BPHZ renormalization of λφ4 field theory in curved spacetime

ANNALS

OF PHYSICS

BPHZ

131, 118-148 (1981)

Renormalization

of h+b4 Field Theory

in Curved

Spacetime

T. S. BUNCH Deparitneni of Applied Mathemafics and Theoretical Physics, University of Liverpool, P.O. Box 147, Liverpool L69 3BX, England

ReceivedJuly 7, 1980

A proof is givento all ordersin perturbationtheory of the renormalizabilityof A4* field theory in curved spa&me. The proof is basedon the BPHZ definitionof a renormalized Feynmanintegrandandusesdimensional regularizationto ensurethat productsof Feynman propagatorsare well-defineddistributions.The explicit structureof the pole termsin the Feynmanintegrandis obtainedusinga local momentumspacerepresentationof the Feynmanpropagatorand is shownto he of a form which can be cancelled by counterterms in the scalar field Lagrangian. The proof given is, technically, only valid for metrics which have been analytically continued to Euclidean (+ + + +) signature.

1. INTRODUCTION

There has recently been some interest in the renormalization of interacting quantum field theories in curved spacetimes. A number of field theories have been shown to be renormalizable in the weak field limit when the spacetime metric is expanded about the Minkowski metric [I, 21 and perturbation theory calculations have been performed for De Sitter spacetime [3,4]. For more general spacetimes, which may have strong curvature, rapidly varying metrics and non-trivial topologies, multiloop diagrams give rise to quantum state-dependent divergences which mutually cancel in all twoloop graphs of X$4 field theory [5--g]. The problem of renormalizability in general has been tackled using non-perturbative techniques [IO] but the appearance of statedependent divergences prevented a complete proof of renormalizability from being given. In this paper, X44 field theory is shown to be renormalizable to all orders in perturbation theory when the spacetime metric, g&), is analytically continued to imaginary time so that its signature becomes(+ + + +). The difficulties posed by the appearance of state-dependent divergencesare overcome by using the BPHZ definition

of a renormalized

Feynman intergrand

[12-151.

Apart from analyticity, no restrictions are imposed on the spacetimemetric and the only assumption of any consequence is that the homogeneous wave equation (2.4)

should possess a complete set of C” solutions. The global topology of the spacetime manifold is arbitrary provided that it does not permit the existence of closed timelike

curves. 118 0003-4916/81/010118-31$05.00/O Copyright JUI rights

Q 1981 by Academic Press, Inc. of reproduction in any form reserved.

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x$4 FIELD THEORY IN CURVED SPACETIME

In Section 2, the detailed structure of the free field Feynman propagator is discussed and the BPHZ definition of the renormalized intergrand, RF, constructed from a Feynman integrand, Ir, for a graph, r, is introduced. The graph will always be assumed to be one-particle irreducible (1PI). The definition of Rr makes use of an intermediate quantity, R, , and an operator, Sr, called the singular part operator and some of the important properties of these objects are obtained in Section 3. Momentum space expressions for I= and W, are introduced in Section 4 enabling further properties of &. to be deduced in Section 5. In Section 6, a number of theorems are proved which imply that X+4 field theory is renormalizable in curved spacetime. These theorems are discussed in Section 7, where it is shown that the removal of divergences can be achieved by including counterterms in the scalar field Lagrangian. Finally a short discussion of the renormalization of vacuum-to-vacuum diagrams is given. Sign conventions used in this paper are those of Ref. [16]. Some essential definitions and results of graph theory appear in an Appendix.

2. THE FEYNMAN PROPAGATORANDTHE

BPHZ RENORMALIZED

FEYNMAN INTEGRAND

Let the spacetime manifold &’ be covered by co-ordinate charts (Q, #) where the open sets 0 are normal neighbourhoods and the co-ordinate functions @ associate to each x E Sz Riemann normal co-ordinates relative to some fixed origin z E !G. For convenience of notation write x” for F(x). Let C?(Q) and Cm(&) denote respectively the sets of C” functions on Q and on J?‘. Let COW(Q) and COm(m denote respectively the sets of C” functions of compact support on 1;2and on &I. The image of 9 under $I, I,@?), is an open subset of W (n = dim .&‘) and so one may define D ($(.n>) to be C,““( $(Q)) with th e usual inductive limit topology. Elements of D’($(Q)), the topological dual of D(#(sZ)), are distributions on $(Q). Some understanding of distributions on manifolds is required for renormalization theory and a useful reference is the book by Friedlander [17] which provides the following DEFINITION. A linear form U: C,,a(&‘) -+ C given by U: f w (uJ) is a distribution on &’ if, for every co-ordinate chart (Q, #), there is a distribution u 0 $-l(x) f D/($(Q)) such that (2.1) (4 f) = @ a P(x), fo yw4 PbN

for allfe Coai(s2), where g(x) = I det g,Jx)I. The vector space of distributions is denoted D’(J&!).

on Jl

In general, it is not possible to define an operation of multiplication between distributions and this difficulty lies at the heart of renormalization theory. However, there are two important special cases in which multiplication can be defined. The first is if u E D’(kf’) and x E Cm(&‘); then a distribution xu E D’(A&) is defined in any co-ordinate chart (Q, $) by

(X% f) = (4 xf)

G.2)

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for all f E Corn(Q). The second is the tensor product of two distributions. Let Q, , Q, be two normal neighbourhoods and let u E D/(52,), u ED’(Q& Suppose that f e C,“(sZ, x Q& and define the tensor product u @ v E D’(& x .QJ by (u 0 VPf(x1 9 x2)) = @4x3, @(x2), f(Xl 3 x2))),

(2.3)

where for each xl E Q, ,f(x, , x2) E C,,m(Qz). It is shown in [17] that (v(x& f(x, , x2)) E C,“(D,). and that (2.3) does indeed define a distribution. The quantum theory of a free scalar field on &! is based on the classical wave equation (-Cl

+ m2 + SR(x))$(x)

= 0,

(2.4)

where IJ denotes the covariant d’Alembertian for the metric g,, and R is the Ricci scalar. It will be assumed that there exists a complete set of Cm solutions of (2.4). The distribution on .&’ which is of interest in the quantum theory is the free Feynman propagator G(x, Y) = (0 I T(~(x)$(Y))

I 0)

(2.5)

which satisfies (-- 0 + m2 + !fR)G(x, y) = g1’2(x)6(x - y)

6”(x- Y),

(2.6)

where 8(x - y) is the covariant delta function defined by 6(x - VI> f(Y)) = f(x).

(2.7)

The operator T in (2.5) is the usual time-ordering operator which is well defined provided that &’ does not contain any closed timelike curves. Because G(x, y) is bilinear in the quantum field b(x) and each quantum field is an operator-valued distribution on test functions in D(&, G(x, y) is a continuous bilinear functional on D(&!) x D(J%‘). By the Nuclear Theorem [I 81 it is therefore an element of D’(.&’ x 4. In curved spacetime, G(x, y) is not unique because there are many possible choices of vacuum ) 0) and the two vacua in (2.5) need not even be the same. However, the difference between two propagators is a solution of (2.4) and hence belongs to Cm(&). Thus the distributional structure of G(x, y) is determined uniquely by (2.6). Now suppose that the metric g,,“(x), which is assumed to be analytic, has been analytically continued to Euclidean signature and consider (2.6) as a differential equation in x for fixed y E 9. Let U be any open set containing y and contained in Q: y E lJ C Q. Then for x 6 U the 6 function in (2.6) vanishes and G(x, y) is a solution of (2.4) and hence is in Cm(~). It follows that any propagator may be written G(x,

Y)

= G’D’(x,

Y)

+ G’R’(~, y),

where the divergent part GtD)(x, y) is a distribution

only in an open neighbourhood

cw

of

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x = y and is a C” function elsewhere, and the remainder GcR)(x, y) is a C” function everywhere, i.e., GfR)(x, y) E P(.A’ x A). Using a smooth function LX(X)which is unity inside U and zero outside Sz one can arrange for GtD)(x, y) to vanish for x $ J2. Then GR)(x, y) contains all information about the quantum states in (2.5) and about the global topology of the spacetime. The structure of the divergences in quantum field theory is determined by GcD)(x, y)? the distributional part of G(x, y). It is therefore necessary to have an explicit expression for this quantity and this can be obtained by solving (2.6) in Riemann normal coordinates [8, 91. Define a distribution G(x, y) by G(x, y) = g”“(x)G(x,

y) g”‘(y).

