# How to determine matching coefficients in a Effective Field Theory?

Assume that I have the Lagrangian $$\mathcal{L}_{UV} =\frac{1}{2}\left[\left(\partial_{\mu} \phi\right)^{2}-m_{L}^{2} \phi^{2}+\left(\partial_{\mu} H\right)^{2}-M^{2} H^{2}\right] -\frac{\lambda_{0}}{4 !} \phi^{4}-\frac{\lambda_{2}}{4} \phi^{2} H^{2},$$ where $$\phi$$ is a light scalar field with mass $$m_L$$ and $$H$$ a heavy one with mass $$M$$. Let the Lagrangian of the effective field theory (EFT) be $$\mathcal{L}_{EFT} = \frac{1}{2}\left[\left(\partial_{\mu} \phi\right)^{2}-m^{2} \phi^{2}\right]-C_{4} \frac{\phi^{4}}{4 !}-\frac{C_{6}}{M^{2}} \frac{\phi^{6}}{6 !}.$$

Assume that I have calculated the $$4$$-point function up to $$1$$-loop order and regularized it correctly (renormalization scale $$\mu$$). The results are: \begin{align*} \mathcal{M}_{4}^{\mathrm{EFT}} &=-C_{4}+\frac{C_{4}^{2}}{32 \pi^{2}}[f(s, m)+f(t, m)+f(u, m)] \\ &+\frac{3 C_{4}^{2}}{32 \pi^{2}}\left(\log \left(\frac{\mu^{2}}{m^{2}}\right)+2\right)+\frac{C_{6} m^{2}}{32 \pi^{2} M^{2}}\left(\log \left(\frac{\mu^{2}}{m^{2}}\right)+1\right)\\\\ \mathcal{M}_{4}^{\mathrm{UV}} & \approx-\lambda_{0}+\frac{3 \lambda_{0}^{2}}{32 \pi^{2}}\left(\log \left(\frac{\mu^{2}}{m^{2}}\right)+2\right)+\frac{3 \lambda_{2}^{2}}{32 \pi^{2}}\left(\log \left(\frac{\mu^{2}}{M^{2}}\right)\right)+\frac{m^{2} \lambda_{2}^{2}}{48 \pi^{2} M^{2}} \\ &+\frac{\lambda_{0}^{2}}{32 \pi^{2}}[f(s, m)+f(t, m)+f(u, m)]. \end{align*}

The matching at tree-level resulted in: $$m^2=m_L^2,\qquad C_4 = \lambda_0,\qquad C_6=0.$$ I would now like to perform the matching at one-loop, i.e. we again demand $$\mathcal{M}_4^{EFT}= \mathcal{M}_4^{UV}+O(M^{-2})$$.

### Problem

We have two unknowns, $$C_4$$ and $$C_6$$, that need to be expressed in terms of $$\lambda_0, \lambda_2, m, M, etc.$$. But $$\mathcal{M}_4^{EFT}= \mathcal{M}_4^{UV}+O(M^{-2})$$ gives us only one equation.. I don't see how we can determine both coefficients with only the above information.

### Notes

I'm reading Adam Falkowski's lecture notes, see here. In section 2.3, p.~24, he performs the matching with only the above information...

We need one observable for each EFT quantity that we want to match to the UV. The physical mass $$m_\text{phys}^2$$ allows to 1-loop match the EFT mass $$m$$ to the UV quantities, and the 2-2 scattering amplitude allows to 1-loop match the EFT coupling $$C_4$$ to the UV quantities. If you wanted to 1-loop match the EFT coupling $$C_6$$, you'd need another equation, as you correctly observe, coming from an observable such as the 3-3 scattering amplitude.
The 3-3 scattering amplitude is very messy and the author doesn't even give the calculations for the tree-level matching with the $$\lambda_1$$ coupling turned on, but only the results (eq. $$2.16$$).
If you want to match the coupling $$C_6$$ without the messy calculations, the author provides in part 3 a functional method to obtain it far more easily: the result is in eq. $$3.47$$.