PRECISION CALCULATIONS FOR FCC-ee selected examples on Γ ( Z ) , Γ ( W ) and Higgs production, mainly from QCD, not a review J. H. K¨ uhn
I) Γ Z and related quantities II) M W from G F , M Z , α III) Higgs production and decay 2
I) Γ Z and related quantities Tera Z: Γ Z aim δΓ Z = 0 . 1 MeV (LEP: 2495 . 2 ± 2 . 3 MeV) present theory error: 0 . 2 MeV from ? [ stated in TLEP-paper ] closer look on QCD and mixed EW ⊗ QCD corrections 3
Mixed electroweak and QCD: light quarks (u,d,c,s) terms of O ( αα s ) , Czarnecki, JK; hep-ph/9608366 W W Z Z Z Z W W (a) (b) W W Z Z Z Z W W (c) (d) ∆Γ ≡ Γ ( two loop (EW ⋆ QCD) ) − Γ Born δ NLO EW δ NLO QCD = − 0 . 59 ( 3 ) MeV three loop: reduction by # · α s π = #0 . 04 # should not exceed 5! corrections of O ( α w α 2 s ) (three loop) difficult 4
Tera Z: Γ ( Z → b ¯ b ) ≡ Γ b aim: δ R b ≡ δΓ b Γ Z = 2 − 5 × 10 − 5 (LEP: R b = 0 . 21629 ± 0 . 00066 , corresponds to δΓ b ≈ 1 . 6 MeV) 2 × 10 − 5 corresponds to 0 . 05 MeV! corrections specific for b ¯ b : m 2 t -enhancement: order G F m 2 t and G F m 2 t α s ∆Γ = G F M 3 w )( 1 − π 2 − 3 α s t ( 1 − 2 16 π 3 G f m 2 3 s 2 π ) (Fleischer et al 1992) 3 Complete α w α s result: Γ b − Γ q = ( − 5 . 69 − 0 . 79 O ( α ) + 0 . 50 + 0 . 06 O ( αα s )) MeV separated into m 2 t -enhanced and rest (Harlander, Seidensticker, Steinhauser dressed with gluons hep-ph/9712228) 5
motivates the evaluation of m 2 t -enhanced corrections of O ( G F m 2 t α 2 s ) (Chetyrkin, Steinhauser, hep-ph/990480) δΓ b ( G F m 2 t α 2 s ) ≈ 0 . 1 MeV (non-singlet) (absent in Z-fitter, G-fitter!) General observation: many top-induced corrections become significantly smaller, if m t is expressed in MS convention � � 4 � � α s � � α s � 2 � α s � 3 � α s m t ( ¯ m t ) = 1 − 1 . 33 − 6 . 46 − 60 . 27 − 704 . 28 ¯ m pole π π π π ր ր (Karlsruhe, 1999) ( Marquard, Smirnov, Smirnov, Steinhauser, 2015) = ( 173 . 34 − 7 . 96 − 1 . 33 − 0 . 43 − 0 . 17 ) GeV � � = 163 . 45 ± 0 . 72 | m t ± 0 . 19 | α s ± ? | th GeV top scan ⇒ m ( potential subtracted ) δ m t ∼ 20 − 30 MeV 6
Tera Z: Γ b ( Z → b ¯ b ) Can we isolate the Zb ¯ b -vertex? R b = 0 . 21629 ± 0 . 00066 (LEP); 3% � = 1 . 65 MeV 2 − 5 × 10 − 5 � = 50 − 120 keV TLEP: conceptual problem: singlet-terms � � 2 � � c b c b � � � � � � + + ... ¯ � c ¯ � b � � � � ¯ c ¯ b c c b b c b + + Im c b mixed contributions, “singlet” � � � α s � 2 ≈ 340 keV G F M 3 Γ singlet Z = √ 0 . 31 b ¯ π bc ¯ c 2 π 8 (total hadronic rate more robust!) 7
