Quest for precision in hadronic cross sections at low energy: Monte Carlo tools vs. experimental data

We present the achievements of the last years of the experimental and theoretical groups working on hadronic cross section measurements at the low-energy e+e− colliders in Beijing, Frascati, Ithaca, Novosibirsk, Stanford and Tsukuba and on τ decays. We sketch the prospects in these fields for the years to come. We emphasise the status and the precision of the Monte Carlo generators used to analyse the hadronic cross section measurements obtained as well with energy scans as with radiative return, to determine luminosities and τ decays. The radiative corrections fully or approximately implemented in the various codes and the contribution of the vacuum polarisation are discussed.

T. Riemann | A. Kalinowski | G. Venanzoni | V. Cherepanov | P. Beltrame | S. Eidelman | S. Jadach | H. Czyż | A. Arbuzov | O. Nicrosini | F. Piccinini | G. Montagna | Z. Was | G. V. Fedotovich | J. H. Kühn | W. Kluge | L. Trentadue | O. Shekhovtsova | G. Pancheri | G. Pakhlova | J. Gluza | P. Lukin | S. Eidelman | V. Cherepanov | A. Kalinowski | R. Bonciani | F. Ignatov | T. Teubner | T. Riemann | J. Gluza | C. Yuan | G. Pakhlova | H. Czyz | D. Nomura | P. Roig | Z. Was | C. Shen | A. Sibidanov | G. Pancheri | F. Jegerlehner | M. Czakon | F. Nguyen | J. Bij | P. Mastrolia | G. Montagna | O. Nicrosini | F. Piccinini | S. Actis | A. Denig | M. Worek | C. Calame | A. Ferroglia | Cheng-ping Shen | L. Trentadue | A. Arbuzov | S. Jadach | W. Placzek | G. Rodrigo | C. M. Carloni Calame | A. Penin | P. Beltrame | A. Korchin | G. Fedotovich | W. Kluge | P. Lukin | S. Müller | M. Passera | E. Remiddi | C. P. Shen | P. Wang | C. Z. Yuan | G. Venanzoni | S. Actis | G. Balossini | C. Bignamini | R. Bonciani | M. Czakon | A. Denig | A. Ferroglia | A. Grzelińska | M. Gunia | A. Hafner | F. Ignatov | F. Jegerlehner | A. Korchin | E. A. Kuraev | P. Mastrolia | S. E. Müller | F. Nguyen | D. Nomura | M. Passera | A. Penin | W. Płaczek | T. Przedzinski | E. Remiddi | G. Rodrigo | P. Roig | A. L. Sibidanov | T. Teubner | J. J. Bij | B. F. L. Ward | M. Worek | Working Group on Radiative Corrections and Monte Carlo Gen Energies | E. Kuraev | Changzheng Yuan | P. Wang | B. Ward | T. Przedzinski | J. Kühn | V. Cherepanov | O. Shekhovtsova | G. Fedotovich | A. Grzelińska | G. Balossini | C. Bignamini | S. Eidelman | M. Gunia | A. Hafner | W. Płaczek

[1]  M. Greco,et al.  Structure functions and initial-final state interference in QED , 1990 .

[2]  Y. Tsai.  Decay Correlations of Heavy Leptons in e+ e- ---> Lepton+ Lepton- , 1971 .

[3]  Master Integrals for the 2-loop QCD virtual corrections to the Forward-Backward Asymmetry , 2003, hep-ph/0311145.

[4]  M. Davier,et al.  The physics of hadronic tau decays , 2005, hep-ph/0507078.

[5]  G. Heinrich,et al.  Analytical evaluation of dimensionally regularized massive on-shell double boxes , 2004, hep-ph/0406053.

[6]  J. Stevens.  Applied Multivariate Statistics for the Social Sciences , 1986 .

[7]  A. Sirlin.  Role of $\sin^{2} \theta_{W}(m_Z)$ at the $Z^0$ Peak , 1989 .

[8]  Alexander A Penin.  Two-loop corrections to Bhabha scattering. , 2005, Physical review letters.

[9]  M.Yu. Kalmykov,et al.  Massive Feynman diagrams and inverse binomial sums , 2004 .

[10]  B. Pietrzyk,et al.  Update of the hadronic contribution to the QED vaccum polarization , 2001 .

[11]  S. Eidelman.  Processes e+e- ? e+e-?, +-?, ??? with emission of final particles at large angles , 1979 .

[12]  F. Nguyen,et al.  Measurement of the pion form factor at KLOE with photons at small angle , 2006 .

[13]  L. Trentadue,et al.  Hadronic Cross Sections in Electron-Positron Annihilation with Tagged Photon , 1998, hep-ph/9804430.

[14]  W. Kluge.  Initial State Radiation: A Success story , 2008, 0805.4708.

[15]  N. Bjerrum-Bohr,et al.  Analytic structure of three-mass triangle coefficients , 2007, 0709.2086.

[16]  A. Sirlin.  Radiative Corrections in the SU(2)L×U(1) Theory a , 1980 .

[17]  A. Vicerè,et al.  Two loop heavy top effects in the Standard Model , 1993 .

[18]  A. Leike,et al.  Analytic final state corrections with cut for e sup + e sup minus yields massive fermions , 1992 .

[19]  M. Steinhauser,et al.  Corrections of order ${\cal O}(G_F M_t^2 \as^2)$ to the $\rho$ parameter , 1995, hep-ph/9502291.

[20]  J. Fleischer,et al.  Two-loop heavy top corrections to the ϱ parameter. A simple formula valid for arbitrary Higgs mass , 1993 .

[21]  R. Bonciani,et al.  Heavy-flavor contribution to Bhabha scattering. , 2007, Physical review letters.

[22]  V. S. Panin,et al.  The measurement ofR ine+e−-annihilation at center-of-mass energies between 7.25 and 10.34 GeV , 1996 .

[23]  A. Vicerè,et al.  Radiative correction effects of a very heavy top , 1992, hep-ph/9205238.

[24]  L. Dixon,et al.  Dimensionally-regulated pentagon integrals☆ , 1993, hep-ph/9306240.

[25]  G. Pancheri,et al.  Final-state radiation and the possibility to test a pion-photon interaction model near the two-pion threshold , 2007, 0706.3027.

[26]  M. Davier,et al.  ANOMALOUS MAGNETIC MOMENT OF THE MUON , 2008 .

