Few-body nuclear physics: working towardsthe transition region

W. PolyzouUniversity of Iowa

Feb 27, 2008

Collaborators

F. Coester (Iowa, ANL), B. Keister (NSF), C. Elster (OhioU), T. Lin (Ohio U), W. Glockle (Bochum - Germany), H.

Wita la (Jagiellonian - Poland), H. Kamada (Krushu - Japan),W. Klink (Iowa), G. L. Payne (Iowa), Y. Huang (Iowa), P.L.

Chung (Iowa), H. C. Jean (Iowa), T. Allen (Iowa), S.Veerasamy (Iowa), M. Tucker (Iowa), E. Sengbusch (Iowa)

Outline

• Problem

• Discussion of scales

• Essential quantum mechanics

• Symmetries - Galilean and Poincare

• Model description

• Some results

• Future

Problem

Nucleus = quarks + gluons

Nucleus = nucleons + mesons

?

Scales

• Mass of nucleon mc2 = 940 MeV

• Size of nucleon r ≈ 1× 10−15 meters = 1 fm.

• Binding energy per nucleon 8 MeV.

• Nuclear excitation energies - a few MeV.

• Range of nuclear force r ≈ 1× 10−15 meters

• Nucleon excitation energies 500 MeV

Elementary analysis

∆x∆p ≥ ~

∆p ≥ ~12nucleon radius

∆pc &1

2GeV ≈ 1

2× nucleon rest energy

Implications

• Maximal cross section ⇒ minimal beam momentum

• Minimal beam momentum ⇒ ∆p∆x ≈ ~

• ∆p∆x ≈ ~ ⇒ quantum mechanical treatment

• Minimal beam momentum + ∆x ≈ 12nucleon radius ⇒

minimal ∆p ≈ 12nucleon mass

• ∆p ≈ 12 nucleon mass ⇒ relativistic treatment necessary

Goals

• Formulate a mathematical model of nuclei that providesa quantitative description of nuclear structure andreaction observables for energies up to a few GeV.

• The model should be quantum mechanical.

• The model should be relativistically invariant.

• The model should be as simple as is necessary.

• The model should be able to describe complete sets ofobservables for a large class of nuclei.

Poincare invariant quantum mechanics of

few-nucleon or

few-quark systems

Minimal quantum theory

• Complex vector space

• Scalar product

(~a)∗ · ~b := 〈a|b〉

Example: nucleon at rest

• Basis vectors

| ↑〉 =

(10

)| ↓〉 =

(01

)• Scalar product

〈↑ | ↑〉 = 〈↓ | ↓〉 = 1 〈↓ | ↑〉 = 〈↑ | ↓〉 = 0

Measurements

The state of a nucleon is represented by a unit vector

|e〉 = cos(θ) | ↑〉+ sin(θ) | ↓〉

P↑ := |〈e| ↑〉|2 = | cos(θ)|2 = cos2(θ)

P↓ := |〈e| ↓〉|2 = | sin(θ)|2 = sin2(θ)

P↓ + P↑ = sin2(θ) + cos2(θ) = 1

Quantum probabilities

and inner products

Pa,b = |〈a|b〉|2

Key observations

• The Schrodinger equation plays no role in quantummeasurements.

• The result of any measurement is a probability.

Symmetries of quantum theories

A vector correspondence

|a〉 → |a′〉

is a symmetry of a quantum theory if

Pab = |〈a|b〉|2 = |〈a′|b′〉|2 = Pa′b′

If |a〉 → |a′〉 is a symmetry

⇓

physics in the unprimed world is indistinguishable fromphysics in the primed world.

Relativity = existence of inertial reference frames

Physics in different inertial reference frames isindistinguishable

|a〉 → |a′〉

⇓

Pab = Pa′b′

Differs from the historical development.

How are inertial reference frames related?

Use classical physics

• By coordinate transforms that preserve the form ofNewton’s second law ? (Galilean relativity)

• By coordinate transforms that preserve the form ofMaxwell’s equations of electricity and magnetism ?(special relativity)

Galilean relativity preserves

|~x − ~y | = |~x ′ − ~y ′| |tx − ty | = |t ′x − t ′y |

~x ′ = R~x + ~vt +~a t ′ = t + c

Special relativity preserves

|~x − ~y |2 − c2|tx − ty |2 = |~x ′ − ~y ′|2 − c2|t ′x − t ′y |2

c = speed of light

Which is the correct symmetry of nature?

Michelson-Morley experiment

⇓

Special relativity

Poincare group

|~x − ~y |2 − c2|tx − ty |2 = |~x ′ − ~y ′|2 − c2|t ′x − t ′y |2

xµ := (ct, x1, x2, x3)

⇓

x ′µ =3∑

ν=0

Λµνx

ν + aµ∑µ,ν

ΛµαηµνΛν

β = ηαβ

ηµν :=

−1 0 0 0

0 1 0 00 0 1 00 0 0 1

Elementary Poincare transformations

Rotations (3):

~x ′ = O~x OtO = I

Translations (4):

~x ′ = ~x +~a, ct ′ = ct + a0

Rotationless Lorentz transformations (3):

x ′ = γ(x + vt) t ′ = γ(t + vx/c2) γ =1√

1− v2/c2

• Λµν and aµ are labels for distinct inertial reference

frames.

