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was subsequently cross–checked by Kataev [150]. A very recent calculation from the class of leading tenth order contributions (singlet (SI) VP insertion diagrams which includes the last diagram of Fig. 13 with two electron LbL loops) yields A (10) 2 (mµ/me) = −1.26344(14) [151]. This result has been reproduced at the 3% level by an asymptotic expansion in [152], where the much larger 4–loop non-singlet (NS) VP insertion with electron loops has also been calculated: A (10),as 2 (mµ/me) = 63.481NS − 1.21429SI = 62.2667. Since the leading terms are included in Eq. (97) already and subleading terms are unknown in general, we will stay with the above result in the following. Thus, taking into account Eq. (53), we arrive at or C5 ∼ 663.0(20.0)(4.6) a (10) QED µ ∼ 663(20)(4.6) � α �5 ≃ 4.483(135)(31) × 10 π −11 as an estimate of the 5–loop QED contribution. In Table 3 we collect the results of the QED calculations. In spite of the fact that the expansion coefficients Ci multiplying (α/π) i grow rapidly with the order, the convergence of the perturbative expansion of a QED µ is good. This suggests that the perturbative truncation error is well under control at the present level of accuracy. Table 3 The QED contributions to aµ. . . Ci (2i) QED a µ × 1011 C1 0.5 a (2) 116140973.289(43) C2 0.765857 410 (27) a (4) 413217.620(14) C3 24.050509 64(46) a (6) 30141.902(1) C4 130.8105(85) a (8) 380.807(25) C5 663.0(20.0)(4.6) a (10) 4.483(135)(31) The universal QED terms have been given in Eq. (54) and together with the mass dependent QED terms of the 3 flavors (e, µ, τ) we obtain a QED µ = 116 584 718.104(.044)(.015)(.025)(.139)[.148] × 10 −11 . (99) The errors are given by the uncertainties in αinput, in the mass ratios, the numerical error on α 4 terms and the guessed uncertainty of the α 5 contribution, respectively. Now we have to address the question what happens beyond QED. What is measured in an experiment includes effects from the real world and we have to include the contributions from all known particles and interactions such that from a possible deviation between theory and experiment we may get a hint of the yet unknown physics. 4. Hadronic Vacuum Polarization Corrections On a perturbative level we may obtain the hadronic vacuum polarization contribution by replacing internal lepton loops in the QED VP contributions by quark loops, adapting charge, color multiplicity and the masses accordingly. Since quarks are, however, confined inside hadrons, a quark mass cannot be defined in the same natural way as a lepton mass and quark mass values depend in various ways on the physical circumstances. Moreover, the running strong coupling “constant” αs(s) becomes large at low energies E = √ s. Therefore perturbative QCD (pQCD) fails to “converge” in any practical sense in this region and pQCD may only be 38 (98)
µ γ had Fig. 19. Leading hadronic contribution to g − 2. trusted above about 2 GeV and away from thresholds and resonances. The low energy structure of QCD with confinement and the spontaneous breaking of chiral symmetry in the chiral limit (on a Lagrangian level characterized by vanishing (current) quark masses) is completely beyond the scope of pQCD. Low energy QCD is characterized by its typical spectrum of low lying hadronic states, the pseudoscalar pions, the Kaons and the η as quasi Goldstone bosons (true ones in the chiral limit), the pseudoscalar singlet η ′ , the spin-1 vector bosons ρ, ω, φ and by the order parameters of chiral symmetry breaking, like the quark condensates 〈¯qq〉 �= 0 (q = u, d, s). For the calculation of the hadronic contributions a had µ to the g − 2 of the muon, baryons like proton and neutron do not play a big role. Quarks contribute to the electromagnetic current according to their charge � � µ had 2 j em = 3 c ūcγ µ uc − 1 3 ¯ dcγ µ dc − 1 3 ¯scγ µ sc + 2 3 ¯ccγ µ cc − 1 3 ¯bcγ µ bc + 2 3 ¯tcγ µ � tc µ . (100) µ had The hadronic electromagnetic current j em is a color singlet and hence includes a sum over colors indexed by c. Its contribution to the electromagnetic current correlator Eq. (64) defines Π ′ had γ (s), which enters the calculation of the leading order hadronic contribution to a had µ , diagrammatically given by Fig. 19. Perturbative QCD fails to be a reliable tool for estimating a had µ and known approaches to low energy QCD like chiral perturbation theory as well as extensions of it which incorporate spin-1 