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DISPERSION RELATIONS IN QUANTUM

arXiv:hep-ph/9205236v1 26 May 1992EFI 92-17DISPERSION RELATIONS IN QUANTUMCHROMODYNAMICS1Reinhard OehmeEnrico Fermi Institute and Department of PhysicsUniversity of Chicago, Chicago, Illinois, 60637ABSTRACTDispersion relations for the scattering of hadrons are consid-ered within the framework of Quantum Chromodynamics. It isargued that the original methods of proof remain applicable.

Thesetting and the spectral conditions are provided by an appropri-ate use of the BRST-cohomology. Confinement arguments areused in order to exclude quarks and gluons from the physicalstate-space.

Local, BRST-invariant hadron fields are consideredas leading terms in operator product expansions for products offundamental fields.The hadronic amplitudes have neither or-dinary nor anomalous thresholds which are directly associatedwith the underlying quark-gluon-structure. Proofs involving theEdge of the Wedge Theorem and analytic completion are dis-cussed briefly.1To appear in the πN-NEWSLETTER No.

7

1. IntroductionDispersion relations for the scattering of hadrons have been formulated [1-3] and proved in the Fifties [4-9].

They are by no means simple generalizationsof the familiar Kramers-Kronig relations for the scattering of light [10,11].The presence of finite masses presents a formidable problem for obtaining therequired analytic continuations. Charges of various kinds give rise to non-trivial crossing properties, which lead to analytic connections of amplitudesfor quite different reactions.Even though they have been introduced a long time ago, dispersion re-lations have continued to play an important rˆole in the analysis of hadronscattering.

In a more general framework, the analytic properties of Green’sfunctions are the foundation for many important results and theorems infield theory.However, this analytic structure has not been discussed indetail within the framework non-Abelian gauge theories like QCD and, inparticular, in the presence of confinement.The original derivations of dispersion relations [4-9] are within the frame-work of the general postulates of relativistic quantum field theory of hadrons[12]. There is no need to specify the theory in detail.

The essential inputis locality, in the form of the existence of Heisenberg field operators, whichcommute or anti-commute at space-like separations, and which interpolatebetween asymptotic fields describing non-interacting physical hadrons. Inaddition, spectral conditions are very important for the proof.

The difficul-ties with multi-particle intermediate states, and with the analytic structureof the corresponding multi-particle amplitudes, are the main reason for thelimitations of general proofs in some interesting cases.It is the purpose of this note to consider hadronic dispersion relationswithin the framework of Quantum Chromodynamics (QCD). As a constraint2

system, this SU(3) gauge field theory of color is best quantized with the helpof the Becchi-Rouet-Stora-Tyutin (BRST) symmetry [13] in a state-space Vof indefinite metric, and in a covariant gauge like the Landau gauge, for ex-ample [14]. A priori, the space V contains quanta like ghosts and longitudinaland time-like gluons, which are unphysical even without confinement.

Usingthe nilpotent BRST operator Q, we can define a subspace of states which sat-isfy QΨ = 0. This is the kernel kerQ = {Ψ : QΨ = 0, Ψ ∈V} of the operatorQ.

For ghost number zero, the space kerQ can provide the basis for a physicalsubspace, provided we have completeness of the BRST operator [15]. This no-tion implies that all states Ψ ∈kerQ with zero norm are of the form Ψ = QΦ,Φ ∈V.

It is then easy to show that kerQ contains no states with negativenorm. We can define a cohomology space H = kerQ/imQ with zero ghostnumber, containing only states with positive definite norm.

All zero normstates in kerQ are contained in the subspace imQ = {Ψ : Ψ = QΦ, Φ ∈V}.As is well known, the cohomology space H provides a Lorentz-invariant def-inition of a physical state-space.Without completeness, states in kerQ with zero norm and zero ghostnumber could be made from ghosts and their conjugates (singlet pair repre-sentations of the BRST algebra) [14,16]. There are arguments for complete-ness [17,18], but we do not know of a general proof for four-dimensional gaugetheories like QCD.

Unless we have completeness, a consistent formulation ofthe theory seems to be impossible. In certain string theories, completenesshas been proven explicitly [15,19,20].In the Hilbert space H, the ghosts, as well as the longitudinal and thetime-like gluons, are eliminated in a kinematical fashion, and this is all thathappens in weak coupling QCD perturbation theory.

But in the full theory,we expect that all quarks and gluons are confined. With certain limitations3

concerning the number of quark flavors, one can give arguments that, fordynamical reasons, transverse gluon states cannot be elements of the co-homology space H [21,22]. Some more preliminary arguments also excludequark states [23].

Under these circumstances, H is a true physical Hilbertspace containing only hadronic states. Here we adopt this algebraic viewof confinement.

