PoS(RAD COR 2007)025 http://pos.sissa.it/ ce. umerical evaluation of these dia- ped and implemented an efficient rder correction, which is urgently nables us to obtain finite amplitudes d the automated code-generating sys- 2 measurements. We focus on a type of on of QED corrections to the anomalous − g lerator Research Organization (KEK) ersity ive Commons Attribution-NonCommercial-ShareAlike Licen ∗ aoym@post..jp 2 Speaker. needed considering the recent improvementof magnetic moment of leptons. Our major concern is the tenth-o This article reports our project on the automated calculati algorithm to perform subtractions of IR divergences. This e tem for the UV-renormalized amplitude. We have newly develo that are free from both UV and IR divergences. Currently the n diagrams that have no internal lepton loops, and have devise grams of tenth order is in progress. ∗ − Copyright owned by the author(s) under the terms of the Creat c

Tatsumi Aoyama Institute of Particle and Nuclear Studies, High Energy Acce Automated calculation of QED corrections tog lepton E-mail: Masashi Hayakawa Department of Physics, Nagoya University Toichiro Kinoshita Laboratory for Elementary-Particle Physics, Cornell Univ Makiko Nio Laboratory, Nishina Center, RIKEN 8th International Symposium on Radiative Corrections October 1-5, 2007 Florence, Italy PoS(RAD COR 2007)025 is as- e a (2.1) (1.1) action y must . We employ ) 10 1 ( ), and it is given last two digits of 1 A A , . This reduces the number ] ) q ) , Penning trap with cylindrical p ν p ( p ( ber and many of them have very 76ppb ν Σ . ∂ evaluated [3]. A particularly diffi- 0 ∂ Λ [ t I–VI) according to their structure t have no closed lepton loops (called − ost stringent test of QED. Recently, [4, 5]. e Ward-Takahashi identity It is turned into a parametric integral tly the numerical integration codes of tion to this obstacle by an automated ndent contribution 0 9. ] in which the precision of the value of t previous value [2]. 12 ical means at present for the eighth and own tenth-order term → − er Set V diagrams, the number is, after q rther improvement of the measurement, it  10 ) q × , 2 TatsumiAoyama ν ) p q is given by an integral over loop momenta of a ( − 76 µ ∂ 2 ( G Λ ∂ 85 1 . The QED correction is evaluated by the perturbation .  2 of the electron is one of the most precisely studied is given by the static limit of the magnetic form factor. α µ − q e g a ≃− ) 2 is explained almost entirely by the electromagnetic inter and the sum of the vertex part q , − ) p g p . To match the precision of the recent measurement, the theor comes from 12672 vertex-type Feynman diagrams, which are cl ( ( 1 159 652 180 th-order diagram ν ) α Σ n Λ = 10 1 ( e A a 2 and the numeral in the parenthesis is the uncertainty in the / ) 2 − g ( ≡ e a ). The difficulty stems from the fact that they are large in num Theoretically, the electron The contribution of a 2 The anomalous magnetic moment The contribution to The anomalous magnetic moment Note added: the Harvard group reported a new measurement [16 1 where the value. It has a 5.5 times smaller uncertainty than the bes between electron and photon alone (referred as mass-indepe as a function of the fine structure constant product of vertices and propagators of leptons and photons. of independent integrals substantially.taking For account of the the tenth-ord time-reversal symmetry, reduced to 38 2.1 Amplitude sified into 32 gauge-invariant(Figure groups 1). within 17 6 groups among super thesecult sets 32 one (Se groups is have Set already V been consistingq-type of 6354 vertex-type diagrams tha complicated divergence structure. We willscheme for present code-generation, our which solu enables usthe renormalized to and create finite swif amplitude of the q-type diagrams 2. Parametric Integration Formalism the numerical integration approach, which ishigher the order only corrections. pract quantities in the particlea physics, new and measurement it was has carriedcavity. provided Their out value the by announced m a in 2006 Harvard is group [1] using the between the self-energy part theory as a series in termsinclude of up to eighth-order corrections. Considering theis fu urgently required to evaluate the actual value of the unkn improved by a factor 2.7 over the value (1.1). Automated calculation of QED corrections to lepton g 1. Introduction In our formulation, we first employ a relation derived from th PoS(RAD COR 2007)025 , V , (2.2) bdia- U gration , j . n renor- A ,  i j B ··· is of self-energy + 1 1 S − Z n + V corrections. 