Asymptotic Behavior in Quantum Field Theory

Chapter 6 / ASYMPTOTIC BEHAVIOR IN QUANTUM FIELD THEORY

The purpose of this chapter is to study the asymptotic behavior of subtracted-out Feynman amplitudes d E ( P ,p), in Euclidean space as defined in (5.3.2), when some of the elements in P and p “take on” asymptotic values. Throughout this chapter we write d ( P , p ) for d E ( P ,p ) to simplify the notation, and no confusion should arise. Before plunging into a general asymptotic analysis of d ( P , p ) we derive in Section 6.1 some elementary asymptotic estimates f o r d directly from the general definition in (2.2.84) (with 9,in the latter identified with RE) which require no detailed knowledge of the structure of RE.We show in particular that if the dimensionality d(G) of the graph G , with which d is associated, is nonnegative, then if the external momenta are led to approach zero, it follows that d vanishes (Theorem 6.1.1). This is consistent with (and is expected from) the overall subtraction at the origin one carries out in defining R . We also show, whether d(G) 2 0 or d ( G ) 0, if all the masses appearing in d are led to approach infinity, then d again vanishes (Theorem 6.1.2). This is a particular case of the so-called decoupling theorem of field theory, which essentially says that Feynman amplitudes involving “very heavy” masses may be “neglected.” In the remaining sections of this chapter we carry out a general asymptotic analysis for d .In Section 6.2 a general dimensional analysis for RE is given on which the remaining sections are based. In Section 6.3 a study of the high-energy behavior of d is given when all the masses are fixed and nonzero. The study in Section 6.4 generalizes the results in Section 6.3 to deal with

-=

I35

6 Asymptotic Behavior in Quantum Field Theory

136

the situation when not only some (or all) of the external momenta of G become large but some (or all) of its masses as well become large. A study of the zero mass behavior of d is given in Section 6.5. The latter is then applied in Section 6.6 to study the low-energy behavior of d when some of the masses in the theory are led to approach zero. Section 6.7 deals with a very general study of d when some of the external momenta become large, some become small,’ and some of the masses become small. In general we let these various asymptotic components “approach” their asymptotic values at dgerent rates. In Section 6.8 a generalized decoupling theorem is proved which establishes the vanishing property of d when any subset of the masses in the theory become large and, in general, at different rates, and gives sufficiency conditions for the validity of the decoupling theorem when any subset of the remaining nonasymptotic masses are scaled to zero as well. The breakdown of this chapter into the above mentioned sections, as just outlined, will, it is hoped, facilitate the reading of this rather difficult chapter. One thing worth noting in this chapter is that one is able to obtain the asymptotic behavior of d without the need of carrying out explicitly the rather complicated integrals defining .d,as usually no closed expression for d may be obtained when one is involved with complicated Feynman graphs. PRELIMINARY ASY MPTOTlC EST1M A T E S

6.1

We write2 (6.1.1)

n

m

and %P, K, P ) =

c P“%(K, P),

(6.1.4)

a

in a notation similar to the one in (2.2.18), where a = (aol,..., aSm),

dPj2apj20,

la1 = aol

j = 1,..., m,

+ + a3m, p=O,1,2,3.

(6.1.5)

’ Of course, some of the external momenta may remain nonasymptotic. ’ Since in this chapter we consider only a Euclidean metric, we omit the E subscript in Q;c.

6 .I

137

Preliminary Asymptotic Estimates

+

. . . , pydr, denote dPj 1 distinct values for ps. Let P: = (plIo,, 0 . . . , pA13,), where 0 I t P j I dPj. Then the generalized Lagrange interpolating formula (4.1.11) states that we may find a constant C,(P:) depending on a and P: such that Let py,,,

PXK, P ) =

1Ca(P:)P(P:, K ,

(6.1.6)

I

where the sum is over all 0 It P j I d P j ,withj = 1 , . . . ,rn, p = 0, 1,2, 3. We note that with Q = k + p , we may write (6.1.7) (6.1.8) and upon using the elementary inequalities +2kP I 21kl I P I ,

-2Qp I 2 I Q I IPI,

2IQIp 5

2lklp I k2

we obtain

Q2

+ p2,

+ p2,

+ p 2 ) I (k2 + p 2 ) A , (k2 + p 2 ) I (Q2 + p2)A,

(Q2

or (Q2

+ p2)A-'

I (k2

+ p 2 ) I (Q2 + p 2 ) A ,

(6.1.9)

(6.1.10) (6.1.11) (6.1.12)

where A = l + - +IPI+ . P 2 P

Let D(P, K , PI =

n I

P

(Q:

+ p:).

(6.1.13)

(6.1.14)

Then from (6.1.6), the inequalities in (6.1.12) and the definition (6.1.14) we may find a strictly positive constant G depending on P and P:, respectively, such that3

D(0, K , p) means that the elements in P are set equal to zero in D(P, K ,p),

6 Asymptotic Behavior in Quantum Field Theory

138

The inequalities in (6.1.15) imply, in particular, that the absolute convergence of fa.. dK P(PF,K , p)D-'(P:, K , p ) imply the convergence of

and of

Let AP = (Ipy, . . . , I&, I > 0; then

d ( I P , p) =

1 Ilalpa J a

dK

E(K,p ) D - ' ( I P , K , p),

(6.1.16)

R4"

where la1 2 d ( G ) + 1 for d(G) 2 0. Suppose I I 1; then the left-hand side of the inequality in (6.1.15) implies that dK I %(K, p ) ID- '(JP, K , PI IG ( P ) S.4.

IR4" I Eb(K I dK

p)

D -'(0, K , PI, (6.1.17)

and its right-hand side is independent of A.4 Now we consider the limit I -+ 0 and apply the Lebesgue dominated convergence theorem [Theorem 1.2.2(ii)] and conclude from (6.1.17) that we may take the limit I --t 0 inside the integral in (6.1.16), and for d(G) 2 0 we have lim d ( I P , p ) = 140

lim Alalpa a

dK Pa(K,p ) lim D- ' ( I P , K , p ) 1-0

1-0

Accordingly, we may state the following theorem.

0,

Theorem 6.1.1 : For d(G) 2

lim d ( I P , p) = 0,

(6.1.19)

1+0

and from the left-hand side inequality in (6.1.15) we have I d ( I P , p ) I IC 0 I d ' G ' + 1 , Note that G(AP) 5: G ( P ) for L

I; 1

[see (6.1.13)].

0 II I1,

(6.1.20)

6.2 Analysis of Subtracted-Out Feynman Integrands

139

where

and, as we have seen above, the integral in (6.1.21) exists. Finally we write qyc = (r]pl, . . .,upp) to obtain (6.1.22) where do(G) = d ( G ) -

P

1 i=

CT~S

1

d(G).5

(6.1.23)

Accordingly upon identifying q-' in the integral in (6.1.22) with 1 in (6.1.18) we obtain lim c d ( P ,rl, PI 4+m

=

1 lirn yldo(')-la1 a

d K g a ( K ,p) lirn D-'

4-00

4-00

(6.1.24)

for all d(G), since if d(G) < 0, then la 1 2 0, and if d(G) 2 0, then la1 2 d(G) + 1. Hence we may state the following theorem. Theorem 6.1.2

(The decoupling theorem) :

For all d(G), i.e., for d(G)

lim d ( P , r]p) = 0

(6.1.25)

< 0 or d ( G ) 2 0,

and

9-00

IJW,?P)l I rl-'co,

r]

2 1,

(6.1.26)

where Co is dejned in (6.1.21). 6.2

GENERAL DIMENSIONAL ANALYSIS OF SU BTRACTED-OUT FEYN M A N INTEG RAN D S

In this section we carry out a general dimensional analysis of subtractedout Feynman integrands. This analysis will then be applied in the remaining part of this chapter to investigate the asymptotic behavior of subtracted-out Feynman amplitudes. See the definition of the positive integers u iin (2.2.17).We assume (without loss ofgenerality) that degr,,, .?p = degrp,K,p.?in (2.2.17) [see also (6.5.4)J

1 40

6 Asymptotic Behavior in Quantum Field Theory

In (5.4.14) and (5.4.17) we have reduced the expression for the renormalized Feynman integrand, associated with a proper and connected graph G, to the form R = (1 where AGIG=

c

0cG’gG

[

-

TG)AGlG,

c

I-I

(6.2.1)

(-T+

S I....,n(G‘) B E S I..., , dG’)

(6.2.2)

with S1,,.,,,(G’)= &G;) u . . . u

b(GL).

(6.2.3)

The sum C Q c G , g G in (6.2.2), for G’# 125, is over all proper subdiagrams G’$L G, with G;, . . . , GL denoting the connected components of G’ and d(G;) 2 0,. .. ,d(G:) 2 0. &G:) is a set with Gf the largest subdiagram in it; if g E D(G:), then g c GY and g is proper and connected with nonnegative dimensionality; if g,, g2 E b(Gi), then either g1 n g2 = 125 or 9, $L g2 or g2 $L 9,. We define S , ,..., = 0 and ( - T O ) = 1. We note that if gl, g2 E S,,...,,(G’),then g,, g2 are proper and connected and if g1 c G : , g2 c GJ, with i # j in [l, . , . , n], then g1 n g2 = 125. Ifg,, g2 c G ; ,for some i in [l, . . . , n], then either g1 n g2 = /a or g, c g2 or g2 c 9,. The proper are called the maximal elements in and connected subdiagrams G;, . . . , C:, S , , ...,,,(G‘) because for any g E S1,...,,,(G’), there is an i E [l, . . . ,n] such that g c G:. Let g E S , , ...,,,(G‘) and suppose g l , . .. ,gmare the maximal elements in S , , ...,,,(G‘), with g,, . . . ,gm$ g. Again the latter means that ifg’ E S , , ...,,,(G’), withg’g g,theng‘c g,forsomeiE[l, ..., m].Wedenoteg/g, u ug,by g. By definition, gl, . . .,gmare proper and connected and pairwise disjoint. As usual, for convenience, we shall remove the restriction on the g in (6.2.2) with d(g) 2 0 by allowing the proper and connected subdiagrams g in (6.2.2) with d(g) < 0 as well by simply setting ( - q)= 0 in the latter case. We may rewrite (6.2.1), (6.2.2) in the form

,,(a)

R

=

X

0cG‘cG

[

C

n

(-q)I1G,

S I , ..., dG‘) B E S I..... n(G‘)

(6.2.4)

where n = 1 for G’= G. We note that a set S , ( G ) = S(G) is of the form S ( G ) = {G}u S1,...,”(G’),where G ; , . . . , GL (the connected components of a proper subdiagram G’)are the maximal elements in S(G) contained in G:G: $L G for i = 1 , . . . ,n. Let I be a 4n-dimensional subspace of R 4 n + 4 m + p associated with the 4n R4n+4m+p integration variables. Let E be a complement of I in R4n+4m+p: = I @ E (see Section 1.3 and Chapter 3). In turn let El be a 4m-dimensional

6.2 Analysis of Subtracted-Out Feynman Integrands

141

subspace of E associated with the components of the independent external momenta. Let E 2 be a complement of El in E ; then we may write R4n+4m+p = I‘ 0 E l = I” 0 E 2 , and we denote by A(E), A(I’), A(I”), A ( I ) the projection operations on the subspaces I , El, E 2 , E along the subspaces E , I‘, I”, I , respectively. We choose E to be the orthogonal complement of I in R4n+4m+p and E 2 to be the orthogonal complement of El in E . The subspace E 2 will be associated with the relevant masses in the theory. Let P’be a vector in R4n+4m+p such that the elements in K , P , and p may be written as some linear combinations of the components of P . Suppose P is of the form

P’ = Liq1q2

“‘qk

+

”‘

+ L k q k + C,

(6.2.5)

where 1 I k I 4n + 4m + p, and Ll, . . . ,L; are k independent vectors in , C is a vector confined to a finite region in R4n+4m+p, such that pi + 0 for all i = 1 . . . , p. Let S: = {Li, . . . , Lk}, where r is a fixed integer in 1 5 r 5 k. Suppose that A(E)S: # {0}, i.e., A(E)S: is not the zero subspace. We carry out an analysis similar to the one in Section 5.3.2 for grouping the Taylor operations in (6.2.4)with respect to the parameter q,. Consider a set { G } u S1,,.,,,,(G’) = S(G). Let g be a (proper and connected) subdiagram in S(G). Let c(g) be the set of all lines in g which have their kfj, independent of v , . ~We may, symbolically, use the convenient notation: 0 = c(B) = 9. [ ~ 4 n 4+m + p

(i) Suppose 0 $ c ( g ) $ g, and introduce the subdiagram g o = g - c(8) as the subdiagram consisting of all the lines L P - c(g) and of the relevant vertices as the end points of these lines. Let g i , . . . ,g h be the maximal elements in S(G) contained in g : gi $ g . Then, by construction, all the kfj, in go = go/g; u . . . u g ; are dependent on q,.’ A similar analysis to that in Section 5.3.2 shows that go is proper and all the k f j in go are dependent on q r . Let g o l , . . . , goN be the connected components of g o . For any subdiagram g E S(G) such that 0 $ c(g) $ we introduce the set O(g) = { g o l , . . . ,gON}. We then define the set M 3 S(G) by (6.2.6) where

u$denotes the union over all sets

O(g) with g E S ( G ) such that

0 $ 4 8 ) $ 3. ‘As usual, we say that a four-vector kf,,depends on q, if at least one of its four components depends on q.. Otherwise we say that the four-vector kfj, is independent of q,. ’ If g is a minimal element in S(G), i.e.,g’, = . . . = gk = 521, then S = g and So = go.

