Analytic Structure of Amplitudes in Gauge Theories with Con Nement 1

Home , Reinhard Oehme

,La al Physics ess of Mathematic International Congr ted at the Conference on New Problems in the General articles of the HEP-TH-9412040 vited talk presen t Address aris, July 1994. ermanen P Based on an in 2 1 Sorb onne, P Theory of Fields and P e v olving or gen- v ust ha EFI 94-39 amplitudes amplitudes 2 al al MPI-Ph/94-52 1 e theory in t wn, that there are physic Unphysic h are asso ciated with con- It is sho y b e asso ciated with con ned artment of Physics ur Physik g-Institut - er hma wn that these functions m t, the analytic structure of amplitudes ago, Il linois, 60637, USA and Abstract ts, whic Reinhard Oehme ago, Chic ysical amplitudes whic 80805 Munich, Germany ysical elds. The corresp onding pro ofs of disp ersion - Werner-Heisenb Max-Planck-Institut f e quarks and gluons in QCD. ermi Institute and Dep , using propagator functions as an example. F wn that the analytic prop erties of Analytic Structure of Amplitudes alid. Anomalous thresholds are considered. They are re- oF ts, lik in Gauge Theories with Con nemen t, linear gauges, it is sho Enric University of Chic arian v h thresholds in ph or gauge theories with con nemen F states. are considered brie y eral, co singularities at nite, real p oin ned constituen no suc only the comp osite, ph relations remain v lated to the comp osite structure of particles. are the same as those obtained on the basis of an e ectiv is explored. It is sho

I. INTRODUCTION

The analytic structure of amplitudes is of considerable imp ortance in all

quantum eld theories, from a physical as well as a conceptional p ointof

view. It has b een studied extensively over manyyears, but mainly for eld

theories with a state space of de nite metric, and in situations where the in-

terp olating Heisenb erg elds are closely related to the observable excitations

of the theory. Within a relativistic framework of this typ e, the commutativ-

ityoranti-commutativity of the Heisenb erg elds, at space-like separations,

gives rise to tub es (wedges) of holomorphy for retarded and advanced ampli-

tudes, which are Fourier transforms of temp ered distributions. Lower b ounds

for the sp ectrum of eigenstates of the energy momentum op erator provide

real domains where these amplitudes coincide at least in the sense of dis-

tributions. One can then use the Edge of the Wedge theorem [1] to show

that there exists an analytic function which is holomorphic in the union of

the wedges and a complex neighb orho o d of the common real domain, and

which coincides with the advanced and retarded amplitudes where they are

de ned. Then the envelop e of holomorphy [2, 1 ] of this initial region of ana-

lyticity gives the largest domain of holomorphy obtainable from the general

and rather limited input. Further extensions require more exhaustive use of

unitarity [3], which is often rather dicult. Although the theory of functions

of several complex variables is the natural framework for the discussion of

the analytic structure of amplitudes, for sp ecial cases, like those involving

one complex four-vector, more conventional metho ds, like di erential equa-

tions and distribution theory, can b e used in order to obtain the region of

holomorphy [4, 5 ]. Many more technical details are involved in the deriva-

tion of analytic prop erties and disp ersion relations [6, 7], [5, 1, 8, 9 ], but the 1

envelop es mentioned ab ove are at the center of the problem.

It is the purp ose of this pap er to discuss essential asp ects of the analytic

structure of amplitudes for gauge theories, which require inde nite metric in

acovariant formulation, and for which the physical sp ectrum is not directly

related to the original Heisenb erg elds. Rather, these elds corresp ond to

unphysical, con ned excitations in the state space of inde nite metric. We

will b e mainly concerned with physical amplitudes, corresp onding to hadronic

amplitudes for QCD. We will often use the language of QCD. As examples

of unphysical Green's functions, we consider the structure functions of the

gluon and quark propagators. In all covariant, linear gauges, we show that

these generally cannot b e entire functions, but must have singularities which

can b e related to unphysical states. A preliminary account of some of our

results may b e found in [10].

In the framework of hadronic eld theory with p ositive de nite metric,

the derivation of analytic prop erties, and of corresp onding disp ersion rela-

tions, is on a quite rigorous basis, and uses only very general asp ects of

the theory.For gauge theories with con nementhowever, the derivation of

disp ersion relations requires several assumptions, whichmay not have b een

proven rigorously in the non-p erturbative framework required in the pres-

ence of con nement. We will discuss these assumptions in the following sec-

tions. They mainly concern the de nition of con nement on the basis of the

BRST-algebra, and the construction of comp osite hadron elds as pro ducts

of unphysical Heisenb erg elds.

In order to provide for the input for pro ofs of disp ersion relations and

other analyticity prop erties of physical amplitudes, we discuss in the following

paragraphs certain results for non-p erturbative gauge theories. For some of

these results, we can refer to the literature for detailed pro ofs, but wehave 2

to explore their relevance for the derivation of analytic prop erties and for the

identi cation of singularities. It is not our aim, to provide reviews of several

asp ects of non-p erturbative gauge theories. What we need is to show that

there is a mathematically well de ned formulation of con nement, whichwe

use in order to derive the sp ectral conditions, and a construction of lo cal,

comp osite elds for hadrons, from whichwe obtain the initial domain of

holomorphy. We also have to exhibit the assumptions made within this

framework.

