nuclear effective field theory
DESCRIPTION
Nuclear Effective Field Theory. Paulo Bedaque Lawrence-Berkeley Laboratory. Extracting low energy information from QCD in a model independent way:. No nucleons a chiral perturbation theory One nucleon a heavy baryon chiral perturbation theory - PowerPoint PPT PresentationTRANSCRIPT
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Nuclear Effective Field TheoryNuclear Effective Field Theory
Paulo BedaquePaulo BedaqueLawrence-Berkeley LaboratoryLawrence-Berkeley Laboratory
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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:
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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)
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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
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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)
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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:
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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
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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
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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
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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
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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
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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
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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))
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' 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
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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
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one 3-body free parameterone 3-body free parameter
one lineone line
Phillips line:Phillips line:
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““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//
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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:
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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
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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