PHYSICAL REVIEW C VOLUME 49, NUMBER 3 MARCH 1994 Self-consistent solution of the Schwinger-Dyson equations for the nucleon and meson propagators M. E. Bracco, A. Eiras, and G. Krein Instituto de Fisica Teorica Uni—versidade Estadual Paulista, Rua Pamplona, 1/5 —01/05-900 Sao Paulo, Sao Paulo, Brazil L. Wilets Department oj Physics, FM 15, -University of Washington, Seattle, Washington 98195 (Received 6 July 1993) The Schwinger-Dyson equations for the nucleon and meson propagators are solved self- consistently in an approximation that goes beyond the Hartree-Fock approximation. The traditional approach consists in solving the nucleon Schwinger-Dyson equation with bare meson propagators and bare meson-nucleon vertices; the corrections to the meson propagators are calculated using the bare nucleon propagator and bare nucleon-meson vertices. It is known that such an approximation scheme produces the appearance of ghost poles in the propagators. In this paper the coupled system of Schwinger-Dyson equations for the nucleon and the meson propagators are solved self-consistently including vertex corrections. The interplay of self-consistency and vertex corrections on the ghosts problem is investigated. It is found that the self-consistency does not affect signi6cantly the spectral properties of the propagators. In particular, it does not affect the appearance of the ghost poles in the propagators. PACS number(s): 21.30.+y, 21.60.Jz, 21.65.+f I. INTRODUCTION The development of relativistic many-body theories for the nucleus is one of the most important goals of contem- porary nuclear theory. Models based on the methods of relativistic quantum field theory have been developed for more than two decades. The starting point for understanding the many-nucleon problem is a description of the elementary processes in vacuum: the nucleon propagator, meson-nucleon scatter- ing, and the N-N interaction. Successes and difhculties with relativistic meson-nucleon field theory have been the subject of papers for more than half a century. We will certainly not detail the history here, but note that a nag- ging inconsistency in (almost) all calculations has been the appearance of ghost poles. Brown, Puff, and Wilets [1], for example', calculated the nucleon propagator by summing all planar meson dia- grams with one nucleon line using x, p, and u mesons. No cutofFs were introduced. The renormalized nucleon prop- agator was well defined and self-consistent, but contained a pair of complex conjugate poles located approximately 1 GeV ofF the real and imaginary axes. The full prop- agator, including these unphysical poles, was used with some success to describe the isovector nucleon magnetic moment, z-nucleon scattering [2], and nucleon-nucleon scattering [3]. (The last did require cutoffs in the NN- interaction, but yielded better chi-squared fits to scat- tering data with fewer parameters than the then current Paris potential). The inclusion of the complex poles was essential. Nevertheless, the occurrence of the complex poles remained an enigma. Several interpretations of the appearance of the poles have been profFered, including the statement that it is a signal of the inconsistency of any local, relativistic field theory, and that a field theory with asymptotic &eedom (e.g. , /CD) is required. The program of the previous section was driven by the interpretation that the appearance of the ghosts is an artifact of the approximations, and that progressively better calculations should lead to the receding or elimi- nation of the ghosts, but that for consistency one must keep the ghosts as they emerge from the calculations at each stage. Another interpretation is that it is an efFective theory, and that one should be prepared to introduce further parameters to ensure physical quantities. In a recent paper [4], the problem of ghosts poles in the nucleon propagator was investigated. The appear- ance of the ghost poles is related to the short distance behavior of the model interactions [1]; asymptotically free theories appear to be free of ghost poles [5]. An interesting possibility to eliminate the complex poles is the regularization of the theory by means of vector me- son dressing of nucleon-meson vertices. It is known that in a theory with neutral vector mesons there are vertex corrections that generate a strongly damped vertex func- tion in the ultraviolet [6]. In quantum electrodynamics (/ED), such corrections give rise to the Sudakov form factor [7]. When the Sudakov form factor, generated by massive vector mesons, is included in the Hartree-Fock approximation to the Schwinger-Dyson equation (SDE) for the nucleon propagator, the ghost poles disappear. A similar result was obtained by Allendes and Serot [8] ear- lier in the study of the ghost pole in the meson propaga- tor. Those authors concluded that the Sudakov corrected propagator is &ee of ghost poles. It is the purpose of the present paper to solve seM- 0556-2813/94/49(3)/1299(10)/$06. 