The Wightman axioms for the weakly coupled Yukawa model in two dimensions

Abstract

We prove the convergence of a cluster expansion for the weakly coupled Yukawa model in two dimensions.

Full text

Communications
in
Comraun. math.
Phys. 51, 297—313 (1976)
Mathematical
Physics
©
by Springer-Verlag 1976
The
Wightman
Axioms
for the
Weakly
Coupled
Yukawa
Model
in
Two
Dimensions
J.
Magnen* and R. Seneor*
ZiF,
University
of
Bielefeld, D-4800 Bielefeld
1,
Federal Republic of Germany
Abstract.
We
prove the convergence
of a
cluster expansion
for
the
weakly
coupled
Yukawa
model in two dimensions.
I.
Introduction
and
Results
The
purpose of this paper
is
to prove the convergence of a cluster expansion [8,
3]
for the
Yukawa
model in two dimensions1. We use here the model
as
defined by
Seiler [13] and McBryan [10], and we shall use the presentation
of
Seiler and
Simon
[14].
The
Yukawa
model has been also studied
by
Glimm [4], Glimm and Jaffe
[5] and [6], Schrader [12], Brydges and Federbush [2] and Brydges [1].
In
this introduction we define the problem and state the main results,
in
the
second chapter we define and
give
the properties
of
our main tool:
a
set
of
de-
coupling functions allowing
to
do the cluster expansion—see also [9] —,
in
the
last chapter we prove the convergence
of
the cluster expansion.
Let
us
give
some definitions, see [14].
The
partition function in
a
volume
A is:
ZA=
j^detren(l+XJ.
(I.I)
The
unnormalized Schwinger functions in
a
volume
A
are:
=
f
dμ |dcttfc SF (iP2
+
m2)-^gi,
(p2^?)3/4
ft*)}
Π
where:
*
On leave
of
absence from Centre de Physique Theorique, Ecole Polytechnique, F-91120 Palai-
seau, France
1
A. Cooper and L. Rosen have shown also the same result [17]

298 J. Magnen and R. Seneor
and
iP is the gradient operator KΛ is the operator in L2 of kernel:
KΛ(x,y)
(P2 +
m2)112
λ
is the coupling constant and is real, m is the mass of the fermion, Γ = iy5 in the
pseudoscalar case and Γ = ί in the scalar case, Λ(x) is the characteristic function
of the volume A.
Also:
with
and
Trreg
:K2
Λ\ is defined as in [14] and we note:
Trreg
:K2
Λ: = j dxdy :φ(x)bieg(x,
y)φ(y):.
Also
dμ is the Gaussian measure of mean zero and covariance
+K)
exp f^ (- l)n- ΎτKn
Finally fl9 gp hk are functions in suitable spaces (defined later), and for convenience
we suppose that their supports are localized in unit squares of a lattice cover oϊR2:
R2=
(J AΛ, Aa is the unit square centered at α.
αez2
Let:
where χ^ is the characteristic function of the unit square A. Then:
^=
Σ
sΛ(fi,...,fn
,8Ίι,~.,ffl Mι,-M'')
<*l,βk
with
QtikSF(Ui,
k) fl
Φ(fi)
detren(l
+KΛ). (1.2)

Wightman Axioms for the
Weakly
Coupled
Yukawa
Model 299
We construct the cluster expansion for SΛ (all the functions, /, g, h having their
support in unit squares); SΛ is obtained by resummation.
Remark. In the sequel we shall take the boson mass m to be equal to m, because
in
the analysis of the convergence of the cluster expansion the two masses play
the
same role.
Let us call Z2* the sides of the squares in the net defined by the lattice cover
of R2. To each
beZ2*
is associated a variable sb9
O^s^l.
For each choice of
{sb}beZ2*wQ
define s-dependant quantities:
C(s) (x, y) = Σ H(s; Aa9 Aβ9 Ay)M(Aa9 Aβ9 Aγ)Cm(x, y)χΔJ,x)χΔβ{y)
α,/?,yeZ2
K{s)A(x9 y) = Σ H(s; 4* Δβ9 Ay)KAynΛ(x, y)xAp)XAfi(y) -
a,β,yeZ2
The
definition of breg(s) (x, y)
follows
in a natural way from the definition of K(s).
Then
if in formula (I.I) and (1.2) we replace all the quantities by the corre-
sponding s-dependant quantities this defines Sis) and Z{s).
In
Chapter II we define H(s; Aa, Aβ, Ay) and E(Aa, Aβ, Ay) and prove:
Lemma
I.I. 0^tf(s; Aa, Aβ, Aγ)^ί9 0^M(Aa9:Aβ9 Ay)^ί9
ΣM(Aa9Aβ9Ay)=l.
b
=
WbthenH{s;Aa9Aβ9Ay)ϊΞl.
Let
DCZ2*
defined
by D={beZ2*\sb = 0} and let R2\D =
X1U...UXp
be the
decomposition
of R2\D as a
disjoint
union of
connected
components
then:
H(s; Aa9 Aβ9 Ay)=ΣH(s; Δa9 Δβ9
Δy)\Xh
k
where
Ws;Aa,Ap9Ay) if
AaCXk9ApCXk,AyCXk
It
is then clear that C(s) (x, y) = 0 C(s) (x, y)\Xk and also that K(s)Λ= ^ K(s)\Xk
ere K
Then
k
k
[where K(s)\Xk means the restriction of K(s) to I?(XJ].
and
this is obviously true also for detren(l-fi£(s)) because ΎrKq decomposes
itself
in a sum ]Γ Ύr(K\Xk)q.
k
Also
i, hk)=±U
tetjiSAgp
ft,)
k
where for k given the determinant is formed with the function:

