boson sector because of γ Z-mixing we have to consider a 2 2 matrix propagator D 1 (p 2 ) = ( p 2 Π γγ (p 2 ) Π γz (p 2 ) ) Π Zγ (p 2 ) p 2 m 2 Z Π ZZ (p 2 ) Position of Z-pole: s P m 2 Z Π ZZ(s P ) Π2 γz (s P ) s P Π γγ (s P ) = 0. Mixing term Π 2 γz starts contributing at two-loops Photon term Π γγ only contributes beyond two-loop Notation for self-energies Π V (V = W, Z) with Π W (p 2, ) = Π W W (p 2, ); Π Z (p 2, ) = Π ZZ (p 2, ) + Π2 γz (p2, ) p 2 Π γγ (p 2, ). Formally, same formulae apply for W and Z.

26 Pole mass master formula By iterative solution of the pole formula to two-loops we obtain our master formula: s P = m 2 +Π (1) (m 2, m 2, ) +Π (2) (m 2, m 2, ) + Π (1) (m 2, m 2, ) Π (1) (m 2, m 2, ) + which yields the pole mass M 2 and the width Γ at this order. Π (L) is the bare (m = m 0 ) or MS -renormalized (m the MS -mass) L-loop contribution to Π, and the prime denotes the derivative with respect to p 2. In this way we need to evaluate propagator type diagrams and their derivatives at p 2 = m 2. Note: the p 2 dependence has disappeared in this solution; it turned into a mass dependence which cannot be disentangled from the original mass dependence of the off-shell amplitude. Remark: the mixed Π (1) (m 2, m 2, ) Π (1) (m 2, m 2, ) term is crucial for getting a gauge invariant result for the two loop mass counter term

27 Diagrams and topologies To be computed on-shell (p 2 = m 2 V ): Π(p2 ) = Π 1 (p 2 ) + Π 2 (p 2 ) + Π (1) = + + H (2) Π = H + + H H + H H H H + H H H H H H H H

28 Bosonic contribution Number of diagrams linearr ξ gauge nonlinearr ξ gauge one-loop : 50 two-loop : 1P I T otal 1P I T otal Z W With one massive fermion family two-loop : 1P I T otal 1P I T otal Z W Approach: QGRAPH DIANA FORM MAPLE high precision numerics required! To get numerically stable results it is necessary to work on MAPLE with sufficiently high accuracy (our experience: we get an accuracy of 40 decimals) (when calculating with 100 decimals). The i0 causal prescription is introduced in program as small number Calculations in R ξ gauge and in non-linear gauge (Fujikawa 73, Dicus & Kao 94)

29 Evaluation of 2 loop self energies There exist a number of programs, which calculate the bubble diagrams (analytical), obtained by low energy expansions, but also arbitrary self-energy diagrams (analytical and 1-dimensional integral-representations): a class of massive 2 loop integrals (Broadhurst 90, Fleischer, Kalmykov, Kotikov 99), which depend on one scale only have been implemented in ONSHELL2 (Fleischer, Tarasov 92, Fleischer, Kalmykov 00), in general exact analytic results are not known and one has to resort to series expansions at low or high energies (Broadhurst, Fleischer, Tarasov, 93, Fleischer, Tarasov, 94, Fleischer, Kalmykov, Veretin, 98), which may be combined with methods of conformal mapping and Padé resummation, a combined analytical-numerical program for 2 loop self-energy functions has been developed by the W urzburg/leiden Collaboration (Bauberger, Berends, Böhm, Buza, 95, Bauberger, Böhm, 95 ), for the reduction of integrals to a basis of standard-integrals there exist packages which solve the systems of recurrence-relations (Tarasov 97).

30 Utilizing relations between integrals in different dimensions D the problem of irreducible numerators could be solved (Tarasov 96b) For integrals showing up in a large mass expansion the package TLAMM (Avdeev et al., 97) is available. Various expansions with respect to small parameters may by utilized (Smirnov, 90, 95, 99, 01)

31 Evaluation by expansion Check of gauge invariance: R ξ gauge (independent gauge parameters ξ W, ξ Z and ξ γ ) Then there are several scales: m W, ξ W m W, m Z, ξ Z m Z, m H We perform expansions in 3 steps: 1. Taylor (naive) expansion in (ξ V 1): i.e., propagators of the vector bosons and associated Higgs scalar ghosts look like where V = W, Z. D V µν(p) = V (p) = i p 2 m 2 V i p 2 m 2 V 2. Expansion in the small parameter p µ p ν g µν + (1 ξ V ) p 2 m 2 V 0 m 2 (1 ξ V ) p 2 m 2 V sin 2 Θ W = 1 m2 W m 2 Z (1 ξ V ) 2 + (1 ξ V ) 2 m 2 V p 2 m 2 V < 0.24 m 2 V p! µp ν (p 2 m V )2! by which m 2 W = m2 Z (1 sin2 Θ W ); no Higgs diagrams then are one scale and can be calculated analytically with the ONSHELL2 package. 3. Diagrams with Higgs lines are expanded for large m H in using the TLAMM package z m 2 Z /m2 H < 0.64 A

