---
# System prepended metadata

title: Analysis ToDo's
tags: [Belle]

---

# Analysis ToDo's
###### tags: `Belle`


### Nakao-san

Dear Markus,

thank you for the complex analysis and detailed manuscript.  I have one
reservation about the title of the paper and the description of Vcb.

A naive reader would read Eq.1 in such a way that you measure the
numerator inside sqrt and take a theory calculation in the denominator
from somewhere else, but what is done here is opposite: take a measured
value from somewhere else and measure the parameters in the theory
calculation in the denominator.

But the measured value which you take is an average from somewhere else
with unexplained assumptions (they also need theory form factors to some
extent) and I feel uncomfortable unless what are included in the average
is clearly explained, if this paper claims Determination of Vcb.

I would suggest to change the title from "Determination of" to
"Implication to", and make it clearer that what you are measuring is the
denominator when you introduce \Gamma of Eq. 1.  Section VIII title then
should be something like "Measurement of form factors and implication to
Vcb".

Dear Nakao-san, Christoph

We understand your reasoning. We adapted the title using the proposed phrasing "Implications" and clarified in the introduction which component of Eq.1 we measure.

The text comments have been addressed in the current version of the draft (1.4). Please find it attached to the E-mail. This draft also contains corrections by Yoshi. 

Cheers
Markus for the team



### Yoshi

Dear Yoshi,

I did not attach a draft in the previous E-mail because I was uncertain about one of your corrections. Now that this is clarified, please find draft version 1.4 attached to this e-mail.

Cheers
Markus for the team

Dear Markus/Florian, and all,

  Thanks for further responses (but I do not find the updated paper
draft). Below are some further follow-up comments and I will check
updated paper draft when available.

9) L183: It would be helpful to give n\sigma (or efficiency) corresponding
to the mass window.
A: Fixed.

YS: Thanks and sorry, I meant to add the corresponding n\sigma
to the mass window values (so, readers can know full information
such as mass resolution, how tight/loose the cut is).

Maybe I misunderstand your follow-up comment. In v1.3 we write e.g.
"Theπ0π0candidates are recombined from photon pairs and selected if
their [reconstructed] invariant mass is within a3σ3σmass window." (Added
"reconstructed") This tells you that we select ~99% of the correct
candidates. Is this what you are requesting, or do you mean something else?

YS2: A usual text is like "The pi0 candidates are recombined from
photon pairs having 0.104 MeV < M_gg < 0.165 MeV, corresponding
to +-3\sigma mass resolution".

14)
YS: I see. I thought that the combinations with incorrect tag would
have larger background fractions and worsen the measurement
accuracy. As fractions of background seem to be low and no such
effect, but have you checked/compare with/without incorrect
combinations (just for my interest) ?

A2: We did not explicitly check this. However, the extractions are almost
background free already (see e.g. Fig 4). Requiring correct tags would
probably have a larger impact on the signal than reducing the
background. ..

YS2: Thanks. I agree that the background is small enough and
better to increase signal yield (so, it is not necessary to check
further). But, I do not think "Further, reducing the background
might cause stability issues in the fit due to lack of constraining
power on the background shapes.

15) L227: "neutral energy deposit" here is different from photons
selected in L179-81 ?
A: No, it is basically the sum of the energy of unassigned photon
candidates (ECL clusters). Unassigned being neither used in the
reconstruction nor that a track is matched to the ECL cluster.

YS: Then, why not use "photons" ?

A2: Because in principle it could also be an ECL cluster generated by a
charged particle where the track matching of the cluster and the track
failed. But this is mis-reconstruction. We could use something like
"E_ECL which is the sum of unassigned photon clusters in the full event
reconstruction." Would this clarify what is meant?

YS2: I understand your reasoning. As "energy depositions in the
ECL without an associated track" are regarded as photons (L118-9),
I think that your proposed phrase above is fine.

23)
YS: .. BTW, only Appendix A is referred
in main text and B and C are not referred (better to be referred).

A2: In version 1.3 of the draft the references to the appendices are
in line 452 (A), line 474 (B), and line 519 (C)

YS2: Yah, I see. I searched "Appendix" and failed to find "App.".
It would be better to unify to "Appendix".

Fixed.




---





# Felix

Branching ratio $B$ is given by 
$B = \sum_i B_{D^*} \times B_{D,i}$ 
$B = B_{D^*}\times \sum_i B_{D,i}$ 

$\epsilon = \frac{N_R}{N}$
$N_R$ and $N$ use the same weights for the branching ratios.

