Multiloop QCD & Crewther identitiesmaa29312/conference/... · 2014. 8. 18. · •massless QCD propagators (e.g. gluon self-energy in the Landau gauge, ... Starting object: the polarization

Multiloop QCD & Crewther identities

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based on works of “Karlsruhe-Moscow group” (2001 – 2014 - . . . )

Pavel Baikov (MSU), Johannes Kühn (KIT)

Konstantin Chetyrkin (KIT)

CONFORMAL SYMMETRY IN FOUR-DIMENSIONAL FIELD THEORIES,Regensburg, 16.07.2014

• 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 verynontrivial 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(αs4)and its compliance with the CBK-relation /new result!/

• open problem: a kind of CBK relation for V V Asi triangle amplitude withAsi =

∑

f ψ̄fγ5γαψf being (anomalous!) flavour singlet axial current? Wouldbe of great use for the Ellis-Jaffe sum rule and < AsiAsi > correlator (do appear in QCD

in the formal limit of mt → ∞)

⋆ ignited by: S.A. Larin, The singlet contribition to the Bjorken SR for polarized DIS, arxiV:1303.4021

multiloop /in pQCD but not only!/ problems reducible to

massless propagators (p-integrals for brevity)

• 2-points correlators at large energies (massless term + O(m2q) corrections)via the optical theorem lead to:

R(s) = σtot(e+e−→ hadrons)/σ(e+e−→ µ+µ−)

semi-leptonic τ -decays

Γ(Z → hadrons)

Γ(H → b̄b) 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)

R(s) from p-integrals

Starting object: the polarization operator of EM quark current jµ = eqq̄γµq

Πµν(q) = i

∫

dxeiqx〈0|T [ jvµ(x)jvν(0) ]|0〉 = (−gµνq

2 + qµqν)Π(q2)

related to R(s) throughR(s) ≈ ℑΠ(s− iδ)

Π is not completely physical due to a divergency of T (jvµ(x)jvν(0)) at x → 0, as a result its

normalization mode and corresponding evolution equation reads ((as ≡ αs/π), massless QCD)

Π = Zem

+ ΠB(−Q

2, α

Bs )

(

µ2 ∂

∂µ2+ β(as)as

∂

∂as

)

Π = γem(as)

At first sight, it would be advantageous to avoid this by considering (obviously RG invariant!) Adler

function defined as D = Q2 ∂∂Q2

Π0 and which is related to R(s) in a unique and simple way

R(s) ↔ D(Q) ⇐= Adler function ≡ Q2 d

dQ2Π(q

2) = Q

2∫

R(s)

(s + Q2)2ds

BUT, this is not true!

The reason: O(αsL) /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 rewritethe RG equation for Π as follows:

For massless (L + 1) loop Π0(L = lnµ2

Q2, as) RG equation amounts to

∂

∂LΠ0 = γem(as)−

(

β(as)as∂

∂as

)

Π0

ր տ

(L+1) loop anom. dim.L-loop integrals only contributedue to the factor of β(as)

If one knows the rhs to αLs , then one could trivially construct the Adler function with the same

accuracy!

Anomalous dimensions (as well as β-functions are simple (no-scale) polynomilasin αs /at least in MS-like schemes/ =⇒ one loop could be always ”undone”with so-called Infrared Rearrangement trick

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 AnomalousDimension) 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 expressedthrough pole and constant terms of some L-loop p-integrals

THEOREM ( Corollary ) is our key tool for multiloop RG calculations as it

reduces the general task of evaluation of (L+1)-loop UV counterterms (absrptivepart of (L+1)-loop 2-point massless correlators) to a well-defined and clearly posedpurely mathematical problem: the calculation of L-loop p-integrals (that is masslesspropagator-type FI’s).

In the following we shall refer to the latter as the L-loop Problem.

1. 1-loop Problem is trivial.

2. the 2-loop Problem was solved after inventing and developing the Gegenbauerpolynomial technique in x-space (GPTX) (K.Ch.,F. Tkachov (1980); further importantdevelopments in works by D. Broadhurst and A. Kotikov ).

GTPX is applicable to analytically compute some quite non-trivial three and evenhigher loop p-integrals. However, in practice calculations quickly get clumsy, especiallyfor diagrams with numerators. . Nevertheless, it proved to be very usefull in cases ofscalar diagrams with many multilinear vertexes /appear frequently in supersymmetrictheories/

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)

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 2000–2012).

