Week 5, Wednesday

Leading Order 4-Point Function

The possible Wick contractions of the $4$ external positions $x_1$, $\dots$, $x_4$ are just products of the leading $2$-point functions, \begin{equation} \begin{split} \langle \phi_1 \phi_2\phi_3 \phi_4 \rangle =&\; \langle \phi_1 \phi_2\phi_3 \phi_4 \rangle_0 + O(\lambda) \\ =&\; \langle \phi_1 \phi_2 \rangle \langle \phi_3 \phi_4 \rangle + \langle \phi_1 \phi_3 \rangle \langle \phi_2 \phi_4 \rangle + \langle \phi_1 \phi_4 \rangle \langle \phi_2 \phi_3 \rangle + O(\lambda) \end{split} \end{equation}

First-Order 4-Point Function

The order-$\lambda$ contribution to the 4-point correlator is \begin{equation} \langle \phi_1 \phi_2 \phi_3 \phi_4 \rangle^{(1)} = – \frac{i}{24} \int d^4y \Big( \langle \phi_1 \phi_2 \phi_3\phi_4 \phi_y^4 \rangle_0 – \langle \phi_1 \phi_2 \phi_3 \phi_4 \rangle_0 \langle \phi_y^4 \rangle_0 \Big) \end{equation} The Wick contractions of the first term are \begin{equation} \begin{split} \langle \phi_1 \phi_2 \phi_3 \phi_4 \phi_y^4 \rangle_0 =&\; 24~ \overline{\phi_1 \phi_y}~ \overline{\phi_2 \phi_y}~ \overline{\phi_3 \phi_y}~ \overline{\phi_4 \phi_y} \\ &\; + 12\Big( \overline{\phi_1 \phi_2}~ \overline{\phi_3 \phi_y}~ \overline{\phi_4 \phi_y}~ \overline{\phi_y \phi_y} + \text{perm.} \Big) \\ &\; + 3\Big( \overline{\phi_1 \phi_2}~ \overline{\phi_3 \phi_4}~ \overline{\phi_y \phi_y}~ \overline{\phi_y \phi_y} + \text{perm.} \Big) \end{split} \end{equation} The Wick contractions of the second term just cancel the last summand above (the terms multiplied by $3$). If you draw the corresponding diagram, you see that they are “vacuum bubbles”, that is, contain a subdiagram that is disconnected from all external positions. Such vacuum bubbles are generally divergent, though fortunately they end up being subtracted off by the second term.

We also notice that many of the terms are just products of 2-point functions, for example \begin{equation} \begin{split} \langle \phi_1 \phi_2 \rangle \langle \phi_3 \phi_4 \rangle =&\; \overline{\phi_1\phi_2}~ \overline{\phi_3\phi_4} \\ &\; – \frac{i\lambda}{2} \int d^4y\; \Big( \overline{\phi_1 \phi_2}~ \overline{\phi_3 \phi_y}~ \overline{\phi_4 \phi_y}~ \overline{\phi_y \phi_y} + \overline{\phi_1 \phi_y}~ \overline{\phi_2 \phi_y}~ \overline{\phi_3 \phi_4}~ \overline{\phi_y \phi_y} \Big) \\ &\; + O(\lambda^2) \end{split} \end{equation} This should also not be surprising, part of amplitude for the scattering process of two ingoing and two outgoing particles is just two particles not interacting at all. In fact, only the single connected diagram at $O(\lambda)$ remains after collecting everything we can into products of two-point functions: \begin{equation} \begin{split} \langle \phi_1 \phi_2\phi_3 \phi_4 \rangle =&\; \langle \phi_1 \phi_2 \rangle \langle \phi_3 \phi_4 \rangle + \langle \phi_1 \phi_3 \rangle \langle \phi_2 \phi_4 \rangle + \langle \phi_1 \phi_4 \rangle \langle \phi_2 \phi_3 \rangle \\ &\; – i \lambda \int d^4y \overline{\phi_1 \phi_y}~ \overline{\phi_2 \phi_y}~ \overline{\phi_3 \phi_y}~ \overline{\phi_4 \phi_y} + O(\lambda^2) \end{split} \end{equation}

In fact, this is true in general:

  • Vacuum bubbles are cancelled by the expansion of the denominator \begin{equation} \frac{1}{\int \mathcal{D}\phi e^{iS_\text{int}}}, \end{equation} which we got because we are generally not able to normalize the path integral measure.

  • $n$-point correlators are sums of products of disconnected correlators plus the connected diagrams.

  • Each internal node is accompanied by a factor of $-i\lambda$ and an integral over its position. We will make these also part of the graphical notation.

  • Each term comes is multiplied with a combinatorial symmetry factor $\frac{1}{|\mathop{Aut} G|}$, where $\mathop{Aut}(G)$ is the automorphism group of the diagram. That is, count all ways to map the vertices to vertices and lines to lines. Note that we divide by the number of automorphisms because each symmetry reduces the number of distinct Wick contractions we can make. In the diagrams so far, we had

    • $ \overline{\phi_1 \phi_y}~ \overline{\phi_2 \phi_y}~ \overline{\phi_3 \phi_y}~ \overline{\phi_4 \phi_y}$, $\mathop{Aut} G = 1$, coefficient $\tfrac{24}{24} = 1$,

    • $ \overline{\phi_1 \phi_y}~ \overline{\phi_2 \phi_y}~ \overline{\phi_y \phi_y}$, $\mathop{Aut} G = \mathbb{Z}_2$, coefficient $\tfrac{12}{24} = \tfrac{1}{2}$,

    • $ \overline{\phi_1 \phi_2}~ \overline{\phi_y \phi_y}~ \overline{\phi_y \phi_y},$ $\mathop{Aut} G = D_8$, that is, the dihedral group with $8$ elements, coefficient $\tfrac{3}{24} = \tfrac{1}{8}$.

Feynman Rules in Position Space

Definition: A Feynman graph in position space, for the $\phi^4$-theory, is a graph (undirected, without edge labels) with

  • 1-valent vertices labelled by $x_i$, called “external”,
  • 4-valent vertices labelled by $y_j$, called “internal”, and
  • without vacuum bubbles.

In particular, disconnected graphs are allowed but each connected component must be attached to at least one external vertex. As we have seen, such a graph translates directly into a summand in the series expansion of correlation functions. Explicitly, the rules are

Definition: Feynman rules for $\phi^4$-theory in position space

  • For each line joining two vertices $u$, $v$ multiply the integrand with a free propagator $\overline{\phi(u)\phi(v)} = \frac{1}{i} \Delta(u-v)$.

  • For each vertex, integrate $-i\lambda \int d^4y_j$
  • Multiply with the symmetry factor $\frac{1}{|\mathop{Aut} G|}$.