For the purp ose of writing Fourier representations for hadronic ampli-

tudes, wemust consider the construction of lo cal, BRST-invariant, com-

p osite Heisenb erg elds corresp onding to hadrons. This problem requires a

discussion of op erator pro ducts [11 , 12 ] of elementary, con ned elds in a

non-p erturbative framework.

Since we need a manifestly covariant formulation of the theory,we con-

sider linear, covariant gauges within the framework of the BRST-algebra

[13]. Assuming the existence and the completeness [14] of a nilp otent BRST-

op erator Q in the state space V of inde nite metric, we de ne an invariant

physical state space H as a cohomology of Q. As a consequence of complete-

ness, which implies that all neutral (zero norm) states satisfying Q =0

are of the form = Q; 2V, the space H has (p ositive) de nite metric

[15, 16 ].

We assume that there are hadronic states in the theory, and take con-

nement to mean that, in a collision of hadrons, only hadrons are pro duced.

Within the BRST-formalism, this implies that only hadron states app ear as

physical states in H. At least at zero temp erature, transverse gluons and

quarks are con ned for dynamical reasons, forming non-singlet representa-

tions of the BRST-algebra in combination with other unphysical elds. In 3

the decomp osition of an inner pro duct of physical states, there app ear then

only hadron states. The same is true for the decomp osition of physical matrix

elements of pro ducts of BRST-invariant op erator elds, b ecause these op er-

ators map physical states into other physical states. In this waywe nd that

the fundamental spectral conditions for hadronic amplitudes are the same as

in the e ective hadronic eld theory. The absorptive thresholds are due only

to hadronic states.

There is, however, another category of singularities, which is related to

the structure of particles as a comp osite system of other particles. These

are the so-called anomalous thresholds or structure singularities. They were

encountered in the pro cess of constructing examples for the limitations of

pro ofs for disp ersion representations [18, 19 ]. These limitations are related

to anomalous thresholds corresp onding to the structure of a given hadron

as a comp osite system of non-existing particles, which are not excluded by

simple sp ectral conditions. But physical anomalous thresholds [20, 19 , 21]

are very common in hadronic amplitudes: the deuteron as a np-system,

and hyp erons as KN systems, etc. In theories like QCD, the imp ortant

question is, whether there are structure singularities of hadronic amplitudes

which are related to the quark-gluon structure of hadrons. We show that

this is not the case. Indep endent of p erturbation theory,we describ e how

anomalous thresholds are due to p oles or absorptive thresholds in crossed

channels of other hadronic amplitudes, which are related to the one under

consideration by analytic continuation into an appropriate lower Riemann

sheet [21]. Since, as explained ab ove, wehave no absorptive singularities in

hadronic amplitudes which are asso ciated with quarks and gluons, we also

have no corresp onding anomalous thresholds.

For form factors of hadrons, whichmay b e considered as lo osely b ound 4

systems of of heavy quarks, there are interesting consequences stemming

from the absence of anomalous thresholds asso ciated with the quark-gluon

structure. In a constituent picture, these hadrons can b e describ ed bya

Schrodinger wave function with a long range due to the small binding energy.

But in QCD, in contrast to the situation for the deuteron, for example, there

are no anomalous thresholds asso ciated with the spread-out quark structure.

However, there is no problem in obtaining a large mean square radius with

an appropriate form of the discontinuities asso ciated with hadronic branch

lines. In addition, there maybe hadronic anomalous thresholds which are

relevant.

Finally,we consider the analytic prop erties and the singularity structure

of unphysical (colored) amplitudes. It is sucient to discuss two-p oint func-

tions as examples. The structure functions of quark and transverse gluon

propagators are analytic in the k -plane, with cuts along the p ositive real

axis. This is a direct consequence of Lorentz covariance and sp ectral condi-

tions. In previous pap ers [22], wehave derived the asymptotic b ehavior of

2 2

these functions for k !1in all directions of the complex k -plane, and

for general, linear, covariant gauges [23 ]. With asymptotic freedom, they

vanish in these limits. Hence the structure functions cannot b e non-trivial,

entire functions. They must have singularities on the p ositive, real k -axis,

which should b e asso ciated with appropriate unphysical (colored) states in

the general state space V with inde nite metric. These states are not ele-

ments of the physical state space, but form non-singlet representations of the

BRST-algebra. 5

I I. CONFINEMENT

In this section, we brie y de ne the general framework for the later dis-

cussion of amplitudes and their analytic structure.

We consider quantum chromo dynamics and similar theories. Since it is

essential to have a manifestly Lorentz-covariant formulation, we use covariant

gauges as de ned by a gauge xing term

L = B (@ A )+ B B; (1)

where B is the Nakanishi-Lautrup auxiliary eld, and is a real parameter.

The theory is de ned in a vector space V with inde nite metric. We as-

sume that the constrained system is quantized in accordance with the BRST-

algebra [13] :

Q =0; i[Q ;Q]= Q; (2)

where Q is the BRST-op erator, and Q the ghost-numb er op erator. On the

basis of this algebra, we de ne the subspaces

kerQ = f : Q =0; 2Vg ; (3)

imQ = f : =Q; 2Vg ; (4)

where imQ ? kerQ, with resp ect to the inde nite inner pro duct ( ; ). We

can write

kerQ = V imQ ; (5)

and de ne the BRST-cohomology

kerQ

(6) H =

imQ 6

as a covariant space of equivalence classes, which is isomorphic to V [15 ].

