can be cast into an integral equation, the Lippann-Schwinger equation (LSE): (H E) = V (.6) j i = j i + V j E + i H i (.7) Let us consider the congura

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1 Chapter The Nucleon-Nucleon Syste. The Lippann-Schwinger Equation for the Scattering Process Let us consider two-nucleon scattering and dene ~ k and ~ k to be the individual nucleon oenta. The relative oentu is then given as = (~ k ~ k ) (.) The oentu eigenstates in the nucleon-nucleon (NN) c.. syste are then and are chosen to be noralized as j i (.) h j i = 3 ( ) : (.3) They are eigenstates to the free Hailtonian for the relative otion H = where is the nucleon ass. Let V be the NN potential, which is assued to be energy independent. The Schrodinger equation for a scattering state (.4) (H + V ) = E 3 (.5)

2 can be cast into an integral equation, the Lippann-Schwinger equation (LSE): (H E) = V (.6) j i = j i + V j E + i H i (.7) Let us consider the conguration space representation. The conjugate variable to is and we choose j ~xi to be noralized as Then the Fourier transfor is given by ~x = ~r ~r ; (.8) h~x j ~x i = 3 (~x ~x ): (.9) h~x j i = The conguration space representation of the free propagator is given as h~x j E + i H j ~x i = with j ~x ~x j Standard residue techniques lead to = = = G () 3= ei~x : (.) E + i H (.) d h~x j i E + i p = h j ~x i d 3 p e i~x E + i p = e i~x () 3= () 3 () 4 dp p j 3 (p) d 3 p e i(~x ~x ) E + i p = E + i p = (.) h~x j G j ~x i = 4 e ip Ej~x ~x j j ~x ~x j ; (.3) 4

3 which exhibits the outgoing wave behavior fro the source point ~x to ~x. It results the conguration space representation of the LSE = h~x j i () 3= ei~x 4 (~x) d 3 x e i p Ej~x ~x j j ~x ~x j V (x ) (~x ) : (.4) Thereby we assued a local potential h~x j V j ~x i = 3 (~x ~x ) V (x ) (.5) In a well known anner one reads o the asyptotic for for j ~x j! : (~x)! () 3 ( e i~x + eipx x f(^x) ); (.6) with the scattering aplitude f(^x) depending on the direction ^x of observation r f(^x) = d 3 x e ip^x~x V (x ) (~x ) : (.7) This can be interpreted in ters of a scattered oentu ^xp ; (.8) and one introduces a transition aplitude h j t j i () 3 d 3 x e i ~x V (x ) (~x ) = h j V j i : (.9) Apparently t is the result of the scattering process and deterines all scattering observables. Is there an integral equation directly for t? Fro (.4) we read o t j i V j i ; (.) and using (.7) we nd t j i = V j i + V G V j i = V j i + V G t j i (.) 5

4 We can strip o the initial state j i and get the operator relation t = V + V G t (.) which is the LSE for the transition operator. Its siple physical interpretation results by iterating that equation: t = V + V G ( V + V G t ) = V + V G V + V G V G V + V G V G V G V + : : : (.3) This is the Born series for scattering on V, a su of ters of increasing order in V. Each ter consists of a sequence of V s with free propagations in between. This is a general structure valid for any nuber of particles. It is useful to visualize that ultiple scattering process in the for t where the dashed lines stand for the action of V and two horizontal lines for the free propagation G between two interactions. Intuitively one can start fro that su of ters (.3) t = V + V G V + V G V G V + : : : (.4) and ask the question: can this series be sued up into an integral equation for t? Obviously it can: t = V + V G ( V + V G V + V G V G V + : : :) ; (.5) and we recover t again on the right hand side and thus get t = V + V G t (.6) which is the LSE. Of course if one starts fro Eq. (.3) in an ad hoc anner one has to know the for of the free propagator G and therefore one has to ake contact to the underlying dynaical equation, in our case the Schrodinger equation. Forally however this ultiple scattering series is quite general and also valid for the Bethe-Salpeter equation, where G is dierent fro the G used in our nonrelativistic context. 6

