Scalar Matter Coupled to Quantum Gravity in the Causal Approach

Annals of Physics 287, 153190 (2001) doi:10.1006aphy.2000.6104, available online at http:www.idealibrary.com on

Scalar Matter Coupled to Quantum Gravity in the Causal Approach One-Loop Calculations and Perturbative Gauge Invariance Nicola Grillo Institut fur Theoretische Physik, Universitat Zurich, Winterthurerstrasse 190, CH-8057 Zurich, Switzerland E-mail: grillophysik.unizh.ch Received February 10, 2000; revised August 8, 2000

Quantum gravity coupled to scalar massive matter fields is investigated in the framework of causal perturbation theory using the EpsteinGlaser regularizationrenormalization scheme. Detailed one-loop calculations include the matter loop graviton self-energy and the matter self-energy. The condition of perturbative operator gauge invariance to second order implies the usual SlavnovWard identities for the graviton two-point connected Green function in the loop graph sector and generates the correct quartic graviton-matter interaction in the tree graph 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 gravitational interactions coupled to scalar matter fields (see the introduction to this subject in [1] and the references therein). This approach allows a quantization of the involved fields, matter and graviton fields, and a Lorentz covariant perturbative expansion of the scattering matrix S. Calculations of matter loop diagrams in this conventional framework led to nonrenormalizable ultraviolet (UV) divergences [2]. These were later confirmed by means of dimensional regularization and back-ground field method, both in the massive [3] and in the massless [4, 5] case. The counterterms needed to cancel the divergences are not of the type present in the original Lagrangian density and therefore cannot be absorbed in the redefinition of physical quantities. According to these findings, quantum gravity (QG) coupled to matter fields does not fulfil the criterion of perturbative renormalizability [6]. We adopt here another approach to investigating these outcomes by applying an improved perturbation scheme which has as central objects the time-ordered products and constructing principle causality. In this natural regularizationrenormalization scheme, called causal perturbation theory, the S-matrix is constructed 153 0003-491601 35.00 Copyright  2001 by Academic Press All rights of reproduction in any form reserved.

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inductively as a sum of smeared operator-valued n-point distributions T n(x 1 , ..., x n ) avoiding UV divergences at each stage of the calculations as a consequence of a deeper mathematical understanding of how loop graph contributions have to be calculated. This idea goes back to Stuckelberg [7], Bogoliubov and Shirkov [8], and the program was carried out successfully by Epstein and Glaser [9] for scalar field theories and subsequently applied to QED by Scharf [10], to non-Abelian gauge theories by Dutsch et al. [11], and to quantum gravity [12] in the past few years. This calculation scheme exploits the concept of causality from the beginning and has a resemblance to the BPHZ-subtraction scheme. Quantum gravity, being a nonrenormalizable theory, unfortunately suffers in our approach from a nonuniqueness in the fixing of the normalization of the timeordered products, which endangers the physical predictability. Here, the nonrenormalizability problem of QG is translated into an increasing ambiguity in the normalization N n of the time-ordered products T n . In addition, QG has considerable gauge properties [13], which are formulated by means of a gauge charge that generates infinitesimal gauge variations of the fundamental free quantum fields (Section 2.3). The present work focuses mainly on three aspects of QG coupled to massive matter 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 order massive 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-free result are obtained using the techniques of causal perturbation theory for the S-matrix and the EpsteinGlaser regularizationrenormalization scheme without introducing nonrenormalizable counterterms. The inherent freedom in the normalization of the time-ordered products is to some extent discussed, but some local normalization terms remain undetermined (Section 3.4). The second aspect consists in the investigation of the gauge properties of the graviton self-energy (Section 2.4). Gauge invariance of the S-matrix implies some identities between the C-number parts of the n-point distributions which yield the gravitational SlavnovWard identities (SWI) [14] (Section 3.3). The third aspect of this work is also connected with gauge invariance: perturbative gauge invariance to second order in the tree graph sector requires the introduction, at a purely quantum level, of a quartic mattergraviton interaction exactly as prescribed by the expansion of the classical mattergravity 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 also the conventions and the notations used here. Calculations involving graviton selfcouplings are not considered here; see [16]. The causal scheme applied to quantum gravity coupled to photon fields leads also to analogous results with regard to loop calculations and gauge invariance [17].

SCALAR MATTER COUPLED TO QUANTUM GRAVITY

155

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 MATTERGRAVITY SYSTEM AND PERTURBATIVE GAUGE 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 power series in the coupling constant is central; namely, S is considered as a sum of smeared operator-valued distributions 

S(g)=1+ : n=1

1 d 4x 1 } } } d 4x n T n(x 1 , ..., x n ) g(x 1 ) } } } } } g(x n ), n!

|

(2.1)

where the Schwartz test function g # S(R 4 ) switches the interaction and provides a natural infrared cutoff. The S-matrix maps the asymptotically incoming free fields on the outgoing ones and it is possible to express the T n 's by means of free fields without introducing interacting quantum fields. Establishing the existence of the so-called adiabatic limit g Ä 1 in theories involving self-coupled massless particles, such as QG, may be very problematic, and there is evidence that the limit may not exist. This aspect will not be considered here. The n-point distribution T n is a well-defined renormalized time-ordered product expressed in terms of Wick monomials of free fields: O(x 1 , ..., x n ): T n(x 1 , ..., x n )=: :O(x 1 , ..., x n ): t On (x 1 &x n , ..., x n&1 &x n ).

(2.2)

O

The t n 's are C-number distributions. T n is constructed inductively from the first order T 1(x), which describes the interaction among the quantum fields, and from the lower orders T j , j=2, ..., n&1, by means of Poincare covariance and causality. The latter leads directly to a UV finite and cutoff-free distribution T n . For the purpose of this paper, we outline briefly the main steps in the construction of T 2(x, y) from a given first-order interaction. Following the inductive scheme, we first calculate the causal operator-valued distribution D 2(x, y) :=R$2(x, y)&A$2(x, y)=[T 1(x), T 1( y)].

(2.3)

In order to obtain D 2(x, y), one has to carry out all possible contractions between the normally ordered T 1 using Wick's lemma, so that D 2(x, y) has the structure D 2(x, y)=: :O(x, y): d O2 (x& y). O

(2.4)

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d O2 (x& y) is a numerical distribution that depends only on the relative coordinate x& y because of translation invariance. D 2(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. D 2(x, y) is causal, i.e., supp(d O2 (z))V +(z) _ V &(z), with z :=x& y. In order to obtain T 2(x, y), we have to split D 2(x, y) into a retarded part, R 2(x, y), and an advanced part, A 2(x, y), with respect to the coincident point z=0, so that supp(R 2(z))V +(z) and supp(A 2(z)) V &(z). This splitting, or improved time-ordering, has to be carried out in the distributional sense so that the retarded and advanced part are mathematically well-defined. The splitting affects only the numerical distribution d O2 (x& y) and must be accomplished according to the correct singular order |(d O2 ) which describes roughly speaking the behavior of d O2 (x& y) near x& y=0 or that of d O2 ( p) in the limit p Ä . If |<0, then the splitting is trivial and agrees with the standard time-ordering. If |0, then the splitting is nontrivial and nonunique, O

|(d 2 )

d (x& y) Ä r (x& y)+ : C a, O D a$ (4)(x& y). O 2

O 2

(2.5)

|a| =0

A retarded part r O2 (x& y) of d O2 (x& y) is usually obtained in momentum space by means of a subtracted dispersion-like integral; see Eq. (3.18). Equation (2.5) contains a local ambiguity in the normalization: the C a, O 's are undetermined finite normalization constants, which multiply terms with point support D a$ (4)(x& y)(D a is a partial differential operator). This freedom in the normalization has to be restricted by physical conditions. Finally, T 2 is obtained by subtracting R$2(x, y) from R 2(x, y) and the whole local normalization coming from (2.5) is called N 2(x, y). 2.2. Quantized MatterGravity 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 background of a Minkowski space-time. The free scalar field of mass m satisfies the KleinGordon wave equation (g+m 2 ) ,(x)=0,

(2.6)

which follows from the free matter Lagrangian density 1 m2 2 ,& ,. L (0) M = ,, & , & 2 2

(2.7)

SCALAR MATTER COUPLED TO QUANTUM GRAVITY

157

The matter energymomentum tensor reads m2 2 1 +& ,+ ,& +& (0) ,+ ,& ,\ +& T +& , , M :=, , &' L M =, , & ' , , \ , +' 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)]=&iD m(x& y),

(2.9)

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

=

i d 4p $( p 2 &m 2 ) sgn( p 0 ) e &ip } x (2?) 3

|

(2.10)

is the causal JordanPauli 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 +&:;D 0(x& y),

(2.12)

where the b-tensor is constructed from the Minkowski metric b +&:; := 12(' +:' &; +' +;' &: &' +&' :; )

(2.13)

and D 0(x) is the JordanPauli distribution of Eq. (2.10) with m=0. The graviton field interacts with the conserved energymomentum tensor of the matter fields. The first-order matter coupling is chosen to be TM 1 (x)=i =i

} :; :h (x) b :;+& T +& M (x): 2 } 2

{

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

m2 :h,,: , 2

=

(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 ## and all Lorentz indices of the fields are written as superscripts,, whereas the derivatives acting on the fields are written as subscripts. All indices occurring twice are

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contracted by the Minkowski metric ' +&. We skip the space-time dependence if the meaning is clear. The presence of the b-tensor in Eq. (2.14) is a consequence of the choice of the graviton variable (see Section 2.3). 2.3. Perturbative Gauge Invariance The classical gauge properties of h +&(x) (which are related to the general covariance of the metric g +&(x) under coordinate transformations [19, 20]) are formulated at the quantum level by the gauge charge [12, 13] Q :=

|

x0 =const

Â Ä d 3x h +&(x) , &  0x u +(x).