142

6 Asymptotic Behavior in Quantum Field Theory

(ii) Let gES(G) with g $Z G and consider the case where c(g) = 0. Suppose g is a maximal element in S(G) contained in a subdiagram g’: g g’. If all the k$ in S’ are independent of q,, then we introduce the set B(g) = {g}, and otherwise we write p(g) = 0. (iii) Let gES(G) and consider the case where c(g) = g; then we write =

0.

We introduce sets Q X and g o as obtained from the set S ( G ) defined by

(6.2.7) 9 0

=

N - 9”v,

(6.2.8)

where ui p(g) is the union over all sets p(g), with g in category (ii) or (iii), i.e., with c(8) = 0or c(8) = ij. By construction, for any g E JV all the kfjl in ij are either all dependent on qr or are all independent of q,. From the set gXin (6.2.7)we may generate any one, all possible sets which contain in addition to the elements in or any two, or . . . or finally all the elements in go.The collection of all these sets together with the set gXwill be denoted X = { g X.,. . , QX u go}. The corresponding Taylor operations in (6.2.4) for the subdiagrams in the set A’”may then be readily combined, and the corresponding sum over all the sets in X is then reduced to the elementary expression FX(G) =

n

- q)IG?

(6.2.9)

if g E g 0 if g E g N .

(6.2.10)

geX

where

d:={

1 0

By summing over all distinct sets JV we then obtain an equivalent expression for R in (6.2.4) given by

(6.2.11) with d$ = 1.’ We introduce the following notation: ./v = XlW) u

X2(.w,

(6.2.12)

where g E Xl(J(T) if all the kTj, in g are dependent on q,, and g E S 2 ( N )if all the kfjl in S are independent of q,. We also write Xl(Jlr)= S l ( J ( T ) u

92(JVh

(6.2.13)

Zimmermann (1969) was the first to reduce his Bogoliubov-type subtraction scheme to a form as in (6.2.11) and obtain the estimate of the form in (6.2.44) and (6.2.51).

6.2 Analysis of Subtracted-Out Feynman Integrands

143

with < F 2 ( N= ) goif G E X l ( N )and R2(N)= g o - {G} if G E X 2 ( N )We . and 8 ; = 0 ifg E s 1 ( Nu) X 2 ( N ) . note that for g $ G, 8; = 1 ifg E R2(N), We note, in particular, that for g $ G , with g E R2(N),if g is a maximal element in N contained in a subdiagram g’ E N :g $ g’, then all the k$ in a’ are independent of qr. We use the following convenient recursion relation as obtained from (6.2.11) :

F 8 ( 4 = (8: where

R

=

- qYg

n F81(J1T).

(6.2.14)

i

cFG(N),

(6.2.15)

.K ’

and {gi}iin (6.2.14) denotes the set of the maximal elements in N contained in g: gi $ g.9 We also write

(6.2.16) and set L(Qr)G 0. A line L, in G joining a vertex vi to a vertex v j is, in Euclidean space, of the form (see (5.1.1)) DGdQijl, PijJ = PijLQijl, PijJ/CQ$

+ ~$1.

(6.2.17)

We assume that degr D$ I - 1,“’

(6.2.18)

Pill

and degr Pij, I dtgr Piif, QiiisPij1

Qijt

degr D;, I degr D;,. Qzjl. ~ 1 1 1

(6.2.19)

QiJl

The reason for using the expression in (6.2.14). (6.2.15) rather than the one in (5.3.48). (5.3.54). (5.3.55) for R in our asympotic analysis is that it turns out that the latter leads, in general, to an overestimate for thedegr R over the former one. The expression (5.3.48),(5.3.54). (5.3.55) for R is, however, by far much simpler in structure that the one in (6.2.14), (6.2.15). l o For example. for spins 0, I, 2: D& = O(p;;), and for spins 4, : D;l = 0(pGl1),and (6.2.19) is true as well. Equations(6.2.18). (6.2.19) will be used when dealing with the asymptotic behavior of .dwhen some of the underlying masses “take o n ” some asymptotic values, and they will not be needed when all the masses have fixed nonzero finite values.

144

6 Asymptotic Behavior in Quantum Field Theory

Throughout this chapter, since we are interested in the asymptotic behavior of d with respect to the external momenta and/or to the masses of G, we assume that A(1)S; # {O}. Now we prove the following very important lemma for subdiagrams g E N - {G}. The case for the graph G will be treated separately. Lemma 6.2.1 : Let r be a $xed integer in 1 I r 5 k. Suppose that A(E)S; # {O}. Let F,(N)be as dejned in (6.2.14) and suppose that g E N - {G}. For g E Tl(N)u X 2 ( N )ifthere , is

(i) a subdiagram g' $ g with g f E T2(N), andlor (ii) a subdiagram g" t g with g" E Xz(N),such that at least one mass pijrin depends on q,,

a''

then

degr F , W ) < d(g) - 4s).

(6.2.20)

9r

Otherwise, i.e., f(i) and (ii) are not truefor g, then we may replace the sign c in (6.2.20)by <. The latter means that in such a case an equality in (6.2.20) may hold.

The proof is by induction. We suppose that the lemma is true for all those maximal elements g i $ g with g i E Pl(N)u X z ( N ) .In addition to this hypothesis, we suppose, as part of the induction hypotheses, that if we scale the k$ in ij' for all g' E .Xl(N),g' c g i , for all i, and scale as well the masses pijl in the g i , depending on q l , by a parameter I, then

I - 1 - a(gJ (6.2.21) 1

for a g i E X z ( N )and ,

1

min[d(gi),

- 11 -

a(gi)

(6.2.22)

for a gi E .F2(N).11 The condition = O in (6.2.21) and an equality in (6.22) may hold if both conditions (i) and (ii) in the lemma are not true for the corresponding gi with g in these conditions (i), (ii) formally replaced by gi. In addition, suppose that for a gi E P 1 ( Nu) X2(N),an Fgi(N)has a structure as in F,,(N)= Pgi(kBi, q8',p)f:,(ke', p)f:i(kg1, P), It

The notation in (6.2.21) and (6.2.22) will be used whenever convenient,

(6.2.23)

145

6.2 Analysis of Subtracted-Out Feynman Integrands

where P,, is a polynomial in the elements in the sets kei, qgi, and p, and, in general, in the ( p i ) - as well. Here we denote by f : ( k g , q8, p ) any function of the form f i ( k , , 48, p) = [(@$,)2 + p$,]-4J1, (6.2.24)

n

Hi'

g ' ~ J l O ~ ( i t ri j) f g'cg i
for n = 1, 2, where @yjl = Qfjf,and for g' $ g, g' E #"(A'-), = k$. The f l y j f are strictly positive integers. We also denote byf"(ke, qg, p ) any function of the form f"(kg, 4'1 = [Qfjf + ~ $ 1 ""li (6.2.25)

ng

ijl i
for n = 1, 2. If the qyjfappearing in (6.2.24) and (6.2.25) are set equal to zero, then we denote the corresponding functions f ; ( k e , p ) and f"(ke, p), respectively. Finally, if a q i e P 2 ( J f / 'then ) , we suppose that the corresponding F , i ( N ) has a structure as in (6.2.26) We prove the above lemma as well as the results in (6.2.21)-(6.2.26) for the subdiagram g as well. [A] Suppose g E Z 2 ( N ) . Then the g i E X 2 ( N ) u P 2 ( NWe ) . write kei = kei(ke),qsi = qg'(ke,4'). Let 91, . . . , g k l E #2(N) and g k r + 1, . . ,s k i + k z E P2(N).From the induction hypotheses we may then write in a compact notation F g i ( 4= PJk,', qgi, p)fji(kgi, qsi, p)f,,(2

F,(Jf/') = -

degr a

A

P).

c (qg)A+BP$(kg,0, P ) f W , n f,z,(k"(k", ki +kz

p)

A. B a.6.c

5;+ degr fji + degr

degr P$

k9'I,

a

+ degr a

f:j,cj

i= 1

PI

I - 1- 4gi) ffi

1

+ degr fi,{ a

(6.2.29)

7

5)

min[d(gj),

- 13 -

u(gj). (6.2.30)

146

6 Asymptotic Behavior in Quantum Field Theory

Accordingly if degr, 1 ; I- 1 and/or at least one of the conditions = O (for some i ) in (6.2.29) or an equality in (6.2.30) (for somej) does not hold, then we obtain degr F , ( N ) I.

-1

(6.2.31)

- o(g).12

1

If k , = 0, (i) and (ii) in the lemma are not true for the g i , i = 1, . . . , k , , i.e., the conditions = O in (6.2.29) hold for the gi,and degr, f i = 0, then (6.2.32)

degr F,(N)= 0, a

and a(g)

=

0. Finally, from (6.2.27)-(6.2.31) we have

{I

degr F,W) or d(g) Ir

-

ds),

(6.2.33)

where an equality in (6.2.33) may hold if the conditions (i) and (ii) in the lemma are not true for g. We also note from (6.2.27)that F , ( N ) has a structure as in (6.2.23). [B] Suppose g E .Fl(N).Then the g iE 9 , ( N )u X 2 ( N )According . to the induction hypotheses we may write

where

+

1-41 IBI

lai/

+ degr P: + degrf; 1

1

I d(8) - 4L(g),

+ degr Pf;+ degr fii + degr ff, a 1

1

(6.2.35)

d(gi)- o(gi). (6.2.36)

Accordingly we have

For degr, F , ( N ) # 0, the expression on the right-hand side of the inequality in (6.2.31) gives an upper bound value for degr, F,(.&").

147

6.2 Analysis of Subtracted-Out Feynman Integrands

where, again, an equality in (6.2.37)may hold if both (i) and (ii) in the lemma are not true for g. From (6.2.34)we also note that F , ( N ) has a structure as in (6.2.23). [C] Finally, suppose that g E S2(N).Then the gi E X 2 ( N u ) Sl(N), and by the induction hypotheses we may write F,(N) =

a

n i

(qeY'~~(kei(ks), q W e , O), p)f

#YW, p )

x f ,2,(k"(k", p)[1 - T p ' - q I g .

For d(g) 2 la 1, Eq. (5.3.26) in Lemma 5.3.3 implies that degr [l - T:'8)-Iat]IgI d(8) - 4L(ij) - d(g) I

(6.2.38)

+ J a l - 1,

(6.2.39)

where we also have (6.2.40)

degr I , I d ( 8 ) - 4L(g). A

From (6.2.38),(6.2.39), and (6.2.36) we then obtain for d(g) 2 0

I:[

degr F , ( N ) or I

-1

-

a(g).

