MASSIVE TADPOLES: Techniques & Applications I. RHO-PARAMETER II. QUARK MASSES III. MISCELLANEOUS J. K¨ uhn / Project A1
I. RHO-PARAMETER 1. Definition, work before 2003 2. Three-loop electroweak results with full m H -dependence (Faisst, JK, Seidensticker, Veretin) � � G F α 3 3. Four-loop QCD contributions, O s (Chetyrkin, Faisst, JK, Maierh¨ ofer, Meier, Sturm) 2
I.1 Definition, work before 2003 central prediction of SM: M W = f ( G F , M Z , α ; M t , M H , . . . ) � �� � � �� � Born radiative corrections [similarly for couplings of fermions: sin 2 θ eff ] � � 1 − M 2 πα M 2 W = √ (1 + ∆ r ) W M 2 2 G F Z ∆ r dominated by: ∆ r = − c 2 s 2 ∆ ρ + ∆ α � � G F M 2 ∆ ρ ∼ + . . . (Veltman) t 3
aim: compete with experiment δM W [MeV] δM t [GeV] 33 5 status 2003 (LEP, TEVATRON) 15 0.76 now (TEVATRON, LHC) 8 → 5 0.6 aim (LHC), theory limited 3, < 1.2 0.1 - 0.2 ILC, TLEP theory: correlation (for α ( M Z ) , M H fixed): δM W ≈ 6 · 10 − 3 δM t � � � LHC 5 MeV ⇒ shifts in M W � are relevant for e + e − collider 1 MeV 4
Theory: Status 2002 � Barbieri, Beccaria, Ciafaloni, Curci, Vicere status: two-loop Fleischer, Jegerlehner, Tarasov approximation: M 2 t ≫ M 2 ⇒ scalar bosons only, gaugeless limit W 2 π 2 = g 2 X t ≡ G F M 2 ∆ ρ = X 2 t Yukawa t f ( M t /M H ) with √ 16 π 2 8 status: three-loop M H = 0 (van der Bij, Chetyrkin, Faisst, Jikia, Seidensticker) poor approximation, leads to tiny corrections for terms of order X 3 t and α s X 2 t 5
M H � = 0 (Faisst, JK, Seidensticker, Veretin, 2003) � � requires: 3-loop tadpoles Π WW (0) , Π ZZ (0) . . � H ; �; � X 3 t f ( m t /M H ) H H � � � � � � � H ; �; � . . . α s X 2 t f ( m t /M H ) H � � � . and two-loop on-shell diagrams: M t ⇐ ⇒ m t ( MS ) . . . . t H , � H , � � H , � � � H , � H , � t t t t t t t t t t t t t t t t b b b . . . . . H , � t t t t t H , � . in general two (!) mass scales, three loops 6
special cases: √ M H = 0 : H ) n mod. log hard mass expansion in ( M 2 t /M 2 M H ≫ M t : M H = M t : one scale M H in neighbourhood of M t : Taylor expansion: δ = ( M H − M t ) /M t excellent approximation ⇒ reduction to one-scale two- or three-loop integrals 7
on-shell result O ( α s X 2 t ) -4 1.5 ⋅ 10 M H /M t = 0 . 726 400 300 up to 1. ord (m t ≈ M H ) -4 1 ⋅ 10 up to 2. ord (m t ≈ M H ) up to 3. ord (m t ≈ M H ) up to 4. ord (m t ≈ M H ) 200 up to 5. ord (m t ≈ M H ) -5 0. ord (m t <M H ) 5 ⋅ 10 up to 1. ord (m t <M H ) 100 up to 3. ord (m t <M H ) up to 5. ord (m t <M H ) 0 0 0 1 2 3 4 5 M H / M t 2 ) 2 ) 2 ∆ρ (α s X t (α s X t ( α s / π )X t ∆ρ -100 Contributions of order α s X 2 t to ∆ ρ in the on-shell definition of the top quark mass. The black squares indicate the points where the exact result is known. M H = (0 | 126) ⇒ ∆ ρ ( α s X 2 t ) = (2 . 9 | 120) 8
