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Fig. 28. Modulus square of the I = 1 pion form factor extracted from τ ± → ντπ ± π 0 which shows the ρ ± –resonance. The ratio |Fπ(E)| 2 (τ)/|Fπ(E)| 2 fit (e+ e − [I = 1]) illustrates the missing consistency of the τ–data relative to a CMD-2 fit. Dashed horizontal lines mark ± 10% (see also [207,208]). Note that the reference fit line represents the e + e − data only below about 1 GeV (see Fig. 21). At higher energies data for e + e − → π + π − are rather poor, but old Orsay DM2 data as well as the new preliminary BaBar radiative return data [176] also exhibit the dip at 1.5 GeV, i.e. our “normalization” above 1 GeV is to be considered as arbitrary. with σ (0) ππ = � � Kσ(s) dΓππ[γ] KΓ(s) ds KΓ(s) = G2 F |Vud| 2 m 3 τ 384π 3 and the isospin breaking correction RIB(s) = 1 GEM(s) β 3 π − π + β 3 π − π 0 RIB(s) × , (121) SEW � 1 − s m2 �2 � 1 + 2 τ s m2 � ; Kσ(s) = τ πα2 3s , � � �FV � (s) � � � f+(s) � 2 includes the QED corrections to τ − → ντπ −π0 decay with virtual plus real soft and hard photon radiation integrated over all phase space. Originating from Eq. (120), β3 π−π +/β3 π−π0 is a phase space correction due to the π ± − π0 mass difference. FV (s) = F 0 π I = 0 contribution. The latter ρ − ω mixing term is due to the SU(2) breaking by the md − mu mass difference. Finally, f+(s) = F − π is the charged current (CC) I = 1 vector form factor. One of the leading isospin breaking effects is the ρ − ω mixing correction included in |FV (s)| 2 . The form–factor corrections, (122) (s) is the neutral current (NC) vector form factor, which exhibits besides the I = 1 part an 48
in principle, also should include the electromagnetic shifts in the masses and the widths of the ρ’s 16 . Up to this last mentioned effect, discussed in [179] (also see [215]), which was considered to be negligible in earlier isospin breaking estimates, all the corrections were applied in [178] but were not able to eliminate the observed discrepancy between v−(s) and v0(s). The deviation starts at the peak of the ρ and increases with energy to about 10-20%. More precisely, Fig. 28 shows a good agreement below about 800 MeV, a 10% enhancement between 800 and 1200 MeV and a pronounced dip around 1500 MeV. The trend shown by the ALEPH 97 and CLEO data is clearly stressed by the new Belle measurement [207]. The Belle data differ substantially from the ALEPH 05 data, however, which lie higher by more than 10% in the intermediate range [+20% relative to e + e − ]. We should mention here once more that photon radiation by hadrons is poorly understood theoretically. The commonly accepted recipe is to treat radiative corrections of the pions by scalar QED, except for the short distance (SD) logarithm proportional to ln MW/mπ which is replaced by the quark parton model result and included in SEW by convention. This SD log is present only in the weak charged current transition W +∗ → π + π 0 (γ), while in the charge neutral electromagnetic current transition γ ∗ → π + π − (γ) this kind of leading log is absent. In any case there is an uncertainty in the correction of the isospin violations by virtual and real photon radiation which is hard to quantify. We also should stress that the possible isospin breaking resonance parameter shifts, like ∆mρ and ∆Γρ, so far have not been determined unambiguously. Note, however, that the new Belle results [207] precisely confirm the earlier observed shifts in mass and width of the ρ [216,179], which could be part of the source of additional isospin violations. A reason for the τ vs e + e− discrepancy could be the way (incoherent) the pure I = 1 part (given by τ– data) is combined with the missing I = 0 contribution (approximately separated out from the e + e−-data). What is needed and what is measured in e + e− is the interference |A1(s) + A0(s)| 2 which in any case must be smaller than |A1(s)| 2 + |A0(s)| 2 . In any case, one has to keep in mind that isospin breaking can only have two origins: the mu − md mass difference and electromagnetic effects and the latter require a small positive mass difference mρ ± − mρ 0 ∼ 1 MeV and similar for ρ ′ and ρ ′′ . In fact, a fit of the data for the ρ yields a factor of 2 larger result, which maybe is a problem. It should be noted that the fits including several masses, widths and mixings are not very stable. In Ref. [217] effects of the ρ–ω–φ mixing on the dipion mass spectrum in e + e− –annihilation and τ–decay were analyzed within the HLS effective model and it was suggested that they could explain the observed isospin breakings. As a possibility one also may consider the case that the τ–data based evaluation of ahad µ is the more reliable one. The integrated data in the range mπ − 1.8 GeV after applying known isospin violation corrections is given by – Belle (τ) [207] a ππ µ = (523.5 ± 1.5(exp) ± 2.6(Br) ± 2.5(iso)) × 10−10 , (123) – ALEPH, CLEO, OPAL (τ) [178] a ππ µ = (520.1 ± 2.4(exp) ± 2.7(Br) ± 2.5(iso)) × 10−10 , (124) which compares to – CMD2, SND (e + e − ) [184] a ππ µ = (504.6 ± 3.1(exp) ± 0.9(rad)) × 10 −10 , (125) where errors are the experimental ones (exp), from the normalizing branching fraction (Br), from isospin breaking corrections (iso) and from radiative corrections (rad). Including τ data shifts the theoretical pre- diction by δa had µ ≃ +17.9 × 10−10 thus would improve the agreement between theory and experiment for aµ to the 1.2 σ level, see Sect. 7. However, using the τ–data would also increase the “gap” between a too low 16 Because of the strong resonance enhancement, especially in the ρ region, a small isospin breaking shift in mass and width between ρ 0 and ρ ± , typically ∆mρ = m ρ ± − m ρ 0 ∼ 2.5 MeV and ∆Γρ = Γ ρ ± − Γ ρ 0 ∼ 1.5 MeV and similar shifts for the higher resonances ρ ′ , ρ ′′ and the mixing amplitudes of these states, causes a large effect in the tails by the kinematical shift this implies. 49
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 and 44: Fig. 22. Experimental results for R
Page 45 and 46: γ γ had ⇔ Π ′ had γ (q 2 )
Page 47: � �� Ï � �Ù � �
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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