O Peuchant

R Thiébaut

S Capdepont

V Lavignolle-Aurillac

D Neau

P Morlat

F Dabis

H Fleury

B Masquelier

Anrs Co3

Aquitaine Cohort

Transmission of HIV-1 minority resistant variants and response to first-line antiretroviral therapy

Keywords: HIV-1 drug resistance, transmission, antiretroviral therapy, minority variants

Background:

The transmission of drug-resistant HIV-1 can impair the virological response to antiretroviral therapy (ART). Minority resistant variants (RV) have been detected in acute seroconverters. We investigated the clinical relevance of the detection of majority and of minority RV in an observational study in ART-naïve, recently infected patients.

Methods:

We included patients infected between 1996 to 2005, with a plasma sample obtained < 18 months after seroconversion and prior to ART initiation. Majority RV were determined by direct population sequencing. Minority RV were searched by allele-specific PCR for the mutations K103N and M184V in RT and L90M in protease. The association between resistance and viro-immunological response to ART was estimated by using a piecewise linear mixed model.

Results: Majority RV were detected in 23/172 (13.4%) patients. Patients with majority RV had a lower mean plasma viral load (VL) and higher mean CD4 cell count at baseline compared to those without resistance. The decrease in VL between one and 6 months on ART was significantly steeper in patients with sensitive viruses compared to patients with majority RV (p=0.029). Minority RV were detected in 21/73 (29%) patients with wild-type viruses at sequencing analysis. The presence of minority RV did not modify baseline VL and CD4 cell count, and did not affect the changes in VL and CD4 cell count.

Conclusion:

The transmission of majority RV, but not of minority RV, influenced the response to ART in this prospective study. The detection of the transmission of minority RV warrants further clinical validation.

Introduction

The selection of HIV-1 variants with resistance to antiretroviral drugs (ARV) is one important factor limiting the effect of antiretroviral therapy (ART). In ART-treated patients with detectable plasma HIV-1 RNA, the prevalence of resistance to at least one drug can reach up to 80% [START_REF] Costagliola | Prevalence of HIV-1 Drug Resistance in Treated Patients: A French Nationwide Study[END_REF][START_REF] Richman | The prevalence of antiretroviral drug resistance in the United States[END_REF]. The transmission of drug resistant HIV-1 has been widely documented and the prevalence of resistance in drug -naïve patients has been shown to reach 10% or more in Western Europe and in North America [START_REF] Masquelier | Prevalence of transmitted HIV-1 drug resistance and the role of resistance algorithms: data from seroconverters in the CASCADE collaboration from 1987 to 2003[END_REF][START_REF] Wensing | Prevalence of drug-resistant HIV-1 variants in untreated individuals in Europe: implications for clinical management[END_REF][START_REF] Weinstock | The epidemiology of antiretroviral drug resistance among drug-naive HIV-1-infected persons in 10 US cities[END_REF]. Phylogenetic analyses have shown that transmitted resistant variants could originate not only from treated patients but also from acutely-infected, drug-naïve patients [START_REF] Brenner | High rates of forward transmission events after acute/early HIV-1 infection[END_REF]. Some studies have shown the impact of the transmission of resistant variants leading to worse virological and immunological responses to first-line ART [START_REF] Chaix | Sub-Optimal response to HAART in patients treated at time of primary HIV-1 infection and infrected with HIV resistant strains[END_REF][START_REF] Little | Antiretroviral-drug resistance among patients recently infected with HIV[END_REF][START_REF] Grant | Time trends in primary HIV-1 drug resistance among recently infected persons[END_REF], but other studies failed to show such an influence [START_REF] Pillay | The impact of transmitted drug resistance on the natural history of HIV infection and response to first-line therapy[END_REF].

The detection of transmitted resistant variants has been processed in the vast majority of studies by using conventional nucleotide sequencing methods after PCR amplification of the viral genome from plasma RNA. However, these bulk sequencing procedures cannot detect minority variants below a detection threshold of 20% of the global population. The detection of minor populations of drug-resistant HIV-1 has been recently reported in acute seroconverters [START_REF] Metzner | Detection of minor populations of drugresistant HIV-1 in acute seroconverters[END_REF]. Thus even in the absence of resistance in the current dominant genotype, the possibility remains that minority species may exist against which the treatment will prove less effective.

In this study we report the detection of majority resistant variants and of minority resistant variants in a cohort of recently HIV-1 infected patients, and we study the influence of majority and of minority resistant variants on subsequent response to first-line ART.

Patients and Methods

Study population

The patients included in this retrospective study were HIV-1 infected patients followed-up at the Bordeaux University Hospital within the ANRS CO 3 Aquitaine Cohort. The Aquitaine Cohort is a prospective hospital-based cohort of HIV-1-infected patients under routine clinical management [START_REF] Lazaro | Change in T-lymphocyte count after initiation of highly active antiretroviral therapy in HIV-infected patients with history of Mycobacterium avium complex infection[END_REF], initiated in 1987 in the Bordeaux University Hospital and four other public hospitals in Aquitaine by the Groupe d'Epidémiologie Clinique du Sida en Aquitaine (GECSA). Inclusion criteria were: all adults who were in-or out-patients of the participating hospital wards with HIV-1 infection confirmed by Western blot testing, regardless of clinical stage, either having at least one follow-up after the first report or with a known date of death, and having given informed consent. Patients were included in the present study if they had a reliably estimated date of HIV-1 seroconversion between 1996 and 2005, either with a laboratory evidence of acute infection (antibody negative with plasma HIV-1 RNA and/or p24 antigen positivity, or evolving antibody response), or with previously seronegative test within 3 years of the first seropositive test. In this latter case, the estimated date of seroconversion was the mid point between the last negative and the first positive serology. Included patients had plasma HIV-1 RNA and CD4 cell count measurements available before therapy and at least at one time point during the first six months on ART.

Among the patients with an accurate estimation of their date of HIV infection between 1996 and 2005 and starting ART, conventional genotypic resistance analysis could be processed in a subgroup of patients with available sample before ART and viral load and CD4 cell counts measurements on ART. The study of minority resistant variants could then be processed on a subgroup of patients originating from this initial study and with available sample and no genotypic resistance detected by sequence analysis.

Genotypic Resistance Analysis

Genotypic resistance analysis was processed on the 1 st available plasma sample within 18 months of the estimated date of seroconversion and before initiation of ART. HIV-1 reverse transcriptase (RT) and protease genes were sequenced according to the Agence Nationale de Recherche sur le SIDA (ANRS) consensus protocol; the details of the methods appear at www.hivfrenchresistance.org. Resistance mutations were reported as listed by the International AIDS Society-USA Panel (www.iasusa.org, update October 2005), including revertant mutations at RT codon 215, and/or the ANRS algorithm version 2006 (www.hivfrenchresistance.org), excluding minor protease mutations and the V118I RT mutation. Genotypic resistance analysis was processed retrospectively for most patients, because these tests are recommended in France for all antiretroviral -naïve patients only since 2006 [START_REF] Yeni | Prise en charge thérapeutique des personnes infectées par le VIH[END_REF] (ref Yeni). Genotypic resistance analysis results were available before the choice of regimen only for patients with diagnosis of HIV-1 primary infection since 2000.

Quantification of minority resistant variants by allele-specific PCR

Minority populations of drug-resistant variants were detected from plasma samples by realtime, allele-specific PCR (ASPCR) for the RT mutations K103N and M184V and the protease mutation L90M. ASPCR was processed on a subgroup of patients originating from the initial group with conventional genotypic resistance analysis, with available plasma samples and/or RT-PCR products and no resistance to the subsequent ART regimen as detected by direct sequencing. We used a method adapted from Metzner et al [START_REF] Metzner | Detection of minor populations of drugresistant HIV-1 in acute seroconverters[END_REF]. RT and protease were first reverse transcribed and PCR-amplified from plasma RNA, as described above for the genotypic resistance analysis. The PCR products were then submitted to nested real-time PCR enabling the specific amplification of wild type or mutated sequences, or whole population for each considered mutation. For the detection of the K103N mutation, the primers were IN The specificity of the detection of the amplified product was checked using the analysis of the melting curve for each reaction. The ASPCR result was expressed as the percentage of mutated sequences reported to the whole (mutated + wild type) sequences ± standard deviation. The discriminatory ability of each assay was tested as previously described [START_REF] Metzner | Emergence of minor populations of human immunodeficiency virus type 1 carrying the M184V and L90M mutations in subjects undergoing structured treatment interruptions[END_REF], in reciprocal experiments by adding 10 6 copies of non complementary DNA (10 6 -10 copies) of either wild type or mutant DNA with the oligonucleotides specific to the wild type or mutated (for L90M) or to the whole population and mutated (for M184V and K103N) population, taken into account the standard deviation.

Samples presenting mutations detected by bulk sequencing analysis and samples scored as wild type by cloning analysis were used as positive and negative controls.

The minority resistant study was done all retrospectively, and could thus not influence the choice of treatment regimens.

Evaluation of the ASPCR assay

The detection of minority resistant variants within the global plasma viral population was assessed by using the ASPCR. This method uses allele-specific oligonucleotides with discriminatory abilities for the detection of either wild-type or mutant sequences. In order to overcome non-specific amplification with wild-type primers at RT codons 103 and 184V, we modified the technique published by Metzner et al [START_REF] Metzner | Detection of minor populations of drugresistant HIV-1 in acute seroconverters[END_REF] by quantifying the global viral population instead of the wild type population. The sensitivities of the assay for detecting a mutated sequence out of the global population were shown to be 0.08%, 0.4% and 0.3% for the mutations L90M, K103N and M184V, respectively. The specificity and sensitivity of the method were always successfully checked on positive and negative controls.

Quantification of plasma HIV-RNA

Plasma HIV-1 RNA was measured using bDNA (Quantiplex ® , Bayer Diagnostics, Eragny, France, threshold 500 or 50 copies/ml) or RT-PCR (Cobas TaqMan ® , Roche Diagnostics, Meylan, France, threshold 40 copies/ml).

Statistical methods

The association between genotypic resistance and plasma HIV-1 RNA or CD4 levels was estimated at treatment initiation and during the first six months on ART. Repeated measurements of HIV RNA and CD4+ cell count were analysed using a piecewise linear mixed model accounting for the intra-patient correlation. Resistance was defined as genotypic resistance to at least one drug of the ART regimen. The effect of resistance was tested on the baseline value at ART initiation, on the short-term slope (until one month) and on the longterm slope (after one month). Furthermore, the estimation of model parameters took into account the left-censoring of HIV RNA due to undetectable measures [START_REF] Thiébaut | Mixed models for longitudinal left-censored repeated measures[END_REF]. Models were adjusted for the type of assay used to quantify plasma HIV RNA. All analyses were performed with SAS Software (SAS v9.0, SAS Institute, Cary, NC, USA). We studied separately the influence of majority resistant variants and of minority resistant variants.

Results

Patients' characteristics and drug resistance in the bulk viral population

Among

Transmission of minority resistant variants

The research of minority resistant variants could be processed for 73 patients originating from the 172 patients with conventional genotypic resistance analysis, who presented similar characteristics to the whole population and were treated by: 2-3 NRTIs (n=21); two NRTIs + one NNRTI (n=14), or two NRTIs + one PI (n=38). Ten of 73 (13.7%) patients were infected by HIV-1 non-B subtypes. Mutations were searched by ASPCR when they were not detected by direct sequencing and had a putative impact on response to ART. Minority variants were detected in 21/73 (29%) patients: K103N was detected in 3/14 (21%) patients, M184V in 18/72 (25%) and L90M in 0/40 (0%) patients. Minority variants represented 2.5% to 7.1% of the total viral population for the mutation K103N and 1.3% to 23.5% for M184V. In patients for whom the M184V was detected as minority population, the subsequent ART regimens were non boosted-PI-based (n=5), boosted-PI-based (n= 5), NRTI-based (n= 5), and efavirenz-based (n= 3).

Association between transmitted resistance and response to ART

Majority variants and response to ART

The association between the transmission of majority resistant variants detected by direct sequencing and the virological and immunological outcome on ART was studied on 172 patients with a 6-month follow-up on ART. The median number of viral load and CD4 measurements in the first 6 months was three. The influence of majority resistant variants on virological response is shown in Figure 1A. At baseline before ART, the baseline plasma HIV-1 RNA was significantly lower in the resistant group compared to the susceptible group (3.81 log 10 copies/ml vs 4.68 log log 10 copies/ml, p=0.002). There was then no difference in the decay of plasma HIV-1 RNA between baseline and month 1 (-1.93 log 10 copies/ml for resistant group vs -2.04 log 10 copies/ml for susceptible group, p= 0.48). However, the continuing decrease in viral load was significantly steeper in the susceptible group compared to the mutated group between month 1 and 6 (-0.43 log 10 copies/ml/month vs -0.11 log 10 copies/ml/month, p= 0.029).

The baseline CD4 (+) cell count was significantly higher in the resistant group compared to the susceptible group (560 cells/µl vs 438 cells/µl, p= 0.04) (Figure 1B). The two groups had then parallel increases in CD4 (+) cell counts on ART leading to the same difference in CD4+ level at 6 months (682 cells/µl vs 533 cells/µl, p= 0.04).

