Multiloop QCD & Crewther identities

Transcription

1 Multiloop QCD & Crewther identities B AC KA SFB TR9 Computational Theoretical Particle Physics based on works of Karlsruhe-Moscow group ( ) Pavel Baikov (MSU), Johannes Kühn (KIT) Konstantin Chetyrkin (KIT) CONFORMAL SYMMETRY IN FOUR-DIMENSIONAL FIELD THEORIES, Regensburg,

2 intro: massless propagators in multiloop QCD: main applications mini-history and current status of the art the problem of reliabilty of 5-loop calculations Conformal Symmetry at work: CBK (Crewther-Broadhurst-Kataev relations) and their implications for very nontrivial checks of the five-loop results on the (singlet and non-singlet) Adler functions a new contribution to the Bjorken SR for polarized scattering at O(α s 4 ) and its compliance with the CBK-relation /new result!/ open problem: a kind of CBK relation for VVA si triangle amplitude with A si = f ψ f γ 5 γ α ψ f being (anomalous!) flavour singlet axial current? Would be of great use for the Ellis-Jaffe sum rule and < A si A si > correlator (do appear in QCD in the formal limit of m t ) ignitedby: S.A. Larin, The singlet contribition to the Bjorken SR for polarized DIS, arxiv:

3 multiloop /in pqcd but not only!/ problems reducible to massless propagators (p-integrals for brevity) 2-points correlators at large energies (massless term + O(m 2 q ) corrections) via the optical theorem lead to: R(s) = σ tot (e + e hadrons)/σ(e + e µ + µ ) semi-leptonic τ-decays Γ(Z hadrons) Γ(H bb) and Γ(H gg) /via a top quark loop/ beta-functions and anomalous dimensions coefficient functions in OPE of 2 local operators (DIS, SVZ sum rules,... ) massless QCD propagators (e.g. gluon self-energy in the Landau gauge, useful for lattice)

4 R(s) from p-integrals Starting object: the polarization operator of EM quark current j µ = e q qγ µ q Π µν (q) = i dxe iqx 0 T[ j v µ (x)jv ν (0) ] 0 = ( g µνq 2 + q µ q ν )Π(q 2 ) related to R(s) through R(s) IΠ(s iδ) Π is not completely physical due to a divergency of T(j v µ (x)jv ν (0)) at x 0, as a result its normalization mode and corresponding evolution equation reads ((a s α s /π), massless QCD) Π = Z em + Π B ( Q 2,α B ( ) s ) µ 2 µ + β(a s)a 2 s Π = γ em (a s ) a s At first sight, it would be advantageous to avoid this by considering (obviously RG invariant!) Adler function defined as D = Q 2 Q 2Π 0 and which is related to R(s) in a unique and simple way R(s) D(Q) = Adler function Q 2 d ) = Q 2 dq 2Π(q2 R(s) (s + Q 2 ) 2 ds

5 BUT, this is not true! The reason: O(α s L ) /that is (L + 1)-loop/ Adler function receives, obviously, contributions from (L + 1) loop p-integrals (including their constant part). In fact, only L-loop integrals are enough HUGE simplification. Indeed, let us rewrite the RG equation for Π as follows: For massless (L + 1) loop Π 0 (L = ln µ2 Q 2,a s ) RG equation amounts to (L+1) loop anom. dim. L Π 0 = γ em (a s ) ր ( ) β(a s )a s Π 0 a s տ L-loop integrals only contribute due to the factor of β(a s ) If one knows the rhs to α L s, then one could trivially construct the Adler function with the same accuracy! Anomalous dimensions (as well as β-functions are simple (no-scale) polynomilas in α s /at least in MS-like schemes/ = one loop could be always undone with so-called Infrared Rearrangement trick

6 Anom. Dim. from p-integrals IRR (Infrared ReaRrangement)/Vladimirov, (78)/ + IR R -operation /K. Ch., Smirnov (1984)/ lead to main THEOREM of RG-calculations: any (L+1) loop UV counterterm (read: any (L+1) loop MS Anomalous Dimension) can be expressed through pole and constant terms of some L-loop p-integrals Corollary: absorptive part of any (L+1) massless 2-point correlator can be expressed through pole and constant terms of some L-loop p-integrals

