A Systematization for One-Loop 4D Feynman Integrals-Different Species of Massive Fields

Affiliation(s)

Departamento de Fsica, Universidade Federal de Santa Maria, Santa Maria, Brazil.

Departamento de Ciências Exatas, Universidade Federal de Lavras, Lavras, Brazil.

Departamento de Fsica, Universidade Federal de Santa Maria, Santa Maria, Brazil.

Departamento de Ciências Exatas, Universidade Federal de Lavras, Lavras, Brazil.

ABSTRACT

A systematization for the manipulations and calculations involving divergent (or not) Feynman integrals, typical of the one loop perturbative solutions of Quantum Field Theory, is proposed. A previous work on the same issue is generalized to treat theories and models having different species of massive fields. An improvement on the strategy is adopted so that no regularization needs to be used. The final results produced, however, can be converted into the ones of reasonable regularizations, especially those belonging to the dimensional regularization (in situations where the method applies). Through an adequate interpretation of the Feynman rules and a convenient representation for involved propagators, the finite and divergent parts are separated before the introduction of the integration in the loop momentum. Only the finite integrals obtained are in fact integrated. The divergent content of the amplitudes are written as a combination of standard mathematical object which are never really integrated. Only very general scale properties of such objects are used. The finite parts, on the other hand, are written in terms of basic functions conveniently introduced. The scale properties of such functions relate them to a well defined way to the basic divergent objects providing simple and transparent connection between both parts in the assintotic regime. All the arbitrariness involved in this type of calculations are preserved in the intermediary steps allowing the identification of universal properties for the divergent integrals, which are required for the maintenance of fundamental symmetries like translational invariance and scale independence in the perturbative amplitudes. Once these consistency relations are imposed no other symmetry is violated in perturbative calculations neither ambiguous terms survive at any theory or model formulated at any space-time dimension including nonrenormalizable cases. Representative examples of perturbative amplitudes involving different species of massive fermions are considered as examples. The referred amplitudes are calculated in detail within the context of the presented strategy (and systematization) and their relations among other Green functions are explicitly verified. At the end a generalization for the finite functions is presented.

A systematization for the manipulations and calculations involving divergent (or not) Feynman integrals, typical of the one loop perturbative solutions of Quantum Field Theory, is proposed. A previous work on the same issue is generalized to treat theories and models having different species of massive fields. An improvement on the strategy is adopted so that no regularization needs to be used. The final results produced, however, can be converted into the ones of reasonable regularizations, especially those belonging to the dimensional regularization (in situations where the method applies). Through an adequate interpretation of the Feynman rules and a convenient representation for involved propagators, the finite and divergent parts are separated before the introduction of the integration in the loop momentum. Only the finite integrals obtained are in fact integrated. The divergent content of the amplitudes are written as a combination of standard mathematical object which are never really integrated. Only very general scale properties of such objects are used. The finite parts, on the other hand, are written in terms of basic functions conveniently introduced. The scale properties of such functions relate them to a well defined way to the basic divergent objects providing simple and transparent connection between both parts in the assintotic regime. All the arbitrariness involved in this type of calculations are preserved in the intermediary steps allowing the identification of universal properties for the divergent integrals, which are required for the maintenance of fundamental symmetries like translational invariance and scale independence in the perturbative amplitudes. Once these consistency relations are imposed no other symmetry is violated in perturbative calculations neither ambiguous terms survive at any theory or model formulated at any space-time dimension including nonrenormalizable cases. Representative examples of perturbative amplitudes involving different species of massive fermions are considered as examples. The referred amplitudes are calculated in detail within the context of the presented strategy (and systematization) and their relations among other Green functions are explicitly verified. At the end a generalization for the finite functions is presented.

Cite this paper

O. Battistel and G. Dallabona, "A Systematization for One-Loop 4D Feynman Integrals-Different Species of Massive Fields,"*Journal of Modern Physics*, Vol. 3 No. 10, 2012, pp. 1408-1449. doi: 10.4236/jmp.2012.310178.

O. Battistel and G. Dallabona, "A Systematization for One-Loop 4D Feynman Integrals-Different Species of Massive Fields,"

References

[1] O. A. Battistel and G. Dallabona, “Scale Ambiguities in Perturbative Calculations and the Value for the Radiatively Induced Chern-Simons Term in Extended QED,” Physical Review D, Vol. 72, No. 4, 2005, Article ID: 045009.

