Comparison of subtraction terms in two renormalization schemes

Vol. 27 (1989)

REPORTS

ON MATHEMATICAL

No. 1

PHYSICS

COMPARISON OF SUBTRACTION TERMS IN TWO RENORMALIZATION SCHEMES

G. Department

of Physics

and Astronomy,

DORFMEEXER University

(Received

March

of Kansas,

Lawrence,

KS 66CM-2151, USA

9, 1988)

We compare in detail the structure of the subtraction terms of the time-ordered distributions of the BPH-theory and the time-ordered distributions constructed in a previous paper [l]. These distributions are graphwise defined and derived in the framework of an LSZ-theory. We find that the subtraction terms of the @“-theory are identical in structure, but differ in the regularizations used.

5 1. Introduction In a previous paper [l] we constructed axiomatically for an LSZ-field theory, the retarded distributions r(yG) associated with a connected graph G, where y is a vertex of G. We showed’ that r(yG) is of the form of the retarded distributions as constructed by Steinmann [2]. From r(yG) we obtained a time-ordered distribution zEG(G) and an advanced distribution a(yG) associated with the graph G by evaluating some well-known [3-S] formulas relating retarded and time-ordered operators graphwise. We denote this time-ordered distribution by zEG(G), since (r(yG), a(yG)) is a solution of Epstein-Glaser’s splitting problem [3, 41 which can be treated also graphwise, see below, Remark 2.6. zEG(G) is constructed from r(yGj via the formula zEG(G) = C'.(Yj,H,)O..

. .Or(Yj,Hl)

(1)

(see Lemma 2.3 below for the details). Hence the “subtraction terms” of Steinmann [2] are carried over into subtraction terms in zEG(G). Let zH(G) be the time-ordered distribution associated with the graph G as constructed in the BPH-approach [6, 71. A simple inductive argument yields that 7EG and ~~ are equivalent [lo] renormalizations. But we will show more, namely that for the Q4-theory, ~~~ and 7” are ’ Ref. Cl], Theorem

4.9. ~1271

128

G. DORFMEISTER

identical after integrating2 over the internal vertices of G, provided.rEG and rH are constructed with minimal subtraction terms [lo] and minimal order of singularity [3, 41 and are correctly normalized [7]. Moreover, we will compare in detail the structure of the subtraction terms in rEG and r” associated with a partition -ty- of the inner vertices of G. It will turn out that they have the same structure. To carry out this investigation it is necessary to prove that the “regularization limit” of Steinmann [2] commutes with the composition “0.” in (1). This will be proven in Theorem 4.3. We conclude this paper with a detailed example to illustrate our results. Notation. We will use the notation of Ref. [l] resp. Ref. [l l] throughout. Especially we use {yr,. . . , y,} = {x1,. . . , x,,, ur,. . . , uO} = X, U,- U, being the inner vertices - to denote the vertices of a c-graph Cl]. ISI denotes the number of elements in the set S. We will exclusively deal with quantities which are amputated with respect to all external variables, i.e. all variables except u1 , . . . , u,. Hence we do not introduce a special notation for amputated objects. The present paper is essentially Section 7 of Ref. [8]. 0 2. Equivalence of 7EG and T” As the headline of this section suggests, we will show now that the two time-ordered distributions are equivalent. We will need various definitions and results which we first list. DEFINITION2.1. Let G be a connected c-graph [l], y a vertex of G. Let r(yG) be a distribution satisfying (rl) r(yG) is a tempered distribution in 9’(R4”‘), (r2) (r/f) is real for real test functions fE Y((R4*), r(y) = 0, (r3)

r(xr, x2) = &(x1 r(xr,...,

xJ=O

-x2),

for n>2,

(r4) supp r(yG) c {y-y’ E V; for all vertices y’ of G} = : r:, (r5) r(yG) is invariant under P$_, (r6) r(yG) satisfies the following equation for y,, y2 vertices

= -i[Cr(y,H,)or(y,H,)-~r(y,H,)or(Y,Ht)]

of G:

=:Z(YtY2GL

(2)

where the first sum extends over subgraphs H,, H, with H,oH, = G and yj E Hj, the second over H,, H, with H,oH, = G and yj E Hj. The composition “0” for graphs as ’ We canintegrate r(yG) over the internal vertices without loosing any of the properties of r(yG) and in such a manner that i(r(X, U,)or(X,U/,))dU,dU, = (jr(X, U,)dUi,)o(Jr(X,U,)dU,). This has been shown in Ref. [S] and Ref. [Y]. Hence z EG(X,UG) = ~...jr(X,U,)dU,o...o~r(X,U,)dU,.

SUBTRACTION

TERMS

IN TWO RENORMALIZATION

SCHEMES

129

well as distributions was given in Ref. [l]. For graphs it simply means to connect certain outer vertices of H, with certain outer vertices of H,, the resulting graph For distributions we have in p-space being G = H,oH,. r”(p,p,G)-%JAG)

= I”(P,P,G) = -ixzrF

SK”k(W)r”(PIP,~~--)r”(p,P,~~W)dW,

where the sum extends over the graphs as in (2), Hj having the vertices pjPjoj. defined

as Ek(W) = ~‘!+(W)-(-l)kI?+(-W),

R:(W)

(3) Rk is

= fi S+(wJ. j=l

We will treat the a theory of “type (,u, v))‘, specifying in 2.15 and later to the G4-theory, i.e. (p, v) = (4, 0). To achieve this we demand

