The effective Lagrangian logic

2.6 The effective Lagrangian logic

With the power-counting results in hand we can see how to calculate predictively – including loops – using the non-renormalizable effective theory. The logic follows these steps:

Choose the accuracy desired in the answer. (For instance an accuracy of 1% might be desired in a particular scattering amplitude.)
Determine the order in the small ratio of scales (i.e., $r = E ∕m ℛ$ in the toy model) which is required in order to achieve the desired accuracy. (For instance if $r = 0.1$ then $𝒪(r2)$ is required to achieve 1% accuracy.)
Use the powercounting results to identify which terms in $ℒeff$ can contribute to the observable of interest to the desired order in $r$ . At any fixed order in $r$ this always requires a finite number (say: $N$ ) of terms in $ℒeff$ which can contribute.
1. If the underlying theory is known, and is calculable, then compute the required coefficients of the $N$ required effective interactions to the accuracy required. (In the toy model this corresponds to calculating the coefficients $a$ , $b$ , $c$ , etc.)
2. If the underlying theory is unknown, or is too complicated to permit the calculation of $ℒe ff$ , then leave the $N$ required coefficients as free parameters. The procedure is nevertheless predictive if more than $N$ observables can be identified whose predictions depend only on these parameters.

Step 4a is required when the low-energy expansion is being used as an efficient means to accurately calculating observables in a well-understood theory. It is the option of choosing instead Step 4b, however, which introduces much of the versatility of effective-Lagrangian methods. Step 4b is useful both when the underlying theory is not known (such as when searching for physics beyond the Standard Model) and when the underlying physics is known but complicated (like when describing the low-energy interactions of pions in quantum chromodynamics).

The effective Lagrangian is in this way seen to be predictive even though it is not renormalizable in the usual sense. In fact, renormalizable theories are simply the special case of Step 4b where one stops at order $0 r$ , and so are the ones which dominate in the limit that the light and heavy scales are very widely separated. We see in this way why renormalizable interactions play ubiquitous roles through physics! These observations have important conceptual implications for the quantum behaviour of other non-renormalizable theories, such as gravity, to which we return in the next Section 3.

2.6.1 The choice of variables

The effective Lagrangian of the toy model seems to carry much more information when $θ$ is used to represent the light particles than it would if $ℐ$ were used. How can physics depend on the fields which are used to parameterize the theory?

Physical quantities do not depend on what variables are used to describe them, and the low-energy scattering amplitude is suppressed by the same power of $r$ in the toy model regardless of whether it is the effective Lagrangian for $ℐ$ or $θ$ which is used at an intermediate stage of the calculation.

The final result would nevertheless appear quite mysterious if $ℐ$ were used as the low-energy variable, since it would emerge as a cancellation only at the end of the calculation. With $θ$ the result is instead manifest at every step. Although the physics does not depend on the variables in terms of which it is expressed, it nevertheless pays mortal physicists to use those variables which make manifest the symmetries of the underlying system.

2.6.2 Regularization dependence

The definition of $ℒe ff$ appears to depend on lots of calculational details, like the value of $Λ$ (or, in dimensional regularization, the matching scale) and the minutae of how the cutoff is implemented. Why doesn’t $ℒeff$ depend on all of these details?

$ℒeff$ generally does depend on all of the regularizational details. But these details all must cancel in final expressions for physical quantities. Thus, some $Λ$ -dependence enters into scattering amplitudes through the explicit dependence which is carried by the couplings of $ℒe ff$ (beyond tree level). But $Λ$ also potentially enters scattering amplitudes because loops over all light degrees of freedom must be cut off at $Λ$ in the effective theory, by definition. The cancellation of these two sources of cutoff-dependence is guaranteed by the observation that $Λ$ enters only as a bookmark, keeping track of the light and heavy degrees of freedom at intermediate steps of the calculation.

This cancellation of $Λ$ in all physical quantities ensures that we are free to make any choice of cutoff which makes the calculation convenient. After all, although all regularization schemes for $ℒeff$ give the same answers, more work is required for some schemes than for others. Again, mere mortal physicists use an inconvenient scheme at their own peril!

2.6.2.1 Dimensional regularization

This freedom to use any convenient scheme is ultimately the reason why dimensional regularization may be used when defining low-energy effective theories, even though the dimensionally-regularized effective theories involve fields with modes of arbitrarily high momentum. At first sight the necessity of having modes of arbitrarily large momenta appear in dimensionally-regularized theories would seem to make dimensional regularization inconsistent with effective field theory. After all, any dimensionally-regularized low-energy theory would appear necessarily to include states having arbitrarily high energies.

In practice this is not a problem, so long as the effective interactions are chosen to properly reproduce the dimensionally-regularized scattering amplitudes of the full theory (order-by-order in $1∕M$ ). This is possible ultimately because the difference between the cutoff- and dimensionally-regularized low-energy theory can itself be parameterized by appropriate local effective couplings within the low-energy theory. Consequently, any regularization-dependent properties will necessarily drop out of final physical results, once the (renormalized) effective couplings are traded for physical observables.

In practice this means that one does not construct a dimensionally-regularized effective theory by explicitly performing a path integral over successively higher-energy momentum modes of all fields in the underlying theory. Instead one defines effective dimensionally regularized theories for which heavy fields are completely removed. For instance, suppose it is the low-energy influence of a heavy particle $h$ having mass $M$ which is of interest. Then the high-energy theory consists of a dimensionally-regularized collection of light fields $ℓi$ and $h$ , while the effective theory is a dimensionally-regularized theory of the light fields $ℓi$ only. The effective couplings of the low-energy theory are obtained by performing a matching calculation, whereby the couplings of the low-energy effective theory are chosen to reproduce scattering amplitudes or Green’s functions of the underlying theory order-by-order in powers of the inverse heavy scale $1∕M$ . Once the couplings of the effective theory are determined in this way in terms of those of the underlying fundamental theory, they may be used to compute any purely low-energy observable.

An important technical point arises if calculations are being done to one-loop accuracy (or more) using dimensional regularization. For these calculations it is convenient to trade the usual minimal-subtraction (or modified-minimal-subtraction) renormalization scheme, for a slightly modified decoupling subtraction scheme [148 , 123 , 124 ]. In this scheme couplings are defined using minimal (or modified-minimal) subtraction between successive particle threshholds, with the couplings matched from the underlying theory to the effective theory as each heavy particle is successively integrated out. This results in a renormalization group evolution of effective couplings which is almost as simple as for minimal subtraction, but with the advantage that the implications of heavy particles in running couplings are explicitly decoupled as one passes to energies below the heavy particle mass. Some textbooks which describe effective Lagrangians are [73 , 58 ]; some reviews articles which treat low-energy effective field theories (mostly focussing on pion interactions) are [116 , 107 , 113 , 132 , 126 , 99 , 74 ].

A great advantage of the dimensionally-regularized effective theory is the absence of the cutoff scale $Λ$ , which implies that the only dimensionful scales which arise are physical particle masses. This was implicitly used in the power-counting arguments given earlier, wherein integrals over loop momenta were replaced by powers of heavy masses on dimensional grounds. This gives a sufficiently accurate estimate despite the ultraviolet divergences in these integrals, provided the integrals are dimensionally regularized. For effective theories it is powers of the arbitrary cutoff scale $Λ$ which would arise in these estimates, and because $Λ$ cancels out of physical quantities, this just obscures how heavy physical masses appear in the final results.