# Multiplicities of neutral K-mesons

## Motivation

The multiplicity is the relevant observable to study the hadronisation mechanism in SIDIS. For hadrons of a specific type h, it is defined as the differential cross section for hadron production normalised to the inclusive DIS cross section: Interpreted in pQCD it depends on parton distribution functions (PDFs) and the fragmentation functions (FFs), and at leading order (LO) in pQCD it reads:

with $q(x,Q^2)$ the quark PDF for the flavour q and $D_{q}^h(z,Q^2)$ the FF for a parton q hadronizing into hadron h. While PDF's are relatively well determined, there are still a lot of uncertainties about FFs. The existing global fits do not agree and are based on assumptions e.g. $D_{\overline{d}}^{\pi^+}=D_s^{\pi^+}$. The charged and neutral kaon multiplicities combined, offer model independent access to FFs without any assumptions except SU(2) and charge invariances~\cite{Albino:2011gq}.

With SU(2) FFs of $K^\pm$ and $K_s^0$ are related:

Thus, no new FFs are introduced. Further it is shown by~\cite{Albino:2011gq} that measuring the difference of charged and neutral kaons

in any of the three processes

always measures the same non-singlet (NS) combination $(D_u-D_d)^{K^+&space;+&space;K^-}$, thus allowing to test the assumption of universality or even the SU(2) invariance of kaon FFs itself. In addition, since $(D_u-D_d)^{K^+&space;+&space;K^-}$ is a NS, no new FFs will enter through its $Q^2$ evolution, hence, providing a test for the $Q^2$ evolution itself.

To determine the hadron multiplicities (averaged number of hadrons per DIS event), one needs to determine the yields of the produced hadrons $N^{h}$ as a function of the kinematical variables x, y and z and to normalise them by the number of DIS events determined as a functions of x and y. To guarantee the quality of the DIS event sample and the hadron sample, several cuts are applied on each sample separately.

The analysis was performed on data collected in the year 2006 using 160 GeV muon beam on a deuteron (6LiD) target. The data have been taken with longitudinally polarised target but in the analysis the data with both polarisations are combined.

## Data sample

### Experimental data

• 06W40 (Slot 2)
• 06W41 (Slot 2)
• 06W42 (Slot 2)
• 06W43 (Slot 2)
• 06W45 (Slot 2)
• 06W46 (Slot 3)

### MC 2006

Monte Carlo sample was produced by Yann Bedfer at IN2P3.FR
• /sps/compass/02/yann/m2006.FL.r507[.xxx]

## Raw multiplicity extraction

The raw multiplicities are calculated as in Ref. \cite{M_hpm:2013April} in bins of the Bjorken scaling variable, x, the lepton energy fraction carried by the virtual photon, y, and the virtual photon energy fraction carried by final-state hadron z:

where $N^{DIS}$ are the number of DIS events after radiative corrections (Section \ref{sec:radcorr}) and $N^{h}$ are the number of hadrons after radiative corrections. In practice, the multiplicity is measured in bins of x, y and z so the expression of the calculated multiplicity is more accurately written as:

where $\Delta&space;z$ is the width of the z -bin.

### Selection criteria

The event kinematics satisfy the conditions Q2 < 1 (GeV/c)2, which select the deep-inelastic scattering (DIS) regime. The range of the fractional energy transfer to the target 0.1<y<0.9 excludes the region where the momentum resolution starts to degrade (lower limit) and the kinematic region where large radiative corrections to inclusive cross section are required (upper limit). Finally the Bjorken scaling variable x is contained to 0.004<x<0.7, the range covered by the spectrometer. Hadrons are required to have a fractional energy z in the range from 0.2 to 0.85 to suppress the target fragmentation region at low z and exclude the region the most affected by diffractive exclusive processes at high z.

#### DIS

• best primary vertex with μ and μ'
• 140 < Ebeam < 180 GeV, Ebeam ≠ 160 GeV
• Q2 > 1(GeV/c)2
• 0.004 < x < 0.7
• 0.1 < y < 0.9
• Cross cells
• in target + strict Z PV cuts
• BMS, i.e. PHAST function PaParticle::Chi2CutFlag() for μ (for RD only)
• Inclusive triggers ONLY, i.e. iMT and OT

### Extraction of Numbers of Neutral Kaons

To separate the kaons from the background we fit the sum of a signal and a background function to the invariant mass distribution.

Then, the number of $K_s^{0}$ can be calculated in three different ways:

• Taking the integral of the signal function and its error as the number and error of kaons.
• Using the Pretz-method''\cite{pretz_paper}

where S[i] and B[i] correspond to numbers of signal and background events from the fit, and H[i] is a number of events in i th bin of a histogram.

• Using sideband subtraction'' method (ss-method). The value of σ is determined with 1 Gaussian fit at the invariant mass distribution of all kaon candidates. The ss-method yields only kaons in a range of 2 σ and therefore must be corrected.

In total we have 106552 K0 events with 1 Gaussian, 110351 events with 2 Gaussians, and 103124 events in 2 σ (i.e. 95 %) with ss-method.

## Acceptance correction

• Double ratio method:
Nr -- reconstructed MC using same method as R
Ng -- generated MC

σ2(Ah) does not take correlation between Nh and NDIS into account

assuming Bayesian approach:

## cτ measurement

To determine a cτ value of K0s from the COMPASS data, one needs to know number of produced hadrons, $N_{K^{0}_{s}}$, as a function of L/γ. Where L is a distance between a primary and a V0 vertices, and γ is a Lorentz factor. After that L/γ distribution is fitted by an exponential function:

where A and B are parameters of the fit. Parameter B corresponds to a cτ value.

## 3D analysis

### Binning

The following multidimensional binning is used for the multiplicity extraction:

• x [0.004, 0.01, 0.020, 0.030, 0.040, 0.060, 0.1, 0.15, 0.7]
• y [0.1, 0.15, 0.2, 0.3, 0.5, 0.7, 0.9]
• z [0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.65, 0.7, 0.75, 0.85]

• ν cut ?

## Presentations

K0s selection studies:

cτ measurement:

## 3D analysis (outdated)

-- ElenaZemlyanichkina - 2013-11-04

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Topic revision: r11 - 2015-10-14 - ElenaZemlyanichkina

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