It is quite consistent with more intuitive pictures for thequark-gluon structure of hadrons. In particular, the existence of an approx-imately linear quark-antiquark potential follows from the same argumentsin a natural fashion [24,25].

Our arguments for confinement make use ofthe renormalization group, and are valid for zero temperature.At finitetemperature, a new, dimensionful parameter comes in, and there may bedeconfinement transitions.We assume here that exact QCD exists as a quantum field theory, or thatpossible embeddings in more comprehensive schemes are not of importancefor confinement and for scattering processes at energies well below the Planckmass. If local field theory is considered as a low energy limit of string theory,we may expect deviations from microscopic causality at very small distances,and hence corresponding corrections to dispersion relations.Since the S-matrix, as an observable operator, is invariant under BRST-transformations, it follows that the unitarity relations involve only statesfrom the subspace H, at least as far as matrix elements with respect tophysical states are concerned [14,16].

With the notion of confinement wehave adopted, this implies that only hadronic states play a direct rˆole inthe physical S-matrix. Furthermore, intermediate state decompositions ofhadronic matrix elements of products of BRST-invariant operators with zeroghost number require only a complete set of hadronic states which span thespace H.For the purpose of deriving hadronic dispersion relations, this4

implies that the spectral conditions remain the same as in the old hadronicfield theory. Of course, we assume here that there exist composite hadronstates in QCD.As we have mentioned, the locality of the Heisenberg field operators isthe basis for obtaining analytic properties of scattering amplitudes.

Theseoperators interpolate between asymptotic fields, which generate states of non-interacting particles [26]. Since we are interested in hadrons, we need to con-struct local operators related to these particles in terms of quark and gluonfields, which are the fundamental fields of QCD.

These hadronic, compositeHeisenberg fields are BRST-invariant, and they are asymptotically relatedto the corresponding non-interacting hadron fields. They can be obtained,under certain conditions, from the leading terms in the operator productexpansion [27] for the product of quark and antiquark operators (mesons),or for three quark operators (baryons).

The construction is not unique, butthere are equivalence classes of interpolating fields which give rise to the sameS-matrix, as in the case of fundamental fields.Local field operators associated with the center-of-mass motion of com-posite particles have been discussed extensively in the literature [28-30]. Itshould not be surprising that such fields exist, because locality does not implya point-like structure for the corresponding particles.

In quantum electrody-namics, the electron has charge and magnetic moment distributions, whichare described by the familiar form factors. Generally, in a relativistic fieldtheory, a given particle can be considered as a composite of an appropriateset of the other particles.

The composite structure manifests itself in theform of characteristic branch points for vertex functions and scattering am-plitudes. These structure singularities are caused by thresholds in crossedchannels of other amplitudes which are related by unitarity to the amplitude5

under consideration [31]. For loosely bound systems, the structure branchpoints can appear as anomalous thresholds [7,31-34] in the physical sheet ofthe relevant variable.In the case of hadrons in QCD, a new element comes in.

We do not haveordinary composite systems where the constituents are observable particles,but we have confinement.In principle, we may consider a picture wherenucleons and mesons are made up of quarks with rather large constituentmasses. But the relevant quark masses appearing in QCD are the currentmasses, which are very small in comparison with hadron masses, at leastas far as u- and d-quarks are concerned.The constituent masses wouldhave to be viewed as generated in connection with the confinement process.In this process, gluons play an important rˆole.

Formally, they come intoconsideration via the anomaly in the trace of the energy-momentum tensor.Estimates indicate, that the gluons actually give the dominant contributionto the nucleon mass [35]. In view of the situation as described, we cannotuse weak-coupling perturbation theory in order to argue for the existence ofcomposite operators for the hadrons, but we must rely on what is known ingeneral about operator products in local field theories [36].As far as structure singularities and corresponding, possible anomalousthresholds are concerned, the situation in the case with confinement alsodiffers from the conventional picture of a composite system with observableconstituents.

As explained above, hadronic amplitudes, as expressed in termsof local hadronic fields, have no thresholds associated with quarks or gluons.Consequently there are no non-hadronic structure singularities.In deriving hadronic dispersion relations on the basis of local, hadronicfield theory, we have generally considered the asymptotic condition as anadditional postulate. We do the same in QCD, where we use the condition6

essentially only in the physical subspace. Since the theory involves gaugefields, there is, a priori, no mass gap, and in its covariant form, QCD operatesin a state-space of indefinite metric.

The Haag-Ruelle arguments [37] forobtaining the asymptotic condition from the other postulates of the theoryare not applicable under these circumstances.In the following Section we give a brief account of the hadronic subspaceand of confinement. In Section 3, we discuss some relevant aspects of com-posite operators and of operator product expansions.

