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Register a new userNew results for the epsilon-expansion of certain one-, two- and three-loop Feynman diagrams
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For certain dimensionally-regulated one-, two- and three-loop diagrams, problems of constructing the epsilon-expansion and the analytic continuation of the results are studied. In some examples, an arbitrary term of the epsilon-expansion can be calculated. For more complicated cases, only a few higher terms in epsilon are obtained. Apart from the one-loop two- and three-point diagrams, the examples include two-loop (mainly on-shell) propagator-type diagrams and three-loop vacuum diagrams. As a by-product, some new relations involving Clausen function, generalized log-sine integrals and certain Euler--Zagier sums are established, and some useful results for the hypergeometric functions of argument 1/4 are presented.
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In arXiv:1909.01269 it was shown that the scaling dimension of the lightest charge $n$ operator in the $U(1)$ model at the Wilson-Fisher fixed point in $d=4-varepsilon$ can be computed semiclassically for arbitrary values of $lambda n$, where $lambda$ is the perturbatively small fixed point coupling. Here we generalize this result to the fixed point of the $U(1)$ model in $3-varepsilon$ dimensions. The result interpolates continuously between diagrammatic calculations and the universal conformal superfluid regime for CFTs at large charge. In particular it reproduces the expectedly universal $O(n^0)$ contribution to the scaling dimension of large charge operators in $3d$ CFTs.
We review the hypergeometric function approach to Feynman diagrams. Special consideration is given to the construction of the Laurent expansion. As an illustration, we describe a collection of physically important one-loop vertex diagrams for which this approach is useful.
We study the scaling dimension $Delta_{phi^n}$ of the operator $phi^n$ where $phi$ is the fundamental complex field of the $U(1)$ model at the Wilson-Fisher fixed point in $d=4-varepsilon$. Even for a perturbatively small fixed point coupling $lambda_*$, standard perturbation theory breaks down for sufficiently large $lambda_*n$. Treating $lambda_* n$ as fixed for small $lambda_*$ we show that $Delta_{phi^n}$ can be successfully computed through a semiclassical expansion around a non-trivial trajectory, resulting in $$ Delta_{phi^n}=frac{1}{lambda_*}Delta_{-1}(lambda_* n)+Delta_{0}(lambda_* n)+lambda_* Delta_{1}(lambda_* n)+ldots $$ We explicitly compute the first two orders in the expansion, $Delta_{-1}(lambda_* n)$ and $Delta_{0}(lambda_* n)$. The result, when expanded at small $lambda_* n$, perfectly agrees with all available diagrammatic computations. The asymptotic at large $lambda_* n$ reproduces instead the systematic large charge expansion, recently derived in CFT. Comparison with Monte Carlo simulations in $d=3$ is compatible with the obvious limitations of taking $varepsilon=1$, but encouraging.
One approach to the calculation of cross sections for infrared-safe observables in high energy collisions at next-to-leading order is to perform all of the integrations, including the virtual loop integration, by Monte Carlo numerical integration. In a previous paper, two of us have shown how one can perform such a virtual loop integration numerically after first introducing a Feynman parameter representation. In this paper, we perform the integration directly, without introducing Feynman parameters, after suitably deforming the integration contour. Our example is the N-photon scattering amplitude with a massless electron loop. We report results for N = 6 and N = 8.
We study the Ising model in $d=2+epsilon$ dimensions using the conformal bootstrap. As a minimal-model Conformal Field Theory (CFT), the critical Ising model is exactly solvable at $d=2$. The deformation to $d=2+epsilon$ with $epsilonll 1$ furnishes a relatively simple system at strong coupling outside of even dimensions. At $d=2+epsilon$, the scaling dimensions and correlation function coefficients receive $epsilon$-dependent corrections. Using numerical and analytical conformal bootstrap methods in Lorentzian signature, we rule out the possibility that the leading corrections are of order $epsilon^{1}$. The essential conflict comes from the $d$-dependence of conformal symmetry, which implies the presence of new states. A resolution is that there exist corrections of order $epsilon^{1/k}$ where $k>1$ is an integer. The linear independence of conformal blocks plays a central role in our analyses. Since our results are not derived from positivity constraints, this bootstrap approach can be extended to the rigorous studies of non-positive systems, such as non-unitary, defect/boundary and thermal CFTs.
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