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Finite system size effects on the effective coupling in scalar quantum field theory
Findings from the Large Hadron Collider (LHC) provide evidence that quark-gluon plasma (QGP) formation , traditionally associated with large nucleus-nucleus collisions, also emerges in high multiplicity proton-proton (p + p) and proton-nucleus (p + A) collisions. It has also been shown that azimutha...
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Department of Mathematics and Applied Mathematics
2024
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| _version_ | 1867613457147232256 |
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| access_status_str | Open Access |
| author | Du Plessis, Jean |
| author2 | Shock, Jonathan |
| author_browse | Du Plessis, Jean Shock, Jonathan |
| author_facet | Shock, Jonathan Du Plessis, Jean |
| author_sort | Du Plessis, Jean |
| collection | Thesis |
| description | Findings from the Large Hadron Collider (LHC) provide evidence that quark-gluon plasma (QGP) formation , traditionally associated with large nucleus-nucleus collisions, also emerges in high multiplicity proton-proton (p + p) and proton-nucleus (p + A) collisions. It has also been shown that azimuthal correlations of low transverse momentum outgoing particles from these high multiplicity p + p and p + A collisions can be modeled well using nearly inviscid relativistic hydrodynamics. One would expect the equation of state (EOS) used in such hydrodynamic simulations to receive significant contributions in systems of size characteristic of p+p collisions. Indeed it has been shown that free scalar thermal field theory receives ∼ 40% corrections to the usual thermodynamic quantities when confined using Dirichlet boundary conditions with characteristic lengths set by central p + p collisions. Furthermore, the potential importance of asymmetric system sizes has also been highlighted in quenched lattice QCD calculations using periodic boundary conditions. In order to analyse the finite system size corrections to the QCD EOS, we need to explore the finite system size corrections to the running of the QCD coupling. This thesis provides a first step toward such a calculation. We consider massive scalar ϕ 4 theory in a spacetime with periodic boundary conditions. We allow characteristic lengths to be asymmetric, or even infinite. We first recount the infinite volume NLO 2 → 2 scattering amplitude calculation using dimensional regularisation, and then introduce and employ denominator regularisation. We then perform the non-trivial calculation of the NLO 2 → 2 scattering amplitude in our compactified spacetime. This requires the derivation of an new analytic continuation of the generalised Epstein zeta function after employing denominator regularisation in order to isolate the UV divergence. Denominator regularisation is necessary, since the usual techniques in dimensional regularisation no longer applies when we allow asymmetric characteristic lengths. We confirm that taking the limit of infinite characteristic lengths yields the usual infinite space-time result. We then perform another non-trivial self-consistency check by verifying that the NLO 2 → 2 Page ii scattering amplitude satisfies unitarity in the form of the optical theorem, regardless of the number of compactified dimensions. In order to show this we derived a generalisation of a formula originally proposed by Hardy and Ramanujan, and interpret its analytic continuation in the context of Abel summation. We numerically explore the scattering amplitude in some special cases, but find the s-channel of the 2 compactified dimension case to be numerically ill-behaved. We then derive and employ a dispersion relation in order to numerically explore the s-channel. Using the Callan-Symanzik equation to derive the beta function, we find it insensitive to finite system effects, which only modifies the IR. We then perform a geometric resummation of bubble diagrams, and show that the running coupling from the beta function agrees to a leading log in energy to the resummed amplitude. Interpreting the modulus of the resummed amplitude as an effective coupling in analogy to QED, we numerically explore its dependence on energy scale as well as the length scale of the system. We find that surprisingly at small length scales the effective coupling decreases, even though the beta function is positive. This thesis has overlap with related work by the author. |
| format | Thesis |
| id | oai:open.uct.ac.za:11427/40348 |
| institution | University of Cape Town (South Africa) |
| language | Eng |
| last_indexed | 2026-06-10T12:36:27.056Z |
| license_str | Not specified — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UCTD — University of Cape Town Open Access Repository |
| publishDate | 2024 |
| publishDateRange | 2024 |
| publishDateSort | 2024 |
| publisher | Department of Mathematics and Applied Mathematics |
| publisherStr | Department of Mathematics and Applied Mathematics |
| record_format | dspace |