300
J.
Magnen
and R.
Seneor
Then
with obvious notation we have:
z=γ\zXk
s=γ\s\Xk.
k
xk
As a consequence, according to the general scheme of the cluster expansion
(see [8] and [3]), one has a convergent cluster expansion if one can prove that for
some values of the parameters λ and m(m^l):
Ax)
A being a unit square. Z{s) Δ>0 for sb = 0bedA.
A2)
Let
ΓcZ2*
and cfnote \\ —. Let X be one of the connected components
ber dsb
ofK2\(Z2*\Γ)then:
s Π
ll/ill-i
Π
Mjhjhjh.oiίf
o(irl\(n(Λ)\)1/2e~QllΓι
ί=l
j=1 J
where \Γ\ is the number of bonds in Γ, \\f\\2_1=j 2 dp9 n(A) is the number
of f{ with support in A and βx is some positive constant large enough.
Also
for a
matrix
\A\= sup \A^.
Indeed,
giving
us /lo>0, there
exists
m(i0) such that for
|A|^/l0,
λ real and
m^m(λo)A1
and ^42 are satisfied2, and even by taking m large enough one can
take Qγ as large as we want.
We now want to show that one can bound the norms of the g's functions by
norms
of the initial functions g.
First
suppose that suppgCA, and that AnAa = 0. Let ηΔ be a CQ function
such
that ηΔ{x)=ί if xeA and such that distΐsupp^, Aa)>j (if
ΔnΔa
=
0).
We
have
\\g ||L2-
defining D: = (P2 +
m2),
then:
where
Using then the Theorem 2.2 of Seiler and Simon [14]:
where QΊ is as big as we want if m is big enough, and d(Δ, Λα) = sup(l, dist(/l, Δa)).
Remark
that
the
theory
depends
only
of
the
ratio
λjm

Wightman
Axioms
for
the
Weakly
Coupled
Yukawa
Model
301
Then
if
zlnzJαΦ0,
we use
ll$αH2L
= ί \(P~llArg){-
We have thus obtained that:
II
L = J \Vυ
g)(X)\
XAa\Λ)UA^
||t/||_i/2
Using then
the
fact that ^^"d(J Jβf)^O(l)
we
obtain
for the
original Schwinger
functions bounds
in
||#||_1/2
and
||/Ϊ||_I/2>
and the exponential decrease between
the
support
of
the
g
and
h
give
exponential decreases between the support
of
the
g
and
h
using
the
exponential decrease:
e-{QΊ-l)d{Δ,Δa)
Then
as a
consequence
of Aγ and A2 we get
Theorem
1. Let
λo>0be
given.
Uniformly
in s,
and
λ,
\λ\<λ0,
these
exists
m
large
enough
such
that:
lim SMΛf'>g;h)
exists
and
is
bounded
by
θ(i)"θ(if
Π
ILΛII-i
Π
iω-i/2llM-i/2ΠW^)01/2
i=l
k=ί A
Moreover,
there
is an
exponential
clustering
which
is as big as we
want
if
m
is
taken
large
enough.
Finally under the same conditions
as for
Theorem 1, and
SΛ
being defined
as in
(I.I),
one
has:
Theorem
2.
The
infinite
volume
limits
lim
Z^
ιSΛ{f\ g\
h)
exists
and
satisfy
all the
A—>oo
Osterwalder-Schrader
axioms,
including
an
exponential
clustering.
As
an
obvious consequence:
Corollary
1.
There
exists
a 2
dimensional
Yukawa
relativistic
theory
satisfying
the
Wightman
axioms
and
possessing
a
mass
gap.
Theorem
2
follows
from Theorem
1.
Indeed
we
proceed
as in
[9].
We
define
new Schwinger functions
SΛ;Y, for Y a big
square union
of
lattice squares
and
containing
Λ, by:
SΛ;Y: =
S{s)tΛ
for
sb=l
if
beBγ
and sb
=
0
otherwise.
Bγ
is the set of
lattice lines strictly contained
in Y.
Then
by the
equivalent
of
Proposition IV. 1.3
of
[9]:

302 J. Magnen and R. Seneor
On
the other hand from Theorem 1, it
follows
that
lim lim ——
Λ-»oo
Y->oo
^Λ Y
exists; this proves the existence part of Theorem 2. Now since this limit is also
the
limit of the theory defined by Seiler [13] and McBryan [10], it is obvious
that
all the Osterwalder Schrader axioms are satisfied.
It
remains thus to prove Aγ and A2. The proof of Aγ results from Seiler [13].
In
fact
C{s)(x,
y)\Δ and K{s)(x, y)\Δ are proportional to C(x, y) and K(x, y)9 x9 ye A.
Thus
the proof of Z{s)Δ>0 reduces
itself
to the Seller's proof of ZΦO for the
volume Λ = A, i.e. to have
||J£|J|4<1.
This condition is obtained by taking λ/m
small enough.
The
remaining of the article is devoted to the proof of A2.
II.
The
Combinatoric
of the
Cluster
Expansion
We
give
now the explicit form of H which is a function of a parameter mί9 and of
M which is a function of another parameter m2:
Definition.
Let mΐ>0 and m2>0:
Hmi(s;Al9A29Δ3) = £ Π sb Π (^~sb) - *'— - 3?— (H l)
γCZ2*bey
bφγ ^m^A
±
A 3)
Cmi(A
3 A2)
and
Mm2(Al9
A29 A3) = Em2(Al9 A29 A3) [£ Em2(Al9 Δ2, zlj"1
Um2\Δl>
Δ2>
ΔV=ze
CJnι(x,y)= je~miT § Y[ J\{z)άzΎ
xydΎ
beγc
where dzτ
xy is the Wiener density for the paths in R2 and
τ
JO if z(τ)eb O^τ^T
b
[1 otherwise
Cy
m,{Δa9
Aβ)= J
Cy
mι(x,
y)χΔΆ{x)χΔβ{y)dxdy, Cmί = C%
Proof
of Lemma LI. It is obvious. In particular H being a convex sum of non
negative quantities smaller than one, we have
O^H^l.
IfsΞl then in formula
(Π.l)
the only contribution to the sum over y is for y = Z2^and so H=l.
Finally remark that if σmι(. .)= Σ^ί )\xk then σmi(Δx A3)σmi(Δ3; A2)
k
is equal to zero if Al9 A2, and A3 don't belong to the same Xk. This finishes the
proof.
We now reduce A2 to a proposition whose proof is given in Chapter III. This
is obtained through a lemma showing that HmιEm2 has essentially the same com-
binatoric properties as Dirichlet covariances, see [8]. Thus let us consider

Wightman Axioms
for the
Weakly
Coupled
Yukawa
Model
303
The
d/ds derivations acting
on C(s) or K(s) are
localized:
ds
tή' a ds
with obvious notations
γK(s)=
Σ
γH(s;A,A',A")χAKΔ,,χA,.
US
A,Δ' ,Δ"
aS
We
then
can
write
(see [8]):
=
Σ Σ
8»H(s;ΔJ,Δ'J,Δ'i)ΣlRdμ
(112)
πe^(Γ)
A\,A\,A'{
R
where
R
contains
L, K, or C. M
localized
in (Δi9 A'i9
Δ") and
SP(T)
is
the
set of all the
partitions
of Γ.
Now
the
following lemma summarizes
the
cluster expansion estimates:
Lemma
II.l. Let bj be an
arbitrary
element
of γj9 for j=
ί,...,
L and let Q4 and Q3
be
any
positive
constants,
then
there
exist
a
positive
constant
Q2 and m1 and m2,
m2>m1
such
that:
Σ
Σ Π
{d>>H{s;Δj,Δ'J,Δ'flE(ΔJ,q,Δ'j)e-teέtoW' Δ*+«b>>W}
πe@(Γ)
Δι,Δ\,Δ'{
j=ί
π
= {yι,» ,yL) : i
ΛL,Δ'L,Δ'£
Se~Mr\.
{113)
Proof. Let
yCyo>
we
define:
SXΔ,
Δ')=
Jί e~^τ J Π
(1
-
Jΐ(z))
Π
Jl(z)dzτ
xyχA(x)χAy)dxdy.
beγ
γ$
Then:
A
A' A»Λ V TΛ Tin \ V
s;Δ,Δ',Δ)=Σ
Π^ΓK1"^)
Σ a
(Δ'Δ")CUΔ'-Δ")
yc
T
yiny2
=
0
Using
A"
and
dγσ^8yC we
obtain:
Σd>H(s;Δ9Δ',Δ")E(Δ,Δ',Δ")^O(l)
Σ sup
^Cmi(Δ;Δif)
A"
yiuy2=y
A"
.
e~
{m2-mi
-
2)[d(A,A")
+
d{A',A")]
ffiQ
iβ' A")
In
formula
(II.3),
for
any
πe^(Γ),
π={yl9
...,yL}
let bt be
any
element
of yt
then:
L
Σ
T]
e~[d{bj
{Ai,Δ[}
j=ί

