1,721,030 research outputs found
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
The analytic value of the sunrise self-mass with two equal masses and the external invariant equal to the third squared mass
No full text
QED vertex form factors at two loops
No full textWe present the closed analytic expression of the form factors of the two-loop QED vertex amplitude for on-shell electrons of finite mass m and arbitrary momentum transfer S = -Q(2). The calculation is carried out within the continuous D-dimensional regularization scheme, with a single continuous parameter D, the dimension of the space-time, which regularizes at the same time ultraviolet (UV) and infrared (IR) divergences. The results are expressed in terms of 1-dimensional harmonic polylogarithms of maximum weight 4. (C) 2003 Elsevier B.V. All rights reserved
Master Integrals for the 2-loop QCD virtual corrections to the Forward-Backward Asymmetry
No full textWe present the Master Integrals needed for the calculation of the two-loop QCD corrections to the forward-backward asymmetry of a quark-antiquark pair produced in electron-positron annihilation events. The abelian diagrams entering in the evaluation of the vector form factors were calculated in a previous paper. We consider here the non-abelian diagrams and the diagrams entering in the computation of the axial form factors, for arbitrary space-like momentum transfer Q^2 and finite heavy quark mass m. Both the UV and IR divergences are regularized in the continuous D-dimensional scheme. The Master Integrals are Laurent-expanded around D=4 and evaluated by the differential equation method; the coefficients of the expansions are expressed as 1-dimensional harmonic polylogarithms of maximum weight 4.We present the Master Integrals needed for the calculation of the two-loop QCD corrections to the forward-backward asymmetry of a quark-antiquark pair produced in electron-positron annihilation events. The abelian diagrams entering in the evaluation of the vector form factors were calculated in a previous paper. We consider here the non-abelian diagrams and the diagrams entering in the computation of the axial form factors, for arbitrary space-like momentum transfer Q^2 and finite heavy quark mass m. Both the UV and IR divergences are regularized in the continuous D-dimensional scheme. The Master Integrals are Laurent-expanded around D=4 and evaluated by the differential equation method: the coefficients of the expansions are expressed as 1-dimensional harmonic polylogarithms of maximum weight 4.We present the master integrals needed for the calculation of the two-loop QCD corrections to the forward–backward asymmetry of a quark–antiquark pair produced in electron–positron annihilation events. The Abelian diagrams entering in the evaluation of the vector form factors were calculated in a previous paper. We consider here the non-Abelian diagrams and the diagrams entering in the computation of the axial form factors, for arbitrary space-like momentum transfer Q 2 and finite heavy quark mass m . Both the UV and IR divergences are regularized in the continuous D -dimensional scheme. The master integrals are Laurent-expanded around D =4 and evaluated by the differential equation method; the coefficients of the expansions are expressed as 1-dimensional harmonic polylogarithms of maximum weight 4
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
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