PHYSICAL REVIEW C VOLUME 47, NUMBER 6 JUNE 1993 Ghost poles in the nucleon propagator: Vertex corrections and form factors G. Krein Instituto de Fx'sica Teorica, Universidade Estadual Paulista, Rua Pamplona, 145 014-05 Sao Paulo, Brazil M. Nielsen Instituto de Fxsica, Universidade de Sao Paulo, Caiza Postal, 80516 02-498 Sao Paulo, Brazil R. D. PufF and L. Wilets Department of Physics, FM 15, V-niversity of Washington, Seattle, Washington 98195 (Received 23 December 1992) Vertex corrections are taken into account in the Schwinger-Dyson equation for the nucleon propa- gator in a relativistic field theory of fermions and mesons. The usual Hartree-Fock approximation for the nucleon propagator is known to produce the appearance of complex (ghost) poles which violate basic theorems of quantum field theory. In a theory with vector mesons there are vertex corrections that produce a strongly damped vertex function in the ultraviolet. One set of such corrections is known as the Sudakov form factor in quantum electrodynamics. When the Sudakov form factor generated by massive neutral vector mesons is included in the Hartree-Fock approximation to the Schwinger-Dyson equation for the nucleon propagator, the ghost poles disappear and consistency with basic requirements of quantum field theory is recovered. PACS number(s): 21.30.+y, 21.60.Jz, 21.65.+f I. INTRODUCTION Nonrelativistic many-body theory has been used with success in the study of ground and excited states of nu- clear systems. Nevertheless, experiments in the near fu- ture will provide data on nuclear systems at extreme con- ditions of density and temperature and obviously a new theoretical approach that goes beyond the nonrelativistic one will be required. This has motivated a great deal of interest in recent years in the development of relativis- tic many-body theories for nuclear physics based on the methods of renormalizable relativistic quantum field the- ory. In this context, there is an extensive literature on calculations of nuclear matter and finite nuclei proper- ties by means of models based on the original Walecka model [I]. Calculations employing mean field and one- loop (Hartree) approximations have achieved consider- able success in the description of bulk properties of nu- clei and of proton-nucleus scattering parameters. How- ever, there are severe difficulties in extending the calcu- lations to include quantum corrections which go beyond the one-loop Hartree approximation. The inclusion of these quantum corrections leads to catastrophic results, with the appearance of complex poles in the baryon and meson propagators which, among other things, introduce a large imaginary part to the nuclear matter energy. Complex poles, or ghosts, have long been noted in lo- cal relativistic field theory [2, 3]. They are physically unacceptable because they correspond to eigenstates of the system with complex energies and probabilities. In the case of quantum electrodynamics (@ED), the appear- ance of a ghost in the one-loop correction to the photon propagator, the so-called Landau ghost, is not taken as a serious drawback of the theory. This is because the mo- mentum scale at which the ghost appears is far from mea- surable and, at this scale @ED should probably be modi- fied to include the effects of other electroweak effects. In the case of a nuclear theory with mesons and baryons, the corresponding ghosts appear at momentum scales of the order of 1 GeV. The appearance of the ghost poles is related to the short distance behavior of the model inter- actions; asymptotically free theories appear to be free of ghosts [4]. It is clear that a description of hadronic mat- ter in terms of mesons and baryons only must break down in the region where short distance properties are involved. However, such a description should provide a reasonable description of the properties which are thought to be in- sensitive to the short distance physics. In this sense, the appearance of ghosts in hadronic theories appears to frus- trate the hope of constructing relativistic models based on