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New realization of Conformal Symmetry Breaking in p erturbative QCD

A. L. Kataev1
2 1

S. V. Mikhailov2

INR, Moscow, Russia BLTPh, Dubna, Russia 19th QFTHEP Workshop, 2010, SINP MSU 100th Anniversary of Sergey Nikolaevich Vernov (1910-1982) Golitsyno, Moscow Region, Septemb er, 8-15 2010

13 September 2010


Plan of Presentation

Introduction review of results of advanced QCD analytical calculations Perturbative violation of the conformal symmetry in the quark-parton model eects of non-zero -function New form of the relation between rules in Euclidean region p ower

e +e

-

series in

-characteristics and DIS sum (a s )/a s term
11 2 A ) and new (2010) results

Banks-Zaks condition 0 (NF ) = 0, (NF = constraints for Baikov, Chetyrkin, Kuhn

C

form of

Conclusions


sntrodution eview of results of dvned gh nlytil lultion


Results of advanced QCD analytical calculation 1

Main quantities for e + e - annihilation process (Novosibirsk- Russia; NS Beijing- China), Adler function (nonsinglet) DA :

D

NS A(

Q

2) =

Q

2 0



(s +

R (s ) ds Q 2 )2
,

=3

f

Qf2 · DA (a Q
2 )/

s)

D A ( as

)=1+

Bjorken polarized sum rule intermediate/low Q 2

Bjp

n

n as dn

a

s = s ( µ2 =

measured at CEBAF at
2 ) - ln ( , 1

Bjp (Q

2) =

1 0

lp g1 (x , Q

g xQ
n

2)

dx

=

1 gA · 6

C

NS (as )

C

NS (as ) = 1 +

n as cn

Gross-Llewellyn Smith sum rule - measured in

n related with as -coecients of pQCD series for

N

DIS

GLS(Q

2) =

1 0

F

3



N (x ,

Q 2 )dx


d d d

4 Chetyrkin,Kataev,Tkachov (1979); d3 Gorishny, Kataev, Larin (1988-90, presented 1990, published 1991);
4

1= 2

CF

3

;

=

dabcd dabcd F A dR C4 4157 + F 2048 C2 T2 N FF C3 CA F C2 TF N F CF T2 N F C2 C2 FA CF TF N CF C3 A



Baikov, Chetyrkin, Kuhn (2010), new results

dabcd dabcd - 13 - + 3 1 5 F - 3 - 5 + N F F 3 16 4 4 dR 16 3 1001 99 125 105 3 + C3 TF NF + 3 - 5 + 7 F 8 384 32 4 4 2 5713 - 581 + 125 + 3 2 - C T3 N3 6131 - 3 5 FFF 3 F 1728 24 6 972 139 2255 1155 253 + 3 - 5 + 7 + 32 128 32 16 32357 10661 5155 33 2 105 + 3 - 5 - - 7 + F CA 13824 96 48 43 8 10453 170 12 340843 2C - 3 - 5 - 3 + FA 5184 288 9 2 592141 43925 6505 1155 - - 3 + 5 + 7 + 18432 384 48 32 4379861 8609 18805 11 2 35 2 + 3 + 5 - + 7 F CA - 20736 72 288 2 3 16 52207039 456223 77995 605 2 385 - 3 - 5 + - 7 248832 3456 1152 32 3 64

5 5 + 2
+

203 5 3 - 5 - 54 3

+


c1 c2

3 = - CF 4



Gorishny,Larin (1986);
=

c3



Larin, Vermaseren (1991)

