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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12 2. Low-momentum effective theories for two nucleons<br />
these constra<strong>in</strong> <strong>in</strong> the same way the ultraviolet divergences. This makes it possible to absorb all<br />
ultraviolet divergences aris<strong>in</strong>g from loops <strong>in</strong> parameters of the Lagrangian and the theory becomes<br />
well-def<strong>in</strong>ed.<br />
At first sight, such non-renormalizable field theories seem to be completely useless, because an<br />
<strong>in</strong>f<strong>in</strong>ite number of vertices <strong>in</strong> the Lagrangian would lead to an <strong>in</strong>f<strong>in</strong>ite set of possible diagrams,<br />
which contribute to a def<strong>in</strong>ite process. In such a case, practical calculations would be, of course, not<br />
possible. However, as we will see below, <strong>in</strong> some cases <strong>in</strong> the low-energy limit non-renormalizable<br />
<strong>in</strong>teractions become weak and are strongly suppressed compared to renormalizable ones. This is<br />
because coupl<strong>in</strong>g constants of non-renormalizable <strong>in</strong>teractions can be expressed like Ci / An, w here<br />
A is so me mass scale, n > 0, and Ci is so me dimensionless number. Thus, if no additional mass<br />
scale appears <strong>in</strong> the correspond<strong>in</strong>g theory or if all mass sc ales are much smaller than A, one<br />
observes <strong>in</strong> the low-energy limit E ---+ 0 a strong suppression of non-renormalizable <strong>in</strong>teractions<br />
due to factors E / A. 3 So, <strong>in</strong> spite of <strong>in</strong>f<strong>in</strong>itely many parameters enter<strong>in</strong>g the Langangian, the<br />
theory possesses a predictive power. Indeed, if calculations are performed to a certa<strong>in</strong> f<strong>in</strong>ite level<br />
of accuracy, only a f<strong>in</strong>ite number of <strong>in</strong>teractions is relevant. As so on as all relevant coupl<strong>in</strong>gs are<br />
fixed from so me processes, predictions can be made for other processes and observables. This is<br />
w hat is usually understood under effective field theories (EFTs).<br />
Figure 2.1: <strong>The</strong> lead<strong>in</strong>g l/m! contribution to photon-photon scatter<strong>in</strong>g at<br />
second order <strong>in</strong> the f<strong>in</strong>e structure constant CI' can be represented by a contact<br />
<strong>in</strong>teraction (filled cirele).<br />
<strong>The</strong>re are many physical situations, <strong>in</strong> which EFTs provide a very useful and sometimes the only<br />
available way of perform<strong>in</strong>g practical calculations. If the fundamental theory is known, EFT can<br />
be useful as its low-energy approximation. For example, the lead<strong>in</strong>g photon-photon scatter<strong>in</strong>g<br />
diagram <strong>in</strong> QED, shown <strong>in</strong> fig. 2.1, can be calculated from the QED Lagrangian at one loop order<br />
where 'I/J denotes the electron fields, Fftv == 8ft AV - 8V Aft is the field strength tensor, me and<br />
CI' = e 2 /(47r) rv 1/137 denote the electron mass and electromagnetic f<strong>in</strong>e structure constant,<br />
respectively. In eq. (2.2) we have not shown gauge fix<strong>in</strong>g terms. Already <strong>in</strong> 1936 Euler, Kockel<br />
3 This statement obviously holds at the tree-level. Quantum corrections <strong>in</strong>clude loop <strong>in</strong>tegrals, which may<br />
be ultraviolet divergent and should be regularized and renormalized. This requires more careful considerations.<br />
We<strong>in</strong>berg has shown that quantum corrections are also suppressed [60).<br />
(2.2)