Section 4 is devotedto a very brief survey of the analytic methods used in proving dispersionrelations.This article can give only a brief sketch of the many problems involvedin deriving analytic properties of hadronic amplitudes in QCD. We hope topresent a more comprehensive report elsewhere.7

2. Hadronic SubspaceIn this Section we give a brief introduction to the construction of a phys-ical subspace with positive definite metric, which, in view of confinement,can serve as a space spanned by hadronic states exclusively.

Under thesecircumstances, the spectral conditions used in the derivation of dispersionrelations for hadrons remain the same as in the generic, hadronic field theoryused in the past.As explained in the Introduction, we consider QCD in a covariant gauge,and quantize in a space V of indefinite metric in accordance with BRST-symmetry. The self-adjoint BRST operator Q, and the corresponding ghostnumber operator Qc, form the algebra [13,14]Q2 = 0,i[Qc, Q] = Q ,(2.1)which can be used to generate a decomposition of V in the formV = kerQ ⊕Vu ,kerQ = Vp ⊕imQ .

(2.2)Here we have introduced the subspaceskerQ = {Ψ : QΨ = 0, Ψ ∈V} ,(2.3)andimQ = {Ψ : Ψ = QΦ, Φ ∈V} . (2.4)We notice that imQ ⊥kerQ with respect to the indefinite inner product(Ψ, Φ) defined in V. A priori, the subspace Vp is a candidate for a physicalstatespace, but it is not invariant under Lorentz transformations, nor underequivalence transformations, which leave the physics unchanged.

As is wellknown, one therefore uses the cohomology space H = kerQ/imQ, which is a8

space of equivalence classes. It is isomorphic with Vp.

A state Ψ ∈H maybe written symbolically as Ψ = Ψp + imQ, Ψp ∈Vp . We have ignored herethe grading due to the ghost number operator, since we are interested in thesector Nc = 0 as far as H is concerned.In order to assure a physical subspace H ≃Vp with positive definite met-ric, we must assume completeness of the BRST operator Q [15].

This notionimplies that all states with zero norm in kerQ are contained in imQ. Givencompleteness, it is easy to see that there cannot be any negative norm statesin kerQ.

It is not enough to have zero ghost number, because the singlet pairrepresentations of the BRST algebra (2.1), which include states of ghosts andanti-ghosts, must also be eliminated. There are arguments for the absence ofsinglet pairs in the dense subspace generated by Heisenberg operators, butin view of the indefinite metric, the extension to the full space V is delicate[17,18].

In certain string theories, completeness has been proven explicitly.In any case, without completeness of the BRST operator, a consistent for-mulation of QCD would seem to be impossible. From a mathematical pointof view, the actual sign of the definite norm in the cohomology space is aconvention.Given completeness, we use a simple matrix notation for the zero ghostnumber sector of V, with components referring to the subspaces Vp, imQ andVu respectively.

We writeΨ =ψ1ψ2ψ3,C =100001010,(2.5)where the self-adjoint involution C may be viewed as a metric matrix. Interms of components, the inner product in V is then given by(Ψ, Φ) = (Ψ, CΦ)C = ψ∗1φ1 + ψ∗2φ3 + ψ∗3φ2 ,(2.6)9

where the subscript C denotes an ordinary inner product. We see that forstates Ψ, Φ ∈kerQ, which are representatives of physical states, only the firstterm on the right-hand side of Eq.

(2.6) remains. Since Vp is a non-degeneratesubspace, we can define a projection operator P(Vp) with P † = P 2 = P. Forthe inner product of two states Ψ, Φ ∈kerQ, and with a complete set ofstates {Ψn} in V, we obtain then the decomposition(Ψ, Φ) =Xn(Ψ, Ψn)(Ψn, Φ) = (Ψ, P(Vp)Φ) =Xn(Ψ, Ψpn)(Ψpn, Φ) .

(2.7)We see that only a complete set of states {Ψpn} in the Hilbert space Vp ≃Happears in the sum. It may be replaced by the equivalent set {ΨHn}, where wecan write symbolically ΨHn = Ψpn + imQ.

Although the projection operatorP(Vp) is not Lorentz invariant by itself, the use in Eq. (2.7) is invariant.In our matrix representation, a BRST-invariant operator A, which com-mutes with Q, and leaves kerQ as well as imQ invariant, is of the formA =A110A13A21A22A2300A33.

(2.8)Given a state Ψ ∈kerQ, it follows that also AΨ ∈kerQ. With Eq.

(2.7),states Ψ, Φ ∈kerQ, and BRST invariant operators A and B, we have there-fore the decomposition(Ψ, ABΦ) =Xn(Ψ, AΨpn)(Ψpn, BΦ) ,(2.9)which involves only physical states [14,21,38,39].Proper unphysical states are characterized by QΨ ̸= 0, or ψ3 ̸= 0 in ourmatrix representation (2.5). They may well have components in Vp, but onecan always find an equivalence transformation which removes this component[38,39].

