Chapter 1

The Nucleon-Nucleon System

1.1 The Lippmann-Schwinger Equation for the Scat-

tering Process

Let us consider two-nucleon scattering and define ~k1 and ~k2 to be the individual nucleonmomenta. The relative momentum is then given as

~p =1

2(~k1 − ~k2) (1.1)

The momentum eigenstates in the nucleon-nucleon (NN) c.m. system are then

| ~p〉 (1.2)

and are chosen to be normalized as

〈~p | ~p ′〉 = δ3(~p − ~p ′) . (1.3)

They are eigenstates to the free Hamiltonian for the relative motion

H0 =~p 2

m(1.4)

where m is the nucleon mass.Let V be the NN potential, which is assumed to be energy independent.The Schrödinger equation for a scattering state Ψ

(+)~p

(H0 + V ) Ψ(+)~p = E Ψ

(+)~p (1.5)

2

can be cast into an integral equation, the Lippmann-Schwinger equation (LSE):

(H0 − E) Ψ(+)~p = −VΨ(+)~p (1.6)

| Ψ(+)~p 〉 = | ~p〉+1

E + iǫ−H0V | Ψ(+)~p 〉 (1.7)

Let us consider the configuration space representation. The conjugate variable to ~p is

~x = ~r1 − ~r2 , (1.8)

and we choose | ~x〉 to be normalized as

〈~x | ~x′〉 = δ3(~x− ~x′). (1.9)

Then the Fourier transform is given by

〈~x | ~p〉 = 1(2π)3/2

ei~p·~x . (1.10)

The configuration space representation of the free propagator

G0 ≡1

E + iǫ−H0(1.11)

is given as

〈~x | 1E + iǫ−H0

| ~x ′〉 =∫

d~p 〈~x | ~p〉 1E + iǫ− p2/m 〈~p | ~x

′〉

=1

(2π)3/2

∫

d3p ei~p·~x1

E + iǫ− p2/me−i~p·~x ′

=1

(2π)3

∫

d3p ei~p·(~x−~x′) 1

E + iǫ− p2/m

=1

(2π)34π

∫ ∞

0

dp p2j0(pρ)1

E + iǫ− p2/m(1.12)

with ρ ≡| ~x− ~x′ |Standard residue techniques lead to

〈~x | G0 | ~x ′〉 = −m

4π

ei√mE|~x−~x′|

| ~x− ~x′ | , (1.13)

3

which exhibits the outgoing wave behavior from the source point ~x′ to ~x. It results theconfiguration space representation of the LSE

〈~x | Ψ(+)~p 〉 ≡ Ψ(+)~p (~x)

=1

(2π)3/2ei~p·~x − m

4π

∫

d3x′ei

√mE|~x−~x′|

| ~x− ~x′ | V (x′) Ψ

(+)~p (~x

′) .

(1.14)

Thereby we assumed a local potential

〈~x | V | ~x′〉 = δ3(~x− ~x ′) V (x′) (1.15)

In a well known manner one reads off the asymptotic form for | ~x |→ ∞ :

Ψ(+)~p (~x) →

1

(2π)3

2

( ei~p~x +eipx

xf(x̂) ), (1.16)

with the scattering amplitude f(x̂) depending on the direction x̂ of observation

f(x̂) = −m√

π

2

∫

d3x′ e−ipx̂·~x′

V (x′) Ψ(+)~p (~x

′) . (1.17)

This can be interpreted in terms of a scattered momentum

~p ′ ≡ x̂p , (1.18)

and one introduces a transition amplitude

〈~p ′ | t | ~p〉 ≡ 1(2π)

3

2

∫

d3x′ e−i~p′~x ′ V (x′) Ψ

(+)~p (~x

′)

= 〈~p ′ | V | Ψ(+)~p 〉 .(1.19)

Apparently t is the result of the scattering process and determines all scattering observ-ables.Is there an integral equation directly for t? From (1.24) we read off

t | ~p〉 ≡ V | Ψ(+)~p 〉 , (1.20)

and using (1.7) we find

t | ~p〉 = V | ~p〉+ V G0 V | Ψ(+)~p 〉= V | ~p〉+ V G0 t | ~p〉

(1.21)

4

We can strip off the initial state | ~p〉 and get the operator relation

t = V + V G0 t (1.22)

which is the LSE for the transition operator. Its simple physical interpretation results byiterating that equation:

t = V + V G0 ( V + V G0t )

= V + V G0V + V G0V G0V + V G0V G0V G0V + . . .

(1.23)

This is the Born series for scattering on V , a sum of terms of increasing order in V . Eachterm consists of a sequence of V ′s with free propagations in between. This is a generalstructure valid for any number of particles.

It is useful to visualize that multiple scattering process in the form

t

where the dashed lines stand for the action of V and two horizontal lines for the freepropagation G0 between two interactions. Intuitively one can start from that sum ofterms (1.23)

t = V + V G0 V + V G0 V G0 V + . . . (1.24)

and ask the question: can this series be summed up into an integral equation for t ?Obviously it can:

t = V + V G0 ( V + V G0V + V G0V G0V + . . .) , (1.25)

and we recover t again on the right hand side and thus get

t = V + V G0 t (1.26)

which is the LSE.Of course if one starts from Eq. (1.23) in an ad hoc manner one has to know the form ofthe free propagator G0 and therefore one has to make contact to the underlying dynamicalequation, in our case the Schrödinger equation. Formally however this multiple scatteringseries is quite general and also valid for the Bethe-Salpeter equation, where G0 is differentfrom the G0 used in our nonrelativistic context.

5

1.2 Alternative Derivation of the Lippmann-Schwinger

Equation

A free momentum eigenstate obeys

H0 | ~p 〉 = Ep | ~p 〉 (1.27)

and a scattering state obeys

H | ~p 〉(+) = Ep | ~p 〉(+) . (1.28)

Here a scattering state is defined via

| ~p 〉(+) = Ω(+) | ~p 〉 = limǫ→0 iǫ G(E + iǫ) | ~p 〉 , (1.29)

where Ω(+) is the Møller operator, which maps a free state | ~p 〉 onto a scattering state| ~p 〉(+).