renormalizable field theories without employing sub- nucleonic degrees of freedom. Achtzehnter and Wilets [5] have shown that quark substructure plays an impor- tant role in the interaction of nucleons with an external (Bose) field at all momentum transfers This involve. s issues which we will not pursue here. Rather, we concen- trate on the construction of some intrinsically consistent field theory. Of course one should keep in mind that the efFective Lagrangians commonly used in nuclear physics are likely not be derivable from the fundamental theory of the strong interactions (@CD). Such an effective La- grangian will probably be very complicated and quite inelegant. In the past, several methods have been proposed to eliminate this short distance sensitivity as, for exam- ple, modifying the analytic structure of the propagators 0556-2813/93/47(6)/2485(7)/$06. 00 47 2485 1993 The American Physical Society 2486 G. KREIN, M. NIELSEN, R. D. PUFF, AND L. WILETS 47 in such a way to remove the unwanted singularities [6]. More recently, in the light of the quark substructure of the nucleons, form factors at the meson-baryon vertices [7] have been used to regulate the theory at short dis- tances. Another perspective on the problem is the reg- ulation of the theory by means of vector meson 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 function in the ultraviolet region [8]. The damping arises from the in- frared structure of the theory, despite the fact that the external nucleon momenta and the momentum transfer to the vertex are large. Physically, the damping arises from the large likelihood of matter fields to radiate soft virtual vector mesons. This phenomenon is a property of theories containing vector mesons. Based on these con- siderations in @ED, Sudakov [9] derived a form factor. In a recent publication, Allendes and Serot [10] included the Sudakov form factor in the calculation of the polarization loop correction to the vector meson propagator. The au- thors concluded that inclusion of the Sudakov form factor regulates the ultraviolet behavior so that the corrected propagator is free from ghost poles. This is a very im- portant achievement for the construction of a consistent relativistic nuclear many-body theory with mesons and baryons only, as it restores the hope for the consistency of such a theory. In this paper we study the effect of the Sudakov and quark-substructure form factors on the ghost problem in the nucleon propagator. We consider a model with nu- cleons, neutral vector mesons, and pions. Although the theory is designed to study nuclear matter, we shall re- strict ourselves in this paper for simplicity to the vacuum only. The appearance of the ghosts does not depend on the presence of matter [11]. We analyze the effect of the vertex corrections by means of form factors in the renormalized Schwinger-Dyson equation for the nucleon propagator. The calculation of the complete vertex func- tion is a complicated problem and therefore we shall use simplifying approximations which we discuss below. G(p) = A(v) P —K + lC (2) A(r) is the spectral function. It represents the probabil- ity that a state of mass lvl is created by Q or g, and as such it must be non-negative. Negative K corresponds to states with opposite parity to the nucleon. Equation (2) can be rewritten conveniently as G( ) =P+( )G( + )+P-( )G(— — ) where P~(p) are projection operators defined as P+(p) =- I 1+1( ~.) with ~pz if p2) 0, ig —p' if p' (0, (4) and G(z) is given by the dispersion integral G(z) = + A(r) (6) It follows from the commutation relations that dr A(r) = 1 . The inverse of the propagator can be written in terms of the projection operators P~(p) as G '(p) = P+(p)G '(uip+ie) + P (&)G '( m„—ic) . — Since A(rc) is supposed to be non-negative, it is simple to show that G(z) can have no poles or zeros off the real axis. This is known as the Herglotz property. Now, if G(z) possesses the Herglotz property, then so does G (z). This permits us to write a spectral representation for G-'(.), II. NUCLEON PROPAGATOR AND GHOST POLES