Measurement of branching fractions, isospin and $Cp$-violating asymmetries for exclusive $b\to d\gamma$ modes

The Belle Collaboration

The $b\to d\gamma$ process, which proceeds via a loop diagram (Fig. 1(a)) in the Standard Model (SM), provides a valuable tool to search for physics beyond the SM, since the loop diagram may also involve virtual heavy non-SM particles bib:rhogam-bsm (). The process has been observed in the exclusive modes $B\to \rho \gamma$ and $B\to \omega \gamma$ by Belle bib:belle-rhogam () and Babar bib:babar-rhogam (). Branching fractions for these modes have been used to constrain the ratio of Cabibbo-Kobayashi-Maskawa (CKM) matrix elements bib:ckm () $|V_{td}/V_{ts}|$; a non-SM effect may be observed as a deviation of $|V_{td}/V_{ts}|$ from the expectation based on measurements of other CKM matrix elements and unitarity of the matrix bib:ckmfitter (). An additional contribution from an annihilation diagram (Fig. 1(b)) may induce a direct $CP$-violating asymmetry in $B^{+}\to {\rho }^{+}\gamma$, and an isospin asymmetry between $B\to \rho \gamma$ modes; the latter can be used to constrain the CKM unitarity triangle angle ${\varphi }_3$ bib:pball (). These quantities are also sensitive to physics beyond the SM bib:alilungi (). In this paper, we report new measurements of the $B\to \rho \gamma$ and $B\to \omega \gamma$ processes using a data sample of $(657\pm 9)\times {10}^6$ $B$ meson pairs accumulated at the ${\rm Y}\left(4S\right)$ resonance. With a data sample almost twice as large and an improved analysis procedure, these results supersede those in bib:belle-rhogam ().

The data are obtained in $e^{+}{e}^{-}$ annihilation at the KEKB energy-asymmetric (3.5 on 8 GeV) collider bib:kekb () and collected with the Belle detector bib:belle-detector (). The Belle detector includes a silicon vertex detector (SVD), a central drift chamber (CDC), aerogel threshold Cherenkov counters (ACC), time-of-flight (TOF) scintillation counters, and an electromagnetic calorimeter (ECL) comprised of CsI(Tl) crystals located inside a 1.5 T superconducting solenoid coil. An iron flux-return located outside of the coil is instrumented to identify $K_L^0$ and muons (KLM).

We reconstruct three signal modes, $B^{+}\to {\rho }^{+}\gamma$, $B^0\to {\rho }^0\gamma$ and $B^0\to \omega \gamma$, and two control samples, $B^{+}\to K^{\ast +}\gamma$ and $B^0\to K^{\ast 0}\gamma$. Charge-conjugate modes are implicitly included unless otherwise stated. The following decay modes are used to reconstruct the intermediate states: ${\rho }^{+}\to {\pi }^{+}{\pi }^0$, ${\rho }^0\to {\pi }^{+}{\pi }^{-}$, $\omega \to {\pi }^{+}{\pi }^{-}{\pi }^0$, $K^{\ast +}\to K^{+}{\pi }^0$, $K^{\ast 0}\to K^{+}{\pi }^{-}$, and ${\pi }^0\to \gamma \gamma$.

Photon candidates are reconstructed from ECL energy clusters having a photon-like shape and no associated charged track. A photon with an ${\rm Y}\left(4S\right)$ center-of-mass (c.m.) energy $\left(E_{\gamma }^{\ast }\right)$ in the range $[1.8,3.4]\text{~{}GeV}$ is selected as the primary photon candidate. Photons detected by the endcap ECL, which were excluded in the previous analysis, are also used. To suppress backgrounds from ${\pi }^0/\eta \to \gamma \gamma$ decays, we apply a veto algorithm based on the likelihood calculated for every photon pair consisting of the primary photon and another photon. We also reject the primary photon candidate if the ratio of the energy in the $3\times 3$ crystal array, centered on the crystal with the maximum energy, to that in the $5\times 5$ array is less than 0.95.

