A NNALES DE L ’I. H. P., SECTION A
J. B ROS
B. D UCOMET
Two-particle structure and renormalization
Annales de l’I. H. P., section A, tome 45, n
o2 (1986), p. 173-230
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173
Two-particle structure and renormalization
J. BROS
Service de Physique Theorique CEA-Saclay, 91191 Gif-sur Yvette Cedex, France
B. DUCOMET
Service de Physique Neutronique et Nucleaire
Centre d’Etudes de Bruyères-le-Châtel,
91680 Bruyeres-le-Chatel, France
Vol. 45, n° 2, 1986, Physique theorique
ABSTRACT. - We present and
study
renormalized forms of the Bethe-Salpeter equation
and of similartwo-particle
structureequations
for theconnected
N-point
Greenfunctions;
theseequations
areimplied by
thestandard BPHZ scheme for renormalized field theories with
positive
mass;only
the scalar neutral case(corresponding
to theories of the1>4 type)
has been
considered,
but ouranalysis
isapplicable
to similar non-scalar cases, such as the two-dimensional massive Gross-Neveu model.Specific asymptotic properties
of the renormalizedBethe-Salpeter
kernel G and of the connected
four-point
function F(resp. N-point
func-tions
F(N»)
emerge as naturalassumptions
and theleading
term of theWilson-Zimmermann
expansion
of the field is exhibited in asimple
way in this renormalizedBethe-Salpeter
formalism.The
following
two constructionproblems
are then treated(in complex
momentum
space) :
i ) Considering
F asgiven
in the axiomaticframework,
construct Gin terms of F.
ii) Reconstruct
F in terms of G.As in the
regularized
version of theBethe-Salpeter equation,
the renor-malized formalism is shown to include the
two-particle
structuralproperties
of the
theory (analyticity, equivalence
between theirreducibility
of G andan
asymptotic completeness equation
forF).
Annales de l’Institut Henri Poincaré - Physique theorique - 0246-0211
Vol. 45,~86/02/173/58/$ 7,80/(~) Gauthier-Villars
174 J. BROS AND B. DUCOMET
RESUME. - On
presente
et on etudie des formes renormalisees de1’equation
deBethe-Salpeter
et,plus generalement,
desequations
destructure a deux
particles
pour les fonctions de Green connexes a Npoints ;
ces
equations
resultent du formalisme standard de B. P. H. Z. pour les theories dechamps
renormalisables de massespositives ;
seulle cas scalaireneutre
(correspondant
aux theories de type1>4)
a ete considereici,
mais1’analyse s’applique
a des cas semblablesnon-scalaires,
tels que Ie modele de Gross-Neveu massif a deux dimensions.Des
proprietes asymptotiques spécifiques
du noyau deBethe-Salpeter
renormalise G et de la fonction a 4
points
connexe F(resp.
des fonctions a Npoints F(N»)
sedegagent
comme deshypotheses naturelles,
et Ie termedominant du
developpement
de Wilson-Zimmermann duchamp
estretrouve de
façon simple
dans ce formalisme deBethe-Salpeter
renormalise.Les
problemes
de construction suivants sont alors traites(dans l’espace
des
impulsions complexes) :
i )
Considerant F comme donne dans Ie cadreaxiomatique,
determiner Gen fonction de F.
ii)
Reconstruire F en fonction de G.On montre en outre, comme dans la version
regularisee
de1’equation
de
Bethe-Salpeter,
que Ie formalisme renormalise contient lesproprietes
de structure a deux
particules
de la theorie(analyticite, equivalence
entrel’irréuctibilité de G et une
equation
decompletude asymptotique
pourF).
1. INTRODUCTION
Although originating
inperturbation theory,
the formalism of irreducible(or Bethe-Salpeter type)
kernels has a very rich content in thegeneral
framework of
Quantum
FieldTheory.
One feature of this formalism(first emphasized
in[1 ],
andrigorously
studied in[2] ] [3 ])
is the fact that irre- ducible kernels are the cornerstone ofMany-Particle
StructureAnalysis (at
least for theories withpositive masses).
A second feature is theimpli-
cation of these kernels in the
asymptotics
of renormalizable field theories.Although already
discovered andexploited
in[4] ]
forobtaining original
results on the behaviour of the Green functions at the so-called « excep- tional momenta », this feature still deserves a
systematic general study.