(6.2.41)

If d(g) < 0, then (6.2.38),(6.2.40),and (6.2.36) imply that (6.2.42) Equations (6.2.41) and (6.2.42) may be combined to yield min[d(g),

- 13

- a(g),

(6.2.43)

I

in the notation in (6.2.22). O n the other hand, (6.2.38) shows that F & N ) has a structure as given in (6.2.26). This completes the proof of the lemma together with the results in (6.2.21)-(6.2.26) for the subdiagram g itself. Now we apply the above lemma to the graph G in question. [I] Suppose G E F2(N). Then (6.2.43) implies that degr F , ( N ) c 1

in the notation of (6.2.22).

- a(C),

(6.2.44)

148

6 Asymptotic Behavior in Quantum Field Theory

Let G , , . . . ,Gm be the maximal elements in N contained in G: Gi $ C . We may then directly apply the estimate in (6.2.37) to obtain for d(G) 2 0 m

degr( - TG)Zafl FG,(N)

(6.2.45)

i= 1

Ir

Also, (6.2.20) and Lemma 6.2.1 imply that degr I a

n

m

m

FG,(N)

[d(Gi) - o(Gi)].

(6.2.46)

i=l

P J ~

To simplify the notation, let G' be that subdiagram of G with c' = G ' / u y = Gi corresponding to all those lines in G depending on qr. Then from (6.2.45) and (6.2.46) we may write

c o(Gi), m

- 4L(Q -

d(G')

i= 1

'Ir

- 4L(G') -

m

1o(Gi)]

(6.2.47)

i= 1

for d(G) 2 0, and (6.2.46) implies that d(G') - 4L(G') -

c o(Gi) m

(6.2.48)

i= 1

'Ir

for d(G) < 0. An equality in (6.2.47), (6.2.48) may hold if there is no subdiagram g' $ G such that g' E S2(N),and there is no subdiagram g" c G in 3Ep2(N)such that at least one of the masses pijl in depends on q r . [11] Suppose that G E 3Ep2(N). Then (6.2.31) implies that for d(G) 2 0

a''

degr(-

TG)lG

1

n i

FG,(N)

< -o(G),

(6.2.49)

where the Gi denote the maximal elements in N contained in G: Gi $ G.i Since degr, 1,- = 0, (6.2.31) and (6.2.43) imply, when applied to the G i , that

ci

degr Ic 1

fli F G I ( ~ <) - O ( G ) ,

(6.2.50)

Hi

where o(G) = o(G!). If d(G) < 0, F G ( N ) coincides with I E FG,(N); accordingly from (6.2.49) and (6.2.50) we always have, i.e., for d(G) < 0 or for d(G) 2 0, degr FG(N)< -o(G). 1

(6.2.51)

149

6.2 Analysis of Subtracted-Out Feynman Integrands

Suppose that the Gi are such that G,, . . ., Gkl, G k l + l , .. ., G k l + k z ~ 9 z ( J V ) with d(Gi) 2 0 for i = 1 , . . ., k,, and d(Gi) < 0 for i = k, + 1 , . . . , k, + k , ; and G k ,+kz+ . . . , G , E sEc;(JV). Let { G i j } be the set of maximal elements in JV contained in G i : Gij $ G i . Let GI be that subdiagram of Gi with Gi = G f / U jGij corresponding to all those lines in Gi depending on q r . Let Gfj be that subdiagram of Gij with GIj corresponding to all those lines in Gij depending on q,. We may use (6.2.33), (6.2.47), and (6.2.48) to write

,,

d(G‘) - 4L(G‘) 9.

-

c a(Gij)] + i

+

i= 1

max[d(Gi) - 4L(Gi), d(GI)

m

1

i=kl+kz+l

(6.2.52)

[d(Gi) - o(Gi)]

for d(G) < 0, and max[d(G), d(G‘) - 4L(G‘)]

-kk1ik2{max[d(Gi)

i=l

4L(Gi),d(G:) - 4L(G;)] -

c a(Gij) i

m

(6.2.53) for d ( G ) 2 0. An equality in (6.2.52) may hold if the conditions (i) and (ii) in Lemma 6.2.1 are not true for all the Gi. An equality in (6.2.53) may hold if (i) and (ii) in Lemma 6.2.1 are not true for all the Gi and, in addition to these constraints, we have

Now consider the situation when A(E)S: = (0). In this case the integration variables are independent of qr. Then we may simply replace Sf by 0 for g $ G in (6.2.14) and, as usual, replace :8 by 1, and JV “becomes” simply Z Z ( J V )= S(G). We may then use directly the estimates in (6.2.52)

6 Asymptotic Behavior in Quantum Field Theory

150

and (6.2.53) by replacing k, and k , in them by zero, as well as setting o(Gi) = 0, and obtain m

d(G') - 4L(G') vr

+ 1 d(Gi)

(6.2.54)

i= 1

for d(G) < 0, and

f<1

rn

max[d(G'), d(G') - 4L(G')]

vr

+ 1 d(Gi)

(6.2.55)

i= 1

for d(G) 2 0. An equality in (6.2.54) and (6.2.55) may hold if the Gi have no masses depending on q, . This completes our dimensional analysis of subtracted-out Feynman integrands R and will be applied in the remaining sections of this chapter. We note in particular that we may write a(G)

+4

L($) = 4L(G),

(6.2.56)

0' E H A X ) g'cG

and quite generally we have dim A(E)S: Ia(G).

(6.2.57)

In particular, to have an equality in (6.2.57) the subspaces S: must be such that dim A(E)S: is a multiple of 4, and all the four components of the kfr,, in a with g E X,(N),depend on q,. Let P be a vector in E, and L,, . . . ,Lk be k independent vectors in E such that

P

=

Llql

* * '

qk

+ + Lrqk + * * *

* * '

+ Lkqk + c,

(6.2.58)

where C is confined to a finite region in E, with the pi # 0 for i = 1, . . . , p, and the external momenta and the masses of the graph G in question may be written as some linear combinations of the components of P. We may then write d ( P , p)

d(Llq1

'*

qk

+ + Lkqk + c). * ' *

(6.2.59)

According to Theorem 3.1.1, the power asymptotic coefficient a,(S,) of d ( P , p ) associated with a subspace S, = { L l , . . . , L,} is given by a,@,) = max [a@) A(I)S = S,

+ dim S - dim S,]

(6.2.60)

with S c R4n+4m+p, where a(S) is the power asymptotic coefficient of the integrand R, and, according to the analysis in Chapter 2, may be identified

6.3 High-Energy Behavior

151

with the degree of R . In the subsequent sections we shall construct the class of the maximizing subspaces A (see Chapter 3) for the various situations at hand directly from the dimensional analysis carried out in this section. This will lead to both the power and logarithmic behavior of d . We recall that if S E A, then a,(S,) = a(S)

+ dim S - dim S,.

Note that for any S c R4n+4m+p ,dim S Idim A(I)S dim S - dim A(I)S 5 dim A(E)S.

6.3

(6.2.61 )

+ dim A(E)S

or

HIGH-ENERGY BEHAVIOR

In this section we are interested in the high-energy behavior of renormalized Feynman amplitudes with all the masses involved in the graph G in question fixed and nonzero. Technically we are interested in the behavior of ed(Llq1 " ' q k

+

' * '

+ Lrqr

* ' '

qk

+ + Lkqk + c), * ' '

(6.3.1)

for q l , q,, . . . , q k + 00 independently, where L,, . . . , Lk are k independent vectors in E , and 1 I k 5 4m. C is a vector confined to a finite region in E such that pi # 0 for all i = 1 , . . . , p. As usual we write S, = { L l , . . ., L,} with 1 5 r Ik. (Recall that R4"+4m+"= I @ E , E = El @ E,, where E , is associated with the external momenta.) We may specialize Lemma 6.2.1 to the problem at hand through the following corollary [see also (6.2.5)]. Let r be a fixed integer in 1 5 r Ik . Suppose that A(E)S: # (0). L e t F , ( N ) beasdefinedin(6.2.14)andsupposethatgEN - { G } . For g E Y l ( N u ) X2(M),if there is a subdiagram g' $ g with g' E Y 2 ( N ) , then

Corollary 6.3.1:

degr F,(J-) < d(g) - 4s). 9r

Otherwise, if there is no subdiagram g' $ g with i g' E P2(N), then the < sign may be replaced by I, which means that an equality may hold for the latter case.

By treating the situation for the graph G in the light of Corollary 6.3.1 we arrive to the estimates for degr, FG(N)in (6.2.47) and (6.2.48) for G E P2(N),and to the estimates (6.2.52) and (6.2.53) for G E JEL2(N)by completely deleting, in the process, condition (ii) in Lemma 6.2.1, as all the pi are fixed (nonzero), i.e., are independent of q,. Similarly, for A(E)S: = (0)

152

6 Asymptotic Behavior in Quantum Field Theory

we have the estimate in (6.2.54) and (6.2.55) with a possible equality holding, as, again, all the masses are kept fixed. For each line el joining a vertex ui to a vertex uj of the graph G , we introduce vectors V0(iJ), . . . , V,(ijl) in R4n+4m+p such that, with P as given in (6.2.5). Vo(ijl) P = Q$l, (6.3.2) V,(ijl) * P’ = Qil.

-

We denote by So($) the subspace generated by the vectors Vo(ijl),. . . , V3(ijl). We also introduce vectors Vb(ijf), . . . , V;(ij/) such that Vb(ijl) * P’ = /&

-

(6.3.3)

V;(ijf) P‘ = k;,.

A close examination of Corollary 6.3.1, together with the estimates (6.2.47),(6.2.48),(6.2.52)-(6.2.55) for the present situation with fixed masses, as discussed above, after summing over JV in (6.2.15), suggests defining the following class d oof subspaces, which turns out to form, in general, a subset of the maximizing subspaces for the I integration of R relative to S, = {Ll, . . . , L,}, with L 1 , .. . , L, as given in (6.3.1). Definition of class M o : We define a class Jt0 = { S ’ , . . .} of subspaces ~4n+4m+p and a set of subdiagrams zo = {G‘, . . .} in such a way that

s

the following are consistent:

(i) A(Z)S = S,. (ii) Let G’ be the subdiagram of G , corresponding to all those lines in G (and, of course, corresponding to the vertices as the end points of these lines), such that all the subspaces So(ijf)of G in G’ are not orthogonal to S’.13 (iii) The proper part Gb of G‘ corresponds to all those lines in G with their Vb(ijI), . . . , V;(ijl)not orthogonal to S’.’“ (iv) If S” c R 4 n + 4 m + p is such that (i)-(iii) above are consistently true, with G”,in particular, corresponding to all the lines in G with the subspaces So(ijr) of G in G” not orthogonal to S”, then d(G”) Id(G’).If d(G”) = d(G‘), then S” E d oand G” E T ~ . In light of this definition, we say that the subdiagram G’ is associated with the subspace S’. l 3 The subspaces S O ( i j 1 ) are defined following Eq. (6.3.2).By all subspaces SO(ijl)ofG in G’ we mean the subspaces So($) for all i, j , and I pertaining to the lines and vertices of G in G’. l4 The condition that all the Vb(ij/),. . . V;(ijl) in the lines of G appearing in G’ be not orthogonal to S’ will be necessary in order to have dim A(E)S’ = 4L(Gb).

.

