Multiplicity Report

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This is a working page which will continuously change. The content is not reviewed and is intended for people working on this specific topic.

This report describes the extraction of charge- and type separated multiplicities for $\pi ^{\pm }$, $\pi ^{0}$ and $K^{\pm }$ on a hydrogen or deuterium target. The results are presented versus the kinematic quantities $Q^{2}$, Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle x_{\text{B}}} , $z$ and Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle p_{h\perp }} . They represent a unique data set of identified hadrons that will significantly enhance our understanding of the fragmentation of quarks into final state hadrons in deep inelastic scattering (DIS).

Introduction

Isolated stable quarks have never been observed in nature. When a quark is ejected from an ensemble of quarks and gluons, for example by absorption of a high energy photon, as it separates from the ensemble additional quark-antiquark pairs are generated from the vacuum. The newly generated quarks and antiquarks combine with the target quarks. This recombination process continues as the separation between the partons increases until a configuration of stable singlet multi-quark states is reached. This hadronization process is an essential feature of the interaction of quarks in QCD. Its understanding is basic to a complete picture of the dynamics of quark-quark and quark-gluon interactions. Experimentally, this hadronization process is described by fragmentation functions (FFs) ${\mathcal {D}}_{q}^{h}$, the number densities for the conversion of a struck quark of a given flavor $q$ into a specific hadron type $h$.

The flavor dependence of FFs provides a powerful tool for probing the flavor structure of the nucleon in semi-inclusive deep-inelastic scattering (SIDIS). In the framework of leading-twist QCD, SIDIS is viewed as the hard scattering of a lepton on a quark, that subsequently hadronizes into e.g. a final-state pion, kaon or proton. According to the factorization theorem CITATIONS [1,2] from DC19v0.8, SIDIS can be described in leading twist QCD by three components: parton distribution functions (PDFs), hard scattering cross sections and FFs. The hard scattering cross section is calculable from perturbative QCD. The PDFs parameterize the quark flavor structure of the initial hadron state. Both the PDFs and the FFs are non-perturbative quantities, but they are believed to be universal, i.e. to not depend on the particular type of process from which they were determined CITATIONS [3, 4] from DC19v0.8. While the evidence is by no means complete, data demonstrate that fragmentation of a struck quark of a specific flavor is favored for a final state hadron that contains that quark as a valence quark. This strong flavor correlation is reflected in the magnitude of FFs for "favored" and "unfavored" fragmentation.

While the knowledge of PDFs is highly developed, the data available to date for FFs have been much more limited in scope, particularly for unfavored fragmentation. Most extractions of FFs rely on information from electron-positron annihilation into charged hadrons that is available in high precision. However, this data does not distinguish between quark and anti-quark contributions because it always involve the charge sum of specific hadron species (e.g. $\pi ^{+}$ + $\pi ^{-}$). In addition, most data is taken at the mass scale of the Z boson, at which weak and electromagnetic couplings become approximately equal and thus only flavor singlet combinations of FFs can be determined. Also, because of the similar energy scale of available data determination of the evolution properties of FFs is difficult. However, accurate measurement of normalized yields of specific final states, i.e. particle multiplicities, provide another means of extracting FFs at much lower energy scales than those of the collider measurements. The HERMES experiment with its highly developed particle identification and pure gas targets is ideally suited for such measurements.

This report presents the extraction of the hadron multiplicity distribution ${\mathcal {M}}_{n}^{h}$ of hadrons of the type $h$ (Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle h=\pi ^{+},\pi ^{-},\pi ^{0},K^{+},K^{-}} ) produced of a target $n$ ($n=p,d)$. ${\mathcal {M}}_{n}^{h}$ is defined as there respective hadron yield $N^{h}$ normalized to the DIS yield. Multiplying and dividing by the luminosity shows that this is equivalent to the ratio of the differential SIDIS cross section Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle \sigma ^{h}} over the differential inclusive cross section Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle \sigma _{\text{DIS}}} .

Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle {\begin{aligned}{\mathcal {M}}_{n}^{h}(Q^{2},x_{\text{B}},z,p_{h\perp })&\equiv {\frac {{\text{d}}x_{\text{B}}{\text{d}}Q^{2}}{{\text{d}}^{2}N_{n}^{\text{DIS}}(Q^{2},x_{\text{B}})}}{\frac {{\text{d}}^{4}N_{n}^{h}(Q^{2},x_{\text{B}},z,p_{h\perp })}{{\text{d}}x_{\text{B}}{\text{d}}Q^{2}{\text{d}}z{\text{d}}p_{h\perp }}},\\&={\frac {{\text{d}}x_{\text{B}}{\text{d}}Q^{2}}{{\text{d}}^{2}\sigma _{n}^{\text{DIS}}(Q^{2},x_{\text{B}})}}{\frac {{\text{d}}^{4}\sigma _{n}^{h}(Q^{2},x_{\text{B}},z,p_{h\perp })}{{\text{d}}x_{\text{B}}{\text{d}}Q^{2}{\text{d}}z{\text{d}}p_{h\perp }}}.\\\end{aligned}}}

The extraction of these charge-separated multiplicities of pions and kaons provides sensitivity to the individual quark and antiquark flavors in the fragmentation process. The results of this analysis are the most precise results for multiplicities currently available at this lower energy scale.

A simple check for isospin symmetry

In the naive LO, leading twist factorized framework, the comparison of the $\pi ^{0}$ to the average charged pion multiplicity provides for a very straightforward check of isospin symmetry. In this framework, the pion multiplicities are given by:

Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle {\begin{aligned}{\mathcal {M}}_{p}^{\pi ^{\pm }}(z)&\propto {\frac {1}{\sigma _{p}^{DIS}}}(4u{\mathcal {D}}_{u}^{\pi ^{\pm }}+d{\mathcal {D}}_{d}^{\pi ^{\pm }}+4{\bar {u}}{\mathcal {D}}_{\bar {u}}^{\pi ^{\pm }}+{\bar {d}}{\mathcal {D}}_{\bar {d}}^{\pi ^{\pm }}),\\{\mathcal {M}}_{p}^{\pi ^{0}}(z)&\propto {\frac {1}{\sigma _{p}^{DIS}}}(4u{\mathcal {D}}_{u}^{\pi ^{0}}+d{\mathcal {D}}_{d}^{\pi ^{0}}+4{\bar {u}}{\mathcal {D}}_{\bar {u}}^{\pi ^{0}}+{\bar {d}}{\mathcal {D}}_{\bar {d}}^{\pi ^{0}}).\end{aligned}}}

Applying $C$-parity invariance, we can also write this as,

Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle {\begin{aligned}{\mathcal {M}}_{p}^{\pi ^{+}}(z)&\propto {\frac {1}{\sigma _{p}^{DIS}}}(4u{\mathcal {D}}_{u}^{\pi ^{+}}+d{\mathcal {D}}_{d}^{\pi ^{+}}+4{\bar {u}}{\mathcal {D}}_{\bar {u}}^{\pi ^{+}}+{\bar {d}}{\mathcal {D}}_{\bar {d}}^{\pi ^{+}}),\\{\mathcal {M}}_{p}^{\pi ^{-}}(z)&\propto {\frac {1}{\sigma _{p}^{DIS}}}(4u{\mathcal {D}}_{d}^{\pi ^{+}}+d{\mathcal {D}}_{u}^{\pi ^{+}}+4{\bar {u}}{\mathcal {D}}_{\bar {d}}^{\pi ^{+}}+{\bar {d}}{\mathcal {D}}_{\bar {u}}^{\pi ^{+}}),\\{\mathcal {M}}_{p}^{\pi ^{0}}(z)&\propto {\frac {1}{\sigma _{p}^{DIS}}}((4u+4{\bar {u}}){\mathcal {D}}_{u}^{\pi ^{0}}+(d+{\bar {d}}){\mathcal {D}}_{d}^{\pi ^{0}}).\end{aligned}}}

If we can make the (flawed) assumption of perfect isospin symmetry, we finally obtain,

Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle {\begin{aligned}{\frac {{\mathcal {M}}_{p}^{\pi ^{+}}(z)+{\mathcal {M}}_{p}^{\pi ^{-}}(z)}{2}}&\propto {\frac {1}{\sigma _{p}^{DIS}}}((4u+d+4{\bar {u}}+{\bar {d}}){\frac {{\mathcal {D}}_{u}^{\pi ^{+}}+{\mathcal {D}}_{d}^{\pi ^{+}}}{2}}),\\{\mathcal {M}}_{p}^{\pi ^{0}}(z)&\propto {\frac {1}{\sigma _{p}^{DIS}}}((4u+d+4{\bar {u}}+{\bar {d}}){\mathcal {D}}_{u}^{\pi ^{0}}).\end{aligned}}}

Data Selection and Cuts

Data Selection

Cuts & DIS Selection

The DIS event and SIDIS hadron track selection are detailed in the following two tables. The selection criteria for the data and tracked MC - used for the migration matrix and background estimation - are by design almost exactly the same. The only differences are the true PID used for the MC, and the lack of the (unsimulated) trigger selection. Because both the RICH unfolding and the correction for trigger efficiencies are performed prior to the unfolding, the corrected data is assumed to have perfect PID and 100% efficiency, and it is therefor correct to treat the MC accordingly - without simulating efficiency effects and without using the simulated PID.

DIS Selection criteria

Data	Tracked MC	Born MC
Inclusive (scattered lepton)
trigger 21 fired $Q^{2}>1$ $W^{2}>10$ Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle 0.1<y<0.85} $P_{\text{tot}}<28$ Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle 0<{\text{PID}}_{2+5}-\log _{10}\Phi <100} standard fiducial cuts	$Q^{2}>1$ $W^{2}>10$ Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle 0.1<y<0.85}   g1MTrack.iLType = ±11 standard fiducial cuts	$Q^{2}>1$ $W^{2}>10$ Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle 0.1<y<0.85}   g1MTrack.icType = 0
Semi-inclusive (hadron track)
$2<p<15$ $-100<PID_{2+5}-\log _{10}\Phi <0$ standard fiducial cuts	$2<p<15$ g1MTrack.iLType = ±211, ±321 standard fiducial cuts	g1MTrack.iLType = ±211, ±321
Semi-inclusive ($\pi ^{0}$)
$E_{\gamma }>0.8$   two-cluster invariant mass spectrum standard photon fiducial cuts	$E_{\gamma }>0.8$ photons have same parent photon g1MTrack.iLParent = 111 standard photon fiducial cuts	g1MTrack.iLType = 111

Fiducial Cuts

Lepton	Hadron	Photon
Front field Clamp fabs(smTrack.rXoff+(172.*tan(theta_x)) < 31
Vertical lower limit (septum plate) fabs(smTrack.ryOff+(181.*tan(theta_y)0 > 7
Rear field clamp fabs(smTrack.ryOff + 383.*(tan(theta_y))) < 54
Rear clamps (long tracks) fabs(smTrack.rXpos+(108.*smTrack.rXslope)) < 100 fabs(smTrack.rYpos+(108.*smTrack.rYslope)) < 54
Calorimeter fabs(smTrack.rXpos+(463.*smTrack.rXslope)) < 175 30 < fabs(smTrack.rYpos+(463.*smTrack.rYslope)) < 108		Calorimeter fabs(smCluster.rXlEw) < 125 33 < fabs(smCluster.rYlEw) < 105
Vertex fabs(g1Track.rVertZ) < 18 g1Track.rVertD < 0.75

Binnings

3D binning used for the unfolding of the multiplicities vs $z$

	0	1	2	3	4	5	6	7	8	9	10
Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle x_{\text{B}}}	0	0.085	1
$z$	0.1	0.15	0.2	0.25	0.3	0.4	0.5	0.6	0.7	0.8	1.1
Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle p_{h\perp }}	0	0.1	0.3	0.45	0.6	3

1D binning used for the unfolding of the $\pi ^{0}$ multiplicities vs $z$

	0	1	2	3	4	5	6	7	8	9	10
$z$	0.1	0.15	0.2	0.25	0.3	0.4	0.5	0.6	0.7	0.8	1.1

3D binning used for the unfolding of the multiplicities vs Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle p_{h\perp }}

	0	1	2	3	4	5	6	7	8	9
Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle x_{\text{B}}}	0	0.085	1
$z$	0	0.2	0.3	0.4	0.6	0.8	1.1
Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle p_{h\perp }}	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	3

3D binning used for the unfolding of the multiplicities vs Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle x_{\text{B}}} and $z$

