1.7 Dimensional Regularization of the One Loop
1.8.1 Two and Multi Loops
Consider the two-loop contribution. 2 is the expansion parameter in the loop expansion. With „ set equal to one, the divergent part in two loops must be proportional to the dimensionless integral:
2 Z
.dx/p
g F.x/; (1.8.19)
50[59]. Unitarity (positivity) of such a theory in a non-perturbative setting has been elaborated upon by Tomboulis [63].
51For the extension of the above alternative to higher order loops with a field redefinition, in the process of renormalization, such that the right behavior of a propagator forp2 ! 1 is maintained thus avoiding non-positivity conditions, by expanding the metric around the Minkowski one:gDCh, see [2]. See also [29].
where, because of dimensional reasons,F.x/must be of dimensionalityŒLength6. Clearly, any invariant of the form proportional toR.x/, orR.x/, i.e., having the structuresR.x/R.x/R.x/R.x/cannot contribute toF.x/on the mass shell. On other hand, any term having a structure such as:
pg
rR rR
; p
g rR
rR p ;
g
rR rR
; p
g
rR rR
; (1.8.20)
may, from Problem1.10, be written as a total derivative@.pg/,onthe mass shell, for some vector, and hence cannot contribute to the integral in (1.8.19).
Accordingly, only invariant products of three Riemann tensors, with each tensor involving two derivatives, may contribute toF.x/ above. In Appendix D of this chapter, we prove the remarkable result that all such invariants, on the mass shell, are proportional to the invariant
RRR; (1.8.21)
from which, we may infer that the divergent part of the effective action in two loops must be proportional to
2 Z
.dx/p
g.x/R.x/R.x/R.x/: (1.8.22) On the other hand, since the one-loop contribution is finite, the divergence, in the two-loop contribution, must come from an overall integration as all the sub- integrations are finite by dimensional regularization, implying that the two-loop divergence must be proportional to a single power of 1="instead of a quadratic expression for the leading term.52Accordingly the general expression for the two- loop divergent part of quantum general relativity may be written to have the explicit form
Œ 2 ˇˇˇ
divD c2 2
"
1 .16 2/2
Z .dx/p
g.x/R.x/R.x/R.x/; (1.8.23) involving only one numerical constantc2. If the latter constant were zero the theory would be two-loop finite. Unfortunately, as has been confirmed by two groups,53 the constantc2turns up to be not zero, implying that perturbative quantum general relativity is non-renormalizable as we now discuss.
We note that the Einstein-Hilbert action contain terms of second order in derivatives in the integrand, whileRR,R2,RRR : : :are of higher
52For the spelling out of this underlying technical detail see [17,18,60].
53Goroff and Sagnotti [30], van de Ven [65]. The computation of the coefficientc2is quite tedious and these two groups had to resort to the computer for its evaluation.
orders. If such terms are added to the original integrand, one may be then tempted to combine the coefficientsNc1;dN1;cN2in
Z .dx/p
gh
cN1RRC Nd1R2C Nc2RRRC i
(1.8.24) with such terms as in (1.8.15), (1.8.23), to define new coefficients which are to be determined experimentally. However, such terms, in turn, will, in general, generate divergent terms in the effective action of arbitrary high order loops, giving rise to an infinite number of parameters that are to be determined experimentally, and the theory would lose its predictive power. This is what is meant that the theory is perturbatively non-renormalizable. It is traced back to the fact that the expansion coupling2 has a dimensionality of positive power of length: ŒLength2. Non- renormalizability still holds when scalars, Yang-Mills fields field as well as the photon, are included in the theory. Here it is important to note thatR ¤ 0 in the presence of such fields and finiteness even in one-loop fails.54
Let us investigate how the divergences in the perturbative loop-expansion changes with the increase of the number of loops. To this end, we note by working in the momentum presentation, that the graviton propagator given in (1.3.18), (an internal line to a graph) carrying a momentumpgoes like1=p2, and a vertex involves two derivatives and goes likep2. Two lines joining two vertices generate a loop with momentum integration goes likep4. Three lines joining two vertices generate two loops. Moreover, a graph55involving`intlines,Lloops, andV vertices satisfy the relation
L D `int V C 1: (1.8.25)
By power counting, we may define the superficial degree of divergence of a graphg by
d.g/ D 4L C 2V2 `int; (1.8.26) where the factor 4 multiplying the number of loopsLcorresponds to the dimension of spacetime. The factor 2 multiplying the number of verticesVcorresponds to the two powers of momenta in a vertex part. The factor2multiplying the number of lines`int corresponds to the two powers of the inverse of the momenta associated with a propagator. From (1.8.25), (1.8.26), we may solve ford.g/in terms of the number of loopsL:
d.g/ D 2 C 2L: (1.8.27)
54See, e.g., [20], also [61].
55Here one is considering a connected proper graph, where proper means that it cannot become disconnected by cutting a single line.
Thus we see that the superficial degree of divergence increases without bound with the number of loops.
The increase of the superficial degree of divergence d.g/ with the number of loops, naturally urges one to investigate the role of the renormalizabilty of a corresponding quantum gravity involving higher order derivatives terms added to the Einstein-Hilbert action. This is discussed next.