3.6. Application to chiral <strong>in</strong>variant Hamiltonians number of pions 12 / , \ I , 11 \ \ , , \ \ 10 / I , 9 / I , 8 \ \ / � 7 6 / , , 5 / I , 4 \ 3 2 1 � 0 , ,\ • • / -- - ' , \ '. / I , \ ., \ / � , , / \ e- , -- / , \ '. / I , \ ., \ / � , , / \ e- , /. / , .... . , \ \ • • • • • • I • • / I ., • , e- -' \ , , • \ , \ / -- , \ '. / , '. / I , \ ., \ / I , \ ., \ / � , , e- / \ � \ e- / . - , , , / , , -- / / -- , , \ ' '::: . \ , , \ .... �. 2 3 , \ \ • • • • • • • • • • • • • • • • • • • • • • • • ., / • • I , / \ e-- , , • • \ , / -- \ , , \ '�. / -- \ , , \ .... �. • , -->-- .. 4 5 6 order r Figure 3.5: Recursive prescription for calculat<strong>in</strong>g the operator ,\4k+i Ar]. <strong>The</strong> dashed l<strong>in</strong>e shows a sequence of recursive steps. <strong>The</strong> large circles correspond to states with 4k pions (i = 0). Thus, we aga<strong>in</strong> count the power of Q for the terms enter<strong>in</strong>g eq. (3.206) start<strong>in</strong>g from its m<strong>in</strong>imal value. In appendix B we check what k<strong>in</strong>d of operators ,\4k+i AIr] contribute to eq. (3.206) at each fixed order r. That equation can be expressed as 85 (3.217) Here, E('\ 4k+i) denotes the free energy of particles <strong>in</strong> the state ,\ 4k+i. In the curly brackets we have written <strong>in</strong> symbolic form all rema<strong>in</strong><strong>in</strong>g terms of eq. (3.206). For i = 1,2 they conta<strong>in</strong> only operators of the type ,\4k +i AIr] with 4k + i < 4k + i for l = r , or l < r (3.218) As an example, we regard ,\4k+iH,\4 k +iAlr] with i = 1, 2. For 4k + i < 4k + i it follows from
86 3. <strong>The</strong> derivation of nuclear forces from chiral Lagrangians eqs. (B.2), (B.3) that l ::; r. For 4k + i = 4k + i, eq. (B.4) leads to l ::; r - 2. F<strong>in</strong>ally, for 4k + i > 4k + i eqs. (B.5)-(B.8) require l ::; r - 2. Eq. (3.218) is also valid for i = 3 apart from the s<strong>in</strong>gle term _,\4k +3 Hl,\4( k+l) Al=r'l]. This is due to eq. (B.7). In the ease i = 0, only the operators ,\4 k + i Al'l], restrieted by the eonditions i = 0, k < k for l = r , or l < r (3.219) ean enter the right hand side of eq. (3.217). Furthermore, the value of k is bounded from above <strong>in</strong> all eases by the <strong>in</strong>equality - 1 k ::; k + 6 (r - l + 8) , (3.220) as ean be deferred from the seeond <strong>in</strong>equality <strong>in</strong> (B.8) and the equality <strong>in</strong> (B.16). Now it is clear how to deal with the system of equations (3.205). <strong>The</strong> equations have to be solved order by order, start<strong>in</strong>g from r = O. At eaeh fixed order r one solves the equations with <strong>in</strong>ereas<strong>in</strong>g number 4k + i start<strong>in</strong>g from 4k + i = 1 to obta<strong>in</strong> all operators ,\4k + iAl=r'l]. This requires the knowledge of operators ,\ 4k + i Al'l], 4k + i < 4k + i, of the same order l = r and a f<strong>in</strong>ite number of operators ,\(4k + i ) Al 'I] at lower orders l < r. <strong>The</strong> only exeeptions of this proeedure are the equations with arbitrary k's and i = 3, that require the knowledge of the operators ,\ 4( k +l) Al=r'l]. Such equations have, therefore, to be solved after those ones with 4( k + 1) external pions. <strong>The</strong> number of equations to be solved at each order can be estimated by use of eq. (3.220) and the <strong>in</strong>equalities of appendix B. After solv<strong>in</strong>g the required number of equations at order r one ean go to the next order r + 1. <strong>The</strong>se rules for the reeursive solution of eqs. (3.205) are summarized onee more graphically <strong>in</strong> fig. 3.5. To justify our ansatz about the strueture of the operator A, eonsider the start<strong>in</strong>g equations at order r = 0, given by E(,\4),\4Ao'l] = ,\4H 2 '1] E(,\l),\l Ao'l] = ,\ 1 Hl'l] (3.221) (3.222) From eq. (3.217) one can see that the unknown operator ,\4k + i Al=r'l] has the structure which we assumed at the beg<strong>in</strong>n<strong>in</strong>g of this seetion, if and only if the already known operators '\A'I] enter<strong>in</strong>g the right hand side of this equation have precisely this form. <strong>The</strong>refore, to proof our ansatz recursively for all operators '\A'I] it is sufficient to see that it holds for the eorrespond<strong>in</strong>g start<strong>in</strong>g operators <strong>in</strong> eqs. (3.221) and (3.222), whieh is obviously the ease. Hav<strong>in</strong>g ealculated the operators ,\ 4k + i A'I] it is straight forward to obta<strong>in</strong> the expansion <strong>in</strong> powers of Q for the transformed Hamiltonian eq. (3.197). To do that we have to estimate the order of all terms on its right hand side. Introduc<strong>in</strong>g the operators and their ehiral power (the power of Q) with L = 'I1At \ 4kt '11 t - ., lt /\ +it A l't " Vt = 3 - 3N + Vt Vt = 4kt + 2it + lt + l' t (3.223) (3.224) (3.225)
