Renormalization of the self-consistent summation of exchange diagrams in a relativistic quantum field theory

Nuclear Physics A404 (1983) 428-442 @ Nosh-Holland Publishing Company

R~O~~ZA~ON EXCHANGE

OF THE S~F-CONSISTS DIAGRAMS

IN A RELATIVISTIC

SUMMATION QUANTUM

OF

J!IELD

THEORY+ ALEX F. AIELAJEW ~nsriiure nf Theoretical Physics, Deparfment of Physics, Stanford Universiiy, Stanford, CA 94305,

USA

and Natianat Research Council of Canada, Division of Physics, Oftawa, Canada KlA

OR6’

Received 2 February 1983 The self-consistent summation of exchange diagrams is renormalized in a relativistic fermionscalar field theory using spectral function methods. This is a non-perturbative approach in which all uncrossed exchange diagrams are summed to all orders in the coupling constant. A detailed exposition of the renormalization technique is presented along with numerical results. Anomalous analytic structure (ghost poles and ghost branch cuts) that arises naturally from the approximation is briefly discussed. We discuss the generalization to a finite density of fcrmions that may have application in nucleon-meson field theories which are currently in vogue.

Abstract:

1. Intfoduc~on Recently, the nuclear many-body problem has been investigated by the use of renormalizable, relativistic quantum field theories (RQFT’s) I-‘). [This list of references is not intended to be comprehensive but represents the essential background for this discussion. A more complete set of references may be found in ref. ‘).I The basic motivation for this approach is that one need never introduce potentials to describe the interactions of the nucleons since they interact via the exchange of virtual mesons. Moreover, the constraints of Lorentz covariance, causality, and retarded propagation are incorporated in the theory from the outset in the formulation. Since we shall restrict ourselves to renormalizable theories we expect that we may render any approximation to the theory finite with a minimum number of renormalization constants which we include via counterterms to the original lagrangian. Therefore, once we have determined the coupling constants and masses and we have renormalized the theory, all other physical quantities may be predicted. The use of RQFl?s begins to become relevant for the consideration of large, stable nuclei where the nucleons at the Fermi surface approach velocities of about 0.25~. Furthermore, if one wishes to study nuclei under extreme conditions such as may be encountered in neutron stars or heavy-ion collisions, a consistent relativistic theoretical framework is essential to understand the physics. ’ Supported in part by US National Science Foundation Grant NSF-PHY81-07395. *A Current address. 428

429

A.F. Bielajew / Renormalization

However, RQFT’s are fraught with mathematical complexities and divergences. These reflect both the richness of the theory and the fact that a particular theory is applicable over a finite distance, or equivalently, energy scale. Although we may extract finite results for any approximation to a renormalizable RQFT, nuclear RQFT’s are burdened by the added complication that the coupling constants must be large to describe the physics and_ the perturbation calculations that are so successful for quantum electrodynamics become less useful. We must, therefore, restrict ourselves to seek non-perturbative approximations to nuclear RQFT’s. One such approximation that has had recent attention is the self-consistent Hartree-Fock approximation (SCHFA) as formulated by Horowitz and Serot 5, in the Walecka RQFT for nuclear matter. In their approximation the renormalization difficulties have been circumvented by neglecting the contribution from the Dirac antibaryon sea. In this paper we develop a renormalization scheme for a class of divergences otherwise encountered in the SCHFA, that of exchange diagrams with uncrossed meson lines to arbitrary order in perturbation theory. In the second-quantized version of non-relativistic quantum mechanics, the SCHFA may be written in terms of Feynman diagrams as depicted in fig. 1. A single solid line represents the non-interacting fermion propagator, the dashed line represents the potential, and the double solid line represents the approximation to the full fermion propagator in the SCHFA. Spin indices and momentum flow information are suppressed in this diagram, and the reader is referred to the literature *) for a more complete explanation of fig. 1. The enormous utility of the SCHFA is that in the non-relativistic regime it provides a variational estimate of the best single-particle wave functions and energies in a properly antisymmetrized many-body system of identical fermions. The literature is rich with applications in molecular, atomic and nuclear physics and, indeed, this approximation may be used in any non-relativistic fermion system both for uniform matter and finite systems. The most obvious way of extrapolating the approximation to the relativistic regime is to allow the fermion propagators to be relativistic, thereby including the constraints of special relativity and including the propagation of particles that may be excited out of the Dirac antiparticle sea. Horowitz and Serot 7, have demonstrated that the relativistic SCHFA diagrams (which includes a similar defining relation for the meson propagator), in the limit that the effects due to the Dirac antiparticle sea are neglected, reproduce the Dirac-Hartree-Fock scheme of Mittleman ‘) and Sucher lo). The potentials are interpreted as arising from particle exchange and in what follows the dashed lines represent exchange particles.