The
present
work aims toinvestigate
the renormalizedBethe-Salpeter equation (and
similarequations
for theN-point functions) implied by
thestandard B. P. H. Z. renormalization
scheme,
and to exhibit therefrom the mergence oftwo-particle
structure andasymptotics,
from ageneral
Annales de l’Institut Henri Poincaré - Physique theorique
axiomatic
viewpoint.
In thecomplex
momentum-space formulation of axiomatic fieldtheory,
the two features mentioned abovecorrespond
to two
complementary
sets ofproperties
of theanalytic N-point
Greenfunctions, namely :
i )
the localsingularity
structure near thecomplex
mass shell.ii)
thesingularity
structure atinfinity.
To a
large
extent, the structuralproperties i )
are of universal nature :they
express the consequences of thegeneral
axiomaticbackground
ofQuantum
FieldTheory (involving
a certain formulation oflocality
andof the
spectral condition) together
with the property ofAsymptotic Comple-
teness of the Fields
(the
latterimplying
theunitarity
of theS-operator).
As it was
proved
in[2 ],
theasymptotic completeness properties
are mostconveniently expressed
in thelanguage
ofanalytic
functionsthrough
theirreducible character of relevant
Bethe-Salpeter
typekernels,
thegeneral (axiomatic)
definition ofirreducibility being
thevanishing
of a certaindiscontinuity
function(or
«absorptive part »)
of the considered kernel below a certain relevant mass threshold. As a consequence, the Bethe-Salpeter
typeequations
haveappeared
as apowerful
tool forworking
out the
complete analytic
and monodromic structure of the Green functions and(by
restriction to thecomplex
massshell)
of themultiparticle scattering amplitudes. Among
variousresults,
thefollowing
outcomes of thisapproach
deserve to be mentioned :
a)
the structuralequations
that one obtains are exact relativistic gene- ralizations of the Faddeev orWeinberg-type equations
formultiparticle scattering
inpotential theory (see [3 ]) ;
b)
theanalysis
of various terms of theseequations
introduces at the axiomatic level the notion of Landausingularity,
whose status was at firstpurely
attached to theexpansion
inFeynman graphs ( 1 ).
’This
conceptual
success of irreducible kernels inMany-Particle-Struc- ture-Analysis (M.
P. S.A.)
suggests that similarinvestigations
of axiomatic type of this formalism should alsoprovide
aninsight
on thesingularity
structure at
inflnity
of the Greenfunctions,
for the class of renor-malizable theories. In
fact,
suchinvestigations
were carried outby Syman-
zik
(see
theAppendices
of[4 ]),
in anapproach
which indicated an ori-ginal
way ofintroducing
theleading
term of the Wilson-Zimmermannexpansion
of the basic field[6 ].
For a clearunderstanding
of this alter- native aspect of theBethe-Salpeter formalism,
it is essential thatspecial
e) It is only in the context of pure S-matrix theory that such a non-perturbative inter- pretation of the occurrence of Landau singularities had been proposed (see [5] ] and refe-
rences therein).
Vol. 45, n° 2-1986.
176 J. BROS AND B. DUCOMET
asymptotic properties (suggested by
thetheory
ofrenormalization)
beincorporated
in thegeneral
axiomatic framework and thiscorresponds
to a crucial distinction in the use of irreducible kernels. The universal character of M. P. S. A. was reflected
by
the use ofregularized
irreduciblekernels,
linked to the Green functionsby Bethe-Salpeter
typeequations involving
a convergent measure atinfinity,
and therefore valid(to
alarge extent) independently
of the behaviour atinfinity
of the Green functions.Mathematically speaking,
thisregularization procedure played
the roleof a «
compactification »
of theintegration
space, which allowed one to focus the formalism on the localanalyticity properties
of the Green functions.On the
contrary,
thestudy
ofproperties
atinfinity requires
the deri-vation and the
exploitation
of a renormalized version of theBethe-Salpeter-
type kernels and
equations.