(r7)

rcq,...,x,,

24)= 6”,D fi

6(Xj+4),

j=l

where 6,, = 0 for n # ,u and 6,, = 1. D is a PI-invariant differential operator, homogeneous of degree v and with constant real coefficients. Distributions r(yG) satisfying (rlHr7) were constructed in Ref. [1] and Ref. [S]. Note that we do not demand r(yG) to be symmetric in the vertices y’ # y of G. This property is not needed in our context and is, in general, not satisfied for an arbitrary c-graph G. From r(yG) we derive a time-ordered distribution rEG. DEFINITION

2.2. Let r(yG) be as in Definition 2.1. (a) Let G, be the connected graph with no inner lines or vertices, but with two outer vertices x1, x2 and a line connecting x1 and x2, that is G, is the graph of the free propagator:

z~~(G,J = r(xlGo)

= r(x,G,)

= -K,$&-x2),

(4)

where K, is the Klein-Gordon operator. (Recall that we treat only objects which are amputated with respect to all outer variables.) Let G, be the connected ,graph with one inner vertex U, no inner lines and ,U outer lines. We set rEG(G,) = y(q G,),

(5)

where r(x,G,) is given by (r7). (b) We define inductively

zEG(G): = r(yG)-

,c2iv-’

~z~~(G~)o.. .oxEG(Gy),

(6)

where 0 is the number of inner vertices of G and the sum extends over subgraphs with G,o . . ..oG.EG. G,#0 for all j.

130

G. DORFMEISTER

It was shown in Ref. [l] LEMMA

2.3.

Let

r(yG)

and zEG(G) be as above. zEG(G) = r(yG)-

(a) where

that rEG has the following

the sum extends

over

subgraphs

we have

zEG(H,)or(yH,),

with H,oH,

(7)

= G, H, f 0.

1 r(yj,H,)o.. HI,...Jf”

zEG(G) = i ( -i)“-1 V=l

(W

iI

Then

properties:

.or(yj,

H,),

(8)

where

(i) H,o.. .oH, = G, (ii) H,#0for allj= l,..., (iii) Let Zk be the set of indices labelling the vertices of Hk. Let j,: = min {I, v . . . v I,}. Then we demand jAEZA or, in other words, yjAE H,. (Note that (i) implies i

IVi(Hj)l

= o = IVi(G)l:

j=l (c)

Define

fEG(G)

by

fEG(G):=

i (-i)“-Y~~EG(H1)o...o~EG(HY), V=l

(9)

where

(i) H,o.. .oH, (ii) Hj#Ofir

= G, all j-

l,...

Then fEG(G) = (-i)“-‘zEG*(G).

(10)

r(yG) = zEG(G) + ix z*EG(Hl)~~EG(H2),

(d) the sum extends

over

subgraphs

with H, OH, = G, y E H,,

(11) H, # 0.

r(yG) = ,EG*(G)-ix~EG*(H1)o~EG(H2),

(e) the sum extends

over

subgraphs

with H, OH, = G, YE H,,

z~~*(G)-z~~(G)

(f) the sum extens

over

subgraphs

H, # 0.

= i~zEG*(Hl)o~EG(HZ),

with H,oH,

z~~*(G)-T~~(G)

(8)

(12)

= G, H, # 0,

(13) H, # 0.

= izzEG(H1)oTEG*(H2),

(14)

the sum extends over subgraphs with H,oH, = G, H, # 0, H, # 0. i. zEG(G) is causal [3, 4, 11, e. let G = H,oH,, where VJH,) k Vi(H,), (h) T~~(G) = - izEG(Hl)orEG(H2).

(i) ~~~ (G) does From

r(yG)

not depend

on the choice

and zEG(G) we obtain

of the vertex

then (15)

y in (6).

SUBTRACTION

DEFINITION 2.4. a(yG):=

TERMS

IN TWO

Let r(yG) and

RENORMALIZATION

7EG(G)

be as above.

SCHEMES

131

We set

r(yG)-~~7*EG(H,)07EG(H,)+i~7EG(H1)~7*EG(HJ,

(16)

where the first sum extends over subgraphs HioH, = G with H, # 0, ~EH~, second sum over HioH, = G with H, # 0 and ye H,. It was shown

in Ref. [l]

that a(yG) has the following

the

properties:

LEMMA2.5. (a) The tempered distribution a(yG) is an advanced distribution, i.e. it (r2), (r3), (r5) and (r6) of Definition 2.1 and hus the advanced support (a4):

satisjies

suppa

c (y-y’~

V_ for all y’~9(G)} =:r,.

(b) The pair (r(yG), a(yG)) 1sa solution oj’the splitting problem of Refs. [3,4], set d(yG): = a(yG)-r(yG),

supp a(yG) c

then suppd(yG)

= r;

u r,

that is

and d(yG) = r(yG) in Cr,T,

r, .

Furthermore, zEG(G) = r(yG)-r’(yG)

= -a’(yG),

(17)

where r’(yG) = i 17

*“‘(Hl)ozEG(HZ),

a’(yG) = i 1 7EG(HI)~7*EG(HZ), the sums extending over subgraphs and ~EH, in (19).

