SM predictions for electroweak precision observables

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1 SM predictions for electroweak precision observables A. Freitas University of Pittsburgh Workshop on Physics at the CEPC, August Introduction 2. Electroweak precision observables 3. Current status of SM loop results 4. Future projections

2 Introduction 1/21 Standard Model after Higgs discovery: Good agreement between measured mass and indirect prediction Very good agreement over large number of observables Γ Z, σ had, R l, R q (1σ) Z pole asymmetries (1σ) M W (1σ) direct m t (1σ) direct M H precision data (90%) Erler 13 Direct measurements: M H = ± 0.4 GeV m t = ± 0.95 GeV M H [GeV] Indirect prediction: M H = ± 2.3 GeV (with LHC BRs) M H = GeV (w/o LHC data) m t [GeV] m t = ± 2.1 GeV

3 Current status of electroweak precision tests 2/21 Surprisingly good agreement: χ 2 /d.o.f. = 18.1/14 (p = 20%) Most quantities measured with 1% 0.1% precision A few interesting deviations: M W σ 0 had ( 1.4σ) ( 1.5σ) A l (SLD) ( 2σ) A b FB ( 2.5σ) (g µ 2) ( > 3σ) GFitter coll. 14

4 Current status of electroweak precision tests 2/21 Surprisingly good agreement: χ 2 /d.o.f. = 18.1/14 (p = 20%) 2.5σ 1.4σ 1.5σ 2.0σ Most quantities measured with 1% 0.1% precision A few interesting deviations: M W σ 0 had ( 1.4σ) ( 1.5σ) A l (SLD) ( 2σ) A b FB ( 2.5σ) (g µ 2) ( > 3σ) GFitter coll. 14

5 Electroweak precision observables 3/21 W mass µ decay in Fermi Model µ decay in Standard Model µ G F ν µ ν e e µ W ν µ ν e µ Z ν µ ν e ν µ e W µ γ e ν e QED corr. (2-loop) G 2 F 2 = e2 8s 2 (1 + r) w M2 W e Γ µ = G2 F m5 µ 192π 3 F ( m 2 ) e (1 + q) m 2 µ electroweak corrections Ritbergen, Stuart 98 Pak, Czarnecki 08

6 Z-pole observables 4/21 e + e f f for s m Z : Γ ee Γ ff σ = R ini [12π (s m 2 Z )2 m 2 (1 + δx)(1 P e A e ) + σ non res ], Z Γ2 Z Γ ff = R f V g2 V f + Rf A g2 Af, Γ Z = f Γ ff, g V f /g Af A f = (g V f /g Af ) 2 = 1 4 Q f sin 2 θ 1 4 Q f sin 2 θ f eff + 8( Q f sin 2 θ f eff )2. f eff A f FB = 3 4 A ea f A LR = A e

7 Z-pole observables 5/21 e + e f f for s m Z : Γ ee Γ ff σ = R ini [12π (s m 2 Z )2 m 2 (1 + δx)(1 P e A e ) + σ non res ], Z Γ2 Z Γ ff = R f V g2 V f + Rf A g2 Af, Γ Z = f Γ ff, g V f /g Af A f = (g V f /g Af ) 2 = 1 4 Q f sin 2 θ 1 4 Q f sin 2 θ f eff + 8( Q f sin 2 θ f eff )2. f eff A f FB = 3 4 A ea f A LR = A e QED/QCD corrections on ext. fermions Chetyrkin, Kataev, Tkachov 79 Dine, Saphirstein 79 Celmaster, Gonsalves 80 Gorishnii, Kataev, Larin 88,91 Chetyrkin, Kühn 90 Surguladze, Samuel 91 Kataev 92 Chetyrkin 93 etc...

8 Z-pole observables 6/21 e + e f f for s m Z : Γ ee Γ ff σ = R ini [12π (s m 2 Z )2 m 2 (1 + δx)(1 P e A e ) + σ non res ], Z Γ2 Z Γ ff = R f V g2 V f + Rf A g2 Af, Γ Z = f Γ ff, g V f /g Af A f = (g V f /g Af ) 2 = 1 4 Q f sin 2 θ 1 4 Q f sin 2 θ f eff + 8( Q f sin 2 θ f eff )2. f eff A f FB = 3 4 A ea f A LR = A e additional initial-state QED corrections Kuraev, Fadin 85 Berends, Burgers, v. Neerven 88 Kniehl, Krawczyk, Kühn, Stuart 88 Beenakker, Berends, v. Neerven 89 Bardin et al. 89,91 Montagna, Nicrosini, Piccinini 97 etc...