In order to make the paper self-contained, at the cost of being a little repetitive, in the following we briefly review the problem of ghosts in the nucleon propagator following the work of Brown, Puff, and Wilets (BPW) [3]. We start with the usual definition of the nucleon propagator where Q represents the nucleon field operator and l0) is the physical vacuum state. The Kallen-Lehmann repre- sentation for the Fourier transform G(p) of Gp~(x —x') can be written as G (z) =z —Mo— G~'(z) = ZzG '(z), is given by (10) G (z) has the Herglotz property only if T(K) is non- negative. In general, the integral in Eq. (9) is divergent and therefore needs renormalization. The usual mass and wave-function renormalizations are performed by impos- ing the condition that the renormalized propagator has a pole at the physical nucleon mass M with unit residue. This implies that the renormalized inverse propagator G& (z), defined as GR'(z) = (z —M) 1 —(z —M) T~(K) (r —M)2(z —r) 47 GHOST POLES IN THE NUCLEON PROPAGATOR: VERTEX. . . 2487 where TR(r) = ZzT(r) and Z2= 1— TR (K) (K —M)2 (12) GR(z) = + AR(r) where AR(z) = A(r)/Zz. In terms of renormalized quan- tities, Z2 can be written as From Eq. (10), it follows that the spectral representation for GR(z) is Z2=1— dK + TR(K) (14)~ —M2 +OO - —1 dr. AR(v) (15) In order to compare with previous work of BPW, we now consider a model with nucleons (g), pions (n), and omegas (w") given by the following Lagrangian density: 8 = g(ip„B" —igo ps~ m —go p„~")g — F„—F" 1 2 1 ~ 1 2——m ~~"+—Om Our ——mm vr (16)2" ~h~~~ +""= 0"~ —8 tu". The Schwinger-Dyson equa- tion for the nucleon propagator, Fig. 1, is given by G '(») =Go'(p)+»go d4 AD (q )G(p —q)I's(p —q, p; q) + igo d4 (2 ),V&D" (q')G(» —q)l' (» —q, »;q), where D and D"~ are the vr and u1 propagators and I's(p —q, p; q) and I'&(p —q, p; q) are the pion-nucleon and omega- nucleon vertex functions, respectively. We do not consider the vector meson tadpole contribution to the nucleon propagator since it drops out in the renormalization procedure. The Hartree-Fock (HF) approximation amounts to use the noninteracting D and D and the bare vertices I's(p —q, p; q) = ps and I'„(p —q, p; q) = p„ in Eq. (17). In the HF approximation the set of equations to be solved self-consistently for the renormalized AR(rc) and TR(K) is given by TR(r) = dK, 'K(r, K') AR(r'), (18) GR'(r(1+ ie)) = (r. —M) 1 —(K —M) TR(r') (M —r')2 [K' —K(1+ ie)] AR(~) = ~(M —K) + IGR'(K(1+ ie))l 'TR(K) (20) where K(r., K') is given by K(v, r') = K (K, r.') + K (K„K'), (21) g 2 - Z/2 K (r, K') =3 K —2K (r.' +m )+(r,' —m ) (r, —r, ') —m 8(K —(lK'l+m ) ),4~ K 2 - 1/2 K~(r, y') = r. —2K (r' + m ) + (r' —m ) s (v —r') —2Kr' —m &(K —(lK'I+ m ) ) . 4m K (23) In the above equations, g and g~ are the renormal- ized coupling constants, defined as g = Z2go and gu Z2gOw. Following BPW, let us initially consider the pion, ne- glecting for the moment the vector meson. Equations (18)—(20) were solved numerically by iteration, beginning in Eq. (18) with the free value for A(K), A(r) = b(M K). — The converged function A(K), for gz/4~ = 14.4 and the FIG. 1. Diagrammatic representation of the Schwinger- Dyson equation for the full nucleon propagator. The wavy (dashed) line represents the ur (vr) meson propagator, and the solid (double solid) line represents the free (full) nucleon prop- agator. 2488 G. KREIN, M. NIELSEN, R. D. PUFF, AND L. FILETS 47 physical nucleon and pion masses, is shown by the dashed line in Fig. 2. In addition to the pole at the nucleon mass z = M (which is fixed by the renormalization procedure), G& (z) in Eq. (11) has zeros at z = (0.73 + 1.25i)M. These complex zeroes mean that the nucleon propagator has poles at those complex masses with corresponding residues of —0.75 +0.32i, respectively. The signal for the presence of ghosts is revealed by the fact that Zz cal- culated from TJt(r) in Eq. (14) gives Z2 = —oo. Since Z2 = 0, it follows from Eq. (15) that the integral of A~ is zero. We must therefore include the pair of complex conjugated poles in GR. GR(z) = A~(~) A, A; dK + + Z —K Z —K