Number of candidates in an event is calculated by
$N = N_{BB} \times ((B (1 - B) + (1 - B) B + 2 B^2)$

* Term1: First B decays
* Term2: Second B decays
* Term3: Both decay (factor of two because both)

$N = N_{BB} \times (B - B^2 + B - B^2 + 2 B^2)$
$N = N_{BB} \times 2B$

$N = 2 B_{D^*}\times \sum_i B_{D,i} \eta_i$ 
with $\eta_i$ given by the new vs old ratio for branching ratio i.

## NHT

* R1, R2 plot for best fit , ava


## NHT w/o Unitarity

### Remove $\rho > 90$

|             | $V_{\mathrm{cb}}$   |   $\chi^2$ |   dof |   N |   $\rho_\mathrm{max}$ |
|:------------|:----------------------|-----------:|------:|----:|----------------------:|
| BGL$_{111}$ | $40.3 \pm 0.8$        |       45.7 |    32 |   3 |                  0.71 |
| BGL$_{121}$ | $40.6 \pm 0.9$        |       45.3 |    31 |   4 |                  0.62 |

### All
|             | $V_{\mathrm{cb}}$   |   $\chi^2$ |   dof |   N |   $\rho_\mathrm{max}$ |
|:------------|:----------------------|-----------:|------:|----:|----------------------:|
| BGL$_{111}$ | $40.3 \pm 0.8$        |       45.7 |    32 |   3 |                  0.71 |
| BGL$_{112}$ | $40.8 \pm 0.9$        |       43.8 |    31 |   4 |                  0.98 |
| BGL$_{121}$ | $40.6 \pm 0.9$        |       45.3 |    31 |   4 |                  0.62 |
| BGL$_{122}$ | $41.3 \pm 1.0$        |       42.8 |    30 |   5 |                  0.98 |
| BGL$_{131}$ | $38.6 \pm 1.5$        |       42.7 |    30 |   5 |                  0.98 |
| BGL$_{132}$ | $39.1 \pm 1.5$        |       38.4 |    29 |   6 |                  0.98 |
| BGL$_{211}$ | $39.8 \pm 0.9$        |       43.5 |    31 |   4 |                  0.99 |
| BGL$_{212}$ | $40.3 \pm 0.9$        |       40.2 |    30 |   5 |                  0.99 |
| BGL$_{221}$ | $37.3 \pm 1.2$        |       39.5 |    30 |   5 |                  0.99 |
| BGL$_{222}$ | $38.4 \pm 1.9$        |       38.8 |    29 |   6 |                  1    |
| BGL$_{231}$ | $38.2 \pm 1.5$        |       40.6 |    29 |   6 |                  0.96 |
| BGL$_{232}$ | $39.0 \pm 1.5$        |       38.1 |    28 |   7 |                  0.98 |
| BGL$_{311}$ | $40.0 \pm 0.9$        |       43.3 |    30 |   5 |                  0.97 |
| BGL$_{312}$ | $39.9 \pm 1.0$        |       37.7 |    29 |   6 |                  0.97 |
| BGL$_{321}$ | $37.4 \pm 1.2$        |       39.2 |    29 |   6 |                  0.96 |
| BGL$_{322}$ | $39.4 \pm 1.8$        |       37.6 |    28 |   7 |                  0.98 |
| BGL$_{331}$ | $38.0 \pm 1.5$        |       38.4 |    28 |   7 |                  0.99 |
| BGL$_{332}$ | $38.8 \pm 2.1$        |       38.1 |    27 |   8 |                  0.99 |

## NHT w/ Unitarity

### Remove $\rho > 90$
|             | $V_{\mathrm{cb}}$   |   $\chi^2$ |   dof |   N |   $\rho_\mathrm{max}$ |
|:------------|:----------------------|-----------:|------:|----:|----------------------:|
| BGL$_{111}$ | $40.3 \pm 0.8$        |       47.1 |    32 |   3 |                  0.71 |
| BGL$_{121}$ | $40.6 \pm 0.9$        |       46.7 |    31 |   4 |                  0.62 |
| BGL$_{131}$ | $40.0 \pm 0.9$        |       45.5 |    30 |   5 |                  0.66 |