Neutral pions are formed from photon pairs with invariant masses within $\pm 16\text{~{}MeV}/c^2$ ($\sim 3\sigma$) of the ${\pi }^0$ mass. We require the energy of each photon to be greater than $50\text{~{}MeV}$, and the cosine of the angle between the two photons in the laboratory frame to be greater than 0.58 (0.40) for the ${\pi }^0$ from ${\rho }^{+}$ ($\omega$). The photon momenta are then recalculated with a ${\pi }^0$ mass constraint.

Charged pions and kaons are selected from tracks in the CDC and SVD. Each track is required to have a transverse momentum greater than $100\text{~{}MeV}/c$ and a distance of closest approach to the interaction point within $0.5\text{~{}cm}$ in radius and within $\pm 3.0\text{~{}cm}$ along the positron beam ($z$-) axis. We use a likelihood ratio $L_{\pi }/(L_{\pi }+L_K)<0.3$ for pions and $>0.6$ for kaons, where the pion and kaon likelihoods $L_{\pi }$ and $L_K$ are determined from ACC, TOF and CDC information. The criteria have efficiencies of $86\%$, $87\%$ and $89\%$ for a pion from ${\rho }^{+}$, ${\rho }^0$ and $\omega$, respectively; the misidentification probability for a kaon is $8.3\%$ ($8.5\%$) for ${\rho }^{+}$ (${\rho }^0$). Kaons for $K^{\ast }$ candidates are selected with an efficiency of $85\%$. Invariant masses for the $\rho$, $\omega$ and $K^{\ast }$ candidates are required to be within windows of $[640,890]\text{~{}MeV}/c^2$, $[760,800]\text{~{}MeV}/c^2$, and $[820,970]\text{~{}MeV}/c^2$, respectively.

Candidate $B$ mesons are reconstructed by combining a $\rho$ or $\omega$ candidate with the primary photon and calculating two variables: the beam-energy constrained mass $M_{bc}=\sqrt{(E_{beam}^{\ast }/c^2{)}^2-|\overrightarrow{p}{}_B^{\ast }/c{|}^2}$, and the energy difference $\Delta E=E_B^{\ast }-E_{beam}^{\ast }$. Here, $\overrightarrow{p}{}_B^{\ast }$ and $E_B^{\ast }$ are the c.m. momentum and energy of the $B$ candidate, and $E_{beam}^{\ast }$ is the c.m. beam energy. To improve resolution, the magnitude of the photon momentum is replaced by $(E_{beam}^{\ast }-E_{\rho /\omega }^{\ast })/c$ when the momentum $\overrightarrow{p}{}_B^{\ast }$ is calculated.

To optimize the event selection, we study Monte Carlo (MC) events in a signal box defined as $5.273\text{~{}GeV}/c^2<M_{bc}<5.285\text{~{}GeV}/c^2$ and $|\Delta E|<0.1\text{~{}GeV}$. For each signal mode, we choose selection criteria to maximize $N_S/\sqrt{N_S+N_B}$, where $N_S$ and $N_B$ are the expected signal and the sum of the background yields.

The dominant background arises from continuum events ($e^{+}{e}^{-}\to q\overline{q}\left(\gamma \right)$, $q=u,d,s,c$), where a random combination of a $\rho$ or $\omega$ candidate with a photon forms a $B$ candidate. We suppress this using a Fisher discriminant ($F$) calculated from modified Fox-Wolfram moments bib:ksfw () and other variables, i.e., the cosine of the polar angle ($\cos {\theta }_B^{\ast }$) of the $B$ direction, the distance along the $z$-axis ($\Delta z$) between the signal vertex and that of the rest of the event and, in addition, $M_{{\pi }^{+}{\pi }^{-}{\pi }^0}$ and Dalitz plot variables for the $\omega \gamma$ mode. For each of these quantities, we construct likelihood distributions for signal and continuum events. The distributions are determined from MC samples.

