Approaches to Quantum Gravity

456 C. Burgess) E ( m) P( ( 1 2L EA(E) ∼ v 2 m 2 v 4πv)m( ) 1 E ( ) E 2L ∏ ∏( ) E (d−2)Vid∼ v 2 E 2 , (23.9)v 4πvmiwhere P = 2 + 2L + ∑ id (d − 2)V id. This is the main result, since it shows whichgraphs contribute to any order in E/m using a nonrenormalizable theory. 3To see how eqs. (23.9) are used, consider the first few powers of E in the toymodel. For any E the leading contributions for small E come from tree graphs,i.e. those having L = 0. The tree graphs that dominate are those for which∑ ′id (d−2)V id takes the smallest possible value. For example, for 2-particle scatteringE = 4 and so precisely one tree graph is possible for which ∑ ′id (d −2)V id = 2,corresponding to V 44 = 1 and all other V id = 0. This identifies the single graphwhich dominates the 4-point function at low energies, and shows that the resultingleading energy dependence in this case is A(E) ∼ E 4 /(v 2 m 2 ), as was alsofound earlier in the full theory. The numerical coefficient can be obtained in termsof the effective couplings by more explicit evaluation of the appropriate Feynmangraph.The next-to-leading behaviour is also easily computed using the same arguments.Order E 6 contributions are achieved if and only if either: (i) L = 1andV i4 = 1, with all others zero; or (ii) L = 0and ∑ )i(4V i6 + 2V i4 = 4. Sincethere are no d = 2 interactions, no one-loop graphs having 4 external lines can bebuilt using precisely one d = 4 vertex and so only tree graphs can contribute. Ofthese, the only two choices allowed by E = 4atorderE 6 are therefore the choices:V 46 = 1, or V 34 = 2. Both of these contribute a result of order A(E) ∼ E 6 /(v 2 m 4 ).Besides showing how to use the effective theory to compute to any order inE/m, eq.(23.9) also shows the domain of approximation of the effective-theorycalculation. The validity of perturbation theory within the effective theory reliesonly on the assumptions E ≪ 4πv and E ≪ m. Inparticular,itdoesnot rely onthe ratio m/4πv = λ/4π being small, even though there is a factor of this orderappearing for each loop. This factor does not count loops in the effective theorybecause it is partially cancelled by another factor, E/m, which also comes withevery loop; λ/4π does count loops within the full theory, of course. This calculationsimply shows that the small-λ approximation is only relevant for predictingthe values of the effective couplings, but are irrelevant to the problem of computingthe energetics of scattering amplitudes given these couplings.d>23 It is here that the convenience of dimensional regularization is clear, since it avoids keeping track of powers ofa cutoff like , which drops out of the final answer for an observable in any case.