### All
|             | $V_{\mathrm{cb}}$   |   $\chi^2$ |   dof |   N |   $\rho_\mathrm{max}$ |
|:------------|:----------------------|-----------:|------:|----:|----------------------:|
| BGL$_{111}$ | $40.3 \pm 0.8$        |       47.1 |    32 |   3 |                  0.71 |
| BGL$_{112}$ | $40.8 \pm 0.9$        |       45.1 |    31 |   4 |                  0.98 |
| BGL$_{121}$ | $40.6 \pm 0.9$        |       46.7 |    31 |   4 |                  0.62 |
| BGL$_{122}$ | $41.3 \pm 1.0$        |       44.2 |    30 |   5 |                  0.98 |
| BGL$_{131}$ | $40.0 \pm 0.9$        |       45.5 |    30 |   5 |                  0.66 |
| BGL$_{132}$ | $40.8 \pm 1.0$        |       42.5 |    29 |   6 |                  0.98 |
| BGL$_{211}$ | $39.8 \pm 0.9$        |       44.9 |    31 |   4 |                  0.99 |
| BGL$_{212}$ | $40.3 \pm 0.9$        |       41.6 |    30 |   5 |                  0.99 |
| BGL$_{221}$ | $38.7 \pm 0.9$        |       42.8 |    30 |   5 |                  0.96 |
| BGL$_{222}$ | $39.4 \pm 1.1$        |       40.7 |    29 |   6 |                  0.98 |
| BGL$_{231}$ | $38.6 \pm 1.5$        |       42.8 |    29 |   6 |                  0.99 |
| BGL$_{232}$ | $39.1 \pm 1.3$        |       40.4 |    28 |   7 |                  0.99 |
| BGL$_{311}$ | $39.8 \pm 0.9$        |       44.9 |    30 |   5 |                  0.95 |
| BGL$_{312}$ | $40.3 \pm 0.9$        |       41.6 |    29 |   6 |                  0.98 |
| BGL$_{321}$ | $38.7 \pm 0.9$        |       42.8 |    29 |   6 |                  0.95 |
| BGL$_{322}$ | $39.4 \pm 1.1$        |       40.7 |    28 |   7 |                  0.97 |
| BGL$_{331}$ | $38.6 \pm 1.5$        |       42.8 |    28 |   7 |                  0.99 |
| BGL$_{332}$ | $39.1 \pm 1.3$        |       40.4 |    27 |   8 |                  0.98 |

## Yoshi

Dear Sakai-san,

thank you for your comments and sorry for the slow reply. One of your questions triggered a bigger study (see below) which took a while.

1) Abstract is missing. this is nor a latest version ?
- We planned to write the abstract last and then simply forgot about it. 

2) L9: Which part of PDG is intended to cite ? not clear.
- The citation [3] here points to the CKM Quark-Mixing Matrix review.

3) 27-32: "hadronic tagging" is better to be explained a bit in more detail here (for general readers).
- We will add a clarification to the text.

4) L 44: "note" -> "paper" or "article"
- Fixed.
6) L138-9: Why PDG2016 [28] is cited for "the latest values" ?
- That is a mistake in our citation. Fixed
7) L139,45,46,47 : "in [xx]" -> "in Ref.[xx]" (as in L123)
- Fixed.
8) L169: z-axis does not seem to be defined.
- Clarification added.
9) L180-1: It would be helpful to readers to give polar angle ranges for barrel and for/backward endcaps.
- We can add this information, however I checked some other Belle publications and these angles are never explicitly given.
10) L183: It would be helpful to give n\sigma (or efficiency) corresponding to the mass window.
- Fixed.
10) L184-5: Please give Ref. No mass requirement is applied ?
- The mass cut was forgotten in the text. It is |delta M| < 100 MeV and |M - M_mc| / sigma_M < 3.0 and now added into the paper. I am unaware of a reference we can use here, I am only aware of the existing internal Belle note.
11) L187-8: It is better to give typical or mass window range (in n\sigma)
- Fixed
13) L199: "D+ pi_slow^-" -> "D0 pi_slow^+"
- Fixed.
14) L202: again, it would be helpful to give n\sigma corresponding to the delta-M requirement.
- Will be added.
14) L225: Why B+_sig B0_tag and B0_sig B+_tag combinations are included ? If B_sig and B_tag are correctly reconstructed, there should be no such combination. On the other hand, you do not include B0_sig B0_tag, which is possible by B0-mixing though the rate is < 1/4 of unmixed B0_sig B0-bar_tag ?
- We do not care about a fully correctly reconstructed tag side, only that we get a tag to form an Upsilon with the described selections. In principle we do not care if a track was missed on the tag sideby the reconstruction. This can generate all the combinations which are given in the text. The mode you are referring to was missed by accident. We added this mode to our reconstruction and updated all results accordingly. The Asimov still yields closure in our analysis, and the updated values for all observables change within their respective uncertainties. The most prominent change is that the nested hypothesis test now yields the BGL 122 parametrization as optimal instead of the BGL 121 parametrization. We attach an updated paper draft with all numerical values and plots updated after including the additional Y(4S) candidates that were missed before.

