Superrenormalizable Gauge and Gravitational Theories^{1}^{1}1Research supported in part by NSF Grant PHY 95310223, and Monbusho, Japan
[2cm] E. T. Tomboulis^{2}^{2}2email address:
Department of Physics
University of California, Los Angeles
Los Angeles, CA 900951547
[2cm]
Abstract
We investigate 4dim gauge theories and gravitational theories with nonpolynomial actions containing an infinite series in covariant derivatives of the fields representing the expansion of a transcendental entire function. A class of entire functions is explicitly constructed such that: (i) the theory is perturbatively superrenormalizable; (ii) no (gaugeinvariant) unphysical poles are introduced in the propagators. The nonpolynomial nature is essential; it is not possible to simultaneously satisfy (i) and (ii) with any polynomial series in derivatives. Cutting equations are derived verifying the absence of unphysical cuts and the Bogoliubov causality condition within the loop expansion. A generalized KL representation for the 2point function is obtained exhibiting the consistency of physical positivity with the improved convergence of the propagators. Some physical effects, such as extended bound excitations in the spectrum, are briefly discussed.
1 Introduction
In this paper we investigate 4dimensional gauge theories defined by nonpolynomial actions with an infinite number of derivatives. Specifically, we consider Langrangians of the form:
where is a transcendental entire function having an infinite series expansion in the covariant D’Alembertian , and some scale. Similarly, we consider gravitational theories with actions including terms of the form
where and are entire functions.^{3}^{3}3Additional structures involving higher than second powers of may be included, but will not be considered in this paper.
When the functions , or , are taken to be polynomials, these are the Lagrangians of the familiar higherderivative (covariant PauliVillars) regularization of gauge theory [1]. As it is wellknown, such regularization renders gauge theory superrenormalizable at the expense of introducing massive ghosts. It is easily shown that this will always be the case for any polynomial . Here we consider the question whether it is possible to choose nonpolynomial so as to obtain good UV behavior while avoiding the introduction of ghosts. Somewhat surprisingly, we find that there is a class of transcendendal entire functions, which can be explicitly constructed, and give a superrenormalizable theory, while, at least formally within the perturbative loop expanssion, maintaining unitarity and causality. Superrenormalizability and unitarity appear interconnected. The requirement that the function be entire, thus possessing no singularities anywhere in the finite complex plane, is absolutely crucial for this to be possible.
To avoid a potential confusion at the outset, let us stress that what is being considered here is not the expansion in derivatives (powers of momenta) of the nonlocal effective action resulting from integration over some of the fields of a local field theory. Note that such an effective action necessarily contains singularities corresponding to the thresholds for production of the integrated out degrees of freedom. By the same token it cannot define a unitary Smatrix solely in terms of the remaining fields appearing in it, since the integratedout fields still can occur in the intermediate state cuts.
Though no nonlocal kernels are explicitly introduced in our actions, the dependence of the argument of the nonpolynomial on derivatives does introduce an effective nonlocality. Actions with general nonlocal kernels are, of course, known to lead to problems with causality. The nonlocality due to transcendental entire functions with derivativedependent argument is, on the other hand, of a rather mild sort sometimes termed ’localizable’ in distribution theory. As we will see, many things work for our actions pretty much as for polynomial actions precisely because of the similar properties of polynomial and nonpolynomial entire functions.
The idea of nonpolynomial entire Langrangians as a natural extension of the usual polynomial ones is certainly not new. Efimov, in particular, pursued such investigations [3], mostly in the context of attempts to obtain finite scalar theories, some time ago.
In the context of relativistic particle mechanics, Kato [4] considered actions analogous to the field theory actions considered here. To the usual (gaugefixed) particle action , he adds a term , with some function. To this one must add Lagrange multiplier terms incorporating the constraints of reparametrization invariance. This is then analogous to the BRS action (2.1) below, with corresponding to . For appropriate choice of meromorphic function , he finds a class of theories which includes the open bosonic string. It might be that the theories considered here have some kind of underlying extended structure associated with them. In this paper, however, we study them as a field theory problem.