0 . 726 -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 2 contribution -10 α s X t -5 6 ⋅ 10 3 contribution X t -15 -5 8 ⋅ 10 -4 1 ⋅ 10 -20 M H / M t 0 1 2 3 4 5 δM W ≈ 2 . 3 MeV δ sin 2 θ eff ≈ 1 . 5 · 10 − 5 9
I.3 Four-loop QCD contributions QCD: Status 2002 . . t one-loop (1977) t; b Z Z W W (Veltman) b . . two-loop (1987) (Djouadi; Kniehl, JK, Stuart) tadpoles; zero scale three-loop (1995) (Chetyrkin, JK, Steinhauser) (Avdeev,. . . ) 10
α 0 α 1 α 2 δM W in MeV s s s M 2 611.9 -61.3 -10.9 t const. 136.6 -6.0 -2.6 1 /M 2 -9.0 -1.0 -0.2 t four loop? 11
Techniques: 3-loop M pole ⇔ ¯ m relation (1999/2000) (Chetyrkin+Steinhauser, Melnikov+van Ritbergen) and 4-loop tadpoles: Laporta algorithm [previously: 3 loop tadpoles ⇒ recursive algorithm (Broadhurst, Steinhauser: MATAD)] analytical and numerical evaluation of ∼ 50 four-loop master integrals: difference equations, semi-numerical integration 12
13
result (2006) (Chetyrkin, Faisst, JK, Maierh¨ ofer, Sturm) = 3 G F m 2 δρ (4 loop) 2 π 2 α 3 t √ s ( − 3 . 2866 +1 . 6067 = − 1 . 6799) t 8 ↑ ↑ singlet piece this result (Schr¨ oder+ Steinhauser, 2005) result immediately independently confirmed (Boughezal+Czakon) conversion to pole mass: = 3 G F M 2 δρ (4 loop) 2 π 2 α 3 t √ s ( − 93 . 1501) ! t 8 corresponds to a shift δM W ∼ 2 MeV (similar to O ( x 2 t α s )) 14
II. QUARK MASSES from relativistic 4 loop moments 1. Why 2. Theory 3. Results, from experiment and from lattice in collaboration with K. Chetyrkin, Y. Kiyo, A. Maier, P. Maierh¨ ofer, P. Marquard, A. Smirnov, M. Steinhauser, C. Sturm and the HPQCD Collaboration 15
II.1 WHY precise masses? B-decays: ν ) ∼ G 2 F m 5 b | V ub | 2 Γ( B → X u l ¯ ν ) ∼ G 2 F m 5 b f ( m 2 c /m 2 b ) | V cb | 2 Γ( B → X c l ¯ B → X s γ comparison with Υ-spectroscopy: � 4 � 2 M b M (Υ(1s)) = 2 M b − 3 α s 4 + ... + excitations (Penin & Zerf, . . . δm b ∼ 9 MeV) 16
H decay (ILC, TLEP) H → b ¯ b dominant decay mode, all branching ratios are affected! status: Γ b = G F M 2 2 π m 2 b ( M H ) R S ( M H ) H √ 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 . 19551 + 0 . 03469 + 0 . 00171 − 0 . 00117 � ↑ (Chetyrkin, Baikov, JK, 2006) Theory uncertainty ( M H / 3 < µ < 3 M H ) : 5 � (four loop) reduced to 1 . 5 � (five loop) present uncertainties from m b 17
m b (10 GeV) = 3610 − α s − 0 . 1189 12 ± 11 MeV (Karlsruhe, arXiv:0907.2110) 0 . 002 running from 10 GeV to M H depends on anomalous mass dimension, β -function and α s =1 . 5 × 10 − 4 ) m b ( M H )2759 ± 8 | m b ± 27 | α s MeV aim ± 4 MeV ( � γ 4 (five loop): Baikov + Chetyrkin, 2013 β 4 under construction δm 2 b ( M H ) b ( M H ) = − 1 . 4 × 10 − 4 ( b 4 = 0) | − 4 . 3 × 10 − 4 ( b 4 = 100) | − 7 . 3 × 10 − 4 ( b 4 = 200) m 2 to be compared with δ Γ / Γ = 2 . 0 × 10 − 3 (TLEP) 18