c4

dabcd dabcd F A dR +CF T3 N3 FF
+C2 F +C2 F +CF +CF +C2 F +C3 F

direct form of Baikov, Chetyrkin, Kuhn (2010) results

dabcd dabcd 13 + - 5 3 1 5 F + 3 + 5 + NF F 3 5 16 4 4 dR 16 2 605 165283 43 5 12 + CF CA T2 N2 - - 3 + 5 - 3 FF 972 20736 144 12 6 29 839 451 145 265 T2 N2 - 576 + 24 3 + C3 TF NF 2304 + 96 3 - 24 5 F FF 87403 CA TF NF - 13824 - 1289 3 + 275 5 + 35 7 144 144 4 1238827 59 1855 11 2 35 2T N CA F F 41472 + 64 3 - 288 5 + 12 3 - 16 7 2 C3 - 8004277 + 1069 3 + 12545 5 - 121 3 + 385 7 A 248832 576 1152 96 64 1071641 1591 1375 385 C2 55296 + 144 3 - 144 5 - 16 7 A 3707 4823 CA - 4608 - 971 3 + 1045 5 + C4 - 2048 - 3 3 F 96 48 8
-


Results of advanced QCD analytical calculation 2

QCD -function a measure of the conformal symmetry breaking eects, (a s )/a s x renormalization of trace of energy momentum tenzor
( as ) 0 2 = = = + 3 = + + + - µ2

11 17 2 1 , 1 = C - 5 CA TF NF - 4 CF TF NF 12 24 A 12 2857 3 79 1 C - 1451 C2 TF NF - 205 CA CF TF NF + 864 CA T2 N2 + 32 C2 TF NF FF F 3456 A 1728 A 576 11 CF T2 N2 ; FF 144 150653 11 39143 17 23 3 - 3 C4 + - + 3 C3 TF NF + C TF NF A A 124416 576 20735 96 128 F 7073 41 11 1051 53 - 3 CF C2 TF NF - - CA C2 TF NF + 7776 CA T3 N3 A F FF 62208 144 1728 72 7 219 7 3965 + 3 C2 T2 N2 + + 3 CF CA T2 N2 FFF FF 10368 72 7776 36

d 2 d µ 2 as = - as 1 CA - 3 TF NF

2 3 0 + 1 as + 2 a s + 3 as

Tarasov,Vladimirov,Zharkov (1980)

77 CF T3 N3 - FF 15552 11 2 - 3 36 3

Notice appearance of 3

dabcd dabcd N2 F F F dR new SU (Nc )-group

5 11 - 3 144 12

dabcd dab A A dR

cd

+

2 13 - 3 9 6

dabcd dabcd N F A dR

F

van Ritbergen,Vermaseren,Larin (1997)
structures


erturtive gh violtion of the onforml symmetry of mssless qurkEprton model E soureX the proedure of renormliztion leds to nonEzero gh Efuntion gonforml symmetry is the generliztion of oinrЎ e symmetry ymmetry under following trnsformtions peEtime trnsltions x

= xµ + vorentz trnsformtions x µ = µ x peil onforml trnsformtions x µ + µ x 2 x µ = 1+2 x + 2 x 2
le trnsformtions x
µ

µ

µ

= x

µ


Generalized CBK relation in the

Conformal symmetry relates e + e - -annihilation Adler D-function with DIS sum rules (Bjorken/Gross-Llewellyn Smith) in the massless quark-parton model - [Crewther (1972)] Explicit role of conformal symmetry breaking eects factorization of QCD -function in the analogous massless QCD relation, discovered 3 at as level [Broadhurst, Kataev (93)]. Shown to be true in all orders by applying operator product expansion to the 3-point AVV triangle diagram in momentum p space [Gabadadze, Kataev (95), Kataev (96)- INR-09296]; Proved in coordinate x -space by [Crewther (97)] and [D. Mueller (97)], this pro of is in the review [V.Braun, G. Korchemsky, D.
Mueller (2003)]

