The 8th- and 10th-order QED Contributions to the g − 2 of the and

and New Determination of α from ae

presented at Third International Symposium on Lepton Moments

Cape Cod, June 19 - 22, 2006

Toichiro Kinoshita Laboratory of Elementary-Particle Physics , Ithaca, NY 14853

1 1. Introduction • Currently the electron g − 2 is by far the best source of α.

• Let me first discuss how α is determined by comparing new 4 Harvard measurement with theory of ae up to order α .

• Then report on our effort to evaluate α5 term in preparation for further improvement of α.

• I discuss in particular diagrams without closed lepton loop (called q − type) which are very large and complicated and hardest to evaluate of all 10th-order diagrams. • To evaluate them we are developing automating algorithm. Initial report is published in T. Aoyama, M. Hayakawa, T. Kinoshita and M. Nio, Nucl. Phys. B 740, 138 (2006).

• Also discuss briefly other diagrams that contribute to ae and aµ. T. Kinoshita and M. Nio, Phys. Rev. D 73, 053007 (2006)

2 2. Electron g − 2: Measurement. • In 1987 the value of electron g-2 was improved over previous best value by three orders of magnitude in a Penning trap experiment:

−12 ae− = 1 159 652 188.4 (4.3)× 10

−12 ae+ = 1 159 652 187.9 (4.3) × 10 Van Dyck et al., PRL 59, 26 (1987)

• Uncertainty was dominated by cavity shift due to interac- tion of electron with hyperboloid cavity which has compli- cated resonance structure. • Several ways to reduce this error examined: (a) Use cavity with smaller Q. (b)Studycavityshiftofmany(∼ 1000)-electron cluster. (c) Use cylindrical cavity, whose resonance property is known. Brown, Gabrielse, RMP 58, 233 (1986)

3 • New Harvard measurement of ae is based on (c). • After many years of hard work it has just been reported:

−12 ae−[HV06]=1 159 652 180.85 (0.76)× 10 (0.66 ppb)

B. Odom, D. Hanneke, B. D’Urso, G. Gabrielse, submitted to PRL (2006)

• 5.5 times more precise than the previous best value.

• Uncertainty due to cavity shift is reduced by 13.

• Remaining uncertainty is mostly from line shape modeling.

4 3. Theory of Electron g − 2 up to Order α4

ae(QED)=A1 + A2(me/mµ)+A2(me/mτ )+A3(me/mµ, me/mτ )

      = (2) α + (4) α 2 + (6) α 3 + = Ai Ai π Ai π Ai π ..., i 1, 2, 3 (2) A1 = 0.51diagram(analytic) (4) A1 = −0.328 478 965 ... 7diagrams (analytic) (6) A1 = 1.181 241 456 ... 72 diagrams (numerical, analytic) Kinoshita, PRL 75, 4728 (1995) Laporta, Remiddi, PLB 379, 283 (1996) (8) A1 = −1.728 3 (35) 891 diagrams (numerical) Kinoshita, Nio, Phys. Rev. D 73, 013003 (2006) (8) • Error of A1 reduced to one-tenth of old one. −12 •A2 contribution to ae is small :∼ 2.72 × 10 . −21 •A3 contribution to ae is even smaller :∼ 2.4 × 10 . •Non − QED term () is small, too : 1.70(2) × 10−12.

• This is why ae provides a very good test of QED.

5 • To compare theory with measurement an independent α is needed. • At present best α available are −1 α (h/MRb) = 137.035 998 78 (91) [6.7 ppb] P. Clad´e et al., PRL 96, 033001 (2006) −1 α (h/MCs) = 137.036 000 00 (110) [8.0 ppb] Wicht, Hensley, Sarajlic, Chu, Physica Scripta T102, 82-88 (2002) V. Gerginov et al., PRA 73, 032504 (2006)

6 (10) • Assuming |A1 | < x we find −12 ae(h/MRb)=1 159 652 188.70 (0.10)(0.08x)(7.71) × 10

−12 ae(h/MCs)=1 159 652 178.40 (0.10)(0.08x)(9.30) × 10 (8th)(10th)(α(h/M))

(10) • Thus, not knowing A1 is not critical as far as 0.08x  8. • For x =3.8 (chosen by Mohr-Taylor), which satisfies this condition, theory and experiment are in good agrement:

−12 ae[HV06] − ae(h/MRb)=−7.9 (7.7) × 10 −12 ae[HV06] − ae(h/MCs)= 2.5 (9.3) × 10

• The conspicuous feature of ae(h/MRb) and ae(h/MCs) is that their errors are mostly from measurement of α.