7 THEOREM ( Corollary ) is our key tool for multiloop RG calculations as it reduces the general task of evaluation of (L+1)-loop UV counterterms (absrptive part of (L+1)-loop 2-point massless correlators) to a well-defined and clearly posed purely mathematical problem: the calculation of L-loop p-integrals (that is massless propagator-type FI s). In the following we shall refer to the latter as the L-loop Problem loop Problem is trivial. 2. the 2-loop Problem was solved after inventing and developing the Gegenbauer polynomial technique in x-space (GPTX) (K.Ch.,F. Tkachov (1980); further important developments in works by D. Broadhurst and A. Kotikov ). GTPX is applicable to analytically compute some quite non-trivial three and even higher loop p-integrals. However, in practice calculations quickly get clumsy, especially for diagrams with numerators.. Nevertheless, it proved to be very usefull in cases of scalar diagrams with many multilinear vertexes /appear frequently in supersymmetric theories/

8 The main breakthrough at the three loop level happened with elaborating the method of integration by parts (IBP) of integrals. Historical references: At one loop, IBP (for DR integrals) was used in, a crucial step an appropriate modification of the integrand before differentiation was undertaken in (in momentum space, 2 and 3 loops) and in (in position space, 2 loops) G. t Hooft and M. Veltman (1979) A. Vasiliev, Yu. Pis mak and Yu. Khonkonen (1981) F. Tkachov (1981); K. Ch. and F. Tkachov (1981)

9 With the use of IBP identities the 3-loop Problem was completely solved and corresponding (manually constructed) algorithm was effectively implemented first in SCHOONSCHIP CAS (Gorishny, Larin, Surguladze, and Tkachov) and then with FORM (Vermaseren, Larin, Tkachov, /1991/... Vermaseren ). This achievement resulted to a host of various important 3- and 4-four loop calculations performed by different teams during 80-th and 90-th. Note that the 4-loop correction to the QCD β function was done only as late as in 1996 and using massive way /van Ritbergen, Vermaseren,and Larin/; the reason was too complicated combinatorics of the IR reduction

10 4-loop Problem has been under study in the Karlsruhe-Moscow group (P. Baikov, K.Ch., J. Kühn...) since late 90th. It is essentially solved by now with the help of 1/D expansion /reduction to masters, implemented as a FORM program BAICER/ and Glue-and-Cut symmetry (analytical evaluation of all necessary masters) As a result during last 12 years in our group the the results for the Adler function, R VV (s) and a closely related quantity Z-decay rate into hadrons have been extended byonemoreloop(thatistoorderα 4 s,whichcorresponds5-loopfortheadlerfunction). These results +some others related to 5 and 4-loop correlators (Higgs decays into hadrons, etc.) can be found in: Phys.Rev.Lett. 88 (2002) Phys.Rev.Lett. 95 (2005) Phys.Rev.Lett. 96 (2006) Phys.Rev.Lett. 97 (2006) Phys.Rev.Lett.101:012002,2008 Phys.Rev.Lett. 102 (2009) Phys.Rev.Lett.104:132004,2010 Phys.Rev.Lett. 108 (2012) JHEP 1207 (2012) 017 Phys.Lett. B714 (2012) 62-65

11 Example of Phenomenological Relevance With previous αs 3 calculation of Γ h Z, the theoretical errors were comparable with the experimental ones and, in despair, everybody was using the famous Kataev&Starshenko /1993/ estimation of the αs 4 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 h Z, where the the theoretical error was reduced by a factor of four! α 4 s correction to the τ decay rate has decreased the theoretical error and improved stability the result wrt the renormalization scale (µ) variation Gorishnii, Kataev, Larin, (1991); Surguladze, Samuel, (1991); (both used Feynman gauge); K. Ch. (1997) (in general covariant gauge)