3 1 N U when ] j + ,  0 S n [ Z adopt subtractive renormaliza- ··· / V G + 2 + 0 M U 2 1 he subtraction term associated with N − C n UV S called building blocks, ameter space. This scheme is called  + B n exact lower-order amplitude and the n one divergent subdiagram, the whole V d away by appropriate integrals which 3 1 + + E U S + / G 1 2 TatsumiAoyama 0 − C M − n -operations of subdiagrams in the forest. 3 + V K UV S 2 0 m E U δ  1 1 − n  G ) of a vertex-renormalization constant when the divergent su dz ( assigned to the propagators. Carrying out the momentum inte UV S are the leading UV-divergent parts of mass and wave-functio i L Z z ! ) UV S 1 B − Classification of diagrams contributing to tenth-order QED n and ( n  UV S 1 4 m − δ Figure 1:  is of vertex type, and into the form = I(a)I(g) I(b) I(h) I(c) I(i) I(d) I(j) I(e) I(f) , which are polynomials of Feynman parameters. ) II(a) II(b) II(c) II(d) II(e) II(f) VI(a) VI(b) VI(c) VI(d) VI(e) VI(f) III(a) III(b) III(c) IV V VI(g) VI(h) VI(i) VI(j) VI(k) n S i j 2 The amplitude constructed above is divergent in general. We ( G C M -operation [6]. The subtraction integral factorizes into a The result is given symbolically as a function of quantities type. Here, 2.2 Subtraction of UV divergence tion with the strategy thatcancel these the divergences singularities are point-by-point subtracte K on the Feynman par and over Feynman parameters Automated calculation of QED corrections to lepton g leading UV-divergent part malization constants, respectively. When thereUV is divergences more are tha identified by thea Zimmermann’s forest forests. is T obtained by successive operations of analytically, we can express the amplitude in a concise form gram PoS(RAD COR 2007)025 - K -operation of the self- I the magnetic way. To this UV S m δ ad hoc -subtraction. The IR R in this IR limit. The latter momenta of photons in the ) t of UV divergence. The k gral of the amplitude of a di- ( ntegration codes for evaluating energy subdiagrams. In the old rgy subdiagram, the IR subtrac- S / numerical integration routine. The Maple. The numerical integrations G re dealt in an lic manipulations required in the in- ts the amplitude and the subtraction se two subtraction operations. This a subtraction integral that cancels it. e L l renormalization. To complete the ighth orders, the subtraction terms are ly the leading part e, VEGAS [10]. Our implementation tions. This step is called the residual asible for the automated treatment. us section. It takes a single-line infor- eynman parameters, called -subtraction and is regarded to introduce a spurious two- the information of distinct types of sub- IR divergence. A remedy to this problem I -subtractions. They are recognized by a ion and the intermediate renormalization R . The whole IR subtraction terms can be UV S m 2 TatsumiAoyama δ − − 4 S - and/or I by constructing an appropriate integral. We call this m δ S and the vertex part ≡ m f δ S S annotated forests m f δ -subtraction. I go to zero the amplitude behaves factorized into a product of S / of the subdiagram G S -subtraction. M R One type of IR divergence stems from our particular treatmen We developed an automated system that generates numerical i Thus far we have constructed the renormalized and finite inte A diagram suffers from IR divergences when it contains self- The other type of divergence arises in such a way that when the For more complicated cases involving more than one self-ene subtraction terms can be identifiedscheme by relies totally the on combination the of diagrammatic the property and is fe operation for the self-energy subdiagram subtracts away on problem, we developed new subtraction schemes called agram, but it iscalculation, not the fully difference equivalent between the toachieved full so the renormalizat far standard must on-shel berenormalization evaluated [8]. by collecting all contribu 3. Automated Calculation identified by finding all possible annotated forests. 