Fig. 1. The full fermion propagator in the self-consistent

Hartree-Fock

approximation.

430

A.F. Bielajew / Renormalization

The difficulties in this relativistic scheme arise from the divergences in the integrals represented by the closed loops of fig. 1. Within this diagrammatic framework, Brown et al. ‘r) have attempted to renormalize this equation in a nucleon-pion theory where, due to the y5 nature of the interaction, the simply closed baryon loop of fig. 1 vanishes when the matrix indices are traced. However, we use a representation of the fermion propagator that permits renormalization in a direct fashion following conventional methods. Moreover, the Lorentz covariance of our equations is manifestly evident through all stages of the calculation. If one neglects the exchange diagrams in fig. 1 we are left with the self-consistent Hartree approximation (SCHA) r2). The relativistic generalization of this approximation has been examined in detail by several authors 3*13).Also known as the “self-consistent summation of tadpole diagrams”, this approximation gives physically reasonable results for the nucleon-scalar meson-vector meson nuclear field theory of Walecka I) and is closely related to the mean-field approximation for the same field theory in which the boson fields are treated as classical functions r4*r3).

=

+ CTC

Fig. 2. The full fermion propagator in the self-consistent Fock approximation.

In this paper we shall investigate the other sector of the relativistic theory represented by the self-consistent summation of exchange diagrams without the tadpoles as depicted graphically in figs. 2a and 2b. Fig. 2a is a graphical representation of Dyson’s equation 14) that sums proper self-energy insertions to all orders in perturbation theory. The cross-hatched circle is the renormalized fermion selfenergy insertion, -iJ*, in this approximation. The double solid line is 8, the renormalized fermion propagator in this approximation, while the single solid line is its non-interacting counterpart, is”. The dashed line denotes iA’, the boson propagator. The self-energy will be shown to be finite after regularization by counterterm correction (CTC). A perturbation expansion of the fermion propagator is presented in fig. 3 for this approximation. We note that all possible disjoint and nested exchange diagrams are generated in which the boson lines do not “cross”. Crossed meson lines, which we know to appear in the full field theory, are neglected in this approximation. For obvious reasons this approbation is dubbed the self-consistent Fock approximation (SCFA). We postpone discussion of the full SCHFA to a future publication. Indeed, the enormous complication of mixed diagrams (including both exchange and tadpole subdiagrams) presents a formidable challenge. However, that is the larger problem to be solved and the SCFA is a stepping stone in that direction. Nevertheless, the SCFA may have some application

431

Fig. 3. A perturbation

expansion of the full fermion propagator in the self-consistent mation.

Fock approxi-

as the SCHFA in the nucleon-peon system or for the investigation of non-p~rturbalive effects in strong-~oupl~ng relativistic systems. The goal of this present work is to establish an unambi~ous renormalization scheme based upon CTc’s that we may define via a finite number of counterterms to the lagrangian. In fact, the procedure we shall develop is technically no more complex than the renormalization of exchange diagrams in lowest-order perturbation theory. Yet, it is remarkable that the properly defined coun~erte~s renormalize nested exchange diagrams to all orders in perturbation theory. Moreover, the technique explicitly exhibits Lore&z covariance. As a vehicle for this study we shal1 use the simplest relativistic fermion-boson field theory, a neutral scalar field (G) coupled in renormalizabIe, Lorentz-covariant fashion to the fermion (I@).For nuclear matter applications, we may extend this model to the Walecka nuclear field theory ‘>by including the neutral vector meson, V,, via the ~~~ interaction. Given the formalism we shall establish, this is a straightforward task I’), However, for brevity, we consider only the fe~ion-neutral scalar meson model. The c~cuIationa1 framework is completely specified by the l?gra~gian density

where ~(~~) is the mass of the fermion(boson), g,, is the dimensionless coupling constant, our system of units is h = c = 1, we use the (11 li) metric, and we follow the conventions and notations of Sakurai 16). The non-interacting fermion propagator is given by

