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Topics in perturbation theory: From IBP identities to integrands
Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Theoretical Physics.ORCID iD: 0000-0002-1601-4218
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis we present different topics in perturbation theory. We start by introducing the method of integration by parts identities, which reduces a generic Feynman integral to a linear combination of a finite basis of master integrals. In our analysis we make use of the Baikov representation as this form gives a nice framework for generating efficiently the identities needed to reduce integrals. In the second part of the thesis we briefly explain recent developments in the integration of Feynman integrals and present a method to bootstrap the value of p-integrals using constraints from certain limits of conformal integrals. We introduce also another method to obtain p-integrals at l-loops by cutting vacuum diagrams at l+1-loops. In the last part of the thesis we present recent developments in N=4 SYM to compute structure constants. We use perturbation theory to obtain new results that can be tested against this new conjecture. Moreover we use integrability based methods to constrain correlation function of protected operators.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2019. , p. 64
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 1805
Keywords [en]
Perturbation theory, Feynman Integrals, Integrable field theories, Correlation functions.
National Category
Subatomic Physics
Research subject
Theoretical Physics
Identifiers
URN: urn:nbn:se:uu:diva-381810ISBN: 978-91-513-0648-3 (print)OAI: oai:DiVA.org:uu-381810DiVA, id: diva2:1304836
Public defence
2019-06-07, Room Å4001, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
Opponent
Supervisors
Available from: 2019-05-17 Created: 2019-04-15 Last updated: 2019-06-17
List of papers
1. AZURITE: An algebraic geometry based package for finding bases of loop integrals
Open this publication in new window or tab >>AZURITE: An algebraic geometry based package for finding bases of loop integrals
2017 (English)In: Computer Physics Communications, ISSN 0010-4655, E-ISSN 1879-2944, Vol. 221, p. 203-215Article in journal (Refereed) Published
Abstract [en]

For any given Feynman graph, the set of integrals with all possible powers of the propagators spans a vector space of finite dimension. We introduce the package AZURITE (A ZURich-bred method for finding master InTEgrals), which efficiently finds a basis of this vector space. It constructs the needed integration by-parts (IBP) identities on a set of generalized-unitarity cuts. It is based on syzygy computations and analyses of the symmetries of the involved Feynman diagrams and is powered by the computer algebra systems SINGULAR and MATHEMATICA. It can moreover analytically calculate the part of the IBP identities that is supported on the cuts. In some cases, the basis obtained by AZURITE may be slightly overcomplete.

Program summary

Program Title: AZURITE

Licensing provisions: GNU General Public License (GPL)

Programming language: Wolfram MATHEMATICA version 10.0 or higher

Supplementary material: A manual in the form of a MATHEMATICA notebook

Nature of problem: Determination of a basis of the space of loop integrals spanned by a given Feynman diagram and all of its subdiagrams

Solution method: MATHEMATICA implementation.

Keywords
Feynman diagrams, Computational algebraic geometry, Integration-by-parts identities
National Category
Computer Sciences
Identifiers
urn:nbn:se:uu:diva-340677 (URN)10.1016/j.cpc.2017.08.013 (DOI)000413376800015 ()
Funder
EU, FP7, Seventh Framework Programme, 627521Knut and Alice Wallenberg Foundation, 2015-0083
Available from: 2018-02-05 Created: 2018-02-05 Last updated: 2019-04-15Bibliographically approved
2. Konishi OPE coefficient at the five loop order
Open this publication in new window or tab >>Konishi OPE coefficient at the five loop order
2018 (English)In: JHEP, Vol. 11, article id 184Article in journal, Letter (Refereed) Published
Abstract [en]

We use the method of asymptotic expansions to study the OPE limit of a fourpoint function of protected operators in N=4 SYM. We use a new method for evaluating the resulting propagator-type integrals and then extract the OPE coefficient with Konishi at the five loop order.

National Category
Other Physics Topics
Identifiers
urn:nbn:se:uu:diva-371272 (URN)10.1007/JHEP11(2018)184 (DOI)000453291200001 ()
Available from: 2018-12-20 Created: 2018-12-20 Last updated: 2019-04-15Bibliographically approved
3. Complete sets of logarithmic vector fields for integration-by-parts identities of Feynman integrals
Open this publication in new window or tab >>Complete sets of logarithmic vector fields for integration-by-parts identities of Feynman integrals
Show others...
2018 (English)In: Physical Review D: covering particles, fields, gravitation, and cosmology, ISSN 2470-0010, E-ISSN 2470-0029, Vol. 98, no 2, article id 025023Article in journal (Refereed) Published
Abstract [en]

Integration-by-parts identities between loop integrals arise from the vanishing integration of total derivatives in dimensional regularization. Generic choices of total derivatives in the Baikov or parametric representations lead to identities which involve dimension shifts. These dimension shifts can be avoided by imposing a certain constraint on the total derivatives. The solutions of this constraint turn out to be a specific type of syzygies which correspond to logarithmic vector fields along the Gram determinant formed of the independent external and loop momenta. We present an explicit generating set of solutions in Baikov representation, valid for any number of loops and external momenta, obtained from the Laplace expansion of the Gram determinant. We provide a rigorous mathematical proof that this set of solutions is complete. This proof relates the logarithmic vector fields in question to ideals of submaximal minors of the Gram matrix and makes use of classical resolutions of such ideals.