304
J.
Magnen
and
R.
Seneor
To
finish the proof it
is
then sufficient to prove
X Σ sup flW'C^AJdtC^AJ
πe^(Γ)
γ
uyί
=yt
Δι,Δ[,Δ0Cι
i
=
l
π
= {yι,...,yL) yιny[^ϋ
,
e~
(m2-mί- 2)[d{Δι,AXι) +
d(A[,AXi)]
e(Q3+ l)[d(bι,Δι) + di^^)]
Q^\
e~
Q2Ί
^
e~
Qo\Γ\
where
bt is
any element
of
yt . Suppose that bfeyί5 we have
If y£ = 0 (or
y'i
= 0) then we use: Cmί(Ai. zJα):g0(l) and we can choose the param-
eters such that:
0(1)e~Qy2e2{Q3 +1^e~{m2~mi~Q3~^d^Δ^Δ^<
\
so that
in
the
remaining
of
the
proof
we can
forget about
the
empty
yt.
We
symmetrize:
let now
bt
(resp.
fo
)
be an
arbitrary element
of
yt
(resp.
y )
then
for
7iΦ0
we
define:
ΔιJι
.
e2{Q3
+1) O(\)e~Q2/2
g^i.^i)
+
^.A)
andforyί
=
0,/(0)
=
l.
To
prove
(II.3)
it is
sufficient
to
prove
that:
Σ
Σ
ϊ\O(ϊ)e-«b"bi)f(vdf(Ϋι)Ze-°<W.
(Π.4)
We assert then that the l.h.s. in (II.4)
is
smaller than3:
Σ
0(1)"2" flJ(γt)
πe0>(Γ)
ί
=
l
π
=
(yί...yP)
the
proof that this
is
smaller than
e~Q4^
for
a
correct choice
of
Q2, m1(m2>mί)
is given in the references [8] and [15] and
is
one
of
the main combinatoric tool
of the cluster expansion. This finishes the proof
of
the lemma.
With this lemma the proof of
Λ2 is
reduced to the proof
of:
Proposition
II. 1.
For
given
Q2, λ0,
and
m2
(large)
there
exists
Q3
and
Q6
(independant
of
the
parameters)
such
that
for m
large
enough
and \λ\<λ0 (see
formula
II.2):
L
sup
sup sup Π
{e-Q3id(Δ''bJ)+d{bJ-ΔWeQ2}
L
bjeΓ
Δi,Δ\,Δ'{
ί
=
l
j=l,...,L
i !
ΓT
^[diΔj^
+
diΔ^Δ])]
ΓT
e
2m2d{Aj'A'j)Y\[Rdμ\
derived
k
derived
C
R
^0(l)"0(lfnW^)!)1/2eQ6|r|
ΓΊ
Il/ίll-i
Π
II^IIL2||A;IIL2
(Π.5)
A
;
=
i j
=
i
Remark. We have used that:
3 It has
been pointed
out by Lon
Rosen that this inequality
in
a
preliminary version
and the
corresponding formula
in [9]
are
incorrect

