Homework:

1) Composite Operator Insertions

How do we calculate diagrams with insertions of composite operators?

Iain:

You can think of an operator insertion as just an additional Feynman rule, where you insert that operator once. The idea behind the Composite Operator Renormalization homework problem is that you can make this very explicit, by simply thinking about the composite operator as an additional term that you add to the Lagrangian. If you do not want to think about it this way, but rather simply as an operator, then the review by Buras (see entry for Chapter 5 on the syllabus) goes through this in detail.

Sam:

This is partially addressed in MQ3 as well, quoted below, which uses inspiration from Peskin and Schroeder as well as Collins' book.

``Doing the source procedure for composite fields is actually a bit tricky, so I hope that one of this problem can help elucidate the procedure. First, we add $\int d^4 x J_{\phi^2}(x) \phi_0^2(x)$ to the action.

I think about this as simply adding an extra Feynman rule: instead of trying to do the path integral exactly, imagine that $J(x)$ behaves like a field.

In this naive picture, we have a new Feynman rule where the legs of a diagram can include $J$s, and we have a 3-point vertex which couples two $\phi$s to a $J$.

In momentum space, this is a bit easier for me to think about, and the Feynman rule must satisfy momentum conservation as usual.

Since $\int d^4 x J_{\phi^2}(x) \phi_0^2(x) \int \frac{d^4 k}{(2\pi)^4} \widetilde{J_{\phi^2}}(-p) \widetilde{\phi_0^2}(p)$, we can find the result of inserting $\phi^2(x)$ by taking functional derivatives with respect to $\widetilde{J_{\phi^2}}(-p)$ to get insertions of $\widetilde{\phi_0^2}(p)$, setting $J_{\phi^2} = 0$, and then Fourier transforming to find the result of inserting $\phi^2(x)$.''

EFT Concepts:

1) Why is the MS scheme (or MS bar) valid? Why can we simply ignore contributions from charged particles in the full theory to effects such as anomalous dimensions, beta functions, etc.

Let's work in the example of QED. Once we integrate out the electron at the scale

m_e

in QED we no longer have charged particles in the theory that can cause the coupling to run. At that stage we have the MS bar coupling

$\begin{array}{l}\overline\alpha(m_e)\end{array}$ and there is no beta function for mu < m_e. One can choose to convert between

$\begin{array}{l}\overline\alpha(m_e)\end{array}$ and the onshell coupling with fixed order perturbation theory if desired.

2) We mentioned that beta functions and $\begin{array}{l}\Lambda_{\rm QCD}\end{array}$ are scheme dependent beyond two loops. Is there an intuitive explanation for this?

You should think of "scheme dependence" as "depends on the definition". From that point of view its not surprising when things are scheme dependent, what is surprising is that there is a universality where different definitions give the same result at low loop orders. If one compares a "mass-independent" scheme like $\begin{array}{l}\overline{MS}\end{array}$ to a "mass-dependent" scheme like off shell momentum subtraction, then a quantity like the coupling or $\begin{array}{l}\Lambda_{\rm QCD}\end{array}$ will differ already at one-loop, because the mass of particles plays a role in one and not the other. If we alternatively consider a range of different "mass-independent" schemes (of which MS and $\begin{array}{l}\overline{MS}\end{array}$ are just two) then we've restricted ourselves to a smaller range of variations, such that the (mathematical) outcome is that the first two beta-function coefficients are the same across these schemes. Why its two loop and not one-loop versus three-loop can be thought of as a mathematical result.

3) We interpret the epsilon poles of dimensional regularization as logarithmic divergences. Why is this?

Heuristics

One way of thinking about this is that if we have power divergences, like Lambda^2 in a hard cutoff regularization scheme, these are usually associated with pieces of Feynman integrals which have no scale dependence. These vanish in dimensional regularization.

For dimensional regularization, we sometimes have to be a bit more careful. For example, consider an integral in dimensional regularization with no scale, but which has logarithmic divergences in both the UV and the IR. These depend on an IR and a UV value of epsilon. In dimensional regularization, the IR and UV parts of a logarithmic divergence are both associated with a 1/epsilon pole, and they cancel! However, when calculating counterterms, we should subtract off only the UV piece. In other words, the UV divergences are, in some sense, ``fake'', and can be renormalized away. The IR divergences are real divergences which have physical effects, and need to be treated more carefully.

Example: phi^4 theory

Let's look a bit more carefully at the example of phi^4 theory, and think about how different regularization schemes can play with the concepts of EFT. In phi^4 theory, or for the Higgs potential in the Standard Model, people do discuss the quadratic cutoff sensitivity for the mass term, which is known as the hierarchy problem.

If you look at either of these problems from the point of view that you know the power counting, and hence know the solution you're perturbing about, then the question becomes whether the power counting is preserved by your treatment of the theory, such as how you regulate it. If you use dimensional regularization, then the power counting is preserved, and you do not see the quadratic divergence that would seem to mess up the power counting in bare quantities, so the power counting is simpler. If you use cutoff regularization, then the power counting is not preserved since there is a quadratic divergence, and dealing with power counting is more complicated (it can be restored for renormalized quantities). In both the dimreg and cutoff pictures, the logs come out the same, so your question is really about how to interpret the power law cutoff dependence in the cutoff picture, given its absence in the standard dim.reg. picture with $\begin{array}{l}\overline{MS}\end{array}$ .

In fact, one can see power law dependence in dimreg for the Higgs mass, if one introduces a new physical scale. In particular, if one has any physics beyond the standard model, such as heavy particles in some GUT theory, then this physics introduces a high energy scale M, and the Higgs mass will have quadratic sensitivity to this physical mass scale M. So one sees the hierarchy problem, even in dimensional regularization, with the cutoff replaced by a physical parameter. If one just uses the standard model with dimensional regularization (ignoring also the high energy scale M_Planck induced by gravity) then one has no problem with the fine tuning reappearing in loops, and indeed this is what people use for precision calculations with SM particles. (You can actually see power law cutoff behavior in dim.reg. schemes other than $\begin{array}{l}\overline{MS}\end{array}$ , and we will talk about this in Chapter 10, in a context of an example where it is useful.)

Thus, the way to think about the quadratic divergence issue for the hierarchy problem is that it challenges whether we have the right to think about using a small Higgs mass in the first place. Once we accept the fine tuning of having a small Higgs mass, then the next question is whether the power counting is maintained at loop level. In the absence of any high scale physics, then the power counting is maintained. But since there are various clues for new physics above the weak scale, it is a puzzle why the Higgs boson is so light compared to those (hypothetical) new physics scales.

4) When integrating out heavy fields, do we integrate out both high-energy ``on-shell'' modes and low-energy ``off-shell'' modes?

Yes indeed, when you integrate out a massive particle in continuum EFT you are removing both the higher energy modes, k > m, and the lower energy off-shell modes, k<m. This is only a valid procedure if the physics you are interested in is for momenta far from the mass, since otherwise you can probe the full propagator for the massive particle, 1/(k^2 - m^2), which is not captured by the truncated Taylor series expansion encoded in the EFT. That is what is meant by the statement that the EFT is only valid up to the mass scale of the particle that is being integrated out.

In the Wilsonian case we indeed could consider only integrating out the modes of the heavy particle with k > m. In this case there would still be a field left in the path integral that describes the off-shell modes of the heavy field which have k < m. This field would not have a standard propagator, but could effect the dynamics of other particles which do beyond the lowest order.

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FAQ: Loops, Renormalization, and Matching