Z —KC C (24) I I & i r I & I i I I I & ~ l I I I where r, and A, are the complex pole and residue, re- spectively. The sum of residues of the complex poles is negative and exactly cancels the integral over real K. Ac- tually, the difBculty lies in the negative sign of Z2, since this destroys the Herglotz property of G i (unrenormal- ized). The ghosts have their origin in the ultraviolet behavior of the interaction, as we shall show in the following. The kernel K (K, r') has the following asymptotic form, for large K or K': 1 K —K K —K K —K 2/v/s (25) Since f dK A(K) is finite [3], it follows from Eqs. (18) and (25) that T(r) for large K is given by (26) Therefore, the integral in Eq. (14) is divergent andZ2: —00. The w meson introduces a new ingredient in the prob- lem, namely, the spectral function AR(r) can be negative for some values of real v. . The dotted line in Fig. 2 repre- sents AR(K) for g~/4vr = 6.36 and m = 780 MeV. The complex poles are located at z = (5.67 6 11.76i)M, with residues —1.0412 6 0.22i, respectively. A(tc) is negative for M+m & K & 3.9. K has a finite negative jump at K = M + m~ due to the term —2vv' in Eq. (23). This introduces a discontinuity in the integrand of Eq. (19) for G& . At the discontinuity, the real part (principal value integral) in Eq. (11) has a logarithmic singularity, implying that AR(K) has a (sharp) zero at r = M + m~. This zero is represented in Fig. 2 by the vertical straight line which hits the K axis at the discontinuity. Although negative A~(r) represents presumably the presence of negative metric states (sometimes also re- ferred to as ghosts), this is not related to the complex ghost poles which motivated this work. We show below that the use of form factors will eliminate the complex poles, but AR(v) may still contain negative regions as before. For QED, the interaction kernel is the same as in Eq. (23), with m~ = 0, giving rise to a negative spectral function as well [12]. In theories with massless vector bosons, the theory is formulated in terms of an indefinite metric [13] and the positivity of the spectral functions is not a necessary requirement [12]. However, in our case where m~ g 0, AR(K) & 0 is a necessary requirement and the only explanation we have at the moment for AR(r) & 0 is the inadequacy of the HF approximation, or the inconsistency of the theory. Including both vr and u mesons, the situation is quali- tatively similar as above. The contribution of the vr dom- inates the one of the w and A~(r) is non-negative for real K. In Fig. 2, the solid line indicates the function A~(K) for the same values of g, m, g, and m as above. The complex poles are located at z = (1.05 ~ 1.26i)M with residues —0.77 + 0.20i, respectively. III. VERTEX CORRECTIONS In this section we discuss the effect of form factors on the problem of ghost poles. We start with the consider- ation of the Sudakov form factor. Let us consider the proper NNu vertex I'4'(pi, p2, q), Fig. 3(a). The pi and p2 are the external nucleon mo- menta and q is the external meson momentum. The Su- dakov form factor is obtained by summing the leading- log contributions of all vertex corrections. For QED, for off-shell spacelike nucleon momenta, the Sudakov form 0.0 x 5.0 ~ ~ ~ ~ 4 ~ ~ ~ ~ ~ ta) c~—= —+ ( + Ii—+ ". x 0.5 -0.2— ~ ~ ~ e ~ ~ o ~ ~ ~ ~ M —10 0 5 10 (b) II + FIG. 2. Spectral function AR(K) for vr (dashed), u (dot- ted), and or+ ~ (solid). I IG. 3. Diagrammatic representation of vertex correc- tions for the (a) NNa vertex and (b) NN~ vertex. 47 GHOST POLES IN THE NUCLEON PROPAGATOR: VERTEX. . . 