The Polarization Function, the QED Beta Function and the Muon - PowerPoint PPT Presentation

The Polarization Function, the QED Beta Function and the Muon Anomalous Magnetic Moment Johann H. K¨ uhn Mainz 27-28 September 2012

I. Four loop polarization function II. QED beta function at five loops III. Anomalous magnetic moment of the muon: selected five- and six-loop terms based on Baikov, Chetyrkin, JHK, J. Rittinger, arxiv: 1206.1284, JHEP 1207(2012)017 Baikov, Chetyrkin, JHK, C. Sturm, arxiv: 1207.2199 2

I.The Polarization Function � ( − g αβ q 2 + q α q β ) Π( L, a s ) = i d 4 xe iq · x � 0 | T j α ( x ) j β (0) | 0 � available in 4 loops (including constant piece) Examples of two non-singlet and two singlet diagrams contributing to the vector correlator. using � � � � β ( a s ) ∂ D ( L, a s ) = 12 π 2 γ ( a s ) − Π( L, a s ) ∂a s with Π in O ( α 3 s ) and anomalous dimension γ in 5 loops ( O ( α 4 s )) ⇒ R ≡ σ (had /σ ( µ + µ − ) in O ( α 4 s ) 3

Results: �� � �� � d R d R NS = SI = i a i i a i NS SI Π p , Π p , s s 16 π 2 16 π 2 i ≥ 0 i ≥ 3 20 NS p = 9 , 0 � 55 � NS p = C F 12 − 4 ζ 3 , 1 � � � 44215 � − 143 72 − 37 2592 − 227 18 ζ 3 − 5 C 2 NS p = 6 ζ 3 + 10 ζ 5 + C F C A 3 ζ 5 2 F � � − 3701 648 + 38 + C F T F n f 9 ζ 3 , � � � 196513 � − 31 192 + 13 8 ζ 3 + 245 23328 − 809 162 ζ 3 − 20 + T 2 n 2 C 3 NS p = 8 ζ 5 − 35 ζ 7 f C F 9 ζ 5 3 F � � − 7505 10368 + 1553 3 + 11 24 ζ 4 − 250 + T n f C 2 54 ζ 3 − 4 ζ 2 9 ζ 5 F � � − 5559937 + 41575 1296 ζ 3 + 2 3 − 11 24 ζ 4 + 515 3 ζ 2 + T n f C F C A 27 ζ 5 93312 � � − 382033 20736 − 46219 864 ζ 3 − 11 48 ζ 4 + 9305 144 ζ 5 + 35 + C 2 F C A 2 ζ 7 � 34499767 � − 147473 2592 ζ 3 + 55 3 + 11 48 ζ 4 − 28295 864 ζ 5 − 35 + C F C 2 6 ζ 2 12 ζ 7 , A 373248 � 431 d abc d abc � 1728 − 21 64 ζ 3 − 1 3 − 1 16 ζ 4 + 5 6 ζ 2 p SI = 16 ζ 5 . 3 d R Can be applied for QCD (corresponding to quark loops) or to pure QED (lepton loops) with properly chosen colour factors. 4

II. QED Beta Function � The QED beta function receives contributions from non-singlet (starting from 1-loop) and from singlet (starting from 4-loop) terms. � RG-equation: perturbative QCD contribution to µ 2 d dµ 2 A = β EM ( A, a s ) = 16 π 2 A 2 γ EM ( a s ) with γ EM = ( � q 2 i ) γ NS + ( � q 2 i ) γ SI and A = α/ 4 π ; a s = α s / 4 π γ = anomalous dimension, evaluated in 5 loops (with the help of massless 4-loop propagator integrals) ⇒ result in MS scheme 5