5 . Alternative Derivation of the Lippann-Schwinger Equation A free oentu eigenstate obeys H j i = E p j i (.7) and a scattering state obeys H j i = E p j i : (.8) Here a scattering state is dened via j i = j i = li! i G(E + i) j i ; (.9) where j i. is the Mller operator, which aps a free state j i onto a scattering state The propagators, or Resolvents, are given by and G (z) = G(z) = z H (.3) z H : (.3) Here G (z) is the free propagator (Resolvent) and G(z) the full propagator (Resolvent). Let us consider G G = (z H ) (z H) = H + H = V ; (.3) where we use that H = H + V. Multiplying Eq. (.3) fro the left with G and fro the right with G yields or G (G G )G = G G = G V G (.33) G = G + G V G : (.34) The above relation, Eq. (.34), is called rst Resolvent Identity. 7

6 Applying the rst Resolvent Identity on a oentu eigenstate and ultiplying both sides of the resulting equation with i yields i G(E + i) j i = i G (E + i) j i + i G (E + i)v G(E + i) j i i = j i + i G (E + i)v G(E + i) j i E + i H = j i + i G (E + i)v G(E + i) j i : (.35) Taking the liit! gives, together with the denition Eq. (.9), the Lippann- Schwinger equation for states If we ultiply Eq. (.36) by V and dene we obtain or j i = j i + G (E + i)v j i (.36) V j i = t j i ; (.37) V j i = V j i + V G (E + i)v j i (.38) t j i = V j i + V G (E + i)t j i : (.39) Since the operators in Eq. (.39) are applied on a general state j i, we can consider this equation as operator equation: t = V + V G (E + i)t : (.4) This equation is also called operator Lippann-Schwinger equation. A next task is to derive fro Eq. (.36) a relation to the scattering wave function + (~r). Let us consider which leads to h~r j i = h~r j i + h~r j G V j i (.4) (~r) h~r j i = h~r j i + h~r j G t j i (.4) = h~r j i + d 3 p h~r j i h j G t j i : (.43) Applying the denition of G leads to h~r j i = h~r j i + d 3 p h~r j i 8 E + i p h j t j i ; (.44)

7 which is the desired equation for the scattering wave function (r). Energy conservation leads to an additional constraint for the t-operator. If a oentu before the scattering event is denoted with, and after the scattering event with, then energy conservation requires = ) = : (.45) This eans that we can extract an energy conserving -function fro the atrix eleent h j T (E) j i = (E p E p ) h^p j t(e) j ^pi : (.46) The latter relation is soeties called on-shell condition. The physical eaning is that the observables of N N scattering only deterine the atrix eleents consistent with the relation (.46)..3 The Lippann-Schwinger Equation for the Bound State Let us assue, that V supports a bound state j bi at E = E b h: Then or (H + V ) j bi = E b j bi (.47) (H E b ) j bi = V j bi (.48) Since E b h there is no regular and square integrable solution to the left hand side alone and j bi obeys the hoogeneous LSE j bi = E b H V j bi (.49) Using the conguration space representation Eq. (.) for E = E b h we see that (.) guarantees the correct exponential fall-o behavior of r h~xj bi b(~x) = d 3 x e p je b jj~x ~x j j~x ~x V (x ) b(~x ) (.5) j 9

8 .4 Connection Between Hoogeneous and Inhoogeneous LSE's It is of interest and iportance to relate the hoogeneous equation, valid at the discrete energy E = E b j bi = G (E b )V j bi (.5) and the inhoogeneous equation, derived for Ei t(e) = V + V G (E)t(E) : (.5) The transition operator t(e) can be evaluated also for Eh. What happens for E! E b? We rewrite (.5) ( V G (E) ) t(e) = V (.53) Let us expand t(e) = ( V G (E) ) V (.54) t(e) = ( + V G + V G V G + : : : ) V = V ( + G V + G V G V + : : : ) : (.55) If we apply t(e) onto j bi and choose E = E b, then we nd, using Eq. (.3) t(e b ) j bi = V ( : : :) j bi ; (.56) which is clearly diverging. More precisely t(e) = [ ( G V ) G ] V = G E H V V = G E H V = G G V (.57) Inserting the copleteness relation j bih b j + d 3 p j ih j= (.58)