In general, the norm of these states is indefinite. It can be shown,10

that some unphysical states with positive norm are needed for the consistencyof the theory [21].In QCD perturbation theory, the space H consists of states correspondingto quarks and transverse gluons. But in the general theory, if we have confine-ment, only colorless states like hadrons should be included.

We understandhere confinement in this algebraic fashion. If the number of flavors NF inQCD is limited (NF ≤9), arguments can be given that gluons cannot be inthe physical subspace [21,22].

These arguments are based upon the existenceof superconvergence relations for the structure function of the gluon propa-gator, which provide a connection between short and long distance proper-ties. Within the same framework, we can also obtain an approximately linearquark-antiquark potential, because a dipole representation can be written forthe propagator, which has a weight function of the appropriate shape [24].While a linear potential may be a phenomenological reality for heavy quarks,we require an algebraic argument also for quark confinement.

A sufficientcondition for general color confinement in terms of the BRST-cohomologyhas been given by Kugo and Ojima [14], and discussed further by Nishijima[40]. So far, only approximate methods have been used in order to arguethat this condition is also necessary [23].

But if we accept the necessity, theconfinement of transverse gluons also implies the confinement of quarks.For the purpose of deriving dispersion relations in QCD, we take it forgranted that we have confinement in the sense that the physical state-spaceH contains only hadronic states. Under these circumstatences, quarks andgluons do not appear as BRST singlets, but form quartet representationsof the algebra, together with other unphysical states.

As we have seen inEq. (2.9), only hadronic states appear then in intermediate state decomposi-tions involving hadronic operators, and we have the same spectral conditions11

as used in the old derivations of dispersion relations within the frameworkof hadronic field theory. Under these circumstances, in the direct or crossedchannels of amplitudes, there appear no thresholds which are associated withthe quark-gluon structure.

Also anomalous thresholds related to this sub-structure do not exist, neither in the physical nor in the unphysical sheets ofthe relevant variable, since they are generated by crossed channel thresholdsof amplitudes related by unitarity to the one under consideration.2 We havea situation, which is quite different from the usual bound state system, wherethe constituents are physical particles which can contribute to intermediatestate decompositions as in Eq. (2.7).Although the quark-gluon structure of hadrons does not give rise to spe-cial singularities of hadronic amplitudes, it is expected to be of importancein the determination of the weight functions (discontinuities) in dispersionrepresentations.

To the extent that perturbative QCD is related to the weakcoupling limit (g2 →+0 , g = gauge coupling) of the full theory, we mayexpect to see evidence of the composite structure in regions of momentumspace where the effective gauge coupling is small as a consequence of asymp-totic freedom. In these regions, pertubation theory may be a reasonable toolfor the approximate determination of weight functions.Since there is noconfinement in perturbation theory, which is the extreme asymptotic limitg2 →+0 , the absence of quarks and gluons from the physical space H ofthe full theory should be related to the required hadronization.Our algebraic description of confinement should be in accordance withother, perhaps more intuitive approaches to the problem.

From the point ofview of covariant field theory, the mathematical question is always whether2A detailed discussion of the mathematical and physical aspects of structure singular-ities may be found in Ref. 31.12

or not a given excitation is in the physical cohomology space H. Only statesin H are observable and contribute directly to the singularity structure ofhadronic amplitudes.13

3. Local Hadronic OperatorsThe problem of dealing with composite particles in quantum field theorywas considered in the late Fifties [28,29].

The methods can be generalizedto gauge field theories with a state space of indefinite metric. It is possibleto define local field operators for stable composite systems, which interpolatebetween the asymptotic fields generating these bound states as incoming oroutgoing particles in a scattering process.Let ψ(x) describe fundamental fields of the theory.

We suppose that thereexists a stable, composite system, which has a rest-mass M, and quantumnumbers in accordance with those of a product like ψψ. Then we may asso-ciate a local operator field B(x) with the composite system.

The field B(x)could be defined as the limitB(x) = limξ→0ψ(x + ξ)ψ(x −ξ)F(ξ),−ξ2 < 0. (3.1)The space-like approach is convenient, but not essential.

The invariant func-tion F(ξ) is only of importance as far as it’s behavior for ξ →0 is concerned.Generally, the function is singular in this limit, in order to compensate forthe expected singularity of the operator product. In fact, it must be as singu-lar as the most singular matrix elements of this product.

Here we make theassumption that such maximal matrix elements exist. Otherwise, we wouldhave a situation where, for every matrix element with a given singularity,there exists a more singular one.

Given the existence of maximal matrix ele-ments, we generally have an equivalence class Kmax of functions F with therequired maximum singularity. Possible oscillations in the limit (3.1) can behandled by an appropriate choice of the sequence of points in the approachto ξ = 0.The limit (3.1) corresponds to the leading term in an operator product ex-14

pansion [27]. Such expansions are known to exist in many lower-dimensionalfield theory models.