� conversion: MOM-scheme Π MOM ( Q 2 , µ 2 ) vanishes at Q 2 = µ 2 (with µ 2 � = 0!) A ( µ ) ˜ ⇒ A ( µ ) = 1 + (4 π ) 2 A ( µ ) Π( L = 0 , a s ( µ )) . with L ≡ ln µ 2 Q 2 and    γ EM ( a s ) − β QCD ( a s ) ∂ A, a s ) = 16 π 2 ˜ A 2 β EM MOM ( ˜ Π EM ( L = 0 , a s )  ∂a s No new calculation needed. 6

� application: pure QED, MS scheme � � 4 A 2 � � 2 n f + 44 +4 n f A 3 − A 4 9 n 2 β QED ( A ) = n f f 3 � � − 46 n f + 760 f − 832 f − 1232 A 5 27 n 2 ζ 3 n 2 243 n 3 + f 9  � 4157 � � � − 7462 A 6 + n 2  n f + + 128 ζ 3 − 992 ζ 3 + 2720 ζ 5 f 6 9 � � � � 856 − 21758 + 16000 ζ 3 − 416 3 ζ 4 − 1280 243 + 128 n 3 + n 4 + ζ 5 27 ζ 3  . f f 81 27 3 � conversion to MOM-scheme: as before 7

� conversion: on-shell scheme Π OS ( Q 2 , M 2 ) vanishes at Q 2 = 0 ( M 2 � = 0!) ⇒ Π MS ( Q 2 = 0 , m 2 , µ 2 ) is required: 4-loop tadpoles! conversion of coupling constant (4-loop) conversion of mass (3-loop) ⇒ Π OS ( Q 2 , M 2 ) Q 2 -dependent (logarithmic) part at 5 loop. ( µ 2 disappears, M 2 appears) 8

term of order α 5 , Q 2 dependent part Results: � 4157 6144 + 1 Π (5) ( ℓ MQ ) = Nℓ MQ 8 ζ 3 � 55 � 96 + 5 96 π 2 + 179 256 ζ 3 − 115 12 ζ 5 + 35 4 ζ 7 + 13 128 ℓ MQ − 1 12 π 2 ln(2) + N � � − 13 12 − 4 3 ζ 3 + 10 + N si 3 ζ 5 N 2 � − 11 432 + 1 36 π 2 − 17089 3 + 125 18 ζ 5 + 35 2304 ζ 3 + ζ 2 + 288 ℓ MQ � − 7 8 ζ 3 ℓ MQ + 5 6 ζ 5 ℓ MQ + 1 72 ℓ 2 MQ � � − 149 108 + 13 6 ζ 3 + 2 3 − 5 3 ζ 5 − 11 72 ℓ MQ + 1 N 2 si 3 ζ 2 + 3 ζ 3 ℓ MQ N 3 � − 6131 2916 + 203 162 ζ 3 + 5 9 ζ 5 − 151 324 ℓ MQ + 19 54 ζ 3 ℓ MQ − 11 216 ℓ 2 + MQ �� + 1 1 27 ζ 3 ℓ 2 432 ℓ 3 MQ − . (1) MQ N = number of leptons; ℓ MQ = ln M 2 /Q 2 ⇒ β -function in OS scheme at 5 loops 9

III. Anomalous magnetic moment of the muon Recall (numbers from Kinoshita et al. 1205.5370) 116592089(63) × 10 − 11 a µ ( exp ) = 63 × 10 − 11 = δ exp Theory: dominant errors (hadronic) (37 . 2) exp + (21 . 0) rad × 10 − 11 δ vacpol = 40 × 10 − 11 δ ll = QED: 2 loop, 3 loop: exact, analytic 4 loop, 5 loop (recently): numerical (Kinoshita). 10