This achievement resulted to a host of various important 3- and 4-four loop calculationsperformed 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 in1996 and using “massive” way /van Ritbergen, Vermaseren,and Larin/; the reasonwas too complicated combinatorics of the IR reduction

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 of1/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,RV V (s) and a closely related quantity – Z-decay rate into hadrons have been extendedby one more loop (that is to order α4s, which corresponds 5-loop for the Adler function).

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) 01200

Phys.Rev.Lett. 95 (2005) 012003

Phys.Rev.Lett. 96 (2006) 012003

Phys.Rev.Lett. 97 (2006) 061803

Phys.Rev.Lett.101:012002,2008

Phys.Rev.Lett. 102 (2009) 212002

Phys.Rev.Lett.104:132004,2010

Phys.Rev.Lett. 108 (2012) 222003

JHEP 1207 (2012) 017

Phys.Lett. B714 (2012) 62-65

Example of Phenomenological Relevance

• With previous α3s calculation⋆ of ΓhZ, the theoretical errors were

comparable with the experimental ones and, in despair, everybodywas using the famous Kataev&Starshenko /1993/ estimation of theα4s term which (incidentally?) has happened to be quite close to the true number!

• After our calculations the situation has become significantly better,especially for ΓhZ, where the the theoretical error was reduced by afactor of four!

• α4s correction to the τ decay rate has decreased the theoretical errorand 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)

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 expandeddeeply enough in ǫ = 2−D/2 (this is exactly was happened with the very first calculation ofthe Adler function in O(α3s) /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 ofthe sorting rather intensively and it showed itself mainly with TFORM witha probability of occurring proportional to at least W 3 if W is the number ofworkers.” (citation of the FORM creator and leading maintainer Jos Vermaseren)

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 α3s: 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

Five loop RG

Here the situation is not so good: since 2002 we have performed many 5-loopRG calculations:

and (almost) no one has yet been confirmed in full by an independentcomputation. An exception is quark and gluon form factors to three loops inmassless QCD: reduction to masters was done in 2 independent ways (withBAICER 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 — allindependent — 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)

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)

DIS Sum Rules

• the polarized Bjorken sum rule (as ≡αsπ )

Bjp(Q2) =

∫ 1

0

[gep1 (x,Q2)− gen1 (x,Q

2)]dx =1

6|gAgV|CBjp(as)

Coefficient function CBjp(as) is fixed by OPE of two non-singlet vector currents (upto power suppressed corrections)

i

∫

TV aα (x)Vbβ (0)e

iqxdx|q2→∞ ≈ CQ,abcαβρ A

cρ(0) + . . . (1)

where

CQ,abcαβρ ∼ idabcǫαβρσ

qσ

Q2CBjp(as)

and Q2 = −q2

• the Gross-Llewellyn Smith sum rule

GLS(Q2) =1

2

∫ 1

0

F νp+νp3 (x,Q2)dx = 3CGLS(as)

the function CGLS(as) comes from operator-product expansion of the axial and vectornon-singlet currents

i

∫

TAaµ(x)Vbν (0)e

iqxdx|q2→∞ ≈ CV,abµνα Vα(0) + . . .

where CV,abµνα ∼ iδabǫµναβ

qβ

Q2CGLS(as)

Note that both sum rules are unambiguous/modulo higher twists!/

predictions of QCD which in principle could be confronted withexperimental data

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 coulduse techniques developed for R(s)

Bjp and GLS GLS only

+ 5

q

+ 3 4 6

q

At order α3s both CF’s were computed in early nineties.The next order is contributedby about 54 thousand of 4-loop diagrams . . . (cmp. to ≈ 20 thousand of 5-loopdiagrams contributing to R(s) at the same order)

The Crewther relation states that in the conformal invariant limit (β ≡ 0) CBjp(as)is related to the (nonsinglet) Adler function via the following beautiful equality

CBjp(as)DNS(as))|c−i = 1

its generalization for real QCD reads:

CBjp(as)DNS(as) = 1 +

β(as)

as

[

KNS = K1 as + K2 a2s + K3 a

3s + . . .

]

Note that similar relation connects also the CF of the Gross-Llewellyn Smith sum ruleto the full Adler function, to be discussed later.

