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2 from second order vacuum polarization insertion
Volume49B, number 5
PHYSICSLETTERS
13 May 1974
SIXTH ORDER ELECTRON AND MUON (g-2)/2 FROM SECOND ORDER VACUUM POLARIZATION INSERTION R.
BARBIERI
Scuola Normale Superiore, h'sa, Italy Istituto Nazionale di Fisica Nucleate, Sezione di Pisa, Italy
and E. REMIDDI* CERN; Geneva, Switzerland
Received 18 February 1974 '7,recompletethe analytic evaluationof sixth order contributions to electron and rouen anomalousmagneticmoments comingfrom second order vacuumpolarizationinsertionsinto all the fourth order graphs. Continuing previous work [1-3] we have completed the analytic evaluation of the electron and muon anomalous magnetic moments ae, a~ due to the insertion of second order vacuum polarization into fourth order graphs. The relevant graphs are shown in fig. 1. They are numbered as in ref. [4]. For ae, both the external and the internal loop leptons in fig. I are electrons. We find a(7) + a(8)= (~) 3 I_ r 3-3-~ 2005--+ 88-~4-{-(2) ---9--154{-(2)log 2 - ~47 {-(3)+ ~5 {-2(2)] = (~)3[._0.00312135] ' a~ll)+a(12)=f°t~3~599\rr] L162
~15{-(2)-~-~ {-(2) log 2 9+7~--~{-(3) + 213 {-2(2)] = (~)3 [0.0274617],
a(13) + a(14)= (~)3 [ 2641 + _~15{-(2)_ ~{-(2) log 2 _ _~41{-(3, +26 T-~{-2(2) + (- 11~- + 4{-(2))1og~e ] = (~)3 [-- 0.0458692 -- 0.0313748 log ~e ] For at, the external lepton is a muon, the lepton of the vacuum polarization loop is an electron. We find (~)3[ 19 317 1~_~_6 a(7) + a(8) = 18 54 {-(2)+ {-(2) log 2 - 5{-(3) + 5£74 + (2+ ~ {-(2)--~ {-(2)log 2+ 5 {-(3))log mumeJ]= (~)3 [2.14230 - 0.623527 log m~ ]mej a(ll) + [-(12) 4 3= ~:-a~3 u ] ~ kT~ _ ~11{'(2) - ~ {-(2)log 2 + ~-8{-(3)- C4 + 12 mlogt a .me ~ + ( - - 29+2 ~ ~{-(2)+4~{-(2)log2 _ 1~.(3)) log m~] me J = (-~)3 [1.280005 +llog2 m~ m--e-- 0.931570 log mu me--] a(13)+_(14) //ot~31937 _4{-(2)_~{-(2)1og2+~.(3)_C4+
a~ =~,-~,/ k-~--~
* On leavefrom Istituto di Fisieadell'Universith,Bologna. 468
(12.~ 21ogmta~log X .~
me~
m----~
(1)
Volume 49B, number 5
PHYSICS LETTERS
13 May 1974
Fig. 1. Second order vacuum polarization insertions into fourth order vertex graphs.
+ l - 36~76+ 2~'(2) + 4~'(2) log 2 - 1 ~ ( 3 ) ) log m~ ] =
(_~)3[
0.619579 +
2(]_2 m_._.U)log X logmU] - - ~ log me/ --mu - 0.376014 me J •
(2)
In eqs. (2) we have introduced 1~
C4 = ~ ' 2 ( 2 )
-
4
~'(2) log 2 2 -
a 4 =_ ~** 2-nn4 = 0.51747906, n=l
1
log 4 2 - ~ a 4 = 0.83376809
~'(2) = -g-. /1"2
Summing up with the results already obtained [ 1 - 3 ] , all the graphs of fig. 1 give
(3) au = =
24
( 110
) m.]
9 ~'(2) + 10~'(2) log 2 -- ~'(3) + 3C 4 + - T2- + -3- ~'(2) - 4~'(2) log 2 + ~'(3) log mee
0.0487202 - 0.458889 log mu = me.]
[ - 2.39789]
(4)
(in all the above results for au, terms vanishing with me/m u are dropped in the righ-hand-side). Quite in the tradition of similar Q.E.D. calculations, the sum looks simpler than separate contributions. Agreement with previous numerical calculations [4] is fairly good*. * In comparing the results of present paper with the results quoted in ref. [4], note that "Class II" of ref. [4] contains also graphs 19), 20), fig. (5.1) of ref. [4], not included in fig. (1), present paper. As an additional cheek, we have made a collapse of the electron bubble in each diagram and we found the fourth order results [5]. 469
Volume 49B, number 5
PHYSICS LETTERS
13 May 1974
The method we used to derive eqs. (1), (2) is a suitable extension of that of ref. [3]. A more complete description of the calculation will be given elsewhere. We thank Dr. M. Caffo for his collaboration in the early stage of this work and Dr. H. Strubbe for his assistance in the use of the Program Schoonschip, by which all algebra was worked out.
References [1] [2] [3] [4] [5]
D. BiUi,M. Caffo and E. Remiddi, Lettere al Nuovo Cimento 4 (!972) 65q.. R. Barbieri, M. Caffo and E. Remiddi, Lettere al Nuovo Cimento 5 (1972) 769. R. Barbieri, M. Caffo and E. Remiddi, preprint TH 1802-CERN (January 1974). B.E. Lautrup, A. Petermann and E. de Rafael, Phys. Reports 3 (1972) 4. A. Petermann, HeN. Phys. Acta 30 (1957) 407; C. Sommerfield, Phys. Rev. 107 (1957) 328.