15) L227: "neutral energy deposit" here is different from photons selected in L179-81 ? 
- No, it is basically the sum of the energy of unassigned photon candidates (ECL clusters). Unassigned being neither used in the reconstruction nor that a track is matched to the ECL cluster.

16) L241-2: "the lowest absolute extra energy on the tag side is selected." is not clear to me. What is an extra energy in tag side ?
- We actually meant the lowest absolute missing energy on the tag side $\min(\|Delta E_\mathrm{tag}|)$. We fixed it in the text.
17) L277: "helicity angle" does not seem to be defined.
- Fixed.
19) L287-8: This binning is result of optimization ? The resolution effects do not seem to be mentioned in systematic uncertainty section. Are they negligible ? (sorry, if I missed)
- There was no optimization done for the binning. The binning is chosen as such to integrate sufficiently over the peak that resolution effects are small, and that the bins next to the peak have sufficient statistics for the fits in the bins of w and the helicity angles to be able to fit the distributions, and that we can use the same binning in mm2 for all fits.
- We checked the effect of the resolution smearing function by varying its parameters within the uncertainties. The uncertainty on the signal yield is <0.1%, and below 10^-4 relative to the statistical uncertainty in the mm2 fit. We therefore treat it as neglible. We will add a statement to the text.

19) Fig.3 caption, L4: "They gray" -> "The gray"
- Fixed
20) L369 (and others): "Tab. I" -> "Table I" (APS style, I guess)
- Fixed
21) Fig.7 caption, L5: "normal interval approximation" is commonly used terminology (i.e. no explanation is needed) ? Is it 64% contained interval corresponding to +-1sigma in Gaussian ?
- Correct.

22) L489,91: Values are to be put in "X" later ?
- TODO We missed them, will be added.
23) L119-21: It would be better to give some comment on the impact of using result of Ref.[15] compared to Ref.[16]. The Ref.[15] result seems to be improved from Ref.[16] extending FF calculation to non-zero recoil region. So, the result with Ref.[16] would be more advanced and better to be advertised ? or still too new to be reliable ?
- The beyond zero-recoil form factors are not published yet (they are only available on arxiv). Thus, we decided to stick to the published zero-recoil value for our nominal result, but still study the impact of the beyond zero-recoil lattice data.

Due to the comprehensive update to all numerical values after updating the analysis with your comment, we attach the updated draft instead of copying all values into this E-mail.
Thanks again for your avaluable comments, I hope we have been able to address them. If you have further questions please let us know.

Cheers
Markus for the team



---
not fully up to date studies

## d'Agostini

We tested explictily for the d'Agostini bias. The impact of this bias on our quoted value of Vcb and the form factor parameters is ca. a factor of 30 smaller than the quoted uncertainties.

For the BGL coefficients

* Nominal fit:

|                        | Value          |   Correlation |       |       |       |       |
|:-----------------------|:---------------|--------------:|------:|------:|------:|------:|
| $a_0 \times 10^3$      | 24.93+/-1.33   |          1    |  0.26 | -0.2  |  0.26 | -0.31 |
| $b_0 \times 10^3$      | 13.11+/-0.18   |          0.26 |  1    | -0.01 | -0.01 | -0.63 |
| $b_1 \times 10^3$      | -11.95+/-12.02 |         -0.2  | -0.01 |  1    |  0.27 | -0.47 |
| $c_1 \times 10^3$      | -0.88+/-0.92   |          0.26 | -0.01 |  0.27 |  1    | -0.47 |
| $V_{cb} \times 10^3$   | 40.77+/-0.90   |         -0.31 | -0.63 | -0.47 | -0.47 |  1    |

