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ANNALES DE L’INSTITUT FOURIER
C. DENSON HILL
MAURO NACINOVICH
The topology of Stein CR manifolds and
the Lefschetz theorem
Annales de l’institut Fourier, tome 43, no 2 (1993), p. 459-468
<http://www.numdam.org/item?id=AIF_1993__43_2_459_0>
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Ann. Inst. Fourier, Grenoble
43, 2 (1993), 459-468
THE TOPOLOGY OF STEIN CR MANIFOLDS
AND THE LEFSCHETZ THEOREM
by C.D. HILL & M. NACINOVICH
1. Preliminaries.
An abstract CR manifold is a triple (M, H, J ) where M is a connected
paracompact smooth real manifold, H is an even dimensional subbundle
of the tangent bundle TM, and J a partial pseudocomplex structure on
H; i.e. J : H —> H is a smooth fiber preserving bundle isomorphism with
J2 = — I . We also require that J be formally integrable; i.e. that we have
(1.1) [r^M.r^M] CT^M,
where
r^M = [x + ux\x e r(M, H)} c r(M, CTM),
with r denoting smooth sections.
Let m be the real dimension of M and 2n be the real dimension of
the fiber of H. Then n is called the CR dimension of M and k = m — 2n
is its CR codimension. In this case we say that M is of type (n, k).
A CR map of a CR manifold (Mi,ifi,Ji) into a CR manifold
(Ms, H^^ J^) is a differentiable map (^ : M\ —> M^ such that
(1.3) d^(^i) C H^
(1.4) d^(JiX) = J^X) f o r X e A T i .
Key words : CR manifold - Topology.
A.M.S. Classification : 32F25.
460 C.D. HILL & M. NACINOVICH
A CR map / : M —> C is called a CR function.
If, in addition, (p is a diffeomorphism and (p~1 is CR, then we say that (p
is a CR isomorphism.
A CR manifold of the form (M, TM, J) is a complex manifold (type
(n, 0)) by the Newlander-Nirenberg theorem, and one of the form (M, 0,0)
is simply a real differentiable manifold (type (0,A;)).
Let M be a real submanifold of a complex manifold M, with complex
structure J . For p C M set
(1.5) HpM = TpM H JTpM.
Then (M, ffM, J\HM) is a CR manifold provided that the spaces HpM
have constant dimension. In this case the embedding map b : M —^ M is a
CR map of (M, ffM, J|7fM) into (M, TM, J).
An embedding (resp. immersion) <p of a CR manifold (M, If, J) into
a complex manifold (M,TM,J) is a CR map (p : M —^ M which is an
embedding (resp. immersion).
An immersion (resp. embedding) of a CR manifold (M, ft, J ) of type
(n, k) into a complex manifold M of complex dimension n 4- k is said to be
generic.
DEFINITION 1. — A Stein CR manifold is a CR manifold
(M, ft, J ) such that
(M, Jf, J) has a CR embedding as a closed CR submanifold
of some Stein manifold X.(1.6)
Because of the known results about the embedding of Stein manifolds
[3] we could just as well replace (1.6) by the equivalent condition
(M, ft, J ) has a CR embedding as a closed CR submanifold
of C^, for some N.(1.7)
For example suppose fl, is a domain ofholomorphy in C^, with complex
structure J, and let (M,HM^J\HM) be a closed CR submanifold of f2.
Then it is a Stein CR manifold. Let (M, -H", J) be a CR manifold of type
(n,k). Let H° C T*M be the annihilator bundle of the bundle H. We
consider the bundle T^M = [X + iJX\X € H\. Then the Levi form of
(M, ft, J) at u} € Jf^ is the Hermitian form on T^M :
(1.8) L(o;, Z) == ido)(Z, Z) = -M[Z, Z]),
where a) € F(M, If0) satisfies U}[P) = cj and Z e r^M satisfies Z(P) = Z.
TOPOLOGY OF STEIN ... 461
The equality of the last two expressions shows that they do not depend
on the choice ofuj and Z, and therefore L is a function defined on the direct
sum of the bundles H° and T^M.
DEFINITION 2. — A weakly q-concave CR manifold (0 ^ q < n)
is a CR manifold (M, H, J ) such that, for every uj G H° — {0}, the Levi
form L{uj^ -) has at least q eigenvalues that are <: 0.
Replacing the requirement < 0 above by < 0 we arrive at the standard
definition of a ^-concave CR manifold. Hence every g-concave CR manifold
is a fortiori weakly q concave. Note that weak 0-concavity involves no
condition at all on the manifold. We adopt the convention that a real
differentiable manifold (type (0, k)) is Stein (Whitney embedding theorem)
and is also weakly 0-concave. A complex manifold (type (n,0)) is weakly
n-concave. Any Levi flat {L = 0) CR manifold of type (n, k) is also weakly
n-concave. It follows from the result of [2] that a weakly 1-concave Stein
CR manifold cannot be compact.
1. The topology of weakly g-concave CR manifolds.
THEOREM 1. — Let (M, H, J ) be a weakly q-concave Stein CR
manifold of type (n,k). Then M has the homotopy type of a CW-complex
of dimension < 2n + k — q. In particular
(2.1) Hj{M', Z) = 0 for j > 2n + k - q
and
(2.2) H^ri+k-q(M;Z) has no torsion.
Note that the above theorem interpolates between two classical
results :
1. (type (0, k)) Any real ^-dimensional differentiable manifold has the
homotopy type of a CW-complex of dimension <,k [4], and
2. (type (n,0)) Any complex n-dimensional Stein manifold has the
homotopy type of a CW-complex of dimension < n. Thus we obtain the
classical result of Andreotti-Frankel [1].
Proof. — Assume for a moment the first assertion of the theorem.
Then we have (2.1) as well as
(2.3) Hj{M; K) = 0 for j > 2n + k - q,
462 C.D. HILL & M. NACINOVICH
where K is an arbitrary field. The universal coefficient theorem
(2.4) ^.(M; K) = ^.(M; 1)^K+ Tor[^_i(M; Z), ̂ ]
then yields (2.2).
Since M is Stein we can assume that M is embedded as a closed
submanifold ofCN. Following Andreotti-Frankel we take, for PQ e C^-M
the square of the Euclidean distance
(2-5) ^(P)=|P-Po|2, PeM.
By a standard argument using Sard's theorem, we choose the point
PO so that (p(P) is a Morse function on M; i.e. (p has only isolated
nondegenerate critical points. By Morse theory (see [4], p. 20) M has the
homotopy type of a CW-complex obtained by attaching an r-cell for each
critical point having Morse index r. Hence it will suffice to show that (p has
no critical point with Morse index r > 2n + k - q.