We are interested in zero ghost numb er, and hence ignore the grading due

to the ghost numb er op erator Q . In order to use H as a physical state

space, it must have de nite metric, whichwe can cho ose to b e p ositive. This

is not assured, a priori, but requires the assumption of \completeness" of

the BRST-op erator Q, which implies that all neutral (zero norm) states in

kerQ are contained in imQ [14 , 15 ]. Then V and hence H must b e de nite,

b ecause every space with inde nite metric contains neutral states. With

completeness, wehaveimQ =(kerQ) , and hence imQ is the isotropic part

of kerQ. It is not enough for the de niteness of H to have ghost numb er zero,

since `singlet pair' representations, containing equivalentnumb ers of ghosts

and anti-ghosts, must also b e eliminated. In view of the inner pro duct for

eigenstates of iQ :

0 0

( ; )= ; (7)

N N N;N

c c

they would give rise to an inde nite metric in V , and hence to neutral states.

There are arguments for the absence of singlet pairs in the dense subspace

generated by the eld op erators. But we are dealing with a space of inde nite

metric, so that the extension to the full space V is delicate [25, 24 , 26 ].

Completeness can b e proven explicitly in certain string theories, however

these are more simple structures than four-dimensional gauge theories. In

any case, without completeness, we cannot get a physical subspace with

de nite norm, and a consistent formulation of the theory would seem to b e

imp ossible.

Given completeness, physical states are BRST-singlets with Q =

p p

0, p ositive norm and ghost numb er zero. Unphysical states form quartet 7

representations of the BRST-algebra [15]:

w ith Q 6=0 ; (8)

N N

c c

= Q ; (9)

N +1 N

c c

w ith Q 6=0 and ( ; ) 6=0 ; (10)

N 1 N 1 N +1 N 1

c c c c

= Q and ( ; ) 6=0 : (11)

N N 1 N N

c c c c

The states and are implied by the non-degeneracy of V , and

N 1 N

c c

the inner pro duct in Eq.(7).

In weak coupling p erturbative theory, the state space H consists of quarks

and transverse gluons. Ghosts, longitudinal- and timelike gluons form quar-

tet representations, and are unphysical. They are con ned in a kinematical

fashion.

In a general non-Ab elian gauge theory like QCD, we can have asymptotic

freedom, and we exp ect that all colored states are con ned, provided the

numb er of matter elds is limited. In the language of QCD, this implies

that quarks and transvers gluons, at zero temp erature, are not elements of

the physical state space H, which then contains only hadrons as colorless,

comp osite systems [25 , 17 , 16 ]. Under these circumstances, only hadrons can

b e pro duced in a collision of hadrons. This algebraic notion of con nement

should b e compatible with more intuitive pictures of quark con nement, and

with two-dimensional mo dels. However, for gluons, two-dimensional mo dels

are useless, b ecause there are no transvers gluons in two dimensions. If the

number of avors in QCD is limited, we can give arguments that gluons 8

are not elements of the physical subspace [16, 17 ]. These arguments are

based up on sup erconvergence relations satis ed by the structure function of

the gluon propagator, which provide a connection b etween short- and long

distance prop erties of the theory [22, 23 , 27].

In our discussion of analytic prop erties of hadronic amplitudes, we take

it for granted that con nement is realized in the sense that the physical

state space H contains only hadronic states. Quarks and gluons are not

BRST-singlets. Together with other unphysical states, they form quartet

representations of the BRST-algebra and remain unobservable.

I I I. LOCAL HADRONIC FIELDS

Having de ned the general state space of the gauge theory with con ne-

ment, wenow turn to the problem of constructing lo cal Heisenb erg op erators,

which can b e used as interp olating elds in amplitudes describing reactions

between physical particles (hadrons), and in form factors of hadrons. The

construction of comp osite op erators, and of op erator pro duct expansions, has

b een discussed extensively in the literature [11, 12 ]. The relatively new as-

p ects in our case are the state space of inde nite metric, and the fact that the

constituents are unobservable. In addition, in view of con nement, we can-

not use p erturbation theory metho ds, and consequently some assumptions

are needed for the non-p erturbative construction of comp osite elds.

In the following, we discuss the problem with the help of a generic exam-

ple. We consider the construction of a meson eld B (x) in terms of funda-

mental elds (x) and (x), ignoring all inessential asp ects like indices etc..

Hence, our formulae in the following are rather symb olic. The eld B (x) 9

must b e lo cal and BRST-invariant, so that B (x) is a representativeofa

physical physical state, provided is one.

Let us rst consider the pro duct

B (x; )= (x+) (x) : (12)

2 2

With jk; M i b eing a one particle hadron state with k = M ,we assume

that this state exists as a comp osite system, so that wehave a non-vanishing

matrix element

h0jB (x; )jk; M i6=0 ; (13)

where j0i denotes the vacuum state, and where the inner pro duct involved

in Eq.(13) is the inde nite pro duct de ned in the general state space V .We

now de ne a Poincarecovariant, lo cal op erator by the weak limit

(x + ) (x )

B (x) = lim : (14)

F ( )

Wemay consider a space-like approach with < 0 , but this is not essential.