00 49 1299 1994 The American Physical Society 1300 M. E. BRACCO, A. EIRAS, G. KREIN, AND L. WILETS 49 consistently the coupled system of Schwinger-Dyson equations for the nucleon and meson propagators and investigate the role of self-consistency on the appearance of ghost poles in the propagators. Vertex corrections are introduced by means of form factors. There is an extensive literature on calculations of nu- clear matter and finite nuclei properties based on the Walecka scalar-vector model [9]. In general, the appli- cations have been performed using Hartree-Fock (HF) type of approximations. In a relativistic HF approxima- tion, the single-nucleon propagator is calculated by solv- ing self-consistently the SDE using bare meson propaga- tors and bare meson-nucleon vertices. An additional ap- proximation has been the neglect of the quantum vacuum of the nucleon propagator. The corrections to the meson propagators are usually calculated considering the vac- uum polarization correction using nucleon propagators with an effective mass. Although the nucleon propagator is solved self-consistently by means of the SDE, the self- consistency is only partial, since the meson propagators used are the bare ones. The meson propagators satisfy their own SDE's, which require for their solution the nu- cleon propagator. A self-consistent solution requires the consideration of the coupled system of nucleon and meson SDE's. Besides the lack of self-consistency, the neglect of the quantum vacuum in the nucleon sector is a major limi- tation. It is exactly the nontrivial nature of the vacuum of a relativistic quantum field theory that motivates the introduction of models which go beyond the usual non- relativistic approach. However, severe diKculties arise in including the vacuum effects beyond the one-loop Hartree approximation. The inclusion of these vacuum correc- tions leads to catastrophic results due to the presence of the ghost poles in the propagators. Among other things, the ghosts lead to a large imaginary part to the nuclear matter energy. Although the primary aim of our studies is the con- struction of an intrinsically consistent relativistic quan- tum field theory for the nuclear many-body problem, these studies have connections to other fields that use the SDE's to study nonperturbative effects in field the- ory. Such fields include the problems of dynamical chiral symmetry breaking and color confinement in QCD, tech- nicolor models and QED in four and lower dimensions. For a recent review on the subject of the SDE's in this context see Ref. [10). It is common practice in QCD and QED to study the solutions of the fermion SDE in Euclidean space, instead of Minkowski space as we do in our studies. In princi- ple, the formulation of the problem either in Minkowski space or in Euclidean space is entirely equivalent; both formulations are connected by an analytic continuation. However, this equivalence holds provided there are no singularities in the complex plane. Since the pioneer- ing works of Fukuda and Kugo [11] and Atkinson and Blatt [12], it is known that the solution of the SDE for the electron propagator in Euclidean space treated in the Hartree-Fock approximation has pairs of complex conju- gate branch points. The same feature was found in recent studies of the SDE in a variety of models of QCD [13]. The existence of the ghost poles in Minkowski space and of the complex branch points in Euclidean space spoils the equivalence of the Minkowski and Euclidean formula- tions. It would be interesting to investigate the possibil- ity that the branch points have their origin in the ultravi- olet behavior of the interaction as in the case of the ghost poles. This could be done by using a Sudakov