Main ingredients of the derivation: the AVV 3-point function and constraints on itfrom (approximate) conformal invariance + Adler-Bardeen anomaly theorem

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:1011.5248; most recent discussion in A. Kataev,Archive: 1305.4605

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)

Which exactly constraints come from the Crewther relation?

CBjp(as)CNSD (as) = 1 +

β(as)

as

[

K1 as +K2 a2s +K3 a

3s + . . .

]

If it is valid at order ans , then at the next order an+1s , we have

(dn+1 − CBjpn+1 + interference terms) a

n+1s = β0 as

[

Kn ans

]

α1s : (d1 − C1) : CF⇐⇒K0 ≡ 0← one constraint

α2s : (d2 − C2) : C2F , T CF , CF CA⇐⇒K1 : CF ← two constraints

α3s : (d3 − C3) : C3F , C

2FCA , CFC

2A , C

2FT ,CFCAT ,CFT

2

m

K2 : C2F , CFCA, CFT← three constraints

At last, at O(α4s) there exist exactly 12 color strtuctures:

C4F , C3FCA , C

2FC

2A , CFC

3A , C

3FTFnf , C

2FCATFnf ,

CFC2ATFnf , C

2FT

2Fn

2f , CFCAT

2Fn

2f , CFT

3Fn

3f , d

abcdF d

abcdA , nfd

abcdF d

abcdF

while the coefficient K3 is contributed by only 6 color structures:

CFT2 , CF C

2A , C

2F T ,CF CA T ,C

2F CA , C

3F

Thus, we have 12-6 = 6 constraints on the difference

d4 − (CBjp)4

3 of them are very simple: the above difference cannot contain

C4F , dabcdF d

abcdA nfd

abcdF d

abcdF

remaining three are a bit more complicated

d4 (1/CBjp)4

C4F41572048 +

38 ζ3

41572048 +

38 ζ3

nfdabcdF d

abcdF

dR−

1316 − ζ3 +

52 ζ5 −

1316 − ζ3 +

52 ζ5

dabcdF dabcdA

dR

316 −

14 ζ3 −

54 ζ5

316 −

14 ζ3 −

54 ζ5

CFT3f −

6131972 +

20354 ζ3 +

53 ζ5 −

605972

C2FT2f

57131728 −

58124 ζ3 +

1256 ζ5 + 3 ζ

23

869576 −

2924 ζ3

CFT2fCA

3408435184 −

10453288 ζ3 −

1709 ζ5 −

12 ζ

23

16528320736 +

43144 ζ3 −

512 ζ5 +

16 ζ

23

C3FTf1001384 +

9932 ζ3 −

1254 ζ5 +

1054 ζ7 −

4732304 −

39196 ζ3 +

14524 ζ5

C2FTfCA3235713824 +

1066196 ζ3 −

515548 ζ5 −

334 ζ

23 −

1058 ζ7 −

1730913824 +

1127144 ζ3 −

95144 ζ5 −

354 ζ7

CFTfC2A −

(··· )(··· ) +

860972 ζ3 +

18805288 ζ5 −

112 ζ

23 +

3516 ζ7 −

(··· )(··· ) −

5964 ζ3 +

1855288 ζ5 −

1112 ζ

23 +

3516 ζ7

C3FCA −25332 −

139128 ζ3 +

225532 ζ5 −

115516 ζ7 −

87014608 +

110396 ζ3 −

104548 ζ5

C2FC2A −

59214118432 −

43925384 ζ3 +

650548 ζ5 +

115532 ζ7 −

43542555296 −

1591144 ζ3 +

559 ζ5 +

38516 ζ7

CFC3A

(··· )(··· ) −

(··· )(··· ) ζ3 −

779951152 ζ5 +

60532 ζ

23 −

38564 ζ7

(··· )(··· ) −

(··· )(··· ) ζ3 −

125451152 ζ5 +

12196 ζ

23 −

38564 ζ7

CBjp

(αs)CNSD (αs) = 1 +

β(αs)

αsCF

[

K1 αs + K2 α2s + K3 α

3s + . . .

