Nuclear Effective Field TheoryNuclear Effective Field Theory

Paulo BedaquePaulo BedaqueLawrence-Berkeley LaboratoryLawrence-Berkeley Laboratory

No nucleons chiral perturbation theory

One nucleon heavy baryon chiral perturbation theory

Two or more nucleons Nuclear effective theory

Extracting low energy information from QCD in a model independent way:

Hierarchy of scales:

8 MeV,45 MeV m , 270 MeV m

QCD scaleFermi momentum

in nuclei

NN scale (spin singlet), momentum in the

deuteron (spin triplet)

Two consequences:Two consequences:

Bound states within the EFT range of validity

Nuclear EFT is non-perturbativeNuclear EFT is non-perturbative

Two possible EFT’s

““pionfull”pionfull” Qm m

““pionless”pionless” Q ~ 1/a << m

0 2

2† 4 2 2 2

0( ) ( ) | | ( ) | | | |2

C CiM

L

Pionless theoryPionless theory: two-nucleons: two-nucleons

0 ( )C 20 ( ) ( )kC i 3 2

0 ( ) ( )iC k

0

4π 14πM + +…-ik

M (Λ

)ΛC

20r1k cotδ - + k +…

a 2

0

4π 1 + Λ

C Λ-

M ( ) a

fine tuned fine tuned cancellationcancellation

That’s why nuclear physics That’s why nuclear physics exists !exists !

10

31

S

S

(1/a = 8 MeV

1/ a = 45 MeV)

0 04( ) ( )Mc C

0 0 0(1 )d c c cd

trivial fixed trivial fixed pointpoint

non-trivial non-trivial fixed pointfixed point

another way of looking at the fine tuning:another way of looking at the fine tuning:

Assuming this is the only fine tuning:Assuming this is the only fine tuning:

• Expansion in powers of Q/m, keep Qa to all orders

20

201

1 1 (1 )1 2 1

2ik

r kTr ik ikk a aa

C2 is NLO, not NNLO

• Naïve dimensional analysis fails

• CC00 is the only non-perturbative operatoris the only non-perturbative operator

A good example: neutrino-deuteron collisions (Butler, Chen)A good example: neutrino-deuteron collisions (Butler, Chen)

Haxton Haxton et al.et al. : no : no exchange currentsexchange currents

Both calculations are reproduced by EFT with two different values ofBoth calculations are reproduced by EFT with two different values of

i † - 32 21A 1A2 2(Nσ σ τ N) Nτ τ , LσL N 5 fm

The same constant appears on pp fusion, The same constant appears on pp fusion,

capture on deuterium, triton beta decaycapture on deuterium, triton beta decay

Kubodera Kubodera et al. : et al. : a model of a model of meson exchange currentsmeson exchange currents

5% difference5% difference

For the three-body (“pionless”) :For the three-body (“pionless”) :

How large is ?How large is ?† 30D (ψ ψ)

4

1

MΛ naïve dimensionalnaïve dimensional

analysisanalysis

would appear would appear only at only at

NNNLONNNLO

ultraviolet finiteultraviolet finite2

1

Λ

2p 1/a,k

1t(k,p)

p

kk ppDD00 would not not run would not not run

and would not needed and would not needed at leading orderat leading order

1For , k p :a

2 2

2 20

dq p + pq+ q4p t(k,p) = ln q t(k,q)q p - pq+ q3

s-1ssin( )8 6t(k,p) p , 1 =s3 s cos( )

2

Two kinds of Two kinds of channels:channels:

L=0, S=1/2: L=0, S=1/2: triton, helium 3, bosonstriton, helium 3, bosons

All others: All others: Pauli principle, centrifugal barrierPauli principle, centrifugal barrier

All others:All others:

3.17…

1 1= - , t(k,p)2 p

Three-body force no Three-body force no needed until very high needed until very high

orders, a lot of orders, a lot of predictive powerpredictive power

1)1)

2)2)' '

'

0

dq ( (q,p) - K(q,p) ) t (q) = - dq ( )

~ 1/Q~ 1/Q2 2 ~ 1/~ 1/22

Neutron-deuteron elastic phase shiftsNeutron-deuteron elastic phase shifts

L=0, S=3/2L=0, S=3/2 L=2, S=1/2L=2, S=1/2

++ = AV18 + UX (Kievski = AV18 + UX (Kievski et alet al.) .) mm=Schmelzbach =Schmelzbach et al.et al.