The propagators, or Resolvents, are given by

G0(z) =1

z −H0(1.30)

and

G(z) =1

z −H . (1.31)

Here G0(z) is the free propagator (Resolvent) and G(z) the full propagator (Resolvent).

Let us consider

G−10 −G−1 = (z −H0)− (z −H) = −H0 +H = V , (1.32)

where we use that H = H0 + V .

Multiplying Eq. (1.32) from the left with G0 and from the right with G yields

G0(G−10 −G−1)G = G−G0 = G0V G (1.33)

or

G = G0 +G0V G . (1.34)

The above relation, Eq. (1.34), is called first Resolvent Identity.

6

Applying the first Resolvent Identity on a momentum eigenstate and multiplying bothsides of the resulting equation with iǫ yields

iǫ G(E + iǫ) | ~p 〉 = iǫ G0(E + iǫ) | ~p 〉 + iǫ G0(E + iǫ)V G(E + iǫ) | ~p 〉

=iǫ

E + iǫ−H0| ~p 〉 + iǫ G0(E + iǫ)V G(E + iǫ) | ~p 〉

= | ~p 〉 + iǫ G0(E + iǫ)V G(E + iǫ) | ~p 〉 .(1.35)

Taking the limit ǫ → 0 gives, together with the definition Eq. (1.29), the Lippmann-Schwinger equation for states

| ~p 〉(+) = | ~p 〉 + G0(E + iǫ)V | ~p 〉(+) (1.36)

If we multiply Eq. (1.36) by V and define

V | ~p 〉(+) = t | ~p 〉 , (1.37)

we obtainV | ~p 〉(+) = V | ~p 〉 + V G0(E + iǫ)V | ~p 〉(+) (1.38)

ort | ~p 〉 = V | ~p 〉 + V G0(E + iǫ)t | ~p 〉 . (1.39)

Since the operators in Eq. (1.39) are applied on a general state | ~p 〉, we can consider thisequation as operator equation:

t = V + V G0(E + iǫ)t . (1.40)

This equation is also called operator Lippmann-Schwinger equation.

A next task is to derive from Eq. (1.36) a relation to the scattering wave function ψ+(~r).Let us consider

〈~r | ~p〉(+) = 〈~r | ~p 〉 + 〈~r | G0 V | ~p〉(+) (1.41)

which leads to

ψ(+)(~r) ≡ 〈~r | ~p 〉(+) = 〈~r | ~p 〉 + 〈~r | G0 t | ~p〉 (1.42)

= 〈~r | ~p〉 +∫

d3p′ 〈~r | ~p′〉 〈~p′ | G0 t | ~p〉 .

(1.43)

Applying the definition of G0 leads to

〈~r | ~p 〉(+) = 〈~r | ~p 〉 +∫

d3p′ 〈~r | ~p′ 〉 1E + iǫ− p′2

m

〈~p′ | t | ~p 〉 , (1.44)

7

which is the desired equation for the scattering wave function ψ(+)(r).

Energy conservation leads to an additional constraint for the t-operator. If a momentumbefore the scattering event is denoted with ~p, and after the scattering event with ~p ′, thenenergy conservation requires

~p ′2

m=

~p 2

m⇒ ~p ′2 = ~p 2 . (1.45)

This means that we can extract an energy conserving δ-function from the matrix element

〈~p ′ | T (E) | ~p 〉 = δ(Ep′ − Ep) 〈p̂′ | t(E) | p̂〉 . (1.46)

The latter relation is sometimes called on-shell condition. The physical meaning is thatthe observables of NN scattering only determine the matrix elements consistent with therelation (1.46).

1.3 The Lippmann-Schwinger Equation for the Bound

State

Let us assume, that V supports a bound state | Ψb〉 at E = Eb〈0. Then

(H0 + V ) | Ψb〉 = Eb | Ψb〉 (1.47)

or(H0 − Eb) | Ψb〉 = −V | Ψb〉 (1.48)

Since Eb〈0 there is no regular and square integrable solution to the left hand side aloneand | Ψb〉 obeys the homogeneous LSE

| Ψb〉 =1

Eb −H0V | Ψb〉 (1.49)

Using the configuration space representation Eq. (1.12) for E = Eb〈0 we see that (1.21)guarantees the correct exponential fall-off behavior of

〈~x|Ψb〉 ≡ Ψb(~x) = −m√

π

2

∫

d3x′e−

√m|Eb||~x−~x ′|

|~x− ~x ′| V (x′)Ψb(~x

′) (1.50)

8

1.4 Connection Between Homogeneous and Inhomo-

geneous LSE’s

It is of interest and importance to relate the homogeneous equation, valid at the discreteenergy E = Eb

| Ψb〉 = G0(Eb)V | Ψb〉 (1.51)and the inhomogeneous equation, derived for E〉0

t(E) = V + V G0(E)t(E) . (1.52)

The transition operator t(E) can be evaluated also for E〈0. What happens for E → Eb?We rewrite (1.52)

( 1 − V G0(E) ) t(E) = V (1.53)t(E) = ( 1 − V G0(E) )−1 V (1.54)

Let us expand

t(E) = ( 1 + V G0 + V G0V G0 + . . . ) V

= V ( 1 +G0V + G0V G0V + . . . ) .

(1.55)

If we apply t(E) onto | Ψb〉 and choose E = Eb, then we find, using Eq. (1.31)

t(Eb) | Ψb〉 = V ( 1 + 1 + 1 + . . .) | Ψb〉 , (1.56)

which is clearly diverging.