Thisstudy
should exhibit in asynthetic
way :i )
thegeometrical
aspects of M. P. S.A., namely (as
in theregularized formalism)
therelationship
betweenirreducibility
andasymptotic
com-pleteness,
and the two-sheetedanalyticity
domains of the Green functions around the relevanttwo-particle
thresholds.ii)
thespecial
structuralproperties
atinfinity, corresponding
to theexistence of a Wilson-Zimmermann type
leading singularity
for the Greenfunctions.
Moreover the links between this renormalized formalism and the
(pre- vious) regularized
one should also be clarified.In the present paper, this program is
developed
underassumptions
at
infinity
that are consistent with theperturbative
B. P. H. Z. renormaliza- tion scheme. Forsimplicity,
weonly
consider the case of an even scalarfield with
positive
mass,corresponding
to theories of theHowever,
the same
analysis
isapplicable
to moregeneral (non-scalar)
cases of renor-malizable field
theories, corresponding
to more realistic models(from
thepoint
of view of constructiveQ.
F.T.),
such as the massive Gross-Neveumodel, recently
constructed in[7 ].
In thisconnection,
ourstudy
was alsomotivated
by
the occurrence of thedisturbing
« Landauphenomenon » [8 ],
which was among the first facts that led
people
toquestion
about theexistence of the
03C64-theory
in dimension four. Let us recall the argument.The series of
Feynman graphs
, renorma-lized
according
to the B. P. H. Z.rules,
can be summed up as a geome- tricalseries,
thusyielding
thefollowing
contribution to thefour-point
function :
Fo(k) =
1 - g ;here g
denotes the renormalizedcoupling
gIo(k)
p gconstant,
supposed
to bepositive,
and is the renormalized one-loonintegral
,(~ being
the total energy-momentum ofAnnales de l’Institut Poincaré - Physique theorique
the
distinguished channel).
SinceIo(k)
behaves likelog ( - k2)
atinfinity,
the function
Fo
has apole
in the euclideanregion (at
k2 ~ - whichis in contradiction with the
primitive analytic
structureimplied by
theaxioms of field
theory.
Although
theargument
isnon-conclusive,
since based on a very smallpart
of theperturbation series,
it deals with thesimplest example
of arenormalized
Bethe-Salpeter equation (with
the kernel Gequal
to theconstant
g)
and thus indicates that themerging
oftwo-particle
structureand
asymptotics
may generate undesirableghosts. Thus,
thequestion
ofthe existence of renormalizable field theories
(or
of the sense in whichthey exist)
has aconnection,
viaanalyticity properties,
with the forma- lism of renormalizedBethe-Salpeter equation.
The
plan
of this article can be summarized as follows : while parts2,
3 and 4 are devoted to the introduction of the renormalizedBethe-Salpeter
formalism
through
a formalexpansion approach
based on the B. P. H. Z.renormalization
scheme,
thesubsequent
sections 5 to 8present
a « semi- axiomatic »exploitation
of this formalism which exhibitssimultaneoulsy properties
of the typesi )
andii)
mentionedabove,
for theN-point
functionsof the
theory.
Section 2 will fix some notations and
give
a recall on the use(in
axio-matic
Q.
F.T.)
ofregularized Bethe-Salpeter equations
of the formF =
G p
+G p,
Fbeing
thefour-point function, 03C1
a certainintegration operation involving
aregularization procedure
andGp
thecorresponding
«
regularized Bethe-Salpeter » (or
«two-particle irreducible » )
kernel.This formalism admits a « formal
expansion » (or
«semi-perturbative » )
aspect, whichsimply
consists inconsidering
F as the formal Neumannseries F
= 03A3 G03C1 03C1... 03C1 G03C1.
For the derivation of a renormalized Bethe-Salpeter (in
short B.S.) formalism,
one is led toinvestigate
theanalog
of the
previous
formalexpansion
ofF, Gp being
nowreplaced by
a«
genuine »
renormalized B. S.kernel, G,
and each termGp op
... ~ pGp being replaced by
a certain « renormalized ~-convolution »product (in
the sense of[9 ])
which we denoteby
... o11
The
justification
of the existence of such anexpansion
relies on theperturbative approach
of fieldtheory,
andspecially
on the B. P. H. Z.scheme. The various
properties
of the terms ... of theexpansion
n
can be
studied, however,
at a moregeneral level, namely
as consequences ofpostulated properties
for G and for thetwo-point
function H(2) ofthe
theory,
withoutreferring
to thesubjacent perturbative
level. These results will be described in section3, supplemented by Appendices
A and B.As a matter of
fact,
a morecomplete study
of theasymptotic
andanalytic
Vol. 45, n° 2-1986.
