Singlet Contributions to the Vector Correlator Computational - PowerPoint PPT Presentation

Singlet Contributions to the Vector Correlator Computational Theoretical Particle Physics B AC KA SFB TR9 Johann H. K¨ uhn in collaboration with P. Baikov, K. Chetyrkin, and J. Rittinger based on P. Baikov, K. Chetyrkin, and J. K., Nucl.Phys.Proc.Suppl.205-206:237-241,2010; P. Baikov, K. Chetyrkin, J. K. and J. Rittinger, in preparation

Outline • status of vector (VV) correlator in massless QCD: singlet versus nonsinglet • tool-box • singlet contribution O ( α s 4 ) for a generic gauge group • QED β -function in five loops • phenomenological implications for σ tot ( e + e − → hadrons ) • Gross-Llewellyn Smith sum rule in O ( α s 4 ) and test of the constraints on the singlet part of the Adler function coming from the Crewther relation • Conclusions

“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 − ) 2 (via unitarity) R ( s ) ≈ ℑ Π( s − iδ ) = � Π( Q 2 ) ≈ e iqx � 0 | T [ j v µ ( x ) j v µ (0) ] | 0 � dx = Adler function ≡ Q 2 d � R ( s ) d Q 2 Π( q 2 ) = Q 2 R ( s ) ↔ D ( Q ) ⇐ ( s + Q 2 ) 2 ds ( a s ≡ α s /π, µ = Q, Q 2 ≡ − q 2 ) � � r i a s ( s ) i , d i a s ( Q ) i , R ( s ) = 1 + D = 1 + i ≥ 1 i ≥ 1

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

Recall non-singlet results (PLR 101 (2008) 012002): R NS = 1 + a s + (1 . 9857 − 0 . 1152 n f ) a 2 s ( − 6 . 63694 − 1 . 20013 n f − 0 . 00518 n 2 f ) a 3 s +( − 156 . 61 + 18 . 77 n f − 0 . 7974 n 2 f + 0 . 0215 n 3 f ) a 4 s Impact on α s from Z − decays: α s ( M Z ) NNLO = 0 . 1185 ± 0 . 0026 exp ± 0 . 002 th O ( α 3 s ) : Including the α 4 s term leads to an increase of δα s ( M Z ) = 0 . 0005 and to four-fold decrease of the theory error! α s ( M Z ) NNNLO = 0 . 1190 ± 0 . 0026 exp ± 0 . 0005 th O ( α 4 s ) : Impact on α s from τ -decays (FO and CI): α s decreased by δα s ( M Z ) = 0 . 0016 δα s ( M Z ) NNNLO = 0 . 1202 ± 0 . 0019 exp

Massless Correlators: Technicalities Π related to the corresponding absorptive part R ( s ) through R jj ( s ) ≈ ℑ Π jj ( s − iδ ) RG equation Π jj = Z jj + Π B are ( − Q 2 , α B are ) s � � µ 2 ∂ ∂µ 2 + β ( a s ) ∂ Π = γ jj ( a s ) ∂a s extremely useful for determining the absorptive part of Π jj

For Π at 5 loop ∂ β ( a s ) ∂ � � ∂ log( µ 2 )Π = γ jj ( a s ) − Π ∂a s ր տ 4-loop integrals at O ( α 3 s ) only anom.dim. at O ( α 4 contribute s ) due to the factor of 5 loop integrals β ( a s ) ∂ ∂a s = O ( a s ) • to find Log-dependent part of Π at 5-loops one needs 5-loop anomalous dimension γ jj and 4-loop Π (BUT! including its constant part) • 5-loop anom.dim. reducible to 4-loop p-integrals

Tool-box for massless correlators at α 4 s : • IRR / Vladimirov, (78)/ + IR R ∗ -operation /Chetyrkin, Smirnov (1984)/ + resolved combinatorics /Chetyrkin, (1997)/ • reduction to Masters: “direct and automatic” construction of CF’s through 1 /D expansion within the Baikov’s representation for Feynman integrals ( Phys. Lett. B385 (1996) 403; B474 (2000) 385; Nucl.Phys.Proc.Suppl.116:378-381,2003 ) • computing: MPI-based (PARFORM) as well as thread-based (TFORM) versions of FORM Vermaseren, Retey, Fliegner, Tentyukov, ...(2000 – . . . )

Results Singlet contribution to the Adler function (Last missing term!) � ∞ � � D SI ( Q 2 ) = d R d SI a i s ( Q 2 ) i i =3 � 11 = d abc d abc = d abc d abc 192 − 1 � d SI , d SI C F d SI 4 , 1 + C A d SI 4 , 2 + T n f d SI � � 8 ζ 3 3 4 4 , 3 d R d R 4 , 1 = = − 13 64 − ζ 3 4 + 5 ζ 5 4 , 3 = − 149 576 − 13 32 ζ 3 + 5 16 ζ 5 + 1 8 ζ 2 d SI d SI 8 , 3 − 3893 4608 + 169 128 ζ 3 − 45 64 ζ 5 − 11 32 ζ 2 d SI = 4 , 2 3

Phenomenological implications for σ tot ( e + e − → hadrons ) Numerically: � Q 2 1 + a s + a 2 � R ( s ) = 3 s (1 . 986 − 0 . 1153 n f ) f f a 3 − 6 . 637 − 1 . 200 n f − 0 . 00518 n 2 � �� + s f 2 �   � � 1 . 2395 a 3 s + ( − 17 . 8277 + 0 . 57489 n f ) a 4 − Q f  s f for n f =5 11 +1 1 + a s + a 2 s 1 . 409 − 12 . 767 a 3 s − 79 . 98 a 4 − 1 . 240 a 3 s − 14 . 95 a 4 � � � � s s 3 9 Extra suppression factor 3 99 ≈ 0 . 03 !