More precisely

t(E) = [ ( G−10 − V ) G0 ]−1V

= G−101

E −H0 − VV

= G−101

E −H V

= G−10 G−1 V

(1.57)

Inserting the completeness relation

| Ψb〉〈Ψb | +∫

d3p | Ψ(+)~p 〉〈Ψ(+)~p |= 1 (1.58)

9

to the left of V gives

t(E) = (E −H0) | Ψb〉1

E − Eb〈Ψb | V

+

∫

d~p (E −H0) | Ψ(+)~p 〉1

E − ~p 2/m〈Ψ(+)~p | V

= V | Ψb〉1

E −Eb〈Ψb | V

+

∫

d~p V | Ψ(+)~p 〉1

E − ~p 2/m〈Ψ(+)~p | V .

(1.59)

We see explicitly that t(E) has a pole at E = Eb

t(E) → V | Ψb〉1

E −Eb〈Ψb | V for E → Eb (1.60)

Thus t(E) has a pole at the energy where the homogeneous LSE has a solution, whichis the same as requiring that the homogeneous part of the inhomogeneous LSE has asolution:

Θ(E) = V G0(E) Θ(E) (1.61)

PutΘ(E) ≡ V χ(E) (1.62)

thenχ(E) = G0(E) V χ(E) (1.63)

This is identical to (1.38) and thus

χ(E) = Ψb at E = Eb (1.64)

This pole in t(E) at the NN bound state will be of decisive importance for describing aninteracting system of 3 or more nucleons.

1.5 Realization in a Partial Wave Representation in

Momentum Space

We introduce the momentum space basis to a fixed orbital angular momentum l andmagnetic quantum number m

| plm〉 (1.65)

10

These states are defined via

〈 ~p ′ | p l m〉 ≡ δ(p′ − p)p p′

Ylm(p̂′) . (1.66)

They are complete and orthonormal

∑

lm

∫

dp p2 | p l m〉〈p l m | = 1 (1.67)

〈p l m | p′ l′ m′〉 = δ(p′ − p)p p′

δll′ δmm′ (1.68)

Let us consider the LSE for t(E) in this basis

〈p′l′m′|t(E)|plm〉 =

〈p′l′m′|V |plm〉+∑

l′′m′′

∫ ∞

0

dp′′ p′′2〈p′l′m′|V |p′′l′′m′′〉

× 1E + iǫ− p′′2/m〈p

′′l′′m′′|t(E)|plm〉

(1.69)

We take V to be rotationally invariant:

〈p′ l′ m′ | V | p l m〉 = δll′ δmm′ Vl(p′, p) (1.70)

which leads to an integral equation in one variable:

tl(p′p) = Vl(p

′p) +

∫ ∞

0

dp′′ p′′ 2 Vl(p′p′′)

1

E + iǫ − p′′ 2/m tl(p′′p) (1.71)

What is Vl, assuming V (r) to be given? Introduce states

| rlm〉 (1.72)

defined analogously to (1.59) via

〈~x | r l m〉 ≡ δ(x− r)xr

Ylm(x̂) (1.73)

Then

〈 p l m | r l m〉 =∫

d~p ′∫

d~x 〈p l m | ~p ′〉 〈~p ′ | ~x〉〈~x | r l m〉

=

∫

d3p′∫

d3xδ(p′ − p)p′p

Y ∗lm(p̂′)

1

(2π)(3/2)e−i~p

′·~x δ(x− r)xr

Ylm(x̂)

=

√

2

πjl(pr) i

l

(1.74)

11

Therefore, assuming a local potential:

Vl(p′p) = 〈p′lm | V | plm〉

=

∫ ∞

0

dr r2∫ ∞

0

dr′r′ 2 〈p′ l m | r′ l m〉

× 〈r′ l m | V | r l m〉〈〈r l m | p l m〉

=2

π

∫ ∞

0

dr r2∫ ∞

0

dr′ r′ 2 jl(p′r′)

δ(r − r′)rr′

V (r) jl(pr)

=2

π

∫ ∞

0

dr r2 jl(p′r) V (r) jl(pr)

(1.75)

This is one way to determine the momentum space representation of a local potential.The LSE for tl can easily be solved by standard methods.Let us now consider the full space for two nucleons including spin and isospin:

|p(ls)jm(12

1

2)tmt〉 ≡

∑

ml

C(lsj,mlm−ml)|plml〉|sm−ml〉

×∑

ν

C(1

2

1

2t, νmt − ν)|

1

2ν〉|1

2|mt −mν〉 (1.76)

Clearly one has s = 0, 1 and t = 0, 1. The antisymmetry (working in isospin formalism)leads to the well known restriction

(−)l+s+t = −1 (1.77)

for the allowed quantum numbers. Thus t = 1 states are

1S0 ,3P0 ,

3P1 ,1D2 ,

3P2 −3 F2, . . . (1.78)

and t = 0 states are1P1 ,

3S1 −3 D1 , 3D2 , . . . (1.79)

The hyphen denotes coupled states, where l is not conserved. A well known mechanismfor that is the tensor force.For a general NN force, which conserves spin and parity one has

〈p′(l′s′)j′m′t′m′t|V |p(ls)jmtmt〉 = δjj′δmm′δtt′δmtmt′δss′Vsjtmtll′ (p

′, p) (1.80)

Because of (1.71) conservation of isospin follows and the indicated t-dependencies for Vis redundant.There is a dependence on mt, the charge state of the two nucleons, in case of charge-independence breaking (CIB) or charge-symmetry breaking (CSB):

12

CIB means: np 6= pp/strong forces CSB means: nn 6= pp/strong forces

It is well established that in the state 1S0 the np force is different from the nn or pp force.This is evident in the different scattering lengths:

anp = −23.48± 0.009fm

app/strong = −17.36± 0.4fm (recommended value)(G.A. Miller et al, Phys. Rep. 194 (1990) 1)

ann = −18.6± 0.3fm

(extracted from π− + d → n + n+ γ; B. Gabioud et al, Phys. Rev. Lett. 42 (1979) 1508;O. Schori et al, Phys. Rev. C35 (1987) 2252). That π− absorption experiment has beenredone at Los Alamos and is presently being analyzed.