32 Gauge invariance As we know, resonant Z and W bosons decay mainly into fermion pairs. Indeed, if we switch off the fermions (as we do here) the gauge bosons are close to stable! For the purely bosonic contributions alone the imaginary part of Π(p 2 ) on the mass-shell is zero at the two-loop level. This is due to the fact that in the bosonic sector we have the physical masses m γ = 0, M Z, M W and M H and by inspection of the possible two and three particle intermediate states one observes that all physical thresholds lie above the mass shells of the W and Z bosons, i.e., the self-energies of the massive gauge bosons develop an imaginary part only at p 2 > M 2 V (to two loops in the SM). On kinematical grounds imaginary parts could show up from the Higgs or Faddeev-Popov ghosts, which have square masses ξ V M 2 V, for small values of the gauge parameter. However, as we have verified, the two-loop on-shell self-energies are gauge independent. This implies that ghost contributions have to cancel and hence cannot contribute to the imaginary part. Thus s P = MV 2 part as soon as p 2 > 0, from diagrams like in our case. In higher orders for the Z propagator one gets an imaginary For the W propagator an imaginary part is only possible for p 2 > M 2 W requires at least one W in any physical intermediate state. Z W γ. γ W Z., because charge conservation Drawback of our choice of expansion about ξ i = 1: analytic structure (ghost thresholds) lost: do not get correct imaginary part from ghost contributions!

33 Examples: 1.) threshold p 2 = 4ξ Z M 2 Z of Z φ 0 φ 0 production, φ 0 the neutral Higgs ghost which is below the Z mass-shell p 2 = MZ 2 when ξ Z < ) threshold p 2 = ξ W MW 2 of W ± φ ± γ production, φ ± the charged Higgs ghosts which is below the W mass-shell p 2 = M 2 W when ξ W < 1. Thus for small values of ξ we do not get correct imaginary part diagram by diagram. However, ghost contributions must cancel on-shell. Thus by gauge invariance, which we check we know that we get the correct result. At one loop one may check this analytically. (Fleischer, Jegerlehner, 81) Cancellation is highly non-trivial: a consequence of the Slavnov-Taylor identities, which tell us how Higgs ghost, Faddeev-Popov ghosts and scalar components contained in the gauge boson fields decouple from physical amplitudes like the physical width. For gauge parameters ξ > 1 the imaginary part of the W and Z self-energies in the bosonic sector up to two-loops is zero, by applying the Cutkowsky rules and inspecting all possible two and three particle intermediate states allowed by the SM Lagrangian. While for ξ > 1 the imaginary part is zero for each individual diagram, for small enough values of the gauge parameters a nontrivial cancellation must take place. An independent direct check of this is possible by considering the problem in the limit ξ 0, for example.

34 UV renormalization is exact (analytic); Results UV singularities (poles 1/ε 2 and 1/ε) are not affected by SSB; check against RG results within unbroken theory (Jones, 82, Machacek, Vaughn, 83) confirms IR finiteness of on shell mass for both Z and W gauge invariance of position of pole s P ; requires taking into account tadpoles large m H expansion breaks down at large m H because of strong coupling problem (gets non-perturbative) relation between MS and pole mass exhibits unphysical terms proportional to m 4 H, which violate Veltman s screening theorem: in observables at L loops: X(m H ) O((G µ m 2 H) L 1 ln(m H /M W ) 2 ) as m H The fake terms drop in sin 2 Θ W = 1 M 2 W M 2 Z as they should. behavior for intermediate Higgs masses: looks O.K. down to about 130 GeV complete 2 loop calculation of fermionic corrections incl. QCD complete one of the main ingredients of full 2-loop corrections to µ decay: Awramik&Czakon, Onishchenko&Veretin