2489 factor is given by I's(pl p2 q) = ysF(pr, p2, q) (30) g2 q2 q2I' (pr, p2, q) = p" exp — ln —ln- 8x2 p2 p2 (27) I'"(pr, p2, q) = W'F (» i, p2, q) (28) with This expression is valid for large nucleon momenta, ~pr ~, ~pq2~ )) M2, and ~q2~ )) ~p2r, ~p2~. Although the rnornenta appearing in Eq. (27) are spacelike, Sullivan and Fish- bane [14] argue that the expression may be freely contin- ued to the timelike region. In the case of massive vector mesons, we have exactly the same expression, for large rnomenta, as in Eq. (27), with e replaced by g2. Our approach consists in replacing the bare vertices by the corresponding vector meson corrected ones. Of course, all values of the loop momentum q~ are formally required in the evaluation of G r(p) in Eq. (17), cor- responding to the exchanged mesons in Fig. 1. The lowest-order (BPW) approximation, using the Hartree- Fock form of the I"s, is correct only for q = 0 and is not asymptotically correct for high q (I'„and I's are en- tirely independent of q in BPW). Since the Sudakov form, Eq. (27), introduces convergence at high momenta, one might hope that our approach will eliminate the ghost problem appearing in the ultraviolet. Although we will see that this is the case, a fully satisfactory analysis re- quires knowing that the asymptotic high q2 behavior of the Sudakov form is correct. Although the integral in Eq. (18) requires all values of q, a correct high q behavior, together with a correct q = 0 limit, should provide at least a "ghost-free" result. Allendes and Serot calculated in first-order perturbation theory the (low q ) on-shell vertex function and then in- terpolated the result to the on-shell Sudakov form factor in the ultraviolet (high q ). Since the calculation of the low momentum behavior of the off-shell vertex function is a tremendous task, even in lowest-order perturbation theory, we prefer not to follow the approach of Allendes and Serot. We, instead, rewrite Eq. (27) as where we have also introduced an infrared regulator. With the introduction of the infrared regulator, the ver- tex functions have the correct zero momentum limit F(0, 0, 0) = 1. By substituting the Sudakov corrected vertex functions in the Schwinger-Dyson equation, Eq. (17), we obtain the following expression for the Sudakov corrected kernels: K " (r. , K') = K (K, r') F(K, K', m ) (31) K " (K, K') = K (K, rc') F(K, K', m ), (32) where the value of A is discussed in the next section. Substitution of this in the Schwinger-Dyson equation, Eq. (17), we obtain Eqs. (31) and (32) for the corrected kernels. IV. NUMERICAL RESULTS We start discussing the Sudakov suppression for the pion. The iterative solution of the equation proceeds as before. We find that the ghost poles disappear for values of A smaller than a critical value A„;t —0.9M. The function Air(rc) is shown by the dashed line in Fig. 4 for where K (r, r') and K~(K, r') are the "bare" interaction kernels of Eqs. (22) and (23). We have also studied the ghost problem in the con- text of purely parametrized form factors, similar to the ones employed in boson exchange nucleon-nucleon poten- tial models [15]. There is one complication in our case, namely, we need oEF-shell form factors, since the external nucleon legs are off shell. Here we follow Ref. [16] and use one suppression factor for each external leg of the vertex 1 j. 1 1+~ ~A ~ 1+~q ~A ~ 1+~ (33) g2 A2 + q2 A2 + q2 F(pr, pg, q) = exp — ln 2 ln 87r' A'+ p', A'+ p', (29) 0.4— 0.2 I I I I I I I I I I I I I where A is an infrared cutoff which will be fixed later. The effect of A is to extend the validity of form factor to the infrared region of the loop integral. The important momentum dependence of the form factor for the elim- ination of the ghosts is the pr (or pq) dependence. We note that the region of large q does not contribute in the loop integral, and so the conditions for the validity of the form factor in the ultraviolet region of the loop integral are satisfied in practice. It is not diKcult to show, following Sudakov's deriva- tion [9], that the vector meson correction to the properN¹r vertex, Fig. 3(b), gives the same exponential sup- pression as given by Eq. (27). We write the NN7r vertex function as 0.0 -0.2—~ ~ ~ ~ ~ ~ ~ ~ ~ o 4) -0.4— I -15 . . . I -10 -5 I I I I 0 . . . l. . . , l. . . 