They are expected to be a general property of local fieldtheories. In four dimensions, the existence of operator product expansions,and of local, composite operators like B(x) in particular, can be proven usingperturbation theory methods of renormalizable field theories [41].

But in thecorresponding exact theories, we still have to make the technical assumptionconcerning the maximal singularity mentioned above [36]. As explained inthe introduction, we should not rely upon QCD perturbation theory for thepurpose of deriving hadronic dispersion relations.Given the existence of local, BRST-invariant operators in QCD whichare associated with hadrons, we can write representations for amplitudes asFourier transforms of time ordered or retarded products of these operators.The Fourier representations are then the starting points for obtaining an-alytic properties.In order to give some more details, we consider brieflyan amplitude for the elastic scattering of hadrons in QCD.

For simplicity,we ignore spin and other quantum numbers, concentrating on the generalstructure of the S-matrix elements. Consequently, the following formulae arerather symbolic.

Let us define time-ordered products of basic fields in theformB(x, ξ) = Tψ(x + ξ)ψ(x −ξ),(3.2)orB(x; ξ1, ξ2, ξ3) = Tψ(x + ξ1)ψ(x + ξ2)ψ(x + ξ3),(3.3)with ξ2 < 0 and the distances ξi −ξj kept space-like. We assume that theseoperators have non-trivial hadronic quantum numbers, so that their vacuumexpectation value vanishes.Considering B(x, ξ), we suppose that there exists a hadron (meson) with15

mass M so that ⟨0|B(x, ξ)|k⟩̸= 0 for −k2 = M2, where |k⟩is the singlehadron state. The free retarded and advanced propagator functions ∆R,A(x−x′, M) can be used to define asymptotic fields Bin(x, ξ) and Bout(x, ξ) withthe help of the Yang-Feldman representation.

With⟨0|B(x, ξ)|k⟩= ⟨0|Bin(x, ξ)|k⟩= eik·xFk(ξ),(3.4)we introduce a function Fk(ξ) = ⟨0|B(0, ξ)|k⟩. Denoting the Fourier trans-form of Bin(x, ξ) by Bin(k, ξ), we can show that there are creation and de-struction operators likeB∗in(k, ξ)Fk(ξ)= b∗in(k),(3.5)which are independent of ξ and satify the usual commutation relations.

Inthis derivation, the completeness of states in V, which are generated by allasymptotic fields, including composite fields, has been assumed [28,29].In principle, we may consider asymptotic fields for unphysical excitationsin the state-space V of indefinite metric.The states generated by thesefields are not elements of the physical space H, the cohomology space of theBRST oprator. We associate asymptotic fields with the poles of time orderedGreen’s functions corresponding to non-negative eigenvalues of −P 2, where Pis the energy-momentum tensor [14].

We do not exclude here the possibilityof multipole fields.With the asymptotic fields (3.5), and the weak asymptotic conditionlimξ→0(Ψ, Bf(x0, ξ)Φ) = (Ψ, Bfin(ξ)Φ)(3.6)for all Ψ, Φ ∈V, whereBf(x0, ξ) = −iZd3xB(x, ξ)↔∂0 f ∗(x) ,(3.7)16

for any normalizable f(x) satisfying Kxf = (✷−M2)f(x) = 0, we can use thereduction formulae of Lehmann, Symanzik and Zimmermann [26] in order toobtain representations for hadronic amplitudes. For example, let us considerthe scattering of mesons with mass M. With the product of basic fields asdefined in Eq.

(3.2), we obtain a formula like⟨k′, p′|S|k, p⟩=1Fk′(ξ′)F k(ξ)1(2π)3Z Zd4x′d4x exp[−ik′x′ + ikx]Kx′Kx⟨p′|TB(x′, ξ′)B(x, ξ)|p⟩,(3.8)where −k2 = −k′2 = M2, and |p⟩, |p′⟩are single hadron in-states.Theright-hand side of Eq. (3.8) is independent of the relative coordinates ξ andξ′.So far, we have not taken the limit ξ, ξ′ →0.

This limit is necessaryin order to have the microscopic causality required for dispersion relations.Furthermore, the operator B(x, ξ) is not BRST-invariant for ξ ̸= 0. Only alocal limit like B(x) in Eq.

(3.1) is invariant. As suggested by representationslike Eq.

(3.8), and the properties of the operators B(x, ξ), we restate theassumption made in connection with Eq. (3.1) and suppose that the limitB(x) = limξ→0B(x, ξ)Fk(ξ)(3.9)exists.

It then defines a local hadronic Heisenberg operator, and it impliesthat the functions Fk(ξ) are elements of the equivalence class Kmax mentionedabove. If we now interchange the local limit and the space-time integrationsin Eq.

