Scalar Matter Coupled to Quantum Gravity in the Causal Approach: One-Loop Calculations and...

Annals of Physics 287, 153�190 (2001)

Scalar Matter Coupled to Quantum Gravity inthe Causal Approach

One-Loop Calculations and Perturbative Gauge Invariance

Nicola Grillo

Institut fu� r Theoretische Physik, Universita� t Zu� rich, Winterthurerstrasse 190,CH-8057 Zu� rich, Switzerland

E-mail: grillo�physik.unizh.ch

Received February 10, 2000; revised August 8, 2000

Quantum gravity coupled to scalar massive matter fields is investigated in the frameworkof causal perturbation theory using the Epstein�Glaser regularization�renormalization scheme.Detailed one-loop calculations include the matter loop graviton self-energy and the matterself-energy. The condition of perturbative operator gauge invariance to second order impliesthe usual Slavnov�Ward identities for the graviton two-point connected Green function in theloop graph sector and generates the correct quartic graviton-matter interaction in the treegraph sector. The mass zero case is also discussed. � 2001 Academic Press

Key Words: quantum gravity.

1. INTRODUCTION

In this paper we follow the quantum field theoretical approach to gravitationalinteractions coupled to scalar matter fields (see the introduction to this subject in[1] and the references therein). This approach allows a quantization of theinvolved fields, matter and graviton fields, and a Lorentz covariant perturbativeexpansion of the scattering matrix S.

Calculations of matter loop diagrams in this conventional framework led to non-renormalizable ultraviolet (UV) divergences [2]. These were later confirmed bymeans of dimensional regularization and back-ground field method, both in themassive [3] and in the massless [4, 5] case.

The counterterms needed to cancel the divergences are not of the type present inthe original Lagrangian density and therefore cannot be absorbed in the redefini-tion of physical quantities. According to these findings, quantum gravity (QG)coupled to matter fields does not fulfil the criterion of perturbative renor-malizability [6].

We adopt here another approach to investigating these outcomes by applying animproved perturbation scheme which has as central objects the time-orderedproducts and constructing principle causality. In this natural regularization�renor-malization scheme, called causal perturbation theory, the S-matrix is constructed

doi:10.1006�aphy.2000.6104, available online at http:��www.idealibrary.com on

1530003-4916�01 �35.00

Copyright � 2001 by Academic PressAll rights of reproduction in any form reserved.

inductively as a sum of smeared operator-valued n-point distributions Tn(x1 , ..., xn)avoiding UV divergences at each stage of the calculations as a consequence of adeeper mathematical understanding of how loop graph contributions have to becalculated.

This idea goes back to Stu� ckelberg [7], Bogoliubov and Shirkov [8], and theprogram was carried out successfully by Epstein and Glaser [9] for scalar fieldtheories and subsequently applied to QED by Scharf [10], to non-Abelian gaugetheories by Du� tsch et al. [11], and to quantum gravity [12] in the past few years.This calculation scheme exploits the concept of causality from the beginning andhas a resemblance to the BPHZ-subtraction scheme.

Quantum gravity, being a nonrenormalizable theory, unfortunately suffers in ourapproach from a nonuniqueness in the fixing of the normalization of the time-ordered products, which endangers the physical predictability. Here, the nonrenor-malizability problem of QG is translated into an increasing ambiguity in thenormalization Nn of the time-ordered products Tn .

In addition, QG has considerable gauge properties [13], which are formulatedby means of a gauge charge that generates infinitesimal gauge variations of thefundamental free quantum fields (Section 2.3).

The present work focuses mainly on three aspects of QG coupled to massivematter fields. Brief remarks for the massless case are given for these aspects.

The first aspect is the calculations of loop graphs, which include the lowest ordermassive and massless scalar matter loop corrections to the graviton propagator(Section 3) and to the matter self-energy (Section 4). The UV finite and cutoff-freeresult are obtained using the techniques of causal perturbation theory for theS-matrix and the Epstein�Glaser regularization�renormalization scheme withoutintroducing nonrenormalizable counterterms. The inherent freedom in the nor-malization of the time-ordered products is to some extent discussed, but some localnormalization terms remain undetermined (Section 3.4).

The second aspect consists in the investigation of the gauge properties of thegraviton self-energy (Section 2.4). Gauge invariance of the S-matrix implies someidentities between the C-number parts of the n-point distributions which yield thegravitational Slavnov�Ward identities (SWI) [14] (Section 3.3).

The third aspect of this work is also connected with gauge invariance: pertur-bative gauge invariance to second order in the tree graph sector requires theintroduction, at a purely quantum level, of a quartic matter�graviton interactionexactly as prescribed by the expansion of the classical matter�gravity Lagrangian(Section 5).

The quantization of the graviton field, the identification of the physical subspace,and the proof of S-matrix unitarity are investigated in [15], which provides alsothe conventions and the notations used here. Calculations involving graviton self-couplings are not considered here; see [16].

The causal scheme applied to quantum gravity coupled to photon fields leadsalso to analogous results with regard to loop calculations and gauge invariance[17].

154 NICOLA GRILLO

We use the unit convention �=c=1; Greek indices :, ;, ... run from 0 to 3,whereas Latin indices i, j, ... run from 1 to 3.

2. QUANTIZED MATTER�GRAVITY SYSTEM AND PERTURBATIVEGAUGE INVARIANCE

2.1. Inductive Construction of Two-Point Distributions in the S-Matrix Expansion

In causal perturbation theory [10, 18], the ansatz for the S-matrix as a powerseries in the coupling constant is central; namely, S is considered as a sum ofsmeared operator-valued distributions

S(g)=1+ :�

n=1

1n ! | d 4x1 } } } d 4xn Tn(x1 , ..., xn) g(x1) } } } } } g(xn), (2.1)

where the Schwartz test function g # S(R4) switches the interaction and provides anatural infrared cutoff. The S-matrix maps the asymptotically incoming free fieldson the outgoing ones and it is possible to express the Tn 's by means of free fieldswithout introducing interacting quantum fields.

Establishing the existence of the so-called adiabatic limit g � 1 in theories involv-ing self-coupled massless particles, such as QG, may be very problematic, and thereis evidence that the limit may not exist. This aspect will not be considered here.

The n-point distribution Tn is a well-defined renormalized time-ordered productexpressed in terms of Wick monomials of free fields: O(x1 , ..., xn):

Tn(x1 , ..., xn)=:O

:O(x1 , ..., xn): tOn (x1&xn , ..., xn&1&xn). (2.2)

The tn 's are C-number distributions. Tn is constructed inductively from the firstorder T1(x), which describes the interaction among the quantum fields, and fromthe lower orders Tj , j=2, ..., n&1, by means of Poincare� covariance and causality.The latter leads directly to a UV finite and cutoff-free distribution Tn .

For the purpose of this paper, we outline briefly the main steps in the construc-tion of T2(x, y) from a given first-order interaction. Following the inductivescheme, we first calculate the causal operator-valued distribution

D2(x, y) :=R$2(x, y)&A$2(x, y)=[T1(x), T1( y)]. (2.3)

In order to obtain D2(x, y), one has to carry out all possible contractions betweenthe normally ordered T1 using Wick's lemma, so that D2(x, y) has the structure

D2(x, y)=:O

:O(x, y): dO2 (x& y). (2.4)

155SCALAR MATTER COUPLED TO QUANTUM GRAVITY

dO2 (x& y) is a numerical distribution that depends only on the relative coordinate

x& y because of translation invariance.D2(x, y) contains tree (one contraction), loop (two contractions), and vacuum

graph (three contractions) contributions. Due to the presence of normal ordering,tadpole diagrams do not show up. In this paper we do not consider vacuum graphs.D2(x, y) is causal, i.e., supp(dO

2 (z))�V +(z) _ V &(z), with z :=x& y.In order to obtain T2(x, y), we have to split D2(x, y) into a retarded part,

R2(x, y), and an advanced part, A2(x, y), with respect to the coincident point z=0,so that supp(R2(z))�V +(z) and supp(A2(z)) V &(z). This splitting, or improvedtime-ordering, has to be carried out in the distributional sense so that the retardedand advanced part are mathematically well-defined.

The splitting affects only the numerical distribution dO2 (x& y) and must be

accomplished according to the correct singular order |(dO2 ) which describes roughly

speaking the behavior of dO2 (x& y) near x& y=0 or that of d� O

2 ( p) in the limitp � �. If |<0, then the splitting is trivial and agrees with the standard time-order-ing. If |�0, then the splitting is nontrivial and nonunique,

dO2 (x& y) � rO

2 (x& y)+ :|(d 2

O)

|a|=0

Ca, ODa$ (4)(x& y). (2.5)

A retarded part rO2 (x& y) of dO

2 (x& y) is usually obtained in momentum space bymeans of a subtracted dispersion-like integral; see Eq. (3.18).

Equation (2.5) contains a local ambiguity in the normalization: the Ca, O 's areundetermined finite normalization constants, which multiply terms with point sup-port Da$(4)(x& y)(Da is a partial differential operator). This freedom in the nor-malization has to be restricted by physical conditions.

Finally, T2 is obtained by subtracting R$2(x, y) from R2(x, y) and the whole localnormalization coming from (2.5) is called N2(x, y).

2.2. Quantized Matter�Gravity Interaction

We consider the coupling between the quantized symmetric tensor field h+&(x),the graviton, and the quantized scalar field ,(x), the matter field, in the backgroundof a Minkowski space-time.

The free scalar field of mass m satisfies the Klein�Gordon wave equation

(g+m2) ,(x)=0, (2.6)

which follows from the free matter Lagrangian density

L (0)M =

12

, , &, , &&m2

2,2. (2.7)

156 NICOLA GRILLO

The matter energy�momentum tensor reads

T +&M :=, , +, , &&'+&L (0)

M =, , +, , &&12

'+&, , \ , , \+'+& m2

2,2, (2.8)

where '+&=diag(+1, &1, &1, &1) and it fulfils T +&M , &=0. Quantization of the

scalar field is accomplished through

[,(x), ,( y)]=&iDm(x& y), (2.9)

where

Dm(x)=D (+)m (x)+D (&)

m (x)

=i

(2?)3 | d 4p $( p2&m2) sgn( p0) e&ip } x (2.10)

is the causal Jordan�Pauli distribution of mass m.The free graviton field satisfies the wave equation

gh+&(x)=0 (2.11)

and is quantized (see [15]) according to

[h+&(x), h:;( y)]=&ib+&:;D0(x& y), (2.12)

where the b-tensor is constructed from the Minkowski metric

b+&:; := 12('+:'&;+'+;'&:&'+&':;) (2.13)

and D0(x) is the Jordan�Pauli distribution of Eq. (2.10) with m=0.The graviton field interacts with the conserved energy�momentum tensor of the

matter fields. The first-order matter coupling is chosen to be

T M1 (x)=i

}2

:h:;(x) b:;+& T +&M(x):

=i}2 { :h:;, , :, , ; : &

m2

2:h,,: = , (2.14)

where }, is the coupling constant (see below for its relation to Newton's constant).To simplify the notation, the trace of the graviton field is written as h=h#

# andall Lorentz indices of the fields are written as superscripts,, whereas the derivativesacting on the fields are written as subscripts. All indices occurring twice are


contracted by the Minkowski metric '+&. We skip the space-time dependence if themeaning is clear.

The presence of the b-tensor in Eq. (2.14) is a consequence of the choice of thegraviton variable (see Section 2.3).

2.3. Perturbative Gauge Invariance

The classical gauge properties of h+&(x) (which are related to the generalcovariance of the metric g+&(x) under coordinate transformations [19, 20]) are for-mulated at the quantum level by the gauge charge [12, 13]

Q :=|x0=const

d 3x h+&(x) , &

��0

x

�u+(x). (2.15)

For the construction of the physical subspace and in order to prove unitarity of theS-matrix on the physical subspace [15], the ghost, field u+(x), together with theantighost field u~ &(x), have to be quantized as free fermionic vector fields,

gu+(x)=0, gu~ &(x)=0, [u+(x), u~ &( y)]=i'+& D0(x& y), (2.16)

whereas all other anticommutators vanish. The gauge charge generates theinfinitesimal operator gauge variations

dQh+&(x) :=[Q, h+&(x)]=&ib+&\_u\(x) , _ ,

dQu:(x) :=[Q, u:(x)]=0,

and

dQu~ :(x) :=[Q, u~ :(x)]=ih:;(x) , ; . (2.17)

The condition of gauge invariance is formulated in terms of the n-point distribu-tions Tn . For the moment, we define the condition of perturbative operator gaugeinvariance to nth order by the requirement

dQTn(x1 , ..., xn)=sum of divergences. (2.18)

Heuristically, this would imply that dQS(g) vanishes in the adiabatic limit g � 1because of partial integration and Gauss' theorem. But this is only purely formal,because this limit may not exist.

Using a simplified notation which keeps track of the field type only, perturbativeinvariance to first order in pure QG [12], namely for a coupling of the formT h

1 t:hhh: without matter fields, requires the introduction of the ghost couplingT u

1 t:u~ hu: , so that dQ(T h1+T u

1)(x)=�x_T _ h+u

1�1 (x). Here, T _ h+u1�1 t:uhh: +:u~ uu: is

the so-called Q-vertex.

158 NICOLA GRILLO

Using Eq. (2.17), the first-order matter coupling (2.14) is gauge invariant,

dQT M1 (x) =

}2

b:;\_ :u\(x) , _ b:;+& T +&M(x): =

/2

:u\(x) , _T \_M(x):

= �x_ \}

2 { :u\, , \ , , _ : &12

:u_, , \, , \ : +m2

2:u_,,:=+

=: �x_T _ M

1�1 (x), (2.19)

because of T \_M , _=0. T _ M

1�1 (x) is the matter Q-vertex. The concept of a Q-vertexallows us to formulate in a more stringent way the condition (2.18) of perturbativegauge invariance to the n th order,

dQTn(x1 , ..., xn)= :n

l=1

��x&

l

T &n�l (x1 , ..., xl , ..., xn), (2.20)

where T &n�l is the renormalized time-ordered product obtained according to the

inductive causal scheme, with one Q-vertex at xl while all other n&1 vertices areordinary T1 -vertices. This condition follows from the formula for the constructionof Tn , namely the time-ordering of n, first-order interactions T1 , and from gaugeinvariance to first order, Eq. (2.19).

The procedure outlined here corresponds to the expansion of the Hilbert�Einstein and matter Lagrangian density [2, 3, 5]

LEH+LM=&2}2 - & g g+&R+&+

12

- & g (g+&,; +, ; &&m2,2) (2.21)

(}2=32?G), written in terms of the Goldberg variable g~ +&, in powers of thecoupling constant } according to the metric decomposition

g~ +& :=- & g g+&='+&+}h+&, (2.22)

which defines the graviton field h+& in the Minkowski background. Then oneobtains

LEH+LM= :�

j=0

} j (L ( j)EH+L ( j)

M ). (2.23)

From L (0)EH , choosing the Hilbert gauge h+&

, &=0, one obtains Eq. (2.11) and thepresence of the b-tensor is made clear [15].

The first-order graviton coupling T h(x)t :hhh: corresponds then to the normallyordered product of i}L (1)

EH (see [13] for a derivation based merely on the principleof perturbative operator gauge invariance) and L (2)

EH thhhh represents the quarticgraviton coupling required by perturbative gauge invariance to second order in thetree graph sector [12].


The expansion of the matter Lagrangian density reads

LM =12

('+&, , +, , &&m2,2)+}2

h+& \, , + , , &&m2

2'+&,2+

LM(0) =}LM

(1)

+m2}2

8 \h:;h:;,,&12

hh,,++O(}3). (2.24)

=}2LM

(2)

From the first term one obtains (2.6) and the matter coupling of Eq. (2.14)corresponds to i} :L (1)

M : . A quantized quartic interaction t:hh,,: which agreeswith L(2)

M will be necessary for reasons of gauge invariance; see Section 5.

2.4. Identities for the Two-Point Functions from Perturbative Gauge Invariance toSecond OrderFrom the structure of T M

1 it is evident that the two-point distribution describingloop graphs has the form (up to noncontributing divergences for the matter self-energy, see Section 4.1)

T2(x, y)loops=:h:;(x) h+&( y): i6(x& y):;+&+:,(x) ,( y): i7(x& y). (2.25)

Here, the first term represents the matter loop graviton self-energy and the secondterm the scalar matter self-energy. The C-number distributions 6(x& y):;+& and7(x& y) will be explicitly calculated in Section 3.3 and in Section 4.2, respectively.

Perturbative gauge invariance to second order, namely Eq. (2.20) with n=2,allows us to derive a set of identities for these numerical distributions by comparingdistributions attached to the same operators on both sides of Eq. (2.20) [21].

We compute dQT2(x, y)loops by meas means of (2.17) and isolate the contribu-tions with external operator of the type :u(x) h( y): . We obtain

dQT2(x, y)loops| :u(x) h( y): =:u\(x) , _ h+&( y): (b:;\_6(x& y):;+&). (2.26)

On the other side, T _2�1(xy) has to be constructed with one Q-vertex at x and one

``normal'' vertex at y. From the structure of both interaction terms, it follows thatthe loop contributions coming from T _

2�1(x, y) can only be of the form

T _2�1(x, y)=:u\(x) h+&( y): t_

uh(x& y)\+&+:u_(x) h+&( y): tuh(x& y)+& , (2.27)

by performing two matter field contractions. The second term T _2�2(x, y) does not

contain terms with Wick monomials of the type :u(x) h( y): . Applying �x_ to the

expression above we find

�x_T _

2�1(x, y)=+:u\(x), _ h+&( y): [t_uh(x& y)\+&+'_

\tuh(x& y)+&]

+:u\(x) h+&( y): �x_[t_

uh(x& y)\+&+'_\tuh(x& y)+&]. (2.28)

160 NICOLA GRILLO

We compare the C-number distributions in (2.26) and in (2.28) attached to theexternal operators

:u\(x) , _ h+&( y): and :u$x) h+&( y): . (2.29)

Therefore, we obtain the two identities

b\_:;6(x& y):;+& =[t_uh(x& y)\

+&+'\_tuh(x& y)+&],(2.30)

0=�x_[t_

uh(x& y)\+&+'\_tuh(x& y)+&].

By applying �x_ to the first identity and inserting the second one, we obtain

b:;\_�x_6(x& y):;+&=0. (2.31)

This identity for the matter loop graviton self-energy tensor has been explicitlychecked and implies the gravitational Slavnov�Ward identities for the two-pointconnected Green function (see Section 3.3).

Gauge invariance to second order in the tree graph sector is much more involvedand requires also the full analysis of the matter�graviton interaction; see Section 5.

3. MATTER LOOP GRAVITON SELF-ENERGY

3.1. Causal D2(x, y)-Distribution

In order to construct D2(x, y), according to Section 2.1 we first need the contrac-tions between field operators. From (2.9) and (2.12), we derive them,

C[,(x) ,( y)] :=[,(x)(&), ,( y) (+)]=&iD (+)m (x& y),

(3.1)C[h:;(x) h+&( y)] :=[h:;(x) (&), h+&( y) (+)]=&ib:;+&D (+)

0 (x& y),

where ($ refers to the positive�negative frequency part of the corresponding quan-tity.