Wightman
Axioms
for the
Weakly
Coupled
Yukawa
Model 305
and
R is defined by what we obtain after performing the derivations up to the
factors dyH and M.
Also
in the formula above 0(1) in 0(1)" depends on Q2.
Then
A2
follows
from the fact that one can take Q4 as large as we want by
choosing m2 (and thus m) large enough and define Q1 =
Q4.-Q6.
III. The
Cluster
Expansion:
Proof
of
Proposition
II. 1
First
let us see what we obtain when we do a derivation
d/dsb,
beΓona Schwinger
function
S(ω)
S(ω)=
Π detjkSF(xp yk; φ) Π Φ&) detren(l +K)
ω(x1...;yί,...;t1,...)dμYldxjdyJY\dti
(III.l)
j i
where SF(x, y\ φ) stands for the kernel of (l+K)~1, and ω is some function with
each
argument localized in some unit square of the lattice cover.
Also
for simplicity
since we look at some algebraic aspects we omit any reference to λ, Γ of A.
Acting on an expression of the form j Rdμ where R and dμ depend on s, the
derivation d/ds produces two categories of terms:
d
R
being of the form of the integrand in
(III.l)
one sees that one has essentially to
know the effect of d/ds or d/dφ on SF(x, y φ) or on detren(l + K).
One
gets:
|detren(l+K)
=
detren(l+K)
h
dxdyχΔi,x)χΔβ{y)
Uβ
Σ J
dwdzχAγ(z)χAό(w)SF(y,
z;
φ)K(z,
w)K(w5
x)ψ (x, y)
y,δ
aS
—
SF(x,y;φ)=
£ j- ίdz —
(x9z)SF(z9y;φ)χ/igί(x)χΔβ(z)
+
\dzdw
χAoc(z)
χAβ(w) K(x, z) — (z, w) SF(w, y φ)
-Σ\dzdwdv du
χAoc(v)
χAβ(u)
χAγ(z)
χAδ(w)
SF(x,
z; φ)K(z, w)K(w, v) -^ (v, u)SF(u, y; φ)
.
Λ C C Λ C
—
.Λ^p —
dFΛ2δF
.

306 J. Magnen and R. Seneor
The
formula of derivations for d/dφ(z) are the same with —— replaced by
as
dK(x,y) =
21~1/4
We note δ/dφ detren(l + K) = detren(l + K) {Tr SFA'2 + A'3}
ASF =
A\SF-SFA'2SF.
By their definition the A{ or A[ are completely localized expressions. In each of
these terms between any two localization squares there is
always
a chain of
"propagators":
or
,... , ... -F+m
(P2+m2)l/2
Each
boson propagator, each boson
field
and each propagator is localized.
The
At and A\ are polynomials in the boson
field
of degree 3 at most, and
between any two such
fields
there is
always
a chain of "propagators". Each At
or
A\ has at most 3K. Finally, acting on SF a derivation generates an expression
or
order 2 in SF, and acting on detren(l + K) an expression of degree 1 in SF.
Before estimating the number of terms produced by derivation we have
first
to
reorder them in
view
of
preserving
the antisymmetric structure since it is
essential for the volume dependance estimate ([13,10]). Therefore after each
derivation we perform the
following
operation:
-
when the derivation acts on detί/ciSFdetren(l+i£) put together the terms
which increase the degree in SF. They form a new determinant of one order higher
(this
can be checked easily).
-
other terms are
left
unchanged.
Now
as a
first
step in proving Proposition Π.l we bound the number of terms
produced
by the derivations. To do this we use the technique of the combinatoric
factors (see [7]).
To
take account of the sum over the localization squares we need exponential
localization factors of the type e
0{l)d{Δ>ΔΊ or e
O{1)d{btA) where A and A' are two
localization squares in an A{ or A[ generated by a derivation
relatively
to sb.
Two localizations squares in an At or A\ are linked by a chain of "propagators".
There
is at most 6 "propagators" by chain. We thus distribute the localization
factors to the "propagators" using:
ii
Δ)+\2
Let L be the number of derived K or C. We take account of these factors by the
following
combinatoric factors:
eO(ί)d{A,A')
derivations
"propagators"
ΓT
eOa)[d(b,Ai)
+
d(b,A2)]
TΊ eO

Wightman
Axioms
for the
Weakly
Coupled
Yukawa
Model 307
where product over the derivations means product over the derived K or C,
Aί,
and A2 being the localizations squares of K(x, y) or C(x, y) and b is one of the
bond
relatively to which K or C is derived.
We
will
not
list
these factors in the
following.
Because of Lemma
II.
1
we don't need to take account of how many times,
and
relatively to what set of bonds a K or C is derived i.e. we count only the
derivations acting on a non derived K or C. In a
given
term we attribute a factor 2
to
each K or C to decide whether it is derived or not. This
gives
a 20(1)L since the
number
of K or C is bounded by 0(1 )L.
Also
for each derived Ai9 A[ we fix each localization square A using a com-
binatoric factor O(l)ed(M).
We are thus ready to compute the number of terms generated by the deriva-
tions.
Giving a factor 2 to each derivation we separate its effect according to the
following
cases.
α)
— acts on everything except the measure dμ.
as
β)
— acts on dμ.
as
We
first
compute the combinatoric factors for the case α. With a factor 0(1)
given
to the derivation, we divide case α in
several
subcases:
oq) — derives SF and we select Λ^p.
as
α2)
We consider the sum of terms of higher degree in SF which form
a
determinant of higher order.
α3)
— derives detren(l + K) and we select A3.
as
α4)
— derives aKorC created by a previous derivation (i.e. a fermion prop-
agator in A{ or Aβ.
•
Let us consider the combinatoric for each case separately (excluding the
localization factors).
Case α4: the combinatoric factor is 1 since the derived propagators have
already been chosen.
Case α3: the combinatoric factor is 1 since there is only one term.
Case α2: the combinatoric factor is 1 since there is only one term.
Case a1: it is the case: SF(x, y;
φ)^(AίSF)
(x, y).
Let i in detikSF(xi9 yk; φ) labels the columns. The term
given
by the determinant
in
case a1 is a sum of determinant each with a column v41iSF. The combinatoric
factors we are looking for control the number of columns (initial or produced by
derivations). With a factor 2 we distinguish 2 subcases:
a) the smearing function for the variable x is a Ai or A[ created by a previous
derivation.
b) the smearing function for the variable x is an original g{ of formula (1.2).

