Wigner Symmetry, Large Nc and Renormalized OBE - GSItheory.gsi.de/Heraeus-410/Talks/RuizArriola.pdf · Symmetries in Nuclear Physics vs QCD “QCD-Evident” symmetries Pauli symmetry:

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.


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)


Outline of the talk

Renormalization

Wigner symmetry

Large Nc

Long distance symmetry

Conclusions


RENORMALIZATION


The need for renormalization in NN

These potentials are different below 1.5 fm.

Formulation where short distance insensitivity is manifest.


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


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


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.


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


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)


WIGNER SYMMETRY


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


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)


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)


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


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


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.


LARGE NC


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


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 ?


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).


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)


LONG DISTANCE SYMMETRY


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)


-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)


-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)


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


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)


-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


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)


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)


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


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

)


-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


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.


CONCLUSIONS


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.


OUTLOOK


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

]


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)



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)



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)



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) %



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)




Documents

Wigner Symmetry, Large Nc and Renormalized OBE - GSItheory.gsi.de/Heraeus-410/Talks/RuizArriola.pdf · Symmetries in Nuclear Physics vs QCD “QCD-Evident” symmetries Pauli symmetry: