Loops in EFTs

Last updated Dec 29, 2024

• Rework

Irrelevant operators have coupling constants with negative mass dimension. Therefore they produce non-renormalizable interactions

1. Mass / momentum dependet scheme

2. Mass / momentum independet scheme

Example: Example EFT lagrangian: $$\mathcal{L} = \overline{\psi}(i\partial \negthinspace\negthinspace\negthickspace/ - m)\psi - \frac{a}{\Lambda^2}(\overline{\psi}\psi)^2 - \frac{b}{\Lambda^4}(\overline{\psi}\partial^2\psi)(\overline{\psi}\psi) + …$$

• Higher order operators are irrelevant
• consider divergent loop contribution due to the higher order operators

1. Mass dependent:

$(\overline{\psi}\psi)^2$-Term: Dim 6: UV cutoff is $\Lambda$ $$\delta m \propto 2ia\frac{m}{\Lambda^2} \int^\Lambda \frac{\text{d}^4k}{(2\pi)^4} \frac{1}{k^2-m^2 + i\epsilon} \propto m$$

$(\overline{\psi}\partial^2\psi)(\overline{\psi}\psi)$-Term: Dim 8: $$\delta m \propto 2ib\frac{m}{\Lambda^4} \int^\Lambda \frac{\text{d}^4k}{(2\pi)^4} \frac{k^2}{k^2-m^2 + i\epsilon} \propto m$$

$\Rightarrow$ Mass corrections of order $m$

• Power divergences in mass dependent regularization scheme are unphysical and will cancel with divergences from the coefficients when they are computed. Therefore the coefficients need to be computed in the same scheme.

• Mass dependet scheme used for lattice

• Less suitable for analytical calculations

2. Mass independent:

• dimensional regularization: $d = 4 - 2\epsilon$, $d=4$ restored for $\epsilon \to 0$

$(\overline{\psi}\psi)^2$-Term: Dim 6: $$\delta m \propto 2 i a \frac{m}{\Lambda^2}\int \frac{\text{d}^{4-2\epsilon}k}{(2\pi)^{4-2\epsilon}}\frac{1}{k^2-m^2 + i\epsilon} = 2\frac{a}{\Lambda^2}m\int \frac{\text{d}^{4-2\epsilon}k}{(2\pi)^{4-2\epsilon}}\frac{1}{k^2+m^2}$$

The integral calculates to

$$\delta m \propto 2 \frac{a}{\Lambda^2}m \frac{m^2}{16 \pi^2} \mu^{-2\epsilon} (-\frac{1}{\epsilon} - 1 + \gamma_E -\log 4\pi + \log\frac{m^2}{\mu^2})$$

Definition: Minimal Subtraction (MS) renormalization scheme In the MS scheme, the $\frac{1}{\epsilon}$ term is absorbed / subtracted into the redefinition.

Definition: Modified MS ($\overline{\text{MS}}$) renormalization scheme In the $\overline{\text{MS}}$ scheme, the $\frac{1}{\epsilon} - \gamma_E + \log 4\pi$ term is absorbed / subtracted into the redefinition.

Therefore in $\overline{\text{MS}}$ we have the mass correction for the dim 6 operator

$$\delta m \propto - m \frac{a}{8\pi^2}\frac{m^2}{\Lambda^2}(1 - \log\frac{m^2}{\mu^2})$$

• power counting:

  • Mass scale: $m$
  • Dim. 6 contribution: $m\frac{m^2}{\Lambda^2}$
  • Dim. 8 contribution: $m\frac{m^4}{\Lambda^4}$

• $\mu$ is the renomalization scale. It cancels out when calculating physical quantities.

• The logarithms $\propto \log\frac{m^2}{\mu^2}$ may become large depending on $\mu$. They can be resummed with Renormalization Group Equations (RGEs)

Example: $\gamma \bar{e} e$ at one loop

Electron loop:

1. V-A EFT operator: $-\frac{G_F}{\sqrt{2}} \bar{e}\gamma_\mu (1-\gamma_5)e\bar{e}\gamma^\mu(1-\gamma_5)e$ Loop contribution with $G_F \propto \frac{1}{M^2_W}$: $\mathcal{M}\propto\frac{1}{M^2_W}\int \frac{d^4k}{(2\pi^4)}\frac{1}{k^2-m^2_e+i\epsilon}$

2. V-A EFT operator with derivatives: $-\frac{G_F}{\sqrt{2}} \bar{e}\gamma_\mu (1-\gamma_5)\frac{\partial^2}{M^2_W}e\bar{e}\gamma^\mu(1-\gamma_5)e$ Loop contribution: $\mathcal{M}\propto\frac{1}{M^4_W}\int\frac{d^4k}{(2\pi^4)}\frac{k^2}{k^2-m^2_e+i\epsilon}$

Mass-dependent renormalization scheme: cut integral at $\Lambda \approx M_W$

1. $\propto \frac{\Lambda^2}{M^2_W} \approx 1$
2. $\propto \frac{\Lambda^4}{M^4_W} \approx 1$

Mass-dependent renormalization scheme ($\overline{\text{MS}}$):

1. $\propto \frac{m^2_e}{M^2_W}(1 + … + \log\frac{\mu}{m_e}) \ll 1$
2. $\propto \frac{m^4_e}{M^4_W}(1 + … + \log\frac{\mu}{m_e}) \ll 1$

• No power divergences in mass independet renormalization scheme

• At fixed order in expansion, EFT behaves as renormalizable theory, as only finite terms
• In mass independent renormalization scheme no power divergences with $\Lambda^n, n>0$ (at most logarithmic)
• In mass dependent scheme power divergences cancel in physical observables with contributions from coefficients