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


Hnning


Outline

EFT

Nucleon-alpha system

Alpha-alpha system

Other systems

Outlook


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!


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


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!


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


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


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


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


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


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


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


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


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


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.


Bedaque, Hammer + v.K. ’03

Haesner et al. ‘83

NNDC, BNL

0

1

2


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


1

mod 1


Haesner et al. ‘83

NNDC, BNL

0



01 0

2

mod 1

1PSA, Arndt et al. ’73

1,0,1

0.80 MeVRE ( ) 0.55 MeVRE

scatt length only


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…


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


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


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


+ + …=

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?

04/22/23 26

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


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


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


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,


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


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


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


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


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)


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, …

Halo Effective Field Theory

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