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Lt_Dax
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I've been looking at how people calculate basic things on the lattice. How would you go about calculating a single number? By this I mean decay rate, cross section, amplitude or contribution to an amplitude.
The reason I ask is that I've been looking at some lattice QCD code up close. The code evaluates the usual path integral by summing over the possible paths (of quark propagators) in colour and spin space and to all possible sink sites (the source site for creation of quarks is fixed). In practise of course this means the use of Monte Carlo techniques to estimate the path integral by creating configurations distributed according to [tex]e^{-S}[/tex], where [tex]S[/tex] is the action, and averaging.
For example, in a simple case where we create a pion correlation function (no fancy amplitudes intended, or anything), we want to evaluate the following integral:
[tex]C_\pi = \frac{\int \mathcal{D}\mathcal{D}[\bar{\psi}]\mathcal{D}[\psi] \; \bar{\psi}_u \gamma_5 \psi_d \bar{\psi}_d \gamma_5 \psi_u
\; e^{-S}}{\mathcal{D}\mathcal{D}[\bar{\psi}]\mathcal{D}[\psi] \; e^{-S}}
[/tex]
where we use two gamma fives to create a state with the correct parity and spin. In practise we also, I understand, integrate out the fermions so that we are left with an action and integral which is in terms of gauge
fields only. Together with a simple manipulation of gamma matrices, we would actually use:
[tex]C_\pi = \frac{\int \mathcal{D} \; Tr\gamma_5 \overrightarrow{M} \gamma_5 \overleftarrow{M} \; detM^2 \; e^{-S_G}}
{\int \mathcal{D} \; detM^2 \; e^{-S_G}}
[/tex]
So now we have these two matrices [tex]M[/tex], forward and backward, which we calculate by looking at all quark propagators in space/colour/spin, and take a sum, although we don't take a complete sum, we take the spin and colour trace of these matrices multiplied by the appropriate interpolating operators (the gammas).
My source of confusion is that the code doesn't add these traces together cleanly over space - the code actually separates out the sum to produce a four-dimensional correlation function rather than a number. Why do this?
I can understand looking at a 2-dimensional correlation function for simple cases like the pion or other hadrons (because that's how you calculate hadron masses, by measuring the rate of decay of these correlation functions in the time direction). But I don't understand why you would do this to calculate the amplitude of a diagram, or similar.
Apparently this 4D correlation function is the correct procedure, and we can take its Fourier transform and use it with contributions from QED, for example (which represents the rest of the diagram). So the lattice calculation represents the hadronic contribution, with line and legs from QED calculated using perturbation theory.
The trouble is, I'm used to the idea of getting single numbers in QED perturbation theory, but I was imagining that in lattice QCD we could just take the complete sum of traces and be done with it. Apart from anything else, you'd also have to calculate a 4D correlation function (in momentum space) for the QED part, and I'm definitely not sure how to do that.
There must be a reason for this four dimensional separation. Can anyone explain this to me?
The reason I ask is that I've been looking at some lattice QCD code up close. The code evaluates the usual path integral by summing over the possible paths (of quark propagators) in colour and spin space and to all possible sink sites (the source site for creation of quarks is fixed). In practise of course this means the use of Monte Carlo techniques to estimate the path integral by creating configurations distributed according to [tex]e^{-S}[/tex], where [tex]S[/tex] is the action, and averaging.
For example, in a simple case where we create a pion correlation function (no fancy amplitudes intended, or anything), we want to evaluate the following integral:
[tex]C_\pi = \frac{\int \mathcal{D}\mathcal{D}[\bar{\psi}]\mathcal{D}[\psi] \; \bar{\psi}_u \gamma_5 \psi_d \bar{\psi}_d \gamma_5 \psi_u
\; e^{-S}}{\mathcal{D}\mathcal{D}[\bar{\psi}]\mathcal{D}[\psi] \; e^{-S}}
[/tex]
where we use two gamma fives to create a state with the correct parity and spin. In practise we also, I understand, integrate out the fermions so that we are left with an action and integral which is in terms of gauge
fields only. Together with a simple manipulation of gamma matrices, we would actually use:
[tex]C_\pi = \frac{\int \mathcal{D} \; Tr\gamma_5 \overrightarrow{M} \gamma_5 \overleftarrow{M} \; detM^2 \; e^{-S_G}}
{\int \mathcal{D} \; detM^2 \; e^{-S_G}}
[/tex]
So now we have these two matrices [tex]M[/tex], forward and backward, which we calculate by looking at all quark propagators in space/colour/spin, and take a sum, although we don't take a complete sum, we take the spin and colour trace of these matrices multiplied by the appropriate interpolating operators (the gammas).
My source of confusion is that the code doesn't add these traces together cleanly over space - the code actually separates out the sum to produce a four-dimensional correlation function rather than a number. Why do this?
I can understand looking at a 2-dimensional correlation function for simple cases like the pion or other hadrons (because that's how you calculate hadron masses, by measuring the rate of decay of these correlation functions in the time direction). But I don't understand why you would do this to calculate the amplitude of a diagram, or similar.
Apparently this 4D correlation function is the correct procedure, and we can take its Fourier transform and use it with contributions from QED, for example (which represents the rest of the diagram). So the lattice calculation represents the hadronic contribution, with line and legs from QED calculated using perturbation theory.
The trouble is, I'm used to the idea of getting single numbers in QED perturbation theory, but I was imagining that in lattice QCD we could just take the complete sum of traces and be done with it. Apart from anything else, you'd also have to calculate a 4D correlation function (in momentum space) for the QED part, and I'm definitely not sure how to do that.
There must be a reason for this four dimensional separation. Can anyone explain this to me?
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