35 Form of results M 2 V m 2 V ( e 2 = π 2 sin 2 θ W ) X (1) V + ( e 2 16π 2 sin 2 θ W ) 2 X (2) V, X (2) V = m4 H m 4 V A V i = 5 sin 2k θ W A V k. k=0 5 j=0 A V i,j ( m 2 V m 2 H ) j All parameters in MS scheme. Six coefficients calculated analytically. Expansion in powers and log s (i.e., is an asymptotic expansion not a naive Taylor expansion). Expansion coefficients A i,j given by a small set of transcendental constants like: S 0 = π , 3 S 1 = π ln , 3 S 2 = 4 ( Cl π ) , 9 3 ( S 3 = πcl π )

36 MS mass in terms of on shell mass Inverse of master formula : express all MS parameters in terms of on-shell ones: m 2 V = M 2 V j ˆΠ (1) V { ( m 2 j) (1) Π (2) V m 2 j + Π(1) V Π(1) V } MS ˆΠ (1) V ( e)(1) e ˆΠ (1) V m 2 j =M 2 j, e=e OS, sum runs over all species of particles j = Z, W, H ( m 2 j) (1) = ReˆΠ (1) j m 2 j =M 2 j, e=e OS M 2 V e 2 OS 16π 2 sin 2 X (1) V θ W m 2 j =M 2 j stands for the self-energy of the jth particle at p 2 = m 2 j in the MS scheme and parameters replaced by the on-shell ones. Includes a change from the MS to the on-shell scheme also for the electric charge e(µ 2 ) = e OS [ 1 + e2 OS 16π 2 ( 7 2 ln ( M 2 W µ 2 ) 1 )] 3 with e 2 OS /4π = α 1/137 ˆΠ (1) depends on e by an overall factor e 2 only, ( ) ] ( e) (1) (1) e ˆΠ V = e2 M 2 16π [7 ln 2 W µ 2 ˆΠ(1) 2 3 V

37 Identifying m 2 V = m2 V,0 = M V 2 + δm V 2 δmv 2 : [ δm 2 V = Re On-shell scheme mass counter-terms ˇΠ (1) V,0 in inverse MF on-shell gauge-boson mass counter-terms + ˇΠ (2) V,0 (1) (1) + ˇΠ V,0 ˇΠ V,0 + j (δm 2 j ) (1) m 2 j,0 ˇΠ (1) V,0 + (δe)(1) e V,0] ˇΠ(1) 0 m 2 j,0 = Mj 2 = ( ) Z MS Z OS 1 M 2 V e 0 = e OS in terms of the original bare on-shell amplitudes ˇΠ (i) V,0 = Π(i) V,0 (p2, m 2 V,0, ) p 2 =M 2 V,m2 j,0 =M 2 j, e 0=e OS and the bare on-shell counter-terms δm 2 j and δe. The second equality gives δmv 2 in terms of the singular factor Z MS = m2 V,0 /m2 V Z OS = m 2 V /M V 2 (µ) and the finite factor. These will be needed in two-loop calculations of observables in the on-shell scheme. Explicit expressions in (Jegerlehner, Kalmykov, Veretin, 01)

38 The quark pole mass The tensor decomposition of the one particle irreducible self energy of a massive fermion Σ(p, m,...) has the form ] Σ(p, m,...) = iˆp [Ã(p 2, m,...) γ 5 C(p 2, m,...) [ ] +m B(p 2, m,...) γ 5 D(p 2, m,...) Ã, B, C, D Lorentz scalar functions depending on all parameters of the SM. At O(αα s ) C = D = 0. The position of the pole M is given the zero of the inverse of the connected full propagator. By iterative solution we have up to 2-loops: M m = 1 + Σ 1 + Σ 2 + Σ 1 Σ 1 Σ L is the bare (m = m 0 ) or MS - renormalized (m the MS -mass) L-loop contribution to fermion self-energy, the prime denotes the derivative with respect to p and Σ(p, m,...) = Σ ˆp=im0 + (iˆp + m 0 ) [ Σ ] ˆp=im0, +

39 and define dimensionless on shell amplitudes Σ, Σ by and [ Σ ] ˆp=im0 = Σ ˆp=im0 = [( Σ (iˆp) [ m 0 Ã + m 0 B )] ˆp=im0 = ] p2= m2 0 [Ã + 2p 2 Ã + 2m 2 0 m 0 Σ(m 0,...) ] B p2= m2 0 where Ẋ(p2,...) denotes the derivative of X(p 2,...) with respect to p 2. Σ (m 0,...) In this way we need to evaluate propagator type diagrams and their derivatives at p 2 = m 2

40 What is the interpretation of the complex mass M M i 2 Γ M. C. Smith and S. S. Willenbrock, Phys. Rev. Lett We define the pole mass M and the on shell width Γ as in the bosonic case by (look at T 2 ) M 2 = M 2 imγ = M 2 Γ 2 /4 im Γ such that M = M 2 Γ 2 /4 ; Γ = M M Γ Since M = M + O(α 2 ) and Γ = Γ + O(α 2 ) for the O(αα s ) terms considered in this paper we can identify M = M and Γ = Γ in the following.