corrected fermion —vector-boson vertex [4]. This would be particu- larly interesting for the case of QCD, were the running of the coupling constant provides extra logarithms in the Sudakov form factor. The paper is organized as follows. In Sec. II we present the model for the interacting nucleon-meson system. We briefly review the spectral representation of the propa- gators and their inverses and discuss the renormalization procedure. In Sec. III we discuss the coupled system of Schwinger-Dyson equations for the nucleon and. meson propagators in terms of their spectral representations. Section IV presents the method of solution of the equa- tions and presents our numerical results. Conclusions are presented in Sec. V. II. THE MODEL In this paper we consider a model field theory with nucleons (g), pions (m), and vector isoscalar mesons (ur). The model Lagrangian density is l: = g(ip„8" —igo~psa. ~ —go~pp~")g 1 „1 2 ~ 1——F F""——m v v" + —0 m 0"m' 4 pv 2 4P p 2 p 1 2——m 7r '7r 7r G-~(~' —~) = —&(01&W-(*')6(*)]lo) (2) where F""= 8"~"—8"ur" This Lagrangian density is not compatible with the re- quirements of the partial conservation of the axial current (PCAC). Although it is true that a consistent hadronic model must be compatible with PCAC, in this paper we are mostly interested in the interplay of self-consistency of the nucleon and meson propagators and the problem of the ghost poles. Chiral symmetry allows the presence of self-interacting meson terms in the Lagrangian, as for instance in the linear sigma model. Such terms will al- ter the low momentum structure of the meson spectral functions, in comparison to those obtained in this paper. However, it is unlikely that the appearance or disappear- ance of the ghost poles, which are an ultraviolet phe- nomenon, will be altered by this. The implementation of chiral symmetry in a renormalizable hadronic model for practical uses in nuclear physics has difBculties due to the many-body forces implied by the self-interacting terms [14]. In this sense, our model is probably an ap- propriate starting point for studies towards a consistent relativistic many-body theory for the nucleus. As usual, the nucleon propagator is defined by 49 SELF-CONSISTENT SOLUTION OF THE SCH%'INGER-DYSON. . . 1301 and D„"(x' —*) = —i(0~T[~'(*')~~(*)]~0) where ~0) represents the physical vacuum state. The m- and u-meson propagators are defined respectively by D""(*'—~) = —i(olr [~ (*')~"(*)]10) (4) The Schwinger-Dyson equations for the nucleon and meson propagators in momentum space are given by the following expressions, Fig. 1: (a) nucleon, G(p) = G'"(p) + G"'(p)~(p)G(p) 4 4 p(p) = —ig02 ps''D (q )G(p —q)I'5(p —q, p; q) + ig0 4p„D""(q )G(p —q)I'„(p —q, p; q); (b) pion, D j (q 2 ) D (0) (q 2 ) + D (0) (q 2 )II a l (q 2 )D lj (q 2 ) u4 II". (q') = igo. T [~5r'G(p)l'5(p p+ q q)G(p+ q)] (c) omega, DP ( 2) DP (o) (q2) + DPP(o) (q2)IIP (q2)D (q2) d4 II"."(q') = -igo. ,T [~"G(p)I'"(J»p+ q q)G(p+ q)] . (a) / 'W W In the above equations, I'5 (p, p+ q; q) and I'"(p, p+ q; q) are the three-point x-nucleon and u-nucleon vertex func- tions, respectively. They satisfy their own Schwinger- Dyson equations. These relate the three-point functions to four-point vertices and so on ad inPnitum. In practice one has to truncate this infinite set. In this paper we I truncate the SDE's by postulating a specific form for the three-point functions (see below). Next, we discuss the spectral representations of the propagators and their inverses. We do not intend to re- view the subject of spectral representations; we simply make use of the relevant equations for the purposes of the present paper. We refer the reader to Refs. [15—18] for an extensive discussion on the subject. Let us start with the nucleon propagator. The spectral representa- tion of the nucleon propagator (in momentum space) can be written as (b) —K+ X6 A(r) is the spectral function. It represents the probabil- ity that a state of mass ~lc~ is created by g or Q, and as such it must be non-negative. Negative z corresponds to states with opposite parity to the nucleon. Defining the projection operators (c) ~~~m, =~ + where FIG. 1. Diagrammatic representation of the Schwinger- Dyson equations for the full (a) nucleon, (b) pion, and (c) omega propagators. The solid, wavy and dashed lines rep- resent respectively the nucleon, the cu, and the vr. The