MS

-scheme

DA (as

)в

NS (a ) = 1 + l s CSB (as ) (as ) CSB (as ) = P (as ), polynomial P (as ) = as m

C

1

m Km as

K1 = K1 [1, 0, 0]CF , K2 = K2 [2, 0, 0]C2 + K F

TF N

2 F -dep endence (!)

[1, 1, 0]CF

C

A

+

K

2

[1, 0, 1]CF

TF N

F

-

notice


Explicit form of CBK relation at

3 Validity of Generalized CBK at as -strong check of dierent as analytical calculations of the Adler function and DIS sum rules [Broadhurst,Kataev (93)]: K1 [1, 0, 0] = - 21 + 33 ; K2 [2, 0, 0] = 397 + 17 3 - 155 ; 8 96 2 K2 [1, 1, 0] = - 629 + 221 3 ; K2 [1, 0, 1] = 163 - 19 3 . Explicit demonstration 32 12 24 3 3
4 of -function factorization at as -level

a

s

3

and

a

s

4

K

[2, 0, 1]C2 TF NF + K3 [1, 1, 1]CF CA TF NF + K3 [1, 0, 2]C F (Contain additional TF NF -terms) K3 [3, 0, 0] = 2471 + 61 3 - 715 5 + 315 7 ; 768 8 8 4 K3 [2, 1, 0] = 99757 + 8285 3 - 1555 5 - 105 7 2304 96 12 8 2 K3 [1, 2, 0] = - 406043 + 18007 3 + 2975 5 - 77 3 2304 144 48 4 917 125 7729 2 K3 [2, 0, 1] = - 1152 - 16 3 + 2 5 + 93 2 K3 [1, 1, 1] = 1055 - 2521 3 - 125 5 - 23 9 36 3 203 307 K3 [1, 0, 2] = - 18 + 18 3 + 55 agrees with BK(93)
3
Validity at

5 New explicit terms+ 1 known [Baikov, Chetyrkin, Kuhn (2010)]: 3 = 3 [3, 0, 0] 3 + 3 [2, 1, 0] 2 A + 3 [1, 2, 0] F 2 F F A

K

K

C

K

CC

K

2 FF

CC TN

2 F

+

New 3 gauge group contributions in 3 will not spoil factorization of the QCD 5 -function in as -they should b e multiplied by K1 [1, 0, 0]-coecient.

a

s

4 - strong check of advanced

a

s

4 analytical calculations!


xew form of the reltion etween e e Ehrteristis nd hs sum rules in iuliden region ! p ower series in (as )/as term Emore detiled understnding of struture of gh generliztion of [grewther reltion [ utevD wikhilov @HWEIHAE guw reltion
+-


Q: Is it p ossible to unravel structure of CSB ( s )term? Guess : Yes! [Kataev, Mikhailov (09-10)] CERN-PH-TH/2009-203; PoS 4 (RADCOR2009) 036, 2010 (prior learning [BChK2010] s results,

a

arXiv:1001.0728)

CSB

(as ) =

n k

1

Pn (as ) = P1 (as ) = as -
47 48

P
1

n (as ) Pn (as ), as (k ) k n as

a

C

F

- +

21 8

+ +

3

(=

K

1

[1, 0, 0] - BK-expansion coe.) +
5

+

3

CA

397 96

17 2 3

- 15

CF a

s+

O (a

3 s)

P2 (as ) = as CF 163 - 19 (P3 (as ) = O (as )- was unxed. 8 3 Relation obtained by: a) requiring TF NF -independence of Pn (as ) and absorption them into k (as )- coe. (leads to unique system of equations, which relates Pn to i and Ki ); b) using expansions of DA (as ) and C NS (as )coe. dn and cn (1 n 3) through 0 , 1 ([S.V. Mikhailov Quarks-2004 and JHEP(2007)]).


P1 (as ) =
(3)

More general structure : CSB = n1 (as ) Pn (as ) with as (r ) (k ) k kmr Pn (as ) = k 1 Pn as = r 1 Pn [k, m]CF CA as where r = k + m After learning the results of [Baikov, Chetyrkin, Kuhn (09-10)] the 4 guess was conrmed at as -level. We get additional 3 contributions:

n

C3 F

2471 768

+

61 8 3

-

715 8 5

+

315 4 7

+C +C

2 FA 2 FA

C

16649 1536 2107 192

-

11183 192 3

+

1015 24 5

-

105 8 7 2 3

+

99 2 4 3

C

+

2503 72 3

-

355 18 5

- 33 +

3 as

P2 (as ) =
(2)

