Cross-Section Measurement of ψ(2S) → J/ψ(→ µ

ATLAS-CONF-2013-094 03/09/2013

ATLAS NOTE

ATLAS-CONF-2013-094

September 3, 2013

Cross-Section Measurement of ψ(2S) → J/ψ(→ µ

µ

)π

π

in

√ s = 7 TeV pp Collisions at ATLAS

The ATLAS Collaboration

Abstract

The prompt and non-prompt production cross-sections forψ(2S) mesons are measured using 2.1 fb⁻¹ ofppcollision data at a centre-of-mass energy of 7 TeV recorded by the ATLAS experiment at the LHC. The measurement studies the ψ(2S) → J/ψ(→ µ⁺µ⁻)π⁺π⁻ decay mode, and probes ψ(2S) with transverse momenta in the range p_T = 10−100 GeV and rapidity|y|<2.0. The results are compared to existing LHCψ(2S) production measurements and theoretical models for prompt and non-prompt production.

c

Copyright 2013 CERN for the benefit of the ATLAS Collaboration.

Reproduction of this article or parts of it is allowed as specified in the CC-BY-3.0 license.

1 Introduction

The production of quarkonium states in hadronic collisions has been the subject of intense theoretical and experimental study for many decades, especially since measurements of prompt

and

production at the Tevatron [1, 2, 3, 4] exposed order-of-magnitude differences between theoretical expectations and data. Despite these being among the most studied heavy quark bound states, there is still no complete understanding of the underlying production mechanisms and properties of formation. Quarkonium pro- duction acts as a unique and important testing ground for Quantum Chromodynamics (QCD) in its own right. While the production of a heavy quark pair occurs at a hard scale and is generally well-described by QCD, its subsequent evolution into a bound state includes many non-perturbative e

ects at much softer scales that pose a challenge to current theoretical techniques. With the data obtained from the Large Hadron Collider (LHC), it is possible to make stringent tests of existing theoretical models across a large range of momentum transfer.

Studies on heavy quarkonium have previously been conducted at ATLAS with production measure- ments in the

µ

µ

[5] and

(nS )

µ

µ

[6, 7] modes. The measurements described here are based on an analysis of 2.1 fb

of

collision data at

s=

7 TeV, studying the prompt and non-prompt production of the ψ(2S) meson in the decay mode

µ

µ

)π

π

, where the prompt production arises from direct QCD production mechanisms and the non-prompt production arises from weak decays of

production, which can occur through either direct QCD production of

or the production of excited

χ

states that subsequently decay into

final states, the prompt production of ψ(2S) only proceeds through direct production. The measurement presented here, combined with a concurrent measurement of the prompt and non-prompt production of χ

(nP) states [8] and with existing measurements of the production cross-section of the

[5], serve to provide an accurate picture of the production of both prompt and non-prompt charmonium production.

The ψ(2S) cross-sections measured here are compared with the results from LHCb [9] and CMS [10]

and with a variety of theoretical models for both prompt and non-prompt production.

2 The ATLAS Detector

The ATLAS detector [11] is composed of an inner tracking system, calorimeters, and a muon spectrom- eter. The inner detector (ID) surrounds the proton-proton collision point and consists of a silicon pixel detector, a silicon microstrip detector, and a transition radiation tracker, all of which are immersed in a 2 T axial magnetic field. The inner detector spans the pseudorapidity

range

< 2.5 and is enclosed by a system of electromagnetic and hadronic calorimeters. Surrounding the calorimeters is the large muon spectrometer (MS). This spectrometer is equipped with monitored drift tubes and cathode strip chambers that provide precision measurements in the bending plane within the pseudorapidity range

< 2.7. Re- sistive plate and thin gap chambers with fast response are additionally used primarily to make fast event data-recording decisions in the ranges

< 1.05 and 1.05 <

< 2.4 respectively, and also provide position measurements in the non-bending plane and improve overall pattern recognition and track re- construction. Momentum measurements in the muon spectrometer are based on track segments formed in at least two of the three precision chamber planes.