The contents of the paper are as follows. In section 2 the action for the gauge theory case is introduced. A brief review of some features of higher derivative regularization provides the motivation for the inroduction of nonpolynomial entire functions. The structure of the resulting interaction vertices is quite complicated, and is examined in section 3, with technical details relegated to Appendix A. Provided that the function satisfies appropriate asymptotic conditions, detailed power counting shows that only 1loop divergences occur in the perturbative loop expansion. These asymptotic conditions are supplemented in section 4 by the requirement of the absence of unphysical poles at tree level. A class of entire functions satisfying all the requirements is then explicitly constructed. After discussing the relation between Euclidean and Minkowski Feynman rules in section 5, we turn to the basic issues of unitarity and causality to any order in the loop expansion in section 6. The special nature of the vertices allows one to obtain a largest time equation and hence generalized Cutkosky rules, which, applied to physical amplitudes, give the unitarity condition equations. No gaugeinvariant unphysical poles occur in the intermediate states, whereas the cancellation of longitudinal and FP ghost gauge dependent excitations occurs as in the standard gauge theories and is explicitly verified. Similarly, the Bogoliubov causality condition equation is shown to hold. Details concerning the derivations are relegated to Appendices B and C. In section 7 we consider the 2point function and obtain a generalized KällenLehmann representation for it. This makes explicit how, in this type of theory, the absence of unphysical excitations can be consistent with the improved convergence of propagators. With slight modifications, the entire development can be repeated for gravitation, which, in fact, provides one of the main motivations for this study. This is done in section 8.
The coupling to matter is discussed, though not in explicit detail, in the concluding section 9. There is a variety of potentially rather interesting physical effects in these gauge theories, such as the appearance of bound extended excitations, due to the modified short distance behavior. These matters are also briefly discussed in section 9.
2 Action
Consider the Lagrangian^{4}^{4}4We use standard notation: .
(2.1) 
and denote the covariant and ordinary D’Alembertian, respectively. is a gaugefixing function, with a gaugefixing weighting function. Note that the corresponding FP ghost term , where is the infinitesimal transformation of with gauge transformation parameter , does not depend on . is a given function to be specified, and an arbitrary mass. The coupling can, of course, be absorbed in the definition of the function , but is more convenient to keep it explicit.
(2.1) is invariant under the BRS transformation:
(2.2) 
With , and the usual rescalings , the bare propagator is then given by
(2.3) 
Further definition of (2.1) hinges on the specification of the function .
Polynomial  Higher Derivative (HD) Regularization. In the HD regularization scheme [1], the function is chosen to be a polynomial, , of degree . With the weight also a polynomial, and if , and , straightforward power counting shows that, at finite , the only divergent diagrams are oneloop diagrams with and external gauge field legs and no external ghost legs. All other oneloop diagrams, and all IPI multiloop diagrams are superficially convergent. Superficially convergent multiloop diagrams may, of course, still contain subdivergences due to oneloop subdiagrams. The theory is thus rendered superrenormalizable.
To completely regulate the theory then, the remaining oneloop divergences must be regulated separately. Dimensional regularization is straightforward to implement and very convenient for this purpose. Alternatively, and perhaps more in the spirit of the original HD scheme, additional PauliVillars (PV) regulators may be used [1].
For discussion of renormalization, it is very convenient to note that by taking sufficiently high all gauge dependent divergences disappear; renormalization may then be performed by the addition of only gauge invariant counterterms. So at finite , the remaining oneloop divergences are formally manifestly gaugeinvariant for , since FP ghost field and vertex renormalizations are finite. They may be removed by adding the oneloop counterterm; it is important to note that the function does not get renormalized. These statements can, of course, be made rigorous only in the presence of appropriate oneloop regularization. Dimensional regularization works well. The introduction of additional PV regulators, on the other hand, requires considerable care to avoid conflicts with gauge or BRS invariance, and has been the subject of several recent investigations; for a review and discussion see Ref.[2], and references therein.