432

A.F. Bielajew / ~eno~al~z~tio~

where E is a smal1 positive infinitesimal. The non-interacting given by

boson propagator

A”(g)=-(q2+m~-i~)-1.

is

(1.3)

Aside from the momentum conservation and integration prescriptions, the only other Feynman rule we require in this approximation is the association of a factor ig, at each fermion-boson vertex. In the numerical work that follows we make the connection to nuclear physics by giving the nucleon mass (939 hCeV) to the fermion and associating the CTwith the broad c - 2rr resonance at 520 MeV as given in the Walecka model ‘). The coupling constant, g,, in this case has been determined experimentally by Bryan and Scott r7) by phase-shift analysis of nucleon-nucleon scattering data and we use their value of g:/4r = 8.19. In sect. 2 of this report we discuss the renormalization technique and derive the explicitly finite equations for the SCFA. In sect. 3 we generalize to a uniform, finite density system of fermions and prove that the approximation remains finite at all densities with the renormalization terms as defined in the vacuum. In sect. 4 we present the numerical analyses and results of computation for the vacuum. Finally, in sect. 5 we discuss some technical difficulties, in particular, the presence of anomalous analytic structure in the fermion propagator that arises as a result of the approximation.

2. The renormalization technique for the SCFA In functional form, the Feynman equations of the SCFA:

diagrams of fig. 2 represent

RP>=S”(P)+sYP)~*bm, I -f:*(P)

=$4

the following

(2.la)

2

d”q s”(q)A”(q -p> +CTC

,

(2Sb)

where we shall use the dimensional regularization scheme of ‘t Hooft and Veltman ‘*) and it is understood that we shall take the limit y1+4 at the end of the calculation. This represents a complicated, non-linear integral equation in which the integral diverges. We proceed in-two steps. First we find CTc’s that render the integral finite and then we solve the self-consistent equations represented above. Before renormalizing the fermion self-energy we must assume a general form for the fermion propagator because we are to solve for it in a self-consistent fashion. Since we are to replace this general form in eq. (2.lb) and subsequently renormalize the integral via the CTC’s, it would be advantageous to use a form that permits

A.F. ~~elaje~ ,i ~en~~al~z~~~an

433

considerable analytical reduction (preferably the integral over 4) using standard techniques (dimensional re~la~zation and parametric integral representation) that are prevalent in the perturbation series c~culations. If such a form for S man~estly exhibits the proper Lorentz structure and the calculation retains this feature through the intermediate steps, we might readily identify the necessary renormalization terms needed to supplement the original Lorentz-covariant lagrangian. Fortunately, such a form for S is known. From the s~rnet~ properties of the la~angian, we may derive a spectral representation of the full fermion propagator in our field theory. This is done in many standard texts [see e.g. ref. I’)], and in our notation we write

where Zz is the wave-function renormalization constant while cy’ and p’ are weight functions that measure the contributions from intermediate states in the fermionboson continuum to the fermion propagator. These quantities have the follo~ng properties:

(2.3a)

(2.34

CY’(--YY~~) and /3’(--m*) are real,

(2.3d)

which are derived in ref. Is>.We may divide through by 2~ to write the renormalized fermion propagator

&P)=

dm2 i~~(-~2)-M~(-~2) p2+M2--iE

’

(2.4)

where we have redefined the spectral functions to absorb the factor of Zz, That is,

a(-m2) = d(-m2}fz2,

(2.5a) (2.5b)

434

A.F. Bielajew / Renormalization

With this representation of the fermion propagator we may calculate the selfenergy by use of eqs. (2.lb), (1.3) and (2.4) and we find that .2

00

if*(p)=3

o

I

dm2

J

d”q

+CTC.

(2.6)

The great advantage -of the use of the spectral representation is evident at this stage. Aside from the added complication of the integral over the spectral functions, the above equation is no more difficult to treat than lowest-order perturbation theory which can be recovered by setting a (-m”) and p (-m 2, equal to zero. By power counting eq. (2.6) is linearly divergent in the limit II + 4 but we assert that it can be regularized by mass and wave-function renormalization of the form CTC=-(Z;l

-l)(@+M)-SM.