The A$2(x, y) gSE distribution for the graviton self-energy by a matter loop isobtained by performing two matter field contractions in &T M

1 (x) T M1 ( y). Using

(3.1) and with �x:=&� y

: , we find that

A$2(x, y) gSE=:h:;(x) h+&( y):}2

4 _&�x: �x

+D (+)m } �x

; �x& D (+)

m

&�x: �x

& D (+)m } �x

;�x+D (+)

m +m2'+&�x: D (+)

m } �x;D (+)

m

+m2':; �x+D (+)

m } �x& D (+)

m &m4

2':;'+& D (+)

m } D (+)m & (x& y). (3.2)


We introduce the functions

D (\), m} | } (x) :=D (\)

m (x) } D (\)m (x), D (\), m

: | ; (x) :=�x: D (\)

m (x) } �x; D (\)

m (x),(3.3)

D (\), m:; | +& (x) :=�x

: �x;D (\)

m (x) } �x+�x

& D (\)m (x),

so that we have

A$2(x, y) gSE=:h:;(x) h+&( y): a$2(x& y) gSE:;+& ,

a$2(x& y) gSE:;+&=

}2

4 _&D (\), m:+ | ;& &D (\), m

:& | ;+ +m2'+&D (\), m: | ; (3.4)

+m2':; D (\), m+ | & &

m4

2':; '+&D (\), m

} | } & (x& y).

Products of Jordan�Pauli distributions are evaluated in momentum space withEq. (2.10), because there products go over into convolutions between the Fouriertransforms. For example, D (\), m

} | } (x) becomes

D� (\), m} | } ( p)=

&1(2?)4 | d 4k $(( p&k)2&m2) 3($ p0&k0))

_$(k2&m2) 3(\k0). (3.5)

Therefore, the basic integrals that remain to be computed are of the form

I ($m ( p)&�:�:;�:;+�:;+& :=| d 4k $(( p&k)2&m2) 3($ p0&k0))

_$(k2&m2) 3(\k0)[1, k: , k:k; , k:k;k+ , k:k;k+k&],

(3.6)

which are calculated in Appendix A. By means of the I ($m ( p) } } } -integrals, the

D(\), m} } } | } } } -functions momentum space are

D� (\), m} | } ( p)=

&1(2?)4 [I (\)

m ( p)],

D� (\), m: | ; ( p)=

+1(2?)4 [+ p:I (\)

m ( p);&I (\)m ( p):;],

(3.7)

D� (\), m:; | +& ( p)=

&1(2?)4 [+ p: p;I (\)

m ( p)+&& p:I (\)m ( p);+&

& p;I (\)m ( p):+&+I (\)

m ( p):;+&].

162 NICOLA GRILLO

Inserting (3.7) and Eqs. (A.6), (A.7), (A.11), (A.15), and (A.19) into (3.4), then thea$2 -distribution in momentum space reads

a$2( p) gSE:;+& =

&}2?960(2?)4 [+Ap: p; p+ p&+Bp2( p: p;'+&+ p+ p&':;)

+Cp2( p: p+ ';&+ p: p& ';++ p; p+ ':&+ p; p&':+)

+Ep4(':+ ';&+':& ';+)+Fp4':;'+&] d� ( p) (+)m , (3.8)

with the coefficients

A :=&8&16m2

p2 &48m4

p4 , B := &4&8m2

p2 &24m4

p4 ,

C :=+1&8m2

p2 +16m4

p4 , E :=&1+8m2

p2 &16m4

p4 , (3.9)

F :=&1&12m2

p2 +4m4

p4 ,

and the distribution d� ( p) (\)m :=- 1&(4m2�p2) 3( p2&4m2) 3(\p0). Performing

the same calculations for R$2(x, y)=&T M1 ( y) T M

1 (x), we obtain

R$2(x, y) gSE=:h:;(x) h+&( y): r$2(x& y) gSE:;+& ,

(3.10)r$2( p) gSE

:;+&=&}2?

960(2?)4 [the same as in Eq. (3.8)] d� ( p) (&)m .

Therefore, with (2.3) the causal D2(x, y)-distribution reads

D2(x, y) gSE=:h:;(x) h+&( y): d2(x& y) gSE:;+& ,

(3.11)d� 2( p) gSE

:;+&=}2?

960(2?)4 [the same as in Eq. (3.8)] d� ( p)m .

Here, d� ( p)m=d� ( p) (+)m &d� ( p) (&)

m =- 1&(4m2�p2) 3( p2&4m2) sgn( p0). The d2 -dis-tribution can be recast into the form

d� 2( p) gSE:;+& = :

3

i=1

d� ( p) (i):;+&

=}2?

960(2?)4

=: (

_P� ( p):;+&+m2

p2 Q� ( p):;+&+m4

p4 R� ( p):;+&& d� ( p)m , (3.12)


where the polynomials of degree four are given by their coefficients

P� ( p):;+& =[&8, &4, +1, &1, &1],

Q� ( p):;+&=[&16, &8, &8, +8, &12], (3.13)

R� ( p):;+&=[&48, &24, +16, &16, +4],

according to the structure given in Eq. (3.8).In the case of massless (m=0) matter coupling, that is, the first-order matter

interaction is chosen to be T M1 (x)=i(}�2) :h:;(x) ,(x) , : ,(x) , ; : , then the d2 -dis-

tribution reads

d� 2( p)m=0:;+& =(P� ( p):;+& 3( p2) sgn( p0). (3.14)

Hence, the limit m � 0 of (3.12) is feasible without problems; see Eq. (3.48) for thesplitting in the m=0 case.

The extension to nonminimally coupled massless matter is also considered. From

LM= 12 - & g g+&,; +,; &+ 1

12 - & g R,2, (3.15)

we derive the first order matter coupling

T M1 (x)=i

}2 {

23

:h:;, , :, , ; : &16

:h, , _ , , _ : &13

:h:;,, , :; : = , (3.16)

which yields (see [16, 17] for the calculations in the m=0 case)

d� 2( p)non-min.:;+& =([&12

9 , &69 , 1, &1, 6

9] 3( p2) sgn( p0). (3.17)

The corresponding t2 -distribution will be given in Eq. (3.49).

3.2. Distribution Splitting for the Graviton Self-Energy Matter LoopWe turn now to the splitting of the D2 -distribution of Eq. (3.12). The leading

singular order is four because the polynomials are of degree four in p. However,since these polynomials act in configuration space as derivatives, the essential struc-ture of the distributions is given by the scalar part. Therefore, neglecting the poly-nomials, the first, second, and third term in (3.12) have singular order 0, &2, and&4, respectively, due to the inverse powers of p. When discussing the normalizationN2(x, y), we will realize that this was the correct choice.

As anticipated in Section 2.1, a retarded part of d2 is obtained in momentumspace by the integral [10]

rc( p0)=i

2?p|+1

0 |+�

&�dk0

d� (k0)(k0&i0)|+1 ( p0&k0+i0)

, (3.18)

for p&( p0 , 0), p0>0. This retarded part is the so-called ``central splitting solution''because the subtraction point is the origin.

164 NICOLA GRILLO

The behavior of the first term in (3.12) is dictated by a scalar distribution of theform

d� (k)(1)=�1&4m2

k2 3(k2&4m2) sgn(k0)=: f (k2) sgn(k0). (3.19)

Therefore, we have to split d� (k) with |(d� )=0, because d� (k0)tconstant for|k0 | � � and k&=(k0 , 0) # V +. From (3.18) we obtain

rc( p0)=i

2?p0 |

+�

&�dk0

f (k20) sgn(k0)

(k0&i0)( p0&k0+i0)

=i

2?p0 _|

�

0dk0

f (k20)

k0( p0&k0+i0)&|

0

&�dk0

f (k20)

k0( p0&k0+i0)&=

i2?

p0 |�

0dk0

f (k20)

k0 \ 1p0&k0+i0

+1

p0+k0+i0+=

i2?

p0 |�

0dk0

f (k20)

k20

2p0k0

p20&k2

0+ip00. (3.20)

With the new variable s :=k20 we find

rc( p0)=i

2?p2

0 |�

0ds

f (s)s( p2

0&s+ip0 0). (3.21)

Inserting the explicit form of f (s) we have

rc( p0)=i

2?p2

0 |�

4m2ds �1&

4m2

s1

s( p20&s+ip00)

. (3.22)

We decompose the integral into real and imaginary parts according to (x+i0)&1=P(x&1)&i?$(x),

rc( p0)=i

2?p2

0 P |�

4m2ds �1&

4m2

s1

s( p20&s)

+12 �1&

4m2

p20

3( p20&4m2) sgn( p0). (3.23)

The T2(x, y)-distribution is obtained from the retarded part R2(x, y) by subtractingR$2(x, y). This subtraction affects only the numerical distributions. Therefore,subtracting

r$( p0)=&�1&4m2

p20

3( p20&4m2) 3(&p0) (3.24)


from (3.23), we obtain the numerical t-distribution belonging to T2(x, y),

t( p0)= rc( p0)&r$( p0)

=i

2?p2

0P |�

4m2ds �1&

4m2

s1

s( p20&s)

+12 �1&

4m2

p20

3( p20&4m2), (3.25)

which can be written in the form

t( p0)=i

2?p2

0 |�

4m2ds �1&

4m2

s1

s( p20&s+i0)

. (3.26)

This result can be generalized for p # V + by introducing the ``inverse momentum''q :=4m2�p2 so that

t( p)=i

2? |�

qds

- s(s&q)s2(1&s+i0)

=:i

2?6� ( p2). (3.27)

Note that, we write for simplicity the p-dependence instead of the q-dependence ofthe basic integral 6� which remains to be calculated. Therefore, the splitting of thefirst term in (3.12) and the subtraction of r$( p) (1)

:;+& yields

t( p) (1):;+&=i5P� ( p):;+& 6� ( p2). (3.28)

Here, 5 :=(�(2?)=}2?�(960(2?)5).Following the same steps as from Eq. (3.20) to Eq. (3.28), we find for the second

term in (3.12) that

t( p) (2):;+&=i5

m2

p2 Q� ( p):;+& 6� ( p2), (3.29)

because in d� ( p) (2):;+&=m2(Q� ( p):;+& d� ( p) (2) the scalar part reads

d� ( p) (2)=1p2 �1&

4m2

p2 3( p2&4m2) sgn( p0). (3.30)

Analogously, for the third term of (3.12) we obtain

t( p) (3):;+&=i5

m4

p4 R� ( p):;+& J� ( p2), (3.31)

because in d� ( p) (3):;+&=m4(R� ( p):;+& d� ( p) (3) the scalar part reads

d� ( p) (3)=1p4 �1&

4m2

p2 3( p2&4m2) sgn( p0), (3.32)

166 NICOLA GRILLO

with the definition

J� ( p2) :=|�

qds

- s(s&q)s3(1&s+i0)

. (3.33)

The difference between J� ( p2) and 6� ( p2) lies in the powers of s in the denominatorsin Eqs. (3.27) and (3.33). This is a consequence of splitting these distributionsaccording to the singular orders of the corresponding scalar distributions.