solid circles represent full propagators and vertices. iQ—p2 if p2 ( 0. G(p) can be rewritten conveniently as G(p) = P+(p)G(mP + ie) + P (p)G( m„—ie), (14)— 1302 M. E. BRACCO, A. EIRAS, G. KREIN, AND L. WILETS 49 where G(z), z = +(w~ + ie), is given by the dispersion integral from this one has that the imaginary part of G(z) is given by A(r. ) Z —K (is) ImG(z+ iy) = —y dr. A(r.) z —e 2+y2 The inverse of the propagator can also be written in terms of the projection operators Py(p) as G '(z) = z —MQ —E(z) +oo = z —Mp— T(r.) Z —K (17) G '(P) = P+(P)G '(iQ„+i~)+P (P)G '( ii)„——ie) . (16) The spectral representation for G i(z) is written as which is nonzero for y g 0. This is a necessary condition for writing a spectral representation for G (z); the ab- sence of poles oK the real axis for G(z) follows then from the absence of zeros for G i(z). This last property can be demonstrated from Eq. (17) for T(r) non-negative. In general, the integral in Eq. (17) needs renormaliza- tion. The usual mass and wave-function renormalizations are performed by imposing the condition that the renor- malized propagator have a pole at the physical nucleon mass M, with unit residue. This implies that the renor- malized propagator GR(z), defined as The function Z(z) is related to the E(q) of Eq. (S) by the projection operators P~(q) as in Eq. (16). Since A(z) is supposed to be non-negative, G(z) can have no poles or zeros off the real axis. This is known as the Herglotz property. The absence of zeros can be demonstrated as follows [ij: G(z) can be written as (z —K —iy) A(r)G z+iy = d~ (z —r)2+ y2 GR(z) = G(z)/Z» is given by the following expression: GR(z) = f AR(r) Z —K The renormalized inverse is given by (20) (2i) +oo GR (z) = (z —M) 1 —(z —M) TR(r) (r. —M)2(z —r) (22) TR (r) (r. —M)' - —1 dr.AR(r) (23) (24) In the above expressions, AR(K) = A(r)/Z2 and TR(r) = Z2T(r). In terms of renormalized quantities, Z2 can be written as where p (02) is the pion spectral function. It represents the probability that a state of mass ~o'2 is created by the pion field and as such it must be non-negative. The meaning of the complex variable z is that the physical propagator D~(q2) is the limit of D~(z) when z ~ q2+ic. Using the SDE for the pion, Eq. (7), the inverse of D (z) can be written in terms of the pion self-energy II (z) as The spectral functions AR(r) and TR(r) are related by D '(z) = z —m —II (z) . (28) AR(r) = b(K —M) + ~GR'(r(1+ ie))~ 'TR(r) —:b(K —M) + AR(r) . (2s) (26) D()= d Z —0 (27) The possibility of writing such an expression, relating the spectral function of the propagator to the spectral function of its inverse, is of course only permissible if the Herglotz property is valid. In the presence of ghosts, this expression is not valid. Let us now consider the spectral representations of the meson propagators. The isospin structure of the m-meson propagator is such that D'~(q ) = b'~D (q2) For D (q ). one can write the spectral representation Similarly to the case of the nucleon, one can write a spec- tral representation for D (z), D-i() Q d2 -( ) Z —0 (29) is then D~R(z) = D (z)/Z ( ) d 2 P R(~') 0 Z —CT (30) Its inverse is given by The renormalized propagator is again obtained by fix- ing the pole position at the physical mass, and the residue at the pole equal to 1. The renormalized propagator D~R(z), defined as 49 SELF-CONSISTENT SOLUTION OF THE SCHWINGER-DYSON. . . 1303 OO S o 0 (32) The renormalized spectral functions are defi.ned as p ~(cr ) = p (o' )/Zs and 8 z(o' ) = Z3 S (o' ). In terms of the renormalized quantities, Z3 is given by Zs ——1 — d(r (33) 0 o2 —m2 2 OQ - —1 dry p ~(o') (34) 0 The spectral functions p ~ and 8 ~ are related by p ~(q') = ~(q' —m.') + ID.~I'~-R(q') (35) —= ~(q' —m'. ) +P R(q') (36) Let us now consider the u-meson propagator. Since the baryon current is conserved, the cu-meson self-energy II~"(q ) must satisfy q„II""(q') = q„II""(q ) = 0 . (37) Therefore, the Lorentz structure of II""must be II""(q ) = (g"" —q"q"/q j II (q ) . (38) Substituting this in the SDE for the ~-meson propa- gator, Eq. (9), D""(q~) can be written as D.""