12 How reliable are available results at 4 loops and 5 loops? A lot of things might go wrong in a multyloop (and usually multi-month) calculation: from a trivial normalization factor buried somewhere in your programs and not expanded deeply enough in ǫ = 2 D/2 (this is exactly was happened with the very first calculation of the Adler function in O(α 3 s ) /Gorishny, Kataev and Larin, 1988/, the result was corrected only by three years after)... to an error in FORM which shows itself irregularly: it affected mainly very big programs that needed the fourth stage of the sorting rather intensively and it showed itself mainly with TFORM with a probability of occurring proportional to at least W 3 if W is the number of workers. (citation of the FORM creator and leading maintainer Jos Vermaseren)

13 Four loop RG At 4 loops every calculation was repeated (and confirmed!) by independent computation(s): 4-loop QED β function (in QCD) + R-ratio at α 3 s: an original (Feynman gauge result) /Gorishny, Larin, Kataev (1991)/ was confirmed 5 years later /K. Ch. (1996), (general covariant gauge)/ 4-loop QCD β function /T. van Ritbergen, J. Vermaseren, S.Larin, (1997)/ was confirmed 8 years later /M. Czakon, (2004)/ (general covariant gauge in both cases) 4-loop quark anomalous dimension was computed 2 times (general covariant gauge in both cases) once with massless and once with massive setups with identical results all master integrals apearing in 4-loop calculations (both massless props and massive tadpoles) have been evaluated many times independently, both analytically and numerically

14 Five loop RG Here the situation is not so good: since 2002 we have performed many 5-loop RG calculations: and (almost) no one has yet been confirmed in full by an independent computation. An exception is quark and gluon form factors to three loops in massless QCD: reduction to masters was done in 2 independent ways (with BAICER and FIRE); the pole part was found first by the Zeuthen group /S. Moch, J.A.M. Vermaseren, A. Vogt (2005)/ All master p-integrals appearing in 5-loop calculations (4-loop massless props) are certainly correct (confirmed by 2 analytical and one numerical all independent evaluations). But how to check two reductions: 1. to masters (performed a sophisticated FORM program) and 2. IR reduction from 5 to 4 loops (human made and also quite complicated)

15 Exactly at this point the conformal symmetry entered to the game and provided us with extremely powerful and highly non-trivial test of the 5-loop Adler function (both, its nonsinglet and singlet terms)

16 DIS Sum Rules the polarized Bjorken sum rule (a s α s π ) Bjp(Q 2 ) = 1 0 [g ep 1 (x,q2 ) g en 1 (x,q 2 )]dx = 1 6 g A g V C Bjp (a s ) Coefficient function C Bjp (a s ) is fixed by OPE of two non-singlet vector currents (up to power suppressed corrections) i TV a α(x)v b β(0)e iqx dx q 2 C Q,abc αβρ Ac ρ(0)+... (1) where and Q 2 = q 2 C Q,abc αβρ id abc ǫ αβρσ q σ Q 2CBjp (a s )

17 the Gross-Llewellyn Smith sum rule 1 GLS(Q 2 ) = 1 F νp+νp 3 (x,q 2 )dx = 3C GLS (a s ) 2 0 the function C GLS (a s ) comes from operator-product expansion of the axial and vector non-singlet currents i TA a µ(x)v b ν(0)e iqx dx q 2 C V,ab µνα V α (0)+... where C V,ab µνα iδ ab ǫ µναβ q β Q 2 C GLS (a s ) Note that both sum rules are unambiguous /modulo higher twists!/ predictions of QCD which in principle could be confronted with experimental data

18 As is well-known, the evaluation of L-loop corrections to a CF of OPE could be done in terms of massless L-loop propagators (S. Gorisny, S. Larin and F. Tkachov (1982)) = one could use techniques developed for R(s) Bjp and GLS GLS only q q At order α 3 s both CF s were computed in early nineties.the next order is contributed by about 54 thousand of 4-loop diagrams... (cmp. to 20 thousand of 5-loop diagrams contributing to R(s) at the same order)

19 The Crewther relation states that in the conformal invariant limit (β 0) C Bjp (a s ) is related to the (nonsinglet) Adler function via the following beautiful equality its generalization for real QCD reads: C Bjp (a s )D NS (a s )) c i = 1 C Bjp (a s )D NS (a s ) = 1 + β(a s) a s [ K NS = K 1 a s + K 2 a 2 s + K 3a 3 s +... ] Note that similar relation connects also the CF of the Gross-Llewellyn Smith sum rule to the full Adler function, to be discussed later. Main ingredients of the derivation: the AVV 3-point function and constraints on it from (approximate) conformal invariance + Adler-Bardeen anomaly theorem

