TASI Lectures on Effective Field Theories

TL;DR

This paper provides an introductory overview of effective field theories, covering recent techniques and applications to help readers understand their foundational concepts and practical uses.

Contribution

It offers a comprehensive introduction to effective field theories, highlighting new methods and recent developments in the field.

Findings

Summarizes key techniques in effective field theories.

Discusses recent applications and developments.

Serves as an educational resource for newcomers.

Abstract

These notes are a written version of a set of lectures given at TASI-02 on the topic of effective field theories. They are meant as an introduction to some of the latest techniques and applications in the field.

Equations

$$-L_{int}=\frac{{\lambda }_0}{4!}{\Phi }_L^4+\frac{{\lambda }_1}{2}{\Phi }_H{\Phi }_L^2+\frac{{\lambda }_2}{4}{\Phi }_L^2{\Phi }_H^2+\frac{{\lambda }_3}{4!}{\Phi }_H^4.$$

$$\delta L_{EFT}=\mathrm{Full}\mathrm{Theory}\mathrm{Result}-\mathrm{Effective}\mathrm{Theory}Result.$$

$$\text{Fig 1}=\frac{-i{\lambda }_1^2}{s-m_H^2}+s\to (t,u).$$

$$\delta L_{int}=C_4^{(1,0)}\frac{{\Phi }_L^4}{4!}$$

$$\delta L_{int}=\frac{C_4^{(1,2)}}{m_H^2}\frac{{\Phi }_L^2{\partial }^2{\Phi }_L^2}{4!}.$$

$$\text{Fig 2a.}=\frac{i{\lambda }_0{m}_L^2}{32{\pi }^2}\left(\frac{1}{\overline{ϵ}}-\ln (\frac{m_L^2}{{\mu }^2})+1\right)$$

$$\frac{1}{\overline{ϵ}}=\frac{1}{ϵ}-\gamma +\ln (4\pi ).$$

$$\text{Fig 2b.}=\frac{i{\lambda }_1^2}{16{\pi }^2}\left(\frac{1}{\overline{ϵ}}-\ln (\frac{m_H^2}{{\mu }^2})+1\right)$$

$$\text{Fig 2d.}=-\frac{i{\lambda }_1^2}{32{\pi }^2}\frac{m_L^2}{m_H^2}\left(\frac{1}{\overline{ϵ}}-\ln (\frac{m_L^2}{{\mu }^2})+1\right).$$

$$C_m^{(1,0)}=\frac{1}{32{\pi }^2}\left[{\lambda }_2{m}_H^2\left(1-\ln (\frac{m_H^2}{{\mu }^2})\right)-{\lambda }_1^2\left(2\ln (\frac{m_H^2}{{\mu }^2})-2-2\frac{m_L^2}{m_H^2}(\ln \frac{m_L^2}{{\mu }^2}+1)\right)\right].$$

$$\text{Fig 2b.}\to \text{(8)}+i\frac{{\lambda }_1^2}{16{\pi }^2}(\frac{m_L^2}{m_H^2}\ln \frac{m_L^2}{m_H^2}).$$

$$C_m^{(1,0)}=\frac{1}{16{\pi }^2}{\lambda }_1^2(1+\frac{m_L^2}{m_H^2})+{\lambda }_2\frac{m_H^2}{32{\pi }^2}.$$

$$m_L^{2(phys)}=m_L^{2\overline{MS}}+O(\lambda /(32{\pi }^2))(m_H^2),$$

$$O_2=C_2^{(1,2)}{\Phi }_L\frac{{\nabla }^4}{m_H^2}{\Phi }_L.$$

$$\text{Fig 3d.}=-\frac{i{\lambda }_2^2}{16{\pi }^2}\left(\ln (\frac{m_H^2}{{\mu }^2})-\frac{\hat{s}}{6}+\cdot \cdot \cdot \right).$$

$$I_{{\Lambda }^2}=\int \frac{d^{4}p}{(2\pi {)}^4}\frac{p^2}{(p^2-m_H^2+iϵ)((p_1+p_2+p{)}^2-m_H^2+iϵ)}.$$

$$\delta {\Phi }_L=-F({\Phi }_L,{\Phi }_H)$$

$$\delta L^{LO}=(\delta {\Phi }_L){\partial }^2{\Phi }_L$$

$$H_t=m_t^2\frac{G_F^2}{16{\pi }^2}{\eta }_2{O}_4,$$

$${\psi }_L\to g_L{\psi }_L,{\psi }_R\to g_R{\psi }_R,\Phi \to g_L\Phi g_R^{†},$$

$$\text{textsl}L=i{\overline{\psi }}_L\partial /{\psi }_L+{\overline{\psi }}_{R}i\partial /{\psi }_R-\lambda {\overline{\psi }}_L\Phi {\psi }_R-\lambda {\overline{\psi }}_R{\Phi }^{†}{\psi }_L,$$

$$W(U)=-i\ln \int d[{\psi }_L]d[{\psi }_R]exp\left(i\int d^{4}x\text{textsl}L\right)$$

$$N_L=U^{†}{\psi }_L,N_R={\psi }_R,$$

\textslL=NL​(i∂/−A/L​)NL​+NR​i∂/N