Introduction to Effective Field Theories in QCD Ubirajara U. van - - PowerPoint PPT Presentation
Introduction to Effective Field Theories in QCD Ubirajara U. van Kolck University of Arizona Supported in part by US DOE 6/2/2009 v. Kolck, Intro to EFTs 1 Background by S. Hossenfelder Outline Effective Field Theories Introduction
6/2/2009
1
Background by S. Hossenfelder
Supported in part by US DOE
6/2/2009
2
6/2/2009
3
Introduction to effective field theories in QCD, in New states of matter in hadronic interactions (Proceedings of the Pan American Advanced Studies Institute, 2002), nucl-th/0205058 D.B. Kaplan, Effective field theories, Lectures at 7th Summer School in Nuclear Physics Symmetries, Seattle, WA, 18-30 Jun 1995, nucl-th/9506035
6/2/2009
4
Midwestern Conference on Theoretical Midwestern Conference on Theoretical Physics, Purdue University, 1960 Physics, Purdue University, 1960
6/2/2009
5
Nuclei are essentially made out of non-relativistic nucleons (protons and neutrons), which interact via a potential The potential is mostly two-nucleon, but there is evidence for smaller three-nucleon forces Isospin is a good symmetry of nuclear forces, except for a sizable breaking in two-nucleon scattering lengths and other, smaller effects External probes --such as photons-- interact mainly with each nucleon but there is evidence for smaller two-nucleon currents
but…
6/2/2009
6
Up, down quarks are relativistic, interacting via multi-gluon exchange The interaction is a multi-quark process Isospin symmetry is not obvious: External probes can interact with collection of quarks difficulty 1 2 3 4
1 3
d u d u
m m m m ε + = − ∼
quarks and gluons not the most convenient degrees of freedom at low energies
e.g.
6/2/2009
7
Ba ry2
QCD
Hadronic Scales
PDG, 2005
(938) (940)
6/2/2009
8
Nuclear Scales
nuc
2 2
QCD nuc
P
MeV E
12
2
fm
ch
r
2H d
3H t
3He
4He α 5He
1 2 + 1 2 +
3 2 − 1 2 1 2 1 2
2.2246 − 8.482 − 7.718 − 28.296 − 0.9 + 2.116(6) 1.755(86) 1.959(34) 1.676(8)
MeV E A
1 fm
F
k
Friar, ‘93
nucc
6/2/2009
9
2 2 2 2 2 2 2 2 2
1 ; 2 4
e e e
p e p H r c c r m m m π ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ = − + ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦
p R
r R ∼
2 2 2
2 4
e
e R R R m E π ⎛ ⎞ − ⎜ ⎟ ⎝ ⎠
dE R dR =
e
R c m α =
1 1 4 137 e c π α ≡ ≅
0.5 MeV
e
m c =
2
3.6 keV
e
c p c m α = ∼
2 2 2
1 13.6 eV 2 2
e e
m c p E m α − = ∼ ∼
H atom
α α
Three scales (from now on, units such that )
1, 1 c = =
6/2/2009
10
no obvious small coupling in nuclear forces. EFFECTIVE FIELD THEORY
0.1 0.2 0.3 1 10 10
2
µ GeV αs(µ)
QCD “fine-structure” constant
Needed: method that does not rely on small couplings
PDG, 2005
6/2/2009
11
1972
6/2/2009
12
6/2/2009
13
Seurat, La Parade (detail)
6/2/2009
14
choose the coordinates that fit the problem
6/2/2009
15
6.4 Mm
E
R 1.7 Mm
m
R
2-body forces 2+3-body forces change in resolution
Wikipedia
6/2/2009
16
choose the coordinates that fit the problem what is not forbidden is compulsory
6/2/2009
17
A farmer is having trouble with a cow whose milk has gone sour. He asks three scientists—a biologist, a chemist, and a physicist—to help him. The biologist figures the cow must be sick or have some kind of infection, but none of the antibiotics he gives the cow work. Then, the chemist supposes that there must be a chemical imbalance affecting the production of milk, but none of the solutions he proposes do any good either. Finally, the physicist comes in and says, “First, we assume a spherical cow…”
ij i j ij
1
ij
δα
u v
no, say,
ij i j ij
amenable to perturbation theory
6/2/2009
18
choose the coordinates that fit the problem what is not forbidden is compulsory
not everything is allowed
6/2/2009
19
After scales have been identified, the remaining, dimensionless parameters are
unless suppressed by a symmetry
‘t Hooft ‘79
simplest assumption, to be revised if necessary
Occam’s razor:
und
fine-tuning
cow non-sphericity…
energy scale of underlying theory energy of probe
6/2/2009
20
A classical example: the flat Earth -- light object near surface of a large body
und
d.o.f.: mass sym:
, ,
eff eff
V x y V h h =
2
const
eff i i i
h m h V g g m h h η
∞ =
= = + + +
…
1 1
1 1
i i i und i
m h E h m E h g R g
+ +
= × = × O O
1 i i
R g g + ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ O
1
eff
V G h M R h m = − +
1
1
i i i g
g R
+ = −
g ≡
parameters
(neglecting quantum corrections…)
naturalness:
h R
1
1
i i i
M h G R R m
− ∞ =
− ⎛ ⎞ ⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠
itself the first term in a low-energy EFT of general relativity…
6/2/2009
21
A short path to quantum mechanics
sum over all paths each path contributes a phase given by the classical action
2 1 2
2 1 2 3
b i a
Feynman ‘48
RULE
i i
6/2/2009
22
( )
i
q t
( )
i n
q t +
( )
2 i n
q t +
i
2 2 2
i
i t
und
scale of fine-structure
coarse-graining scale (cutoff)
scale of variation
EFFECTIVE THEORY
j
i
i t
i
1 j
2 j
i
1 i
i n
2 i n
6/2/2009
23
3 4
6/2/2009
24
Partition function
4 free int 2 4 4 4 int int free
2 2
i p m iε = − + iλ =
4 int
4 4 2 2 2 2 1 2
(2 ) ( ) ( ) d l i i i i p l m i p l m i λ λ π ε ε = + − + − − +
1
p
2
p
1
p l +
2
p l −
1
' p
2
' p
momentum space (skip many steps…)
6/2/2009
25
4 , 4
H L und H L L EFT
Λ
Euler + Heisenberg ’36 Weinberg ’67 … ’79 Wilson, early 70s …
H Q
L
( , )
n EFT i i d i d n
∞ =
renormalization-group invariance underlying symmetries details of the underlying dynamics local 1 1 :
H
Q x M φ Δ < ∼ 1 1 :
L
Q x M φ Δ > ∼
6/2/2009
26
min
( ) , ,
i i i
ν ν ν ν ν ∞ ∞ =
( ) ( )
ν ν
1 ( )
ν ν +
For Q ~ m, truncate consistently with RG invariance so as to allow systematic improvement (perturbation theory): “power counting” e.g. # loops L
non-analytic, from loops normalization
characteristic external momentum
6/2/2009
27
Why is this useful? Because in general the appropriate degrees of freedom below M are not the same as above
,
H L
φ φ φ =
Examples:
virtual exchange in coefficients (Appelquist-Carazzone decoupling theorem)
appearance of massless Goldstone bosons or gauge-boson mass (Goldstone’s theorem, Higgs mechanism)
i
L
φ
H
φ
L
φ
L
φ
L
φ
i
ϕ ϕ ϕ ϕ
6/2/2009
28
How can we do it? Two possibilities: know and can solve underlying theory -- get ‘s in terms of parameters in know but cannot solve, or
use Weinberg’s “theorem”:
Weinberg ‘79
i
c
“The quantum field theory generated by the most general Lagrangian with some assumed symmetries will produce the most general S matrix incorporating quantum mechanics, Lorentz invariance, unitarity, cluster decomposition and those symmetries, with no further physical content.” Note: proven only for scalar field with symmetry in , but no known counterexamples
Ball + Thorne ‘94
( )
,
und H L
φ φ L
2
Z
4
E
6/2/2009
29
(“regularization”)
(“renormalization”)
Q < Λ
i
c Λ Λ Λ
not a model form factor
controlled expansion in
1 Q M ×O
“naturalness”: what else? unless suppressed by symmetry… contrast to models, which have fewer, but ad hoc, interactions; useful in the identification of relevant degrees of freedom and symmetries, but plagued with uncontrolled errors
what is not forbidden is mandatory!