H,oH,

= G, H, # 0 and ~EH,

(18) (19) in (18), H, # 0

Remark 2.6. We would like to point out that one can construct inductively graphwise defined rEG(yG), aEG(yG), 71EG(G) via the graphwise evaluated splitting procedure of Epstein and Glaser [3, 41. It was shown in Ref. [S] that these distributions can be constructed in such a way that rEG(yG)* = r(yG), aEG(yG)* = u(yG), 7’EG(G) = (-i)“-* f’(G) where o is the number of inner vertices of G and TEG is given in equation (9). Furthermore, 71EGsatisfies the unitarity equations (13), (14) and rEG and aEG satisfy the GLZ-equation (r6). As a consequence [ 1 l] of the last statement, we obtain

rEG(yG) = r(yG)+D,

nd(y-y’)

(20)

and therefore 71EG(G) = zEG(G)+D2f18(y-y’),

(21)

aEG(yG) = a(yG)+&nJ(,-Y’),

(22)

where the products extend over all vertices y’ # y of G. Dj are differential operators, D, and D, with constant real coefficients and i”-’ D, with constant real coefficients.

G. DORFMEISTER

132

(r(yG), a(yG)) and (rEG(yG), aEG(yG)) are

They are Poincare invariant. In conclusion solutions of the same splitting problem:

r(yG) - a(yG) = rEG(yG) - aEG(yG),

(23)

hence D, = D, in (20), (22). Consider now the time-ordered distribution zH(G) of the BPH-approach which is constructed via the “9-operation” of Bogoliubov-Parasiuk [6]: DEFINITION

2.7.

Let

{Vi,.

. . , V:} be a generalized

1,

s=

0

if G(I/;,...,

{

where

.&Q”(l/;,...,

vertex of G. Then set [7]:

1,

- M@&“(l/;,...,

for one particle

1PI is an abbreviation

[6, 71

V,‘) is lP1,

(24)

Vk),

irreducible

and

‘(25)

V@:=

the sum extends over ali partitions W” = {{ J$, . . . , Qrcn}, 1 5 j s k(W)), 14 k(W), i.e. k(W) is the number of elements in W. The product “corm” is to be taken over all lines of G which connect different elements v, q in W. The operation M is defined in p-space: there RdE is of the form s(x~$F(p~, . . . , p;), F being in 0,(R4”), i.e. C” of slow increase, and ML%%&” is then s(zp(i,T(p;, . . . , p:), where T is the Taylor polynomial of F around pi = ...= p:= 0,up to a certain order which is given by the topological structure of G( Vi,. . . , V:): the superficial degree of divergence d(G) [7]. Finally, Xd”(T/;,...,

VL):= s&y

r, E are the usual regularization following THEOREM 2.8 [7,

define

a time-ordered

v;

)

.

.

parameters

.

)

I$) + _%-y(v; ,. . . , Vi), for A,.

The results

(26) of BPH

The limits r -+ 0, E -+O in 9°C” exist as limits in distribution zH(G) with the usual [lo] properties.

lo].

is the

9"(R4s) and

Our first main goal is now to show that zEG(G) and rH(G) are equivalent renormalizations [lo], that is they differ only by a finite renormalization [lo] F(G). F(G) depends only one the graph G and has in x-space the form F(G) = Dfld(y-_y’),

(27)

where the product extends over all vertices of G and where i”- ’ D is a Poincareinvariant differential operator with constant real coefficients. This statement will be clear once we prove

SUBTRACTION TERMS IN TWO RENORMALIZATION SCHEMES LEMMA 2.9.

133

Define rIEG, arEG as in (18) (19) and r’“, a’” analogously. Define rEG,

aEG as in Remark 1.6 and define r” and a” from i” with the formulas (11)or (12). Then (r”, a”) and (rEG, aEG) are solutions of the same splitting problem. Proof: We proceed by induction on c = jVi(G)l. For c = 0, 1 there is nothing to show, since r”(GJ = -&6(x-x’), recall Definition 2.2. Note that in r’, a’ only subgraphs with less than u inner vertices appear. Hence by the induction hypothesis r” = ‘Eo and a’” = a’so_ Therefore, rH(G) = z”(G)-r’“(G) = zH(G)-r’EG(G) and

lH(G)r= z”(G)-a’“(G) = rEG(G)-aEG(G).

whence r”(G)-a”(G) Ref. [3], p. 225, we obtain

= zH(G)-a’EG(G),

From

a”(G) = aEG(G)+Dn6(y-y’)

r”(G) = rEG(G)+Dn8(y-y’),

and as a corollary

= rrEG(G)-arEG(G)

(28)

we get

COROLLARY 2.10. zH(G) = zEG(G)+D,,(y-y’)

= z’EG(G)+D’,,(y-y’,,

(29)

where YEG was introduced in Remark 2.6 and the products extend over all vertices of G. D and D’ satisfy the conditions of (27) above. Hence z”, ~~~ and 71EG d@er only by a finite renormalization.

Let yblv be the following DEFINITION2.11.

space:

For NE iV we set

Y0N(,4cn+b)) : = {f E Y(R4@‘+a)): Df = 0 whenever i#j

Then we have immediately

xi = xj,

for all D with IDjsN}.

as another

corolarry

to Lemma

2.9.

