Halo Effective Field Theory
description
Transcript of Halo Effective Field Theory
04/22/23 v. Kolck, Halo EFT 1
Background by S. Hossenfelder
Halo Effective Field Theory
U. van Kolck
University of Arizona
Supported in part by US DOE
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Hnning
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Outline
EFT
Nucleon-alpha system
Alpha-alpha system
Other systems
Outlook
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Nuclear physics scales
expansion in
,4,,~ fmmM NQCD
~ , 1/ ,nuc NNM f r m
perturbative QCD
Qln
~1 GeV
~100 MeV
hadronic theory Chiral EFT
unknown;use brute force
(lattice, …)
QCDQ M
s
no small coupling constants!
04/22/23 v. Kolck, Halo EFT 5
tripletscattering
length
Deuteron bindingenergy
Fukugita et al. ‘95Lattice QCD:quenched
EFT:
(incomplete) NLO
Beane, Bedaque, Savage + v.K. ’02
…
Large deuteron size because
cf. Beane et al ‘06
QCDm M
QCDMm m
nuc
m mM
m
unitarity limit
2a
1 4.5 fma
1/ 1 fmr m New scale
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Nuclear physics scales
expansion in
,4,,~ fmmM NQCD
~ , 1/ ,nuc NNM f r m
NNa1~
perturbative QCD
Qln
~1 GeV
~100 MeV
~30 MeV
hadronic theory
Contact EFT
Chiral EFT
unknown;use brute force
(lattice, …)
QCDQ M
nucQ M
s
no small coupling constants!
04/22/23 v. Kolck, Halo EFT 7
r
r
3 0d Ar rV r
0r
0 3 0 3 0 30 0 0i i ji i jr r r r r r r rA d A d A d r
0 0qA 0D E 21
03 ij ij i jq r Q E
r
O 2
2r
O
Expansion in powers ofr
distance scale of underlying distribution
distance scale of interest
All possible interactions allowed by gauge invariance
04/22/23 v. Kolck, Halo EFT 8
pionless EFT
• degrees of freedom: nucleons
• symmetries: Lorentz, B, P, T
• expansion in:
,N
nuc
QQQ
m
mM
multipole
non-relativistic
~ nucQ M
simplest formulation: auxiliary field for two-nucleon bound states
31d S
N N vector field d
10d S isovector field d
Kaplan ’97
v.K. ’99
4 2
3
2
0
0
2 2
8 2
EFN
T
N N
gN i N d d d NN N N d hd d N N
N N d i d
m
m m
L
omitting spin, isospinsign
04/22/23 v. Kolck, Halo EFT 9
First orders applyalso to atoms 1nuc vdWM l
4
6( )
2vdWV rmr
l from
describes structure and reactions of bound states --
deuteron, triton, alpha particle
can be extended to p-shell nuclei with No-Core Shell Model
makes evident new phenomena --
from one-parameter three-body force at LO:
SO(4) invariance, limit-cycle behavior, Phillips line, Efimov spectrum
6A
Bedaque, Hammer + vK ’98, ’99, ‘00
Hammer, Platter + Meissner ’04Stetcu, Barrett + v.K. ’07
…
- many-body systems get complicated rapidly, just as for models
04/22/23 v. Kolck, Halo EFT 10
Halo/Cluster states
loosely bound nucleons around tightly bound cores
new scale leads to proliferation of shallow states (near driplines):
2 2
2 2c
coreseN
pN m
EM
Em
O =Oseparation
energy
core excitation
energy
pn
n
n
pp
1
1 cM
pn
np
core
2
2 N
m
m
O
04/22/23 v. Kolck, Halo EFT 11
He5
He6
resonance at 0.8 MeVnE 2/3p
0.97 MeVnnE
e.g.
8 Be 0.09 MeVE resonance at0s
12C
bound state at0s
0.38 MeVE 0s resonance at
He4 *
*
8 MeV 20 MeV
28 MeV
BE B B
B
18 MeVRk Em
38 V2 MeR nNmk E
1 eV2 95 Mc NM m E
m
1.6 MeVnE bound state at2/3p
5 Li resonance at 1.7 MeVpE 2/3p 56 V2 MeR pNmk E
6 Be 1.4 MeVppE 0s resonance at
0.19 MeVpE 9 B 2/3p resonance at
9 Be
04/22/23 v. Kolck, Halo EFT 12
halo EFT
• degrees of freedom: nucleons, cores
• symmetries: Lorentz, B, P, T
• expansion in: non-relativistic
multipole
simplest formulation: auxiliary fields for core + nucleon states
e.g.