QED β -function in five loops By a proper change of color factors we arrive at the full Adler function of QED in five loops = ⇒ the QED β -function; e 2 for a QED with one charged fermion we get ( A ≡ 16 π 2 ) Gorishny, Kataev, β QED = 4 3 A +4 A 2 − 62 � 5570 243 + 832 � 9 A 3 − A 4 9 ζ 3 Larin, Surguladze, 1991 − A 5 � 195067 + 800 3 ζ 3 + 416 3 ζ 4 − 6880 � 3 ζ 5 486 Numerically ( A = α 4 π ≈ 5 . 81 · 10 − 4 ) β QED = 4 1 + 3 A − 5 . 1667 A 3 − 100 . 534 A 4 + 1129 . 51 A 5 � � 3 A

To check reduction to masters, two more calculations in O ( α s 4 ) , all for general gauge group! : 1. Perturbative factor C Bjp ( a s ) in Bjorken sum rule: � 1 1 ( x, Q 2 )] dx = 1 6 | g A [ g ep 1 ( x, Q 2 ) − g en | C Bjp ( a s ) g V 0 2. Perturbative factor C GLS ( a s ) in Gross-Llewellyn Smith sum rule: � 1 1 F νp + νp ( x, Q 2 ) dx = 3 C GLS ( a s ) 3 2 0 Both sum rules are unambiguous QCD predictions /modulo higher twists!/ confrontable with data

Typical diagrams at α s 3 (computed in early nineties /Larin & Vermaseren/), Bjp and GLS GLS only (1) (non-singlet) (singlet) q q + 5 + 3 4 6 a s + a 2 C NS GLS ≡ C BJp = 1 − s [ − 4 . 583 + 0 . 3333 n f ] a 3 − 41 . 44 + 7 . 607 n f − 0 . 1775 n 2 � � + s f a 4 − 479 . 4 + 123 . 4 n f − 7 . 697 n 2 f + 0 . 1037 n 3 � � + s f GLS = 0 . 4132 n f a 3 s + a 4 C SI s n f (5 . 80157 − 0 . 233185 n f ) Note: C SI GLS ≪ C NS GLS as expected (MS-scheme):

(Generalized) Crewther relation for D NS C Bjp ( a s ) D NS ( a s ) = 1 + β ( a s ) � K NS = K 1 a s + K 2 a 2 � s + K 3 a 3 s + . . . ( ⋆ ) a s T f n f with β ( α s ) β 0 = 11 ≡ − β 0 a s + . . . , 12 C A − 3 α s ( ⋆ ) implies 6 constraints on 12 color structures C 4 F , C 3 F C A , C 2 F C 2 A , C F C 3 A , C 3 F T F n f , C 2 F C A T F n f , C F C 2 A T F n f , C 2 F T 2 F n 2 f , C F C A T 2 F n 2 f , C F T 3 F n 3 f , d abcd d abcd , n f d abcd d abcd F A F F appearing at O ( α 4 s ) in the difference D NS − 1 /C Bjp All 6 constraints are met identically! (which means 6 · 7 = 42 separate constraints on coefficients of ζ 3 , ζ 2 3 , . . .

Crewther relation between D = D NS + D SI and C GLS D NS + d SI 3 a 3 4 a 4 3 a 3 4 a 4 s + d SI C NS GLS + c SI s + c SI � � � � = s s d abc d abc 1 + β ( α s ) � K NS + a 3 � s K SI 3 n f ⋆ α s d R T f with β ( α s ) β 0 = 11 ≡ − β 0 a s + . . . , 12 C A − 3 α s = n f d abc d abc = n f d abc d abc d SI d SI d SI � C F d SI 4 , 1 + C A d SI 4 , 2 + T F d SI � 3 , 1 , 3 4 4 , 3 d R d R = n f d abc d abc = n f d abc d abc c SI c SI c SI � C F c SI 4 , 1 + C A c SI 4 , 2 + T F c SI � 3 , 1 , 3 4 4 , 3 d R d R rhs of ⋆ depends on only 1 unknown parameter, K SI 3 , thus 3-1 =2 constraints on three coefficients in d SI 4

Obvious solution of these constraints reads: 4 , 1 = − 3 4 , 1 = − 13 64 − ζ 3 4 + 5 ζ 5 d SI 2 c SI 3 , 1 − c SI 8 4 , 2 + 11 4 , 3 + 1 d SI 4 , 2 = − c SI 12 K SI d SI 4 , 3 = − c SI 3 K SI 3 , 1 3 , 1 All 2 constraints are met identically! (which means 2*7=14 separate constraints on coefficients of front of ζ 3 , ζ 2 3 , ζ 4 , ζ 4 ζ 3 , ζ 5 , ζ 7 , n m )

CONCLUSIONS • Singlet and non-singlet parts of the Adler function and the perturbative factors C ( a s ) of Bjorken and GLS sum rule have been both analytically evaluated for generic gauge group at O ( α s 4 ) • The generalized Crewther relation puts 42 constraints on the non- singlet result and 14 constraints on the difference d SI 4 − C CLS which 4 are all fulfilled! • Numerically, the singlet contribution to the VV-correlator is tiny