Reduction with degenerate Gram matrix for one-loop integrals
A bstract An improved PV-reduction (Passarino-Veltman) method for one-loop integrals with auxiliary vector R has been proposed in [ 1 , 2 ]. It has also been shown that the new method is a self-completed method in [ 3 ]. Analytic reduction coefficients can be easily produced by recursion relations i...
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| Published in | The journal of high energy physics Vol. 2022; no. 8; pp. 110 - 46 |
|---|---|
| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Berlin/Heidelberg
Springer Berlin Heidelberg
09.08.2022
Springer Nature B.V SpringerOpen |
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| Online Access | Get full text |
| ISSN | 1029-8479 1126-6708 1127-2236 1029-8479 |
| DOI | 10.1007/JHEP08(2022)110 |
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| Abstract | A
bstract
An improved PV-reduction (Passarino-Veltman) method for one-loop integrals with auxiliary vector
R
has been proposed in [
1
,
2
]. It has also been shown that the new method is a self-completed method in [
3
]. Analytic reduction coefficients can be easily produced by recursion relations in this method, where the Gram determinant appears in denominators. The singularity caused by Gram determinant is a well-known fact and it is important to address these divergences in a given frame. In this paper, we propose a systematical algorithm to deal with this problem in our method. The key idea is that now the master integral of the highest topology will be decomposed into combinations of master integrals of lower topologies. By demanding the cancellation of divergence for obtained general reduction coefficients, we solve decomposition coefficients as a Taylor series of the Gram determinant. Moreover, the same idea can be applied to other kinds of divergences. |
|---|---|
| AbstractList | Abstract An improved PV-reduction (Passarino-Veltman) method for one-loop integrals with auxiliary vector R has been proposed in [1, 2]. It has also been shown that the new method is a self-completed method in [3]. Analytic reduction coefficients can be easily produced by recursion relations in this method, where the Gram determinant appears in denominators. The singularity caused by Gram determinant is a well-known fact and it is important to address these divergences in a given frame. In this paper, we propose a systematical algorithm to deal with this problem in our method. The key idea is that now the master integral of the highest topology will be decomposed into combinations of master integrals of lower topologies. By demanding the cancellation of divergence for obtained general reduction coefficients, we solve decomposition coefficients as a Taylor series of the Gram determinant. Moreover, the same idea can be applied to other kinds of divergences. An improved PV-reduction (Passarino-Veltman) method for one-loop integrals with auxiliary vector R has been proposed in [1, 2]. It has also been shown that the new method is a self-completed method in [3]. Analytic reduction coefficients can be easily produced by recursion relations in this method, where the Gram determinant appears in denominators. The singularity caused by Gram determinant is a well-known fact and it is important to address these divergences in a given frame. In this paper, we propose a systematical algorithm to deal with this problem in our method. The key idea is that now the master integral of the highest topology will be decomposed into combinations of master integrals of lower topologies. By demanding the cancellation of divergence for obtained general reduction coefficients, we solve decomposition coefficients as a Taylor series of the Gram determinant. Moreover, the same idea can be applied to other kinds of divergences. An improved PV-reduction (Passarino-Veltman) method for one-loop integrals with auxiliary vector R has been proposed in [1, 2]. It has also been shown that the new method is a self-completed method in [3]. Analytic reduction coefficients can be easily produced by recursion relations in this method, where the Gram determinant appears in denominators. The singularity caused by Gram determinant is a well-known fact and it is important to address these divergences in a given frame. In this paper, we propose a systematical algorithm to deal with this problem in our method. The key idea is that now the master integral of the highest topology will be decomposed