9 to the left of V gives t(e) = (E H ) j bi h b j V E E b + d (E H ) j i = V j bi h b j V E E b + d V j i E = h j V E = h j V : (.59) We see explicitly that t(e) has a pole at E = E b t(e)! V j bi h b j V for E! E b (.6) E E b Thus t(e) has a pole at the energy where the hoogeneous LSE has a solution, which is the sae as requiring that the hoogeneous part of the inhoogeneous LSE has a solution: (E) = V G (E) (E) (.6) Put then This is identical to (.38) and thus (E) V (E) (.6) (E) = G (E) V (E) (.63) (E) = b at E = E b (.64) This pole in t(e) at the NN bound state will be of decisive iportance for describing an interacting syste of 3 or ore nucleons..5 Realization in a Partial Wave Representation in Moentu Space We introduce the oentu space basis to a xed orbital angular oentu l and agnetic quantu nuber j pli (.65)

10 These states are dened via They are coplete and orthonoral h j p l i (p p) p p Y l (^p ) : (.66) Let us consider the LSE for t(e) in this basis l dp p j p l ihp l j = (.67) hp l j p l i = (p p) p p ll (.68) hp l jt(e)jpli = hp l jv jpli + l E + i p = hp l jt(e)jpli dp p hp l jv jp l i (.69) We take V to be rotationally invariant: hp l j V j p l i = ll V l (p ; p) (.7) which leads to an integral equation in one variable: t l (p p) = V l (p p) + dp p V l (p p ) What is V l, assuing V (r) to be given? Introduce states dened analogously to (.59) via E + i p = t l(p p) (.7) j rli (.7) Then h p l j r l i = = = h~x j r l i d d 3 x (p p) p p (x r) xr Y l (^x) (.73) d~x hp l j i h j ~xih~x j r l i Y l(^p ) () (3=) e i ~x (x r) Y l (^x) xr d 3 p s j l(pr) i l (.74)

11 Therefore, assuing a local potential: V l (p p) = hp l j V j pli = dr r dr r hp l j r l i hr l j V j r l ihhr l j p l i = = dr r dr r j l (p r ) (r r ) rr dr r j l (p r) V (r) j l (pr) V (r) j l (pr) (.75) This is one way to deterine the oentu space representation of a local potential. The LSE for t l can easily be solved by standard ethods. Let us now consider the full space for two nucleons including spin and isospin: jp(ls)j( )t ti l C(lsj; l l )jpl l ijs l i C( t; t )j ij j t i (.76) Clearly one has s = ; and t = ;. The antisyetry (working in isospin foralis) leads to the well known restriction for the allowed quantu nubers. Thus t = states are ( ) l+s+t = (.77) S ; 3 P ; 3 P ; D ; 3 P 3 F ; : : : (.78) and t = states are P ; 3 S 3 D ; 3 D ; : : : (.79) The hyphen denotes coupled states, where l is not conserved. A well known echanis for that is the tensor force. For a general NN force, which conserves spin and parity one has hp (l s )j t tjv jp(ls)jt t i = jj tt t t ss V sjtt ll (p ; p) (.8) Because of (.7) conservation of isospin follows and the indicated t-dependencies for V is redundant. There is a dependence on t, the charge state of the two nucleons, in case of chargeindependence breaking (CIB) or charge-syetry breaking (CSB): 3

12 CIB eans: np 6= pp= strong forces CSB eans: nn 6= pp= strong forces It is well established that in the state S the np force is dierent fro the nn or pp force. This is evident in the dierent scattering lengths: a np = 3:48 :9f a pp = strong = 7:36 :4f (recoended value) (G.A. Miller et al, Phys. Rep. 94 (99) ) a nn = 8:6 :3f (extracted fro + d! n + n + ; B. Gabioud et al, Phys. Rev. Lett. 4 (979) 58; O. Schori et al, Phys. Rev. C35 (987) 5). That absorption experient has been redone at Los Alaos and is presently being analyzed. In addition, nd breakup experients are being presently perfored (W. Tornow, TUNL), in order to extract a nn using odern Faddeev calculations. In t = states dierent fro S CIB or CSB is not yet convincingly established, though sall eects at least have to be there, siply because of the dierent pion asses. We shall drop in the following the possible t -dependence in the notation. In this ost general basis, the LSE for t is represented as or hp (l s )jtjt(e)jp(ls)jti = hp (l s)jtjv jp(ls)jti t sj l l (p ; p) = V sj l l (p p) + l + dp p hp (l s)jtjv jp (l s)jti l E + i p = hp (l s)jtjt(e)jp(ls)jti (.8) dp p V sj l l (pp ) E + i p = tsj l l (p p) (.8) Since s is at ost and parity is conserved l = l or l = l (.83) Thus one has either a single equation or two coupled equations. A proinent exaple for the coupled case is 3 S 3 D (.84) acting in the deuteron. 4