The invariant function F ( ) is only of relevance as far as its singularity for

! 0 is concerned. It is the purp ose of F ( ) to comp ensate the singularity

of the op erator pro duct. Writing

F ( )=( ;B(0;)) ; ; 2V ; (15)

wewanttocho ose these states so that they b elong to a class K , for which

max

the matrix element (15) is most singular, assuming that such most singular

matrix elements exist [28]. Possible oscillations in the limit (14) may require

the choice of an appropriate sequence f g in the approachto =0. By

construction, the op erator B (x) is lo cal with resp ect to the constituent

elds (x) and (x), and with resp ect to itself. 10

In view of the requirement (13), wehave

ik x ik x

h0jB(x; )jk; M i = e h0jB (0;)jk; M i = e F ( ) ; (16)

with F ( ) 6= 0. Then the op erator eld

B (x; )

B (x) = lim (17)

F ( )

has a nite matrix element. Wemay assume that F ( ) 2K , so that

k max

B (x) app ears as the leading term in the general op erator pro duct expansion

of B (x; ). However, by construction, the eld B (x) should b e a BRST-

invariant op erator. Since we are dealing only with matrix elements of B (x)

with resp ect to states in the physical state space H, it is sucient to assume

0 0

that F ( ) 2K , where K refers only to states in H.We then use the

max max

eld B (x) in Eq.(17) as the Heisenb erg eld interp olating b etween the corre-

sp onding asymptotic states. Weintro duce asymptotic elds B (x) using the

free retarded function (x x ;M) in the Yang-Feldman representation,

and apply the conventional LSZ-reduction formalism [29] in order to obtain

representations of physical amplitudes in terms of pro ducts of B (x) elds.

An example would b e the S-matrix element

4 04 0 0 0 0

dxdxexp[ik x ik x] hk ;M;p ;MjSjk; M ; p; M i =

(2 )

0 0

K K hp ;MjTB(x )B(x)jp; M i ; (18)

x x

or corresp onding expressions in terms of retarded or advanced pro ducts. We

can also make further reductions as required for the pro ofs of disp ersion

representations. The reduction metho d is used here only within the space H

with de nite metric and in a framework without infrared problems.

In four dimensions, the existence of op erator expansions, and of comp osite

op erators like B (x), can b e proven within the framework of renormalized 11

p erturbation theory [28], but not yet in the general theory, as required for

our purp ose. Hence wehave to make the technical assumptions describ ed

ab ove. In manylower dimensional theories, op erator pro duct expansions are

known to exist indep endent of p erturbation theory. They are exp ected to b e

a general prop erty of lo cal eld theories.

The construction of interp olating, hadronic Heisenb erg op erators, like

B (x) in our example, is of course not unique. But the di erent p ossibilities

b elong to the same Borchers class [30]. Di erent elds in a given class, which

have the same asymptotic elds, de ne the same S-matrix. It can b e shown

that lo cality is a transitive prop erty: two elds, which commute with a given

lo cal eld, are lo cal themselves and with resp ect to each other. Wehave

equivalence classes of lo cal elds. Whatever the construction of a comp osite

op erator like B (x), the resulting elds all are lo cal with resp ect to the fun-

damental elds, and hence b elong to the same Borchers class. Although we

use Borchers theorem here essentially only in the physical subspace, it can

b e generalized to spaces with inde nite metric. The pro of involves the equiv-

alence of weak lo cal commutativity and CPT-invariance, as well as the Edge

of the Wedge Theorem. Intro ducing appropriate rules for the transformation

of ghost eld under CPT, we can de ne an anti-unitary CPT-op erator in the

state space V .Together with the p ostulates of inde nite metric eld theory,

we then get equivalence classes of lo cal Heisenb erg elds in gauge theories

like QCD.

The construction of comp osite hadron elds, as describ ed ab ove, can b e

generalized to other pro ducts of fundamental elds which form color singlets. 12

IV. SPECTRAL CONDITIONS

In the previous section, wehave describ ed howwe can obtain represen-

tations of hadronic amplitudes in QCD in terms of lo cal Heisenb erg elds,

which are BRST-invariant and interp olate b etween asymptotic states of non-

interaction hadrons. While the lo cal commutativity of the hadron elds im-

plies supp ort prop erties of Fourier representations which give rise to tub es

(wedges) as regions of holomorphy, the sp ectral conditions de ne the real

domain where retarded and advanced amplitudes coincide, generally in the

sense of distributions. Given completeness of the BRST-op erator Q,itis

convenient for our further discussion to intro duce a self-adjointinvolution

C in V , which converts the inde nite inner pro duct into a de nite pro duct

denoted by

( ; ) =( ;C) ; (19)

y 2

where C = C and C = 1 [15, 14, 16]. With resp ect to the de nite pro duct,

we obtain a decomp osition of V in the form

V = V imQ imQ ; (20)

where Q = C QC and Q = 0. With completeness of Q, the subspace V has

(p ositive) de nite metric, while imQ and imQ contain conjugate pairs of

neutral (zero norm) states, so that for every 2 imQ, there is a 2 imQ

with ( ; ) 6= 0, while b oth states are orthogonal to V . Here and in the

following we ignore the grading due to the ghost numb er op erator, since we

are mainly interested in N =0. It is convenienttointro duce a matrix

c 13

notation, writing a vector 2V in the form

1 0

C B

; (21) =

A @

with comp onents referring to the subspaces V , imQ and imQ of the de-

comp osition (20). Then

0 1

1 0 0

B C

0 0 1

; (22) C =

@ A

0 1 0

and the inner pro duct is given by

( ; ) = ( ; C ) = + + : (23)

C 1 3 2

1 2 3

The BRST-op erator can b e written as

1 0

0 0 0

C B

; (24) Q = 0 0 q

A @

0 0 0

with q b eing an invertible sub op erator [16]. 2 kerQ is characterized by

= 0, and a representativeofa physical state 2H by =0, 6=0.

