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
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
14 2. Low-momentum effective theories for two nucleons<br />
fundamental theories like QED or the Standard Model are really fundamental <strong>in</strong> the sense that they<br />
rema<strong>in</strong> valid for arbitrary high energies. Rather, these renormalizable theories may be low-energy<br />
approximations of some other and more fundamental underly<strong>in</strong>g theories. As already po<strong>in</strong>ted<br />
out before, non-renormalizable <strong>in</strong>teractions are usually suppressed by <strong>in</strong>verse powers of energy,<br />
at which new physics appear. For example, the <strong>in</strong>teractions <strong>in</strong> the effective QED Lagrangian<br />
(2.3) are suppressed by four powers of the <strong>in</strong>verse electron mass. In pr<strong>in</strong>ciple, there are two<br />
ways of test<strong>in</strong>g new physical effects beyond the Standard Model: either one can try to perform<br />
more precise measurements of observables to test effects of non-renormalizable <strong>in</strong>teractions or one<br />
can <strong>in</strong>crease the energy. At present, no experimental <strong>in</strong>consistencies of the Standard Model to<br />
data have been observed. This <strong>in</strong>dicates that the energies at which new physics might become<br />
significant are quite high and, probably, <strong>in</strong> the TeV region [140].<br />
Up to now we have only considered effective field theories. <strong>The</strong> discussed ideas f<strong>in</strong>d also many<br />
<strong>in</strong>terest<strong>in</strong>g applications <strong>in</strong> quantum mechanics and <strong>in</strong> other fields of physics. For example, the<br />
standard multipole expansion of a complicated current source J(r, t) of size d, which generates<br />
radiation with large wavelengths A » d, already uses the basic idea of effective theories: large<br />
distance physics should be not very sensitive to details of the structure at short distances. That<br />
is why one usually observes quick convergence of such multipole expansions.<br />
In the next sections of this chapter we will concentrate, basically, on purely quantum mechanical<br />
problems. All ideas and methods of effective theories, that we have discussed <strong>in</strong> the context of<br />
relativistic quantum field theories, rema<strong>in</strong>, of course, valid <strong>in</strong> this case. In particular, as already<br />
stressed above, the low-energy behavior of a theory is not sensitive to details of the short-distance<br />
<strong>in</strong>teraction. 6 Follow<strong>in</strong>g Lepage's lecture [141], we would like to enumerate here the basic steps of<br />
construct<strong>in</strong>g the effective theory:<br />
• First, one should correctly <strong>in</strong>corporate the low-energy physics <strong>in</strong>to the effective theory. This<br />
must be known from the underly<strong>in</strong>g theory. Note that if the underly<strong>in</strong>g <strong>in</strong>teraction has a<br />
f<strong>in</strong>ite range, one can always restrict an effective theory to very low energies, at which the<br />
details of the <strong>in</strong>teraction are not important.7 In such a case one does not need to know<br />
anyth<strong>in</strong>g about the underly<strong>in</strong>g theory.<br />
• Secondly, one should <strong>in</strong>troduce an ultraviolet cut-off to exclude high-moment um states. <strong>The</strong><br />
cut-off furthermore provides f<strong>in</strong>iteness of various <strong>in</strong>tegrals aris<strong>in</strong>g by perform<strong>in</strong>g calculations.<br />
• F<strong>in</strong>ally, the missed short-range physics is represented <strong>in</strong> form of local correction terms with<br />
arbitrary coefficients to the effective Hamiltonian. <strong>The</strong>se coefficients have to be fixed from<br />
the data. It is important that one should keep all local terms allowed by the symmetries of the<br />
underly<strong>in</strong>g theory. Putt<strong>in</strong>g new correction terms systematically removes the dependence of<br />
the low-energy observables on the cut-off, which is compensated by "runn<strong>in</strong>g" of parameters<br />
<strong>in</strong> the effective Hamiltonian. Any given precision can be achieved with a f<strong>in</strong>ite number of<br />
correction terms.<br />
60ne needs, <strong>in</strong> general, higher energies to probe shorter-distance physics. This follows immediately from Heisenberg's<br />
uncerta<strong>in</strong>ty relation.<br />
7 A counter example is the Coulomb potential which has an <strong>in</strong>f<strong>in</strong>ite range. In the language of quantum field<br />
theory, the Coulomb force results from the exchange of photons. S<strong>in</strong>ce photons are massless, they can not be<br />
completely <strong>in</strong>tegrated out even at very low energies. One can, however, separate photons <strong>in</strong>to "soft" and "hard"<br />
on es and keep only "soft" photons explicitly with<strong>in</strong> an effective theory.