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High-precision ϵ-expansions of three-loop master integrals contributing to the electron g–2 in QED
No full textAbstractIn this Letter we calculate at high-precision the Laurent expansions in ϵ=(4−D)/2 of the 17 master integrals which appeared in the analytical calculation of 3-loop QED contribution to the electron g–2, using difference and differential equations. The coefficients of the expansions so obtained are in perfect agreement with all the analytical expressions already known. The values of coefficients not previously known will be used in the high-precision calculation of the 4-loop QED contribution to the electron g–2
The analytic value of a 4-loop sunrise graph in a particular kinematical configuration
No full textThe 4-loop sunrise graph with two massless lines, two lines of equal mass M and a line of mass m, for external invariant timelike and equal to m^2 is considered. We write differential equations in x=m/M for the Master Integrals of the problem, which we Laurent-expand in the regularizing continuous dimension d around d=4, and then solve exactly in x up to order (d-4)^3 included; the result is expressed in terms of Harmonic PolyLogarithms of argument x and maximum weight 7. As a by product, we obtain the x=1 value, expected to be relevant in QED 4-loop static quantities like the electron (g-2). The analytic results were checked by an independent precise numerical calculationThe 4-loop sunrise graph with two massless lines, two lines of equal mass M and a line of mass m, for external invariant timelike and equal to m^2 is considered. We write differential equations in x=m/M for the Master Integrals of the problem, which we Laurent-expand in the regularizing continuous dimension d around d=4, and then solve exactly in x up to order (d-4)^3 included: the result is expressed in terms of Harmonic PolyLogarithms of argument x and maximum weight 7. As a by product, we obtain the x=1 value, expected to be relevant in QED 4-loop static quantities like the electron (g-2). The analytic results were checked by an independent precise numerical calculationThe 4-loop sunrise graph with two massless lines, two lines of equal mass M and a line of mass m , for external invariant timelike and equal to m 2 is considered. We write differential equations in x = m / M for the Master Integrals of the problem, which we Laurent-expand in the regularizing continuous dimension d around d =4, and then solve exactly in x up to order ( d −4) 3 included; the result is expressed in terms of harmonic polylogarithms of argument x and maximum weight 7. As a by product, we obtain the x =1 value, expected to be relevant in QED 4-loop static quantities like the electron ( g −2). The analytic results were checked by an independent precise numerical calculation
Analytic treatment of the two loop equal mass sunrise graph
No full textThe two loop equal mass sunrise graph is considered in the continuous d-dimensional regularisation for arbitrary values of the momentum transfer. After recalling the equivalence of the expansions at d=2 and d=4, the second order differential equation for the scalar Master Integral is expanded in (d-2) and solved by the variation of the constants method of Euler up to first order in (d-2) included. That requires the knowledge of the two independent solutions of the associated homogeneous equation, which are found to be related to the complete elliptic integrals of the first kind of suitable arguments. The behaviour and expansions of all the solutions at all the singular points of the equation are exhaustively discussed and written down explicitly.The two loop equal mass sunrise graph is considered in the continuous d-dimensional regularisation for arbitrary values of the momentum transfer. After recalling the equivalence of the expansions at d=2 and d=4, the second order differential equation for the scalar Master Integral is expanded in (d-2) and solved by the variation of the constants method of Euler up to first order in (d-2) included. That requires the knowledge of the two independent solutions of the associated homogeneous equation, which are found to be related to the complete elliptic integrals of the first kind of suitable arguments. The behaviour and expansions of all the solutions at all the singular points of the equation are exhaustively discussed and written down explicitly.The two loop equal mass sunrise graph is considered in the continuous d -dimensional regularisation for arbitrary values of the momentum transfer. After recalling the equivalence of the expansions at d = 2 and d = 4 , the second order differential equation for the scalar master integral is expanded in ( d − 2 ) and solved by the variation of the constants method of Euler up to first order in ( d − 2 ) included. That requires the knowledge of the two independent solutions of the associated homogeneous equation, which are found to be related to the complete elliptic integrals of the first kind of suitable arguments. The behaviour and expansions of all the solutions at all the singular points of the equation are exhaustively discussed and written down explicitly
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