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Π ′ γ ren(q 2 ) = − α �1 π 0 = α π � 0 1 dz 2z (1 − z) ln(1 − z (1 − z) q 2 /m 2 ℓ) dt t 2 (1 − t 2 /3) and performing the integral yields Π ′ γ ren(q 2 ) = − α 3π 1 4m2 ℓ /q2 − (1 − t2 , (73) ) � 8 3 − β2 ℓ + 1 2 βℓ (3 − β 2 ℓ) ln βℓ − 1 βℓ + 1 � , (74) where βℓ = � 1 − 4m2 ℓ /q2 is the lepton velocity. The imaginary part is given by the simple formula Im Π ′ γ (q2 ) = α � 1 + 3 2m2 ℓ q2 � βℓ . (75) For q2 < 0 the amplitude Π ′ γ ren(q2 ) is negative definite and what is needed in Eq. (69) is −Π ′ (ℓ) γ (− x2 1−x m2 µ) or Eq. (74) with βℓ = � 1 + 4 x2 ℓ (1 − x)/x2 , where xℓ = mℓ/mµ and mℓ is the mass of the virtual lepton in the vacuum polarization subgraph. Using the representation Eq. (69) together with Eq. (73) the VP insertion was computed in the late 1950s [131] for mℓ = me and neglecting terms of O(me/mµ). Its exact expression was calculated in 1966 [132] and may be written in compact form as [35] A (4) 2 vap (1/x)=−25 lnx − 36 3 + x2 (4 + 3 lnx) + x 4 � � � 2 π 1 − 2 lnx ln − x − Li2(x 3 x 2 � ) + x � 2 1 − 5x 2 ��π2 � � � 1 − x − lnx ln − Li2(x) + Li2(−x) 2 1 + x =− 25 lnx − 36 3 + x2 (4 + 3 lnx) + x 4 � 2 ln 2 � (x) − 2 lnx ln x − 1 � + Li2(1/x x 2 � ) + x � 2 1 − 5x 2 �� x − 1 − lnx ln + Li2(1/x) − Li2(−1/x) (x > 1) . (76) x + 1 The first form is valid for arbitrary x. For x > 1 some of the logs as well as Li2(x) develop a cut and a corresponding imaginary part like that of ln(1 − x). Therefore, for the numerical evaluation in terms of a series expansion, it is an advantage to rewrite the Li2(x)’s in terms of Li2(1/x)’s, according to Eq. (A.11), which leads to the second form. There are two different regimes for the mass dependent effects, the light electron loops and the heavy tau loops [131,132]: • Light internal masses give rise to potentially large logarithms of mass ratios which get singular in the limit mlight → 0 µ γ e γ γ a (4) � 1 mµ µ (vap, e) = ln − 3 me 25 + O 36 � me mµ �� α Here we have a typical result for a light field which produces a large logarithm ln mµ ≃ 5.3, such that the me first term ∼ 2.095 is large relative to a typical constant second term −0.6944. Here the exact 2–loop result is a (4) � α �2 µ (vap, e) ≃ 1.094 258 3111(84) = 5.90406007(5) × 10 π −6 . (77) 30 π � 2 .