From these likelihood distributions we form likelihoods $L_s$ and $L_c$ for signal and continuum background, respectively. In addition, we use a flavor-tagging quality variable $r$ that indicates the level of confidence in the $B$-flavor determination as described in Ref. bib:hamlet (). In the $(r,R)$ plane defined by the tagging quality $r$ and the likelihood ratio $R=L_s/(L_s+L_c)$, signal tends to populate the edges at $r=1$ and $R=1$, while continuum preferentially populates the edges at $r=0$ and $R=0$. We divide the events into six bins of $r$ (two bins between 0 and 0.5, and four bins between 0.5 and 1) and determine the minimum $R$ requirement for each bin. In the ${\rho }^{+}\gamma$ mode, if the tagging-side flavor is the same as that of the signal side, we assign the events to the lowest bin $0\le r<0.25$. The $R$ criteria reject 98$\%$ of continuum background while retaining $35\%$, $51\%$ and $43\%$ of the ${\rho }^0\gamma$, ${\rho }^{+}\gamma$ and $\omega \gamma$ signals, respectively. For the $K^{\ast +}\gamma$ ($K^{\ast 0}\gamma$) mode, we use the criteria for the ${\rho }^{+}\gamma$ (${\rho }^0\gamma$) mode.

We consider the following backgrounds from $B$ decays: $B\to K^{\ast }\gamma$, other $B\to X_s\gamma$ processes, decays with a ${\pi }^0/\eta$ ($B\to \rho {\pi }^0$, $\omega {\pi }^0$, $\rho \eta$ and $\omega \eta$), other charmless hadronic $B$ decays, and $b\to c$ decay modes. The $B\to K^{\ast }\gamma$ background can mimic the $B\to \rho \gamma$ signal if the kaon from the $K^{\ast }$ is misidentified as a pion. To suppress $B\to K^{\ast }\gamma$ events we calculate $M_{K\pi }$, where the kaon mass is assigned to the charged pion candidate; for ${\rho }^0\gamma$, the lower of the two $M_{K\pi }$ values is taken (misassignment tends to give a higher $M_{K\pi }$). For the ${\rho }^{+}\gamma$ mode, we reject the candidate if $M_{K\pi }<0.90\text{~{}GeV}/c^2$, while for the ${\rho }^0\gamma$ mode we use $M_{K\pi }$ in the fit procedure to extract the signal (note: $M_{K\pi }>0.92\text{~{}GeV}/c^2$ is required when optimizing selection criteria). The $b\to s\gamma$ modes ($B\to K^{\ast }\gamma$ and other $B\to X_s\gamma$ decays) contribute to the background when the $\rho$ and $\omega$ candidates are formed from random combinations of particles. Decays with a ${\pi }^0/\eta$ can mimic the signal if one of its daughter photons is not detected. To suppress this background, we reject the candidate if $|\cos {\theta }_{hel}|>0.80$, $0.75$ and $0.83$ for the ${\rho }^{+}\gamma$, ${\rho }^0\gamma$ and $\omega \gamma$ mode, respectively, where the helicity angle ${\theta }_{hel}$ is the angle between the ${\pi }^{+}$ track (the normal to the $\omega$ decay plane) and the $B$ momentum vector in the $\rho$ ($\omega$) rest frame. We study large MC samples and find no other distinctive hadronic $B$ decay background sources.