Let P e M be a critical point of (p. By an affine orthogonal change
of coordinates we may assume P = 0 and that M is described in a
neighborhood of 0 by equations of the form
^^ { V 3 = h j ( X l , . . ^ X k , W ^ . . . , W n ) KJ^k
[ Cs =^(^l,...,^Wi,...,Wn) 1 < S < C
where x = (a- i , . . . ,^) and y = (z / i , . . . ,? / / , ) are real coordinates, and
z = x + iy, w = ( w i , . . . , Wn), C = ( C i , . . . , Q) are complex holomorphic
coordinates, with N = k + n + £. Here the smooth functions hj are
real valued, the gs are complex valued, and all vanish to second order
at 0. Thejangent space to M at 0 is ToM = {(a;,w,0)}. As the line
segment PPo is orthogonal to M, the point ?o has coordinates PQ =
(mi , . . . , idk, 0 , . . . , 0, A i , . . . , A^) with the a^ e H and the \s € C. A point
Q € M near P = 0 has coordinates Q == (x + ih(x, w), w, g{x, w)). We have
" k k
^(^E^+E^--^)2i i
+EK12+I>-V
1 1
^w+^+f^l2^^^,
1 1 1
i
-2Re^A,^+0(3).
TOPOLOGY OF STEIN ... 463
The gs are CR functions on M near 0 since they are the restrictions to M
of the holomorphic coordinates <^. Therefore the formal Taylor expansion
of each gs, about 0 is an expansion in terms of the xj + ihj and the Wr.
Thus
(2.8) gs = ̂  a^XiX, 4- ̂  f^ W^x^r + f^ c^w,Wt + 0(3).
i i i i
We write
(2.9) <p(Q) = (^(0) + B(x, w, w) + 0(3)
where B is a quadratic form in re, w, w. To prove our contention, it is enough
to show that the quadratic form B(0,w,w) has at most 2n — q negative
eigenvalues. In view of (2.8) we can write
n
(2.10) B(0,w,w)= ReA(w)+L(o;,w)+^|w^|2
i
where A is a holomorphic quadratic form in w, and L is the Levi form of
k
M at uj = ±2^djdxj\Q.
i
Let W be a maximal real subspace of C71 on which B(0,w,w) is
n
negative definite. On W H V^^IW the quadratic form L(o;, w) + ̂  |wy.|2 is
i
negative definite. Indeed if w and y^lw belong to W we have
n
(2.11) ReA(w)+L(cc;,w)-h^|wr|2 <0
i
and
n
(2.11) -ReA(w)+L(o;,w)+^|wy.|2 < 0,
i
as ReA(\/:rlw) = —ReA(w). It follows, using our hypothesis of weak q
concavity, that W H ̂ /^IW has complex dimension <, n — q and therefore
W has dimension < 2n — q. The proof of the theorem is complete.
3. The Lefschetz theorem on hyperplane sections.
DEFINITION 3. — A project! ve CR manifold is a CR manifold
(M, H, J ) which has a closed CR embedding in CP^, for some N .
464 C.D. HILL & M. NACINOVICH
It follows from the results of [2] that a weakly 1-concave projective CR
manifold, when embedded in CP^, for some N^ intersects every hyperplane.
Let S be such a hyperplane, and set MQ = MUE. We call the closed subset
MQ a hyperplane section of M.
THEOREM 2. — Let (M, H, J ) be an orientable weakly q-concave
projective CR manifold of type (n, k), and MQ be a hyperplane section of
M. Then the natural homomorphism
(3.1) ^(M;Z)^^(Mo;Z)
is an isomorphism for j < q — 1. It is injective for j = q — 1.
THEOREM 2 \ — Dropping the assumption of orientability, the
same results are valid with Zs coefficients.
Proof. — Since Mo is closed in M, we have the exact cohomology
sequence
• • • -^ H^{M - Mo;Z) -. ̂ '(M;Z) -^ ^'(Mo;Z) -
-^^(M-A^Z)-^...,
where the subscript K denotes compact supports.
By Poincare duality we obtain H^{M — Mo;Z) ^ H^n+k-j^M —
Mo; Z) = 0 for j < q. Hence the result follows by (3.2). For the case where
M — MQ is not orientable, we apply Poincare duality for Za coefficients and
argue as above.
THEOREM 3. — Under the same hypotheses as Theorem 2, the
natural homomorphism
(3.3) ^(Mo;Z)-^(M;Z)
is an isomorphism for j < q — 1 and is surjective for j = q — 1. For the
homomorphism
(3.4) J^(M;Z)-^(M,M-Mo;Z),
we obtain an isomorphism if j > 2n + k — q + 1, and an injection for
j = 2 n + A : - g + 1.
Proof. — We consider the exact homology sequence for the pair
(M,Mo)
...-^i(M,Mo;Z)-^(Mo;Z)-^(M;Z)-
^^(M,Mo;Z)^...
TOPOLOGY OF STEIN ... 465
But the Lefschetz duality theorem asserts that
Hj(M, Mo; Z) ^ Jf^^-^M - Mo; Z),
and the latter group is zero for j < q^ again by Theorem 1.
Next we consider the exact homology sequence for the pair (M, M —
Mo)
...-^(M-Mo;Z)-^(M;Z)-.^(M,M-Mo;Z)-.
(o.b)
^^-i(M-Mo;Z)-....
By Theorem 1, Hj{M - Mo;Z) = 0 for j > 2n + k - q. Hence the
desired conclusion follows.
Remark. — When M is a smooth projective algebraic variety (type
(n,0)) we recover the classical Lefschetz theorem on hyperplane sections
along the lines of the Morse theoretic proof given by Andreotti-Frankel [1].
4. Homotopy of projective CR manifolds.
Following Milnor [4] we prove
THEOREM 4. — Let (M.ff.J) be a weakly q-concave projective
CR manifold of type (n,A;), and MQ be a hyperplane section of M. Then
(4.1) 7^(M,Mo)=0 for j < q.
Proof. — (We fix a base point XQ e Mo.) We have a closed CR
embedding of M in CP^, for some N^ and MQ = M D S with E given by
ZQ = 0 in homogeneous coordinates. Let A/^S, 6) and A/^M, e) be tubular
neighborhoods of E and M, of radius S and £, respectively, with respect
to the Fubini-Study metric. For So > 0 sufficiently small, the geodesic flow
determines a deformation retract
(4.2) F6:IxM^8)-^Af(^6)
of A/'(S,<5) —-)• S if 0 < (5 < (5o. For e > 0 sufficiently small, we have a
retraction
(4.3) g:Af{M,e)^M.