2.4 Residual renormalization q-type diagrams following the stepsmation described specifying in the the form previo ofterms, a and diagram produces as FORTRAN an codes input, readilysystem integrated construc is by implemented the as a set oftermediate Perl steps programs, are and performed the with symbo the helpsare of processed FORM by [9] an and adaptive Monteis applicable Carlo to integration arbitrary routin order of q-type diagrams. point vertex effectively. This causes linear oris more severe to subtract away the contribution of [7]. However, some diagrams have linear divergences that we Automated calculation of QED corrections to lepton g 2.3 Subtraction of IR divergence approach employed in the numerical evaluation ofidentified sixth by and e power-counting rules in a particular limit of F mass term, and the remaining part residual diagram operation as forest-like structure of self-energy subdiagramstractions to which is assigned. We call them accounts for the logarithmic IR divergence.We call Thus this we operation prepare as moment part tion terms are given by the combinations of PoS(RAD COR 2007)025 , ) 343 ( (4.1) (4.2) 179 16 . [11, 12], , respec- 2 ) . − 53 ··· ( 344 166 905 26 . . 0 ed for automation. We , 0 − (1987) 26–29. ) (2006) 030801. (2006) 138–180. 53 59 ( 97 luation of known diagrams of d encourages us to proceed to 740 type diagrams, but it can be ex- inconsistency in the treatment of . ulation of QED corrections to the ) to the new value [14] consistent evaluations. It should be 98 343 95 ained by inserting vacuum polariza- hth-order term. The value of the fine . ( rams that have no closed lepton loops. 0 nd the theory is revised to [15] , III(a), III(b) and IV (see Figure 1) have rth-order term of Set V have already been obtained by er contributions obtained by our codes ) nce. The numerical calculation codes of − corrected, the old calculation, 35 Phys. Rev. Lett. ( Nucl. Phys. B arXiv:0709.1568 [hep-ph] Phys. Rev. Lett. 2 TatsumiAoyama − (1974) 3991. (1974) 3978. 914 4 5 . 035 999 070 . 1 10 10 − (2006) 053007. 137 = ) 73 8 ( 1 )= [13], respectively. e A , agree within the numerical precision employed [5]. a ) ( 1 ··· 53 Phys. Rev. D Phys. Rev. D − ( α 219 Phys. Rev. D . 2 904 979 . − ´ ´ c and T. Kinoshita, c and T. Kinoshita, 2. We developed a new treatment of IR divergences that is suit − g The coefficient of the eighth-order contribution is revised These tests confirmed the validity of our automated system an In this article we reported our project on the automated calc For the eighth-order case, we have unexpectedly revealed an Our code-generating system is primarily designed for the q- For the purpose of testing our system, we applied it to the eva [1] B. Odom, D. Hanneke, B. D’Urso, and G. Gabrielse, [2] R. S. Van Dyck, P. B. Schwinberg, and H. G. Dehmelt, [3] T. Kinoshita and M. Nio, [7] P. Cvitanovi [4] T. Aoyama, M. Hayakawa, T. Kinoshita, and M. Nio, [5] T. Aoyama, M. Hayakawa, T. Kinoshita, and[6] M. Nio, P. Cvitanovi and it is firmly establishednoted by that the this two is independent, the mutually firststructure constant independent determined check from of the the measurement whole (1.1) eig a implemented the code-generating system for the type of diag lepton the evaluation of the tenth-order contributionall with 389 confide independent integrals that representour 6354 system, diagrams and the numerical evaluation is now in progress. 4. Concluding Remarks and the new calculation, tively [5], which reproduce well the exact results of the fou IR divergences in the previous calculation. With this error and the sixth-order term 0 tended with slight modifications totions the to q-type types diagrams. of The diagrams numerical obt been evaluations of carried Sets out. The results will be reported elsewhere. References fourth, sixth, and eighth orders.are, after The taking fourth- account and of sixth-ord the residual renormalization, Automated calculation of QED corrections to lepton g PoS(RAD COR 2007)025 . (2007) 039902. 98 (2007) 110406. 99 Phys. Rev. Lett. rgy physics, 7). om, Phys. Rev. Lett. 2 TatsumiAoyama − 6 (1996) 283–291. . arXiv:0801.1134 [physics.atom-ph] (T. Kinoshita, ed.), pp. 218–321. World Scientific, B379 (1957) 328–329. (1957) 407–408. (1978) 192. 30 27 107 Phys. Lett. math-ph/0010025 Phys. Rev. Helv. Phys. Acta J. Comput. Phys. Singapore, 1990. (Advanced series on directions in high ene [8] T. Kinoshita in [9] J. A. M. Vermaseren, [15] G. Gabrielse, D. Hanneke, T. Kinoshita, M. Nio, and B. Od [16] D. Hanneke, S. Fogwell, and G. Gabrielse, [14] T. Aoyama, M. Hayakawa, T. Kinoshita, and M. Nio, [13] S. Laporta and E. Remiddi, [10] G. P. Lepage, [11] A. Petermann, [12] C. M. Sommerfield, Automated calculation of QED corrections to lepton g