[27]  E. Richter-Was,et al.  Higher order QED corrections to Bhabha scattering at low angles , 1991 .

[28]  E. al.,et al.  E+e-→K+K-π+π-, K+K-π0π0 and K+K-K+K- cross sections measured with initial-state radiation , 2007, 0704.0630.

[29]  A. Vicini,et al.  Precision electroweak calculation of the production of a high transverse-momentum lepton pair at hadron colliders , 2007, 0710.1722.

[30]  B. Pietrzyk,et al.  Low energy hadronic contribution to the QED vacuum polarization , 2003, hep-ph/0506323.

[31]  Z. Was,et al.  Probing the CP nature of the Higgs boson at linear colliders with τ spin correlations; the case of mixed scalar–pseudoscalar couplings☆ , 2003, hep-ph/0307331.

[32]  K. Mahanthappa.  MULTIPLE PRODUCTION OF PHOTONS IN QUANTUM ELECTRODYNAMICS , 1962 .

[33]  et al,et al.  Exclusive branching-fraction measurements of semileptonic τ decays into three charged hadrons, into φπ-ντ, and into φk-ντ , 2007, 0707.2981.

[34]  J. Kühn,et al.  Structure functions in τ decays , 1992 .

[35]  J. Bij,et al.  Two Loop Correction to Weak Interaction Parameters Due to a Heavy Fermion Doublet , 1987 .

[36]  B. Valeriani.  The hadronic cross section measurement at KLOE , 2004 .

[37]  Z. Was,et al.  TAUOLA - a library of Monte Carlo programs to simulate decays of polarized τ leptons , 1991 .

[38]  P. G. Verdini,et al.  A study of τ decays involving η, and ω mesons , 1997 .

[39]  F. A. Berends,et al.  Higher Order Radiative Corrections at LEP Energies , 1988 .

[40]  Mikel L. Forcada,et al.  ATLAS , 2011, EAMT.

[41]  D. Yennie,et al.  The infrared divergence phenomena and high-energy processes , 1961 .

[42]  Zbigniew Was,et al.  The τ decay library TAUOLA, update with exact (α) QED corrections in τ → μ(e) vv decay modes , 1992 .

[43]  M. Bohm,et al.  Radiative corrections to Bhabha scattering in SU(2) × U(1) , 1984 .

[44]  S. Brodsky,et al.  SCALING LAWS AT LARGE TRANSVERSE MOMENTUM , 1973 .

[45]  T. Sack,et al.  Electroweak radiative corrections toe+e−→γZ0 , 1986 .

[46]  Complete two-loop electroweak fermionic corrections to $\sin^{2} \theta^{\rm lept}_{\rm eff}$ and indirect determination of the Higgs boson mass , 2004, hep-ph/0407317.

[48]  G. Colangelo,et al.  φ→π+π−γ at Φ-factories , 1992 .

[49]  F. Tkachov,et al.  Integration by parts: The algorithm to calculate β-functions in 4 loops , 1981 .

[50]  W. Marciano,et al.  The muon g-2 and the bounds on the Higgs boson mass , 2008, 0804.1142.

[51]  Connection between Feynman integrals having different values of the space-time dimension. , 1996, Physical review. D, Particles and fields.

[52]  R. Kass,et al.  Observation of a broad structure in the pi+ pi- J/psi mass spectrum around 4.26 GeV/c2. , 2005, Physical review letters.

[53]  R. Barbieri,et al.  Electron form factors up to fourth order. - II , 1972 .

[54]  B. Kniehl,et al.  Hadronic contributions to O(α2) radiative corrections in e+e− annihilation , 1988 .

[55]  Rene Brun,et al.  Developments in ROOT I/O and trees , 2009, ArXiv.

[56]  A. Penin.  Two-loop photonic corrections to massive Bhabha scattering , 2005, hep-ph/0508127.

[57]  J. Vermaseren,et al.  Harmonic Polylogarithms , 1999, hep-ph/9905237.

[58]  J. Bij,et al.  Three Loop Leading Top Mass Contributions to the ρ Parameter , 2000, hep-ph/0011373.

[59]  Barbara Clerbaux,et al.  Measurement of the Tau Polarisation at LEP , 2001 .

[60]  H. Czyz,et al.  Four pion final states with tagged photons at electron positron colliders , 2000, hep-ph/0008262.

[61]  Durham,et al.  New results on the theoretical precision of the LEP/SLC luminosity 1 Work supported in part by the U , 1998, hep-ph/9811245.

[62]  P. Lukin,et al.  BaBar results on hadronic cross sections with initial state radiation , 2008 .

[63]  P. Mastrolia,et al.  NLO QED Corrections to Hard-Bremsstrahlung Emission in Bhabha Scattering , 2009, 0909.1750.

[64]  J. Kuhn.  tau polarimetry with multimeson states. , 1995 .

[65]  A. Arbuzov,et al.  Radiative corrections for pion and kaon production at e + e - colliders of energies below 2 GeV , 1997, hep-ph/9703456.

[66]  S. Mandelstam.  DETERMINATION OF THE PION-NUCLEON SCATTERING AMPLITUDE FROM DISPERSION RELATIONS AND UNITARITY. GENERAL THEORY , 1958 .

[67]  F. Ceradini,et al.  Storage ring luminosity measurements by small angle e + e - scattering , 1975 .

[68]  N. Usyukina.  On a representation for the three-point function , 1975 .

[69]  Aulchenko,et al.  First observation of the decay τ−→ϕK−νττ−→ϕK−ντ , 2006 .

[70]  Isospin violating effects in e+e− vs. τ measurements of the pion form factor |Fπ|2(s) , 2003, hep-ph/0310181.

[71]  A. Kotikov.  Differential equation method: The Calculation of N point Feynman diagrams , 1991 .

[72]  H. J. Bhabha,et al.  The scattering of positrons by electrons with exchange on Dirac's theory of the positron , 1936 .

[73]  A. I. Davydychev.  A simple formula for reducing Feynman diagrams to scalar integrals , 1991 .

[74]  S. Jadach,et al.  Monte Carlo simulation of the process e+e-→τ+τ-, τ±→X± including radiative O(α3) QED corrections, mass and spin effects , 1985 .

[75]  Marciano,et al.  Electroweak radiative corrections to tau decay. , 1988, Physical review letters.

[76]  S. Mandelstam.  ANALYTIC PROPERTIES OF TRANSITION AMPLITUDES IN PERTURBATION THEORY , 1959 .