The two integrals 6� ( p2) and J� ( p2) have the same structure and the last can beexpressed by means of the first. We decompose 6� ( p2) into real and imaginaryparts,

6� ( p2)=P |�

qds

- s(s&q)s2(1&s)

&i? - 1&q 3(1&q), (3.34)

and concentrate our attention on the principal value part. With the substitution1

s(s&q)=(s&x)2 the real part of 6� ( p2) reads

6� r( p2)=&2P |�

qdx

(x&q)2

x2(x2&2x+q)

=&2P |�

qdx { q

x2+1&q

x2&2x+q= , (3.35)

having factorized the integrand. The J� ( p2)-integral yields a real part

J� r( p2)=&2P |�

qdx

(x&q)2 (2x&q)x4(x2&2x+q)

=&2P |�

qdx {q&1

x2 +2qx3&

q3

x4+1&q

x2&2x+q= . (3.36)

A look at Eqs. (3.35) and (3.36) enables us to isolate in the expression for J� r( p2)the terms appearing also in (3.35). The others can be easily integrated and weobtain

J� r( p2)=2

3q+6� r( p2)=

p2

6m2+6� r( p2), (3.37)


1 We choose x(s)=s+- s(s&q), so that x(s) goes from q to � for s going also from q to �.

The imaginary parts always have the same form as in Eq. (3.34). Gathering theresults in (3.28), (3.29), and (3.31) together with (3.37), we can write the distribu-tion describing the matter loop graviton self-energy,

t2( p) gSE:;+& = :

3

i=1

t( p) (i):;+&

=i5 _{P� ( p):;+&+m2

p2 Q� ( p):;+&+m4

p2 R� ( p):;+&= 6� ( p2)

+m2

6p2 R� ( p):;+&& . (3.38)

Therefore, the two-point operator-valued distribution T2(x, y) for the graviton self-energy reads

T2(x, y) gSE=:h:;(x) h+&( y): i6(x& y):;+& , (3.39)

where 6(x& y):;+& is the graviton self-energy tensor. Its Fourier representation isgiven by &i_(3.38).

We still must calculate explicitly the integral representation for 6� ( p2), given inEq. (3.27). There are three different regimes, depending on the value of q. For q=1,namely p2=4m2, we obtain by means of the partial decomposition (3.35),

6� a( p2=4m2)=&2 |�

1dx

1x2=&2. (3.40)

For q<1, namely, p2>4m2, the integration of the partial decomposition (3.35),taking into account also the imaginary part from (3.34), yields

6� b( p2)=&2+- 1&q log } q&1&- 1&q

q&1+- 1&q }&i? - 1&q 3(1&q). (3.41)

For q>1, namely 0<p2<4m2, the integration of (3.34) gives

6� c( p2)=&2+2 - q&1 \?2

&arc tanq&1

q&1+=&2+2 - q&1 arc tan \ 1

- q&1+ . (3.42)

Note that these three results are connected by

limp2z4m2

6� b( p2)=6� a( p2=4m2), limp2Z4m2

6� c( p2)=6� a( p2=4m2). (3.43)

168 NICOLA GRILLO

Writing the p-dependence explicitly, the final form for 6� ( p2) is

6� ( p2)=&2&{+�1&4m2

p2 _log }1&�1&

4m2

p2

1+�1&4m2

p2}+i?& 3( p2&4m2)

&2 �4m2

p2 &1 arc tan1

�4m2

p2 &1

3(4m2& p2)= . (3.44)

Two limits of this result will be used in the discussion of the normalization N2 ofthe T2 -distribution (Section 3.4), the limit of 6� ( p2) for p2z0 and the limit of6� ( p2)�p2 for p2z0, too. In the first case we have

limp2z0

6� ( p2)= limq � �

6� c( p2)

= limq � � {&2+2 - q&1 \?

2&\?

2&

1

q&1+O \ 1

q3�2+++== lim

q � � {&2+2&2

3(q&1)+O \ 1

q2+==0. (3.45)

For the second limit, we obtain

pp2z0

6� ( p2)p2 = lim

q � �

q6� c( p2)4m2

= limq � �

q4m2 {&2+2 - q&1 \ 1

- q&1&

1

3 - q&13+O \ 1

q5�2++==

&16m2 . (3.46)

The last consideration concerns the retarded part in Eq. (3.23), given also by(3.44) up to the step-function in p0 : this retarded part is the boundary value of theanalytic function

r( p)an=&2&�1&4m2

p2log

�1&4m2

p2 &1

�1&4m2

p2 +1

. (3.47)


Summing up the whole calculation, we have found the two-point distribution(3.39) for the graviton self-energy contribution. For the corresponding tensor, thestructure is given by (3.38) and the integral by (3.44).

During the calculation, we never resorted to a regularization of potentially UVdivergent expressions (for example, dimensional regularization as in [3]). This wasmade possible by using the correct starting point, namely Eq. (3.18), which is, soto say, a careful multiplication by a step-function in the time argument. If this hadbeen done naively, then it would have corresponded to the choice |=&1 in (3.18)when splitting the first term of Eq. (3.12), a choice which is manifestly wrong, |being =0.

Choosing |=&1 in Eq. (3.20), one obtains a UV logarithmic divergence.For the sake of completeness, we briefly report also the results in the case of

massless matter coupling, Eq. (3.14), and in the case of nonminimally coupledmassless matter, Eq. (3.17).

The splitting of the scalar distribution 3( p2) sgn( p0) requires some modificationsif one tries to use the splitting formula (3.18); see [11, 16]. The retarded part isgiven by (i�2?) log((&p2&ip0 0)�M2), so that the m=0 matter self-energy tensorreads

6� ( p)m=0:;+& =5P� ( p):;+& log \& p2&i0

M 2 + , (3.48)

where M>0 is a scale invariance breaking normalization constant and not a cutoff.For the nonminimally coupled case we find analogously

6� ( p)non-min.:;+& =5 _&

129

, &69

, 1, &1,69& log \& p2&i0

M2 + . (3.49)

This graviton self-energy tensor is traceless: ':;6� ( p)non-min.:;+& =0, because in this case

the graviton is coupled to a traceless matter�energy-momentum tensor. The lattercorresponds to the so-called improved energy-momentum tensor [22].

In addition, it is transversal: p:6� ( p)non-min.:;+& =0. This property follows from the

gauge identity (2.31), namely b:;\_p_6� ( p):;+&=0 (see Section 3.3), and from itsvanishing trace.

In these two cases we have found graviton self-energy contributions withoutintroducing counterterms. This is in contrast to the calculations carried out formassless scalar matter fields coupled to QG in the background field method withdimensional regularization [4, 5].

3.3. Graviton Self-Energy Tensor and Perturbative Gauge Invariance

The gauge properties of T2(x, y) gSE are contained in the identity coming fromEq. (2.31): b:;\_�x

_6(x& y):;+&=0. This identity implies the conditions

A&2B=0, C+E=0, B&2E&2F=0, (3.50)

170 NICOLA GRILLO

for the coefficients of the self-energy tensor. These conditions are satisfied by ourresult of Eq. (3.9) and therefore 6(x& y):;+& is gauge invariant. This is certainly thecase at the level of the D2(x, y) gSE-distribution before distribution splitting. In thecausal scheme, UV finiteness and gauge invariance are established separately. Thelatter is not used to reach the former. The identity (2.31) implies the Slavnov�Wardidentities (SWI) for the two-point connected Green function. The latter is definedas

G� ( p)[2]:;+& :=b:;#$D� F

0( p) 6� ( p)#$\_b\_+& D� F0( p), (3.51)

where D� F0( p)=(2?)&2 (&p2&i0)&1 is the scalar Feynman propagator. The two

attached lines represent free graviton Feynman propagators. The SWI reads [3, 14]

p:G� ( p)[2]:;+&=0, (3.52)

namely, that the two-point connected Green function is transversal. In terms of thecoefficients A, ..., F as in Eq. (3.8) we have

A&2B=0, C+E=0,A4

+C&F=0. (3.53)

These are equivalent to the conditions (3.50). This conclusion is valid also in themassless case, Eqs. (3.48) and (3.49).

In [23], the gauge invariance of the massless matter loop graviton self-energytensor also was investigated, but there it was not realized that the correct mattercoupling is the one in Eq. (2.14), namely, that with the b-tensor, when one uses theGoldberg variable expansion. This deficiency does not have consequences if thegraviton is coupled to a traceless matter energy-momentum tensor as in the non-minimal coupling case, Eq. (3.16).

The condition of perturbative gauge invariance to second order in the loop graphsector,

dQT2(x, y) gSE=�x_(:u\(x) h+&( y): [b:;\_6(x& y):;+&])+(x W y)

=�x_(:u\(x) h+&( y): [t_

uh(x& y)\+&+'\_tuh(x& y)+&])+(x W y),

(3.54)

has been explicitly checked by calculating also the distributions t_uh(x& y)\+& and

tuh(x& y)+& with one Q-vertex from Eq. (2.27).

