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Hadronic light-by-light corrections to the muon g-2: the pion-pole contribution  [PDF]
Marc Knecht,Andreas Nyffeler
Physics , 2001, DOI: 10.1103/PhysRevD.65.073034
Abstract: The correction to the muon anomalous magnetic moment from the pion-pole contribution to the hadronic light-by-light scattering is considered using a description of the pi0 - gamma* - gamma* transition form factor based on the large-Nc and short-distance properties of QCD. The resulting two-loop integrals are treated by first performing the angular integration analytically, using the method of Gegenbauer polynomials, followed by a numerical evaluation of the remaining two-dimensional integration over the moduli of the Euclidean loop momenta. The value obtained, a_{mu}(LbyL;pi0) = +5.8 (1.0) x 10^{-10}, disagrees with other recent calculations. In the case of the vector meson dominance form factor, the result obtained by following the same procedure reads a_{mu}(LbyL;pi0)_{VMD} = +5.6 x 10^{-10}, and differs only by its overall sign from the value obtained by previous authors. Inclusion of the eta and eta-prime poles gives a total value a_{mu}(LbyL;PS) = +8.3 (1.2) x 10^{-10} for the three pseudoscalar states. This result substantially reduces the difference between the experimental value of a_{mu} and its theoretical counterpart in the standard model.
Scalar and pseudoscalar meson pole terms in the hadronic light-by-light contributions to $a_μ^{had}$  [PDF]
E. Bartos,A. -Z. Dubnickova,S. Dubnicka,E. Kuraev,E. Zemlyanaya
Physics , 2001, DOI: 10.1016/S0550-3213(02)00242-0
Abstract: Third QED order hadronic light-by-light (LBL) contributions $a_\mu^{LBL}(M)$ to the anomalous magnetic moment of the muon $a_\mu^{had}$ from the pole terms of scalar $\sigma$, $a_0(980)$ and pseudoscalar $\pi^0$, $\eta$, $\eta '$ mesons (M) in the framework of the linearized extended Nambu-Jona-Lasinio model are evaluated. The off-shell structure of the photon-photon-meson vertices is taken into account by means of constituent quark triangle loops. The mass of the quark is taken to be $m_u=m_d=m_q=(280 \pm 20)$ MeV. The unknown strong coupling constants of $\pi^0, \eta, \eta '$ and $a_0$ mesons with quarks are evaluated in a comparison of the corresponding theoretical two-photon widths calculated in the framework of our approach with experimental ones. The $\sigma$-meson coupling constant is taken to be equal to $\pi_0$-meson coupling constant as it follows from the linearized Nambu-Jona-Lasinio model Lagrangian. Then one obtains $a_\mu^{LBL}(\pi_0)$=$(81.83 \pm 16.50) \times 10^{-11}$, $a_\mu^{LBL}(\eta)$=$(5.62 \pm 1.25) \times 10^{-11}$, $a_\mu^{LBL}(\eta ')$=$(8.00 \pm 1.74) \times 10^{-11}$, $a_\mu^{LBL}(\sigma)$=$(11.67 \pm 2.38) \times 10^{-11}$ and $a_\mu^{LBL}(a_0)$=$(0.62 \pm 0.24) \times 10^{-11}$. The total contribution of meson poles in LBL is $a_\mu^{LBL}(M)$=$(107.74 \pm 16.81) \times 10^{-11}$.
Pion pole contribution to hadronic light-by-light scattering and muon anomalous magnetic moment  [PDF]
Ian Blokland,Andrzej Czarnecki,Kirill Melnikov
Physics , 2001, DOI: 10.1103/PhysRevLett.88.071803
Abstract: We derive an analytic result for the pion pole contribution to the light-by-light scattering correction to the anomalous magnetic moment of the muon, $a_\mu = (g_\mu-2)/2$. Using the vector meson dominance model (VMD) for the pion transition form factor, we obtain $a_\mu^{{\rm LBL},\pi^0} = +56 \times 10^{-11}$.
Comment on the sign of the pseudoscalar pole contribution to the muon g-2  [PDF]
Masashi Hayakawa,Toichiro Kinoshita
Physics , 2001,
Abstract: We correct the error in the sign of the pseudoscalar pole contribution to the muon g-2, which dominates the O(alpha^3) hadronic light-by-light scattering effect. The error originates from our oversight of a feature of the algebraic manipulation program FORM which defines the epsilon-tensor in such a way that it satisfies the relation epsilon_{mu_1 mu_2 mu_3 mu_4} epsilon_{nu_1 nu_2 nu_3 nu_4} eta^{mu_1 nu_1} eta^{mu_2 nu_2} eta^{mu_3 nu_3} eta^{mu_4 nu_4} = 24, irrespective of space-time metric. To circumvent this problem, we replaced the product epsilon_{mu_1 mu_2 mu_3 mu_4} epsilon_{nu_1 nu_2 nu_3 nu_4} by - eta_{mu_1 nu_1} eta_{mu_2 nu_2} eta_{mu_3 nu_3} eta_{mu_4 nu_4} \pm cdots in the FORM-formatted program, and obtained a positive value for the pseudoscalar pole contribution, in agreement with the recent result obtained by Knecht {\it et al}.