COROLLARY 2.13.

~~~(G)lyoN = ~‘~~(G)lyoN = n dF(Ui-Uf) n 6(X-U),

(30)

where thefirst product extends over all inner lines 1 of G with l(u,, uf) = Ui-u/ and the second over all external vertices x where u is the inner vertex x is attached to.

Lemma 2.9 together with Lemma 2.3 (d), (f) and Corollary 2.13 yield the following theorem which is of course another way of expressing the contents of Corollary 2.10: THEOREM 2.14. zEG(G)and z’“~(G) satisfy the Renormalization Axioms of Hepp [lo]. Remark 2.15. We would like to point out that as a Consequence of r” = rEG+ D n 6 we obtain for rn the existence of the integrals r”(G,)o.. .oz”(G.) for any finite j: we know [9] that rEG(G, o . . .orEG(Gj) exists and the ditrerence term D dS is of the form treated in

Ref. [9] for c = 1. We also obtain the existence of r~,(Xl)o. . ..o$.(Xj), where $JXJ denotes the integrated jz”(GJ, where one has integrated over the inner

134

G. DORFMEISTER

variables of G. As a consequence, we have shown that in perturbation theory the Reduction Formula 1121 holds for zH(G) without the restriction to non-overlapping states. (With the same arguments as in Ref. [ll], pp. 12-13.) We conclude this section by investigating the normalizations of the two- and four-point functions of r EG. From now on we restrict ourselves to the G4-theory since for this theory there exists a uniqueness theorem for the time-ordered distributions [7]. Our result is THEOREM 2.16. Let f,,(j,p,), ?,,(pl.. . p4) be the integrated retarded distributions as constructed in Refs. [3, 4, S] or [ll]. Set as usual r”,(p,. . . p,) = 6(cpj)Y^,(p1.. . p,). Then r^,(PlP,) = @PI +P2){(27v(Pt-m2)+(P:-m2)2F0(P:))Y where F, is analytic

(31)

in the region pf < 4m2 and

rl(xl..

. x,) = an4c fi

6(x, -xi),

c = i,(OOOO).

(32)

j=2

Construct rEG from r(G) as described

in 2.2(b) or (8), (11). Then we obtain

(p~-m2)z^~G(p1p2) = (2~)~~ z^yG(p,.. . p4) = g(27c)-2 z^,EG(p,...p4)=0

for pf = m2 and CJ2 2, for pj = 0, j = 1,. . . , 4,

for pj=O,

j=l,...,

4 and o>

(33) (34)

1.

(35)

Proof: The proof of (33) follows closely the arguments of Ref. [ll], p. 68, using 2.3 (a) and 2.2 (a) which yields fFG(p,p2) = r”,(plp2) and hence equation (33). To show (34) and (35) we use 2.3 (a) and obtain ftG(pl.. . p4) = r”,(pl.. . p4) which yields the desired result by choosing the constant c appropriately. 0 3. Orders of singularity and scaling degrees In this section we will investigate the “singularity behaviour” of our distributions. Together with Hepp’s uniqueness theorem [7] for the G4-theory we will find that after integrating over the internal variables rH(X,) and rEG(X,) are identical, provided both distributions are constructed “correctly”, i.e. with the correct normalizations (33)-(35) and the correct singularity behaviour (see below). First we recall the definitions of “order of singularity”4 and “scaling degree” [ll] which measure the “singularities” of distributions, DEFINITION 3.1. (a) Let T be a distribution in 9’(RN). T is singular of order CL)at 0 - briefly, so T = o -, if there are M, PER such that for all E > 0 there is a K(E) > 0 so that for all cp~9’ we have I(Tl~)l

5 K(s)

c sup0 + fal
II~Il~PII~II~-W+‘a’-e~+ P”dX)I,

(36)

SUBTRACTION

where llXj/ = ( :

xf)“‘,

TERMS

IN TWO

RENORMALIZATION

, Q) a multiindex,

a = (a,,...

SCHEMES

ltlj = c 01~and (-0

135 + Ial -E)+

j=l =

max{O, --o+lal-8)). (b) Let TE Y’(RN). T has scaling degree lim ldvETA = 0 J.+ao

d -

briefly, sd T = d -,

if

for all E > 0 but not for any E c 0.

(37)

Here GYd = ~NG%,), where cpl(X) = q(iX), XE RN, 1~ R+. From this definition we obtain by straightforward computation3 LEMMA 3.2. (a) Let T be in Y’(RN) with so T = co. Then sd T 2 -w-N. (b) Let T be in Y’(RN) with sd T = d. Then so T 2 -d-N.

In the next lemma we list some properties we will need later: LEMMA

3.3.

of the sd and so of a distribution

. Then

(a) Let T E 9”(RN), so T = w, sd T = d. Let D”: = soD”T

and

= o+lal so(D” fi

(b)

sdD”T

which

= d-lal.

(38)

S(xj)) = lal.

(39)

sd(D” fi c!?(x~))= -N-Ial.

(40)

j=l

03

j=l

(d) Let THEY,

j = 1, 2. Let so q = wj, sd17;. = dj. Then

so(T,+T,)Smax{cL,,, w2) and sd(T, + T,) >=min{d,,

d2}.

(41)

Proof: (a) Ref. [3], p. 241 for the first statement, the second can be shown by straightforward calculation.4 (b) Note first that so 6(x) = 0 and apply (a). (c) Ref. [ll], p. 56. (d) The first claim is shown by a simple calculation, the second can be found in Ref. [l 11, p. 56. The next result is an immediate generalization of the computation in Ref. [3], pp. 243-247, to our setting. It will be very important for our considerations. Recall [l] that IV,, I is the number of lines connecting G, with G, in G, oG,. 3 Ref. [S], Lemma 5.15. 4 Ref. [S], Lemma 5.17.