In HD regularization, where , the theory (2.1) is rendered superrenormalizable at the expense of introducing ghosts. Indeed, as it is evident from (2.3), the transverse part of the propagator acquires additional poles from the zeroes of the polynomial . Note that some of these will, in general, be complex. The residues of some of these poles will necessarily be negative (more generally, have a negative real part). This follows from the improved UV behavior of (2.3). Indeed, by the factorization theorem for polynomials and partial fraction decomposition one may write:
(2.4) 
The assertion that at least one must be negative follows immediately by multiplying (2.4) by and taking the large limit. More generally, the spectral function in the KällenLehmann representation for the dressed propagator must contain negative contributions and satisfy a superconvergence relation.
Entire transcendendental . The question we consider in this paper then is: is it possible to choose the function in (2.1) so that no unphysical poles are introduced while at the same time maintaining the (super)renormalizability of the theory?
It is, of course, clear from the above argument that the answer is no as long as is taken to be a polynomial of any finite order (fundamental theorem of algebra!). One, therefore, has to consider non polynomial functions. Now a polynomial is an entire function, i.e. holomorphic anywhere in the finite complex plane. This property is necessary for the action to be welldefined everywhere (including the complex domain needed for analyticity and unitarity considerations). The natural generalization of a polynomial possessing this property is a transcendental (i.e. nonpolynomial) entire function, which we will take to be. This means that it can be represented by an everywhere convergent power series about any point, in particular the origin:
(2.5) 
with . Infinite radius of convergence (in fact, absolute convergence) implies . The operator function is then defined through (2.5) as a power series in the covariant D’Alembertian , and gives a welldefined nonpolynomial action (2.1).
Recall [5] that the standard growth scale for entire functions is based on exponentials of powers as comparison functions: if is of order , then for arbitrary positive , and sufficiently large. (Polynomials are of order zero.) It may thus at first appear that controllable UV behavior would not be possible. The overall growth scale provided by , however, ignores any dependence of growth on the direction in which grows large. A more refined growth measure is obtained by defining^{5}^{5}5Let
3 Perturbative expansion and renormalization
With a transcendental entire function, as in eq.
(2.5), the action (2.1) now possesses, in
addition to the usual YM vertices,
an infinite set of interaction vertices. In an
obvious notation, suppressing spacetime and group indices
and with
, an point
dependent vertex is given by:
(3.1)  
where
(3.2)  
In (3.2), stands for the sum of all possible ways of distributing powers of in the positions among an ordered sequence, indexed by , of factors of , (see A.1). The total number of ’s among these factors is , and or since is at most bilinear in . The total number of such ordered sequences is then
(3.3) 
The structure of is examined in Appendix A, where it is explicitly reexpressed in terms of the function and its derivatives . (It is of course important that one be able to do this, so that the asymptotic behavior of (3.1) can be related to that of .) For arbitrary configuration of momenta carried by the legs of the vertex (3.1), is given through (A.9), (A.11)(A.14) as a sum of products of rational functions of momenta and or its derivatives. Our fundamental requirement is that behaves asymptotically for real values of its argument as a polynomial. The assumptions of the power counting theorem [6] are then satisfied.^{6}^{6}6The vertices (3.1), and hence integrands of graphs, are, in the terminology of reference [6], functions in the class . Let for some set of constants and fixed finite momenta , and with k growing arbitrarily large. By choosing the ’s and the growth of the vertex (3.1) along every hyperplane in the space of the vertex momenta can then be examined. As shown in Appendix A, in all cases the leading asymptotic behavior of is given by a sum of terms that grow at most either as^{7}^{7}7In the following, to avoid cluttering the notation we often write
(3.4) 
(3.5) 
where is the number among the legs of carrying momentum of order . Let
(3.6) 
Note that finiteness of the limit defined in (3.6) implies the requirement that exhibit at most polynomial behavior at infinity on the real axis. Detailed power counting (Appendix A) shows that UV divergences arise solely from terms with growth of type (3.4) provided . Then the superficial degree of divergence of a 1PI graph with loops, external gauge boson lines and no external ghost lines is:
(3.7) 
Thus only 1loop diagrams with or are superficially divergent. All other diagrams, i.e. 1loop graphs with , and all graphs with any number of external legs are superficially finite. Also, graphs with any number of external ghost legs are convergent for all . The theory is then superrenormalizable by power counting, and the 1loop divergences present are gaugeinvariant. Renormalization of (2.1) is thus performed very simply by the addition of only the gaugeinvariant 1loop counterterm:
(3.8) 
where is now the renormalized coupling fixed at some renormalization scale . With the customary rescaling
It is crucial for what follows that the function does not get renormalized, or, more precisely, the functional dependence on its argument is not altered under renormalization.
