# $\tau^{\pm} \rightarrow \rho^{\pm}(\pi^{\pm} \pi^0) \nu_{\tau}$
###### tags: `Belle` `tau`

## Motivation
### make $a_{\mu}$ precise
* Pion form factor is one of the major factor in $e^{+}e^{-}$ spectral
function, and this function is an important parameter in predicting hadronic
vacuum polarization term $a^{had,LO}_{\mu}$. Because this parameter
contributes the most significant uncertainty in $a_{\mu}$,
well-measured of pion form factor could lower the
uncertainty of muon anomalous magnetic moment.
\vspace{5mm} %5mm

* Experimentally, both $Br(\tau^{\pm} \rightarrow \pi^{\pm} \pi^{\pm} \pi^0)$ and $\rho$ mass
spectrum are two parameters in this measurement. In Belle2 tau
group, we make efforts on measuring these two parameters to
improve pion form factor.
* The theoretical uncertainty comes from the **hadronic vacuum polarization** $a^{had,LO}_{\nu}$
* $a^{had,LO}_{\nu}=\frac{1}{4\pi}\int^{\infty}_{4m^2_{\pi}}\sigma^{0}_{had}(s)(\frac{m^2_\mu}{3s})K(s)ds$
1. $\sigma^{0}_{had}(s)$ is the total cross section of $e^+e^- \rightarrow hadrons$ at C.M. energy $\sqrt{s}$ which also includes $e^+e^- \rightarrow \pi^+\pi^-$
2. \begin{aligned}K(s)&=\frac{3s}{m^2_{\mu}}\{x^2(1-\frac{x^2}{2}) \\ & +(1+x)^2(1+\frac{1}{x^2})[ln(1+x)-x+\frac{x^2}{2}]+(\frac{1+x}{1-x})x^2lnx\}\\ & x=\frac{1-\beta_{\mu}}{1+\beta_{\mu}}, B_{\mu}=\sqrt{1-4m_{\mu}^2/s}\end{aligned}
* the $\sigma(e^+e^- \rightarrow \pi^+ \pi^-)$ is part of $\sigma^{0}_{had}(s)$. $\sigma(e^+e^- \rightarrow \pi^+ \pi^-)=\frac{4\pi^2\alpha_0^2}{s}v_0(s)$, $\alpha_0$ is fine-structure constant at s=0, $v_{0}(s)$ is $\pi^+\pi^-$ spectral function(proportional to structure function)
* Conservation of Vector Current theorem(CVC) $\rightarrow$ conserve U(1) gauge symmetry(Noether theorem). This theorem combined with isospin-violating effects helps us relate the $\pi\pi^0$ mode to $\pi^+\pi^-$ by equation $v_{-}(s) = v_0^{I=1}(s)$.
* Also, the spectral function is related to structure function(form factor): $v_{-}(s)=\frac{\beta_{-}^3(s)}{12\pi}|F^{j}_{\pi}(s)|$, $\beta_{-}=1-4m_{\pi^2}/s$(pion velocity in $\pi^-\pi^0$ rest frame)
* **Our Experiment aims to give high-precision form factor, and this can be accomplished by correctly finding the $Br(\tau \rightarrow \pi\pi^0\nu)$ and normalized mass spectrum** by equation: $s=M^2_{\pi\pi^0}$, so $\frac{dN_{\pi\pi}}{ds}$ is the $\rho$ mass spectrum
$F^{-}_{\pi}(s) = \frac{2m_{\tau}^2}{|V_{ud}|^2(1-\frac{s}{m_{\tau}^2})^2(1+\frac{2s}{m_{\tau}^2})S_{EW}\beta^3_{-}}\frac{\mathcal{B_{\pi\pi}}}{\mathcal{B_{e}}}(\frac{1}{N_{\pi\pi}}\frac{dN_{\pi\pi}}{ds})$
Finally, if aquiring this form factor, I can put this into spectral function and by CVC theorem(two spectral function are equal), then acquire $\sigma(e^+e^- \rightarrow \pi^+ \pi^-)$ to influence $a_{\mu}^{had,LO}$ precision.