6/2/2009
30
T.Y. Cao, in Renormalization, From Lorentz to Landau (and Beyond), L.M. Brown (ed) 1993
6/2/2009
31
Time for a paradigm change, perhaps?
6/2/2009
32
(GeV) E
20
16
3
2
15
19
General Relativity + higher-curvature terms
Chiral EFT ? QCD
(2 or 3 flavors)
QCD
(6 flavors)
Electroweak Th + higher-dim ops
QED Fermi Th
(SUSY)
?
GUT? nuclear physics atomic physics molecular physics condensed-matter physics and beyond
NRQED
1
6/2/2009
33
und
μν μν
F A A
μν μ ν ν μ
= ∂ − ∂
μ μ μ
ie
μ
γ =
μ
2
p i i
μν
η ε − = + p p single fermion ψ Lorentz, parity, time-reversal, and U(1) gauge invariance
M i p iε = − +
1 4 3 e πα ∝ = ∼
interactions
perturbation theory
How do E&M bound states arise? , massless spin-1 boson Aμ
μ ν
6/2/2009
34
p q
p q +
p q Q =
O
q q Q = =
2 2 2
M M p p M Q ⎛ ⎞ = + = + ⎜ ⎟ ⎝ ⎠
02 2 2 2 2
2 2 1 2 p q q p q p q q q p q i i p q i i i i i p M M p M p p q i M q γ γ ε ε γ γ ε γ ε − ⋅ + + = = + − + + − + − + + − ⋅ + = + − ⋅ − + + = + +
1 2 P γ
±
± ≡ , P P P P P
± ± ± ±
= =
∓
projector onto energy states
“heavy-fermion formalism”
Georgi ’90
M i t
± ±
i t M
− + − + −
particles: annihilates creates antiparticles: creates annihilates
6/2/2009
35
anomalous magnetic moment =O(1) Pauli term
1 2 4
und
iD i D i D i M D F F μν
μν
γ γ
+ + − + + − − −
= Ψ Ψ − Ψ ⋅ Ψ + Ψ ⋅ Ψ − Ψ + Ψ −
4 , 4
und EFT
+ − + − +
2
1 1 2 2 4
jk EFT i ijk
e i F F F M M D D
μν μν
σ ε = Ψ Ψ + Ψ Ψ + Ψ Ψ + − + … …
complete square, do Gaussian integral
+
2
jk i ijk
M e F κ σ ε + Ψ Ψ +…
most general Lag with Ψ, A invariant under U(1) gauge, parity, time-reversal, and Lorentz transformations non-relativistic expansion
6/2/2009
36
ievμ =
( ) ( )
2 1 ' 2 p p q M e i v S
β ν α μναβ μ
κ ε = + + + q
μ μ
1,0 v ≡
S σ ⎛ ⎞ ≡ ⎜ ⎟ ⎝ ⎠
2 2
1 1 2 e iv v v D S F D D M M
μναβ α β μν
κ ε +Ψ ⋅ Ψ + Ψ ⋅ − Ψ + + Ψ Ψ +…
(0) (1) 2 2
M M S S γ γ + ΨΨ ΨΨ + Ψ Ψ ⋅Ψ Ψ +…
(0) (1) 1 2 2
i S M S γ γ = + ⋅
2
p i i
μν
η ε − = + p
2
e i v M v
μν μ ν
η = −
μ ν
etc.
2 2 4 4
1 4
EFT
a b F F M F F M F F
μν μν μν μν μν μν
= − + + +
L
p p′
μ ν ρ σ
1 3 4 4 2
i a p p p M p b
μρ νσ
η η ⎡ ⎤ = ⋅ ⋅ + + ⎣ ⎦ … …
1
p
2
p
3
p
4
p
Euler + Heisenberg ‘36
μ ν
( )
2 2
1 2M v p v i p p i ε = ⋅ + − ⋅ + + …
p
6/2/2009
37
no explicit fermion-antifermion pair creation!
e.g.
+ …
light-by-light scattering
Various processes at low energies:
T
γγ
= + …
T
γψ
= + +
Compton scattering
Thompson limit
no change in heavy-fermion number!
6/2/2009
38
Back to atomic bound states: the NRQED perspective = + + + + + … + …
higher powers of
Q M
T
ψψ
( )
' p p Q =
O
2
' Q p p M ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ ∼ O
( )
0,
p p
)
0,
p p −
( )
' , ' p p
)
' , ' p p −
( )
( ) ( )
2 2 2 2 2 2 2 2 2
' ' ' ' ie ie ie e Q p p p p p p i p i p i ε ε ε − − − − = = + + − − −
2
4 e V r r π → =
CoM frame
6/2/2009
39
( )
4 4 4 2 2 2 2 02 2
' ' ' 1 1 ( ) ( ) 2 2 2 1 1 ' p p p M d l p p p p p e l l l i l i l l i l l i M π ε ε ε ε = + − + − + + − + − + + + + − − −
)
3 4 3 2 2 2 2
1 1 ( ) ( ) 2 2 2 1 1 2 ' ' ' ' d l ie l l l i l i l i M p p p M p p p l l i p p π ε ε ε ε = + − − + − − + − − − − + − − + +
( )
2 2 2 4 3 2
1 1 1 1 4 4 Q Q Q Q Q e e Q α π π ∼ ∼
( )
0,
l l
)
0,
p l p l + +
)
' , ' p l p l + − +
l
1 2 3 4 4 3 3 4 3 4
2
Q M Q Q
just as expected…
6/2/2009
40
( )
4 4 4 2 2 2 2 02 2
1 1 ( ) ( ) 2 2 1 ' ' 2 1 p p d l e l l l i l i l l i l p p p M M p p l i p π ε ε ε ε − = + + + − + − + − + − + + + − − +
)
3 4 3 2 2 2 2 2 2 2
1 1 ( ) 2 ( ) 2 2 1 ( 2 ' ' ) 2 p p d l ie l l i l i l l i M M M p p p p p p p π ε ε ε = + ⎛ ⎞ + − + − − − + ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ + − − − + + ⎜ ⎟ ⎝ ⎠ + −
3 2 2 2 2 2 2 4 2
1 1 4 4 e Q Q Q Q e e M M Q Q Q α π π α ⎛ ⎞ + + ⎜ ⎟ ⎝ ⎠ ∼ ∼ …
( )
0,
l l
)
0,
p l p l + +
)
0,
( ) p l p l − − +
2
l
1 2 3 4 2 3 3 4 3 4
2
Q M Q Q
infrared enhancement!
