Multiloop QCD: 33 years of continuous development

Transcription

1 B 1 / 35 Multiloop QCD: 33 years of continuous development (on example of R(s) and inclusive Z decay hadrons) Konstantin Chetyrkin KIT, Karlsruhe & INR, Moscow SFB TR9 Computational Theoretical Particle Physics KA AC "Quantum Chromodynamics: History and Prospects" September 3-8, 2012, Oberwoelz, Austria

2 Last 30 years revolution in our ability to deal with multiloop Feynman Integrals 2 / 35 Main (but not all) ingredients: Dim. Reg. /G. t Hooft & M. Veltman (72)/ IBP method (K. Ch. & Tkachov (81)+... Laporta ) effective theory, Eucliudean and non-eucldean expansion of FI s, method of regions (M. Beneke & V. Smirnov (1998) +... ) summation, (new!) special functions /see talk by C. Schneider/ Computer Algebra: from legendary SCHOONSCHIP (M. Veltman) to Mathematica and, especially, FORM (J. Vermaseren)

3 3 / 35 Integration By Parts in DR [IBP] known for ages under active development since 1981 at present one of most versatile and useful methods Main identity: d D p f (p,...) = 0. p µ E.g. applying ( / p 5 ) p 5 to the integrand of a 2-loop massless φ 3 propagator I(n 1,..., n 5 ) = and after the differentiation and using momentum conservation at each vertex one arrives at = 1 ε [ 5 3 ] 1 / 1

4 4 / 35 Examlpe: for all non-trivial 4-loop massless propagators (masters) m61 m62 m63 m51 m41 m42 m44 m45 m34 m35 m36 m52 first 10 terms of ǫ expansion have been computed first numerically (around 500 significant digits!) and then their full anaytical structure has been constructed (V. Smirnov & R. Lee, 2011). Tools: IBP + recurrence relations in the space-time dimension D (O. Tarasov,1996) + a lot of ingenuity

5 Now: Evaluation of FI = established, fastly developing part of math. physics 5 / 35

6 Now: Expansion of FI = established, fastly developing part of math. physics 6 / 35

7 Multiloop START: JINR (Dubna) & INR (Moscow) 7 / 35 Volume 85B, number 2,3 PHYSICS LETTERS 13 August 1979 HIGHER-ORDER CORRECTIONS TO σ tot (e + e HADRONS ) IN QUANTUM CHROMODYNAMICS K.G. CHETYRKIN, A.L. KATAEV and F.V. TKACHOV Institute for Nuclear Research of the Academy of Sciences of the USSR, Moscow, USSR Received 24 May 1979 We present the α 2 s corrections to σ tot (e + e hadrons) in massless QCD... Volume 93B, number 4 PHYSICS LETTERS 30 June 1980 THE GELL-MANN-LOW FUNCTION OF QCD IN THE THREE-LOOP APPROXIMATION O.V. TARASOV, A.A. VLADIMIROV and A.Yu. ZHARKOV Joint Institute for Nuclear Research, Dubna, USSR Received 28 March 1980

8 Final results of a Large Karlsruhe based project: 8 / years of work of the KIT-MSU-INR multiloop group: + starting from the first publication Baikov, K. Ch. and Kühn, Five-loop vacuum polarization in pqcd: O(αs 4 Nf 2 ) results, Phys.Rev.Lett. B559 (2003) and finishing with recent works Phys.Rev.Lett.101:012002,2008; arxiv: Phys.Rev.Lett.104:132004,2010; arxiv: v1 Nucl.Phys.B837: ,2010; arxiv: Phys.Rev.Lett.108:222003,2012; arxiv: P. Baikov, K. Ch., J. Kühn and J. Rittinger, JHEP 1207 (2012) 017 ; arxiv: P. Baikov, K. Ch., J. Kühn and J. Rittinger, Phys.Lett. B714 (2012) 62-65; arxiv: P. Baikov, K. Ch., J. Kühn and C. Sturm, arxiv:

9 9 / 35 Aim of the project Extending to NNNNLO (which means α 4 s!) the well-known NNNLO O(α 3 s ) results by Goirisny, Kataev, Larin, (1991); Samuel, Surguladze (1991) on the three gold plated (Bjorken, 1979) QCD observables: R Z = Γ(Z 0 hadrons)/σ(z 0 µ + µ ) R τ = Γ(τ hadrons + ν τ )/Γ(τ l + ν l + ν τ ) R(s) = σ tot (e + e hadrons)/σ(e + e µ + µ )

10 Why gold plated? 10 / 35 As predictions are very clean theoretically: inclusive observables, expressible completely through two-point correlators. As R Z, R τ and, to a lesser extent, R(s) are very precisely measured. At last, the ratio R(s) is related in unique way with the vacuum polarization operator, which contribute to the few important physical quantities. For example, the running of the effective α QED in the SM as well as the hadronic contribution to the muon anomalous magnetic moment.

11 Why NNNNLO could be of any importance at all? 11 / 35 Because, with the NNNLO accuracy the theoretical errors were comparable with the experimental ones and, in despair, everybody was using the famous Kataev&Starshenko /1993/ estimation of the α 4 s term which (incidentally?) has happened to be quite close to the true number! After our calculations the situation has become significantly better, especially for Γ Z, where the the theoretical error was reduced by a factor of four! (As for R τ see the talk by M. Jamin.)

12 12 / 35 New Results to report (published in 2012) (new!) result for the singlet contribution into the (massless) VV corelator = final complete O(α 4 s) result for the R(s) (new!) as spin-off: the complete QED β-function in five loops (new!) final complete result for Z hadrons at O(α 4 s) (the latter including in full power-not-suppressed top-mass dependence)

13 13 / 35 Z Boson Decay Rate into Hadrons Γ(Z hadrons) = f QCD Z = dφ 2 dφ M(Z f QCD ) +... Opt.Th. Im 2 [ Z Z ] +... Z Z QCD µ ν = Π µν = i e iqx 0 T j µ Z (x)jν Z(0) 0 dx = g µν Π 1 ( q 2 ) + q µ q ν Π 2 ( q 2 ) Γ(Z hadrons) = Γ 0 2πi ( ) Π 1 (s i ε) Π 1 (s + i ε) s ( = Γ 0 R(s) Γ 0 = G FMZ 3 ) 8π 2 7 / 28

14 14 / 35 Vector and Axial Contributions to Z-decay Z current to fermion i: j µ Z i = g V i j µ V i + g A i j µ A i = g V i =V i {}}{ ψ i γ µ ψ i + g A i =A i {}}{ ψ i γ µ γ 5 ψ i Z QCD µ ν = i Z (g V i ) 2 V i V i + i,j g V i g V j V i V j + i (g A i ) 2 A i A i + i,j g A i g A j A i A j i, j = {t, b, c, s, u, d} NO V i A i and V i A i 8 / 28

15 15 / 35 status of theory (in the massless limit) R NS = 3 i Q 2 i ( 1 + α ( s π + # αs π ) 2 + # ( αs π ) 3 + # ( αs π ) 4 + ) parton model QED Källen+ Sabry 1955 Chetyrkin, Kataev, Tkachov; Dine, Sapirstein; Celmaster 1979 Gorishny, Kataev, Larin; Surguladze, Samuel 1991 Chetyrkin /gen. gauge/ 1996 Baikov, Chetyrkin, Kühn 2008; Baikov, Chetyrkin, Kühn 2010 (Feynman Gauge only) R SI = ( i Q i ) 2 ( # ( αs π ) 3 ( +?? αs ) 4 π + ) D NS D SI