We still have to show that functions with the required properties can be found.
4 Construction of the entire function
In view of the form of the denominator in (2.3), it is convenient to define
(4.1) 
We require that the function be an entire transcendental function with the following properties:

is real and positive on the real axis, and has no zeroes anywhere in the complex plane, .

has the same asymptotic behavior along the real axis at .

There exists such that
for arguments in the cones:
(4.2)
Condition (i) is the requirement that no poles appear in the transverse bare propagator (2.3) other than the physical (positive residue) massless gauge boson pole. Reality of the action (2.1) is ensured. Condition (iii) ensures that the power counting requirements for superrenormalizability of the previous section are satisfied. The appropriate asymptotic behavior is imposed in compliance with (ii), and not only on the real axis but in conelike regions surrounding it. This is in fact necessary since amplitudes are defined as boundary values of complex functions on the real axis. Condition (ii) is not strictly necessary as far as power counting requirements go. Rigorous power counting is considered to be performed in Euclidean space.^{8}^{8}8But it can actually also be done directly in Minkowski space using Zimmermann’s trick [7]. It is, however, necessary if we are to obtain the usual formal identity in (asymptotic behavior of) Feynman rules for Euclidean and Minkowski . This is important in the derivation of unitarity cutting rules. (The relation between Euclidean and Minkowski amplitudes is discussed in the following section.)
The conditions (i)  (iii) lead directly to a general
form of . It is a basic result that an entire
function with no zeroes anywhere in the complex plane
can only be the exponential of an
entire function.Thus, if (i) were to be satisfied,
we must have:
where is entire,
and, from (iii), should exhibit logarithmic asymptotic
behavior in the region . Thus we arrive at
the form:
(4.3) 
with:
(a) a real polynomial of degree , and
,
(b) entire and real on the real axis, and
,
(c)
(4.4) 
We take for in (2.1):
(4.5) 
where is given by (4.3). Since is bounded in domains extending to infinity, it must be of order (Wiman’s theorem), and is of infinite order.
The absence of zeros requirement in (i) is now satisfied if we set . Since is an arbitrary parameter, this is always possible. We should, however, consider its meaning under changes of renormalization scale.
Assume that, at a given scale , we have . Now evolve to the value of where . (We make the ’naturalness’ assumption that, with (4.5) in (2.1), one of the couplings and is not unnaturally small or large relative to the other.) By RG invariance, physical quantities are unchanged under this change of the parametrization of the theory in terms of to one in terms of .
Another way of looking at this is to note that a specification of at one value of is equivalent to a specification of a RGinvariant scale corresponding to the renormalization of . Suppose a different specification is made, and evolving from one obtains . Then, by a familiar argument, ; and the scale must be rescaled by the same amount to keep the same numerical value. So two versions of the theory specified by and , respectively, for given , differ only by a change in mass scale.
We may indeed always assume, with no loss of generality, that we work with a renormalization prescription such that at some convenient renormalization scale . In fact, note that any split between a renormalized and a constant part (coefficient in (2.5)) in is a renormalization prescription. Since the modification to the action is relevant only in an UV regime set by the scale , it is natural to fix of order .
