The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

160 5. Isospin violation in the two-nucleon system
�AII
1,-2
.. I ..
$ + 2
.. ..
+ +
Figure 5.3: Relevant graphs contributing to charge independence breaking at leading order aQ -2.
The open (filled) circle denotes a pion mass insertion rv 8M2 (an insertion of the leading fournucleon
operators rv aEb 1 )). For charge symmetry breaking, the leading contribution is given by
the last diagram with the filled circle denoting an insertionrv aEb2) .
From this we conclude that OPE diagrams with different pion masses have the isospin structure
T(I)i��T(I)j and lead to CIB since
012 =
= (a + b) T(I) . T(2) - bT(31) T(3 2)
(ppIOdpp)cs
(npIOdnp)
(nnI012Inn) = a ,
+ (npI012Ipn) = a + 2b , (5.34)
for the various isospin components of the two-nucleon system and 'Cs' stands for Coulombsubtracted.
Obviously, these effects are of order aQ-2. The np amplitude was already calculated
by KSW. We have worked out the leading corrections �A = Ann - Anp = A�: - Anp due to the
pion mass difference. The pertinent diagrams are shown in fig. 5.3. We follow the notation of
KSW and call these corresponding three amplitudes Ai�_,.IiI,IV where the first (second) subscript
refers to the power in a (Q) and the superscripts to the first three diagrams of the figure. We find
r =
r [
4 � 2 In (1 + �;) - M; : 4p2] ,
(mMnA_l) [ 1 2p i ( 4p2) 1 + � ]
47r pMn Mn 2pMn M; M; + 4p2
r -arctan -+ -- In 1 +- _ 7T
r (mMnA_l) 2 [ i 2p 1 ( M; + 4p2) I 1 1 1 + � ]
47r M2 M 2M2 1/2 M2 2 M2 + 4p2
2
n n n ,.., n n
- arctan - - -- n + - - - ---- -c:------"---;o-
-8M2 �;; , 8M2 = M;± -M;o , M; = M;o + 28M2 , (5.35)
with A-1 == AO,-1 the leading term in the expansion of the np I SO amplitude [91], see eq. (5.31)
Here, J.l is the PDS regularization scale. Note while the diagrams II and III are finite, the corresponding
integral in the �A{v:..2 diverges logarithmically. Therefore, the Lagrangian must contain
,