Seven patients presented at least one of three mutations 184V, 103N or 90M as a majority mutation population and tended to have a lower baseline HIV RNA load (difference -0.67, p=0.08) and the change in HIV RNA at months 1 and 6 was similar (p=0.65 and p=0.79) compared to the patients without mutations.

Minority resistant variants and response to ART

The influence of the minority resistant variants was studied in a subgroup of 73 patients. The resistant group (with minority resistant variants) and the susceptible group (with no resistance to the ART regimen) had comparable plasma HIV-1 RNA levels at baseline (4.71 log 10 copies/ml vs 4.81 log 10 copies/ml, p= 0.68) (Figure 2A). There was then no difference in the evolution of the plasma HIV-1 RNA in the two groups between baseline and month 1 (-1.99 log 10 copies/ml in susceptible group vs -2.17 log 10 copies/ml in resistant group, p=0.49) and between month 1 and month 6 (-0.54 log 10 copies/ml in susceptible group vs -0.35 log 10 copies/ml in resistant group, p=0.23)

The comparison of the baseline CD4 (+) cell counts in the two groups showed a non significant trend towards higher CD4 (+) cell counts in the resistant group compared to the susceptible group (451 cells/µl vs 534 cells/µl, p= 0.14) (Figure 2B). No significant difference in the evolution of the CD4 (+) cell counts could was then shown between the two groups (p=0.50 during the first month and p=0.52 later on).

Discussion

In our cohort of patients with recent HIV-1 infection and subsequent ART, enrolled between 1996 to 2005, we could detect the transmission of resistant HIV-1 in 13.4% of patients by using a conventional sequencing method. This prevalence is comparable to other studies in France or in Europe in the same period [START_REF] Masquelier | Prevalence of transmitted HIV-1 drug resistance and the role of resistance algorithms: data from seroconverters in the CASCADE collaboration from 1987 to 2003[END_REF][START_REF] Wensing | Prevalence of drug-resistant HIV-1 variants in untreated individuals in Europe: implications for clinical management[END_REF][START_REF] Chaix | Sub-Optimal response to HAART in patients treated at time of primary HIV-1 infection and infrected with HIV resistant strains[END_REF]. We then investigated the transmission of minority resistant variants in a subgroup of patients with no resistance detected by bulk sequencing. We used a modified version of the ASPCR method in order to detect accurately both the minority mutated genomes, with a limit of detection at < 0.5% of the total population, and the total number of amplified genomes. We detected minority mutated variants associated with resistance to the subsequent ART in 29% of patients. This finding is in accordance with previous data obtained in acute seroconverters [START_REF] Metzner | Detection of minor populations of drugresistant HIV-1 in acute seroconverters[END_REF] and suggests that minority resistant variants can be detected not only at the onset of the primary infection but also later on. Similar data were recently reported from an American cohort of antiretroviralnaïve patients [START_REF] Johnson | Sensitive testing demonstrates a high prevalence of transmitted drug resistance among conventionally genotyped wildtype HIV-1 infections[END_REF] showing that low frequency mutations were equally found in recentlyinfected persons and infections of longer duration. This result is also in agreement with model predictions according to the rapid turnover, the high mutation rate and large viral population size [START_REF] Coffin | HIV population dynamics in vivo: implications for genetic variation, pathogenesis, and therapy[END_REF]. We cannot rule out the possibility that the minority variants could have been generated de novo after transmission, even if, considering the loss of viral fitness conferred by M184V there is no reason that this mutation could have been selected in absence of drug, even at a minority level.

It is important to note that our results were obtained by investigating only three mutations (K103N and M184V in the RT and L90M in the protease), thus the estimation of the frequency of minority resistant variants would have been certainly higher by searching for more mutations. When we studied the virological response in the group of patients having K103N, M184V or L90M as majority population, we could not show any difference in comparison with the group of patients without mutations. However, according to the restricted number of patients with the mutations of interest, the statistical power for finding a significant difference was limited (about 60% for finding a difference of 0.5 log10 copies/mL at a given time).

The mutation more often detected by ASPCR in our study was the M184V mutation. We have two explanations for this finding: first, we showed previously that M184V was the most frequent mutation in patients on ART with detectable viral load [START_REF] Costagliola | Prevalence of HIV-1 Drug Resistance in Treated Patients: A French Nationwide Study[END_REF], thus this mutation is likely to be frequently transmitted; second, the M184V mutation confers an impaired replicative capacity to the virus [START_REF] Wainberg | The impact of the M184V substitution on drug resistance and viral fitness[END_REF], and in absence of 3TC or FTC, the M184V variants rapidly disappear in the majority population.

We could investigate the virological and immunological responses to ART in our patients according to the presence or absence of resistant variants in the majority or only in the minority population. The detection of a baseline resistance in the majority population was shown to be associated with a less pronounced decrease in plasma HIV-1 RNA between one and six months on therapy. This finding is in accordance with previous studies, suggesting that transmitted resistance could lead to sub-optimal response to first-line therapy, supporting the guidelines which recommend performing genotypic resistance analysis prospectively in all patients at the time of diagnosis of the HIV infection [START_REF] Yeni | Prise en charge thérapeutique des personnes infectées par le VIH[END_REF][START_REF] Hirsch | Antiretroviral drug resistance testing in adults infected with human immunodeficiency virus type 1: 2003 recommendations of an International AIDS Society-USA Panel[END_REF]. Interestingly, the detection of transmitted majority resistant variants was associated with a lower baseline plasma viral load and higher CD4 cell counts, suggesting a lower replicative capacity of transmitted resistant variants. Of note, this difference in the CD4 cell count was not determined by the clinician decision of when to start ART. The higher CD4 cell counts in the resistant group were conserved over six months on therapy, suggesting an immunological benefit in the short term.

The detection of minority resistant variants by ASPCR was not associated to a worse virological response to ART in our study, contrary to the association found with the majority resistant variants. This finding could be explained by several hypotheses; the most commonly detected mutation by ASPCR was M184V. This mutation codes the resistance to 3TC and FTC but can reverse the resistance to AZT and tenofovir; Moreover, the decreased replicative capacity conferred by M184V is likely to result in lower viral loads and the continuation of 3TC-containining regimens despite the presence of M184V is not necessarily associated with a worse virological outcome [START_REF] Castagna | Lamivudine monotherapy in HIV-1-infected patients harbouring a lamivudine-resistant virus: a randomized pilot study (E-184V study)[END_REF]. It is also possible that the putative negative effect of the M184V on the virological response -if it exits-could have been masked by the antiviral effect of the other drugs, in majority protease inhibitors. Finally, we cannot exclude that the limited statistical power could have lead to the non-detection of the influence of minority variants in our study, since we could only perform ASPCR in a subgroup of patients.

However, we maximised the statistical analyses by estimating slopes using repeated measurements of markers and taking into account undetectable viral load [START_REF] Metzner | Emergence of minor populations of human immunodeficiency virus type 1 carrying the M184V and L90M mutations in subjects undergoing structured treatment interruptions[END_REF]. Furthermore, our data are very concordant with those recently reported from the Zurich-PHI Study [START_REF] Metzner | Prevalence of minority quasispecies of drug-resistant HIV-1 in patients with primary HIV-1 infection in Zurich in the years 2002-2006[END_REF] that found minority M184V variants in 14.9% of patients and minority K103N variants in 5.4% of patients without association to virological failure on early therapy. By contrast, Johnson et al [START_REF] Johnson | Low-frequency mutations substantially increase the prevalence of transmitted drug resistance and greatly strengthen the relationship between resistance mutations and virological failure. 14th conference on retroviruses and opportunistic infections[END_REF] found an association between the detection of baseline minority resistant variants and virological response in patients enrolled in an abacavir/lamivudine/efavirenz trial. Hence, the heterogeneity due to the various treatment regimens used in our study population may explain the discrepant results with Johnston et al. by reducing the statistical power of our analysis.

In conclusion, our study confirms the need for the detection of transmitted resistant variants by conventional sequencing methods in routine practice. The detection of minority resistant variants in nearly one-third of the investigated patients was not associated with the virological outcome on therapy but warrants further clinical validation in order to precise it's potential interest for the management of ART. Abbreviations: ART: antiretroviral therapy; NRTI: nucleoside reverse transcriptase inhibitor; NNRTI: non-nucleoside reverse transcriptase inhibitor; PI: protease inhibitor

Legends to the figures

K103N 5 '

 5 -CCGCAGGGTTAAAAAAGAIC-3' (nt 2839-2858) and Pol 3002 5'-CTGTGGAAGCACATTGTACTG-3'(nt 2982-3002) for detection of the mutated population, and Pol 2801 5'-TCAAGACTTCTGGGAAGTTCA-3' (nt 2801-2821) and Pol 3122 5'-TGCTGCCCTATTTCTAAGTCA-3' (nt 3122-3134) for the amplification of a 334 bp fragment corresponding to the whole viral population and comprising RT codons 103 and 184. For the detection of the mutation M184V, the primers IN M184V 5nt 3300-3333) enabled the detection of the mutated population, and Pol 2801 and Pol 3122 the detection of the whole population. For the detection of the L90M protease mutation, the primers were Pol 2316 5'-GCTCTATTAGATACAGGAGCAG-3' (nt 2316-2337) and either INL90M 5'-TGCAACCAATCTGAGTCIT-3' (nt 2520-2538) for the detection of the mutated population, or INL90 5'-TGCAACCAATCTGAGTCIA-3'for the detection of the wild type population. In order to overcome the possibility of non-amplification due to subtype-specific orunspecific polymorphic variations in the HIV genome, when we designed primers screened by ASPCR we took into account the polymorphic variations observed in the bulk sequencing results. The real-time PCR reactions were performed on the Light Cycler 1.5 with the LightCycler FastStart DNA Master SYBR Green I ® kit (Roche Diagnostics, Meylan, France). For the quantitation of the different viral populations, DNA standards were prepared by PCR amplification of the RT or protease gene of different pNL4-3 HIV-1 plasmids carrying or not the mutations L90M, K103N and M184V (gift from Dr François Clavel, INSERM U552, Paris, France). The presence or absence of each mutation in the corresponding PCR products was confirmed by sequencing. The standards were tested in duplicate for each experiment, and the standards were prepared by serial dilution from 10 6 to 10 2 copies per reaction. The viral PCR products were tested in triplicate at three different dilutions (10 -1 , 10 -2 and 10 -3 ).

  295 patients with an accurate estimation of their date of HIV infection between 1996 and 2005 and starting ART, 172 patients had an available genotype before ART and viral load and CD4 cell counts measurements on ART and were therefore included in the study. The characteristics of the included patients are shown in table 1 and were not different from those of the initial population. The median time from estimated seroconversion to the date of sampling for resistance genotypic analysis was 5.2 months (IQR: 1.7-10.4). Genotypic resistance was detected by sequence analysis in 23/172 (13.4%) HIV-1 strains. Resistance to NRTIs was detected in 20 patients, resistance to NNRTIs in 11 patients and resistance to PIs in seven patients. Fifty patients were treated by 2-3 NRTIs, 83 patients received PI-containing ART, and 39 received NNRTI-based ART. Twenty-two (12.8%) patients received less than three ARVs. The characteristics of patients, as shown in table 1, did not significantly differ according to the presence or absence of genotypic resistance. The median delay between the date of seroconversion and the date of beginning ART was 9.7 months (IQR 2-16) and 11.9 months (IQR: 5-15) in patients with sensitive and in patients with resistant viruses, respectively (p = 0.41). Finally, 16 (9.3%) patients were infected by an HIV-1 strain presenting genotypic resistance to at least one drug comprised in the subsequent ART regimen.

Figure 1 :

 1 Figure 1: Mean levels of plasma HIV RNA (A) and CD4 cell count (B) predicted by a

Figure 2 :

 2 Figure 2: Mean levels of plasma HIV RNA (A) and CD4 cell count (B) predicted by a

Figure 1

 1 Figure 1 A

Table 1 :

 1 Characteristics of the 172 patients included in the study

	N (%)

Acknowledgements

The authors thank all patients included in the study. We are indebted to Pascal Bonot and Marie-Hélène Shrive for excellent technical assistance. Authors' contributions: O Peuchant developed and processed the allele-specific PCR; R Thiébaut developed the statistical model and participated in the writing of the manuscript; S Capdepont made the sequence analysis and phylogeny; V Lavignolle-Aurillac made the statistical analysis; D Neau and P Morlat were responsible for the clinical diagnosis and follow-up of the patients; F Dabis coordinated the Aquitaine Cohort; H Fleury directed the virology laboratory; and B Masquelier coordinated the study and wrote the manuscript.


