During the last lecture we looked at N-N scattering in the
low-energy regime, and determined what square well
potentials were required to reproduce the data.
We found that the potential had very different depths and ranges,
but had a volume integral that was
approximately the same.
Also, for homework this week you are looking at the low energy phase shifts
induced by a Yukawa potential of
range
and coupling of
.
The phase shift will be the same for spin triplet and
singlet channels by construction.
In the extreme low energy limit, we would expect that the long-range part
of the potential between
nucleons is dominated by the exchange of the lightest strongly interacting
particle (as we are requiring all interactions are local), which we know to be the
pion.
Let us consider the dynamics of non-relativistic nucleons with 's.
As we are dealing with low energy processes, with a characteristic momentum
transfer of order
,
we expect that virtual pair production of nucleons is
suppressed by
and that
production doesn't occur for energies
.
Therefore, we can use, honest to god, two-component spinors to describe
the theory and use a lagrange density
of the form
We are interested in the long-range behavour of the nuclear force and this
will be induced by the exchange of 's.
Let us construct the potential arising from one-pion exchange (OPE), it arises
from the tree-level graphs
with four external nucleons and an off-shell pion, as shown in
fig. (
).
The matrix element for nucleons scattering by pion exchange is
given by the graph, using our
usual Feynman rules, at lowest order in perturbation theory
(and recalling that a derivative acting on a scalar
field with incoming momentum q gives
)
The spatial potential
is recovered from the momentum
space potential
via a fourier
transform
Let us start by doing the integral, without all the factors out the front,
This looks like a fairly innocent expression, but now we have to evaluate the
partial derivatives acting on .
Lets set up the partial derivatives so that it is clear where the
various parts of the final expression come
from, we have that
The final result for the potential between two nucleons induced by
the exchange of a virtual pion is given
by
Notice that we have recovered the
function we
discussed earlier,
however, it was a bit subtle, as these things usually are!
We had to add it in by hand in order to ensure that the action of the
laplacian is
reproduced.
As we are not attempting to describes the short-distance part of the
potential at this stage, and will not ask
questions about operators that depend on short distance physics we can
forget about the
contribution. This gets renormalized away into the short
distance part of the potential,
described by
and
,
which in the language of meson exchange
models correspond to
and
exchange. In these models, which have many free parameters
fit to reproduce
nucleon-nucleon scattering, the singular part of the pion potential is
"renormalized away" by the fitting of
the heavy meson parameters.
This fact is usually not stated in this way, but this is infact what
is going on.
Let us now examine the behavour of each of the components
of the pion exchange potential.
The potential is strongly isospin dependent due to the
factor
(where
is the total isospin of the N-N
system).
The interactions is 3 times stronger in the
channel than in
the
channel.
The contribution from the
operator depends on which angular
momentum states are involved, it is a
non-central force.
Notice that
is a traceless and symmetric
tensor with 2 indices, and as such must
transform as a
object under rotations.
For a more pedestrian view of this consider inserting different values for
.
For
we have
,
and
or
gives
.
Both of which are written entirely in terms of the
tensors.
Therefore, this term can only contribute when the angular momentum of
the initial and final states differs by
.
So for, initial and final state S-waves, it doesn't
contribute.
However, in the case of the deuteron, where we have a total
state,
this operator can induce and
admixture of
into our initially
state. It is now clear that
we must consider both components
from the beginning and perform a coupled channels analysis.
Physically, the form of this potential indicates that it is strongest when the
separation vector
is aligned or antialigned in the direction of the spins.
The sign of the contribution depends upon the isospin channel under consideration,
but for the deuteron with isospin
,
the noncentral force deforms the deuteron to be cigar-shaped.
We should note that this form of interaction preserves spin,
i.e. it commutes with the
operator.
It is this interaction that is responsible for the quadrupole moment of the deuteron.
The central part of the potential induced by pion exchange
(preserves angular momentum by definition)
depends on the spin and isospin channel. It is attractive for
,
but repulsive for the
and
channels, to name a few.