4
1
6/2/2009
41
+ … + + +
2
1 M Q ∝ 1 Q ∝
+ + …
1 Q ∝
“time-ordered perturbation theory”
dl
dl
6/2/2009
42
= + + …
(0)
T
ψψ
2 2 2 2
1 1 1 e e M M Q Q Q Q α α ⎧ ⎫ ⎛ ⎞ + + ⎨ ⎬ ⎜ ⎟ ⎛ ⎞ ⎝ ⎠ ⎩ ⎭ − ⎜ ⎟ ⎝ ⎠ ∼ … ∼ O O
bound state at
Q M α ∼
2 2M
M Q E α − ∼ ∼
= + + …
(0)
V
ψψ
= +
(0)
T
ψψ
=
Lippmann-Schwinger eq. Coulomb potential = Schroedinger eq.
known results…
2 2
Q e ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ O
(0) (0) (0 ) 2 ) (
ˆ 2M E p V
ψψ
ψ ψ ⎛ ⎞ + = ⎜ ⎟ ⎝ ⎠
(0)
V
ψψ (0)
V
ψψ (0)
V
ψψ (0)
V
ψψ (0)
V
ψψ
6/2/2009
43
But more: = + + …
2 2
4 Q e α π ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ O
= + + +
(0) (1 (1) (0) ) (0)
V E E
ψψ
ψ ψ = +
(0)
4 E α π ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ O
+ + …
(1)
V
ψψ (1)
V
ψψ (1)
V
ψψ (1)
V
ψψ (1)
V
ψψ (1)
T
ψψ (0)
T
ψψ (0)
T
ψψ (0)
T
ψψ (0)
T
ψψ
6/2/2009
44
+ + … =
2 2 2 2
Q Q M e ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ O
= …
(0) (2) ( (1) (2 ) )
V E E
ψψ
ψ ψ = + +… ( ) ( ) ( ) ( )
2 3 (0)* (3) (0) (0) 1 2 1 2
d r r r r μ μ ψ δ ψ μ μ ψ ∝ ⋅ = ⋅
magnetic interaction
(0) 2 2
Q M E ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ O
(2)
V
ψψ (2)
T
ψψ
6/2/2009
45
N O T E starting at , sufficiently many derivatives appear at vertices that loops bring positive powers of , which need to be compensated by and higher-order “counterterms”
(3)
ψψ
Λ
( ) i
γ Λ
( ) 2 i
γ α = O
Etc. + + … = +
2 2 2 2
4 e Q Q M π α ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ O
2 2
ln M α Λ ∝
(3)
V
ψψ
6/2/2009
46
Most precise determination of electron mass Pachucki, Jentschura + Yerokhin ‘04 electron Larmor frequency trapped-ion cyclotron frequency ion mass electron mass ion charge
Example: g factor for electron bound in H-like atoms
measured known measured
(expt)(th)
2 1 g κ = +
12 ( )
12
C gs
m u ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠
= + + …
(0)
ψ
(0)
ψ ψ ψ
6/2/2009
47
EFT is a general framework to deal with a multi-scale problem using the small ratio of scales as an expansion parameter Nuclear systems involve multiple scales but no obvious small coupling constant Applied to low-energy QED, EFT reproduces well-known facts and also provides a systematic expansion for the potential -- NRQED is in fact the framework used in state-of-the-art QED bound-state calculations Stay tuned: next, how we can make nuclear physics as systematic as QED
6/2/2009
48
Supported in part by US DOE
6/2/2009
49
6/2/2009
50
Phenomenological Lagrangians, Physica A96:327,1979
Effective chiral Lagrangians for nucleon-pion interactions and nuclear forces, Nucl.Phys.B363:3-18,1991 S.R. Beane, P.F. Bedaque, L. Childress, A. Kryjevski,
Singular potentials and limit cycles, Phys.Rev.A64:042103,2001, quant-ph/0010073
Renormalization of one-pion exchange and power counting, Phys.Rev.C72:054006,2005, nucl-th/0506005
6/2/2009
51
(GeV) E
20
16
3
2
15
19
General Relativity + higher-curvature terms
Chiral EFT ? QCD
(2 or 3 flavors)
QCD
(6 flavors)
Electroweak Th + higher-dim ops
QED Fermi Th
(SUSY)
?
GUT? nuclear physics atomic physics molecular physics condensed-matter physics and beyond
NRQED
1
6/2/2009
52
lattice
NCSM,…
necessary to extrapolate to large small
mπ
want model independence
lattice,… ?
6/2/2009
53
3 1 3 5
u d u d und f f f f s u d u d
μν μν μν μν
=
u q d ⎛ ⎞ =⎜ ⎟ ⎝ ⎠
f f
l l ν
+
⎛ ⎞ =⎜ ⎟ ⎝ ⎠
1 0 0 0 C ⎛ ⎞ =⎜ ⎟ ⎝ ⎠
23 3 13
1 3 6 Q τ
−
⎛ ⎞ + = = ⎜ ⎟ ⎝ ⎠ leptons: quarks: photon: Aμ
a
Gμ
gluons:
higher-dimension interactions: suppressed by larger masses unnaturally small T violation (strong CP problem)
d.o.f.s symmetries:
(3,1) global, (1) gauge, (3) gauge
em c
SO U SU
9
10 θ
−
<
2 ,
1
F W Z
M G ∝
QED + QCD
6/2/2009
54
4 4
1 Tr 2
QCD s
d x d x q i g G q G G
μν μν
⎧ ⎫ = ∂ / + / − ⎨ ⎬ ⎩ ⎭
L
u d
1
x x λ − →
32
q q λ → G G λ →
invariant under scale transformations
single, dimensionless parameter
but in
4
QCD
scale invariance “anomalously broken” by dimensionful regulator “chiral limit”
0.1 0.2 0.3 1 10 10
2
µ GeV αs(µ)
coupling runs
1GeV 1
s Q
α ∼ ∼
(“dimensional transmutation”)
6/2/2009
55
QCD
Non-perturbative physics at Assumption 1: confinement
Observation: (almost) all hadron masses Assumption 2: naturalness masses are determined by characteristic scale
1 GeV >
Q ∼
Observation: pion mass
QCD
π
NO! “spontaneous breaking” of chiral symmetry
6/2/2009
56
QCD L s L R s R
μν μν
5
1 2 q γ −
5
1 2 q γ +
invariant under
( ) ( ) ( )
exp
L R L R L R
q i q α τ → ⋅
L R
chiral symmetry by vacuum down to
L R
+ ∼
isospin
exp q i q α τ → ⋅
σ π
N
− +
6/2/2009
57
QCD
5 1
two isospin axis not shown chiral circle
fπ
pion decay constant (in chiral limit)
σ π
EFT =
piece invariant under
[function of
2 2
1 4 f
μ π
⎛ ⎞ − + ⎜ ⎝ ∂ ⎟ ⎠ π π …
μ
∂ π → + π π ε
Chiral Limit
6/2/2009
58
QCD s
μν μν
3
u d u d
4th component of 3rd component of
vector vector
5
5
(explicit chiral-symmetry breaking) (isospin violation)
u d
v.K. ’93
6/2/2009
59
QCD
5 1
two isospin axis not shown slightly-tilted chiral circle pion decay constant
σ π
EFT =
piece invariant under
[function of
]
μ
∂ π → + π π ε
+ piece in direction [function of explicitly]
π
( )
u d
m m ∝ +
+ isospin breaking CHIRAL SYMMETRY WEAK PION INTERACTIONS
Q ∝
( )
u d
m m ∝ −
92 MeV fπ ≅
Chiral Limit
6/2/2009
60
Two types of interactions: “soft” photons – explicit d.o.f. in the EFT “hard” photons – “integrated out” of EFT
μ μ μ
F A A
μν μ ν ν μ
= ∂ − ∂
2 2 und
μν μ ν
34 comp of antisymmetric tensor
5 ijk k j i
μ μ μ μ
4π α ∝
EFT =
soft photons + further isospin breaking
(and in particular)
e ∝
v.K. ’93
6/2/2009
61
5 u d u un d d
m m m m qi q θ γ + = + + … … L
4th component of vector
5
3 4
1 2
u d u d u d
m m m m m m P P θ − + − +
Hockings, Mereghetti + v.K., in preparation
e.g. T-violating quark EDM and color-EDM P-violating four-quark operators
… Kaplan + Savage ’96 Zhu, Maekawa, Holstein, Musolf + v.K. ’02 De Vries, Mereghetti, Timmermans + v.K., in progress
6/2/2009
62
“His scales are His pride”, Book of Job
QCD N
ρ π
nuc NN
π π …
NN
perturbative QCD
~1 GeV ~100 MeV ~30 MeV hadronic th with chiral symm
this talk
halo nuclei
brute force (lattice), …?