G. DORFMEISTER

136 LEMMA 3.4.

Let G, G, be connected so(r(y,GJor(y,GJ)

c-graphs.

Let sor(yjGj)

= oj.

Then

= ~,+~,+2KRI-4~

where the r are written in diflerence

(42)

variables

COROLLARY 3.5. Let G, ,..., G, be connected c-graphs of the Q4-theory. Let SOr(yjGj) = -nj+4, where nj = IV,(G,)l, i.e. nj is the number of external vertices of Gj. Then

so(r(y, G,)o..

Next we investigate

. or(y,G,))

= -n+4,

the so of solutions

where n = ~~JG,o...oG,J.

of equation

(2):

Let d, be the supremum of the s-degrees of all solutions of r(y, y, G) -r(y,y, G) = I(y, y,G). Let co0 be the minimum of all orders of singularity of all solutions of (2). Then there exists a solution r’(yG) of (2) satisfying sdr’ = do and so r” = oo. As a consequence of Lemma 3.2, we have w. 2 -do - 4(n + o). D Proof: Let r” be a solution of (2) with sdr’ = sdl(G). Such a solution exists LEMMA 3.6.

since all the arguments of Ref. [ll], 9 5, can be carried out graphwise. Suppose o. $ so r”. Let roe be a solution of (2) with sorwo = coo. Since rwo and r” are both solutions of (2) they differ only by an expression O(Y) = D n 6(y, - yj) where the product extends over the vertices of G.’ Hence so r” s max{so 0, oo} by (41). Next we consider the sd’s. Since do was chosen maximal, we clearly have sd rwo 5 do. If sdrWo = d, then the distribution r@ satisfies the assertions. ‘Hence we only have to consider the case sdr?$ do. Note that ildo-‘OA = ~do-Erj:-IZdo-Er~ and that do = sd r”. Hence Ado-‘OA and I.do-Er~ diverge or converge simultaneou$y, hence sd O(Y) = sd roe. Now we obtain with (39) and (40) sd roe = sd O(Y) = - 4(n + a)- (~1 = -4(n + a)-so O(Y). With Lemma 3.2(a) this yields o. 2 so O(Y) which implies sor’ 5 max{soO, wo} = oo. But this is a contradiction to the assumption o. < so r”.

LEMMA 3.7.

Let r” and coo be as above. Then

co0 5 -n+4,

n = IVJG)l.

Proof:

We proceed by induction on Q = lVl’i(G)l,the number of inner vertices of G. o=l: sor’=sons=O= -4+4 by (39). r~ > 1: Let rw be a graphwise solution of Epstein-Glaser’s splitting problem, graphwise evaluated (compare Remark 2.6), with so rw = co. Then o = so(a’-r’), where r’ and a’ are as in (17)-(19). With (S)‘[lO) we write a’-r’ as a sum over terms or&H,), H, 0.. . OH, = G. By Corollary 3.5, the induction hypothesis r(yA)o... and (41) we obtain 0‘2 max(r(y, H,)o...or(y,H,)), where the maximum of H1o...oHk, k= 2 ,..., cr, has to be taken. Recall that r” is a solution of (2), hence o. 5 o and we obtain the claim. We are now able to calculate a4-theory: 5 Ref. [S], Corollary 6.29.

the=order

of singularity

for our rEG(G) for the

SUBTRACTION TERMS IN TWO RENORMALIZATION SCHEMES

137

THEOREM 3.8. Let zEG(G) be defined by (6) where the inductively appearing retarded distributions have maximal s-degree (or minimal order of singularity respectively). Then zEG(G) is singular of order -n +4, where n =,IYJG)l. Proof:

We use (8), (41), Lemma

3.7 and Corollary

3.5.

Recall that in the BPH-approach the Taylor polynomials in the 6%operation of minimal possible degree. 6 Hence we obtain for the @‘-theory THEOREM 3.9. zH(G) is singular of order

-n+4,

are

n = IV,(G)j.

Proof: From (29) we obtain so zEG(G) = so [zH(G) - D n S(y-y’)]. Suppose there -is an CQ,with (01~1 > so z“‘(G) = -n +4. Then this term Da0116(y- y’) has to be cancelled by a subtraction term from zH(G) of the same structure. Such a subtraction term can only arise from the “overall” subtraction of G itself, not from the renormalization of a subgraph, since there would be only the vertices of this subgraph in n 6, whereas n S(y - y') extends over all vertices of G. But the subtraction terms in zH(G) are all of minimal degree 5 d(G), where d(G) is the superficial degree of divergence of the graph G. It is well known [3, 43 that d(G) = -n+4 for the Q4-theory. Hence no subtraction term of rH(G) can cancel Duon 6, and all th e appearing a satisfy Ial 5 -n+ 4. Hence the assertion follows. As a consequence we obtain a sharper version of Corollary 2.10. This was the reason for investigating the order of singularity of our distributions. (We had to deal with the s-degree too, since Steinmann [ 1 l] classified the singularity behaviour of his distributions with the concept of the s-degree and not in terms of the singular order.) We obtain for the G4-theory THEOREM 3.10. T'(G) and T~~(G) ure equivalent minimal renormalizations, i.e. they differ only by a finite minimal ‘Yenormalization: \ T~(G)-z~~(G) = D”n6(y-y’), (43) where lcll S -n+4