20 Crewther Relation: (short) bibliography discovered: R.J. Crewther, Phys. Rev. Lett. 28, 1421 (1972). S.L. Adler, C.G. Callan, D.J. Gross and R. Jackiw, Phys. Rev. D 6, 2982 (1972). generalized for real QCD: D.J. Broadhurst and A.L. Kataev, Phys. Lett. B 315, 179 (1993). further developed: G.T. Gabadadze and A.L. Kataev,JETP Lett. 61, 448 (1995). S.J. Brodsky, G.T. Gabadadze, A.L. Kataev and H.J. Lu, Phys. Lett. B 372, 133 (1996);... A. Kataev and S. Mikhailov, Archive: ; most recent discussion in A. Kataev, Archive: proven: R.J. Crewther, Phys. Lett. B 397, 137 (1997). V. M. Braun, G. P. Korchemsky and D. Müller, Prog. Part. Nucl. Phys. 51, 311 (2003)

21 Which exactly constraints come from the Crewther relation? C Bjp (a s )CD NS (a s ) = 1+ β(a [ ] s) K 1 a s +K 2 a 2 s +K 3 a 3 s +... a s If it is valid at order a n s, then at the next order a n+1 s, we have (d n+1 C Bjp n+1 ] +interference terms)an+1 s = β 0 a s [K n a n s α 1 s : (d 1 C 1 ) : C F K 0 0 one constraint α 2 s : (d 2 C 2 ) : C 2 F,T C F,C F C A K 1 : C F two constraints α 3 s : (d 3 C 3 ) : C 3 F,C2 F C A,C F C 2 A,C2 F T,C FC A T,C F T 2 K 2 : C 2 F,C FC A,C F T three constraints

22 At last, at O(α 4 s) there exist exactly 12 color strtuctures: C 4 F, C3 F C A, C 2 F C2 A, C FC 3 A, C3 F T Fn f, C 2 F C AT F n f, C F CA 2T Fn f, CF 2T2 F n2 f, C FC A TF 2n2 f, C FTF 3n3 f, dabcd F d abcd A while the coefficient K 3 is contributed by only 6 color structures:, n fd abcd F d abcd F C F T 2,C F C 2 A,C 2 F T,C F C A T,C 2 F C A,C 3 F Thus, we have 12-6 = 6 constraints on the difference d 4 (C Bjp ) 4 3 of them are very simple: the above difference cannot contain CF, 4 d abcd F d abcd A remaining three are a bit more complicated n f d abcd F d abcd F

23 C 4 F d n abcd F d abcd F f d 13 R d abcd F d 4 (1/C Bjp ) ζ ζ 3 16 ζ ζ ζ ζ 5 d abcd A d R ζ ζ 5 C F T 3 f ζ ζ C 2 F T f ζ ζ 5 + 3ζ C F T 2 f C A C 3 F T f ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ 5 C 2 F T fc A ζ ζ ζ ζ ζ ζ ζ 7 C F T f C 2 A ( ) ( ) ζ ζ ζ ζ 7 ( ) ( ) ζ ζ ζ ζ 7 C 3 F C A ζ ζ ζ ζ ζ 5 C 2 F C2 A ζ ζ ζ ζ ζ ζ 7 C F C 3 ( ) A ( ) ( ) ( ) ζ ζ ζ ζ ( ) 7 ( ) ( ) ( ) ζ ζ ζ ζ 7

24 C Bjp (α s )C NS D (α s) = 1 + β(α s) α s C F [ K 1 α s + K 2 α 2 s + K 3α 3 s +... ] K 1 = ζ 3 K 2 = n f T ( ζ 3) +C A ( ζ 3) +C F ( ζ 3 15ζ 5 ) K 3 = n 2 f T2 ( ζ 3+5ζ 5 ) +CA 2 ( ζ ζ C F n f T ( ζ ) 2 ζ 5+9ζ3) 2 +C A n f T ( (2521) 36 ζ 3 125) 3 ζ 5 2ζ3) 2 +C A C F ( CF 2 ( ζ ζ ) 4 ζ2 3) 96 ζ 3 ( ζ ζ 7) 4 ζ 7)