(q') = -g""D-(q') (39) where (4O)D-(q') = 1 q —m —II (q ) +is Terms proportional to q"q" in Eq. (39) can be neglected when using D"" in Eq. (6), because of current conserva- tion. The spectral representation of D is 2 p~(~ ) (41) Z —CT As in the case of the pion, one can write the Cauchy representation for the inverse of the u-meson propagator as I ones for the pion, Eqs. (31, 32, 34, 36), with the m index replaced by (d index. III. SCHWINGER-DYSON EQUATIONS We start with the nucleon SDE, Eq. (5). It can be written as G '(p) =& '(p) —~(p) (43) where Z(p) is given by Eq. (6). To proceed, we need to specify the form of the vertex functions I's(p, p+q; q) and I'"(p, p+ q; q). In the usual HF approximation, I' s (p, p + q;q) = 7'ps and I'"(p, p+ q;q) = p". In this paper we consider vertex functions written as I" (pi, p2 q) = ~'~ Fs(pi p2 q) I'"(pi J 2' q) = y"Fv (pi, p2; q), (44) (45) where Fs(pq, p2, q) and Fv(pq, p2', q) are scalar functions. It is important to note the inconsistency of our ansatz for the NNur vertex function with the (first) Ward- Takahashi identity. This identity is an exact statement for the conservation of the baryon current; it relates (the longitudinal part of) I'"(p, p+ q; q) to the nucleon self- energy. There are attempts [19] to incorporate vertices consistent with this identity in studies of model SDE's. It would be very interesting to pursue such an approach in hadronic models, mainly in connection with the ultra- violet behavior of the vertex function. For the purposes of the present paper, we use the above ansatz and reserve for a future publication the study of an ansatz consistent with the Ward- Takahashi identity. Substituting Eqs. (44,45) and the spectral representa- tions for G(q), D, and D, in the integral for Z(q), Eq. (6), and applying the projection operators P~(p) to Eq. (43), one obtains D.— (.) =.— ."- 0 Z —CT (42) +OO TR(r) = dr'K(r, K') A~(~'), (46) Renormalization proceeds as for the pion. The renor- malized quantities are given by expressions similar to the where K(r, r') is given by K(e, e') =K (r, r';m )+2K (e, r';m ) der p ~(o')K (e, e';'o )+2 do p ~(o )K (m, r';0 ) . 0 0 (47) K (r, r'; m ) and K (r, e', m ) are respectively the n-nucleon and u-nucleon scattering kernels, 2 - X/2 K (r, K';m ) = 3 — r —2r (r' + m ) + (r' —m ) 4m x [(rc —~') —m j 8(K —(~~'~+m) )Fs(~, ~';m) 2/r fs (4S) 1304 M. E. BRACCO, A. EIRAS, G. KREIN, AND L. WILETS 49 and 2 2 2K (r, r';m ) = — e —2r (r' + m ) + (r' —m )4' x (r. —r') —2Kr' —m ] 0(r —(lK'l + m) )Fv. (r, r'; m) . 2lr. l'- (49) p R(0. ) is related to S R as shown in Eqs. (35—36) [similarly for p R(0 )]. The meson self-energies S R(q ) and S R(q ) are obtained using the spectral representation of the nucleon propa- gator in Eqs. (8, 10). S R(q2) is given by S R( q) = S (M M S ) + 2 f d~Aa(~)S (M ~ q ) +f d~d~'Az(~)Az(~')S (~ a';S ) (50) where A(r) is defined in Eq. (26), and I 2 [q' —2q'(K'+ ~") + (K' —~")']'o-(q' —(l~l+ l~'I)')Fs(& K' q) q (51) with g = Z2go and g = Z2go For the u meson, we have the same expression as in Eq. (50), with the index z replaced by (d and ( g' ') r. —r.' ' 1 ( 87I.2 ) q2 3q4 x[q —2q (K + r' ) + (K —r' ) ]&O(q —(lrl + lr'l) )Fv(r, r'; q) . (52) IV. NUMERICAL RESULTS The problem consists in solving for the spectral func- tions AR(r), p R(0' ), and p R(o ). The equations re lated to AR(ds) are Eqs. (46)—(49), (22), and (26). For p R(0 ), the relevant equations are Eqs. (50), (51), (32), and (36). For the u meson, the equations are the equiv- alent ones of Eqs. (50), (51), and (32), with the index 7r replaced by ur, and Eq. (52). These represent a set of coupled nonlinear integral equations, which we solve by iteration. We start solving for A.~ with the bare vr and u propa- gators, for which the spectral functions are given by p- ( ') =&( ' — .'), p- ( ') =~( ' — .') (53) This is the usual Hartree-Fock approximation for the nu- cleon propagator including vertex corrections by means of the form factors of Eqs. (44), (45). The solution for AR(r. ) is obtained, as in Refs. [1] and [4], by iteration Rom the perturbative solution. This AR(r) is used to obtain the spectral function of the inverse of the pion propagator S R(q2), Eq. (50), and the equivalent one with n. —i ur. Using Eq. (36) one obtains p ~ and similarly p ~. This completes the first iteration. The next iteration starts calculating the fermion kernel K(r, r') of Eq. (47) using the spectral functions p R and p R obtained in the first iteration. With this K(r, , r, '), we solve for AR by iteration starting from the AR(r) ob- tained in the first iteration. The process is then repeated to convergence, for