4.4 Scattering and the LSZ formula

From a physical point of view, the most relevant observables that are associated with a QFT are scattering amplitudes. Loosely speaking, scattering means that one prepares in the far past \(t\to -\infty \) a suitable state \(\ket {\psi ;\mathrm {in}}\in \HH \) that describes a collection of incoming particles and asks the question what’s the probability for these particles to interact with each other to produce in the far future \(t\to +\infty \) another state \(\ket {\phi ;\mathrm {out}}\in \HH \) that describes an in general different collection of outgoing particles. The following picture visualizes this scenario:

The probability for this scattering process to happen can be derived from the scattering amplitude

\begin{flalign} \braket {\phi ;\mathrm {out}}{\psi ;\mathrm {in}}\,\in \,\bbC \end{flalign} that is defined in terms of the inner product of the Hilbert space. The aim of this section is to explain how such scattering amplitudes can be computed perturbatively in QFT.

The first aspect we have to clarify is how to define suitable asymptotic states \(\ket {\psi ;\mathrm {in}}\in \HH \) and \(\ket {\phi ;\mathrm {out}}\in \HH \) that admit an interpretation in terms of multiparticle states. Since the interacting field operator \(\Phi (x)\) from (4.9) is not simply a Fourier integral over annihilation and creation operators (as it would be the case for a free field operator (4.4b)), it is not immediately obvious how to extract from it interacting multiparticle states. Assuming that all particles of interest are well separated in the far future/past such that interactions between them are negligible, it is reasonable to expect that the interacting field operator \(\Phi (x)\) approaches a free field operator in the limit \(t\to \pm \infty \). Heuristically, we may write this as

\begin{flalign} \label {eqn:outinfields} \Phi (x) ~\xrightarrow {t\to +\infty }~\sqrt {Z}~\Phi _{\mathrm {out}}(x) \quad ,\qquad \Phi (x) ~\xrightarrow {t\to -\infty }~ \sqrt {Z}~\Phi _{\mathrm {in}}(x) \quad , \end{flalign} where \(\Phi _{\mathrm {out}}(x)\) and \(\Phi _{\mathrm {in}}(x)\) are free quantum fields, but some mathematical care is needed to make precise the sense in which one takes limits of operators. The relevant concept here is called weak limits of operators, see e.g. Chapter 9.2 of the textbook by Greiner/Reinhardt for some details. The constant \(Z\) (called wave function renormalization) will in general be different from \(1\), which is due to the phenomenon from Observation 4.8 that self-interactions may modify the normalization of fields. Furthermore, the out and in quantum fields \(\Phi _{\mathrm {out}/\mathrm {in}}(x)\) will in general have a different mass \(m^2_{\mathrm {int}}\neq m^2\) than the parameter \(m^2\) from the classical action, which is due to the phenomenon from Observation 4.7 that self-interactions may modify the mass of fields. Since the out and in quantum fields are by hypothesis free, we can write them in terms of annihilation and creation operators

\begin{flalign} \Phi _{\mathrm {out}/\mathrm {in}}(x) = \int _{\bbR ^{d-1}} \frac {1}{\sqrt {2\omega _{\mathbf {k}}}}~ \bigg (a_{\mathrm {out}/\mathrm {in}}(\mathbf {k}) \,e^{\ii \,k\,x}+ a_{\mathrm {out}/\mathrm {in}}^\dagger (\mathbf {k})\, e^{-\ii \,k\,x}\bigg )\,\frac {\dd \mathbf {k}}{(2\pi )^{d-1}}\quad , \end{flalign} where \(\omega _{\mathbf {k}} = \sqrt {\mathbf {k}^2 + m^2_{\mathrm {int}}}\) involves the interacting mass. We can now use these out/in annihilation and creation operators to define in analogy with (3.41) a state

\begin{flalign} \ket {k_1,\dots ,k_n;\mathrm {in}} \,:=\, \sqrt {2\omega _{\mathbf {k}_1}}\cdots \sqrt {2\omega _{\mathbf {k}_n}} \,a^\dagger _{\mathrm {in}}(\mathbf {k}_1)\cdots a^\dagger _{\mathrm {in}}(\mathbf {k}_n)\ket {\Omega } \end{flalign} that describes \(n\) incoming particles with relativistic momenta \(k_a\) satisfying the on-shell condition \(k_a^2 = -m^2_{\mathrm {int}}\) and a state