In addition, nd breakup experiments are being presently performed (W. Tornow, TUNL),in order to extract ann using modern Faddeev calculations.

In t = 1 states different from 1S0 CIB or CSB is not yet convincingly established, thoughsmall effects at least have to be there, simply because of the different pion masses.We shall drop in the following the possible mt-dependence in the notation.

In this most general basis, the LSE for t is represented as

〈p′(l′s′)jt|t(E)|p(ls)jt〉 = 〈p′(l′s)jt|V |p(ls)jt〉

+∑

l′′

∫ ∞

0

dp′′2 p′′2〈p′ (l′s)jt|V |p′′(l′′s)jt〉

× 1E + iǫ− p′′2/m〈p

′′(l′′s)jt|t(E)|p(ls)jt〉 (1.81)

or

tsjl′l(p′, p) = V sjl′l (p

′p) +∑

l′′

∫ ∞

0

dp′′ p′′2V sjl′l′′(pp′′)

1

E + iǫ− p′′2/m tsjl′′l(p

′′p) (1.82)

Since s is at most 1 and parity is conserved

l = l′ or l = l′ ± 2 (1.83)

Thus one has either a single equation or two coupled equations. A prominent examplefor the coupled case is

3S1 −3 D1 (1.84)acting in the deuteron.

13

1.6 NN Phase-Shifts

The t-matrix generated by the coupled or uncoupled LSE is unitary. Let us choose amatrix notation

t ≡ tsjl′l(p′p) etc. (1.85)Then

t = V + V G0 t = V + t G0 V (1.86)

The adjoint of that ist† = V + V G0

∗ t† (1.87)

since V † = V. This is valid on physical grounds. Subtraction yields

t − t† = V G0t− V G∗0t†= V G0(t− t†) + V (G0 −G∗0)t†

(1− V G0)(t− t†) = V (G0 −G∗0) t†t− t† = (1− V G0)−1V (G0 −G∗0 ) t†

= t ( G0 − G∗0 ) t†

(1.88)

Now

G0 =1

E + iǫ − H01 (1.89)

thusG0 − G∗0 = −2πi δ(E −H0) 1 (1.90)

and we get, back in explicit notation

tl′l(p′p)− t∗ll′(pp′) =

∫ ∞

0

dp′′ p′′ 2∑

l′′

tl′l′′(p′p′′)(−2πi) δ(E − p

′′ 2

m) t∗ll′′(pp

′′)

= −2πi m√mE

2

∑

l′′

tl′l′′(p′√mE) t∗ll′′(p

√mE)

(1.91)

Let us choose the on-the-energy shell values p = p′ =√mE :

tl′l(pp) − t∗ll′(pp) = −πimp∑

l′′

tl′l′′(pp)t∗ll′′(pp) (1.92)

Back in matrix notation this is

t− t† = −πimp t t† (1.93)

14

Now we introduce a S-matrixS = 1 − iπmp t (1.94)

and find

S S† = (1 − iπmp t) (1 + iπmp t†)= 1 − iπmp (t − t† − iπmp t t†) = 1 (1.95)

Thus S is unitary and can be parameterized in the coupled case by 3 parameters:

S =

(

cos2ǫ e2iδ1 isin2ǫ ei(δ1+δ2)

isin2ǫ ei(δ1+δ2) cos2ǫ e2iδ2

)

(1.96)

which is the ”Stapp” or ”bar”-phase shift parameterization(H.P. Stapp et al, Phys. Rev. 105 (1957) 302).In the uncoupled case S is simply

S = e2iδ (1.97)

with δ real.

The well known NN phase-shift parameters by the Nijmegen group(V.G.J. Stoks et al, Phys. Rev C48 (1993) 792)can be viewed on-line at

http://nn-online.sci.kun.nl/

and by George Washington University INS Data Analysis Center (R. A. Arndt et al,Phys. Rev. D45 (1992) 3995)

http://http://gwdac.phys.gwu.edu/

which also links several other partial wave analysis for e.g. pion-nucleon scattering orphotoproduction.

1.7 Deuteron Properties

The homogeneous LSE Eq. (1.51) is now projected onto the basis given in Eq. (1.73).Thus for

Ψl(p) ≡ 〈p (ls) j t | Ψb〉 (1.98)

15

with l = 0, 2, s = j = 1, t = 0 one gets the set of two coupled equations

Ψl(p) =1

Eb − p2

m

∑

l′=0,2

∫ ∞

0

dp′ p′ 2 Vll′(pp′) Ψl′(p

′) (1.99)

This can be solved numerically by standard techniques. Realistic forces are adjusted toreproduce various measurable quantities:

• Eb = −2.2246 MeV

• Q = 0.2859 fm2 (there are theoretical uncertainties in the description of that ex-perimental value caused by MEC)

• As = 0.8883 fm−1/2 (asymptotic normalization constant for the s-wave component)

• η = AD/As = 0.02564 (asymptotic d/s ratio)

The deuteron d-state probability

pd ≡∫∞0

Ψ22(p) p2dp

∫∞0

Ψ20(p) p2 dp +

∫∞0

Ψ22(p) p2 dp

(1.100)

is not a measurable quantity, but strongly correlated to nuclear binding energies, as weshall see later. In general, the smaller pd the larger the triton and α-particle bindingenergies.Let us now consider the single nucleon momentum distribution

n(k) ≡ 12

1

3

∑

m

〈Ψb m |2∑

i=1

δ(~k − ~kcmi ) | Ψb m〉 (1.101)

=1

3

∑

m

〈Ψb m | δ(~k − ~kcm1 ) | Ψb m〉 (1.102)

We have

~p =1

2(~kcm1 − ~kcm2 ) = ~kcm1 (1.103)

and thus

n(k) =1

3

∑

m

∫

d3p 〈Ψb m | ~p〉 δ(~k − ~p) 〈~p | Ψbm〉. (1.104)

One has

〈~k|Ψbm〉 =∑

l

∫ ∞

0

dp p2〈~k|p(ls)jm〉Ψl(p)