| source_str | UCTD — University of Cape Town Open Access Repository |
| spelling | oai:open.uct.ac.za:11427/40348 Finite system size effects on the effective coupling in scalar quantum field theory Du Plessis, Jean Shock, Jonathan Horowitz William Mathematics and Applied Mathematics Findings from the Large Hadron Collider (LHC) provide evidence that quark-gluon plasma (QGP) formation , traditionally associated with large nucleus-nucleus collisions, also emerges in high multiplicity proton-proton (p + p) and proton-nucleus (p + A) collisions. It has also been shown that azimuthal correlations of low transverse momentum outgoing particles from these high multiplicity p + p and p + A collisions can be modeled well using nearly inviscid relativistic hydrodynamics. One would expect the equation of state (EOS) used in such hydrodynamic simulations to receive significant contributions in systems of size characteristic of p+p collisions. Indeed it has been shown that free scalar thermal field theory receives ∼ 40% corrections to the usual thermodynamic quantities when confined using Dirichlet boundary conditions with characteristic lengths set by central p + p collisions. Furthermore, the potential importance of asymmetric system sizes has also been highlighted in quenched lattice QCD calculations using periodic boundary conditions. In order to analyse the finite system size corrections to the QCD EOS, we need to explore the finite system size corrections to the running of the QCD coupling. This thesis provides a first step toward such a calculation. We consider massive scalar ϕ 4 theory in a spacetime with periodic boundary conditions. We allow characteristic lengths to be asymmetric, or even infinite. We first recount the infinite volume NLO 2 → 2 scattering amplitude calculation using dimensional regularisation, and then introduce and employ denominator regularisation. We then perform the non-trivial calculation of the NLO 2 → 2 scattering amplitude in our compactified spacetime. This requires the derivation of an new analytic continuation of the generalised Epstein zeta function after employing denominator regularisation in order to isolate the UV divergence. Denominator regularisation is necessary, since the usual techniques in dimensional regularisation no longer applies when we allow asymmetric characteristic lengths. We confirm that taking the limit of infinite characteristic lengths yields the usual infinite space-time result. We then perform another non-trivial self-consistency check by verifying that the NLO 2 → 2 Page ii scattering amplitude satisfies unitarity in the form of the optical theorem, regardless of the number of compactified dimensions. In order to show this we derived a generalisation of a formula originally proposed by Hardy and Ramanujan, and interpret its analytic continuation in the context of Abel summation. We numerically explore the scattering amplitude in some special cases, but find the s-channel of the 2 compactified dimension case to be numerically ill-behaved. We then derive and employ a dispersion relation in order to numerically explore the s-channel. Using the Callan-Symanzik equation to derive the beta function, we find it insensitive to finite system effects, which only modifies the IR. We then perform a geometric resummation of bubble diagrams, and show that the running coupling from the beta function agrees to a leading log in energy to the resummed amplitude. Interpreting the modulus of the resummed amplitude as an effective coupling in analogy to QED, we numerically explore its dependence on energy scale as well as the length scale of the system. We find that surprisingly at small length scales the effective coupling decreases, even though the beta function is positive. This thesis has overlap with related work by the author. 2024-07-04T14:09:46Z 2024-07-04T14:09:46Z 2024 2024-07-02T14:13:47Z Thesis / Dissertation Masters MSc http://hdl.handle.net/11427/40348 Eng application/pdf Department of Mathematics and Applied Mathematics Faculty of Science |
| spellingShingle | Mathematics and Applied Mathematics Du Plessis, Jean Finite system size effects on the effective coupling in scalar quantum field theory |
| thesis_degree_str | Master's |
| title | Finite system size effects on the effective coupling in scalar quantum field theory |
| title_full | Finite system size effects on the effective coupling in scalar quantum field theory |
| title_fullStr | Finite system size effects on the effective coupling in scalar quantum field theory |
| title_full_unstemmed | Finite system size effects on the effective coupling in scalar quantum field theory |
| title_short | Finite system size effects on the effective coupling in scalar quantum field theory |
| title_sort | finite system size effects on the effective coupling in scalar quantum field theory |
| topic | Mathematics and Applied Mathematics |
| url | http://hdl.handle.net/11427/40348 |
| work_keys_str_mv | AT duplessisjean finitesystemsizeeffectsontheeffectivecouplinginscalarquantumfieldtheory |