(2.15)

For the construction of the physical subspace and in order to prove unitarity of the S-matrix on the physical subspace [15], the ghost, field u +(x), together with the antighost field u~ &(x), have to be quantized as free fermionic vector fields, gu +(x)=0,

gu~ &(x)=0,

[u +(x), u~ &( y)]=i' +& D 0(x& y),

(2.16)

whereas all other anticommutators vanish. The gauge charge generates the infinitesimal operator gauge variations d Q h +&(x) :=[Q, h +&(x)]=&ib +&\_u \(x) , _ , d Q u :(x) :=[Q, u :(x)]=0, and d Q u~ :(x) :=[Q, u~ :(x)]=ih :;(x) , ; .

(2.17)

The condition of gauge invariance is formulated in terms of the n-point distributions T n . For the moment, we define the condition of perturbative operator gauge invariance to n th order by the requirement d Q T n(x 1 , ..., x n )=sum of divergences.

(2.18)

Heuristically, this would imply that d Q S( g) vanishes in the adiabatic limit g Ä 1 because 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, perturbative invariance to first order in pure QG [12], namely for a coupling of the form T h1 t:hhh: without matter fields, requires the introduction of the ghost coupling T u1 t:u~hu: , so that d Q(T h1 +T u1 )(x)= x_ T _11h+u(x). Here, T _11h+u t:uhh: +:u~uu: is the so-called Q-vertex.

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SCALAR MATTER COUPLED TO QUANTUM GRAVITY

Using Eq. (2.17), the first-order matter coupling (2.14) is gauge invariant, dQ T M 1 (x) =

} :;\_ \ / \ \_ :u (x) , _ b :;+& T +& b M(x): = :u (x) , _ T M (x): 2 2 } 2

=  x_

\{

x _

_M 11

=:  T

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

=+

(x),

(2.19)

_M because of T \_ M , _ =0. T 11 (x) is the matter Q-vertex. The concept of a Q-vertex allows us to formulate in a more stringent way the condition (2.18) of perturbative gauge invariance to the n th order, n

d Q T n(x 1 , ..., x n )= : l=1

 T & (x 1 , ..., x l , ..., x n ), x &l nl

(2.20)

where T &nl is the renormalized time-ordered product obtained according to the inductive causal scheme, with one Q-vertex at x l while all other n&1 vertices are ordinary T 1 -vertices. This condition follows from the formula for the construction of T n , namely the time-ordering of n, first-order interactions T 1 , and from gauge invariance 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 1 - & g g +&R +& + - & g ( g +&, ; + , ; & &m 2, 2 ) 2 } 2

(2.21)

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

(2.22)

which defines the graviton field h +& in the Minkowski background. Then one obtains  j) +L (Mj) ). LEH +LM = : } j (L (EH

(2.23)

j=0 +& From L (0) EH , choosing the Hilbert gauge h , & =0, one obtains Eq. (2.11) and the presence of the b-tensor is made clear [15]. The first-order graviton coupling T h(x)t :hhh: corresponds then to the normally ordered product of i}L (1) EH (see [13] for a derivation based merely on the principle of perturbative operator gauge invariance) and L (2) EH thhhh represents the quartic graviton coupling required by perturbative gauge invariance to second order in the tree graph sector [12].

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NICOLA GRILLO

The expansion of the matter Lagrangian density reads 1 } m2 LM = (' +&, , + , , & &m 2, 2 )+ h +& , , + , , & & ' +& , 2 2 2 2

\

(0)

(1)

=} LM

LM 2 2

+

m} 8

\

+

1 h :;h :;,,& hh,, +O(} 3 ). 2

+

(2.24)

(2)

=} 2 LM

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 agrees with 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 to Second Order From the structure of T M 1 it is evident that the two-point distribution describing loop graphs has the form (up to noncontributing divergences for the matter selfenergy, see Section 4.1) T 2(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 second term the scalar matter self-energy. The C-number distributions 6(x& y) :;+& and 7(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 comparing distributions attached to the same operators on both sides of Eq. (2.20) [21]. We compute d Q T 2(x, y) loops by meas means of (2.17) and isolate the contributions with external operator of the type :u(x) h( y): . We obtain d Q T 2(x, y) loops | :u(x) h( y): =:u \(x) , _ h +&( y): (b :;\_6(x& y) :;+& ).

(2.26)

On the other side, T _21(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 that the loop contributions coming from T _21(x, y) can only be of the form T _21(x, y)=:u \(x) h +&( y): t _uh(x& y) \+& +:u _(x) h +&( y): t uh(x& y) +& ,

(2.27)

by performing two matter field contractions. The second term T _22(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 _21(x, y)= +:u \(x) , _ h +&( y): [t _uh(x& y) \+& +' _\ t uh(x& y) +& ] +:u \(x) h +&( y):  x_[t _uh(x& y) \+& +' _\ t uh(x& y) +& ].

(2.28)

SCALAR MATTER COUPLED TO QUANTUM GRAVITY

161

We compare the C-number distributions in (2.26) and in (2.28) attached to the external 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) \ +& +' \_t uh(x& y) +& ], 0= x_[t _uh(x& y) \ +& +' \_t uh(x& y) +& ].

(2.30)

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 explicitly checked and implies the gravitational SlavnovWard identities for the two-point connected Green function (see Section 3.3). Gauge invariance to second order in the tree graph sector is much more involved and requires also the full analysis of the mattergraviton interaction; see Section 5.

3. MATTER LOOP GRAVITON SELF-ENERGY

3.1. Causal D 2(x, y)-Distribution In order to construct D 2(x, y), according to Section 2.1 we first need the contractions between field operators. From (2.9) and (2.12), we derive them, C[,(x) ,( y)] :=[,(x) (&), ,( y) (+) ]=&iD (+) m (x& y), C[h :;(x) h +&( y)] :=[h :;(x) (&), h +&( y) (+) ]=&ib :;+&D (+) 0 (x& y),

(3.1)

where (\) refers to the positivenegative frequency part of the corresponding quantity. The A$2(x, y) g SE distribution for the graviton self-energy by a matter loop is M obtained by performing two matter field contractions in &T M 1 (x) T 1 ( y). Using x y (3.1) and with  : =& : , we find that A$2(x, y) g SE =:h :;(x) h +&( y):

}2 & x:  x+ D (+) }  x;  x& D (+) m m 4

_

2 x (+) & x:  x& D (+) }  x;  x+ D (+) }  x; D (+) m m +m ' +&  : D m m

+m 2' :;  x+ D (+) }  x& D (+) m m &

m4 } D (+) (x& y). ' :; ' +& D (+) m m 2

&

(3.2)

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NICOLA GRILLO

We introduce the functions m (\) D (\), (x) :=D (\) }|} m (x) } D m (x),

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

m x x (\) x x (\) D (\), :; | +& (x) := :  ; D m (x) }  +  & D m (x),

(3.3)

so that we have g SE A$2(x, y) g SE =:h :;(x) h +&( y): a$2(x& y) :;+& , g SE = a$2(x& y) :;+&

}2 m (\), m 2 (\), m &D (\), :+ | ;& &D :& | ;+ +m ' +& D : | ; 4

_

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

(3.4)

m4 m (x& y). ' :; ' +& D (\), }|} 2

&

Products of JordanPauli distributions are evaluated in momentum space with Eq. (2.10), because there products go over into convolutions between the Fourier m transforms. For example, D (\), (x) becomes }|} m ( p)= D (\), }|}

&1 (2?) 4

| d k $(( p&k) &m ) 3(\( p &k )) 4

2

2

0

0

_$(k 2 &m 2 ) 3(\k 0 ).

