# Belle $\tau \rightarrow \mu \gamma$ selection

###### tags: `Belle` `Steering File` `Selection`

## Total Selection (*not* identification)
| Alias| Selection | Reasons|
| -------- | -------- | -------- |
| $p^{CM}<4.5 GeV/c$ | Track CMS frame<4.5GeV | Reduce Bhabha & $\mu^+$$\mu^-$ events |
| $P_t<0.1 GeV/c$ , $-0.866<\cos(\theta)<0.956$ | In CDC acceptable region cut transverse momentum | Suppress two photons BG |
| $E_\gamma>0.1 GeV$ , $-0.866<\cos(\theta)<0.956$ | In CDC acceptable region cut gamma energy | |
| thrust vector | 0.90~0.98 | >0.90 : Supress $q\bar{q}$ Bg , <0.98 : Supress $\mu^+$$\mu^-$ |

## $\mu \gamma$ Identification and Selection
| Alias | Selection | Reasons |
| --------------------------------------------------------- | ---------------------------------------------------- | ----------------------------------------------- |
| $0.4<cos\theta^{CM}_{\mu \gamma}<0.8$ | Cosine Angle between $\mu^{\pm}, \gamma$ in CM frame | Reject $\tau^+\tau^-$ events containing $\pi^0$ |
| $\mathcal{L_{\mu^{\pm}}}>0.95$, $p_{\mu^{\pm}}>1.0 GeV/c$ | | |
| $E_{\gamma}>0.5 GeV$ | | |
| $-0.602<\cos\gamma<0.829$ | | Remove spurious combinations of $\gamma$ |
| $E_{Sum}^{CM}<9.0 GeV$ | Total energies(two tracks+gamma) | reject $\mu^{+}\mu_{-}$ events |
|$\theta^{Two\;Tracks}_{CM}>90^{o}$|two tracks opening angles > 90 degree|Remove multiplicity(combinations) of $\mu$.$\gamma$
|$\cos{\theta_{\mu\tau}^{CM}}<0.4$||Remove multiplicity(combinations) of $\mu$.$\gamma$

## Tag Selection

| Alias | Selection | Reasons |
| ----------------------- | --------- | --------------------------------------------- |
| $\mathcal{L_{\mu}}<0.8$ | | Tag side doesn't want $\mu^{\pm}$ as daughter |
|$p_{miss}>0.4 GeV/c$, $-0.866<\cos{\theta_{miss}}<0.956$ | | In CDC acceptable regions & missing momentum >0.4 where $p_{miss}$=$\mid\vec{P_{beam}}-(\vec{P_{Two\;Tracks}}+P_{E})\mid$
| $0.4<\cos{\theta^{CM}_{miss_\neq\mu}<0.98}$ | tag side track & missing particle opening angle in CM frame | Reduce $\tau^+\tau^-$ Bg events
|$-1.0GeV^2/c^4<m^2_{miss}<2.0GeV^2/c^4$ & $-5m^2_{miss}-1<p_{miss}<1.5m^{2}_{miss}-1$ |2-D figure $(m^2_{miss}-P_{miss})$|$m_{miss}^2 = (E_{\mu\gamma}^{CM}-E_{tag}^{CM})^2-(P^{CM}_{miss})$

## Main Background Contribution
* $\tau^{\pm} \rightarrow \mu^{\pm} \nu_{\mu} \nu_{\tau}$
### $\pi$ misidentified + beam Bg photon or initail state radiation
* $\tau^{\pm} \rightarrow \pi^{\pm} \mu_{\tau}$

* $e^+e^- \rightarrow \mu^+\mu^-$

* $e^+e^- \rightarrow \gamma \gamma$

### fitting variables $\quad \Delta E \quad M_{inv}$
#### 1. PDF: Signal Monte Carlo Assymetric Gaussian $\sigma^{high/low} = 14.49\pm0.10/24.24\pm0.13 MeV/c^2$ $\\ \Delta E = 35.29\pm0.49/28.14\pm0.94 MeV$
#### 2. Use Binned Likelihood Fitting Background: Landau + Gaussian

#### 3. Fitting Region: Assymetric Gaussian $5\sigma$ region $\rightarrow$ ****blind test****: Data $3\sigma$ region is not examined . Only examiing sideband(ex:3~5$\sigma$) to modify background fitting size owing to simulation and real data discrepancy.

#### 4. Final Data Sideband fitting(almost Background) is also using Binned Likelihood Fitting

## Signal Region
#### 1.Fitting Method: Unbinned Extended Maximum Likelihood
$$\cal{L} = \frac{e^{-(s+b)}}{N!} \prod_{i}^{N}(sS_i + bB_i)$$
N: total Events s: the number of signal $S_i$: ith value from Signal MC PDF
b: the number of BG $B_i$: ith value from Bg MC PDF

#### Concentrate on 2$\sigma$ fit