(3.8), we obtain a representation of the S-matrix element in terms oflocal hadron fields :⟨k′, p′|S|k, p⟩=1(2π)3Z Zd4x′d4x exp[−ik′x′ + ikx]17

Kx′Kx⟨p′|TB(x′)B(x)|p⟩. (3.10)We can make further reductions in Eq.

(3.10) in order to get the formulaeneeded for the derivation of non-forward dispersion relations, and of forwardrelations for amplitudes with unphysical continuum contributions.Instead of taking the limit ξ →0 in Eq. (3.8), we can use directly the localoperator (3.9) and its asymptotic limit in order to construct the scatteringamplitude (3.10) with the help of the reduction formula involving the localcomposite field B(x).

Under these circumstances, the reduction method isused only within the physical subspace, where there should be no problemsresulting from the infra-red singularities of the theory. But even though thepath via Eq.

(3.8) appears to involve more assumptions, we think that it maybe of interest for the understanding of the hadronic, local limit.The reduction described above for the product (3.2) can be generalizedto operator products like (3.3), as well as to other products of fundamentalfields which can form color singlets. In this connection, it is important to notethat the Heisenberg fields interpolating between given asymptotic, hadronicfields are not unique.

There are equivalence classes of fields giving rise to thesame S-matrix.For field theories with a state space of positive definite metric, it canbe shown that locality is a transitive property: two fields, which commutewith a given local field, are local themselves and with respect to each other.We have equivalence classes of local fields (Borchers classes) [42]. The proofinvolves the equivalence of weak local commutativity and CPT-invariance[43], as well as the Edge of the Wedge Theorem [7].

It is then possible toshow that different fields in a given class, which have the same asymptotic18

fields, define the same S-matrix.Given special rules for the transformation of ghost fields under CPT, wecan define an anti-unitary CPT-operator in the state-space V of QCD [14].Together with the other postulates of indefinite metric field theory, this thenleads to the existence of equivalence classes of local Heisenberg fields in QCD.In particular, the hadron fields B(x), defined by different versions of thelocal limit, are in the same equivalence class, as are corresponding productsinvolving basic fields as factors. This is a consequence of the locality of thebasic fields in QCD.

As long as the different composite fields B(x) have thesame quantum numbers and the same in-fields, they give rise to the samephysical S-matrix in the subspace H of hadron states.19

4. Methods of ProofIn previous Sections, we have explained that one can define local Heisen-berg Operators for hadrons in QCD.

We have seen that BRST methods allowfor the definition of a physical subspace H of the general state-space V ofQCD. Given confinement, the Hilbert space H contains only hadronic states.With these features of QCD, we can proceed to derive dispersion relationsusing the methods developed within the general framework of local, hadronicfield theory.

In the following, we recall briefly some of the essential mathe-matical steps in the proof of dispersion relations for forward scattering, andfor finite values of the momentum transfer.Dispersion relations for the forward scattering amplitudes of reactionslike pion-pion scattering and pion-nucleon scattering can be derived rathersimply by using the gap method.3 As a simple model, let us consider thescattering of massive scalar particles. In terms of local Heisenberg fields, theforward amplitude has the representationF(ω) =Zd4xeiωx0−i√ω2−µ2ˆe·⃗xθ(x0)χ(x0, |⃗x|) ,(4.1)whereχ(x0, |⃗x|) =i(2π)3⟨p|j(x2), j(−x2)|p⟩,(4.2)with j ≡(✷−µ2)φ.If we write, with r = |⃗x|,F(ω) =Z ∞0drF(ω, r) ,(4.3)we find that, for the relevant values of r, the function F(ω, r) is analytic inthe upper half of the complex ω-plane, because the integrand in Eq.

(4.1)3The gap method was introduced in Ref. 4 (see the appendix, in particular).20

has support only in the future cone. As a Fourier transform of a tempereddistribution, it is bounded by a polynomial.

We ignore a possible polynomial,which can be taken care of by subtractions, and write a Hilbert representationfor F(ω, r). This representation involves an integral with the weight functionImF(ω + i0, r) along the real ω - axsis.

As a consequence of the spectralconditions, the weight function vanishes in the gap −µ < ω < +µ. But for|ω| ≥µ, we can perform the r - integration on both sides of the Hilberttransform.

Using the crossing symmetry for the neutral, scalar model, weobtain the dispersion relationF(ω) = 2ωπZ ∞µ2 dω′ImF(ω′ + i0)ω′2 −ω2. (4.4)Although the arguments sketched above ignore many fine-points, theyshow directly how locality (microscopic causality) and simple spectral con-ditions translate into the analytic properties required for the validity of dis-persion relations.

The generalization of the gap method to cases with singleparticle states, like the nucleon in pion-nucleon amplitudes, is straightfor-ward. We simply remove the one-particle contribution by the appropriatefactor, and later regain it as a pole term in the once-subtracted dispersionrelation.The real coefficient of the single-nucleon term can be identifiedwith the pion-nucleon vertex function on the mass shell [5].