(2.7)

Z2 and SM are uniquely specified by requiring that the self-energy have the following structure: f*(P)

= ($rtM2NP,P2)

be finite and

(2.8)

3

where R is finite and, by Lorentz covariance, it can only depend on @ and p2. This corresponds to a subtraction of the first two terms in a Taylor series expansion of f* in powers of (ip+M). The first subtraction, SM, guarantees that we measure M to be the physical fermion mass while the second subtraction with coefficient (2,’ - 1) guarantees that the pole of the renormalized fermion propagator at ig = -A4 has a residue of -1. Also, by Lorentz covariance, the self-energy must have the form T*(p)

= i@(p2)+Mu(p2),

and we find, by application of dimensional regularization ation I’), that 2

a(p2)+C=s

r

J

J

and integral parameteriz-

1

00

0

(2.9)

dm2

dx [~(m2-M2)+f?(m2-(M+m,)2)a(-m2)]

0

M2(1-x)+m%+p2x(1-x) X10gm2(1-x)+m2~-M2x(1-x)’ b(p2)+C=

-g

2 t?cr

J

cc

o dm2

J

(2.10a)

1

dxx[6(m2-M2)+O(m2-(M2+m,)2)/3(-m2)]

0

M2(1-x)+ms +p2x(1-x) xlog m 2 (l-x)+mb--M’x(l-xX)’

(2.10b)

A.F. BiE~ajew / ~en5~a&izaii~n m

435

1

dm2 dx C=&M2 J’ 50 0 ~x(l-x)[(l+x)S(m~-M~)+~(m~-(M+m~)~)(cu(--m~)+xp~--m~~)1 m2(1--n)+& -M”x(l -X)

(2SOc)

These represent an explicitly finite set of integral representations for _I?*in terms of the spectral functions, CIIand p, that are assumed to be convergent enough to allow the integrals to be done over the semi-infinite range. This is, indeed, the case and we discuss their asymptotic forms shortly. We may now write self-consistent integral equations for the spectral functions by use of Dyson’s equation as given by eq. (2.la) and, therefore, s”(P, a,@)=

wwl-l-~*(P, ff,m-"9

(2.11)

where, for emphasis, S and j* are written as functionals of (Yand /3. The renormalizations guarantee the existence of the physical pole at ip = -M and proper behavior there and it can be easily demonstrated that the self-energy functions, a(p”) and b (p”), develop imaginary components for p2 c -(M + RZ~)~.What this accomplishes is the summation, to all orders, of nested and improper exchange diagrams. By improper is meant that the self-energy can be separated into two distinct pieces by cutting a single fermion line. For example, the 3rd, 5th, 6th and 7th diagrams on the right-hand side of the equation in fig. (3) are improper while the others represented are “proper”. Using -(&+M) for (So)-* and eq. (2.9) for f*, eq. (2.11) may be rewritten (2.12) Using the symbolic relation (X*k-l - 9(x-‘)~i?rs(x) and equating the two expressions for s” in eqs. (2.4) and eq. (2.12) yields self-consistent relations for the spectral functions, cy and p : I+ dq2> =AIm ( Pk2) > s-

4q24fm ( m23 fy?PI >

42(1+b(q2,~,p))2+M2~1+~(qz,,,p))2~

(2.13)

Again, for emphasis, the self-energy functions, a and b, are written as functionals of a and ~3 and eqs. (2.10a)-(2.10c) demonstrate the non-linear integral character of the self-consistent relations. We may obtain the asymptotic forms of CYand /3 from eq. (2.13) and it can be shown that the initial rise after the threshold at q* = -(A4 + mm)* behaves as (- [q2+ (M-I- rr~,)*])~‘~, while the large q2 behavior is qp2 log (q2). This large q2 behavior is sufficiently damped so that there is no additional ultraviolet divergence in eqs. (2.10a)-(2.10~). We have succeeded in deriving explicitly finite integral equations for the fermion self-energy and propagator in the SCFA, a summation to all orders in perturbation