16 16 / 35 Result for the R NS (s) /Ch., Baikov, Kühn, / R = 1 + a s + ( n f ) a 2 s ( n f n 2 f ) a 3 s +( n f n 2 f n3 f ) a4 s Impact on α s from Z decays: α s (M Z ) NNLO = ± exp + ±0.002 th Including the α 4 s term leads to a shift of δα s (M Z ) = and to four-fold decrease of the theory error! α s (M Z ) NNNLO = ± exp + ± th

17 17 / 35 Singlet contribution to the (vector) Adler function (Last missing term!) D SI (Q 2 ) = d R ( i=3 d SI i a i s(q 2 ) ) d3 SI = dabc d abc ( 11 d R ) 8 ζ 3, d4 SI = dabc d abc ( CF d4,1 SI d + C A d4,2 SI + T n f d SI ) 4,3 R d SI 4,1 = = ζ ζ 5 8, dsi 4,3 = ζ ζ ζ2 3 d SI 4,2 = ζ ζ ζ2 3

18 Extra suppression factor ! 18 / 35 Phenomenological implications for σ tot (e + e hadrons) Numerically: 11 3 R(s) = 3 { Q 2 f 1 + as + a 2 s ( n f ) f ( )} + a 3 s nf n 2 f ( ) 2 ) Q f ( a 3 s + ( n f ) a 4 s f for n f =5 [ 1 + as + a 2 s a 3 s as] 4 1 [ a 3 9 s as] 4

19 19 / 35 Adler Function = QED β-function for a theory with n f identical fermions [ ] [ 4 A 2 β QED (A) = n f +4 n f A 3 A 4 2 n f + 44 ] 3 9 n2 f [ + A 5 46 n f n2 f ζ 3 n 2 f 1232 ] 243 n3 f ( [ ] A 6 n f + 128ζ 3 + n 2 f 6 + n 3 f [ 7462 ] 992ζ ζ 5 9 ] [ ζ ζ ζ 5 + n 4 f ( Baikov, K. Ch., Kühn and Rittinger, JHEP 1207 (2012) 017) [ ] ) 27 ζ 3

20 20 / 35 If we set n f = 1, then the above result takes the form: β QED (A) = 4 3 A2 + 4A 3 62 [ 9 A4 +A ] 9 ζ 3 Numerically, [ + A ζ ζ ζ 5 ] β QED (A) = 4 3 A2 + 4 A A A A 6

21 Quenched QED puzzle: β-function is completely rational in 1,2,3, and 4 loops! β qqed (A) = n f [ 4 A 2 3 ] [ +4 n f A 3 A 4 2 n f + 44 ] 9 n2 f +A 5 [ 46 n f ] some people understood the observed rationality of β qqed at 1,2,3 and 4 loops and strongly believed that it is not a pure coincidence. For instance: David Broadhurst: (in hep-th/ ) Noting the profound work of Alain Connes and Dirk Kreimer [1], one arrives at the nub of the rationality of quenched QED: dimensional regularization of the derivative of the scheme-independent single-fermion-loop Gell-Mann-Low function, via Fock-Feynman-Schwinger formalism 21 / 35

22 22 / 35 Quenched QED puzzle: rationality spell is broken at 5 loops! (Baikov, K. Ch., Kühn, ) [ ] [ 4 A 2 β qqed (A) = n f +4 n f A 3 A 4 2 n f + 44 ] 3 9 n2 f [ ] A 5 [ 46 n f ] +A 6 n f + 128ζ 3 6

23 23 / 35 Vector and Axial Contributions to the Z-decay Z current to fermion i: j µ Z i = g V i j µ V i + g A i j µ A i = g V i =V i {}}{ ψ i γ µ ψ i + g A i =A i {}}{ ψ i γ µ γ 5 ψ i Z QCD µ ν = i Z (g V i ) 2 V i V i + i,j g V i g V j V i V j + i (g A i )2 A i A i + i,j g A i g A j A i A j i, j = {t, b, c, s, u, d} scale s = M z m l = 0 (l = {b, c, s, u, d}) 2 scales s = M Z and m t (but MZ 2/(4m2 t ) 1): Decoupling of top (factorization of top effects) 15 / 28

