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Maximal Unitarity at Two Loops: A New Method for Computing Two-Loop Scattering Amplitudes
Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Theoretical Physics. (Institut de Physique Théorique, CEA-Saclay, F-91191 Gif-sur-Yvette cedex, France)
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The study of scattering amplitudes beyond one loop is necessary for precision phenomenology for the Large Hadron Collider and may also provide deeper insights into the theoretical foundations of quantum field theory. In this thesis we develop a new method for computing two-loop amplitudes, based on unitarity rather than Feynman diagrams. In this approach, the two-loop amplitude is first expanded in a linearly independent basis of integrals. The process dependence thereby resides in the coefficients of the integrals. These expansion coefficients are then the object of calculation.

Our main results include explicit formulas for a subset of the integral coefficients, expressing them as products of tree-level amplitudes integrated over specific contours in the complex plane. We give a general selection principle for determining these contours. This principle is then applied to obtain the coefficients of integrals with the topology of a double box. We show that, for four-particle scattering, each double-box integral in the two-loop basis is associated with a uniquely defined complex contour, referred to as its master contour. We provide a classification of the solutions to setting all propagators of the general double-box integral on-shell. Depending on the number of external momenta at the vertices of the graph, these solutions are given as a chain of pointwise intersecting Riemann spheres, or a torus. This classification is needed to define master contours for amplitudes with arbitrary multiplicities.

We point out that a basis of two-loop integrals with as many infrared finite elements as possible allows substantial technical simplications, in terms of obtaining the coefficients of the integrals, as well as for the analytic evaluation of the integrals themselves. We compute two such integrals at four points, obtaining remarkably compact expressions. Finally, we provide a check on a recently developed recursion relation for the all-loop integrand of the amplitudes of N=4 supersymmetric Yang-Mills theory, examining the two-loop six-gluon MHV amplitude and finding agreement. The validity of the approach to two-loop amplitudes developed in this thesis extends to all four-dimensional gauge theories, in particular QCD. The approach is suited for obtaining compact analytical expressions as well as for numerical implementations.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2012. , 135 p.
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 952
Keyword [en]
Amplitudes, NNLO calculations, Quantum Chromodynamics, Unitarity
National Category
Subatomic Physics
Research subject
Physics with specialization in Elementary Particle Physics
Identifiers
URN: urn:nbn:se:uu:diva-179203ISBN: 978-91-554-8423-1 (print)OAI: oai:DiVA.org:uu-179203DiVA: diva2:543742
Public defence
2012-09-21, Å2001, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 09:15 (English)
Opponent
Supervisors
Available from: 2012-08-31 Created: 2012-08-09 Last updated: 2013-01-22Bibliographically approved
List of papers
1. Two-Loop Maximal Unitarity with External Masses
Open this publication in new window or tab >>Two-Loop Maximal Unitarity with External Masses
2013 (English)In: Physical Review D, ISSN 1550-7998, E-ISSN 1550-2368, Vol. 87, no 2, 025030- p.Article in journal (Refereed) Published
Abstract [en]

We extend the maximal unitarity method at two loops to double-box basis integrals with up to three external massive legs. We use consistency equations based on the requirement that integrals of total derivatives vanish. We obtain unique formulae for the coefficients of the master double-box integrals. These formulae can be used either analytically or numerically.

Keyword
Amplitudes, Form factors, NNLO calculations, QCD
National Category
Subatomic Physics
Research subject
Physics with specialization in Elementary Particle Physics
Identifiers
urn:nbn:se:uu:diva-179184 (URN)10.1103/PhysRevD.87.025030 (DOI)000314336700003 ()
Available from: 2012-08-09 Created: 2012-08-09 Last updated: 2017-12-07Bibliographically approved
2. Global Poles of the Two-Loop Six-Point N=4 SYM integrand
Open this publication in new window or tab >>Global Poles of the Two-Loop Six-Point N=4 SYM integrand
2012 (English)In: Physical Review D, ISSN 1550-7998, E-ISSN 1550-2368, Vol. 86, no 8, 085032- p.Article in journal (Refereed) Published
Abstract [en]

Recently, a recursion relation has been developed, generating the four-dimensional integrand of the amplitudes of N=4 supersymmetric Yang-Mills theory for any number of loops and legs. In this paper, I provide a comparison of the prediction for the two-loop six-point maximally helicity violating (MHV) integrand against the result obtained by use of the leading singularity method. The comparison is performed numerically for a large number of randomly selected momenta and in all cases finds agreement between the two results to high numerical accuracy.

Keyword
Amplitudes, NNLO calculations
National Category
Subatomic Physics
Research subject
Physics with specialization in Elementary Particle Physics
Identifiers
urn:nbn:se:uu:diva-179165 (URN)10.1103/PhysRevD.86.085032 (DOI)000309999700006 ()
Available from: 2012-08-08 Created: 2012-08-08 Last updated: 2017-12-07Bibliographically approved
3. Uniqueness of Two-Loop Master Contours
Open this publication in new window or tab >>Uniqueness of Two-Loop Master Contours
2012 (English)In: Journal of High Energy Physics (JHEP), ISSN 1126-6708, E-ISSN 1029-8479, no 10, 026- p.Article in journal (Refereed) Published
Abstract [en]

Generalized-unitarity calculations of two-loop amplitudes are performed by expanding the amplitude in a basis of master integrals and then determining the coefficients by taking a number of generalized cuts. In this paper, we present a complete classification of the solutions to the maximal cut of integrals with the double-box topology. The ideas presented here are expected to be relevant for all two-loop topologies as well. We find that these maximal-cut solutions are naturally associated with Riemann surfaces whose topology is determined by the number of states at the vertices of the double-box graph. In the case of four massless external momenta we find that, once the geometry of these Riemann surfaces is properly understood, there are uniquely defined master contours producing the coefficients of the double-box integrals in the basis decomposition of the two-loop amplitude. This is in perfect analogy with the situation in one-loop generalized unitarity. In addition, we point out that the chiral integrals recently introduced by Arkani-Hamed et al. can be used as master integrals for the double-box contributions to the two-loop amplitudes in any gauge theory. The infrared finiteness of these integrals allow for their coefficients as well as their integrated expressions to be evaluated in strictly four dimensions, providing significant technical simplification. We evaluate these integrals at four points and obtain remarkably compact results.

Keyword
Scattering Amplitudes, Gauge Symmetry
National Category
Subatomic Physics
Research subject
Physics with specialization in Elementary Particle Physics
Identifiers
urn:nbn:se:uu:diva-179169 (URN)10.1007/JHEP10(2012)026 (DOI)000310851800041 ()
Available from: 2012-08-08 Created: 2012-08-08 Last updated: 2017-12-07Bibliographically approved
4. Maximal unitarity at two loops
Open this publication in new window or tab >>Maximal unitarity at two loops
2012 (English)In: Physical Review D, ISSN 1550-7998, E-ISSN 1550-2368, Vol. 85, no 4, 045017- p.Article in journal (Refereed) Published
Abstract [en]

We show how to compute the coefficients of the double-box basis integrals in a massless four-point amplitude in terms of tree amplitudes. We show how to choose suitable multidimensional contours for performing the required cuts, and derive consistency equations from the requirement that integrals of total derivatives vanish. Our formulas for the coefficients can be used either analytically or numerically.

National Category
Physical Sciences
Identifiers
urn:nbn:se:uu:diva-170345 (URN)10.1103/PhysRevD.85.045017 (DOI)000300241200004 ()
Available from: 2012-03-12 Created: 2012-03-12 Last updated: 2017-12-07Bibliographically approved

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