Introduction Effective Field Theories
In nature there are different energy scales. At different scales, the behavior can be described by different degrees of freedom. At a lower energy $\Lambda_L$ e.g. some degrees of freedom of a theory from a higher scale $\Lambda_H$ might not be resolved. The behavior at a lower energy scale can be characterized by an Effective Field Theory (EFT)
Definition: Effective Field Theory (EFT) is a field theory where the expansion in $\frac{\Lambda_L}{\Lambda_H}$ is implemented at the level of the lagrangian, which is build out of the effective d.o.f.
The general EFT lagrangian $\mathcal{L}_\text{EFT}$ is suitable to describe the system $H$ at energies lower than $\Lambda_H$. It is characterized by
- A cutoff $\mu$ with $\Lambda_H \gg \mu \gg \Lambda_L$
- Some d.o.f. that exist below $\mu$
The lagrangian $\mathcal{L}_\text{EFT}$ consists of all operators $\mathcal{O}_n$ that may be built of the effective d.o.f and obey the symmetries of $\mathcal{L}$ $$\mathcal{L}_\text{EFT} = \sum_n c_n(\Lambda_H, \mu) \frac{\mathcal{O_n}(\mu, \Lambda_L)}{\Lambda^{d_n-4}_H}$$.
$d_n$ is the dimension of the respective operator.
Question: How do effective interactions and effective d.o.f. emerge at energies close to $\Lambda_L$ ?
Effective interactions: $\Rightarrow$ Expand fundamental interaction in $\frac{\Lambda_L}{\Lambda_H}$. Valid as long as $\frac{\Lambda_L}{\Lambda_H}«1$.
Effective d.o.f.: $\Rightarrow$ The d.o.f. at the low scale $\Lambda_L$ are the ones with wavelength $\lambda \lessapprox \Lambda_L$. “High energy d.o.f.s” with $\lambda\in\mathcal{O}(\Lambda^{-1}_H)$ disappear and are absorbed into the parameters of the low energy effective theory.
Example: Gravity on earth: Let $M_E$ be the earths mass and $R_E$ the radius. At height $h$ from the surface we have $$V_g = -G_N \frac{M_E}{R_E + h} = - G_N \frac{M_E}{R_E}(1-\frac{h}{R_E}+\mathcal{O}(\frac{h^2}{R^2_E}))$$. We want to go from the high energy $\Lambda_H = \frac{1}{h}$ to $\Lambda_L = \frac{1}{R_E}$ by matching the expansion in $\frac{h}{R_E}$ to an effective gravity field which is constant on the surface of the earth $$V^\text{eff} = \frac{G_N M_E}{R^2_E} (h-R_E)(1+\mathcal{O}(\frac{h^2}{R^2_E})$$. This is valid if $h\ll R_E$.
- For $\mu \approx \Lambda_L$ the expectation value of the operator is $\braket{\mathcal{O}_n,\Lambda_L}\approx \Lambda^{d_n}_L$. This value is called power counting.
- Since $\mathcal{L}_\text{EFT}$ is an expansion in $\frac{\Lambda_L}{\Lambda_H}$ it has infinite terms. For a given prescision $\epsilon$ however only terms with $(\frac{\Lambda_L}{\Lambda_H})^N > \epsilon$ are needed. Therefore the worse the the hirachy of $\Lambda_L \ll \Lambda_H$ the more terms are needed.
- The Wilson coefficents $c_n(\Lambda_H, \mu)$ contain the behavior from $\Lambda_H$. They are computed by imposing $\mathcal{L}_\text{EFT}$ equals the full theory up to a given prescision, which is called matching.
- $\mathcal{L}_\text{EFT}$ is not renormalizable, because of couplings with negative mass dimension.
- Naturalness of EFT when power counting is “fullfilled”: $c_n \appox 1$ $\braket{\mathcal{O}_n} \approx \Lambda^{d_n}_L$