24 t Hooft-Veltman-Larin Treatment of γ 5 Problem: naive γ 5 ([γ 5, γ α ] = 0) is not applicable for the singlet diagrams. Solution t Hooft-Veltman 72: treat ɛ µµ 1µ 2 µ 3 as 4-dimensional object γ 5 = i 4! ɛµ 1µ 2 µ 3 µ 4 γ µ1 γ µ2 γ µ3 γ µ4 (i = {t, b, c, s, u, d}) Larin 93: introduce a special prefactor ζ A = a s +... and define the axial vector current as: A i = ψ i γ µ i γ 5 ψ i = ζ A 6 ɛµµ 1µ 2 µ 3 ψ i γ µ1 γ µ2 γ µ3 ψ i ( ζ A effectively restores antisymmetry of γ 5 in NS diagrams; it is fixed uniquely from the (non-anomalous) Ward identity,or, equivalently, ) by i requiring scale inv. of the current ζ A 6 ɛµµ 1µ 2 µ 3 [ψ i γ µ1 γ µ2 γ µ3 ψ i ] MS 24 / 35

25 Decoupling of Top 25 / 35 Decoupling for QCD fields and coupling constant (ζ 2, ζ 3, ζ 3 and ζ g ): 6 flavour 5 flavour Vector current: naive decoupling (due to Ward Identity) V (6) t m t = 0 + O(1/m 2 t ), V (6) b m t = V (5) b + O(1/m 2 t ) Axial vector current: no naive decoupling (due to potential anomaly if it it would!) A (6) t m t = C h A (5) S + O(1/m 2 t ), A (6) b m t = A (5) b + C ψ A (5) S + O(1/m 2 t A (5) S = l A (5) l, (A (6) t A (6) b ) mt = (C h C ψ ) A (5) S A b + O(1/m 2 t )

26 26 / 35 Decoupling of Top Calculation of C h with method of projectors (Gorishny, Larin 86): A t (6) m t = Ch ( a (6) (µ), µ/m t ) AS (5) t A t A u A u l A l Tū A t u : Tū A S u :,, u u u u Tū A i u p=0 : TAD s Born, 0, 0 Calculation of massive tadpoles: matching to topologies with EXP (Seidensticker and Steinhauser) reduction with Laporta alg. via CRUSHER (Marquard, Seidel) all master integrals up to 4 loop are known (Schröder, Vuorinen 05) 18 / 28

27 27 / 35 Decoupling of Top Calculation of C ψ : (6) m A t (5) ( b = A b + C ψ a (6) ) (5) (µ), µ/m t A S b A b T ū A b u : t u 19 / 28

28 Decoupling of Top 28 / 35 C h = C ψ = + + A (6) t ( ) α (6) 2 [ (µ) π ) 3 [ ( α (6) (µ) π ( α (6) (µ) π ( ) α (6) 3 [ (µ) π ( α (6) (µ) π m t = Ch A (5) S, A (6) b ln ( µ 2 )] m 2 t ( µ ln ) 4 [ ln ln ) 4 [ ln m 2 t ( µ 2 ( µ 2 m 2 t ( µ 2 2 loop Collins, Wilczek and Zee 78 m 2 t m t = A (5) b + C ψ A (5) S. ) ( )] ln 2 µ 2 m 2 t m 2 t ) ( ) ( )] ln 2 µ ln 3 µ 2 m 2 t ) ( )] ln 2 µ 2 3 loop log enhanced terms Chetyrkin, Kühn 93 m 2 t ) ( ) ( )] ln 2 µ ln 3 µ 2 3 loop Larin, Ritbergen, Vermaseren 93; Chetyrkin, Tarasov 94 4 loop is new m 2 t m 2 t m 2 t

