4,093 research outputs found
Rotation method for accelerating multiple-spherical Bessel function integrals against a numerical source function
A common problem in cosmology is to integrate the product of two or more
spherical Bessel functions (sBFs) with different configuration-space arguments
against the power spectrum or its square, weighted by powers of wavenumber.
Naively computing them scales as with the number of
configuration space arguments and the grid size, and they cannot be
done with Fast Fourier Transforms (FFTs). Here we show that by rewriting the
sBFs as sums of products of sine and cosine and then using the product to sum
identities, these integrals can then be performed using 1-D FFTs with scaling. This "rotation" method has the potential to
accelerate significantly a number of calculations in cosmology, such as
perturbation theory predictions of loop integrals, higher order correlation
functions, and analytic templates for correlation function covariance matrices.
We implement this approach numerically both in a free-standing,
publicly-available \textsc{Python} code and within the larger,
publicly-available package \texttt{mcfit}. The rotation method evaluated with
direct integrations already offers a factor of 6-10 speed-up over the
naive approach in our test cases. Using FFTs, which the rotation method
enables, then further improves this to a speed-up of
over the naive approach. The rotation method should be useful in light of
upcoming large datasets such as DESI or LSST. In analysing these datasets
recomputation of these integrals a substantial number of times, for instance to
update perturbation theory predictions or covariance matrices as the input
linear power spectrum is changed, will be one piece in a Monte Carlo Markov
Chain cosmological parameter search: thus the overall savings from our method
should be significant
The 1/2 BPS Wilson loop in ABJ(M) at two loops: The details
We compute the expectation value of the 1/2 BPS circular Wilson loop operator
in ABJ(M) theory at two loops in perturbation theory. Our result turns out to
be in exact agreement with the weak coupling limit of the prediction coming
from localization, including finite N contributions associated to non-planar
diagrams. It also confirms the identification of the correct framing factor
that connects framing-zero and framing-one expressions, previously proposed.
The evaluation of the 1/2 BPS operator is made technically difficult in
comparison with other observables of ABJ(M) theory by the appearance of
integrals involving the coupling between fermions and gauge fields, which are
absent for instance in the 1/6 BPS case. We describe in detail how to
analytically solve these integrals in dimensional regularization with
dimensional reduction (DRED). By suitably performing the physical limit to
three dimensions we clarify the role played by short distance divergences on
the final result and the mechanism of their cancellation.Comment: 54 pages, 2 figure
Perturbative quantum gauge fields on the noncommutative torus
Using standard field theoretical techniques, we survey pure Yang-Mills theory
on the noncommutative torus, including Feynman rules and BRS symmetry. Although
in general free of any infrared singularity, the theory is ultraviolet
divergent. Because of an invariant regularization scheme, this theory turns out
to be renormalizable and the detailed computation of the one loop counterterms
is given, leading to an asymptoticaly free theory. Besides, it turns out that
non planar diagrams are overall convergent when is irrational.Comment: Latex 2e, 19 pages 5 eps figures, typos corrected and 1 reference
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Vacuum birefringence in strong magnetic fields: (I) Photon polarization tensor with all the Landau levels
Photons propagating in strong magnetic fields are subject to a phenomenon
called the "vacuum birefringence" where refractive indices of two physical
modes both deviate from unity and are different from each other. We compute the
vacuum polarization tensor of a photon in a static and homogeneous magnetic
field by utilizing Schwinger's proper-time method, and obtain a series
representation as a result of double integrals analytically performed with
respect to proper-time variables. The outcome is expressed in terms of an
infinite sum of known functions which is plausibly interpreted as summation
over all the Landau levels of fermions. Each contribution from infinitely many
Landau levels yields a kinematical condition above which the contribution has
an imaginary part. This indicates decay of a sufficiently energetic photon into
a fermion-antifermion pair with corresponding Landau level indices. Since we do
not resort to any approximation, our result is applicable to the calculation of
refractive indices in the whole kinematical region of a photon momentum and in
any magnitude of the external magnetic field.Comment: To appear in Ann. Phys., 47 pages, 7 figure
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