A proof involvesapplying the gap method also to this vertex function in a nucleon channel.For reactions involving charged particles, like π±p - scattering, we have non-trivial crossing relations in the sense that the physical amplitudes for π+p- and π−p - scattering are different boundary values of the same analyticfunction, which is regular in the cut ω-plane, except for the single nucleonpole.For reactions like ππ- and πN- scattering, we can prove near-forward21

dispersion relations with the help of the gap method. These relations in-volve the derivatives of amplitudes with respect to the momentum transfert, evaluated at t = 0 [2].

But for amplitudes with fixed, finite momentumtransfer, more sophisticated methods must be used. In these cases, we havecontinuous unphysical regions.

The same is true for forward amplitudes forreactions like nucleon-nucleon scattering, where the crossed channel involvesnucleon-antinucleon scattering, and has continuous contributions from stateswith two or more mesons.The natural mathematical framework for the derivation of these disper-sion relations is the theory of functions of several complex variables. Twoaspects of this theory are of fundamental importance for our purpose: 1.

TheEdge of the Wedge Theorem, and 2. the existence of Envelopes of Holomor-phy.In order to describe the Edge of the Wedge Theorem [7], we use an ex-ample involving one complex four-vector.Suppose amplitudes, like thosefor π+N- and π−N- scattering, are given as Fourier transforms of tempereddistributions with support in the future or the past light-cone respectively:F±(K) = ±i(2π)3Zd4x e−iK·xθ(±x0) ⟨p′|j†(x2), j(−x2|p⟩. (4.5)Here 2K = k + k′, and k + p = k′ + p′.As a consequence of locality,the functions F±(K) are analytic in the tubes -(ImK)2 > 0, ImK0 > 0or ImK0 < 0 respectively.

For real values of the four-vector K, outside ofthe physical regions for both reactions, there is a domain R where the twofunctions coincide. This is a consequence of the spectral conditions, as maybe seen by making a decomposition of the absorptive parts with respect to acomplete set of hadron states.

Given the situation as described, the Edge ofthe Wedge Theorem implies that there exists a complex neighborhood N(R)22

of the real domain R, where both functions are analytic and coincide. Hencethere exists a unique analytic function F(K), which is regular, at least, inthe union of N(R) and the region -(ImK)2 > 0.

It is important to note thatN(R) contains all points with sufficiently small, space-like imaginary part,which are not points of the original tubes. The physical amplitudes F±(K)are boundary values of the general analytic function F(K) in the appropriatereal regions.

In many cases, the domain of analyticity obtained from the Edgeof the Wedge Theorem is not yet large enough for dispersion relations, but itgives an analytic connection between the two physical amplitudes, and hencea meaning to the crossing relations.The case described above is only a simple example of the Edge of theWedge theorem.It has been generalized in many ways.4For the proofof dispersion relations at fixed momentum transfer t, we have used it forfunctions of two complex four-vectors, together with an original domain ofanalyticity of the form W ⊗W, where W is the tube -(ImK)2 > 0 usedabove [7].The other essential tool for the derivation of non-forward dispersion re-lations is analytic completion. For functions with two or more complex vari-ables, we have the remarkable situation that, for many domains D, all func-tions, which are holomorphic in D, can be continued into a larger domainE(D), the envelope of holomorphyof D. This envelope is a purely geo-metrical notion.

The basic, generic tool for the construction of envelopes ofholomorphy is the Continuity Theorem, which has been used in Ref.44 to givea complete construction of E(W ∪N(R)), where W ∪N(R) is the domain of4The Edge of the Wedge Theorem has many applications beyond the problem of dis-persion relations. In the literature, one can find elaborate explanations concerning theorigin of the name.

In fact, while working on the problem in Princeton in 1956-57, we(BOT) called it Keilkanten Theorem, which was simply translated for the publication [7].23

analyticity described in the example for the Edge of the Wedge theorem givenabove. On the other hand, in Ref.7, we have used a subdomain, which is ageneralized semitube, and for which the envelope is well known.

This gives aregion of analyticity which is large enough for most purposes. For example,it touches the full envelope at points of interest for the nucleon-nucleon scat-tering amplitude.

The boundary of an envelope of holomorphy can often beexplored with the help of properly constructed examples of analytic functions[7].For problems involving one complex four-vector, and domains of the formW ∪R considered above, one can obtain the region of analyticity correspond-ing to the envelope of holomorphy of W ∪N(R) with the help of methodsfrom the theory of distributions and of partial differential equations. Theresulting Jost-Lehmann-Dyson representation has been discussed widely inthe literature [45,46].