=∑

l

∑

ml

C(lsj,ml, m−ml)Ylml(k̂)|sm−ml〉Ψl(k) (1.105)

16

and therefore

n(k) =1

3

∑

m

∑

ll′

∑

ml

C(l′ s j,ml m−ml) (1.106)

× C(l s j,ml m−ml) Y ∗l′ml(k̂) Ylml(k̂)× Ψl′(k) Ψl(k)

Now, with â being defined as â ≡ 2a+ 1 we have

C(l s j,ml m−ml) = (−)s+m−ml√

ĵ

l̂C(j s l ,−m,m−ml) (1.107)

Using the above relation we find

n(k) =1

3

∑

ml

∑

ll′

Y ∗l′ml(k̂) Ylml(k̂) (1.108)

× Ψl′(k) Ψl(k)∑

m

√

ĵ

l̂

√

ĵ

l̂ ′

× C(j s l,−m,m−ml) C(j s l′,−m,m−ml)

=1

3

∑

ml

∑

l

Y ∗lml(k̂) Ylml(k̂)ĵ

l̂Ψ2l (k) (1.109)

=ĵ

3

1

4π

∑

l

Ψ2l (k) =1

4π

∑

l=0,2

Ψ2l (k) (1.110)

This is displayed for several realistic NN forces in the next figure, where different short

range behavior of NN forces is reflected for k>∼ 1fm−1.

17

There is hope to measure these quantities in electron scattering on deuterons.

Of interest is also the NN correlation function, the probability to find 2 nucleons at a

18

distance r:

C(r) ≡ 13

∑

m

〈Ψbm|δ(~r − ~x)|Ψb m〉

=1

3

∑

m

〈Ψbm|~r〉〈~r|Ψb m〉

=1

4π

∑

l=0,2

Ψ2l (r) (1.111)

The connection between configuration and momentum space is given by

Ψl(r) =

√

2

π

∫ ∞

0

dp p2jl(pr)Ψl(p) (1.112)

The l = 0 and 2 parts of C(r) together with their sum are displayed below. We seedifferences at short distances, depending on the strengths of the short range repulsions,as shown in the next figure

19

20

1.8 Nuclear Forces

The determination of the nuclear force is a longstanding and still unsolved basic problem.The whole issue on how to set up a framework for deriving a nuclear force is not touchedhere. Simply a list of so called ”realistic” NN forces is given:

• Paris potential (dispersion theoretical background) by M. Lacombe et al, Phys. Rev.C21 (1980) 861

• Nijmegen 78 potential (one-boson-exchange background) by M.M. Nagels et al,Phys. Rev. D17 (1978) 768

• AV14 potential (one-pion tail, otherwise phenomenological) by R. B. Wiringa et al,Phys. Rev. C29 (1984) 1207

• Bonn potential (meson exchange potential (multiple meson), based on time-orderedperturbation theory by R. Machleidt, K. Holinde, Ch. Elster, Phys. Rep. 149 (1987)1 and R. Machleidt, Adv. Nucl. Phys. 19 (1989) 189

All those potential have χ2/Ndata ≥ 2 with respect to the Nijmegen data base.

Most recent NN potential, however all phenomenological with about 30-50 parameters fitthe Nijmegen data base with a χ2/Ndata ∼ 1 and are

• Nijmegen I (includes ∇2-term)

• Nijmegen II (local)

• Reid 93 (local) by V.G.J. Stoks et al, Phys. Rev. C49 (1994) 2950

• AV18 (updated AV14, local, as operators defined) by R.B. Wiringa et al, Phys. Rev.C51 (1995) 38

• CD-Bonn (nonlocal) by R. Machleidt, F. Sammarruca, Y. Song, Phys. Rev. C53,(1996), R1483.

They come in charge-dependent versions and describe the NN data up to 350 MeV per-fectly well with χ2/Ndata ∼ 1

This is the first time that one has a set of ”realistic” nearly phase-equivalent NN forces.They cover a certain range of properties, a NN force can have:

21

• local versus nonlocal

• soft or hard core

What is still missing in that family are potentials with dynamical nonlocalities at veryshort distances r ≤ 0.8fm, say, resulting from the overlap regions of extended nucleons.They might be good for surprises.

1.9 Construction of the NN Potential From Invari-

ance Requirements

We want to investigate to what extent the form of the potential VNN(1, 2) acting betweentwo nucleons is determined by the requirement that the Hamiltonian describing the sys-tem be invariant under various symmetry transformations. This analysis will be madeconsidering the two nucleons as identical particles, i.e., disregarding the difference of themass and charge between the neutron and proton. The Hamiltonian has then the form

H =1

2m(p21 + p

22) + VNN(1, 2) , (1.113)

m being the nucleon mass.

Regarding the symmetry properties of H , we shall assume first of all invariance under therestricted Galilei group. Then we shall assume invariance under the discrete transforma-tions of space reflection, time invariance and permutation of the two nucleons. Finally,we shall assume invariance under the isospin transformations of the group SM(2).

The operators we have at our disposal to build up the potential are the coordinates ~r1, ~r2,the momenta ~p1, ~p2, the spin vector operators ~σ

(1), ~σ(2) and the isospin vector operators~τ (1), ~τ (2). Going to the two nucleon c.m. frame gives

~r = ~r2 − ~r1~R =

1

2(~r1 + ~r2)

~p = ~p2 − ~p1~P = ~p2 + ~p1 . (1.114)

1. Assume that the potential operator is hermitian

VNN = V†NN . (1.115)

22

2. Using time-translation invariance, which makes VNN not explicitly dependent onthe time t, gives

VNN ≡ VNN (~r, ~R, ~p, ~P , ~σ(1), ~σ(2), ~τ (1), ~τ (2)) . (1.116)

Let us now discuss the implications of the assumed invariance on the dependenciesof VNN on the indicated variables.