Since A/'(M, e) is open, F^^M^M^ e)) is open in I x A/^E, <?), and contains
I x Mo. Then we can find 77, 0 < rj < 6, such that Fs(I x M D A/"(S, rf)) C
A^(M,£).
466 C.D. HILL & M. NACINOVICH
We have CP^ - E ̂  C^ and take (p as in (2.5). Set
f O on E
^= i on M-E.{Î v
Then ^ is continuous on M and is a Morse function on -0 l([ry0o)) for
r > 0. The critical points of ^ all have Morse index > q. Therefore M has
the homotopy type of '^^([O,?'']) with finitely many cells of dimension > q
attached. Then, for j < g, every continuous pointed map / : ( P , 9 I 3 ) —>
(M,Mo) can be deformed to a continuous pointed map /i : {I^QI3) —>•
(^([O.r^Mo). I f r is sufficiently small, then ^([O.r]) C M n.A/'(E,^),
and g o Ff(t^f^{p)) gives a homotopy of /i to a continuous pointed map
/2 : (P ,9P) —> (M,Mo). The proof is complete.
We consider next the exact homotopy sequence of the pair (M, Mo) :
-^ 7T,(Mo) ̂  7T,(M) ̂  7T,(M, Mo) -. 7r,-i(Mo) -. • • •
(4.5)
—> 71-1 (M,Mo) —> Tro(Mo) -^ 7ro(M).
Therefore if Mo is a hyperplane section of a weakly ^-concave projective
CR manifold (M, H, J), then the natural map
(4.6) 7r,(Mo)-^7r,(M)
is an isomorphism for j < q— 1, and is surjective for j = q— 1. In particular,
for q > 2, every hyperplane section of M is arcwise connected (as M is
connected), and for q > 3, every hyperplane section of M has the same
fundamental group as M. Finally we remark that a generically chosen
hyperplane section Mo of M is a smooth submanifold.
5. A remark on the embedding dimension
of projective CR manifolds.
Let (M, AT, J ) be a projective CR manifold of type (n, k). Consider a
closed CR embedding of it into CP^, for some N. Then it may be possible
to reduce the embedding dimension N as follows :
THEOREM 5. — With (M, H, J ) as above we have
{k = 1) : It has a global closed CR embedding in CP271"^2, and a global
closed CR immersion in CP2714"1.
(k > 2) : It has a global closed CR embedding in CP771, where m =
[2n + (3/2)fc] (greatest integer in).
TOPOLOGY OF STEIN ... 467
Remark. — For compact Stein CR manifolds of type (n, A:), precisely
the same results hold, with complex projective space replaced by complex
Euclidean space. For non compact Stein CR manifolds of type (n,fc),
the same results hold, with the word "closed" removed, and the word
"embedding" replaced by "one-to-one immersion".
Proof. — Let M'=={(p, q) eM x CP^IgCCP^ tangent to M at p}.
This is a smooth submanifold of M x CP^ of real dimension 4n -j- 3k. The
map
(5.1) M' ^^q)^qeCPN
is smooth. By Sard's theorem its image has measure zero in CP^ if
N > 2n + (3/2)k. Choosing a point QQ ^ {its range} U M, and projecting
from this point into a hyperplane S not containing Qo, we obtain a CR
closed immersion into a CP^"1.
Next we consider M" = {(p, q, r)|(p, q) G M x M - A, r e CPN and
p, g, r are collinear}. It is a smooth manifold of real dimension 4n + 2k + 2.
The map
(5.2) M'^^r^rCCP^
is smooth and, again by Sard's theorem, its image has measure zero if
N>fln-[-k-\-l.liN satisfies both inequalities, the above CR immersion
can be chosen to be globally one-to-one.
468 C.D. HILL & M. NACINOVICH
BIBLIOGRAPHY
[1] A. ANDREOTTI and T. FRANKEL, The Lefschetz theorem on hyperplane sections,
Ann. Math., 69 (1959), 713-717.
[2] C.D. HILL and M. NACINOVICH, A necessary condition for global Stein immersion
of compact CR manifolds, to appear in Riv. Mat. Parma.
[3] L. HORMANDER, An Introduction to Complex Analysis in Several Complex Vari-
ables, Van Nostrand, Princeton, 1966.
[4] J. MILNOR, Morse Theory, Ann. Math. Studies, Princeton, 51 (1963).
Manuscrit recu Ie 28 avril 1992,
revise Ie 15 juillet 1992.
C.D. HILL,
Department of Mathematics
SUNY at Stony Brook
Stony Brook, NY 11794 (U.S.A.)
&
M. NACINOVICH,
Dipartimento di Matematica
Universita di Pisa
Via F. Buonarroti, 2
56127 Pisa (Italy).


The topology of Stein CR manifolds and the Lefschetz theorem

Grabner, Peter J.

ANNALES DE L’INSTITUT FOURIER
PETER J. GRABNER
Block distribution in random strings
Annales de l’institut Fourier, tome 43, no 2 (1993), p. 539-549
<http://www.numdam.org/item?id=AIF_1993__43_2_539_0>
© Annales de l’institut Fourier, 1993, tous droits réservés.
L’accès aux archives de la revue « Annales de l’institut Fourier »
(http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions gé-
nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa-
tion commerciale ou impression systématique est constitutive d’une in-
fraction pénale. Toute copie ou impression de ce fichier doit conte-
nir la présente mention de copyright.
Article numérisé dans le cadre du programme
Numérisation de documents anciens mathématiques
http://www.numdam.org/
Ann. Inst. Fourier, Grenoble
43, 2 (1993), 539-549
BLOCK-DISTRIBUTION IN RANDOM STRINGS
by Peter J. GRABNER
1. Introduction.
We investigate some properties of infinite sequences of independent
random variables, which take the values 0 and 1 with probabilities p and q
respectively (Bernoulli's scheme). It is one of the basic results of probability
theory that the limit relation
^ #{1 < n <, N - k : XnXn+l . . . Xn-^-k = CLj • • ' Qfc} ^ / ^
holds in probability for all blocks A == a\... a/c of a given constant length
k (/^fc(A) is the A:-fold product measure generated by /^({O}) = p and
/^({l}) = ^)- This result can also be naturally imbedded into ergodic
theory : consider the infinite product space X = {0,1}N equipped with
the infinite product measure /^oo generated by fi. Then the shift operator S
(Bernoulli shift) on X defined by S{x\, x ^ ^ . . . ) = (^2^3?. • •) is an ergodic
transformation on X (cf. e.g. [Wa]) and the above relation is a consequence
of BirkhofTs ergodic theorem.
