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ρ, ω Υ ψ 0.0 GeV, ∞ 9.5 GeV 3.1 GeV φ, . . . 2.0 GeV 1.0 GeV 1.0 GeV φ, . . . ρ, ω 0.0 GeV, ∞ 3.1 GeV 2.0 GeV Fig. 23. The distribution of contributions (left) and errors (right) in % for a (4) µ (vap, had) from different energy regions. The error of a contribution i shown is δ2 i tot /�i δ2 i tot in %. The total error combines statistical and systematic errors in quadrature. Table 4 Some recent evaluations of a (4) µ (vap, had). a (4) µ (vap, had) × 10 10 data Ref. 696.3[7.2] e + e − [178] 711.0[5.8] e + e − + τ [178] 694.8[8.6] e + e − [179] 684.6[6.4] e + e − TH [180] 699.6[8.9] e + e − [181] 692.4[6.4] e + e − [182] a (4) µ (vap, had) × 10 10 data Ref. 693.5[5.9] e + e − [183] 701.8[5.8] e + e − + τ [183] 690.9[4.4] e + e −∗∗ [184] 689.4[4.6] e + e −∗∗ [185] 692.1[5.6] e + e −∗∗ [161] 690.3[5.3] e + e −∗∗ [175] Fig. 24. History of evaluations before 2000 (left) [73]–[76],[186]–[189],[79]–[81],[190,191,83], [162]–[167], and some more recent ones (right) [178]–[185], [161,175]; (e + e − ) = e + e − –data based, (e + e − ,τ) = in addition include data from τ spectral functions (see Sect. 4.1.2). ones are listed in Table 4. Fig. 24 gives a fairly complete history of the evaluations based on e + e − –data. Before we will continue with a discussion of the higher order hadronic contributions, we first present additional details about what precisely goes into the DR Eq. (109) and briefly discuss some issues concerning the determination of the required hadronic cross–sections. 4.1.1. Dispersion Relations and Hadronic e + e− –Annihilation Cross Sections To leading order in α, the hadronic “blob” in Fig. 19 has to be identified with the photon self–energy function Π ′ had γ (s). The latter we may relate to the cross–section e + e− → hadrons by means of the DR Eq. (101) which derives from the correspondence Fig. 25 based on unitarity (optical theorem) and causality (analyticity), as elaborated earlier. Note that Π ′ had γ (q2 ) is a one particle irreducible (1PI) object, represented by diagrams which cannot be cut into two disconnected parts by cutting a single photon line. At low energies the imaginary part is related to intermediate hadronic states like π0γ, ρ, ω, φ, · · ·, ππ, 3π, 4π, · · ·, 44
γ γ had ⇔ Π ′ had γ (q 2 ) γ had ∼ σhad tot (q2 ) Fig. 25. Optical theorem for the hadronic contribution to the photon propagator. ππγ, , · · · , KK, KKπ · · · which in the DR correspond to the states produced in e + e − –annihilation via a virtual photon. At least one hadron plus any strong, electromagnetic or weak interaction contribution counts. e + e − –data in principle may be used up to energies where γ − Z interference comes into play above about 40 GeV. Experimentally, what is determined is of the form R exp had (s) = Nhad (1 + δRC) Nnorm ε σnorm(s) σµµ, 0(s) , where Nhad is the number of observed hadronic events, Nnorm is the number of observed normalizing events, ε is the detector efficiency–acceptance product of hadronic events while δRC are radiative corrections to hadron production. σnorm(s) is the physical cross–section for normalizing events, including all radiative corrections integrated over the acceptance used for the luminosity measurement, and σµµ, 0(s) = 4πα 2 /3s is the normalization. This also shows that a precise measurement of R(s) requires precise knowledge of the relevant radiative corrections. Radiation effects may be used to measure σhad(s ′ ) at all energies √ s ′ lower than the fixed energy √ s at which an accelerator is running [192]. This is possible due to initial state radiation (ISR), which can lead to huge effects for kinematical reasons. The relevant radiative return (RR) mechanism is illustrated in Fig. 26: in the radiative process e + e − → π + π − γ, photon radiation from the initial state reduces the invariant mass from s to s ′ = s (1 − k) of the produced final state, where k is the fraction of energy carried away by the photon radiated from the initial state. Such RR cross-section measurements are particularly interesting for machines running on–resonance like the φ– and B–factories, which have enhanced event rates as they are running on top of a peak [193,194,195]. The first dedicated RR experiment has been performed by KLOE at DAΦNE/Frascati, by measuring the π + π − cross–section [86,171] (see Fig. 21 and Refs. [196,197]). Results for exclusive multi–hadron production channels from BaBar play an important role in the energy range between 1.4 to 2 GeV. In fact new data became available for most of the channels of the exclusive measurements in this region. In contrast the inclusive measurements date back to the early 1980’s and show much larger uncertainties. It is important to note that what we need in the DR is the 1PI “blob” which by itself is not what is measured, i.e. it is not a physical observable. In reality the virtual photon lines attached to the hadronic “blob” are dressed photons (full photon propagators which include all possible radiative corrections) and in order to obtain the 1PI part one has to undress the cross section by amputation of the full photon lines. The γ hard γ π + π− φ , ρ0 hadrons s = M2 φ ; s′ = s (1 − k), k = Eγ/Ebeam a) Fig. 26. a) Principle of the radiative return determination of the π + π − cross–section by KLOE at the φ–factory DAΦNE. At the B–factory at SLAC, using the same mechanism, BaBar has measured many other channels at higher energies. b) Standard measurement of σhad in an energy scan, by tuning the beam energy. 45 e + e − b) 2
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 and 38: 2 IVa IVb IVc IVd Fig. 18. Typical
Page 39 and 40: µ γ had Fig. 19. Leading hadronic
Page 41 and 42: a (4) µ (vap, had) = � αmµ 3π
Page 43: Fig. 22. Experimental results for R
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
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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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