(2.9)

For a smooth (analytic) metric, g(x) E: P(A) and so G(x, y) is defined as a distribution by (2.2). Corresponding to (2.8) there is the decomposition

qx,

y) = cw(x,

y) + GCR’(x, y).

(2.10)

If y = z (the origin of normal co-ordinates) one finds [8] (2.1 I)

where k2 r: 77,,k”kfi,

kx = q,,k”xfi;

(2.12) (2.13)

M2 = m2 + (.$ - Q)R(z)

with vas representing the flat space metric (actually the unit matrix in the Euclideanized theory). More generally, if X, y E 52, P)(x, y) may be expressed in Riemann normal co-ordinates about z E Q as [9]

dnk &k(o-y) 1 k&v G(D'(x, Y>= jw k2+ M2 1 - 3 RuavBxayep_tMP

3~ ...

I

(2.14)

where RuaYBare the components of the Riemann tensor at the point z in normal coordinates. The terms not written explicitly in (2.14) have geometric coefficients evaluated at z which are of adiabatic order A > 2 (that is are homogeneous of degree A in derivatives of the metric), are homogeneous polynomials of degree 5 in X~ and y*, are of order K in powers of k, and satisfy A +

K -

f

=:

O
The quantity

> -4

-2,

(2.15)

(2.16) (2.17)

.$ in (2.15)-(2.16) should not be confused with 5 in (2.4). (2.15) is a

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consequence of the dimensions of each term being fixed to the dimensions of the first term which has A = 0, 5 = 0 and K = -2; (2.16) is a consequence of the power series expansion of gUV(x) used to derive (2.14) (since g,,, is assumed analytic, this power series expansion exists); and (2.17) holds because terms with K < -4 are nonsingular when n = 4 and are therefore included in GtR)(x, y). An important property of GcD)(x, y) is that it is a sum of terms of the form P&c”, y”) D,.@ - y”) where P&P, y”) is a polynomial in xa and y” (and hence a Cm function of its arguments) and D.&X” - y”) is a distribution in the co-ordinate difference xa - y”. For any f E D(Q), it follows from Theorem 2.2.5 of [I 71 that (Da(xa -y”), f(x”)) is a C” function of ya. A simple consequence of this result is that (G(x, y), f(x)) E C”(Q) for eachfo D(Q). It should be noted that the series in (2.14) is strictly only an asymptotic series valid in the limit xa - y” + 0. A rigorous discussion of GcD)(x, y), such as appears in [17], would include convergence factors in the higherorder terms to make the series well defined. These factors are Cm functions which do not essentially alter the distributional structure of GtD)(x, y). Now let r be any connected Feynman graph with vertices x1 ,..., xN and lines 11 ,a*., IL (see Appendix for a summary of graph theoretical definitions and results). To each vertex xK there corresponds a spacetime point, also denoted xk , and to each line Z, there corresponds a Feynman propagator GA($)xi , E$‘x~). To the graph, r, there corresponds the Feynaman integrand ,.*., xN) = fi G,(r,‘:‘x, , $‘xJ A=1

Wl

(2.18)

= ,a*-, xN) is to be regarded as a continuous multilinear functional on D”(Jf) x **. x D(4) [Nfactors]. In practice it is often convenient to fix one argument, say xN , and regard I& ,..., x,,,) as a distribution in x, ,..., xNel , i.e., as a continuous multilinear functional on DN-l(=k’). The vertices of r are of two kinds, internal and external. In h4* theory, any vertex which has four lines incident on it (a loop line counts as two incident lines) is internal and any vertex with fewer than four is external. The external vertices act on arbitrary test functions representing wave packets of real external particles but the internal vertices do not and so may be integrated over immediately. Let xl ,..., x&l, XN be external vertices and & ,..., xN-r internal vertices. Then integrating over the internal vertices leads to

Z&l

D(Ji’)

N-l

I

MXl

,***,

XN)

fl

gl"txk)

d"xk

k:=E N-l =

GA($)xi

, $xj)

n g1’2(xk) d”x,

(2.19)

k=E

Note. In most texts on quantum field theory all the vertices x1 ,..., XN are called internal and integrations over each of x, ,..., XN are performed. This makes it difficult to distinguish vertices at which real (external) particles enter from vertices which

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involve only virtual (internal) particles. Expression (2.19) is to be regarded as a distribution in -u, ... xEMl , x N acting on elements of DE(A) or as a distribution in x1 ,..., x~-~ for fixed xN acting on elements of 0E-‘(JZ). This action as a distribution corresponds to “integration” over the remaining vertices of r. If xN is fixed, the x,-integration cannot be performed, but for the purposes of renormalization this does not matter since the counterterms needed to cancel divergences in I’ are formally expressed as integrals over a single spacetime point. (See Section 7). In most of what follows, it will be convenient to work with (2.18), the integrations implied by (2.19) being left until Section 6. Now (2.18) is a product of distributions and so is not necessarily well defined as a continuous linear functional on V(J&‘) when n = 4, although it will always be defined on some proper subspace of W’(g and can therefore be extended (not uniquely) to a continuous linear functional on all of II,“(&), by the Hahn-Banach theorem. (For a discussion of this extension, see [lo].) Actually, one can always find N - 1 propagators whose product is well defined as a tensor product of distributions and whose lines correspond to a maximal tree. The difficulty comes from the remaining L - N + 1 propagators which may give divergent convolution integrals in momentum space (loop integrals). In h44 field theory there exist values of n for which (2.18) is always well defined on all of DN(~ and so it can be evaluated explicitly in terms of known distributions with coefficients that depend on IZ. Expanding these coefficients in Laurent series about n = 4 gives an explicit representation of the divergences as poles at n = 4. Dimensional regularization thus provides a natural method of handling the divergences that arise when attempting to define products of distributions such as (2.18).

Now let D(x, ,..., xN) be any distribution in x1 ,..., xN , the vertices of r, which is well defined for some values of n, but not necessarily for fl = 4. Define Sr, the singular part operator for r, by S,D(x, ,..., x,) = explicit pole terms obtained when coefficients of D(x, ,..., xN) are expanded about n = 4.

(2.20)

A more detailed discussion of S, will be given in Section 4. The only properties of STneeded at the moment are (i) (ii) (iii)

S,D(X~ ,..., xN) = 0 * D(x, ,..., xN) is well defined on DN(JK) when n = 4, Srz y= Sr, SJf(.u,)D(x, ,,.., XN)] = Sy[f(Xg) S,D(x, ,..., SN)] Vf E c?(.,fl).

(2.21)

(2.22) (2.23)

These properties are trivia1 consequences of the definition of Sr . Property (ii) shows that Sr is a projection operator, projecting out the pole terms in D(x, ,..., x,~). In general, the structure of SJ,(x, ,..., xN) is very complicated and consists of nonlocal products of distributions whose detailed form would be impossible to calculate for a general spacetime. The complications arise because not only is the product (2.18)

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T. S. BUNCH

likely to be ill defined but subsets of the set of all propagators in (2.18) may have ill defined products. In the language of Feynman graphs this means that there may be divergent subgraphs, a subgraph of r being defined as a subset of the lines of I’ together with their initial and final vertices. To avoid this difficulty, it is convenient to introduce a distribution &(x1 ,..., xN) by the recursive definition: (2.24)

where the sum is over all sets {n ,..., ye> of mutually disjoint, IPI, proper subgraphs of I’ and hyl,...,y,~ is defined by

Mutually disjoint means that if c # r then yU and y, have no lines or vertices in common. Proper means that y7 # r for each T. Proper inclusion will be denoted C; thus y7 C r for each T. It will be shown in Section 3 that Sr&(x, ,..., xN) is actually a local quantity and most of this paper will be concerned with analyzing its structure in detail. Given i&- one may define Rr(xI ,..., xN) by R,=(l

-ST)&.

