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The majority of beam-beam collisions occur in a tiny re- gion in the center of the detectors, the interaction region or beamspot. The size of the interaction region is determined by beam optics and has varied through the B Factory runs. It is typically 1 mm along the beam (z), 100μm in the horizontal direction (x) and a few μm in the vertical direction (y).

The position and size of the beamspot are used as a constraint in the reconstruction of the B0_B0 _{decay time}

difference Δt. Since the beamspot is smallest in the verti- cal plane, the vertical coordinate is the most constraining. In the directions alongxandzthe beamspot is not smaller than a typical B decay length, which is about25 μm in the transverse plane and about200 μm along thez-axis, and its constraint plays a marginal role.

The position and shape of the interaction region vary with time and needs to be carefully calibrated and moni- tored. The calibration is based on the spatial distribution of reconstructed primary vertices (PVs). In the produc- tion of a B0_B0 _{or B}+_B− _{pair at the Υ(4S) resonance}

there are no particles originating from the primary col- lision point other than the B mesons themselves. Con- sequently, the primary vertex cannot be directly recon- structed in these decays and the beamspot calibration in- stead relies on continuum events. Bhabha and di-muon events have the advantage that there are only two tracks in the event, that have both relatively high momentum and are guaranteed to originate from the PV. Hadronic events have more tracks and consequently a smaller sta- tistical per-event uncertainty on the vertex position, but they are polluted by a b¯b contribution. The calibration

x 0 500 1000 -1.5 -1 -0.5 0 0.5 1 1.5 Entries Mean RMS 7282 -0.5021 0.1718 [mm] [entries/0.06 mm ] y 0 500 1000 1500 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Entries Mean RMS 7282 0.4819 0.1305 [mm] [entries/0.04 mm ] z 0 500 1000 -30 -20 -10 0 10 20 30 Entries Mean RMS 7282 -1.913 3.169 [mm] [entries/1.2 mm ]

Figure 6.4.1.Distribution of thex(top),y(middle), andz

(bottom) position of reconstructed primary vertices in a typical Belle run (Exp. 5, run 333). From (Tomura, 2002a).

inBABAR relies both on two-prong events and on multi- hadron events with at least 5 tracks. The calibration pro- cedure in Belle uses only multi-hadron events (Tomura, 2002a).

An example of the distribution of the position of recon- structed primary vertices in hadronic events in a typical Belle run is shown in Figure 6.4.1. In they direction the RMS of the distribution is dominated by the vertex reso- lution. In thezdirection it is dominated by the beamspot size, while in thexdirection it is a combination of both.

The distribution of PV positions is characterized by an average position, the direction of its three principal axes (which are close, but not identical to thex,y andz axis; see Chapter 2) and the RMS along each axis. The cali- brated position, rotation and sizes are determined from moments of (BABAR) or fits to (Belle) the (x, y, z) distri- bution of PVs.

To determine the size of the beamspot the vertex reso- lution must be ‘subtracted’. In the vertical direction since the resolution is so much wider than the beam size, the beam spread must be estimated by other means. InBABAR

the size inyis computed from the luminosity reported by the accelerator (Chapter 1). In Belle it is obtained from measurements of the size of the HER and LER beams by the accelerator (Tomura,2002a). When the beamspot is used as a constraint in vertex fits, its size always appears in quadrature with the actual vertex resolution. Hence, it is important to know the size in the vertical direction precisely.

Figure 6.4.2. Average primary vertex position inx(top), y

(center) andz(bottom) as a function of run number in Belle data. From (Tomura, 2002a).

To accommodate variations over time the calibration procedure is performed in time slices. Belle fits the mean position with the other parameters (the widths and the rotation angles) fixed for everyO(104_{) events.BABARup-}

dates all parameters every ∼ 10 minute interval, corre- sponding to approximately the same number of selected events. Figure 6.4.2shows the average primary vertex po- sition as a function of run number in the early days of Belle. In this period the typical duration of a run was about 2 hours. Under stable conditions, the variation of the position within a run is much smaller, typically of the order of 10 μm inx, 1 μm inyand 100 μm inzin both experiments.