into combinations of master integrals of lower topologies. By demanding the cancellation of divergence for obtained general reduction coefficients, we solve decomposition coefficients as a Taylor series of the Gram determinant. Moreover, the same idea can be applied to other kinds of divergences. A bstract An improved PV-reduction (Passarino-Veltman) method for one-loop integrals with auxiliary vector R has been proposed in [ 1 , 2 ]. It has also been shown that the new method is a self-completed method in [ 3 ]. Analytic reduction coefficients can be easily produced by recursion relations in this method, where the Gram determinant appears in denominators. The singularity caused by Gram determinant is a well-known fact and it is important to address these divergences in a given frame. In this paper, we propose a systematical algorithm to deal with this problem in our method. The key idea is that now the master integral of the highest topology will be decomposed into combinations of master integrals of lower topologies. By demanding the cancellation of divergence for obtained general reduction coefficients, we solve decomposition coefficients as a Taylor series of the Gram determinant. Moreover, the same idea can be applied to other kinds of divergences. |
| ArticleNumber | 110 |
| Author | Li, Tingfei Hu, Chang Feng, Bo Song, Yuekai |
| Author_xml | – sequence: 1 givenname: Bo surname: Feng fullname: Feng, Bo organization: Zhejiang Institute of Modern Physics, Zhejiang University, Beijing Computational Science Research Center, Center of Mathematical Science, Zhejiang University, Peng Huanwu Center for Fundamental Theory – sequence: 2 givenname: Chang surname: Hu fullname: Hu, Chang organization: Hangzhou Institute of Advanced Study, UCAS, University of Chinese Academy of Sciences – sequence: 3 givenname: Tingfei surname: Li fullname: Li, Tingfei email: tfli@zju.edu.cn organization: Zhejiang Institute of Modern Physics, Zhejiang University – sequence: 4 givenname: Yuekai surname: Song fullname: Song, Yuekai organization: Zhejiang Institute of Modern Physics, Zhejiang University |
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| References_xml | – reference: G. ’t Hooft and M.J.G. Veltman, Scalar One Loop Integrals, Nucl. Phys. B153 (1979) 365 [INSPIRE]. – reference: C. Anastasiou, R. Britto, B. Feng, Z. Kunszt and P. Mastrolia, Unitarity cuts and Reduction to master integrals in d dimensions for one-loop amplitudes, JHEP03 (2007) 111 [hep-ph/0612277] [INSPIRE]. – reference: G. Duplancic and B. Nizic, Reduction method for dimensionally regulated one loop N point Feynman integrals, Eur. Phys. J. C35 (2004) 105 [hep-ph/0303184] [INSPIRE]. – reference: BrittoRCachazoFFengBGeneralized unitarity and one-loop amplitudes in N = 4 super-Yang-MillsNucl. Phys. B20057252752005NuPhB.725..275B216429310.1016/j.nuclphysb.2005.07.014[hep-th/0412103] [INSPIRE] – reference: B. Feng and T. Li, PV-Reduction of Sunset Topology with Auxiliary Vector, arXiv:2203.16881 [INSPIRE]. – reference: C. Hu, T. Li and X. Li, One-loop Feynman integral reduction by differential operators, Phys. Rev. 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bstract
An improved PV-reduction (Passarino-Veltman) method for one-loop integrals with auxiliary vector
R
has been proposed in [
1
,
2
]. It has also been... An improved PV-reduction (Passarino-Veltman) method for one-loop integrals with auxiliary vector R has been proposed in [1, 2]. It has also been shown that the... An improved PV-reduction (Passarino-Veltman) method for one-loop integrals with auxiliary vector R has been proposed in [1, 2]. It has also been shown that the... Abstract An improved PV-reduction (Passarino-Veltman) method for one-loop integrals with auxiliary vector R has been proposed in [1, 2]. It has also been shown... |
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| SubjectTerms | Algebra Algorithms Classical and Quantum Gravitation Coefficients Decomposition Divergence Elementary Particles High energy physics Integrals Physics Physics and Astronomy Quantum Field Theories Quantum Field Theory Quantum Physics Reduction Regular Article - Theoretical Physics Relativity Theory Renormalization and Regularization Scattering Amplitudes Singularity (mathematics) String Theory Taylor series Topology |
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| Title | Reduction with degenerate Gram matrix for one-loop integrals |
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