(2.26)

Expressions (2.24) and (2.25) constitute the BPHZ definition of the renormalized Feynman integrand, Rr, for the graph r [12-151. A trivial consequence of (2.22) and (2.26) is that SrRr = 0.

(2.27)

Thus Rr is well-defined when n = 4 and so it is reasonable to call it the renormalized Feynman integrand. The BPHZ definition of Rr can be given for any field theory. However, a field theory is said to be renormahzable if the subtraction of divergences by the operators -Sy, and -S, in (2.24) and (2.26) can be formally implemented by the inclusion of a finite number of counterterms in the scalar field action. Whether or not this can be done depends critically on the structure of S&- for an arbitrary Feynman graph. In subsequent sections, properties of I,, &, and Rr will be obtained when x1 ,..., xN E Q. In this special case each propagator G(x, y) is a sum of terms of the form P,(xU, y”) D,,&x~ - y”). Therefore, I&, ,..., xN) is a sum of distributions in the N - I independent variables ehkxba multiplied by Cm functions of x, ,..., xN. The structure of i?, and Rr is similar. 3. PRELIMINARY

PROPOSITIONS

PROPOSITION 1. Let r be a Feynman graph with vertices x1 ,..., xN . Let 0, ,..., Qn, (1 < r & N) be a set of mutually disjoint normal neighbourhooa!son .A’. Restrict the

A+’

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points x1 ,..., xN so that eachpoint lies in one and only one normal neighbourhoodand each of the normal neighbourhoodshas at least one point restricted to it. Then for all 5 1 ,..., xN satisfying theseconditions: S&-(x,

)...)

XN)

=

0.

Proof. By induction in the number of loops of r, the result being true for ail one-loop graphs which satisfy the hypotheses of the proposition. Assume that it is true for any proper subgraph y C r. Each of the normal neighbourhoods a, (t I,..., r) defines a proper subgraph g, of r given by the vertices which have been restricted to lie in Qn, together with all lines in F which connect these vertices. By construction, the subgraphs g, are mutually disjoint. Consider the sum in (2.24) over all sets {rI ,..., rcj. There are two cases to be considered:

(i) each y7 is a subgraph of some gt : yT C g, , each T, some t; (ii) at least one yr is not a subgraph of any g, : y7 g g, , some 7, each c. In case (ii) consider a subgraph yT satisfying yr g g, for any t. Then y7 satisfies the hypotheses of the proposition and so, by the inductive hypothesis: S.JzY7 = 0. So (2.24)

becomes

where the sum is over all sets of mutually disjoint, (i):above and (I,, ... &,)I(~,,....~,) is defined by

IPI subgraphs of r satisfying

I 111*-* 19, = ug, -** L,,).IVl ,...,., ,.I 1,,1-*- I;,,. .

(3.7)

Now (3. I) may be expressed

where for each t = I,..., r in the product, the sum is over all sets of mutually disjoint, 1PI subgraphs y7 C g, . lf g, is IPI then the sum includes a contribution from y7 g, and the result is

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If g, is not IPI suppose that g, has two IPI components. (Actually it can have any number but the argument given when it has only two 1PI components also applies to the more general case). In fact, suppose

gt = g u 1,u g,

(3.5)

where g and g denote the two IPI components and IA is a line connecting them. Then (3.4) is replaced by R,, = &GA&,

(3.6)

where RVt appears on the left because the sum in (3.4) does not include y7 = g, in this case. Now suppose that the vertices of gt are labelled so that

WV4 = R&I ,..., xa) GA&~,x,+1)R&a+1 ,...>&cb).

(3.7)

Each vertex xi (i = I,..., b) lies in Qt and by the discussion at the end of Section 2, the distributions R, and i& are, up to multiplication by Cm functions, distributions in the co-ordinate differences x1” - xaa,..., xzW1- x,= for R, and Y,+l - xba,..., xi-‘_, - xba for Rg . In addition, GA(xa, x,+r) is essentially a distribution in the co-ordinate difference x,” - x;+r . Therefore the product R,G,R, is a well-defined tensor product of distributions. It follows from (2.21) that, whenever g, is IPR: SJ?,, = 0.

(3.8)

it,, = R,, .

(3.9)

Equivalently:

Therefore, whether g, is 1PI or IPR, a contribution is obtained. Hence (3.3) becomes

R,‘ to the product over t in (3.3)

K- = Ir/(q,...,g~ii R, .

(3.10)

t=1

Now & Rgt is a well-defined Rgt (t = I,..., r) so, by (2.21):

tensor product of the well-defined

S,nRRgt=O.

distributions (3.11)

kl

Furthermore, each propagator in Irllv,, . .. ,,,I has its two arguments restricted to two disjoint normal neighbourhoods. Thus each is a C” function with this restriction on arguments and so, by (2.23): S&(x, Hence the proposition

is proved.

)...) XN) = 0.

(3.12)

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Comment. Because (3.6) and (3.9) hold for 1PR graphs any 1PR graph is renormalized by renormalizing its IPI components. For this reason it is only necessary to consider IPI graphs, r, in order to prove renormalizability. COROLLARY 1. Consider Sri?r(~l ,..., x,,,) as a distribution in x1 ,.,., xNeI for fixed xhr . This distribution has support only at the point x1 = x2 = ..’ -= xNdl = xN .

Proof. If xi # xj for some i, j, surround xi and xj by disjoint normal neighbourhoods (spacetime is Hausdorff). Then by Proposition I, SrRr = 0. Thus the support of Srar must be at x1 = .. = .Y.,~-~= xN . ,..,, xN) is a linear combination of products of 6 functions COROLLARY 2. S&-(x, and derivatives of 6 functions. Each product is a tensor product of N - 1 distributions, one 6 function or derivative of a 6 function for each x1 ,..., .xN-l . Proqfi

Given Corollary

1. this is a standard result in distribution

theory [19].

Comment. It is not necessary to fix xN in Corollaries

1 and 2. As a distribution in -v] ,..., XN , SrRr is still a linear combination of tensor products of N - 1 distributions. In any co-ordinate chart with co-ordinates x1=,..., x,,?, the arguments of the 8 functions (or their derivatives) may be taken to be xlcl - xNn,..., x$-1 - xNn. The importance of Proposition 1 is that it reduces the global problem of defining the divergent part of 1, to the local problem of defining the divergent part of W, . In particular, S&-(x, ,..., xN) can be defined by taking all xlc f S for some normal neighbourhood. This enables the local momentum space representation (2.14) to be used for the propagators. For any Feynman graph r and any line IA of I’ define (3.13) (3.14)

(3.15)

(3.16)

where @D,A) YT

=

WY7

=

jp’.

(3.17)

if I,, is not a line of y7, and (3.18) 595/‘31/1-9

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T. S. BUNCH

if IA is a line of ‘y-rfor some 7. The following two results are immediate of the definitions (3.13)-(3.18):

consequences

LEMMA 1. zr = p)

+ z$-QA).

(3.19)

LEMMA 2.

R, = ,$-%A) + $/.A). PROPOSITION 2.

(3.20)

Let RkD) denote RF with everypropagator G,, replacedby its divergent

part GiD’. Then srRr = Sri@?

(3.21)

Proof: The method of proof is to eliminate the contribution line in turn. The first thing to prove therefore is

from GiR’ for each

sj-R, = sJ?Ip~A’

(3.22)

for an arbitrary choice of line In E r. By Lemma 2, (3.22) is equivalent to &.@*“’

= 0.

(3.23)

This is proved by induction, the result being trivially true for one-loop graphs. Assume it is true for any y C r for which ZAE y. Then

*

,?)

=

G?)

Ii-/h [

+

c

{Yj... ..YJ

If r/h is IPI the sum includes a contribution

ZmA.vl,....v,)

fi

7=1

(--S,&$

1 -

(3.25)

from I’/A and (3.25) becomes

jp.A) = GA (R)&-,A .