The ATLAS detector employs a three-level trigger system [12], which reduces the 20 MHz proton bunch collision rate to the several hundred Hz transfer rate to mass storage. The Level-1 muon trigger

1ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the centre of the detector and thez-axis along the beam pipe. Thex-axis points from the IP to the centre of the LHC ring, and they-axis points upward. Cylindrical coordinates (r, φ) are used in the transverse plane,φbeing the azimuthal angle around the beam pipe. The pseudorapidity and the transverse momentum are defined in terms of the polar angleθasη=ln tan(θ/2) and pT =p sinθ, respectively. Theη−φdistance between two particles is defined as∆R=p

∆η²+ ∆φ².

searches for hit coincidences between different muon trigger detector layers inside pre-programmed ge- ometrical windows that bound the path of muon candidates over a given transverse momentum threshold and provide a rough estimate of its position within the pseudorapidity range

< 2.4. At Level-1, muon candidates are reported in “regions of interest” (RoI). Only a single muon can be associated with a given RoI of spatial extent

φ

η

0.1

0.1. This limitation has a small e

ect on the trigger e

ciency for ψ(2S) mesons, corrected for in the analysis using a data-driven method based on analysis of

µ

µ

and Υ

µ

µ

decays. There are two subsequent higher-level, software-based trigger selection stages.

Muon candidates reconstructed at these higher levels incorporate, with increasing precision, information from both the muon spectrometer and the inner detector, reaching position and momentum resolution close to that provided by the offline muon reconstruction.

In this analysis combined muons were used, in which the reconstruction relies on a statistical com- bination of both a MS track and an ID track, whereby, due to ID coverage, the combined reconstruction covers

< 2.5. The muons selected for this analysis are restricted to

< 2.3, thereby ensuring high quality tracking and a reduction of fake muon candidates, and to remove regions of strongly varying e

ciency and acceptance corrections.

3 Data and Event Selection

Data for this analysis were taken during 2011 LHC proton-proton collisions at a centre-of-mass energy of 7 TeV. The data sample was collected using a trigger requiring two oppositely-charged muon candidates each with transverse momentum satisfying

> 4 GeV, fulfilling additional quality requirements and with both muons being consistent with having originated from a common vertex. This data selection resulted in a total integrated luminosity of 2.09±0.04 fb

[13].

The ψ(2S) meson candidates are reconstructed using a similar technique as for

[14] can- didates in ATLAS. Events are chosen containing at least two muons identified by the muon spectrometer with tracks reconstructed in the inner detector. The two muon tracks are considered

µ

µ

candi- dates if they can be fitted to a common vertex with a di-muon invariant mass between 2.8 and 3.4 GeV.

The muon track parameters are taken from the ID measurement alone, since the MS does not noticeably improve the precision in the momentum range relevant for the ψ(2S) measurements presented here. To ensure accurate inner detector measurements, each muon track must contain at least six silicon microstrip detector hits and at least one pixel detector hit. Muon candidates passing these criteria are required to have opposite charges,

> 4 GeV and

< 2.3 and a successful fit to a common vertex. Good spatial matching,

(∆ η)

(∆ φ)

) < 0.01, between each reconstructed muon candidate and the trigger identified candidates is required to accurately correct for trigger ine

ciencies. The di-muon pair is fur- ther required to satisfy

> 8 GeV and

< 2.0 to ensure that the

candidates are reconstructed in a fiducial region where acceptance and efficiency corrections do not vary too rapidly. An additional requirement on the di-muon vertex χ

< 200 helps to remove spurious di-muon combinations.

The other two particles in the ψ(2S)

π

decay are reconstructed by taking all pairs of the remaining oppositely-charged tracks with

> 0.5 GeV and

< 2.5 and assigning the pion mass hypothesis to each reconstructed track. A constrained four-particle vertex fit is performed to all ψ(2S) candidates, where the

µ

µ

candidates have their invariant mass constrained to the world average value for the

(3096.916 MeV) [15]. A χ

probability cut of

) > 0.005 is applied to the vertex fit quality, which considerably reduces combinatorial background from incorrect di-pion candidate assign- ment. The constrained vertex fit also allows for significantly improved invariant mass resolution for the

system over what would be expected from momentum resolution alone. The distribution of the reconstructed di-pion mass (m

) was found to be in agreement with previous measurements[16, 9].