6.3 High-Energy Behavior

153

We note that G’/G’,(if not empty), with G’ E z0 associated with a subspace S’ E d oand Gb being the proper part of G’, corresponds to the external and improper (if any) lines of G’. An analysis very similar to the one given in Section 5.3.2 in reference to Fig. 5.11 shows that all the vectors Vb(ijl), . . . , V;(ijl) of the external and improper (if any) lines of G in G are orthogonal to S’. This is consistent with condition (iii) in the definition. Note, however, that condition (iii) requires that the whole proper part of G , not just a proper subdiagram of G’,have the Vb(ijl),. . . , V;(ijl) of G in its lines not orthogonal to S’. We also note that for any proper subdiagram g 2 Gb, we may write k$ - (qYjJ0(kC,0) = (kYjJo, or conveniently as Vb(ijr) - (qfjI)O(Vb(ijr),. . . , V;(ijl)) = VS,(ijl)’and V$(ijr)‘ P = ( k f j I ) O . Here(qfj,)’ is a linear combination of the Vb(ijl),. . . , V;(ijl) with i, j , and I pertaining to the lines and vertices in G/g [see (5.1.117)]. Since these latter vectors are orthogonal to S’, and Vb(ijf),with i, j , and I pertaining to G/Gb, is orthogonal to S’ as well, it follows that the V$(ijl)’in g/Gb are orthogonal to S’. Repeating the same analysis for the V!(ijl)’,. . . ,Vq(ijr)’, similarly defined, as well, we arrive to the con. . . ,Vg(ijr)’in g/Gb are clusion that for any proper y $ Gb, all the VS,(ijf)’, orthogonal to S’ since the Vb(ijl),. . . ,V;(ijl) in G/Gb are orthogonal to S‘. If the whole graph GET,, and is associated with a subspace S E ~ ~ , then (6.2.47), (6.2.48), applied to the problem at hand, imply for the set A” = { G } that we may take for an estimated degree of F c ( N ) , with respect to q r , the expression d(G) - 4L(G), i.e., degr F G ( N ) = d ( G ) - 4L(G),

(6.3.4)

,r

with 4L(G) = dim A(E)S. More generally, if G‘ E t o ,with G‘ $ G, and is associated with a subspace S’ E d othen , (6.2.52), (6.2.53), applied to the problem at hand, imply for the set A” = { G , Gb,, . . ., Gb,}, where Gb,, . . . , Gb, are the connected parts of the proper part Gb of G’, that we may take

degr F G ( N ) = d(G’) - dim A(E)S’, ,r

(6.3.5)

,

where dim A(E)S’ = 4 L(G;,) = 4L(Gb). On the other hand, if Gb, , . . . , Gb, are contained (as maximal elements) in some maximal elements G,,.. . , rn I n, with the latter contained in G : Gi $ G , in some set N, then according to Corollary 6.3.1 and the definition of (do, z0) [in particular, condition (iv)], degr,, Fc(J)T) for such a set A” will not be greater than the one in (6.3.5). A moment’s reflection then shows that the power asymptotic coefficient of R itself [see (6.2.15)] for a subspace S’ E d omay be taken to be

c,,,,

ct(S’) = d(G’) - dim A(E)S’,

(6.3.6)

154

6 Asymptotic Behavior in Quantum Field Theory

and that all the subspaces S’€Ao are maximizing subspaces for the I integration relation to S, = {Ll,. . . , L,}. The latter, in particular, follows from conditions (i)-(iv) in the definition of Ao,the power counting conditions (6.2.44), (6.2.51) [see criterion [A], (3.1.3) in Theorem 3.1.11, and (3.1.4), which imply that the power asymptotic coefficient a,(&) of d in reference to the parameter qr in (6.3.1) may be taken for the bound of Id1 [see (6.2.60),(6.2.61),(6.3.6)]:

(6.3.7)

a,@,) = d(G’).

Accordingly, we may state the following theorem. Theorem 6.3.1 : The power asymptotic coe@cient

amplitude

Cd(L141 ’ ” V k

u,(S,) of the renormalized

+ + L,qr***)lk + + Lkqk + c) ”’

’ * *

is simply given b y ~ , ( S R= ) d(G’),

Sr

c

El,

(6.3.8)

where G’ is any subdiagram in 7 0 , respectively in r, with 1 Ir Ik I4m. We note that when some (or all) of the external independent momenta of the graph G becomes large, specified by a parameter, say, q, -+ 00 in (6.3.1), then in reference to this parameter a subdiagram G’ E zo cannot have an extral vertex at which all the momenta carried by all the external lines t o G, at this vertex, are nonasymptotic.15 As a matter of fact, if G’ has an extral vertex, then the total external momentum at that vertex must be asymptotic. This follows from the fact that an extral vertex u i of G’ is necessarily an external vertex of G and an external line t , of G‘ attached to this vertex has its k$ independent of q,, and the corresponding q:; is dependent on q,, by definition of G’. Momentum conservation then requires that qy must depend on q, (q, -+ a).Conversely, G’ contains all those vertices of G at which the total momentum carried by the external lines to G, at each of these vertices, is asymptotic. Note also (for G’ G), G‘ cannot have a subdiagram, say, G i ,as one of its connected components with all the external momenta of Gi being nonasymptotic. Because, whether d(Gi) < 0 or d(Gi) 2 0, this can only “decrease the value” of a($) below d(G’), since then degrqrFG,(N) < - a(Gi) [see (6.2.43)]. Therefore the determination of the subdiagrams G’ E zo is not difficult. Finally, we recall that if any other subdiagram G” similarly defined as G is such that d(G”) 5 d(G’), then G” E 70 only if d(G”) = d(G’). Is

Such a point was also emphasized by Weinberg (1960, p. 847).

6.3 High-Energy Behavior

I55

The subspaces in A. do not, in general, constitute all of the maximizing subspaces for the I integration relative to S , = { L l , . . . , L,} due to the simple fact that if there is a proper and connected subdiagram g’ in a set Jf, with d(g‘) 2 0 and g‘ $ g , g’ E X2(Jf), in Corollary 6.3.1, then degr,, F , ( J f ) may still coincide with d(g) - a(g). Accordingly we may readily extend the definition of the class A. as follows. Consider a subdiagram G’ in t oassociated with a subspace S’ in Ao. Let J,. = {g’,, g i , . . . ,g;.} be the set of all proper, but not necessarily connected, subdiagrams of the proper part Gb of G’ such that the connected part of each of the subdiagrams in .Icp has a nonnegative dimensionality. In particular, we note that if each of the connected parts Gb has a nonnegative dimensionality, then Gb E J , , , by definition of J , , . Let g ; E J,. . We define a generalized subdiagram (G‘lg;) obtained from G’ by shrinking gIl in it to a point and replacing the analytical expression I,,,, in the unrenormalized integrand I,. for G’, by a polynomial in the external uariables of g’, of degree I d(g’,).Therefore (G’ 18;) is nothing but the subdiagram G’ with g’, in it replaced by a vertex, which we call a generalized vertex, with the corresponding analytical expression for the latter as a polynomial in the external varibles of g’,. In this respect, we also note from (5.1.1 17), that the external variables of g‘, may be expressed as linear combinations of the Qijrin G’/g’,,with the vi being vertices in G’/g‘,,but not in g’,, and the uj being external vertices of g’,, and also, in general, as linear combinations of external variables of G’.Accordingly we may formally define an integrand in the same way as we defined I , . . By considering all the elements in J,. , we may generate the following generalized subdiagrams: (G’lg’,), . . . ,(G’ Ig’,.). Finally, we repeat the above construction by considering all of the remaining subdiagrams in t oand generate: (G”I g i ) , . . . , (G” I &-), . . .; G”,. . . E to. We extend the definition of the class d oto the class A by simultaneously enlarging the set t o to include as well all the generalized subdiagrams (G’lg’,),. . . , (G’Ig;.); (G”Ig;‘),. . . ,(G”)g;;.,);. . . for all G‘, G”, . . .E t o ,as follows. To this end we define (G’ 10)= G’. Definition of class M :

We define the class

.I= {S,s;, . . ., s;,; S“,s;, . . . , She,;. . .},

where d o= { S ’ , S”, . . .), and the set of subdiagrams t = {G’,(G’lg‘l),. . . , (G’lg;), G”,(G”Ig‘;),. . . ,(G”(g;;..); . . .} such that the following are consistent: For any 3 E A (i) A(I)s = S,. (ii) Let be the subdiagram of G, corresponding to all the lines in G , such that all the subspaces So($) of G in G are not orthogonal to 3.

e

156

6 Asymptotic Behavior in Quantum Field Theory

e e,

(iii) The set of all the lines of G in having all their Vb(ijl), .. . , V;(ijl) not orthogonal to $ coincide with the set of all the lines in the proper part of the generalized subdiagram (GI& = where 4 is either empty, or otherwise 4 is a proper subdiagram of such that each of the connected components of 4 has nonnegative dimensionality. (iv) G belongs to zo (and obviously to z) and G belongs to z.16 As before, we also say that the generalized subdiagrams (G’lg;),. . . , (G’Igb.); (G”lg:), . . . ,(G”Ig&);.. . are associated with the subspaces S’,, . . . , Sb,; S‘;, . . . , S’h,,; . . . , re~pective1y.l~ The basic difference between a subspace S’ and S;, say, is that

e

a(S’) = d(G’) - dim A(E)S’

(6.3.9)

a@‘,) = d(G’) - dim A(E)S;

(6.3.10)

and where 4L(Gb) = dim A(E)S’, 4L(Gb/g;) = dim A(E)S;, and hence dim A(E)S; < dim A(E)S’. Both subspaces S’ abd Y1are obviously, however, maximizing subspaces for the 1 integration relative to S,, and from the definition of the set t otogether with (6.2.60), (6.2.61), we may take (6.3.11)

a,(S,) = 4 G ’ )

when considering both subspaces S’ and S;. Now with the class A of the maximizing subspaces for the 1 integration of R relative to the subspace S , = { L , , . . ., L,}, with L,, . . . , L, as given in (6.3. l), we determine the logarithmic asymptotic coefficients p,(S,) of d . We decompose 1 as a direct sum of 4n one-dimensional subspaces: 1 = I , @ l , @ . . . @., l Lemma 6.3.1 : All the maximizing subspaces for I, integration relative to S,, afer performing the l 2 @ . . . @ 14,,,a w given in the set

A’ = { A(12 @

. @ 14,,)S: all S E A}.

(6.3.12)

All the muximizing subspaces for the 1, integration relative to any one of the subspaces in A’,say, S’ E A’,a f e r performing the I 3 @ @ I,,,, are given in the set A2= {A(l, @ . . . @ 14”)S;S E A and A(1, @

0 14,,)S = S ’ } . (6.3.13)

Of course the classes d oand .M depend on the integer r. Note that with the connected components g‘liE%2(.N) of a g;, all the F&,(.h.) are some polynomials of degree
”

157

6.3 High-Energy Behavior

More generally we have recursively that all the maximizing subspaces for the I i integration relative to any one of the maximizing subspaces in M i - ' ,say, Si- E Ai- after performing the I i + Q - QI 14, integration are given in the set

'

',

.Ai= { A ( I i + Q . . . @ 14,)S : S E A and A(Ii Q

. . Q 14,)S = S i - ' } ,

'

'

(6.3.14)

for i < 4n. W e set Ao= {S,} and we define for some S4"- E A4,A4"= {S : S E A and A(14,)S = S4"- '}.

(6.3.15)

Suppose that the lemma is true for all i in 1 I i I k < 4n. Let Sk be any given subspace in Ak.The latter means, in particular, that there is some subspace S' E A such that A(Ik+ 1 @

* *

0 14,)s

(6.3.16)

Sk.

We now show that for any subspace S E A such that A(Ik+ 1 6 . .. @ 14,)s= Sk

-

(6.3.17)

it follows that A(Ik + Q . Q 14,,)S is a maximizing subspace for the I k + integration relative to Sk after performing the I k + @ . . @ I,, integration. This follows from the following chain of inequalities: 'Ik+

I @ "'@14n(Sk)

-

max

-

[aIk+2e...e14n($ + dim

A(lk+I)S=Sk

2 -

'Ik+

z@

...@14n(A(lk+

+ dim A(Ik

+2

2

s - dim SkJ

Q * * * Q I4n)V

Q . . . Q 14,,)S- dim Sk

max

[a(S")

A ( I k + z@ . ' . @ I 4 n ) S " = h ( l k + l @ '"@/4,)s

+ dim S - dim Sk = a,(S,) + dim S, - dim Sk

+ dim S

- dim SkJ

2 a(S)

-

max

[alk

A(ll@."@Ik)S=s,

2 -

[email protected],(Sk)

+ 1 @ "'@ 14"(Sk),

+

@.