* Single iteration d'Agostini yields:

|                        | Value          |   Correlation |       |       |       |       |
|:-----------------------|:---------------|--------------:|------:|------:|------:|------:|
| $a_0 \times 10^3$      | 24.93+/-1.41   |          1    |  0.25 | -0.21 |  0.26 | -0.3  |
| $b_0 \times 10^3$      | 13.11+/-0.18   |          0.25 |  1    | -0.01 | -0.01 | -0.62 |
| $b_1 \times 10^3$      | -12.35+/-12.73 |         -0.21 | -0.01 |  1    |  0.25 | -0.48 |
| $c_1 \times 10^3$      | -0.92+/-0.97   |          0.26 | -0.01 |  0.25 |  1    | -0.49 |
| $V_{cb} \times 10^3$   | 40.80+/-0.92   |         -0.3  | -0.62 | -0.48 | -0.49 |  1    |

For the CLN coefficients:

* Nominal Fit

|                        | Value        |   Correlation |       |       |       |
|:-----------------------|:-------------|--------------:|------:|------:|------:|
| $\rho^2$               | 1.26+/-0.09  |          1    |  0.56 | -0.89 |  0.38 |
| $R_1(1)$               | 1.32+/-0.08  |          0.56 |  1    | -0.63 | -0.03 |
| $R_2(1)$               | 0.85+/-0.07  |         -0.89 | -0.63 |  1    | -0.15 |
| $V_{cb} \times 10^3$   | 40.33+/-0.86 |          0.38 | -0.03 | -0.15 |  1    |

* Single Iteration

|                        | Value        |   Correlation |       |       |       |
|:-----------------------|:-------------|--------------:|------:|------:|------:|
| $\rho^2$               | 1.26+/-0.09  |          1    |  0.57 | -0.9  |  0.37 |
| $R_1(1)$               | 1.32+/-0.08  |          0.57 |  1    | -0.64 | -0.02 |
| $R_2(1)$               | 0.85+/-0.07  |         -0.9  | -0.64 |  1    | -0.16 |
| $V_{cb} \times 10^3$   | 40.30+/-0.85 |          0.37 | -0.02 | -0.16 |  1    |

## Vcb B+/B0 Tension

The discrepancy we are seeing between the B0 and B+ mode is approximately 1.56 standard deviations considering the correlations between the two values. As Vcb also strongly relies on our external branching ratio input, we want to point out that there is already a 0.94 standard deviation tension between the external B0 -> D* l nu and B+ -> D* l nu branching ratios, so to some extent we can attribute the tension in Vcb already to the external inputs.

As a cross-check, we re-run our fits and use for the B0 and B+ case the same external input, the isospin averaged branching ratio. This way, we can attribute the tension solely on the measured shapes:

This is the table from the paper draft, where we use the external input depending on which shape we look at:
|                                | BGL$_{121}$   | CLN        |
|:-------------------------------|:--------------|:-----------|
| $B^+ \to D^{*0} \ell \nu_\ell$ | 41.8+/-1.2    | 41.3+/-1.2 |
| $B^0 \to D^{*+} \ell \nu_\ell$ | 39.3+/-1.4    | 38.9+/-1.2 |
| $B \to D^{*} \ell \nu_\ell$    | 40.8+/-0.9    | 40.3+/-0.9 |

This is the same table, where we use the external branching ratio of the isospon averaged case, and fit then only the B0 or B+ case (The uncertainties change because the isospin averaged external branching ratio has a smaller uncertainty than the individuals, driven by the BR(B0)):

|                                | BGL$_{121}$   | CLN        |
|:-------------------------------|:--------------|:-----------|
| $B^+ \to D^{*0} \ell \nu_\ell$ | 41.2+/-1.0    | 40.7+/-0.9 |
| $B^0 \to D^{*+} \ell \nu_\ell$ | 39.5+/-1.4    | 39.2+/-1.2 |
| $B \to D^{*} \ell \nu_\ell$    | 40.8+/-0.9    | 40.3+/-0.9 |

We can see that the central values for Vcb move closer together, indicating that the tension is created by the external input to some extent. The leftover tension is 1.17 standard deviation (taking into account the correlations between both determinations). We can make a statement about this in the draft, as other readers will probably think about the same question.