bosons or lattice QCD are far from being able to make precise predictions. We therefore have to resort to a semi-phenomenological approach using dispersion relations together with the optical theorem and experimental data. The basic relations are – analyticity (deriving from causality), which allows one to write the DR Π ′ γ(k 2 ) − Π ′ γ(0) = k2 π � 0 ∞ ImΠ ds ′ γ(s) s (s − k2 − iε) . (101) – optical theorem (deriving from unitarity), which relates the imaginary part of the vacuum polarization amplitude to the total cross section in e + e − –annihilation with ImΠ ′ γ(s) = R(s) = σtot/ 4πα(s)2 3s s 4πα(s) σtot(e + e − → anything) := α(s) R(s), (102) 3 . (103) The normalization factor is the point cross–section (tree level) σµµ(e + e − → γ ∗ → µ + µ − ) in the limit s ≫ 4m 2 µ. We obtain the hadronic contribution if we restrict “anything” to hadrons. The complementary leptonic part may be calculated reliably in perturbation theory and the production of a lepton pair at lowest order is given by � Rℓ(s) = 1 − 4m2 ℓ s � 1 + 2m2 � ℓ , (ℓ = e, µ, τ), (104) s 39
Page 1 and 2: Abstract The Muon g-2 Fred Jegerleh
Page 3 and 4: to mt ≃ 173 GeV for the top quark
Page 5 and 6: transformations C, P and T, which a
Page 7 and 8: magnetic moment by Schwinger in 194
Page 9 and 10: 1.2.1. Spin Transfer in Production
Page 11 and 12: and typically is strongly peaked at
Page 13 and 14: ⇒ ⇒spin ⇒ momentum µ ⇒ ⇒
Page 15 and 16: µ + → e + + νe + ¯νµ µ +
Page 17 and 18: x = A cos( √ 1 − n ωc t), z =
Page 19 and 20: The two uncertainties given are the
Page 21 and 22: and later we will denote by CL = 3
Page 23 and 24: 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11)
Page 25 and 26: 3.2. Electron Anomalous Magnetic Mo
Page 27 and 28: Table 2 Contributions to ae(h/M) in
Page 29 and 30: The last simple representation in t
Page 31 and 32: The error is due to the uncertainty
Page 33 and 34: loops yield contributions which are
Page 35 and 36: which together with Eq. (73) leads
Page 37: 2 IVa IVb IVc IVd Fig. 18. Typical
Page 41 and 42: a (4) µ (vap, had) = � αmµ 3π
Page 43 and 44: Fig. 22. Experimental results for R
Page 45 and 46: γ γ had ⇔ Π ′ had γ (q 2 )
Page 47 and 48: � �� Ï � �Ù � �
Page 49 and 50: in principle, also should include t
Page 51 and 52: c1 = 1, c2 = C2(R) = 365 24 � −
Page 53 and 54: At the 3-loop level all diagrams of
Page 55 and 56: Table 5 Higher order contributions
Page 57 and 58: µ(p) q3λ γ(k) kρ q2ν had Fig.
Page 59 and 60: π 0 , η, η ′ µ γ (a) [L.D.]
Page 61 and 62: 5.1. Pseudoscalar-exchange Contribu
Page 63 and 64: where P6 = 1/(Q2 2 +m2π ), and P7
Page 65 and 66: The second situation of interest co
Page 67 and 68: with the photon momenta qi (incomin
Page 69 and 70: ehavior which qualitatively agrees
Page 71 and 72: the ρ and the ρ ′ (LMD+V). Both
Page 73 and 74: Melnikov and Vainshtein [258], 27 t
Page 75 and 76: Table 9 Results for the axial-vecto
Page 77 and 78: Finally, the last entry in Table 6
Page 79 and 80: The leading contribution to aLbL;ha
Page 81 and 82: a (2) EW µ (Z) = √ 2Gµm 2 µ 16
Page 83 and 84: µ γ γ (a) [L.D.] π Z 0 , η, η
Page 85 and 86: a (4) EW µ ([f])VVA ≃ K2 Λ 2
Page 87 and 88: O αβ 1 F = 4π2 ˜ F αβ = 1 4π
Page 89 and 90:
� i gi = 1 , � i gim 2 i = 0 ,
Page 91 and 92:
with typical values ∆C tH = (5.84
Page 93 and 94:
The high value −40.98 corresponds
Page 95 and 96:
The total weak contribution thus is
Page 97 and 98:
some small effect has been overlook
Page 99 and 100:
YU ∼ (3, ¯3, 1) SU(3) 3 q , YD
Page 101 and 102:
(a) Case: mµ = M ≪ M0 (b) Case:
Page 103 and 104:
M0[S,P] H ± M0[V,A] mµ mµ f M f
Page 105 and 106:
µ γ h, A a) γ b) γ c) γ d) τ,
Page 107 and 108:
higgsinos, respectively. In additio
Page 109 and 110:
a) ˜χ ˜χ ˜ν b) ˜µ ˜µ Fig.
Page 111 and 112:
Fig. 48. Constraint on large tanβ
Page 113 and 114:
m 0 (GeV) 800 700 600 500 400 300 2
Page 115 and 116:
the parameter space. As shown in Re
Page 117 and 118:
fields on M 4 , which is called the
Page 119 and 120:
for D = 5 requires a cut-off. Using
Page 121 and 122:
inconsistencies between the differe
Page 123 and 124:
The same four-point function 〈V V
Page 125 and 126:
In calculations of hadronic contrib
Page 127 and 128:
[49] L. E. Kinster, W. V. Houston,
Page 129 and 130:
[151] T. Aoyama, M. Hayakawa, T. Ki
Page 131 and 132:
[242] H. Kolanoski, P. Zerwas, Two-
Page 133 and 134:
[337] S. Ritt [MEG Collaboration],
Page 135:
[440] T. Blum, Phys. Rev. Lett. 91
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