for all a, n = IV,(G)1 and the product extends over all vertices of G.

It is well known’ that zH(G) is uniquely defined via the %‘-operation provided zH(G) is minimal and is normalized as in (33H35). As above, we denote by 7(X,) the integrated distribution Jz(G)dU,, where we Have integrated over the internal vertices of G. We obtain the main result of this section: THEOREM 3.11. zH(X,)= zEG(XG). Proof: Since rEG(G) is minimal, correctly normalized and differs from zH(G) only by a minimal finite renormalization, it satisfies the same properties as zH(G), hence zEG(XG) = zH(XG) by the uniqueness theorem mentioned above. I

6 A finite renormalization (27) is called minimal in Ref. [lo] if IDI 5 d(G) where d(G) is the superficial degree of divergence of thee-graph G. ’ Compare e.g. Refs. [7], [lo], [13-173.

G. DORFMEISTER

138

0 4. The structure of the subtraction terms in zEG(G) In this section we will investigate the subtraction terms in rH(G). So far we have not mentioned explicitly that rEG(G) is - as zH(G) - of the form rEG(G) = lim{$G(G)-Sg}, d

(44)

where fEG(G) = rEG(G)ly, = limz”fG. We recall [l, S] that r(y&) is of the form II+0

I

?I+0

= lim[ 1 ljx(Y)~(YlYjG)- C R&D I-I SCYl-Yj)] x

j=2

IDI
(45)

j=2

that these x-limits are carried over into zEG(G) by (8). The &limit in (44) stands for the collection of these x-limits. Before we go into the details of the structure of the subtraction terms S, we have to show that (44) is indeed correct, i.e. that the x-limits commute with the operation “o”, that is that so

r(y, G,)or(y,

G,) = [lim . . .] o[lim..

x1 holds (here we have similar methods as references. To make well as the concept

.] = lim ([. . .]o[.

x2

x1.x2

. .])

(46)

indicated by . . . the expressions occurring in (45)). We will use in Ref. [S] and Ref. [9] and the notation will be as in those the reading easier we give the definition of the space H&4; B) as of strong convergence:

DEFINITION 4.1. (a) Let H&4; B) be the space of functions f(M) for which there exists an X~E Rf such that for all NE IV, N > 0 the supremum A s~p~~~i(l+ sup

ll~I12)NClf(~~)I+lf(~+X~~

B+X&-+oB)I

Wll-“II existsand

is polynomially bounded in B, X = X,, X2. We callfEH,(A; B) strongly decreasing in A and slowly increasing in B. Here I/AI(2 = 1 laj12. (b) Let ri = (n,,..., n,), nj E N. Let f; be a sequence of functions in H,(A; B). We say that f; converges strongly to a function f (AB) in H&4; B) if for all NE N there P,(B) such that for all j = 1,. . . , r exists a sequence C;E R+ and a polynomial with nj 2 MEN we have c; < u, a arbitrarily small, and If;(AB)-f(AB)( 5 c,$‘,(B)l(l + IIA112)-N.The polynomial P, does not depend on fi but may depend on N. The next lemma

will yield the desired

equation

(46):

LEMMA4.2. Letf (KPQUV) be in H,(KPU; P-QV), E < f. Here we use the notation o(p) = (p2+m)“‘, P = @cl7Pl? P2, P3 ) = (PO,@)fir a 4-vector p and p- = pO-o(p),

SUBTRACTION

P- = (pl,...,

p;).

L(P-QUV:=

TERMS

IN TWO RENORMALIZATION

139

SCHEMES

Set 2 K = c k, and nx(-qj)SdKd~ns(pi)[C(~j,*I”(.G))(KPQ)-~(~K+CP+CQ)A,(KPQ)I

where A,(P) = S(c P)c R&Y(P), compare (45). Then we have (1) L,HH,,(P-QU; V)for E’ < E. (2) f: converges strongly tof(P-QUV),= nX(-qj)jdKdpn&+)f(. x f(KPQUV).

~(KPQUV,

G)(KPQ) x

Proof: (1) This is proven in Theorem 7.1 of Ref. [ll]. (2) To show strong convergence we note that by the argument of Ref. [ 111, p. 94, which we can apply directly, we know that Y(.G)(KPQ)- C rj,*I”(G)is of the form c?(~K+~P+~Q)[G,(KPQ)-~(CK+CP+CQ)CR~,S(KPQ) w h ere G, is an entire function which is bounded by a polynomial, -C R?,~wPQ)] the degree of this polynomial does not depend on x. Hence IG,(KPQ)-C Rz,.6?\ 5 FJKPQ) where F, is a polynomial with degree independent of 3t. Since xrjX*I”--A, approximates r”(.G)(KPQ) in Y’ we have F,(KPQ)+O in Y(KPQ) for x+ co. We thus get

I InX(-_j)S’Kd~n,~i)6(CK+CP+CQ)F,(KPQ)f(KPQUV)(. Recall [8, 93 that we interpret the inner variables as K-variables hence lKl >=1 and we can integrate over one of the K-variables and obtain IL(P- QUV)-f(PQUV)l 5 I~x(-~~)~~~K’~PI~&+)~IF~(K”K’PQ)~(K”K’PQUV)I. Here K’ = (k,,... , k&, IKI K” = - c kj-xP-xQ. F: is the polynomial F written in the variables K”, K’. j=2

We also have FX -+O in Y’(K”K’PQ). Since the degree of FX does not depend on x there exist d,~ R+ and a polynomial H independent of x such that JF:(K”K’PQ)J S d,IH(K”K’PQ)I holds. Since ~(K”K’PQUV)EH,(K”K’PU; P-QV) we know that If(K”K’PQUV)l 5 IP,(QP-V)l(l+)I(K”, K’)I12+IIP112+IIU112)-N with a polynomial P, and any NEN. This yields [8] IL(P-QUV)-f(P-QUv)l 5 d,lnx(-qj)l