[77]  S. Brodsky,et al.  Exclusive Processes in Perturbative Quantum Chromodynamics , 1980 .

[78]  M. Napsuciale,et al.  Loop effects in Φ→π+π−γ , 1994 .

[79]  M. Bohm,et al.  Radiative corrections to Bhabha scattering at high energies (I): Virtual and soft photon corrections , 1988 .

[80]  A. Mitov,et al.  The singular behavior of massive QCD amplitudes , 2006, hep-ph/0612149.

[81]  Ciulli,et al.  Measurement of the axial-vector , 1998 .

[82]  Z. Was,et al.  Gauge-invariant sub-structures of tree-level double-emission exact QCD spin amplitudes , 2008, 0802.2182.

[83]  V. Smakhtin,et al.  Study of ϕ → π + π − π 0 with CMD-2 detector , 2006 .

[84]  Zbigniew Was,et al.  PHOTOS : a universal Monte Carlo for QED radiative corrections in decays , 1991 .

[85]  Heinrich Leutwyler,et al.  Chiral Perturbation Theory: Expansions in the Mass of the Strange Quark , 1985 .

[86]  E. al.,et al.  The e+ e- --> pi+ p- pi+ pi-, k+ k- pi+ pi-, and k+ k- k+ k- cross sections at center-of-mass energies 0.5 gev to 4.5 gev measured with initial-state radiation , 2005, hep-ex/0502025.

[87]  A. Bogdanchikov,et al.  Study of the process e+e− → π+π− in the energy region $$400 < \sqrt s < 1000 MeV^\P $$ , 2005 .

[88]  S. Jadach,et al.  The precision Monte Carlo event generator KK for two-fermion final states in e+e− collisions☆ , 2000 .

[89]  J. H. Kang,et al.  Precise measurement of hadronic τ-decays with an η meson , 2009 .

[90]  A. Denner,et al.  Techniques for one-loop tensor integrals in many-particle processes , 2006 .

[91]  T. Riemann,et al.  Two-loop fermionic corrections to massive Bhabha scattering , 2007, 0704.2400.

[92]  K. Kołodziej,et al.  Helicity amplitudes for spin-0 or spin-1 boson production in e+e- annihilation. , 1991, Physical review. D, Particles and fields.

[93]  Heinrich Leutwyler,et al.  Chiral perturbation theory to one loop , 1984 .

[94]  S. K. Dhawan,et al.  Measurement of e+e−→π+π− cross-section with CMD-2 around ρ-meson , 2002 .

[95]  R. Pittau,et al.  On the rational terms of the one-loop amplitudes , 2008, 0802.1876.

[96]  Hadronic contributions to the photon vacuum polarization and their role in precision physics , 2001, hep-ph/0104304.

[97]  V. Gribov,et al.  e+ e- pair annihilation and deep inelastic e p scattering in perturbation theory , 1972 .

[98]  Alessandro Cafarella,et al.  Helac-Phegas: A generator for all parton level processes , 2007, Comput. Phys. Commun..

[99]  T. Binoth,et al.  Reduction formalism for dimensionally regulated one loop N point integrals , 1999, hep-ph/9911342.

[100]  G Gabrielse,et al.  New determination of the fine structure constant from the electron value and QED. , 2006, Physical review letters.

[101]  R. Pittau,et al.  Recursive Numerical Calculus of One-loop Tensor Integrals , 2008 .

[102]  Richard Phillips Feynman,et al.  Radiative corrections to Compton scattering , 1952 .

[103]  J. Frenkel,et al.  Exponentiation of leading infrared divergences in massless Yang-Mills theories , 1976 .

[104]  D. Melrose.  Reduction of feynman diagrams , 1965 .

[105]  V. F. Dmitriev,et al.  Constraints on NN¯ interaction from the process e+e−→NN¯ near threshold , 2008 .

[106]  R. Cutkosky.  Singularities and Discontinuities of Feynman Amplitudes , 1960 .

[107]  A. Bogdanchikov,et al.  Update of the e+e− → π+π− cross section measured by the spherical neutral detector in the energy region 400 < √s < 1000 MeV , 2006 .

[108]  E. Eichten,et al.  Charmonium: Comparison with Experiment , 1980 .

[109]  V. Cirigliano,et al.  International Europhysics Conference on HEP PROCEEDINGS Isospin Violation and the Magnetic Moment of the , 2001 .

[110]  V. Smakhtin,et al.  Measurement of the e+e− → π+π− cross section with the CMD-2 detector in the 370–520-MeV energy range , 2006 .

[111]  A. Kotikov,et al.  Differential equations method: the calculation of vertex-type Feynman diagrams , 1991 .

[112]  K. Hikasa,et al.  Scalar top production in e+e− annihilation , 1990 .

[113]  R. Decker,et al.  Tau decays into four pions , 1994, hep-ph/9410260.

[114]  A. Pich,et al.  Vector form factor of the pion from unitarity and analyticity: a model-independent approach , 2001, hep-ph/0101194.

[115]  Zeuthen,et al.  On the tensor reduction of one-loop pentagons and hexagons , 2008, 0807.2984.

[116]  Measurement of the negative muon anomalous magnetic moment to 0.7 ppm. , 2002, Physical review letters.

[117]  S. Uccirati,et al.  The effective electroweak mixing angle sin2θeff with two-loop fermionic contributions , 2005, hep-ph/0507158.

[118]  F. Jegerlehner.  Hadronic contributions to electroweak parameter shifts , 1986 .

[119]  A. Pich,et al.  Vector form factor of the pion : A model-independent approach , 2002, hep-ph/0209224.

[121]  E. al.,et al.  Total cross section of the process e+e−→π+π−π+π− in the c.m. energy range 980–1380 MeV , 2004, hep-ex/0404019.

[122]  A. Pich,et al.  Chiral lagrangians for massive spin-1 fields , 1989 .

[123]  J. Kühn,et al.  τ Decays to pions , 1990 .

[124]  Precision electroweak calculation of the charged current Drell-Yan process , 2006, hep-ph/0609170.

[125]  E. al.,et al.  Measurement of the e+e-→D0D*-π+ cross section using initial-state radiation , 2009, 0908.0231.

[126]  N. Achasov,et al.  Interference in the reactione+e−→γπ+π−and the search for the decayφ→γf0→γπ+π− , 1997 .