Tera Z: Γ had and Γ had / Γ lept corrections known to O ( α 4 N 3 LO s ) , (Baikov, Chetyrkin, JK, Rittinger, arxiv: 0801.1821, 1201.5804) 0 non-singlet & singlet, vector & axial correlators � 0.00001 V � M Z , Μ � � 0.00002 t,b t,b � 0.00003 r S � 0.00004 � 0.00005 0.5 1.0 1.5 2.0 2.5 3.0 1.041 Μ � M Z � 0.003 1.040 � 0.004 1.039 � M Z , Μ � � 0.005 A � M Z , Μ � 1.038 r N S � 0.006 1.037 � 0.007 r S 1.036 � 0.008 � 0.009 1.035 0.5 1.0 1.5 2.0 2.5 3.0 � 0.010 Μ � M Z 0.5 1.0 1.5 2.0 2.5 3.0 Μ � M Z 8
� theory uncertainty from M Z / 3 < µ < 3 M Z ⇒ δΓ NS = 101 keV ; Σ = 145 . 7 keV δΓ V = 2 . 7 keV ; S ( corresponds to δα s ∼ 3 × 10 − 4 ) δΓ A = 42 keV ; S TLEP: δΓ had � = 100 keV � similar analysis of Γ ( W → had) only affected by non-singlet corrections! � b-mass corrections under control: m 2 b α 4 s ; m 4 b α 3 s ; ... � one more loop? α 2 s ( 1979 ) , α 3 s ( 1991 ) , α 4 s ( 2008 ) , α 5 s ( ? ) , guesses on α 5 s based on ... . 9
II) M W from G F , M Z , α LEP: δ M W ≃ 30 MeV; TLEP: δ M W ≃ 0 . 5 − 1 MeV Theory � �� � � M 2 1 − 4 πα ( 1 − δρ ) M 2 1 Z W = f ( G F , M Z , m t , ∆α ,... ) = 1 + √ 1 − ∆α + ... ; 2 ( 1 − δρ ) 2 G F M 2 Z m t -dependence through δρ t cos 2 θ w δ M W ≈ M W 1 cos 2 θ w − sin 2 θ w δρ ≈ 5 . 7 × 10 4 δρ [MeV] 2 � � 3 � � α s � � α s � 2 − 93 . 1 � α s δρ t = 3 X t 1 − 2 . 8599 − 14 . 594 π π π ↓ ↓ δ M W = 9 . 5 MeV δ M W = 2 . 1 MeV α 3 s : 4 loop (Chetyrkin, JK, Maierh¨ ofer, Sturm; Boughezal, Czakon, 2006) 10
mixed QCD ⋆ electroweak -5 -4 ⋅ 10 δ M W [MeV] 2 θ eff δ sin 5 -5 -2 ⋅ 10 0 0 -5 2 ⋅ 10 -5 2 contribution X t -5 2 X t contribution 4 ⋅ 10 α s α s X t 2 contribution -10 -5 6 ⋅ 10 3 contribution X t -15 -5 8 ⋅ 10 1 ⋅ 10 -4 -20 M H / M t 0 1 2 3 4 5 ( X t ≡ G F m 2 three loop t ) X 3 ⇒ 200 eV (purely weak) t α s X 2 ⇒ 2 . 5 MeV (mixed) t α 2 ⇒ − 9 . 5 MeV s X t (QCD three loop) α 3 ⇒ 2 . 1 MeV s X t (QCD four loop) 11
the future individual uncalculated higher orders below 0 . 5 MeV, examples: α 2 s X 2 t presumably feasible (4 loop tadpoles), α 4 s X t 5 loop tadpoles? dominant contribution from m t ( pole ) ⇒ ¯ m t crucial input: m t also for stability of the universe 0.05 δ M W ≈ 6 × 10 − 3 δ m t 3 loop 0.04 2 loop δ m t = 1 GeV 0.03 M t � 173.34 � 0.76 GeV 0.02 M t � 173.34 � 0.76 GeV ⇒ δ M W ≈ 6 MeV (status) Λ � Μ � 0.01 conversely: 0.00 TLEP: δ M W = 0 . 5 MeV � 0.01 requires δ m t = 100 MeV � 0.02 6 8 10 12 14 16 18 Log 10 � Μ � GeV � (Zoller) 12
TLEP: δ m t = 10 − 20 MeV based on bold extrapolation of ILC study (ILC: 35 MeV, no theory error) momentum distribution etc: LO only σ tot in N 3 LO just completed (Beneke, Kiyo Marquard, Piclum, Penin, Steinhauser) 1.4 1.10 Γ t +100 MeV 1.2 Γ t − 100 MeV NNLO R/R( µ = 80 GeV) 1.05 1.0 0.8 NNNLO NLO 1.00 R 0.6 0.4 0.95 0.2 0.0 0.90 340 342 344 346 348 340 342 344 346 348 � s (GeV) � s (GeV) robust location of threshold, extraction of λ Yuk requires normalization! 13