9 Z-pole observables 7/21 e + e f f for s m Z : Γ ee Γ ff σ = R ini [12π (s m 2 Z )2 m 2 (1 + δx)(1 P e A e ) + σ non res ], Z Γ2 Z Γ ff = R f V g2 V f + Rf A g2 Af, Γ Z = f Γ ff, g V f /g Af A f = (g V f /g Af ) 2 = 1 4 Q f sin 2 θ 1 4 Q f sin 2 θ f eff + 8( Q f sin 2 θ f eff )2. f eff A f FB = 3 4 A ea f A LR = A e electroweak corrections Correction term first at NNLO: Grassi, Kniehl, Sirlin 01 δx (2) = (ImΣ Z(1) )2 2Γ Z M Z ImΣ Z(1) Freitas 13

10 Low-energy observables 8/21 Test of running MS weak mixing angle sin 2 θ(µ) Polarized ee, ep, ed scattering (Q W (e), Q W (p), edis) E158 05; Qweak 13; JLab Hall A 13 νn/ νn scattering NuTeV 02 e ν/ ν Z e ν/ ν e, p, N e, p, N sin 2 θ W (µ) SM published planned Q W (Cs) Q W (Ra) KVI Q W (p) JLab Q W (p) Mainz Q W (e) JLab Q W (e) SLAC edis JLab µ [GeV] NuTeV screening Tevatron LEP 1 SLD SoLID JLab antiscreening LHC Erler 14 JE 2014 Atomic parity violation (Q W ( 133 Cs)) Wood et al. 97 Guéna, Lintz, Bouchiat 05

11 Current status of SM loop results 9/21 Known corrections to r, sin 2 θ f eff, g V f, g Af : γ,z,w W W H γ,z γ,z,w W Z Complete NNLO corrections ( r, sin 2 θeff l ) Freitas, Hollik, Walter, Weiglein 00 Awramik, Czakon 02; Onishchenko, Veretin 02 Awramik, Czakon, Freitas, Weiglein 04; Awramik, Czakon, Freitas 06 Hollik, Meier, Uccirati 05,07; Degrassi, Gambino, Giardino 14 Fermionic NNLO corrections (g V f, g Af ) Czarnecki, Kühn 96 Harlander, Seidensticker, Steinhauser 98 Freitas 13,14 Partial 3/4-loop corrections to ρ/t -parameter O(α t α 2 s), O(α 2 t α s), O(α t α 3 s) Chetyrkin, Kühn, Steinhauser 95 Faisst, Kühn, Seidensticker, Veretin 03 Boughezal, Tausk, v. d. Bij 05 Schröder, Steinhauser 05; Chetyrkin et al. 06 (α t y2 t 4π ) Boughezal, Czakon 06

12 Current uncertainties 10/21 Experiment Theory error Main source M W ± MeV 4 MeV α 3, α 2 α s Γ Z ± 2.3 MeV 0.5 MeV α 2 bos, α3, α 2 α s, αα 2 s σ 0 had ± 37 pb 6 pb α 2 bos, α3, α 2 α s R b Γ b Z /Γhad Z ± α 2 bos, α3, α 2 α s sin 2 θeff l ± α 3, α 2 α s Methods for theory error estimates: Parametric factors, i. e. factors of α, N c, N f,... Geometric progression, e. g. O(α3 ) O(α 2 ) O(α2 ) O(α) Renormalization scale dependence (often underestimates error) Renormalization scheme dependence (may underestimate error)

13 Renormalization scheme dependence 11/21 Use of MS renormalization for m t reduces h.o. QCD corrections of O(α t α n s): /( loops 3GF (n+1) ρms m 2 t (n) 8 2π 2 ) ( α s π ) /( ρ OS 3GF m 2 t (n) 8 2π 2 ) ( α s π ) Djouadi, Verzegnassi 87 Kniehl ( α s π ( α s π ) ( ) α s 2 π Avdeev, Fleischer, et al. 94 Chetyrkin, Kühn, Steinhauser 95 ) ( ) α s 3 π Schröder, Steinhauser 05 Chetyrkin, Faisst, Kühn, et al. 06 Boughezal, Czakon 06 No clear pattern of this kind known for O(α n ) Only few results available that allow direct comparison e.g. Faisst, Kühn, Seidensticker, Veretin 03

14 Future projections 12/21 ILC: FCC-ee: CEPC: s MZ with 30 fb 1, s MZ with fb 1, s MZ with fb 1, s 2MW with 100 fb 1 s 2MW with fb 1 s 2MW with 100 fb 1 Current exp. ILC FCC-ee CEPC Current perturb. M W [MeV] Γ Z [MeV] R b [10 5 ] < sin 2 θeff l [10 5 ] Existing theoretical calculations adequate for LEP/SLC/LHC, but not future e + e machines!