178 J. BROS AND B. DUCOMET
properties
of the terms ... oG ]r,
based on aprecise
formulation ofn
asymptotic properties
to be satisfiedby
G and will begiven
in separate papers[10 ] :
ourappendix
A willonly
present a sketch of theproof
of[10 ].
while the
appendix
B is devoted to a derivation(of purely algebraic nature)
of «
asymptotic completeness equations »
for the various terms... ~ G]r
of the
expansion
ofF ;
theseequations
were firstpresented
in[11] (in
thelatter, they
were established for similarquantities
... in aregularized
version of the renormalized B. S.formalism).
In section
4,
two kinds ofBethe-Salpeter-type
identities will be derived from theprevious
formalexpansion approach.
On the one hand a « renor-malized
Bethe-Salpeter equation »
will be obtained for thefour-point
function F : it contains an extra-term that involves an
auxiliary
functionA,
defined
through
a formalexpansion
in terms of G which is similar to that of F. On the otherhand,
various relations which link F to the increments of G betweencouples
ofpoints
of momentum space are also derived(for making
ourpresentation simpler,
theproof
of one of these has beenrejected
- in
Appendix C).
These relations areindependent
of A and turn out to beformally
identical with relations that can be derived from the « naive »or
regularized
B. S.equation.
This fact had beenessentially
taken as anansatz in the
approach
of[4 ],
where all theequations involving
« G »only through
its increments had been considered as reliableequations
forthe renormalized
quantities.
Section 4 alsogives
a set of renormalizedequations
for theN-point
Green functions in terms of thecorresponding
renormalized
two-particle
irreducibleN-point functions;
as in the four-point
case,auxiliary
functionsAN (of
the same nature asA)
are involved.The
subsequent
sections of ourarticle,
devoted to thestudy
of the renor-malized
two-particle
structureequations
from ageneral viewpoint,
arebased on the
following
considerations.The
problem
ofsolving
the renormalized B. S.equation
either with respect to F(considering
G asgiven
in a constructiveapproach)
or conver-sely
with respect to G(considering
F asgiven
in an axiomaticapproach)
is not
obviously
tractable via the Fredholmtheory,
as it was the casefor
regularized
B. S.equations.
As a matter of
fact,
the difference between theasymptotic properties
of F and G
(in
euclidean directions of momentumspace)
turns out to breakthe symmetry of their
respective
roles in the B. S.equation.
Theregularized equations,
unsensitive to theasymptotic properties
of F andGP,
werepure Fredholm resolvent
equations, algebraically symmetric
with respectto F and
( - Gp).
On the contrary, acomplete dissymetry
appears between F and G in the renormalized B. S.equation,
with the presence of the auxi-liary
function A: thisdissymetry
isprecisely supposed
to express the dis-l’Institut Henri Poincaré - Physique theorique
crepancy between the
asymptotic
behaviours of F andG,
taken into account in the renormalizedequation.
Thisanalysis
of theconsistency
of relevantasymptotic properties
of F and G with thealgebraic
form of the renor-malized B. S.
equation
is carried out in section 5.To be more
specific,
we can say that the existence of a renormalized B. S.formalism is linked to the
possibility
ofextracting
from F and G suitable«
regular
parts »having integrability properties
atinfinity (in
euclideandirections of momentum
space)
in theories where F and G themselvesare
bounded,
but notintegrable (namely,
the twoparticle
~-convolutionintegral
F o G = does notexist).
As far as G is
concerned,
our basicasymptotic assumption (suggested by
theperturbative
framework and formulated in section 3 as « property B» )
asserts that the derivatives of G with
respect
to four-momentum coordi- nates have decreaseproperties
atinfinity
whichimply integrability (It
is
indeed,
underassumptions
of this type, formulated withappropriate inequalities,
that the results of the formalexpansion approach
of section 3 and[70] can
beestablished).
