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
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
2. Secondly, we considered the realistic case of the nuclear <strong>in</strong>teraction and presented a novel<br />
approach, the projection formalism, to the problem of deriv<strong>in</strong>g the forces between few (two,<br />
three, ... ) nucleons from effective chiral Lagrangians. For the case at hand, we first had to<br />
modify the power count<strong>in</strong>g rules proposed orig<strong>in</strong>ally by We<strong>in</strong>berg [73], s<strong>in</strong>ce <strong>in</strong> the projection<br />
formalism one decomposes the Fock space of nucleons and pions <strong>in</strong>to subspaces with def<strong>in</strong>ite<br />
nucleon and pion numbers. While <strong>in</strong> old-fashioned time-ordered perturbation theory the<br />
result<strong>in</strong>g wave functions are only orthonormal to a certa<strong>in</strong> order <strong>in</strong> the chiral expansion,<br />
this problem does not occur <strong>in</strong> the projection formalism. Furthermore, <strong>in</strong> the previous<br />
calculations based on time-ordered perturbation theory, the two-nucleon potentials turn<br />
out to be energy-dependent, which is a severe complication for apply<strong>in</strong>g these <strong>in</strong> systems<br />
with three or more nucleons (although to lead<strong>in</strong>g order, these recoil corrections are cancelled<br />
by certa<strong>in</strong> N-body <strong>in</strong>teractions). We now summarize the results obta<strong>in</strong>ed <strong>in</strong> chapter 3.<br />
• We start from the most general chiral <strong>in</strong>variant Hamiltonian for nonrelativistic nucleons<br />
and pions and divide the full Fock space <strong>in</strong>to the two subspaces T} and )... <strong>The</strong> first one<br />
conta<strong>in</strong>s only purely nucleonic states, whereas the second one <strong>in</strong>volves all rema<strong>in</strong><strong>in</strong>g<br />
states with nucleons and pions. To obta<strong>in</strong> an effective Hamiltonian act<strong>in</strong>g on the T}space<br />
we perform an appropriate unitary transformation, parametrized <strong>in</strong> terms of<br />
the operator )"AT} via eq. (3.190). After project<strong>in</strong>g onto the states )..i with i pions,<br />
the decoupl<strong>in</strong>g equation (3.192) turns <strong>in</strong>to an <strong>in</strong>f<strong>in</strong>ite system of coupled equations<br />
(3.205). We have proved that this system of equations can be solved perturbatively <strong>in</strong><br />
powers of the small momentum scale Q. For that we have developed an appropriate<br />
power count<strong>in</strong>g rule and analyzed the power of Q for all terms <strong>in</strong> eq. (3.206). We<br />
have worked out the recursive prescription for solv<strong>in</strong>g the system of equations (3.205),<br />
which allows to determ<strong>in</strong>e the operators )..i AT} for any f<strong>in</strong>ite i to any required order<br />
<strong>in</strong> the Q-expansion. <strong>The</strong> effective Hamiltonian Heff act<strong>in</strong>g on the T}-subspace can be<br />
obta<strong>in</strong>ed via eq. (3.197) us<strong>in</strong>g the power count<strong>in</strong>g rules (3.223)-(3.229).<br />
• We have applied the projection formalism and have given the explicit expressions of<br />
the two-nucleon potential to next-to-lead<strong>in</strong>g order (<strong>The</strong> lowest order potential comprises<br />
the lead<strong>in</strong>g one-pion exchange and contact terms). In particular, we have also<br />
calculated self-energy type corrections and one-loop corrections to these four-fermion<br />
operators not considered before.<br />
• We have discussed <strong>in</strong> detail the similarities and differences of the result<strong>in</strong>g potential<br />
with those obta<strong>in</strong>ed from time-ordered perturbation theory. In particular, it turns<br />
out that <strong>in</strong> our approach the isoscalar sp<strong>in</strong> <strong>in</strong>dependent central and the isovector sp<strong>in</strong><br />
dependent parts of the two-nucleon potential correspond<strong>in</strong>g to the two-pion exchange<br />
add up to zero. This was first noted <strong>in</strong> ref. [108] us<strong>in</strong>g a different scheme. Note that<br />
<strong>in</strong> the energy dependent potential of the time-ordered perturbation theory there is no<br />
such cancellation. However, the most salient features of the potential <strong>in</strong> the projection<br />
formalism is its energy--<strong>in</strong>dependence and the orthonormality of the correspond<strong>in</strong>g wave<br />
functions.<br />
• We have performed renormalization of the two-nucleon potential at NLO. In particular,<br />
we have shown that all ultraviolet divergences can be removed by an appropriate<br />
redef<strong>in</strong>ition of the axial coupl<strong>in</strong>g constant gA and various coupl<strong>in</strong>gs of the contact <strong>in</strong>teractions.<br />
<strong>The</strong> renormalized expressions for the N N potential at NLO agree with the<br />
ones given by the Munich group which were obta<strong>in</strong>ed us<strong>in</strong>g the Feynmann diagram<br />
technique.<br />
167