3.4. Reduction of the Freedom in the Normalization

We turn now to the normalization of the T2-distribution of Eq. (3.39). The totalsingular order is four because the polynomials are of degree four in p. Therefore, wehave to add normalization terms up to the singular order four.


The freedom in the normalization due to the splitting procedure is contained inthe local term N2(x, y) gSE,

N2(x, y) gSE=:h:;(x) h+&( y): iN(�x , �y):;+& $(4)(x& y). (3.55)

From (2.5), we can write in momentum space this normalization as a sum ofpolynomials of degree 2i�4=|,

N� ( p):;+&= :2

i=0

N� ( p) (2i):;+& . (3.56)

Normalization terms with odd | are absent due to parity and Lorentz covariance.Gauge invariance b:;\_p_N� ( p) (2i)

:;+&=0 (i=0, 1, 2) and symmetries reduce thefreedom in the normalization in such a way that the polynomials have to be of theform

N� ( p) (0):;+&=0, N� ( p) (2)

:;+&=5[0, 0, &a, a, &a]1p2 ,

(3.57)

N� ( p) (4):;+&=5[4(b+c), 2(b+c), &b, b, c],

in the usual representation given by Eq. (3.8). The constants a, b, c # R should befixed by requiring the appropriate mass- and coupling constant-normalizations forthe corrections of order }2 to the graviton propagator. Letting formally g � 1 inEq. (2.1), we write the order }2 corrected propagator as

&iD(x& y)[2]:;+& =&ib:;+&DF

0(x& y)+| d 4x1 d 4x2(&ib:;\_ DF0(x&x1))

_i(6(x1&x2)\_#$+N(x1&x2)\_#$)(&ib#$+& DF0(x2& y)). (3.58)

In momentum space, this becomes

D� ( p)[2]:;+& =&

b:;+&

(2?)2 ( p2+i0)

+&1

( p2+i0)b:;\_(6� +N� )( p)\_#$ b#$+&

&1( p2+i0)

. (3.59)

=: >� (p):;+&

After a little work, we find in the form of Eq. (3.8)

6� ( p):;+&=5[ fA( p2), fB( p2), fC( p2), fE ( p2), fF ( p2)], (3.60)

172 NICOLA GRILLO

with

fA( p2)=\&8&16m2

p2 &48m4

p4 + 6� ( p2)&48m2

6p2 +4(b+c),

fB( p2)=\+6+32m2

p2 +16m4

p4 + 6� ( p2)+16m2

6p2 &2b&4c+2ap2

= & fF ( p2), (3.61)

fC( p2)=\+1&8m2

p2 +16m4

p4 + 6� ( p2)+16m2

6p2 &b&ap2

= & fE ( p2).

With the formula (if g( p2)t}2)

1& p2&i0

�1

& p2&i0+

1& p2&i0

g( p2)1

& p2&i0

=1

& p2&i0& g( p2)+O(}4), (3.62)

we obtain the order }2 corrected graviton propagator

D� ( p)[2]:;+& =

1(2?)2 _1

2(':+ ';&+':&';+)

1& p2&i0&2(2?)2 5p4fE ( p2)

&12

(':;'+&)1

& p2&i0+2(2?)2 5p4fF ( p2)&+ } } } , (3.63)

where terms that do not contribute between conserved matter�energy-momentumtensors have been neglected. The corrected propagator above has the correct limitlim} � 0 D� ( p)[2]

:;+&=b:;+& D� F0( p).

Mass normalization (the graviton mass remains zero under quantum corrections)yields

p4fE ( p2)| p2=0=0= p4fF ( p2)| p2=0 . (3.64)

Since 6� ( p2=0)=0 from Eq. (3.45), these conditions always hold.Coupling constant normalization (} is not shifted by the quantum corrections)

implies

p2fE ( p2)| p2=0=0= p2fF ( p2)| p2=0 . (3.65)


Analysis of the first condition,

p2fE ( p2)| p2=0=&16m4 6� ( p2)p2 } p2=0

&16m2

6+a=0, (3.66)

=&1�6 m2

yields a=0. Analysis of the second condition,

p2fF ( p2)|p2=0=&16m4 6� ( p2)p2 }p2=0

&16m2

6&2a=0, (3.67)

=&1�6 m2

yields also a=0. The compensation between the first two terms in (3.66) and (3.67)is decisive. This is due to the presence of the term (m2�6p2) R� ( p):;+& in Eq. (3.38).

A remark about the splitting is appropriate: if we had split the distributionsd� 2( p) (i)

:;+& , i=2, 3, in Eq. (3.12) according to their true singular orders, namely 2and 0 (because of the presence of the polynomials), respectively, then the term(m2�6p2) R� ( p):;+& would have been missing from (3.38). Working out the conse-quences for what concerns the normalization question, Eq. (3.66) would haverequired the choice a=&8m2�3, whereas Eq. (3.67) would have required the choicea=4m2�3. This would have meant the impossibility of a consistent normalization.Therefore, the splitting, as carried out in Section 3.2, is justified.

The origin of the above mentioned problem lies in the fact that the central split-ting solution (3.18) is not applicable in that case and one has to choose a subtrac-tion point, different from the origin.

The remaining constants b and c are not fixed by these requirements. The totalgraviton self-energy tensor including its normalization then has the form

6� ( p) tot:;+& =5 _{P� ( p):;+&+

m2

p2 Q� ( p):;+&+m4

p4 R� ( p:;+&)= 6� ( p2)

+m2

6p2 R� ( p):;+&+ :2

i=1

ziZ� ( p) (i):;+&& , (3.68)

where zi # R, i=1, 2, are still undetermined constants. The Z� ( p) (i):;+& 's are basis

elements in the two-dimensional space of gauge invariant polynomials of degreefour. They can be chosen to be Z� ( p) (1)

:;+&=[4, 2, &1, 1, 0] and Z� ( p) (2):;+&=

[4, 2, 0, 0, 1].Analysis of the issue of normalization with the method used in [16, 17] leads to

the same conclusions.

174 NICOLA GRILLO

4. MATTER SELF-ENERGY

The two-point distribution describing the matter self-energy graph is not interest-ing from the point of view of its gauge properties. However, calculation of thecorresponding distribution is carried out in order to investigate within theEpstein�Glaser scheme its UV structure.

4.1. Causal D2(x, y)-Distribution and Distribution SplittingThe D2 -distribution is obtained here by performing one matter field contraction

and one graviton contraction (3.1). The result reads

D2(x, y)mSE= +:,(x) , : ,( y) , :: da(x& y)+:,(x) ,( y) , : : db(x& y):

+:,(x) , : ,( y): dc(x& y):+:,(x) ,( y): dd (x& y), (4.1)

where the numerical distributions

da(x& y) :=}2m2

2(C (+)

} | } &C (&)} | } )(x& y),

db(x& y): :=&}2m2

2(C (+)

} | :&C (&)} | :)(x& y), (4.2)

dc(x& y): :=}2m2

2(C (+)

} | :&C (&)} | :)(x& y),

and

dd (x& y) := &}2m4(C (+)} | } &C (&)

} | } )(x& y),

are expressed in terms of the C (\)} } } -functions,

C (\)} | } (x) :=D (\)

0 (x) } D (\)m (x),

(4.3)C (\)

} | :(x) :=D (\)0 (x) } �x

: D (\)m (x).

These products are calculated in momentum space, see Appendix B, so that thedistributions di , i=a, b, c, d, read

d� a( p)=&}2m2?4(2?)4 \1&

m2

p2 + 3( p2&m2) sgn( p0),

d� b( p):=&i}2m2?

8(2?)4 \1&m4

p4+ p:3( p2&m2) sgn( p0)=&d� c( p):, (4.4)

d� d ( p)=}2m4?2(2?)4 \1&

m2

p2 + 3( p2&m2) sgn( p0).


From power-counting arguments, one could expect that d� a will behave as p2 forlarge momenta. This is not the case, because the wave equation (g+m2)D (\)

m (x)=0 lowers the power of p coming from the product of contractions. In order toshorten the calculation, we bring D2(x, y)mSE into the form

D2(x, y)mSE=:,(x) ,( y): d2(x& y)mSE+divergences, (4.5)

with

d� 2( p)mSE=+ p2d� a( p)&ip:d� b( p):+ip:d� c( p):+d� d ( p)

=&}2m2?2(2?)4

=: 1

_p2&3m2

2+

m4

2p2& 3( p2&m2) sgn( p0). (4.6)

Truly, this simplification can only be made for the corresponding T2-distribution,because divergences do not formally contribute in the adiabatic limit g � 1 ofEq. (2.1). Therefore, one should split the four distributions di , i=a, b, c, d,separately and then recast the ti 's in a form similar to Eq. (4.5) for the T2 -distribu-tion. The final result would be the same. Note that in the m=0 caseD2(x, y)mSE=0.