436

A.F. Bielajew / Renormalization

theory of all exchange graphs in which the boson lines do not “cross”. Eqs. (2.4), (2.9), (2.10) and (2.13) completely determine the numerical problem to be solved. The calculation is self-consistent in the following sense - the spectral functions characterize the fermion propagator which itself determines the selfenergy, and the propagator and self-energy are themselves related through Dyson’s equation. The numerical solution of this equation shall be discussed in sect. 4. We now generalize our equations to a system with a finite fermion density. 3. The generalization

to a finite fermion density

In this section we generalize the SCFA to a uniform system with a finite density of fermions. We shall argue that to any order in the coupling constant, for any Feynman diagram (including all others not encompassed by this approximation), that if one has regulated all the divergences in the vacuum (at zero fermion density) then, as one generalizes to a finite fermion density, all the Feynman amplitudes must remain finite. We then prove this is true by making use of Weinberg’s theorem on asymptotic Feynman amplitudes ‘“). In the generalization to finite fermion density, the non-interacting (zeroth-order) fermion propagator is SO(P) =S%)+S!&) s;(p)

(3.la)

9

= (@-M)/(g2+M2-iE)

)

SOD(p)= -2~i(i~-M)S(p2+M2)eePo)8(kF-IpI).

(3.lb) (3.lc)

This form has been derived by Walecka ‘) and the meaning is as follows. S”, is the usual fermion propagator of the vacuum theory. Sk accounts for the propagation of real fermions in the Fermi sea (lp / s kF) and corrects for the Pauli exclusion principle in the propagation of virtual fermions in the Fermi sea. We see that the additional “density” piece cuts off 4-momentum flow by evaluating the frequency at the fermjon mass shell via the S (p* +M2)t9(po) factor and restricting 3-momentum flow to magnitudes below the Fermi momentum, kF. To prove our main point, consider an arbitrary Feynman diagram in the vacuum in which we have regulated all the divergences. As we generalize to finite fermion density, the SD’s can only restrict the momentum flow to the subdiagrams of the original. It follows, therefore, that if the overall vacuum diagram is regulated, then all internal divergences (associated with the subdiagrams) are also finite, and restricting momentum flow to these subdiagrams does not affect the convergence. This completes the proof. It is useful to see in detail how this works for an example and we demonstrate it for our approximation, the SCFA. Recall the vacuum exchange self-energy we have regulated in sect. 2. Now, for example, consider a typical fermion vacuum self-energy insertion as given in fig. 4. (We denote vacuum amplitudes by a “V”

437

Fig. 4. An O(g8) example of tbe fermion self-energy in the vacuum. A “V” represents amplitudes and the “x"'s indicate that this diagram has been renormaIized.

vacuum

and represent regulated amplitudes by an inscribed “x”,) If we now generalize to finite density, the self-energy is as given in fig. 5. (A single solid line represents iSo and a single solid line with a “D” is iSg.) Terms with factors of S~&J vanish because they are proportional to (~2+~2)~(~2+~z) by virtue of the renormalization of the vacuum self-energy as seen in eq. (2.8). This example may be extended to arbitrary order in the coupling constant and demonstrates the finiteness of the fermion self -energy at all densities.

Fig. 5. The generalization of fig. 4 to a finite fermion density. The solid line with a “D” is i$,.

We can determine the leading asymptotic form of the finite fermion density contribution to the self-energy at large momenta from Weinberg’s theorem on asymptotic Feynman amplitudes 20). In fig. 6 is a graphical representation of the finite density fermion self-energy defined with the vacuum amplitude subtracted. Since fermion current is conserved in this theory the fermion line runs continuously through it. However, since we have subtracted the vacuum amplitude from &, it must have at least one factor of Sg included in the fermion line. For the purpose of establishing the asymptotic behavior, since SL cuts off momentum as previously discussed, the large momentum flow through & is forced to run through the meson lines. Effectively, it is as if the fermion line is “snipped”. Weinberg’s theorem then predicts a large momentum behavior of O(pe2) for _$,. Of course, this may be multiplied by logarithms of p* but we neglect these for determining the leading behavior. Logarithmic factors do not affect the convergence of Feynman amplitudes. We now consider SD = S -Sv, the finite fermion density contribution to the fermion propagator. This is depicted in fig. 7. Again terms with factors of Szv

\ 1 I

I

Fig. 6. The fermion self-energy with the vacuum amplitude subtracted.