Jean-Philippe Aguilar
email: aguilar@cpt.univ-mrs.fr

Eduardo De Rafael

David Greynat
email: david.greynat@ific.uv.es

Muon Anomaly from Lepton Vacuum Polarization and the Mellin-Barnes Representation

Keywords:

We evaluate, analytically, a specific class of eighth-order and tenth-order QED contributions to the anomalous magnetic moment of the muon. They are generated by Feynman diagrams involving lowest order vacuum polarization insertions of leptons l = e, µ, and τ . The results are given in the form of analytic expansions in terms of the mass ratios me/mµ and mµ/mτ . We compute as many terms as required by the error induced by the present experimental uncertainty on the lepton masses. We show how the Mellin-Barnes integral representation of Feynman parametric integrals allows for an easy analytic evaluation of as many terms as wanted in these expansions and how its underlying algebraic structure generalizes the standard renormalization group properties. We also discuss the generalization of this technique to the case where two independent mass ratios appear. Comparison with previous numerical and analytic evaluations made in the literature, whenever pertinent, are also made.

I. INTRODUCTION

The present experimental world average of the anomalous magnetic moment of the muon a µ , assuming CPTinvariance, viz. a µ + = a µ -, is a (exp) µ = 116 592 080 (63) × 10 -11 (0.54 ppm) ,

where the total uncertainty includes a 0.46 ppm statistical uncertainty and a 0.28 ppm systematic uncertainty, combined in quadrature. This result is largely dominated by a series of precise measurements carried out at the Brookhaven National Laboratory (BNL) during the last few years, by the E821 collaboration, with results reported in ref. [START_REF] Bennett | Muon (g-2) Collaboration[END_REF] and references therein. The prediction of the Standard Model, as a result of contributions from many physicists is where the error here is dominated at present by the lowest order hadronic vacuum polarization contribution uncertainty (±46.6 × 10 -11 ), as well as by the theoretical uncertainty in the hadronic light-by-light scattering contribution, estimated to be ±40 × 10 -11 . Errors here have also been combined in quadrature. The results quoted in (1.1) and (1.2) imply a 3.4 standard deviation in the difference a (exp) µ -a (SM) µ = (295 ± 88) × 10 -11 .

(1.3) This 3.4 σ deviation deserves attention. Ideally, and before one can attribute the present discrepancy to new Physics, one would like to reduce the theoretical uncertainties as much as possible, parallel to a new experimental effort toward an even more precise measurement of a µ [START_REF] Hertzog | The Physics case for the New Muon (g-2) Experiment[END_REF][START_REF] Farley | [END_REF]. It is also important to reexamine critically the various theoretical contributions to Eq. (1.2); primarily the hadronic contributions of course, but also the higher order QED and electroweak contributions. It would be reassuring to have, at least, two independent calculations of some of these contributions, as well as of the higher order estimates. The purpose of this article is a first step (albeit a small one) in this direction.

We shall be concerned with a specific class of eighth-order and tenth-order QED contributions to a µ illustrated by the Feynman diagrams shown in Fig. 1 and Fig. 2 below. The contribution from the diagrams in these figures with no τ vacuum polarization insertions i.e., diagrams (A), (B) and (C) in Fig. 1 and diagrams (A), (B), (C) and (D) in Fig. 2, have been estimated numerically by Kinoshita and Nio in refs. [5] and [6] (see also ref. [7]). There are also analytic results of the same diagrams, published by Laporta [8,9], in the form of analytic expansions in terms of the electron to muon mass ratio me mµ . To our knowledge, the diagrams in Fig. 1 (D) and Fig. 2 (E) involving a tau loop vacuum polarization insertion have not been considered before. We find for these contributions the following results: where the errors are the ones induced by the present accuracy in the determination of the lepton masses. As one can see from the other results reported in the Appendix, the contribution from the diagrams in Fig. 2 (E) is of the same size as the one from the diagrams in Fig. 2 (D).

In a previous paper [10], it has been shown how the Mellin-Barnes integral representation of Feynman parametric integrals combined with the converse mapping theorem [11], allow for an easy evaluation of as many terms as wanted in the asymptotic behaviour of Feynman diagrams in terms of a mass ratio. We shall show how this technique applies to the evaluation of the class of diagrams considered here, and how to generalize it to the case where two independent mass ratios, like me mµ and mµ mτ in the diagrams in Fig. 1 (D) and Fig. 2 (E), enter into consideration. We shall do that in a rather detailed way for various reasons:

• We find, regretfully, that many analytic calculations found in the literature provide very few details about the techniques employed; yet it would be useful to compare the efficiency of different methods when planning new more complex calculations.

• Giving details on some of the intermediate steps of complex calculations can only help to see the qualities and limitations of the methods employed and, occasionally, to spot potential errors.

• As explained in ref. [12] many years ago, there is a large class of higher order contributions which can be estimated using renormalization group arguments: powers of log mµ me terms at a given order are algebraically related to lower order contributions. Subsequent applications have been made by many other authors [START_REF]For a recent application to O(α 4 ) and O(α 5 ) in QED see e[END_REF]; however, very little is known about how to extend renormalization group arguments to subleading terms: constant terms and powers of log mµ me suppressed by me mµ powers. The Mellin-Barnes technique that we are advocating provides a precise answer to that question: as we shall see, in the Mellin-Barnes representation, the asymptotic contributions at a given order factorize in terms of well defined moments of lower order contributions.

• Having a powerful technique to obtain asymptotic expansions provides a useful alternative to the computation of exact analytic expressions, which are often very complicated and cumbersome. After all, ratios of masses are known from experiment only to a fixed accuracy; the associated error propagates into a numerical uncertainty of the exact analytic result in any case. Computing as many terms in the corresponding asymptotic expansion, as required by the experimental precision in the masses involved, provides an easier alternative to the computation of an exact analytic result, with the same practical accuracy. The calculations in this paper have all been done within this spirit.

• Finally, we think that it is important to have an independent way to check the precision of the contributions evaluated numerically. Many of the multidimensional integrals involved in higher order calculations are far from trivial, which has obliged the experts to develop skillful methods. The numerical results are often dominated by a statistical error which is larger than the error induced by the experimental determination of the lepton masses.

As we shall see, within this limited precision, we find that all the results of Kinoshita and collaborators [5,6,7] checked in this paper are correct, within less than one standard deviation of their estimated error, which is a remarkable performance.

The paper has been organized as follows. The next section summarizes well known results about lepton vacuum polarization and the way we choose to express the various contributions to the muon anomalous magnetic moment. Section III gives a detailed description of the Mellin-Barnes technique and the converse mapping theorem as applied to the eighth-order and tenth-order contributions that we are considering. We also discuss in this section the underlying algebraic properties which generalize the usual renormalization group constraints previously discussed in the literature. Section IV contains a discussion of the evaluation of the various moment integrals which appear in the intermediate steps. They can all be done analytically and we give explicit expression for all of them; but we also show how in practice, the evaluation of a finite number of terms in the ultimate expansion, only requires the explicit knowledge of approximate expressions for these moments. This is important to know, in view of more complicated situations that one may very well encounter when considering other Feynman diagrams. The final evaluation of the eighth-order diagrams is done in Section V and of the tenth-order diagrams in Section VI. Section VII is dedicated to the generalization of the techniques discussed in previous sections to the case where three mass scales appear. The calculation of the diagrams in Figs. 1 (D) and 2 (E) are non-trivial examples of applications of this new technology. Finally, we collect in an Appendix all the results that we have computed in this paper.

II. PHOTON PROPAGATOR AND THE MUON ANOMALY

We are interested in the replacement of a free photon-propagator by the dressed propagator

D αβ (q) = -i g αβ - q α q β q 2 1 q 2 1 1 + l Π (l) (q 2 ) -ia q α q β q 4 , (2.1)
where Π (l) (q 2 ) denotes the proper vacuum polarization self-energy contribution induced by a lepton loop l = e, µ, τ and a is a parameter reflecting the gauge freedom in the free-field propagator (a = 1 in the Feynman gauge). In fact, all the diagrams which we shall be considering here are gauge independent and, therefore, the term i(1 -a)

qαq β q 4 does not contribute to their evaluation.

The perturbation theory expansion generates a series in powers of the functions Π (l) (q 2 ). Here we shall be concerned with a particular selection of the terms in this series, namely those with three and four powers of the lowest order Π (l) (q 2 ) self-energies; more precisely with the terms:

1 1 + l Π (l) (q 2 ) . = -Π (e) (q 2 ) 3 -3 Π (e) (q 2 ) 2 Π (µ) (q 2 ) -3 Π (e) (q 2 ) Π (µ) (q 2 ) 2 (2.2) -3 Π (e) (q 2 ) 2 Π (τ ) (q 2 ) (2.3) + Π (e) (q 2 ) 4 + 4 Π (e) (q 2 ) 3 Π (µ) (q 2 ) + 6 Π (e) (q 2 ) 2 Π (µ) (q 2 ) 2 + 4 Π (e) (q 2 ) Π (µ) (q 2 ) 3 (2.4) +4 Π (e) (q 2 ) 3 Π (τ ) (q 2 ) . (2.5)
When inserted in the free-photon propagator of the lowest order one loop muon vertex, these terms generate the four types of Feynman diagrams collected in Fig. 1 and the five types of diagrams collected in Fig. 2. The Feynman diagrams in these figures give, respectively, eighth order and tenth order contributions to the anomalous magnetic moment of the muon, which are enhanced by powers of log mµ me factors. This is why we are interested in these terms. The reason why we only keep the terms in (2.3) and (2.5) with one power of Π (τ ) (q 2 ) is because, in spite of the enhancement by log . Contributions from two or more powers of Π (τ ) (q 2 ) factors to the muon anomaly will be, therefore, neglected.

A very convenient integral representation for the contribution to the muon anomaly from the graphs in Fig. 1 and Fig. 2 can be obtained as follows. First we write a dispersion relation for the on-shell renormalized photon propagator induced by electron self-energy loops only. This generates the following set of relations:

Π (e) (q 2 ) j = ∞ 0 dt t q 2 t -q 2 -iǫ ρ j 4m 2 e t , j = 1 , 2 , 3 , 4 , (2.6)
with

ρ 1 4m 2 e t = 1 π ImΠ (e) (t) , (2.7)
ρ 2 4m 2 e t = 1 π 2 ReΠ (e) (t) ImΠ (e) (t) , (2.8)
ρ 3 4m 2 e t = 1 π 3 ReΠ (e) (t) 2 ImΠ (e) (t) -ImΠ (e) (t) 3 , (2.9)
ρ 4 4m 2 e t = 1 π 4 ReΠ (e) (t) 3
ImΠ (e) (t) -4ReΠ (e) (t) ImΠ (e) (t)

3

.

(2.10)

Using these representations, the contribution to the muon anomaly from the various terms in the series in Eqs. (2.2), (2.3) and (2.4), (2.5) can be viewed as the convolution of electron spectral functions, modulated by powers of the muon or tau self-energy functions, with a free-photon propagator replaced by a fictitious massive photon propagator:

-ig αβ 1 q 2 ⇒ -ig αβ (-1) q 2 -t + iǫ . (2.11)
The muon vertex loop integral over the virtual q-momenta can then be traded by a Feynman x-parameter integral (see the review article in ref. [15] and the earlier references therein), with a net contribution to the muon anomaly, up to an overall combinatorial factor F (j,p) which can be read from the expansion in Eqs. (2.2) to (2.5):

a (j,p) µ = α π ∞ 0 dt t 1 0 dx x 2 (1 -x) x 2 + t m 2 µ (1 -x) F (j,p) Π (l=µ ,τ ) -x 2 1 -x m 2 µ p ρ j 4m 2 e t , (2.12)
where p = 0, 1, 2, 3 when l = µ and p = 1 when l = τ , while the index j counts the number of electron loops in the vacuum polarization; therefore

F (3,0) = 1 , F (2,1) = 3 , F (1,2) = 3 , (2.13)
and

F (4,0) = -1 , F (3,1) = -4 , F (2,2) = -6 , F (1,3) = -4 . (2.14)
Notice that in the process of trading the integral over the virtual q-momenta by the Feynman parameter x, one has obtained the effective replacement:

q 2 ⇒ -x 2 1 -x m 2 µ , (2.15)
in the muon and tau self-energy functions.