+

.. @ 14m(g) dim

- dim Sk]

+ dim Sk - dim Sk (6.3.18)

where we have used the facts that S' and S in (6.3.16) and (6.3.17) are in A and the fact that (6.3.19)

158

6 Asymptotic Behavior in Quantum Field Theory

Since the extreme left-hand side and the extreme right-hand side of the inequalities in (6.3.18) are identical, we may replace all the inequality signs in it by equalities. Therefore we obtain, in particular, that 'II, + I

@

..' @14.(Sk)

=

'1,'

z@

..' @14n(A('k

+2 @

' *

0 I4t1)~)

+ dim A(fk+2 @ . - .@ I4,)S - dim S k ; +

(6.3.20)

i.e., I \ ( I k + @ . . . @ 14")Sis a maximizing subspace for the I k + integration relative to Sk after performing the I k + Z @ . . . Q 14, integration. Now we prove that any maximizing subspace for the I,+ integration relative to Sk after performing the l k + @ . . . Q 14, integration is in the set A k + l

= { ~ ( ~ k + 2 $ ' . ' @ ~ 4 , ) S : S E ~ a n d A (@ I k* + . *,@ I , , , ) S =

sk},

(6.3.21)

thus completing the proof of the lemma by induction.'* integration Suppose that So is a maximizing subspace for the relative to Sk, after performing the 1, + @ . Q 14,, integration, and that So 9 A k + l .We shall then reach a contradiction. By hypothesis I @'" @14n(Sk)

= 'Ik+ z@

"'

@14.(So)

A(lk+ 1)So = sk.

Let

+ dim '0

- dim S k , (6.3.22)

s be a maximizing subspace for the @ . @ 14,,relative to S o , i.e., aIkt2@...@f4,(So) = a(S) + dim S - dim S o , Ik+2

A(I~+ 2 0 ...

(6.3.23)

14,)s = SO.

From (6.3.22) and (6.3.23) we have = a(s) aIk+ ... e14m(Sk) A ( I k + 1 @ * * @ 14,)s = Sk.

Equation (6.3.18),however, implies that [email protected](Sk) = a,&)

+ dim s - dim Sk, (6.3.24)

+ dim S , - dim Sk,

(6.3.25)

which upon comparison with (6.3.24) shows that E A and hence, by A ( ] k + 2 @ . - .@ 14,,)S = So EA'", thus leading to a definition of Ak+', Note that the proof contradiction of the initial hypothesis that So $ Ak+'. does not depend on the fact that the I i are one dimensional, and the same Wenotethat theset.1'" (6.3.16). belongs to .XktI .

isnot emptysinceA(lkt2 @ . . . @ I,,)S',withS'introducedin

159

6.3 High-Energy Behavior

analysis as above shows that all the maximizing subspaces for the I , integration relative to S, after performing the I 2 Q . . . Q I,, integration are in A’.This completes the proof of the lemma by induction. Since the dimension numbers p j , j = 1, . . . , 4 n , in the expression for the logarithmic asymptotic coefficients fi,(S,) of A in (3.1.6) may be computed relative to any one of the maximizing subspaces for the Ij-l integration, after performing the I j Q - . Q I,, integration, we may use the results in Lemma 6.3.1 and (3.1.6) to state +

Theorem 6.3.2: The logarithmic asymptotic coeficients fi,(Sr) of the renormalized amplitude .d are given b y 4n

PdSr) =

CPj,

(6.3.26)

j= 1

where p j , j = 1 , . . . ,4n, is equal to zero same dimension, and p j = 1 otherwise.

if all the subspaces in Aj have the

We shall simplify Theorem 6.3.2 further to a form more suitable for applications. Choose some subspace in Ao;call it So. Let A2in (6.3.13) be the set of all the maximizing subspaces for the I 2 integration relative t o A(12 0 . . Q I,,)SO after performing the I 3 Q . . Q I,, integration. By definition, we observe that A(I3 0 . . . 0 14,)S0 is necessarily in A2 [see (6.3.13)]. Recursively, then, let Aj be the set of all the maximizing subspaces for the I j integration relative to A(Ij Q . . . @ 14,)S0 after performing the I j , , 0 . . . 0 14” integration. Obviously A ( I j + Q 0 I,,)SO belongs to 4’ since A ( I j ) A ( j + Q . . . Q 14,)S0 = A(Ij Q . . . Q 14,)S0. Accordingly we may replace Theorem 6.3.2 by the following simpler version : Theorem 6.3.3: Choose So to be any subspace in d o Let . {S,S”,

set of all those subspaces in A such that A(Ij 0 . . * Q 14,)s’ = A(Ij Q *

.

*

. . .} be the

l9

Q 14,)s” =

-

* *

= A(Ij Q *

.

*

@ 14,)S0,

(6.3.27) then

(6.3.28) where p j = 0 ifall the elements in

{dim A(Ij+ 0 . . Q 14,,)S - dim S,, dim A ( I j + Q . . . @ I,,,)S” - dim S,, . . . ,dim A ( I j + Q . . . Q 14,)S0- dim S r } (6.3.29) l9

Obviously this set is not empty sincc it contains the subspace So itself.

160

6 Asymptotic Behavior in Quantum Field Theory

are equal, and p j = 1 otherwise. The subspaces S', S", . . . are given by (6.3.27). The dimensions of the A(Ij+ Q . . Q 14,,)S',. . . have been measured relative to dim S,. The chosen subspace So in d owill be called a reference subspace. In many applications, d oconsists of only one element d o= { S O } and Theorem 6.3.3 may be readily applied. From (2,1.1), the work in Chapter 2, Theorem 6.3.1, and Theorem 6.3.3 we may then state Theorem 6.3.4 d(L1ql

"'qk

-

+ + L r V r " ' q r + + Lkqk + c) . . qid{Lls...sLk)) (In qnl))'I - (In q n k ) y k } , " *

" '

(6.3.30) ....Y k where { L l , ..., Lr} = S , c E l , 1 I r I k I 4m, C is confined to a finite region in E , with the masses pi # 0 for all i = 1, . . . , p. T h e sum in (6.3.30) is over all nonnegative integers y l , . . . , y k such that o { q a 1I ( ( L I ) ) .

?I.

I

for all 1 5 t Ik , and the logarithmic coeficients

increasing order

(6.3.3 1)

P

have been arranged in

(6.3.32) P({L,, . . . , Ln,H I - - * 5 P((L1, - L n k h where {?rl,. . . , nk} is a permutation of the integers in { l , . . . , k } . The power al(Sr) and the logarithmic Pl(Sr) asymptotic coeficients are, respectively, given in Theorem 6.3.1 and Theorem 6.3.3. 7

Consider the self-energy graph G of a fermion shown in Fig. 6.1. Suppose we let the external momentum q become large. Consider the amplitude d ( q 4 , m, p ) and let q + co. Here m denotes the mass of the fermion and p denotes the mass of the boson, with both masses assumed to be nonzero.2o Let 1, = I , 0 ... Q I4 be associated with integration variables k,, and 1, = I, @ . . @ I s be associated with integration variables k , . I . . . , I , denote one-dimensional subspaces. We note that in Fig. 6.1, degr, D:3 = degro D14 = - 2 for the spin 0 propagation, and degr, D:2 = degrv D t 3 = degrQD14 = - 1 for the spin 4 particle. Also the dimensionalities of G, g , , and 9 , are as follows: d(G) = 1, d ( g l ) = d(g2) = 0. Obviously zo = (G}. As a matter of fact any other subdiagram g $C G has d(g) c 1. Canonical decompositions of the Q i j ,for G in Fig. 6.1 are given in (5.1.40), and the corresponding expressions for k$,, q$, k $ , qf$ are given in (5.1.81)-(5.1.84). Example 6.1 :

,,

2"

Further generalizations to this will be given in subsequent sections,

6.3 High-Energy Behavior

161

Fig. 6.1 A self-energy graph G of a fermion with a $+#I coupling contributing to i t with the dashed lines representing a spin 0 and the fermion is of spin f.

The graph G E T , , is associated with the subspace So = S ( q ) 0 1, 0 I,, where S ( q ) is a subspace associated with the momentum q. We readily infer from Theorem 6.3.1 with no further work that (6.3.33)

a,(S(q)) = d(G) = 1.

From (5.1.82) we note that 1, is associated with k$, and 1, is associated with k f j . From the definition of (A,T ) we may write

.x = {S', s;, s;,s;} 7 =

where

(6.3.34)

{ G , ( G I g A (Gig,), (GIG)},

(6.3.35)

s = so= S(q) 0 i, 0 I;, (6.3.36)

s; = S(q). The subdiagrams G , ( G l g J , ( G i g , ) , (GIG) are associated, respectively, with the subspaces S', S ; , S,, S ; . Hence in the notation of Theorem 6.3.2 and Lemma 6.3.1 we have .N' = (A(I2 0 .. @ 1 6 ) s : s = so, s;, s 2 , &}, (6.3.37) j=2,3,4,6,7,8,

J T J = { A ( l j + l 0 . . . 0 I 6 ) S ",)

JT5 = {A(Z6 @ 17 0 1 6 ) s : S = So, S;}.

(6.3.38) (6.3.39)

Accordingly by the application of Theorem 6.3.3, using {dim A(Z2 0 ... 0 1,)s - dim S ( q ) : S

=

So, S ; , S2,S3} = {l,O, l,O}, (6.3.40)

we obtain p ! = 1, and from (6.3.38) we obtain p j = 0 f o r j = 2, 3,4,6, 7, 8 since the A',for suchj, constitute only one element. Finally, from {dim A(16 0 Z7 0 1 6 ) s - dim S(q) : S

=

So, S2}= {5,4},

(6.3.41)

we have p s = 1. From (6.3.28) we then obtain (6.3.42)

162

6 Asymptotic Behavior in Quantum Field Theory

Therefore we may write (6.3.43)

6.4

GENERAL ASYMPTOTIC BEHAVIOR I

In this section we generalize the results given in Section 6.3 and consider the asymptotic behavior of d when not only some of the momenta become large but also some of the masses in the theory become large as well. The analysis here is similar to the one carried out in Section 6.3, and we shall be brief. Applications of this analysis will be given in the remaining part of this chapter. As in (6.3.2), we introduce for each line e, joining a vertex ui to a vertex v j of G, vectors V0(ijr), .. .,V3(ijr) and now an additional vector V,(ijr) such that, with P as given in (6.2.5),

Vo(ijr) P = QP,,,

P V,(ijr) P V3(ijr)

(6.4.1)

= Q&, = pt,,.

We denote by S(ijr) the subspace generated by the vectors Vo(ijr), . . ., V3(ij0, V,(ijr). We also introduce vectors Vo(ijr), .. , ,V;(ijr) as in Section 6.3 satisfying (6.3.3). Technically we are interested in the behavior of d(LlV1 ‘ * *

tlk

+ + Lrqr ’ * *

*.*

vk

+ + Lkqk + c) ‘ * *

(6.4.2)

for ql, q 2 , . . .,tlk + 00, independently, where L 1 , .. . , Lk are k independent vectors in E = El 6 E 2, 1 5 k I 4m + p, C is a vector confined to a finite region in E, such that pi # 0 for all i = 1, . . . , p.

a class A. = {S’, . . .} of subspaces and a set of subdiagrams r0 = {G’,. . .} in such a way that the following are consistent: Definition of class .MAY,: We define

s

~4n+4m+p

(i) A(Z)S’ = S,. (ii) Let G’ be the subdiagram of G, corresponding to all the lines in G (and, of course, corresponding to the vertices as their end points) such that all the subspaces S(ijr) of G in G‘ are not orthogonal to S’. (iii) The proper part Go of G’ corresponds to all those lines in G with all their Vb(ijl),. ..,V;(ijl) [see (6.3.3)] not orthogonal to S’.