[127]  Fred Jegerlehner.  Muon g 2 update , 2008 .

[128]  B. Kniehl,et al.  QCD corrections, virtual heavy quark effects and electroweak precision measurements , 1988 .

[129]  P. G. Verdini,et al.  Branching ratios and spectral functions of tau decays: final ALEPH measurements and physics implications , 2005, hep-ex/0506072.

[130]  R. Pittau,et al.  Numerical evaluation of six-photon amplitudes , 2007, 0704.1271.

[131]  M. Czakon,et al.  Bosonic corrections to the effective weak mixing angle at O(α2) , 2006, hep-ph/0605339.

[132]  Stefan Weinzierl,et al.  Numerical evaluation of multiple polylogarithms , 2005, Comput. Phys. Commun..

[133]  R. Bonciani,et al.  Two-loop Bhabha scattering in QED: Vertex and one-loop by one-loop contributions , 2005, hep-ph/0507047.

[134]  S. Jadach,et al.  The present theoretical error on the Bhabha scattering cross section in the luminometry region at LEP , 1996, hep-ph/9605239.

[135]  A. Kotikov.  Differential equations method. New technique for massive Feynman diagram calculation , 1991 .

[136]  A. Denner,et al.  Reduction schemes for one-loop tensor integrals , 2005, hep-ph/0509141.

[137]  O. Nicrosini,et al.  Soft photons and second order radiative corrections to e+e−→Z0 , 1987 .

[138]  Daniel Maître.  HPL, a Mathematica implementation of the harmonic polylogarithms , 2006, Comput. Phys. Commun..

[139]  E. Vlachos,et al.  Bulletin , 2003 .

[140]  H. Czyz,et al.  Charge asymmetry and radiative ϕ decays , 2004, hep-ph/0412239.

[141]  Aggeliki Kanaki,et al.  HELAC: A package to compute electroweak helicity amplitudes , 2000, hep-ph/0002082.

[142]  Observables with τ-leptons at LHC and LC structure of event records and Monte Carlo algorithms , 2004, hep-ph/0402129.

[143]  Large-angle Bhabha scattering and luminosity at flavour factories , 2000, hep-ph/0003268.

[144]  A. Pich,et al.  Hadronic off-shell width of meson resonances , 2000, hep-ph/0003320.

[145]  Confirmation of the Y(4260) resonance production in initial state radiation , 2006, hep-ex/0611021.

[146]  H. Czyz,et al.  Electron–positron annihilation into three pions and the radiative return , 2005, hep-ph/0512180.

[147]  F. Maas.  Electromagnetic Form Factors in the Time Like Domain , 2008 .

[148]  J. Fleischer,et al.  Two-loop two-point functions with masses: asymptotic expansions and Taylor series, in any dimension , 1993 .

[149]  M. Steinhauser.  Four-loop Singlet Contribution to the Ρ Parameter , 2005 .

[150]  G. Nanava,et al.  Scalar QED, NLO and PHOTOS Monte Carlo , 2007 .

[151]  G. Montagna,et al.  Multiple photon corrections to the neutral-current Drell-Yan process , 2005, hep-ph/0502218.

[152]  Coherent exclusive exponentiation for precision Monte Carlo calculations , 2000, hep-ph/0006359.

[153]  Hayes,et al.  Review of Particle Physics. , 1996, Physical review. D, Particles and fields.

[154]  K. Chao,et al.  Higher Charmonia and X,Y,Z states with Screened Potential , 2009, 0903.5506.

[155]  R. Pittau,et al.  TOPAZ0 - a program for computing observables and for fitting cross sections and forward-backward asymmetries around the Z0 peak , 1993 .

[156]  M. Jamin,et al.  Spectral distribution for the decay τ→ντKπ , 2006 .

[157]  O. Shekhovtsova,et al.  FASTERD: A Monte Carlo event generator for the study of final state radiation in the process e+e-->pipigamma at DAPhiNE , 2009, Comput. Phys. Commun..

[158]  K. Ackerstaff.  Measurement of the strong coupling constant alpha(s) and the vector and axial-vector spectral functions in hadronic tau decays , 1999 .

[159]  T.Gehrmann,et al.  Numerical evaluation of two-dimensional harmonic polylogarithms , 2002 .

[160]  A. Korol,et al.  Experimental study of the reaction e+e− → KSKL in the energy range √S = 1.04−1.38 GeV , 2006, hep-ex/0606057.

[161]  P. Privitera.  Structure functions in τ → 3πντ and the τ polarization measurement , 1993 .

[162]  David Hanneke,et al.  Erratum: New Determination of the Fine Structure Constant from the Electron g Value and QED [Phys. Rev. Lett. 97, 030802 (2006)] , 2007 .

[163]  Z. Was,et al.  PHOTOS Monte Carlo: a precision tool for QED correctionsin Z and W decays , 2005, hep-ph/0506026.

[164]  A. Djouadi,et al.  Erratum: Electroweak gauge boson self-energies: Complete QCD corrections , 1994, Physical review. D, Particles and fields.

[165]  F. Gasparini,et al.  Determination of the electric and magnetic form factors of the proton in the time-like region , 1994 .

[166]  R. Pittau,et al.  Optimizing the Reduction of One-Loop Amplitudes , 2008, 0803.3964.

[167]  Marciano,et al.  Erratum: Radiative corrections to neutrino-induced neutral-current phenomena in the SU(2)L x U(1) theory , 1980, Physical review. D, Particles and fields.

[168]  R. Pittau,et al.  On a semi-analytical and realistic approach to e+e− annihilation into fermion pairs and to Bhabha scattering within the minimal standard model at LEP energies , 1993 .

[169]  W. T. Ford,et al.  Study of tau decays to six pions and a neutrino. , 2000, Physical review letters.

[170]  T. Feldmann,et al.  Model-independent analysis of lepton flavour violating τ decays , 2007, 0707.0988.

[171]  L. Trentadue,et al.  Small-angle electron-positron scattering with a per mille accuracy☆ , 1995, hep-ph/9512344.

[172]  Raymond Gastmans,et al.  Hard photon corrections for Bhabha scattering , 1974 .

[173]  G. Burgers.  On the two-loop QED vertex correction in the high energy limit , 1985 .

[174]  V. A. Smirnov.  Analytical result for dimensionally regularized massless on shell double box , 1999 .

[175]  K. Gaemers,et al.  Hard photon corrections for the process e+e−→μ±μ∓ , 1973 .