(3.26)

If r/h is not IPI it must be expressed in terms of its 1PI components. As in the proof of Proposition 1, this procedure also leads to (3.26). Now GiR’ is a C” function and so (2.23) and (2.27) lead immediately to (3.23). The proof of the proposition is completed by repeating the arguments above leading to (3.23) for each line of I’ in turn. 4. MOMENTUM SPACE REPRESENTATION OF A FEYNMAN INTEGRAND The propositions proved in Section 3 have shown that S&(x, ,..., xN) may be evaluated by restricting all the points xk to a single normal neighbourhood, 52, and by disregarding all contributions from the Cm functions GiR). As a result one can

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immediately use the momentum space representation (2.14) to obtain detailed information about the structure of Srar(xl ,..., xN). For the time being, it will be convenient to omit the factors of g-l/a(x) and g-““(y) contained in G(x, y) (see (2.9)) and to take the integrand for a graph r to be

with G, given by (2.14). The g-1/4 factors will be restored later. Because they are CL’ functions, their exclusion or inclusion does not affect the distributional nature of I&l ,..., x,,,). Indeed, the propositions of Section 3 could have been proved taking (4.1) as the starting point rather than (2.18). Now let x1 ,..., x,v E Q and take some z E 52 to be the origin of Riemann normal co-ordinates. Substituting (2.14) into (4.1) gives

(4.2)

where the terms not written explicitly are of order A > 2 and satisfy (cf. (2.15)-(2.17)) A + K - ( = -2L,

(4.3)

O
(4.4)

K 3 -4~5.

(4.5)

Now let ril be a path matrix relative to xN . Then, by (Al 1): xi = xN + riAeAkxk .

Since eA,,has rank N - 1, one can define N - 1 independent ki (i -= I,..., N - 1) by wAkXk = k&i

(4.6)

momentum

- xN),

variables (4.7)

where X is summed over I,..., L; k is summed over I,..., N; and i is summed over 1,..., N - 1. Hence (q, - kin-,) e,,xk = 0.

(4.8)

Only N - 1 of the L quantities ehkxk are linearly independent and the linear dependence of the remainder is expressed by the L - N + 1 constraints qjAeAkxk

=

o

(j = l,..., C = L - N + 1).

(4.9)

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Introduce a set of C Lagrange multipliers h - kini,

pj to obtain

(4.10)

- PEA) e,vcxk = 0.

Now choose the pj so that q,t = kirih + Pjvj,+

(4. I 1)

for h = I,..., L. Then (4.2) becomes

where &-(pj,

k,; x,J = fi

’

A=1 4A2+ M2

1 - ; RuiB xi” x; ;

&‘d2

1

qA’uqr’V + .-a

A’=1 A* I’i q;, + M2

(4.13)

with qn = qA( pr , kJ given by (4.11). In obtaining (4.12) and (4.13) from (4.2) it has been assumed that the Jacobian determinant of the transformation (4.11) is unity. In Lemma A in the Appendix, it is shown that the determinant is 51. The sign can be changed by changing the orientation of one of the circuits which corresponds to changing the sign of one of the momenta pi in (4.11). So one can always arrange for (4.11) to be a rotation without reflection in momentum space. It is instructive to apply the above construction to particular Feynman graphs. It will be found that the usual momentum conservation rules apply with momentum ki flowing out of each vertex xi (i = I,..., N - 1) and momentum k, + ..a + k,-, flowing into vertex xN . If a vertex is internal it will be integrated over according to (2.19). This will have the effect of introducing a 6 function for the corresponding momentum ki (see Section 6). Integrating over ki then sets it to zero throughout the integrand and hence no k,-momentum flows out of an internal vertex. Therefore the momenta ki will be called external momenta and the momentapj will be called internal or loop momenta. The divergences in (4.12) appear in the loop integrals and so the operator Sr, which picks out the poles at n = 4, operates directly on these integrals as follows:

&4-(x1 ,***,

exp[--iki(xi

- xN)]

&<,j

, k; x,d,

x si- fJ 1%

(4.14)

where srfJJw

dnpi f?p(pj p ki; XJ = terms with poles at n = 4 obtained by

performing

p,-integrals for n E Z and continuing

analytically

in II.

(4.15)

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The momentum integrals in (4.15) are standard flat space integrals which can be evaluated using any of the usual techniques such as combining (qA2 + M3-l factors using Feynman parameters. Note that the form of (4.13) implies that (4.15) is an infinite series with terms of all adiabatic orders appearing. It will be shown in Section 6 that the only terms which contribute to SrRr are those given explicitly in (4.13). which have A -< 2. Now suppose y C Z’ and define S,Z, by (4.16)

SJr = Ii-,&ZY .

It is important to know how S, acts in momentum space on Zr( pj , X-i ; .Y~). Not all the momenta pi will be internal momenta for y so the action of S,, will be given once a set of internal momenta for y has been identified. Suppose that y is defined by lines Z,,for h F A C {I,..., L}; vertices xk for k E KC {I,..., N}; and a set of independent loops labelled by j’ = I,..., C, = L, - N, + 1, where L, and IV, are the number of elements in fl and I<, respectively. If a set of loops has already been chosen for Z’ it may not be possible to choose the loops of y to coincide with those of r. This does not matter. It will be convenient to write (4.13) as L

h4P, 3h; Xk) = Afj 4A2 i M2 [l +CJh. A’~-1

I-...

1

.

Because (4.18)

one has 1 + i J,, 4. . . . == 1 + c Jn, -c ... A’- 1 A’E‘I

1 + c J,, fm.*. . j d’B.1

(4.19)

Define the incidence matrix for y, e’$, by restricting eAlzto y: (Y)= e,, eAk

hEA,liEK.

(4.20)

Note that (4.21)

Thus: (4.22)

So, from (4.2) and (4.19):

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Thus Ir is decomposed into the product of Irlv and I, and one may proceed to define the action of S, directly on the loop momentum integrals of I, as in (4.14) and (4.15). Let 7717;(A E A) be a circuit matrix for y and let rr!Yl be a path matrix relative to any fixed vertex of y. Thus i’ runs over NY - 1 values. Introduce internal momenta $j, and external momenta b,t for y by

The two sets of momenta {k, , pi} for r and (& , ji, , qn : h $ A> for y and I’/y are equivalent and are related by an orthogonal transformation (since each is related to the set (qA : A = l,..., L} by an orthogonal transformation). The relationship between them can be obtained by introducing a matrix Cg! , defined to be e$) restricted to N, - 1 columns, the omitted column being the column for the vertex fixed in defining the path matrix ri?‘l . Then, by (A19):

But

and so

These N,, - 1 relations determine & = &(k, , pJ. Eliminating using (4.27) gives C,, = Ly - N,, + 1 relations which determine These relations are given implicitly by

Ai, from (4.26) jj, = jj,(ki , pi).

Finally, the relations qa =

hia

+P~TO

(he.4

(4.29)

give qA = qn(ki , pi) for A 4 A. It is not difficult to obtain the inverse relations to (4.27), (4.28) and (4.29). Equations (4.26) and (4.29) are a set of L equations labelled by X = I,..., L. Multiplying be t?Aiand summing over all h gives k, = 1 &~~~~‘C~i f C qA&Ai aefl hbA

(4.30)

7r&

(4.3 I)

since = &k;

%A

= 0

and (4.32)

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This determines ki = k,(&, , jj, , q,J and it is important to note that ki is actually independent of jj, . Eliminating ki from (4.26) and (4.29) gives an implicit relation for = 2$.3’ , q,$ Its precise form is not needed in what follows.

pj

pj(Xi'

5. FURTHER

PROPOSITIONS

A typical term appearing in (4.2) which is of adiabatic order A, of order E in powers of xp and of order K in powers of qAuhas degree of divergence d defined by counting powers of momenta in the integral when n = 4: d=4Cftc.

(5.1)

Using (A25) and (4.3): d==2L-4N+4+tJ-A.

(5.2)

Let Y = 4N - 2L.

(5.3)

For X+4 field theory, Y is the number of external lines of a graph. From (5.2) and (5.3): d-4-

Y-+5-A.