3. Consider the invariance for rotations in charge space, which determines the depen-dence of VNN on the isospin vectors ~τ

(1) and ~τ (2). The unitary operator representinga rotation in charge space is given by

UI(w) = ei~I ·~w , (1.117)

where ~I is the total isospin and ~w = ~n w. The required invariance is expressed by

U †I VNN UI = VNN (1.118)

with arbitrary ~n and w. (1.118) will be satisfied if VNN is a scalar in isospin space.In order to construct all possible scalars from ~τ (1) and ~τ (2), it is remarked that anypolynomial expression in ~τ (i) can be reduced to a linear expression by using

[τ(i)j , τ

(i)k ] = iεjkℓ τ

(i)ℓ

(τ(i)j )

2 = 1 ,

(1.119)

so that, e.g.,

(~τ (1) · ~τ (2))2 = 3− 2(~τ (1) · ~τ (2)) . (1.120)

Hence, the most general expression that has to be considered is linear, both in ~τ (1)

and ~τ (2). The only scalar quantity obtained in this way is ~τ (1) · ~τ (2). It follows thatVNN is a function only of this quantity as regards its dependence on the isospinvariables of the two particles:

VNN ≡ VNN(~r, ~R, ~p, ~P , ~σ(1), ~σ(2), [~τ (1) · ~τ (2)]) . (1.121)

Expanding VNN in a power series of ~τ(1) ·~τ (2) and expressing (~τ (1) ·~τ (2))n with (1.120)

in terms of ~τ (1) · ~τ (2) and the identity operator in isospin space, one obtains

VNN = V1(~r, ~R, ~p, ~P , ~σ(1), ~σ(2)) + (~τ (1) · ~τ (2)) V2(~r, ~R, ~p, ~P , ~σ(1), ~σ(2)) . (1.122)

We can now limit ourselves to study the implications of invariance an each termVi separately, since all other symmetry transformations commute with the isospinoperators. For convenience we drop the index i from now on.

23

4. Invariance under space translations is expressed by

U †a V Ua = V (1.123)

with Ua = exp(

i~

~P · ~a)

. We get

U †a V (~r,~R, ~p, ~P , ~σ(1), ~σ(2)) Ua

= V (U †a ~r Ua, U†a~R Ua, U

†a ~p Ua, U

†a~P Ua, U

†a ~σ

(1) Ua, U†a ~σ

(2) Ua)

= V (~r, ~R− ~a, ~p, ~P , ~σ(1), ~σ(2)) .(1.124)

The condition (1.123) then implies that V does not depend on ~R:

V (~r, ~p, ~P , ~σ(1), ~σ(2)) . (1.125)

5. Invariance under proper Galilei transformations is expressed by

U †G V UG = V (1.126)

with

UG ≡ exp(

i

~

~P · ~v0t)

exp

(

− i~m~R · ~v0

)

(1.127)

with ~v0 being the c.m. velocity. It follows that

U †G V UG = V (~r, ~p, U†G~P UG, ~σ

(1), ~σ(2))

= V (~r, ~p, ~P − ~v0m,~σ(1), ~σ(2)) .(1.128)

The condition (1.126) then implies that V is independent of ~P :

V = V (~r, ~p, ~σ(1), ~σ(2)) . (1.129)

6. Invariance under space reflections implies in the normal way that

V (~r, ~p, ~σ(1), ~σ(2)) = V (−~r,−~p, ~σ(1), ~σ(2)) . (1.130)

7. Invariance under the permutation of the two nucleons gives

V (~r, ~p, ~σ(1), ~σ(2)) = V (−~r,−~p, ~σ(2), ~σ(1)) . (1.131)

Invariance under the combined transformations (6) and (7) gives

V (~r, ~p, ~σ(1), ~σ(2)) = V (~r, ~p, ~σ(2), ~σ(1)) . (1.132)

24

8. Invariance under time reversal means

V (~r, ~p, ~σ(1), ~σ(2)) = V ∗(~r,−~p,−~σ(1),−~σ(2))= V (~r,−~p,−~σ(1),−~σ(2))

(1.133)

Since V is assumed to be hermitian.

9. Invariance under spatial rotations is expressed by

U †R V UR = V (1.134)

with UR = exp(

i~

~J · ~n w)

, with ~J being the total angular momentum of the

system, ~J = ~L+ ~S. Requiring rotational invariance means that

V (~r, ~p, ~σ(1), ~σ(2)) = V (R~r,R~p, R~σ(1), R~σ2)) , (1.135)

where R~a gives the rotated of the vector ~a.

Let us first take into account the dependence of V on the spin variables. Here the proce-dure is not so straightforward as it was for the isospin, since spin, position and momentumvectors can be combined to build rotational invariant quantities. Using spin identifies sim-ilar to (1.120), one can show that V can be expressed as

V = Vα + ~σ(1)~V

(1)β + ~σ

(2)~V(2)β + Vγ(~r, ~p, ~σ

(1), ~σ(2)) . (1.136)

Vγ is linear in both ~σ(1) and ~σ(2) but contains only bilinear combinations of these two

operators. From rotation and space-reflection invariance, Vα and Vγ must be scalars, ~V(1)β

and ~V(2)β pseudovectors. Combination of space reflection and particle exchange [(6) and

(7)] implies that

Vα + ~σ(1) · ~V (1)β + ~σ(2) · ~V

(2)β + Vγ(~r, ~p, ~σ

(1), ~σ(2))

= Vα + ~σ(2) · ~V (1)β + ~σ(1) · ~V

(2)β + Vγ(~r, ~p, ~σ

(2), ~σ(1)) .