The lowest order vacuum polarization self-energy functions in QED are well known. We shall use the representations given in ref. [16]. With

δ = 1 - 4m 2 e t , (2.16)
the lowest order spectral function for the electron is

1 π ImΠ (e) (t) = α π δ 1 2 - 1 6 δ 2 θ t -4m 2 e , (2.17)
and the real part δ = 1 -

4m 2 e q 2 is ReΠ (e) (q 2 ) = α π 8 9 - 1 3 δ 2 + δ 1 2 - 1 6 δ 2 log |1 -δ| 1 + δ . (2.18)
We shall also need the expression for the vacuum polarization self-energy induced by a muon loop, in the euclidean region, and as a function of the Feynman x-parameter:

Π (µ) -x 2 1 -x m 2 µ = α π 5 9 + 4 3x - 4 3x 2 + - 1 3 + 2 x 2 - 4 3x 3 log(1 -x) ; (2.19)
as well as the one induced by a tau loop which, for our purposes, it is more convenient to keep in the Feynman parametric integral representation:

Π (τ ) -x 2 1 -x m 2 µ = - α π 1 0 dz2z(1 -z) log 1 + x 2 1 -x z(1 -z) m 2 µ m 2 τ .
(2.20)

III. THE MELLIN-BARNES REPRESENTATION AND THE RENORMALIZATION GROUP

It is now useful to introduce the two Mellin-Barnes integral representations [33] :

x 2 (1 -x) x 2 + t m 2 µ (1 -x) = 1 2πi cs+i∞ cs-i∞ ds 4m 2 e t s 4m 2 e m 2 µ -s x 2s (1 -x) 1-s Γ(s)Γ(1 -s) , (3.1)
where the threshold scale 4m 2 e is chosen for later convenience, and [START_REF]The Mellin variable t should not be confused with the dispersive invariant mass squared t[END_REF]

log 1 + x 2 1 -x z(1 -z) m 2 µ m 2 τ = 1 2πi ct+i∞ ct-i∞ dt m 2 µ m 2 τ -t x 2 1 -x z(1 -z) -t Γ(t) t Γ(1 -t) , (3.2)
where the integration paths along the imaginary axis are defined in the fundamental strips [11]:

c s = Re(s) ∈ ]0, 1[ and c t = Re(t) ∈ ] -1, 0[ . (3.3)
The interest of these representations lies in the property that the dependence on the physical mass ratios is then fully factorized and the Feynman parametric integrals one is left with are then those of a massless case. The only new feature is that they have to be computed as functions of the Mellin s-complex variable, in the case of electron and muon loops only; and as functions of the (s , t)-complex manifold in the case of electron, muon and tau loops. In all the cases we shall be concerned with here, it is possible to obtain analytic expressions for these functions in a rather straightforward way. In fact, they turn out to be rational functions of products of Gamma functions and Polygamma functions which depend linearly on the Mellin variables s, or s and t.

Let us first discuss the case where we only have one Mellin s-complex variable. It corresponds to the Feynman diagrams where only two mass scales are involved (i.e. the case where l = µ in Eq. (2.12)). Using the representation in Eq. (3.1) one can rewrite Eq. (2.12), for a fixed p and j, as follows

a µ = α π 1 2πi cs+i∞ cs-i∞ ds 4m 2 e m 2 µ -s M(s) , (3.4)
with

M(s) = Γ(s)Γ(1 -s) 1 0 dx x 2s (1 -x) 1-s F (j,p) Π (µ) -x 2 1 -x m 2 µ p ∞ 0 dt t 4m 2 e t s ρ j 4m 2 e t . (3.5)
The converse mapping theorem relates the asymptotic behaviour of a µ as a function of the small mass ratio 4m 2 e /m 2 µ , to the singularities of the integrand M(s) as a function of the Mellin s-complex variable. For m 2 e ≪ m 2 µ the appropriate s-singularities are those in the left-hand-side of the fundamental strip and they are all on the negative real axis. The precise relation goes as follows: with p ∈ R and k ∈ N, the function M(s) in the l.h.s. of the fundamental strip has a singular expansion of the type (ordered in increasing values of p):

M(s) ≍ p k a p,k (s + p) k+1 . (3.6)
The corresponding asymptotic behaviour of a µ (ordered in increasing powers of p) is then:

a µ ∼ 4m 2 e m 2 µ → 0 α π p k (-1) k k! a p,k 4m 2 e m 2 µ p log k 4m 2 e m 2 µ , (3.7)
and the problem is then reduced to the calculation of the a p,k residues. This, in turn, requires the evaluation of the moment integrals One can immediately see that these moments can also be seen as the Mellin transform of the spectral functions ρ j with respect to the variable ξ = 4m 2 e t , since

∞ 0 dt t 4m 2 e t s ρ j 4m 2 e t and 1 0 dx x 2s (1 -x) 1-s Π (µ) -x 2 1 -x m 2 µ p , (3.8
α π j R j (s) = ∞ 0 dξ ξ s-1 ρ j (ξ) , ξ = 4m 2 e t , (3.10)
where in fact the integral over ξ only runs from zero to one, because of the threshold factor θ(t-4m 2 e ) in the lowest order spectral function in Eq. (2.17). The interest of this point of view lies in the fact that the Mellin-Barnes representation relates the singular expansion of any of these R j (s) moments to the asymptotic expansion of the spectral functions ρ j (ξ). Indeed, from Eq. (3.10) and the asymptotic behaviour of ρ j (ξ) at small ξ and at large ξ, there follows that .11) In other words, the direct mapping (in the sense of ref. [11]) establishes a precise relation between the asymptotic expansion of the spectral functions ρ j (ξ) at small ξ (which is known from their explicit analytic expression) and the singular series of the moments R j (s), for s ≤ 0, that we are interested in. We shall come back later on to these relations in somewhat more detail.

ρ j (ξ) = α π j 1 2πi rs+i∞ rs-i∞ ds 1 ξ s R j (s) , with r s = Re(s) ∈ ]0, ∞[ . ( 3
The second type of moments in Eq. (3.8) are also well defined Mellin transforms; in this case with respect to the invariant photon momenta

ω ≡ Q 2 m 2 µ = x 2 1 -x , (3.12)
flowing in the basic muon vertex diagram. Indeed, one can easily verify that

α π p Ω p (s) ≡ 1 0 dx x 2s (1 -x) 1-s Π (µ) -x 2 1 -x m 2 µ p = ∞ 0 dω ω s-1 ω 4 + ω √ 4 + ω - √ ω √ 4 + ω + √ ω 2 Π (µ) (-ω m 2 µ ) p . (3.13)
Again, the evaluation of the singular series associated to the moments Ω p (s), for s ≤ 0, is very much facilitated by the direct mapping which relates them to the asymptotic expansion of Π (µ) (-ω m 2 µ ) p for small ω.

The remarkable property of the Mellin-Barnes representation is precisely the factorization in terms of moment integrals as shown in Eq. (3.5). It is in fact this factorization which is at the basis of the renormalization group properties discussed by many authors. The algebraic factorization above, however, is more general and it also shows the full underlying renormalization group structure at work. The classical renormalization group constrains which have been exploited in the literature only apply to the evaluation of leading asymptotic behaviours (powers of logarithms and constant terms). In the example above, this is encoded by the properties of the Mellin singularity at s = 0. This singularity governs the spectral function moments R j (0) on the one hand, as well as the contribution to the muon anomaly from vacuum polarization muon loop insertions (p loops): the moments Ω p (0). Notice that the moments R j (0) are UV-singular, hence the need of the s-regularization. The singularity is related to charge renormalization and therefore to the QED β-function at a given order in perturbation theory [12], in our case the lowest order. The predictive power of the renormalization group structure lies in the fact that once we know the two types of moments in Eq. (3.8) at a given order in perturbation theory we have a prediction at a higher order for the convolution which in our case gives a µ in Eq. (3.4). What is new here is that this factorization extends as well to the subleading terms in the expansion modulated by inverse m 2 µ /m 2 e -powers. What governs these terms is nothing but the residues of the successive Mellin singularities in the negative real axis. Here, the expansion in inverse m 2 µ /m 2 e -powers is analogous to the 1/Q 2 -expansion in the Operator Product Expansion of Green's functions in QCD, while the moments R j (s), for s < 0, are the equivalent of the so called vacuum condensates. The big difference, of course, is that in QED these condensates are moments of spectral functions which are known at all values of t and, therefore, can be calculated explicitly at a given order in perturbation theory. Notice, however, that also in QED these moments (condensates) are a priori ill defined because they are singular. The Mellin-Barnes technique provides a systematic regularization and hence a precise separation of short-distance and long-distance effects.

The case of two Mellin (s , t)-complex variables, corresponding to Feynman diagrams with both electron loops and tau loops, is conceptually more complicated and it requires a specific treatment which we shall provide in Section VII. A detailed discussion of the very interesting underlying mathematics which governs this case will be given in a forthcoming publication [21].

We finally give below the exact explicit expressions, in the Mellin-Barnes integral representation, of the contribution to the muon anomaly from each of the specific set of Feynman diagrams in Fig. 1 and Fig. 2. • Three Electron Loops, Fig. 1(A) [one diagram]:

A. Eighth Order Contributions

a (eee) µ = α π 4 1 2πi cs+i∞ cs-i∞ ds 4m 2 e m 2 µ -s Γ(s)Γ(1 -s) Ω 0 (s) R 3 (s) . (3.14)
• Two Electron Loops and One Muon Loop, Fig. 1(B) [three diagrams]:

a (eeµ) µ = α π 4 3 2πi cs+i∞ cs-i∞ ds 4m 2 e m 2 µ -s Γ(s)Γ(1 -s) Ω 1 (s) R 2 (s) . (3.15)
• One Electron Loop and Two Muon Loops, Fig. 1(C) [three diagrams]:

a (eµµ) µ = α π 4 3 2πi cs+i∞ cs-i∞ ds 4m 2 e m 2 µ -s Γ(s)Γ(1 -s) Ω 2 (s) R 1 (s) . ( 3
a (eeτ ) µ = α π 4 3 2πi cs+i∞ cs-i∞ ds 4m 2 e m 2 µ -s 1 2πi ct+i∞ ct-i∞ dt m 2 µ m 2 τ -t × Γ(s)Γ(1 -s) Γ(t) t Γ(1 -t) Θ(s, t) R 2 (s) , (3.17)
where Θ(s, t) is the Feynman parametric integral (see Eqs.

Θ(s, t) = 1 0 dxx 2s (1 -x) 1-s 1 0 dz x 2 1 -x z(1 -z) -t = (-2) Γ(2 -t)Γ(2 -t) Γ(4 -2t) Γ(1 + 2s -2t)Γ(2 -s + t) Γ(3 + s -t) . (3.18)
Notice that the dependence on the variables s and t in the function Θ(s, t) is not factorized. It is this fact that requires new technical considerations which we shall discuss in Section VII.

B. Tenth Order Contributions

• Four Electron Loops, Fig. 2(A) [one diagram]:

a (eeee) µ = α π 5 (-1) 2πi cs+i∞ cs-i∞ ds 4m 2 e m 2 µ -s Γ(s)Γ(1 -s)Ω 0 (s) R 4 (s) . (3.19)
• Three Electron Loops and One Muon Loop, Fig. 2(B) [four diagrams]:

a (eeeµ) µ = α π 5 (-4) (-4) 2πi cs+i∞ cs-i∞ ds 4m 2 e m 2 µ -s Γ(s)Γ(1 -s) Ω 1 (s) R 3 (s) . (3.20)
a (eeµµ) µ = α π 5 (-6) 2πi cs+i∞ cs-i∞ ds 4m 2 e m 2 µ -s Γ(s)Γ(1 -s) Ω 2 (s) R 2 (s) . ( 3
a (eµµµ) µ = α π 5 (-4) 2πi cs+i∞ cs-i∞ ds 4m 2 e m 2 µ -s Γ(s)Γ(1 -s) Ω 3 (s) R 1 (s) . (3.22)
• Three Electron Loops and One Tau Loop, Fig. 2

(E) [four diagrams]:

The only change here, with respect to the representation in Eq. (3.17), is the convolution with the moment R 3 (s) (instead of R 2 (s)) and the combinatorial factor (-4) [instead of (3)]:

a (eeeτ ) µ = α π 5 (-4) 2πi cs+i∞ cs-i∞ ds 4m 2 e m 2 µ -s 1 2πi ct+i∞ ct-i∞ dt m 2 µ m 2 τ -t × Γ(s)Γ(1 -s) Γ(t) t Γ(1 -t) Θ(s, t) R 3 (s) , (3.23)
where Θ(s, t) is the function defined in Eq. (3.18).

IV. THE MOMENT INTEGRALS

We are now left with two types of moment integrals: the R j (s) moments of the electron-positron spectral functions in Eq. (3.10), (the integrals over the variable ξ, or equivalently the variable δ in Eq. (2.16)); and the Ω p (s) moments of the euclidean muon vacuum polarization, (the integrals over the invariant photon momenta ω, or equivalently the Feynman x-parameter). It is a luxury that these integrals can all be done analytically. Here we shall give the results and comment on a few technical points; in particular on their relation to the underlying renormalization group properties.

• The Ω 0 (s) Moment.

