The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

72 3. The derivation o{ nuclear {orces {rom chiral Lagrangians
Thus, the function A(4) is suppressed by two powers of external momenta Q as compared to
the leading order result A (2). Apart from a polynomial part with undetermined constants, the
amplitude contains also logarithmic terms, that have the coefficients given by eq. (3.155) as definite
functions of frr. In fact, the structure ofthe non-polynomial part of A(4) can be obtained using the
renormalization group technique even without performing any explicit one-Ioop calculations [60].
Requiring that the amplitude A at each order does not depend on f-l allows to uniquely predict
the non-polynomial terms. To find the numerical coefficients of the logarithms one needs to
perform complete loop calculations.29 Note further, that an additional overall numerical coefficient
'" 1j(47r)2 appears in A(4), that arises from the loop integrals. Therefore, the pertinent expansion
parameter is Qj(47rfn.), that is remarkably smaller then the naive expected one Qjfn. Manohar
and Georgi have shown [166] that the scale which enters the values of the renormalized coupling
constants is Ax ::; 47r f 7r '" 1200 MeV. Their argumentation is based on the observation, that
the change in the renormalization scale of order one would change the effective couplings by
o(ljA�) '" 1j(47rf7r)2. Therefore, if the scale Ax would be very large for one particular value
f-lo of the renormalization scale, 1jA� « 1j(47rf7r)2 , it would be no more the case for another
choice of f-l. A consistent power counting requires the assumption Ax '" 47r f 7r. In that case the
quantum corrections are of the same order of magnitude as the contributions from the renormalized
interaction terms.
One can straightforwardly extend all these results to account for the explicit chiral symmetry
breaking. The amplitude A becomes also a function of the pion mass and does not vanish at
the threshold s = 4M;. Already in 1966 Weinberg calculated the 7r7r scattering lengths from the
leading order amplitude A(2) [167] using the current algebra techniques. Corrections of higher
order in m7rj(47rf7r) seem to improve the agreement with the experimental values [168].
To end this section we would like to make a remark concerning the so-called chiral anomaly. It
turns out that the effective Lagrangian [)2) + .c(4) implies a stronger symmetry than QCD does.
The absence of the terms with an insertion of the total antisymmetric tensor EIl-Vpu together with
the parity conservation requires that all terms in the effective Lagrangian are even in the pion
fields. This rules out such experimentally observed processes like, for instance, 7r0 ----7 rr. This
problem for the SU(3) case has been solved in 1971 by Wess and Zumino [169]. They have found
a term in the action of the effective field theory, the so-called Wess-Zumino-Witten term, that
preserves the chiral symmetry of the action but cannot be expressed as the integral of the chiral
invariant density over spacetime. Its explicit form (in the absence of external sources) is given by
(3.156)
where the integration goes over the five-dimensional sphere whose surface is spacetime, Eijklm
is the totally antisymmetric tensor in 5 dimensions and Ne = 3 is the number of colors. The
geometrical interpretation of this anomalous term was given by Witten [170], who had also shown
that the coefficient of this term is not a freely adjustable parameter. A further discussion of the
chiral anomaly goes beyond the scope of this manuscript. The interested reader is referred to the
original publications [169], [170], [171].
29 These coefficients have been derived in 1972 using unitarity instead of the phenomenological Lagrangian [164],
[165].