[2] O. A. Battistel and G. Dallabona, “A Systematization for One-Loop 4D Feynman Integrals,” The European Physical Journal C, Vol. 45, No. 3, 2006, pp. 721-743. doi:10.1140/epjc/s2005-02437-0

[3] G. Passarino and M. Veltman, “One Loop Corrections for e^{+}e^{-} Annihilation into μ^{+} μ^{-} in the Weinberg Model,” Nuclear Physics B, Vol. 160, No. 1, 1979, pp. 151-207.
doi:10.1016/0550-3213(79)90234-7

[4] W. L. van Neerven and J. A. Vermaseren, “Large Loop Integrals,” Physics Letters B, Vol. 137, No. 3-4, 1984, pp. 241-244.

[5] G. J. Oldenborgh and J. A. Vermaseren, “New Algorithms for One Loop Integrals,” Zeitschrift fur Physik C Particles and Fields, Vol. 46, No. 3, 1990, pp. 425-437.

[6] A. I. Davydychev, “A Simple Formula for Reducing Feynman Diagrams to Scalar Integrals,” Physics Letters B, Vol. 263, No. 1, 1991, pp. 107-111.

[7] Z. Bern, L. J. Dixon and D. A. Kosower, “Dimensionally Regulated One Loop Integrals,” Physics Letters B, Vol. 302, No. 2-3, 1993, pp. 299-308.

[8] O. V. Tasarov, “Connection between Feynman Integrals Having Different Values of the Space-Time Dimension,” Physical Review D, Vol. 54, No. 10, 1996, pp. 6479-6490.

[9] R. G. Stuart, “Algebraic Reduction of One Loop Feynman Diagrams to Scalar Integrals,” Computer Physics Communications, Vol. 48, No. 3, 1988, pp. 367-389. doi:10.1016/0010-4655(88)90202-0

[10] J. Campbell, E. Glover and D. Miller, “One Loop Tensor Integrals in Dimensional Regularization,” Nuclear Physics B, Vol. 498, No. 1-2, 1997, pp. 397-442. doi:10.1016/S0550-3213(97)00268-X

[11] G. Devaraj and R. G. Stuart, “Reduction of One Loop Tensor Form-Factors to Scalar Integrals: A General Scheme,” Nuclear Physics B, Vol. 519, No. 1-2, 1998, pp. 483-513. doi:10.1016/S0550-3213(98)00035-2

[12] J. Fleischer, F. Jegerlehner and O. V. Tasarov, “Algebraic Reduction of One Loop Feynman Graph Amplitudes,” Nuclear Physics B, Vol. 566, No. 1-2, 2000, pp. 423-440. doi:10.1016/S0550-3213(99)00678-1

[13] G.’t Hooft and M. Veltman, “Scalar One Loop Integrals,” Nuclear Physics B, Vol. 153, 1979, pp. 365-401. doi:10.1016/0550-3213(79)90605-9

[14] T. Binoth, J. P. Guillet and G. Heinrich, “Reduction Formalism for Dimensionally Regulated One-Loop N-Point Integrals,” Nuclear Physics B, Vol. 572, No. 1-2, 2000, pp. 361-386. doi:10.1016/S0550-3213(00)00040-7

[15] T. Binoth, J. P. Guillet, G. Heinrich and C. Schubert, “Calculation of One Loop Hexagon Amplitudes in the Yukawa Model,” Nuclear Physics B, Vol. 615, No. 1-3, 2001, pp. 385-401. doi:10.1016/S0550-3213(01)00436-9

[16] A. Denner and S. Dittmaier, “Reduction of One Loop Tensor Five Point Integrals,” Nuclear Physics B, Vol. 658, No. 1-2, 2003, pp. 175-202. doi:10.1016/S0550-3213(03)00184-6

[17] G. Duplancic and B. Nizic, “Reduction Method for Dimensionally Regulated One Loop N Point Feynman Integrals,” European Physical Journal, Vol. 35, 2004, pp. 105-118.

[18] G. Duplancic and B. Nizic, “Dimensionally Regulated One Loop Box Scalar Integrals with Massless Internal Lines,” European Physical Journal, Vol. 20, 2001, pp. 357-370.

[19] G. Duplancic and B. Nizic, “IR Finite One Loop Box Scalar Integral with Massless Internal Lines,” European Physical Journal, Vol. 24, 2002, pp. 385-391.

[20] F. del Aguila and R. Pittau, “Recursive Numerical Calculus of One-Loop Tensor Integrals,” Journal of High Energy Physics, Vol. 7, 2004, p. 17.

[21] W. T. Giele and E. W. N. Glover, “A Calculational Formalism for One Loop Integrals,” Journal of High Energy Physics, Vol. 8, 2004, p. 29.