AR(r), p~R((T ), and p~R(o). Initially, we considered bare vertices Fs(pi, p2, q) Fv(pi, p2, q) = 1, and investigated the role of the self- consistency on the spectral functions. We used the fol- lowing values for the coupling constants and masses: 2 —14.6, m —0.144M,4' 2—= 6.36, m = 0.833M,4' (54) (55) where M is the nucleon mass. The converged spectral functions A~, p R, and p ~ are shown (without the delta functions) in Figs. 2—4. The solid (dashed) lines represent the self-consistent (not self-consistent) solutions. The not self-consistent meson spectral functions are the ones obtained by calculating the nucleon loop in Figs. 1(b), 1(c) using the bare nu- cleon propagators (AR = 0); i.e. , these are the first order perturbative spectral functions. As discussed in Ref. [4], the contribution of the (d meson to the kernel K(r, r') in Eq. (47) has a finite jump at K = M+ m . This intro- duces a discontinuity in the integrand of Eq. (22). At the discontinuity, the real part (principal value integral) of Eq. (22) has a logarithmic singularity, implying that AR(r) has a (sharp) zero at r. = M + m . This zero is represented in Figs. 2 and 5 by the vertical straight line which hits the K, axis at the discontinuity. The self-consistency does not acct the fermion spec- tral function perceptively; therefore we have plotted only the self-consistent one. However, the self-consistency does acct the meson spectral functions, although not very importantly (see Figs. 3 and 4). It is interesting to note that the eKect of the self-consistency is opposite in 49 SELF-CONSISTENT SOLUTION OF THE SCHWINGER-DYSON. . . 1305 0.40 0.30 0.20 0.10 FIG. 2. Self-consistent (solid curve) and not self-consistent (dashed curve) nucleon spectral function AR(~). s, is in units of the nucleon mass M and AR(e) is in units of M . The curves are multiplied by 5 for negative ~. 0.00—10.0 I -7.5 —5.0 -2.5 0.0 2.5 5.0 7.5 10.0 p ~ and p ~, it increases the former and decreases the last. Next, we search for ghost poles. This is done by search- ing the zeros in the complex variable z of the term in square brackets in Eq. (22). The search is done using a Newton-Raphson method. Once complex zeros of G& (z) are located, the residues of the corresponding poles in GR = [G~(z)] i are easily computed regarding this func- tion as the ratio of two analytic functions. The role of the self-consistency on the appearance of ghost poles is shown in Table I. Clearly, the self-consistency does not change much the position of the poles and residues of the nucleon propagator, although it changes somewhat the ones of the meson propagators. As discussed in Refs. [1] and [4], the signal for the presence of ghosts in the nucleon propagator is revealed by the fact that the renormalization constant Z2 calcu- lated via the spectral function of the nucleon self-energy, TJt(e), gives Zq ———oo. The minus sign is the indication of a ghost. This happens because, for large K or ~', one has K (~, ~') (~2 —rc")(~ —~')'9(~' —~'2), (56) 2/~/s and since the integral of AR(z) is finite, one has [1] TR(~) (57) and the integral for Z2 is therefore logarithmically diver- gent On t.he other hand, the integral over AR(K) is not zero, and therefore Z2 calculated via A~, Eq. (24), does not give Z2 ——0. However, as shown in Ref. [1], con- sistency is recovered if one includes the pair of complex conjugate poles in GR [note that the real parts of the residues are negative (see Table I)]. For the case of the renormalization constants of the x and u mesons, we obtain exactly the same result: The Z3's calculated via the spectral function of the self- 5 0 I I ~ 15~0 s s s s ~ 40 3.0 cv b 3 10.0 2.0c) c) 5.0 1.0 0.0 10 logio (o ) 100 ~ ~ ~ I ~ l~ ~ I 1000 0.0 10 ~og~o (o') 100 I 1000 FIG. 3. Self-consistent (solid curve) and not self-consistent (dashed curve) s. spectral function p n(o. ). o is in units of M and p~R(o ) is in units of M FIG. 4. Self-consistent (solid curve) and not self-consistent (dashed curve) ~ spectral function p n(o ) The units are. the same as in Fig. 3. 