Chapter 5 E3

Operators:

• \(O_3= \phi(0)\phi(0)\phi(0)\)

• \(O_3^{P_x}= \phi(p_x)\phi(0)\phi(0)\)

• \(O_3^{P_{xy}}= \phi(p_{xy})\phi(0)\phi(0)\)

• \(O_3^{P_{xyz}}= \phi(p_{xyz})\phi(0)\phi(0)\)

• \(O_3^{A1}= \sum_{i=x,y,z}\phi(0)\phi(p_i)\phi(-p_i)\)

Correlators:

• \(C_3=\langle O_3(0)O_3(t)\rangle\)

  \[ c_3(t)=|A_{3-0}|^2 \exp(-E_3 \frac{T}{2}) \cosh\left(E_3 (t-\frac{T}{2})\right)+\\ |A_{2-1}|^2 \exp(-(E_2+M) \frac{T}{2}) \cosh\left((E_2-M) (t-\frac{T}{2})\right). \]

• \(C_3^{P1}=\sum_{i=x,y,z} \langle O_3^{P_i}(0)O_3^{P_i}(t)\rangle\)

\[ c_3^{P1}(t)=|A_{3-0}|^2 \exp(-E_3 \frac{T}{2}) \cosh\left(E_3 (t-\frac{T}{2})\right)+\\ |A_{2p-1}|^2 \exp(-(E_2(p)+M) \frac{T}{2}) \cosh\left((E_2(p)-M) (t-\frac{T}{2})\right)+\\ |A_{2-1p}|^2 \exp(-(E_2+E_1(p)) \frac{T}{2}) \cosh\left((E_2-E_1(p)) (t-\frac{T}{2})\right)+\\ \]

• \(C_3^{P11}=\sum_{i=xy,yz,zx} \langle O_3^{P_i}(0)O_3^{P_i}(t)\rangle\)

• \(C_3^{P11}= \langle O_3^{P_{xyz}}(0)O_3^{P_{xyz}}(t)\rangle\)

• \(O_3^{A1}=\langle O_3^{A1}(0)O_3^{A1}(t)\rangle\)

\[ c_3^{A1}(t)=|A_{3-0}|^2 \exp(-E_3 \frac{T}{2}) \cosh\left(E_3 (t-\frac{T}{2})\right)+\\ |A_{2A1-1}|^2 \exp(-(E_{2A1}+M) \frac{T}{2}) \cosh\left((E_{2A1}-M) (t-\frac{T}{2})\right)+\\ |A_{2p-1p}|^2 \exp(-(E_2(p)+E_1(p)) \frac{T}{2}) \cosh\left((E_2(p)-E_1(p)) (t-\frac{T}{2})\right) %+\\ |A_{1-0}|^2 \exp(-M \frac{T}{2}) \cosh\left( M (t-\frac{T}{2})\right) \]