xS~K’~P~~~(~,)(IH(K”K’PQ)P,(QPwi + IIW, WII~+ IIPV+ IIWP 5 ‘xIH,(P-Q)IIP,(QPV(l+ IIUl12)-MI nX(-qj)lS’K’d~l nX(PJH2(K”K’P)I(l+ + IIK’l12+ llPl12)-M, where H, , H, are suitable polynomials and 2M 5 N. Because of the form of support x
140

G. DORFMEISTER

We have thus found that the x-limit in & is strong in the sense of Definition 4.1 (b). Hence we can apply the results of Ref. [9], Theorem 2.1 (or Ref. [S], Theorem 4.13) to 7% and obtain THEOREM4.3. Let G, , . . . , G, be connected ICY, GAO.. .or(~,G,)

=

lim [I x1....,x*

c-graphs,


yjeGj.

0.. .o[c

Then

5jx,I(G,)-kJGr)].

(47) (Note that $ G contains only one inner vertex given by (r7).)

v then A(u) = r(xl,

x2, x3, x4, u) as

This result (47) enables us to investigate the structure of the subtraction terms for a given finite x. The limit x-+ cc of a subtraction term does not exist in general, of course. Let G be a given Feynman graph with inner vertices U = (ul,. . . u,). Let YY = {V,,..., I$} be a partition of U. This partition #‘- determines a partition X,} of the external vertices X = (x1,. . . , x,} of G, k s m, by 3 = {Xi,..., Xj = {x E X: x is attached to a vertex u E I$]. In other words, the Xi are the free [ 1, S] external vertices of G(Vj). If k ft m, then some of the G(Vj) do not have free external vertices. Since X is completely determined by “Iy- and since the subtraction terms depend only in a trivial manner on the external vertices, we simply neglect the external vertices and write shortly zEG(G) = c...r(Uj)o.. ...

.or(U,),

(48)

where Uj v .. . u U, = U = Y,(G). Consider now a summand r(Uj)o...or(U,) in (48). The conditions on the summation are: (i) Gjo.. .oG, = G, (ii) G, # 0 for all i, (iii) Let Ii be the set of indices labelling the vertices of Gj. Let j, : = min{I, u . . . u Ij>. Then we demand j, E I, or, in other words, YjkE G,. Because of (i) we distinguish two possibilities for each, Ui: (1) There’ exists an s with Ui c V,. (2) For all s we have Ui + VS. In the first case we use the expression . . .r(U,). . . for the treatment of our subtraction terms. In the second case we pick a certain part of r(Uj)o.. .or(U,). Note that in that case there exist at least two different sets V,, V,E YY with V, n Ui # 0 and V, n Ui # 0. Consider now the space DEFINITION4.4. Y$&R40):=

two arguments

ui, uj coincide,

(f~9’~(R~“): IIf = 0 for all D with IDI 5 N whenever i # j,Z+E v, USE 5, K and I$E YV, i, j = 1,. . . , m}.

SUBTRACTION TERMS IN TWO RENORMALIZATION

For f”~9& (qJj)O..

SCHEMES

141

we have

. OTq.Ji)O.. .or(UJfW) = (@Jj)O..

. dim

= (rqUj)O..

. o~C,‘(“i)o...or(u,)i/“)

[C 5sx1(uJ-Ax(Ui)]

O*. .oW~)lf”) (4%

because A,(U,) contains points from two different sets V, and T/,E w: sincefvanishes strongly when points from two different sets of w fall together we see that the subtraction term A,(U,) in r(Uj)o.. .olim[. . . -A,(U,)]o.. .or(U,) disappears when applied to f”‘ and the x-limit can be carried out and gives precisely c <,Z(UJ. Thus for every Ui which is not in a certain V, we substitute c &Z(U,) for r(U,). By iteration (r(Uj)o..

.o~(U~)lf")

=

C(hj(k;.)r(~):-)o...oh,(Y,)r(Y,)lf")

(50)

with Yk= Ui n V, for suitable i, r. The hk(Yk) contain all the [, and all the appearing factors (- 1) and i, arising from the expressions Z(abU) = c r(aU,)or(bU,) - r(bU,)o or(aU,) substituted iteratively. Recall that m was the number of sets in 9+‘-.Hence, if j > m, then 9): = {Y,,..., q} is a subpartition of w: VJ c w. Ifj = m then yk = V, for all k = 1,. . . , j = m, because the ordering of the inner vertices of G is fixed and no permutations are possible in the “0”-product. In the case j > m we therefore get subpartitions which are different from w. For the discussion of the subtraction terms associated with -Iy- we will not consider such summands. In other words, our subtraction term associated with 71y-will only consist of terms involving the partition w but none of its subpartitions or refinements. In the case j = m we get the expressions . . .oh,(l/,)r(V,) and (i) kK,,)r(KJo (ii) r(V,)o.. .or(V,). The first term (i) comes from an r(Uj)o... or(U,) as described above, (ii) is a summand itself in zEG(G). We treat both terms simultaneously by allowing hk(Vk) = 1. We have h,(~)r(~~)o...oh,(~)r(~~) =

lim h,(V,)[C5sx,Z(v,)-A,,(V,)]o...ohl(~~)[CSSX1Z(I/1)-AX1(~)] XI.....Xr

=

lim h,(T/,)CSsx,Z(V,)oh,-,(l/,-,)Crsx,_lZ(l/m-1)o...ohl(l/l)C5,,, XI....,X, xZ(I/,)--h,(~)Cgsx,Z(V,)oA,m_l(l/,-l)o... +(-

l)“k#LJ~(KJo.~

.o~,(~‘,)~,,(I/,)]~

x

* **. (51)

All terms containing an I(&,) can be decomposed again, leading to subpartitions of -tY and are not considered here any more. Therefore the following definition seems natural:

14’

G. DORFMEISTER

DEFINITION 4.5. Let G be a connected c-graph (or a connected Feynman graph). Let $V = (V,,..., I$} be a partition of the inner vertices of G. Then the subtraction term in rEG(G) belonging to nly- is

A:,...,,,,, : = c ~,OY&L,(W,)O..

.oh,

W-&L, W-A

(52)

where the sum extends over all permutations WI,. . . , W, of VI,. . . , V,. The h, contain all the factors (- 1) and i as well as the retarddon factors <,, determined by the summands r(Uj)o.. .or(U,). Let us turn now to the subtraction BPH-approach. Clearly, the following

terms associated with a partition YV in the definition is suited to the situation:

DEFINITION 4.6. Let G be a connected of the inner vertices of G. We set

Feynman

graph and let $Y be a partition

(53) (For the notation We first note

compare

Definition

2.7.)

Remark 4.7. Note that the subtraction term (53) has the same structure as the subtraction term (52): both contain - in p-space - polynomials in the variables associated with the Vj and the factors 6( 1 The product “0” and the product over ujk).

k

“corm” connect vertices from different sets in w. The A,-factors in (53) are “hidden” in the “0’‘-product and the hk(Vk). The next example will clarify this statement: EXAMPLE 4.8.

Consider

the following

graph:

Consider the partition S/ = [ [L.~, L’~,1.~1, 1~~;I. The subtraction belonging to this partition is Av = fj

%‘~(u,,

term

in zH(G)

u2, u&l~~(u3 - u‘&ly(vZ -II‘+)

j=l

since X~(uJ = 1. In zEG(G) a subtraction term associated with YY comes from two summands: in r(ul v2 v3 uJ the summand lim [9,( u1 -Vq)(-i)(r(v,v,v,)or(v,)--r(u,)o We want to show now or(u1u2vg))-AA,( u1v2v3u4)] and the term -ir(v,)or(v,u,u,).

SUBTRACTION

TERMS

IN TWO RENORMALIZATION

SCHEMES

143

that the connecting lines 1, and 1, turn out indeed to be A,(u, - uq) and A,(u, - u,J. Recall that A,(u) = r(u), compare the remark after (47). We obviously have to calculate

Consider r(u,)o9,,(u, -uZ)AU2(u1u2u3) = ~S(U,-~~)G(U~-W~)S,,(W~ x [6(u,-w;)6(u,-w;)+6(u,-w~)6(u,-w;)]6(u,-u,)6(u,-u,)A+(w,-w;)A+

-w$)Dx2(wY;wi) x x

x(w,-w’,)dw,dw,dw;dw’, = ~(u,-u3)[Qx2(~1-u2)+QX2(~2-u1)]A+(u4-ul)A+ x x (uq--u2)Dx2(u1 uz). The other terms are treated analogously. We proceed now on a formal level since the limits 3c1, x2 --+ 00 cannot be performed for single subtraction

terms. But we want only to show that “combinatorically” the connecting lines become A,. Recall that $,(a-b)+ O(a-b) and that f3(a-b)+O(b-a) = 1. Hence we obtain as contribution to the connecting lines I, and 1, the expression

+A+(u,-~,)A+(~,-u,)~(~,-~,)~(~,-~,) = @, - 04)CA:(~2-~q)-A:(uq-~z)l~(u~-~z)~(~~-ug)+ -u2) = +A2,(uq-u2)8(~2-~3)8(~1 and therefore the lines 1, and 1, carry A,-expressions. indeed BPH-type subtraction terms in zEG(G).

iA;(u,-u,)6(u,

-u2)6(u2-u3)

This example shows that we get

REFERENCES [l]

Dorfmeister, Reports

[2]

[6] [7] [S]

[9] [lo] [11]

Theory (to appear

in

Phl,\ir,s).

O., Perturbution Expansions in Axiomatic Fir/d Theory, Springer Lecture Notes in (1971) (formula (4.58)). and Glaser, V., Ann. de l’lnstitut Henri Poincare, Sect. A, Vol. XIX, no. 3 (1973), 221. and Glaser, V., Le role de la localite duns la renormalisution perturbatiue en theorie quantique des chumps, Les Houches 1970. Epstein, H., Glaser, V., Stora, R., General Properties of the n-point Function in Local Quantum Field Theory, Les Houches 1975. Bogoliubov, N. and Parasiuk, O., Acta Math. 97 (1957) 227. Hepp, K., Comm. Math. Phys. 2 (1966), 301. Dorfmeister, G. Axiomatic Derivation of Graphwise Defined Retarded Distributions and Related Topics, to be submitted as a Ph.D. Thesis. Dorfmeister, G., Reports Math. Phys. 27 (1989), 257. Hepp, K., Renormalization Theory, Los Houches 1970, 473. Steinmann, O., Perturbation Expansions in Axiomatic Field Theory, Springer Lecture Notes in Physics, 11 (1971).

Steinmann, Physics 11 [3] Epstein, H. [4] Epstein, H. [S]

G., Graphwise Defined Rrt.arded Distributions ,for an LSZ-Field

on Moth.

144

G. DORFMEISTER

[12] Bogoliubov, N., Logunov, A., Todorov, I., Introduction to Axiomatic Quantum Field Theory, Benjamin, Reading 1975, Sec. 14.3. [13] Callan, C. G. Jr., Introduction to Renormalization Theory, Les Houches 1970. [14] Coleman, S. Renormalization and Symmetry: A Review for Non-specialists, Erice 1971. [15] Hepp, K., Theorie de renormalisation, Springer Lecture Notes in Physics 2, 1969. Lowenstein, J., BPHZ-Renormalization, Erice 1975. [16] Zimmermann, W., Local Operator Products and Renormalization in Quantum Field Theory, Brandeis 1970. [17] Zimmerman, W., Comm. Math. Phys. 6 (1967), 161.