wigner symmetry, large nc and renormalized obe -...
TRANSCRIPT
Wigner Symmetry, Large Nc andRenormalized OBE
E. Ruiz Arriola (with A. Calle Cordon)arXiv:0804.2350 [nucl-th]; arXiv:0807.2918 [nucl-th]
Departmento de Fısica Atomica, Molecular y NuclearUniversidad de Granada (Spain)
410. WE-Heraeus-Seminar:”Ab-Initio Nuclear Structure - Where do we stand?”
Bad-Honnef, July 27-30, 2008.
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
Introduction
Nuclear force is non-perturbative and among heavyparticles (non-relativistic potentials)
Nuclear potentials are unknown at short distances butpotentials are short distance sensitive.
Symmetries and features of potentials should be explainedfrom QCD.
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
Symmetries in Nuclear Physics vs QCD
“QCD-Evident” symmetries
Pauli symmetry: Nucleons are spin 1/2 fermions iff quarksare spin 1/2 fermions
Isospin symmetry: Mn = Mp iff mu = md
Chiral Symmetry:1) MN 6= 0 iff 〈qq〉 6= 02) mπ = 0 iff mu = md = 0
Heavy Quark Symmetry
Large Nc symmetry
“QCD-accidental” symmetries
Wigner SU(4)
Elliot SU(3)
Draayer Sp(3,R)
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
Outline of the talk
Renormalization
Wigner symmetry
Large Nc
Long distance symmetry
Conclusions
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
The need for renormalization in NN
These potentials are different below 1.5 fm.
Formulation where short distance insensitivity is manifest.
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
The traditional approach
Solve Schrodinger equation
−u′′
p(r) + MN V (r) up(r) = p2 up(r)
with a regular boundary condition at the origin up(0) = 0and
up(r) →sin (pr + δ0(p))
sin δ0(p)
OBE-potential in the 1S0 channel
V (r) = −g2πNNm2
π
16πM2N
e−mπr
r︸ ︷︷ ︸
OPE
−gσNN2
4π
e−mσr
r+
gωNN2
4π
e−mωr
r+ · · ·
︸ ︷︷ ︸
OBE
We expect that for p < pmax = 400MeV, heavier mesonsshould not be crucials, and ω itself marginally important
1/mω = 0.25fm << 1/pmax = 0.5fm
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
Fit mσ, gσNN , gωNN to the 1S0 phase-shift
Fit 1 χ2/DOF = 0.8
mσ = 477.0(5)MeV , gσNN = 8.76(4) , gωNN = 7.72(4)
Fit 2 χ2/DOF = 0.5
mσ = 556.34(4)MeV , gσNN = 13.04(2) , gωNN = 12.952(2)
Fit 1 and Fit 2 are two incompatible scenarios
Extreme fine tuning !!!. The 1S0 scattering length isunnaturally large α0 = −23.74(2)fm. Hence V → V + ∆Vhas a dramatic effect
∆α0 = α20MN
∫∞
0∆V (r)u0(r)
2dr
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
Renormalization in Coordinate space
Solve Schrodinger equation with the boundary condition
uk (r) → sin(kr + δ0)
sin δ0
Impose orthogonality of solutions in rc ≤ r ≤ ∞
δ(k − p) =
∫∞
rc
uk(r)up(r)dr
which implies
u′
k (rc)
uk (rc)=
u′
p(rc)
up(rc)=
u′
0(rc)
u0(rc)
Zero energy wave function
u0(r) → 1 − rα0
Remove the cut-off rc → 0. This is NOT the regularsolution , u(0) = 0.
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
Fit with χ2/DOF = 0.26 yields
mσ = 493(12)MeV , gσNN = 8.8(2) , gωNN = 0(5)
Short distance insensitive !!!Potential always attractive (no hard-core).Spurious bound states at EB = −.7,−1.1GeV
-20
-10
0
10
20
30
40
50
60
70
0 50 100 150 200 250 300 350 400 450
δ 0 [
deg]
kcm [MeV]
Renormalized neutron-proton phase shifts
α0 = - 23.74 r0 (Nijmegen) = 2.67mσ = 493 (12) MeVgσNN = 8.8 (2)gωNN = 0 (5)χ2/DOF = 0.26
δ0( kcm; mσ; gσNN; gωNN)Nijmegen pseudo-data
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
Coordinate space renormalization has been used for chiraland singular potentials, (Pavon Valderrama, ERA)
V (r) → 1f nπ
MmN
1rn+m+1
Results converge for practical cut-offs rc ∼ 0.5fm which is∼ 1/pmax.