This moment, which appears in a (eee) µ and a (eeee) µ , corresponds to the trivial x-integral

Ω 0 (s) = 1 0 dx(1 -x) x 2 1 -x s = Γ(1 + 2s)Γ(2 -s) Γ(3 + s) . (4.1)
The particular value Ω 0 (0) = 1 2 is precisely the coefficient of the lowest order muon anomaly, the Schwinger term:

a µ = α π Ω 0 (0) = 1 2 . Notice that Ω 0 (s) is IR-singular at s = -1/2, -3/2, • • • .
It is precisely these IR-singularities that are at the origin of the contributions in odd powers of m e /m µ in a

R 1 (s) = 1 0 dδ 2δ(1 -δ 2 ) s-1 1 2 δ 1 - 1 3 δ 2 , (4.2)
which can be trivially computed, with the simple result:

R 1 (s) = √ π 4 1 s Γ(2 + s) Γ 5 2 + s , (4.3)
explicitly showing the singularities in the negative real axis (the origin s = 0 included). Notice that, because of the zeros provided by the factor

1 Γ( 5 2 +s) , there cannot be terms of O m 2 e m 2 µ p for p = 5/2 + n, n = 0, 1, 2, • • • in vacuum polarization contributions to a µ from one electron loop.
The two moments Ω 0 (s) and R 1 (s) are the only quantities required to evaluate the contribution to a (e) µ from the fourth order Feynman diagram in Fig. 3, which, in our representation, is given by the integral

a (e) µ = α π 2 1 2πi cs+i∞ cs-i∞ ds 4m 2 e m 2 µ -s Γ(s)Γ(1 -s) Ω 0 (s) R 1 (s) . (4.4)
Since the moments Ω 0 (s) and R 1 (s) are known analytically, the singular series expansion of the integrand can be computed up to as many terms as one wishes; e.g.,

Γ(s)Γ(1 -s) Ω 0 (s) R 1 (s) ≍ 1 6 1 s 2 + 1 3 log 2 - 25 36
1 s + π 2 8 1 1/2 + s - 1 2 1 (1 + s) 2 + 3 4 -log 2 1 1 + s + • • • , (4.5)
and the converse mapping theorem tells us how to read in a straightforward way the corresponding asymptotic expansion contribution [35]: The reason why we consider this well known contribution here is that it allows us to discuss the factorization properties of the Mellin-Barnes representation and its relation to the underlying renormalization group structure in a case which, in spite of its simplicity, illustrates the generic features rather well. Notice that the singular expansion in Eq. (4.5) results from the combination of the Laurent series of three factors: the geometric series factor Γ(s)Γ(1 -s), which appears in the original Mellin-Barnes representation in Eq. (3.1), and the two moments Ω 0 (s) and R 1 (s). The residue of the leading singularity in 1/s 2 , which provides the coefficient of the log

a (e) µ ∼ m 2 e m 2 µ → 0 α π 2 1 6 log m 2 µ m 2 e - 25 36
m 2 µ m 2
e term in Eq. (4.6), can be read off directly from the asymptotic behaviour of the lowest order spectral function ρ

4m 2 e t ∼ t→∞ 1
3 , (which in turn is correlated to the residue of the leading term in the singular expansion of R 1 (s) ≍ 1 3 1 s ) and the lowest order result Ω 0 = 1/2 ; the geometric factor Γ(s)Γ(1 -s) providing the extra 1/s factor. This is precisely the leading prediction of the renormalization group in this case [12].

The next-to-next-to-leading term in the asymptotic expansion in Eq. (4.6) (the second line) is governed by the 1 s+1/2 term in the singular expansion in Eq. (4.5) (the second line in this equation). The origin of this singularity is the moment Ω 0 (s), which is singular at s = -1/2; the other factors are regular: Γ(-

1/2)Γ(1 + 1/2) = -π and R 1 (1/2) = -π 4 .
Again, the residue of the 1 s+1/2 -singularity of Ω 0 (s) can be read off from the leading term in the asymptotic expansion of the integrand function in Eq. (3.13) i.e.,

ω 4 + ω √ 4 + ω - √ ω √ 4 + ω + √ ω 2 ∼ ω→ 0 1 2 √ ω + O(ω) . (4.7)
We see, therefore, how the factorization of the Mellin-Barnes representation allows one to fix a non-trivial coefficient of the asymptotic expansion of an O α π 2 quantity from the knowledge of two numbers which appear at a lower O α π and have been very easily identified. What about the next-to-leading term? This refers to the factor - 25 36 in the first line of Eq. (4.6), which is not fixed by simple renormalization group arguments. The reason why simple renormalization group arguments fail to fix this term is due to the fact that it does not originate from products of leading terms of singular expansions only. More precisely, the coefficient of the 1 s term in Eq. (4.5) also depends on the next-to-leading terms of the Laurent series of each of the individual factors Ω 0 (s) , and R 1 (s). These terms, however, can also be easily obtained without having to calculate explicitly the functions Ω 0 (s) and R 1 (s). Indeed, since Ω 0 (s) is regular at s → 0 we can simply Taylor expand the integrand and find

Ω 0 (s) ∼ s→ 0 1 2 - 5 4 s + O(s 2 ) . (4.8)
By contrast, the moment R 1 (s) is singular at s → 0; but here we know that the residue of the singularity originates in the leading term of the asymptotic expansion of the lowest order spectral function in the integrand. Therefore, subtracting this leading term from the spectral function itself produces a regular integral from which one can extract the regular series in the Laurent expansion at s → 0 by simple Taylor expansion. The only integral we have to do is then

α π R 1 (s) ∼ s→ 0 1 3s + ∞ 0 dξ ξ -1 ρ 1 (ξ) - 1 3 + O(s) = 1 3s + 1 9 (6 log 2 -5) + O(s) . (4.9)
Combining the results in Eqs. (4.8) and (4.9) with the fact that Γ(s)Γ(1 -s)

∼ s→ 0 1 s + O(s)
, one easily gets the coefficient of the 1 s -term in Eq. (4.5) and hence the term - 25 36 in the asymptotic expansion in Eq. (4.6). We leave to the studious reader the pleasure of reproducing the residues of the

α π 2 R 2 (s) = ∞ 0 dξ ξ s-1 ρ 2 (ξ) , (4.10)
associated to the spectral function in the r.h.s. of Eq. (2.8). In terms of the variable δ, it gives rise to the integral

R 2 (s) = 2 1 0 dδ 2δ(1 -δ 2 ) s-1 1 2 δ 1 - 1 3 δ 2 8 9 - 1 3 δ 2 + δ 1 2 - 1 6 δ 2 log 1 -δ 1 + δ , (4.11)
which can be done analytically rather easily with the result

R 2 (s) = √ π 9 (-1 + s)(6 + 13s + 4s 2 ) s 2 (2 + s)(3 + s) Γ(1 + s) Γ 3 2 + s , (4.12)
explicitly showing the singularities in the negative real axis. Notice that, because of the zeros provided by the factor

1 Γ( 3 2 +s) , there cannot be terms of O m 2 e m 2 µ p for p = 3/2 + n, n = 0, 1, 2, • • • in vacuum polarization contributions to a µ from two electron loops. • The Ω 1 (s) Moment.
It corresponds to the Mellin transform (see Eq. (3.13))

α π Ω 1 (s) = ∞ 0 dω ω s-1 ω 4 + ω √ 4 + ω - √ ω √ 4 + ω + √ ω 2 Π (µ) (-ω m 2 µ ) = 1 0 dx (1 -x) x 2 1 -x s Π (µ) -x 2 1 -x m 2 µ . (4.13)
This integral can also be easily done, using the expression in Eq. (2.18) in terms of a subtracted logarithm:

log(1 -x) → log(1 -x) --x - x 2 2 - x 3 3 + -x - x 2 2 - x 3 3 . (4.14)
The integral over x is then convergent at x = 0 term by term, with the result

Ω 1 (s) = Γ(2 -s) - 4 3 Γ(-1 + 2s) Γ(1 + s) + 4 3 Γ(2s) Γ(2 + s) + 5 9 Γ(1 + 2s) Γ(3 + s) + - 4 3 Γ(-2 + 2s) Γ(s) + 2 Γ(-1 + 2s) Γ(1 + s) - 1 3 Γ(1 + 2s) Γ(3 + s) H 1-s + 4 3 Γ(-2 + 2s) Γ(s) H -1+s -2 Γ(-1 + 2s) Γ(1 + s) H s + 1 3 Γ(1 + 2s) Γ(3 + s) H 2+s , (4.15)
where H s denotes the function

H s = ψ(1 + s) + γ E , (4.16)
related to the ψ-function [START_REF]Recall that the ψ(z)-function is meromorphic with simple poles at z = 0[END_REF] ψ(z) = d dz log Γ(z), as follows

ψ(z) = -γ E + ∞ m=0 1 m + 1 - 1 z + m , (4.17)
and, therefore, when z is an integer n, H n corresponds to the Harmonic sum

H n = 1 + 1 2 + 1 3 + 1 4 + • • • + 1 n . (4.18)
Notice that Ω 1 (0) = -119 36 + π 2 2 is precisely the well known coefficient of the fourth order contribution to a (µ) µ from the Feynman graph in Fig. 4:

a (µ) µ = α π 2 - 119 36 + π 2 2 . ( 4

.19)

Fig. 4 Fourth-order Feynman diagram with a lowest order vacuum polarization muon-loop.

Another property of Ω 1 (s), which is also valid for all Ω p (s) with p ≥ 1, is the fact that they are no longer singular at s = -1/2. This can be readily seen from the fact that Π (µ) -x 2 1-x m 2 µ in Eq. (4.13) vanishes at x = 0:

Π (µ) -x 2 1 -x m 2 µ ∼ x→ 0 α π - 1 15 x 2 + O(x 3 ) . (4.20)
This is why Feynman graphs with electron loops and at least one muon loop insertion in the vacuum polarization

have no O m 2 e m 2 µ 1/2
terms in their asymptotic contributions. However, from the asymptotic behaviour

ω 4 + ω √ 4 + ω - √ ω √ 4 + ω + √ ω 2 Π (µ) (-ω m 2 µ ) ∼ ω→ 0 α π - 1 30 ω 3/2 + O(ω 2 ) , (4.21)
in Eq. (4.13), there follows that .

Ω 1 (s) ≍ - 1 30
1 3 2 + s + • • • , ( 4
The moments Ω 0 (s) and R 2 (s) fix entirely the evaluation of the contribution to a µ from the sixth order Feynman diagram in Fig. 5, which, in our representation, is given by the integral

a (ee) µ = α π 3 (-1) 1 2πi cs+i∞ cs-i∞ ds 4m 2 e m 2 µ -s Γ(s)Γ(1 -s) Ω 0 (s) R 2 (s) . ( 4

.23)

Fig. 5 Sixth-order Feynman diagram with two lowest order electron-loop vacuum polarization insertions.

Since we know Ω 0 (s) and R 2 (s) analytically we can evaluate the full singular series of their product and, therefore, by the converse mapping theorem, the asymptotic expansion of a (ee) µ to as high a degree of accuracy as we wish. As an illustration, we give the results for the first few terms: where the contribution from the leading singularity at s = 0 is the one in the first line, which agrees with the earlier calculations in refs. [16,22]. The exact analytic evaluation of a (ee) µ , as well as of the full sixth-order contribution to a µ from electron loop insertions, including light-by-light scattering loops, can be found in the papers by Laporta and Remiddi [23,24]. On the other hand, the moments Ω 1 (s) and R 1 (s) fix entirely the evaluation of the contribution to a µ from the mixed sixth order Feynman diagram in Fig. 6, which, in our representation, is given by the integral

a (ee) µ ∼ m 2 µ m 2 e → ∞ α π 3 1 18 log 2 m 2 µ m 2 e - 25
a (eµ) µ = α π 3 (-2) 1 2πi cs+i∞ cs-i∞ ds 4m 2 e m 2 µ -s Γ(s)Γ(1 -s) Ω 1 (s) R 1 (s) . ( 4

.25)

Fig. 6 Sixth-order Feynman diagram with one electron-loop and one muon loop vacuum polarization insertions.

Again, since we know Ω 1 (s) and R 1 (s) we can compute as many terms as we wish of the asymptotic expansion of a (eµ) µ

. We give a few terms below:

a (eµ) µ ∼ m 2 µ m 2 e → ∞ α π 3 119 54 - 2 9 π 2 log m 2 µ m 2 e - 61 162 + π 2 27 - m 2 e m 2 µ 115 27 + 4 9 π 2 + O m 2 e m 2 µ 2 log 2 m 2 µ m 2 e , (4.26)
in agreement with earlier calculations [24,25]. Finally, we observe that once the moments Ω 1 (s) and R 2 (s) are known, we have all the ingredients to evaluate the eighth-order contribution a (eeµ) µ in Eq. (3.15) which we shall discuss in the next section. With R 2 (s) known we can also attempt the evaluation of a (eeτ ) µ in Eq. (3.17), which we shall do in Section VII.

• The R 3 (s) Moment.
This moment appears in the expressions of a

α π 3 R 3 (s) = ∞ 0 dξ ξ s-1 ρ 3 (ξ) , (4.27)
of the spectral function in the r.h.s. of Eq. (2.9). It gives rise to the δ-integral

R 3 (s) = 1 0 dδ 2δ(1 -δ 2 ) s-1 3δ 1 2 - 1 6 δ 2 8 9 - 1 3 δ 2 + δ 1 2 - 1 6 δ 2 log 1 -δ 1 + δ 2 -π 2 δ 1 2 - 1 6 δ 2 3 . (4.28)
Only the terms proportional to log 2 1-δ 1+δ require special attention. They give rise to integrals of the type

f (σ) = 1 0 dδ(1 -δ 2 ) σ log 2 1 -δ 1 + δ with σ = s + n -1 and n = 0, 1, 2, 3, 4, 5 . (4.29)
Integrating by parts, we observe that f (σ) obeys a simple functional relation

(1 + 2σ)f (σ) -2σf (σ -1) + 2 √ π σ Γ(σ) Γ 1 2 + σ = 0 , (4.30)
from which, and the boundary condition f (0) = π 2 3 , there follows that

f (σ) = √ π Γ(1 + σ) Γ 3 2 + σ
ψ (1) (1 + σ) , where ψ (1) (z) = d dz ψ(z) . (4.31)

One can then do the integral in Eq. (4.28). After some rearrangement so as to exhibit explicitly the singular structure, we get the following expression

R 3 (s) = √ π 864 Γ(s) Γ 11 3 + s P 7 (s) s(1 + s)(2 + s) -(1 + s)(35 + 21s + 3s 2 ) 27π 2 -162 ψ (1) (s) , (4.32)
with P 7 (s) the seventh degree polynomial P 7 (s) = 3492 -8748s -26575s 2 -9214s 3 + 18395s 4 + 17018s 5 + 5120s 6 + 512s 7 .