Analysis of the Hadronic Light-by-Light Contributions to the Muon $g-2$  [PDF]
Johan Bijnens,Elisabetta Pallante,Joaquim Prades
Physics , 1995, DOI: 10.1016/0550-3213(96)00288-X
Abstract: We calculate the hadronic light-by-light contributions to the muon $g-2$. We use both $1/N_c$ and chiral counting to organize the calculation. Then we calculate the leading and next-to-leading order in the $1/N_c$ expansion low energy contributions using the Extended Nambu--Jona-Lasinio model as hadronic model. We do that to all orders in the external momenta and quark masses expansion. Although the hadronic light-by-light contributions to muon $g-2$ are not saturated by these low energy contributions we estimate them conservatively. A detailed analysis of the different hadronic light-by-light contributions to muon $g-2$ is done. The dominant contribution is the twice anomalous pseudoscalar exchange diagram. The final result we get is $a_\mu^{\rm light-by-light}= (-9.2\pm3.2 ) \cdot 10^{-10}$. This is between two and three times the expected experimental uncertainty at the forthcoming BNL muon $g-2$ experiment.
Hadronic light-by-light scattering contribution to the muon g-2: an effective field theory approach  [PDF]
M. Knecht,A. Nyffeler,M. Perrottet,E. de Rafael
Physics , 2001, DOI: 10.1103/PhysRevLett.88.071802
Abstract: The hadronic light-by-light contribution to a_{mu}, the anomalous magnetic moment of the muon, is discussed from the point of view of an effective low-energy theory. As an application, the coefficient of the leading logarithm arising from the two-loop graphs involving two anomalous vertices is computed, and found to be positive. This corresponds to a positive sign for the pion-pole contribution to the hadronic light-by-light correction to a_{mu}, and to a sizeable reduction of the discrepancy between the present experimental value of a_{mu} and its theoretical counterpart in the standard model.
Hadronic Light-by-Light Contribution to Muon g-2 in Chiral Perturbation Theory  [PDF]
M. Ramsey-Musolf,Mark B. Wise
Physics , 2002, DOI: 10.1103/PhysRevLett.89.041601
Abstract: We compute the hadronic light-by-light scattering contributions to the muon anomalous magnetic moment, $\amulbl$, in chiral perturbation theory that are enhanced by large logarithms and a factor of $N_C$. They depend on a low-energy constant entering pseudoscalar meson decay into a charged lepton pair. The uncertainty introduced by this constant is $\pm 60\times 10^{-11}$, which is comparable in magnitude to the present uncertainty entering the leading-order vacuum polarization contributions to the anomalous moment. It may be reduced to some extent through an improved measurement of the $\pi^0\to e^+ e^-$ branching ratio. However, the dependence of $\amulbl$ on non-logarithmically enhanced effects cannot be constrained except through the measurement of the anomalous moment itself. The extraction of information on new physics would require a future experimental value for the anomalous moment differing significantly from the 2001 result reported by the E821 collaboration.
Hadronic light-by-light scattering in the muon g-2: a Dyson-Schwinger equation approach  [PDF]
Tobias Goecke,Christian S. Fischer,Richard Williams
Physics , 2010, DOI: 10.1103/PhysRevD.83.094006
Abstract: We determine the hadronic light-by-light scattering contribution to the anomalous magnetic moment of the muon using the framework of Dyson-Schwinger and Bethe-Salpeter equations of QCD. Our result for the pseudoscalar ($\pi^0, \eta, \eta'$) meson exchange diagram is commensurate with previous calculations. In our calculation of the quark loop contribution we improve previous approaches by implementing constraints due to gauge invariance. As a consequence, our value $a_\mu^{\textrm{LBL;quarkloop}} = (136 \pm 59)\times 10^{-11}$ is significantly larger. Taken at face value, this then leads to a revised estimate of the total $a_\mu=116\,591\,891.0(105.0)\times 10^{-11}$.
Comment on the Pion Pole Part of the Light-by-Light Contribution to the Muon $g-2$  [PDF]
Johan Bijnens,Elisabetta Pallante,Joaquim Prades
Physics , 2001, DOI: 10.1016/S0550-3213(02)00074-3
Abstract: We comment on the recent calculations of the pion pole part of the light-by-light contribution to the muon anomalous magnetic moment and we point out where the analysis in our previous work was mistaken.
Status of the Hadronic Light-by-Light Contribution to the Muon Anomalous Magnetic Moment
Bijnens, Johan;Prades, Joaquim
High Energy Physics - Phenomenology , 2007,
Abstract: We review the present status of the hadronic light-by-light contribution to muon g-2 and critically compare recent calculations.
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