Total decay and transition rates from LQCD

We present a new technique for extracting total transition rates into final states with any number of hadrons from lattice QCD. The method involves constructing a finite-volume Euclidean four-point function whose corresponding infinite-volume spectral function gives access to the decay and transition rates into all allowed final states. The inverse problem of calculating the spectral function is solved via the Backus-Gilbert method, which automatically includes a smoothing procedure. This smoothing is in fact required so that an infinite-volume limit of the spectral function exists. Using a numerical toy example we find that reasonable precision can be achieved with realistic lattice data. In addition, we discuss possible extensions of our approach and, as an example application, prospects for applying the formalism to study the onset of deep-inelastic scattering. More details are given in the published version of this work, Ref. [1].


Treatment of scattering in QM
h 2 |S| 1 i = h 2 |U I (1, 1)| 1 i unstable states are formally nothing but poles in the S matrix in the complex plane.
This is the easy and straightforward way but not too much insight… Consider the following Euclidean two-point function: 3) The first two diagrams in Eq. (A.3) do not have any p 2 dependence and are fully included in the mass counterterm. The remaining two give contributions at order g 2 and g 2 k . The associated spectral function can be calculated via analytical continuation Obtain the spectral function: The spectral function has the form of a relativistic Breit-Wigner Agrees with my previous result and this relation is valid to all orders in PT.
The point is that widths can be calculated from spectral functions! (which is not very surprising since EVERYTHING is encoded in spectral functions).
The problem is that the extraction from the lattice is not an easy task.
To finish this story … To finish this story … To finish this story … To finish this story … To finish this story … To finish this story … To finish this story … Suppose H = H QCD + V and V is a small perturbation.
Consider a QCD-stable single-particle state |D, Pi which can decay through V into a multi particle state |E, p, ↵; outi This is the total width into ALL allowed multi particle states with quantum numbers .
We can generalise the previous relation: be a local current that can inject or carry away energy and/or momentum.
(This is useful for describing scattering with leptons or photons).
We define the transition spectral function: and we obtain the less general result if we back-substitute: again: widths can be obtained from spectral functions! so far this was a continuum Minkowski discussion.
Possible issues… • Finite volume does not allow the definition of in/out states.
• Finite volume energy-levels are discrete.
• Multi particle states in finite volume have power-law corrections instead of exponential.
• As the energy is increased, the density of finite volume level is very high. No possible resolving of those levels.
No L-L approach.
• Minkowski real time is not Euclidean imaginary time. No real-time evolution study is possible.
• In many cases a lot of multi particle channels are open with more than two particles in the final state.
We can rewrite it as: ?! a lot of progress has been done since Lellouch, Lüscher (2001) We aim at total decay rates! What about the infinite volume limit?
This inverse Laplace transformation is an numerically ill-defined problem.

Fermi Golden Rule motivation
The smearing idea goes back to Fermis Golden rule: if interested in TOTAL decay rates, one can use other regularised delta functions. This is a key aspect that we exploit.
( (open big parenthesis …) • Developed in 1967 by geophysicists Bakus and Gilbert to study the propagation of earthquakes on the Earth.
• It is a linear method: • It converges to the exact solution: • No a priori Ansatz. Model Independent estimator.
• No free lunch: The regularisation of the problem is translated into a trade off between resolving power and error estimation.
Some remarks on the BG method • The optimal coefficients C j (!, ) are calculated by minimizing  Residue estimation with the BG method: assume the existence of a state at a given omega define an estimator for the residue The order of the double limit Take the scalar toy model again and consider only what does the smoothing when ?
for now we consider a normalised gaussian resolution function to study the effect of smoothing and the double limit.
b (!, !) = 1 p 2⇡ 2 e (! !)/(2 2 ) We can look at the same effect but doing a BG and starting with the finite volume Euclidean corrector itself.