The splitting of (4.6) is accomplished by means of the splitting formula (3.18)with |(dmSE

2 )=2,

rc( p0)mSE=i

2?p3

0 |+�

&�dk0

d� 2(k0)mSE

(k0&i0)3 ( p0&k0+i0), (4.7)

for p&=( p0 , 0), p0>0. The retarded part then reads

rc( p0)mSE=i12?

p30 |

+�

&�dk0

3(k20&m2) sgn(k0)

k0( p0&k0+i0) {1&3m2

2k20

+m4

2k40= . (4.8)

With s=k20 , ds=2k0 dk0 , we obtain

rc( p0)mSE=i12?

p40 |

�

0ds

3(s&m2)( p2

0&s+ip00) {1s

&3m2

2s2 +m4

2s3= , (4.9)

which can be decomposed into real and imaginary parts,

rc( p0)mSE=i12?

p40P |

�

0ds

3(s&m2)( p2

0&s) {1s&

3m2

2s2 +m4

2s3=+

12

3( p20&m2) sgn( p0) {p2

0&3m2

2+

m4

2p20= . (4.10)

176 NICOLA GRILLO

Subtracting the distribution

r$2( p0)mSE=&1 _p20&

3m2

2+

m4

2p20& 3( p2

0&m2) 3(&p0) (4.11)

(coming from R$2(x, y)mSE) from rc( p0)mSE, we find the two-point distribution

t2( p0)mSE= rc( p0)mSE& r$2( p0)mSE

=i12?

p40 |

�

0ds

3(s&m2)( p2

0&s+i0) {1s

&3m2

2s2 +m4

2s3==: i7� ( p0). (4.12)

As a next task, we compute the integral of the principal value part of (4.12),denoted by X( p0),

X( p0)=&i14?

p40P |

�

m2ds

2s2&3m2s+m4

s3(s& p20)

=&i14?

p40P |

�

m2ds {:

s+

;s2+

#s3+

&:s& p2

0= , (4.13)

with

: :=3m2

p40

&m4

p60

&2

p20

, ; :=3m2

p20

&m4

p40

, # := &m4

p20

. (4.14)

Integration of the partial fractions in (4.13) yields

X( p0)=i12? _\p2

0&3m2

2+

m4

2p20+ log } p

20&m2

m2 }+m2

2&

5p20

4 & . (4.15)

4.2. Matter Self-Energy Two-Point Distribution and Freedom in Its Normalization

From (4.5) with (4.12) and (4.15), we can derive the two-point distribution forthe matter self-energy

T2(x, y)mSE=:,(x) ,( y): i7(x& y), (4.16)

and in an arbitrary Lorentz system the matter self-energy distribution for p # V + is

7� ( p)=12? _\p2&

3m2

2+

m4

2p2+_log } p2&m2

m2 }&i?3( p2&m2)&+

m2

2&

5p2

4 & . (4.17)


This loop contribution was also calculated in [24] within the operatorregularization scheme. Parts of their result agree with our expression in (4.17),whereas differences concern the explicit presence of parameters, other p-dependentlogarithms, and terms which go as p4 in their expression for 7� ( p). In the causalscheme, these latter cannot appear, because the singular order remains the sameafter distribution splitting.

The retarded part in (4.10) is the boundary value of the analytic function of com-plex momentum p+i', '=(=, 0), =>0:

r( p)an=i12? _\p2&

3m2

2+

m4

2p2+ log \1&m2

p2 ++m2

2&

5p2

4 & . (4.18)

Having split dmSE2 with |=2, an ambiguity in the normalization of 7� ( p) of the type

N� ( p)=12?

(c0+c2 p2) (4.19)

must be taken into account. In order to fix the constants c0 and c2 , radiativecorrections to the matter Feynman propagator by matter self-energy loops areconsidered: formally letting g � 1, they are of the form

&iDFm(x& y)+| d 4x1 d 4x2(&iDF

m(x&x1))

_i[7(x1&x2)+N(x1&x2)](&iDFm(x2& y))+ } } } . (4.20)

In momentum space the series becomes

D� Fm( p)+D� F

m( p)(2?)4 (7� ( p)+N� ( p)) D� Fm( p)+D� F

m( p)

_(2?)4 (7� ( p)+N� ( p)) D� Fm( p)(2?)4 (7� ( p)+N� ( p)) D� F

m( p)+ } } }

=: 7� ( p)tot, (4.21)

where D� Fm( p)=(2?)&2 (&p2+m2&i0)&1. The geometric series in (4.21) leads to

7� ( p)tot=D� Fm( p)(1+(2?)4 (7� ( p)+N� ( p)) 7� ( p)tot)

=1

(2?)2

1& p2+m2&i0&(2?)2 (7� ( p)+N� ( p))

. (4.22)

Mass corrections are avoided by requiring

[7� ( p)+N� ( p)]|p2=m2=0. (4.23)

178 NICOLA GRILLO

This implies

c0=m2( 34&c2), (4.24)

so that the matter self-energy distribution including its normalization reads

7� ( p)+N� ( p)=12? _\p2&

3m2

2+

m4

2p2+ _log } p2&m2

m2 }&i?3( p2&m2)&+( p2&m2) \c2&

54+& . (4.25)

For the remaining constant c2 , we cannot provide a value here. A restriction shouldcome from the vertex corrections 4� ( p, q):; to third order in the three-point dis-tribution T3(x, y, z)=:,(x) ,( y) h:;(z): 4(x, y, z):; .

5. PERTURBATIVE GAUGE INVARIANCE TO SECOND ORDER IN THETREE GRAPH SECTOR

In this section we show that the condition of perturbative gauge invariance tosecond order in the tree graph sector generates a local quartic interaction of theform t}2 :hh,,: $(4)(x& y), which agrees with the second-order L (2)

M in the expan-sion of the matter Lagrangian density LM , Eq. (2.24).

5.1. MethodologySince R$2(x, y) is trivially gauge invariant due to its definition and the gauge

invariance of T1(x), instead of Eq. (2.20) for n=2 we have to examine whether

dQR2(x, y)+dQN2(x, y)=�x& R&

2�1(x, y)+� y& R&

2�2(x, y)

+�x& N &

2�1(x, y)+� y& N &

2�2(x, y) (5.1)

can be satisfied by a suitable choice of the free constants in the normalization termsN2 , N &

2�1 , and N &2�2 of the retarded parts R2 , R&

2�1 , and R&2�2 . Here, R&

2�1 and R&2�2 are

the retarded distributions obtained by splitting the inductively constructed distribu-tions

D&2�1(x, y) :=[T &

1�1(x), T1( y)] and D&2�2(x, y) :=[T1(x), T &

1�1( y)]. (5.2)

This procedure has been described in [12] for pure QG. It turned out that Eq. (5.1)can be spoiled by terms with point support t$(4)(x& y), only. N2 , N &

2�1 , and N &2�2

are by definition local terms, but there is another source of ``local anomalies,''namely the splitting procedure for tree graphs: in the inductive construction of R&

2�1 ,the Q-vertex T &

1�1 can give rise to expressions of the form

D&2�1(x, y)=:O(x, y): �&

xDm(x& y). (5.3)


:O(x, y): is a normally ordered product of four fields, being the other two fields inthe commutator (5.2). m can also be zero, if the commutator contains two gravitonfields. The retarded part is simply

R&2�1(x, y)=:O(x, y): �&

xDretm (x& y), (5.4)

because the trivial splitting for tree graphs follows from Dm=D retm &Dav

m with thecorrect support properties as explained in Section 2.1. Applying the derivative �x

& ,which forms the divergence in �x

& R&2�1 , we get

:O(x, y): gx Dretm (x& y)=:O(x, y): [$(4)(x& y)&m2D ret

m ]. (5.5)

The expression :O(x, y): $(4)(x& y) is a local anomaly. A corresponding mechanismworks for R&

2�2 also.We denote by an(�x

& R&2�1+� y

& R&2�2) the set of all local anomalies generated by the

described mechanism.Following [12], gauge invariance is preserved if we can choose N2 , N &

2�1 , andN &

2�2 so that the condition

dQN2(x, y)=an(�x& R&

2�1+� y& R&

2�2)+�x& N &

2�1(x, y)+� y& N &

2�2(x, y) (5.6)

involving the local terms of (5.1) is satisfied.Note that in QG coupled to matter dQR2 does not generate local terms with mat-

ter fields involved, namely of the type :uh,,: . This is in contrast, to the much moreinvolved pure QG case.

At this point we realize that, it is not sufficient to consider T M1 only. Also, the

graviton and ghost first-order couplings T h1 and T u

1 have to be taken into account,because they yield also local anomalies with external operators t:uh,,: whensplitting the commutators of (5.2).

Using a simplified notation which keeps track of the structure of the couplingonly, gauge invariance to first order then becomes

dQ(:hhh: +:u~ hu:+:h,,: )

T 1h+u+M

=�x&(:[uhh]&: +:[u~ uu]&: +:[u,,]&:)

T 1�1& h+u+T 1�1

& M

. (5.7)

With the ``extended'' Q-vertex T & h+u1�1 +T & M

1�1 , we decompose the commutator defin-ing D&

2�1 in the following way:

D&2�1(x, y)=pure QG sector+[:[uhh]&:, :h,,: ]

+[:[u,,]&:, :u~ hu: ]+[:[u,,]&:, :h,,: ]. (5.8)

In the pure QG sector, which involves terms of the type :uhhh: and :u~ uuh:, pertur-bative gauge invariance has been shown in [12].

180 NICOLA GRILLO

5.2. Explicit CalculationsOur task consists in the investigation of the three remaining sectors in Eq. (5.8)

in which matter and graviton fields are mixed together. We denote the three com-mutators in (5.8) as the graviton-, ghost-, and matter sectors, respectively.

Performing one contraction, they lead to

D& h2�1(x, y)=( :uh

x

,,:y

[h(x), h( y)])&,

D& u2�1(x, y)=( :,,

x

hu:y

[u(x), u~ ( y)])&, (5.9)

D& M2�1 (x, y)=( :u,

x

h,:y

[,(x), ,( y)])&,

respectively. By considering the explicit form of T u1 in [12] and T &M

1�1 in (2.19), wefind that no local anomalies arise in the ghost sector.