1306 M. E. BRACCO, A. EIRAS, G. KREIN, AND L. WILETS 49 TABLE I. Ghost poles. The 6rst value is the pole position and the second is the residue at the pole. The nucleon poles are in units of M and of the mesons are in units of M . Self-consistent 1.06 + 1.25i —0.77 + 0.20i —1.04 —1.08 —3.50 —1.30 Not self-consistent 1.05 + 1.26i —0.77 + 0.20i —1.44 —1.13 —5.68 —1.49 energy, S~, give Z3 ', —oo. To obtain zero for the integral over the spectral function of the propagator, p~, the residue of the ghost pole has to be included. In Ref. [4], the problem of ghosts poles in the nu- cleon propagator was investigated using form factors at the nucleon-meson vertices. Two types of form factors were used: (a) a Sudakov form factor, which is generated by vector meson dressing of the vertices, and (b) a phe- nomenological form factor, of the monopole type. The conclusion there was that both types of form factors are able to kill the ghosts. However, as remarked in that ref- erence, a proper extension of the Sudakov form factor to lower momenta is necessary for a better study of these is- sues. In this paper we use only the simple monopole form factor to investigate the interplay of self-consistency and vertex corrections on the spectral functions. As in Ref. [4], we use for Fs(pq, p2, q) and F~ (pq, p2, q) the following expressions: Fs(pi, S2, q)=Fv(Ji S2, q) 1 1 1 1+ ]p2~/A'! 1+ ]q'/A ] 1+!p',/&'! (58) where A is an ultraviolet cutoff. The calculated spectral functions with use of the form factors are plotted in Figs. 5—7. We plotted A~, p ~, and p R (again without the delta functions) for three representative values of cutouts, A = M (solid curves), 1.25M (long-dashed curves), and A = oo (short-dashed curves). The eff'ect of the form factor is to increase AR(K) for negative K, , a result already found in [4]. The eff'ect on the meson spectral functions is to increase (decrease) vR~ (vR~) In Ref. [4], it was found that for a A ( A„;t = 1.75M the ghost poles in the nucleon propagator disappear. In the present case, we found that the self-consistency does not alter significantly this critical value; the ghost poles disappear in all propagators for a A & 1.60M. We have also investigated the effect of the self- consistency on the ghost-&ee spectral functions; i.e., we compared the self-consistent and not self-consistent spec- tral functions for several values of A's smaller than A„;t. We found the surprising result that the effect of the self- consistency is negligible in all spectral functions; the ef- fect is almost invisible when one plots the spectral func- tions. Although on physical grounds one expects that the cut- offs for the m and u vertices have different values, we used the same value for both, since in this work we are mostly interested in the qualitative effects. The consequences of the modifications induced by the form factors on physi- cal observables deserves a separate study. Work in this direction is in progress. We conclude this section with the general remark that the self-consistency does not affect the spectral properties of the propagators. V. CONCLUSIONS AND PERSPECTIVES In this paper we have solved self-consistently the cou- pled set of Schwinger-Dyson equations for the nucleon and ~ and u mesons in the vacuum. The set of equa- tions was truncated by postulating a three-point meson- nucleon vertex function. The understanding of the vac- uum properties of the nucleon and meson propagators is a 0.40 0.30 CL 0.20 0.10 FIG. 5. AR(r) for difFerent values of the cutoff'. A = M (solid curve), 1.25M (long-dashed curve), and oo (short-dashed curve). Units are the same as in Fig. 2. The short-dashed curve is multiplied by 5 for neg- ative K. 0.00—10.0 —75 —5.0 —2.5 0.0 2.5 5.0 7.5 10.0 49 SELF-CONSISTENT SOLUTION OF THE SCHWINGER-DYSON. . . 1307 250 I I I I I 1 5.0 20.0 15.0 b 3 10.0 I I I I I ' ~ I I \ 10.0 C) 5.0 5.0 10 ~o9«(o') 100 I I I I I I I I 1000 0.0 10 ~o910 (o ) '1e s il I O'W~LLLI I 100 1000 FIG. 6. p R(o ) for same A's as in Fig. 5. Units are the same as in Fig. 3. FIG. 7. Same as in Fig. 6 for p R(o' ). necessary first step towards the study of the properties of nucleon and meson in nuclear rnatter, as well as those of nuclear matter and finite nuclei. Although many of such properties have been studied using relativistic quantum field models, the vacuum polarization effects in medium have invariably been neglected. The main conclusion of our investigation is the sur- prising result that the self-consistency does not modify significantly the spectral properties of the propagators. The appearance or disappearance of the ghost poles in the propagators is not affected by the self-consistency. One important