## Select $\tau^+ \tau^{-}$ Events

| Column 1 | Column 2 |
| ---------------------------------------------- | ----------------------------- |
| 2 or 4 Tracks | |
| Each Tracks $\quad p_{T}>0.1GeV$ | |
| $\lvert dz \rvert< 2.5cm$ | |
| $dr <0.5cm$ | |
| $p^*_{1st} + p^*_{2st} < 9.0 GeV/c$ | Supress Bhabha & $\mu^+\mu^-$ |
| $p_{Tmax}>0.5GeV/c$ |
|<font color="#FF0000">$35<\theta^*<145$</font>||
|barrel:$E_{\gamma}> 0.05GeV$, endcap:$E_{\gamma}> 0.15GeV$||
|$M_{miss}-\theta^*_{miss}$ plot ( fig1 )|$M_{miss}\approx 0:$ Bhabha $M_{miss}\geq 7: 2\gamma$ |
|$X_{part}=(n_{tr}+n_{\gamma})_{1}\times{(n_{tr}+n_{\gamma})_{2}}\leq25$|num of tracks, $\gamma$ in respective hemisphere divided by perpendicular to $p_{tr max}$ in CM frame(low multiplicity)|
|$\varepsilon=\lvert \lvert\phi_1-\phi_2\rvert-\pi \rvert>1^{。}$|$p_{1st},p_{2st}$ acollinearity in azimuth<font color="#FF0000"> I think is in CM frame</font>|
<iframe width="90%" height="300" src="https://i.imgur.com/UggXLVJ.jpg" frameborder="10"></iframe>

## $\pi_0\rightarrow \gamma \gamma$

$S_{\gamma\gamma}=\frac{(m_{\gamma\gamma}-m_{\pi_0})}{\sigma_{\gamma\gamma}}$,<font color="#FF0000"> $\sigma_{\gamma\gamma}=\sigma_{\pi_0}(p_{\pi_0},\phi)$</font>
| Column 1 | Column 2 |
| -------- | -------- |
| $p^{CM}_{\pi_0}>0.25 GeV/c$ | neglect electronic bg |
|$E_{\gamma}>0.08 GeV$|neglect electronic bg
|\begin{split} Signal\:Region:-6<S_{\gamma\gamma}<5\\Sideband\:Region: 7<\lvert S_{\gamma\gamma}\rvert<9\end{split}||
|$Reject\: E_{\gamma}>0.2GeV$ *for the additional $\gamma$ in the same hemisphere* |reduce the final states $\tau^- \rightarrow \pi^-(n\pi^0)\nu_{\tau}$|

## $\rho^-(770)\rightarrow \pi^-\pi^0\quad\rho^{'}(1450)\rightarrow\pi^-\pi^0\quad \rho^{''}(1700)\rightarrow\pi^-\pi^0$

| Column 1 | Column 2 |
| -------- | -------- |
| $\pi_0 \:Mass\: Constraint$ | Improve $\pi_0$ energy resolution |
|*Tag Restriction: 1 Track & 0 $\gamma$* | Improve S/N in high mass region$M_{\pi\pi_0}^2\geq2.0(GeV/c^2)^2$ |

## Dominant Background ( First assuming that both $\tau^{\pm} \rightarrow h^{\pm}(\rho^{\pm},K^{\pm}) \nu_{\tau}$ are signal)
1. $(6.02\pm0.08\%)\quad\tau^-\rightarrow h^-(n\pi^0)\nu_{\tau}$: peaking at low mass signal region
2. $(0.48\pm0.04\%)\quad\tau^-\rightarrow K_Lh^-\pi_0\nu_{\tau}$: peaking at low mass signal region
3. $(0.10\pm0.01\%)\quad\tau^-\rightarrow \omega(\pi^0\gamma)\pi^-\nu_{\tau}$
4. $(2.22\pm0.05\%)\quad q\bar{q}$: slightly peaking at low mass
5. 5.total tau + qq Bg = (7.02+2.22)%

----
## Questions
### in Belle Paper
1. in Belle paper for selecting $\tau\tau$, it shows a large difference compared to nowadays tauskim in Belle2, should I do both for selecting $\tau\tau$

> [name=Michel] As you pointed out, the Belle and Belle II skim selections are quite different. The Belle paper is just a guide on what cuts may be used for removing any non-taupair background contributions, but you don't have to use exactly the same.
> You have to find a selection that works with the Belle II conditions. The steering file that I wrote is also just a proposal.