Choosing to renormalize at then, (i) is satisfied, and the bare transverse propagator has no additional poles. This will be extremely convenient in the following, in particular in deriving cutting rules and equations for unitarity and causality. It is clearly not essential, however, and we may choose a different renormalization point. The technical nuisance then would be that, if we want to avoid dealing with fictitious poles, we must work with cutting equations in terms of dressed propagators.^{9}^{9}9In general, cutting rules in terms of dressed propagators often become necessary in order to accomodate complications due to mass renormalization pole shifts, unstable particles etc., see [8]. Indeed, consider any other where (2.3) will have additional poles at such that:
(4.6) 
In fact, it will have an infinite number of (complex) poles since (4.6) must have an infinite number of roots (Picard’s little theorem). They are all, however, clearly unphysical since their position moves with and is actually driven off to infinity as . We know, of course, that they must cancel, at any fixed , by RG invariance. Thus, if in analogy to , occuring in the transverse bare 2point function, we consider the dressed counterpart
(4.7) 
occuring in the RG invariant transverse inverse full 2point function, cp. eq. (7.3) below^{10}^{10}10An explicit factor of has been factored out in the definition of the selfenergy in (4.7) compared to that in (7.1), (7.3)., we have:
(4.8)  
Here is the finite difference between the counterterms renormalizing the selfenergy at scales and , respectively  it provides the finite renormalization between and , and the shift responsible for the zeroes in the bare part, (eq. (4.6)), at scale . The equality (4.8) shows how these zeros in the bare part at scale cancel against the self energy contribution to reproduce their absence at . In short, the requirement (i) above may be replaced by the RG invariant statement that no zeros (4.6) survive in . But this is automatically satisfied once (i) is fulfilled at some renormalization scale . From now on we will assume that the coupling has been renormalized at scale .
Clearly, many examples of functions that satisfy the stated conditions on can be given. An obvious choice is to assume exponential falloff and take:
(4.9) 
Define equal angular sectors with common vertex at the origin:
(4.10) 
with . The sectors then divide the plane in sectors. Now elementary estimates show that, for any (arbitrarily) small , and larger than some number , one has:
(4.11) 
The cones in (4.2) then are the sectors and , with . Note that as increases decreases, but the total area of the angular sectors in which always occupies half of the total plane area.
5 Relation between Minkowski and Euclidean formulation
By construction, the function exhibits polynomial asymptotic behavior in certain directions in the complex plane, in particular the cones , eq. (4.2), surrounding the real axis. Being an entire function of infinite order, however, it must grow doubly exponenentially in some other directions, such as the evennumbered sectors (4.10) for as in (4.9). This raises the issue of the relation between Minkowski and Euclidean formulations.
We may follow the standard path of Euclidean field theory: the theory is defined through the Euclidean path integral for the action (2.1), correlation functions are computed, and finally continued to Minkowski space by analytic continuation in the external momenta.
For ordinary local (gauge) field theories (), this is sometimes purely formally justified as ’Wick rotation of the integration contour’ of the functional integral. But, in fact, actual rigorous justification at the nonperturbative level has only been obtained for ’simple’ theories.^{11}^{11}11These are the ’reconstruction theorems’ showing that Minkowski correlation functions obtained by analytic continuation will obey the Wightman axioms if the theory obeys the usual Euclidean axioms such as reflection positivity. There are, of course, no such rigorous results for dim gauge theories. Within the perturbative loop expansion, however, the procedure is indeed justified on a graph by graph basis. It may at first appear that this is no longer the case when we allow . But this is not so. Analytic continuation in the external momenta of the result of computation with Euclidean Feynman rules again formally agrees with the result of using Minkowski Feynman rules and Wickrotating integration contours according to the following procedure.
