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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68 3. <strong>The</strong> derivation of nuc1ear forces from chiral Lagrangians<br />
<strong>The</strong> corrections O(M) <strong>in</strong> eq. (3.134) correspond to higher order symmetry break<strong>in</strong>g terms that<br />
<strong>in</strong>clude two and more <strong>in</strong>sert ions of the quark mass matrix M. Eq. (3.134) allows one to express<br />
the pion mass <strong>in</strong> terms of the quark condensate as follows<br />
This expression is known as the Gell-Mann-Oakes-Renner relation [163].<br />
(3.135)<br />
Let us f<strong>in</strong>ally make aremark concern<strong>in</strong>g the constant <strong>in</strong>: . For that we can calculate the axial-vector<br />
current correspond<strong>in</strong>g to the Lagrangian (3.124), that can be equivalently rewritten as<br />
(3.136)<br />
To calculate the symmetry current we use eq. (3.13). Note that <strong>in</strong> the case of a SU(2)A transformation<br />
one has (Jv = 0, (JR = (JA and (JL = -(JA, see eqs. (3.92), (3.93). Consequently, an<br />
<strong>in</strong>f<strong>in</strong>itesimal axial-vector transformation 90 E SU(2)A of the matrices U, ut is given by<br />
<strong>The</strong> related axial-vector current iS21<br />
U � U' = hRUhR = U - �(J� (TiU + UTi) ,<br />
ut' - h-1Uth-1 R R - ut + �(Ji A (TiUt + UtTi) ,<br />
2<br />
(3.137)<br />
(3.138)<br />
where the ellipses stay for terms with more pion fields. Clearly, the axial current connects the<br />
vacuum state with the one-pion state:22<br />
(3.139)<br />
Here, n hys 23 denotes the experimentally observed value of the pion decay constant, which can be<br />
calculated from the rate for pion decay. <strong>The</strong> dots correspond to corrections, which are proportional<br />
to powers of p2 / i; . 24 Consequently, these corrections are proportional to the pion mass squared<br />
M; , which is zero <strong>in</strong> chiral limit. Thus, <strong>in</strong> the chiral limit <strong>in</strong>: is not renormalized and <strong>in</strong>: = n hys.<br />
To perform later the expansion <strong>in</strong> powers of small momenta Q we need an order<strong>in</strong>g scheme for<br />
various <strong>in</strong>teractions <strong>in</strong> the effective Lagrangian, which can be viewed as the derivative expansion<br />
and the expansion <strong>in</strong> powers of quark masses correspond<strong>in</strong>g to the symmetry break<strong>in</strong>g terms.<br />
One commonly counts the squared pion mass M; rv (140MeV)2 or, equivalently, the quark mass<br />
<strong>in</strong>sertion as two powers of small external momenta (two derivatives). <strong>The</strong>refore, the complete<br />
Lagrangian to lead<strong>in</strong>g order is given by<br />
(3.140)<br />
21<br />
Note that BA corresponds to _ca <strong>in</strong> the notation of eq. (3.12).<br />
22This is a general consequence of Goldstone theorem.<br />
23<br />
0ne commonly uses a different notation, <strong>in</strong> which OUf Irr (I!:hys) is denoted by F or I (Irr).<br />
24This can be demonstrated from Lorentz <strong>in</strong>variance and the power count<strong>in</strong>g scheme, that will be <strong>in</strong>troduced <strong>in</strong><br />
the follow<strong>in</strong>g section.