	0	1	2	3	4	5	6	7	8	9
Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle x_{\text{B}}}	0.023	0.04	0.055	0.075	0.1	0.14	0.2	0.3	0.4	0.6
$z$	0	0.2	0.3	0.4	0.6	0.8	1.1
Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle p_{h\perp }}	0	0.3	0.5	0.7	3

3D binning used for the unfolding of the multiplicities vs $Q^{2}$ and $z$

	0	1	2	3	4	5	6	7	8	9
$Q^{2}$	1	1.25	1.5	1.75	2	2.25	2.5	3	5	15
$z$	0	0.2	0.3	0.4	0.6	0.8	1.1
Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle p_{h\perp }}	0	0.3	0.5	0.7	3

Data corrections and unfolding

Charge-Symmetric Background Correction

In a very small fraction of the events it is to be expected that an electron (positron) from an $e^{+}e^{-}$ pair is mistakenly identified as the DIS electron (positron). Due to symmetry considerations of the charge-symmetric background, this can easily be corrected for by also accepting positrons (electrons) as DIS leptons, but with an additional event weight of -1. This way, the contributions to the charge symmetric background from the electrons and positrons cancel each other.

Comment: NEEDNUMBERS, NEEDPLOT

Unfolding for radiative smearing and acceptance correction

In order to correct for bin-to-bin migration due to radiative and detector effects, we use an unfolding formalism originally based on NEEDCITATION. During this step, we also correct for the limited detector acceptance.

Formalism

In general, due to limited detector resolution and acceptance, and due to radiative effect, one cannot directly measure the true ("Born level") value $\mu _{i}$ of Physics distribution $f(...)$ (eg. a cross section) in bin $i$. The measurable quantity Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle \nu _{i}} is related to $\mu _{i}$ by,

Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle {\begin{aligned}\nu _{i}&=\mu _{\text{tot}}\sum _{j=1}^{M}{\frac {\int _{{\text{bin}}\,i}{\text{d}}X\int _{{\text{bin}}\,j}{\text{d}}Ys(X_Y)\epsilon (Y)f(Y)}{\int _{{\text{bin}}\,j}{\text{d}}Yf(Y)}}\mu _{j}+\beta _{i},\end{aligned}}}

where $X$ and Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle Y} are the experimental resp. true kinematic variables, $M$ is the total number of bins, $\beta _{i}$ the background contribution and the overal normalization Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle \mu _{\text{tot}}} is given by $\mu _{\text{tot}}\equiv \sum _{j}\mu _{j}$. The resolution function Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle s(X_Y)} is the probability that an event at the kinematic variables $X$ is observed providing the true variables were Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle Y} , and the acceptance function $\epsilon (Y)$ is the probability that an event with true variables Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle Y} is observed.

Defining the smearing matrix $S$, this equation can be rewritten as,

$\nu _{i}=\sum _{j=1}^{M}S_{ij}\mu _{j}+\beta _{i},$

or in matrix notation,

${\vec {\nu }}=S.{\vec {\mu }}+{\vec {\beta }}$

Assuming that the binning is chosen in such a way, that that Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle f(Y)} is approximately constant inside of each bin, the occurence of $f$ in the numerator and denominator of the smearing matrix $S$ cancel, and the smearing matrix will be independent of Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle f(Y)} . This property is essential, because it enables us to obtain a model-independent estimate of the smearing matrix from a MC sample.

Application

Starting from the definition of the differential multiplicity,

${\begin{aligned}{\mathcal {M}}_{n}^{h}(Q^{2},x_{\text{B}},z,p_{h\perp })&\equiv {\frac {{\text{d}}x_{\text{B}}{\text{d}}Q^{2}}{{\text{d}}^{2}N_{n}^{\text{DIS}}(Q^{2},x_{\text{B}})}}{\frac {{\text{d}}^{4}N_{n}^{h}(Q^{2},x_{\text{B}},z,p_{h\perp })}{{\text{d}}x_{\text{B}}{\text{d}}Q^{2}{\text{d}}z{\text{d}}p_{h\perp }}},\end{aligned}}$

we would naively attempt to unfold the DIS and SIDIS cross sections seperately and then combine them to obtain the unfolded multiplicity. In this case, the resulting multiplicity would be influenced by the additional systematics due to the luminosity measurement, and the inevitable influence not simulated detector inefficiencies further complicates matters,

Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle {\frac {N_{i}}{\mathcal {L}}}=k_{i}\left(\sum _{j=1}^{M}S_{ij}\sigma _{j}^{\text{B}}+\beta _{i}\right),}

with $\sigma _{j}^{B}$ the true ("Born level") cross section we want to obtain, ${\mathcal {L}}$ the total integrated experimental luminosity and Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle k_{i}} representing the not-simulated detector inefficiencies.

