Physics Letters B 802 (2020) 135247
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Physics Letters B
www.elsevier.com/locate/physletb
Dilute Fermi gas at fourth order in effective field theory
C. Wellenhofer
a,b,∗
, C. Drischler
c,d
, A. Schwenk
a,b,e
a
Institut für Kernphysik, Technische Universität Darmstadt, 64289 Darmstadt, Germany
b
ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, 64291 Darmstadt, Germany
c
Department of Physics, University of California, Berkeley, CA 94720, United States of America
d
Lawrence Berkeley National Laboratory, Berkeley, CA 94720, United States of America
e
Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany
a r t i c l e i n f o a b s t r a c t
Article history:
Received
13 December 2019
Received
in revised form 16 January 2020
Accepted
16 January 2020
Available
online 27 January 2020
Editor:
W. Haxton
Using effective field theory methods, we calculate for the first time the complete fourth-order term in the
Fermi-momentum or k
F
a
s
expansion for the ground-state energy of a dilute Fermi gas. The convergence
behavior of the expansion is examined for the case of spin one-half fermions and compared against
quantum Monte-Carlo results, showing that the Fermi-momentum expansion is well-converged at this
order for |k
F
a
s
| 0.5.
© 2020 The Author(s). 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
.
The dilute Fermi gas has been a central problem for many-
body
calculations for decades [1–11]. Renewed interest in this
problem has been triggered by striking progress with ultracold
atomic gases. In particular, by employing so-called Feshbach res-
onances [12]one
can tune inter-atomic interactions and thereby
probe Fermi systems over a wide range of many-body dynam-
ics [13].
On the theoretical side, a systematic approach towards
the dynamics of fermions (or bosons) at low energies has emerged
in the form of effective field theory (EFT) [14–19]. Motivated by
this, we revisit the expansion in the Fermi momentum k
F
of the
ground-state energy density E(k
F
) of a dilute gas of one species of
interacting fermions. Using perturbative EFT methods, we calculate
E(k
F
) up to fourth order in the expansion, including for the first
time the complete fourth-order term. From this, we analyze the
convergence behavior of the expansion, and obtain precise predic-
tions
for E(k
F
) with systematic uncertainty estimates. Our analytic
results have important applications for various problems in many-
body
physics, including benchmarks for experimental and theoret-
ical
studies of cold atoms, the construction of improved models of
neutron star crusts, and for constraining nuclear many-body calcu-
lations
at low densities.
Short-ranged
EFT represents a systematic framework for the dy-
namics
of fermions (or bosons) at low momenta Q <
b
, where
*
Corresponding author.
E-mail
addresses: wellenhofer@theorie.ikp.physik.tu-darmstadt.de
(C. Wellenhofer),
cdrischler@berkeley.edu (C. Drischler),
schwenk@physik.tu-darmstadt.de (A. Schwenk).
b
denotes the breakdown scale. At low momenta, details of the
underlying interactions are not resolved and can be replaced by
a series of contact interactions. Few- and many-body observables
are then expressed in terms of a systematic expansion in Q /
b
(called “power counting”). The EFT Lagrangian is given by the
most general operators consistent with Galilean invariance, par-
ity,
and time-reversal invariance. The low-energy constants of the
Lagrangian have to be fitted to experimental data or (if possi-
ble)
can be matched to the underlying theory. Assuming spin-
independent
interactions, the (unrenormalized) Lagrangian reads
(see, e.g., Refs. [14–19])
L
EFT
= ψ
†
i∂
t
+
−→
∇
2
2M
ψ −
C
0
2
(ψ
†
ψ)
2
+
C
2
16
(ψψ)
†
(ψ
←→
∇
2
ψ)+h.c.
+
C
2
8
(ψ
←→
∇
ψ)
†
·(ψ
←→
∇
ψ)−
D
0
6
(ψ
†
ψ)
3
+..., (1)
where ψ are nonrelativistic fermion fields,
←→
∇
=
←−
∇
−
−→
∇
is the
Galilean invariant derivative, h.c. the Hermitian conjugate, and M
the
fermion mass.
The
ultraviolet (UV) divergences that appear beyond tree level
in perturbation theory can be regularized by introducing a cutoff
for relative momenta p
()
and Jacobi momenta q
()
. The two-
and
three-body potentials emerging from L
EFT
are then given by
https://doi.org/10.1016/j.physletb.2020.135247
0370-2693/
© 2020 The Author(s). 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
.