308
J
Magnen
and R.
Seneor
We consider
first
the
case
a). The
function
A{ (or
A[)
has
been produced
by a
ivation
—
and
thus contains:
as
(w> v)χΛ2(v)
or has
been produced
by
χΔl{z)
—— .
localized derivation
—
and
thus contains:
as
The square
A2 is
chosen with
a
localization factor exp{O(l)d(b,
A2)}.
Let Ra;C(A)
be the
number
of
times that
in a) A2 =
A.
In a
given
term
at the end
of
the
expansion,
let nc(A) be the sum of the
number
of
times that
in
——
(u,
υ),
v
is
localized
in A
and
of
the number
of
times that
in
-—
(u,
v\ u or v is
localized
in A.
as
Then
the
number
of At or A\
with
A2
= A
is
bounded
by
nc(A). Doing this choice
Rac
times
we
obtain
as
combinatoric factor:
ac S
Π n
A
A A
We then "attribute"
to
each derivation
in
case
a)
with
A2 = A,a
localization factor
e
2d{b>A\ this attribution
allows
us to use
the
following
lemma:
Lemma
III.l.
\\\nc{A)nc{Δ)
A
[ b su
ch
that
Δ2 = Δ
Π
e
•>
such that
Δ2 = Δ
Proof.
The
first
inequality
is
just
Lemma (10.2)
of [8]. The
second inequality
follows
also
from this lemma
if
one
remarks that: Rac(A)^2nc(A).
The overall combinatoric factor
for
case
a) is
therefore O(l)O(l)L.
Consider
now
case
b). Let
NC(AO)
be the
number
of
functions
gi9
ί=ί,...,
JV
which have support
in Ao (it is
also
in the
determinant
the
number
of
columns
with functions
gt
localized
in Ao. We
choose with
a
localization factor gWMMo)
the
square
Ao
support
of
the
function
g{.
Let
now
RbtC(A)
be the
number
of
times that
in
case
b) Ao =
A.
We
thus
get a
combinatoric factor:
Y[Nc(A)Rb>ciA).
A
Attributing
[as in
case
a)]
to
each derivation
a
factor
ed(b'A\
we
have
at
our
disposal
a factor
e~d{b'A)
that
we
used
in the
following
lemma.
Lemma
III.2.
Δ
b
such that
Δ0 =
Δ
Proof.
One has (see [8],
Lemma 10.2):
TΊ
e-d(b,Δ)
b such that
Δo =
Δ

Wightman Axioms for the
Weakly
Coupled
Yukawa
Model 309
so
that:
iVc(zj)
I [ e =e
b
< 0(l)eO{1){lnNc{A))3 < 0(1)
eO{1)NciA)
but
now Y\ oxp{0(l)Nc(A)} = 0(l)N, this proves the lemma.
A
The
total combinatoric factor for case αx is thus 0(1)0(1^0(1)^.
We now compute the combinatoric factors for case β. Since in this case each
d/ds derivation generates two d/dφ derivations we compute the combinatoric
factor of d/dφ derivations.
With a factor 0(1) by d/dφ derivation we divide the effect of d/dφ in
several
cases in analogy to case α:
oίi) d/dφ derives SF and we select A\SF.
a!2) We consider the sum of therms of higher degree in SF which form a deter-
minant
of higher order.
α3)
d/dφ derives detren(l +K) and we select Ά3.
α4)
d/dφ derives
fields
φ created by previous derivations (i.e.
fields
in Ai or
A[). n
α'5)
d/dφ derives \\ φ(ft) (of formula 1.2).
The
derivation d/dφ is localized in some square Ao (already chosen). Let us
now consider the combinatoric factor for each case. Cases α'l5 α'2, and α3 are as
above.
Case α4: the
field
φ(z), zeA0, which is derived is in some A1 or A[ produced by
We choose Δ2 with a factor e°(^d(Δo,Λ2)t ^ow ajj ^ or ^ generated by deriva-
tions
localized in A2 have at most 3nc(A2) fields, since there is at most 3
fields
in
each
A{ or A[. The combinatoric factor is then:
We deal with this factor as above, see Lemma
III.l,
this
gives
a bound: O(l)O(l)L.
Case α;5: let R(A0) be the number of times that derivations d/dφ, acting on
n
Y[ φ{fi% are localized in Ao. Each time the number of choices is n(A0) (remember
ί = l
it is the number of ft with support in Ao). Thus the total combinatoric factor is
Π
n{A)R{A).
A
Attributing a factor
ed{b>Ao)
to each derivation relative to b, localized in Ao, we have
by Lemma
III.2:
n(A)R{A)
Π e~dib'
b deriving in A

