Getting a matrix element in Lattice QCD

In summary, a matrix element in Lattice QCD is a numerical representation of the interactions between particles in a quantum field theory. It is calculated through numerical simulations on a discrete lattice and is crucial for understanding subatomic particles, testing the Standard Model, and predicting experimental measurements. However, obtaining these matrix elements is challenging due to the need for high-performance computers and the presence of statistical and systematic errors in the simulations.
  • #1
Lt_Dax
77
0
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?
 
Last edited:
Physics news on Phys.org
  • #2


I can provide some insight into the reasoning behind this four-dimensional separation in lattice QCD calculations.

Firstly, it is important to understand that lattice QCD calculations are based on numerical simulations rather than analytical calculations. This means that we are limited by the computational power and resources available to us. Therefore, in order to obtain accurate results, we need to make some approximations and simplifications in our calculations.

In the case of calculating a single number, such as a decay rate or cross section, we would need to perform a sum over all possible paths and contributions. However, this would require a huge amount of computational resources and would not be feasible. Instead, the four-dimensional correlation function allows us to extract the desired information by performing a Fourier transform, which greatly simplifies the calculations.

Moreover, in lattice QCD calculations, we are interested in studying the behavior of particles in the four-dimensional spacetime. This is why we need to perform calculations in four dimensions rather than just a single number. The four-dimensional correlation function provides us with a more comprehensive understanding of the behavior of particles and their interactions.

Additionally, the four-dimensional separation also allows us to separate the contributions from different sources, such as the hadronic contribution and the QED contribution. This is important because it allows us to isolate and study the individual contributions, which can help us to better understand the overall behavior of the system.

In summary, the four-dimensional separation in lattice QCD calculations is necessary due to the limitations of numerical simulations and allows us to obtain accurate results while also providing a more comprehensive understanding of the system.
 

1. What is a matrix element in Lattice QCD?

A matrix element in Lattice QCD is a fundamental quantity that describes the interactions between particles in a quantum field theory called Quantum Chromodynamics (QCD). It is represented as a matrix of numerical values that relate the initial and final states of a particle system.

2. How is a matrix element calculated in Lattice QCD?

A matrix element in Lattice QCD is calculated using numerical simulations on a discrete lattice of points in space and time. These simulations use sophisticated algorithms to solve the equations of QCD and extract the desired matrix element.

3. What is the significance of calculating matrix elements in Lattice QCD?

Calculating matrix elements in Lattice QCD is crucial for understanding the behavior of subatomic particles and their interactions. It can provide insights into the fundamental forces of nature and help test the predictions of the Standard Model of particle physics.

4. What challenges are associated with obtaining matrix elements in Lattice QCD?

Obtaining matrix elements in Lattice QCD is a computationally intensive process that requires high-performance computers and advanced numerical techniques. Additionally, the presence of statistical and systematic errors in the simulations can also complicate the process.

5. How are matrix elements in Lattice QCD used in practical applications?

Matrix elements in Lattice QCD are used to make predictions for experimental measurements in particle physics, such as the decay rates of particles. They are also used in theoretical calculations to test the consistency of the Standard Model and search for new physics beyond it.

Similar threads

  • High Energy, Nuclear, Particle Physics
Replies
1
Views
1K
  • High Energy, Nuclear, Particle Physics
2
Replies
46
Views
3K
  • High Energy, Nuclear, Particle Physics
Replies
1
Views
1K
  • High Energy, Nuclear, Particle Physics
Replies
13
Views
2K
  • High Energy, Nuclear, Particle Physics
Replies
2
Views
885
  • High Energy, Nuclear, Particle Physics
Replies
6
Views
2K
  • High Energy, Nuclear, Particle Physics
Replies
1
Views
2K
  • High Energy, Nuclear, Particle Physics
Replies
3
Views
1K
  • High Energy, Nuclear, Particle Physics
Replies
13
Views
2K
  • High Energy, Nuclear, Particle Physics
Replies
11
Views
2K
Back
Top