The epsilon expansion meets semiclassics

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Abstract: We study the scaling dimension

D e l t a_{p h i^{n}}

of the operator

p h i^{n}

where

p h i

is the fundamental complex field of the

U (1)

model at the Wilson-Fisher fixed point in

d = 4 - v a r e p s i l o n

. Even for a perturbatively small fixed point coupling

l a m b d a_{*}

, standard perturbation theory breaks down for sufficiently large

l a m b d a_{*} n

. Treating

l a m b d a_{*} n

as fixed for small

l a m b d a_{*}

we show that

D e l t a_{p h i^{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,

D e l t a_{- 1} (l a m b d a_{*} n)

and

D e l t a_{0} (l a m b d a_{*} n)

. The result, when expanded at small

l a m b d a_{*} n

, perfectly agrees with all available diagrammatic computations. The asymptotic at large

l a m b d a_{*} 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

v a r e p s i l o n = 1

, but encouraging.

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