[176]  Updated estimate of the muon magnetic momentusing revised results from e + e-annihilation , 2003, hep-ph/0308213.

[177]  Z. Was,et al.  Tau and muon pair production cross-sections in electron-positron annihilations at s**(1/2) = 10.58-GeV , 2007, 0706.3235.

[178]  Z. Was,et al.  Novosibirsk hadronic currents for τ → 4 π channels of τ decay library TAUOLA † , 2002 .

[179]  L. Dixon,et al.  Two-loop correction to Bhabha scattering , 2000, hep-ph/0010075.

[180]  W. Beenakker,et al.  Large angle Bhabha scattering , 1991 .

[181]  J. Fleischer,et al.  Algebraic reduction of one-loop Feynman graph amplitudes , 1999, hep-ph/9907327.

[182]  A. Deandrea,et al.  QED Structure Functions: A Systematic Approach , 1992 .

[183]  E. al.,et al.  Study of the Exclusive Initial-State-RadiationProduction of the DDbar System , 2006, hep-ex/0607083.

[184]  Piotr Golonka,et al.  The tauola-photos-F environment for the TAUOLA and PHOTOS packages, release II , 2006, Comput. Phys. Commun..

[185]  D. Bisello,et al.  Measurement of thee+e−→π+π−π0 ande+e−→ωπ+π− reactions in the energy interval 1350–2400 MeV , 1992 .

[186]  Z. Bern,et al.  Dimensionally regulated one-loop integrals , 1993 .

[187]  S. Laporta,et al.  difference equations , 2001 .

[188]  Michal Czakon,et al.  ZFITTER: a semi-analytical program for fermion pair production in e+e- annihilation, from version 6.21 to version 6.42 , 2006, Comput. Phys. Commun..

[189]  M. Veltman,et al.  One-loop corrections for e + e - annihilation into μ + μ - in the Weinberg model , 1979 .

[190]  J. Kuhn,et al.  Two-Loop Electroweak Logarithms in Four-Fermion Processes at High Energy , 2005 .

[191]  Improved predictions for g-2 of the muon and \alpha_{\rm QED}(M_Z^2) , 2006, hep-ph/0611102.

[192]  T. Riemann,et al.  Virtual hadronic and leptonic contributions to Bhabha scattering. , 2007, Physical review letters.

[193]  A. Pich,et al.  Effective field theory description of the pion form factor , 1997, hep-ph/9707347.

[194]  J. Kuhn,et al.  Two-loop QED hadronic corrections to Bhabha scattering , 2008, 0807.1284.

[195]  B. Bakker,et al.  The S-D mixing and dielectron widths of higher charmonium 1−− states , 2008, 0805.2291.

[196]  A. Khodjamirian,et al.  Modeling the pion and kaon form factors in the timelike region , 2004, hep-ph/0409080.

[197]  K. Ackerstaff.  Measurement of the strong coupling constant , 1999 .

[198]  Y. Ye,et al.  Determination of the ψ(3770), ψ(4040), ψ(4160) and ψ(4415) resonance parameters , 2007, 0705.4500.

[199]  H. Burkhardt,et al.  Uncertainties in the hadronic contribution to the QED vacuum polarization , 1989 .

[200]  L. Dixon,et al.  PROGRESS IN ONE-LOOP QCD COMPUTATIONS , 1996, hep-ph/9602280.

[201]  D. J. Broadhurst.  Massive 3-loop Feynman diagrams reducible to SC$^*$ primitives of algebras of the sixth root of unity , 1999 .

[202]  G. Montagna,et al.  Photon pair production at flavour factories with per mille accuracy , 2008, 0801.3360.

[203]  E. Dally.  Measurement of the Pion Form-Factor , 1976 .

[204]  D. Hertzog.  Measurement of Muon (g-2) and Future Prospects , 2008 .

[205]  S. Eidelman.  Standard Model Predictions for the Muon (g−2)/2 , 2009 .

[206]  Fleischer,et al.  Two-loop large top mass corrections to electroweak parameters: Analytic results valid for arbitrary Higgs boson mass. , 1995, Physical review. D, Particles and fields.

[207]  Ettore Remiddi,et al.  Electron form factors up to fourth order. - I , 1972 .

[208]  Marcia Begalli,et al.  Measurement and interpretation of fermion-pair production at LEP energies above the Z resonance , 2005 .

[209]  A. Arbuzov.  On a novel relativistic quasi-potential equation for two scalar particles , 1994 .

[210]  R. Bonciani,et al.  Master integrals with one massive propagator for the two-loop electroweak form factor , 2003, hep-ph/0304028.

[211]  F. Scuri,et al.  Measurement of the DAΦNE luminosity with the KLOE detector using large angle Bhabha scattering , 2006 .

[212]  M. Jamin,et al.  What can be learned from the Belle spectrum for the decay tau- -> nu_tau K_S pi- , 2008, 0803.1786.

[213]  G. Gabrielse,et al.  New measurement of the electron magnetic moment and the fine structure constant. , 2006, Physical review letters.

[214]  A. Djouadi,et al.  Virtual Very Heavy Top Effects in LEP / SLC Precision Measurements , 1987 .

[215]  V. Smakhtin,et al.  Update: A reanalysis of hadronic cross section measurements at CMD-2 , 2004 .

[216]  Giovanni Ossola,et al.  Reducing full one-loop amplitudes to scalar integrals at the integrand level , 2006, hep-ph/0609007.

[217]  S. E. Baru,et al.  Study of the process e+e- →ωπ0→π0π0γ in c.m. energy range 920-1380 MeV at CMD-2 , 2003 .

[218]  T. Riemann,et al.  Master integrals for massive two-loop Bhabha scattering in QED , 2004, hep-ph/0412164.

[219]  Albert Einstein,et al.  Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? , 1935 .

[220]  R. Bonciani,et al.  QED vertex form factors at two loops , 2003, hep-ph/0307295.

[221]  Fa Berends,et al.  Hard photon corrections for e+e− → γγ , 1973 .

[222]  Fry,et al.  e(+)e(-)-> K+K-pi(+)pi(-), K+K-pi(0)pi(0) and K+K-K+K- cross sections measured with initial-state radiation RID C-5223-2009 RID C-5719-2008 RID D-1055-2009 , 2007 .

[223]  A. Denner,et al.  Two-loop electroweak next-to-leading logarithms for processes involving heavy quarks , 2008, 0809.0800.