It is now natural to ask how fast (depending on N) k could grow such
that this relation persists. In order to answer this question we introduce a
The author is supported by the Austrian Science Foundation (project n°P8274PHY).
Key words : Coin tossing — Uniform distribution.
A.M.S. Classification : 60F15 - 60G50 - 60F10.
540 PETER J. GRABNER
special notion of discrepancy (cf. [HI], [KN]) :
(1.1)
D ^ ( X I , . . . , X N )
- 1 #{Kn<7V-fc :XnXn+i.^Xn^-k=ai'-'ak} ( x v—.r^ax.^ rr—T^ ————————————————————-^(A)
A6{0,l}^p^(A)| ^V
The following calculations will show that this is a proper measure for the
distribution behaviour of the sequence x\^ x ^ ^ . . . . Note that this definition
agrees with the definition in [FKT] for p == q = -.
DEFINITION. — A sequence x ^ ^ x ^ , . . . is called k(N)-distributed
with respect to ^ if
lim D^N\x^...^XN)=^
N-—>oo
Our Theorem will show under which conditions almost all sequences
are A;(7V)-distributed. Without loss of generality assume that p < q. The
notation Ipn is the logarithm to base - : Ipn == logi n.p P
THEOREM. — Let k(N) be a non-decreasing sequence of positive
integers. Then the following 0-1-Jaw holds
^(lim D^(^,.,^)=0)={1 if^-lplpn-kW
\N-^oo / [ O otherwise.
oo
It clearly follows from KolmogorofTs 0-1-law or the fact that the set
hm^D^N\x^...^N)=o}
is invariant under the (ergodic) shift S, that the only possible values for
the above probability are 0 and 1. The proof of this theorem will use
bivariate correlation polynomials, which are a generalization of Guibas' and
Odiyzko's correlation polynomials in one variable (cf. [GO]). Using these
polynomials we are able to compute the probability generating functions
of the events we are interested in.
2. Generating Functions.
Throughout this section let A = 0102 . . . OA; be a 0-1-string of length
k. We are interested in the cardinalities of the following subsets of the set
BLOCK-DISTRIBUTION IN RANDOM STRINGS 541
Sr,s of strings containing r digits 0 and s digits 1 :
(2.1)
/A^, s)=#{B e <S^s : B contains A only at the end}
9A(r,s)=#{Be S r , s ' ' B contains A only at the beginning and at the end}
hA(r,s,)=#{BeSr,s''Bdoes not containA}.
In order to compute the generating functions of these quantities we intro-
duce the bivariate autocorrelation polynomial [AA](^,w) :
(1 if a\a^... dk-r-s = ̂ r+s+i^r+s+2 " ' Cik and the\zrw8]\AA](z w}= strm§ ala2 . • • a'r+s contains r digits 0 and s[ J L J v ' / digits 1
0 otherwise,
where [^w5]?^, w) a^ usual denotes the coefficient of ^w5 in P{z, w). We
are now ready to formulate
PROPOSITION 1. — The generating functions of the combinatorial
expressions (2.1) are given by
^ ^0(A)^1(A)
FA{Z^ = ̂ ^s}zrws = ̂ ^^l-.-^AA]^)
G.(z w) = z°^w1^ + ^+W-1)^(A)^1(A)v ' / ^O(A)^KA) + (i - ̂  - w)[AA](^w)
H,^ ^ ̂  _______[AA]^^)_______
A v ' / ^A )w l (A )+( l -^-w)[AA](^w) ?
where 0(A) and 1(A) denote the number of 0's and Fs in A respectively.
The proof of this proposition is analogous to the proof of the corre-
sponding results for ordinary generating functions (cf. [GO]).
Remark 1. — Obviously these results can be generalized to any finite
alphabet.
As in [FKT] we use these functions to compute the probability
generating function (p.g.f.) of all strings containing the substring A exactly
r times :
^A\^ = ———^(P^q^GAipz^zY-1 for r ^ 1
^\z)=HA{pz^qz).
Inserting the results of Proposition 1 and setting
(2.2) p^)=^[AA](p^)
542 PETER J. GRABNER
yields
r-l^ _ .fc ((^(^-^^r
A v / ^(A) ((l-^P^+z^)^1
^(0)^^ p^)
A v / (1-2)P(2)+^'
3. Proof of the Theorem.
We split the proof into two parts; first we show that almost all
sequences are k(N) distributed if Ipn — Iplpn — k(n) —> oo. Using our
p.g.f. results we can write
(3.1)
P-oc (#{0 <, n ̂  N - k : a;n+i... Xn+k = 01 • . • a*;} = r) = p^\N) =
f.(r)f^ dzw< î̂ '?^
In order to be able to estimate the integral we need information on the the
zeros of the polynomial (1 — z)P(z) + zk.
LEMMA 1. — The zero of smallest modulus ZQ of (1 — z)P(z) + zk
is real and positive and satisfies the estimate
ZQ >1+C/^(A)
for a positive constant C only depending on p.
Proof. — As FA.(pz^qz) is a p.g.f. and (1 — z)P(z) + z1^ is the
denominator of this rational function the zero of smallest modulus has
'to be positive and > 1. Investigation of the derivative shows the existence
of the constant C.
Let now
k{n) = Ipn — Iplpn — lp'0(n),
where ^(n) —> oo. We need estimates for the probability that the number
of occurrences ZN (A) of a block A deviates too far from the mean value :
(3.2)
LN(SA) = A<oo(^v(A) < Np,k(A)(l - SA)) and
UN(SA) = /U^v(A) > AT^(A)(1 + SA)).
BLOCK-DISTRIBUTION IN RANDOM STRINGS 543
These probabilities are sums of the p^\N) defined in (3.1) :
LN{SA)= Y^ P^W and
/^ ^ r<A^(A)(l-<!)A)
UN{SA)= ^ P^\N).
r>N^(A)(l+<5A)
We will use the integral representation (3.1) to estimate these quantities.
For convenience we now introduce some notations
Q(z) = (1 - z)P(z) + zk
(3.4) ^ ^ _ ^
^'Q^5 ^'^..(AjoO-
This gives
^) = ̂ a(2)6(')r-l
for r > 1. Observe further that
^^^G^)
(3.5) b{l±£}=l±^+o{-^
V(l ±e)= exp f±-i— + 0 f-^—))V /Xfe(A) \^{A)2}}
(1 ± e)-" = exp (rne + 0(n£2)).