[22] R. Britto and B. Feng, “Integral Coefficients for One-Loop Amplitudes,” Journal of High Energy Physics, Vol. 2, 2008, p. 95.

[23] A. Denner and S. Dittmaier, “Reduction Schemes for One-Loop Tensor Integrals,” Nuclear Physics B, Vol. 734, No. 1-2, 2006, pp. 62-115. doi:10.1016/j.nuclphysb.2005.11.007

[24] T. Binoth, J. Ph. Guillet, G. Heinrich, E. Pilon and C. Schubert, “An Algebraic/Numerical Formalism for One-Loop Multi-Leg Amplitudes,” Journal of High Energy Physics, Vol. 10, 2005, p. 15.

[25] Y. Kurihara, “Dimensionally Regularized One-Loop Tensor-Integrals with Massless Internal Particles,” European Physical Journal, Vol. 45, No. 2, 2006, pp. 427-444. doi:10.1140/epjc/s2005-02428-1

[26] C. Anastasiou, E. W. Nigel Glover and C. Oleari, “Scalar One Loop Integrals Using the Negative Dimension Approach,” Nuclear Physics B, Vol. 572, No. 1-2, 2000, pp. 307-360. doi:10.1016/S0550-3213(99)00637-9

[27] P. Mastrolia, G. Ossola, C. G. Papadopoulos and R. Pittau, “Optimizing the Reduction of One-Loop Amplitudes,” Journal of High Energy Physics, Vol. 6, 2008, p. 30.

[28] R. Keith Ellis and G. Zanderighi, “Scalar One-Loop Integrals for QCD,” Journal of High Energy Physics, Vol. 2, 2008, p. 2.

[29] G. Ossola, C. G. Papadopoulos and R. Pittau, “On the Rational Terms of the One-Loop Amplitudes,” Journal of High Energy Physics, Vol. 5, 2008, p. 4.

[30] O. A. Battistel, “Uma Nova Estrategia Para Manipula?oes e Cálculos Envolvendo Divergências em T.Q.C.,” Ph.D. Thesis, Universidade Federal de Minas Gerais, Belo Horizonte, 1999.

[31] Y. Sun and H.-R. Chang, “One Loop Integrals Reduction,” 2012.

[1] O. A. Battistel and G. Dallabona, “Scale Ambiguities in Perturbative Calculations and the Value for the Radiatively Induced Chern-Simons Term in Extended QED,” Physical Review D, Vol. 72, No. 4, 2005, Article ID: 045009.

[2] O. A. Battistel and G. Dallabona, “A Systematization for One-Loop 4D Feynman Integrals,” The European Physical Journal C, Vol. 45, No. 3, 2006, pp. 721-743. doi:10.1140/epjc/s2005-02437-0

[3] G. Passarino and M. Veltman, “One Loop Corrections for e

[4] W. L. van Neerven and J. A. Vermaseren, “Large Loop Integrals,” Physics Letters B, Vol. 137, No. 3-4, 1984, pp. 241-244.

[5] G. J. Oldenborgh and J. A. Vermaseren, “New Algorithms for One Loop Integrals,” Zeitschrift fur Physik C Particles and Fields, Vol. 46, No. 3, 1990, pp. 425-437.

[6] A. I. Davydychev, “A Simple Formula for Reducing Feynman Diagrams to Scalar Integrals,” Physics Letters B, Vol. 263, No. 1, 1991, pp. 107-111.

[7] Z. Bern, L. J. Dixon and D. A. Kosower, “Dimensionally Regulated One Loop Integrals,” Physics Letters B, Vol. 302, No. 2-3, 1993, pp. 299-308.

[8] O. V. Tasarov, “Connection between Feynman Integrals Having Different Values of the Space-Time Dimension,” Physical Review D, Vol. 54, No. 10, 1996, pp. 6479-6490.

[9] R. G. Stuart, “Algebraic Reduction of One Loop Feynman Diagrams to Scalar Integrals,” Computer Physics Communications, Vol. 48, No. 3, 1988, pp. 367-389. doi:10.1016/0010-4655(88)90202-0

[10] J. Campbell, E. Glover and D. Miller, “One Loop Tensor Integrals in Dimensional Regularization,” Nuclear Physics B, Vol. 498, No. 1-2, 1997, pp. 397-442. doi:10.1016/S0550-3213(97)00268-X

[11] G. Devaraj and R. G. Stuart, “Reduction of One Loop Tensor Form-Factors to Scalar Integrals: A General Scheme,” Nuclear Physics B, Vol. 519, No. 1-2, 1998, pp. 483-513. doi:10.1016/S0550-3213(98)00035-2