25 Comments: The CBK test is highly non-trivial: four-loop box-type diagrams (in propagator kinematics) versus five loop propagators No IR-trickery is neccessary in calculation of C Bjp (a s ) final 4-loop p-integrals are much simpler for OPE (2 instead of 3 squared propagators inside) As a result we have been able to check that C Bjp (a s ) is indeed gauge-independent (the Adler finction was computed in the simplest Feynman gauge only!) Technical note: in the course of our calculations we have had to extend the Larin treatment of Hooft-Veltman γ 5 at 4-loop level (a natural object for the dim. reg., which really appears in the course of calculations, is γ [µνα] instead of γ 5 γ µ with anticommuting γ 5 ; the mismatch should be corrected by the Larin factor)

26 CBK relation between D = D NS +D SI and C GLS ( D NS + d SI 3 a3 s + ) ( dsi 4 a4 s C NS GLS + csi 3 a3 s + csi 4 s) a4 = 1 + β(α [ s) K NS + a 3 s α KSI 3 n d abc F dabc ] F f s d R note that CGLS NS = CBjp due to the chiral invariance. with β(α s) αs d SI 3 = n f d abc c SI 3 = n f d abc β 0 a s +..., β 0 = C A T f 3 F dabc F d R F dabc F d R d SI 3,1, dsi c SI 3,1, csi 4 = n f d abc 4 = n f d abc ( F dabc F d R ( F dabc F d R ) C F d SI 4,1 + C Ad SI 4,2 + T Fd SI 4,3 ) C F c SI 4,1 + C Ac SI 4,2 + T Fc SI 4,3 rhs of depends on only 1 unknown parameter, K SI 3, thus we have 3-1 =2 constraints on three coefficients in d SI 4. The coeffcients csi 3 and dc SI 3 and are known from nineties, the results for O(α 4 s ) contributions D S and C SI GLS were obiained by P. Baikov, K.Ch. and J. Kühn and J. Rettinger in (the second calculation) in

27 CBK relation at work: (a historical piece of evidence from a talk at an internal meeting in December of 2010) Obvious solution of these constraints reads: d SI 4,1 = 3 2 csi 3,1 c SI 4,1 = ζ ζ 5 8 d SI 4,2 = c SI 4, KSI 3,1 d SI 4,3 = c SI 4, KSI 3,1 All 2 constraints are met identically! (which means as many as 2*7=14 separate constraints on numbers in front of ζ 3,ζ3,ζ 2 4,ζ 4 ζ 3,ζ 5,ζ 7, n 13 m one: 64! ). For the moment we use the prediction from the CBK relation and testing our calculation...

28 CBK relation at work: (a historical piece of evidence from a talk at an internal meeting in December of 2010) Obvious solution of these constraints reads: d SI 4,1 = 3 2 csi 3,1 c SI 4,1 = ζ ζ 5 8 d SI 4,2 = c SI 4, KSI 3,1 d SI 4,3 = c SI 4, KSI 3,1 All 2 constraints are met identically! (which means as many as 2*7=14 separate constraints on numbers in front of ζ 3,ζ3,ζ 2 4,ζ 4 ζ 3,ζ 5,ζ 7, n 13 m one: 64! ). For the moment we use the prediction from the CBK relation and testing our calculation of D SI Needless to say, that the problem with the coefficent in our calculation of DSI at O(α 4 s ) was found and fixed in full agreement with the result obtained from the CBK prediction

29 Conclusions conformal symmetry based CBK relations do provide higly non-trivial and very usefull constraints on VVA NS triangle amplitude these constraints have been successfully tested at five loops a kind of CBK relation for VVA SI triangle amplitude (with the non-abeliean anomaly inside!) would be very useful. If exists it would connect the Ellis- Jaffe sum rule and the (anomalous) 2-point correlator A SI α (x)a SI β (0) with A si = f ψ f γ 5 γ α ψ f being (anomalous!) flavour singlet axial current (appears in QCD after decoupling the top quark from the weak neutral current)