We can now write
TT { K \ 1 J 1 t \ y(/^) dz
^^^^^MA)^!-^)?^'
where j = [7V/^(A)(1 + ^A)]. As all the power series involved have positive
coefficients and because of Lemma 1 we can estimatec'^>^a(•-c)I^)(l-£'-"
for every positive e < C7^(A). Using (3.5) yields
^(^ ̂ ^Lp ((N-———} .+o(^)+0(^)) .
£l+0(-_) V\ ^(A)7 WA)2; ^
\ P ' k y - ^ - ) /
( \pN\^
Inserting e = ^ ^^(A)—— j into the above inequality yields
(3.6) UN^A) < exp(-^ (Niik{A) lp7v)^ + Ci log TV).
544 PETER J. GRABNER
In the same way we treat the lower tail. Let now j = [^fc(A)(l-^)].
Thus we obtain
L^) =±{ ( p ^ + a ^ ^)-n ^_
"v A' 2m Jc \Q(z) • ^(A) b(z) - 1 ) z^+i •
We can now estimate
LN(6A) ^ ̂ (1 + £)-' + ̂ W + ̂  + ̂  + £)-"
Using the same value for e as above yields
(3.7) LN^A) < exp (-(^A (N^(A) \pN)^ + C^ logA^) .
Combining this with (3.6) yields
^ ^(ZN^1-^(A) >^.(A))
(6^) \ l y }
<exp(-6A{N^{A)\pN)^C3\ogN).
^ \ i
now 6 A = 6 [ —-—Let ( p } and observe that ^ = ^^^^(N}
\^(A)/ ' N " /)y
Therefore we have
^oc (^w^) > 6) < 2 /CWexp(-^(7V)ilp^4-C3log7V)
^ exp(-(^(AO^ IpA^ + C'togA^).
(3.9)
We now choose 6 as a function of TV
6=^(N)~^
and observe that
00
^ exp(-^(7v)i Ip TV + C' log TV) < oo.
N=1
Thus by the Borel-Cantelli lemma (cf. [Fe]), we obtain the first part of our
Theorem.
We now have to prove that almost no series are A;(7V)-distributed if
Ipn - Iplpn - k{n) /> oo (we confine ourselves to the case p < -, because
1 2
the case p = ^ has been treated by Grill [Gr]). We introduce a set A
of strings of length k, which have only trivial autocorrelation and do not
overlap each other :
A= ^ 0 . . . 0 A 1...1
[ I Z+d(fc)-2 I
BLOCK-DISTRIBUTION IN RANDOM STRINGS 545
f k}
where ^ = < . >+1 and d(fc) = A; mod 3. We need the p.g.f. (p(z) of all strings
I ° J
not containing an element of A. This function satisfies the equations
ip(z) + <^Ai (^) + . . . + ̂ Am (^) = ̂ {z) + 1
^Ai(^)=^^fc(Ai)^)
^A^)=^WArnM<),
where A i , . . . , Ay^ are the elements of A and (J)AI (z)(l = 1,.. . , m) is the
p.g.f. of the blocks ending with Ai but containing no further occurrence of
any element of A. Solving these equations yields
t3-10' ^i-.^W
Note that the simplicity of these equations comes from the trivial overlap
structure of the elements of A.
Because of this simple overlap structure it is easy to see that
(3.11)
<^l...Jm(2;)
^Oi+...+^)!^^^ .^^ymzk^•••^m^{zyl^•^m+l
Jl- ' ' ' 3m-
is the p.g.f. of all blocks containing A; exactly ji times (I = 1... m). As in
the first part of the proof we use
MN(S) = ̂ oo (l^v(Ai) - N^(Ai)\ < N^(Ai)6A.,l= 1 . . .m)
^ =^i E .̂..̂ )̂
1/0 |j;-N^(A;)|^Ar^(A^^
/ pk Y
where 6 A i = 6 ( ———\^k{Ai)j
We want to treat (3.12) exactly like the corresponding expressions in
the first part of the proof. For this purpose we need information on the
zeros of the polynomial 1 — z + ^.Lk{A)zk.
LEMMA 2. — The zero of smallest modulus ZQ of 1 — z + ^k{A)zk
is real and satisfies
ZQ> l-^^(A).
Proof. — The proof of the first statement is as in the proof of
Lemma 1. For the proof of the inequality insert z = 1 + l^k{A) into the
polynomial.
546 PETER J. GRABNER
Observe now that
^ ̂  ̂ ^^TTT ^ ̂  + ̂ (l + ̂ J-N
for $- ^ ^fc(^)- Inserting e = /^(.4)—— and performing similar calculations
as in the first part of the proof yields
(3.13)
1 { , . j . i dz
^i^' ?""-̂
.̂ <(^M-< )̂
Let now n == N^kW, J = Ji + • • • + 3m and pi = -fc—— and insert^k(A)
(3.13) into (3.12) to obtain
(3.14)
MNW
j , TTt / / j \^ \
5 ̂  ,,,-̂ ,., ̂ -^ ̂ p;'exp (-4^ ̂ ^"t-4'")) •
m
where ^ p^ = 1. Thus we have arrived at an expression that we can treat
^==1
by the normal approximation of the multinomial distribution.
Assume that N runs through a subsequence of N such that
\pN -lp\pN - k ( N ) -^limsup(lp7V-lplpJV-fc(AO) =lpC < oo.
N-^oo
It will suffice to prove our theorem for the case that lim sup Ip N — Ip Ip N -
k(N) > —oo, such that 0 < C < oo. Observe now that N^k{^i}^Ai :
6 ̂ /CNp^k (Ai) Ip N and use Stirling's formula
/n^ ,—— 11 /n\n i——
(-) ^/27m<n! < — (-) \/27m
\e) ~ ~ 1 0 \ e /
BLOCK-DISTRIBUTION IN RANDOM STRINGS 547
to obtain
(3.15)
. . 11 exp(0(fc/^(.4)n)) ^ ^7
( / 10 ^(A)^-1 ^—————~n~
• ' ' \Jl-npt\^S^/CN^(Ai)\pN
33 FT 3. ( ^-n?\x —.———— I I v1 exp I —-————
^•••^m n \ 2n )
11 exp(0{kp,k(A)n)) ^ ____^/TT^____
10n--i2^(A)(27r)^^pi——^ i,2^ ^/( l+^) . . . ( l+^)^
=l,...,m
nx exp -^ ̂ p^? + 0(nrf) + 0 n^p^?
\ ^=1 \ 1=1 ) )
where ji =npi{l+xi) and J = n(l + 77). The terms in the last exponential
( x2 \come from (1 + a;)1"^ = exp ( x + — + 0{x3) ) for x —^ 0 and the
observation that j\ + • • • + jm = J transforms to ^pixi = r]. In the
following we will use p < . which yields ^A; —> 0 for our choice of A (in
1 2
the case p = . we have &A; = 6 and the following arguments cannot bez^
used).