Small ine

ciencies for signal events (within

1% level) coming from the vertex requirements on di-

muon candidates are corrected for, as are the slightly larger inefficiencies (∼ 3

7%) from the cut on

constrained fit quality of the four-particle fit.

Figure 1 shows the

invariant mass distribution after the above selection cuts. A clear peak of the ψ(2S) is observed near 3.69 GeV. At larger invariant mass a further structure is also observed, identified as the

[GeV]

π π ψ

mJ/

3.6 3.65 3.7 3.75 3.8 3.85 3.9

Entries / 4 MeV

0 20 40 60 80 100

103

×

Data Fit

Background (2S) Signal ψ

ATLAS Preliminary Ldt=2.1fb-1

∫

=7TeV, s

Figure 1: The uncorrected

mass spectrum between 3.586 and 3.946 GeV. Superimposed on the data points is the result of a fit using a double Gaussian to describe the

signal peak, and a 2nd order Chebyshev polynomial to model the background, where the region within

(m

3.872 GeV) is excluded from the fit.

The cross-section measurements presented in this note are presented in three ψ(2S) rapidity intervals:

|y|

< 0.75, 0.75 <

< 1.5, and 1.5 <

< 2.0. Figure 2 illustrates the uncorrected yields and the resolutions of the di-muons (left) and the

system (right) in the three rapidity regions, which comprise about 96,000, 66,000 and 41,000 ψ(2S) candidates respectively.

For both the di-muon and the

invariant mass fits, a double Gaussian is used to describe the signal shape, and a second-order Chebyshev polynomial to model the background. The two-dimensional distribution in

and

of all

candidates contributing to Figure 2, is shown in Figure 3.

4 Cross-Section Determination

The differential production cross-section for ψ(2S) can be categorised into prompt production and non-

prompt production. The prompt ψ(2S) has no significant feed-down from higher mass charmonium

states, since they decay predominantly to

¯

charm-quark pair which subsequently binds to form the ψ(2S). Non-prompt ψ(2S) production processes

are distinguished from prompt processes by their longer apparent lifetimes, with production occurring

through the decay of a

possible to use its lifetime to isolate prompt from non-prompt production. Instead, to distinguish between

these prompt and non-prompt processes, a parameter called pseudo-proper lifetime τ is constructed using

[GeV]

µ

mµ

2.8 2.9 3 3.1 3.2 3.3 3.4

Entries / 12 MeV

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

106

×

|y| < 0.75 Data Fit Background

Signal ψ J/

Signal ~ 2.8 M ~ 37 MeV σ Peak

ATLASPreliminary Ldt=2.1fb-1

∫

=7TeV, s

(a)|y|<0.75

[GeV]

π π ψ

mJ/

3.6 3.62 3.64 3.66 3.68 3.7 3.72 3.74 3.76 3.78

Entries / 4 MeV

0 10 20 30 40 50

103

×

|y| < 0.75 Data Fit Background

(2S) Signal ψ

Signal ~ 96 k ~ 5.6 MeV σ Peak

ATLAS Preliminary Ldt=2.1fb-1

∫

=7TeV, s

(b)|y|<0.75

[GeV]

µ

mµ

2.8 2.9 3 3.1 3.2 3.3 3.4

Entries / 12 MeV

0 0.05 0.1 0.15 0.2 0.25

106

×

0.75 < |y| < 1.5 Data Fit Background

Signal ψ J/

Signal ~ 2.3 M ~ 53 MeV σ Peak

ATLASPreliminary Ldt=2.1fb-1

∫

=7TeV, s

(c) 0.75<|y|<1.5

[GeV]

π π ψ

mJ/

3.6 3.62 3.64 3.66 3.68 3.7 3.72 3.74 3.76 3.78

Entries / 4 MeV

0 5 10 15 20 25 30

103

×

0.75 < |y| < 1.5 Data Fit Background

(2S) Signal ψ

Signal ~ 66 k ~ 7.9 MeV σ Peak

ATLAS Preliminary Ldt=2.1fb-1

∫

=7TeV, s

(d) 0.75<|y|<1.5

[GeV]

µ

mµ

2.8 2.9 3 3.1 3.2 3.3 3.4

Entries / 12 MeV

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

106

×

1.5 < |y| < 2 Data Fit Background

Signal ψ J/

Signal ~ 2 M ~ 72 MeV σ Peak

ATLASPreliminary Ldt=2.1fb-1

∫

=7TeV, s

(e) 1.5<|y|<2.0

[GeV]

