Last week I introduced the basics of EFT from the perspective of renormalization; today, I’m considering a practical example : getting an EFT for QCD without the top quark, that is, trying to simplify QCD at energy scales $\lesssim m_t\simeq 172$ GeV.

## Integrating out the top quark

Rewrite the QCD Lagrangian, isolating the heavy $Q$ (top quark) field :

and assume $p_ip_j\ll m_t^2$ for all external momenta (there is no external $Q$ line). Now we can decompose the other gluon and quark fields into low- and high-frequency modes, expressing the partition function as

with

(The LHS of the previous equation can be expanded out in local operators, while the RHS shows how we “integrated out” the heavy fields and high energy modes.) In most cases, $S_\text{eff}$ can be constructed only perturbatively : ## Matching

Consider for instance the gluon 2-point function : It is exactly reproduced for $\mathcal{L}_\text{eff}=-\frac{1}{4}G^2+\sum_f\bar{\psi}_f(i\cancel{D}-m)\psi_f$, having used dimensional regularization followed by $\overline{\bf MS}$ renormalization. There is no explicit high-frequency cut-off, as the high-frequency modes appear only in diagrams with $Q$-lines that contain the scale $m_t$. Matching the third diagram with its expansion (in $q^2/m_t^2$) in terms of local operators $\mathcal{O}_i$, we get

with

The effective Lagrangian is then

and we see right away that the $d_i=4$ term has been modified, while a new $d_i=6$ term has been generated! Rescaling the gluon fields, we recover the canonical kinetic term, but obtain a new strong coupling in the effective theory :

## Scales

$\mathbf{E>m_t}$

use $\mathcal{L}$ : $\mu^2\dfrac{\mathrm{d}\alpha_s}{\mathrm{d}\mu^2}=-\beta_0\dfrac{\alpha_s}{4\pi},\quad\beta_0=11-\dfrac{4}{3}\cdot 6$.

$\mathbf{E\sim m_t}$

define $\hat{\alpha}_s$ from $\alpha$ as above and write : $\mu^2\dfrac{\mathrm{d}\hat{\alpha}_s}{\mathrm{d}\mu^2}=\mu^2\dfrac{\mathrm{d}\alpha_s}{\mathrm{d}\mu^2}-\dfrac{\alpha^2_sT_f}{3\pi}=-\beta_0^{(5)}\dfrac{\hat{\alpha}_s^2}{4\pi}+\mathcal{O}(\alpha_s^3),\quad\beta_0^{(5)}=11-\dfrac{4}{3}\cdot 5$.

$\mathbf{E<m_t}$

here we must use $\mathcal{L}_\text{eff}$, otherwise for $\mu\sim p\ll m_t$ we would get $\alpha_s\ln\frac{m_t^2}{p^2}$ and perturbation theory would break down! On the other hand, these high-energy logarithms are easily absorbed in $\hat{\alpha}_s(p)$ when working with $\mathcal{L}_\text{eff}$.

## Effective theories of QCD… coming soon!

Starting with good ol’ QCD, there are a number of EFTs we can develop. Adding new fields (non-perturbatively), one can focus on symmetries and consider the weak coupling case – interesting results concern the chiral Lagrangian and effective theories for nucleons. Alternatively, one could go down the perturbative matching road (like we started doing). There are distinctions to be made between the weakly and strongly coupled approximations, but one can come up with theories like perturbative non-relativistic QCD (PNRQCD), heavy quark effective theory (HQET) or soft-collinear effective theory (SCET), which apply to high-temperature/density situations.

The latter two are of particular interest to me. I’ll say a bit more about HQET in a future post (soon, I promise…), while keeping SCET for the end (and perhaps at that point I’ll actually come up with a good idea or two about jets that I could apply to my day-to-day research).

Here’s a teaser, while I think about the next post – now that we’ve integrated the top quark out of QCD, can we do the same for the bottom quark? That is, discarding Feynman diagrams with external bottom lines, can we build up a valid EFT? You can start thinking about the physics of such a theory, but remember we haven’t mentioned the electroweak bosons yet… What issue might arise then?