QCD
nuc
no small coupling expansion in
6/2/2009
63
pionful EFT
( ~ 2 )
N
m m mπ
Δ −
QCD
π
N n ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠
++ + −
⎛ ⎞ Δ ⎜ ⎟ Δ ⎜ ⎟ Δ = ⎜ ⎟ Δ ⎜ ⎟ ⎜ ⎟ Δ ⎝ ⎠
1 2 3
2 2 i π π π π π π π π
+ − + −
⎛ ⎞ + ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = = − − ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ π
2
1 4 f f
μ μ π π
⎛ ⎞ ⎛ ⎞ ≡ − + ⎜ ⎟ ⎟ ⎝ ⎠ ⎝ ⎠ ∂ ⎜ … D
2
π π
i
μ μ μ
≡ − ⋅ ∂ D t E
(chiral) covariant derivatives
f
μ μ π
≡ × E D π
pion fermions
4 3 34
's, 's, 's S P F +
( )
2 u d QCD
m m m M
π
+ = O
2 QC u d D
m m m M
π
+ ⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ O chiral invariants
Weinberg ’68 Callan, Coleman, Wess + Zumino ‘69
Non-linear realization of chiral symmetry
6/2/2009
64
2 2
2 { , , } { , , 2 2 2 2 } 2 2
, ,
p f
QCD QCD Q n EFT n p f n p f D QCD N C
M M M m m m f M c f f
π π π π
ψ ψ
Δ +
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ −
π D L D
( ) ∞ Δ Δ=
2 2 2 2 f f n p d Δ ≡ + + − ≡ + − ≥
chiral symmetry
calculated from QCD: lattice, … fitted to data
(1) = O
(NDA: naïve dimensional analysis)
isospin conserving isospin breaking
Schematically,
“chiral index”
6/2/2009
65
[ ]
( )
2 2 2 2 2 (0) 2 2 2 2 2 2 2
1 1 ( ) 1 1 2 2 2 4 1 ( ) ( ) 1 4 2 4 ( ) Η.c. ( ) 1 2
A A S T N
g N i N N N h i m f f f f f N S C N N C N f N m m
μ π π π π π π π
σ σ
+ + + + + Δ +
⎛ ⎞ ⎛ ⎞ = − + − − + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎡ ⎤ ⎛ ⎞ + − ⋅ × + + ⋅⋅ − + ⎜ ⎟ ⎢ ⎥ ⎣ ⎦ ⎝ ⎠ + Δ − + Δ + + Δ + ∂ ∂ ∂ ∇ ∂ − ⋅ + − ∇ ⋅ − π π π π π τ π π τ π T π … …
…
… …
( )
(1) 3 3 2 2 2 2 2 2 2 2 3 1 2 4
1 1 1 1 ( ) ( ) 2 4 2 2 1 ( ) ( ) 2 ( )( ) Η.c ( ) 1 4 ( )
ijk ab i j c k c b p N c A n N
N N N b b b m m ib N g iN N d N f f N N m f N m f f m
π π π π π π
τ π ε ε σ τ π π σ σ
+ + + + +
⎡ ⎤ ⎛ ⎞ ⎛ ⎞ ⎢ ⎥ = + ⋅ × + + − ⋅ + ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ ⎝ ⎠ ⎝ ⎣ ∇ ∇ ∂ ∇ ∂ ∂ ∇ ∂ ∇ ⎦ ⎡ ⎤ + − − + + ⎣ ⎦ ⎡ ⎤ − ⋅ + ⋅ + ⎣ ⎦ ⋅ − + ⋅ τ π π π τ π π π τ π τ π
…
( )
3
1 E N N
+
+ − …
(2) =…
L ⎠ Form of pion interactions determined by chiral symmetry … … … … … …
6/2/2009
66
Weinberg ’79 Gasser + Leutwyler ’84 … Gasser, Sainio + Svarc ’87 Jenkins + Manohar ’91 …
1 0.3 fm
QCD
M ≈
1 1.4 fm mπ ≅
dense but short-ranged long-ranged but sparse
1 1 E Q Δ ∼
QCD
π ν ν ν ν
min
2 2 2
i i i
A L V A ν ν = − + + Δ ≥ = −
# vertices of type i # loops
non-relativistic multipole pion loop
, 4
N QCD
Q Q m Q m Q M f
ρ π
π ⎧ ⎪ ⎨ ⎪ ⎩ ∼ …
expansion in
6/2/2009
67
Analogous to NRQED… = + + …
T
ππ
+ +
Weinberg ’79 Gasser + Leutwyler ’84 …
N
T
π
= + + + + … + +
Gasser, Sainio + Svarc ’87 Jenkins + Manohar ’91 …
current algebra
Weinberg ’66 …
( )
3 2 N QCD
M Q E m m
Δ
⎛ ⎞ − < ⎜ ⎟ ⎜ ⎝ − ⎟ ⎠ O
N.B. For a resummation is necessary
Phillips + Pascalutsa ’02 Long + v.K. , in preparation
Etc.
6/2/2009
68
Weinberg ‘90, ‘91
2
1
N
Q E m Δ ∼
3 3 2 2
(2 )
N
m l l k d V V π = + −
…
infrared infrared enhancement! enhancement!
V V
V V
e.g.