(1.137)

Taking the average of these two expressions for V , one gets

V = Vα + ~S · ~Vβ + Vγ(~r, ~p, ~σ(1), ~σ(2)) (1.138)

where ~S = ~2(~σ(1) + ~σ(2)), ~Vβ =

1~(~V

(1)β , ~σ

(2)), and Vγ now being symmetric under the

exchange of the spin operators. The vector we can use to construct ~Vβ are ~r, ~p and

25

~L = ~r × ~p, but only ~L is a pseudovector. Thus, ~Vβ must then be ~L× (scalar quantity).Then (1.138) reads

V = Vα(~r, ~p) + ~S · ~L Vβ(~r, ~p) + Vγ(~r, ~p, ~σ(1), ~σ(2)) . (1.139)

Since Vα and Vβ are scalars, they can only be functions of r2, p2, L2, ~r · ~p and ~p · ~r. Since

the operators ~r ·~p and ~p ·~r are non-hermitian, it is convenient to consider their hermitiancombinations (~r · ~p+ ~p ·~r) and i(~r · ~p− ~p ·~r). The latter is a constant and can be dropped.The former can only appear quadratically in Vα and Vβ due to time-reversal invariance.With

(~p · ~r + ~r · ~p)2 = 2(r2p2 + p2r2) − 4L2 + 3~2 , (1.140)

we get

V = Vα(r2, p2, L2) + ~S · ~L Vβ(r2, p2, L2) + Vγ(~r, ~p, ~σ(1), ~σ(2)) . (1.141)

From the requirements on Vγ follows that it can only contains terms of the type

~σ(1) · ~σ(2), (~σ(1) · ~r)(~σ(2) · ~r), (~σ(1) · ~p)(~σ(2) · ~p),(~σ(1) · ~L)(~σ(2) · ~L) + (~σ(2) · ~L)(~σ(1) · ~L),

(~σ(1) · ~p)(~σ(2) · ~r) + (~σ(2) · ~r)(~σ(1) · ~p) + 1 ↔ 2 . (1.142)

The last expression changes sign under time reversal and must be replaced by

[(~σ(1) · ~p)(~σ(2)~r) + (~σ(2) · ~r)(~σ(1) · ~p) + 1 ↔ 2](~p · ~r + ~r · ~p) . (1.143)

It can be shown that (1.142) is de facto a function of the other quantities appearing in(1.141) and thus not independent. We have, therefore, for Vγ

Vγ = (~σ(1) · ~σ(2)) V (I)γ (r2, p2, L2)

+ (~σ(1) · ~σ(2)) V (II)γ (r2, p2, L2)+ (~σ(1) · ~p)(~σ(2) · ~p) V (III)γ (r2, p2, L2)+ [(~σ(1) · ~L)(~σ(2) · ~L) + (~σ(2) · ~L)(~σ(1) · ~L)] V (IV )γ (r2, p2, L2)

(1.144)

as most general form of Vγ compliant with all symmetry requirements.

Concluding, the most general, velocity-dependent, non-relativistic NN potential has theform (1.122) with Vi given by

Vi = Vci (r

2, p2, L2) + ~σ(1) · ~σ(2) V σi (r2, p2, L2)+ S12 V

Ti (r

2, p2, L2) + ~S · ~L V LSi (r2, p2, L2)+ [(~σ(1) · ~L)(~σ(2) · ~L) + (~σ(2) · ~L)(~σ(1) · ~L)] V σLi (r2, p2, L2) ,

(1.145)

26

where S12 is the tensor force operator

S12 =3

r2(~σ(1) · ~r)(~σ(2) · ~r) − (~σ(1) · ~σ(2)) . (1.146)

Remark: As shown in the Appendix of S. Okubo and R.E. Marshak, Ann. Phys. 4,166 (1958), the potential of (1.145) gives an S-matrix that on-shell is identical to theone obtained from a potential in which the term V σp is dropped. Therefore, if one isonly interested in NN scattering, it can be neglected. The same cannot be said for thebound states or for the off-shell S-matrix. Often the term V σL, is also neglected. Somearguments are given in Machleidt, Holinde, Elster, Phys. Rep. 149, 1 (1987).

1.10 Simple Introduction to One Boson-Exchange Po-

tential (OBEP)

The basic idea of OBE models is to represent the NN interaction as superposition oftree-diagrams (born terms) which represent the exchange of single mesons, namely scalar(s), pseudoscalar (ps), vector (v) bosons (Jp = 0+, 0−, 1−, respectively), with massesup to 1 GeV between two nucleons. Mesons with masses larger than 1 GeV would onlygive very short-ranged exchange contributions and contribute in a region where the OBEmodel is no longer valid.

The couplings for the various mesons are given in terms of their interaction Lagrangiandensities by

LNNps = gps ψ̄ iγ5 ψφps (1.147)

LNNs = gs ψ̄ ψ φs (1.148)

LNNv = gv ψ̄ γµ ψ φµv +fv4m

ψ̄ σµv ψ(∂µφvv − ∂vφµv ) (1.149)

for pseudoscalar (π, η), scalar (σ, δ) and vector mesons (ρ, ω), respectively. m is thenuclear mass, ψ the nucleon and φα the meson field operators. For isospin T = 1 meansφα is to be replaced by ~τ · ~φα, with τi being the usual Pauli matrices. Furthermore,σµν =

i2[γµ, γν ], where γµ are the usual Dirac-matrices (see, e.g., Bjorken-Drell). The

coupling constants gα(α = s, ps, v) and fv and the meson masses mα are at least partiallydetermined from high-energy experiments or symmetry relations. The Lagrangian densityfor vector mesons contains Dirac (gv) as well as Pauli coupling (fv). An OBE-potential

27

V (~q ′, ~q) is obtained through the superposition of exchange contributions of the differentmesons

V (~q ′, ~q) =∑

α=s,ps,v

Vα(~q′, ~q) (1.150)

with

Vα(~q′, ~q) =

√

m

Eq′

√

m

Eqū (−~q ′) Γ(2)α u(−~q) Pα ū(~q ′) Γ(1)α u(~q) . (1.151)

The factors√

mEq′

√

mEq

are the so-called minimal relativity factors, which take into con-

sideration the relativistic unitarity condition (see K. Erkelenz, Phys. Rep. 13C, 191(1974)). They certainly contribute to the nonlocality of V (~q ′, q). (Their effect has beenstudied in a simple model in Ch. Elster, E.E. Evans, H. Kamada, W. Glöckle, Few-BodySystems 21, 25 (1996).