(5.4)

Y

(5.5)

BY (4.41, d<4-

and d is maximised by terms having c = A. Let the maximum be D(r) = 4 -

Y = 2L - 4N + 4.

value of dfor a graph I’ (5.6)

Y and D are both even and one finds (assuming Y =k 0): Y=2

D = 2.

(5.7)

Y=4

D = 0,

(5.8)

Y=6

D = -2,

(5.9)

. .. . etc. All this is familiar from flat space. Also familiar is Weinberg’s Theorem [20] which states that a momentum space Feynman integral which has negative degree of divergence and for which every subintegration has negative degree of divergence is convergent. The essential property of i?, is that it should have no divergent subintegrations and this result has been proved in flat space by Zimmerman. The result should therefore hold in curved spacetime because the degree of divergence d is

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maximised by the leading flat space contribution. Unfortunately it is rather difficult to translate Zimmerman’s proof to a form that applies to this paper because Zimmerman has a different definition of R, from that used here and it is simpler to prove corresponding, though less rigorous, results directly. LEMMA 3. Let y1 and y2 be any two connected subgraphs of a graph. Then NY1 ” YJ = D(Yl) + D(Y2.l - WY1 * Yz). Prooj

A straightforward

(5.10)

consequence of definition (5.6).

PROPOSITION 3. Let r be a 1PI graph with D(I’)

-=c 0. Then Sri?, = 0.

Proof. By induction in the number of loops, the result being true for one-loop graphs by Weinberg’s theorem. Assume that it is true for all y C r with D(y) < 0. Consider all lP1 subgraphs of r satisfying D(y) > 0. Let g, ,..., g, denote the maximal such subgraphs. Thus if g, C y _Cr, one has D(y) < 0. The subgraphs g, ,..,, g, are mutually disjoint. For, suppose not and that g, n g, # @. By Lemma 3:

D(glngz) =D(g,)+D(g,)--D(glug,). Now D( g,) 3 0, D( gz) > 0 and by maximality

(5.11)

of g, , g, :

D(glugz) < 0.

(5.12)

Hence D(g, n gJ > 0, which forces D(g, n gJ = 2. Hence, by (5.7), g, n g, has Y = 2. Thus g, n g, is attached to the rest of r by just two lines in r/(gl n gJ. One line must be in g, and one in g, . (They cannot both be in g, , say, for then g2 C g, , which contradicts the maximality of g, .) But then g, and g, must be 1PR because each has a single line connecting g, n g2 to the rest of the subgraph. This is a contradiction and so the subgraphs g, ,..., g, are mutually disjoint. Now consider the sum over sets (yl ,..., yc} in the definition 8,. As in the proof of Proposition 1 there are two cases to be considered. In case (ii) in which at least one y7 is not contained in any g, one must have for such yI: D(yr) < 0. The inductive hypothesis ensures that there is no contribution to i’?, in this case. Consideration of the remaining sets {yl ,..., yc} then leads to K- = Ii-/tsn...,s,~ ]i &t .

(5.13)

t=1

Now choose any line IA E r not contained in any g, (t = l,..., r). Let y = F//A . If r has 1 lines and C - 1 loops. Although y is connected, it need not be 1PI. However, g, ,..., g, are the maximal 1PI subgraphs of y with D( gt) 3 0. Since D(r) < 0 and y has the same number of vertices as I’ but one line fewer: L lines and C loops, y has L -

D(Y) -=c0.

(5.14)

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Equation (5.13) may be written as (5.15) But the argument which derived (5.13) may be applied to y to give &

= GARY = GAR;, ,

(5.16)

the second equality holding because y satisfies (5.14) and so, by the inductive hypothesis, S,,R,, vanishes. If y is 1PR the inductive hypothesis cannot be used but in this case 8, = R, automatically (see (3.9)). Since R, is well defined when n = 4 the C -- I loop momentum integrals may be performed. Then (5.16) gives Br as a single-loop momentum integral and since D(r) < 0, Weinberg’s theorem ensures that S,R,, is zero. Hence Proposition 3 is proved. Now consider graphs r for which D(r) > 0. In momentum space a propagator has the form (5.17) with qh = k,nih -t pj~jA. It is convenient to perform an expansion about ki = 0 or equivalently qn = pjvjA :

(5.18)

Now Gh( ~7) is of order J+, (8GA/%qAu),~,, is of order p-3 and so on. More generally. an expansion of this kind can be obtained by expanding about ki = constant. An important consequence of (4.30) is that ki = constant for a graph r is equivalent to & = constant for a subgraph y since the momenta qA for h g/l may be treated as constants in the loop integrals for y. For a line IA f I’define Gy’ to be quantity obtained by expanding G,(q,J in Taylor series about ki = 0 up to order D(r) in powers of k, . Thus, if LI(I’) = 0: (5.19) ,I” = G>(pjyj>). If D(r) = 2, Gi” is given by the terms appearing explicitly in (5.18). Define FJk. ZYp J) = G A(q A) - G’*‘(q A A) -

(5.20)

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Define also i?kFp” and R$TV”’ in the same way as RkD*‘) and RkRIAJwere defined in (3.15) and (3.16). Notice that (5.21) and (5.22) and the corresponding definitions of RhFVh) and RpYX’ are perfectly good configuration space definitions, although explicitly formulated in terms of momentum space objects. The methods described in Section 4 allow translation to be made easily between configuration and momentum space for a graph p or a subgraph y C I? PROPOSITION 4. Let r be a 1PI graph with D(r) > 0. Let RsT) denote R, with every propagator GAreplaced by GkT’. Then S,i?kT’ = S,ir, .

Proof.

The first thing to show is that for any line IA E I’ SJp

= sj-it,

&@J)

zzz0.

(5.23)

or, equivalently,

The proof of (5.24) is very similar to the proof of Proposition 3. Because the propagator Gh has been replaced by FA, the maximum degree of divergence of any term in the Feynman integrand is actually d(p) = -1. If y is any subgraph of r its maximum degree of divergence is

d(r) = D(Y) - D(r) - 1 d(r) = D(Y)

if if

IA E y, lA # y.

(5.25) (5.26)

The proposition is trivially true for one-loop graphs by Weinberg’s theorem. Assume it is true for any y C r for which I, E y. The following analogue of Lemma 3 is easily proved: (5.27)

Let g, ,..., g, denote the maximal subgraphs of r having d(gJ > 0. Then the intersection of any pair, say g, and g, , satisfies

d(g,n gz)> 0.

(5.28)

DW-W

(5.29)

Hence, a fortiori,

and by the argument given in Proposition derive

> 0

3, g, n g, is empty. Therefore one may

The derivation assumes that sets (yI ,..., yc} for which at least one y7 is not contained -(FY’). Such a subgraph y7 must have d(rr) < 0. If in any g, do not contribute to R,

hb4 FIELD THEORY IN CURVED SPACETIME

1, $ y7 then D(yT) < 0 and so SY7RY7= 0 by Proposition hypothesis gives s $FA = 0 y7 Y7

137

3. If /,, E yr then the inductive (5.31)

In either case there is no contribution to & (F9A’. Now suppose that G is the only line in I’ which is not in any of the subgraphs g, (t = I,..., Y). Equation (5.30) becomes p-A I-

= F, fi Rjl:,“.

(5.32)

t=1

Then JJ=, RE.“’ IS a well-defined tensor product. Performing the C - I loop integrals in this tensor product leaves a one-loop integral which has d(r) m= -- 1. Thus (5.32) satisfies (5.24). More generally, let IA, be any line different from IA which is not in any of the subgraphs gt (t = l,..., I). Write (5.30) as

where y == r/l,,, . Arguing as in Proposition p’

3 leads to

= GA,@+

(5.34)

By the inductive hypothesis, i?y*” .IS a well-defined distribution if y is 1PI. If y is I PR, iT, is well defined and so I?:*” is certainly well defined. Finally (5.24) follows because (5.34) is a one-loop momentum integral with a(r) = - 1. To complete the proof of the proposition requires showing c#p..

.A,) = 0

(5.35)

where propagators G,, ... G, have been replaced by FAX,..., FA for any set of a lines. The method of proof iiven ibove for a = 1 extends easily to’the more general case.