π π J/ψ

m

3.6 3.62 3.64 3.66 3.68 3.7 3.72 3.74 3.76 3.78

Entries / 4 MeV

0 2 4 6 8 10 12 14 16 18

103

×

1.5 < |y| < 2 Data Fit Background

(2S) Signal ψ

Signal ~ 41 k ~ 10 MeV σ Peak

ATLAS Preliminary Ldt=2.1fb-1

∫

=7TeV, s

(f) 1.5<|y|<2.0

Figure 2: Uncorrected yields and the invariant mass resolutions for the di-muon (left) and

π

system (right) in the three rapidity ranges of the measurement. The data distributions are fitted with a

combination of two Gaussians (for the signal peak) and a second order polynomials (for backgrounds).

π |y|

π ψ J/

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

[GeV] T pππψJ/

10 20 30 40 50 60 70 80 90 102

0 10 20 30 40 50 103

ATLAS Preliminary ^s=7TeV,

∫

Figure 3: Distribution of ψ(2S)

π

candidates in the mass interval 3.586 <

<

3.786 GeV as a function of ψ(2S) transverse momentum and rapidity, before correction for e

ciencies, acceptance, and detector resolution effects.

the

transverse momentum:

τ

p_T

, (1)

with

defined by the equation:

L_xy≡

~

~

/p

, (2)

where ~

is the vector from the primary vertex to the

decay vertex and ~

is the transverse mo- mentum vector of the

charged-particle track

scalar sum, which we identify as the location of the primary proton-proton in- teraction. The presence of additional simultaneous proton-proton collisions, and the e

ect of associating the final state particles to the wrong collision was found to have a negligible impact on the decay length determination or the signal extraction.

To obtain a measurement of the production cross-sections, the reconstructed candidates are first in- dividually weighted to correct for detector e

ects, such as acceptance, muon reconstruction e

ciency, pion reconstruction efficiency and trigger efficiency. The distribution of candidates in ψ(2S)

and

|y|

intervals are then fitted using a weighted two-dimensional unbinned maximum likelihood method, performed on the invariant mass and pseudo-proper lifetime distributions to isolate signal candidates from the backgrounds and separate prompt signal from non-prompt signal. The corrected prompt and non-prompt signal yields (N

,

) are then used to calculate the di

erential cross-section times branching ratio, using the equation:

Br

ψ(2S)

µ

)π

π

×

d

σ

(ψ(2S))

d

∆p_T∆

y

Ldt

, (3)

where

Ldt

is the total integrated luminosity,

and

y are the sizes of the intervals in ψ(2S) trans- verse momentum and rapidity, respectively, and

ψ(2S)

µ

)π

π

is the total branching

ratio of the signal decay, taken to be 0.0202

0.004, obtained by combining the world average values for

µ

µ

and

ψ(2S)

π

[15].

In addition to the prompt and non-prompt production cross-sections, the non-prompt ψ(2S) produc- tion fraction

is simultaneously extracted from the maximum likelihood fits to the same kinematic intervals. This fraction is defined as the corrected yield of non-prompt ψ(2S) divided by the corrected total yield of produced ψ(2S), as seen in the equation:

f_B^ψ(2S) ≡ N_NP^ψ(2S) N_P^ψ(2S)+N_NP^ψ(2S)

. (4)

Determination of this fraction has the advantage that many acceptance and efficiency effects cancel or are reduced in the ratio and so systematic uncertainties on this quantity are minimised.

Acceptance

The acceptance

, y,

) is defined as the probability that the decay products in ψ(2S)

µ

µ

)π

π

fall within the fiducial volume (

(µ

) > 4 GeV,

)| < 2.3,

(π

) > 0.5 GeV,

)| <

2.5). It has been shown [16] that the final state is largely dominated by the simplest angular momentum configuration, where the two pions are in a relative

-wave state, and the

and di-pion system are in

-wave as well. Consequently, the spin-alignment state of the ψ(2S) is directly transferred to the

ψ(2S). For the central results obtained in this analysis, the ψ(2S) was assumed to be isotropic, with variations corresponding to a number of extreme spin-alignment scenarios detailed below, in Table 3.