Inserting the definition of ^A; yields the estimate
(3.16)
^ 3 (log^)i^ 1n > pix] < v 0- / > ——
^ ^ ^v^
=0(7V-^^^+^lp(^+^)(log7V)i-^^-^p(7.+7.))
and a similar estimate holds for nrj3. Using an exponential estimate yields
V / ( l+a ; l ) • • • ( l+^ )
=exp(0(A^-^+^^+^ lp(^+^)(log^)^~A^
Inserting these inequalites into (3.15) and setting a = - +
6 log?
^(^T^)^8
MNJS) < exp^N-^^pgN^-^n1^1^'' -p^
^(^)(27^)lr¥A
m \ ^n v-^ n \ 1E l iexp -^E^ (^)...(^)-X
\Xl\<6A, \ ~ ^=1
54^ PETER J.GRABNER
The sum in the last line can be interpreted as a lower Riemann sum for the
integral
/* / m \
ii^^l-^^T'1'"^
using the lattice
{(^i,...,^) 1 ^ = ^ - 1 , \xi\^6^ 1 = 1 , . . ..ml.
Thus we obtain
(3.17)
MN(S) ̂  exp^^-i+^logAQi-") + O^ogN^^S^/ClpN))"1,
where
W = —— F e-^dt ~ 1 - ̂ -e-^
V2n J-x V TT x
for x —> oo. Therefore we can estimate
(3.18)
M^)
^ exp(~vl^mp7^^2g'p^+o(A'"5+a(logjv)t~a)+o(logjv)) •v /
Observe now
/ N \^P2
"llog^J
(3.19) /log^V^^^
^(-4) ^ [-^-)
n ̂  N1^^ ^^(log N)~^ ̂ ^.
Inserting these estimates into (3.18) yields
/ ^jlp2-j<$2C'lpe \
M^(^)^exp^-D^^^^^^+Q(7V-i^(log7V)t--)+0(log7V)j,
where D > 0 is a constant implied by (3.19). The right hand side tends to
0 for sufficiently small 6 > 0, because - l p 2 > - - + a holds for p < 1.
^ o 2
Note that
^(D^N\^)<6)^MN(6).
Thus the proof is complete. Q
BLOCK-DISTRIBUTION IN RANDOM STRINGS 549
Remark 2. — Modifying (1.1) one can also investigate discrepancies
D^^) _
/ W #{1 < U < N - k : XnXn-^-l . . . Xn+k = A}= max \———————————————————————————--LAfc(A),AG{O,I}^ V /^(A) N ^\ ^
where </> is a monotonically increasing function. Then the same calculations
as above yield
,» ( .in, '̂"•*(") = 0) ' { ' if ̂  "̂  - °°VN-'00 / l O otherwise.
This answers a question posed by Flajolet, Kirschenhofer and Tichy [FKT],
Remark 2.
BIBLIOGRAPHY
[Fe] W. FELLER, An Introduction to Probability Theory and Its Applications, J. Wi-
ley, New York, 1965.
[FKT] P. FLAJOLET, P. KIRSCHENHOFER and R.F. TICHY, Deviations from Uniformity
in Random Strings, Probab. Th. Rel. Fields, vol 80 (1988), 139-150.
[Gr] K. GRILL, A Note on Randomness, Stat. and Probab. Letters, to appear.
[GO] L. GUIBAS and A.M. ODLYZKO, String Overlaps, Pattern Matching and Non-
transitive Games, J. Comb. Th., Ser A, vol 30 (1981), 183-208.
[HI] E. HLAWKA, Theorie der Gleichverteilung, Bibliographisches Institut, Mann-
heim,1979.
[KN] L. KUIPERS and H. NIEDERREITER, Uniform Distribution of Sequences, J. Wiley,
New York, 1974.
[Od] A.M. ODLYZKO, Enumeration of Strings, in Combinatorial Algorithms on Words,
A. Apostolico and Z. Galil eds., Springer, Berlin, Heidelberg New York, 1984.
[Wa] P. WALTERS, An Introduction to Ergodic Theory, Springer Verlag, New York,
1982.
Manuscrit recu Ie 13 avril 1992,
revise Ie 23 novembre 1992.
Peter J. GRABNER,
Institut fur Mathematik
Technische Universitat
Steyrergasse 30
8010 Graz (Austria).


Block distribution in random strings

Palamodov, Victor P.

Harmonic synthesis of solutions of elliptic equation with periodic coefficients

Ehrenpreis, Leon

Exotic parametrization problems

Kawai, T.

Stapp, H. P.

ANNALES DE L’INSTITUT FOURIER
T. KAWAI
H. P. STAPP
Infrared catastrophe in a massless Feynman function
Annales de l’institut Fourier, tome 43, no 5 (1993), p. 1301-1310
<http://www.numdam.org/item?id=AIF_1993__43_5_1301_0>
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Ann. Inst. Fourier, Grenoble
43, 5 (1993), 1301-1310
INFRARED CATASTROPHE IN A MASSLESS
FEYNMAN FUNCTION (*)
by T. KAWAI and H.P. STAPP
In our recent article [KS3] we examined the effect of using Q-couplings
(in the sense of [S2]) in some Feynman functions, and found that the
resulting functions have singularities of the physically required degree. This
basic property of Q-coupled functions is not enjoyed by ordinary Feynman
functions, some of which exhibit stronger singularities. We will verify this
latter fact by a detailed study of the Feynman function associated with the
following diagram D' :
D'=
/-A
.'R +
Figure 1
Here the wiggly line represents a massless propagator l/(fc2 + zO), and
each straight segment with + sign represents a massive propagator
l/(p2 — m2 4- ^O)- As we are concerned with the singularity structure of
the integrals in question, we ignore the numerator factor in the integrands;
i.e., several terms relevant to the -/-matrices are replaced by 1 here and in
what follows. In connection with this, the external lines of the diagram are
(*) This work was supported by the director, Office of Energy Research, Office of
High Energy and Nuclear Physics, Division of High Energy Physics of the U.S.
Department of Energy under Contract DE-AC03-76SF0098, and by the Japanese
Ministry of Education, Science and Culture under a Grant-in-Aid for Scientific
Research (International Scientific Research Program 03044078).
Key words : Infrared catastrophe - Feynman function - Landau-Nakanishi singularity
surface — Massless propagator — Microlocal analysis — Holonomic system.
A.M.S. Classification : 81T18 - 81V10 - 46F99 - 32S40.