Renormalization with a boundary condition is equivalent toput counterterms in the Lippmann-Schwinger equation inmomentum space (Entem’s talk).
Many results are OK (Pavon Valderrama’s talk)
Causality, completeness and dispersion relations (spuriousdeeply bound states with no important effect)
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
SU(4) Wigner symmetry
Generators
T a =12
∑
A
τaA , ISOSPIN
Si =12
∑
A
σiA , SPIN
Gia =12
∑
A
σiAτa
A , GAMOW − TELLER
Casimir operator
CSU(4) = T aTa + SiSi + GiaGia ,
Irreducible representations (λ, µ, ν)
CSU(4) = µ(µ + 4) + ν(ν + 2) + λ2
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
Selection rules in Gamow-Teller weak decays betweensupermultiplets
〈λµν|Gia|λ′µ′ν ′〉 = 0
SU(4) mass formula (Franzini+Radicatti 63)
E = c1A(A + 1) + c2
[
µ(µ + 4) + ν(ν + 2) + λ2 − 154
A]
Anomalously large double binding energy for even-evenN = Z nuclei (Van Isaacker,Warnerr,Brenner,1995).
SU(4) inequalities for nuclei on the lattice (Chen, Lee,Schaffer 2004)
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
One nucleon state
4 = (p ↑, p ↓, n ↑, n ↓) = (S = 1/2, T = 1/2) Quartet
Two nucleon states
CSTSU(4) =
12
(σ + τ + στ) +152
,
τ = τ1 · τ2 = 2T (T + 1) − 3 ,σ = σ1 · σ2 = 2S(S + 1) − 3
Sextet and decuplet (−1)S+L+T = −1
6A = (1, 0) ⊕ (1, 0) L = 0, 2, . . . (1S0,3 S1), (
1D2,3 D1,2,3)
10S = (0, 0) ⊕ (1, 1) L = 1, 3, . . . (1P1,3 P0,1,2)
Symmetry of the potential → Symmetry of the S-matrix
V3S1(r) = V1S0(r) → δ1S0(p) = δ3S1(p)
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
Wigner symmetry from Lattice QCD calculations
(S. Aoki, T. Hatsuda, N. Ishii) → V1SO(r) = V3S1(r)
0
100
200
300
400
500
600
0.0 0.5 1.0 1.5 2.0
VC
(r)
[MeV
]
r [fm]
-50
0
50
100
0.0 0.5 1.0 1.5 2.0
1S03S1
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
S-wave phase shifts (Nijmegen)
δ1S0(p) 6= δ3S1(p)
0 50 100 150 200 250 300 350-20
0
20
40
60
80
PWA93ESC96 potentialNijmI potentialNijmII potentialReid93 potential
Tlab(MeV)
1S0
np phaseshift 1S0
0 50 100 150 200 250 300 350-40
0
40
80
120
160
200
PWA93ESC96 potentialNijmI potentialNijmII potentialReid93 potential
Tlab(MeV)
3S1
np phaseshift 3S1
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
FIRST PUZZLE: V3S1(r) = V1S0(r) but δ3S1(p) 6= δ1S0(p)
It has been noted (Braaten and Hammer 2003) that ifmπ ∼ 200MeV then one might have Wigner symmetry.
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
Large Nc
Nc → ∞ with αNc fixed (t’Hoot, Witten)
Hadronic spectrum (baryons and mesons are stable)
mπ,ρ,ω,σ ∼ N0c Γσ,ρ,ω ∼ 1/Nc mN,∆ ∼ Nc Γ∆ ∼ 1/Nc
Couplings
gMMM ∼ 1/√
Nc gMMMM ∼ 1/Nc gBMM ∼√
Nc
Scattering
T (ππ → ππ) ∼ 1/Nc , T (πN → πN) ∼ N0c , T (NN → NN) ∼ Nc
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
Large Nc vs Wigner symmetry
The potential is well defined since MN ∼ Nc
(Kaplan,Savage,Manohar, 1996-97)
V (r) = VC(r) + τ1 · τ2[σ1 · σ2WS(r) + S12WT (r)] ∼ Nc
Leading terms correspond to π,σ,ρ, ω exchange (OBE)
Corrections (spin-orbit, meson widths, rel) are ∼ 1/Nc.