(3.5)

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

|

4 2 2 0 0 I (\) m ( p) &::;:;+:;+& := d k $(( p&k) &m ) 3(\( p &k ))

_$(k 2 &m 2 ) 3(\k 0 )[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 m D (\), -functions momentum space are }}} | }}} m D (\), ( p)= }|}

&1 [I (\) ( p)], (2?) 4 m

m D (\), ( p)= :|;

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

&1 m (\) D (\), [ + p : p ; I (\) :; | +& ( p)= m ( p) +& & p : I m ( p) ;+& (2?) 4 (\) & p ; I (\) m ( p) :+& +I m ( p) :;+& ].

(3.7)

SCALAR MATTER COUPLED TO QUANTUM GRAVITY

163

Inserting (3.7) and Eqs. (A.6), (A.7), (A.11), (A.15), and (A.19) into (3.4), then the a$2 -distribution in momentum space reads g SE = a^$2( p) :;+&

&} 2? [ +Ap : p ; p + p & +Bp 2( p : p ; ' +& + p + p & ' :; ) 960(2?) 4 +Cp 2( p : p + ' ;& + p : p & ' ;+ + p ; p + ' :& + p ; p & ' :+ ) +Ep 4(' :+ ' ;& +' :& ' ;+ )+Fp 4' :; ' +& ] d( p) (+) m ,

(3.8)

with the coefficients A := &8&16 C := +1&8

m4 m2 &48 , p2 p4

m2 m4 +16 , p2 p4

F := &1&12

B := &4&8 E := &1+8

m2 m4 &24 , p2 p4

m2 m4 &16 , p2 p4

(3.9)

m2 m4 +4 4 , 2 p p

:=- 1&(4m 2p 2 ) 3( p 2 &4m 2 ) 3(\p 0 ). Performing and the distribution d( p) (\) m M the same calculations for R$2(x, y)=&T M 1 ( y) T 1 (x), we obtain g SE , R$2(x, y) g SE =:h :;(x) h +&( y): r$2(x& y) :;+& g SE r^ $2( p) :;+& =

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

(3.10)

Therefore, with (2.3) the causal D 2(x, y)-distribution reads g SE D 2(x, y) g SE =:h :;(x) h +&( y): d 2(x& y) :;+& ,

g SE = d 2( p) :;+&

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

(3.11)

(&) 2 2 2 2 0  Here, d( p) m =d( p) (+) m &d( p) m =- 1&(4m p ) 3( p &4m ) sgn( p ). The d 2 -distribution can be recast into the form 3 g SE d 2( p) :;+& = : d( p) (i) :;+& i=1

=

} 2? m2 m4 P( p) :;+& + 2 Q( p) :;+& + 4 R( p) :;+& d( p) m , (3.12) 4 p p 960(2?)

_

=: (

&

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NICOLA GRILLO

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 M 1 (x)=i(}2) :h (x) ,(x) , : ,(x) , ; : , then the d 2 -distribution reads d 2( p) m=0  ( p) :;+& 3( p 2 ) sgn( p 0 ). :;+& =(P

(3.14)

Hence, the limit m Ä 0 of (3.12) is feasible without problems; see Eq. (3.48) for the splitting in the m=0 case. The extension to nonminimally coupled massless matter is also considered. From 1 LM = 12 - & g g +&, ; + , ; & + 12 - & g R, 2,

(3.15)

we derive the first order matter coupling TM 1 (x)=i

} 2 :; 1 1 :h , , : , , ; : & :h, , _ , , _ : & :h :;,, , :; : , 2 3 6 3

{

=

(3.16)

which yields (see [16, 17] for the calculations in the m=0 case) 6 6 2 =([& 12 d 2( p) non-min. :;+& 9 , & 9 , 1, &1, 9 ] 3( p ) sgn( p 0 ).

(3.17)

The corresponding t 2 -distribution will be given in Eq. (3.49). 3.2. Distribution Splitting for the Graviton Self-Energy Matter Loop We turn now to the splitting of the D 2 -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 structure of the distributions is given by the scalar part. Therefore, neglecting the polynomials, 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 normalization N 2(x, y), we will realize that this was the correct choice. As anticipated in Section 2.1, a retarded part of d 2 is obtained in momentum space by the integral [10] r^ c( p 0 )=

i |+1 p 2? 0

|

+

&

dk 0

d(k 0 ) , (k 0 &i 0) |+1 ( p 0 &k 0 +i 0)

(3.18)

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

165

SCALAR MATTER COUPLED TO QUANTUM GRAVITY

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



d(k) (1) = 1&

4m 2 3(k 2 &4m 2 ) sgn(k 0 )=: f (k 2 ) sgn(k 0 ). k2

(3.19)

Therefore, we have to split d(k) with |(d )=0, because d(k 0 )tconstant for |k 0 | Ä  and k & =(k 0 , 0) # V +. From (3.18) we obtain r^ c( p 0 )= = = =

i p0 2? i p0 2? i p0 2? i p0 2?

|

dk 0

f (k 20 ) sgn(k 0 ) (k 0 &i 0)( p 0 &k 0 +i 0)

dk 0

f (k 20 ) & k 0( p 0 &k 0 +i 0)

+

&

_| |

 0

&

dk 0

f (k 20 ) k 0( p 0 &k 0 +i 0)

dk 0

f (k 20 ) 1 1 + k 0 p 0 &k 0 +i 0 p 0 +k 0 +i 0

dk 0

f (k 20 ) 2 p0 k0 . 2 2 k 0 p 0 &k 20 +ip 0 0



0

|

|

0



0

\

&

+ (3.20)

With the new variable s :=k 20 we find r^ c( p 0 )=

i 2 p 2? 0

|



ds 0

f (s) . s( p 20 &s+ip 0 0)

(3.21)

Inserting the explicit form of f (s) we have r^ c( p 0 )=

i 2 p 2? 0

|



ds 4m2



1&

4m 2 1 . s s( p 20 &s+ip 0 0)

(3.22)

We decompose the integral into real and imaginary parts according to (x+i 0) &1 = P(x &1 )&i?$(x), r^ c( p 0 )=

i 2 p P 2? 0 +

1 2

|



ds 4m2



1&



1&

4m 2 1 2 s s( p 0 &s)

4m 2 3( p 20 &4m 2 ) sgn( p 0 ). p 20

(3.23)

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



r^ $( p 0 )=& 1&

4m 2 3( p 20 &4m 2 ) 3(&p 0 ) p 20

(3.24)

166

NICOLA GRILLO

from (3.23), we obtain the numerical t^-distribution belonging to T 2(x, y), t^( p 0 )=r^ c( p 0 )&r^ $( p 0 ) =

i 2 p P 2? 0

|



ds

4m2



1&

4m 2 1 1 + 2 s s( p 0 &s) 2



1&

4m 2 3( p 20 &4m 2 ), (3.25) p 20

which can be written in the form t^( p 0 )=

i 2 p 2? 0



|

ds

4m2



1&

4m 2 1 . 2 s s( p 0 &s+i 0)

(3.26)

This result can be generalized for p # V + by introducing the ``inverse momentum'' q :=4m 2 p 2 so that t^( p)=

i 2?

|



ds q

- s(s&q) i =: 6( p 2 ). s (1&s+i 0) 2? 2

(3.27)

Note that, we write for simplicity the p-dependence instead of the q-dependence of the basic integral 6 which remains to be calculated. Therefore, the splitting of the first term in (3.12) and the subtraction of r^ $( p) (1) :;+& yields t^( p) (1)  ( p) :;+& 6( p 2 ). :;+& =i5P

(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 Q( p) :;+& 6( p 2 ), p2

(3.29)

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

d( p) (2) =

1 p2



1&

4m 2 3( p 2 &4m 2 ) sgn( p 0 ). p2

(3.30)

Analogously, for the third term of (3.12) we obtain t^( p) (3) :;+& =i5

m4 R( p) :;+& J( p 2 ), p4

(3.31)

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

d( p) (3) =

1 p4



1&

4m 2 3( p 2 &4m 2 ) sgn( p 0 ), p2

(3.32)

167

SCALAR MATTER COUPLED TO QUANTUM GRAVITY

with the definition J( p 2 ) :=



|

ds

q

- s(s&q) . s 3(1&s+i 0)

(3.33)