For convenience, we choose the gaugefixing weight function , and adopt the Feynman gauge . The basic point is that the vertices (3.1) and the factor in the propagator (2.3) do not contain any singularities. In the computation of some arbitrary graph then we proceed in a standard fashion introducing Schwinger parameters for the scalar propagator in each propagator (2.3). This allows one to successively perform the momentum integrals. The integral over internal momentum is of the form
(5.1) 
Here depends only on the Schwinger parameters , while depends on and also linearly on the other loop momenta , and external momenta . stands for the product of all vertex and factors, and thus consists of sums of products of polynomials and entire functions. We may then reexpress (5.1) as
(5.2) 
The Gaussian integral can now be performed by translation , scaling , and finally Wick rotation to obtain:
(5.3) 
The momenta again appear only quadratically and linearly in the exponent. Performing the differentiations and picking another loop momentum , we have the integration in the form (5.1). Continuing in this way all momenta integrations are then performed.^{12}^{12}12Some regularization, e.g. dimensional, or cutting an off the Schwinger parameter integration region, is, of course, implicitly used so that this series of steps be always well defined. Since in the UV regions all vertices behave as polynomials, any of a number of conventional schemes may be used.
We stress that the above is nothing but the standard use of Schwinger parameters. It goes through in the present context because the usual polynomial vertices are generalized to entire functions which again do not introduce any singularities in the integrands. The steps (5.1)  (5.3) may in fact be viewed here as a definition of the computational rules relating Minkowski to Euclidean Feynman rules.
6 Unitarity and Causality
As discussed in section 4, the treelevel propagator is arranged to have, at the chosen renormalization scale, no gaugeinvariant unphysical poles. We now turn to the issue of unitarity and causality to any order in the loop expansion. Again, with the convenient choice for the gauge weight function, the propagator (2.3) in configuration space is:
(6.1) 
where is the usual bare massless scalar propagator. has the decomposition
(6.2) 
where
(6.3) 
are the usual energy functions. (6.2) implies that obeys the KL representation.
no longer satisfies this decomposition at where the r.h.s. of (6.4) differs from by contact terms, i.e. terms proportional to , and its derivatives, resulting from the action of the derivatives in (6.1) on the functions in (6.2). Now the derivation of results such as unitarity and causality conditions via largest time equations rely on the decomposition (6.4). Equal times regions may then, in some cases, require special consideration as the contact terms present a technical, but for the most part innocuous, complication.^{13}^{13}13The precise treatment of such contact terms in field theory is, in general, regularization dependent. Actually (6.2), and hence (6.1) with given as (6.2), are strictly derived only for . This reflects, in the operator formalism, the arbitrariness in the definition of the product, which is completely specified only when its arguments are diffferent. The standard convention is that the l.h.s. and r.h.s. of (6.2), (6.1) also coincide in an infinitesimal neighborhood of equal times, with the l.h.s. defined independently as the appropriate Green’s function. Alternative conventions amount to the addition of local counterterms in the Lagrangian. For detailed discussion see [10].
Recall first the case of ordinary gauge theories, i.e. take . We write for (6.1) with . When the theory possesses gauge invariance, explicit consideration of equal time contact terms becomes unnecessary since the contribution of such terms must cancel in any physical amplitude, as indicated by the presence of gauges where they are absent. In particular, in the Feynman gauge, , all derivatives are eliminated, and one indeed has
(6.6) 
where , valid for all . For the ghost propagator , the corresponding equation follows trivially from (6.2) with . Given (6.6), one may proceed to derive [8],[9] the largest time equation, and hence cutting rules leading to unitarity conditions for physical amplitudes; and then note that, by virtue of the WI, these equations continue to hold if one replaces with (6.5) for arbitrary .
To follow the same procedure in the case we are considering here, when , appears at first somewhat problematic. Since the action of induces contact terms in all parts of the propagator, these cannot be cancelled by gauge invariance alone. (This, of course, reflects the fact that is an actual, gauge invariant modification of the usual gauge theory action.)
It is, however, not difficult to circumvent this problem. The trick is to use as the bare propagator, and include the factors in the vertices where the propagator line ends. More precisely, write (2.3) as
(6.7) 
and define vertices:
(6.8) 
Here are the perturbative vertices from the expansion of the action (1) (Section 3), with the ghostghostgauge boson vertex, and we redefined external sources by inserting factors. For this to work, it is, of course, crucial that and singularityfree for all ; we set . This assignment of factors to vertices is, of course not unique, but a convenient, symmetric choice.