The reconstruction efficiency for each mode is defined as the fraction of the signal remaining after all selection criteria are applied, where the signal yield is determined from a fit to the sum of the signal and continuum MC samples using the procedure described below. We take the pion identification efficiency from a data sample of $D^{\ast +}\to D^0{\pi }^{+}$, $D^0\to K^{-}{\pi }^{+}$. The total efficiencies are listed in Table 1. The systematic error in the efficiency is the quadratic sum of the following contributions, estimated using control samples: the uncertainty in the photon detection efficiency (2.4%) as measured in radiative Bhabha events; charged tracking efficiency (1.0% per track) from partially reconstructed $D^{\ast +}\to D^0{\pi }^{+}$, $D^0\to K_S^0{\pi }^{+}{\pi }^{-}$, $K_S^0\to {\pi }^{+}\left({\pi }^{-}\right)$; charged pion and kaon identification (0.5 to 0.6% per track) from $D^{\ast +}\to D^0{\pi }^{+}$, $D^0\to K^{-}{\pi }^{+}$; neutral pion detection (4.6%) from $\eta$ decays to $\gamma \gamma$, ${\pi }^{+}{\pi }^{-}{\pi }^0$ and $3{\pi }^0$; and $R$-$r$ and ${\pi }^0/\eta$ veto requirements (2.0 to 8.4%) from $B\to D{\pi }^{-}$ with $D^0\to K^{-}{\pi }^{+}$, $K_S^0\omega$ and $D^{+}\to K^{-}{\pi }^{+}{\pi }^{+}$.

We perform an unbinned extended maximum likelihood fit to $M_{bc}$ and $\Delta E$ (and $M_{K\pi }$ for the ${\rho }^0\gamma$ mode) for candidates satisfying $|\Delta E|<0.5\text{~{}GeV}$ and $M_{bc}>5.2\text{~{}GeV}/c^2$. The fit is performed individually for the three $\rho \gamma /\omega \gamma$ signal modes and the two $K^{\ast }\gamma$ modes. We describe the events in the fit region using the sum of probability density functions for the signal, continuum, $K^{\ast }\gamma$ (for the $\rho \gamma$ modes only), and other background hypotheses. We use the distributions of MC events in histograms to model the $M_{bc}$-$\Delta E$ shapes of $B$ decay background components and the $M_{K\pi }$ shapes for all components.

The signal distribution for the ${\rho }^0\gamma$ and $K^{\ast 0}\gamma$ modes is modeled as the product of a Crystal Ball lineshape bib:cbls () in $\Delta E$ to reproduce the asymmetric ECL energy response, a Gaussian in $M_{bc}$, and an MC histogram distribution for $M_{K\pi }$. For the ${\rho }^{+}\gamma$, $K^{\ast +}\gamma$ and $\omega \gamma$ modes, we use the product of a Crystal Ball lineshape for $\Delta E$ and another Crystal Ball lineshape for $M_{bc}$. The signal parameters of $M_{bc}$ and $\Delta E$ shapes for $K^{\ast }\gamma$ modes are determined from fitting the data; for the $\rho \gamma /\omega \gamma$ modes, they are taken from MC and calibrated using the data/MC difference of the fits to the $K^{\ast +}\gamma$ and $K^{\ast 0}\gamma$ samples for the modes with and without a neutral pion, respectively.

The continuum background component is modeled as the product of a linear function in $\Delta E$, an ARGUS function bib:argus-function () in $M_{bc}$, and, for ${\rho }^0\gamma$, an MC histogram for $M_{K\pi }$. The continuum shape parameters and normalizations are mode dependent and allowed to float.

There is significant $K^{\ast 0}\gamma$ background in the ${\rho }^0\gamma$ sample. This background is modeled by the product of a two-dimensional $M_{bc}$-$\Delta E$ histogram and an $M_{K\pi }$ histogram. Similarly, the $K^{\ast +}\gamma$ background for ${\rho }^{+}\gamma$ is modeled by a two-dimensional $M_{bc}$-$\Delta E$ histogram. In both cases, the $\Delta E$ peak position is shifted from the $\rho \gamma$ signal peak; this offset is determined from fitting the MC histogram shape to a $K^{\ast }\gamma$ data sample in which the pion mass is assigned to kaons. The same $K^{\ast }\gamma$ sample together with the known kaon to pion misidentification probability is also used to determine the size of the $K^{\ast }\gamma$ background.