C2 F CF

-

13597 384

-

2523 16 3 2 3

375 2 5

+ 27

2 3

+

CF C
(1)

A

1433 32

- 1 3 - 4 -

170 4 5

- 6

2 as 5

P3 (as ) =

307 2

+

203 2 3

+ 45

as


Higher order parts: the terms, leading in large p owers of

N

F

n n<10 Sn x = - 21 + 123 x + 2
2760448 243

-

17920

7

x

304 2 + - 9824 + 6496 + 320 3+ 5 3 3 9 93 1268480 48640 4 + - 280736320 + 89300480 + 5196800 3 5 243 3 - 9 5 2187 2187 81 5 + 10320047360 - 2327111680 - 507392000 - 1361920 6 3 5 7 6561 6561 729 3

326 3

-

x

x

x

+ +

x

Large NF -dependence of polynom, multiplied by (as )/as factor - BK (93); Here x = TF NF as /4. This gives us rst coecients of the expansion:

3723517199360 + 611395563520 3 + 50008268800 5 + 203714560 7 + 177147 177147 6561 27 48742400 7 + 485484017500160 - 59933178265600 - 5212730163200 - 9 3 5 27 1594323 1594323 59049 79559065600 14817689600 7616109282344960 726735764193280 7 - 9 8 + - + 3 729 243 1594323 1594323 195646580326400 5 + 1120185221120 7 + 316630630400 9 + 7821721600 11 9 177147 729 243 27

-

x

x

+

x

ЎCSB =

n1

( as )

n

a

s

(1) s , where n

(1)

n

=

Sn (n-1) F 4n Q

C

.


Is it possible to apply this NEW EXPRESSION for the conformal symmetry breaking term in practical QCD applications ? give NEW constraints between 5-loop results of advanced complicated computer calculations by [BChK (09-10)] Consider Mikhailov (04/07) representations for the DA (as ) coecients d 2 = 0 d2 [1] + d 2 [H] 2 d 3 = 0 d3 [2, 0] + 1 d3 [0, 1] + 0 d3 [1, 0] + d 3 [H] 3 2 d 4 = 0 d4 [3] + 1 0 d4 [1, 1] + 2 d4 [0, 0, 1] + 0 d4 [2] + 1 d4 [0, 1] + 0 d4 [1] + d 4 [H] and the similar representations for the C NS (as ) coecients cn . At the 3 order as it is possible to dene all coecients for d 2 , d 3 ( [Mikhalov Using similar representations for cn and the original Crewther relation, valid in the conformal-invariant limit of (as ) = 0 we get all coecients for c 2 , c 3 . [Kataev-Mikhailov (this work)]
(04/07)] Answer:

Question:


Concrete new relations:

for 3,4 and 5 -loop results from the lower loops ones: c 2 [0] + d 2 [0] = d 2 1

D A ( as c c

From the original
)в

C

NS |ci = 1 we get constraint l

Crewther
d 1d d 1d

relation

3 [ 0] +

4 [ 0] +

d d

3 [ 0] = 4

2 [ 0] = 2
- as - as

2 [ 0] -

3 [ 0] -

d4 1 3d 2 d 1

2 2 [0] + ( 2 [0]) +

d

d

4 1

P 1 ( as )

= = +

P

(1) 1+

as P

(2) 2 (3) 1 +s1

aP c

c2

[1] + d2 [1] + as

3 [1] + 3 [1] + 1 ( 2 [1] - 2 [1])

d

dc

d

2 as c4

[1] + d4 [1] + d1 (c3 [1] - d3 [1]) + d2 [0]c2 [1] + d2 [1]c2 [0]

P 2 ( as ) P 3 ( as )

= =

as P as P

(1) 2+ (1 ) 3

as P

(2) 2

= as

c d

2 3 [2] + 3 [2] + s

d

ac

4 [2] + 4 [2] + 1 ( 3 [2] - 3 [2])

d

dk

c

= - as

c

4 [3] + 4 [3]) ; Pn (

a

s ) = (-1)n as

cn [ n

- 1] + dn [n - 1]