The acceptance maps are created using a high statistics generator-level Monte Carlo simulation, which randomly creates and decays ψ(2S)

µ

µ

)π

π

, as a function of the ψ(2S) trans- verse momentum and rapidity, in finely-binned intervals of the di-pion invariant mass

covering the allowed range, 2m

<

<

. An example of the acceptance map for the lowest di-pion mass (m

2m

) is shown in Figure 4(a) for the isotropic ψ(2S) assumption. The variation of acceptance with di-pion mass is illustrated by the ratio of the acceptance at the lowest di-pion mass

2m

to the acceptance at the highest di-pion mass

shown in Figure 4(b).

The largest variations are observed at low

and at high rapidity, where relative acceptance can vary by up to 50%.

The spin-alignment of

from ψ(2S) decay is, as mentioned above, assumed to be fully transferred from the spin-alignment of ψ(2S) and hence, in general, in its decay frame the angular dependence of the decay

µ

µ

is given by

d

d cos θ

dφ

1 3

λ

!

1

λ

cos

θ

λ

sin

θ

cos 2φ

λ

sin 2θ

cos φ

, (5)

where the λ

are coe

cients related to the spin density matrix elements of the ψ(2S) wavefunction. Here, in the helicity frame, θ

is the angle between the momentum of the µ

and the direction of the

momentum in the lab frame, while φ

is the angle between the production and decay planes of the

In the default case of isotropic ψ(2S), all three λ

coe

cients in Eq. 5 are equal to zero. This assumption is compatible with recent measurements [17].

In certain areas of the phase space, the acceptance

may depend quite strongly on the values of the λ coe

cients in Eq. 5 and we have identified seven extreme cases that lead to the largest possible variations of acceptance within the phase space of this measurement. These are used to define a range in which the results may vary under any physically-allowed spin-alignment assumptions:

•

isotropic distribution independent of θ

and φ

:

λ

λ

λ

0, Isotropic;

(2S) |y|

ψ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 [GeV] T(2S) pψ

9 10 20 30 40 50 60 70 80 90 102

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 = 0.2831 GeV

π

Isotropic mπ ATLASPreliminary

(a) Lowest di-pion mass interval

(2S) |y|

ψ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 [GeV] T(2S) pψ

9 10 20 30 40 50 60 70 80 90 102

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

Isotropic Ratio ATLASPreliminary

(b) Ratio of acceptances for lowest to highest di-pion masses

Figure 4: Example of ψ(2S)

µ

)π

π

acceptance for (a) the lowest di-pion mass,

2m

, and (b) the acceptance ratio at lowest di-pion masses to the highest di-pion masses. The

µ

µ

decay is assumed to be isotropic.

•

longitudinal alignment:

λ

λ

0, Longitudinal;

•

three types of transverse alignment:

λ

1, λ

λ

0, Transverse zero λ

1, λ

1, λ

0, Transverse positive λ

1, λ

0, Transverse negative;

•

two types of o

-(λ

λ

)-plane alignment:

λ

0, λ

0, λ

λ

0, λ

0, λ

-plane negative.

Figure 5 illustrates the variation of the acceptance correction weights with

and rapidity, for the six anisotropic spin-alignment scenarios described above, relative to the isotropic case.

Muon Reconstruction E ffi ciency

The muon reconstruction e

ciency, determined using a data-driven tag-and-probe method [7] from

µ

µ

decays is given by:

(

, η

)

(

, η

)

(

,

η

)

(

,

η

), (6)

where

is the charge of the muon,

is the muon track reconstruction e

ciency in the ID, while

is

the efficiency of the combined muon reconstruction given that the muon track has been reconstructed in

the ID. The muon track reconstruction efficiency

is determined [7] to be 99

1% within the kinematic

range of interest.

