The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

5.3 The leading eIB and eSB effects in the np 1 So channel 159
+ + 2
+ +
Figure 5.2: The subleading amplitude Ao in the isospin symmetrie ease. The shaded ovals are
defined in fig. 5.1. The filled square (diamond) eorresponds to the eontaet vertices with two
derivatives (one insertion of M;).
where the leading term in the expansion of the np I SO amplitude [91] is given by:
5.3 The leading CIB and CSB effects in the N N lS0 channel
(5.30)
(5.31)
Consider now the effeet of the eharged to neutral pion mass differenee 8M2, see the upper left
diagram in fig. 5.3. OPE between two neutrons, neutron and proton or two protons ean obviously
involve eharged and neutral pions. The mass differenee ean be treated in the following way: as
proposed in ref. [224], one ean modify the pion propagator,
(5.32)
with e the pion four-momentum and 'i,j' isospin indices. Sinee we are interested only in the
leading eorreetions rv 8M2 rv ü, it suffiees to work with the expanded form of eq.(5.32),
(5.33)