In vertex reconstruction the average beamspot can be used as a constraint on theproductionvertex of theB(or D, orτ) particle. Theχ2_{contribution takes the form,cf.}

Eq. (6.3.7), Δχ2 ₌ ⎛ ⎝x_ypp−−xyIPIP zp−zIP ⎞ ⎠ T V−1 IP ⎛ ⎝x_ypp−−xyIPIP zp−zIP ⎞ ⎠ (6.4.1) wherexp are the parameters of the production vertex in

the vertex fit, xIP is the position of the center of the

beamspot andVIP is a 3×3 covariance matrix, represen-

tative of the size of the beamspot. In Belle the constraint is only applied to the coordinates in the transverse plane; in BABAR both the 2D and 3D constraint are used, de- pending on the vertex algorithm. Figure 6.4.3 shows the D∗+_−D0 _{mass difference in e}+_e− _{→ D}∗+_{X continuum}

events where we have selectedD∗+_→D0_π+_{decays with}

D0 _{→ K}−_π+ _{with and without the constraint that the}

D∗+ _{originates from the beamspot. Due to its low mo-}

mentum the direction of the soft pion is very sensitive to multiple scattering. Requiring it to originate from the interaction region substantially improves the mass resolu- tion. ) 2 ) (MeV/c 0 )-M(D + M= M(D* Δ 142 144 146 148 ) 2 Events/(70 keV/c 0 20 40 60 80 100 120 140 160 180 200 220 3 10 × no constraint beam spot con.

Figure 6.4.3. Distribution of the reconstructed D∗+₋ D0 mass difference in D∗+ _→ D0 π+ decays with D0 _→ K−π+

fromBABARcontinuum data with and without a primary vertex constraint.

In some applications, such as forD∗fromBdecays or the reconstruction of the associatedBvertex for Δtrecon- struction in Belle, the beamspot is used as a constraint on adecay vertex. In this case the size of the beamspot must be increased with the effective width of the decay length distribution of the (mother) particle, schematically,

VIP,tot = VIP + Vflight. (6.4.2)

Both experiments add the RMS of theBdecay length dis- tribution in the transverse plane (about25μm, see Fig- ure 6.4.4) in quadrature with the calibrated beamspot size to obtain an effective size appropriate forB decay prod- ucts. This mostly affects the size iny.

Finally, although these quantities do not directly per- tain to the vertex algorithms, it is convenient in the char- acterization of the beamspot, to mention the calibration of the beam kinematics. The beam energies are used in the computation ofe.g.the beam-energy-constrained mass (Chapter 9) and the proper decay time. In principle, there are six unknown parameters related to the incident beams, namely the 3-momenta of the electron and positron beam. In practice, the beam-directions are close enough to their nominal direction that only the relative direction matters, reducing the number of degrees of freedom to four. These are parameterized by the center-of-momentum energy√s and by the boost vector.

Both experiments calibrate√swith the kinematics of fully reconstructed hadronic B decays. In particular, a

0 1000 2000 3000 4000 5000 6000 -50 0 50 100 150 -100 -150

B meson flight length in y (μm)

Figure 6.4.4.Distribution of theBmeson flight length in the

ydirection in Belle simulated data. A fit to a single Gaussian (red), with a width of 25μm, is superimposed.

deviation of√sfrom nominal can be directly inferred from a shift of the beam-energy-constrained mass relative to the nominalBmass. The uncertainty is dominated by the uncertainty on the nominalB mass.

In Belle the boost vector is sufficiently constant that it has been fixed to its nominal value for the entire period of data taking. In BABAR the boost vector is calibrated on a run-by-run basis using the four-momentum sum in dimuon events. Note that due to effects of initial and final state radiation, the latter is not a very sensitive probe of √_s

.