Relative accuracy 1/N2c ∼ 10% !!!!
Wigner symmetry implies WT = 0
Large Nc supports Wigner symmetry for L = 0, 2, 4, . . .
Large Nc violates Wigner symmetry for L = 1, 3, 5, . . .
SECOND PUZZLE: If Wigner and Large Nc contradict eachother, which one is best ?
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
The long distance (large Nc) OBE potential
V1S0(r) = V3S1(r) = −g2πNNm2
π
16πM2N
e−mπr
r− g2
σNN
4π
e−mσr
r
+g2
ωNN
4π
e−mωr
r−
f 2ρNNm2
ρ
8πM2N
e−mρr
r+ O
(
Nc−1
)
,
Values of couplings from other sources
gπNN = 13.1 ∼ gAMN/fπ, (Goldberger-Treiman)
gσNN = 10.1 ∼ MN/fπ, (Goldberger-Treiman’)
gωNN = 9 ∼ 3gρNN , (SU(3))
gρNN = gρππ/2 = mρ/(√
2fπ) = 2.9, (Universality+KSFR)
fρNN = 15 − 17 ∼ (µp − µn − 1)gρNN , (VMD).
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
The scalar meson mass
Johnson+Teller (1951) Saturation and binding
mσ ∼ 500MeV
Roy equations (chiral symmetry + analyticity+ crossing +unitarity) (Caprini-Colangelo-Leutwyler,2006)
mσ − iΓσ
2= 441+16
−8 − i272+9−12MeV
ππ scattering in (I, J) = (0, 0) channel
t IIππ
(s) → g2σππ
s − (mσ − iΓσ/2)2 → g2σππ
s − m2σ
we get mσ → 507MeV.Contribution to leading Nc potential
V CNN(r) → −g2
σNN
4π
e−mσr
r+ O(1/Nc)
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
Superposition principle of boundary conditions
Finite energy
uk (r) = uk ,c(r) + k cot δ0(k) uk ,s(r) →sin(kr + δ0(k))
sin δ0(k)
Zero energy (k → 0, and δ0(k) → −α0k)
u0(r) = u0,c(r) − u0,s(r)/α0 → 1 − r/α0 ,
Orthogonality ( rc short distance cut-off)
0 =
∫∞
rc
dr[
u0,c(r) −1α0
u0,s(r)] [
uk ,c(r) + k cot δ0(k) uk ,s(r)]
.
Renormalize (rc → 0) → Potential and α0 independent
k cot δ0(k) =α0A(k) + B(k)
α0C(k) + D(k)
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0 50 100 150 200 250 300 350 400 450
A(p
cm
) [f
m-2
]
pcm [MeV]
Universal function A
A(pcm)
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
0 50 100 150 200 250 300 350 400 450
B(p
cm
) [f
m-1
]
pcm [MeV]
Universal function B
B(pcm)
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0 50 100 150 200 250 300 350 400 450
C(p
cm
) [f
m-2
]
pcm [MeV]
Universal function C
C(pcm)
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0 50 100 150 200 250 300 350 400 450
D(p
cm
) [f
m-1
]
pcm [MeV]
Universal function D
D(pcm)
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
-20
-10
0
10
20
30
40
50
60
70
80
0 50 100 150 200 250 300 350 400 450
Ph
ase
Sh
ifts
[d
eg]
pcm [MeV]
1S0 phase shifts
Nijmegenπ-exchangeσ-exchange
(π+σ)-exchange
0
20
40
60
80
100
120
140
160
180
0 50 100 150 200 250 300 350 400 450
Ph
ase
Sh
ifts
[d
eg]
pcm [MeV]
3S1 phase shifts
Nijmegenπ-exchangeσ-exchange
(π+σ)-exchange
k cot δ1S0(k) =
α1S0A(k) + B(k)
α1S0C(k) + D(k)
, k cot δ3S1(k) =
α3S1A(k) + B(k)
α3S1C(k) + D(k)
α1S0= −23.74fm α3S1
= 5.42fm
A symmetry of the potential IS NOT a symmetry of the S-matrix(Anomalies in QFT)
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
Effective range (low energy theorem)
r0 = 2∫
∞
0dr
[(
1 − rα0
)2
− u0(r)2
]
= A +Bα0
+Cα2
0
,
0
0.5
1
1.5
2
2.5
3
3.5
4
-0.4 -0.2 0 0.2 0.4 0.6 0.8
r 0 [
fm]
1/α0 [fm-1]