He4
4 He N
fieldscalar
3
1
field 3/2-spin1
field 1/2-spin1
field 0-spin0
23
21
21
Tp
Tp
ss
~ cQ M
,
,N c
c
m mQ QQ
M Q m
04/22/23 v. Kolck, Halo EFT 13
0 0
2 2
2
3 3
33
0
11 1 1
3
0
1
3 0
2 2
2( )
( ) H.c.2
( ) H.c.2
( ) H.c.2
EFN
N
T N i N i
T i T
gT S N N
gs
m m
m m
s s N
gT T T N N
L
Bertulani, Hammer + v.K. ’02
Bedaque, Hammer + v.K. ’03
spin transition operator
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nT =n + + …
=
3 3
3 33 32
2
2E
i i
k
reduced mass
2/3presonance at
and
if
~Q
33 0 22
323
1~ , ~ ,
4 cM
g
+ + …=
2 2~Q
2 2 3 2
2 2 2 2 2 2 2 2
4~ ~
4c c
Q Q Q Q
Q Q QM MQ
1 12
1~ , ~ ,c
c
a r MM
1
width
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0 0
1 13
1 1~ , ~ ,
1~ , ~ ,
c c
cc
a r
a
M
MM
r
M
2 20 1
1
2
0 ~ ~ ~ ,
1 1~ , ~ ,
4 4
c
c
g
M
g
M
other waves:
+ + …=
1~
cM
2 3
2
1 4 1~ ~
4c c c
Q Q
M M M
Q
2 2 3 3
2 2 2
1 4 1~ ~
4c c c cM
Q Q
M M
Q
MQ
21s
21p 1
~cM
04/22/23 v. Kolck, Halo EFT 16
22 32 2 2
2 2 2 2 2 2
24~ 1 0
0 0 0
c
cn
c
c
M M
M MT
Q Q Q Q Q
Q Q Q
Q
2/1p
2/1s
2/3p
0 21
21
1~ cMa
10 ~ ca M
1 ~ cr M
11 ~ cM
P
10 ~ cr M
2 2 4 2 1
12 1
, 2 1 cos2 4l l
ll l
ll
rT lk k k kP i
ak
P
31 ~ ca M
etc.
04/22/23 v. Kolck, Halo EFT 17
Bedaque, Hammer + v.K. ’03
Haesner et al. ‘83
NNDC, BNL
0
1
2
04/22/23 v. Kolck, Halo EFT 18
except at2
cMQ
O where
3~ cM
+ + …
=
2 2 3
3 2 2 3
4~ ~
4c c
c
M M
M
2/3p
enhanced by cM
resum self-energy
2 ( ) 2
( ) 22 Rn
EiT
iE E E E
04/22/23 v. Kolck, Halo EFT 19
1
mod 1
Bertulani, Hammer + v.K. ’02
Haesner et al. ‘83
NNDC, BNL
0
04/22/23 v. Kolck, Halo EFT 20
Bertulani, Hammer + v.K. ’02
01 0
2
mod 1
1PSA, Arndt et al. ’73
1,0,1
0.80 MeVRE ( ) 0.55 MeVRE
scatt length only
04/22/23 v. Kolck, Halo EFT 21
21
1~ cMa
10 ~ ca M
1 ~ cr M1
1 ~ cMP
10 ~ cr M
31 ~ ca M
Arndt et al ‘73
1 ~ cr Mcf.
100MeVcM
30MeVconsistent…
04/22/23 v. Kolck, Halo EFT 22
p + electromagnetic interactionsEFT L
iZ eA
F
Coulomb p em Ckk
Z
k
kZ
Sommerfeldparameter
corrections
2
,pm
Q
m
O
transverse photons
NN iZ eA N
Higa, Bertulani + v.K. in progress
CG CG= + + + …
= +
3
22 2 24
4
1 4
4p
Qe
Q QZ Z
Q
non-perturbative for 1k
2
2
4p
Q
Z Z e
Q
04/22/23 v. Kolck, Halo EFT 23
CT = CG 2 2 2csc exp ln csc 22 2C
ClT i
ki
11
ln2 1
l i
i l i
Coulombphase shift
pT = CT CST+
pureCoulomb
Coulomb/short-rangeinterference
CST = + + …
= CG
CG
CG
04/22/23 v. Kolck, Halo EFT 24
0
2
H.c.4EFT d i d dm
g
L
Higa, Hammer + v.K. ’080 0 spin-0 field s d
3Rk deep non-perturbative Coulomb region!
20exp 22
2CS
C
C iT
Hk K
2 2
exp 2 1C
Sommerfeldfactor 2Re 1 ln
2
iH i C
2
0cot2
CK i H
CST =
0s
CG
CG
Landau-Smorodinskyfunction
Coulomb-corrected phase shift
04/22/23 v. Kolck, Halo EFT 25
+ + …=
0Ck
2 2~Q
2 3 2
2 2 2 2 2 2 2 2
4~ ~
4c cQ Q
Q Q Q
M M
Q Q
1
unitarity limitin LO
( )2
2 3R g
D
2
( )
2
2
2 2
2 22
2C
C
R
Kg g
kk
0
1
a
0
2
2rk
2 Ck H ik
2
~cM
2
c
Q
M
Q
2( ) ~R
2
2
~4
cg M
: renorm scale4
0 4
k P
3
4
c
Q
M
2cM
O2
O
“usual” fine-tuning?