470
PHYSICSLETTERS
13 May 1974
SIXTH ORDER ELECTRON AND MUON (g-2)/2 FROM SECOND ORDER VACUUM POLARIZATION INSERTION R.
BARBIERI
Scuola Normale Superiore, h'sa, Italy Istituto Nazionale di Fisica Nucleate, Sezione di Pisa, Italy
and E. REMIDDI* CERN; Geneva, Switzerland
Received 18 February 1974 '7,recompletethe analytic evaluationof sixth order contributions to electron and rouen anomalousmagneticmoments comingfrom second order vacuumpolarizationinsertionsinto all the fourth order graphs. Continuing previous work [1-3] we have completed the analytic evaluation of the electron and muon anomalous magnetic moments ae, a~ due to the insertion of second order vacuum polarization into fourth order graphs. The relevant graphs are shown in fig. 1. They are numbered as in ref. [4]. For ae, both the external and the internal loop leptons in fig. I are electrons. We find a(7) + a(8)= (~) 3 I_ r 3-3-~ 2005--+ 88-~4-{-(2) ---9--154{-(2)log 2 - ~47 {-(3)+ ~5 {-2(2)] = (~)3[._0.00312135] ' a~ll)+a(12)=f°t~3~599\rr] L162
~15{-(2)-~-~ {-(2) log 2 9+7~--~{-(3) + 213 {-2(2)] = (~)3 [0.0274617],
a(13) + a(14)= (~)3 [ 2641 + _~15{-(2)_ ~{-(2) log 2 _ _~41{-(3, +26 T-~{-2(2) + (- 11~- + 4{-(2))1og~e ] = (~)3 [-- 0.0458692 -- 0.0313748 log ~e ] For at, the external lepton is a muon, the lepton of the vacuum polarization loop is an electron. We find (~)3[ 19 317 1~_~_6 a(7) + a(8) = 18 54 {-(2)+ {-(2) log 2 - 5{-(3) + 5£74 + (2+ ~ {-(2)--~ {-(2)log 2+ 5 {-(3))log mumeJ]= (~)3 [2.14230 - 0.623527 log m~ ]mej a(ll) + [-(12) 4 3= ~:-a~3 u ] ~ kT~ _ ~11{'(2) - ~ {-(2)log 2 + ~-8{-(3)- C4 + 12 mlogt a .me ~ + ( - - 29+2 ~ ~{-(2)+4~{-(2)log2 _ 1~.(3)) log m~] me J = (-~)3 [1.280005 +llog2 m~ m--e-- 0.931570 log mu me--] a(13)+_(14) //ot~31937 _4{-(2)_~{-(2)1og2+~.(3)_C4+
a~ =~,-~,/ k-~--~
* On leavefrom Istituto di Fisieadell'Universith,Bologna. 468
(12.~ 21ogmta~log X .~
me~
m----~
(1)
Volume 49B, number 5
PHYSICS LETTERS
13 May 1974
Fig. 1. Second order vacuum polarization insertions into fourth order vertex graphs.
+ l - 36~76+ 2~'(2) + 4~'(2) log 2 - 1 ~ ( 3 ) ) log m~ ] =
(_~)3[
0.619579 +
2(]_2 m_._.U)log X logmU] - - ~ log me/ --mu - 0.376014 me J •
(2)
In eqs. (2) we have introduced 1~
C4 = ~ ' 2 ( 2 )
-
4
~'(2) log 2 2 -
a 4 =_ ~** 2-nn4 = 0.51747906, n=l
1
log 4 2 - ~ a 4 = 0.83376809
~'(2) = -g-. /1"2
Summing up with the results already obtained [ 1 - 3 ] , all the graphs of fig. 1 give
(3) au = =
24
( 110
) m.]
9 ~'(2) + 10~'(2) log 2 -- ~'(3) + 3C 4 + - T2- + -3- ~'(2) - 4~'(2) log 2 + ~'(3) log mee
0.0487202 - 0.458889 log mu = me.]
[ - 2.39789]
(4)
(in all the above results for au, terms vanishing with me/m u are dropped in the righ-hand-side). Quite in the tradition of similar Q.E.D. calculations, the sum looks simpler than separate contributions. Agreement with previous numerical calculations [4] is fairly good*. * In comparing the results of present paper with the results quoted in ref. [4], note that "Class II" of ref. [4] contains also graphs 19), 20), fig. (5.1) of ref. [4], not included in fig. (1), present paper. As an additional cheek, we have made a collapse of the electron bubble in each diagram and we found the fourth order results [5]. 469
Volume 49B, number 5
PHYSICS LETTERS
13 May 1974
The method we used to derive eqs. (1), (2) is a suitable extension of that of ref. [3]. A more complete description of the calculation will be given elsewhere. We thank Dr. M. Caffo for his collaboration in the early stage of this work and Dr. H. Strubbe for his assistance in the use of the Program Schoonschip, by which all algebra was worked out.
References [1] [2] [3] [4] [5]
D. BiUi,M. Caffo and E. Remiddi, Lettere al Nuovo Cimento 4 (!972) 65q.. R. Barbieri, M. Caffo and E. Remiddi, Lettere al Nuovo Cimento 5 (1972) 769. R. Barbieri, M. Caffo and E. Remiddi, preprint TH 1802-CERN (January 1974). B.E. Lautrup, A. Petermann and E. de Rafael, Phys. Reports 3 (1972) 4. A. Petermann, HeN. Phys. Acta 30 (1957) 407; C. Sommerfield, Phys. Rev. 107 (1957) 328.
470