m t ( ¯ m t ) ⇔ m pole important ingredient: ¯ α s ≡ α ( 6 ) example: m pole = 173 . 340 ± 0 . 87 GeV, s ( m t ) = 0 . 1088 4 loop term is just completed ( Marquard, Smirnov, Smirnov, Steinhauser, 2015) � � 1 + 0 . 4244 α s + 0 . 8345 α 2 s + 2 . 365 α 3 s +( 8 . 49 ± 0 . 25 ) α 4 = m t ( ¯ m t ) m pole ¯ s = ( 163 . 643 + 7 . 557 + 1 . 617 + 0 . 501 + 0 . 195 ± 0 . 05 ) GeV four-loop term matters! 14
III) Higgs production and decay ✰ ❡ ❍ ❩ ✲ ❡ ❩ � ✰ ⑧ ❡ ❲ ❍ ❲ ✰ ✰ ✲ ❡ ❡ ❡ ⑧ ✁ ❍ ✁ ✲ ✲ ❡ ❡ Cross sections for the three major Higgs production processes as a function of center of mass energy, from arXiv:1306.6352 15
example: H → b ¯ b dominant decay mode, all branching ratios are affected! TLEP: σ HZ × Br ( H → b ¯ b ) : aim 0 . 2% Higgs WG, arXiv:1307.1347 (Table 1) assumes α s = 0 . 119 ± 0 . 002 , m b | pole = 4 . 49 ± 0 . 06 GeV: δΓ ( H → b ¯ b ) b ) = ± 2 . 3% | α s ± 3 . 2% | m b ± 2 . 0% | th ⇒ 7 . 5% Γ ( H → b ¯ H , µ 2 = M 2 b ) = G F M H Γ ( H → b ¯ 2 π m 2 b ( M H ) R S ( s = M 2 √ H ) Our estimate: 4 � α s � � α s � 2 � α s � 3 � α s � 4 R S ( M H ) = 1 + 5 . 667 + 29 . 147 + 41 . 758 − 825 . 7 π π π π = 1 + 0 . 1948 + 0 . 03444 + 0 . 0017 − 0 . 0012 = 1 . 2298 (Chetyrkin, Baikov, JK, 2006) for α s ( M Z ) = 0 . 118 , α s ( M H ) = 0 . 108 Theory uncertainty ( M H / 3 < µ < 3 M H ) : 5 � (four loop) reduced to 1 . 5 � (five loop) 16
present parametric uncertainties: m b ( 10 GeV ) = 3610 − α s − 0 . 1189 12 ± 11 MeV (Karlsruhe, arXiv:0907.2110) 0 . 002 � � Bodenstein+Dominguez: 3623 ( 9 ) MeV 3617 ( 25 ) MeV HPQCD ( α s uncertainties are presently dominant, assuming δ = 0 . 002 they influence m b -determination; runnung to M H ; R S ) running from 10 GeV to M H depends on anomalous mass dimension, β -function and α s m b ( M H ) = 2759 ± 8 | m b ± 27 | α s MeV γ 4 (five loop): Baikov + Chetyrkin, 2012 β 4 under construction δ m 2 b ( M H ) b ( M H ) = − 1 . 4 × 10 − 4 ( β 4 − 4 . 3 × 10 − 4 ( β 4 − 7 . 3 × 10 − 4 ( β 4 β 0 = 0 ) | β 0 = 100 ) | β 0 = 200 ) m 2 b ) = 2 . 0 × 10 − 4 (FCC-ee) to be compared with δΓ ( H → b ¯ b ) / Γ ( H → b ¯ 17
(assume δα s = 2 × 10 − 4 ) perspectives: δ m b ( 10 GeV ) / m b ∼ 10 − 3 conceivable (dominated by δΓ ( ϒ → e + e − ) ) ⇒ δΓ H → b ¯ b = ± 2 × 10 − 3 | m b ± 1 . 3 × 10 − 3 | α s , running ± 1 × 10 − 3 | theory b Γ H → b ¯ similarly: Γ c δ m c ( 3 GeV ) / m c ( 3 GeV ) = 13 MeV / 986 MeV (now) = 5 MeV / 986 MeV (conceivable) m c ( M H ) = ( 609 ± 8 | m c ± 9 | α s ) MeV (now) ± 3 MeV (conceivable) ⇒ δΓ c ± 5 . 5 × 10 − 2 = (now) Γ c ± 1 × 10 − 2 = (conceivable) Starting from order α 3 s the separation of H → gg and H → b ¯ b is no longer unambiguously possible. (Chetyrkin, Steinhauser, 1997) 18
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