The error is due to the uncertainty in the mass ratio (me/mµ). The leading term as we have shown is due to the charge screening according to the RG. For comparison we next consider the • equal internal mass case, which yields the pure number µ γ μ γ γ a (4) µ (vap, µ) = � � 119 �α π2 − 36 3 π and is already included in the universal part Eq. (50). The result is typical for these kinds of radiative correction calculation: a rational term of size 3.3055... and a transcendental π2 term of very similar magnitude 3.2899... but of opposite sign largely cancel. The result is only 0.5% of the individual terms: a (4) µ (vap, µ) ≃ 0.015 687 4219 � α π � 2 � 2 , = 8.464 13320 × 10 −8 . (78) • Heavy internal masses decouple in the limit mheavy → ∞ and thus only yield small power corrections µ γ τ γ γ a (4) µ (vap, τ) = � 1 45 � �2 mµ mτ + O � m 4 µ m 4 τ ln mτ mµ �� α Here we have a typical “heavy physics” contribution, from a state of mass M ≫ mµ, yielding a term proportional to m 2 µ/M 2 . This means that besides the order in α there is an extra suppression factor, e.g. O(α 2 ) → Q(α 2 m2 µ M 2 ) in our case. To unveil new heavy states thus requires a corresponding high precision in theory and experiment. For the τ the contribution is relatively tiny a (4) µ (vap, τ) ≃ 0.000 078 064(25) � α π � 2 π � 2 . = 4.2120(13) × 10 −10 , (79) with the error from the mass ratio (mµ/mτ). Note that at the level of accuracy reached by the Brookhaven experiment (63 × 10 −11 ), the contribution is non–negligible. At the 2–loop level a e − τ mixed contribution is not possible, and hence A (4) 3 (mµ/me, mµ/mτ) = 0. The complete 2–loop QED contribution from the diagrams displayed in Fig. 9 is given by C2 = A (4) 1 uni + A(4) 2 vap (mµ/me) + A (4) 2 vap (mµ/mτ) = 0.765 857 410 (27) , and we have a (4) QED µ = 0.765 857 410 (27) � α �2 ≃ 413217.620(14) × 10 π −11 for the complete 2–loop QED contribution to aµ. The errors of A (4) 2 (mµ/me) and A (4) 2 (mµ/mτ) have been added in quadrature as the errors of the different measurements of the lepton masses may be treated as independent. The combined error δC2 = 2.7 × 10 −8 is negligible by the standards 1 × 10 −5 estimated after Eq. (48). 3.3.2. 3–loop: Light-by-Light Scattering and Vacuum Polarization Insertions At three loops, in addition to photon vacuum polarization corrections, a new kind of contribution shows up exhibiting the so called light–by–light scattering (LbL) insertions: closed fermion loops with four photons attached. Note that the physical process γγ → γγ of light–by–light scattering involves real on–shell photons. There are 6 diagrams which follow from the first one in Fig. 14, by permutation of the photon vertices on the external muon line, plus the ones obtained by reversing the direction of the fermion loop. Remember 31 (80)
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: The last simple representation in t
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 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
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a (2) EW µ (Z) = √ 2Gµm 2 µ 16
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µ γ γ (a) [L.D.] π Z 0 , η, η
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a (4) EW µ ([f])VVA ≃ K2 Λ 2
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O αβ 1 F = 4π2 ˜ F αβ = 1 4π
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� i gi = 1 , � i gim 2 i = 0 ,
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with typical values ∆C tH = (5.84
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The high value −40.98 corresponds
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The total weak contribution thus is
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some small effect has been overlook
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YU ∼ (3, ¯3, 1) SU(3) 3 q , YD
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(a) Case: mµ = M ≪ M0 (b) Case:
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M0[S,P] H ± M0[V,A] mµ mµ f M f
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µ γ h, A a) γ b) γ c) γ d) τ,
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higgsinos, respectively. In additio
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a) ˜χ ˜χ ˜ν b) ˜µ ˜µ Fig.
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Fig. 48. Constraint on large tanβ
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m 0 (GeV) 800 700 600 500 400 300 2
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the parameter space. As shown in Re
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fields on M 4 , which is called the
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for D = 5 requires a cut-off. Using
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inconsistencies between the differe
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The same four-point function 〈V V
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In calculations of hadronic contrib
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[49] L. E. Kinster, W. V. Houston,
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[151] T. Aoyama, M. Hayakawa, T. Ki
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[242] H. Kolanoski, P. Zerwas, Two-
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[337] S. Ritt [MEG Collaboration],
Page 135:
[440] T. Blum, Phys. Rev. Lett. 91
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