### B --> D* l nu with fitted B0/B+ shapes
|                        | Value          |   Correlation |       |       |       |       |
|:-----------------------|:---------------|--------------:|------:|------:|------:|------:|
| $a_0 \times 10^3$      | 24.93+/-1.41   |          1    |  0.25 | -0.21 |  0.26 | -0.3  |
| $b_0 \times 10^3$      | 13.11+/-0.18   |          0.25 |  1    | -0.01 | -0.01 | -0.62 |
| $b_1 \times 10^3$      | -12.35+/-12.73 |         -0.21 | -0.01 |  1    |  0.25 | -0.48 |
| $c_1 \times 10^3$      | -0.92+/-0.97   |          0.26 | -0.01 |  0.25 |  1    | -0.49 |
| $V_{cb} \times 10^3$   | 40.80+/-0.92   |         -0.3  | -0.62 | -0.48 | -0.49 |  1    |

|                        | Value        |   Correlation |       |       |       |
|:-----------------------|:-------------|--------------:|------:|------:|------:|
| $\rho^2$               | 1.26+/-0.09  |          1    |  0.56 | -0.89 |  0.38 |
| $R_1(1)$               | 1.32+/-0.08  |          0.56 |  1    | -0.63 | -0.03 |
| $R_2(1)$               | 0.85+/-0.07  |         -0.89 | -0.63 |  1    | -0.15 |
| $V_{cb} \times 10^3$   | 40.33+/-0.86 |          0.38 | -0.03 | -0.15 |  1    |


### B --> D* l nu with fitted B0 shapes
|                        | Value         |   Correlation |       |       |       |       |
|:-----------------------|:--------------|--------------:|------:|------:|------:|------:|
| $a_0 \times 10^3$      | 26.99+/-2.59  |          1    |  0.15 | -0.14 |  0.29 | -0.29 |
| $b_0 \times 10^3$      | 13.11+/-0.18  |          0.15 |  1    |  0.01 |  0    | -0.41 |
| $b_1 \times 10^3$      | 10.22+/-24.77 |         -0.14 |  0.01 |  1    |  0.46 | -0.72 |
| $c_1 \times 10^3$      | 0.25+/-1.87   |          0.29 |  0    |  0.46 |  1    | -0.73 |
| $V_{cb} \times 10^3$   | 39.53+/-1.35  |         -0.29 | -0.41 | -0.72 | -0.73 |  1    |

|                        | Value        |   Correlation |       |       |       |
|:-----------------------|:-------------|--------------:|------:|------:|------:|
| $\rho^2$               | 1.07+/-0.19  |          1    |  0.56 | -0.89 |  0.64 |
| $R_1(1)$               | 1.36+/-0.15  |          0.56 |  1    | -0.65 |  0.05 |
| $R_2(1)$               | 0.95+/-0.12  |         -0.89 | -0.65 |  1    | -0.35 |
| $V_{cb} \times 10^3$   | 39.17+/-1.21 |          0.64 |  0.05 | -0.35 |  1    |


### B --> D* l nu with fitted B+ shapes

|                        | Value          |   Correlation |       |       |       |       |
|:-----------------------|:---------------|--------------:|------:|------:|------:|------:|
| $a_0 \times 10^3$      | 24.25+/-1.66   |          1    |  0.2  | -0.25 |  0.28 | -0.28 |
| $b_0 \times 10^3$      | 13.11+/-0.18   |          0.2  |  1    | -0.02 | -0.01 | -0.58 |
| $b_1 \times 10^3$      | -21.31+/-15.21 |         -0.25 | -0.02 |  1    |  0.15 | -0.49 |
| $c_1 \times 10^3$      | -1.07+/-1.15   |          0.28 | -0.01 |  0.15 |  1    | -0.5  |
| $V_{cb} \times 10^3$   | 41.20+/-0.99   |         -0.28 | -0.58 | -0.49 | -0.5  |  1    |

|                        | Value        |   Correlation |       |       |       |
|:-----------------------|:-------------|--------------:|------:|------:|------:|
| $\rho^2$               | 1.33+/-0.11  |          1    |  0.57 | -0.9  |  0.37 |
| $R_1(1)$               | 1.31+/-0.10  |          0.57 |  1    | -0.65 | -0.06 |
| $R_2(1)$               | 0.80+/-0.10  |         -0.9  | -0.65 |  1    | -0.12 |
| $V_{cb} \times 10^3$   | 40.67+/-0.91 |          0.37 | -0.06 | -0.12 |  1    |