[12] J. Fleischer, F. Jegerlehner and O. V. Tasarov, “Algebraic Reduction of One Loop Feynman Graph Amplitudes,” Nuclear Physics B, Vol. 566, No. 1-2, 2000, pp. 423-440. doi:10.1016/S0550-3213(99)00678-1

[13] G.’t Hooft and M. Veltman, “Scalar One Loop Integrals,” Nuclear Physics B, Vol. 153, 1979, pp. 365-401. doi:10.1016/0550-3213(79)90605-9

[14] T. Binoth, J. P. Guillet and G. Heinrich, “Reduction Formalism for Dimensionally Regulated One-Loop N-Point Integrals,” Nuclear Physics B, Vol. 572, No. 1-2, 2000, pp. 361-386. doi:10.1016/S0550-3213(00)00040-7

[15] T. Binoth, J. P. Guillet, G. Heinrich and C. Schubert, “Calculation of One Loop Hexagon Amplitudes in the Yukawa Model,” Nuclear Physics B, Vol. 615, No. 1-3, 2001, pp. 385-401. doi:10.1016/S0550-3213(01)00436-9

[16] A. Denner and S. Dittmaier, “Reduction of One Loop Tensor Five Point Integrals,” Nuclear Physics B, Vol. 658, No. 1-2, 2003, pp. 175-202. doi:10.1016/S0550-3213(03)00184-6

[17] G. Duplancic and B. Nizic, “Reduction Method for Dimensionally Regulated One Loop N Point Feynman Integrals,” European Physical Journal, Vol. 35, 2004, pp. 105-118.

[18] G. Duplancic and B. Nizic, “Dimensionally Regulated One Loop Box Scalar Integrals with Massless Internal Lines,” European Physical Journal, Vol. 20, 2001, pp. 357-370.

[19] G. Duplancic and B. Nizic, “IR Finite One Loop Box Scalar Integral with Massless Internal Lines,” European Physical Journal, Vol. 24, 2002, pp. 385-391.

[20] F. del Aguila and R. Pittau, “Recursive Numerical Calculus of One-Loop Tensor Integrals,” Journal of High Energy Physics, Vol. 7, 2004, p. 17.

[21] W. T. Giele and E. W. N. Glover, “A Calculational Formalism for One Loop Integrals,” Journal of High Energy Physics, Vol. 8, 2004, p. 29.

[22] R. Britto and B. Feng, “Integral Coefficients for One-Loop Amplitudes,” Journal of High Energy Physics, Vol. 2, 2008, p. 95.

[23] A. Denner and S. Dittmaier, “Reduction Schemes for One-Loop Tensor Integrals,” Nuclear Physics B, Vol. 734, No. 1-2, 2006, pp. 62-115. doi:10.1016/j.nuclphysb.2005.11.007

[24] T. Binoth, J. Ph. Guillet, G. Heinrich, E. Pilon and C. Schubert, “An Algebraic/Numerical Formalism for One-Loop Multi-Leg Amplitudes,” Journal of High Energy Physics, Vol. 10, 2005, p. 15.

[25] Y. Kurihara, “Dimensionally Regularized One-Loop Tensor-Integrals with Massless Internal Particles,” European Physical Journal, Vol. 45, No. 2, 2006, pp. 427-444. doi:10.1140/epjc/s2005-02428-1

[26] C. Anastasiou, E. W. Nigel Glover and C. Oleari, “Scalar One Loop Integrals Using the Negative Dimension Approach,” Nuclear Physics B, Vol. 572, No. 1-2, 2000, pp. 307-360. doi:10.1016/S0550-3213(99)00637-9

[27] P. Mastrolia, G. Ossola, C. G. Papadopoulos and R. Pittau, “Optimizing the Reduction of One-Loop Amplitudes,” Journal of High Energy Physics, Vol. 6, 2008, p. 30.

[28] R. Keith Ellis and G. Zanderighi, “Scalar One-Loop Integrals for QCD,” Journal of High Energy Physics, Vol. 2, 2008, p. 2.

[29] G. Ossola, C. G. Papadopoulos and R. Pittau, “On the Rational Terms of the One-Loop Amplitudes,” Journal of High Energy Physics, Vol. 5, 2008, p. 4.

[30] O. A. Battistel, “Uma Nova Estrategia Para Manipula?oes e Cálculos Envolvendo Divergências em T.Q.C.,” Ph.D. Thesis, Universidade Federal de Minas Gerais, Belo Horizonte, 1999.

[31] Y. Sun and H.-R. Chang, “One Loop Integrals Reduction,” 2012.