3 3 1

Hence, for ; 2H,wehave( ;) = in Eq.(23). Since V is a

1 p

non-degenerate subspace, we can intro duce a pro jection op erator P (V ) with

2 y

P = P = P .

For the purp ose of sp ectral condition, we are interested in the decomp o-

sition of an inner pro duct with resp ect to a complete set f g of states in V ,

in particular eigenstates of the energy momentum op erator. For ; 2V,

wehave then

( ; ) = ( ; )( ; ) : (25)

n n

n 14

But if we consider only states ; 2 kerQ ,we obtain ( ; ) = =

( ;P(V )), so that we can write

( ; ) = ( ;P(V ) )(P (V ) ; )

p n p n

= ( ; )( ; ) ; (26)

pn pn

with a complete set of states f g in the Hilb ert space V .Writing sym-

pn p

b olically = + imQ,wehave( ; )=( ; ) for 2H, and

Hn pn Hn pn

hence obtain the decomp osition

( ; ) = ( ; )( ; ) ; (27)

Hn Hn

with ; 2H. The expression (27) is manifestly Lorentz invariant, even

though the pro jection P (V )by itself is not invariant. In the full state space

V of inde nite metric, Lorentz transformations are realized by unitary map-

pings U with U = C U C . They are BRST-invariant, and consequently of

the form (28) in our matrix representation. It is then easy to see that only

U app ears in the transformation of physical quantities. Transformations U

with U = 1 are equivalence transformations which do not change physical

matrix elements. Unphysical states, written as vectors like in Eq.(21 ), may

well have a comp onentin V , but we can always nd an equivalence trans-

formation which removes this comp onent, b ecause 6= 0 for these states.

As wehave seen in the previous section, we can obtain hadronic am-

plitudes as Fourier Transforms of matrix elements involving only BRST-

invariant, lo cal hadronic elds and hadron states. All sp ectral conditions

result from decomp ositions of these pro ducts with resp ect to intermediate

states, which are eigenstates of the energy momentum op erator. A BRST-

invariant op erator O commutes with Q, and leaves the subspace kerQ in- 15

variant. In our matrix notation, it is of the form

1 0

O 0 O

11 13

C B

O O O

; (28) O =

A @

21 22 23

0 0 O

with O q = qO , where q is de ned in Eq.(24). Since O 2H if 2H,

22 33

we can use Eq.(27) to write decomp ositions of the form

( ;XY ) = ( ;X )( ;Y ) ; (29)

Hn Hn

where ; 2Hand X; Y are BRST-invariant op erators ( elds). We see

again, that only physical states app ear in the decomp osition.

Eq.(29) is generic for all sp ectral decomp ositions used in the derivation

of analytic prop erties of physical amplitudes. It shows that these sp ectral

conditions involve only hadrons, and it guarantees the unitarity of the S-

matrix [15] in the physical (hadronic) state space H. The describ ed features

of hadronic amplitudes are, of course, a direct consequence of our assumption

of con nement for transvers gluons and quarks.

With the lo cal hadronic op erator and hadronic sp ectral conditions, we

have reached the conclusion, that the derivation of analytic prop erties, and of

disp ersion representations for gauge theories with con nement, can pro ceed

along the same lines as in the old hadron eld theory. The starting p oint are

Fourier transforms of matrix elements of retarded and advanced pro ducts of

the BRST-invariant, comp osite, lo cal hadron elds.

However, one imp ortant asp ect remains to b e discussed: the question of

anomalous thresholds or structure singularities, which will b e considered in

the following section. 16

V. ANOMALOUS THRESHOLDS

In the literature, anomalous thresholds are often considered in connec-

tion with appropriate Feynman graphs [20, 19, 21 ]. However, they can b e

understo o d, completely indep endent of p erturbation metho ds, on the ba-

sis of analyticity and unitarity [21 ]. Within the framework of theories with

con nement, it is essential to have a nonp erturbative approach.

Anomalous thresholds are branch p oints which app ear in a given channel

of an amplitude. They are not directly related to the p ossible intermediate

states in this channel, whichintro duce only \absorptive" singularities. They

are rather \structure singularities", which describ e e ects due to the p ossi-

bility that a given particle can b e considered as a comp osite system of other

particles. They app ear in the physical sheet of the amplitude, in the channel

considered, only if a lo osely b ound comp osite system is involved. Otherwise,

they remain in a secondary Riemann sheet.

In the following we brie y show that anomalous thresholds, in a given

channel of an amplitude, are due to ordinary (absorptive) thresholds in

crossed channels of other amplitudes, which are related to the one under

consideration via unitarity. Since wehave seen b efore that hadronic am-

plitudes have only absorption thresholds related to hadron states, it follows

that the only anomalous thresholds which app ear are due to the structure of

hadrons as comp osite systems of hadronic constituents. There are no such

thresholds asso ciated with the quark-gluon structure of hadrons, even for

lo osely b ound comp osite systems with quarks as constituents.