(4.33)

The function ψ (1) (z) is called the Polygamma function of index one. In general, the Polygamma function of index n is defined as:

ψ (n) (z) = d n dz n ψ(z) , with ψ (0) (z) = ψ(z) for n = 1, 2, 3, . . . . (4.34)
These functions are also related to the Hurwitz function (also called the generalized zeta function)

ζ(s, z) = ∞ m=0 (m + z) -s z = 0, -1, -2, . . . , Res > 1 , (4.35)
as follows

ψ (n) (z) = (-1) n+1 n! ζ(n + 1, z) . (4.36)
The Polygamma functions are therefore meromorphic with poles at z = 0, -1, -2, . . . , with multiplicities n + 1. In fact, the Hurwitz function and therefore the Polygamma functions, have the following Mellin representation:

ζ(s, z) = 1 Γ(s) ∞ 0 dt t s-1 e -zt 1 1 -e -t
Res > 1 , Rez > 0 .

(4.37)

The moments Ω 0 (s) and R 3 (s) fix entirely the evaluation of the contribution to a µ from the eighth order Feynman diagram in Fig. 1(A), while the moments Ω 1 (s) and R 3 (s) fix entirely the evaluation of the contribution to a µ from the tenth order Feynman diagram in Fig. 2(B). We discuss these results in the next section. With R 3 (s) known one can also attempt the evaluation of a (eeeτ ) µ in Eq. (3.23), which we shall do in Section VII.

• The Ω 2 (s) Moment.

It corresponds to the Mellin transform (see Eq. (3.13))

α π 2 Ω 2 (s) = ∞ 0 dω ω s-1 ω 4 + ω √ 4 + ω - √ ω √ 4 + ω + √ ω 2 Π (µ) (-ω m 2 µ ) 2 = 1 0 dx (1 -x) x 2 1 -x s Π (µ) -x 2 1 -x m 2 µ 2 . ( 4

.38)

This integral can also be done, using the expression in Eq. (2.18) and replacing the logarithms with subtracted logarithms, as in Eq. (4.14). One is then left with three types of integrals S (j) p (s, n) defined below, where the upper index j refers to the power of the logarithm in the integrand , the lower index p to the degree of the subtracted polynomial and k ∈ N :

S (0) (s, n) ≡ 1 0 dx x 2s+n (1 -x) 1-s = Γ(2 -s)Γ(1 + n + 2s) Γ(3 + n + s) , (4.39)

S

(1)

5 (s, n) ≡ 1 0 dx x 2s+n (1 -x) 1-s log(1 -x) --x - x 2 2 - x 3 3 - x 4 4 - x 5 5 = 1 60 Γ(2 -
+ 180 Γ(3 + n + s) Γ(1 + n + 2s) ψ 2 (2 -s) -2ψ(2 -s)ψ(3 + n + s)+ +ψ 2 (3 + n + s) + ψ (1) (2 -s) -ψ (1) (3 + n + s) . (4.41)
The analytic result for Ω 2 (s) can then be expressed in terms of these three integrals as follows:

5 (s, -5)

+ 1 9 S (2)
6 (s, 0) -

4 3 S (2) 6 (s, -2) + 8 9 S (2)
6 (s, -3) + 4 S Knowing Ω 2 (s) and R 1 (s) we have all the ingredients to evaluate a (eµµ) µ in Fig. 1C and knowing Ω 2 (s) and R 2 (s) we can also evaluate a (eeµµ) µ in Fig. 2C. We discuss these calculations in the next section.

• The R 4 (s) Moment.

This moment appears in the expression of a (eeee) µ

and it is given by the Mellin transform

α π 4 R 4 (s) = ∞ 0 dξ ξ s-1 ρ 4 (ξ) , (4.44)
of the spectral function in the r.h.s. of Eq. (2.10). It gives rise to the δ-integral

R 4 (s) = 1 0 dδ 2δ(1 -δ 2 ) s-1 4δ 1 2 - 1 6 δ 2 8 9 - 1 3 δ 2 + δ 1 2 - 1 6 δ 2 log 1 -δ 1 + δ 3 -4π 2 8 9 - 1 3 δ 2 + δ 1 2 - 1 6 δ 2 log 1 -δ 1 + δ δ 1 2 - 1 6 δ 2 3 . (4.45)
Here only the integrals proportional to log 3 1-δ 1+δ are new with respect to the integrals which already appeared in the evaluation of R 2 (s). By integration by parts, they can be reduced to integrals proportional to log 2 1-δ 1+δ and hence to the function f (s) in Eq. (4.31). In terms of f (s) and the auxiliary function which we do in the next section.

g(σ) = 2 √ π Γ(σ) Γ σ + 1 2 = 2B 1 2 , σ , (4
• The Ω 3 (s) Moment.

It corresponds to the Mellin transform (see Eq. (3.13))

α π 2 Ω 3 (s) = ∞ 0 dω ω s-1 ω 4 + ω √ 4 + ω - √ ω √ 4 + ω + √ ω 2 Π (µ) (-ω m 2 µ ) 3 = 1 0 dx (1 -x) x 2 1 -x s Π (µ) -x 2 1 -x m 2 µ 3 . (4.48)
As in the evaluation of the integral in Eq. (4.38), the first step consists in replacing the logarithms and their powers by subtracted logarithms and subtracted power logarithms. One is then left with integrals of the type S (j) p (s, n) like those already introduced in the evaluation of Ω 2 (s). The new ones are:

S (1) 7 (s, n) ≡ 1 0 dx x 2s+n (1 -x) 1-s log(1 -x) --x - x 2 2 - x 3 3 - x 4 4 - x 5 5 - x 6 6 - x 7 7 = S (
9 (s, n) ≡

1 0 dx x 2s+n (1 -x) 1-s log 3 (1 -x) - --x 3 - 3x 4 2 - 7x 5
+ 15120 Γ(3 + n + s) Γ(1 + n + 2s) ψ 3 (2 -s) -3ψ(2 -s) 2 ψ(3 + n + s) -ψ 3 (3 + n + s) -3ψ(3 + n + s) ψ (1) (2 -s) -ψ (1) (3 + n + s) +3ψ(2 -s) ψ 2 (3 + n + s) + ψ (1) (2 -s) -ψ (1) (3 + n + s) +ψ (2) (2 -s) -ψ (2) (3 + n + s) . ( 4

.51)

The analytic result for Ω 3 (s) can then be expressed in terms of the S (j) p (s, n) integrals as follows: (2) 8 (s, -5) -

Ω 3 (s) = - 482 405 S (
In particular, the value of Ω 3 (s) at s = 0 fixes the contribution to the muon anomaly from the Feynman diagram in Fig. 8, i.e. Knowing Ω 3 (s) and R 1 (s) we have all the ingredients to evaluate a (eµµµ) µ in Fig. 2D. We discuss these calculations in the next section.

V. EIGHTH ORDER RESULTS FROM ELECTRON AND MUON VACUUM POLARIZATION LOOPS

We have now all the ingredients to proceed to the calculation of the eighth order contributions illustrated by the Feynman diagrams in Fig. 1.

• Three Electron Loops, Fig. 1

(A) [one diagram]:

We recall the expression in Eq. (3.14)

a (eee) µ = α π 4 1 2πi cs+i∞ cs-i∞ ds 4m 2 e m 2 µ -s Γ(s)Γ(1 -s) Ω 0 (s) R 3 (s) . (5.1)
The converse mapping theorem relates the asymptotic behaviour of a (eee) µ as a function of the small mass ratio 4m 2 e /m 2 µ to the singularities of the integrand in this equation as a function of the Mellin s-complex variable. Using the explicit expressions for Ω 0 (s) and R 3 (s) given in Eqs. (5.

2)

The next to leading singularity at s → -1/2 is a simple pole, induced by Ω 0 (-1/2), which governs the contribution of O me mµ :

a (eee) µ ∼ [s → -1 2 ]
α π

4 m e m µ 101 1536 π 4 .

(5.

3)

The next singularity at s → -1 in the Mellin plane is again a multipole type singularity which, by the inverse mapping theorem, governs the terms (

a (eee) µ ∼ [s → -1] α π 4 m 2 e m 2 µ - 2

5.4)

There is no problem in evaluating as many terms as we wish. In the Appendix we give the result of the asymptotic contribution to a agree with those obtained by Laporta [8] using a very different method. The agreement with the numerical determination by Kinoshita and Nio [5] is quite remarkable, although our result is of course more precise.

• Two Electron Loops and One Muon Loop, Fig. 1

(B) [three diagrams]:

The corresponding expression is the one in Eq. (3.15)

a (eeµ) µ = α π 4 3 1 2πi cs+i∞ cs-i∞ ds 4m 2 e m 2 µ -s Γ(s)Γ(1 -s) Ω 1 (s) R 2 (s) , (5.5)
with Ω 1 (s) and R 2 (s) given in Eqs. (4.15) and (4.12). The relevant singularities in the Mellin s-plane occur at s = 0, -1, -2, -3, • • • as multipoles. The singularity at s = 0 governs the terms in the asymptotic expansion. The next-to -leading singularity is at s = -1 and it is responsible for the higher order terms (5.7)

a (eeµ) µ ∼ [s → 0] α π 4 119 108 - π 2 9 log 2 m 2 µ m 2 e - 61
a (eeµ) µ ∼ [s → -1]
The analytic results in Eqs. (5.6) and (5.7) agree with those given by Laporta in ref. [8]. As discussed previously, there is no singularity at s = -1/2 because of the presence of a muon loop self-energy (see the discussion around

Eq. (4.20) ). There could be a priori a singularity at s = -3/2 because it appears in the singular expansion of Ω 1 (s); however, it is screened by the fact that R 2 (s) has a zero at s = -3/2. In the Appendix we give the result for a e ratio in the previous terms. Numerically, our result is more precise than the one obtained by Kinoshita and Nio [5], though it agrees very well within their quoted error.

• One Electron Loop and Two Muon Loops, Fig. 1

(C) [three diagrams]:

The corresponding expression is the one in Eq. (3.16)

a (eµµ) µ = α π 4 3 1 2πi cs+i∞ cs-i∞ ds 4m 2 e m 2 µ -s Γ(s)Γ(1 -s) Ω 2 (s) R 1 (s) , (5.8)
with Ω 2 (s) and R 1 (s) given in Eqs. (4.42) and (4.3). The relevant singularities in the Mellin s-plane occur at s = 0, -1, -2, -3, • • • . The singularity at s = 0 is a double pole and governs the terms

a (eµµ) µ ∼ [s → 0] α π 4 - 943 324 - 4 135 π 2 + 8 3 ζ(3) log m 2 µ m 2 e + 57899 9720 - 5383 4050 π 2 + 2 27 π 4 - 2 45 ζ (3) .
(5.9)

The next-to-leading singularity at s = -1 is a single pole which governs the next-to-leading term

a (eµµ) µ ∼ [s → -1] α π 4 m 2 e m 2 µ 458 81 - 26 105 π 2 - 8 3 ζ (3) ; (5.10)
and so on. The results in Eqs. (5.9) and (5.10) agree with those of Laporta [8]. In the appendix we have also mass ratio. It is not surprising that this analytic evaluation allows for a more accurate determination of this term than the numerical estimate by Kinoshita and Nio [5], though it agrees with it within their given error.

VI. TENTH ORDER RESULTS FROM ELECTRON AND MUON VACUUM POLARIZATION LOOPS

We also have all the ingredients to proceed to the calculation of the tenth order contributions illustrated by the Feynman diagrams in Fig. 2.

• Four Electron Loops, Fig. 2

(A) [one diagram]:

We recall the expression in Eq. (3.19)

a (eeee) µ = α π 5 (-1) 1 2πi cs+i∞ cs-i∞ ds 4m 2 e m 2 µ -s Γ(s)Γ(1 -s) Ω 0 (s) R 4 (s) , (6.1)
with Ω 0 (s) and R 4 (s) given in Eqs. (4.1) and (4.47). The converse mapping theorem relates the asymptotic behaviour of a (eee) µ as a function of the small mass ratio 4m 2 e /m 2 µ , to the singular series expansion of the integrand in this equation. The relevant singularities are those in the left-hand side of the Mellin s-plane. They occur as multipoles at s = 0, -1, -2, -3, . . . , because of the factors Γ(s) and R 4 (s); and as single poles at s = -1/2, -3/2, -5/2, . . . because of the factor Ω 0 (s). The multipoles at s → 0 govern the asymptotic contributions:

a (eeee) µ ∼ [s → 0] α π 5 1 162 log 4 m 2 µ m 2 e - 25
There is only a simple pole at s = -1/2. It governs the next-to-leading asymptotic contribution

a (eeee) µ ∼ [s → -1/2] α π 5 m 2 e m 2 µ 1/2 - 18203 374220 π 4 . (6.3)
The contributions in Eqs. (6.2) and ( 6.3) agree with those given by Laporta in ref. [9]. We have calculated further contributions, up to the required accuracy governed by the present knowledge of the

m 2 e m 2 µ
ratio. They are given in the Appendix. Numerically, although our result is much more precise than the one given by Kinoshita and Nio [5], it agrees very well with it within their quoted errors.