310 J. Magnen and R. Seneor
Finally
we have got that for the cluster expansion, the combinatoric factors are
(see Proposition
II.
1):
O(l)O(l)LO(l)nO(lf Π
eoa)idih*M
+ «bt Δm Π
e°il)d
derivations "propagators"
where d stands for d(A, A'), A and A' being the localization squares of the
"propagator".
Define X0 =
sup{l,/l0}.
Now the following lemma is sufficient to
prove Proposition
II.
1:
Lemma
III.3. Fix mx and m2 as for Lemma II.
1
and let R be an element in the
expansion
of
Proposition
I LI, then
there
exist Q3
large
(depending on m2 and mj,
m
large
enough
and β5>0
such
that:
sup sup sup JΠ e-iQ
bi,ί=l,...,L
Aι,Δ'uΔ'\
R [i=ί
f] e°w" Π eOWd0{\)L\\Rdμ\
derived "propagators"
boson propagators
Δ
ί= 1 j=ί
(III.2)
Q5 can be taken as
large
as we want if m is taken
sufficiently
large
and is independant
of Γ. The 0(1)
factors
in the
right
hand
side
and Q6 are independant of ml5 m2, m,
and Γ.
This lemma includes the combinatoric factors of ]Γ in Proposition II.1, thus
taking Q2 + 3 log/l0 ^ Q5 and the supremum over L, one has proved this proposi-
tion.
Let
us now prove Lemma
III.3.
The integrand
JR
has the general form:
R=
$detikSF(xi,yk;φ)w(x1,..r,y1,...)detren(l+K)dx1...dyί...
where w(x1,
...;y1?...)
is a product or integral of product of gi9 hk, C, Ai9 Λ[ and
To
bound |J Rdμ\ we use:
Proposition
III.
1.
Let ί^i, k^N + 2L, and M (in the definition o/detren(l + K) be
large
enough
depending
on λ0 then
there
exists
Q6>0:
Proof.
Applying twice Schwarz inequality the proposition follows from the work
of Seiler and Simon [14]: their proof applies here since they have also localized
each
K(x, y) in unit squares, the fact of multiplying it by 0^H(A,
A9Af)^
ye A') leaves the proof available up to obvious modifications.

Wightman
Axioms
for the
Weakly
Coupled
Yukawa
Model
311
Note:
Our preprint version [16] was
self
contained and in particular included
a
third proof
of
the linear lower bound, which
differs
from those
of
McBryan
[10] and [11] and
of
Seiler and Simon [14].
It
seems also to
us
that the nice proof
of
McBryan can also be extended to the
s-dependant models described here.
Under
the conditions
of A29
\Λ\
=
\X\^\Γ\
+
ί
(\X\ = surface
of
X). The next
step
is to
bound
J
\\w\\l2dμ.
First
we
estimate
the
effect
of the
functional integration
dμ on
||w||4
by
computing the combinatoric factors corresponding
to
the contractions between
the
fields
φ.
We characterize the
fields
φ by
squares:
1) For
a
field
φ(f) of
formula
(I.I),
the "characterizing square"
is A if
support
of
f{ is in A.
2)
For
a
field
belonging
to
some
At
(resp. A[) the characterizing square
is A
d
t , d
if
A{
(resp.
Aβ
has been generated
by
χA(x)
γκ(^ Ί
resP
bY
XA(X)-ΓC(X>
')
or
by
χΔ(y) — C(
, y)
The
number
of
fields
characterized
by A is
less
than
4n{A)+12nc(A) (since
there
is at
most
3
fields
φ by Ai or
Ά^).
Attributing
a
factor
e°wd{<Δ>Δ')
to each contraction between
a
field
characterized
by
A
with
a
field
characterized
by A', we get as a
combinatoric factor
for the
contractions:
Π
0(l)4n{A)+
12n°<A)\[4n(Δ)+ l2nc{Δ)]
!|1/2
A
Attributing factors e
0{1)d{b'Δ) to the derivations we treat the last term
as
before.
Finally the (localization) factors e
O{1)d{AtAf) can
be
decomposed
in a
product
of localization factors
by
boson propagator and
by
"propagator".
The
total combinatoric factor
for
the contractions
is
thus
for |J
||w||jj>dμ|1/4.
Γί(λ
\nΠ(λ \L TT
(vt(
ΛϊtW2
TT
/,O(l)[d(Aι,bι)
+
d(bί,A'ι)]
ΓT
,yθ{l)d
U{L)
U{ϊ)
[ [
{n{Δ)i)
j|£VLVl
[[ e
A
derivations
boson
propagators
Let
us
consider
a
fully
contracted term:
we
call such
a
term
a big
graph.
A big
graph
is
decomposed
in
small graphs, and each small graph
will
be
estimated
by
its Hilbert-Schmidt norm,
see
[7].
A big
graph
is
formed with:
vertices: the
gt, hk,
and
ft
functions and also the functions
χA, '
boson propagators noted
—
and "propagators" noted
—h—.
The
small graphs are:
^
Ϋ r Y
X
X Z X Z
and
also:
φbτQgφ:
We
first
apply the H.S. norm
to
the vertices
in g
and h, this
gives
a
bound:
7=1