• All Poincare transformations can be generated from 10elementary transformations.

• Compositions of Poincare transformations are Poincaretransforms.

• Poincare transformations are elements of a group(closed under composition, inverse, identity, associative).

Special relativity and quantum mechanics

m

The group of Poincare transformations is a symmetry ofquantum mechanics

|v〉 → |vΛ,a〉

⇓

Puv = PuΛ,a,vΛ,a

Eugene P. Wigner

Puv = PuΛ,avΛ,a

⇓

〈uΛ,a|vΛ,a〉 = 〈u|v〉

⇓

U(Λ2, a2)U(Λ1, a1) = U(Λ2Λ1,Λ2a1 + a2)

Note that U(Λ, a) has the multiplication properties of thePoincare group but it acts on a vector space with different

dimension than four.

U(Λ, a) is a unitary representation of the Poincare group.

How do we construct U(Λ, a) ?

Rotating a nucleon at rest

U(R)| ↑〉 = | ↑〉u↑↑ + | ↓〉u↓↑

U(R)| ↓〉 = | ↑〉u↑↓ + | ↓〉u↓↓

⇓

U(R)|µ〉 =∑

ν=↑,↓|ν〉D1/2

νµ (R)

D1/2νµ (R) is a 2× 2 matrix representation of the rotation group.

Translating a nucleon at rest

U(a0,~a)|µ〉 = |µ〉e−imca0

Nucleon moving with velocity ~v , (~p = mnγ~v)

|~p, µ〉 := U(~v)|µ〉

Every Poincare transformation can be expressed as

• Rotationless Lorentz transform to rest frame

• Rotation

• Translation of a rest state

• Rotationless Lorentz transform to final frame

Transformation law for a free particle

U(Λ, a)|~p, µ〉 =∑µ′

|~p ′, µ′〉D1/2µ′µ(R(Λ, p))e−ip′·a

This transformation law is a mass mn spin 1/2 irreduciblerepresentation for the Poincare group.

This can be done for particles with any mass and spin.

Two free nucleons in zero total momentum frame

|~k, µ1;−~k, µ2〉 := |~0, ~k, µ1, µ2〉 =

• Decompose into linear combinations that have definiteangular momenta, J2, ~J · ~z

• Resulting states look just like free particle resteigenstates with mass and spin

M = 2√

k2 + m2n J

• Simultaneous eigenstates of M, ~P, J2 and Jz transformlike free particles of mass M and spin J:

|~p, µ(M(k), J, d)〉

U(Λ, a)|~p, µ(M(k), J, d) =

∑µ′

|~p ′, µ′(M(k), J, d)〉DJµ′µ[R(Λ, p)]e−ip′·a

Two interacting nucleons

Consistent initial value problem?

A

B

C

••

•

x = 0

t = t1

t = t0

t =√

x2 + t2

0

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...........................................................................................................................................................................................................................................................................................................................................................................................................................................

..........

..........

..........

..........

..........

..........

.

1

Solving the non-linear problem?

• Add interactions to mass

M = 2√

k2 + m21 + V

• Require that V commutes with J2

• R that V commutes with and is independent of ~P and Jz .

• Diagonalize the matrix M in the basis |~p, µ(M(k), J, d)〉.

(2√

k2 + m2n + V )|~p, µ, (M ′, J)〉 = M ′|~p, µ, (M ′, J)〉

UI (Λ, a)|~p, µ(M ′, J)〉 =

∑µ′

|~p ′, µ′(M ′, J)〉DJµ′µ(R(Λ, p))e−ip′·a

Simultaneous eigenstates of M, ~P J2 and Jz complete.

This defines a relativistic interacting two-nucleon quantumtheory

Relation to non-relativistic problem ?

M2 = 4(m2n + ~k2) + 4mnvnn = 4mn(

~k2

m+ vnn) + 4m2

n =

4mnhnr + 4m2n

• Has same eigenvectors as non-relativistic problem

• Has same scattering probabilities as non-relativisticproblem.