The difference between J( p 2 ) and 6( p 2 ) lies in the powers of s in the denominators in Eqs. (3.27) and (3.33). This is a consequence of splitting these distributions according to the singular orders of the corresponding scalar distributions. The two integrals 6( p 2 ) and J( p 2 ) have the same structure and the last can be expressed by means of the first. We decompose 6( p 2 ) into real and imaginary parts, 6( p 2 )=P

|



ds

q

- s(s&q) &i? - 1&q 3(1&q), s 2(1&s)

(3.34)

and concentrate our attention on the principal value part. With the substitution 1 s(s&q)=(s&x) 2 the real part of 6( p 2 ) reads 6 r( p 2 )= &2P

|

dx

(x&q) 2 x (x 2 &2x+q)

dx

{x +x &2x+q= ,



q

= &2P

|

 q

2

q

2

1&q

(3.35)

2

having factorized the integrand. The J( p 2 )-integral yields a real part J r( p 2 )= &2P

|



dx

q

= &2P

|



q

dx

(x&q) 2 (2x&q) x 4(x 2 &2x+q)

{

q&1 2q q 3 1&q + 3 & 4+ 2 . 2 x x x x &2x+q

=

(3.36)

A look at Eqs. (3.35) and (3.36) enables us to isolate in the expression for J r( p 2 ) the terms appearing also in (3.35). The others can be easily integrated and we obtain J r( p 2 )=

1

2 p2 +6 r( p 2 )= 2 +6 r( p 2 ), 3q 6m

(3.37)

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

168

NICOLA GRILLO

The imaginary parts always have the same form as in Eq. (3.34). Gathering the results in (3.28), (3.29), and (3.31) together with (3.37), we can write the distribution describing the matter loop graviton self-energy, 3 g SE t^ 2( p) :;+& = : t^( p) (i) :;+& i=1

=i5 +

_{

P( p) :;+& +

m2 m4 Q( p) :;+& + 2 R( p) :;+& 6( p 2 ) 2 p p

=

m2 R( p) :;+& . 6 p2

&

(3.38)

Therefore, the two-point operator-valued distribution T 2(x, y) for the graviton selfenergy reads T 2(x, y) g SE =:h :;(x) h +&( y): i6(x& y) :;+& ,

(3.39)

where 6(x& y) :;+& is the graviton self-energy tensor. Its Fourier representation is given by &i_(3.38). We still must calculate explicitly the integral representation for 6( p 2 ), given in Eq. (3.27). There are three different regimes, depending on the value of q. For q=1, namely p 2 =4m 2, we obtain by means of the partial decomposition (3.35), 6 a ( p 2 =4m 2 )=&2

|



dx

1

1 =&2. x2

(3.40)

For q<1, namely, p 2 >4m 2, the integration of the partial decomposition (3.35), taking into account also the imaginary part from (3.34), yields 6 b( p 2 )=&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

\

q&1 ? &arc tan 2 q&1

= &2+2 - q&1 arc tan

+

1

\- q&1+ .

(3.42)

Note that these three results are connected by lim 6 b( p 2 )=6 a ( p 2 =4m 2 ),

p2z4m2

lim 6 c( p 2 )=6 a ( p 2 =4m 2 ).

p2Z4m2

(3.43)

169

SCALAR MATTER COUPLED TO QUANTUM GRAVITY

Writing the p-dependence explicitly, the final form for 6( p 2 ) is

{



6( p 2 )= &2& + 1&

4m p2

2

_} log

2

&2

 4mp 4m 1+ 1& p

1& 1&

4mp &1 arc tan

2

1

2



2

2

}& = +i?

2

3( p 2 &4m 2 )

3(4m 2 & p 2 ) .

2

4m &1 p2

(3.44)

Two limits of this result will be used in the discussion of the normalization N 2 of the T 2 -distribution (Section 3.4), the limit of 6( p 2 ) for p 2z0 and the limit of 6( p 2 ) p 2 for p 2z0, too. In the first case we have lim 6( p 2 )= lim 6 c( p 2 )

p2z0

qÄ

?

?

1

1

{ &2+2 - q&1 \ 2& \ 2 &q&1+O \q +++= 1 2 +O = lim &2+2& { \q += =0. 3(q&1)

= lim

32

qÄ

2

qÄ

(3.45)

For the second limit, we obtain p p2z0

6( p 2 ) q6 c( p 2 ) = lim 2 p qÄ 4m 2 = lim

qÄ

=

1 q 1 1 & &2+2 - q&1 +O 52 3 4m 2 q - q&1 3 - q&1

{

\

\ ++=

&1 . 6m 2

(3.46)

The last consideration concerns the retarded part in Eq. (3.23), given also by (3.44) up to the step-function in p 0 : this retarded part is the boundary value of the analytic function 2

r^ ( p) an

4m 1& &1  p 4m =&2& 1&  p log 4m . 1& p +1 2

2

2

2

2

(3.47)

170

NICOLA GRILLO

Summing up the whole calculation, we have found the two-point distribution (3.39) for the graviton self-energy contribution. For the corresponding tensor, the structure is given by (3.38) and the integral by (3.44). During the calculation, we never resorted to a regularization of potentially UV divergent expressions (for example, dimensional regularization as in [3]). This was made possible by using the correct starting point, namely Eq. (3.18), which is, so to say, a careful multiplication by a step-function in the time argument. If this had been 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 coupled massless matter, Eq. (3.17). The splitting of the scalar distribution 3( p 2 ) sgn( p 0 ) requires some modifications if one tries to use the splitting formula (3.18); see [11, 16]. The retarded part is given by (i2?) log((& p 2 &ip 0 0)M 2 ), so that the m=0 matter self-energy tensor reads  ( p) :;+& log 6( p) m=0 :;+& =5P

\

& p 2 &i 0 , M2

+

(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 & :;+&

& p 2 &i 0 12 6 6 log . , & , 1, &1, 9 9 9 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 matterenergy-momentum tensor. The latter corresponds to the so-called improved energy-momentum tensor [22]. =0. This property follows from the In addition, it is transversal: p :6( p) non-min. :;+& gauge identity (2.31), namely b :;\_p _ 6( p) :;+& =0 (see Section 3.3), and from its vanishing trace. In these two cases we have found graviton self-energy contributions without introducing counterterms. This is in contrast to the calculations carried out for massless scalar matter fields coupled to QG in the background field method with dimensional regularization [4, 5]. 3.3. Graviton Self-Energy Tensor and Perturbative Gauge Invariance The gauge properties of T 2(x, y) g SE are contained in the identity coming from Eq. (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)

SCALAR MATTER COUPLED TO QUANTUM GRAVITY

171

for the coefficients of the self-energy tensor. These conditions are satisfied by our result of Eq. (3.9) and therefore 6(x& y) :;+& is gauge invariant. This is certainly the case at the level of the D 2(x, y) g SE-distribution before distribution splitting. In the causal scheme, UV finiteness and gauge invariance are established separately. The latter is not used to reach the former. The identity (2.31) implies the SlavnovWard identities (SWI) for the two-point connected Green function. The latter is defined as G( p) [2]  F0 ( p) 6( p) #$\_b \_+& D F0 ( p), :;+& :=b :;#$ D

(3.51)

where D F0 ( p)=(2?) &2 (& p 2 &i 0) &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 the coefficients A, ..., F as in Eq. (3.8) we have A&2B=0,

C+E=0,

A +C&F=0. 4

(3.53)

These are equivalent to the conditions (3.50). This conclusion is valid also in the massless case, Eqs. (3.48) and (3.49). In [23], the gauge invariance of the massless matter loop graviton self-energy tensor also was investigated, but there it was not realized that the correct matter coupling is the one in Eq. (2.14), namely, that with the b-tensor, when one uses the Goldberg variable expansion. This deficiency does not have consequences if the graviton is coupled to a traceless matter energy-momentum tensor as in the nonminimal coupling case, Eq. (3.16). The condition of perturbative gauge invariance to second order in the loop graph sector, d Q T 2(x, y) g SE = x_(:u \(x) h +&( y): [b :;\_6(x& y) :;+& ])+(x W y) = x_(:u \(x) h +&( y): [t _uh(x& y) \+& +' \_t uh(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 T 2 -distribution of Eq. (3.39). The total singular order is four because the polynomials are of degree four in p. Therefore, we have to add normalization terms up to the singular order four.

172

NICOLA GRILLO

The freedom in the normalization due to the splitting procedure is contained in the local term N 2(x, y) g SE, N 2(x, y) g SE =: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 of polynomials of degree 2i4=|, 2

N( p) :;+& = : N( p) (2i) :;+& .

