The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
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78 3. <strong>The</strong> derivation o{ nuclear {orces {rom chiral Lagrangians<br />
scale as l/Q. Let us now consider the two different cases shown <strong>in</strong> fig. 3.3. In the two-nucleon<br />
center-of-mass system the energy denom<strong>in</strong>ator <strong>in</strong> the graph a) <strong>in</strong> fig. 3.3 is given by<br />
1 1<br />
Ei - E p2 Im -pl2 Im -vi q2 + M; (3.179)<br />
where Ei is the <strong>in</strong>itial energy of two nucleons and q = p' -p and p = 1P1, p' = Ip'l, q = 1iJ1. This<br />
estimation is valid modulo 11m corrections. Consider now the second diagram. For the energy<br />
denom<strong>in</strong>ator correspond<strong>in</strong>g to the energy cut <strong>in</strong> fig. 3.3 b) we have:<br />
1 1 m m .<br />
Ei - E p2 Im -[2 Im + iE [2 (3.180)<br />
p2 _<br />
+ iE' rv Q2<br />
Thus, the energy denom<strong>in</strong>ator <strong>in</strong> the second case is mlQ times larger than for the first diagram.<br />
In the limit m ---+ (X) it becomes even <strong>in</strong>f<strong>in</strong>itely large. <strong>The</strong>refore, the power count<strong>in</strong>g (3.177) is<br />
not valid any more. To solve the problem with the large energy denom<strong>in</strong>ators aris<strong>in</strong>g from the<br />
<strong>in</strong>termediate states with only two nucleons, We<strong>in</strong>berg proposed to apply the technique of CHPT<br />
not directly to the scatter<strong>in</strong>g amplitude but to the effective potential. <strong>The</strong> latter is def<strong>in</strong>ed as a sum<br />
of all irreducible diagrams, i. e. those diagrams without pure nucleonic <strong>in</strong>termediate states. <strong>The</strong><br />
S-matrix elements are obta<strong>in</strong>ed by putt<strong>in</strong>g this potential <strong>in</strong>to a Lippmann-Schw<strong>in</strong>ger equation.<br />
In fact, many equivalent schemes for deriv<strong>in</strong>g the effective potentials <strong>in</strong> nuclear and many-body<br />
physics are known like that due to Bloch and Horowitz [183] or the Ta<strong>in</strong>m-Dancoff approximation<br />
[184]. <strong>The</strong> effective potential, derived us<strong>in</strong>g old-fashioned time-ordered perturbation theory,<br />
possesses one unpleasant property: it is, <strong>in</strong> general, explicitly dependent on the energy of the<br />
<strong>in</strong>com<strong>in</strong>g nucleons and, as a consequence of this, not hermitian. Furthermore, the nucleonic wave<br />
functions are not orthonormal <strong>in</strong> this approach [185].<br />
One can avoid these problems by us<strong>in</strong>g the method of unitary transformation. It was already<br />
applied successfully <strong>in</strong> cases where one has an expansion <strong>in</strong> a coupl<strong>in</strong>g constant, such as the<br />
pion-nucleon coupl<strong>in</strong>g, see e.g. refs. [186]. In the follow<strong>in</strong>g sections we will show how to apply<br />
this method to the case of chiral perturbation theory for pions and nucleons, <strong>in</strong> which the small<br />
momenta of extern al particles play the role of the expansion parameter. In fact, our considerations<br />
are more general s<strong>in</strong>ce they can be applied to any effective field theory of Goldstone bosons coupled<br />
to some massive matter fields.<br />
3.5 Bloch-Horowitz scheme and the method of unitary transfor<br />
mation<br />
<strong>The</strong> method of unitary transformation (projection formalism) was already applied <strong>in</strong> the chapter 2<br />
to decouple the spaces of small and large momenta <strong>in</strong> the quantum mechanical two-body system.<br />
Here we would like to repeat the ma<strong>in</strong> po<strong>in</strong>ts, mostly to establish our notation and keep the<br />
section self-conta<strong>in</strong>ed. We will also compare it with the Bloch-Horowitz scheme of deriv<strong>in</strong>g the<br />
effective <strong>in</strong>teractions.<br />
A system of an arbitrary number of <strong>in</strong>teract<strong>in</strong>g pions and nucleons can be completely described<br />
by a Schröd<strong>in</strong>ger equation<br />
(3.181)<br />
with Ho (Hf) denot<strong>in</strong>g the free (<strong>in</strong>teraction) part of the Hamiltonian. In order to solve this<br />
equation for nucleon-nucleon scatter<strong>in</strong>g, it is advantageous to project it onto a subspace<br />
{IN), INN), INNN),<br />
I