312 J. Magnen and R. Seneor
To
bound this new big graph, we
first
use the technique of [7, 9] and we extract
from each "propagator" and each boson propagator localized in A and A' a
factor
g-ww^')
where 0(1) is taken as
large
as we want by taking m
large
enough.
We then use the
following
bounds:
\χA(pM0(ί)F(p),
{l
[for any 0<ε<l/3 if m is taken
large
enough depending on Q8], and
1(p2
+
m2)1/2
Finally if a boson propagator is attached to a /Γfunction of formula (1.2) we use
•~
~ <
-^—-eQse~Q8.
The bounds for the small graphs are:
pz
+ m
ll*^"
II
H.s.
IIH.S.
= I
For
the small graph with 3 boson lines, because the boson line can
possibly
contract
between themselves we are obliged to consider:
|1H.S.>
These 3 norms are bounded by O(l)(6Γβ8)3 + 1 + iX^.
Finally if A and A' are neighbours or identic it is proved in [14] see also [16]
that:
If A and A' have no intersection then it is trivial that:
From
that we get:
We have then obtained for each term a bound consisting of:
Exponentially decreasing factors for all the localizations, and the decrease
is as strong as we want provided that we take m
large
enough.
A product of norm of small graphs, and for each term the number of small
graphs is smaller than O(1)L non counting the small graphs associated with g,
h and /.
So that each contracted terms is bounded by:
O(l)L\e-e*\Lλ3
0
L\eQr
fl
llϋll-i
Π
ll&MlMz
'Γ

Wightman
Axioms
for the
Weakly
Coupled
Yukawa
Model
313
Thus collecting the various bound and taking
O(l)e~Q8
=
e~Qs we
have:
Π
e°^^+"^^γ[\\M^f[ ll^ll^lb Π
derivations
ί
= 1
j=ί
"propagators"
This finishes the proof
og
Lemma III.3.
References
1.
Brydges,
D.: Boundedness
below
for fermion model theories. Part I. J. Math. Phys.
16,1649—1661
(1975); Part II. Commun. math. Phys. 47, 1—24 (1976)
2.
Brydges,
D., Federbush, P.:
A
semi-euclidean approach
to
Boson-Fermion model theories.
J.
Math.
Phys. 15,
730—732
(1974)
3. Eckmann,J.P., Magnen,J., Seneor,R.: Decay properties and Borel summability
for
the Schwin-
ger functions
in
P(φ)2 theories. Commun. math. Phys. 39,
4
(1975)
4. Glimm,J.:
Yukawa
coupling
of
quantum
fields
in
two dimensions. Commun. math. Phys.
5,
343—386
(1967)
6,
61—76
(1967)
5. Glimm, J., Jaffe, A.:
Self
adjointness of the
Yukawa2
Hamiltonian. Ann. Phys. 60,
321—383
(1970)
6. Glimm, J., Jaffe, A.: The
Yukawa2
field
theory without cutoffs.
J.
Funct.
Anal. 7,
323—357
(1971)
7. Glimm, J., Jaffe, A.: Positivity
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φ%
Hamiltonian. Fortschr. Physik 21,
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(1973)
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Jaffe,A.,
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φ*
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Yukawa2
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quantum
field
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Phys. 70,
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Seiler,E.: Schwinger functions
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Communicated
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A.
S.
Wightman
Received March 8, 1976