6.4

General Asymptotic Behavior I

163

(iv) If S“ c 5!4n+4m+p,with which a subdiagram G” is associated, is such that (i)-(iii) are true, then d ( G ) I d(G’). If d(G”) = d(G’), then S” E A. and G E to. By the same analysis leading to Theorem 6.3.1 we have Theorem 6.4.1 : The power asymptotic coeficients a,(Sr) of the renormalized

amplitude

d(LIV1 with 1 I r I k I4m

”’

qk

+ .” + L r q r

* * *

Vk

+ + L k V k + c), ’ * *

+ p are given, respectively in r, by = d(G’),

(6.4.3) (6.4.4)

where G‘ is any subdiagram in to. The subdiagrams G’ E to are determined as in the case for the high-energy behavior given in Section 6.3. There are, however, some differences in the nature of the diagrams in this case. For example, a subdiagram may contain an extra1vertex at which the total external momentum carried by the external line to G’ at that vertex is nonasymptotic as long as the external line of G’ attached to this vertex carries an asymptotic mass.” Definition of class A :

We define the class

A = {S’,s;, . . . ,SN,; S”,s;, . . . , s;,.;. . .},

where A. = {S’,S”.. .}, and the set of subdiagrams z = {G’, (G’lg’l),. . . , (G’lgh,);G”, (G”Ig);),. . . , (G”IgK,,);. . .} with to = { G’, G”, . . .}, such that the following are consistent: For any 5 E A (i) A(1)S = S,. (ii) Let be the subdiagram of G, corresponding to all the lines in G, such that all the subspaces S(ijl) of G in are not orthogonal to 5. (iii) The set of all the lines of G in G having all their Vo(ijr), . . .,V3(ijr) not orthogonal to coincide with the set of all the lines in the proper part of the generalized subdiagram 14) = G,where 4 is either empty or otherwise 8 is a proper subdiagram of such that each of the connected components of 4 has a nonnegative dimensionality, and all the masses in the lines in 4 are independent of q,. In the notation in (6.4.1) the latter means that the V4(ijl) of the lines in 4 are orthogonal to 5. (iv) G E to c t and G E T for all nonempty 0 as defined above.

e

e

(e e

I ’ In such a case. of course, the momentum-dependent part of this external line is nonasymptotic by momentum conservation.

164

6 Asymptotic Behavior in Quantum Field Theory

Theorem 6.4.2

L , ~ , ” ‘ ~ k + ‘ *+’L k q k

d(Llq1”’qk+“’+

..

- O { ~ 1~ I ( { L I.) ) ,,,;I((LI..-.L~H 71.

1 (In vJ1 ...

+ c, . . . (In q*,)YkI,

(6.4.5)

.Yk

where { L l , . . . ,L,} = S, c E, 1 Ir I k I 4m + p , C is conjned to a j n i t e region in E, with the masses pi # 0 for all i = 1, . . . , p. The sum in (6.4.5) is over all nonnegative integers y l , . . . , Yk such that I

1

i= 1

Yi

5

P({L,,

* * 3

Lzt}),

1 I t I k,

(6.4.6)

and the logarithmic coeficients P have been ordered in increasing order

P(L

* * * 9

LIDI

*

*. I P((L1, . . . I Ln&

(6.4.7)

where {n,,. . . , nk} is a permutation of the integers in { 1, . . . ,A}. a,(S,) is given in Theorem 6.4.1, and the P,(S,) may be obtained from Theorem 6.3.2 or dejned above.22 Theorem 6.3.3from the classes A,and 6.5

ZERO-MASS BEHAVIOR

The purpose of this section is to study the zero-mass behavior of renormalized Feynman amplitudes (Section 6.5.1) and finally give sufficient conditions to guarantee the existence of the zero-mass limit of renormalized Feynman amplitudes (Section 6.5.2).The study is general enough to deal with the most general cases when some (not necessarily all) of the masses of a Feynman graph G become small and, in general, at different rates. That we have to consider (i) the most general cases when some of the masses as well and not necessarily all the masses become small and (ii) the approach of such masses to zero at different rates as well are clearly physical requirements. In quantum electrodynamics, for example, one would be interested in the behavior of a renormalized Feynman amplitude for p -+ 0, m + 0, and (p/m) -+ 0, where p is a photon “mass” and m is the mass of the electron. For a propagator D&, carrying a mass pijrthat we wish to scale to zero, we write (in Euclidean space) D$I = P i j L Q i j r , P i j J / ( Q & + P&), (6.5.1) where Bijl(Qij,,p i j r )is a polynomial in Q i j r and pijl but not in (l/pij,) such that DGdQijls

0) = B i j L Q i j r ,

o)/Q?j~

(6.5.2)

’’ Of course. Theorems 6.3.2 and 6.3.3, as they stand, apply to the present situation as well.

165

Zero-Mass Behavior

6.5

. ~ ~ the propagator in (6.5.1) we denotes the mass p i p = 0 p r ~ p a g a t o r With then carry out the subtractions of renormalization as usual to obtain the final expression for R . All those propagators carrying masses which we do not wish to approach zero will be written as in (5.1.1). We note that in general we may rewrite (51.1) as (in Euclidean space with E = 0) (6.5.3) where d i j , is some nonnegative integer, and Pij, is a polynomial in Qijrand p i j r ,but not in l/pij,. Also quite generally we suppose that degr J‘ijdQijr?

PijJ

=

degr

PijLQijl, PijJ

Qlif. Ptil

Qiir

=

degr P i j / ( Q i j / ,

(6.5.4)

PijA

Qijr

and with D;, = ( p i j l ) - ” i i l D & we , have degr fi;, = degr Qtjfv@#Jl

DGI.

(6.5.5)

QtJf

The expressions (6.5.4) and (6.5.5) will be assumed explicitly. By working in (6.5.5) we may introduce an with the propagators 06,in (6.5.1) and unrenormalized integrand T, given by

I,

n P

=

-

(6.5.6)

(pJ)-OJlc

j= 1

in a form as in (2.2.17), where the oj are some positive integers. 6.5.1

Zero-Mass Behavior of d

The structure of a Feynman amplitude associated with a proper and connected graph G is, from (2.2.3), of the form

”

For example, in quantum electrodynamics we write for the spin f propagator (up

-

m)/

+ m2)and for the photon propagator (in the Feynman gauge)g,,/(Q’ + p2).Quitegenerally, wemayalsoallow higher powersofthedenominator in(6,5,I):(Q$ + @“,with integersn 2 I .

(p’

.

The latter does not necessarily mean that one isallowinga double,triple, . . . etc.. pole term in the propagator. as this depends very much on the structure of the polynomial P i j f .In any case we always have to take the correct dimensionality degru,,, D;f(Qijr, pijJ of D;r(Qijf, p i j f )when carrying out thc subtractions of renormalization.

166

6 Asymptotic Behavior in Quantum Field Theory

where [see (2.2.14), (2.2.17), (2.2.18)-(2.2.20), and (6.5.6)] P

R

=

fl

(6.5.8)

(p')-"jR,

j= 1

RPY,.

3. kO ., prn, 1 3 . . , k:; P ' , - - .

9

= pf' =

P'

xi

L

pP) =

1 A f , , , t , i P ' ~ ' i ~ "CQ:i l n+ d 1 9 i

1= 1

(6.5.9) ( k y l

* . . (k+,

SLj

2 0,

(~!)'61

. . . (p:)f$m,

tij

2 0,

. . . (/p)4,,

u; 2 0,

p"'

(6.5.10)

with Q,= afki + C j bjp,. The Af,,,,, are some suitable coefficients. Suppose that { p ' , . . . ,ps}, with s I p, denotes the subset of the masses that we wish to approach zero. We scale the masses in the set { p ' , . . . ,,us}as follows: (6.5.11) ps + 1112

* * .

AspS.

The masses in the subset { p ' , .. . , p'} have been arbitrarily labeled from 1 to s for convenience of notation. Without loss of generality, we assume that all those masses that we wish to approach zero at the same rate have been identified with pl, or p2, or . . . , ,usdepending on the rate we wish them to approach zero, etc. By the definition in (6.5.2), the factor (pj)-"J in (6.5.8) is invariant under the scaling in (6.5.1 1); in particular, o1 = o2 = . - . = os = 0. Quite generally we have under the scaling (6.5.11).

mZl

k'fp''p"'

+

( I , . . . A,)d(N)(k')"'(P')'i(p')"',

(6.5.12)

where

(6.5.13)

6.5 Zero-Mass Behavior

I67

and d ( N ) is the dimensionality of the expression on the left-hand side of (6.5.12),i.e., d ( N ) = (sb1 + . . . + s i n )

+ ( t i , , + . . + ti,) + (u; + + u;), * * *

(6.5.14)

and is a fixed number of all i, for which A:,,,,, # 0, and coincides with the dimensionality of the numerator in (6.5.9).Similarly the denominator in (6.5.9)is transformed as

n CQ: L

1=1

+ d1

n CQ;’ L

+

(A1

*

I= 1

+ ~;’l,

(6.5.15)

where Q; = Q,/Al . ..A,. d(D) is the dimensionality of the denominator on the left-hand side of (6.5.15),i.e., d(D) = 2L.

(6.5.16)

Accordingly i? in (6.5.9)is transformed to

( A , . . . A,)d(R)d(p;o, . . . ,p z ; k;’, . . . , kk3; p’l, . ..,P ’ ~ ) ,

(6.5.17)

d(R) = d ( N ) - d(D).

(6.5.18)

where Hence finally the renormalized amplitude d formed to

E

npZ1

( p j ) - u J g is trans-

(6.5.19) where d(G) = d ( N ) - 2L

+ 4n,

(6.5.20)

and we infer from (6.5.5)that d(G) coincides with the dimensionality of the graph G in question. We choose the external (independent) momenta pl, . ..,pm such that no partial sums of these momenta vanish; i.e., p i , + . + pi, # 0 for all subsets {il, . . . ,i t ) c ( 1 , . . ., This, in particular, 24 Such external Euclidean momenta have been called nonexceptional momenta (cf. Symanzik, 1971). We recall that these external momenta are momenta carried by the external lines to the graph G taken, by convention, in a direction away from the external vertices. In general, at each external vertex, the total external momentum carried away from that vertex may be written in the form f ( p i , . . . + pi,).

+

168

6 Asymptotic Behavior in Quantum Field Theory

means that the total external momentum carried away from each exernal vertex is nonzero. As the elements in p are also chosen to be fixed and nonL(pLi)-uJ is independent of the parameters A,, . . .,As. zero, the factor The amplitude J$ in (6.5.19) is of a particular form of the amplitude d analized in Section 6.4, where in the former the propagators are of the form in (6.5.1) and b$, introduced below Eq. (6.5.4). We decompose E 2 as E2 = E$ @ E i , where E i is an (s - 1)-dimensional subspace and E: is its orthogonal complement in E 2 ,We introduce orthogonal vectors L2, . . . , L, in E i with nonvanishing components p l , .. . ,ps- I , respectively. We may also introduce a vector L, in El @ E i with nonvanishing components py, . . ., p i , ps' ', .. . ,pp and a vector C E E: orthogonal to L1 with nonvanishing component ps. Finally, we may write .d in (6.5.19) in the form s?(P), where

m=

P=(A1*..AS)-'L,+(A2-**3Ls)-1L2 +

*

*

a

+

&'L,+C.

(6.5.21)

The conditions I,, . . . ,A,+ 0 in (6.5.21) mean, in particular, that all the external momenta become asymptotic and the total external momenta at all the external vertices are asymptotic. Obviously the vectors L1,. .., L, are s independent vectors in IW4n'4m'p. We may now apply Theorem 6.4.2 (see Section 6.4 for details) and infer from (6.5.19),together with (6.5.21), that

for Al, A,,

...,As -+

0, and where

(6.5.23)

d i = d(G) - a,(&)

for i = 1 , . . . ,s, Si= {Ll,...,Li}, with adsi) being the power asymptotic coefficients of 2 ( P ) with respect to the parameters l/&. The sum in (6.5.22) is over all nonnegative integers y,, . . . , ys such that (6.5.24)

for all t in 1 I t Is, and the B have been arranged in an increasing order:

B(IL1, - - *

9

L,})I I B({L,, * * *

* * 3

LZJ.

(6.5.25)

From the definition of the classes 4, and A in Section 6.4 and (6.5.22) we may then readily give sufficient conditions for the existence, with nonexceptional external momenta, of the zero-mass limit of renormalized Feynman amplitudes d for A,, .. . ,A, + 0.