In order to study the emergence of anomalous thresholds on the physical

sheet of an hadronic amplitude, we consider a form factor as an example. We

ignore all inessential complications and use the structure function W (s), s = 17

k of a deuteron-like particle with variable mass, as indicated in Fig.1. For

2 2 2

x<2m , where x =(mass) m , the function W (s) has branch p oints

N D

on the right hand, real k -axis, starting with those due to pion intermediate

states at s . However, we concentrate on the N N -threshold at s =4m .

The discontinuity due to this threshold alone is

ImW (s + i0) = (s + i0)G(s + i0)V (s + i0) ; (30)

2 1 1=2

(s)=[(s4m )s ] : (31)

Here G(s) is the appropriate partial wave pro jective of the amplitude G(s; t)

pictured in Fig.2a. We consider S-wave pro jections for simplicity.Further-

more V (s + i0) = V (s + i0), where V (s) is the nucleon form factor, with

branch p oints analogous to those of W (s). The continuation of V (s)into

sheet II of the N N -threshold is given by [21]

V(s)

V (s)= ; (32)

1+2i(s)F (s)

where F (s) is the partial wave pro jection of the scattering amplitude N N !

N N in the s-channel. With Eqs. (30) and (32), we get for the continuation

of W (s) through the N N cut:

G(s)V (s)

: (33) W (s)=W(s)2i(s)

1+2i(s)F (s)

While W (s) has no left-hand branch lines, W (s) do es, due to left-hand cuts

of G(s) and F (s). For our purp ose, the imp ortant left-hand cut is the one of

G(s), which is caused by the p ole term at t = m , as illustrated in Fig.2b :

(x)

G(s; t)= + : (34)

t m

N 18

The branch p oint is due to the end p ointatcos = 1 in the partial-wave

pro jection, with t = t(s; cos ). It is lo cated at s = g (x), where

g (x)= 4x 1 : (35)

2 2 2

For 0

N N N

4m . The branch p oint in sheet I I at s = g (x) is pictured in Fig.3 for

x<2m .

Let us now increase x to x =2m and ab ove. The p osition g (x) of the

2 2

branch p ointmoves to 4m at x =2m , and then decreases again. Giving

N N

x an imaginary part, we get

y y

2i (x2m ) ; (36) g (x + iy )= g(x)+

2 2

m m

N N

and we see that g (x + iy ) encircles the branch p oint s =4m of W (s),

moving therebyinto the rst and \physical" sheet of the Riemann surface

of this function. There it b ecomes an anomalous threshold. The situation is

illustrated in Fig.4, where the meson cuts have b een omitted. For suciently

large values of x, this branch p oint can movewell b elow the lowest absorption

threshold s .Writing m =2m B,we get, for small values of the binding

D N

energy B ,

2 2

)=4m 1 16m B; (37) g (m

D D

which can givea very long maximal range of the distribution in con guration

space, just as exp ected from the Schrodinger wave function.

There are other anomalous thresholds asso ciated with the N N branch

p ointofW(s). For instance, there are those due to the probability distribu-

tion of the proton in a deuteron, considered as a comp osite system of two 19

nucleons and a limited numb er of pions. Their p osition can easily b e calcu-

lated exactly.For small values of B ,we get g (m ) 16m (B + m ), if one

1 N

pion is added. Anomalous thresholds can also come out of higher absorptive

branch p oints in the s-channel of the form factor W .

Finally,we remark that the ab ove considerations can b e generalized to

other amplitudes. Essentially only kinematics, crossing, and some analytic

prop erties are needed. In all cases the anomalous thresholds are related

to ordinary, absorptive thresholds in other amplitudes, which app ear in the

continuation into secondary Riemann sheets [21].

As wehave p ointed out b efore, due to the fact that anomalous thresholds

are indirectly related to absorptive thresholds, there are no such singularities

which are asso ciated with the quark-gluon structure of hadrons, since there

are no absorptive thresholds related to this structure. However, for hadrons

whichmay b e considered as lo osely b ound systems of heavy quarks, we can

get a large mean square radius on the basis of appropriate weight functions

along hadronic cuts [10, 31 ], even though they maybemuch higher in mass,

and also as a consequence of p ossible hadronic anomalous thresholds.

VI. COLORED APMLITUDES

Having discussed only hadronic amplitudes describing observable conse-

quences of the theory,wewould like to add here some remarks ab out the

analytic structure and the singularities of general Green's functions with col-

ored channels. In particular, we will show that these colored amplitudes

must have singularities at nite p oints, which can b e asso ciated with con-

ned states in V like quarks and gluons [22, 23 ]. Even though quarks and 20

transverse gluons are con ned, we can have asymptotic states asso ciated with

these excitations, as well as corresp onding p oles in colored Green's functions.

In our formulation of con nement, all colored states form quartet represen-

tations of BRST-algebra, and hence are not elements of physical space H,

which contains only singlets.