Consider now diagrams constructed from propagators , and vertices , the FP propagator remaining unchanged. We will refer to these rules as the ’alternative’ rules. It is immediately seen that for any diagram contributing to any point function between arbitrary sources the same result is obtained with these rules as with the original rules ( propagator , vertices ):
(6.9) 
(Here the label stands for all polarization, Lorentz and group indices pertaining to the th leg; and different sources may be chosen for each leg.)
For matrix elements all legs, in addition to be truncated, must be put on massshell,^{14}^{14}14As usual with massless poles, to avoid onshell IR divergences, the massshell condition is taken with an infinitesimal mass, denoted . and all wavefunctions appropriately chosen. Now
(6.10) 
so all factors on external onshell legs are actually irrelevant; and the residue of the (dressed) gauge boson propagator at the pole:
(6.11)  
is the same for both sets of rules. (In (6.11) and denote the full propagators in the two sets of rules.) The residue defines the wavefunction renormalization constant . Each physical external wavefunction, in addition to being of physical polarization, must be normalized by a factor of in order to correctly account for the contribution of selfenergy insertions on external legs.^{15}^{15}15Note that these factors are needed to make the matrix gauge invariant whether or not the ’s are UV finite. Truncating and putting all legs on shell then, we have the equality for amplitudes:
(6.12)  
Note that (6.12) holds for arbitrary ’s of any polarization; and by the equality (6.11), it continues to hold for correctly normalized physical wave functions. Furthermore, (6.9)  (6.12) are also valid directly in the renormalized theory, since the counterterms in (3.8) are included in the set of vertices . Once more, the absence of singularities in , and the fact that its functional form does not change under renormalization, are crucial for obtaining the simple relations (6.9)  (6.12). These relations allow one to view any diagram as constructed according to either set of rules. One may then proceed pretty much as in the case of ordinary gauge theories.
Unitarity
We construct amplitudes using the alternative
rules in the Feynman gauge. The bare propagator
then satisfies the decomposition (6.6). The vertices
, eq. (6.8), are real for real momenta,
and given by entire functions possessing no singularities,
in particular no poles or cuts anywhere in the finite
complex plane. It follows (Appendix C) that the Veltman
largest time equation [8] holds, which in turn implies
the cutting equation (generalized Cutkosky rule):
(6.13) is a general cutting rule that applies to a single diagram, or to any collection of diagrams represented by the blob, and for arbitrary external wavefunctions and momenta. On the shaded side of the ’cuts’, each explicit factor of assigned to each vertex and propagator is changed to , and each in each propagator to . The cut blob on the right hand side stands for the sum over all possible cuts of the diagram, or of each diagram in the collection of diagrams represented by the blob. A possible cut is obtained by cutting propagator lines so that the vertices on the shaded (unshaded) side of the cut form a connected region containing at least one outgoing (ingoing) external line. The rules for cut lines are given by:
for gauge field and ghost lines, respectively. Although conveniently derived in terms of the alternative rules, the result (6.13), once obtained, may be trivially viewed either in the alternative or the original rules by use of (6.9). Note, in particular, that cut legs are on shell, so, from (6.10)
and there is no distinction between a cut and a cut propagator.