(3.56)

i=0

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 the freedom in the normalization in such a way that the polynomials have to be of the form N( p) (0) :;+& =0,

N( p) (2) :;+& =5[0, 0, &a, a, &a]

1 , p2

(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 be fixed by requiring the appropriate mass- and coupling constant-normalizations for the corrections of order } 2 to the graviton propagator. Letting formally g Ä 1 in Eq. (2.1), we write the order } 2 corrected propagator as

|

F 4 4 F &iD(x& y) [2] :;+& = &ib :;+& D 0 (x& y)+ d x 1 d x 2(&ib :;\_ D 0 (x&x 1 ))

_i(6(x 1 &x 2 ) \_#$ +N(x 1 &x 2 ) \_#$ )(&ib #$+& D F0 (x 2 & y)).

(3.58)

In momentum space, this becomes D( p) [2] :;+& = & +

b :;+& (2?) 2 ( p 2 +i 0) &1 &1 b . (6 +N )( p) \_#$ b #$+& 2 ( p 2 +i 0) :;\_ ( p +i 0)

(3.59)

 (p):;+& =: >

After a little work, we find in the form of Eq. (3.8) 6( p) :;+& =5[ f A( p 2 ), f B ( p 2 ), f C ( p 2 ), f E ( p 2 ), f F ( p 2 )],

(3.60)

173

SCALAR MATTER COUPLED TO QUANTUM GRAVITY

with m2 m4 48m 2 2 &48 6  ( p )& +4(b+c), p2 p4 6 p2

\ + m m 16m 2a 6( p )+ &2b&4c+ f ( p )= +6+32 +16 \ p p + 6p p f A( p 2 )= &8&16

2

4

2

4

2

2

2

B

2

= & f F ( p 2 ),

\

f C ( p 2 )= +1&8

2

(3.61)

m2 m4 16m 2 a 2 +16 6  ( p )+ &b& 2 2 4 2 p p 6p p

+

= & f E ( p 2 ). With the formula (if g( p 2 )t} 2 ) 1 1 1 1 Ä + g( p 2 ) 2 2 2 & p &i 0 & p &i 0 & p &i 0 & p &i 0 2

=

1 +O(} 4 ), & p 2 &i 0& g( p 2 )

(3.62)

we obtain the order } 2 corrected graviton propagator D( p) [2] :;+& =

1 1 1 (' :+ ' ;& +' :& ' ;+ ) 2 2 (2?) 2 & p &i 0&2(2?) 2 5p 4f E ( p 2 )

_

1 1 + }}}, & (' :; ' +& ) 2 & p 2 &i 0+2(2?) 2 5p 4f F ( p 2 )

&

(3.63)

where terms that do not contribute between conserved matterenergy-momentum tensors have been neglected. The corrected propagator above has the correct limit  F0 ( p). lim } Ä 0 D( p) [2] :;+& =b :;+& D Mass normalization (the graviton mass remains zero under quantum corrections) yields p 4f E ( p 2 )| p2 =0 =0= p 4f F ( p 2 )| p2 =0 .

(3.64)

Since 6( p 2 =0)=0 from Eq. (3.45), these conditions always hold. Coupling constant normalization (} is not shifted by the quantum corrections) implies p 2f E ( p 2 )| p2 =0 =0= p 2f F ( p 2 )| p2 =0 .

(3.65)

174

NICOLA GRILLO

Analysis of the first condition, p 2f E ( p 2 )| p2 =0 =&16m 4

6( p 2 ) p2

}

16m 2 +a=0, 6

(3.66)

16m 2 &2a=0, 6

(3.67)

& p2 =0

=&16 m2

yields a=0. Analysis of the second condition, p 2f F ( p 2 )| p2 =0 =&16m 4

6( p 2 ) p2

}

& p2 =0

=&16 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 (m 26 p 2 ) R( p) :;+& in Eq. (3.38). A remark about the splitting is appropriate: if we had split the distributions d 2( p) (i) :;+& , i=2, 3, in Eq. (3.12) according to their true singular orders, namely 2 and 0 (because of the presence of the polynomials), respectively, then the term (m 26 p 2 ) R( p) :;+& would have been missing from (3.38). Working out the consequences for what concerns the normalization question, Eq. (3.66) would have required the choice a=&8m 23, whereas Eq. (3.67) would have required the choice a=4m 23. 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 splitting solution (3.18) is not applicable in that case and one has to choose a subtraction point, different from the origin. The remaining constants b and c are not fixed by these requirements. The total graviton self-energy tensor including its normalization then has the form 6( p) tot :;+& =5

_{

+

P( p) :;+& +

m2 m4 Q  ( p) + R( p :;+& ) 6( p 2 ) :;+& p2 p4

=

2 m2 R  ( p) + : z i Z( p) (i) :;+& :;+& , 6 p2 i=1

&

(3.68)

where z i # 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 degree  ( p) (2) four. They can be chosen to be Z( p) (1) :;+& =[4, 2, &1, 1, 0] and Z :;+& = [4, 2, 0, 0, 1]. Analysis of the issue of normalization with the method used in [16, 17] leads to the same conclusions.

175

SCALAR MATTER COUPLED TO QUANTUM GRAVITY

4. MATTER SELF-ENERGY The two-point distribution describing the matter self-energy graph is not interesting from the point of view of its gauge properties. However, calculation of the corresponding distribution is carried out in order to investigate within the EpsteinGlaser scheme its UV structure. 4.1. Causal D 2(x, y)-Distribution and Distribution Splitting The D 2 -distribution is obtained here by performing one matter field contraction and one graviton contraction (3.1). The result reads D 2(x, y) m SE = +:,(x) , : ,( y) , :: d a (x& y)+:,(x) ,( y) , : : d b(x& y) : +:,(x) , : ,( y): d c(x& y) : +:,(x) ,( y): d d (x& y),

(4.1)

where the numerical distributions d a (x& y) :=

} 2m 2 (+) (C } | } &C (&) } | } )(x& y), 2

d b(x& y) : :=

&} 2m 2 (+) (C } | : &C (&) } | : )(x& y), 2

d c(x& y) : :=

} 2m 2 (+) (C } | : &C (&) } | : )(x& y), 2

(4.2)

and (&) d d (x& y) := &} 2m 4(C (+) } | } &C } | } )(x& y),

are expressed in terms of the C (\) } } } -functions, (\) (\) C (\) } | } (x) :=D 0 (x) } D m (x),

(4.3)

(\) x (\) C (\) } | : (x) :=D 0 (x) }  : D m (x).

These products are calculated in momentum space, see Appendix B, so that the distributions d i , i=a, b, c, d, read &} 2m 2? m2 1& 2 3( p 2 &m 2 ) sgn( p 0 ), 4 4(2?) p

\ + &i} m ? m d ( p) = 1& p 3( p &m ) sgn( p )=&d ( p) , 8(2?) \ p + } m? m 1& 3( p &m ) sgn( p ). d ( p)= 2(2?) \ p + d a ( p)=

2

2

4

4

4

:

:

b

2

4

:

2

0

2

2

d

2

4

2

2

0

c

(4.4)

176

NICOLA GRILLO

From power-counting arguments, one could expect that d a will behave as p 2 for large momenta. This is not the case, because the wave equation (g+m 2 )D (\) m (x)= 0 lowers the power of p coming from the product of contractions. In order to shorten the calculation, we bring D 2(x, y) m SE into the form D 2(x, y) m SE =:,(x) ,( y): d 2(x& y) m SE +divergences,

(4.5)

with d 2( p) m SE = + p 2d a ( p)&ip : d b( p) : +ip : d c( p) : +d d ( p) =

&} 2m 2? 2 3m 2 m 4 p & + 2 3( p 2 &m 2 ) sgn( p 0 ). 2 2p 2(2?) 4

_

&

(4.6)

=: 1

Truly, this simplification can only be made for the corresponding T 2 -distribution, because divergences do not formally contribute in the adiabatic limit g Ä 1 of Eq. (2.1). Therefore, one should split the four distributions d i , i=a, b, c, d, separately and then recast the t i 's in a form similar to Eq. (4.5) for the T 2 -distribution. The final result would be the same. Note that in the m=0 case D 2(x, y) m SE =0. The splitting of (4.6) is accomplished by means of the splitting formula (3.18) SE )=2, with |(d m 2 r^ c( p 0 ) m SE =

i 3 p 2? 0

|

+

dk 0

&

d 2(k 0 ) m SE , (k 0 &i 0) 3 ( p 0 &k 0 +i0)