\begin{flalign} \ket {q_1,\dots ,q_m;\mathrm {out}} \,:=\, \sqrt {2\omega _{\mathbf {q}_1}}\cdots \sqrt {2\omega _{\mathbf {q}_m}} \,a^\dagger _{\mathrm {out}}(\mathbf {q}_1)\cdots a^\dagger _{\mathrm {out}}(\mathbf {q}_m)\ket {\Omega } \end{flalign} that describes \(m\) outgoing particles with relativistic momenta \(q_b\) satisfying the on-shell condition \(q_b^2 = -m^2_{\mathrm {int}}\). (Here \(\ket {\Omega }\) denotes the interacting vacuum state from Section 4.1, which we recall is in general different from the free vacuum state \(\ket {0}\).)

The goal for the rest of this section is to express the scattering amplitude

\begin{flalign} \label {eqn:scatteringtmp} \braket {q_1,\dots ,q_m;\mathrm {out}}{k_1,\dots ,k_n;\mathrm {in}} \end{flalign} in terms of the time-ordered \(n+m\)-point function of the interacting quantum field \(\Phi (x)\). To simplify our arguments, we make the technical assumption that the outgoing relativistic momenta are all different from the incoming ones, i.e. \(q_b \neq k_a\) for all \(b=1,\dots , m\) and \(a=1,\dots ,n\). Physically speaking, this means that we exclude the uninteresting case where one or more of the incoming particles do not participate in the interaction and simply act as spectators for the interaction among the other particles. (Physicists would say that we exclude forward scattering.) To achieve our goal, we require some preparations. First, we note that the out/in annihilation and creation operators can be determined by the following integrals

\begin{flalign} \sqrt {2\omega _{\mathbf {k}}}\,a_{\mathrm {out}/\mathrm {in}}(\mathbf {k}) \,&=\, \ii \int _{\bbR ^{d-1}}e^{-\ii \,k\,x} \,\overleftrightarrow {\partial _0}\,\Phi _{\mathrm {out}/\mathrm {in}}(x)~\dd \mathbf {x}\quad ,\\ \sqrt {2\omega _{\mathbf {k}}}\,a^\dagger _{\mathrm {out}/\mathrm {in}}(\mathbf {k}) \,&=\, -\ii \int _{\bbR ^{d-1}}e^{\ii \,k\,x} \,\overleftrightarrow {\partial _0}\,\Phi _{\mathrm {out}/\mathrm {in}}(x)~\dd \mathbf {x}\quad , \end{flalign} where the right-left derivative operator \(\overleftrightarrow {\partial _0}\) is defined, for any two functions \(f(x)\) and \(g(x)\) on the Minkowski spacetime \(x\in \bbR ^d\), by

\begin{flalign} f(x) \overleftrightarrow {\partial _0} g(x) \,:=\, f(x)\,\frac {\partial g(x)}{\partial t} - \frac {\partial f(x)}{\partial t} \,g(x)\quad . \end{flalign} Combining this with our hypothesis in (4.56), we can extract the out/in annihilation and creation operators directly from the interacting field operator \(\Phi (x)\) by taking suitable limits

\begin{flalign} \sqrt {2\omega _{\mathbf {k}}}\,a_{\mathrm {out}/\mathrm {in}}(\mathbf {k}) \,&=\, \frac {\ii }{\sqrt {Z}} \lim _{t\to \pm \infty } \int _{\bbR ^{d-1}}e^{-\ii \,k\,x} \,\overleftrightarrow {\partial _0}\,\Phi (x)~\dd \mathbf {x}\quad ,\\ \sqrt {2\omega _{\mathbf {k}}}\,a^\dagger _{\mathrm {out}/\mathrm {in}}(\mathbf {k}) \,&=\, -\frac {\ii }{\sqrt {Z}} \lim _{t\to \pm \infty } \int _{\bbR ^{d-1}}e^{\ii \,k\,x} \,\overleftrightarrow {\partial _0}\,\Phi (x)~\dd \mathbf {x}\quad . \end{flalign} For later use, we observe that the difference between the out and in annihilation operator can be written in a particularly useful form