To establish unitarity equations one lets the blobs in
(6.13) include all diagrams to a certain order
that contribute to a given process, with all external
legs truncated and on shell. The physical
Smatrix is furthermore defined only between external
physical gauge bosons:
The label denotes physically polarized onshell gauge bosons. Note that since the Lagrangian (2.1) is real (hermitean), the rules for diagrams on the shaded side in (6.15) are indeed those for . In the sum over intermediate state cuts on the r.h.s. of (6.15), however, the cuts, as given by (6.14), include cuts over gauge bosons of unphysical polarizations, as well as ghosts. Therefore, physical unitarity will hold only if these unphysical contributions cancel leaving only a sum over physical transverse cuts given by:
arises from summation over only the two physical polarizations satisfying , and
(6.17) 
where is a timelike unit vector used to fix a timelike polarization direction, and
(6.18) 
Having established the cutting equation (6.15), as well as the Ward identities, eq. (B.1), however, the demonstration of the cancellation of unphysical cuts is actually no different than that in the case of ordinary local gauge theory. This is clear since this demonstration relies only on the gauge, or BRS, invariance of the action. It is usually stated at the formal level of the path integral independence of the gaugefixing term, which, of course, holds here as well. Since, however, we derived the cutting equation by means of the split (6.7)(6.8) on a graph by graph basis, one should, for completeness, check the cancellation also diagrammatically. We outline the derivation in Appendix B, which, starting from the WI (B.1), verifies explicitly (see (B.7)) that indeed the sum over cuts on the r.h.s. of (6.15) reduces to the sum over only physical cuts (6.16).
Causality
Again, working in terms of the
alternative rules, the validity of the largest time
equation implies (Appendix C) that the Bogoliubov
Causality Condition (BCC) [10], [8]
is satisfied:
The blob represents a diagram, or a collection of diagrams, contributing to the  point Green’s function, with external (truncated) legs, and with and legs joined at the spacetime points and by  and  point vertices , respectively.^{16}^{16}16(6.19) is actually shown with . and/or is the case of external source vertices at and/or . The cut blobs stand for the sum over all cuts with the positions of the two vertices at and as shown. (Again, though the equation is conveniently derived in the alternative rules, it is equally well viewed in terms of the original rules by (6.9) and (6.12).)
The physical meaning of (6.19) is as follows. The first term on the r.h.s. is a (set of) cut diagram(s) representing a contribution to the product , with the vertices at and both in the diagram(s) making up the factor of the product. We may now apply equation (6.19) to this diagram(s) for the factor. Iterating this procedure, the r.h.s. of (6.19) can be reduced entirely to a sum of two groups of terms: one group multiplied by and containing only cuts forcing positive energy flow from to , the other group involving the opposite combination.
The vertices at and act as (multileg) external sources and the legs emanating from them correspond in general to particles offshell. This is actually what gives the obvious intuitive meaning to the above physical interpretation of future directed positive flow since, of course, spacetime points cannot be precisely pinpointed by wavepackets representing particles near massshell.
Integrating (6.19) over and converts (6.19) to an equation (now entirely in momentum space) for a (set of) diagram(s) contributing to an  point amplitude.^{17}^{17}17The two vertices at and are now internal vertices. They may also be taken to be external vertices: if they are originally chosen as  and point vertices, respectively, then multiplication by the appropriate external wavefunctions and integration over and converts (6.19) to an equation for an  point amplitude. The l.h.s. is precisely the (set of) diagram(s) for the amplitude in question and is expressed by the r.h.s. in terms of cut graphs in what is in fact a dispersion relation in noncovariant form.^{18}^{18}18It is noteworthy that such a dispersion relation can be written for any individual diagram. In some cases this relation may be converted to a more conventional dispersion relation in some external Lorentz invariant as the dispersed variable. For scalar theories this is developed in [11]. In the presence of derivative interactions, however, some care must be exercised in converting (6.19) into a dispersion relation by integration over and . This is because (6.19) was strictly derived for . The action of derivatives at on the function factors, resulting into functions and function derivatives which give a finite measure contribution to the equal times integration region, must then be properly taken into account as explained in Appendix C. The important special case of the point function is considered below.
7 The 2point function
Consider (6.19) for , i.e. for the two point function betwen sources at . To lowest approximation, where the blob is a single bare propagator line joining and , (6.19) is nothing but eq. (6.4), sandwiched between arbitrary sources, in Feynman gauge (). Consider next (6.19) for the point function between sources at and including an arbitrary number of insertions of the selfenergy
(7.1) 
Summing over all insertions between sources of physical
polarization, so as to eliminate the physically inessential
longitudinal terms at the outset, one arrives at the
BCC equation:
In (7.2) the crosses indicate the sourses, and the shaded blob stands for the physical transverse full propagator given by