[224]  S. Eidelman,et al.  e+e− Annihilation into two and three photons at high energy , 1978 .

[225]  A. Bondar,et al.  Study of the processes e + e - ? ? ? , p 0 ? ? 3 ? in the c.m. energy range 6001380 MeV at CMD-2 , 2005 .

[226]  G. Komen,et al.  Radiative corrections to Bhabha scattering and mupair production from the hadronic vacuum polarization , 1976 .

[227]  G. Duplančić,et al.  Reduction method for dimensionally regulatedone-loop N-point Feynman integrals , 2003, hep-ph/0303184.

[229]  D. Forde.  Direct extraction of one-loop integral coefficients , 2007, 0704.1835.

[230]  Hadronic contributions to (g−2) of the leptons and to the effective fine structure constant α(MZ2) , 1995, hep-ph/9502298.

[231]  A. Penin,et al.  Two-loop Sudakov form factor in a theory with a mass gap. , 2004, Physical review letters.

[232]  A. Arbuzov,et al.  QED collinear radiation factors in the next-to-leading logarithmic approximation , 2007, 0706.2984.

[233]  S. E. Baru,et al.  Study of the process e+e−→K0LK0S in the c.m. energy range 1.05–1.38 GeV with CMD-2 , 2002, hep-ex/0211004.

[234]  Aulchenko,et al.  Study of τ−→KSπ−νττ−→KSπ−ντ decay at Belle , 2007 .

[235]  Costas G. Papadopoulos,et al.  PHEGAS : A phase-space generator for automatic cross-section computation , 2000, hep-ph/0007335.

[236]  M. Davier,et al.  Improved determination of the hadronic contribution to the muon ( ${g-2}$) and to ${\alpha(M_{\mathrm Z}^2)}$ Using new data from hadronic $\tau$ decays , 1997, hep-ph/9703220.

[237]  P. Zerwas,et al.  SECONDARY REACTIONS IN ELECTRON-POSITRON (ELECTRON) COLLISIONS* , 1975 .

[238]  D. Leone.  Measuring the Pion From Factor via Radiative Return at large photon angles with KLOE , 2006 .

[239]  E. al.,et al.  Measurement of the total hadronic cross section in e+e- annihilation below 10.56GeV , 2007, 0706.2813.

[240]  C. Schubert,et al.  An algebraic/numerical formalism for one-loop multi-leg amplitudes , 2005 .

[241]  S. Eidelman,et al.  International Journal of Modern Physics A, ❢c World Scientific Publishing Company SPECTROSCOPY AT B-FACTORIES USING HARD PHOTON EMISSION , 1999 .

[242]  P. Mastrolia,et al.  FEYNMAN DIAGRAMS AND DIFFERENTIAL EQUATIONS , 2007, 0707.4037.

[243]  A. Djouadi.  $$O\left( {\alpha \alpha _S } \right)$$ vacuum polarization functions of the standard-model gauge bosonsvacuum polarization functions of the standard-model gauge bosons , 1988 .

[244]  J. J. Sakurai,et al.  Lifetimes and branching ratios of heavy leptons , 1971 .

[245]  J. Bernstein,et al.  Particles, Sources and Fields , 1971 .

[246]  H. Czyz,et al.  The Master differential equations for the two loop sunrise selfmass amplitudes , 1998, hep-th/9805118.

[247]  G. Montagna,et al.  Matching perturbative and parton shower corrections to Bhabha process at flavour factories , 2006, hep-ph/0607181.

[248]  G. Weiglein,et al.  Precise prediction for the W boson mass in the standard model , 2003, hep-ph/0311148.

[249]  J. Kuhn,et al.  Two-loop high-energy electroweak logarithmic corrections in a spontaneously broken SU(2) gauge model , 2005 .

[250]  Hadronic contributions to the muon lifetime , 1998, hep-ph/9802341.

[251]  M. Bilenky,et al.  ZFITTER v.6.21: A semi-analytical program for fermion pair production in e+e- annihilation , 1999, hep-ph/9908433.

[252]  Dennis V. Perepelitsa,et al.  Muon ( g − 2 ) , 2008 .

[253]  M. Greco,et al.  RADIATIVE CORRECTIONS FOR COLLIDING BEAM RESONANCES , 1975, Coherent States in Gauge Theories and Applications in Collider Physics.

[254]  Asymptotics of Feynman diagrams and the Mellin–Barnes representation , 2005, hep-ph/0505038.

[255]  T. Teubner.  Hadronic Contributions to the theoretical value of ( g - 2 ) and a ( q 2 ) , 2008 .

[256]  E. al.,et al.  R value measurements for e+e− annihilation at 2.60, 3.07 and 3.65 GeV , 2009, 0903.0900.

[257]  S. E. Baru,et al.  Spherical neutral detector for VEPP-2M collider , 2000 .

[258]  R. Baldini,et al.  Unexpected features of e+e-$ \rightarrow$ p$ \overline{{p}}$ and e+e-$ \rightarrow$$ \Lambda$$ \overline{{\Lambda}}$ cross-sections near threshold , 2009 .

[259]  H. Czyz.  Status of PHOKHARA and its theoretical accuracy , 2008 .

[260]  F. Jegerlehner.  The running fine structure constant alpha(E) via the Adler function , 2008, 0807.4206.

[261]  T. Becher,et al.  Two-loop QED corrections to Bhabha scattering , 2007, 0704.3582.

[262]  BHWIDE 1.00: O(α) YFS exponentiated Monte Carlo for Bhabha scattering at wide angles for LEP1/SLC and LEP2 , 1996, hep-ph/9608412.

[263]  H. Czyz,et al.  Radiative return at $\Phi$- and B-factories: FSR for muon pair production at next-to-leading order , 2004, hep-ph/0404078.

[264]  A. Sirlin.  Role of at the Z° peak , 1989 .

[265]  T. Riemann,et al.  Bhabha scattering with higher order weak loop corrections , 1991 .

[266]  F. Tkachov.  A theorem on analytical calculability of 4-loop renormalization group functions , 1981 .

[267]  S. Uccirati,et al.  The effective electroweak mixing angle sin2θeff with two-loop bosonic contributions , 2006, hep-ph/0610312.

[268]  J. H. Kang,et al.  Observation of two resonant structures in e+ e- -->pi+ pi- psi(2S) via initial-state radiation at Belle. , 2007, Physical review letters.