As an example for colored amplitudes, we consider the gluon propagator,

which has b een studied extensively. The structure function is de ned as a

Fourier Transform by

ik x 2

dxe h0jTA (x)A (0)j0i = i D (k + i0)

a b

% % % %

(k k g k k g + k k g k k g ) (38)

with A @ A @ A . As b efore, we consider the linear, covariant gauges

de ned in Eq.(1). In the state space V with inde nite metric, we write the

sp ectral condition in the form [32 ]

4 ipa

d ae ( ;U(a)) = 0 ; (39)

0 2 4

for values of p outside of W = fp : p 0;p 0; p 2 R g , and for all

; 2V.

Lorentz covariance and sp ectral condition are sucient to show that

2 2

D (k + i0) is the b oundary value of an analytic function D (k ), whichis

regular in the cut k -plane, with a cuts along the p ositive real axis only.Itis

then an essential question to obtain the asymptotic b ehavior for k !1 in

all directions of the complex k -plane. In view of the asymptotic freedom of

the theory, the asymptotic terms can b e obtained with the help of renormal-

ization group metho ds. In using the renormalization group, an assumption is

made, whichwehave not used so far. We require that the general amplitude

! +0, where g is the gauge connects with the p erturbative expression for g 21

coupling parameter. The connection is needed only for the leading term.

With this assumption, we nd for k !1 in all directions [23]:

00 0

2 2 2 2

k D (k ; ;g; ) ' + C (g ; ) ln + : (40)

R 0

The corresp onding asymptotic terms for the discontinuity along the p ositive,

real k {axis are then given by

= 1

00 0

2 2 2 2

k (k ; ;g; ) ' C (g ; ) ln + : (41)

R 0

j j

In these relations, wehave used the following de nitions: The anomalous

dimension of the gauge eld is given by

2 2

(g ; )=( + )g + (42)

00 01

for g ! +0, and for the renormalization group function we write, in the

same limit,

2 4 6

(g )= g + g + : (43)

0 1

Furthermore, we use the notation

= = : (44)

0 00 01

For QCD, wehave

2 13

2 1

2 3

; = (16 ) = ; (45)

N 11

where N is the numb er of quark avours. We assume < 0 corresp onding

F 0

to asymptotic freedom. Consequently, the exp onent = in Eqs.(40) and

00 0

(41) varies from 13=22 for N = 0 to 1/10 for N = 9, and from 1=16 for

F F 22

N =10to 15=2 for N = 16. Wehave0< = < 1 for N 9 and

F F 00 0 F

= < 0 for 10 N 16; for = = 1, our relations (40) and (41)

00 0 F 00 0

would require mo di cations.

The parameter < 0 is a normalization p oint. We generally chose to

normalize D so that

2 2 2 2 2

k D (k ; ;g; )=1 for k = : (46)

With this normalization, the co ecient C (g ; ) for = 0 is given by

[22, 23 ]

2 2 =

00 0

exp dx (x); C (g ; 0) = (g )

0 R

(x; 0)

(x) ; (47)

(x) x

2 2

and hence C (g ; 0) > 0. Certainly C (g ; ) is not identically zero. If

R R

there should b e zero surfaces, a term prop ortional to ( ln ) b ecomes

relevant in Eq.(41).

The remarkable prop erty of the asymptotic terms in Eqs. (40) and (41)

is their gauge indep endence except for the co ecients. Furthermore, their

functional form is determined by one lo op expressions.

2 2

From the asymptotic limit (40), we see that D (k )vanishes for k !1in

all directions of the complex k -plane. Hence it cannot b e a nontrivial entire

function, at least for 0

1 1

1 2 2

singularityof (g ). There must b e singularities on the p ositive real k -

axis , and it is natural that these are asso ciated with con rmed, unphysical

states. Similar arguments can b e given for the structure functions of the

quark propagator. 23

We can write an unsubtracted disp ersion representation for D (k ):

02 2

(k ; ;g; )

2 2 02

D(k ; ;g; ) = dk ; (48)

02 2

k k

and even a dip ole representation exists

02 2

(k ; ;g; )

2 2 02

D (k ; ;g; ) = dk ;

02 2 2

(k k )

2 2 02 02 2

(k ; ;g; ) = dk (k ; ;g; ): (49)

For = 0, the dip ole representation has b een used in order to give arguments

for an approximately linear quark-antiquark p otential under the condition

2 0 2 2

= > 0, where (1) = 0, and (k ) > 0, (k)=(k)<0 for

00 0

suciently large values of k [33, 34 ].

Under the restriction = > 0(N 9 for QCD), we nd that D (k )

00 0 F

vanishes faster than k at in nity, so that wehave the imp ortant sum

rule [23] :

2 2 2

dk (k ; ;g; ) = : (50)

For =0, = > 0, wehave a sup erconvergence relation [22]. It gives a

00 0

rather direct connection b etween short and long distance prop erties of the

theory, and has b een used in order to give arguments for gluon con nement

[16, 17 ].

APPENDIX: REMARKS ABOUT PROOFS

Wehave seen that we can construct lo cal hadronic elds as BRST-

invariant op erators in V , and write Fourier representation of hadronic am- 24

plituds in terms of matrix elements of pro ducts of these elds. With BRST-

metho ds, we de ne an invariantphysical state space H which, as a conse-

quence of con nement, contains only hadrons. With the sp ectral conditions

also referring to hadrons only,wehave the imput required in order to use the

old metho ds for the derivation of disp ersion representations as formulated in

hadronic eld theory.For completeness, we give in this app endix a very brief

sketch of the essential ideas of these pro ofs, which are often hidden b ehind

technical details.