41 To be computed on-shell (p 2 = m 2 t ): Diagrams and topologies H γ Z, W The two-loop one-particle irreducible diagrams contributing to the pole mass of a quark. φ 0 is the neutral pseudo-goldstone boson and φ is charge pseudo-goldstone boson.

42 Tadpoles g t t H t H g The two-loop tadpole diagrams to be included for gauge and renormalization group invariance.

43 Reduction to a set of master-integrals In order to check gauge invariance we perform all calculations in the R ξ gauge with three independent gauge parameters ξ W, ξ Z, ξ γ. Using Tarasov s recurrence relations we reduce all diagrams to a minimal set of master-integrals M α λ m m β σ β m α M σ m m M J012 V0012 (λ,α,β,σ) V1112 New master diagrams appearing in this two-loop calculation. Bold, thin and dashed lines correspond to off-shell massive, on-shell massive and to massless propagator, respectively.

44 Numerical illustration Electroweak O(αα s ) correction to M t /m t (m t ) 1 [left] and m t (M t )/M t 1 [right], in comparison with O(α 2 s) and O(α 3 s) QCD corrections as a function of the Higgs boson mass M H.

45 Connection with RG functions of unbroken phase In the SM it is interesting to compare the RG equations calculated in broken phase with the ones obtained in the unbroken phase. Let us remind that at the tree-level the vacuum expectation value v 2 is given by v 2 m2 λ, where m 2 and λ are the parameters of the symmetric scalar potential V = m 2 φ + φ + λ (φ + φ) 2 γ W 2 β g g = γ m 2 β λ γ Z γ m 2 + β λ λ = 2 λ, ( cos 2 β g θ W g + β sin2 g θ W g where the 2-loop RG functions β g, β g, β λ, γ m 2 have been calculated in the unbroken phase (Jones 82, Machacek & Vaughn 83, Ford, Jack & Jones 92.) We have verified in the MS scheme, that these relations are valid up to 2-loop order in the broken phase with the same RG functions. Thus the RG equations for the MS masses in the broken theory can be written m 2 W (µ 2 ) = 1 4 m 2 Z(µ 2 ) = 1 4 m 2 H (µ2 ) = 2m 2 (µ 2 ), m 2 t (µ2 ) = 1 2 g 2 (µ 2 ) λ(µ 2 ) m2 (µ 2 ), g 2 (µ 2 ) + g (µ 2 ) λ(µ 2 m 2 (µ 2 ), ) y 2 t (µ2 ) λ(µ 2 ) m2 (µ 2 ). ),

46 Together with δe e = Z 1 e = Charge renormalization: { (Z γγ ) 1/2 + s0 W c 0 W } 1 2 Z Zγ. Only Π γγ (0) and Π Zγ (0) needed (bubble diagrams) Degrassi& Vicini 03 Note: in on-shell scheme all renormalization counter terms involve self-energy diagrams only! (this also applies to non-physical sector) [not needed for S-matrix elements] ingredients for two loop renormalization available except for two-loop higher point functions need also O(ε) of one loop higher point functions generic d-dimensional solution for one-loop integrals known (J. Fleischer, F.J., O. Tarasov 03)

47 ❻ Outlook Precise test of the limitations of the SM require a LC like the ECFA/DESY TESLA (TEV Superconducting Linear Accelerator) machine Such accelerators can be built now! ICFA recommendation: world wide effort to built ILC (International Linear Collider) based on cold technology Physics: Higgs properties, Higgs couplings SUSY spectrum, SUSY Higgs, scenarios exploring the extra dimensions many other possibilities Theory: Extraordinary challenge: must learn to handle precise and efficient computations of s of diagrams.

48 computer algebra; automatization of calculations numerical methods; multi precision calculations computing resources; massive parallel computers

49 ECFA/DESY LC working group: The Loop Verein similar American initiative: U. Bauer, D. Wackeroth, S. Dawson (LoopFests I,II at BNL, III, April 1-3, 2004, Santa Barbara) Aim: developing new tools and techniques for calculating precise rates for Standard Model and supersymmetric processes that match the expected experimental precision. Motivation field which got a lot of momentum from precision physics at LEP/SLC we must keep momentum and improve calculations to meet ILC requirements of HIGHER ENERGY and HIGHER LUMINOSITY Coordinate activities, identify problems and their solution. Time scale: a long term project. (