6. FINAL THEOREMS THEOREM I. Let I’ be any 1PI graph with Y =~ 4. Consider R,(x, ,..., xN) as a distribution in x, ,..., xN-l for fixed x.” . Let p be an arbitrary parameter having dimensionsof mass.Then, if x, ,. .., xN-1 have Riemannnormal co-ordinates.u,~,.... .u;-~ about origin s,\ :

S&(x,

)...) XN) = /.F4)!yf(n)

6(x?) ... 6(X”,&,),

(6.1)

wherej’(n) is a sum of terms having poles at n = 4 with residuesthat are dimensionless numerical constants independentof all quantities in the theory that have dimension. Proof: From (5.4), terms having d = 0 must have f = A. All others have d < 0 and by Proposition 3 do not contribute to SrRr . By Corollary 2 to Proposition I,

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Sri?, is a sum of products of 6 functions and derivatives of 6 functions. By Proposition 4, S,& can be evaluated in momentum space by putting all external momenta ki = 0 in Ir(k, , pi , xk), This allows the ki integrals to be performed giving a product of 6 functions as in (6.1). There are no derivatives of 6 functions which would come from retaining some powers of ki in the expansion about ki = 0. Now any term in S$r having E > 0 must vanish since xJ(xJ = 0, i = I,..., N - 1 and xNa = 0 since X, is the origin. But 4 = A and so all terms having A > 0 vanish. Consequently no geometrical quantities appear in S,R,. The momentum space loop integral in Ir now has the form (6.2)

Changing variables to jj = pi/M gives

where (A29 and (5.3) have been used, taking Y = 4. The integral in (6.3) gives terms having poles at II = 4 with dimensionless numerical residues. However, the factor (M/p)L+QP can be expanded: (M/p)L(n--l)P

m

1 -t

$L(n

-

4)log(M2/f.2)

+

. . ..

16.4)

Thus 1, may contain pole terms involving residues that are powers of log(Mz/p2). To see that no such terms appear in S,Rr, consider how S,R, changes under the transformation M2 --+ MyI Under (6.9, a momentum

(6.5)

space propagator transforms:

1 qf+

+ E).

1 M2 ‘q;+M2-

EM2 (9,” + M2Mq,2 + MY1 + 41 ’

(6.6)

The additional term is of order q;’ and the proof of (5.24) shows that it makes no contribution to S,RF. So SrRr is invariant under (6.5) and therefore it cannot contain terms of the form (n - 4)-l x log(M2/p2) which transform:

tn-

4)-llog(M2/$)

-+ (n - 4)-‘10g(M2/p2) + (n - 4)-‘log E.

(6.7)

Hence the theorem is proved. Let I’ be a 1PI graph with Y = 4. Let x1 ,..., xEel , xN be its E external vertices and let xE ,. ., xNWl be its internal vertices Consider J &(x1 ,..., xN) nys2 gl12(xi) d*xi as a distribution in xl ,..., xEmlfor fixed xN which is taken as origin COROLLARY.

k#~’

FIELD

THEORY

for Riemann normal co-ordinates.

(4.1)

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Restore to R, the factors

of g-lpi removed in dejning

: &&(.Y 1

1 ,...,

xN)

N-l n

g"'(q)

@xi

=

/~~('-)'~f.(n)

8(x;)

...

(6.8)

~(xU,-~)

i=E

blith f (n) as in Theorem 1 and 8 denoting a covariant 6 function. Proof. The inclusion of powers of g(xi) in the integration and the use of covariant 6 functions represents a trivial modification of (6.1) since the delta functions 8(x,“) ... 8(x;-,) set each g(xJ to g(xN) = 1 in normal co-ordinates. lntegration over internal vertices then yields (6.8). Comment. (6.8) is a generally ax, - x,,,),. .., 8(X&l - XN).

covariant

statement

since the 6 functions

are

THEOREM 2. Let r be any 1PI graph with 1’ =z 2. Suppose that x1 , xN arc’ external vertices and x, ,..., xNel are internal vertices and consider s K,(x, ,..., x,~) J-J;:; glP(xi) d” x, as a distribution in x1 for fixed xN . Then

[&&(x1

)...) XN) N-1 n g1’2(xJ d”Xj is2

= P N(n-4’K(fl) &xl

- xd + f(n) 0 Rx, - .ydl,

(6.9)

where fi(n) is a sum of terms having poles at n = 4 with residues that are linear combinations of m2 and R(x,) andf (n) is as in Theorem 1. Proof. Applying the argument used at the beginning of the proof of Theorem 1 shows that for a graph with Y = 2, Sr& is a product of 6 functions and at most two derivatives of 8 functions in the variables xl=,..., x>=~ which are Riemann normal co-ordinates relative to a fixed origin xN . Only terms with A < 2 can contribute to Sri?, . Therefore, one may take

with 1 -

f

RulsXy

x;

i

A’=1

&!&2! qA'* qn'v A'?. A'3 q;, + M2

1 .

(6.11)

Consider first the A = 0 term in (6.11): (6.12)

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Taking (6.12) in (6.10) leads to

Now only one external momentum b(Pj , kJl,,=

remains and one can expand a&-

. ..=kNel=O

=

I&j

7 ‘1

+

with higher-order terms not contributing leading term in (6.13) as 8(q) times

%,$$

4Pj 90) = p N-M2

(yfn-4)

kr

q

kl”O

to Sri&-,

(6.14)

+;k;k;&j

1 by Proposition

fi JS

;

ki=O

4. Express the

fjl (pj.?,,;2 + l *

(6-l 5)

Now pole terms appear in (6.15) with residues proportional to MZ and expanding (M/P)~(‘+~) about n = 4 may give pole terms with residues proportional to Ma log (M2/p2). However terms involving log (M2/p2) cannot appear in S& . To see this, perform a transformation (6.5) and note that 1 q;+

a --+ 1+&l2---[ aw M2

1q;+

1 M2 + (4; + My

g:

W(1 +

l )I

@.16)

and (6.17)

Now the terms of order qA-2L--p have d = -2 and so by the arguments used to prove Propositions 3 and 4 do not contribute to S$, . Hence &it,

+

[

1 + EM2 -&]

s,w, .

(6.18)

Now M2 transforms according to (6.18) under (6.5) but M2 log (M21p2) does not. Hence the logarithmic terms do not appear in S& . Now return to (6.14). The second term in the expansion is linear in a/ak,” and so its numerator is linear in plu. Thus it is an odd function of plu and will not contribute to SrR, when the loop integrations are performed. The third term in (6.14) does not vanish when integrated over loop momenta but must, by Lorentz invariance, be proportional to the flat space metric qlly. So: i fi s$

s

II =. = --F(n) $‘(a-4)q,y + regular terms, 1

f

(6.19)

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where F(n) is a dimensionless function of n having poles at II :- 4. (6.19) gives a contribution to (6.13) of the form pN(n-4) times exp [ik,(x, - xN)] kPk,= = F(n) ~as~XP,&x,).

(6.20)

The contribution to j S$& ,..., xN) dnxz ... dnxNpl is similar but with F(n) replaced byf(n) which is of the same form asf(n) in Theorem 1. The right-hand side of (6.20) is not generally covariant and this is no surprise since definition (4.1) on which the analysis has so far been based is not covariant. The correct covariant definition of Ir is (2.18) which has covariant propagators Gh . To obtain a covariant expression, factors of g1i4(xJ must be restored. Consider first the factors g(xJ. Since x1 is an external vertex with one external line there must be three internal lines at x1 . Thus a factor g-3/4(x,) must be included. But, in Riemann normal coordinates: g-yx,) Ppa,a&qx,) = c&x, - XN) $ ~R8(x, - XN). (6.21) Now the right-hand side is covariant and compatible with the form of (6.9). There are also factors of g-1/4(xi) for i = 2,..., N - 1 to be considered as well as the integration measure ny:.’ gl/z(xi). Only contributions up to order A = 2 in the normal co-ordinate expansion g(x+) == 1 - gR,,x;xf + ..a (6.22) need be included and the contribution of these terms to (6.9) is of the same form as the contribution from the A = 2 terms in (6.11). All these A = 2 terms have ,!J= 2 and it is simpler to postpone the xz ,..., xNel integrations in (6.13) and to consider 1,(x, ,..., xN) and Br(x, ,..., xN). Evaluating the momentum space integrals gives a sum of products of 6 functions and up to two derivatives of 6 functions for R, . Since the terms being considered have f = 2 only second derivatives of S functions can give a non-zero contribution and the result is a collection of terms of the form pCLN(n-l) times (for example) (6.23)

multiplied by pole terms having the form of f(n) in Theorem I. Hence, integrating over internal vertices, such terms are of the form offi(n)&x, - xN) in (6.9). This concludes the proof of Theorem 2. THEOREM 3. Let r be any 1PI graph with vertex xN . Then

s

S,R,(x,

wheref,(n;

Y = 2 and having only one external

N-l

,..., xN) fl g1j2(xi) dnxi =fi(n; i=l

xN) is of the same form asfi(n) in Theorem 2.