National Category
Astronomy, Astrophysics and Cosmology
Identifiers
urn:nbn:se:uu:diva-361998 (URN)10.1103/PhysRevD.98.025023 (DOI)000439976300007 ()
Funder
EU, Horizon 2020, 725110Knut and Alice Wallenberg Foundation, 2015-0083EU, European Research Council, 648630 IQFT
Available from: 2018-10-05 Created: 2018-10-05 Last updated: 2019-04-15Bibliographically approved
4. Five-loop massless propagator integrals
Open this publication in new window or tab >>Five-loop massless propagator integrals
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We develop a method to obtain ϵ-expansions of massless two-point integrals in position space, based on the constraints implied by symmetries of the asymptotic expansion of conformal four-point integrals. Together with parametric integration, we are able to fix the expansions of 170 genuine five-loop master integrals. In particular, we computed the expansions of all planar master integrals up to transcendental weight 9.

National Category
Other Physics Topics
Research subject
Physics and Astronomy specializing in Theoretical Physics
Identifiers
urn:nbn:se:uu:diva-381808 (URN)
Available from: 2019-04-15 Created: 2019-04-15 Last updated: 2019-04-15
5. Complete integration-by-parts reductions of the non-planar hexagon-box via module intersections
Open this publication in new window or tab >>Complete integration-by-parts reductions of the non-planar hexagon-box via module intersections
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2018 (English)In: Journal of High Energy Physics (JHEP), ISSN 1126-6708, E-ISSN 1029-8479, no 9, article id 24Article in journal (Refereed) Published
Abstract [en]

We present the powerful module-intersection integration-by-parts (IBP) method, suitable for multi-loop and multi-scale Feynman integral reduction. Utilizing modern computational algebraic geometry techniques, this new method successfully trims traditional IBP systems dramatically to much simpler integral-relation systems on unitarity cuts. We demonstrate the power of this method by explicitly carrying out the complete analytic reduction of two-loop five-point non-planar hexagon-box integrals, with degree-four numerators, to a basis of 73 master integrals.

Place, publisher, year, edition, pages
SPRINGER, 2018
Keywords
Differential and Algebraic Geometry, Scattering Amplitudes, Perturbative QCD
National Category
Subatomic Physics
Identifiers
urn:nbn:se:uu:diva-364245 (URN)10.1007/JHEP09(2018)024 (DOI)000443818800006 ()
Funder
EU, Horizon 2020, 725110Knut and Alice Wallenberg Foundation, 2015-0083EU, European Research Council, 648630 IQFTGerman Research Foundation (DFG), SFB-TRR 195
Available from: 2018-10-29 Created: 2018-10-29 Last updated: 2019-04-15Bibliographically approved
6. All five-loop planar four-point functions of half-BPS operators in N = 4 SYM
Open this publication in new window or tab >>All five-loop planar four-point functions of half-BPS operators in N = 4 SYM
2018 (English)In: Journal of High Energy Physics (JHEP), ISSN 1126-6708, E-ISSN 1029-8479, Vol. 11, p. 1-26, article id 69Article in journal (Refereed) Published
Abstract [en]

We obtain all planar four-point correlators of half-BPS operators in N = 4 SYM up to five loops. The ansatz for the integrand is fixed partially by imposing lightcone OPE relations between different correlators. We then fix the integrated correlators by comparing their asymptotic expansions with simple data obtained from integrability. We extract OPE coefficients and find a prediction for the triple wrapping correction of the hexagon form factors, which contributes already at the five-loop order.

Keywords
Conformal Field Theory, Integrable Field Theories, Supersymmetric Gauge Theory
National Category
Subatomic Physics
Identifiers
urn:nbn:se:uu:diva-371300 (URN)10.1007/JHEP11(2018)069 (DOI)000449817300008 ()
Funder
Knut and Alice Wallenberg Foundation, 2015-0083
Note

ArXiv ePrint: 1809.00551

Available from: 2018-12-20 Created: 2018-12-20 Last updated: 2019-04-15Bibliographically approved

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