438

D

=

D

Fig. 7. The fermion propagator with the vacuum amplitude subtracted.

have vanished due to the (P’+M~)S(P~+M~) factor. Since & is O(p) for large momenta (again by Weinberg’s theorem) and .& is O(pP2), it follows from fig. 7 that So is O(gs4). Finally, this permits a more elegant proof that zr, is finite. By the expansion of SD in fig. 7 we obtain an equivalent expression for & as given in fig. 8. By the above stated asymptotic forms, .& is again seen to be finite. Again Weinberg’s theorem applied to the above equation yields an O(p-‘) behavior and this verifies our original assertion.

Fig. 8. The fermion self-energy rewritten.

Therefore, our approximation remains finite at a finite density of fermions and, indeed, we have asserted that this must be true for any appro~mation to the theory, no matter how complicated. 4, Numerical analysis and results The integrals over the Feynman parameter, x, in the eqs. (2.10a)-(2.10c) may be performed exactly and written as a finite sum of elementary functions. They may be found in standard tables of integrals. [See e.g. ref. “).] This was found to be necessary for numerical stability, especially in the threshold region, p’~ -(M + RZ~)~.These equations are too lengthy to state here and the expressions are strai~tforward, if tedious, to obtain 15). The self-consistent equations (2.13) for the spectral functions may be written symbolically as a=A(a,P),

(4.la)

P=B@,P),

(4.lb)

and the following iteration scheme was chosen fY"'=A(O,O),

(4.2a)

@‘O’=B(O, 0))

(4.2b)

A.F. Bielajew / Renormalization a (l) = A (a (‘I, p”‘)

p(1) =B(p,

439

,

(4.2~)

p(o)) )

(4.2d)

,@“-“+(1_y)p’“-2’),

(4.2e)

(y(n)=A(ya!(“-f)+(1-y)a(“-2),

p(n)=g(ycu(n-l)+(l-y)a,(“-2),yp(n-l)+(1_y)P(”-*)),

(4.2f)

for n 2 2. The initial values, o(O) and p(O), correspond to a summation of all the “simple” exchange diagrams, that is, the self-energy is calculated to lowest order, g;, in perturbation theory. However, even this approximation is non-perturbative in nature since the simple self-energies are summed to all orders in perturbation theory by Dyson’s equation (2.11). The higher-order spectral functions then use the lower order estimates as input. The “stability” parameter, y, in eqs. (4.2e) and (4.2f) was not necessary for convergence but a choice of y = 0.5 made convergence to one part in 10’ possible in 20 iterations. The choice of parameters was motivated in the introduction and in this case gtj167r2 = 0.6517 and m,/M = 0.5538. Fig. 9 depicts the numerical results, both the initial values and the converged results after 20 iterations. The initial rise after the threshold behaves as (m*(A4 + m,)*)l’*. The large m* behavior is me2 log (m2). A detailed systematic study of the structure of the spectral functions for varying parameters has yet to be undertaken. However, the utility of the spectral representation in deriving finite results in a complicated, non-perturbative approximation for a variety of field theories is evident.

Fig. 9. The spectral

functions-initial mJA4

values and converged = 0.5538.

results. In this case &162=

m is given in units of M

0.6517

and

440

A.F. Bielajew 5.

It is well-known structure

/ Renormalization

Technical difficulties: the problem that

in the complex

approximations

of ghosts

of this sort exhibit

p2 plane of the fermion

propagator,

anomalous

analytic

Brown et al. 11) found

an additional conjugate pair of complex poles, called “ghost” poles, in their nucleon-pion Hartree-Fock approximation and, indeed, they surface here as well. That ghosts appear is not unreasonable in view of Dyson’s equation (2.11). Ghosts must appear in conjugate pairs to preserve the real analyticity of the self-energy on the real p2 axis away from the boson production cut p2 < -(M + mm)2 and hence, they do not contribute to the spectral functions, a and p. Wilets 22) has discussed the ramifications of the appearance of ghosts and his discussion will not be restated here. However, it is apparent that ghosts must be included from the outset in a self-consistent calculation. This may or may not affect the renormalization arguments stated herein but we note that it is possible to construct a finite field theory if one associates the ghosts with an indefinite metric. [See e.g. Lee and Wick “).I However, the incorporation of ghosts introduces tremendous complications. If one wishes to allow for ghosts in this self-consistent approximation then, as demonstrated by Fishman and Gersten 24), these ghost poles generate “ghost cuts” which, upon iteration, generate other ghost cuts ad infinitum. Moreover, these ghost cuts are not branch cuts in the usual connotation for they delimit complex p2 plane that are non-analytic in nature.