The Gap Method [6] is applicable in cases where there is no unphysical

region. Examples are -, N - forward and near-forward scattering, some

form factors like , N N in the N -channel, etc. [35 , 36, 1]. As an example,

0 0

let us consider - forward scattering. We can write the amplitude as

F (! )= dr F (!; r) ; (51)

with

2 2

r sin( ! )

F (!; r) = 4

2 2

x x

0 i! x

dx e hpj[j ( );j( )]jpi ; (52)

2 2

and j =(+ ).For xed r , F (!; r) is analytic in the upp er half ! -plane,

and ImF (! + i0;r)=0for j!j< due to the sp ectral conditions. Ignoring

subtraction, we can write a Hilb ert representation

2! ImF (! + i0;r)

F (!; r)= : (53) d!

0 2

! !

For real j! j >,we can p erform the r-integration (51) on b oth sides, and get

the corresp onding disp ersion relation for F (! ). Although some re nements

are required, the metho d shows in a very simple wayhow lo cal commutativity 25

and sp ectral conditions lead to a disp ersion representation. Pole terms, like

in N - scattering, can also b e handled by this metho d [5, 7, 1].

The General Method is required in the presence of unphysical regions, like

NN-scattering [37] (even for t=0), N N -form factors in the N N -channel,

for xed t amplitudes [38], to obtain t-analyticity (Lehmann ellipses) [8],

and for st-analyticity [39]. There are many technical details involved in the

derivation of disp ersion representations, like continuations in mass variables,

for example, but the main problem is to construct the largest region of holo-

morphy obtainable on the basis of retarded and advanced functions like

i x x

4 iK x 0 0 y

F (K )= d xe (x ) hp j j ( );j( jpi ; (54)

(2 ) 2 2

0 0 0

with K = (k + k ), k + p = k + p .

Due to lo cal commutativity, the functions F (K ) are analytic in the

wedges

0 2 4

W = fK : ImK > 0 or < 0; (ImK) > 0; ReK 2 R g : (55)

From the sp ectral conditions, we nd that F (K )=F (K) for D 2 R ,

where D is a real domain, and where this equalitymay b e in the sense of

distributions. As a sp ecial case of the Edge of the Wedge Theorem [1], we

can then show that there exists an analytic function F (K ), which coincides

with F (K ) in the wedges W resp ectively, and which is holomorphic in the

domain W [ N (D ), with W = W [ W . Here N (D ) is a nite, complex

neighb orho o d of D .Ifwe then construct the Envelope of Holomorphy E (W [

N (D )), we get the largest p ossible region of analyticity given the assumptions

made. In the original pap er [1], a generalized semitub e has b een used, for

which the envelop e was known., This metho d gives b oudary p oints of the 26

envelop e in imp ortant cases. A complete construction of the envelop e, using

the continuitys theorem, was given in [2].

Indep endently, in [5], elab orate distribution and analytic metho ds were

used in order to get a sub domain of E , directly on the basis of W and D .

For the sp ecial problem with one four-vector considered here, one can use

metho ds from the theory of distributions and di erential equations in order

to give a representation of functions which are holomorphic in E [4].

The limitations of the pro ofs for disp ersion representation are due to

the lack of input from unitarity, and often can b e related to conditions for

the absence of unphysical anomalous thresholds. Some improvements are

p ossible using asp ects of two-particle unitarity, but in general multiparticle

unitarity and analytic prop erties of multiparticle amplitudes are required for

further enlargements of the domain of holomorphy.

For any xed t< 0, and for arbitrary binary reactions, it can b e shown

that the amplitude is holomorphic outside of a large circle in the cut s-plane,

so that one can always prove crossing relations [40].

As is evident from the preceding discussion, the interesting prop osal of

double disp ersion relations [41] has not b een proven. They are compatible

with hadronic p erturbation theory in lower orders. Although it may not b e

avalid approach in QCD, hadronic p erturbation theory is a useful to ol for

lo cating certain singularities of physical amplitudes.

ACKNOWLEDGEMENTS

I am indebted to Gerhard Hohler for insisting that I should write an 27

article ab out the validity of disp ersion relations in QCD. For conversations

or remarks, I would like to thank H.-J. Borchers, G. Hohler, J. Kub o, H.

Lehmann, K. Nishijima, B. Schro er, K. Sib old, F. Stro cchi and W. Zimmer-

mann. It is a pleasure thank Wolfhart Zimmermann, and the Theory Group

of the Max Planck Institut fur Physik, Werner Heisenb erg Institut, for their

kind hospitalityinMunchen.

This work has b een supp orted in part by the National Science Foundation,

grant PHY 91-23780. 28

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Figure Captions

Fig. 1. Vertex Function W (s).

Fig. 2. Inelastic amplitude G(s; t) (a) and relevant p ole term (b)

in the t-channel.

Fig. 3. Branch p oints of W (s) and W (s). The continuation is

with resp ect to the N N -threshold at s =4m .

Fig. 4. Anomalous threshold of W (s)at s=g(x) for x> 2m .

The branch line runs from g (x)to4m in sheet I (physi-

cal sheet), and then from 4m to 1 in sheet I I (dotted

line). The meson branch lines starting at s < 4m have

not b een drawn. 33