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I am trying to learn how to calculate scattering amplitudes in a Klein-Gordon theory. I am getting stuck with the simplest of the examples: $\phi\to\phi$ in a free scalar-field theory.

If I calculate the amplitude for the process $\phi\to\phi$, I get two different results depending on whether I use Feynman rules or the LSZ formula.

Let's say that the incoming particle has momentum $p_\mathrm{in}$ and that the outgoing one has momentum $p_\mathrm{out}$. Then a simple Feynman diagram gives amplitude = propagator, that is,

$$ p_\mathrm{in} ---\stackrel{p}{\blacktriangleright} ---p_\mathrm{out} \qquad=\qquad \frac{1}{p^2-m^2} \tag{1} $$ where $p$ is the momentum of a virtual particle connecting the in and out states. From momentum conservation, $p_\mathrm{in}=p=p_\mathrm{out}$.

From $(1)$ we get that the amplitude for $\phi\to\phi$ is $$ \mathcal A=\frac{1}{p^2_\mathrm{in}-m^2}=\frac{1}{p^2_\mathrm{out}-m^2} $$

If, on the other hand, we calculate the amplitude for the process with the LSZ reduction formula, we get

$$ \mathcal A(2\pi)^4\delta(p_\mathrm{in}-p_\mathrm{out})=i\int\mathrm dx\,\mathrm dy\ e^{ip_\mathrm{in}x} e^{-ip_\mathrm{out}y}\, \square_x \square_y \langle\phi(x)\phi(y)\rangle $$ where $\square_x\equiv \partial_x^2+m^2$, and $\langle \phi(x)\phi(y)\rangle$ is the propagator. As the KG derivative, when acting on the propagator, results in a delta, i.e., $i\square \langle\phi(x)\phi(y)\rangle=\delta(x-y)$, this integral evaluates to $$ \begin{align} \mathcal A (2\pi)^4\delta(p_\mathrm{in}-p_\mathrm{out})&=\int\mathrm dx\,\mathrm dy\ e^{ip_\mathrm{in}x} e^{-ip_\mathrm{out}y}\, \square_x \delta(x-y)\\ &=(2\pi)^4\delta(p_\mathrm{in}-p_\mathrm{out})(p_\mathrm{in}^2-m^2) \end{align} \tag{2} $$

From $(2)$ we get $$ \mathcal A=p_\mathrm{in}^2-m^2=p_\mathrm{out}^2-m^2 $$

Comparing this to the amplitude from the diagram, we find that the results disagree. Can anybody tell me where is my mistake? I believe that the right result is the first one, so maybe my formula for LSZ is wrong, or cannot be used in a one-particle process?

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2 Answers 2

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My own attempt at this: the first result is wrong, and the second one is right but incomplete.

  • Feynman: in the diagram, all the lines are external, so there is no propagator in the diagram. Therefore, Feynman rules give $\mathcal A=1$.

Check up of this:

In canonical quantisation, the amplitude is given by $\langle p|q\rangle=\langle 0|a_p a^\dagger_q|0\rangle=\langle 0|[a_p,a^\dagger_q]|0\rangle\propto\delta(\boldsymbol p-\boldsymbol q)\langle 0|0\rangle$. Therefore, $\mathcal A$ is momenta-independent, and can be made equal to $1$ by an appropriate choice of the one-particle states normalisation. So far so good.

  • LSZ: In the OP, I proved that LSZ gives $\mathcal A_\text{LSZ}= p_\mathrm{in}^2-m^2$; but as the external momenta are on-shell, this means that $\mathcal A_\text{LSZ}=0$. But $\mathcal A_\mathrm{LSZ}$ is not the total amplitude: it is just the connected contribution! In the process $\phi\to\phi$, the tree diagram is disconnected, which means that LSZ has to evaluate to zero, and this is expected behavior.

The total amplitude for any process is$^1$ $\mathcal A=\mathcal A_\mathrm{dis}+\mathcal A_\mathrm{LSZ}$. For this particular process, $\mathcal A=1$ is a disconnected contribution, and $\mathcal A_\mathrm{LSZ}=0$.


$^1$ e.g., see see Timo Weigand's notes on QFT, page 50.

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  • $\begingroup$ How does one calculate these 'disconnected pieces in general (for situations greater than 1 in 1 out particle) - e.g. for a 8 in 8 out particle situation, with disconnected pieces where an ingoing external leg is directly connected to an outgoing external leg. Can you use Feynman rules, or do you always have to revert to commutations of creation/annihilation operators like you did here? $\endgroup$
    – Alex Gower
    Jan 26, 2022 at 22:57
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You are actually not calculating the same thing.

The momentum space Feynman of the propagator is used for internal momentum, which is off-shell. So your first calculation is correct. While in your second case, you are using the LSZ formula to force the momentum to be on-shell. One way to get the same result is as follow:

Consider the following diagram:

enter image description here

(1)Using momentum space Feynman rule, since there is no internal propagator, you should get 1. So the S-matrix element is $\sim \delta^4(p_{in}-p_{out})$

(2) Use LSZ formula: \begin{equation} (-i\int d^4xe^{-ip_{in}x}\Box_x)(-i\int d^4 ye^{ip_{out}y}\Box_y)\int d^4x_1<\phi(x)\phi(x_1)><\phi(x_1)\phi(y)>\sim \delta^4(p_{in}-p_{out}) \end{equation}

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    $\begingroup$ Actually, I have to say, the first calculation is not correct: the amplitude is not $\frac{1}{p^2-m^2}$, because the momenta are on-shell, which would make the amplitude divergent. The result from LSZ is right, but incomplete. For more details, see my own answer above (anyway, thank you for your answer :) ) $\endgroup$ Jan 18, 2016 at 21:32

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