15 Theory and parametric uncertainties 13/21 perturb. error ILC CEPC Param. error Param. error with 3-loop ILC* CEPC** M W [MeV] Γ Z [MeV] < R b [10 5 ] < 1 < 1 sin 2 θ l eff [10 5 ] Theory scenario: O(αα 2 s), O(N f α 2 α s ), O(N 2 f α2 α s ) (N n f = at least n closed fermion loops) Parametric inputs: * ILC: δm t = 100 MeV, δα s = 0.001, δm Z = 2.1 MeV **CEPC: δm t = 600 MeV, δα s = , δm Z = 0.5 MeV also: δ( α) =

16 Theory uncertainties in extraction of pseudo-observables 14/21 Subtraction of QED radiation contributions LEP EWWG 05 Known to O(α 2 ), O(α 3 L 3 ) for ISR, O(α 2 ) for FSR and O(α 2 L 2 ) for A FB (L = log s m 2 e) Berends, Burgers, v.neerven 88 Kniehl, Krawczyk, Kühn, Stuart 88 Beenakker, Berends, v.neerven 89 Skrzypek 92; Montagna, Nicrosini, Piccinini 97 O(0.1%) uncertainty on σ Z, A FB Improvement needed for ILC/FCC-ee σ had [nb] ALEPH DELPHI L3 OPAL measurements (error bars increased by factor 10) σ from fit QED corrected E cm [GeV] e + f Γ Z M Z σ 0 Subtaction of non-resonant γ-exchange, γ Z interf., box contributions, Bhabha scattering e e + γ f f see, e.g., Bardin, Grünewald, Passarino 99 O(0.01%) uncertainty within SM (improvements may be needed) Sensitivity to some NP beyond EWPO e e + e γ W W f f f

17 Non-factorizable contributions 15/21 Factorization of massive and QED/QCD FSR: Γ ff R f V g2 V f + Rf A g2 Af Z Z γ = Z γ + finite, with = Z Z Additional non-factorizable contributions, e.g. Z Z γ Known at O(αα s ) Czarnecki, Kühn 96 Harlander, Seidensticker, Steinhauser 98 Currently not known at O(α 2 ) and beyond O(0.01%) uncertainty on Γ Z, σ Z, (improvements may be needed) maybe larger for A b

18 Theory challenges 16/21 Full SM corrections at >2-loop: Large number of diagrams and tensor integrals, O(1000) O(10000) Use of computer algebra tools Many different scales (masses and ext. momenta) In general not possible analytically Numerical methods must be automizable, stable, fastly converging Need procedure for isolating divergent pieces

19 Analytic calculations 17/21 Useful for diagrams with up to two scales (e. g. M W & m t or M W & M Z ) Reduce to master integrals with integration-by-parts and Lorentz-invariance identities Chetyrkin, Tkachov 81; Gehrmann, Remiddi 00; Laporta 00;... Evaluate master integrals with differential equations or Mellin-Barnes representations Kotikov 91; Remiddi 97; Smirnov 00,01;... Current status: Single-scale problems: Zf f QED/QCD vertex corrections up to 4-loop Gorishnii, Kataev, Larin 88,91; Chetyrkin, Kühn, Kwiatkowski 96 Baikov, Chetyrkin, Kühn, Rittinger 12 Two-scale problems: Zf f electroweak 2-loop vertex diagrams with m f = 0 Awramik, Czakon, Freitas, Weiglein 04 Extendability: Possible, but much work needed

20 Asymptotic expansions 18/21 Exploit large mass ratios, e. g. M 2 Z /m2 t 1/4 Fast numerical evaluation Current status: Two-scale problems: O(αα n s) corr. to ρ, r,... Several expansion terms up to 3-loop, leading term up to 4-loop Djouadi, Verzegnassi 87; Bardin, Chizhov 88 Chetyrkin, Kühn, Steinhauser 95 Faisst, Kühn, Seidensticker, Veretin Up to three-scale problems: Zf f ew. 2-loop vertex corrections Barbieri et al. 92,93; Fleischer, Tarasov, Jegerlehner 93,95 Degrassi, Gambino, Sirlin 97; Awramik, Czakon, Freitas, Weiglein 04 Extendability: Promising, mostly limited by computing/algorithmic power