]

K1 = −218 + 3ζ3

K2 = nfT (16324 −

193 ζ3)

+CA (−62932 +

22112 ζ3)

+CF (39796 +

172 ζ3 − 15ζ5)

K3 = n2fT

2 (−30718 +20318 ζ3 + 5ζ5)

+C2A (−4060432304 +

18007144 ζ3 +

297548 ζ5 −

774 ζ

23)

+CFnfT (−77291152 −

91716 ζ3 +

125)2 ζ5 + 9ζ

23)

+CAnfT (10559 −

(2521)36 ζ3 −

125)3 ζ5 − 2ζ

23)

+CACF (997572304 +

8285)96 ζ3 −

(155512 ζ5 −

1058 ζ7)

+C2F (2471768 +

618 ζ3 −

7158 ζ5 +

3154 ζ7)

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 CBjp(as)

• final 4-loop p-integrals are much simpler for OPE (2 instead of 3 squaredpropagators inside)

• As a result we have been able to check that CBjp(as) 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 Larintreatment of Hooft-Veltman γ5 at 4-loop level (a natural object for the dim. reg., whichreally appears in the course of calculations, is γ[µνα] instead of γ5γ

µ with anticommuting γ5; the

mismatch should be corrected by the Larin factor)

CBK relation between D = DNS +DSI and CGLS

(

DNS + dSI3 a3s + d

SI4 a

4s

) (

CNSGLS + cSI3 a

3s + c

SI4 a

4s

)

=

1 +β(αs)

αs

[

KNS + a3s KSI3 nf

dabcF dabcF

dR

]

⋆

note that CNSGLS ≡= CBjp due to the chiral invariance.

with β(αs)αs ≡ −β0as + . . . , β0 =1112CA −

Tf3

dSI3 = nfdabcF d

abcF

dRdSI3,1, d

SI4 = nf

dabcF dabcF

dR

(

CFdSI4,1 + CAd

SI4,2 + TFd

SI4,3

)

cSI3 = nfdabcF d

abcF

dRcSI3,1, c

SI4 = nf

dabcF dabcF

dR

(

CFcSI4,1 + CAc

SI4,2 + TFc

SI4,3

)

rhs of ⋆ depends on only 1 unknown parameter, KSI3 , thus we have 3-1 =2 constraints on three

coefficients in dSI4 . The coeffcients cSI3 and dc

SI3 and are known from nineties, the results for O(α

4s)

contributions DS and CSIGLS were obiained by P. Baikov, K.Ch. and J. Kühn and J. Rettinger in

2010-2011 (the second calculation) in 2010-2011.

CBK relation at work:

(a historical piece of evidence from a talk at an internal meeting inDecember of 2010)

Obvious solution of these constraints reads:

dSI4,1 = −3

2cSI3,1 − c

SI4,1 = −

13

64−ζ34+

5ζ58

dSI4,2 = −cSI4,2 +

11

12KSI3,1 d

SI4,3 = −c

SI4,3 +

1

3KSI3,1

All 2 constraints are met identically! (which means as many as 2*7=14 separate

constraints on numbers in front of ζ3, ζ23 , ζ4, ζ4ζ3, ζ5, ζ7,

nm one:

1364! ). For the moment

we use the prediction from the CBK relation and testing our calculation . . .

CBK relation at work:

(a historical piece of evidence from a talk at an internal meeting inDecember of 2010)

Obvious solution of these constraints reads:

dSI4,1 = −3

2cSI3,1 − c

SI4,1 = −

13

64−ζ34+

5ζ58

dSI4,2 = −cSI4,2 +

11

12KSI3,1 d

SI4,3 = −c

SI4,3 +

1

3KSI3,1

All 2 constraints are met identically! (which means as many as 2*7=14 separate

constraints on numbers in front of ζ3, ζ23 , ζ4, ζ4ζ3, ζ5, ζ7,

nm one:

1364! ). For the moment

we use the prediction from the CBK relation and testing our calculation of DSI

Needless to say, that the problem with the coefficent 1364 in our calculation of DSI at O(α4s) was found

and fixed in full agreement with the result obtained from the CBK prediction

Conclusions

• conformal symmetry based CBK relations do provide higly non-trivial andvery usefull constraints on V V ANS triangle amplitude

• these constraints have been successfully tested at five loops

• a kind of CBK relation for V V ASI triangle amplitude (with the non-abelieananomaly inside!) would be very useful. If exists it would connect the Ellis-Jaffe sum rule and the (anomalous) 2-point correlator

〈ASIα (x)ASIβ (0)〉

with Asi =∑

f ψ̄fγ5γαψf being (anomalous!) flavour singlet axial current(appears in QCD after decoupling the top quark from the weak neutralcurrent)