Let us compute the local anomalies in the graviton sector. From the expressionfor T & h+u

1�1 in [12] we isolate the terms that generate local anomalies according tothe mechanism described in Section 5.1. They are

T & h+u1�1 =}[+ 1

2 :u+, +h\_h\_

, & : +12 :u+h\_

, + h\_, & : &1

4 :u+, +hh , & :

& 14 :u+h , +h , & : +1

2 :u+, \h+\h , & : &:u+

, \ h\_h_+, & : + } } } ]. (5.10)

Then D& h2�1 contains the following terms which generate anomalies:

D& h2�1(x, y)=

}2

4 {+:u+, +h\_, , _, , \ : &

m2

2:u+

, +h,,: +:u+h\_, + , , _, , \ :

&m2

2:u+h , +,,: &2 :u+

, \h\_, , _, , + : +m2 :u+, \h+\,,: =

_$&xD0(x& y). (5.11)

The first, two fields in the normally ordered products depend on x, whereas thematter fields depend on y. The retarded part R& h

2�1 has exactly the same form as in(5.11) with the replacement D0 � D ret

0 . Applying �x& to R& h

2�1 (5.11), we obtain thelocal anomalies (local anomalies coming from � y

& R& h2�2 are just the same, therefore

we get a factor two)

an(�x& R& h

2�1+� y& R& h

2�2)=}2

2 _:u+, +h\_, , _, , \ : &

m2

2:u+

, +h,,:

+:u+h\_, + , , _, , \ : &

m2

2:u+h , +,,:

&2 :u+, \h\_, , _, , + : +m2 :u+

, \h+\,,: & $(4)(x& y). (5.12)

Because of the $-function, all fields now have the same space-time dependence.


In the matter sector, the first term in the matter Q-vertex of Eq. (2.19) is the onlyone that can generate local anomalies, because it carries the &-index as a derivative.Then, D& M

2�1 becomes

D& M2�1 (x, y)=&

}2

2 {:u+, , +h\_, , \ : �x_+

m2

2:u+, , +h,: = �&

xDm(x& y). (5.13)

The contribution coming from D& M2�2 has x W y and the derivative attached to the

first term is � y_ . The retarded part R&M

2�1 has the same form as in (5.13) with thereplacement D0 � D ret

0 . Since

:A(x) B( y): �x: $(4)(x& y)+:A( y) B(x): � y

: $(4)(x& y)

=&:A(x) , : B(x): $(4)(x& y)+:A(x) B(x) , : : $(4)(x& y), (5.14)

by applying �&x to R&M

2�1 and �&y to R& M

2�2 we obtain

an(�x& R& M

2�1 +� y& R& M

2�1 )

=}2

2[&m2 :u+h, , +,: +:u+

, _h\_, , \, , + :

+:u+h\_, , \, , +_ : &:u+h\_, _ , , \ , , + : &:u+h\_, , \_, , + : ] $(4)(x& y). (5.15)

Because of the $-function, all fields now have the same space-time dependence andhave been recast in the form :uh,,: .

5.3. Quartic Normalization Terms

Summing up, we have found all the local anomalies arising from �x& R&

2�1+� y& R&

2�2

attached to normally ordered products of the type :uh,,: . These can be organizedinto three different types according to their Lorentz structure apart fromderivatives: type I, :u+h\_,,: with derivatives �+ , �\ , �_ ; type II, :u+h,,: withderivatives �+ , �\ , �\ ; and type III, :u+h+\,,: with derivatives �\ , �_ , �_ . DifferentLorentz types do not interfere.

From Eqs. (5.12) and (5.15), the local anomalies of type I are

an(�x& R&

2�1+� y& R&

2�2)| type I =}2

2[+:u+

, +h\_, , \ , , _ : +:u+h\_, + , , \, , _ :

+:u+h\_, , \, , +_ : &:u+, \h\_, , \ ,, + :

&:u+h\_, _ , , \, , + : &:u+h\_, , \_ , , + : ] $(4)(x& y). (5.16)

182 NICOLA GRILLO

From Eqs. (5.12) and (5.15), the local anomalies of type II are

an(�x& R&

2�1+� y& R&

2�2)| type II =&}2

2 _m2

2:u+

, +h,,: +m2

2:u+h , +,,:

+m2 :u+h, , +,: & $(4)(x& y), (5.17)

and those of type III are simply

an(�x& R&

2�1+� y& R&

2�2)type III=}2

2[m2 :u+

, \ h+\,,: ] $(4)(x& y). (5.18)

According to Eq. (5.6), the question is whether these local anomalies can bewritten as divergences and therefore can be compensated by corresponding localdivergence terms coming from �x

& N &2�1+� y

& N &2�2 . If it is not possible to reach such

a compensation, normalization terms on the left side of (5.6) have to be introducedin order to preserve gauge invariance.

Due to the identity

�x&(:A(x) B( y): $(4)(x& y))+� y

& (:A( y) B(x): $(4)(x& y))

=:A(x) , & B(x): $(4)(x& y)+:A(x) B(x) , & : $(4)(x& y), (5.19)

the local anomalies of type I can indeed be indeed written as a divergence:

Eq. (5.16) � +�x+(:u+h\_

x

, , \, , _

y

: $(4)(x& y))

+� y+(:u+h\_

y

, , \, , _

x

: $(4)(x& y))

&[�x\(:u+h\_

x

, , +, , _

y

: $(4)(x& y))+� y+(:u+h\_

y

, , +, , _ :x

$(4)(x& y))].

(5.20)

The same is true for the local anomalies of type II,

Eq. (5.17) � +m2�x+( :u+h

x

,,:y

$(4)(x& y))

+m2� y+( :u+h

y

,,:x

$(4)(x& y)). (5.21)

Thus, by choosing the appropriate local divergence normalization terms �x& N &

2�1+� y

& N &2�2 , Eq. (5.6) restricted to the Lorentz types I and II holds.

The local anomaly of type III, Eq. (5.18), cannot be written as a divergence.Equation (5.6) forces us to consider normalization terms N2 . Then, we require that


the local anomaly of type III has to be coboundary, namely, a gauge variation ofa normalization term N2 ,

dQN2(x, y)=}2m2

2:u+(x) , \ h+\(x) ,(x) ,(x): $(4)(x& y). (5.22)

The only possible N2 that satisfies this requirement is

N2(x, y)=i}2m2

4 {:h:;h:;,,: &12

:hh,,: = $(4)(x& y). (5.23)

Taking the factor 12 for the second-order S-matrix expansion (2.1) into account, the

quartic interaction in (5.23), quadratic in }, generated by gauge invariance agreesexactly with the term of order }2 in the expansion of the matter Lagrangian densityLM in (2.24). But in our case, this mechanism of generation works in a purelyquantum framework.

The only objection to this result is that this N2 is not a ``proper'' normalizationof a tree graph in T2 , obtained starting with T h+u

1 and T M1 . But it can be con-

sidered as a normalization term of box-graphs in fourth order with external legst:hh,,: constructed with two T h+u

1 and two T M1 or with four T M

1 .

5.4. Massless Matter Case

For the massless matter coupling T M1 (x)=i(}�2) :h:;, , :, , ; : , perturbative gauge

invariance to first order reads

dQT M1 (x)=�x

& \}2 {:u\, , \, , & : &

12

:u&, , \ , , \ : =+=: �x& T & M

1�1 (x), (5.24)

namely, the matter Q-vertex of (2.19) in the m � 0 limit. Performing the samecalculation as for the massive case, it turns out that the local anomalies are thoseof type I only, Eq. (5.16) which can be written as a divergence and therefore com-pensated for by proper normalization terms �x

& N &2�1+� y

& N &2�2 as in (5.20). From

Eq. (5.6), it follows that dQN2=0, which is certainly satisfied by N2=0. This agreeswith the fact that even classically there are no hh,, couplings in the m=0 case: theexpansion of LM written in terms of the Goldberg variable reads

LM=12

, , \, , \+}2

h+&, , +, , & . (5.25)

Therefore, L (2)M =0. This concludes our discussion of the condition of perturbative

gauge invariance to second order for tree graphs.

184 NICOLA GRILLO

APPENDIX A: THE I (\)m (p) } } } -INTEGRALS

The I (\)m ( p) ... -integrals are defined by

I (\)m ( p)&�+�+&�+&\�+&\_ :=| d 4q $(( p&q)2&m2) 3(\( p0&q0))

_$(q2&m2) 3(\q0)[1, q+ , q+q& , q+q&q\ , q+ q&q\q_]. (A.1)

Let us investigate I (+)m ( p) in detail, because it contains the main feature of the

calculation of the other integrals. p is time-like due to the presence of the 3- and$-distributions in the integrand. Therefore, we may choose a Lorentz frame suchthat p&=( p0 , 0), p0>0, then

I (+)m ( p0)=| d 4q $( p2

0&2p0q0+q20&|q|2&m2) 3( p0&q0)

_$(q2&m2) 3(+q0)

=| d 3q$( p2

0&2p0Eq ) 3( p0&Eq )2Eq

, (A.2)

with Eq=q0=- q2+m2. Then

I (+)m ( p0)=4? |

�

0d |q|

|q| 2

2Eq

12p0

$ \p0

2&Eq+ 3( p0&Eq )

=?p0

3( p20&4m2) 3( p0) |

�

0d |q|

|q| 2

Eq$ \p0

2&Eq+ , (A.3)

because of the $-distribution

Eq =- q2+m2=p0

2O |q|=�p2

0

4&m2. (A.4)

Therefore

I (+)m ( p0)=

?p0

3( p20&4m2) 3( p0) �p2

0

4&m2

=?2 �1&

4m2

p20

3( p20&4m2) 3( p0). (A.5)


I (&)m can be calculated analogously and the result in an arbitrary Lorentz frame

reads

I (\)m ( p)=

?2 �1&

4m2

p2 3( p2&4m2) 3(\p0)=:?2

d� ( p) (\)m . (A.6)

Computing I (\)m ( p)+ for p&=( p0 , 0), p0>0, we have a nonvanishing contribution

only for +=0. An additional factor q0 is therefore present in the integrand in (A.3),which is set equal to Eq and then to p0 �2. This leads to

I (\)m ( p)+= p+

?4 �1&

4m2

p2 3( p2&4m2) 3(\p0)=?4

p+ d� ( p) (\)m , (A.7)

in an arbitrary Lorentz frame.For I (\)

m ( p)+& , we have to take the two conditions

I (\)m ( p)+

+=m2I (\)m ( p),

(A.8)p+p&I (\)

m ( p)+&=p4

4I (\)

m ( p),

into account. The first condition is imposed by the presence of $(q2&m2) in (A.1).Computing p+p&I (+)

m ( p)+& for p&=( p0 , 0), p0>0, we get an additional factor( p0Eq )2 in the integrand (A.3) which gives p4

0�4 due to $( p0 �2&Eq ). In a generalLorentz frame, the second condition in (A.8) must hold. Therefore, making theansatz

I (\)m ( p)+&=(a( p2) p+ p&+b( p2) p2'+&) I (\)

m ( p), (A.9)

the conditions (A.8) imply

a( p2)=13

&m2

3p2 , b( p2)=&112

+m2

3p2 , (A.10)

so that

I (\)m ( p)+&={\1&

m2

p2 + p+ p&&\14

&m2

p2 + p2'+&= ?6

d� ( p) (\)m . (A.11)

For I (\)m ( p)+&\ , we have to take the two conditions

I (\)m ( p)+&

&=m2I (\)m ( p)+ ,

(A.12)p+p&p\I (\)

m ( p)+&\=p6

8I (\)

m ( p)

186 NICOLA GRILLO

into account. The first condition follows from the definitions of the I (\)m -integrals.