4 2 2 4 2 2
1 1 (2 )
N N N N
l l k l l k m m m d i V V i m i l π ε ε + − − − + − −
A-nucleon irreducible
2
4
N
V Q m π ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ ∼ O
2 2
(4 ) Q π
instead of
1 2 1 2
2 N
k E m =
l
6/2/2009
69
2 2 2 2 12 1 2 1 2 2
( ) 1 2 3 ˆ
A
S g i q q f q m f
π π π
σ σ + ⋅ ⎛ ⎞ ⎛ ⎞ ⋅ ⎜ ⎟ ⎜ ⎟ + ⎝ ⎠ ⎝ ⎠ τ τ
∼
1 2 1 2
( ) 3 ˆ ˆ ˆ q q S q σ σ σ σ = ⋅ ⋅ − ⋅
3 2 2 2 4 2 2 2 2
1 1 1 1 4 4 Q Q Q Q f Q Q Q f Q fπ
π π
π π ∼ ∼
3 2 2 4 2 2 2 2 2
4 1 1 1 4
N N
m m f Q Q Q Q Q f Q Q Q f f f
π π π π π π
π μ π ∼ ∼ ∼
2 2 QCD
Q M ⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ O
1 = O
for Q
π
μ ∼
3 2 2 2 4 2 2 2 2
1 1 1 4 1 4
N N
Q Q Q Q Q f m m Q Q Q m m f f
π π π
π π
Δ Δ
− − ∼ ∼
1 = O
tensor force
1
π
μ ≡
6/2/2009
70
= + + …
(0)
T
2 2
1 1 1 1 1 Q f f Q
π π π π
μ μ ⎧ ⎫ ⎛ ⎞ ⎪ ⎪ + + ⎨ ⎬ ⎜ ⎟ ⎛ ⎞ ⎪ ⎪ ⎝ ⎠ ⎩ ⎭ − ⎜ ⎟ ⎝ ⎠ ∼ … ∼ O O
bound state at
Q
π
μ ∼
2 2 N QCD
m Q M E
π
μ − ∼ ∼
4
nuc N
M f f f m
π π π π
π μ = ≈ ∼
Is 1PE all there is in leading order? That is, are observables cutoff independent with 1PE alone?
(0)
V
= +
= + + …
(0)
V
= +
(0)
T
(0)
V
(0)
V
(0)
V
(0)
V
Nuclear scale arises naturally from chiral symmetry
6/2/2009
71
2 (3) 1 2 2 2 2 2 1
1 1 1 1 ˆ ( ) ( ) ( ) 2 3 4 4 ( ) 3
r A m m r
m g V m f m m r r e e S r r r r r
π π
π π π π π
δ π π
− −
⎫ ⎧ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎪ ⎪ = ⋅ − + + + ⎨ ⎬ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎪ ⎪ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎩ ⎭ τ τ
( )
2 12 1 2 2 2 2 2 1
( ) ˆ 4 2 3
N A
m q q m S f q g i
π π π
σ σ μ π + ⋅ ⎛ ⎞ ⎛ ⎞ ⋅ ⎜ ⎟ ⎜ ⎟ + ⎝ ⎠ ⎝ ⎠ τ τ
∼
2 (3) 1 2 2
( ) ( ) 2 4
m r A
g V r r m f e r
π
π π
δ π
−
⎛ ⎞ ⎛ ⎞ = ⋅ − + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ τ τ
12
1 1 ( 1) 1 1 2 6 2 1 2 1 2 ( 1) 2 1 6 2 2 1 2 1 S j j j j j j j j j j j j j j j j − + + − − − + + + + + − + +
( )
2 2 2 2
' ie e i Q p p ε − − ∼ ∼
2
4 e V r r π → =
much more singular --and complicated!-- than
6/2/2009
72
Weinberg ’90, ‘91 Ordonez + v.K. ’92 Ordonez, Ray + v.K. ‘96 … Entem + Machleidt ’03… Epelbaum, Gloeckle + Meissner ’04 ...
i π
μ μ ∼
Assume contact interactions are driven by short-range physics, and scale with according to naïve dimensional analysis
QCD
M
( ) i
C
in LO
4
N
m π μ ≡
1
4
N
m π μ ≡
(0) (1) 1 2 1 2
( 1) ( 3) 4 4 C C σ σ σ σ ⋅ + ⋅ − −
1 (3)
4 ( ) ( )
N
m V r r μ π δ =
4 ( ) ( )
N
m V r r μ π δ =
2 2
4
N QCD
Q m M
π
π μ ∼
in NLO
(terms linear in break P, T )
QCD
Q M
etc. (W power counting)
6/2/2009
73
2N
V
= + + + + + + + … + … + … higher powers of
3N
V
= + + + … + … + … + Etc. more nucleons
Ordonez + v.K. ’92 v.K. ’94 …
Q
6/2/2009
74
2 2 2
1
QCD
Q f M
π
⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ O
2
1 fπ ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ O
3 2 3
1
QCD
Q f M
π
⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ O
4 2 4
1
QCD
Q f M
π
⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ O
LO 2-body 3-body 4-body NLO
2
1
QCD
Q M fπ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ O
NNLO NNNLO (parity violating) ETC.
6/2/2009
75
2 3 4 N N N
A canon emerges! Similar explanation for
IS IV CSB
2 3 N N N
Weinberg ’90, ‘91 v.K. ’93 Rho ’92
many-body forces
isospin-breaking forces external currents
6/2/2009
76
similar to phenomenological potential models, e.g. AV18 – (OPE)^2 + non-local terms
2N
V
= + + + + + + + … + … + …
Ordonez + v.K. ’92 v.K. ’94 … Stoks, Wiringa + Pieper ‘94
6/2/2009
77
6/2/2009
78
Nijmegen PSA of 1951 pp data But: NOT your usual potential!
Ordonez + v.K. ’92 (cf. Stony Brook TPE)
e.g., + + + …
chiral v.d. Waals force
6 2
1 for m r
π →
∼
Rentmeester et al. ’01, ‘03
σ ω +
2 min
long-range pot #bc OPE 31 2026.2 OPE TPE ( ) 28 1984.7 OPE TPE ( ) 23 1934.5 Nijm78 19 1968.7 lo nlo χ + +
parameters found consistent with πN data!
2.0 2.5 3.0 3.5 4.0
r [fm]
VC(r) [MeV]
2π σ + ω
Isoscalar Central Potential
Kaiser, Brockmann + Weise ’97
at least as good!
Similar results in other channels, e.g. spin-orbit force!
6/2/2009
79
Tucson-Melbourne pot with
3N
V
= + + … + +
two unknown parameters
v.K. ’94 Friar, Hueber + v.K. ‘99 Coon + Han ’99 ...
2
2 a c c m a
π
→ − →
3 2 2
, ' ' ' '
N q q
q t a b c q q q d q q
π αβ αβγ γ αβ
δ ε τ σ ⎡ ⎤ = + ⋅ + + − ⋅ × + ⎣ ⎦
TM’ potential
Coon et al. ‘78 Fujita + Miyazawa ‘58
6/2/2009
80
Many successes of Weinberg’s counting, e.g., To N3LO (w/o deltas), fit to NN phase shifts comparable to those of “realistic” phenomenological potentials
Ordonez, Ray + v.K. ’96 … Epelbaum, Gloeckle + Meissner ’02 Entem + Machleidt ’03 … Entem + Machleidt ’03
NLO NNLO NNNLO Nijmegen PSA VPI PSA
6/2/2009
81
With N3LO 2N and N2LO 3N potentials (w/o deltas), good description of
Epelbaum et al. ’02 Gueorguiev, Navratil, Nogga, Ormand + Vary ’07
6/2/2009
82
measured: Illinois ‘94, SAL ‘00, Lund ‘03 extracted nucleon polarizabilities: Beane, Malheiro, McGovern, Phillips + v.K. ‘04 threshold amplitude predicted: Beane, Bernard, Lee, Meissner + v.K. ’97 confirmed: SAL ’98, Mainz ‘01 measured: IUCF ’90-…, TRIUMF ’91-…, Uppsala ’95-… S waves sensitive to high orders: Miller, Riska + v.K. ‘96 P waves converge, fix 3BF LEC: Hanhart, Miller + v.K. ‘00 CSB asymmetry sign predicted: Miller, Niskanen + v.K. ’00 confirmed: TRIUMF ‘03 measured: IUCF ’03 mechanisms surveyed: Fonseca, Gardestig, Hanhart, Horowitz, Miller, Niskanen, Nogga +v.K. ’04 ’06 + PARITY, TIME-REVERSAL VIOLATION, etc.