6.5 Zero-Mass Behavior 6.5.2

I69

Rules (Sufficiency Conditions) for the Existence of lim .a?

Consider the amplitude d,as given in (6.5.19), associated with a proper and connected graph G. Let i be fixed in 1 I i I s. Let IT;. be the set of all the subdiagrams of G such that the following are true. If Gi E IT;., then Gi ( cG) contains all of the external vertices of G but not necessarily all of its lines. The lines in G/Gi (if not empty) do not carry any external momenta, and all their masses are from the set { p i ,pi+ . . . , p”}.z5Any external line of Gi depends on the elements from the set P and/or { p l , . . . ,p i - 1 , p S + l , .. . ,P P } . * ~ We repeat the definition of IT;. for all i = 1 , . . . , s, thus generating the sets T,, . . . , T,. If the following are true, for all i = 1, . . . , s, then limyl,...,As-,o d exists.” (i) for any Gi E T , d(Gi) I d(G). (ii) If d(Gi) = d(G), then Gi does not have a proper subdiagram gi c Gi in it such that the masses in the lines in gi are from the set { p i , pi+ . . . ,p”} (if not empty), and the dimensionality of each of the connected components of gi is nonnegative. The above rules are very simple to apply and one may, in general, infer the existence of limA,,...,A,+o d by a mere examination of the graph G from these rules with almost no extra work. In particular, the rules state as one of the sufficiency conditions for the existence of the latter limit that the graph G itself is not to contain a proper subdiagram g such that all the masses in g are from the set { p ’ , . . . , p”}, and that the dimensionality of each of the connected components of g is to be nonnegative. Before giving some examples applying these rules we wish to note the following. This will save us time in applications. The above conditions if satisfied imply that di 2 0 for all i = 1, . . ., s and for the corresponding i for which di = 0 we have fl,(Si) = 0. If fl,(Sj) = 0 for some j, then no In(l/Aj) terms will appear in (6.5.22). The reason is that with the ordering as in (6.5.25) with n1 = j, i.e., 0 = B(S,) I . . . I B({L,,. . . ,Lz,}), then from (6.5.24), we have 0 I y1 I fl(S,) = 0, and hence we obtain y1 = 0.

’,

’,

Example 6.2 : Consider the graph in Fig. 6.2 representing the lowest-order contribution to the self-energy of the electron in quantum electrodynamics. We write the photon propagator (in the Feynman gauge) as D,,.(Q) = g,,,/(Q2 + p2). We consider the behavior of the amplitude d ( q , m,Ap) for A -, 0, where m is the electron mass and q # 0. Any subdiagram G’ c G is such that d(G’) I d(G). We also have d(G‘) = d(G) only if G’ is the graph G zs This simply means that the lines in GIGi,in reference to the amplitude .d,are independen1

0r,ir1.

’’This simply means that the external lines of Gi depend on A:’. 27

More precisely, we should say that .dremains bounded in this limit.

170

6 Asymptotic Behavior in Quantum Field Theory

Fig. 6.2 Lowest-order electron self-energy graph in quantum electrodynamics. The wavy line represents a photon, and the straight line represents the spin particle.

itself. Also G does not contain a proper subdiagram having all its masses from the set { p } . Accordingly the limA+od ( q , m, Ap) exists. This example demonstrates the simplicity of the application of the rules for the existence of lim d given before. For the convenience of the reader we give the explicit expression for the renormalized amplitude corresponding to Fig. 6.2 with subtractions performed at the origin and with p = 0: d ( q , m, 0) = -

211

yq Jolx dx In( 1

+

2

x)

(6.5.26)

where a is the fine-structure constant a = e2/4a. The expression d(q,m, 0) in (6.5.26)obviously exists for q2 > 0 (also for q = 0) and m # 0. Example 6.3: Consider the photon self-energy graphs in quantum electrodynamics. Some of these graphs are shown in Fig. 6.3. For the photon propagator we write DPv= g,,,/(QZ + p2). Let G be any graph in Fig. 6.3. We note that for any g c G,we have d@) 5 d(G). Also, G does not contain any proper subdiagram having all its masses equal to p. Accordingly the limit 1 + 0 of the renormalized amplitudes d(q,m, Ap), corresponding to all the graphs in Fig. 6.3. exist in the limit A 0. Finally, we give an example where the behavior of d may be studied as of one of the masses of the graph in question becomes small and another one goes to zero.

Fig. 6.3 Some low- and high-order photon self-energy graphs in quantum electrodynamics.

6.6 Low-Energy Behavior

171

Fig. 6.4 A Fourth-order electron self-energy graph in quantum electrodynamics.

Consider the behavior of the renormalized amplitude d(q,A,m, AIA,p) associated with a fourth-order self-energy graph of the electron in quantum electrodynamics, shown in Fig. 6.4., for A,, A, + 0. Consider the amplitude d ( q / A l ,m, A,p). We refer to Example 6.1 to infer, in the notation in (6.3.33) and (6.3.42), that a,(S(q)) = 1, b,(S(q)) = 2. On the other hand, repeating the same analysis as the one given in the previous two examples shows that the limit A, + 0 of d ( q / A , , m,A2p) exists. Accordingly we have for A,, A, + 0 Example 6.4:

(6.5.27) where we have used the fact that d(G) = 1. 6.6

LOW-ENERGY BEHAVIOR

In Theorem 6.1.1 we have seen that d ( A P , p ) + 0 (A + 0) for d ( G ) 2 0, with fixed nonzero masses, as expected. The same analysis leading to this theorem shows that d ( A P , p ) remains bounded for A + 0 when d ( G ) < 0 since Ia 1, in (6.1.18), is positive." In this section we generalize these results to the cases when some of the underlying masses approach zero as well, and some, not necessarily all, of the external momenta become small. On physical grounds we let these masses approach zero at a rate faster than the corresponding external momenta become For generality we let these vanishing external momentum components and those vanishing masses become small at different rates. We write P=PuP, (6.6.1) Note also that (a1 is bounded above. Applications may be also given when some of the external momenta become small at a faster rate than some of the vanishing masses; however, these are of less interest than the general cases given here and will not be discussed. 29

172

6 Asymptotic Behavior in Quantum Field Theory

where P" is that subset of P containing those elements we wish to scale to zero, and P' constitutes the remaining elements in P. We decompose El as (6.6.2)

El = E ; 8 E:,

where E: is, say, a k-dimensional subspace of El associated with the vanishing momenta, and E i is the orthogonal complement of E: in El. We also decompose E 2 as E2

=

(6.6.3)

E: 8 E ; ,

where E: is, say, of s dimensions and will be associated with the masses that we wish to become small, and E t is the orthogonal complement of E: in E 2 . Let P1be a vector in E : such that the elements in P' may be written as some linear combinations of the components of P,.Let P2 be a vector in E: of the form

+ +

P2 =

' '

,

11

' * *

AkLk+ 1,

(6.6.4)

where L 2 , . . . ,Lk+ are k independent vectors in E:, and suppose that the elements in P" may be written as some linear combinations of the components of P2 such that every element pfl in P" may be conveniently written as pfl = I, Aj(il)jfl for some j(il) Ik, and where jf1 is independent of 11, 1,. Let P, be a vector in E, of the form 9

p3

= I1 *"1k(1k+lLk+2

= P, + L,

+

" *

+ Ik+l

"'Ak+sLk+s+l)+ L (6.6.5)

,

with P3E E : , L E EZ, where Lk+2,. .. ,Lk+s+ are s independent vectors in E: such that the masses that we wish to become small may be written as some linear combinations of the components of P, in such a way that every mass pi' we wish to approach zero may be written in the form p i t = 1, ... & A k + l Ak+j(il)pil for some 1 I j(il) Is, and where pi' is independent of I , , . . .,& + s , and the remaining nonasymptotic masses may be written as some linear combinations of the components of L. Accordingly the subtracted-out amplitude d ( P , p) may be written as d ( P ) , with

P = P, + P, + P,. Introducing the vector P by

(6.6.6)

6.6 Low-Energy Behavior

173

defining

c= L k + s + L1

=

P1

(6.6.8)

1,

+ L,

(6.6.9)

and hence writing

(6.6.10) we obtain the following expression for d ( P ) :

d ( P )=

P

fl ( p J ) - ' J ( I l

' ' '

&+s)dcG)d(p')

(6.6.11)

j= 1

[see (6.5.19)], and for all those masses that we wish to become small the corresponding aj = 0 (see Section 6.5.1). From (6.6.9) and (6.6.10) we note are proportional to (&+ . . . Ik+s)- '. that all the external momenta in J(P) For I , = . . = I k + s = 1, we choose the external momenta p l , . . . ,pm such + pi,# 0 for that no partial sums of these momenta vanish, i.e., pil + all {i,,. . . , i t } c (1,. . . , m } . For 11,. . . , I , + , + 0 independently, the total external momenta carried away from each external vertex is asymptotic. We may then apply Theorem 6.4.2 to investigate the behavior of d ( P ) from (6.6.11). We may also apply the rules for the existence of lim d for I k + l , . . . , I k + s + 0, given in Section 6.5.2, to infer the behavior of d at low energies, in the zero-mass limit. Example 6.5 : Consider the elementary graph G in Fig. 6.2. We are interested in the behavior of d ( I , q , m, I,I,,u) for I,, I, + 0, where ,u is the mass of the

photon and m is the mass of the electron. We write for the photon propagator D J Q ) = y,,,/(Q2 p'). From Example 6.2 we know that lim d for I, + 0 exists. We may write

+

d ( I , q , m, I I I 2 , u ) = I , d ( q , m / 4 , &PO

=

I , & d ( q / I , , m / 4 f b 9PI.

In reference to the parameter l/I,, it is easy to show that for the amplitude d ( q / I , , m / I I I z ,p), we have 7 0 = { G } = t, a,(S(q, m)) = d(G), and flr(S(q, m)) = 0. In reference to the parameter l/I,, we note that for d ( q / I , , m / I I I , , p), as in Example 6.2, to = { G } and hence a,(S(m)) = d(G) = 1. From the definition of class A in Section 6.4 we also note that G has no proper subdiagram g [with d(g) 2 01 dependent only on the mass p [i.e., independent of 1/11,in reference to the amplitude d ( q / 1 2 ,rn/IIIz,p ) ] . Accordingly fl,(S(m)) = 0. Hence we obtain from (6.6.11) and (6.4.5) lim d(l,q,m,AlA,,u) = 0 11,12+0

(6.6.12)

174

6 Asymptotic Behavior in Quantum Field Theory

It is instructive to study the expression explicitly for d ( q , m, p), corresponding to Fig. 6.2, with p = 0 and to determine its behavior for q + 0. The expression for d ( q , m,0) is given in (6.5.26).We use the identity (x 2 0)

(

5)

In 1 + - x

= -q2 x - - x q4 m2

m4

and hence the bound

’s’

Y dY [ I + (q2/m2)xyl’

(6.6.13)

(6.6.14) to bound the expressions

(5)_ - - 2*, (5)

a -

-

= -

5 +5

Jolx dx ln( 1

Jol dx In( 1

+

x)

x),

(6.6.15) (6.6.16)

as (6.6.17) (6.6.18) Accordingly we have

and for A, --* 0, we have from (6.6.17)-(6.6.19) that d remains bounded (actually, it vanishes) consistent with the result in (6.6.12), as expected. Example 6.6:

Consider the behavior of d ( A l q , A1A2m, A l A Z A 3 p ) for

A1,A,, A3 -+ 0 corresponding to the graph in Fig. 6.4. We may write

d ( A , q , A1A2m, A1A2A3p)= A l A z A 3 d ( q / A 2 A 3 , m / A 3 , p). In reference to the latter amplitude we have from Example 6.1, a I ( S ( q ) ) = 1, flI(S(q)) = 2. From Example 6.4.,we may infer that aXS(q, m)) = 1, fl,(S(q, m)) = 0. Accordingly we have from (6.6.11) and (6.4.5)

2

(6.6.20)