.xN),

(6.24)

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ProoJ Similar to Theorem 2. First consider (6.10) and (6.12). Perform integrations over x1 ,..., xNel . This sets all external momenta ki to zero, so the expansion (6.14) is unnecessary. Evaluate pj integrals as before and obtain a contribution to fi(n; x,,,). Now consider terms of order A = 2, ,!j = 2. Expand (6.11) about ki = 0 up to second order and perform ki integrals. Because E = 2, only terms quadratic in ki , and hence quadratic in derivatives acting on flk 8(x3, contribute. Finally use (6.23) and integrate over x1 ,..., xNdI to obtain another contribution to fi(n; xiv). Hence the theorem is proved.

7.

DISCUSSION

In this paper, a detailed account of the BPHZ renormalization of X4* field theory in curved spacetime has been given. The main results proved are contained in Proposition 3 and Theorems 1-3, which give the detailed structure ofS& as a function of Y, the number of external lines in a graph. For Y = 6, 8,..., Proposition 3 says that S$, is identically zero; for Y = 4, ‘Theorem 1 says that S,Rr is proportional to a product of 8 functions; and for Y = 2, Propositions 2 and 3 show that S,ir, depends linearly on m2, R(xN) and the differential operator q acting on a 6 function. This simple structure of S,Rr can be generated by counterterms in the Lagrangian density for the theory, which is 9 = -+

di

[g@“a,+a,4 + (Z,mz + Z&R)

4”] -+

I& hp4-nZ4$4.

Counterterms are generated by expressing the renormalization Z, as power series in h:

(7.1)

constants Z, , Z, and

z+ = 1 + t Zi(W.

(7.2)

7=1

A fourth series of counterterms is generated by renormalizing the field operator 4. This has the effect of renormalizing the multi-point Green’s functions of the theory which become 4Yl ,‘..Y Vu> = Z;y’ziy’p (0 I V4(YJ

**. Wr))l

O>,

(7.3)

where Y is the number of external lines and the renormalization constant Z, has an expansion of the form (7.2). There are two stages involved in the BPHZ renormalization of a Feynman integrand. The first is the construction of Br and the second is the removal of Sr& to give Rr . To show that these procedures can be formally implemented by including counterterms requires showing two things. First that the structure of Sr& can, as claimed, be generated by the four types of counterterms and, secondly, that once a series of constants Zjr) corresponding to subdiagrams y C r have been fixed in constructing R, from &, , these fixed values are precisely the correct values needed to produce all contributions to the sum over sets (yI ,..., yJ of subdiagrams in the definition of & . The second point is a purely combinatorial property of Feynman

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graphs and has nothing whatever to do with the manifold of spacetime. It will therefore not be discussed as it has been treated elsewhere [21]. Consider a diagram I’ with four external lines (Y = 4). Such a diagram appears when considering the four-point Green’s function T( y1 ,..., y4) and the external lines are formally represented by Feynman propagators. Hence, by (6.1): P

N(4--n) f

SlmXl

,...,

x,v)G(y,

= p4-“ftn> @Y, 3xiv)

, x1)

Gty,,

G(y,

, x2)

XN)G(Y,,

Gty,

XN)

, x,)

G(Y,,

G(.Y~

3 JCN)

fi Ptx,) P=l

XN)P~~XN)~~XN.

d%

(7.4)

The factor pCN(J-n) is included because by (7.1) each vertex provides a factor pl--“. The right-hand side of (7.4) is identical in form to a term generated by p4-nZ444 in 2. Now consider a diagram r with two external lines (Y = 2) and suppose that there are two external vertices. A similar argument applies if there is only one external vertex. By (6.9)

pNC4-) )..,-~A~) G(Y, ,x1) Gty2 ,xiv) fiP2(x3 d"xk: sS&-(x, k=l

= -f(n) G(Y, >~3 + s Km” + ~NxN)) f(n) + fi(n; dl x

G(Y,

,x1)

@Y,

,x~)P~(xdd"xiv.

(7.5)

The first term on the right-hand side is a wavefunction renormalization generated by Z,lG(y, , y2) in (7.3). The second term renormalizes m and LJand is generated by V2m2 + -&CR) 4 2 in 2. This completes the demonstration that counterterms in L can be used to generate the terms SrRr subtracted in defining a BPHZ renormalized integrand. A complete discussion of renormalization in curved spacetime should include an analysis of vacuum-to-vacuum diagrams, i.e., diagrams which have Y = 0. In flat spacetime such diagrams contribute an unobservable phase to particle creation amplitudes and so can be ignored. However, in curved spacetime the sum of all such diagrams gives the vacuum persistence amplitude which is a quantity of physical interest since it need not have unit magnitude. In other words, the external gravitational field can create real particles. In principle a BPHZ renormalization of vacuum-tovacuum diagrams can be performed. On dimensional grounds, terms of order A < 4 can give rise to divergences. Thus a generalization of (2.14) is required to include higher-order terms. Given an integrand Ir, the quantity W, is constructed according to (2.24). As before, R, is purely local and independent of terms of order A ‘2 4 and of all global information in GCR). For vacuum-to-vacuum diagrams, all vertices are internal and must be integrated over. This will yield a scalar quantity expressed as a spacetime integral of scalar curvature invariants of order A < 4. The only invariants allowed aretherefore m4, m2R, R2, RaBRmBandRaflyGR,Bvsandtheseleadtorenormalization of coupling constants in a modified Einstein gravitational action. This brief discussion

144

T.

S. BUNCH

of Y = 0 diagrams provides an outline of a method of proving their renormalizability. The details would be similar to those given for the Y = 2 and Y = 4 case but rather more complicated.

APPENDIX:

GRAPH

THEORY DEFINITIONS

AND RESULTS

A Feynman graph, f, consists of a set of vertices, labelled x1 ,..., x, , and a set of lines, labelled I1 ,..., 1, , together with rules assigning to each line 1, a unique pair of vertices, called initial and final vertices, given by connection matrices E$) and ~1:’ according to (1) = 1

if xk is the initial vertex of line Z, ,

EAk

z= 0

(Al)

otherwise;

E$ = 1

if xk is the final vertex of line I,, ,

zzz0

642)

otherwise.