29 29 / 35 Axial Vector Correlator Optical Theorem: Γ(Z hadrons) = Γ 0 R(s) = Γ 0 (R V (s) + R A (s) ) Z QCD µ ν = i Z (g V i )2 V i V i + i,j g V i g V j V i V j + i (g A i )2 A i A i + i,j g A i g A j A i A j R NS R A,S i,j Decoupling: A (6) t m t = Ch A (5) S, A (6) b m t = A (5) b + C ψ A (5) S. R A O(α4 ) = ( 5 R NS + Ch 2 2(C h C ψ )(R NS + R A,S b,s ) + ) RA,S b,b 21 / 28

30 Master Formula for Γ h Z Γ h Z = Γ 0 ( ) { } ( ) g V 2 ( ) i + g V 2 i 1 + a s a 2 s a 3 s a 4 s i ( + i g V i i ) 2( ) a 3 s a 4 s + [ ( lnmt ) a 2 s + ( ln Mt ln 2 M t ) a 3 s + ( ln Mt ln 2 M t ln 3 M t ) a 4 s ] } ( ) Γ 0 = G F MZ 3 8 π 2, a s = α (5) s π, ln M pole 2 t M t = ln M Z 30 / 35

31 31 / 35 Effective Master Formula for Γ h Z Using as inputs M t = 172.0, s 2 W = and α s(m Z ) =.1190 we arrive to an effective Master Formula illustrating the total effect of subleading singlet contributions in comparison to the non-singlet ones: Γ Z = Γ 0 ( i i ) { ( ) g V 2 ( ) i + g A 2 i 1 + a s a 2 s a 3 s a 4 s = NS a 3 s a 4 s = V SI } a 2 s a 3 s a 4 s = A SI

32 Conclusions I 32 / 35 The 10 years + project of computing R(s) and Γ(Z hadrons) at order α 4 s is finished! The last missing ingredients singlet contributions to the (massless) VV and AA as well as (massive) A t A t, A t A b and A b A b correlators at O(α 4 s) are now available singlet O(α 4 s) contributions are numerically tiny the net effects of O(α 4 s) term in Γ h Z are: an increase of δα s (M Z ) = O(α 3 s) : α s (M Z ) NNLO = ± exp ± th O(α 4 s) : α s (M Z ) NNNLO = ± exp ± th and four-fold decrease of the theory error! / K.Ch, Baikov and Kühn, PLR 101 (2008) /

33 Conclusions II 33 / 35 the 5-loop QED β-function is computed = the first example of 5-loop RG function in a (normal) 4-D gauge theory: almost exactly thirty yeas after similar result in a φ 4 model /K.Ch, Gorishnii, Larin, and Tkachov, Phys.Lett. B132 (1983) 351/; D.I. Kazakov,Phys.Lett. B133 (1983) 406; Kleinert, Neu, Schulte-Frohlinde, K.Ch., and Larin, Phys.Lett. B272 (1991) 39/ All our methods and tools are equally well applicable to evaluation of the 5-loop β-functions and anomalous dimensions in general non-abelian gauge theories. A couple of interesting examples could be: 1 QCD β-function (important for a better understanding of the τ-lepton decay rate within the so-called contour-improved method) 2 the quark mass anom. dim. in QCD

34 34 / 35 If we set n f = 1, then the above result takes the form: β QED (A) = 4 3 A2 + 4A 3 62 [ 9 A4 +A ] 9 ζ 3 Numerically, [ + A ζ ζ ζ 5 ] β QED (A) = 4 3 A2 + 4 A A A A 6

35 35 / 35 World Summary of s 2011: -decays Lattice DIS e + e - e.w. fits s(mz)= ± s(mz)= ± s(mz)= ± s(mz)= ± s(mz)= ± s > s (M Z )= ± ( > RPP 2012 ) MS (5) = (214 ±10) MeV MS (4) = (298 ±12) MeV S. Bethke: as (2011) summary Ringberg workshop on HERA physics Sep. 26,

36 Plan 36 / 35