= LO, = NLO, = NNLO= LO, = NLO, = NNLO

L=0, S=3/2 scattering length: a(EFT)=L=0, S=3/2 scattering length: a(EFT)=5.09 + 0.89 + 0.35 + …=6.335.09 + 0.89 + 0.35 + …=6.330.05 0.05 fmfm

a(Exp)=a(Exp)=6.356.350.02 0.02 fmfm

33H, H, 33He (and bosons):He (and bosons):

00is +1

11: t(k,p) , s 1.006…p

harder in the UVharder in the UV1)1)

2)2)' '

'

0

dq ( (q,p) - K(q,p) ) t (q) = - dq ( )

~1/Q~1/Q22 or ~1/Q or ~1/Q (zero mode) ~1/Q (zero mode) ~1/Q

change in change in on-shell on-shell

amplitudeamplitude

' 0t (p) = t (p) +Csin(s log(p))

' 2

2Qdq ( + H( )) t (q) = 0 + ( )O

three-body three-body force:force:

2 20 0

0 2 20 0

sin( log( / ) arctg )a a( ) H( )sin( log( / ) arctg )

s sD

s s

Adjust H(Adjust H() so:) so:

limit cycle: limit cycle: e e /s/s00

At higher orders:At higher orders:

SUSUWW(4) invariant three-body force (4) invariant three-body force

terms are enhancedterms are enhanced

Neutron-deuteron elastic phase shifts: L=0, S=1/2Neutron-deuteron elastic phase shifts: L=0, S=1/2

xx = AV18 + UX (Kievski = AV18 + UX (Kievski et alet al.) .) = Schmelzbach = Schmelzbach et al.et al.

= LO, = NLO, = NNLO= LO, = NLO, = NNLO

blueblue band describes the variation between band describes the variation between =200 =200 600 MeV 600 MeV

one 3-body free parameterone 3-body free parameter

one lineone line

Phillips line:Phillips line:

““Pionfull” EFTPionfull” EFT2 2

† † † † 4 2 20 0 2

2( ) ( ) ( ) | | ( ) | | | |2 4

iiA

fg B Ctr Ci tr

M

L

2 02

0

2,

2

i

fe

Restrictions from Restrictions from symmetrysymmetry

Potential:Potential:

AmplitudeAmplitude::

dependence ?dependence ?

(expansion on Q/(expansion on Q/ andand m m//

Perturbatively this is inconsistent, but we now know betterPerturbatively this is inconsistent, but we now know better

2m ln 2 lnk

destroys chiral expansiondestroys chiral expansion destroys the momentum expansiondestroys the momentum expansion

momentum expansion is momentum expansion is consistentconsistent

still inconsistentstill inconsistent

lattice extrapolations, lattice extrapolations, isospin breaking, isospin breaking,

cosmologycosmology

perturbative:perturbative:

non-perturbative:non-perturbative:

Some NN phase shifts (Epelbaum Some NN phase shifts (Epelbaum et alet al.):.):

33SS1111

=LO =NLO =NNLO* =Nijmegen PWA=LO =NLO =NNLO* =Nijmegen PWA

N couplings fitN couplings fit

500<500<<600<600

Neutral pion photoproduction (Beane, Lee, van Kolck)Neutral pion photoproduction (Beane, Lee, van Kolck)

2 2

03 |8 q d L d

k d E Lq d

-3

d

-3EXP

10E (0) (1.79 0.2)m

10E (0) (1.45 0.09)md

3m