13 .6 NN Phase-Shifts The t-atrix generated by the coupled or uncoupled LSE is unitary. Let us choose a atrix notation t t sj l l(p p) etc. (.85) Then The adjoint of that is since V y t = V + V G t = V + t G V (.86) t y = V + V G t y (.87) = V: This is valid on physical grounds. Subtraction yields t t y = V G t V G t y = V G (t t y ) + V (G G )t y ( V G )(t t y ) = V (G G ) t y t t y = ( V G ) V (G G ) t y = t ( G G ) t y (.88) Now thus G = E + i H (.89) G G = i (E H ) (.9) and we get, back in explicit notation t l l(p p) t ll (pp ) = dp p l p E = i l t l l (p p )( i) (E p Let us choose the on-the-energy shell values p = p = p E : ) t ll (pp ) t l l (pp E) t ll (p p E) (.9) t l l(pp) t ll (pp) = ip l t l l (pp)t ll(pp) (.9) Back in atrix notation this is t t y = ip t t y (.93) 5

14 Now we introduce a S-atrix and nd S = ip t (.94) S S y = ( ip t) ( + ip t y ) = ip (t t y ip t t y ) = (.95) Thus S is unitary and can be paraeterized in the coupled case by 3 paraeters: S = cos e i isin e i( + ) isin e i( + ) cos e i! (.96) which is the "Stapp" or "bar"-phase shift paraeterization (H.P. Stapp et al, Phys. Rev. 5 (957) 3). In the uncoupled case S is siply S = e i (.97) with real. The ost recent NN phase-shift paraeters by the Nijegen group (V.G.J. Stoks et al, Phys. Rev C48 (993) 79) can be viewed on-line at and by the Arndt group (R. A. Arndt et al, Phys. Rev. D45 (99) 3995) Their on-line facility is called SAID and can be accessed via telnet said.phys.vt.edu using the login said. You need an xter if you want to do the graphics..7 Deuteron Properties The hoogeneous LSE Eq. (.5) is now projected onto the basis given in Eq. (.73). Thus for l(p) hp (ls) j t j bi (.98) 6

15 with l = ;, s = j =, t = one gets the set of two coupled equations l(p) = E b p l =; dp p V ll (pp ) l (p ) (.99) This can be solved nuerically by standard techniques. Realistic forces are adjusted to reproduce various easurable quantities: E b = :46 MeV Q = :859 f (there are theoretical uncertainties in the description of that experiental value caused by MEC) A s =.8883 f = (asyptotic noralization constant for the s-wave coponent) = A D =A s = :564 (asyptotic d=s ratio) The deuteron d-state probability p d R R (p) p dp (p) p dp + R (p) p dp (.) is not a easurable quantity, but strongly correlated to nuclear binding energies, as we shall see later. In general, the saller p d the larger the triton and -particle binding energies. Let us now consider the single nucleon oentu distribution n(k) = 3 3 h b j i= ( ~ k ~ k c i ) j b i (.) h b j ( ~ k ~ k c ) j b i (.) We have and thus One has n(k) = 3 h ~ kj bi = l = l = (~ k c ~ k c ) = ~ k c (.3) d 3 p h b j i ( ~ k ) h j bi: (.4) dp p h ~ kjp(ls)ji l(p) l C(lsj; l ; l )Y ll (^k)js l i l(k) (.5) 7

16 and therefore n(k) = 3 ll l C(l s j; l l ) (.6) C(l s j; l l ) Y l l (^k) Y ll (^k) l (k) l(k) Now, with ^a being dened as ^a a + we have C(l s j; l l ) = ( ) s+ l Using the above relation we nd n(k) = 3 l ll Y vu u t^j ^l C(j s l ; ; l) (.7) l l (^k) Y ll (^k) (.8) vu u t^j ^l vu u t ^j ^l l (k) l(k) C(j s l; ; l ) C(j s l ; ; l ) = 3 = ^j 3 l 4 l l Y l l (^k) Y ll (^k) ^j ^l l (k) = 4 l=; l (k) (.9) l (k) (.) This is displayed for several realistic NN forces in the next gure, where dierent short range behavior of NN forces is reected for k > f : 8

17 There is hope to easure these quantities in electron scattering on deuterons. Of interest is also the NN correlation function, the probability to nd nucleons at a distance r: C(r) 3 = 3 h bj(~r ~x)j b i h bj~rih~rj b i 9