[GeV]

(2S) pT

ψ

10 20 30 40 50 60 70 80 90 100

Spin-alignment Ratio

0.6 0.8 1 1.2 1.4 1.6 1.8 2

|y| < 0.75 Isotropic Longitudinal

Transverse 0 Transverse + Transverse -

Off-plane + Off-plane - Spin-alignment envelope

ATLASPreliminary

(a)|y|<0.75

[GeV]

(2S) pT

ψ

10 20 30 40 50 60 70 80 90 100

Spin-alignment Ratio

0.6 0.8 1 1.2 1.4 1.6 1.8 2

0.75 < |y| < 1.5 Isotropic Longitudinal

Transverse 0 Transverse + Transverse -

Off-plane + Off-plane - Spin-alignment envelope

ATLASPreliminary

(b) 0.75<|y|<1.5

[GeV]

(2S) pT

ψ

10 20 30 40 50 60 70 80 90 100

Spin-alignment Ratio

0.6 0.8 1 1.2 1.4 1.6 1.8 2

1.5 < |y| < 2.0 Isotropic Longitudinal

Transverse 0 Transverse + Transverse -

Off-plane + Off-plane - Spin-alignment envelope

ATLASPreliminary

(c) 1.5<|y|<2.0

(2S) |y|

ψ

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Spin-alignment Ratio

0.6 0.8 1 1.2 1.4 1.6 1.8 2

< 100 GeV

10 < pT Isotropic Longitudinal

Transverse 0 Transverse + Transverse -

Off-plane + Off-plane - Spin-alignment envelope

ATLASPreliminary

(d) 10<pT <100 [GeV]

Figure 5: Average acceptance correction relative to the isotropic scenario for the seven extreme spin- alignment scenarios described in the text, as a function of ψ(2S) transverse momentum in the three rapidity regions (a)-(c), and versus ψ(2S ) rapidity for

> 10 GeV (d).

Pion Reconstruction E ffi ciency

The pion reconstruction e

ciency

is given by:

(p

,

η

)

(

,

η

), (7)

where the two

were extracted from the map, shown in Figure 6, which was determined using tech-

niques derived for existing tracking efficiency measurements [18]. Pions produced in Monte Carlo event

simulation (fully simulated PYTHIA [19] sample of ψ(2S) production, using the ATLAS 2011 MC tun-

ing [20]) were used to determine the efficiencies in the interval

> 0.5 GeV and

< 2.5. The pion

track reconstruction efficiencies are calculated in intervals of charge-signed pseudorapidity

η and track

transverse momentum

from Monte Carlo. In addition to the statistical uncertainties on the e

ciency

due to limited Monte Carlo simulation, each efficiency value also contains an additional uncertainty to

account for any possible mis-modelling in the simulations.

) (π q*η

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

) [GeV]π(Tp

0 2 4 6 8 10 12

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

ATLASPreliminary

Figure 6: Pion reconstruction e

ciency map, as a function of pion charge-signed pseudorapidity and transverse momentum, obtained from simulation.

Trigger E ffi ciency

The trigger e

ciency for the di-muon trigger used in this analysis has been measured in Ref. [7] from

µ

µ

and

µ

µ

decays using a data-driven method. The trigger e

ciency is the e

ciency for the trigger system to select signal events that also pass the reconstruction-level analysis selection, and is parameterised as:

(p

,

, η

)

(p

,

, η

)

(∆

(8) where

is the efficiency of the trigger system to find a RoI for a single muon and

is a correction term taking into account muon-muon correlations, dependent on the angular separation

between the two muons, and the absolute rapidity of the di-muon system,

Total Weight

The total weight w for each

candidate was calculated as the inverse of the product of acceptance and efficiency corrections, as described by:

w

(

, y,

)

. (9) The average of the total weight and its breakdown into components is illustrated in Figure 7, shown for the three rapidity regions for each

bin of the measurement, and also shown as an average over the full transverse momentum range (10 <

< 100 GeV) versus rapidity.

Fitting Procedure

A two-dimensional weighted unbinned maximum likelihood fit is performed on the

invariant mass

and pseudo-proper lifetime for each

and

bin. The Probability Density Function (PDF) is defined

as a normalised sum of terms, where each term was factorized into a mass-dependent and lifetime-

dependent function. The PDF can be written in a compact form as:

[GeV]

(2S) pT

10 20 30 40 50 ψ60 70 80 90 100

Weight

1 10

102 |y| < 0.75 Total Weight

Acceptance Muon Reco.