1302 T. KAWAI AND H.P. STAPP
simplified so that one external line represents a pair of external lines, and
the vertex to which it is connected represents a pair of vertices connected
by a line that represents a charged particle that is far from its mass-shell,
and hence a non-singular function.
One can easily verify that the Feynman function FD'{Q,R) is
singular along the triangle-diagram singularity surface; i.e., along the
strictly positive-a Landau-Nakanishi singularity surface L^(D) = {(p = 0}
(hereafter referred to as the Landau surface for short) determined by the
triangle diagram D :
D
Figure 2
Our principal conclusion is that FD' contains a singularity of the
form (log^)2, despite the well-known fact that FD behaves like logy?
near L^(D). Such an increase of the degree of singularities would be as
catastrophic as a divergence (i.e. non-well-definedness as a (hyper) function
of (Q^R)) of the function involved, because it would be contradictory to
the known large-distance fall-off properties for stable charged particles.
Note that in our analysis we do not associate a fictitious mass to
a photon, as is usual in the treatment of mass singularities of Feynman
functions. (See Kinoshita [Ki], for example.) The integral is nevertheless
well defined after ultraviolet cut-off. (In what follows we assume that all
integrals are defined with suitable ultraviolet cut-off factors.) We also note
that we really encounter a (non-trivial) function of this sort in QED as a
consequence of replacing each external line of D' by a pair of photon lines
connected to two vertices separated by a far-off-mass-shell charged particle
line. (This makes the number of photons incident upon the closed loop
even, so that the function does not vanish by virtue of Furry^s theorem.)
Our result pertains only to the single indicated Feynman diagram.
It prevents the vertex part of that diagram from being defined as essentially
a finite correction factor to the basic triangle diagram function, in the
way that is normally done in theories in which all particles are massive.
Renormalization does not help.
INFRARED CATASTROPHE IN A MASSLESS FEYNMAN FUNCTION 1303
We have not confirmed that a summation over all diagrams of the
same order in e2 could not lead to a cancellation of the (log y?)2 singularity,
but see no reason for such cancellation to occur. A rigorous proof of
noncancellation would require a generalization of the present proof that
would yield not only the (log (p)2 term, but its coefficient as well. Here it
is only shown that the coefficient is nonzero for the single diagram, apart
from a trace factor that arises from the numerator. For the case of an odd
number of photons this trace factor is odd under p i—^ —p, k \—> —k^ and
hence would (as noted by Furry) vanish under integration over p and k. For
our case, with an even number of photons, this trace factor is even, and will
not vanish in general.
In what follows we use the same symbols and notations as in [KS2];
for example :
• the function R(k) denotes the retarded propagator
l/((fco + 20)2 - k2) and it is diagramatically represented by --->-- ;
+
• the function A(k) denotes the advanced propagator
V((^o — ^O)2 — k2) and it is diagramatically represented by
• the symbol I > represents
(-2m)^{p2 - m2) = (-2m)6(po)6(p2 - m2).
We also freely use several (microlocal) properties of functions related to
the retarded and advanced propagators that are obtained in [KS2]; in
particular,
(l.a) ^^=(_2^)^(^)+AW
(l.b) =(^2m)6-(k2)^R{k).
(See [KS2] for the definition of 6±(k2).)
The main result of this paper is the following
THEOREM 1. — Let D be the triangle diagram, and let L^(D) denote
the strictly positive-a Landau surface. Let (Q°,R°) be a point in L^(D),
and let D/ denote the diagram given in Fig.l. Then, on a neighborhood
of(Q°,R°), FD'(Q,R) has the form
a(Q, R) (log(^(Q, R) + z0))2 + 6(Q, R) log(^(Q, R) + zO) + c(Q, R)
with a, b and c being analytic on the neighborhood of(Q°, R°). Furthermore
a does not vanish identically.
1304 T. KAWAI AND H.P. STAPP
Proof. — Using (l.b), we first decompose F^' into the sum of two
terms, F^ and F^, where Do and Di are respectively given as follows :
Figure 3
Our purpose is to show the (log (^-character of the integral F^ and
the (log (^-character of F^. Let us begin our computation with Fr>o. Since
the integrand of Fpp contains 6~(k2), we find
f3) 1 1
v / p2 - m2 + iO (p - k)2 - m2 + zO
= ( 1 _ 1 ^ 1
\(p - k)2 - m2 + iO p2 - m2 +10/ 2DA; -(p A;)2 p  zO
1 1
(4) g2 - m2 + zO (9 + k)2 - m2 + lO
= f 1 - 1 ^ 1
V(?2 - m2 + zO (g + A;)2 - m2 + z0/ 2^ + zO
in the integrand of FDQ .
Let us introduce the polar coordinate (r, fl.) in fc-space. With A; = r^
(^, a unit Euclidean vector) neither pfl, nor ^^ vanishes for massive p and g
when f22 = 0. Hence we find Foo is expressed as
(5) I'd4^ 6(^-1)6-(fl2)
X/^[{FA(0,A)-FA(0^+^)}
- {F^(Q - r^R) - F^(Q - r^,A+ rf2)}]
= fd^6(^-l)6~(fl2)
fit _i
x y -^-[{^A(Q,-R)-FA(0,^+rn)}
-{F^Q-rSl,R)-F^(Q,R)}
{F^(Q-rft,R+rS})-F^(Q,R)}],+
INFRARED CATASTROPHE IN A MASSLESS FEYNMAN FUNCTION 1305
where f2 = (f2o, -^2), and FA(O, R) is an integral of the form
(6) S^^l^n^
(£ + Q)2 - m2 + iO
X _______1_______-i4<
(fi - ̂ )2 - m2 + z0
with an analytic factor A(C,Q,R,k) (if (^ + Q)2 ^ 0 and (7i - ̂ )2 ^ 0).
Hence .FA is of the form
(7) B(Q, R, k) log(y>(Q, R) + i0) + C(Q, R, k)
with 5 and C analytic if (Q, R) is-close to (Q°, fi°). Hence our first task is
to calculate the r-integral
(8) y d^(log(y(Q,R+rfl)+iO)-\og(y(Q,R)+iO))
etc. Using the Landau-Nakanishi equations, we find
^R+r^^9'^^^^-^
on y(Q, K) = 0 with QI > 0. Since (R - if = m2 holds, to realize
the triangle singularity, and since fl2 = 0, we find (R - ̂ )n ^ 0, i.e.,
9y(Q,R+rSl}
———a———— _ / u- HsQce the implicit function theorem guarantees
the unique existence of a non-vanishing holomorphic function \ (Q, R, r, f2)
on a neighborhood of (Q°, R°) for sufficiently small r so that
W ,(0.^^)=^.^^)(^^^).
