arXiv subjects

Ian Jack

Publications and source records attributed to Ian Jack.

6 recordsLinked to original sources

arXiv2026

$\phi^6$ at $6$ (and some $8$) loops in $3d$

We recalculate the contributions of individual six loop graphs to the $\beta$-function for a three dimensional scalar theory with an arbitrary sextic scalar potential. Previously this was calculated by Hager who specialised to a theory with maximal $O(N)$ symmetry. Our results differ in some contributions to the overall $\beta$-function but agree with a recent calculation \cite{Kompaniets2}. At large $N$ three eight loop diagrams which are relevant are calculated. At the $O(N)$ fixed point some critical exponents are determined to $\rm O(\varepsilon^3)$. Imposing that the $\beta$-function satisfies a gradient flow equation is shown to require linear relations between some $\beta$-function coefficients. The curvature for the associated metric is also determined. Detailed results for the Feynman integrals are described in the appendices.

hep-th↗

arXiv2024

No-$\pi$ schemes for multi-coupling theories

We show that even $\zeta$-functions may be removed from the $\beta$-functions of general multi-coupling theories up to high loop order by means of coupling redefinitions. For theories whose $\beta$-function is determined by the anomalous dimensions of the fields, such as supersymmetric theories, this corresponds to a renormalisation scheme change to a momentum subtraction scheme.

hep-th↗

arXiv2023

Explorations in Scalar Fermion Theories: $\beta$-functions, Supersymmetry and Fixed Points

Results for $\beta$-functions and anomalous dimensions in general scalar fermion theories are presented to three loops. Various constraints on the individual coefficients for each diagram following from supersymmetry are analysed. The results are used to discuss potential fixed points in the $\varepsilon$-expansion for scalar fermion theories, with arbitrary numbers of scalar fields, and where there are just two scalar couplings and one Yukawa coupling. For different examples the fixed points follow a similar pattern as the numbers of fermions is varied. For diagrams with subdivergences there are extensive consistency constraints arising from the existence of a perturbative $a$-function and these are analysed in detail. Further arbitrary scheme variations which preserve the form of $\beta$ functions and anomalous dimensions in terms of 1PI diagrams are also discussed. The existence of linear and quadratic scheme invariants is demonstrated and the consistency condition are shown to be expressible in terms of these invariants.

hep-th↗

arXiv2016

Scheme Dependence and Multiple Couplings

For theories with multiple couplings the perturbative $β$-functions for scalar, Yukawa couplings are expressible in terms of contributions corresponding to one particle irreducible graphs and also contributions which are one particle reducible depending on the anomalous dimension. Here we discuss redefinitions, or changes of scheme, which preserve this structure. The redefinitions allow for IPR contributions of a specific form, as is necessary to encompass the relation between MS and momentum subtraction renormalisation schemes. Many multiply 1PR terms in the transformed $β$-function are generated but these can all be absorbed into antisymmetric contributions to the anomalous dimensions which are essentially arbitrary and can be discarded. As an illustration the results are applied to the scheme dependence of the anomalous dimension, which determines the $β$-function, for ${\cal N}=1$ supersymmetric scalar fermion theories in four dimensions up to four loops.

hep-th↗

arXiv2007

The four-loop DRED gauge beta-function and fermion mass anomalous dimension for general gauge groups

We present four-loop results for the gauge beta-function and the fermion mass anomalous dimension for a gauge theory with a general gauge group and a multiplet of fermions transforming according to an arbitrary representation, calculated using the dimensional reduction scheme. In the special case of a supersymmetric theory we confirm previous calculations of both the gauge beta-function and the gaugino mass beta-function.

hep-th↗

arXiv1994

Decoupling of the $ε$-scalar mass in softly broken supersymmetry

It has been shown recently that the introduction of an unphysical $ε$-scalar mass $\tilde{m}$ is necessary for the proper renormalization of softly broken supersymmetric theories by dimensional reduction ($\drbar$). In these theories, both the two-loop $β$-functions of the scalar masses and their one-loop finite corrections depend on $\tilde{m}^2$. We find, however, that the dependence on $\tilde{m}^2$ can be completely removed by slightly modifying the \drbar renormalization scheme. We also show that previous \drbar calculations of one-loop corrections in supersymmetry which ignored the $\tilde{m}^2$ contribution correspond to using this modified scheme.

hep-ph↗