2. For $S_{\gamma\gamma}$, how do I calculate the $\sigma_{\gamma\gamma}$

> [name=Michel] This is the resolution of $\pi^0$. For now you can work with the invariant mass of $\gamma \gamma$ (the $\pi^0$ mass) and later check with the neutrals group what are the recomendations for $\pi^0$ reconstruction.
> But again, the Belle paper is just a guide. Their selection was based on studies done at Belle. It is better to check what has been done in Belle II to take a decision, but for now you can go ahead. $\sigma_{\gamma\gamma}$ is relevant once you switch to data and start making comparisons with the simulation, and for now you will only work with MC.

3. should I have to contrain tag side only 1 track+ 0 $\gamma$ first, or like paper did - put it in later

> [name=Michel] In Belle II, we have seen that removing the photons from tag side helps with a better agreement between data and MC. For this particular steering file, we are testing first with 3 tracks in the tag side (see my reply below).

4. I have no idea about $\theta^*$ in the $\tau\tau$ event selection

> [name=Michel] Acording to the paper, this is the polar angle of the leading particle with respect to the beam axis. It means, the theta angle of the track with highest momentum.
> But once more, this is just a guide, if you want to test if this helps go ahead.

### in Simulation, Steering file
1. In Tauolla2, the decay table only generate the $\rho^+\rightarrow \pi^+\pi^0$ and not generate $\rho^-,\rho^{"}$. How can I generate the same Signal Monte Carlo with Belle Paper?

> [name=Michel] Good question. TauolaBelle2 works very diferent to generate $\tau^+\rightarrow \pi^+\pi^0\nu$ (mode 193 in the table I at https://docs.belle2.org/record/1982/files/BELLE2-NOTE-PH-2020-055_v2.pdf). It uses a parametrization based on models that already contains multiple resonances, then you should be able to observe the effects of such resonances in MC. See for example https://indico.belle2.org/event/3072/contributions/14857/attachments/7639/11834/UpdatesInTauolaOct2020.pdf, slide 3.

2. In Belle paper, they finally argued that they only pick the events whose tag side only decay to one track + no photons, which was distinct with your steering file(Tag Side has three pions is scarce in tau decay), is there any benifits motivating you to do so?

> [name=Michel] We decided with Tsovinar to start with 3-1 topology since it is the one that we understand better right now at Belle II (it is the topology for tau mass measurement, tau --> lepton alpha, the early tau studies at Belle II). With 3 tracks on the tag side, we reduce backgrounds from Bhabha events that has not been very well validated at Belle II.
>
> Of course, studying 1-1 topology is also very important for this study. It increase the statistics, but more important, it reduces qqbar contamination. One thing that we may try is spliting the work, with Tsovinar continuing with 3-1, and you working with 1-1, but this is just a proposal and not an strict rule.

3. In your given file(generic), I want to make sure that all the event passed your steering file already? If it is correct, I don't think it's quite precise to do the selection cuts before passing $\tau\tau$ event selection. Should I do this for myself first?

> [name=Michel] Sorry, I don't understand the question.
> If you want to write your own steering file and submit jobs to grid, of course you can. In fact, this you will have to do it for getting 1-1 events.

----
## Branching Ratio Measurement
**$B_{h\pi^0} =\frac{(N_{\pi\pi^0})_{True}}{(2N_{\tau\tau})_{True}\times{\epsilon_{\tau\rightarrow \pi\pi^0}}}=\frac{N_{h\pi^{0}}\times(1-b^{feed-down}-b^{non-\tau})}{2N_{\tau\tau}\times(1-b_{\tau\tau})}\times\frac{\epsilon^{\tau}_{\pi\pi^0}}{\epsilon_{\tau\tau}}\times\frac{1}{\epsilon^{ID}_{\pi\pi^0}}$**
* $b^{feed-down}:$ the fraction of $\tau$ Bg
* $b^{non-\tau}:$ the fraction coming from $non-\tau$ process($q\bar{q}$ continuum)
* $b_{\tau\tau}:$ background fraction for selecting $\tau\tau$
* $\epsilon_{\tau\tau}:$ efficiency for selecting $\tau\tau$ events
* $\epsilon_{h\pi^0}^{\tau}:$ efficiency for $h\pi^0$ pass the $\tau\tau$ event selection
* $\epsilon^{ID}_{h\pi^0}:\tau \rightarrow \pi\pi^0 \nu_{\tau}$ pass the $h^-\pi^0$ selection after passing $\pi\pi_{0}\nu$ selection
----

## Efficiency