6/2/2009
83
2 12 3 1 2 3
ˆ ( ) 2 4 ( )
m r A
m f r e m g S r
π
π π π
π
−
⎛ ⎞ ⎧ ⎫ ⋅ + ⎨ ⎬ ⎜ ⎟ ⎝ ⎠ ⎩ ⎭ τ τ ∼ …
singular potential not enough contact interactions for renormalization-group invariance even at LO BUT attractive in some channels
6/2/2009
84
6/2/2009
85
1
n
r
1 R Λ ≡
(3)
( ) ( ) ( ) 1 C r r V R R δ θ
Λ
Λ ⎛ ⎞ → − ⎜ ⎟ ⎝ ⎠
n
OPE:
2 1 ( ) exp( )
N
r m m r r f r r m m
π π π
μ λ = ⎧ ⎪ = ⎪ ⎨ = ⎪ ⎪ = − ⎩
( )
n n
r r r R u r ψ ≡ ∼
2 2
2 cot 2 , ,
n
m R m R V V F r R λ − − =
so that matching
s s
6/2/2009
86
( )( )
1 4 2 1
( ) cos + ( ) 2 1
n n n n
r r r u r r n r λ λ δ
− −
⎛ ⎞ ⎛ ⎞ ⎜ ⎟ = + ⎜ ⎟ ⎜ ⎟ − ⎝ ⎠ ⎝ ⎠
Beane, Bedaque, Childress, Kryjevski, McGuire + v.K. ’02
r ur4 exact 1st 2nd 3rd
R
model dependence
Two regular solutions that oscillate!
if no counterterm, will depend on cutoff
0.03 0.04 0.05 0.06 0.07 0.08 0.09 1 2 3 4 5 6
H2
2 V m R = −
limit-cycle-like behavior
( ) ( )( )
1 2 2 1
, tan 4 2 1 ,
n n n n
R R R r n r n r F λ λ λ δ
− −
⎛ ⎞ ⎛ ⎞ ⎜ ⎟ = − + + ⎜ ⎟ ⎜ ⎟ − ⎝ ⎠ ⎝ ⎠ … determined by low-energy data
6/2/2009
87
Same is true in all channels were attractive singular potential is iterated
3 2
( 1) l l r r r λ + − +
2
( 1) l l r +
3
r r λ −
3 2 ( 1)
l
r l l r λ = +
but
( 1) l l λ +
l
r r M ∼
where counterterms are needed for
Beane, Bedaque, Savage + v.K. ‘02 Nogga, Timmermans + v.K. ’05 Pavon Valderrama + Ruiz-Arriola, ’06 Birse, ’06, ’07 Long + v.K., ‘07
[ ] ( )
3 2
( 1) 4 27 l l r λ +
3 ( 1) 5 2
QCD
M l l
π
μ + < ∼
l <
6/2/2009
88
certain counterterms that in Weinberg’s counting were assumed suppressed by powers of
QCD
Q M Q lfπ
are in fact suppressed by powers of short-range physics more important than assumed by Weinberg’s; most qualitative conclusions unchanged, but quantitative results need improvement ACTIVE RESEARCH AREA
6/2/2009
89
1
100 200 300 20 40 60
| P | (MeV)
δ
100 200 300 20 40 60
| P | (MeV)
δ
Nijmegen PSA
985 MeV Λ = 492 MeV Λ = 140 MeV Λ =
LO EFT NLO EFT
δ [deg]
2 4 6 8 50 100 150 200
TL [MeV]
δ [deg]
50 100 150 200
TL [MeV]
0.2 0.4 0.6 0.8 1 1P1 1D2 1G4 1F3
Other singlet channels
1
( 20fm )
−
= Λ
Nijmegen PSA LO EFT
Nogga, Timmermans + v.K. ’05 (cf. Birse + McGovern ’04) Beane, Bedaque, Savage + v.K. ‘02
6/2/2009
90
50 100 150 200
TL [MeV]
δ [deg]
50 100 150 200
TL [MeV]
3F3 3P1
1
( 20fm )
−
= Λ
Nijmegen PSA LO EFT
Repulsive triplet channels
Nogga, Timmermans + v.K. ’05 (cf. Pavon Valderrama + Ruiz-Arriola, ’06)
2 4 6 8 10 12
δ [deg]
5 10 15 20 25 30 50 100 150 200
TL [MeV]
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
δ [deg]
50 100 150 200
TL [MeV]
3P0 3P2 3F2
ε2
Attractive triplet channels
100 200 50 100 150
| P | (MeV)
δ χ-limit
π
m 3 1
S
LO EFT
Nijmegen PSA Beane, Bedaque, Savage + v.K. ‘02
6/2/2009
91
Chiral symmetry plays an important role, in particular setting the scale for nuclear bound states A low-energy EFT of QCD has been constructed and used to describe nuclear systems A new power counting has been formulated: more counterterms at each order relative to Weinberg’s; expect even better description of observables Stay tuned: next, how to extend EFT to larger systems Nuclear physics canons emerge from chiral potential
6/2/2009
92
Supported in part by US DOE
6/2/2009
93
6/2/2009
94
Effective field theory of short-range forces, Nucl.Phys.A645:273-302,1999, nucl-th/9808007 P.F. Bedaque, H.-W. Hammer, and U. van Kolck, The three-boson system with short-range interactions, Nucl.Phys.A646:444-466,1999, nucl-th/9811046
No-core shell model in an effective-field-theory framework, Phys.Lett.B(to appear),2007, nucl-th/0609023 P.F. Bedaque, H.-W. Hammer, and U. van Kolck, Narrow resonances in effective field theory, Phys.Lett.B569:159-167,2003, nucl-th/0304007
6/2/2009
95
“His scales are His pride”, Book of Job
no small coupling expansion in
π ρ
N QCD
π π
NN nuc
NN
perturbative QCD
~1 GeV ~100 MeV ~30 MeV
this talk
Chiral EFT
brute force (lattice), …?
QCD
nuc
hadronic th with chiral symm
6/2/2009
96
6/2/2009
97
1
(real) bound state = deuteron
1 ~
d NB
(virtual) bound state
1
1 4.5 fm ≅ ℵ
e.g. NN
*
0 ~
d N
π
6/2/2009
98
pionless EFT
3 2 4 2 2 2
N EFT N
+ + + + + + + +
spin, isospin
nuc
6/2/2009
99
4 2 4 2 2
1 1 ( ) ( ) ( ) 2 2 2
N N
m d l C l l l i m p i p p p l π ε ε + + + − + − + Λ − +
1
3 2 3 2 2
1 ( ) 2
N
m d l i l i k C ε π Λ = − − −
2 2
1 1 ( ) 2
N
i C dl dl l m l i k k k i π ε ε
Λ Λ
⎧ ⎫ ⎪ ⎪ = − + ⎨ ⎬ + + − − ⎪ ⎪ ⎩ Λ ⎭
2 2 2
1 ( ) 2 4 4
N
m i C k i k π π π ⎧ ⎫ ⎛ ⎞ ⎪ ⎪ = − + + ⎨ ⎬ ⎜ ⎟ ⎪ ⎪ Λ ⎠ ⎩ Λ ⎝ Λ ⎭ O
1 2 1 2
l
2
Q M
1 2
l
2
Q M
~ ( ) iC Λ
2
2
N
k p m ≡
absorbed in
0( )
absorbed in
2( )
2 2
~ ( ) iC k Λ
non- analytic in E
2
( ) ( ) iC I Λ ≡ Λ −
etc.