An immediate consequence of thisassumption
is that various increments of G between
couples
ofpoints
of momentumspace have themselves
integrability properties
atinfinity
and canplay
the roles of
regular
parts of G in the renormalizedequations.
The
asymptotic
behaviour of F turns out to becompletely
differentfrom that of
G,
and one of the purposes of sections 5 and 6 is to exhibit variousfour-point kernels,
functionals of F(or
F andG)
which mustenjoy regularity
conditions atinfinity,
if the renormalizedequations
ofsection 4 are reliable.
In section
5,
twoasymptotic properties
of F(resp.
of theN-point
connectedGreen functions
F~),
calledproperties
A and A’(resp. A
andA’),
are inferredfrom the renormalized B. S.
equation (resp.
from thetwo-particle
structureequations
forN-point functions). Property
A(resp. A )
amounts to define suitableregular
parts of F(resp. F(N») by subtracting
from the latter asingular factorized
contributioninvolving
the function A(resp. A~).
Property
A’(resp. A’)
expresses A(resp. A (N»)
in terms of F(resp.
via an
appropriate limiting procedure. Moreover,
thissingular part
of F(resp. F(N») involving
A(resp. A (N»)
is shown to beinterpretable
as theWilson-Zimmermann
leading singularity
of F(resp. F~);
a clear accountof the connection of the latter with the renormalized B. S. formalism is thus
given.
In section
6,
the incrementalequations
derived in section 4 areexploited,
and various «
regular
kernels », functionals of F andG,
are introduced.In this
study
(ofalgehraic
nature). we use as an « ansatz » the fact thatVol. 2-1986.
180 J. BROS AND B. DUCOMET
each
four-point
function withappropriate integrability properties
atinfinity
admits a resolvent kernel whichenjoys
the sameintegrability properties.
Various relations are thenestablished,
between theregular
parts of F and the resolvents of
appropriate
incrementsof G; a
connectionwith the formalism of
[4] is displayed
in the course of thisanalysis (see
§ 6 . 2).
On the otherhand,
aregularized
version of the renormalized B. S.formalism
(introduced by
D.Iagolnitzer during
the course of thepresent work,
in[77])
is summarized(see § 6.3);
thisregularized
version is shown togive
asimple
account of the property(used implicity
in[4 ]) according
to which the kernel G involved in the incremental
equations
can be indiffe-rently
considered as a renormalized or non-renormalized B. S. kernel(before
theregularization
isremoved).
The various results obtained in sections 5 and 6 then allow us to treat,
respectively
in sections 7 and8,
thefollowing
twoproblems :
i ) Considering
F(resp.
all theN-point
functionsF(N»)
asgiven
in theaxiomatic
framework,
and such that theasymptotic properties
A and A’(resp. A
andA’) hold,
construct the renormalized B. S. kernel G(resp.
a
complete
set of irreducible functions in terms of F(resp.
of thefunctions
F~),
and show that all therequisite properties
of G(resp.
ofthe functions are
satisfied, namely : analyticity, irreducibility
andasymptotic properties
of the type B.ii)
G andH(2) being
considered asgiven,
with all therequisite
pro-perties (in particular
withasymptotic properties
BandAo),
reconstruct F and show that it satisfies thetwo-particle
structuralproperties (analy- ticity
and A. C.equations) together
with theasymptotic properties
A and A’.Each of these
problems
isgiven
aunique solution,
whosevalidity
ishowever submitted to the « ansatz » that the resolvent kernels of «
regular
parts » of F or G
(or
of moregeneral
functionals withregularity properties
at
infinity)
are well-defined :here,
the « Landauphenomenon »
may occur...Moreover,
in thecomputation
of G in terms ofF, given
in section7,
an additional
asymptotic assumption
has to be satisfiedby
F : it isexpressed
as the existence of the limit of a certain functional of
F,
when a suitableregularization parameter
is removed.On the other
hand,
it is worthnoticing
that in thisgeneral approach,
a
proof
isgiven (in
sections 7 and8)
of thealgebraic equivalence
betweenthe
irreducibility
of G(resp.
and thevalidity
oftwo-particle
asymp- toticcompleteness equations
for F(resp. F~).