\begin{flalign} \nn \sqrt {2\omega _{\mathbf {k}}}\,\Big (a_{\mathrm {out}}(\mathbf {k}) - a_{\mathrm {in}}(\mathbf {k})\Big ) \,&= \, \frac {\ii }{\sqrt {Z}} \Big (\lim _{t\to +\infty }-\lim _{t\to -\infty }\Big ) \int _{\bbR ^{d-1}}e^{-\ii \,k\,x} \,\overleftrightarrow {\partial _0}\,\Phi (x)~\dd \mathbf {x}\\ \nn \,&=\,\frac {\ii }{\sqrt {Z}} \int _{\bbR ^{d}}\partial _0\Big ( e^{-\ii \,k\,x} \,\overleftrightarrow {\partial _0}\,\Phi (x)\Big )~\dd x\\ \nn \,&=\, \frac {\ii }{\sqrt {Z}} \int _{\bbR ^{d}}\Big ( e^{-\ii \,k\,x} \,\partial _0^2 \Phi (x) - \partial _0^2\Big (e^{-\ii \,k\,x}\Big ) \, \Phi (x)\Big )~\dd x\\ \nn \,&=\, \frac {\ii }{\sqrt {Z}} \int _{\bbR ^{d}}\Big ( e^{-\ii \,k\,x} \,\partial _0^2 \Phi (x) - (\nabla ^2-m^2_\mathrm {int})\Big (e^{-\ii \,k\,x}\Big ) \, \Phi (x)\Big )~\dd x\\ \,&=\,\frac {\ii }{\sqrt {Z}} \int _{\bbR ^{d}} e^{-\ii \,k\,x} \,\big (-\partial ^2 + m^2_{\mathrm {int}}\big ) \Phi (x)~\dd x\quad .\label {eqn:inoutannihilation} \end{flalign} In the second line we have used that \(\int _{-\infty }^{\infty } \partial _0\big (\cdots \big )\,\dd t = \lim _{t\to +\infty }\big (\cdots \big ) - \lim _{t\to -\infty }\big (\cdots \big )\). The fourth line makes use of the Klein-Gordon equation \((-\partial ^2 + m^2_{\mathrm {int}})e^{-\ii \,k\,x} = 0\) and the last line follows via integration by parts. Taking the adjoint of this expression, we obtain a similar formula

\begin{flalign} \label {eqn:inoutcreation} \sqrt {2\omega _{\mathbf {k}}}\,\Big (a^\dagger _{\mathrm {out}}(\mathbf {k}) - a^\dagger _{\mathrm {in}}(\mathbf {k})\Big ) = -\frac {\ii }{\sqrt {Z}} \int _{\bbR ^{d}} e^{\ii \,k\,x} \,\big (-\partial ^2 + m^2_{\mathrm {int}}\big ) \Phi (x)~\dd x \end{flalign} for the creation operators.

With these preparations, we can now rewrite the scattering amplitude (4.60) in a more useful way. This is an iterative procedure, starting with the observation that we can use (4.65) to write

\begin{flalign} \nn &\braket {q_1,\dots ,q_m;\mathrm {out}}{k_1,\dots ,k_n;\mathrm {in}} \,=\, \sqrt {2\omega _{\mathbf {k}_1}}\,\expect {q_1,\dots ,q_m;\mathrm {out}}{a^\dagger _{\mathrm {in}}(\mathbf {k}_1)}{k_2,\dots ,k_n;\mathrm {in}} \\ \nn \,&\qquad =\,\frac {\ii }{\sqrt {Z}} \int _{\bbR ^{d}} e^{\ii \,k_1\,x_1} \,\big (-\partial _{x_1}^2 + m^2_{\mathrm {int}}\big ) \expect {q_1,\dots ,q_m;\mathrm {out}}{\Phi (x_1)}{k_2,\dots ,k_n;\mathrm {in}}~\dd x_1 \\ \nn \,&\qquad ~\qquad +\, \sqrt {2\omega _{\mathbf {k}_1}}\, \expect {q_1,\dots ,q_m;\mathrm {out}}{a^\dagger _{\mathrm {out}}(\mathbf {k}_1)}{k_2,\dots ,k_n;\mathrm {in}}\\ \,&\qquad =\,\frac {\ii }{\sqrt {Z}} \int _{\bbR ^{d}} e^{\ii \,k_1\,x_1} \,\big (-\partial _{x_1}^2 + m^2_{\mathrm {int}}\big ) \expect {q_1,\dots ,q_m;\mathrm {out}}{\Phi (x_1)}{k_2,\dots ,k_n;\mathrm {in}}~\dd x_1 \quad , \end{flalign} where in the last step we have used our hypothesis that \(k_1\neq q_b\) for all \(b=1,\dots ,m\), which means that we can pull \(a^\dagger _{\mathrm {out}}(\mathbf {k}_1)\) through all the annihilation operators \(a_{\mathrm {out}}(\mathbf {q}_b)\) on its left to annihilate the vacuum \(\bra {\Omega }a^\dagger _{\mathrm {out}}(\mathbf {k}_1) = \bra {a_{\mathrm {out}}(\mathbf {k}_1)\Omega } =0\). To perform the next iterative steps, let us note that