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2 4
3
22
6 60 exp 2 1C
CC C
kk
k
k
k
k iH
0Ck
2 120 MeVC ck M
0
0
40
212
2 4C a
krKk
k P
2
~cM
2
c
Q
M
0 0
1
3 Ckr r 0 30
1
15 Ck P P
2
c
Q
M
3
4
c
Q
M
3
4
c
Q
M
since 1 1R Ck kCoulomb “short-
ranged”
+ + …
= CG
exponentially suppressed
04/22/23 v. Kolck, Halo EFT 27
0exp 2 22
22CS
R
E
E
i
E ET
iE
2 ( )
2 ( )02
2
11 1
1 4
R
R
k
C
kR R Rk
ee
eE E E E E
P
02 ( )
0
2
0
4 11
1R
CR
Rk
k
r e r
kE
P
02
0 0 0
2
0
11
2RR
a r a r
kE
P
0 0a
0 0r 0
1
3 C
rk
0
Expansion around pole:
exponential suppression
04/22/23 v. Kolck, Halo EFT 28
Higa, Hammer + v.K. ‘08
92.07 0.03 keVRE
5.57 0.25 eVRE
fitted with 0 0,a r
Extrafitting parameters
0P
none
Wuestenbecker et al. ‘92
‘69
04/22/23 v. Kolck, Halo EFT 29
Rasche ‘67
Higa,
Hammer
+ v.K. ‘08
30 ~ cM P1
0 ~ cr M cf.
150MeVcM
consistent…
0 0
10.13fm
3 Ckr r
fine-tuning of 1 in 10!
but
1~ cM 11.21fm
3 Ck
1~ 10 cM
RE
also,
04/22/23 v. Kolck, Halo EFT 30
0
12 fm
2 Cka
O
1
0 2000fm 1000 2 Ca k
naturalness
Fine-tuning of 1 in a 1000between strong and electromagnetic
interactions!!
( )2 1 3
2 ln 12 3 4 2 2C
CE
R
Dk
k
gC
D
2
0 ( )2 R
ga
2cM
O2
O
2 cCk M
O
Higa,
Hammer
+ v.K. ‘08
20 200fmcaM
O
previousfine-tuning
Extra fine-tuning of 1 in 100!
Rasche ‘67
RE
04/22/23 v. Kolck, Halo EFT 31
6 4He b.s. He n n
12 4 4 4C b.s. He He He
9 4 4Be b.s. He He n
Next: three-body states
6 4Be res. He p p
9 4 4B res. He He p
Rotureau + v.K., in progress
Main issue: three-body force in LO?
3He b.s. p p n
3H b.s. p n n
cf.
in pionless EFT
Bedaque, Hammer + v.K., ‘98
Ando + Birse, ’10Koenig + Hammer, ‘11
: yes (preliminary)6 He
04/22/23 v. Kolck, Halo EFT 32
Other cores 8 7Li b.s. Li n
7 Li2.0 MeV
nE 7 Li
2 60 MeVR N nmk E
Rupak + Higa, ’11Fernando, Higa + Rupak,
in preparation
7 7Li Li 2.5 MeVtE B E
n p d cf.in pionless EFT
Chen, Rupak + Savage, ’99
Rupak, ‘00
7 Li95 M V4 ec NM m E
7 7Li Li* 0.48 MeVB B
other s-, p-waveparameters fit toscattering data,binding energy
one-parameter fit
7 8Li , Lin
7 8Be , Bp Goal:
field included for excited core state
04/22/23 v. Kolck, Halo EFT 33
11 10Be b.s. Be n
10 Be
0.2 MeV
0.5 MeVnE
10 Be
2 30 MeVR nNk Em
Hammer + Phillips, ‘11
Coulomb dissociation of 11Be
s-, p-wave
parameters fit to
1 2
1 2
binding energies,
B(E1) transition strength
10 10 10 *Be Be Be 3.4 MeVE B B 10 Be
2 80 MeVc NM m E
04/22/23 v. Kolck, Halo EFT 34
2Z b.s. ZA A n n Canham + Hammer, ’08, ’10
with spin
0
s-wave interaction
( 1)3 0nB
( 1)3 0nB
( 1)3 0nB
at least one
Efimov state
(negative energies:virtual states)
04/22/23 v. Kolck, Halo EFT 35
ForecastQCD
Pionful
EFT
lattice
Pionless
EFT
Halo/cluster
EFT
Extrapolates torealistically small
r
m
Low-energy
reactions
SM
Extrapolateto larger
and larger
, ,
, ,
QCD
QCD
nuc u d
u d
M m m
Mm m
M
m
,nucM m
,cc nuM M NCSM, …
Faddeev* eqs, …