As aby-product
of thisproof,
the function A(resp. A (N»)
and theregular parts
of F(resp.
are shown to
satisfy
themselvesappropriate asymptotic completeness (or unitarity-type) equations.
Henri Poincaré - Physique theorique
2. THE BETHE-SALPETER
EQUATION:
REGULARIZED VERSIONS
AND THE PROBLEM OF RENORMALIZATION
§
2.1. - Let us fix some notations. In thefollowing,
F willalways
denote the
amputated analytic four-point function,
of a certain even scalarfield, expressed
in the variables of agiven
channel(12, 1’ 2’).
More pre-cisely,
letk 1, k2 (resp. ~, k2)
be theincoming (resp. outgoing)
four-momenta of thischannel,
we put :Then if
H(2)
andH(4)
denoterespectively
theanalytic two-point
andfour-point
functions(see [2a, b ]),
F is defined as :We also define for convenience the « two-line propagator function » :
In the axiomatic
framework,
F isanalytic
in a standard domainD4
ofcomplex (k,
z,z’)-space (2),
which contains inparticular
the euclideansubspace
relative to anygiven frame; D4
isspecified by
thespectrum
ofthe
theory,
which is heresupposed
to contain asingle positive
mass ,u in its discretepart.
If the field satisfies the
Asymptotic Completeness property (A. C.),
thediscontinuity
AF of F across thetwo-particle
cut 03C3:(2 ) k2 (4,u)2 (with k2
=k~°~2 - k2, k
=(k~°~, k)) real,
>0)
fulfils thefollowing equation :
where
~(~)
=,u2),
the constant Z is the residue ofat
k2
= andF +,
F -respectively
refer to theboundary
values of F from the sides Im >0,
Im 0 in itsprimitive analyticity
domain(AF being by
definition F + -F - ).
-(2) See e. g. [2~] ] [79~] [79b] and the original references therein.
Vol. 45, n° 2-1986. ~
182 J. BROS AND B. DUCOMET
The mass shell
integral
at the r. h. s. of(5)
runs over theS2-sphere:
2
k ~
-0, - 2 - k - 4 2 > 0;
> > a s in2 [2d], we denote b
we denoteby
Y * this « mass shell convolution» and rewrite(5)
in short notation:§
2.2. - Theprimary
ideaconcerning
theBethe-Salpeter equation [1]
expressed
the ambition ofconsidering
in ageneral (axiomatic)
sense thefollowing identity
between formal series ofFeynman graphs
Fpert denoting
theperturbative
series of connectedgraphs
andGpert
thecorresponding
subseries oftwo-particle
irreduciblegraphs
in the channel( 12, F2’).
If one substitutes F toFpert
and ahypothetic
kernel G toGpert,
and if one
applies formally
theFeynman
rules toone obtains a
Fredholm-type
relation between F andG, depending
onthe
parameter k, namely :
ror in short :
However,
for theoriesinvolving
renormalization(of
the type4>4
in dimen-sion
4), equation (7)
ismeaningless
as it iswritten;
theintegral
at the r. h. s.is indeed
supposed
to present alogarithmic divergence,
since all thegraphs underlying
F and Gproduce
adegree
0 in the powercounting
atinfinity,
while OJ
= -H~~i)
xH~(~)
would contribute with ade g ree -
4. Thisfact was taken into account in
[1] ] and [4 ],
where it was said that the iden-tity (7)
had to be understood aspertaining
to a «regularized »
version ofperturbation theory.
In the
rigorous approach
ofMany-Particle
StructureAnalysis,
thisdifficulty
was overcome as follows[2 ] :
a new measure~)d ~
is substi-tuted to where :
l’Institut Henri Poincaré - Physique théorique
and the function xp is an
analytic cut-off,
which has sufficient decreaseproperties
atinfinity
in euclidean space and isequal
to 1 on the(complex)
mass shell. A choice which offers a
large possibility
of behaviour atinfinity (it applies
as well toWightman-type
theories as to local observable theories of theAraki-Haag-Kastler type)
is : = with p > 0.The formal
Bethe-Salpeter equation (7)
is thenreplaced by
thefollowing
«
regularized Bethe-Salpeter equation » (where
~p denotes theintegration operation
similar to that involved inEq. (7),
with coreplaced by (Dp):
The latter is a
genuine k-dependent
Fredholmequation
which allowsone to define
G .A. (k;
z,z’
as a ratio ofanalytic
functionsp( ’ ’ z’)
where
N p(
> >)
Yp( )
and
Dp
are Fredholm series with respect to F.Np
has beenproved
to beanalytic,
andGP meromorphic,
in the domain ofF,
the crucial propertybeing
thefollowing
one[2 ] :
PROPERTY
(I).