18 = 4 l=; l (r) (.) The connection between conguration and oentu space is given by s l(r) = dp p j l (pr) l(p) (.) The l = and parts of C(r) together with their su are displayed below. We see dierences at short distances, depending on the strengths of the short range repulsions, as shown in the next gure

19

20 .8 Nuclear Forces The deterination of the nuclear force is a longstanding and still unsolved basic proble. The whole issue on how to set up a fraework for deriving a nuclear force is not touched here. Siply a list of so called "realistic" NN forces is given: Paris potential (dispersion theoretical background) by M. Lacobe et al, Phys. Rev. C (98) 86 Nijegen 78 potential (one-boson-exchange background) by M.M. Nagels et al, Phys. Rev. D7 (978) 768 AV4 potential (one-pion tail, otherwise phenoenological) by R. B. Wiringa et al, Phys. Rev. C9 (984) 7 Bonn potential (eson exchange potential (ultiple eson), based on tie-ordered perturbation theory by R. Machleidt, K. Holinde, Ch. Elster, Phys. Rep. 49 (987) and R. Machleidt, Adv. Nucl. Phys. 9 (989) 89 All those potential have =N data with respect to the Nijegen data base. Most recent NN potential, however all phenoenological with about 3-5 paraeters t the Nijegen data base with a =N data and are Nijegen I (includes r -ter) Nijegen II (local) Reid 93 (local) by V.G.J. Stoks et al, Phys. Rev. C49 (994) 95 AV8 (updated AV4, local, as operators dened) by R.B. Wiringa et al, Phys. Rev. C5 (995) 38 CD-Bonn (nonlocal) by R. Machleidt, F. Saarruca, Y. Song, Phys. Rev. C53, (996), R483. They coe in charge-dependent versions and describe the NN data up to 35 MeV perfectly well with =N data This is the rst tie that one has a set of "realistic" nearly phase-equivalent NN forces. They cover a certain range of properties, a NN force can have:

21 local versus nonlocal soft or hard core What is still issing in that faily are potentials with dynaical nonlocalities at very short distances r :8f, say, resulting fro the overlap regions of extended nucleons. They ight be good for surprises..9 Construction of the NN Potential Fro Invariance Requireents We want to investigate to what extent the for of the potential V NN (; ) acting between two nucleons is deterined by the requireent that the Hailtonian describing the syste be invariant under various syetry transforations. This analysis will be ade considering the two nucleons as identical particles, i.e., disregarding the dierence of the ass and charge between the neutron and proton. The Hailtonian has then the for being the nucleon ass. H = (p + p ) + V NN (; ) ; (.3) Regarding the syetry properties of H, we shall assue rst of all invariance under the restricted Galilei group. Then we shall assue invariance under the discrete transforations of space reection, tie invariance and perutation of the two nucleons. Finally, we shall assue invariance under the isospin transforations of the group SM(). The operators we have at our disposal to build up the potential are the coordinates ~r ; ~r, the oenta ;, the spin vector operators ~ () ; ~ () and the isospin vector operators ~ () ; ~ (). Going to the two nucleon c.. frae gives ~r = ~r ~r ~R = (~r + ~r ) = ~P = + : (.4). Assue that the potential operator is heritian V NN = V y NN : (.5) 3

22 . Using tie-translation invariance, which akes V NN not explicitly dependent on the tie t, gives V NN V NN (~r; ~R; ; ~P ; ~ () ; ~ () ; ~ () ; ~ () ) : (.6) Let us now discuss the iplications of the assued invariance on the dependencies of V NN on the indicated variables. 3. Consider the invariance for rotations in charge space, which deterines the dependence of V NN on the isospin vectors ~ () and ~ (). The unitary operator representing a rotation in charge space is given by U I (w) = e i~ I ~w ; (.7) where ~ I is the total isospin and ~w = ~n w. The required invariance is expressed by U y I V NN U I = V NN (.8) with arbitrary ~n and w. (.8) will be satised if V NN is a scalar in isospin space. In order to construct all possible scalars fro ~ () and ~ (), it is rearked that any polynoial expression in ~ (i) can be reduced to a linear expression by using [ (i) j ; (i) k ] = i" jk` (i) ` ( (i) j ) = ; (.9) so that, e.g., (~ () ~ () ) = 3 (~ () ~ () ) : (.) Hence, the ost general expression that has to be considered is linear, both in ~ () and ~ (). The only scalar quantity obtained in this way is ~ () ~ (). It follows that V NN is a function only of this quantity as regards its dependence on the isospin variables of the two particles: V NN V NN (~r; ~ R; ; ~ P ; ~ () ; ~ () ; [~ () ~ () ]) : (.) Expanding V NN in a power series of ~ () ~ () and expressing (~ () ~ () ) n with (.) in ters of ~ () ~ () and the identity operator in isospin space, one obtains V NN = V (~r; ~R; ; ~P ; ~ () ; ~ () ) + (~ () ~ () ) V (~r; ~R; ; ~P ; ~ () ; ~ () ) : (.) We can now liit ourselves to study the iplications of invariance an each ter V i separately, since all other syetry transforations coute with the isospin operators. For convenience we drop the index i fro now on. 4