Other $B$ decays are considered as an additional background component when we extract the signal yield. The levels of these backgrounds are fixed using known branching fractions or upper limits bib:hfag2006 ().

The systematic error in the signal yield due to the fitting procedure is estimated by varying each of the fixed parameters by $\pm 1\sigma$ and then taking the quadratic sum of the deviations in the branching fraction from the nominal value. The varied parameters are the signal shape parameters, branching fractions of the background components, $\Delta E$ shift of the $K^{\ast }\gamma$ component, and the kaon to pion misidentification probability determined from a control sample. The results of the fits are shown in Fig. 2 and listed in Table 1.

The systematic error in the branching fraction has contributions from the efficiency, fitting, and the number of $B$ meson pairs; we add these together in quadrature. The significance is defined as $\sqrt{-2\ln \left(L_0/L_{max}\right)}$, where $L_{max}$ ($L_0$) is the value of the likelihood function when the signal yield is floated (set to zero). To include systematic uncertainty, the likelihood function from the fit is convolved with a Gaussian systematic error function.

Table 1 also lists combined branching fractions, which are calculated from the products of likelihoods from individual fits. We combine ${\rho }^{+}\gamma$ and ${\rho }^0\gamma$ modes (referred to as $\rho \gamma$) and three $\rho \gamma$ and $\omega \gamma$ modes (referred to as $(\rho ,\omega )\gamma$) assuming a single branching fraction $B(B\to \rho \gamma )\equiv B(B\to (\rho ,\omega \left)\gamma \right)\equiv B(B^{+}\to {\rho }^{+}\gamma )=2\times \frac{{\tau }_{B^{+}}}{{\tau }_{B^0}}B(B^0\to {\rho }^0\gamma )=2\times \frac{{\tau }_{B^{+}}}{{\tau }_{B^0}}B(B^0\to \omega \gamma )$ bib:ali-1994 (); bib:ali-cdlu (), where $\frac{{\tau }_{B^{+}}}{{\tau }_{B^0}}=1.071\pm 0.009$ bib:pdg2006 (). The results are consistent with the previous measurements bib:belle-rhogam (); bib:babar-rhogam () and have smaller errors. They are also in agreement with SM predictions bib:ali-cdlu (); bib:pball (); bib:bosch-buchalla ().

Using the prescription in Ref. bib:pball (), Eq. 3 for example gives $|V_{td}/V_{ts}|=0.195{}_{-0.019}^{+0.020}(exp.)\pm 0.015(th.)$. This is consistent with determinations from $B_s^0$ mixing  bib:hfag2006 (), which involve box diagrams rather than penguin loops. We also find $B(B^{+}\to K^{\ast +}\gamma )=(384\pm 17)\times {10}^{-7}$ and $B(B^0\to K^{\ast 0}\gamma )=(378\pm 8)\times {10}^{-7}$ (statistical error only), in agreement with the world average.

In conclusion, we present a new measurement of branching fractions for $B\to \rho \gamma$ and $B\to \omega \gamma$, a measurement of the isospin asymmetry, and the first measurement of the direct $CP$-violating asymmetry for $B^{+}\to {\rho }^{+}\gamma$. The results are consistent with SM predictions. We improve the experimental precision on $|V_{td}/V_{ts}|$ determined from penguin loops, finding good agreement with the value determined from box diagrams bib:hfag2006 ().

We thank the KEKB group for excellent operation of the accelerator, the KEK cryogenics group for efficient solenoid operations, and the KEK computer group and the NII for valuable computing and SINET3 network support. We acknowledge support from MEXT and JSPS (Japan); ARC and DEST (Australia); NSFC and KIP of CAS (China); DST (India); MOEHRD, KOSEF and KRF (Korea); KBN (Poland); MES and RFAAE (Russia); ARRS (Slovenia); SNSF (Switzerland); NSC and MOE (Taiwan); and DOE (USA).