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UNITARITY
AND
DISCRETE SCALE INVARIANCE
U. van Kolck
Institut de Physique Nucléaire d’Orsay
and
University of Arizona
Supported by CNRS and US DOE
1
Outline
What is essential?
Pionless EFT
with
S. König,
H.W. Grießhammer
& H.-W. Hammer
Unitarity: light nuclei
Unitarity: bosonic clusters
Unitarity: matter
Conclusion
plus
B. Bazak
& M. Eliyahu
S. Gandolfi,
J. Carlson,
& S.A. Vitiello
L. Contessi,
A. Lovato,
F. Pederiva,
A. Roggero
& J. Kirscher
What is essential in nuclear physics?
Traditional
approach
1) Describe NN precisely to some high energy
2) Append 3N forces as needed
Lost in details:
e.g., NLO, N2LO, N3LO, N4LO, …
Here
1) Describe NN and 3N approximately
2) Treat everything else in perturbation theory
cf. atomic systems in QED
“simplicity emerging
from complexity”
Expansion
around unitarity
König, Grießhammer,
Hammer + v.K. ’15 ’16
König ’16
vK ‘17
A=2
T2 (k =
R −1 )
−1
4π
m
r2 2 P2 4
−1
+
−
+
+
a
i
k
k
k
2
+ (l > 0)
2
4
scattering
length
unitarity
limit
(
a2−1 ≈ mB2 → 0
−1
2
T2 a
−1
R mπ
a2,=I
1
S0
shape
parameter
r2
P
R, 2 R 3 , typically
2
4
1
4π
−1
=
kR
( ik ) 1 + , kR
m
ka2
−1
)
unitarity window
nucleons
effective
range
a2,=I
a2,=I
1,=
I3
0 mπ
1,=
I3
+1mπ
1,=
I3
−1mπ
no parameter!
−1
“universality”
0.06
−1
−1
−1
− a2, I =
0.12
1,=
I 3 0 mπ
−1
− a2, I =
0.02
1,=
I 3 0 mπ
3
S1
a2, I =0 mπ
−1
0.26
unitarity limit
unitarity
limit
mπ∗ ( M QCD )
Feshbach resonance
mπ 140 MeV
MIT webpage
Chiral EFT,
(incomplete) NLO
Beane, Bedaque,
Savage + v.K. ’02
Beane + Savage ’03’04
Epelbaum, Glöckle
+ Meißner ‘03
…
or “accidentally”, e.g.
a2 mπ−1 r2
4He
a2 lvdW r2
atoms
ground
states
Q A 2 mN B A A
A
QA mπ
2
3
4
5
6
0.3
0.5
0.8
0.7
56
Q2 a2,−1I =0 ≡ ℵ1 M lo
3
S1 unitarity
König, Grießhammer,
Hammer + v.K. ’16
v.K. ‘17
QA≥3 M lo
0.7
0.9
König, Grießhammer,
Hammer + v.K. ’15
König ’16
1
S0 unitarity
a2,=I
−1
1,=
I3 0
≡ ℵ0 M lo
−1
−1
−1
−1
a2, I =
− a2, I =
α mN ℵ0 M lo
1, I 3 =
+1
1, I 3 =
0
a2, I =
− a2, I =
md − mu ℵ0
1, I 3 =
1, I 3 =
0
−1
Effective Field Theory
momentum
scales
M hi
Λ
UV regulator
T
M lo
non-analytic functions,
from solution of dynamical eq.
(e.g. Lippmann-Schwinger)
arbitrary
Q
(ν )
ν
Q Q (ν ) Λ M lo
Q
(ν )
(Q M lo M hi ) ∝ ∑
, ; γi
,
F
ν = 0 M hi
M lo M hi
M lo Λ
ν
ν
N LO
(unfortunately not the usage
by potential modelers)
RG invariance
(absent in “chiral potentials”)
Q
Λ ∂T
= ν
(ν )
T
∂Λ
M hi Λ
(ν )
c
ν +1
model independent
Qν +1 Qν +1
+ ν +1 , ν
M hi M hi Λ
“low-energy
constants”
controlled
(OTHERWISE, NOT ERROR ESTIMATE)
to minimize cutoff errors, Λ > M hi
for realistic error estimate, Λ ∈ [ M hi , ∞ )
(OTHERWISE, SENSITIVE TO HIGH-MOM DETAILS)
Pionless EFT
Q M lo M hi mπ
• d.o.f.: nucleons
• symmetries: Lorentz, P, T, B, SU(3)c, U(1)em
(trivial)
dt
3
d
∫ 2m ∫ r
S EFT
most general
action
+
∂ 2
ψ 2im + ∇ ψ
∂t
projector on isospin I
+ +
− 4π ∑ C0 I ψ ψ PI ψψ
I = 0,1
(4π ) 2
D0 ψ +ψ +ψ + ψψψ
−
3
more derivatives,
+ } more bodies,
isospin violation
Universality:
first orders
apply also to
neutral atoms
mπ → 1 lvdW
4
vdW
l
where V ( r ) =
−
+
6
2mr
Bedaque, Hammer
+ v.K. ’99’00
Bedaque, Braaten
+ Hammer ’01
…
Now,
two expansions:
ℵ1 M lo ≡ Q2 Q3 0.4
M lo M hi Q3 mπ 0.5
around two-body unitarity
standard Pionless EFT
similar:
ℵ1 M lo2 M hi
For simplicity, also:
starts at NLO
α mN ℵ0 M l2o M hi
md − mu M lo3 M h2i
starts at NLO
starts at N2LO
Bira vK.pdf (PDF, 1.14 MB)
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