This definition is general enough to apply to any scalar field theory and the definitions and results of this appendix are based on it. However, when specialising to @* field theory, one requires that no more than four lines are incident on any one vertex. If the initial and final vertices of a line are the same, the line is called a loop line and a loop line counts as two incident lines. Any vertex which has a maximal set of incident lines is called an internal vertex. All other vertices are called external vertices. The incidence matrix for .T’, e,# , is defined to be 643)

r can be regarded as a l-cell complex with the O-chain group, C, , and l-chain group, C, , being respectively the free abelian groups generated by the vertices x1 ,..., XN is d: C, -j C,, given by and the lines 1, ,..., IL . The boundary homomorphism d(aah)

=

%eAkxk

(or,

E 0

G44)

where summation conventions are being used. A circuit, gj , in r is a minimal set of lines with the property that every vertex in r has either zero or two lines of gj incident upon it. The circuit matrix, Q> , for circuits Vj and lines I,, is defined by if In E 59); ?jA = 51 zzz 0 otherwise, with the relative sign given by weak

=

-%S%k

(01,/3 not summed)

(-46)

A44

FIELD

THEORY

IN

CURVED

145

SPACETIME

whenever 1, and la are in qi and are both incident on vertex .Y~. Then %n%k

with summation

=

(A7)

0

over X implied. %‘j is an element of C1 by the definition %j

=

Tjj,jlA

(A@

By (A4) and (A7): Vj E ker A = Z, ,

(A‘4

where Z, denotes the group of l-cycles of r. By the definition of circuits, all l-cycles are linear combinations of circuits. A path in rfrom xi to xj is a sequence of lines &tl),..., fAcp)such that fncl) is incident upon xi , IA(P) is incident on xj and, for every pair Inca)and r’A(ail) there exists a vertex of r on which both lAt,, and lA(a+l) are incident. r is decomposed into disjoint equivalence classes by the equivalence relation “xi and xj can be connected by a path in r.” The equivalence classes are called connectivity components and a general graph will contain K of them. If K = I, r is said to be connected. r is strongly connected. or one-particle irreducible (IPI), if it remains connected on removal of any one line. Suppose K = I and consider any path with initial vertex x,,, which passes through every vertex of l? Define a path matrix relative to .xN by: xi,, == number of times iA is traversed in direction of its orientation minus the number of times lA is traversed in the opposite direction in going from xN to xi via the path. The direction of orientation tion of 7ri,,

(AlO)

of a line is from its initial to its final vertex. By the delinixi = x&I + RiAPAk.Y,< .

(Al I)

For each i := I...., N - 1, ai,I, is an element of C, which represents a path .4; from xh. to .x-‘i .Yi == iTiAIA,

CA13

A(Pi) = riAeAkxk = xi - xX .

(A13)

having boundary

Thus if 9, and @i are two distinct paths from x,,, to xi

and it follows that two paths differ by a linear combination

of circuits. Therefore the

146 freedom r,~~may Now for two

T. S. BUNCH

in the choice of a path matrix 7~~~is that arbitrary rows of the circuit matrix be added to any row of nib. suppose that a particular choice of n*,+ has been made and consider (Al 1) distinct vertices xi and xi’ . Then

Also, from (Al 1): xi - XN = niAeAkxk.

G416)

Therefore, the subgroup of C, generated by expressions of the form xi - xi’ (i, i’ = I,..., N), which is of rank N - 1, iS generated by e,$& (h = I,..., L). Since enk annihilates the column vector having 1 in every position, the rank of e,,k is p(e) = N-

1.

(A17)

Define a matrix 2Ai (i = l,..., N - 1) by ” f?Ai = e,i

(A = l,..., L; i = l,..., N -

1).

(A19

Then & also has rank N - 1, so it is one-one and hence it has a left inverse. Indeed, putting XN = 0 in (All) gives v&j6?Ak= 6ik *

(A19)

Now suppose that K is arbitrary, so r is not necessarily connected. Then, summing (Al 7) over connectivity components gives p(e) = N - K. The zeroth homology

GW

group is H,, = &,/B,, = C’/Im

where B, = Im A is the group of zero-boundaries. Betti number:

A,

The rank of H,, is the zeroth

b,=N-(N-K)=K. The first homology

0422)

group is HI = Z, and by the first group isomorphism G/Z,

G421)

z 4,

theorem (A23)

Hence the first Betti number is b,=L-N+K.

(A24)

@4

FIELD

THEORY

IN

CURVED

SPACETIME

147

By definition, this is the maximal number of linearly independent circuits and will be denoted by C. The linearly independent circuits of r are often called the loops of I’. In the renormalisability proof, the circuit matrix qjh is restricted to a set of C linearly independent circuits so that j = I,..., C. Moreover the graph r is taken to be connected so that C=L--N+1.

W5)

LEMMA A. Let r be an arbitrary connected graph. Choose a path matrix rriA (i = l,..., N - I) and a circuit matrix qiA ( j = 1,..., C) and form the L x L matrix:

N-l L-N+1 t------f L Then det J fl. Proof. First note that det J is a property of the graph, l-‘, and not of the way that n and 71are chosen. A different choice of independent circuits re-expresses some rows of 7 as linear combinations of the rows of 7) and a different choice of path adds some linear combinations of rows of 7 to the rows of 7r. Neither operation has any effect on det J. To prove det J = fl proceed by induction in C starting with graphs for which C = 0 (tree graphs). It is straightforward to prove that det J = fl for such graphs by induction in L, the number of lines. The result is trivially true when L = 1 and if proved for L follows immediately for L + 1 by making a trivial modification to the path. Now suppose det J = il for graphs with C = L - N + 1 loops and consider a graph, r, with C + 1 loops. It is always possible, since I’ is not a tree, to delete a single line from l’ leaving a connected graph, y, having the same number of vertices as r, say N. Thus I’ has L + 1 lines and C + 1 loops and y has L lines and C loops. Choose a path matrix ‘TT~,, for y. This is a path matrix for r since y and r have the same vertices. Choose C basic circuits for y. These are basic circuits for I’and the (C + 1)th basic circuit for r can be chosen arbitrarily but must contain the line lL+l not in y. Then, if

one finds

J(r) =

TiA

%A

. . . 0 . . . +1 . ..o . . . fl

. ..

N-l I L-N+1 5 1

148

T. S. BUNCH

Because of the form of the final column of J(r):

Hence det J(r) = ~1 by the inductive hypothesis. This proves the lemma.

ACKNOWLEDGMENTS I wish to thank N. D. Birrell for several valuable discussions about this work.

REFERENCES

1. D. Z. FREEDMAN, I. J. MUZLNICH, AND E. J. WEINBERG, Ann. P&X. (N.Y.) 87 (1974), 95. 2. D. 2. FREEDMAN AND E. J. WEINBERG, Ann. Phys. (iV. Y.) 87 (1974), 354. 3. I. T. DRUMMOND, Nucl. Phys. B 94 (1975), 115. 4. I. T. DRUMMOND AND G. M. SHORE,Phys. Rev. D 19 (1979), 1134. 5. N. D. BIRRELL, J. Phys. A 13 (1980), 569. 6. T. S. BUNCH, P. PANANGADEN, AND L. PA-R, J. Phys. A 13 (1980), 901. 7. T. S. BUNCH AND P. PANANGADEN, J. Phys. A 13 (1980), 919. 8. T. S. BUNCH AND L. PARKER, Phys. Rev. D 20 (1979), 2499. 9. T. S. BUNCH, Local momentum space and two-loop renormalizability of Ad4 field theory in CUN~ spacetime, to appear. 10. N. D. BIRRELL AND J. G. TAYLOR, Analysis of interacting quantum iield theory in curved spacetime, to appear. 11. R. BANACH, Topology and renormalizability, to appear. 12. N. N. BOGOLUBOV ANIJ 0. PARASIUK, Acta Math. 97 (1957), 227. to the Theory of Quantized Fields,” 13. N. N. B~G~LUBOV AND D. W. SHIKKOV, “Introduction Interscience, New York, 1959. 14. K. HEPP, Comm. Math. Phys. 2 (1966), 301. 15. W. ZIMMERMANN, Comm. Math. Phys. 15 (1969), 208. 16. C. W. MISNER, K. S. THORNE, AND J. A. WHEELER, “Gravitation,” Freeman, San Francisco, 1973. 17. F. G. FRIEDLANDER, “The Wave Equation on a Curved Space-Time,” Cambridge Univ. Press, London/New York, 1975. 18. I. M. GEL’FAND AND N. YA. VILENKIN, “Generalized Functions,” Vol. 4, “Applications of Harmonic Analysis,” Academic Press, New York, 1964. 19. I. M. GEL’FAM) AND G. E. SHILOV, “Generalized Functions,” Vol. 2, “Spaces of Fundamental and Generalized Functions,” Academic Press, New York, 1968. 20. S. WEINBERG, Phys. Rev. 118 (1960), 838. 21. E. B. MANOUKIAN, Nuovo Cimento A 53 (1970), 345.