23 4. Invariance under space translations is expressed by U y a V U a = V (.3) with U a = exp i h ~ P ~a. We get U y a V (~r; ~R; ; ~P ; ~ () ; ~ () ) U a = V (U y a ~r U a ; U y ~ a R U a ; U y a U a ; U y ~ a P U a ; U y a ~ () U a ; U y a ~ () U a ) = V (~r; ~R ~a; ; ~P ; ~ () ; ~ () ) : (.4) The condition (.3) then iplies that V does not depend on ~R: V (~r; ; ~P ; ~ () ; ~ () ) : (.5) 5. Invariance under proper Galilei transforations is expressed by U y G V U G = V (.6) with U G exp i h ~ P ~v t exp i h ~R ~v (.7) with ~v being the c.. velocity. It follows that U y G V U G = V (~r; ; U y G ~ P U G ; ~ () ; ~ () ) = V (~r; ; ~P ~v ; ~ () ; ~ () ) : (.8) The condition (.6) then iplies that V is independent of P ~ : V = V (~r; ; ~ () ; ~ () ) : (.9) 6. Invariance under space reections iplies in the noral way that V (~r; ; ~ () ; ~ () ) = V ( ~r; ; ~ () ; ~ () ) : (.3) 7. Invariance under the perutation of the two nucleons gives V (~r; ; ~ () ; ~ () ) = V ( ~r; ; ~ () ; ~ () ) : (.3) Invariance under the cobined transforations (6) and (7) gives V (~r; ; ~ () ; ~ () ) = V (~r; ; ~ () ; ~ () ) : (.3) 5

24 8. Invariance under tie reversal eans V (~r; ; ~ () ; ~ () ) = V (~r; ; ~ () ; ~ () ) = V (~r; ; ~ () ; ~ () ) (.33) Since V is assued to be heritian. 9. Invariance under spatial rotations is expressed by U y R V U R = V (.34) i with U R = exp ~ J ~n w, with ~J being the total angular oentu of the h syste, ~J = ~L + ~S. Requiring rotational invariance eans that V (~r; ; ~ () ; ~ () ) = V (R~r; R; R~ () ; R~ ) ) ; (.35) where R~a gives the rotated of the vector ~a. Let us rst take into account the dependence of V on the spin variables. Here the procedure is not so straightforward as it was for the isospin, since spin, position and oentu vectors can be cobined to build rotational invariant quantities. Using spin identies siilar to (.), one can show that V can be expressed as V = V + ~ () ~ V () + ~ () ~ V () + V (~r; ; ~ () ; ~ () ) : (.36) V is linear in both ~ () and ~ () but contains only bilinear cobinations of these two operators. Fro rotation and space-reection invariance, V and V ust be scalars, ~V () and ~V () pseudovectors. Cobination of space reection and particle exchange [(6) and (7)] iplies that V + ~ () ~V () + ~ () ~V () + V (~r; ; ~ () ; ~ () ) = V + ~ () ~V () + ~ () ~V () + V (~r; ; ~ () ; ~ () ) : (.37) Taking the average of these two expressions for V, one gets V = V + ~ S ~ V + V (~r; ; ~ () ; ~ () ) (.38) where ~ S = h (~() + ~ () ); ~V = h ( ~V () ; ~ () ), and V now being syetric under the exchange of the spin operators. The vector we can use to construct ~ V are ~r; and 6