(4.7)

for p & =( p 0 , 0), p 0 >0. The retarded part then reads r^ c( p 0 ) m SE =

i1 3 p 2? 0

|

+

dk 0 &

3m 2 m 4 3(k 20 &m 2 ) sgn(k 0 ) 1& 2 + 4 . k 0( p 0 &k 0 +i 0) 2k 0 2k 0

{

=

(4.8)

With s=k 20 , ds=2k 0 dk 0 , we obtain r^ c( p 0 ) m SE =

i1 4 p 2? 0

|



ds 0

1 3m 2 m 4 3(s&m 2 ) & + , 2 ( p 0 &s+ip 0 0) s 2s 2 2s 3

{

=

(4.9)

which can be decomposed into real and imaginary parts, r^ c( p 0 ) m SE =

i1 4 p P 2? 0



3(s&m 2 ) 1 3m 2 m 4 + & ( p 20 &s) s 2s 2 2s 3

{ = 1 3m m + 3( p &m ) sgn( p ) p & { 2 + 2p = . 2 |

ds

0

2

2 0

2

0

2 0

4

2 0

(4.10)

177

SCALAR MATTER COUPLED TO QUANTUM GRAVITY

Subtracting the distribution

_

r^ $2( p 0 ) m SE =&1 p 20 &

3m 2 m 4 + 2 3( p 20 &m 2 ) 3(&p 0 ) 2 2p0

&

(4.11)

(coming from R$2(x, y) m SE ) from r^ c( p 0 ) m SE, we find the two-point distribution t^ 2( p 0 ) m SE =r^ c( p 0 ) m SE &r^ $2( p 0 ) m SE =

i1 4 p 2? 0

|



ds 0

3(s&m 2 ) 1 3m 2 m 4 + =: i7( p 0 ). & ( p 20 &s+i 0) s 2s 2 2s 3

{

=

(4.12)

As a next task, we compute the integral of the principal value part of (4.12), denoted by X( p 0 ), X( p 0 )=

&i1 4 p P 4? 0

|

=

&i1 4 p P 4? 0

|



ds m2 

m2

ds

2s 2 &3m 2s+m 4 s 3(s& p 20 )

{

: ; # &: + 2+ 3+ , s s s s& p 20

=

(4.13)

with : :=

3m 2 m 4 2 & 6& 2, p 40 p0 p0

; :=

3m 2 m 4 & 4, p 20 p0

# := &

m4 . p 20

(4.14)

Integration of the partial fractions in (4.13) yields X( p 0 )=

i1 2?

_\

p 20 &

p 2 &m 2 3m 2 m 4 m2 5 p2 + 2 log 0 2 + & 0 . 2 2 p0 m 2 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 for the matter self-energy T 2(x, y) m SE =:,(x) ,( y): i7(x& y),

(4.16)

and in an arbitrary Lorentz system the matter self-energy distribution for p # V + is 7( p)=

3m 2 m 4 + 2 2 2p

1 2?

_\

+

m2 5 p2 . & 2 4

p2&

&

+_ } log

p 2 &m 2 &i?3( p 2 &m 2 ) m2

}

& (4.17)

178

NICOLA GRILLO

This loop contribution was also calculated in [24] within the operator regularization scheme. Parts of their result agree with our expression in (4.17), whereas differences concern the explicit presence of parameters, other p-dependent logarithms, and terms which go as p 4 in their expression for 7( p). In the causal scheme, these latter cannot appear, because the singular order remains the same after distribution splitting. The retarded part in (4.10) is the boundary value of the analytic function of complex momentum p+i', '=(=, 0), =>0: r^ ( p) an =

i1 2?

_\

p2&

3m 2 m 4 m2 m2 5 p2 + 2 log 1& 2 + & . 2 2p p 2 4

+ \

+

&

(4.18)

Having split d mSE with |=2, an ambiguity in the normalization of 7( p) of the type 2 N( p)=

1 (c 0 +c 2 p 2 ) 2?

(4.19)

must be taken into account. In order to fix the constants c 0 and c 2 , radiative corrections to the matter Feynman propagator by matter self-energy loops are considered: formally letting g Ä 1, they are of the form

|

&iD Fm(x& y)+ d 4x 1 d 4x 2(&iD Fm(x&x 1 )) _i[7(x 1 &x 2 )+N(x 1 &x 2 )](&iD Fm(x 2 & y))+ } } } .

(4.20)

In momentum space the series becomes D Fm( p)+D Fm( p)(2?) 4 (7( p)+N( p)) D Fm( p)+D Fm( p) _(2?) 4 (7( p)+N( p)) D Fm( p)(2?) 4 (7( p)+N( p)) D Fm( p)+ } } } =: 7( p) tot,

(4.21)

where D Fm( p)=(2?) &2 (& p 2 +m 2 &i 0) &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 1 . (2?) 2 & p 2 +m 2 &i 0&(2?) 2 (7( p)+N( p))

(4.22)

Mass corrections are avoided by requiring [7( p)+N( p)]| p2 =m2 =0.

(4.23)

179

SCALAR MATTER COUPLED TO QUANTUM GRAVITY

This implies c 0 =m 2( 34 &c 2 ),

(4.24)

so that the matter self-energy distribution including its normalization reads 3m 2 m 4 p 2 &m 2 &i?3( p 2 &m 2 ) + 2 log 2 2p m2 5 +( p 2 &m 2 ) c 2 & . (4.25) 4 For the remaining constant c 2 , we cannot provide a value here. A restriction should come from the vertex corrections 4( p, q) :; to third order in the three-point distribution T 3(x, y, z)=:,(x) ,( y) h :;(z): 4(x, y, z) :; . 7( p)+N( p)=

1 2?

_\

p2 &

\

+_ } +&

}

&

5. PERTURBATIVE GAUGE INVARIANCE TO SECOND ORDER IN THE TREE GRAPH SECTOR In this section we show that the condition of perturbative gauge invariance to second order in the tree graph sector generates a local quartic interaction of the form t} 2 :hh,,: $ (4)(x& y), which agrees with the second-order L (2) M in the expansion of the matter Lagrangian density LM , Eq. (2.24). 5.1. Methodology Since R$2(x, y) is trivially gauge invariant due to its definition and the gauge invariance of T 1(x), instead of Eq. (2.20) for n=2 we have to examine whether d Q R 2(x, y)+d Q N 2(x, y)= x& R &21(x, y)+ &y R &22(x, y) + x& N &21(x, y)+ &y N &22(x, y)

(5.1)

can be satisfied by a suitable choice of the free constants in the normalization terms N 2 , N &21 , and N &22 of the retarded parts R 2 , R &21 , and R &22 . Here, R &21 and R &22 are the retarded distributions obtained by splitting the inductively constructed distributions D &21(x, y) :=[T &11(x), T 1( y)]

and

D &22(x, y) :=[T 1(x), T &11( 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. N 2 , N &21 , and N &22 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 &21 , the Q-vertex T &11 can give rise to expressions of the form D &21(x, y)=:O(x, y):  &x D m(x& y).

(5.3)

180

NICOLA GRILLO

:O(x, y): is a normally ordered product of four fields, being the other two fields in the commutator (5.2). m can also be zero, if the commutator contains two graviton fields. The retarded part is simply R &21(x, y)=:O(x, y):  &x D ret m (x& y),

(5.4)

av because the trivial splitting for tree graphs follows from D m =D ret m &D m with the correct support properties as explained in Section 2.1. Applying the derivative  x& , which forms the divergence in  x& R &21 , we get (4) (x& y)&m 2D ret :O(x, y): g x D ret m (x& y)=:O(x, y): [$ m ].

(5.5)

The expression :O(x, y): $ (4)(x& y) is a local anomaly. A corresponding mechanism works for R &22 also. We denote by an( x& R &21 + &y R &22 ) the set of all local anomalies generated by the described mechanism. Following [12], gauge invariance is preserved if we can choose N 2 , N &21 , and & N 22 so that the condition d Q N 2(x, y)=an( x& R &21 + &y R &22 )+ x& N &21(x, y)+ &y N &22(x, y)

(5.6)

involving the local terms of (5.1) is satisfied. Note that in QG coupled to matter d Q R 2 does not generate local terms with matter fields involved, namely of the type :uh,,: . This is in contrast, to the much more involved pure QG case. At this point we realize that, it is not sufficient to consider T M 1 only. Also, the graviton and ghost first-order couplings T h1 and T u1 have to be taken into account, because they yield also local anomalies with external operators t:uh,,: when splitting the commutators of (5.2). Using a simplified notation which keeps track of the structure of the coupling only, gauge invariance to first order then becomes d Q(:hhh: +:u~hu:+:h,,: ) = x&(:[uhh] &: +:[u~uu] &: +:[u,,] &:). h+u+M

T1

& h+u

T 11

(5.7)

&M

+T 11

With the ``extended'' Q-vertex T &11h+u +T &11M , we decompose the commutator defining D &21 in the following way: D &21(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:, perturbative gauge invariance has been shown in [12].