7 • In other words, non-QED α,eventhebestones,istoo crude to test QED to the precision achieved by theory and measurement of ae. • Instead we can turn the argument around and calculate α assuming that QED is still valid, which yields α−1(x) = 137.035 999 710 (12)(8x)(90)

• For a (safe) choice x =3.8, which satisfies 8x<90,weobtain −1 α (ae[HV 06]) = 137.035 999 710 (96) [0.70 ppb] G. Gabrielse, D. Hanneke, T. Kinoshita, M. Nio, B. Odom, submitted to PRL (2006) • Fig. 1 shows graphic comparison of some precise α’s. • To show finer details of lower half, the horizontal scale is enlarged by 10 in Fig. 2.

8 (α-1 - 137.036) × 107

Muonium H.F.S.

h/mn

ac Josephson

Quantum Hall

h/m(Cs)

h/m(Rb)

ae UW87

ae HV06

-200 -100 0 +100 +200

−1 Figure 1: Comparison of various α . α(h/mCs) may be improved by factor 2.

9 (α-1 - 137.036) × 107

h/m(Cs)

h/m(Rb)

ae UW87

ae HV06

-20 -10 0 +10

Figure 2: Magnification of the lower half of Fig. 4 by factor 10.

10 4. Tenth-order term: Why needed ?

• Uncertainty in α(ae[HV06]) is only factor 3 larger than that of theory, which is mostly from the ”unknown” α5 term. • Thus, when measurement improves, an actual value of α5 term becomes necessary to improve α(ae) further. • No. of contributing Feynman diagrams: 12672. • Question: Is such a calculation feasible ? • Answer: YES. • First step:

Classify all diagrams into gauge-invariant sets.

• 32 g-i sets within 6 supersets as shown:

11 I(a) I(b) I(c)

I(d) I(e) I(f)

I(g) I(h) I(i)

I(j)

Figure 3: Self-energy-like diagrams representing 208 vertex dgrms of set I.

12 II(a) II(b) II(c)

II(d) II(e) II(f)

Figure 4: Diagrams of Set II which consists of 600 vertex diagrams.

III(a) III(b) III(c)

Figure 5: Diagrams of Set III which consists of 1140 vertex diagrams.

13 Figure 6: Diagrams of Set IV which consists of 2072 vertex diagrams.

Figure 7: Diagrams of Set V which consists of 6354 vertex diagrams.

14 VI(a) VI(b) VI(c)

VI(d) VI(e) VI(f)

VI(g) VI(h) VI(i)

VI(j) VI(k)

Figure 8: Diagrams of Set VI which consists of 2298 vertex diagrams.

15 • Largest and most difficult is the Set V, which consists of 6354 Feynman diagrams of ”q-type”.

• Fortunately, Set V has simplifying feature that sum of nine vertex diagrams is related to one self-energy diagram by

ν ∂Λµ(p,q) ∂Σ(p) Λ (p, q) −q [ ] = − µ ∂qν q 0 ∂pν derived from the Ward-Takahashi identity.

• This enables us to cut no. of independent integrals to 706.

• Time-reversal invariance reduces it further to 389.

16 Figure 9: Overview of 389 diagrams contributing to Set V.

17 • Analytic integration is likely to be very far in the future.

• Numerical integration is the only viable option at present.

• Fortunately, our renormalization scheme developed for α3 and extended to α4, which generates FORTRAN codes as an exact algebraic realization of the standard on-shell renormalization, applies to α5,too.

• For α5, however, everything is so huge that all steps must be automated to be practically feasible.