25 ~L = ~r, but only ~ L is a pseudovector. Thus, ~ V ust then be ~ L (scalar quantity). Then (.38) reads V = V (~r; ) + ~S ~L V (~r; ) + V (~r; ; ~ () ; ~ () ) : (.39) Since V and V are scalars, they can only be functions of r ; p ; L ; ~r and ~r. Since the operators ~r and ~r are non-heritian, it is convenient to consider their heritian cobinations (~r + ~r) and i(~r ~r). The latter is a constant and can be dropped. The forer can only appear quadratically in V and V due to tie-reversal invariance. With we get ( ~r + ~r ) = (r p + p r ) 4L + 3h ; (.4) V = V (r ; p ; L ) + ~ S ~L V (r ; p ; L ) + V (~r; ; ~ () ; ~ () ) : (.4) Fro the requireents on V follows that it can only contains ters of the type ~ () ~ () ; (~ () ~r)(~ () ~r); (~ () )(~ () ); (~ () ~L)(~ () ~L) + (~ () ~L)(~ () ~L); (~ () )(~ () ~r) + (~ () ~r)(~ () ) + $ : (.4) The last expression changes sign under tie reversal and ust be replaced by [(~ () )(~ () ~r) + (~ () ~r)(~ () ) + $ ]( ~r + ~r ) : (.43) It can be shown that (.4) is de facto a function of the other quantities appearing in (.4) and thus not independent. We have, therefore, for V V = (~ () ~ () ) V (I) (r ; p ; L ) + (~ () ~ () ) V (II) (r ; p ; L ) + (~ () )(~ () ) V (III) (r ; p ; L ) + [(~ () ~L)(~ () ~L) + (~ () ~L)(~ () ~L)] V (IV ) (r ; p ; L ) (.44) as ost general for of V copliant with all syetry requireents. Concluding, the ost general, velocity-dependent, non-relativistic N N potential has the for (.) with V i given by V i = V c i (r ; p ; L ) + ~ () ~ () V i (r ; p ; L ) + S V T i (r ; p ; L ) + ~S ~L V LS i (r ; p ; L ) + [(~ () ~L)(~ () ~L) + (~ () ~L)(~ () ~L)] V L i (r ; p ; L ) ; (.45) 7

26 where S is the tensor force operator S = 3 r (~() ~r)(~ () ~r) (~ () ~ () ) : (.46) Reark: As shown in the Appendix of S. Okubo and R.E. Marshak, Ann. Phys. 4, 66 (958), the potential of (.45) gives an S-atrix that on-shell is identical to the one obtained fro a potential in which the ter V p is dropped. Therefore, if one is only interested in NN scattering, it can be neglected. The sae cannot be said for the bound states or for the o-shell S-atrix. Often the ter V L, is also neglected. Soe arguents are given in Machleidt, Holinde, Elster, Phys. Rep. 49, (987).. Siple Introduction to One Boson-Exchange Potential (OBEP) The basic idea of OBE odels is to represent the NN interaction as superposition of tree-diagras (born ters) which represent the exchange of single esons, naely scalar (s), pseudoscalar (ps), vector (v) bosons (J p = + ; ;, respectively), with asses up to GeV between two nucleons. Mesons with asses larger than GeV would only give very short-ranged exchange contributions and contribute in a region where the OBE odel is no longer valid. The couplings for the various esons are given in ters of their interaction Lagrangian densities by L NNps = g ps i5 ps (.47) L NNs = g s s (.48) L NNv = g v v + f v 4 v (@ v v v ) (.49) for pseudoscalar (; ), scalar (; ) and vector esons (;!), respectively. is the nuclear ass, the nucleon and the eson eld operators. For isospin T = eans is to be replaced by ~ ~, with i being the usual Pauli atrices. Furtherore, = i [ ; ], where are the usual Dirac-atrices (see, e.g., Bjorken-Drell). The coupling constants g ( = s; ps; v) and f v and the eson asses are at least partially deterined fro high-energy experients or syetry relations. The Lagrangian density for vector esons contains Dirac (g v ) as well as Pauli coupling (f v ). An OBE-potential 8

