R. MachleidtNuclear Forces - Lecture 2 Meson Theory (2013) 1 Nuclear Forces - Lecture 2 - R. Machleidt University of Idaho The Meson Theory of Nuclear

R. Machleidt Nuclear Forces - Lecture 2 Meson Theory (2013)

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Nuclear Forces- Lecture 2 -

R. MachleidtUniversity of Idaho

The Meson Theory of Nuclear Forces


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Lecture 2: The Meson Theory of Nuclear Forces

• The mesons• How do those mesons contribute to the

NN interaction?• The One-Boson-Exchange Potential• Closing remarks


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The mesons:Have a look at the Particle Data Group

(PDG) Table


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5

STOP

pseudo scalar

scalar

vector


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What do those mesons do to the NN interaction?

To find out, we have to do some calculations. Proper calculations are done in the framework of Quantum Field Theory. That means, we have to take the following steps:

Write down appropriate Lagrangians for the interaction of the mesons with nucleons.

Using those interaction Lagrangians, calculate Feynman diagrams that contribute to NN scattering.


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Lowest orderFeynman diagram for NN scattering

2p

1'p

1p

2 'p

1 1( ' )q p p 1 2

2 2

with Dirac spinor ( , ) 02

where and is a two-component Pauli spinor.

s ss

s s

s

E Mu p s p p

ME M E M

E p M

1 1 1 2 2 22 2

' 'Amplitude: ( ', )

u u P u uF p p

q m


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Pseudo-vector coupling of a pseudo-scalar meson

( )5

NNNN

fm

L

( )NNfq

m

i

Lagrangian:

Vertex: times the Lagrangian stripped off the fields (for an incoming pion)

Potential: times the amplitude

i

2 1 21 22 2 2

( )( )NN q qfV iF

m q m

2 2( , )P i q q

25( ) NN

NNf

i qm


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Pseudo-vector coupling of a pseudo-scalar meson, cont’d

2

1 2 1 2 12

12 1 2 1 2

ˆ( )( ) ( )3

ˆ ˆ ˆ( ) 3( )( )

qq q S q

S q q q

2

2 2

1 12 1 22 2 2ˆ( )

3NNf q

V S qm q m

Using the operator identity

with (“Tensor operator”),

the one-pion exchange potential (OPEP) can be written as


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Scalar coupling

( )NN g

L

2 2

2

1' ( ' ) 4 1 1

( ' )( ) 4

k q Lp p i p pig ig

E M E M M

2 21 2

1keeping all terms up to / , ( ' ), ( ) ( )

2 2i

Q M P i k p p L S q k

2 2 2

2 2 2 2 212 8 2

g k q L SV iF

q m M M M

Potential:

Lagrangian:

Vertex: ( ') ( ) ( ') ( )NNu p u p ig u p u p ( ')( )1

( ' )( )p p

igE M E M


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Vector coupling of a vector meson

( )NN g

L

2 keeping only1 ,4

the term.L

i LgM

2

2 2 21 32

g L SV iF

q m M

ncluding also the terms: P ig

Potential, i

Lagrangian:

Vertex:0 0 ( ') ( ) ( ') ( )0: NNu p u p ig u p u p ( ')( )

1( ' )( )

p pig

E M E M


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Tensor coupling of a vector meson

(tensor) ( ) ( )( )4NNf

M

L

(tensor) ( ) ( )4 2 2NNff f

q q q qM M M

2(tensor) (tensor) 1 2

1 22 2 2( )( )

4

f q qV iF

M q m

Lagrangian:

Vertex (incoming meson)

Potential: P ig

221 2 1 2

1 22 2 2

( )( )

4

q q qf

M q m

2 2

1 2 12 1 22 2 2ˆ2 ( )

12

f qS q

M q m


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STOP

pseudo scalar

scalar

vector

Recall: We found the mesons below in PDG Table and asked:

What do they do?

Now, we have the answer. Let’s summarize.