\begin{flalign} \nn \sqrt {2\omega _{\mathbf {k}_2}}\,\Phi (x_1) \, a^\dagger _{\mathrm {in}}(\mathbf {k}_2) &=\sqrt {2\omega _{\mathbf {k}_2}}\,\TO \Big (\Phi (x_1) \, a^\dagger _{\mathrm {in}}(\mathbf {k}_2)\Big ) \\ \nn &= \frac {\ii }{\sqrt {Z}} \int _{\bbR ^{d}} e^{\ii \,k_2\,x_2} \,\big (-\partial _{x_2}^2 + m^2_{\mathrm {int}}\big )\TO \Big (\Phi (x_1) \,\Phi (x_2)\Big ) \,\dd x_2\\ \nn &\qquad ~\qquad + \sqrt {2\omega _{\mathbf {k}_2}}\,\TO \Big (\Phi (x_1) \, a^\dagger _{\mathrm {out}}(\mathbf {k}_2)\Big ) \\ \nn &= \frac {\ii }{\sqrt {Z}} \int _{\bbR ^{d}} e^{\ii \,k_2\,x_2} \,\big (-\partial _{x_2}^2 + m^2_{\mathrm {int}}\big )\TO \Big (\Phi (x_1) \,\Phi (x_2)\Big ) \,\dd x_2\\ &\qquad ~\qquad + \sqrt {2\omega _{\mathbf {k}_2}}\,a^\dagger _{\mathrm {out}}(\mathbf {k}_2)\, \Phi (x_1)\quad . \end{flalign} The last term will again not contribute to the scattering amplitude because \(a^\dagger _{\mathrm {out}}(\mathbf {k}_2)\) can be pulled through all the annihilation operators \(a_{\mathrm {out}}(\mathbf {q}_b)\) on its left to annihilate the vacuum. By a similar computation (using (4.64) instead of (4.65)), one can show that also the out annihilation operators arising from \(\bra {q_1,\dots ,q_m;\mathrm {out}} = \bra {q_2,\dots ,q_m;\mathrm {out}}\,\sqrt {2\omega _{\mathbf {q}_1}}\,a_{\mathrm {out}}(\mathbf {q}_1)\) add interacting field operators to the time-ordered product, up to in annihilation operators that annihilate the vacuum to their right. (It is a good exercise for you to check this!) Hence, we can iteratively take care of all \(k_a\) and all \(q_b\) in the scattering amplitude and obtain

which is called the LSZ reduction formula, named after Lehmann, Symanzik and Zimmermann. This formula is indeed very useful because it reduces the description of scattering amplitudes to the interacting time-ordered \(n+m\)-point functions, which we already know how to determine thanks to the Gell-Mann and Low reduction formula (4.16) and Wick’s theorem (4.32).

It is often more convenient to write the LSZ reduction formula in Fourier space. Denoting the Fourier transform of the time-ordered \(n\)-point function by

\begin{flalign} \widetilde {G}_n(k_1,\dots ,k_n) := \int _{\bbR ^{nd}} e^{-\ii \,\sum _{a=1}^n k_a\,x_a}\, \expect {\Omega }{\TO \Big (\Phi (x_1)\cdots \Phi (x_n)\Big )}{\Omega }~\dd x_1\cdots \dd x_n\quad , \end{flalign} one finds that