• Finally, a master code is needed to run all steps smoothly and consecutively.

• Thesestepsaresketchedbelow.

18 • Step I: Diagram generation

• Express each (q-type) diagram by single-line statement which specifies pairing of vertices by virtual photons.

• Name these statements as Xabc (abc = 001, 002, ..., 389) and stored them as plain-text files. • The most important point: This file not only defines the diagram itself but also identifies all UV divergent subdiagrams.

• Step I was carried out manually in α3 and α4 cases which were simple enough to handle.

• But, for α5, Step I is crucial for providing automatic control of subsequent steps.

• Implemented by both Perl and C++.

19 • Step II: Construction of unrenormalized integrand

• Express ”Xabc” as momentum integral by Feynman-Dyson rule. • Carry out momentum integration analytically, which gives integral over Feynman parameters z1,z2, ···,zN ,and“building blocks” Bij,Ai,U,V :

 (dz)GJG, where ( ) ( ) ( ) ≡ N (1 − N ) = F0 Bij,Ai + F1 Bij,Ai + ··· dz G dziδ zi ,JG 2 −1 3 −2 . i=1 i=1 U V n U V n

• Previously JG was obtained by FORM using home-made integration table written in FORM.

• Now Step II is fully automated as follows: Xabc → input for FORM ( by Perl) → analytic integration by integration table in FORM.

20 • Step III: Construction of building blocks

• Express Bij,Ci,j,Ai,U,V as polynomials of z1,z2,...,zN . U,Bij are related to loop momenta, and determined by the topology of diagram. Ai are related to flow of external momenta, and satisfy Kirchhoff”s loop law and junction law for “currents”. 3 4 • Easy to obtain Bij, U,etc.byhandinα and α cases. Much harder for α5. • We now calculate them automatically:

Xabc → Bij,Ci,j, U, ... by MAPLE and FORM.

• Also by C++ for check.

• V has form common to all diagrams:

 photons 2 V = zi(1 − Ai)+ ziλ , i i where electron mass is put to one and λ is infrared cutoff.

21 • Step IV: Construction of UV subtraction terms • PROBLEM: Carry out renormalization exactly to eighth order. • Textbook renormalization is not suitable for putting on computer and known only to lowest order anyway. • We solved renormalization for α3 and α4 by subtractive approach. Fortunately, it works for α5,too. • Subtracting integrand is derived from original integrand by K-operation, defined for each divergent subdiagram based on simple power-counting rule.

• Properties of K-operation:  Pointwise subtraction of UV divergence.  Subtraction term factorizes analytically as product of lower-order quantities, an important feature for cross-checking with other diagrams. UV UV UV  It gives only UV-div. parts δmn ,Ln ,Bn of renorm. consts. δmn,Ln,Bn. • Thus additional (finite) renormalization is required.

22 • Step V: Residual renormalization

• Outputs of Steps I - IV are UV-finite integrals. But they are not standard renormalized amplitudes. • (Finite) residual renorm. needed to get observable g-2.

• This is easy for Ln and Bn, but more complicated for δmn.

• ∆ ≡ − UV This is because δmn δmn δmn must be subtracted in addition to K-operation so that mass renormalization is completed.

• Actually this is not a problem for n =2since ∆δm2 =0.

• For n>2 this can be done in Step IV or Step V.

• Previously this was done in Step V, but there may be some advantage to do it in Step IV, since it makes handling of IR divergence somewhat easier.

23 • IR divergence can be handled by I-operation analogous to K-operation for UV divergence.

• However, IR divergence is more complicated and subtle and requires very careful treatment.

• This is why it has not yet been fully automated.

• The residual renormalization was easy for α3 and still man- ageable by hand for α4. • For α5, however, number of UV subtraction terms (each being integral of up to 8th-order) is 13150 so that sum- ming up residual renormalization terms becomes a huge operation.

• Now they will be fully automated:

”Xabc” → subtraction terms in FORTRAN (implemented by Perl and FORM).

24 • Controling the whole steps: • Flow of entire process governed by shell script.

(a) Find the input information from data ”Xabc” prepared in Step I.