• Three Electron Loops and One Muon Loop, Fig. 2

(B) [four diagrams]:

We recall the expression in Eq. (3.20)

a (eeeµ) µ = α π 5 (-4) 1 2πi cs+i∞ cs-i∞ ds 4m 2 e m 2 µ -s Γ(s)Γ(1 -s) Ω 1 (s) R 3 (s) , (6.4)
with Ω 1 (s) and R 3 (s) given in Eqs. (4.15) and (4.28). The relevant singularities here occur as multipoles at s = 0, -1, -2, -3, . . . , because of the factors Γ(s) and R 3 (s); and as single poles at s = -3/2, -5/2, . . . because of the factor Ω 1 (s). The multipoles at s → 0 govern the asymptotic contributions:

a (eeeµ) µ ∼ [s → 0] α π 5 119 243 - 4 81 π 2 log 3 m 2 µ m 2 e - 61
This expression agrees with the one given by Laporta [9]. We have also calculated the corresponding terms

up to O m 2 e m 2 µ 5/2
, as required by the wanted accuracy. Again,although numerically our result is much more precise than the one given by Kinoshita and Nio [5], it agrees very well with it within their quoted errors.

• Two Electron Loops and Two Muon Loops, Fig. 2

(C) [six diagrams]:

We recall the expression in Eq. (3.21)

a (eeµµ) µ = α π 4 (-6) 1 2πi cs+i∞ cs-i∞ ds 4m 2 e m 2 µ -s Γ(s)Γ(1 -s) Ω 2 (s) R 2 (s) , (6.6)
with Ω 2 (s) and R 2 (s) in Eqs. (4.42) and (4.12). Here, the multipoles at s = 0 generate the terms

a (eeµµ) µ ∼ [s → 0] α π 5 - 943
which agree with the expression found in Laporta [9]. We have also calculated the contributions up to

O m 2 e m 2 µ 3 log 3 m 2 µ m 2 e
, which can be found in the Appendix. Our numerical result agrees with the one given by Kinoshita and Nio [5] within their quoted errors.

• One Electron Loop and Three Muon Loops, Fig 2

.(D) [four diagrams]:

We recall the expression in Eq. (3.22)

a (eµµµ) µ = α π 5 (-4) 1 2πi cs+i∞ cs-i∞ ds 4m 2 e m 2 µ -s Γ(s)Γ(1 -s) Ω 3 (s) R 1 (s) , (6.8)
with Ω 3 (s) and R 1 (s) in Eqs. (4.42) and (4.12). Here, there is a double pole at s = 0 which generates the terms

a (eµµµ) µ ∼ [s → 0]
α π

in agreement wit the expression found in Laporta [9]. We have also calculated the contributions up to

O m 2 e m 2 µ 3 log 3 m 2 µ m 2 e
, which are given in the Appendix. Our numerical result also agrees very well with the one given by Kinoshita and Nio [5] within their quoted errors.

VII. CONTRIBUTIONS FROM ELECTRON LOOPS AND ONE TAU LOOP

We shall finally discuss the calculation of vacuum polarization contributions involving electron loops and a tau loop. The corresponding expressions are the ones in Eqs. (3.17) and (3.23). As already mentioned, the problem here is the non factorization of the dependence in the the two Mellin variables s and t because of the presence in the integrand of the function Θ(s, t) defined in Eq. (3.18).

A. Two Electron Loops and One Tau Loop

The corresponding expression in this case is the one given in Eq. (3.17). Using the result obtained for R 2 (s) in Eq. (4.12) and the expression for Θ(s, t) in Eq. (3.18), we have:

a (eeτ ) µ = α π 4 3 (2iπ) 2 cs+i∞ cs-i∞ ds cs+i∞ cs-i∞ dt 4m 2 e m 2 µ -s m 2 µ m 2 τ -t Γ(1 + 2s -2t) Γ(2 -s + t) Γ(3 + s -t) × 2 9 √ π 6 + 13s + 4s 2 s 3 (2 + s)(3 + s) Γ 2 (1 + s) Γ(2 -s) Γ( 3 2 + s) Γ(t) Γ(1 -t) Γ 2 (2 -t) t Γ(4 -2t) . (7.1)
The evaluation of the asymptotic behaviour of this type of integral calls for a more sophisticated material than the inverse mapping theorem applied to the previous cases where there was only one ratio of masses. Indeed, the generalization of the inverse mapping theorem to this case is a typical problem of calculus of residues in C 2 . Let us then adopt the standard notation of multidimensional complex analysis [37] and denote by ω (eeτ ) the following 2-form: τ are small, the asymptotic behaviour we are looking for in Eq. (7.1), will be governed by the residues associated to the singularities of ω (eeτ ) in the cone {Re(s) ≤ 0, Re(t) ≤ -1}, denoted by Π in Fig. 9. Formally, the solution to our problem is then given by the sum [38]

ω (eeτ ) = 2 √ π 3 4m 2 e m 2 µ -s m 2 µ m 2 τ -t Γ(t) Γ(1 -t) Γ 2 (2 -t) t Γ(4 -2t) × (6 + 13s + 4s 2 ) s 3 (2 + s)(3 + s) Γ 2 (s + 1) Γ(2 -s) Γ(s + 3 2 ) × Γ(1 + 2s -2t) Γ(2 -s + t) Γ(3 + s -t) ds ∧ dt . ( 7
a (eeτ ) µ = α π 4 (s0,t0)∈Π
Res (s0,t0) ω (eeτ ) . (7.3)

From the expression in Eq. ( 7.2), one can easily see that the singularities in the plane [Re(s), Re(t)] associated to ω (eeτ ) , are defined by a set of straight lines. For example, Γ(2 -s + t) induces a family of singular lines parameterized by the affine equation: Re(t) = Re(s) -(2 + n), n ≥ 0. Each of the singular lines is called a divisor. As discussed in ref. [START_REF] Passare | [END_REF], it is sufficient for our purposes to consider the singular set of points defined by the intersections of all the divisors in the appropriate cone. In our case, a subset of the nearest singular points to the origin of the cone Π, is illustrated by the red dots in Fig. 9. There are in fact two kinds of singularities to consider:

1. Singularities of the first kind, where only vertical and horizontal divisors (two or more) cross each other.

In order to obtain the residue associated to such a singular point (s 0 , t 0 ), one proceeds as follows:

• First perform the change of variables s ↔ s 0 + s, t ↔ t 0 + t so as to bring the singularity to the origin (0, 0).

• Use the functional relation Γ(z + 1) = zΓ(z) for each singular Gamma function until reaching an expression of the type

ω (eeτ ) = h(s, t) s n t m ds ∧ dt , (7.4)
explicitly showing the full singular behaviour at the origin, with h(s, t) an analytic function at (0, 0) and n and m positive integers.

• Then, use the Cauchy formula

Res (0,0) ω (eeτ ) = 1 (n -1)!(m -1)! ∂ n+m-2 h(s, t) ∂s n-1 ∂t m-1 (0,0) . (7.5)
2. Singularities of the second kind, where two or more divisors cross each other, one or more of them being oblique lines. This is the case, for instance, at the point (0, -2) in Fig. 9. In fact, in our case, the singularities of the second kind at a point (s 0 , t 0 ) are induced either by the oblique divisors generated by Γ(1 + 2s + 2t) or the oblique divisors generated by Γ(2 -s + t). In order to obtain the residue associated to a singularity of the second kind at a generic point (s 0 , t 0 ), one proceeds as follows:

• Again, first perform the change of variables s ↔ s 0 + s, t ↔ t 0 + t so as to bring the problem back to a singularity at the origin (0, 0); and then apply the functional relation Γ(z + 1) = zΓ(z) to the singular Gamma functions in question, until able to write ω (eeτ ) in a way which explicitly shows the full singular behaviour in question:

ω (eeτ ) = h(s, t) s n t m (-s + t) ds ∧ dt , (7.6)
with h(s, t) an analytic function at (0, 0) and n and m positive integers. The factor t-s in the denominator is specific to the class of oblique divisors in our case which are all parallel lines to s = t with multiplicity one.

• Apply the Transformation Law for Residues [START_REF] Griffiths | Principles of Algebraic Geometry[END_REF], to the form ω (eeτ ) in Eq. (7.6) so as to fully decouple the s and t dependence in the denominator. This, of course, requires some explanation which we next provide.

To be precise, let us discuss the case of the singular point (0, -2) in Fig. 9. There is a very important theorem in multidimensional complex analysis, known as the Transformation Law for Residues [START_REF] Griffiths | Principles of Algebraic Geometry[END_REF] which, in the case of C 2 , states that if U is an open set containing (0, 0); if f = f 1 (s, t) f 2 (s, t) and g = g 1 (s, t) g 2 (s, t) are two analytic mappings from U to C 2 such that f -1 (0, 0) = g -1 (0, 0) = (0, 0); and if there exists an analytic two-by-two matrix A such that g = Af , then

Res (0,0) h(s, t) f 1 (s, t) f 2 (s, t) ds ∧ dt = Res (0,0) h(s, t) det A(s, t) g 1 (s, t) g 2 (s, t) ds ∧ dt . (7.7)
The application of this theorem to our case goes as follows:

• First note that with the change of variables s → s -t, t → t,

Res (0,0) h(s, t) s 3 t(-s + t) ds ∧ dt = Res (0,0) h(-s + t, t) st(-s + t) 3 ds ∧ dt . (7.8)
We do this because in this form and as shown in the next step, we can then find easily a matrix A to apply the Transformation Law for Residues.

• Indeed, with f the vector:

f = st (-s + t) 3 , (7.9)
we can find a matrix A which does the transformation:

-s 4 t 4 . =g = -t 2 + 3st -3s 2 s s 2 -3st + 3t 2 t .
=A st (-s + t) 3 .

=f , (7.10) and det A = -(s 3 + t 3 ) . (7.11)

• By virtue of the theorem quoted above, we can then assert that Res (0,0) h(-s + t, t) st(-s + t) 3 ds ∧ dt = Res (0,0) h(-s + t, t)

s 3 + t 3 s 4 t 4 ds ∧ dt = Res (0,0) h(-s + t, t) 1 st 4 + 1 s 4 t ds ∧ dt = 1 6 ∂ 3 [h(-s + t, t)] ∂t 3 + ∂ 3 [h(-s + t, t)] ∂s 3 (0,0) , (7.12)
where in going from the second line to the third, we have applied the Cauchy formula in Eq. (7.5). • Finally, from Eq. (7.8), and applying the formula for chain-derivation in Eq. (7.12), we get the result

Res (0,0) h(s, t) s 3 t(-s + t) ds ∧ dt = 1 2 ∂ 3 h(s, t) ∂s 2 ∂t (0,0) + 1 2 ∂ 3 h(s, t) ∂s∂t 2 (0,0) + 1 6 ∂ 3 h(s, t) ∂t 3 (0,0) . (7.13)
We shall now proceed to the calculation of a (eeτ ) µ in Eq. ( 7.3), following the procedure which we have just outlined.

• Singularities on the line t = -1.

-The leading singularity is at the point (s 0 , t 0 ) = (0, -1), and it is a singularity of the first kind. After translation to the origin and the reduction to an explicit singular form, we can write ω (eeτ ) in the following way:

ω (eeτ ) = h (0,-1) (s, t) s 3 t ds ∧ dt , (7.14)
with

h (0,-1) (s, t) = 2 √ π 3 4m 2 e m 2 µ -s m 2 µ m 2 τ 1-t Γ(3 + 2s -2t) Γ(1 -s + t) Γ(4 + s -t) × Γ 2 (1 + s)Γ(2 -s)(6 + 13s + 4s 2 ) (2 + s)(s + 3)Γ 3 2 + s Γ 2 (3 -t)Γ(1 + t)Γ(2 -t) (-1 + t) 2 Γ(6 -2t) . (7.15)
Then, using Eq. 7.5, we get -The next singularity on the line t = -1 is at the point (s 0 , t 0 ) = (-1, -1), and it is also a singularity of the first kind. In a similar way to the singularity in (0, -1), one can easily find that the residue associated to this singularity is given by the expression • Singularities on the line t = -2 -The singularity nearest to the origin on the line t = -2 is at (s 0 , t 0 ) = (0, -2) and it is of the second kind.