Nuclear potentials vnn

vnn =∑α

Vα(r)Oα

Oα ∈ {I , s1 · s2, L · S, (r · s1)(r · s2), (s1 · L)(s2 · L), L2, (s1 · s2)L2,

(τ 1 · τ 2), (s1 · s2)(τ 1 · τ 2), (L ·S)(τ 1 · τ 2), (r · s1)(r · s2)(τ 1 · τ 2), · · ·

• τ i = isospin of particle i

• L = orbital angular momentum of system

• si spin of particle i

• S = sum of nucleon spins

• Vα(r) = Yukawa-like interactions ( e−mπcr/~

r )

N > 2

Cluster properties - separate region A from region B

⇓

U(Λ, a)− U(Λ, a)B ⊗ U(Λ, a)B → 0

⇓

H =∑

i

Hi +∑ij

Vij +∑ijk

Vijk + · · ·

Specific case: N = 3

Structure of M dictated by two-body M and clusterproperties

M = W (V )M(V )W †(V ) M(V ) = M0 + V12 + V23 + V31

V12 =√

M212 + q2

3 −√

M2012 + q2

3 · · ·

M212 = 4(k2 + m2

n) + 4mnvnn M2012 = 4(k2 + m2

n)

W (V ) not required to calculate S!

(Coester 1965, Sokolov 1977, F.C & W.P. 1982)

Predictions

Binding energies

M(V )|Ψ〉 = λ|Ψ〉

|Ψ〉 = W (V )|Ψ〉 M(V )|Ψ〉 = λ|Ψ〉

Scattering probabilities

|Sfi |2 = |〈Ψ+f |Ψ

ii 〉|2 = |〈Ψ+

f |Ψii 〉|2

Electromagnetic and weak current matrix elements

〈Ψf |I ν(0)|Ψi 〉 = 〈Ψf |W †(V )I ν(0)W (V )|Ψi 〉

Three-body scattering

Sab = δab − 2πiδ(Ma −Mb)Tab

Faddeev equation

T ab(z) = δab(z −M0) +∑c 6=a

Vc(z −M0 − Vc)−1T cb(z)

a, b, c ∈ {(12)(3), (23)(1), (31)(2)}

0.00

0.02

0.04

0.06

0.08

0.10

0.00

0.05

0.10

0.15

AY

(N)

0 30 60 90 120 150 180

Θc.m. [deg]

0.00

0.05

0.10

0.15

0.20

90 120

0.06

0.08

105 120 135

0.08

0.12

120 1300.10

0.15

E=5 MeV

E=8.5 MeV

E=13 MeV

nd TUNL data

........ nrel- - - - rel noWig____ rel withWig

nd TUNL data

nd Erlangen data

-20

0

20

40

60

80

300 600 900 1200 1500

relative difference [%]

Elab [MeV]

1st

-20

0

20

40

60

80full

101

102

103

104

105σ e

l [mb]

R 1stNRcm 1stRkin 1stR full

NRcm fullRkin full

0

1

2

3

4

130 140 150 160 170 180

200 MeV

0

0.2

0.4

0.6

340 370 400 430 460

500 MeV

0

0.5

1

1.5

650 700 750 800 850 900

1000 MeV

0

0.5

1

1.5

520 570 620 670 720

d2σ b

rlab/dE1d

Ω1 [mb/(MeV sr)]

E1 [MeV]

800 MeVRkin 1st

R 1stRkin full

R full

10-1

100

200 220 240 260 280 300

θ1=38.1o,θ2=38.0

o, φ12=180o

10-1

100

101

200 220 240 260 280 300

θ1=41.5o,θ2=41.4

o, φ12=180o

10-2

10-1

100

101

200 220 240 260 280 300

θ1=47.1o,θ2=47.0

o, φ12=180o

NR 1R 1NR FR Fexp

10-1

100

101

200 220 240 260 280 300

d5σ/

dΩ1d

Ω2dE1 [mb/(MeV sr2)]

E1 [MeV]

θ1=44.1o,θ2=44.0

o, φ12=180o

10-2

10-1

100

101

102

103

104

53 54 55 56 57 58 59 60 61 62

1.6 GeV/c

10-2

10-1

100

101

102

103

104

53 54 55 56 57 58 59 60 61

1.7 GeV/c

10-3

10-2

10-1

100

101

102

103

53 54 55 56 57 58

1.9 GeV/c

10-2

10-1

100

101

102

103

53 54 55 56 57 58 59 60

d5σ b

rlab/d

Ω1d

Ω2dE1 [mb/(GeV sr2)]

θ2 [deg]

1.8 GeV/c

R 1stR 2ndR 3rd

R fullexp

Summary

• Poincare invariant quantum mechanics provides amathematical framework for modeling nuclei that can beused effectively for energies up to a few GeV.

• Calculations have been performed for three-nucleonbound-state and scattering observables.

• Spin has been treated up to 250 MeV andspin-independent observable have been computed up to2 GeV.

• Two-body electromagnetic observables have beencomputed for bound systems of quarks and nucleons.

Conclusions

• A Poincare invariant model is needed to computescattering observables above 250 MeV.

• Kinematic corrections alone lead to misleading results.

• Many standard approximations are not accurate in allkinematic regions.

• There are some surprising low-energy effects in spinobservables.

Future

• Include explicit particle production.

• Include baryon resonances.

• More theory - cluster properties + particle production.

• Effects of cluster properties on electroweak current.

• Models with confined quark degrees of freedom.