Physics Letters B 755 (2016) 253–260
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
The Coulomb interaction in Helium-3: Interplay of strong short-range
and weak long-range potentials
J. Kirscher
∗
, D. Gazit
Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel
a r t i c l e i n f o a b s t r a c t
Article history:
Received
1 October 2015
Received
in revised form 28 January 2016
Accepted
9 February 2016
Available
online 11 February 2016
Editor:
W. Haxton
Quantum chromodynamics and the electroweak theory at low energies are prominent instances of the
combination of a short-range and a long-range interaction. For the description of light nuclei, the large
nucleon–nucleon scattering lengths produced by the strong interaction, and the reduction of the weak
interaction to the Coulomb potential, play a crucial role. Helium-3 is the first bound nucleus comprised
of more than one proton in which this combination of forces can be studied.
We
demonstrate a proper renormalization of Helium-3 using the pionless effective field theory as
the formal representation of the nuclear regime as strongly interacting fermions. The theory is
found consistent at leading and next-to-leading order without isospin-symmetry-breaking 3-nucleon
interactions and a non-perturbative treatment of the Coulomb interaction. The conclusion highlights the
significance of the regularization method since a comparison to previous work is contradictory if the
difference in those methods is not considered.
With
a perturbative Coulomb interaction, as suggested by dimensional analysis, we find the Helium-3
system properly renormalized, too.
For
both treatments, renormalization-scheme independence of the effective field theory is demonstrated
by regulating the potential and a variation of the associated cutoff.
© 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
Quantum chromodynamics (QCD) is non-perturbative at low
energies where it is characterized by a scale separation. These two
facts facilitate an approximate solution of low-energy QCD, i.e., nu-
clear
physics, with renormalization group (RG) and effective field
theory (EFT) techniques [1–4].
The
strongly interacting character of QCD is of particular in-
terest
at very low-energies. There, the nuclear regime can be de-
scribed
solely by nucleons interacting at the same space–time
point, since the excitation of other degrees of freedom is dynam-
ically
forbidden. This “pionless” EFT (EFT(/π)) of nuclear physics is
characterized by the large nucleon–nucleon (NN) scattering lengths
relative to the effective range of the nuclear force, as indicated by
the unnaturally small deuteron binding energy [5,6]. EFT(/π ) repro-
duces
Bethe’s effective range theory [7] as an expansion of the
NN amplitude about a non-trivial fixed point of the RG, i.e., uni-
*
Corresponding author.
E-mail
addresses: j.kirscher@mail.huji.ac.il (J. Kirscher),
doron.gazit@phys.huji.ac.il (D. Gazit).
tary fixed point. Thereby, it describes strongly interacting fermionic
systems with infinite scattering lengths (original formulation [8,9]
and
RG emphasis [10]) at its leading order (LO). For its usefulness
in larger systems, EFT(/π ) includes a 3-body contact interaction at
LO, abandoning the naturalness assumption and naïve dimensional
analysis for this operator. The enhancement of the 3-body-contact
interaction
is related to a limit cycle found in the RG analysis of
the triton [11] and thereby also an expression of the specific regu-
larization
that facilitated the limit cycle and gave it its shape. The
appearance of 3-body bound states at threshold associated with
the limit cycle is a reminiscence of Efimov physics in the unitary
limit [12]. Naïve power counting fails due to proximity to the non-
trivial
unitary fixed point. Perturbations at higher orders include
effective-range corrections, as well as relativistic effects. Additional
counter terms, needed to renormalize 3-nucleon forces in the tri-
ton,
appear only at next-to-next-to leading order (NNLO) [13,14].
The
accidental separation of scales inducing the above EFT(/π )
power counting is realized, in particular, between the large scat-
tering
lengths and the mass of the pion, as the lightest mesonic
degree of freedom. The scale m
π
suggests that the EFT(/π) ap-
proach
is limited to light nuclei (A 4), since the scale set by
http://dx.doi.org/10.1016/j.physletb.2016.02.011
0370-2693/
© 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.