entire

regions

in the

We reserve the treatment of that since the ghosts occur in discontinuity along the boson obtained the correct behavior of

ghosts for future consideration but we recognize conjugate pairs they do not contribute to the production cut, p2< -(M + RI,,)‘, and we have the spectral functions. We have, however, located

a conjugate pair of ghost poles which corresponds to

in the lowest

S”‘(p) where

the self-energy

= ([so(p)].

is calculated

order

I-i:“(p,

only to order

0,

non-perturbative

O))_’)

gt but is summed

calculation (5.1) to all orders

by the above. The results for the same coupling constant and boson mass are that the anomalous poles occur at p2= -2.526*i3.810 in units of A4. We may justify the approximation that we neglect the ghosts in our calculations by observing that they are somewhat removed from the physical areas of interest, the single-particle pole at p2 = -M2 and the fermion-boson continuum, p2 d -(M + m,)2. However, the solution of the ghost problem, or at least, the demonstration that they do not contribute appreciably to physically measured quantities is work that has yet to be undertaken. 6. Conclusion We have presented a technique that renormalizes the self-consistent summation of exchange diagrams, a non-perturbative approximation that sums, to all orders

441

A.F. Bielujcw / Renormalization

in perturbation

theory,

cross. By spectral

all the exchange

function

techniques

and wave-function counterterm relativistic fermion-scalar boson ized to other in many-body

renormalizable Hartree-Fock

strong coupling field theories constant may be an inadequate

diagrams

in which the boson

we are able to effect renormalization

lines do not by mass

correction. Although we have considered only a field theory, the technique may be easily general-

field theories. This technique may have application theory and may be useful for the investigation of in which a perturbation expansion in the coupling approximation to describe the physics. Specifically,

the renormalized amplitudes presented in this paper are similar to those of relativistic Hartree-Fock theory for the nucleon-pion system where there are no tadpoles or they may be generalized to study non-perturbative affects in QCD. We have generalized our approximation to a many-body relativistic fermion system. We have argued that as one generalizes any approximation to a finite fermion density, that the approximation remains finite if the vacuum has been properly regulated. We have demonstrated this by examples from our approximation. We have worked in a formalism that allows a representation of the fermion propagator in terms of spectral amplitudes based upon the real and assumed symmetries of the theory. The spectral functions can be obtained by simple numerical techniques. The appearance of ghosts signifies that either our approximation is inadequate or that we must generalize the theory to allow for the anomalous analytic structure that has no obvious physical significance. Until this technical difficulty is overcome, we are encouraged by the possibility that ghosts may not contribute significantly to measurable processes and that the discussion of higher-order approximations is more than an academic exercise. The author wishes to acknowledge the excellent tutelage Dirk Walecka and Brian Serot during his stay at Stanford their useful comments in the preparation of this paper.

and support of Profs. University as well as

References 1) 2) 3) 4j 5) 6) 7) 8) 9) 10) 11)

J.D. Walccka, Ann. of Phys. 83 (1974) 491 S.A. Chin and J.D. Walecka, Phys. Lett. 52B (1974) 24 S.A. Chin, Phys. Lett. 62B (1976) 263 S.A. Chin, Ann. of Phys. 108 (1977) 301 C.J. Horowitz and RD. Serot, Phys. Lett. 108B (1982) 377 C.J. Horowitz and B.D. Serot, Phys. Lctt. 109B (1982) 341 C.J. Horowitz and RD. Scrot, Stanford University, Institute of Theoretical Physics, Preprint ITP-718 A.L. Fetter and J.D. Walecka, Quantum theory of many-particle systems (McGraw-Hill, KY, 1971) M.H. Mittleman, Phys. Rev. A5 (1972) 2395; A24 (1981) 1167 J. Sucher, Phys. Rev. A22 (1980) 348 W.D. Brown, R.D. Puff and L. Wile& Phys. Rev. C2 (1970) 331

442

A.F. Bielajew / Renormalization

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