- LetGp
andGP
be theboundary
valuesofGp,
at kreal,
from
therespective
sides Im > 0 and Im 0 in the domainD4 ;
and letOGp
=Gp - Gp
be thecorresponding discontinuity.
Then theasymptotic completeness equation (5) for
F isalgebraically equivalent
tothe
irreducibility of Gp, namely
to thefollowing
condition:It is worth
noticing
that this property is sharedby
all kernelsGp
definedby (9)
for any choice of the function xp. As a matter offact,
any such irre- ducible kernelGp
is suitable forstudying
thetwo-particle
structure ofF, namely
theanalytic
continuationproperties
of F around thetwo-particle
threshold
k2
=(2,u)2 (see [2 ]).
Weemphasize
that such afamily
of regu- larizedBethe-Salpeter
kernelsGp
can be associated with anytheory
inthe
general
axiomatic framework mentioned above.Now,
it isonly
intheories without u. v.
divergences
that theregularized equation (9)
has ameaningful
limit(7)
when theregularization
is removed : the limit(p ~ 0)
of the
regularized Bethe-Salpeter
kernel is thenuniquely
defined and canbe called the « exact
two-particle
irreducible kernel ». As shown in the in dimension2,
this exact kernel is evendirectly
accessiblein the functional formalism
[12 ],
and its irreducible character can be used to establish(for
thesemodels),
the property ofasymptotic completeness
in the
two-particle region [7~]:
the property(I) quoted
above is indeedapplicable
to thislimiting equation
as well.In theories
involving renormalization,
one can set thefollowing problem:
define suitable conditions so that a renormalized
Bethe-Salpeter equation exists, namely
a relation between the connected(amputated) four-point
Vol. 45, n° 2-1986.
184 J. BROS AND B. DUCOMET
function
F,
thetwo-point
functionH(2)
and a « renormalizedtwo-particle
irreducible kernel » G. The fact that F is a functional of
H(2)
and G(only)
in renormalizable theories of a certain restricted
type
will be established in section3,
in the framework ofperturbation theory (see proposition 1).
3. THE FORMAL EXPANSION OF F WITH RESPECT TO A RENORMALIZED BETHE-SALPETER KERNEL G
§ 3 .1.
- For anytheory involving
asingle
field with even interaction(3),
the
perturbative expansion
of the connected andamputated four-point
function F admits the
following decomposition,
withrespect
to agiven
v
where the
Fn correspond
to thefollowing
classification ofFeynman graphs.
G denotes the
perturbative
subseriescorresponding
to allFeynman graphs
which are «two-particle
irreducible » withrespect
to the channel(1, 2, l’ 2’);
and for each n2, F"
denotes theperturbative
subseries cor-responding
to allgraphs
which are «two-particle
reducible of order n » withrespect
to the channel(1, 2; 1’ 2’) : by definition,
such agraph ~
admitsa structural
chain,
which is agraph
containing (n -1 )
two-linedloops ll,
..., and n « bubble vertices »..., at each vertex v~ a
two-particle
irreduciblesubgraph
issitting,
and
arbitrary propagator subgraphs
are carriedby
the internal lines of~n.
Considered as a formal series of
Feynman graphs, Fn
hasobviously
thefactorized structure
however,
it isonly
in aregularized
formalism(or
in atheory
withoutrenormalization)
thatF"
isgenuinely
a « ~-convolutionintegral » [2] ] [14 ], namely
the ~-iterated kernel associated with G :(3) All the N-point Green functions with N odd o thus vanish.