Similarly we can find non-vanishing holomorphic functions ^.(Q,I?,r,^)
(j = 2,3) so that the following hold :
(10) ^-^R)=^Q,H,^)(r+^S^),
(11) ,(Q-^^).̂ ,̂ ,n)(^^^).
Using these normalizations of y?(Q, R + r^) etc. we can reduce the
computation of the r-integration in (5) to the computation of the integral
discussed in the following
1306 T. KAWAI AND H.P. STAPP
LEMMA 2. — Let I = Ic(t, K) denote the following integral:
C^ dr
(12) / — (\og(t + cr + ^0) - \og(t + z0))
where c is a non-zero real constant. Then Ic(t, K,) is well-defined and it has
the form
(13) ^ (log(t + %0))2 + a\og(t + zO) + b(t) near t = 0
with a constant a and a holomorphic function b(t) denned near t = 0.
-Here a and & depend on c and K.
Proof. — We may suppose c > 0 without loss of generality. By
decomposing the integral (12) into the sum of two terms
{ t l ( l c dr r^ dr
(14) / — (log(t + cr) - log t) + / — (log(t + cr) - log t),
^O */t/2c r
we can readily verify the well-definedness of the integral in the domain
1m t > 0; in fact, the Taylor expansion of log(l + cr/t) in the first term
of (14) shows the well-definedness of the term, while the well-definedness
of the second term of (14) is clear. As in Appendix I of [KS3], where more
general cases are discussed, we find
(15) t^I^Ft^ 1 --J-)
dt Jo r U+cr+iO t ^ r i Q )
r -c
70 t + cr + zO dr = w + zo) ~ log(^ + CK + i0^
Hence we conclude that t -^ (J- j (log^+^O))2) is holomorphic near t = 0.
This means that I - \ (log(t + i0))2 is of the form a \og(t + %0) + b(t) with
a constant a and a holomorphic function b near t = 0. This completes the
proof of Lemma 2.
Returning to the computation of the integral (5), we use Lemma 2
to find
FDQ = fd^W - ̂ -{^(-^BiQ^R^^og^Q^R) +z0))2
+ /d4^ ($(nn -!)($- (n2) {Ci (Q, Ji, n, /.) iog(^(Q, .R) + io)
+Co(Q,fi,^)},
with Go and C'i holomorphic. Here we have used the fact that the part of
B(Q, R, k) (and C) in (5) which contains a factor of k cancels the 1/r-factor
INFRARED CATASTROPHE IN A MASSLESS FEYNMAN FUNCTION 1307
in (5), hence leading to logy? singularities at worst. Since
/d4^^--!)^-^2)^,
we find that the singularity of Fi)o consists of (logy?)2 and logy? and that
(log y^-singularity does not vanish identically.
Next we show the (log y?)-character of F^ in Fig. 3. One important
fact in the analysis of F^ is the following : the cotangential component
of the singularity spectrum of the retarded propagator is confined to the
positive light cone even at k = 0. (Cf. [KS2], (1.3); see [K3] for the basic
notions in microlocal analysis such as singularity spectrum etc.) We shall
refer to this fact as the light-cone property for short. Using this fact, we
first show that the following diagram D^ does not have the so-called u = 0
points. (See [I] and [Sl] for the definition of a u = 0 points.)
D2=
In fact, the existence of a u = 0 point would entail the realization of
the diagram
A=B=C^D
but this would contradict the light-cone property. Furthermore the light-
cone property also entails that there is no contribution to the singularity
near (Q°, R°) coming from D^. In fact, the closed loop conditions for both
A ^
D
> D
' + and + /
E
1308 T. KAWAI AND H.P. STAPP
that is, the realization of the configuration
A
E B C
with the segment ED being a forward light-cone vector, would imply that
the segment BA should be outside the forward light-cone, leading to a
contradiction. Hence one of the above two closed loops should collapse.
However, the collapsing of the first loop forces D^ to become
A = B = D = E <^
+
and that of the second forces D^ to become
Neither of them contributes to the singularity near (Q0, R°) that lies
in L^(D). Hence we may consider the following D^ in place of Di, as we
are interested only in the region where \k\ is sufficiently small.
Ds=
Here l ) represents (-2m)6^(q2 - m2), as usual. To analyze the
integral associated with 1)3, let us consider the function I(p,q) associated
with the following diagram D^ with keeping q2 = m2, qo > 0 :
D.=
Then the propagator associated with the segment •—?—• reducesE ~ c
to -l/(2qk-k2-^i0). Hence the closed-loop condition for the triangle DCE
INFRARED CATASTROPHE IN A MASSLESS FEYNMAN FUNCTION 1309
is w + a(p + k) + /3{q - k) = 0 with a, /? ^ 0 and w a forward light-cone
vector. As | k \ < 1 by the assumption, this closed-loop condition forces
the triangle DCE to collapse. Otherwise stated, J(p, q) is analytic if q is
confined to the mass-shell manifold q2 = m2. Since g is confined to the
mass-shell manifold in the diagram Ds, we may replace D^ by
^5=
.••̂
On the other hand, the integral associated with
A>=
is known to be analytic. (See [KS2], (6.2); to be precise the integral
considered here is the integral (6.2) with the deletion of r2?^^^ in the
numerator.) Therefore the integral FD^ has the form
W /W ̂  pd^ (oTtT^rfTio ̂ -^ -m2)d4(
with Ai being analytic if £2 ~ m2, (Q + ̂ )2 ~ m2 and (R - £)2 = m2. As it
is well-known that the integral (16) is of the form
ai (Q, R) log(^(Q, R) + zO) + &i(Q, .R)
with ai and &i being analytic functions near L^(D), we have verified
the (log (^-character of F^. This completes the proof of Theorem 1.
1310 T. KAWAI AND H.P. STAPP
BIBLIOGRAPHY
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[KS2] T. KAWAI and H.P. STAPP, Microlocal analysis of infrared singularities, Academic
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[KS3] T. KAWAI and H.P. STAPP, Quantum electrodynamics at large distances, LBL
Technical Report LBL-25819 (1993) and RIMS 957 (1994).
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(1962), 650-677.
[L] L.D. LANDAU, On analytic properties of vertex parts in quantum field theory,
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T. KAWAI,
Research Institute for Mathematical Sciences
Kyoto University
Kyoto 606-01 (Japan)
&
H.P. STAPP,
Theoretical Physics Group
Physics Division
Lawrence Berkeley Laboratory
1 Cyclotron Road
Berkeley CA 94720 (USA).


Infrared catastrophe in a massless Feynman function

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