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Summary

2

2 2

1 12 1 22 2 2ˆ( )

3NNf q

V S qm q m

2

2 2 212

g L SV

q m M

2

2 2 21 32

g L SV

q m M

2 2

1 2 12 1 22 2 2ˆ2 ( )

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f qV S q

M q m

(138)(600)(782)(770)

Long-ranged tensor force

intermediate-ranged, attractive central force

plus LS force

short-ranged, repulsive central force plus strong LS force

short-ranged tensor force,

opposite to pion

It’s EVERYTHING we need to describe the nuclear force!


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Summary:Most important parts of the nuclear force

ShortInter-mediate Long range

Tensor force

Central force

Spin-orbit force


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The One-Boson Exchange Potential (OBEP)

0

OBEP

, , , , , ,a

V V

0

1 2

0

(548) is a pseudo-scalar meson with 0, therefore, is given by

the same expression as , except that carries no ( ) factor.

(980) is a scalar meson with 1, therefore, is given by

th

a

I V

V V

a I V

0 1 2e same expression as , except that carries a ( ) factor.aV V


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Some comments

• Note that the mathematical expressions for the various given on previous slides are simplified (many approximations) --- for pedagogical reasons.

• For a serious OBEP, one should make few approximations. In fact, it is quite possible to apply essentially no approximations. This is known as the relativistic (momentum-space) OBEP. Examples are the OBEPs constructed by the “Bonn Group”, the latest one being the “CD-Bonn potential” (R. M., PRC 63, 024001 (2001)).

• If one wants to represent the OBE potential in r-space, then the momentum-space OBE amplitudes must be Fourier transformed into r-space. The complete, relativistic momentum-space expressions do not yield analytic expressions in r-space after Fourier transform, i.e., it can be done only numerically. However, it is desirable to have analytic expressions. For this, the momentum-space expressions have to be approximated first, e.g., expanded up to , after which an analytic Fourier transform is possible. The expressions one gets by such a procedure are shown on the next slide. Traditionally, the Nijmegen group has taken this approach; their latest r-space OBEPs are published in: V. G. J. Stoks, PRC 49, 2950 (1994).

V

2 2/Q M


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OBEP expressions in r-space(All terms up to are included.)2 2/Q M


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Does the OBE model contain “everything”?

• NO! It contains only the so-called iterative diagrams.

• However: There are also non-iterative diagrams which contribute to the nuclear force (see next slide).

1Lippmann-Schwinger eqn:

1 1 1

T V V Te

T V V V V V Ve e e

“i-t-e-r-a-t-i-v-e”

In diagrams: T = + + + …


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Some examples for non-iterative meson-exchange contributions not included in the OBE model (or OBEP).

The “Bonn Full Model” (or “Bonn Potential”) contains these and other non-iterative contributions. It is the most comprehensive meson-model ever developed (R. M. et al., Phys. Reports 149, 1 (1987)).

The “Paris Potential” is based upon dispersion theory and not on field theory. However, one may claim that, implicitly, the Paris Potential also includes these diagrams; M. Lacombe et al., Phys. Rev. C 21, 861 (1980).


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Reviews on Meson Theory

• Pedagogical introduction which also includes a lot of history: R. M., Advances in Nuclear Physics 19, 189-376 (1989).

• The derivation of the meson-exchange potentials in all mathematical details is contained in: R. M., “The Meson Theory of Nuclear Forces and Nuclear Matter”, in: Relativistic Dynamics and Quark-Nuclear Physics, M. B. Johnson and A. Picklesimer, eds. (Wiley, New York, 1986) pp. 71-173.

• Computer codes for relativistic OBEPs and phase-shift calculations in momentum-space are published in: R. M., “One-Boson Exchange Potentials and Nucleon-Nucleon Scattering”, in: Computational Nuclear Physics 2 – Nuclear Reactions, K. Langanke, J.A. Maruhn, and S.E. Koonin, eds. (Springer, New York, 1993) pp. 1-29.


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End Lecture 2

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R. MachleidtNuclear Forces - Lecture 2 Meson Theory (2013) 1 Nuclear Forces - Lecture 2 - R. Machleidt University of Idaho The Meson Theory of Nuclear