Last two equations result from the chain of "renormalon"graphs These equations allow to get by another method the NF -independent coecients ( ( ( ( ( P11) , P12) , P21) . However to check these constraints for P13) , P22) some additional 5-loop calculations are needed (contribution from gluino - element of ( SUSY QCD ); Pn1) can extracted from cn [n - 1] and dn [n - 1]-results of [Broadhurst, Kataev (93)]


results from condition: 0 (TF N ) = 0 gives relation TF N = 11 CA F F 4 Our new representation for CSB , which is POLYNOMIAL in ( (as )/as ), leads to the identity:
Application: [BChK(2010)] Banks-Zaks (1982)

New cross-checks for

d 4 + c 4 |BZ = d 4 [H] + c 4 [H](gs) -1 (BZ ) [d 4 [H, I] + c 4 [H, I]] - 2 (BZ ) [d 4 [H, H, I] +

c

4

[ H, H, I] ]

1 where 1 (BZ ) = - 16 7C2 + 11CF CA A 11 77 2 (BZ ) = -C3 1127 - CF C2 192 + 128 C2 CA A 1536 A F 333 4 Thus d 4 [H] + c 4 [H](gs) = - 1024 CF + C2 C2 525 - 81 3 + F A 512 16 (2) Next: d 4 [H, I] + c 4 [H, I] = P1 - (c 3 [H, I] - d 3 [H, I]) d1

C3 C F

99 A 64

=

CF C

A

-

47 48

+

3

+

C

2 1117 F 96
(1)

+ 7 3 - 15 4
F
21 8

5 3

d

4

[H, H, I] +

c

4

[H, H, I] = -P1 = -C

- 3


-1 (BZ ) [d 4 [H, I] +

c

4

[H, I]] =

CF C
c
4

(no 5 but they exist in

C2 C F CF C

d
4

3 7 A 16 3

-

329 768

+
671 64 3 165 16 5 33 128 3

4

and

!)

-2 (BZ ) [d 4 [H, H, I] +
3 A

2 A

-

3295 1536

+

471 64 3

-

c

105 16 5

+
2 A

C3 C F
-

A

-

3553 1536

+

-

[H, H, I]] =

-

7889 4069

+

1127 512 3

+

C2 C F C

539 512

+

77 64 3

+

C3 CA F
-

231 1024

-

d 4 (BZ C2 C2 AF

Finally:

) + c 4 (BZ ) = - 7 - 3337 + 2 3 - 1536

333 4 3 1024 F + A F 105 3 16 5 + A F

CC CC

1661 - 3072 + 28931 - 12288 +

1309 128 3 1351 512 3

165 16 5

+

These results are satised for Baikov-Chetyrkin-Kuhn (2010) 5-loop symbolic analytical results - additional conrmation of the correctness of the results of advanced analytic calculations


Conclusions:

1. We present new QFT (QCD and QED) expression for the conformal symmetry breaking term in the relation between e + e - annihilation and DIS sum rules - addition to Crewther unity: n (r ) km CSB = n1 (as ) r 1 P n [k, m]C F C A as where r = k + m CKM-relation 4 and x there coecients at as - level and in the large NF expansion 9. up to as 2. Applications: new constraints between d 4 + c 4 [BChK(2010)] new results. for 0 = 0 Banks-Zaks condition. 3. Odd -function studies- conrm that at BZ condition 2 7 and 3 disappear reason: proportional to 0 -term conrmation of observation of Baikov, Chetyrkin, Kuhn (10) 2n+1 -studies - possible link to N = 4 SUSY YM oriented theoretical multiloop studies ? 4. In QED diagrammatic representations of new generalizations is straightforward. 5. QCD applications- form-factors ? Summations of power series with expansion parameter being RG -function, other possible applications ?
Analytical exp eriments detect detailed structures of QFT mo dels