Res (0,-1) ω (eeτ ) = 1 2 ∂ 2 h (0,-1) ∂s 2 (0,0) = m 2 µ m 2 τ 1 135 log 2 m 2 µ m 2 e - 1
After translation to the origin and the reduction to an explicit singular form, we can write ω (eeτ ) in the following way:

ω (eeτ ) = h (0,-2) (s, t) s 3 t(-s + t) ds ∧ dt , (7.18)
where

h (0,-2) (s, t) = 4m 2 e m 2 µ -s m 2 µ m 2 τ 2-t 2 √ π 3 Γ(5 + 2s -2t)Γ(1 -s + t) Γ(5 + s -t) × (6 + 13s + 4s 2 )Γ 2 (1 + s)Γ(2 -s) (2 + s)(3 + s)Γ 3 2 + s Γ(1 + t)Γ(3 -t) (-1 + t)(-2 + t) 2 Γ 2 (4 -t) Γ(8 -2t) . (7.19)
Then, using the result in Eq. 7.13,we get

Res (0,-2) ω (eeτ ) = 1 2 ∂ 3 h (0,-2) ∂s 2 ∂t | (0,0) + 1 2 ∂ 3 h (0,-2) ∂s∂t 2 | (0,0) + 1 6 ∂ 3 h (0,-2) ∂t 3 | (0,0) = m 2 µ m 2 τ 2 1 1260 log 3 m 2 µ m 2 τ - 1
which for convenience we have ordered in decreasing powers of

m 2 µ m 2 τ 2 log n m 2 µ m 2 τ with n = 3, 2, 1, 0.
The other singularities on the line t = -2 (for s = -1, -2 and further) turn out to give contributions which are too small to be of physical relevance.

• Other Singularities

In fact there is only one more singularity which can give rise to a term of the order of the present experimental error limitations. This is the singularity at (0, -3). Its residue can be computed exactly in the same way as the one for the singularity at (0, -2) with the result, ordered in decreasing powers of Residues at the singular points in Fig. 10 are computed following the strategy discussed in the previous subsection, where one uses the functional relations: Γ(z + 1) = zΓ(z) and ψ (1) (1 + z) = ψ (1) (z) -1 z 2 . For the singularities of the first kind one can then use Eq. (7.5). The new type of singularities of the second kind are reducible to a form of the type

ω (eeeτ ) = h(s, t) s 4 t (-s + t) ds ∧ dt , (7.23)
with h some analytic function at the origin. In this case, the appropriate A-matrix to implement the Transformation Law for Residues reads:

-t 3 + 4st 2 -6s 2 t + 4s 3 s -s 3 + 4s 2 t -6st 2 + 4t 3 t , (7.24)
and proceeding as explained in the previous subsection one finally obtains:

Res (0,0) h(s, t) s 4 t (-s + t) ds ∧ dt = 1 4! ∂ 4 h(s, t) ∂t 4 + 4 ∂ 4 h(s, t) ∂s∂t 3 + 6 ∂ 4 h(s, t) ∂s 2 ∂t 2 + 4 ∂ 3 hs, t) ∂s 3 ∂t (0,0) . (7.25)
Let us now discuss the explicit calculation of a (eeeτ ) µ .

• Singularities on the line t = -1.

-As in the case of a (eeτ ) µ

, the leading singularity is the one at (s 0 , t 0 ) = (0, -1) and it is a singularity of the first kind. Performing the change of variables t ↔ -1 + t we write:

ω (eeeτ ) = h (0,-1) (s, t) s 4 t ds ∧ dt , (7.26)
where

h (0,-1) (s, t) = 8 √ π 864 4m 2 e m 2 µ -s m 2 µ m 2 τ 1-t Γ(1 + t) Γ(2 -t) Γ 2 (3 -t) (-1 + t) 2 Γ(6 -2t) Γ 2 (1 + s)Γ(1 -s) Γ 11 2 + s × s P 7 (s) (1 + s)(2 + s) -(1 + s)(35 + 21s + 3s 2 ) -162 + 27π 2 s 2 -162 ψ (1) (1 + s) × Γ(3 + 2s -2t) Γ(1 -s + t) Γ(4 + s -t) . (7.27)
From this expression we obtain -The next singularity is at (s 0 , t 0 ) = (-1, -1). Also a singularity of the first kind, whose residue we find to be • Singularities on the line t = -2.

Res (0,-1) ω (eeeτ ) = 1 3! ∂ 3 h (0,-1) ∂s 3 (0,0) = m 2 µ m 2 τ 4 1215 log 3 m 2 µ m 2 e - 2
-The singularity at (s 0 , t 0 ) = (0, -2) is of the second kind. After the change of variables t ↔ -2 + t, and using standard functional relations, one can write ω (eeeτ ) in the following way

ω (eeeτ ) = h (0,-2) (s, t) s 4 t(-s + t) ds ∧ dt (7.30) with h (0,-2) (s, t) = 8 √ π 864 4m 2 e m 2 µ -s m 2 µ m 2 τ 2-t Γ(1 + t) Γ(3 -t) Γ 2 (4 -t) (-1 + t)(-2 + t) 2 Γ(8 -2t) Γ 2 (1 + s)Γ(1 -s) Γ 11 2 + s × s P 7 (s) (1 + s)(2 + s) -(1 + s)(35 + 21s + 3s 2 ) -162 + 27π 2 s 2 -162 ψ (1) (1 + s) × Γ(5 + 2s -2t) Γ(1 -s + t) Γ(5 + s -t) . ( 7

.31)

According to Eq. (7.25), we have: • Singularities on the line t = -3.

Res (0,-2) ω (eeeτ ) = 1 4! ∂ 4 h (0,-2) ∂t 4 + 4 ∂ 4 h (0,-2) ∂s∂t 3 + 6 ∂ 4 h (0,-2) ∂s 2 ∂t 2 + 4 ∂ 3 h (0,-2) ∂s 3 ∂t (0,0) = m 2 µ m 2 τ 2 - 1
-The singularity at (0, -3) is also of the second kind, and its residue can be computed in the same way as the one for the singularity at (0, -2). One finds:

B. Tenth Order Results

These are the results corresponding to the Feynman diagrams in Fig. 2 with the combinatoric factors included. Our numerical results for a , and a (eµµµ) µ agree, within errors, with those of Laporta [9] which, however, he obtained using an older determination of the mass ratio mµ me = 206.768 262 [START_REF] Yao | [END_REF]. The number in italics are the results from the numerical evaluation of Kinoshita and Nio in ref. [6].

  mµ me factors, the τ -loop induces a suppression factor ∼

Fig. 1

 1 Fig. 1 Eighth-order Feynman diagrams with lowest order vacuum polarization electron-loops and a τ -loop which contribute to the Muon Anomaly and are enhanced by powers of log mµ me factors. The dots indicate the other diagrams with different permutations of the lepton-loops.

. 16 )•

 16 Two Electron Loops and One Tau Loop, Fig. 1(D) [three diagrams]: Using the representation in Eq. (2.20)for the tau self-energy function as well as the Mellin-Barnes representation in Eq. (3.2) one easily gets the expression

  (3.1) and (3.2))

Fig. 2

 2 Fig. 2 Tenth-order Feynman diagrams with lowest order vacuum polarization electron-loops and a τ -loop which contribute to the Muon Anomaly and are enhanced by powers of log mµ me factors. The dots indicate the other diagrams with different permutations of the lepton-loops.

. 21 )•

 21 One Electron Loop and Three Muon Loops, Fig 2.(D) [four diagrams]:

  in the explicit calculations.• The R 1 (s) Moment.This moment appears in the expressions for a (eµµ) µ and a (eµµµ) µ in Eqs. (3.16) and (3.22) and it corresponds to the integral

Fig. 3

 3 Fig. 3 Fourth-order Feynman diagram with a lowest order vacuum polarization electron-loop.

1 ( 2 e m 2 µ

 122 s+1) 2 and 1 (s+1) singularities in Eq. (4.5), and hence the terms of O m in Eq. (4.6), using similar simple arguments.• The R 2 (s) Moment.This moment appears in the expressions of a (eeµ) µ , a (eeτ ) µ and a (eeµµ) µ in Eqs. (3.17), (3.15) and (3.21) and it is given by the integral

.

  That this is not the case is due to the fact that as we have seen R 2 (s) has zeros at all s = -1/2 + n for n = -1, -2, -3, • • • . However, as we shall later see, R 3 (s) is finite at s = -3/2 and this explains why a term of O

  µ in Eqs. (3.14) (3.20) and (3.23) and it is given by the Mellin transform

Ω 2 (

 2

α π 3 Ω 2 (Fig. 7

 327 Fig. 7 Sixth-order Feynman diagram with two lowest order muon-loop vacuum polarization insertions.

Fig. 8

 8 Fig.8Eighth-order Feynman diagram with three lowest order muon-loop vacuum polarization insertions.

2 e

 2 (4.1) and (4.28) we can proceed to the calculation of the singular series expansion of the integrand in question. The relevant singularities are those in the left-hand side of the Mellin s-plane. They occur as multipoles at s = 0, -1, -2, -3, . . . , because of the factors Γ(s) and R 3 (s); and as single poles at s = -1/2, -3/2, -5/2, . . . because of the factor Ω 0 (s). The leading singularity at s → 0 has a quadruple pole, a triple pole, a double pole and a single pole. Their residues govern the successive terms of O(log 3 m 2 ) and of O(Cte.) in the leading contributions to a

2 . 3 does

 23 Because of the present experimental error in the mµ me ratio, there is no need to go beyond this approximation, but we have checked that including higher order terms up to O α π not change our final numerical result in Table 1 in the Appendix. The terms up to O α π

2 . 2 µ m 2

 222 Taking into account higher order terms brings in contributions which numerically are of the order of the error induced by the present value of the m

included terms up to O α π 4 m 2 e m 2 µ 2 .

 22 These are the terms one needs to fix a

Fig. 9

 9 Fig.9Plot of the divisors and the singular set of points of ω (eeτ ) in Eq. (7.2). The small red dots are singularities of the first kind; the larger red dots are singularities of the second kind. Notice that the singularities on the line s = -2k+1 2 , k ≥ 1, are screened by the presence of the factor Γ(s +3 2 ) in the denominator of ω (eeτ ) .

Res (- 1 ,- 1 )

 11

3 log n m 2 µ m 2 τ

 32 with n = 3, 2, 1, 0:

Res (- 1 ,- 1 )

 11

2 µ m 2 τ 2 log n m 2 µ m 2 τ

 2222 we have ordered in decreasing powers of m with n = 4, 3, 2, 1, 0.

  .2)Recall that the Mellin-Barnes representation in Eq. (7.1) is valid for all s and t such that Re(s) ∈]0, 1[ and Re(t) ∈ ] -1, 0[; these two conditions resulting in the grey fundamental square in the plane [Re(s), Re(t)], as illustrated in Fig.9. In the case of two Mellin variables, this fundamental square, which in general may become a fundamental polyhedra, generalizes the concept of the fundamental strip in the case of one variable. Since both

	4m 2 e m 2 µ	and	m 2 µ m 2

Acknowledgements

The authors are grateful to Santi Peris for a careful reading of the manuscript. The work of E. de R. and D. G. has been supported in part by the European Community's Marie Curie Research Training Network program under contract No. MRTN-CT-2006-035482, Flavianet. The work of D. G. has also been supported by MEC (Spain) under grant FPA2007-60323, by Spanish Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042) and he gratefully acknowledges an Experienced Researcher position supported by the EU-RTN Programme, contract No.MRTN-CT-2006-035482, Flavianet.

, ordered in a decreasing order of magnitude, is given in Eq. (3.17) in the Appendix.

B. Three Electron Loops and One Tau Loop

The corresponding expression in this case is the one given in Eq. (3.23). Using the result obtained for R 3 (s) in Eq. (4.28) and the expression for Θ(s, t) in Eq. (3.18), we can write explicitly the two form associated to this integral as follows:

where P 7 (s) is the polynomial given in Eq. (4.33). When comparing the two forms ω The other singularities have residues which give contributions smaller than the error induced by the leading term. The total contribution from Eqs. (7.28), (7.29), (7.32) and ( 7.33) to a

, ordered in a decreasing order of magnitude, is given in Eq. (B.9) in the Appendix. Notice that, numerically, the contribution with one tau loop and three electron loops: a (eeeτ ) µ , is of the same size as the contribution from three muon loops and one electron loop: a (eµµµ) µ .

APPENDIX

The purpose of this appendix is to collect systematically the various results discussed in the previous sections. This may be useful to readers who are only interested in the final analytic expressions and the numerical results. The values that we have used for the lepton masses are the ones in the 2006 PDG booklet [START_REF] Yao | [END_REF]: m e = 0.510 998 92(04) MeV , m µ = 105.658 369 (9) MeV and m τ = 1776.99 [START_REF] Passare | [END_REF] MeV .

(0.34)

A. Eighth Order Results

These are the results corresponding to the Feynman diagrams in Fig. 1 with the combinatoric factors included. The numbers in italics are the results from the numerical evaluation of Kinoshita and Nio in ref. [5].


Radouane Affani

Philippe Auzeloux

Jean-Claude Madelmont

Denise Dugat

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