181

SCALAR MATTER COUPLED TO QUANTUM GRAVITY

5.2. Explicit Calculations Our 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 commutators in (5.8) as the graviton-, ghost-, and matter sectors, respectively. Performing one contraction, they lead to D &21h (x, y)=( :uh ,,: [h(x), h( y)]) &, x

y

D &21u (x, y)=( :,, hu: [u(x), u~( y)]) &, x

D &21M(x, y)=( :u, x

(5.9)

y

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

M in (2.19), we respectively. By considering the explicit form of T u1 in [12] and T &11 find that no local anomalies arise in the ghost sector. Let us compute the local anomalies in the graviton sector. From the expression for T &11h+u in [12] we isolate the terms that generate local anomalies according to the mechanism described in Section 5.1. They are 1 + \_ \_ 1 + T &11h+u =}[ + 12 :u +, + h \_h \_ , & : + 2 :u h , + h , & : & 4 :u , + hh , & :

& 14 :u +h , + h , & : + 12 :u +, \ h +\h , & : &:u +, \ h \_h _+ , & : + } } } ].

(5.10)

Then D &21h contains the following terms which generate anomalies: D &21h (x, y)=

}2 m2 +:u +, + h \_, , _ , , \ : & :u +, + h,,: +:u +h \_ , + ,, _ ,, \ : 4 2

{

&

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

_$ D 0(x& y).

= (5.11)

The first, two fields in the normally ordered products depend on x, whereas the matter fields depend on y. The retarded part R &21h has exactly the same form as in x &h (5.11) with the replacement D 0 Ä D ret 0 . Applying  & to R 21 (5.11), we obtain the y &h local anomalies (local anomalies coming from  & R 22 are just the same, therefore we get a factor two) an( x& R &21h + &y R &22h )=

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

_

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

&$

(4)

(x& y).

(5.12)

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

182

NICOLA GRILLO

In the matter sector, the first term in the matter Q-vertex of Eq. (2.19) is the only one that can generate local anomalies, because it carries the &-index as a derivative. Then, D &21M becomes }2 m2 :u +, , + h \_, , \ :  x_ + :u +, , + h,: D &21M(x, y)=& 2 2

{

=  D (x& y). & x

m

(5.13)

The contribution coming from D &22M has x W y and the derivative attached to the first term is  _y . The retarded part R &M 21 has the same form as in (5.13) with the replacement D 0 Ä 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)

& &M by applying  &x to R &M 21 and  y to R 22 we obtain

an( x& R &21M + &y R &21M ) =

}2 [&m 2 :u +h, , + ,: +:u +, _ h \_, , \ , , + : 2

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

(5.15)

Because of the $-function, all fields now have the same space-time dependence and have been recast in the form :uh,,: . 5.3. Quartic Normalization Terms Summing up, we have found all the local anomalies arising from  x& R &21 + &y R &22 attached to normally ordered products of the type :uh,,: . These can be organized into three different types according to their Lorentz structure apart from derivatives: type I, :u +h \_,,: with derivatives  + ,  \ ,  _ ; type II, :u +h,,: with derivatives  + ,  \ ,  \ ; and type III, :u +h +\,,: with derivatives  \ ,  _ ,  _ . Different Lorentz types do not interfere. From Eqs. (5.12) and (5.15), the local anomalies of type I are an( x& R &21 + &y R &22 )| type I =

}2 [ +:u +, + h \_, , \ , , _ : +:u +h \_ , + ,, \ ,, _ : 2 +:u +h \_, , \ , , +_ : &:u +, \ h \_, , \ , , + : + \_ (4) &:u +h \_ (x& y). , _ , , \ , , + : &:u h , , \_ , , + : ] $

(5.16)

SCALAR MATTER COUPLED TO QUANTUM GRAVITY

183

From Eqs. (5.12) and (5.15), the local anomalies of type II are an( x& R &21 + &y R &22 )| type II = &

} 2 m2 + m2 :u , + h,,: + :u +h , + ,,: 2 2 2

_

+m 2 :u +h, , + ,:

&$

(4)

(x& y),

(5.17)

and those of type III are simply an( x& R &21 + &y R &22 ) type III =

}2 [m 2 :u +, \ h +\,,: ] $ (4)(x& y). 2

(5.18)

According to Eq. (5.6), the question is whether these local anomalies can be written as divergences and therefore can be compensated by corresponding local divergence terms coming from  x& N &21 + &y N &22 . If it is not possible to reach such a compensation, normalization terms on the left side of (5.6) have to be introduced in 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) Ä + x+(:u +h \_ , , \ , , _ : $ (4)(x& y)) x y +

+ \_

+ (:u h y

y

, , \ , , _ : $ (4)(x& y)) x

&[ x\(:u +h \_ , , + , , _ : $ (4)(x& y))+ +y(:u +h \_ , , + , , _ : $ (4)(x& y))]. x y x y (5.20) The same is true for the local anomalies of type II, Eq. (5.17) Ä +m 2 x+( :u +h ,,: $ (4)(x& y)) x

y

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

(5.21)

x

Thus, by choosing the appropriate local divergence normalization terms  x& N &21 +  &y N &22 , 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 N 2 . Then, we require that

184

NICOLA GRILLO

the local anomaly of type III has to be coboundary, namely, a gauge variation of a normalization term N 2 , d Q N 2(x, y)=

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

(5.22)

The only possible N 2 that satisfies this requirement is N 2(x, y)=

i} 2m 2 1 :h :;h :;,,: & :hh,,: 4 2

{

=$

(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 agrees exactly with the term of order } 2 in the expansion of the matter Lagrangian density LM in (2.24). But in our case, this mechanism of generation works in a purely quantum framework. The only objection to this result is that this N 2 is not a ``proper'' normalization of a tree graph in T 2 , obtained starting with T h+u and T M 1 1 . But it can be considered as a normalization term of box-graphs in fourth order with external legs M and two T M t:hh,,: constructed with two T h+u 1 1 or with four T 1 . 5.4. Massless Matter Case :; For the massless matter coupling T M 1 (x)=i(}2) :h , , : , , ; : , perturbative gauge invariance to first order reads

x dQ T M 1 (x)= &

} 1 :u \, , \ , , & : & :u &, , \ , , \ : 2 2

\{

=+ =:  T x &

&M 11

(x),

(5.24)

namely, the matter Q-vertex of (2.19) in the m Ä 0 limit. Performing the same calculation as for the massive case, it turns out that the local anomalies are those of type I only, Eq. (5.16) which can be written as a divergence and therefore compensated for by proper normalization terms  x& N &21 + &y N &22 as in (5.20). From Eq. (5.6), it follows that d Q N 2 =0, which is certainly satisfied by N 2 =0. This agrees with the fact that even classically there are no hh,, couplings in the m=0 case: the expansion of LM written in terms of the Goldberg variable reads 1 } LM = , , \ , , \ + h +&, , + , , & . 2 2

(5.25)

Therefore, L (2) M =0. This concludes our discussion of the condition of perturbative gauge invariance to second order for tree graphs.

185

SCALAR MATTER COUPLED TO QUANTUM GRAVITY

APPENDIX A: THE I (\) m (p) } } } -INTEGRALS The I (\) m ( p) ... -integrals are defined by

|

4 2 2 0 0 I (\) m ( p) &++&+&\+&\_ := d q $(( p&q) &m ) 3(\( p &q ))

_$(q 2 &m 2 ) 3(\q 0 )[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 such that p & =( p 0 , 0), p 0 >0, then

|

4 2 2 2 2 I (+) m ( p 0 )= d q $( p 0 &2 p 0 q 0 +q 0 & |q| &m ) 3( p 0 &q 0 )

_$(q 2 &m 2 ) 3(+q 0 )

|

= d 3q

$( p 20 &2 p 0 E q ) 3( p 0 &E q ) , 2E q

(A.2)

with E q =q 0 =- q 2 +m 2. Then I (+) m ( p 0 )=4? =

|



d |q| 0

p0 |q| 2 1 $ &E q 3( p 0 &E q ) 2E q 2 p 0 2

\

? 3( p 20 &4m 2 ) 3( p 0 ) p0

|

+



d |q|

0

p0 |q| 2 &E q , $ Eq 2

\

+

(A.3)

because of the $-distribution E q =- q 2 +m 2 =

p 20 p0 O |q| = &m 2. 2 4



(A.4)

Therefore I (+) m ( p 0 )= =

? 3( p 20 &4m 2 ) 3( p 0 ) p0 ? 2



1&



p 20 &m 2 4

4m 2 3( p 20 &4m 2 ) 3( p 0 ). p 20

(A.5)

186

NICOLA GRILLO

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

? 2

2

1&4mp 3( p &4m ) 3(\p )=: ?2 d( p) 2

2

0

2

(\) m

.