Computing p+p&p\I (+)m ( p)+&\ for p&=( p0 , 0), p0>0, we obtain an additional factor

( p0Eq )3 in the integrand (A.3) which yields p60 �8. Making the ansatz

I (\)m ( p)+&\=(c( p2) p+ p& p\+d( p2) p2( p\'+&+ p+'\&+ p&'\+)) I (\)

m ( p), (A.13)

then Eqs. (A.12) imply

c( p2)=14

&m2

2p2 , c( p2)=&124

+m2

6p2 , (A.14)

and therefore

I (\)m ( p)+&\ ={+\1&

2m2

p2 + p+ p& p\

&\16

&2m2

3p2+ p2( p\'+&+ p+'\&+ p&'\+)= ?8

d� ( p) (\)m . (A.15)

For I(\)m( p)+&\_ , we repeat this calculational scheme again. For the reasonspointed out above, now three conditions,

I (\)m ( p)+&\

\=m2I (\)m ( p)+& ,

I (\)m ( p)+

+\

\=m4I (\)m ( p), (A.16)

and

p+p&p\p_I (\)m ( p)+&\_=

p8

16I (\)

m ( p),

must hold. Therefore, the ansatz

I (\)m ( p)+&\_ =(e( p2) p+ p& p\ p_+ f ( p2) p2( p\ p_'+&+ p\ p+'_&

+ p\ p&'_++ p+ p_'\&+ p& p_'\++ p+ p&'\_)

+ g( p2) p4('+\ '&_+'+_'&\+'+&'\_)) I (\)m ( p) (A.17)

leads to

e( p2)=15

&3m2

5p2 +m4

5p4 , f ( p2)=&140

+7m2

60p2&m4

15p4 ,

(A.18)

g( p2)=115 \

116

&m2

2p2+m4

p4 + ,


so that

I (\)m ( p)+&\_ ={\1&

3m2

p2 +m4

p4 + p+ p& p\ p_&\18

&7m2

12p2+m4

3p4+ p2

_( p\ p_'+&+ p\ p+'_&+ p\ p&'_++ p+ p_'\&+ p& p_'\++ p+ p& '\_)

+\ 148

&m2

6p2+m4

3p4+ p4('+\'&_+'+_ '&\+'+&'\_)= ?10

d� ( p) (\)m .

(A.19)

APPENDIX B: THE I� (\)(p) } } } -INTEGRAL

The products of Jordan�Pauli distributions of Eq. (4.3) are evaluated in momen-tum space,

C� (\)} | } ( p) =

1(2?)2 | d 4q D� (\)

0 ( p&q) D� (\)m (q)

=&1

(2?)4 | d 4q $(( p&q)2) 3($ p0&q0)) $(q2&m2) 3(\q0)

=:&1

(2?)4 I� ($( p) (B.1)

and

C� (\)} | :( p) =

1(2?)2 | d 4q D� (\)

0 ( p&q)(&iq:) D� (\)m (q)

=+i

(2?)4 | d 4q q: $(( p&q)2) 3($ p0&q0)) $(q2&m2) 3(\q0)

=:+i

(2?)4 I� ($( p): . (B.2)

Let us calculate I� (+)( p). From the $- and 3-distributions, it follows that p is time-like. We choose a Lorentz frame in which p&=( p0 , 0), p0>0. Then

I� (+)( p0)=| d 4q $( p20&2p0q0+q2

0&q2) 3( p0&q0) 3(q0)$(q0&Eq )

2Eq

=|d 3q2Eq

$( p20&2p0Eq +m2) 3( p0&Eq ), (B.3)

188 NICOLA GRILLO

with Eq=- q2+m2. The $-distribution implies p0=Eq +|q|. From E 2q =q2+m2=

( p0&|q| )2, we obtain |q|=( p20&m2)�2p0 , which, due to p0>0, yields 3( p2

0&m2)and

I� (+)( p0)=2?3( p20&m2) 3( p0) |

�

0d |q|

|q|2

Eq

$( p20&2p0Eq +m2)

=2?3( p20&m2) 3( p0) |

�

mdEq

|q|2p0

$ \Eq &p2

0+m2

2p0 +=

?p0

3( p20&m2) 3( p0) - E 2

q &m2 |Eq=( p20+m2)�2p0

=?

2p20

3( p20&m2) 3( p0)( p2

0&m2)

=?2

3( p20&m2) 3( p0) \1&

m2

p20 + . (B.4)

Therefore, in an arbitrary Lorentz frame,

I� (\)( p)=?2

3( p2&m2) 3(\p0) \1&m2

p2 + . (B.5)

The second integral can be computed in a similar manner: in the Lorentz framewith p&=( p0 , 0), p0>0, I� (+)( p0) i , i=1, 2, 3, vanishes for symmetry reasons. Then

I� (+)( p0)0 =|d 3q2Eq

Eq $( p20&2p0 Eq +m2) 3( p0&Eq )

=2?3( p20&m2) 3( p0) |

�

0d |q|

|q|2

2p0Eq

Eq $ \Eq &p2

0+m2

2p0 +=

?p0

3( p20&m2) 3( p0) |

�

mdEq |q| Eq $ \Eq &

p20+m2

2p0 +=

?p0

3( p20&m2) 3( p0)

p20&m2

2p0

p20+m2

2p0

=?4

p03( p20&m2) 3( p0) \1&

m2

p20 +\1+

m2

p20 + . (B.6)


Therefore, in an arbitrary Lorentz frame we have

I� (\)( p):=?4

p: \1&m4

p4 + 3( p2&m2) 3(\p0). (B.7)

ACKNOWLEDGMENT

I thank Professor G. Scharf, Adrian Mu� ller and Mark Wellmann for valuable discussions andcomments regarding these topics.

REFERENCES

1. R. P. Feynman, F. B. Morinigo, W. G. Wagner, and B. Hatfield, ``Feynman Lectures on Gravita-tion,'' Addison�Wesley, Reading, MA, 1995.

2. R. P. Feynman, Acta Phys. Polon. 24 (1963), 697.3. D. M. Capper, Nuovo Cim. A 25 (1975), 29.4. G. 't Hooft and M. Veltman, Ann. Poincare� Phys. Theor. A 20 (1974), 69.5. M. J. Veltman, in ``Les Houches, 1975: Proceedings, Methods in Fields Theory,'' pp. 265�327,

North-Holland, Amsterdam, 1976.6. S. Deser, P. Van Nieuwenhuizen, and D. Boulware, in ``Tel-Aviv, 1974, Proceedings, General

Relativity and Gravitation,'' pp. 1�18, Wiley, New York, 1975.7. E. C. G. Stu� ckelberg and D. Rivier, Helv. Phys. Acta 23 (1950), 215.8. N. N. Bogoliubov and D. V. Shirkov, ``Introduction to the Theory of Quantized Fields,'' Wiley, New

York, 3rd ed., 1976.9. H. Epstein and V. Glasser, Ann. Poincare� Phys. Theor. A 19 (1973), 211.

10. G. Scharf, ``Finite Quantum Electrodynamics: The Causal Approach,'' Springer, Berlin, 1995.11. M. Dutch, T. Hurth, F. Krahe, and G. Scharf, Nuovo Cim. A 106 (1993), 1029; Nuovo Cim. A 107

(1994), 375.12. I. Schorn, Class. Quant. Grav. 14 (1997), 653; Class. Quant. Grav. 14 (1997), 671.13. G. Scharf and M. Wellmann, General Relativity and Gravitation, preprint hep-th�9903055.14. D. M. Capper and M. Ramon Medrano, Phys. Rev. D 9 (1974), 1641.15. N. Grillo, preprint hep-th�9911118.16. N. Grillo, Classical and Quantum Gravity, preprint hep-th�9912097.17. N. Grillo, preprint hep-th�9912114.18. A. Aste, Ann. Phys. (N. Y.) 257 (1997), 158; preprint hep-th�9608193.19. W. Wyss, Helv. Phys. Acta 38 (1965), 469.20. R. M. Wald, Phys. Rev. D 33 (1986), 3613.21. M. Dutch, Int. J. Mod. Phys. A 12 (1997), 3205; preprint hep-th�9606105.22. C. G. Callan, S. Coleman, and R. Jackiw, Ann. Phys. (N. Y.) 59 (1970), 42.23. S. A. Zaidi, J. Phys. A: Math. Gen. 24 (1991), 4325.24. A. Y. Shiekh, Can. J. Phys. 74 (1986), 172.

190 NICOLA GRILLO

Scalar Matter Coupled to Quantum Gravity in the Causal Approach: One-Loop Calculations and...

Documents

Transcript of Scalar Matter Coupled to Quantum Gravity in the Causal Approach: One-Loop Calculations and...