Todo list from Aida

1. h_c prop: your E,Z vs n and prior width graphs don't give me enough detail to answer my questions about your fits. My questions are:
   1. How do the excited prior widths affect the ground state fit results? I'll need graphs which show this at several select values of n. I don't think that your groundstate fit results should depend on the excited state prior widths. If they do, then maybe you still have a bug in your code.
      There is an effect which I don't understand. The effect is completely independent of N_states, so I'll just show it for one value, N_states=6. Look at my plots of the ground state and its coefficient to see. The different points vary the excited state prior widths (both energy spacing and coefficient, in tandem) by factors of sqrt(2), where my "standard" prior is shown by the point with a burst (*).
      My guess for what's happening here is that the priors have an effect which is fairly small, until they get extremely large. Then the fitter apparently switches into a different mode of operation. I think it might be related to the SVD cut, but at this point that's just an educated guess.
   2. excited state h_c: Look at E1,Z1 vs n and prior width to determine how much information there is in the propagator about the excited states. Again, the graph you have put on your web site doesn't give enough information. I asked you to summarize the situation, which means, please describe it in words. If you want you can illustrate this with a plot. Your current plot doesn't answer my question: Do the fit errors always follow the prior width? All your plot tells me is that the errors increase with prior width.
      For the E1, the central value is similar to the prior central value, and the errorbar is almost exactly (to two digits) the same as the prior width. So we have no information on that state (at least, not using 1S-1S data only). The Z1 is stranger. Its errorbar is 85% of the prior width until the priors get very wide. Then it the state disappears, presumably joined by the E2 state. Here are plots of the first excited state and its coefficient.
   3. Finally, my last question concerning the h_c fits is: How do the fit results depend on the ground state prior if you keep the excited state prior widths fixed. Here, I'd like to see some sort of plateau or at least a region of stability in E, Z vs prior width and n.
      There is a very large plateau here, indicating the ground state energy and coefficient are not dependent on the ground state prior widths. Here are plots of the ground state and its coefficient under variations of the ground state priors (with the excited state priors fixed). In case you're interested, I've also generated plots of the first excited state and its coefficient. I'm including all values of N_states since the variations within each state are minor, but the error bars on E0 decrease slightly as the number of states is increased. It's worth noting, however, that the X² increases as more states are added, though it's still better than 1, as this plot shows.
2. Fit results for the other p-wave states: Do you treat them the same as the h_c prop ? For example: For the h_c prop, the 1S channel gives the best results, adding in the local and 2S sources, just adds more noise. If you are quoting results from combined fits, you should compare them to fits from just the 1S props, to see if that reduces the errors. Have you updated your other p-wave results using your corrected fit program ?
   I've previously treated them the same in the sense of increasing t_min to 2. But I've typically used all (diagonal) correlators. I just looked at the following combinations of diagonal correlators for the a0: local, local+1S, local+1S+2S, 1S, 1S+2S, 2S. The "best" results (as defined by having the smallest erros for the E0 state) come from using the 1S only. But using the local+1S has errors that are larger by an insignificant amount. Here's a table sorted by the size of the error-bar:

   Correlators	Central value	Errorbar
   1S	1.925434	0.006798
   local+1S	1.927117	0.006902
   local+1S+2S	1.924614	0.007179
   1S+2S	1.923601	0.007399
   local	1.935930	0.012318
   2S	1.918057	0.019383

   Note that as long as the 1S state is included, including other states is irrelevant. In all cases, the X²/dof is less than 1 for all cases if fitting to N_states>=3. In the future, I'll plan to include only the 1S correlator for all P-wave states.
3. Status of your other fit results: Do any of the bugs you found while investigating the h_c props affect your other fit results for M2's, Ds props etc...? If yes (or even just potentially yes), then you need to regenerate those fit results, and repost them on your web site.
   Anything's possible. I think the error, if any, should be insignificant, but I can repeat those runs. I'd rather wait till the rest of the analysis is complete, though, to ensure that I'm not running with parameters that we're just going to change the next day.
4. 2S props: Here the question is: Do you understand your fit results and errors? In more detail, you should try to answer the following questions:
   1. Do different choices of priors, n lead to different ansers for the 2S ? If yes, what's the range?
      If I include all diagonal correlators (local, 1S, 2S) then the results stabilize for N_states>=4. The prior width still has an effect, though.
   2. How do the errors change with n and with prior width (ground, excited)?
      
   3. Which correlators contain the most information about the 1S, 2S states ? To answer this question, you'll want to compare your combined fit results for the 1S and for the 2S states with single correlator fits, and with fits to other combinations, such as l+1S, 1S+2S, ...
      I did a similar analysis to what I did with the h_c. The required correlators are summarized below:
      • E₀: local and 1S
      • E₁: the different correlators disagree. With local only, it gives 2.17. If local+1S or 1S only it gives 2.14. And with a combination including the 2S it gives 2.08. The tightest errorbar comes from using all three, but that also results in a large X²/dof: 1.5.
      • Z₀:
      • Z₁:
5. Paul told me that his fits to the h_c propagator aren't sensitive to the starting guess, unlike your and Jim's fits. I want you to study this issue further:
   1. If you adjust the initial starting guess to the fitted value, but keep the central value for the ln(Z0) prior at zero (instead of one), do reproduce the fit which gives you the correct state, or the one where you get the spurius state?
      
   2. The one difference between your (and Jim's) and Paul's program is that Paul doesn't rescale. If you omit the rescaling in your program, how are the h_c fits affected ? Do you get better stability and less sensitivity to the spurious state (like Paul, who doesn't see the spurious state with his fits)?
      
6. In addition to fitting the masses for the Ds propagators, you need to fit the Z's and determine the decay constant. Consult our decay constant paper (Phys. Rev. D 58 (1998) 014506) to learn about this analysis. I think that point 7) is also relevant for this analysis.
   
7. Your fitting program should allow for simple (non exponential) prior parametrizations. You need this for the Z's to include off-diagonal correlator combinations anyway. When you are done with this, please tell me how you tested your new program, to make sure that it is correct.