6/2/2009
100
( ) 2 2
( ) ( ) ( ) 1 ( ) 2 1 ( ) 2
N N R
C C C C m m C C π π Λ Λ Λ Λ Λ ⎧ ⎫ → ≡ − + = ⎨ Λ ⎬ ⎩ ⎭ + Λ …
( )
R N nuc
( ) 2 2 2 2
( ) ( ) ( ) 4
N R
m C C C C π Λ Λ Λ ≡ − Λ → +…
( ) ( )2 2
N R R nuc
( )
R N
nuc
2 ( 2 2 )
n R N uc
2
Naïve dimensional analysis etc.
6/2/2009
101 2
N nuc
But in this case:
( ) R
( )2
N R
) 2 ( 2 R
1
nuc
if
nuc
etc. no b.s. at , no good: just perturbation theory
nuc
assume no other, e.g. still , etc.
2
M M ∼
N M
N
N
N
1 ∼
2
N nuc
need one fine-tuning: for Q
6/2/2009
102
= + + …
(0)
T
bound state
k i = ℵ
2
2
N
E m − = ℵ
2 0 0 0 0 2
4 1 1 4 2 ( )
N N
m i i iC C I C I I C k k i C m π π π Λ Λ = − + + = = ⎛ ⎞ + + + ⎠ Λ + ⎜ ⎟ ⎝ … O
( )
4
R NC
m π = ≡ℵ
scattering length
1 a = ℵ
s wave
2
4 1 ,
N
k k i ik m π Λ Λ ⎡ ⎤ ⎛ ⎞ = + ⎢ ⎥ ⎜ ⎟ ⎠ ℵ + ⎝ ⎣ ℵ ⎦ O k ℵ ∼
v.K. ’97 ’99 Kaplan, Savage + Wise ’98 Gegelia ’98
6/2/2009
103
(1)
V
=
(0) nuc
T M Q ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ O
(1)
T
(1)
V
=
(1)
V
(0)
T
(0)
T
(1)
V
(0)
T
(1)
V
(0)
T
+ + +
2
4 ~
nuc N nuc NN
M m M Q T i i Q Q π ℵ ℵ ⎧ ⎫ ⎛ ⎞ ⎪ ⎪ + + ⎨ ⎬ ⎜ ⎟ + + ⎝ ⎠ ⎪ ⎪ ⎩ ⎭ …
etc. scattering length effective range
0 ~ 1
a ℵ
0 ~ 1 nuc
s wave
p, other waves
1 − = ν
6/2/2009
104
Example: square well
2 2
( ) 1 r V mR R r α θ ⎛ ⎞ = − − ⎜ ⎟ ⎝ ⎠
2 2 2 2 2 2 2 2 2
cot ( ) 1 cot
R NN k i
i r k k k k k k R R R i k T e i r R α α α α
−
⎡ ⎤ + + + ⎢ ⎥ = − − ⎢ ⎥ + + − ⎣ ⎦
(2 1) 2
c
n α π ≡ + tan 1 a R α α ⎛ ⎞ = − ⎜ ⎟ ⎝ ⎠
2 2 2
1 3 R R a R r a α ⎛ ⎞ = − − ⎜ ⎟ ⎝ ⎠
generic
c
1 2
1 1 1
c c
R a α α α
−
⎛ ⎞ = − − + ⎜ ⎟ ⎝ ⎠ ℵ … ∼
1 R R r = +… ∼
zero-energy poles when
1 1
c
α α −
fine-tuning
6/2/2009
105
R
r r ψ
2 2 c
mR α −
r a
e
−
∝
1 a = ℵ
2
1 ma −
N
1
e N
−
2 2
mR α −
In the quantum world,
size much larger than the range of the force provided there is fine-tuning
6/2/2009
106
200 400 600
10
mπ ( MeV )
3S1 (fm)
50 100 150 200 1 2 3 4
mπ ( MeV ) Bd ( MeV )
triplet scattering length Deuteron binding energy Fukugita et al. ‘95
Lattice QCD: quenched
EFT: (incomplete) NLO
Beane, Bedaque, Savage + v.K. ’02 …
Large deuteron size because
QCD
m M
π ∗
QCD
M m m
π π ∗
∼
nuc
π π π ∗ ∗
unitarity limit
2
a → ∞
6/2/2009
107
Feshbach resonance
2
a → ∞
unitarity limit
MIT group webpage
quark masses analog to magnetic field: close to critical values
2 2
200 MeV
u Q d CD
M m m m
π ∗ ∗ ∗
+ =
6/2/2009
108
Kaplan ’97 v.K. ’99
2 4 2 3
EFT N N N
+ + + + + + +
= + + …
NN
T
2
, g C = Δ …
= integrate out auxiliary field: same Lag as before with
2
N
sign
6/2/2009
109
1
100 200 300 45 90
k (MeV) δ
100 200 300 45 90 135 180
k (MeV) δ Chen, Rupak + Savage ’99
*
d
fitted
predicted
1 − = ν 1 − = ν
1 + = ν 1 + = ν 2 + = ν
Nijmegen PSA Nijmegen PSA
1 1
d
fitted
d
predicted
6/2/2009
110
Nd
T
= + + …
3 3 0 (2 ) ONE Nd Nd ONE
Λ
Nd
T
= +
2 2
~
N
g Q m
2 3 2 2 2 2
~ ~ 4
N N
m Q g Q Q m g Q π ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ ℵ
6/2/2009
111
0.02 0.04 0.06
k
2 [fm −2]
−0.2 −0.1
k cotδ [fm
−1]
π λ λ 4 3 3 ≡ ≤
c
2 3
EFT
+ + +
2 3
4 1 ~
N nuc
m M D π ⎛ ⎞ ⎜ ⎟ ⎝ ⎠
1) (
naïve dimensional analysis
+ = ν
1) ( fm 10 . 33 . 6
2 3
+ = ± = ν a (exp) fm 02 . 35 . 6
2 3
± = a
Bedaque + v.K. ’97 Bedaque, Hammer + v.K. ’98 … v.Oers + Seagrave ‘67 Dilg et al. ‘71
predicted
1 − = ν 1 + = ν
no three-body force up to
3 + = ν
2
1
Nd p
p T
ℵ
⎯⎯⎯ →
p Nd
T
ℵ
∂ Λ ⎯⎯⎯ → ∂
3-body interaction
6/2/2009
112
2 1
π λ λ 4 3 3 ≡ >
c
Bedaque, Hammer + v.K. ’99 ’00 Hammer + Mehen ’01 Bedaque et al. ’03 …
cos ln
p Nd
T A s p δ
ℵ
⎛ ⎞ ⎯⎯⎯ → + Λ ⎜ ⎟ ⎝ ⎠
p Nd
T
ℵ
∂ Λ ⎯⎯⎯ →≠ ∂
2 2
4 1 ~
N nuc
D m M π ⎛ ⎞ ℵ ⎜ ⎟ ⎝ ⎠
6/2/2009
113
Nd
T
= + + …
3 3 0 (2 ) TBF Nd TBF
Λ
Nd
T
= + +
Nd
T
+ + + …
2
4 ~ π ℵ
2 2
~
N
g Q m
3 3 0 (2 ) ONE Nd Nd ONE
Λ
6/2/2009
114
2 1
π λ λ 4 3 3 ≡ >
c
(limit cycle!)