Wigner correlation r0(1/α0)
π-exch(π + σ)-exch
rs(th) = 2.695rs(exp) = 2.770
rt(th) = 1.628rt(exp) = 1.753
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
The deuteron state
Orthogonality between bound state and zero energy
M(γ, α0) =
∫∞
0dr uγ(r)u0(r)
where
u0(r) → ASe−γr Ed = −γ2
M= −2.2MeV
yields both the orthogonality relation as well as MM1
(neutron capture, n + p → dγ)
M(γ, αt) = 0 ,
M(γ, αs) = MM1 . (1)
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
-1
0
1
2
3
4
5
-0.1 -0.05 0 0.05 0.1 0.15 0.2
M1
[fm
]
1/α0 [fm-1]
M1 matrix element
(π + σ)-exchπ-exch
no potentialM1th(αs) = 4.047
M1exp(αs) = 3.979M1(αt) = 0
3.6
3.8
4
4.2
4.4
-0.06 -0.04 -0.02
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
RG in coordinate space (alias Vhigh−R)
Universality of Vlow−k at corresponds to a boundarycondition obtained from integrating in the OPE potentialfrom realistic phase shifts
cp(R) = Ru′
p(R)
up(R)∼ c0(R) + p2c2(R) + . . .
RG equation
Rc′
0(R) = c0(R)(1 − c0(R)) + MR2V (R) ,
Scale invariant regime
c0(R) =α0
α0 − R∼ 1 , 1/mπ ≪ R ≪ α0 ,
Thus c1S0(R) ∼ c3S1
(R) (Kaplan+Savage 1996,Mehen+Stewart+Wise,Elster et. al 2002)
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
Visualizing the symmetry
u1S0(r) ∼ u3S1
(r) for 1/mπ ≪ R ≪ α0
-0.5
0
0.5
1
1.5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Wav
e F
un
ctio
ns
[fm
-1/2
]
r [fm]
u0,1S0(r)
u0,3S1(r)
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
Symmetry breaking for S-waves
More counterterms
rt − rs ∼ r shortt − r short
s ∼ 0.1fm
Tensor force (S-D) mixing
r tensor3S1
= 2∫
∞
0
[(
1 − rα3S1
)2
− u0,α(r)2 − w0,α(r)2
]
dr
= r3S1+ 0.1fm
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
Symmetry breaking for non-central waves
Short distances are suppressed uL(r) ∼ rL+1. Wignersymmetry V1L(r) = V3L(r) become more evident
The symmetry implies that δ1L(r) = δ3L(r).
Symmetry breaking: Spin orbit and tensor force → sumrules
δ1P1=
19
(δ3P0
+ 3δ3P1+ 5δ3P2
)
δ1D2=
115
(3δ3D1
+ 5δ3D2+ 7δ3D3
)
δ1F3=
121
(5δ3F2
+ 7δ3F3+ 9δ3F4
)
δ1G4=
127
(7δ3G3
+ 9δ3G4+ 11δ3G5
)
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
-30
-25
-20
-15
-10
-5
0
5
10
0 50 100 150 200 250 300 350 400
Pha
se S
hift
s [d
eg]
pcm [MeV]
P-wave relation
(δ3P0 + 3δ3P1
+ 5δ3P2)/9
δ1P1
0
2
4
6
8
10
0 50 100 150 200 250 300 350 400
Pha
se S
hift
s [d
eg]
pcm [MeV]
D-wave relation
(3δ3D1 + 5δ3D2
+ 7δ3D3)/15
δ1D2
-6
-5
-4
-3
-2
-1
0
1
0 50 100 150 200 250 300 350 400
Pha
se S
hift
s [d
eg]
pcm [MeV]
F-wave relation
(5δ3F2 + 7δ3F3
+ 9δ3F4)/21
δ1F3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 50 100 150 200 250 300 350 400
Pha
se S
hift
s [d
eg]
pcm [MeV]
G-wave relation
(7δ3G3 + 9δ3G4
+ 11δ3G5)/27
δ1G4
Wigner symmetry is fulfilled for L-even and is violated for L-odd
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
The large Nc pattern is (neglecting tensor force)
V (r) = VC(r) + στWS(r) + O(1/Nc) ,
Lower L-channels
V1S = V3S = VC(r) − 3WS(r) + O(1/Nc)
V1P = VC(r) + 9WS(r) + O(1/Nc)
V3P = VC(r) + WS(r) + O(1/Nc)
V1D = V3D = VC(r) − 3WS(r) + O(1/Nc)
Symmetry breaking is compatible with large Nc !!!!
