1.7 Dimensional Regularization of the One Loop
1.8.3 The Low Energy Regime: Quantum GR as an Effective
One is led to believe that Einstein’s general relativity is a low energy effective theory as the low energy limit of a more complicated theory, and as such it provides a reliable description of gravitation at low energies.
64This is in the spirit of Wilson’s [68] (see also [69] ) non-perturbative analysis of renormalizability of a theory which treats renormalizable and non-renormalizable theories on equal footings.
65This is treated in Chap.2.
66See, e.g., Sect.2.10.
67See, e.g., [19].
One may argue that the non-renormalizability of a quantum theory based on GR is due to the fact that one is trying to use it at energies which are far beyond its range of validity. As a matter of fact the derivatives occurring in the theory, in a momentum description, may be treated as small at sufficiently low energies treating it as an effective theory. Several applications on such an approach have been carried out as determining corrections to Newton’s gravitational potential.
The modified terms beyond the Einstein-Hilbert action as counter terms needed to redefine coupling parameters in a perturbative setting, through the process of renormalization discussed above in this section, involve higher order derivatives, and in a momentum description involve powers of momenta. In this sense, one may expect that such modifications are not important at low energies or equivalently in computations dealing with the long distance behavior of the theory, and one may expect, in turn, that the Einstein-Hilbert action to dominate in this regime.
As ultraviolet divergences are associated with the small distance behavior of the theory, one tries to separate low energy effects from high energy ones, in view of applications in the low energy regime even if the theory has unfavorable ultraviolet behavior such as in quantum gravity. For interesting applications of such effective theories, in long distance applications, see, e.g., Bjerrum-Bohr et al. [15, 16];
Donoghue [25–27].
The Einstein-Hilbert action, in the presence of matter, may be written as W D 1
2
Z pg.x/ .dx/R.x/C Z p
g.x/ .dx/Lmatter.x/; (1.8.30) where for a scalar field, for example, the expression for Lmatter.x/ is given in Problem 1.11. To treat pure GR theory, as an effective action, one includes allpossible higher order couplings involving the Riemann tensor R˛ˇ and its contractions, and make the replacement
1 2
pg R!p gh R
2 Ca1R2Ca2RRC i
; (1.8.31)
the coefficients of which are then adjusted after eliminating the divergences in each loop computation. With the divergent term of the Einstein-Hilbert action given in (1.8.15), the elimination of the divergence in the loop level is not difficult. One must, however include higher derivative contributions to the matter Lagrangian density as well.
For a scalar field contribution, one must also simultaneously carry out a similar generalization leading to such a modification:
p gh1
2
g@@Cm22i
! pgh
1 2
g@ @Cm22
Cb1@ @RC C
c1g@ @Cc2m22
RC i
; (1.8.32)
in discussing finiteness problems in loop expansions of the theory in the presence of a scalar field.
At this stage assuming the validity of a perturbation expansion to second order in Newton’s coupling constant68GN, we may use a dimensional analysis to write down corrections that may arise in the modification of Newton’s static potential UN.r/D GNm1m2=rfor the interaction between two massive spin-0 particles of massesm1 andm2separated by a finite distancer. The two possible corrections to Newton’s potential to second order in GN, based on dimensional analysis alone, are then
G2Nm1m2.m1Cm2/
c2r2 ; G2Nm1m2„
c3r3 ; (1.8.33)
conveniently parametrized. We note that a quantum correction naturally arises here proportional to„. Accordingly, to second order in GN, the modified static potential between the two particles becomes
U.r/D GNm1m2 r
h1C˛GN.m1Cm2/
c2r CˇGN„ c3r2
i; (1.8.34)
where˛andˇare dimensionless constants.
As computations of amplitudes are done in the momentum description, and for a static potential computations are carried out in 3 dimensional space, we may perform a 3 dimensional Fourier transforms of the1=r2and1=r3terms in (1.8.33) to investigate the nature of contributions that arise in a long distance treatment of the above problem.
To the above end, we recall that Z
d3reikr r D 4
k2 : (1.8.35)
From dimensional analysis Z
d3reikr r2 Da 1
jkj; (1.8.36)
whereais a constant to be determined. Using the well known relation
@
@ki
@
@ki 1
jkj D 4 ı3.k/; (1.8.37)
68That is, in particular, no corrections are assumed having the structures GNln.GN/, G2Nln.GN/.
and applying the.@=@ki/.@=@ki/to both sides of (1.8.36) gives .2 /3ı3.k/ D 4 aı3.k/, from whichaD2 2. That is
Z
d3reikr
r2 D 2 2
jkj 2 p 2
k2: (1.8.38)
The interesting case by far is the Fourier transform of1=r3. Here we are dealing with a dimensionless integral and hence the answer must be logarithmic ink2:
Z
d3reikr
r3 Dbln.k2/: (1.8.39)
Upon applying.@=@ki/.@=@ki/to both sides of this equation gives
Z
d3reik:r r Db 2
k2; (1.8.40)
which from (1.8.35) givesbD 2 . That is69 Z
d3reikr
r3 D 2 ln.k2/: (1.8.41)
The moral of the above Fourier transforms is that only non-analytic terms such as ln.k2/; 1=p
k2, contribute to the corrections to the potential at low energy (long distance). As a matter of fact analytic terms as in
Z d3k .2 /3eikr
1;k2; : : :DŒ ı3.r/;r2ı3.r/; : : :; (1.8.42) which vanish far away from the origin, and originate from the ultraviolet (short distance) behavior of the theory.
Thus by trying to separate low energy effects from high energy ones, in view of applications in the low energy regime even if the theory has unfavorable ultraviolet behavior, one has been able to evaluate expressions like in (1.8.34) at long distances.70
69We note that the integral may not be of a higher power in ln.k2/.
70For further details, see the above mentioned references. Recent recorded values of ˛ andˇ in (1.8.34) are˛D3, andˇD41=10 [15].