0.02 0.04 0.06
k
2 [fm −2]
−2 −1
k cotδ [fm
−1]
Bedaque, Hammer + v.K. ’99 ’00 Hammer + Mehen ’01 Bedaque et al. ’03 …
v.Oers + Seagrave ‘67 Dilg et al. ‘71 Kievsky et al. ‘96
2 1 =
fitted
t
t
predicted
1 − = ν
) 1 ( − = ν
cos ln
p Nd
T A s p δ
ℵ
⎛ ⎞ ⎯⎯⎯ → + Λ ⎜ ⎟ ⎝ ⎠
p Nd
T
ℵ
∂ Λ ⎯⎯⎯ →≠ ∂
2 2
4 1 ~
N nuc
D m M π ⎛ ⎞ ℵ ⎜ ⎟ ⎝ ⎠
6/2/2009
115
potential models varying
*
Λ
Bedaque, Hammer + v.K. ’99 ’00
6/2/2009
116
Hammer, Meissner + Platter ‘04
7,5 8 8,5 9 Bt [MeV] 20 25 30 35 Bα [MeV]
potential models
varying
*
Λ
Tjon line
exp
6/2/2009
117
nuc nuc
(0) (1)
Barrett, Vary + Zhang ’93 …
6/2/2009
118
For a given computational power, as number of nucleons grows, number of one-nucleon states gets more limited What are the “effective interactions” in the model space?
max
2 , n l N n l
+ ≤
max
ω
“model space” “excluded space”
6/2/2009
119
arbitrary truncation (“cluster approximation”)
1 for fixed P a b → +
convergence:
for fixed a b A P + →
eff 2 ' ' ' ' 1
1
a a a a a a a b A
O PO P PO P PHQ QO P E QH Q O O O O
+ +
→ = + + − = + + + + + … … …
Feshbach projection
Barrett, Vary + Zhang ’93 …
start with god-given (can be non-local!) potential, and run the RG in a harmonic-oscillator basis
e.g., chiral pot from last lecture
6/2/2009
120
max
Stetcu, Barrett +v.K., ’06 Stetcu, Barrett, Vary + v.K., ’07 Vary + v.K., in progress
IR UV
cutoffs
max
N
N
6/2/2009
121
12
2 2 ( )
ˆ exp 2 2
N l l nl s j nl n l s j N
r N L r m r m r Y φ χ ω ω
+ ⎛
⎞ ⎡ ⎤ = − ⊗ ⎜ ⎟ ⎣ ⎦ ⎝ ⎠
1 2 1 ' ' ' 2
,
nlj n l j JI
ψ ξ ξ φ ξ φ ξ ⎡ ⎤ = ⎣ ⎦
1
: relative coordinates
(reduced dimensions, but difficult antisymmetrization)
: Slater-determinant basis
Navratil, Kamuntavicius + Barrett ‘00
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3
1 1 1 1 2 3 2 2 2 3 3 3
, ,
n l j n l j n l j n l j n l j n l j n l j n l j n l j
r r r r r r r r r r r r φ φ φ ψ φ φ φ φ φ φ =
single particle
1
code: REDSTICK
6/2/2009
122
1 1 2 2 1 2
1 2 1 2 1 2 1 2
! ! , ( ) , 3 3 2 2
n n
n n n l r n l L L n n δ ⎛ ⎞ ⎜ ⎟ = = ⎜ ⎟ ⎜ ⎟ Γ + Γ + ⎝ ⎠ ∼
matrix elements of 2-, 3-body delta-functions, e.g. parameters fitted to d, t, α ground-state binding energies
(0) (1)
, , C C D Λ Λ Λ
6/2/2009
123 100 200 300 400 Λ [MeV] 20 25 30 35 First 0
+0 [MeV]
hw = 1 MeV hw = 2 MeV hw = 3 MeV hw = 4 MeV hw = 5 MeV hw = 6 MeV 2 4 6 hω [MeV] 20 25 30 35 40 45 50 First 0
+0 [MeV]
170 250 300 350 400
0.3 0.6 0.9 19.0 20.0 21.0 22.0
preliminary 4He
Exp
Stetcu, Barrett +v.K., ‘06
( ) ( ) b a ω ω + Λ
fits
Λ
2
( ) α β γ ω ω + +
16 N ≤
6/2/2009
124 50 100 150 200 250 300 Λ [MeV]
Eg.s. [MeV] hω=1 MeV hω=2 MeV hω=3 MeV hω=4 MeV hω=5 MeV 1 2 3 4 5 hω [MeV]
Eg.s [MeV] Λ=100 MeV Λ=200 MeV Λ=300 MeV Λ=400 MeV Λ=500 MeV
6Li
preliminary
Stetcu, Barrett +v.K., ‘06
fits fits
( ) ( ) b a ω ω + Λ
( ) α β γ ω ω + +
gs
B ≅
max
8 N ≤
6/2/2009
125
N c p N c
4
5
6
* *
8 MeV 20 MeV 28 MeV B E B B B
α α α α α
≅ ⎫ = − ≅ ⎬ ≅ ⎭
resonance at
~ 1 MeV
n
E
p n n n n p
1 ℵ 1
c
M
2 / 3
p
core excitation energy particle separation energy
2 ~ 1 MeV n
E
bound state at (esp. near driplines)
6/2/2009
126
6/2/2009
127
halo EFT
non-relativistic multipole simplest formulation: auxiliary fields for core + nucleon states e.g.
4
4
1
2 3 2 1 2 1
c
N c c
6/2/2009
128 0.0 2.0 4.0 EN [MeV] 40 80 120 δ1+ [degrees] 0.0 2.0 4.0 EN [MeV] −50 −30 −10 δ0+ [degrees] 0.0 2.0 4.0 EN [MeV] 10 20 30 40 50 δ1− [degrees]
Bertulani, Hammer + v.K. ’02 PSA, Arndt et al. ’73
0.80 MeV
R
E ≅ ( ) 0.55 MeV
R
E Γ ≅
scatt length only
LO NLO LO NLO LO, NLO, NNLO NNNLO
6/2/2009
129
Higa, Hammer + v.K. ‘08
90 120 150 180 1 2 3 δ0 [degrees] ELAB [MeV] LO NLO Afzal et.al.
92.07 0.03 keV
R
E = ±
( )
5.57 0.25 eV
R
E = ± Γ
fitted with
Extra fitting parameters none
1 3 C k r r = −
1 15 C k P P = +
2
1 1 3.6 fm
e C m
Z k
αα
μ ≡
r = −
&
1 2 1.8 fm
C E M
k a =
fine-tuning of 1 in 1000!
2 3 c
a M ℵ ∼
0 ~1 c
r M
fine-tuning of 1 in 10
6/2/2009
130
8 7
4 6
3
Bedaque, Hammer + v.K. ’99 Chen et al. ’00
Hammer + v.K., in progress
4 4
Bertulani, Higa + v.K., in preparation Kong + Ravndal ’99
12 4 4 4
6/2/2009
131