6.7 General Asymptotic Behavior 11

- IP1

+

P2 + P 3

I

175 1P3

I

I

I

II

I

I

I

I

4Fig. 6.5

I

p2

A fermion-fermion (spin f ) scattering graph with a

$@ coupling

Example 6 . 7 : Consider the process depicted in Fig. 6.5, with a &,h@ coupling, where the dashed lines denote scalar bosons, and for simplicity the fermion and the scalar bosons are assumed to have the same mass p. Here d ( G ) = -2. We are interested in the behavior of &(Alpl, Alp,, p 3 , A l l z p ) for 11,A, + 0. As in Example 6.4, the limit A, --t 0 may be taken. In reference to the parameter 1/11, we readily obtain for the amplitude J4PlIAZ

9

P2/&

9

P 3 I A l f b1

that a,(S(p3)) = - 1. Also, G does not contain any proper subdiagram g [with d(g) 2 01, and hence P,(S(p,)) = 0. Accordingly we have

for nonexceptional momenta. 6.7

GENERAL ASYMPTOTIC BEHAVIOR II

In this section we generalize our applications to cases when, in general, some (or all) of the external momenta become large, some (or all) become small, and some of the masses are led to approach zero. We repeat the definition of P in (6.6.1), with the subset P' now consisting of those elements in P becoming either large or remaining nonasymptotic. Let El and E , be written as in (6.6.2) and (6.6.3), respectively. Let PI be a vector in E : of the form P1

=

L1q1 . * . q,

+ + LJ, + Ll+ * * *

1,

(6.7.1)

where L 1 , .. ., L,, with a + 1 I4m - k, are a + 1 independent vectors in E : . As in (6.6.4) and in a notation similar to it, let P, E E: be such that

Pz = AlL,+Z

+ + ' * '

11

."&La+k+l-

(6.7.2)

As in (6.6.5), let P3E E , be such that p3 =

= P;

"*Ak(Ak+lLp+k+Z + "'

+ Lb',,.

+

&+I

"'Lk+sLa+k+l+s)+ G+l, (6.7.3)

176

6 Asymptotic Behavior in Quantum Field Theory

We may then write the amplitude d ( P , p ) as d ( P ) , with

+ P,,

(6.7.4)

lk+s)P,

(6.7.5)

P = P1 + P2 or

P = (A1

' ' '

with (6.7.6)

l/Ai = v . + ~ ,

i = 1,. . ., k

+s

(6.7.7)

The amplitude d ( P ) may be then written as in (6.6.11), and the limits ql, . . . ,u a + k + s + co may be then studied directly from (6.4.5) for the amplitude d(P') with P now defined in (6.7.6). We note that L,, . . .,La+,+, are pi, # 0 for all { i l , . . . ,i,} c independent vectors. Again, with p i , { 1, . . . ,m},all the external momenta at the external vertices then become asymptotic for ql, . . .,)l,+k+s -,co for the amplitude d(P).

+

+

Example 6.8 : Consider the renormalized amplitude d ( q , p, m,p ) associated with the graph G in Fig. 6.6, where a dashed line represents a scalar particle of mass p, and a solid line represents a spin-4 particle of mass m. Let (k;, .. .,k:) be integration variables, with ke = k z , associated with the subdiagram g, and (ky, . . . , k:) be the integration variables associated with the graph G. We note that d(g) = 2, d(G) = 0. We write R'* = I @ E. We also write I = f l $ ~ 2 , f l = l l @ ~ ~ ~ @ I , , ~ z = Z , $ ~ ~ ~ ~ I ~ ( d1,..., imIj=l,j= 8), and E = El $ E 2 . The subspace 1, is associated with the integration variable kZ. We make the further decomposition El = E : @ E: and

Fig. 6.6 A vertex correction with a @ $, coupling, with the dashed line representing a scalar boson with mass p, and a solid line representing a spin 4 particle of mass m.

6.7 General Asymptotic Behavior II

177

E2 = E: Q E : , with E : , E:, E : , E: associated, respectively, with q, p , m,p. We define the subspaces

(6.7.8)

We are interested in the behavior of the amplitude

We consider the parameter q l first. In reference to this parameter, it is readily seen, as in Example 6.1, that aI(S(q))= d(G) = 0, as 70 = {G}. From the definition of ( M 7) , in Section 6.3, we have 7 = {G, (Gig), (GI G)} with the latter subdiagrams associated with the subspaces in A = {So, S', S" = S ( q ) } . By noting that A(I)So = A(Z)S' = A(I)S" = S(q), A(Ij Q . . * Q 18)s' = A(Ij Q *

*

Q 18)s' = I1 Q *

*

Q Ij- 1 @ Ei, j = 2 , . . . , 5,

(6.7.9)

we obtain from Theorem 6 . 3 . 2 ~ '= p s = 1, and p j = 0 forj # 1,5, and hence

B,(S(d) = 2.

We note that d ( v ] l q , Alp,

J1A2m,p) = d ( q l A ;

'A; ' 4 , A; ' p , m, A; '&'p).

Consider the parameter A;'. We note that 70 = {G}, i.e., a,(S(q, p)) = 0, and 7 = {G, (G Is)}. Note that (GIG)4 7 since G (trivially) contains the mass p. The subdiagrams in 7 are associated with the subspaces in A = {Sl, S , } . By repeating an analysis similar to the one given before, we obtain p s = 1 and pi = 0 for i # 5, thus leading to B,(S(q, p)) = 1. Finally, we note that the subdiagram g does not depend on the mass p, and as before we obtain a,(S(q, p, p ) ) = 0 and P,(S(q, p, p ) ) = 1 in reference to the parameter I ; .

'

178

6 Asymptotic Behavior in Quantum Field Theory

Accordingly we have from Theorem 6.4.2, for nonexceptional momenta,

for ql + 00, A, + 0, A2 Y1, Y 2 9 Y 3 such that

-,0, and the sum is over all nonnegative integers 71 I 1,

+ Y2 71 + Y2 + Y 3

1,

Y1

(6.7.1 1)

2-

Other examples may be also carried out where, for example, only some (or all) of the external momenta, of the graph in question, become large, and some (or all) of the masses become small. 6.8

GENERALIZED DECOUPLING THEOREM

In this section we generalize the decoupling theorem given in Section 6.1. In Section 6.8.1 we consider the behavior of d when any subset of the masses in d become large, and in general, at different rates. In section 6.8.2 this theorem is generalized further to cases when some of the remaining masses in the theory are led to become small as well, corresponding to theories which, on experimental grounds, may contain zero-mass particles. 6.8.1

Generalized Decoupling Theorem I

Consider the renormalized integrand R with argument P' as defined in (6.2.5). In this section all the external momenta are kept fixed and hence we require that A(I @ E2)Si = (0). For the problem at hand we suppose that N r 0 E1)Si # (0). First suppose that # (0). Then we may apply (6.2.21) and (6.2.22) to the graph G itself. If G E F2(N)[see (6.2.13)], then (6.2.22) applied to G implies that min[d(G),

- 13

- o(G).jo

(6.8.1)

1r

30 Since the external momenta of G are independent of q,, we took the liberty of replacing degr, by degrq, in (63.1).

6.8 Generalized Decoupling Theorem

179

An equality in (6.8.1) may hold if there is no subdiagram g c G in H 2 ( N ) such that at least one of the masses in the lines in S is dependent on q,, and there is no subdiagram g' $Z G in .F2(N). Finally suppose that G E H2(N). Then we may apply (6.2.21), (6.2.22) [and (6.2.18), in general, applied to l ~to] conclude that degr 1,9r

fl FG,(.N)I i

k

min[d(Gi),

- 13

- a(G),

(6.8.2)

i= 1

where G , , . . . , Gklare those maximal elements in N contained in G : Gi $ G such that Gi E .F2(N), with i = 1, . . . , k,. We denote the remaining maximal elements in N contained in G by Gkl+ . . . , G,. We may also apply (6.2.21) directly with gi in it simply replaced by G, for d(G) 2 0, and use the estimate in (6.8.2) to conclude for d ( G ) 2 0 or d ( G ) < 0 that

,,

degr FG(N)I - 1 - a(G),

(6.8.3)

since k , 2 1, for A(E)S: # (0). From (6.8.1) or (6.8.3) and the fact that dim A(E)S: 5 a(G),

(6.8.4)

we obtain upon summing over the sets N in (6.2.11) that degr R I

-1 -

dim A(E)Si.

(6.8.5)

9r

If A(E)S: = {0},then directly from (6.2.4) and (6.2.18) we conclude that degr R I - 1.

(6.8.6)

4r

Now consider the renormalized amplitude -d(f', 41 . . . qSp1, ~2 . . * qsp2,. . . , q,pS, pS+', . . . , p'),

(6.8.7)

with s I p. We conveniently decompose the subspace E 2 into s one-dimensional orthogonal subspaces Ei and introduce s vectors L1 E El, . . . , L, E Es with nonvanishing components p ' , . . . , ps, respectively. We then rewrite (6.8.7) as d ( L l q , ...qs

+ ... + L,qs + C),

(6.8.8)

where C is confined to a finite region in E , with the pi # 0. We are mainly interested in the vanishing property of .d for q,, . . . , qs --+ CQ, and we shall not carry out an analysis regarding the logarithmic asymptotic coefficients fi,(S,), as the latter is quite cumbersome.

180

6 Asymptotic Behavior in Quantum Field Theory

From (6.2.44), (6.2.51), (6.2.60), (6.8.5), and (6.8.6) we may then state Theorem 0.8.1 : For I ] , , .. . , I],+ oo, d in (6.8.8) vanishes, as the power

asymptotic coeficients al(S,) are bounded above as

(6.8.9)

a,(S,) I- 1.

In particular, since the pl(S,) are finite, we can always find positive integers N , , . . . , N , and a real constant C, > 0 such that

6.8.2

Generalized Decoupling Theorem II

Here we are interested in generalizing Theorem 6.8.1 to the cases when some of the remaining masses p S + l ,. . . , pp vanish; i.e., we are interested in the behavior of (6.8.11)

where s + k Ip, for I ] ~ .,. . , I],+ co and A,, . . . , 2, + 0. We have already established sufficiency conditions for taking the limits A,, . . . ,1 , + 0 of d in Section 6.5.2. Accordingly Theorem 6.8.1 remains true if the sufficiency conditions stated in Section 6.5.2 are satisfied with respect to the parameters 11,.

..

9

Aka

The renormalized amplitudes d(q,m, p ) corresponding to the graphs in Figs. 6.2 and 6.3, for example, all’vanish form + 00, p =fixed and m + co, p + 0. We note, in particular, from the estimates in (6.6.17) and (6.6.18) that for the graph G in Fig. 6.2 we have for p = 0 (6.8.12) with (6.8.13) and (6.8.14) which show the vanishing property of d ( q , qm, 0) for q consistent with our conclusions.

+

oo-a result that is

Notes

181

NOTES

Section 6.1 is based on Manoukian (1981a) and on some analysis, in particular, estimate (6.1.12), due to Hahn and Zimmermann (1968). Section 6.2 is based on Manoukian (1980a, 1981b), Section 6.3 on Manoukian (1978, see also 1980c), Section 6.4 on Manoukian (198Oc, 1981c), Section 6.5 on Manoukian (1980b, 1981c), Section 6.6 on Manoukian (198Od, see also 1979a), Section 6.7 on Manoukian (1981c), and Section 6.8 on Manoukian (1981d). Asymptotics were also studied by completely different methods (in the so-called a-parameter representation) by, e.g., Bergere et al. (1978). The decoupling theorem with only one mass scale becoming large and with no zero mass particles was proved by Ambjgfrn (1979) (again in the a-parameter representation). A proof of the general case with several mass scales becoming large and with zero-mass-particle limit was given in Manoukian (1981d). Interesting applications of the decoupling theorem have been carried out [see Appelquist and Carrazzonne (1975), Poggio et al. (1977), Collins et al. (1978), Toussaint (1978), Kazama and Yao (1979), Ovrut and Schnitzer (1981), and Hagiwara and Nakazawa (1981)], and many other papers on the subject are still appearing.