Pion Reco. Trigger ATLAS Preliminary

Ldt=2.1fb-1

∫

=7TeV, s

(a)|y|<0.75

[GeV]

(2S) pT

10 20 30 40 50ψ60 70 80 90 100

Weight

1 10 102

0.75 < |y| < 1.5 Total Weight

Acceptance Muon Reco.

Pion Reco. Trigger

ATLAS Preliminary Ldt=2.1fb-1

∫

=7TeV, s

(b) 0.75<|y|<1.5

[GeV]

(2S) pT

10 20 30 40 50 ψ60 70 80 90 100

Weight

1 10 102

1.5 < |y| < 2.0 Total Weight

Acceptance Muon Reco.

Pion Reco. Trigger

ATLAS Preliminary Ldt=2.1fb-1

∫

=7TeV, s

(c) 1.5<|y|<2.0

(2S) |y|

ψ

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Weight

1 10 102 103

< 100 GeV

10 < pT Total Weight

Acceptance Muon Reco.

Pion Reco. Trigger

ATLAS Preliminary Ldt=2.1fb-1

∫

=7TeV, s

(d) 10<pT <100 [GeV]

Figure 7: Average correction weights for the three rapidity regions versus

(a)-(c) and for the full

region versus

(d).

PDF(m, τ)

i=1

⊕fi

(m)

(τ)

(10) Here the symbol

stands for a normalised sum of various terms, while

denotes a convolution between two functions. The functions

and

for the mass and lifetime components of the sum,

1, . . . , 5 are shown in Table 1, where

,

, and

stand for Gaussians, second order Chebyshev polynomials and exponential functions, respectively, and the index

corresponds to different sets of parameters, while δ stands for the Dirac delta function. The lifetime resolution function,

Table 1: Components of the probability density function.

i Type Source

(m)

(τ)

1 Signal Prompt

(m)

(m) δ(τ)

2 Signal Non-prompt

(m)

(m)

(τ)

3 Background Prompt

(m) δ(τ)

4 Background Non-prompt

(m)

(τ)

(τ) 5 Background Non-prompt

(m)

(

)

with a mean fixed to zero and a width that is free to be determined from each of the fits.

The two signal Gaussians

and

are required to have a common mean, while the respective widths σ

are required to satisfy the relation σ

1.5

σ

, so

is always fitted to the wider part of the signal peak (the factor of 1.5 was determined from test fits). To constrain the fit model better at high

, the widths (σ

) of the mass signal Gaussians (for prompt and non-prompt) are obtained for each bin from separate fits to the invariant mass distributions, using the same

(m) terms. The resultant fitted σ values of the one-dimensional fits are parameterised as a linear function of

for each of the three rapidity regions, and used to fix the widths in each of the two-dimensional fits. Figure 8 illustrates the results of the unbinned maximum likelihood fits to the data for three representative kinematic intervals.

5 Systematic Uncertainties

Various sources of systematic uncertainties on the measurement are considered and are outlined below.

Acceptance Corrections

The acceptance maps were generated using high statistics Monte Carlo simulation, with negligible sta- tistical and systematic uncertainties.

Fit Model Variations

The uncertainty on the fit was determined by changing one component at a time of the fit model described

in Section 4, creating a set of new fit models. For each new fit model the cross-section was recalculated,

and in each

and

interval the maximal variation from the central fit model was used as its systematic

uncertainty. The full set of fit models used are documented in Table 2. The central fit model is the

double Gaussian mass signal and second order Chebyshev polynomial for mass background (Model 0

in Table 2), where G, C and E are the same as defined in Section 4. DG is a double Gaussian, CB is a

crystal ball function [21], with the α and

parameters fixed to 2 (determined from test fits), L is a linear

[GeV]

π π ψ

mJ/

3.6 3.62 3.64 3.66 3.68 3.7 3.72 3.74 3.76 3.78

Entries / 4 MeV

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 106

× Data Fit Background Prompt Signal Non Prompt Signal

ATLASPreliminary Ldt=2.1fb-1

∫

=7TeV, s

[ps]

π π ψ

τJ/

-2 0 2 4 6 8 10

Entries / 0.12 ps

1 10 102 103 104 105

106 Data

Fit Prompt Signal Prompt Background Non Prompt Signal Non Prompt Background

ATLASPreliminary Ldt=2.1fb-1

∫

=7TeV, s

(a) |y|<0.75, 11<pT<12 GeV

[GeV]