27 V (~q ; ~q) is obtained through the superposition of exchange contributions of the dierent esons with V (~q ; ~q) = =s;ps;v V (~q ; ~q) (.5) V (~q ; ~q) = s Eq s Eq u ( ~q ) () u( ~q) P u(~q ) () u(~q) : (.5) The factors q Eq q Eq are the so-called inial relativity factors, which take into consideration the relativistic unitarity condition (see K. Erkelenz, Phys. Rep. 3C, 9 (974)). They certainly contribute to the nonlocality of V (~q ; q). (Their eect has been studied in a siple odel in Ch. Elster, E.E. Evans, H. Kaada, W. Glockle, Few-Body Systes, 5 (996). The eson propagators are usually given by P = ((~q ~q ) + ) (.5) and the vertex functions for the eson-nucleon vertices (i) (i = ; ) are given by (i) s = g s (.53) (i) ps = g ps i 5 (.54) (i) v (direct) = (g v + f v ) (.55) (i) v (gradient) = f v (~q + ~q ) : (.56) In order to take into account the nite extension of the nucleon and to be able to solve the dynaical equations, the coupling constants get odied with for factors. This is essentially achieved by replacing where F (~q ; ~q) can be, e.g., of dipole type g! g F (~q ; ~q) (.57) F [(~q ; ~q ) ] =! n + (~q ~q ) : (.58) 9

28 The exponent n is usually taken as n = ; is the cuto paraeter and usually of the order GeV. The positive energy Dirac spinors are given by s! Eq + u (i) (~q ) = ~~q j ii (.59) E q+!! where j ii denote the Pauli spinors and. Inserting (.59), (.53), (.5) into (.5) gives for the scalar contribution of the potential (Pauli spinors are oitted): V s (~q ; ~q ) = g s s E q s (E q + )(E q + ) E q 4 (~q ~q ) + s ~q ~q + i~ (~q ~q ) (E q + )(E q + )! ~q ~q + i~ (~q ~q ) (E q + )(E q + )! :(.6) This expression has, due to the ~q and E q dependencies, a strong nonlocality. In order to arrive at expressions, which can be transfored to coordinate space, one changes variables to ~ k = ~q ~q = (~q + ~q ) (.6) and in addition has to introduce the following approxiations:. On-shell approxiation: E q = E q. Expansion of E in powers of q : E = (~q + ~q ) + = + 4 (q + q ) + = + p + k 8 + (.6) 3. Keeping only the lowest order in p and k. 3

29 With these approxiations, the scalar potential becoes V c s ( ~ gs k; ) = " p + k k + s 8 i where ~S = (~ + ~ ). # ~S ( ~ k ) (.63) This expression still contains nonlocalities due to as well as ( ~ k ) ters. The latter leads to the angular oentu operator L ~ = i~r r ~ in r-space, whereas the forer provides r ters. After a Fourier transfor, the coordinate space expression of the scalar potential is given by V c s (r) = g s 4 s + (" 4 s # Y( s r) h i r Y( s r) + Y( s r) r + 4 ( s r) ~L ~S (.64) where Y(x) = e x =x and (x) = (=x + =x ) Y(x). The treatent of the Schrodinger equation with a oentu dependent potential is given by O. Rojo, L.M. Sions, Phys. Rev. 5, 73 (96). The expressions for the other potential ters shall only be given here: V c ps( ~ k ; ) = g ps 4 (~ ~ k)(~ ~ k) k + ps (.65) V c v ( ~ k; ) = k + v ( g v k " + 3p k 8 + 3i ~S ( ~ k ) (~ ~ ) (~ ~ k)(~ ~ k) + g vf v " k + 4i S ~ ( ~ k k ) ~ ~ + (~ ~ k)(~ ~ k) # # + f v 4 h ~ ~ k + (~ ~ k)(~ ~ k) i) : (.66) The structure of the expression (.6) already suggests that one would prefer to work with OBE potentials in oentu space. Even the already approxiated expressions (.63), (.65), (.66) are still coplicated functions of the oenta, though they 3

30 can be Fourier transfored analytically to coordinate space. The corresponding r-space expressions to (.65) and (.66) are V c ps(r) = g ps 4 ps " ps Y(psr) ~ ~ + ( psr) S # (.67) V c v (r) = g v v v g v f v 4 (" + v Y( v r) 3 Y(v r) ~ ~ 3 ( v r) ~L ~S v # ( v Y(v r) ( v r) ~L ~S 3 ( vr) S ( 6 + f v 4 v 4 [r Y( v r) + Y( v r)r ] ) ( vr) S v Y(v r) ~ ~ v Y(v r) ~ ~ ) ( vr) S Here the tensor operator S is given by (.46) and (x) = ( =) ( + 3=x + 3=x )Y(x). Details on OBE potentials are given in the references quoted in Section.8. : (.68) 3