Annales de l’Institut Henri Poincaré - Physique ’ theorique "
in this case,
Fn
issimply
the term of the Neumann series of the Fredholmequation (7);
it ispurely
a functional of G and(through co)
of thecomplete propagator
H(2).In the framework of renormalized
perturbation theory,
eachFn
is definedas the formal series of all renormalized
Feynman amplitudes corresponding
to the relevant
graphs (i.
e. irreducible for n =1,
reducible of order n for n2);
each individualamplitude
isgiven by
the(convergent) integral
of the
corresponding
renormalizedintegrand,
in the sense of[15 ].
Inparticular FI ==
G is the renormalizedBethe-Salpeter
kernel(in
the per- turbativesense).
We now state the
following :
PROPOSITION 1. 2014 ~
sufficient condition for
each termFn (n > 2)
tobe
purely a functional of
G and H(2)(defined
in the senseof perturbation series) is
that the « renormalization parts »of the theory (in
the[15])
are all the proper
(4) graphs of the two-point and four-point functions.
Underthis
condition, F" is formally a
renormalized y-convolutionintegral
asso-ciated with the graph yn
= whose nvertex
factors
areequal to
G and whose linefactors
are allequal to
H(2).Proof
2014 Let y be anarbitrary
reduciblegraph
oforder n,
7i, ..., Yn itstwo-particle
irreduciblecomponents, sitting
at the vertices ... , vn of the associatedchain ~; 6~11~, ... , 6,~,°‘n ~ 1~
will denote the pro- pagatorsubgraphs
carriedrespectively by
the lines ai of theloops li (1
i ~ ~ -1) of n (each
aibeing
a two-valued index : at = 1 or2).
The
corresponding Feynman amplitude FJy] ]
which contributes toFn
admits a renormalizedintegrand (see [15 ])
of the formwhere each term is labelled
by
a « forest » U of y,namely
a set of sub-graphs
of y which arenon-overlapping (i.
e. either nested ordisjoint)
renormalization
parts;
the forestcorresponding
to the empty set labels the term which is the non-renormalizedintegrand
ofFn [y ].
If the renor-malization parts of the
theory
aresupposed
to be all the propergraphs
(4) We mean all the graphs which are completely one-particle irreducible, at the exclusion of the single vertex graph
X
2-1986.
186 J. BROS AND B. DUCOMET
with two or four external lines and no
others,
then the renormalization parts in~n
are(5) :
i )
all propersubgraphs
of with two or four external lines(including
thegraphs yi (1 ~
i ~n)
which are notsingle-vertex graphs).
ii)
all propersubgraphs
with two or four external lines of all the pro- pagatorsubgraphs Qi"1~, Q2"2~, ...,~1~ (including possibly
thelatter).
iii)
All propersubgraphs
of y with four external lines whichcontain p
consecutive components Yr,
yr + 1, ...~,
yr + p -1(2 ~ ~ - ~
+1 ) ;
any suchgraph
admits a structural chainisomorphic
tonamely
the sub-graph
of~n
with vertices vr, ...,(with
1 ~ r n -1).
It then follows that each forest U of Y can be seen as a certain forest
U(~n)
ofé#" (namely
the subset of elements of U which are of typeiii)), completed by
internal forestsUint
of thesubgraphs
7i,...,~ and7~
...,7~i~ (i.
e. elements of U of typesi)
andii)).
By writing = 03A3.03A3
in(13),
andperforming
first the summa-U U(Gn) Vint
tion
one obtainspartial
renormalizedintegrands
for thesubgraphs
Uint
03B31 ... y", ... which can be
integrated
in thecorresponding
internal variables and
yield
vertex factorsF(Y 1),
...,F(yJ,
and line factors~~)...7r(~Y~).
The
amplitude Fn [Y ] = Ry
then reduces to anintegral
over theloop
certain renormalized ~-convolution
integrand (see [9 ])
associated with~n (namely
a certain sum of termslabelled
by
the forestsU(~n)
of and definedby
a directgeneralization
of Zimmermann’s
prescription [15 ],
in which the various propagators and vertexcoupling
constants arerespectively replaced by general
two-point
andfour-point functions) :
in the r. h. s. of
(14),
the notation expresses the fact that the renormalizedintegrand
is a(multilinear)
functional of the vertex factorsF(yi),....
and of the
loop
factors :e) Note that no four-legged subgraph can be overlapping with Yb
...,y,,,
since thelatter are two-particle irreducible.
l’Institut Henri Poincaré - Physique theorique