For Spin-Triplets one has Serber symmetry which isobserved in pp but has no QCD explanation.
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
Renormalization is necessary since it implements shortdistance insensitivity of the unknown short distanceinteraction. It avoids the hard core.
Some symmetries in nuclear physics may be interpreted aslong distance ones, broken only by counterterms.
Wigner SU(4) symmetry and Large Nc are closely relatedbut they are not the same. For odd-L partial waves they areinconsistent.
We view large Nc as a competitive and QCD relatedsymmetry.
Chiral potentials do not embody large Nc constraintsunless ∆ is included and all 2π and 3π effects arere-summed.
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
The full large-Nc OBE potential
V (r) = VC(r) + σ1 · σ2τ1 · τ2WS(r) + S12τ1 · τ2WT (r)
Leading long distance components
VC(r) = −g2σNN
4π
e−mσr
r+
g2ωNN
4π
e−mωr
r
WS(r) =112
g2πNN
4π
m2π
Λ2N
e−mπr
r+
16
f 2ρNN
4π
m2ρ
Λ2N
e−mρr
r
WT (r) =112
g2πNN
4π
m2π
Λ2N
e−mπr
r
[
1 +3
mπr+
3(mπr)2
]
− 112
f 2ρNN
4π
m2ρ
Λ2N
e−mρr
r
[
1 +3
mρr+
3(mρr)2
]
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE
0.025
0.0252
0.0254
0.0256
0.0258
0.026
0.0262
0.0264
0.0266
0.0268
0.027
0 5 10 15 20 25
η
fρNN
η dependence with fρNN
η(exp) = 0.0256(4)
g*ωNN = 0g*ωNN = 2g*ωNN = 6
g*ωNN = 10g*ωNN = 12
0.87
0.875
0.88
0.885
0.89
0.895
0.9
0.905
0 5 10 15 20 25
As
[fm
-1/2
]
fρNN
As dependence with fρNN
ΑS(exp) = 0.8846(9)
g*ωNN = 0g*ωNN = 2g*ωNN = 6
g*ωNN = 10g*ωNN = 12
1.94
1.95
1.96
1.97
1.98
1.99
2
2.01
0 5 10 15 20 25
r m [
fm]
fρNN
rm dependence with fρNN
rm(exp) = 1.9754(9)
g*ωNN = 0g*ωNN = 2g*ωNN = 6
g*ωNN = 10g*ωNN = 12
0.278
0.28
0.282
0.284
0.286
0.288
0.29
0.292
0.294
0 5 10 15 20 25
Qd [
fm2]
fρNN
Qd dependence with fρNN
Qd (exp) = 0.2859(3)
g*ωNN = 0g*ωNN = 2g*ωNN = 6
g*ωNN = 10g*ωNN = 12
5
5.5
6
6.5
7
7.5
8
0 5 10 15 20 25
PD
[%
]
fρNN
PD dependence with fρNN
PD(exp) = 5.67(4) %
g*ωNN = 0g*ωNN = 2g*ωNN = 6
g*ωNN = 10g*ωNN = 12
0.4
0.45
0.5
0.55
0.6
0.65
0 5 10 15 20 25
<r-1
>
fρNN
<r-1> dependence with fρNN
<r-1>(nij) = 0.445(4)
g*ωNN = 0g*ωNN = 2g*ωNN = 6
g*ωNN = 10g*ωNN = 12
Enrique Ruiz Arriola Wigner Symmetry, Large Nc and Renormalized OBE