The natural way to circumvent these complications, is by using the fact that the multiplicity is a ratio of (differential) cross sections. The luminosity of the SIDIS numerator will exactly cancel the luminosity of the DIS denominator, and the unsimulated detector inefficiencies also cancel in a very good approximation. Because of this, we can prevent to introduce these unnecessary uncertainties in the first place by unfolding the cross section ratio $R_{i}^{h}$,

Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle R_{i}^{h}\equiv {\frac {N_{i}^{h}}{N_{i}^{\text{DIS}}}},}

where the index i runs over all kinematic bins in Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle (Q^{2},x_{B},z,p_{h\perp })} for the SIDIS numerator, and for the corresponding bins in $(Q^{2},x_{B})$ for the inclusive denominator. Please note that the experimental multiplicity Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle {\mathcal {M}}_{i}^{h}} is equal to $R_{i}^{h}$ up to a bin width correction.

Restating the unfolding problem then yields,

$R_{i}^{h}={\frac {\sum _{j=1}^{M}S_{ij}\sigma _{j}^{{\text{B}},h}+\beta _{i}}{\sum _{j=1}^{M}S_{ij}^{\text{DIS}}\sigma _{j}^{\text{B,DIS}}+\beta _{i}^{\text{DIS}}}},$

where the indices in the DIS denominator are restricted to the corresponding inclusive variables. The MC describes the experimental DIS cross section very well, and because of the cancellation of the not simulated detector efficiency factor, the denominator is exactly the same as the simulated ("tracked") DIS cross section Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle \sigma _{i}^{\text{T,DIS}}} ,

Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle R_{i}^{h}={\frac {\sum _{j=1}^{M}S_{ij}\sigma _{j}^{{\text{B}},h}+\beta _{i}}{\sigma _{i}^{\text{T,DIS}}}}.}

Left multiplication with the inverted smearing matrix and a slight reshuffling of the equation yields for the true SIDIS cross section,

Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle \sigma _{j}^{{\text{B}},h}=\sum _{i=1}^{M}S_{ji}^{-1}(R_{i}^{h}\sigma _{i}^{\text{T,DIS}}-\beta _{i}).}

Dividing LHS and RHS by $\sigma _{j}^{\text{B,DIS}}$ finally yields an expression that is symmetric in number of MC estimations used in numerator and denominator,

$R_{j}^{B,h}={\frac {1}{\sigma _{j}^{\text{B,DIS}}}}\sum _{i=1}^{M}S_{ji}^{-1}(R_{i}^{h}\sigma _{i}^{\text{T,DIS}}-\beta _{i}).$

A couple of remarks:

• $\sigma _{j}^{\text{B,DIS}}$, Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle S_{ij}} and Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle \beta_i} are calculated from MC, this is explained in more detail in the next subsection.
• The background contribution Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle \beta_i} is directly given by the MC, and this introduces a (small) dependence on the background model in the MC in the unfolded result.
• The equations in this section are all relevant for the infinite statistics limit. In practice, we cannot measure Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle R^h_i} , but rather the estimator Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle \hat{R}^h_i} of this quantity. In a general case with limited statistics and a high degree of bin-to-bin migration, this may cause very big fluctuations in the unfolded result, making it necessary to use some form of regularization (see eg NEEDLINK). This is not an issue due to the very high statistics, and the relatively low degree of kinematic smearing in this analysis. Therefor it is in fact correct to treat the estimator Fehler beim Parsen (Konvertierungsfehler. Der Server („https://wikimedia.org/api/rest_“) hat berichtet: „Cannot get mml. Server problem.“): {\displaystyle {\hat {R}}_{i}^{h}} as Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle R^h_i} , and use the simple smearing matrix inversion described above, without the need to further complicate matters with an unnecessary regularization.