[269]  A. Bogdanchikov,et al.  Analysis of e+e−→π+π−π+π− and e+e−→π+π−π0π0 processes in the energy range of $$\sqrt 2 = 0.98 - 1.38 GeV$$ in experiments with a spherical neutral detector , 2003 .

[270]  L. Dixon,et al.  On-shell methods in perturbative QCD , 2007, 0704.2798.

[271]  Harold S. Park,et al.  Measurement of the total cross section for e^+e^-→hadrons at √s=10.52 GeV , 1997 .

[272]  R. Bonciani,et al.  Planar box diagram for the (NF=1) 2-loop QED virtual corrections to Bhabha scattering , 2003, hep-ph/0310333.

[273]  Fermion pair production in e+e- collisions at 189–209 GeV and constraints on physics beyond the standard model , 2006, hep-ex/0609051.

[274]  F. Ignatov.  CMD-2 and SND results on e+e−→ hadrons cross sections , 2008 .

[275]  G. Venanzoni.  Status report on the KLOE measurement of the hadronic cross section with ISR , 2007 .

[276]  R. Kleiss,et al.  Distributions in the process e+e−→e+e−(γ) , 1981 .

[277]  A. Aloisio,et al.  Study of the decay $\phi \to \pi^+ \pi^- \pi^0$ with the KLOE detector , 2003 .

[278]  V. Smirnov,et al.  Analytical results for dimensionally regularized massless on-shell double boxes with arbitrary indices and numerators , 1999 .

[279]  R. Gatto,et al.  Electron Positron Colliding Beam Experiments , 1961 .

[280]  E. Rafael,et al.  Hadronic contributions to the muon g-factor , 1969 .

[281]  P. Roig.  Hadronic matrix elements for TAUOLA: 3π and KKπ channels , 2008, 0810.5764.

[282]  M. Davier,et al.  THE THEORETICAL PREDICTION FOR THE MUON ANOMALOUS MAGNETIC MOMENT , 2004 .

[283]  F. Filthaut,et al.  Measurement of hadron and lepton-pair production in e+e - collisions at √s = 192-208 GeV at LEP , 2006 .

[284]  H. Czyz,et al.  Order-α2 leading-logarithmic corrections in Bhabha scattering at LEP/SLC energies , 1994 .

[285]  M. Redhead.  Radiative corrections to the scattering of electrons and positrons by electrons , 1953, Proceedings of the Royal Society of London. Series A, Mathematical and physical sciences.

[286]  Zbigniew Was,et al.  PHOTOS - a universal Monte Carlo for QED radiative corrections: version 2.0 , 1994 .

[287]  S. Eidelman,et al.  e+e− annihilation into hadrons and exclusive τ decays , 1991 .

[288]  Wei,et al.  Measurement of the total cross section for hadronic production by e(+)e(-) annihilation at energies between 2.6-5 GeV , 2000, Physical review letters.

[289]  V. Smakhtin,et al.  High-statistics measurement of the pion form factor in the ρ-meson energy range with the CMD-2 detector , 2007 .

[290]  M. Czakon,et al.  Electroweak two-loop corrections to the effective weak mixing angle , 2006, hep-ph/0608099.

[291]  R. Bonciani,et al.  Two-loop NF=1 QED Bhabha scattering: Soft emission and numerical evaluation of the differential cross-section , 2005 .

[292]  Tests of the standard model and constraints on new physics from measurements of fermion-pair production at 189 GeV at LEP , 1998 .

[293]  P. Mastrolia,et al.  Closed-form decomposition of one-loop massive amplitudes , 2008, 0803.1989.

[294]  A. G. Chertovskikh,et al.  a1(1260)π dominance in the process e+e−→4π at energies 1.05–1.38 GeV , 1999 .

[295]  V. Cirigliano,et al.  Radiative τ decay and the magnetic moment of the muon , 2002, hep-ph/0207310.

[296]  R. Bonciani,et al.  Two-Loop NF = 1 QED Bhabha Scattering Differential Cross Section , 2004, hep-ph/0405275.

[297]  P. D. Causmaecker,et al.  Multiple bremsstrahlung in gauge theories at high energies (II). Single bremsstrahlung , 1982 .

[298]  T. Riemann,et al.  AMBRE - a Mathematica package for the construction of Mellin-Barnes representations for Feynman integrals , 2007, Comput. Phys. Commun..

[299]  G. Zanderighi,et al.  Scalar one-loop integrals for QCD , 2007, 0712.1851.

[300]  Charmonium Decays of Y(4260), psi(4160), and psi(4040). , 2006, Physical review letters.

[301]  M. Consoli.  One-loop corrections to e+e− → e+e− in the weinberg model☆ , 1979 .

[302]  Multi-parton cross sections at hadron colliders , 2005, hep-ph/0512150.

[303]  Final report of the E821 muon anomalous magnetic moment measurement at BNL , 2006, hep-ex/0602035.

[304]  X. Zhang,et al.  Synthesis of Trifluoromethyltriazoles via 1,3-DipolarCycloaddition , 2008 .

[305]  G. Pancheri.  Infra-red radiative corrections for resonant processes , 1969 .

[306]  M. Casarsa,et al.  Study of the decay φ→π+π−π0 with the KLOE detector , 2002 .

[307]  V. Smakhtin,et al.  Measurement of the pion form factor in the range 1.04–1.38 GeV with the CMD-2 detector , 2005, hep-ex/0603021.

[308]  H. Czyz,et al.  Status of Phokhara , 2008, 0812.1939.

[309]  W. Kluge.  Physics with DAΦNE , 2004 .

[310]  T. Riemann,et al.  J ul 2 00 8 DESY 08-101 PITHA 0818 Virtual Hadronic and Heavy-Fermion O ( α 2 ) Corrections to Bhabha Scattering , 2008 .

[311]  M. Preger,et al.  Luminosity measurement at Adone by single and double bremsstrahlung , 1974 .

[312]  H. Czyz,et al.  Radiative return at NLO and the measurement of the hadronic cross-section in electron–positron annihilation , 2001, hep-ph/0112184.

[313]  J. Bij,et al.  Three-loop electroweak corrections to the W-boson mass and sin2θefflept in the large Higgs mass limit , 2005, hep-ph/0504092.

[314]  A. Denig.  MEASUREMENT OF THE HADRONIC CROSS SECTION VIA RADIATIVE RETURN AT DAΦNE , 2005 .