(b) Construct components of integration code in FORTRAN by Steps II, III, IV.

(c) Gather all FORTRAN codes from Step IV.

• Step V can be attached at the end of Step IV to make the entire process automatic. • But we are treating Step V separately for the momemt.

25 • Thus far we completed Steps I, II, III, IV for 135 diagrams which have only UV-divergent vertex subdiagrams.

• For 254 diagrams containing self-energy subdiagrams Steps I - III have been completed.

• At present Step IV is not fully automated. Inclusion of subtraction term ∆δmn is done manually. IR divergence has been treated manually, too.

• We are trying to automated them.

26 • While waiting for full automation code, we decided to deal with IR problem temporarily by giving a finite cutoff to photon mass.

• To obtain a good result by numerical integration it is impor- tant to subtract explicitly the residual (finite) mass renor- malization terms beforehand. • If it is left unsubtracted, it becomes the source of linear IR divergence which is harder to handle numerically.

• At present all this is done by hand, but is being automated.

• Once linear IR divergence is removed, logarithmic diver- gence can be handled easily by photon mass cutoff.

27 • As warm-up, we tested this approach for α3 and α4 cases.

• α3 case: q-type only Photon cutoff λ2 =10−6 Exact treatment

0.8941 (272) 0.904979...

 All diagrams generated in 39 seconds on DEC Alpha.  107 sampling points 50 iterations took 25 - 45 min on DEC Alpha.

• Effect of cutoff seems to be within errorbars. • Good agreement shows that our automating algorithm is bug-free and gives good approximate answer.

28 • α4 case: q-type only cutoff λ2 =10−4 numerical with λ =0

-2.1005 (1216) -1.9931 (35)

 All 47 diagrams generated in 1240 seconds on DEC Alpha.  107 sampling points 50 iterations using 64 CPU took 8 min to 101 min.  Final results required up to 150 iterations.

• Crude but good agreement provides tentative confirmation of previous eighth-order code.

• Definitive confirmation will be attempted shortly.

29 • Current status of numerical integration:

• Crude evaluation by VEGAS of all 389 integrals has been carried out. • Statistics of running α5 code:

 10 - 20 minutes for generation of a FORTRAN code for each diagram on DEC Alpha.

 Typical integral consists of 90,000 lines of FORTRAN code occupying more than 6 Megabytes.

 107 sampling points × 20 iterations takes 5 - 7 hours on 32 CPU PC cluster. • Step V for residual renormalization is being carried out.

• We will soon have a crude value of Set V for λ2 =10−4. • Next on schedule is treatment of IR divergence by IR div. subtraction method, which enables us to put λ =0.

30 5. Other diagrams

• We have already evaluated the contribution of 17 gauge invariant subsets Sets I(a, b, c, d, e, f), II(a, b, f), VI(a, b, c, e, f, i, j, k) for both ae and aµ. T. Kinoshita and M. Nio, Phys. Rev. D 73, 053007 (2006)

• The result for the electron from 958 vertex diagrams is (10) A2 [part]=−1.8239 (63),

• This comes from less than 8 % of entire diagrams so that it is not significant numerically except that it is not very large.

31 • On the other hand, for muon these 17 g-i sets (consisting of 2958 vertex diagrams) include all dominant terms, and give

(10) A2 (mµ/me)[part]=662.50 (27).

• Since contribution of other diagrams are not likely to be large, we may chose as best estimate (10) A2 (mµ/me)[estimate]=663 (20).

• This is 8.5 times more precise than the old estimate (10) A2 (mµ/me)[old estimate]=930 (170). S. Karshenboim, Yad. Phys. 56, 252 (1993) (10) and downgrades A2 as the serious source of theoretical uncertainty.

32 6. Remaining task

• The method developed for Set V enables us to evaluate Set III(a), Set III(b), and Set IV very speedily.

• Remaining sets I(g, h), II(c, d), III(c), and VI(d, g, h) do not seem to present particular complication.

• Some extra work may be needed for I(i), I(j), II(e).

• We will have a complete α5 term within few years.

33