aspect regarding the vacuum of meson- nucleon effective theories that was not yet satisfactorily investigated is the role of the three-point meson-nucleon vertex functions. In particular, the interplay of the in- frared and ultraviolet sectors of the u-nucleon three-point vertex is extremely important to the problem of ghosts poles, as shown in the recent studies of Refs. [8,4]. The constraint of current conservation on the three-point function is certainly an important aspect of the problem which also deserves more study. In this respect, an in- teresting possibility to implement the Ward-Takahashi identity is the use of the so called gauge technique, in- vented by Salam a long time ago [20]. This technique is particularly well suited for our formulation of the prob- lem since it postulates a spectral function for the vertex which contains the spectral function of the fermion prop- agator. In this formulation, the Ward- Takahashi identity is automatically satisfied. In the past, this technique has been employed to investigate the problem of ghosts in the electron propagator [21]. Work in this direction is in progress. The effects of the self-consistency on the nucleon and meson propagators in nuclear matter, in connection to the problem of ghost poles, remains an open problem, although work in this direction has recently been com- municated [22]. Another important aspect is the role of chiral symme- try in hadronic models. This is a separate subject, with its own problems. Much remains to be done in this re- spect, both in vacuum and in nuclear matter. ACKNOWLEDGMENTS The work of M.E.B., A.E. and G.K. was partially sup- ported by the Brazilian agencies CNPq and FAPESP. The work of I.W. was partially supported by the U.S. Department of Energy. [1] W.D. Brown, R.D. Puff, and L. Wilets, Phys. Rev. C 2, 331 (1970). [2] W.T. Nutt and L. Wilets, Phys. Rev. D ll, 110 (1975). [3] W.T. Nutt, Ann. Phys. (N.Y.) 100, 490 (1976). [4] G. Krein, M. Nielsen, R.D. PufF, and L. Wilets, Phys. Rev. C 47, 2485 (1993). [5] R.J. Perry, Phys. Lett. B 199, 489(1987); T.D. Cohen, M.K. Banerjee, and C.-Y. Ren, Phys. Rev. C 36, 1653 (1987); K. Wehrberger, .R. Wittman, and B.D. Serot, ibid. 42, 2680 (1990). [6] G.E. Brown and A.D. Jackson, The Nucleon Nucleon In- teraction (North-Holland, Amsterdam, 1976); J. Milana, Phys. Rev. C 44, 527 (1991). [7] V.V. Sudakov, Zh. Eksp. Teor. Fiz. 30, 87 (1956) [Sov. Phys. JETP 3, 65 (1956)]. [8] M.P. Allendes and B.D. Serot, Phys. Rev. C 45, 2975 (1992). [9] J.D. Walecka, Ann. Phys. (N.Y.) 83, 491 (1974); L.S. 1308 M. E. BRACCO, A. EIRAS, G. KREIN, AND L. WILETS 49 [10] [11] [13] [14] [15] [16] Celenza and C.M. Shakin, Relativistic Nuclear Physics: Theories of structure and scattering (World Scientific, Singapore, 1986). C.D. Roberts, in Proceedings of the Workshop on QCD Vacuum Structure, Paris, France, 1992, edited by H. M. Fried and B. Miiller (World Scientific, Singapore, 1993). R. Fukuda and T. Kugo, Nucl. Phys. B117,250 (1976). D. Atkinson and D.W.E. Blatt, Nucl. Phys. B151, 342 (1979). S.J. Stainsby and R.T. Cahill, Phys. Lett. A 14B, 467 (1990); U. Habel, R. Konning, H.-G. Reusch, M. Stingl, and S. Wigard, Z. Phys. A 336, 423 (1990); 336, 435 (1990); P. Maris and H. Holties, Int. J. Mod. Phys. A 7, 5369 (1992); S.J. Stainsby and R.T. Cahill, ibid. 7, 7541 (1992). B.D. Serot, Rep. Prog. Phys. 55, 1855 (1992). S.S. Schweber, An Introduction to Relativistic Quantum Field Theory (Harper lk Row, New York, 1962). P. Roman, Introduction to Quantum Field Theory (John [17] [18] [19] [20] [21] [22] Wiley 8c Sons, Inc. , New York, 1969). L. Wilets, in Mesons in Nuclei, edited by M. Rho and D. Wilkinson (North-Holland, Amsterdam, 1979). N.N. Bogolubov, A.A. Logunov, and I.T. Todorov, Ax- iomatic Quantum Field Theory (Benjamin, Reading, MA, 1975), pp. 269—270, 330ff, Appendix F. H. Pagels and S. Stokar, Phys. Rev. D 20, 2947 (1979); J.S. Ball and T.-W. Chiu, ibid 22., 2542 (1980); D.C. Curtis and M.R. Pennington, ibid. 42, 2542 (1990); C.J. Burden and C.D. Roberts, ibid 47., 5581 (1993). A. Salam, Phys. Rev. 130, 1287 (1963); R. Delbourgo and A. Salam, ibid. 135, 1398 (1964). D. Atkinson and H.A. Slim, Nuovo Cimento 50, 555 (1979). C.L. Korpa, communication presented at the Interna- tional Workshop of Gross Properties of Nuclei and Exci- tations XXI, January 1993, Hirschegg, Austria (unpub- lished) .