5 10 15 FIG. 4. Spectral function AR(K) including the Sudakov form factor. The meaning of the different lines is the same as in Fig. 2. For values of A see text. 2490 G. KREIN, M. NIELSEN, R. D. PUFF, AND L. WILETS 47 A = „;I, .Besides the fact that we do not find complex poles, we also find that 2P I I I I ) I I I I I I I I i I I I I Re(z) Tir(r) (Ir, —M)~ dr A~(r) = 0.16 . (34) Thhe important point about Eq. (34) is that we get a the r positive 2, and so we do not have th l e presence of ghosts. Moreover, we obtain the same value for Zz calculated either with A~(K) or with T (r' ' g a we do not have missing strength and so, G& (z) given by Eq. (11) is the inverse of GR iven b a ear. Th For u, we find that for A„;t —0.55M the h appear. e dashed line in Fig. 4 represents the function ~(K) for A = 0.55M. Figure 4 shows that, as men- tioned in Sec. II, although the ghosts poles have disap- peare, R(K) contains a negative region as before. The obvious conclusion is that the negative metric states are not related to the BPW ghost poles. s, e si uation is quali-Including both vr and u mesons th 't t' tative y similar as for the x and u in isolation. The solid curve in Fig. 4 refers to A = 0.6M for 7r and A = 0.55M. The diferent va uesalues for A„;t for ~ and w are understood on the basis of their range in the NN force. Th' clearl ol ce. is is the su r y manifest in combining the case f d, ho 7tan u, w ere e suppression is governed entirely by A close to A for u. c ose 0 crit In Fig. 5 we present the results for the spectral function w en using the monopole corrected vertices of Eq. (33). The situation is similar to the Sudakov suppression with respect to the role of the cutoff d th d' of an e isappear ance o t e ghost poles. For A's smaller th an a critical value ~ ~ 1 „;t, the complex poles disappear and Z2 calculated via Air and TR give identical results. For the plots in Fig. 5 we have used A = M for all cases. In Fig. 6, w 'll 15— ———— r. m(z) Z0 —10— (f) C) CL Ld 5— o CL A „;4=1.75 p . . . , bI, 0 2 I I I I I A/M I I I ] I I I I 8 10 FIG. 6. Trajectory of the real (solid dso i an imaginary ( as e ) parts of the upper complex pole of the n 1e o e nuc eon prop- g or or e vr contribution as a function of the cutofF pa- the critical value rameter in the monopole form fact Thac ors. e arrow indicates t e critical value of A for the disappearance of the ghost poles. V. CONCLUSIONS the trajectory of the complex poles as a function of A, or the case of 7r. The critical value of A for the disap- pearance of the ghosts is of the same order (Fig. 6) as used in potential models [15]. One interesting feature of the different form factors is that the Sudakov form factor "squeezes" the spectral function around rc = M + m while the monopole form factor has the effect of decreas- vi es a strong suppression for high momenta only, while the monopo e form factors provide a uniform ' orm suppression starting at momenta of the order of the nucleon mass 0 1P I I I I I I l I 0.05 ~ 0.00" -0.05— 0 ~ ~ ~ ~ ~ ~ ~ ~ ~ Q) -0.10 -10 I I I I I I I -5 0 I I I I I I I I 5 10 FIG. 5. Spectral function Ari(K) including the monopole form factors. The meaning of the different lines is the same as in Fig. 4. For values of A see text. The nucleon propagator treated in the Hartree-Fock the r approximation in a hadronic field theor ' 1 d ' h t e presence of ghost poles. The appearance of the ghosts is related to the ultraviolet behavior of the nucleon-meson interaction. In a theory containing vector mesons, the infrared structure of such a theory introduces corrections to t e various nucleon-meson vertex functions which are strongly damped in the ultraviolet. The damping arises from the dressing of these vertices by the vector rnesons n t is work we have shown that such a damping of the vertex function can eliminate the appearance of the ghost poles of the usual Hartree-Fock approximation. improved on several aspects before definite conclusions can be drawn for realistic calculations in nuclear physics. First, although the ghost problem has been shown not to be related to the presence of a Fermi sea, the eKects of the Sudakov form factor in this case is worth investigating. In our work we have simply used the Sudakov form factor in the loop integral of the Schwinger-Dyson equation. 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