The meson propagators are usually given by

Pα = ((~q′ − ~q )2 +m2α)−1 (1.152)

and the vertex functions for the meson-nucleon vertices Γ(i)α (i = 1, 2) are given by

Γ(i)s = gs (1.153)

Γ(i)ps = gps i γ5 (1.154)

Γ(i)v (direct) = (gv + fv)γµ (1.155)

Γ(i)v (gradient) = −fv2m

(~q ′ + ~q )µ . (1.156)

In order to take into account the finite extension of the nucleon and to be able to solvethe dynamical equations, the coupling constants get modified with form factors. This isessentially achieved by replacing

gα −→ gαFα(~q ′, ~q) (1.157)

where Fα(~q′, ~q) can be, e.g., of dipole type

Fα[(~q′, ~q )2] =

(

Λ2α −m2αΛ2α + (~q

′ − ~q )2)nα

. (1.158)

28

The exponent nα is usually taken as nα = 1,Λα is the cutoff parameter and usually of theorder 1− 2 GeV. The positive energy Dirac spinors are given by

u(i)(~q ) =

√

Eq +m

2m

(

1~σ·~q

Eq+m

)

| i〉 (1.159)

where | i〉 denote the Pauli spinors(

10

)

and

(

01

)

. Inserting (1.159), (1.153), (1.152)

into (1.151) gives for the scalar contribution of the potential (Pauli spinors are omitted):

Vs(~q′, ~q ) = −g2s

√

m

E ′q

√

m

Eq

(E ′q +m)(Eq +m)

4m21

(~q ′ − ~q )2 +m2s

×(

1− ~q′ · ~q + i~σ2 · (~q′ × ~q )(E ′q +m)(Eq +m)

)(

1− ~q′ · ~q + i~σ1 · (~q ′ × ~q )(E ′q +m)(Eq +m)

)

.(1.160)

This expression has, due to the ~q and Eq dependencies, a strong nonlocality. In order toarrive at expressions, which can be transformed to coordinate space, one changes variablesto

~k = ~q ′ − ~q~p =

1

2(~q′ + ~q )

(1.161)

and in addition has to introduce the following approximations:

1. On-shell approximation: E ′q = Eq

2. Expansion of E in powers of q2

m2:

E =

(

1

2(~q ′ + ~q )2 +m2

)1

2

= m+1

4m(q′2 + q2) + · · ·

= m+p2

2m+

k2

8m+ · · ·

(1.162)

3. Keeping only the lowest order in p2 and k2.

29

With these approximations, the scalar potential becomes

V cs (~k, ~p) = − g

2s

k2 +m2s

[

1− p2

2m2+

k2

8m2− i

2m2~S · (~k × ~p)

]

(1.163)

where ~S = 12(~σ1 + ~σ2).

This expression still contains nonlocalities due to ~p 2 as well as (~k × ~p) terms. The latterleads to the angular momentum operator ~L = −i~r × ~∇ in r-space, whereas the formerprovides ∇2 terms. After a Fourier transform, the coordinate space expression of thescalar potential is given by

V cs (r) = −g2s4π

ms

{[

1− 14

(ms

m

)2]

Y(msr)

+1

4m2[

∇2 Y(msr) + Y(msr) ∇2]

+1

2Z1(msr) ~L · ~S

}

(1.164)

where Y(x) = e−x/x and Z1(x) =(

mαm

)2(1/x + 1/x2) Y(x).

The treatment of the Schrödinger equation with a momentum dependent potential is givenby O. Rojo, L.M. Simmons, Phys. Rev. 125, 273 (1962). The expressions for the otherpotential terms shall only be given here:

V cps(~k , ~p ) = −

g2ps4m2

(~σ1 · ~k)(~σ2 · ~k)k2 +m2ps

(1.165)

V cv (~k, ~p ) =

1

k2 +m2v

{

g2v

[

1 +3p2

2m2− k

2

8m2+

3i

2m2~S · (~k × ~p)

− (~σ1 · ~σ2)k2

4m2+

1

4m2(~σ1 · ~k)(~σ2 · ~k)

]

+gvfv2m

[

− k2

m+

4i

m~S · (~k × ~p) − ~σ1 · ~σ2

k2

m+

1

m(~σ1 · ~k)(~σ2 · ~k)

]

+f 2v4m2

[

−~σ1 · ~σ2 k2 + (~σ1 · ~k)(~σ2 · ~k)]

}

.

(1.166)

The structure of the expression (1.160) already suggests that one would prefer to workwith OBE potentials in momentum space. Even the already approximated expressions(1.163), (1.165), (1.166) are still complicated functions of the momenta, though they

30

can be Fourier transformed analytically to coordinate space. The corresponding r-spaceexpressions to (1.165) and (1.166) are

V cps(r) =1

12

g2ps4π

mps

[

(mpsm

)2

Y(mpsr) ~σ1 · ~σ2 + Z(mpsr) S12]

(1.167)

V cv (r) =g2v4π

mv

{[

1 +1

2

(mvm

)2]

Y(mvr) −3

4m2[∇2 Y(mvr) + Y(mvr)∇2]

+1

6

(mvm

)2

Y(mvr) ~σ1 · ~σ2 −3

2Z1(mvr) ~L · ~S −

1

12Z(mvr) S12

}

+1

2

gvfv4π

mv

{

(mvm

)2

Y(mvr) +2

3

(mvm

)2

Y(mvr) ~σ1 · ~σ2

− 4Z1(mvr) ~L · ~S −1

3Z(mvr) S12

}

+f 2v4π

mv

{

1

6

(mvm

)2

Y(mvr) ~σ1 · ~σ2 −1

12Z(mvr) S12

}

. (1.168)

Here the tensor operator S12 is given by (1.146) andZ(x) = (mα/m)

2(1 + 3/x+ 3/x2)Y(x).

Details on OBE potentials are given in the references quoted in Section 1.8.

31