### Remove correlation >0.90, sort by N (depcrecated)
|             | $V_{\mathrm{cb}}$     |   $\chi^2$ |   dof |   N |   $\rho_\mathrm{max}$ |
|:------------|:----------------------|-----------:|------:|----:|----------------------:|
| BGL$_{110}$ | $40.0 \pm 0.7$        |       49.2 |    33 |   2 |                  0.79 |
| BGL$_{111}$ | $40.3 \pm 0.8$        |       48.5 |    32 |   3 |                  0.71 |
| BGL$_{120}$ | $40.1 \pm 0.8$        |       49   |    32 |   3 |                  0.72 |
| BGL$_{211}$ | $39.8 \pm 0.9$        |       46.3 |    31 |   4 |                  0.8  |
| BGL$_{121}$ | $40.6 \pm 0.9$        |       48.1 |    31 |   4 |                  0.62 |
| BGL$_{130}$ | $40.0 \pm 0.8$        |       46.9 |    31 |   4 |                  0.72 |
| BGL$_{131}$ | $40.0 \pm 0.9$        |       46.9 |    30 |   5 |                  0.66 |

### All (depcrecated)

|             | $V_{\mathrm{cb}}$   |   $\chi^2$ |   dof |   N |   $\rho_\mathrm{max}$ |
|:------------|:----------------------|-----------:|------:|----:|----------------------:|
| BGL$_{110}$ | $40.0 \pm 0.7$        |       49.2 |    33 |   2 |                  0.79 |
| BGL$_{111}$ | $40.3 \pm 0.8$        |       48.5 |    32 |   3 |                  0.71 |
| BGL$_{112}$ | $40.8 \pm 0.9$        |       46.5 |    31 |   4 |                  0.98 |
| BGL$_{120}$ | $40.1 \pm 0.8$        |       49   |    32 |   3 |                  0.72 |
| BGL$_{121}$ | $40.6 \pm 0.9$        |       48.1 |    31 |   4 |                  0.62 |
| BGL$_{122}$ | $41.3 \pm 1.0$        |       45.6 |    30 |   5 |                  0.98 |
| BGL$_{130}$ | $40.0 \pm 0.8$        |       46.9 |    31 |   4 |                  0.72 |
| BGL$_{131}$ | $40.0 \pm 0.9$        |       46.9 |    30 |   5 |                  0.66 |
| BGL$_{132}$ | $40.8 \pm 1.0$        |       43.8 |    29 |   6 |                  0.98 |
| BGL$_{210}$ | $40.1 \pm 0.7$        |       46.6 |    32 |   3 |                  0.98 |
| BGL$_{211}$ | $39.8 \pm 0.9$        |       46.3 |    31 |   4 |                  0.8  |
| BGL$_{212}$ | $40.3 \pm 0.9$        |       43   |    30 |   5 |                  0.99 |
| BGL$_{220}$ | $39.9 \pm 0.8$        |       46.5 |    31 |   4 |                  0.99 |
| BGL$_{221}$ | $38.7 \pm 0.9$        |       44.2 |    30 |   5 |                  0.96 |
| BGL$_{222}$ | $39.4 \pm 1.1$        |       42.1 |    29 |   6 |                  0.98 |
| BGL$_{230}$ | $39.9 \pm 0.8$        |       45.7 |    30 |   5 |                  0.99 |
| BGL$_{231}$ | $38.6 \pm 1.5$        |       44.1 |    29 |   6 |                  0.99 |
| BGL$_{232}$ | $39.1 \pm 1.3$        |       41.7 |    28 |   7 |                  0.99 |
| BGL$_{310}$ | $40.1 \pm 0.7$        |       46.6 |    31 |   4 |                  0.96 |
| BGL$_{311}$ | $39.8 \pm 0.9$        |       46.3 |    30 |   5 |                  0.95 |
| BGL$_{312}$ | $40.3 \pm 0.9$        |       42.9 |    29 |   6 |                  0.98 |
| BGL$_{320}$ | $39.9 \pm 0.8$        |       46.4 |    30 |   5 |                  0.97 |
| BGL$_{321}$ | $38.7 \pm 0.9$        |       44.2 |    29 |   6 |                  0.95 |
| BGL$_{322}$ | $39.4 \pm 1.1$        |       42.1 |    28 |   7 |                  0.97 |
| BGL$_{330}$ | $39.9 \pm 0.8$        |       45.7 |    29 |   6 |                  0.96 |
| BGL$_{331}$ | $38.6 \pm 1.5$        |       44.1 |    28 |   7 |                  0.99 |
| BGL$_{332}$ | $39.1 \pm 1.3$        |       41.7 |    27 |   8 |                  0.98 |