� 3 ≈ 3 · 10 − 7 � α a (6) 3 loop: = (1 . 181 . . . + 22 . 868 . . . ) µ π [log( m µ /m e )] n const � 4 ≈ 382 · 10 − 11 � α a (8) 4 loop: = ( − 1 . 9106(20) + 132 . 6852(60)) µ π [log( m µ /m e )] n const (theory error: 1 . 7 · 10 − 13 ) � 5 ≈ 5 · 10 − 11 � α a (10) 5 loop: = (9 . 168(571) + 742 . 18(87) + . . . ) µ π [log( m µ /m e )] n const factor 10 below experimental uncertainty. Nevertheless: should be checked: 11

master formulae � x 2 � 1 � � � M 2 α d asymp a asymp µ = dx (1 − x ) , α − 1 , µ R M 2 π 1 − x 0 e 1 d asymp ( Q 2 /M 2 , α ) = 1 + Πasymp( Q 2 /M 2 , α ) . R with Π evaluated in the OS scheme e e e e e e e e e e e e e e I ( a ) I ( b ) I ( c ) I ( d ) I ( e ) e e e e e e e e e e I ( f ) I ( g ) I ( h ) I ( i ) I ( j ) The ten gauge invariant subsets contributing to the muon anomaly which originate from inserting the vacuum polarization up to four-loop order into the first order QED vertex. For each diagram class only one typical representative is shown. Wavy lines denote photons( γ ), solid lines denote electrons( e ) or muons ( µ ). The last five diagrams { I ( f ), I ( g ), I ( h ), I ( i ), I ( j ) } are non-factorizable insertions of the vacuum polarization function; the first five diagrams { I ( a ), I ( b ), I ( c ), I ( d ), I ( e ) } are factorizable ones. 12

Result for coefficients: Subset analytical numerical Ref. num.-ana. I ( a ) 20.1832 + O ( M e /M µ ) 20.14293(23) (Kinoshita) ≈ -0.04 I ( b ) 27.7188 + O ( M e /M µ ) 27.69038(30) (Kinoshita) ≈ -0.03 I ( c ) 4.81759 + O ( M e /M µ ) 4.74212(14) (Kinoshita) ≈ -0.08 I ( d ) 7.44918 + O ( M e /M µ ) 7.45173(101) (Kinoshita) ≈ 0.003 I ( e ) -1.33141 + O ( M e /M µ ) -1.20841(70) (Kinoshita) 0.12 ≈ I ( f ) 2.89019 + O ( M e /M µ ) 2.88598(9) (Kinoshita) ≈ -0.004 I ( g ) + I ( h ) 1.50112 + O ( M e /M µ ) 1.56070(64) (Kinoshita) ≈ 0.06 I ( i ) 0.25237 + O ( M e /M µ ) 0.0871(59) (Kinoshita) ≈ -0.17 I ( j ) -1.21429 + O ( M e /M µ ) -1.24726(12) (Kinoshita) ≈ -0.03 The first column shows the different gauge invariant subsets of diagrams. The second column contains the corresponding results evaluated numerically, where we have used for the mass ratio M µ /M e = 206 . 7682843(52). This result is correct only up to power corrections in the small mass ratio M e /M µ . The third column contains the numerical result obtained by Kinoshita et al. . The last column shows the difference between the numerical and asymptotic analytical results. The subsets { I ( a ), I ( b ), I ( c ), I ( d ), I ( e ) } originate from Feynman diagrams with factorizable vacuum polarization insertions, whereas the subsets { I ( f ), I ( g ), I ( h ), I ( i ), I ( j ) } are non-factorizable. good overall agreement! sum: vacpol= � I = 62 . 26675 to be compared with 751 . 35 for the total 13

lessons from 5-loop logarithmically enhanced terms and factorizable terms dominate: � I = 62 . 26675 = 58 . 8374 + 1 . 915 + 1 . 514 � �� � � �� � � �� � factorizable irreducible irreducible 4 loop vacpol 4 loop vacpol logs const prediction for 6 loops (vacpol-subset) + small irreducible � I = 246 . 381 + 10 . 8647 ≈ 257 5 loop vacpol � �� � � �� � factorizable irreducible const 5 loop vacpol logs still missing (and dominant): light by light! 14