21 Direct numerical integration 19/21 General form of Feynman integral: I = 1 0 dx 1... dx n δ(1 i x i ) N(x i ) D(x i ) r+ε Can be integrated numerically (if finite) Alternatives: Integration in momentum space, Mellin-Barnes space Treatment of divergencies: Sector decomposition: Sub-divide integration space such that divergent terms factorize Binoth, Heinrich 00,03 1 x 2 0 x 1 1 Subtraction terms: Remove divergencies with simple terms that can be integrated analytically Nagy, Soper 03 Becker, Reuschle, Weinzierl 10; Freitas 12

22 Direct numerical integration 20/21 Automizable, but computing intensive Internal thresholds reduce numerical convergence Current status: Several 2-loop applications with many scales Anastasiou et al., Petriello et al., Borowka et al.,... Individual 3-loop integrals Extendability: Likely, but more work needed

23 Conclusions 21/21 Current SM predictions for electroweak precision observables under good control (compared to experimental uncertainties) LHC will provide independent results for sin 2 θ eff and M W, but overall precision not substantially improved ILC/CEPC/FCC-ee with s M Z will reduce exp. error of some EWPO by O(10) 3-loop (and maybe some 4-loop) corrections needed! Asymptotic expansion and numerical integration techniques are promising but more work needed Open questions in evaluation of theory errors, resummation and optimal choice of inputs

24 Backup slides

25 Example: Error estimation for Γ Z 21/21 Goemetric perturbative series α t = αm 2 t O(α 3 ) O(α 3 t ) O(α2 ) O(α 2 t ) O(α 2 ) 0.26 MeV O(α) O(α 2 α s ) O(α 2 t α s) O(α2 ) O(α 2 t ) O(αα s ) 0.30 MeV O(α) O(αα 2 s ) O(α tα 2 s ) O(αα s) O(α t α s ) O(αα s ) 0.23 MeV O(α) O(αα 3 s) O(α t α 3 s) O(αα s) O(α t α s ) O(αα 2 s) MeV O(α) Parametric prefactors: Total: δγ Z 0.5 MeV O(α 2 bos ) O(α bos) MeV O(α 2 bos ) Γ Zα MeV O(αα 2 s ) O(α tα 2 s ) αn lq π α2 s 0.29 MeV

26 Example: Error estimation for M W 21/21 Renormalization scheme dependence: a) Uncertainty of O(α 2 ) corrections beyond leading α 2 m 4 t and α 2 m 2 t from comparison of MS and OS schemes: Degrassi, Gambino, Sirlin 96 δm W 2 MeV (for M H 100 GeV) Actual remaining O(α 2 ) corrections: Freitas, Hollik, Walter, Weiglein 00 δm W 3 MeV (for M H 100 GeV) b) Estimate of missing O(α 3 ) corrections from comparison of MS and OS results: Awramik, Czakon, Freitas, Weiglein 03 Degrassi, Gambino, Giardino 14 δm W MeV (after accounting for O(α t α 3 s) corrections) Saturates previous δm W estimate! Note: Differences in (implicitly) resummed higher-order contributions

27 Parametrization: Resummation 21/21 Parametrization of perturbation series: α vs. G F? G F can resum some leading one-loop terms α 1 α(0) α(m Z ) ρ = 3α 16πs 2 c 2 m 2 t M 2 Z But: Strong cancellations between α and ρ terms beyond one-loop: r (3) res = ( α) 3 3( α) 2( c 2 s 2 ρ) +6( α) ( c 2 s 2 ρ) 2 5 ( c 2 s 2 ρ) 3 ( ) 10 4 = Not the numerically leading contribution anymore

28 Combination of theory errors 21/21 Add theory errors from each source linearly: Idea: each value within error range in equally likely Use flat prior in global fits Add theory errors from each source quadratically: Idea: different error sources are uncorrelated Use Gaussian prior in global fits (central limit theorm) σ +σ σ +σ

Precision EW measurements at Run 2 and beyond

Precision EW measurements at Run 2 and beyond 52 nd Rencontres de Moriond 2017 Session on Electroweak Interactions and Unified Theories Jens Erler (IF-UNAM) La Thuile Aosta Valley Italy March 18 25, 2017