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The Nucleon-Nucleon Interaction in
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11 vom ursprünglichen wesentlich u
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IV Ordnung des Potentials besteht a
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VI sagen für die Schwerpunktsimpul
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Vlll betrachten, um eine komplette
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x 5 Isospin violation in the two-nu
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2 1. Introduction nuclear force of
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4 1. Introduction the Thcson-Melbou
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6 1. Introduction pion-nucleon syst
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8 1. Introduction completely domina
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10 1. Introduction such interaction
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12 2. Low-momentum effective theori
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136 4. The two-nucleon system: nume
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138 4. The two-nuc1eon system: nume
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140 4. The two-nucleon system: nume
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142 4. The two-nuc1eon system: nume
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144 4. The two-nuc1eon system: nume
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146 --- .... . � - --- .... . ...
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148 � .... --- � � .... --- :
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4.5. Results for the NNLO-ß approa
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5.1 Isospin violating effective Lag
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5.1 Isospin violating effective Lag
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5.2 Brief introduction into the KSW
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5.3 The leading eIB and eSB effects
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5.3 The leading CIB and CSB effects
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5.5 Classification scheme 163 scatt
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Chapter 6 Summary and outlook In th
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2. Secondly, we considered the real
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NNLO, this range is larger and exte
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might be responsible for solving th
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derivative. Further helpful equalit
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Appendix B Operators contributing t
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Appendix C Small scale expansion wi
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179 (C.16) l�r-2. (C.17) h + l2 =
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Appendix E Anti-symmetrization of t
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Consequently, only four of the eigh
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where qJ-! is chosen to satisfy (v
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To keep the presentation self-conta
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- i�03{ [(NtVN) . (VNt x ifN) + (
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Consequently, it is not possible to
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(j ± 1, 1jIVIJ =f 1, 1j) Here, Pj
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Bibliography [1] E. Rutherford, Phi
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[62] J. Goldstone, A. Salam and S.
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[137] V. Bernard, N. Kaiser and Ulf
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[204] J.M. Blatt and L.C. Biedenhar