[315]  Z. Kunszt,et al.  Unitarity cuts and reduction to master integrals in d dimensions for one-loop amplitudes , 2006, hep-ph/0612277.

[316]  Z. Was,et al.  Measuring the Higgs boson's parity using τ→ρν , 2002 .

[317]  D. Leith,et al.  Precise measurement of the e+e- --> pi+pi-(gamma) cross section with the initial state radiation method at BABAR. , 2009, Physical review letters.

[318]  Z. Was.  TAUOLA, TAUOLA universal interface PHOTOS and MC-TESTER: Status Report , 2009, 0901.1865.

[319]  M. Antonelli,et al.  Measurement of σ (e + e − → π + π − γ (γ )) and the dipion contribution to the muon anomaly with the KLOE detector , 2009 .

[320]  M. Davier,et al.  Confronting spectral functions from ee annihilation and $\tau$ decays: consequences for the muon magnetic moment , 2002, hep-ph/0208177.

[321]  F. Jegerlehner,et al.  The muon g ― 2 , 2009, 0902.3360.

[322]  V. Smirnov.  Evaluating Feynman Integrals , 2005 .

[323]  L. Trentadue,et al.  Hard pair production in large angle Bhabha scattering , 1996 .

[324]  J. Kuhn,et al.  Three loop top quark contributions to the ρ parameter , 2003, hep-ph/0302275.

[325]  Gerard 't Hooft,et al.  Scalar One Loop Integrals , 1979 .

[326]  F. Jegerlehner.  Theoretical precision in estimates of the hadronic contributions to (g − 2)μ and αQED(Mz) , 2003, hep-ph/0310234.

[327]  M. Hayakawa,et al.  Revised value of the eighth-order QED contribution to the anomalous magnetic moment of the electron , 2007, 0712.2607.

[328]  J. S. Whitaker,et al.  Hadron production by e+e- annihilation at center-of-mass energies between 2.6 and 7.8 GeV. I. Total cross section, multiplicities, and inclusive momentum distributions , 1982 .

[329]  Y. Tsai.  Decay Correlations of Heavy Leptons in e + + e − → l + + l − , 1971 .

[330]  R. Bonciani,et al.  Master integrals with 2 and 3 massive propagators for the 2-loop electroweak form factor—planar case , 2004 .

[331]  T. Riemann,et al.  Fermionic NNLO contributions to Bhabha scattering , 2007, 0710.5111.

[332]  M. Veltman.  Limit on mass differences in the Weinberg model , 1977 .

[333]  L. Landau.  On analytic properties of vertex parts in quantum field theory , 1959 .

[334]  R. Tarrach.  Invariant amplitudes for virtual compton scattering off polarized nucleons free from kinematical singularities, zeros and constraints , 1975 .

[335]  J. Kuhn,et al.  QCD CORRECTIONS TO THE E+E- CROSS-SECTION AND THE Z BOSON DECAY RATE : CONCEPTS AND RESULTS , 1995, hep-ph/9503396.

[336]  R. Bonciani,et al.  Vertex diagrams for the QED form factors at the 2-loop level , 2003 .

[337]  D. Bisello,et al.  Baryon pairs production ine+e− annihilation at $$\sqrt s = 2.4 GeV$$ , 1990 .

[338]  R. Gastmans,et al.  Hard photon corrections for the process $e^+ e^- \to \mu^\pm \mu^\mp$ , 1973 .

[339]  Ciaran Williams,et al.  One-Loop Gluonic Amplitudes from Single Unitarity Cuts , 2008, 0810.2964.

[340]  J. Fleischer,et al.  Analytic two-loop results for self-energy- and vertex-type diagrams with one non-zero mass , 1999 .

[341]  E. Remiddi,et al.  The analytical value of the electron (g − 2) at order α3 in QED , 1996 .

[342]  A. Pich,et al.  The Role of Resonances in Chiral Perturbation Theory , 1989 .

[343]  J. B. Tausk.  Non-planar massless two-loop Feynman diagrams with four on-shell legs , 1999 .

[344]  O. Nicrosini,et al.  Second order electromagnetic radiative corrections toe+e−→γ*Z0→μ+μ− , 1988 .

[345]  Kuehn,et al.  QCD corrections from the top quark to relations between electroweak parameters up to order alpha 2s. , 1995, Physical review letters.

[346]  M. Steinhauser.  Leptonic contribution to the effective electromagnetic coupling constant up to three loops , 1998, hep-ph/9803313.

[347]  F. Nguyen,et al.  Contribution of the direct decay φ→π+π−γ to the process e+e−→π+π−γ at DAΦNE , 2000 .

[348]  R. Boughezal,et al.  Three-loop electroweak correction to the Rho parameter in the large Higgs mass limit , 2004, hep-ph/0410216.

[349]  S. Aoki,et al.  Lattice study of the vacuum polarization function and determination of the strong coupling constant , 2008, 0807.0556.

[350]  M. Davier,et al.  Measurement of the e+e−→π+π−(γ) Cross Section with the ISR Method with BaBar , 2009 .

[351]  Improved determination of alpha (M(Z)**2) and the anomalous magnetic moment of the muon , 1997, hep-ph/9801361.

[352]  A. Goncharov,et al.  Multiple polylogarithms, cyclotomy and modular complexes , 2011, 1105.2076.

[353]  Two-loop electroweak next-to-leading logarithmic corrections to massless fermionic processes , 2006, hep-ph/0608326.

[354]  K. Cho,et al.  Study of the five-charged-pion decay of the τ lepton , 1994 .

[355]  Michal Czakon,et al.  Automatized analytic continuation of Mellin-Barnes integrals , 2005, Computer Physics Communications.

[356]  G. Parisi,et al.  Asymptotic Freedom in Parton Language , 1977 .

[357]  Zbigniew Was,et al.  The τ decay library TAUOLA, version 2.4 , 1993 .

[358]  T. Sack,et al.  Electroweak radiative corrections toe+e−→γγ , 1986 .

[359]  Ronald Kleiss,et al.  Distributions for electron-positron annihilation into two and three photons , 1981 .

[360]  S. Spagnolo.  The hadronic contribution to the muon $g-2$ from hadron production in initial state radiation events at the $e^+e^-$ collider DAΦNE , 1999 .

[361]  P. Golonka,et al.  Next-to-leading logarithms and the PHOTOS Monte Carlo , 2006 .