Yukawa Unification λ τ ∼ λ b or λ τ ∼ λ b ∼ λ t at GUT scale � top-bottom → m t m b ∼ ratio of vacuum expectation values request δm b m b ∼ δm t m t ⇒ δm t ≈ 0 . 5 GeV ⇒ δm b ≈ 12 MeV 19
II.2 Theory m Q from SVZ Sum Rules, Moments and Tadpoles Main Idea (SVZ) 20
Some definitions: � � � − q 2 g µν + q µ q ν Π( q 2 ) d x e iqx � Tj µ ( x ) j ν (0) � ≡ i with the electromagnetic current j µ . � � Π( q 2 = s + iǫ ) R ( s ) = 12 π Im � 3 Taylor expansion: Π Q ( q 2 ) Q 2 C n z n ¯ = Q 16 π 2 n ≥ 0 with z = q 2 / (4 m 2 Q ) and m Q = m Q ( µ ) the MS mass. � α s � 2 ¯ � α s � 3 ¯ + α s C (0) C (1) C (2) C (3) C n = ¯ ¯ ¯ + + + . . . n n n n π π π 21
generic form C (0) C n = ¯ ¯ n � � + α s C (10) C (11) ¯ + ¯ l m c n n π � α s � 2 � � C (20) C (21) C (22) l 2 ¯ + ¯ l m c + ¯ + n n n m c π � α s � 3 � � C (30) C (31) C (32) C (33) l 2 l 3 ¯ + ¯ l m c + ¯ m c + ¯ + n n n n m c π + . . . 22
Analysis in NNLO • FORM program MATAD • Coefficients ¯ C n up to n = 8 • (also for axial, scalar and pseudoscalar correlators) • (Chetyrkin, JK, Steinhauser, 1996) 23
Analysis in N 3 LO Algebraic reduction to 13 master integrals (Laporta algorithm); numerical and analytical evaluation of master integrals n 2 n 1 n 0 f -contributions f -contributions f -contributions · · · · · · · · · :heavy quarks, :light quarks, n f :number of active quarks ⇒ About 700 Feynman-diagrams 24
➪ Reduction to master integrals C 1 in order α 3 C 0 and ¯ ¯ s (four loops!) Program “Sturman” (Sturm) (2006) (Chetyrkin, JK, Sturm; Boughezal, Czakon, Schutzmeier) C 2 and ¯ ¯ C 3 (2008) Program “Crusher”, Marquard & Seidel (Maier, Maierh¨ ofer, Marquard, A. Smirnov) All master integrals known analytically and double checked. (Schr¨ oder + Vuorinen, Chetyrkin et al., Schr¨ oder + Steinhauser, Laporta, Broadhurst, Kniehl et al.) C 4 – ¯ ¯ C 10 : extension to higher moments by Pad´ e method, using analytic information from low energy ( q 2 = 0), threshold ( q 2 = 4 m 2 ), high energy ( q 2 = −∞ ) (Kiyo, Maier, Maierh¨ ofer, Marquard, 2009) (Also: q 2 -dependence of scalar, vector,... correlator) 25
Relation to measurements � � � n � � n � n ≡ 12 π 2 d � = 9 1 M th Π c ( q 2 ) 4 Q 2 � ¯ C n � c d q 2 4 m 2 n ! � q 2 =0 c C n is function of α s and ln m 2 Perturbation theory: ¯ c µ 2 dispersion relation: � q 2 R c ( s ) Π c ( q 2 ) = d s s ( s − q 2 ) + subtraction 12 π 2 � d s ➪ M exp = s n +1 R c ( s ) n constraint: M exp = M th n n ➪ m c 26
Recommend
More recommend
