Checking the B -> D* FF parameterization

[yipengsun]

We are checking $B \rightarrow D^{*}$ FF parameterization again

The parameter $c_3, d_2$ are not used

We were stealing $R(D^+)$'s parameterization, in which they claimed that

The $c3$ and $d2$ parameters are not implemented [...]

However, since these are just z-expansions, and Hammer's handling of these are just looping them over in a for loop, so they should have been implemented!

for $c_n$:

https://gitlab.com/mpapucci/Hammer/-/blob/development/src/FormFactors/BGL/FFBtoDstarBGL.cc#L176-178

which means starting from $z = 1$ (so $c_0$ is skipped)

for $d_n$:

https://gitlab.com/mpapucci/Hammer/-/blob/development/src/FormFactors/BGL/FFBtoDstarBGL.cc#L180-183

which starts from $z = 0$

Related issue: #14

• [ ] Confirm with Dean that $c_3$ and $d_2$ are indeed implemented

The parameter $c_0$ is not used

As discussed above, for $c_n$, the loop start at $z = 1$. Don't know why.

• [x] Ask Dean about handling of $c_n$
  • $c_0$ is exactly determined by $b_0$ so no need to specify

Different parameterizations between Hammer and our source paper

• Hammer: https://hammer.physics.lbl.gov/HammerManual.pdf
  • See Conventions -> Form Factors and Maps
• Source paper: https://arxiv.org/pdf/2105.14019v1.pdf
  • See V -> A

Phoebe pointed out that in Hammer, the FFs are {f,g,F1,F2/P1}, but source uses {g,f,F1,F2}.

Phoebe worries that if in this case avec and bvec is swapped.

We checked the following:

• avec in both cases are describing g
• In Hammer, $g = \frac{hV}{2 \sqrt{r{D^{*}}} m_B}$
• In source, $g = \frac{h_V}{M_B \sqrt{r}}$

Assuming $h_V$ are the same (which is probably A BAD IDEA!) They differ by a constant factor, which has been taken into account. There's an additional factor of 2 difference, which is taken account internally in Hammer:

        result.element({2}) = g/2.;

Different value of ChimL

In source, that is defined as 1.9421e-2:

In Hammer, it is 0.002466.

• [ ] Fix: Define ChimL correctly

[yipengsun]

@manuelfs @CoffeeIntoScience @hadjichris @afernez Here's my not-so-well-organized summary of trying to validate $D^*$ FF parameterization yet again.

Feel free to edit the top post, and don't forget to put a link to this issue in our docs regarding FF validation!

[yipengsun]

If we decide to USE $c_3, d_2$, instead of manually setting them to 0, we'll have the problem of having 12 parameters thus 12 sets of FF variations. need alpha_uu1B, alpha_uu1C for example.

[CoffeeIntoScience]

As discussed above, for $c_n$, the loop start at $z = 1$. Don't know why.

No need, there is an exact analytic relationship between f(0) and F_1(0) (b_0 and c_0) that completely fixes it given b_0. In fact, it's better that Dean did it this way because you want to avoid introducing precision errors in the values or (more importantly maybe) correlations.

You can have ratios of form factors diverge in funny ways too from rounding errors, which was a huge pain for me when I had to implement BGL

[CoffeeIntoScience]

Phoebe worries that if in this case avec and bvec is swapped.

I think we are reasonably converged here as this bit seems to agree that avec->g and bvec->f

[CoffeeIntoScience]

Final comment:

Different value of ChimL

This telling bit of code here gives the hint that changes in ChimL <-> rescaling of the d_i coefficients. The difference is such a large factor (goes like sqrt(chi) so ~2.8?) may be the large difference we were looking for in the implementation of the scalar form factor as we (@manuelfs and @afernez and I) discussed after the group meeting 5/24