(A.6)

Computing I (\) m ( p) + for p & =( p 0 , 0), p 0 >0, we have a nonvanishing contribution only for +=0. An additional factor q 0 is therefore present in the integrand in (A.3), which is set equal to E q and then to p 0 2. This leads to I (\) m ( p) + = p +

? 4



1&

4m 2 ? 3( p 2 &4m 2 ) 3(\p 0 )= p + d( p) (\) m , 2 p 4

(A.7)

in an arbitrary Lorentz frame. For I (\) m ( p) +& , we have to take the two conditions 2 (\) + I (\) m ( p) + =m I m ( p),

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

(A.8)

p 4 (\) I ( p), 4 m

into account. The first condition is imposed by the presence of $(q 2 &m 2 ) in (A.1). Computing p +p &I (+) m ( p) +& for p & =( p 0 , 0), p 0 >0, we get an additional factor ( p 0 E q ) 2 in the integrand (A.3) which gives p 40 4 due to $( p 0 2&E q ). In a general Lorentz frame, the second condition in (A.8) must hold. Therefore, making the ansatz 2 2 2 (\) I (\) m ( p) +& =(a( p ) p + p & +b( p ) p ' +& ) I m ( p),

(A.9)

the conditions (A.8) imply 1 m2 a( p 2 )= & 2 , 3 3p

b( p 2 )=

&1 m 2 , + 12 3p 2

(A.10)

so that I (\) m ( p) +& =

{\

1&

1 m2 ? m2 p p & & 2 p 2' +& d( p) (\) + & m . 2 p 4 p 6

+

\

+

=

(A.11)

For I (\) m ( p) +&\ , we have to take the two conditions 2 (\) & I (\) m ( p) +& =m I m ( p) + ,

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

p 6 (\) I ( p) 8 m

(A.12)

187

SCALAR MATTER COUPLED TO QUANTUM GRAVITY

into account. The first condition follows from the definitions of the I (\) m -integrals. Computing p +p &p \I (+) ( p) for p =( p , 0), p >0, we obtain an additional factor +&\ & 0 0 m ( p 0 E q ) 3 in the integrand (A.3) which yields p 60 8. Making the ansatz 2 2 2 (\) I (\) m ( p) +&\ =(c( p ) p + p & p \ +d( p ) p ( p \ ' +& + p + ' \& + p & ' \+ )) I m ( p),

(A.13)

then Eqs. (A.12) imply 1 m2 c( p 2 )= & 2 , 4 2p

c( p 2 )=

&1 m 2 , + 24 6 p 2

(A.14)

and therefore 2m 2 p + p& p\ p2

{ \ + 1 2m & & \6 3p + p ( p '

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

2

2

2

\

+&

+ p + ' \& + p & ' \+ )

?

= 8 d( p)

(\) m

.

(A.15)

For I(\) m( p) +&\_ , we repeat this calculational scheme again. For the reasons pointed out above, now three conditions, 2 (\) \ I (\) m ( p) +&\ =m I m ( p) +& , 4 (\) + \ I (\) m ( p) + \ =m I m ( p),

(A.16)

and p +p &p \p _I (\) m ( p) +&\_ =

p 8 (\) I ( p), 16 m

must hold. Therefore, the ansatz 2 2 2 I (\) m ( p) +&\_ =(e( p ) p + p & p \ p _ + f ( p ) p ( p \ p _ ' +& + p \ p + ' _&

+ p \ p & ' _+ + p + p _ ' \& + p & p _ ' \+ + p + p & ' \_ ) + g( p 2 ) p 4(' +\ ' &_ +' +_ ' &\ +' +& ' \_ )) I (\) m ( p)

(A.17)

leads to 1 3m 2 m 4 e( p 2 )= & 2 + 4 , 5 5p 5p

f ( p 2 )=

1 1 m2 m4 g( p 2 )= & 2+ 4 , 15 16 2 p p

\

+

&1 7m 2 m4 & , + 2 40 60 p 15p 4

(A.18)

188

NICOLA GRILLO

so that I (\) m ( p) +&\_ =

{\

1&

1 7m 2 m 4 3m 2 m 4 + 4 p + p & p\ p_ & & + p2 2 p p 8 12p 2 3 p 4

+

\

+

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

\

1 ? m2 m4 & 2 + 4 p 4(' +\ ' &_ +' +_ ' &\ +' +& ' \_ ) d( p) (\) m . 48 6 p 3p 10

+

=

(A.19)

APPENDIX B: THE I

(\)

(p) } } } -INTEGRAL

The products of JordanPauli distributions of Eq. (4.3) are evaluated in momentum space, C (\) } | } ( p) =

1 (2?) 2

| d q D 4

(\) 0

( p&q) D (\) m (q)

=

&1 d 4q $(( p&q) 2 ) 3(\( p 0 &q 0 )) $(q 2 &m 2 ) 3(\q 0 ) (2?) 4

=:

&1 I (2?) 4

|

(\)

( p)

(B.1)

and C (\) } | : ( p) =

1 (2?) 2

| d q D 4

(\) 0

( p&q)(&iq : ) D (\) m (q)

=

+i d 4q q : $(( p&q) 2 ) 3(\( p 0 &q 0 )) $(q 2 &m 2 ) 3(\q 0 ) (2?) 4

=:

+i I (2?) 4

|

(\)

( p) : .

(B.2)

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

(+)

|

( p 0 )= d 4q $( p 20 &2 p 0 q 0 +q 20 &q 2 ) 3( p 0 &q 0 ) 3(q 0 ) =

|

d 3q $( p 20 &2 p 0 E q +m 2 ) 3( p 0 &E q ), 2E q

$(q 0 &E q ) 2E q (B.3)

189

SCALAR MATTER COUPLED TO QUANTUM GRAVITY

with E q =- q 2 +m 2. The $-distribution implies p 0 =E q + |q|. From E 2q =q 2 +m 2 = ( p 0 & |q| ) 2, we obtain |q| =( p 20 &m 2 )2 p 0 , which, due to p 0 >0, yields 3( p 20 &m 2 ) and

I

(+)

|

( p 0 )=2?3( p 20 &m 2 ) 3( p 0 )

d |q|

|q| 2 $( p 20 &2 p 0 E q +m 2 ) Eq

dE q

p 2 +m 2 |q| $ Eq & 0 2 p0 2 p0



0

|

=2?3( p 20 &m 2 ) 3( p 0 )



m

\

=

? 3( p 20 &m 2 ) 3( p 0 ) - E 2q &m 2 | Eq = ( p20 +m2 )2 p0 p0

=

? 3( p 20 &m 2 ) 3( p 0 )( p 20 &m 2 ) 2 p 20

+

? m2 = 3( p 20 &m 2 ) 3( p 0 ) 1& 2 . 2 p0

\

+

(B.4)

Therefore, in an arbitrary Lorentz frame,

I

(\)

? m2 ( p)= 3( p 2 &m 2 ) 3(\p 0 ) 1& 2 . 2 p

\

+

(B.5)

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

(+)

( p0)0 =

d 3q

| 2E

E q $( p 20 &2 p 0 E q +m 2 ) 3( p 0 &E q )

q

=2?3( p 20 &m 2 ) 3( p 0 )

=

=

? 3( p 20 &m 2 ) 3( p 0 ) p0

|



d |q|

0

|

|q| 2 p 2 +m 2 Eq $ Eq & 0 2 p0 Eq 2p 0

\



\

dE q |q| E q $ E q &

m

p 20 +m 2 2 p0

+

+

? p 2 &m 2 p 20 +m 2 3( p 20 &m 2 ) 3( p 0 ) 0 p0 2 p0 2 p0

? m2 = p 0 3( p 20 &m 2 ) 3( p 0 ) 1& 2 4 p0

\

+\

1+

m2 . p 20

+

(B.6)

190

NICOLA GRILLO

Therefore, in an arbitrary Lorentz frame we have I

(\)

? m4 ( p) : = p : 1& 4 3( p 2 &m 2 ) 3(\p 0 ). 4 p

\

+

(B.7)

ACKNOWLEDGMENT I thank Professor G. Scharf, Adrian Muller and Mark Wellmann for valuable discussions and comments regarding these topics.

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