Beyond renormalization: a first look at EFT

I’ve started learning effective field theory (EFT), with a long-term view of finding possible applications to the experimental study of data excesses connected to simplified SUSY models and jet physics (substructure), my current areas of research. Here I want to lay out some basic concepts of EFT, as presented e.g. in Beneke’s notes; I might give a summary of the rest of the paper (which concerns heavy quark effective theory) in a subsequent blog post, as that seems like a good introductory example (or maybe even look at the Minimal Composite Higgs Model, following Ben Gripaios’ notes).

Dimensions

Imagine we’re doing a perturbative analysis of some process beyond tree-level, and rescale all loop momenta $k_i$ to $\lambda k_i$, ultimately taking $\lambda\to\infty$ — which should be the same as integrating $I(k)$ over all $k$. Then we have , where $D$ is the so-called superficial degree of divergence. A standard example would be

where I’ve indicated the mass dimensions of the relevant terms. Logarithmic divergence is achieved for $D=0$, while for $D>0$ the quantity $I$ would certainly diverge. Let’s now recast the Lagrangian of our theory in terms of operators $\mathcal{O}$, together with their respective couplings $\lambda$ and proper dimensions $d$ (and some mass scale cut-off $M$) :

For a field of type $f$ (whatever that is), a propagator would look like

for $k\ll m$, and where $s_f$ is a spin factor. In fact, $s_f=0$ for scalar or massless vector fields, $s_f=1/2$ for fermions and $s_f=1$ for massive vector fields. The latter causes problems and so we decide to simply exclude it from our theory. Furthermore, calling $L$ the number of loops in a given diagram, $I_f$ the number of internal $f$-type lines, $E_f$ the external ones, $V_i$ the number of $i$-type vertices, $a_i$ the number of derivatives in and finally $n_{if}$ the number of $f$-type fields in , the following relations hold (and can be verified by staring at Feynman diagrams long enough) :

Putting them all together, we obtain an interesting expression for $D$, in the form $D=4-\epsilon(i,f)$ :

Renormalizability

Why is this formula interesting? Because now, we can characterize the renormalizability of our theory (and the degree of divergence of relevant loops) in terms of the dimensions of the operators that go into it.

Super-renormalizable : $d_i<4$

In this case, $D$ decreases with increasing $V_i$, that is, only a finite number of diagrams are actually divergent.

Renormalizable : $d_i=4$

$D$ is now independent of $V_i$, and can be driven to negative values simply by adding enough external lines : only a finite number of couplings is needed.

Non-renormalizable : $d_i>4$

Now diagrams with any number of external lines are susceptible to divergences if vertex $i$ occurs sufficiently often : we must include all $\mathcal{O}_i$ as counterterms, up to arbitrary dimension.

Excelsior (Finite) ad Theoriam!

What have we learnt? Well, for one thing, we’ve recovered (thankfully) the well-known QFT result that only theories with $d_i\le 4$ (for all operators) are potential candidates for a fundamental theory valid at all energy scales. Indeed, these operators come with physically-sensible couplings that are either dimensionless or have positive mass dimension.

But consider now a scattering amplitude with mass dimension $A$ and a diagram with $n$ insertions of $\mathcal{O}_i$ ($d_i>4$). Then the contribution is of order

where $E$ is the scale of external momenta (roughly the energy scale of our experiment). Only a finite number of non-renormalizable interactions are relevant for $E\ll M$. And so we have our first import result concerning EFTs :

EFTs are non-renormalizable theories valid for $E\ll M\approx$ scale of non-renormalizable interactions.

The effect of states lying at energies $\gtrsim M$ is local in experiments at energies $\lesssim E$, and so a change in UV physics is translated as a change in the $\lambda_i$. Yet, since $\mathcal{L}$ contains all possible $\mathcal{O}_i$ and since $\lambda_i$ is determined from experimental data, the description is UV-insensitive (high-energy fluctuations are “integrated out” and reside in the values of $\lambda_i$).

In the absence of a full, fundamental “Theory of Everything”, we now have a tool to probe deviations from well-accepted partial models (such as the Standard Model of particle physics) through the introduction of higher-order operators up to some arbitrary dimension cut-off. Symmetries become all the more important since they protect our theories from super-renormalizable interactions (for instance, the mass term of a scalar field is $\sim\lambda_2M^2$ and so would either be non-dynamical, or require unnatural fine-tuning $\lambda\ll 1$).

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