Leading-order hadronic contribution to g-2 from lattice QCD Page: 3 of 4
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Leading-order hadronic contribution to g-2 from lattice QCD Dru B. Renner
-0.1 I - - o * a=0.079 fm L=1.6 fm
4 - 4.5 6 " a=0.079 fm L=1.9 fm
-0.16- ' - a=0.079 fm L=2.5 fm
Sa=0.063 fm L=1.5 fm
r 4 A a=0.063 fm L=2.0 fm
-0.15 - - linear
-0.2 + 3.5 - measured
0 1 2 3 4
-0.20 10 20 30 40 50 60 70 80 90 0 0 '.1 0.2 0.3 0.4
Q2 [GeV2] rs [GeV2]
Figure 1: Vacuum polarization function, II(Q2). Figure 2: Leading-order hadronic correction for the
This is an example of our calculation of II(Q2) and muon, ah'p. This is our calculation of ah7p, using the
the interpolation of the entire range of Q2 that is cal- modified method described in the text, compared to
culated on the lattice. the two-flavor contribution to the experimental value.
in effective field theories of vector-mesons  and then add a polynomial expansion to provide a
complete basis of functions. The resulting expression is
M 2 N
low 2 9 E Q2+m2 + an (Q2n
i=1 Vi n=0
The sum over i gives the tree-level contribution for M vector mesons with masses mvz and couplings
gyn. These masses and couplings have precise meanings of their own and are calculated in the same
lattice computation. The form for Hlow is then matched to Hhigh to provide a complete description
that is suitable for numerical integration. An example of this interpolation and extrapolation is
shown in Fig. 1. This method gives a fully non-perturbative determination of ahp that does not rely
on QCD perturbation theory.
4. Modified method
We introduce a class of quantities that have the same physical limit as ahi but approach that
limit more smoothly as a function of the pseudo-scalar meson mass mPs. For any hadronic quantity
H, we define
hvp = a2j ;Q2 $w2/mJ. Hhy/H2) FHR(Q2)
where H is understood to be calculated at each value of mPs and Hphys = H(mps -> mr). By
construction ("p= hVp as mPs approaches ms. Any choice of H leads to a valid definition but
H = my, the lightest vector-meson mass, leads to a mild mPs dependence and results in a well-
controlled lattice calculation of ahVp in the physical limit. The results for ahKp are shown in Fig. 2.
The two-flavor contribution to the experimentally measured value of ap [2, 4] is also shown in
the plot and we find good agreement between the linearly extrapolated lattice calculation and the
measured value. As a check of the method, we perform the same calculation and comparison for
the electron in Fig. 3 and for the tau in Fig. 4.
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Dru Renner, Xu Feng, Karl Jansen, Marcus Petschlies. Leading-order hadronic contribution to g-2 from lattice QCD, article, March 1, 2011; Newport News, Virginia. (digital.library.unt.edu/ark:/67531/metadc837388/m1/3/: accessed January 24, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.