It can be viewed as an elegant method to obtain theenvelope of holomorphy for the example we have considered.As is well known, an elaborate proof of dispersion relations for amplitudeswith fixed values of the momemtum transfer has been given by Bogoliubov,Medvedev and Polivanov [6]. This proof also makes use of distribution meth-ods and other tools.The actual proof of non-forward dispersion relations starts with the Fourierrepresentations (4.5).

A new variable ζ is introduced, which corresponds tothe squared mass of the projectile for the actual physical amplitude [47]. Forreal, and sufficiently negative values of ζ, the amplitude has cut-plane ana-lyticity in the energy variable, so that we can write a Hilbert representation.The problem is then to show that both sides are analytic functions of ζ, andthat the domain of analyticity includes the physical point ζ = µ2.

For theleft-hand side, we can use the domain E(W ∪N(R)) discussed before. For24

the right-hand side, the required analytic properties can be obtained by amore extensive use of locality and spectral conditions for the absorptive partoccurring as a weight function in the Hilbert representation. The tools areagain the Edge of the Wedge Theorem, and the envelope of holomorphy ofthe domain D ⊗D, where D = W ∪N(R) is the domain described above.The methods of analytic completion make it possible to prove dispersionrelations for many binary reactions and vertex functions.

For processes likeππ- and πN-scattering, for example, the proofs are valid for restricted valuesof the momentum transfer: −t = ∆2 < ∆2max, with∆2max = 7µ2 and ∆2max = 8µ232m + µ2m −µ(4.6)respectively.5These limitations have no real physical meaning, as can beseen with the help of models which are unphysical, but satify all the assump-tions we have made [7,31,33]. Nevertheless, it is difficult to incorporate theinformation contained in the detailed structure of the intermediate state spec-trum.

Of course, the missing features are naturally contained in a generic,hadronic perturbation theory, but in QCD we may not want to rely on that.There are similar problems for forward scattering amplitudes with un-physical, continuous contributions. An important example is elastic nucleon-nucleon scattering, where the envelope of holomorphy leads to the limitationµ > (√2 −1)m, which is not satisfied for pion (µ) and nucleon (m) masses.The same limitation is obtained for the pion-nucleon vertex function in thepion channel, and for electromagnetic form factors of the nucleon.

Usingformal perturbation theory simulations, we find that the restriction is due tosingularities describing the composite structure of the nucleon with respect5Tables describing the limitations of proofs for many amplitudes, which we have pre-pared for the 1958 Rochester Conference at CERN, are still applicable. See Ref.

48. Thelimits are also listed in the appendix of Ref.

49.25

to physically non-existent particles with masses such that the simple spectralconditions are satisfied [31,33]. Again, a more exhaustive use of the unitaritycondition is required, but difficult to implement, in particular for intermedi-ate states with more than two particles.

In contrast, dispersion relations forthe pion-nucleon vertex function in a nucleon channel can be proven usingthe gap method [4,7]. As we have mentioned, they are of importance for acomplete derivation of the pion-nucleon relations.For amplitudes involving strong and electromagnetic interactions, we mayconsider dispersion relations involving their hadronic structure, treating theelectromagnetic interaction in lowest, non-trivial order.

Within this frame-work, we can prove dispersion relations for pion photoproduction and sim-ilar reactions [49]. The limitations in momentum transfer may be found inRefs.49 and 48.

There is also no difficulty in deriving a dispersion represen-tation for the electromagnetic form factor of the pion.The envelope of holomorphy E(W ∪N) for the amplitudes F(K) can beused in order to show that the real and imaginary parts of the correspondingamplitudes are analytic functions in momentum transfer, or in cosθ = 1 −2tK2.They are regular in the small or large Lehmann-ellipses respectively[50]. Consequently, there are convergent partial-wave expansions.

For theabsorptive parts of reactions like ππ- or πN- scattering, these expansionsprovide a representation of the weight function in the unphysical region,which is always present in dispersion relations for finite momentum transfer.Further discussions of pion-nucleon dispersion relations may be found inthe papers [51].An interesting proposal for the analytic structure of binary amplitudeshas been made by Mandelstam [52].The double dispersion relations areessentially based on the assumption that the singularities of the amplitudes26

are restricted to those expected on the basis of physical intermediate statesin the three channels s, t and u, where s + t + u = Σm2. As is evidentfrom our previous discussion, these representations have not been provenin general hadronic field theory, and hence in QCD.

They are known to becompatible with hadronic perturbation theory in lower orders. As mentionedbefore, hadronic perturbation theory may not be a valid approach as far asQCD is concerned.

However, it could provide a hint for the analytic structureof hadronic amplitudes.ACKNOWLEDGMENTSThis article owes its existence to the persistent, friendly persuasion byGerhard H¨ohler.We also would like to thank Harry Lehmann, YoichiroNambu and Wolfhart Zimmermann for helpful conversations and remarks.This work has been supported in part by the National Science Foundation,grant PHY 91-23780.27

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