π π ψ

mJ/

3.6 3.62 3.64 3.66 3.68 3.7 3.72 3.74 3.76 3.78

Entries / 4 MeV

0 5 10 15 20 25 30 35 103

× Data Fit Background Prompt Signal Non Prompt Signal

ATLASPreliminary Ldt=2.1fb-1

∫

=7TeV, s

[ps]

π π J/ψ

-2 0 2 4 6 8τ 10

Entries / 0.12 ps

1 10 102 103 104 105

Data Fit Prompt Signal Prompt Background Non Prompt Signal Non Prompt Background

ATLASPreliminary Ldt=2.1fb-1

∫

=7TeV, s

(b) 0.75<|y|<1.5, 16<pT<18 GeV

[GeV]

π π ψ

mJ/

3.6 3.62 3.64 3.66 3.68 3.7 3.72 3.74 3.76 3.78

Entries / 4 MeV

0 50 100 150 200 250 300 350

Data Fit Background Prompt Signal Non Prompt Signal

ATLASPreliminary Ldt=2.1fb-1

∫

=7TeV, s

[ps]

π π ψ

τJ/

-2 0 2 4 6 8 10

Entries / 0.12 ps

1 10 102 103

Data Fit Prompt Signal Prompt Background Non Prompt Signal Non Prompt Background

ATLASPreliminary Ldt=2.1fb-1

∫

=7TeV, s

(c) 1.5<|y|<2.0, 40<pT<60 GeV

Figure 8: Unbinned maximum likelihood fit and data projections onto the invariant mass and pseudo-

proper lifetimes of the ψ(2S) candidates for three representative kinematic intervals studied in this

measurement. Total signal plus background fits to the data are shown, along with the breakdown by

prompt/non-prompt production for the ψ(2S) signal.

Table 2: Fit models used to test sensitivity of extracted ψ(2S) yields to the signal and background mod- elling. The symbols are described in the text.

i Signal Background

0 DG(Fit σ

&σ

) C

1 G(Free σ) C

2 G(Fit σ) C

3 DG(Fit σ

&σ

) L

4 CB(Fit σ, Fixed α & n) C

5 DG(Fit σ

&σ

) C (τ background reduce to 1 E)

Chebyshev polynomial, Fit σ is the result of the fitted σ (defined in Section 4) and Free σ is when the σ is completely free. For all these models the lifetime component was varied by modifying the exponential terms for the background.

ID Tracking E ffi ciency for Muons

The ID tracking e

ciency for muon tracks varies as a function of track transverse momentum and pseu- dorapidity in the kinematic intervals studied in this note. The tracking efficiency also has a small de- pendence on the number of simultaneous proton-proton collisions that occur in the event in question.

These variations are contained within a band of

efficiency, and this band is directly assigned as a systematic uncertainty on measured the cross-sections.

Muon Reconstruction and Trigger E ffi ciencies

Uncertainties on the muon reconstruction and muon trigger efficiencies arise predominantly from sta- tistical uncertainties due to the finite size of the data samples used to determine the e

ciencies. The uncertainties on the ψ(2S) yields are determined by randomly and independently (bin to bin) varying the values of these efficiencies within their uncertainties and recalculating the weights used to extract cross-sections. A Gaussian fit to the distribution of the average weight from many trials is used to extract the width of the weight distribution, which is assigned as the systematic uncertainty on the cross-section due to efficiency uncertainties.

Pion Track Reconstruction E ffi ciency

The pion track reconstruction uncertainty contains the contributions from statistical uncertainties in the pion e

ciency maps, which are estimated using the same procedure as for the muon e

ciency maps.

With the addition of uncertainties that are due to the data simulation, these are estimated to be 2%

3%

per pion in the

ranges considered, varying with rapidity; and an additional 1% contribution per pion for the hadronic interaction uncertainties.

Selection Criteria

For the constrained

π

fit quality cut, the e

ciency was measured to vary between 93% and 97% as a function of

, with an uncertainty of about 2% estimated from the bin-to-bin variation of the efficiency.

Other selection ine

ciencies and their corresponding uncertainties were estimated using simulations to

be at or below the 1% level. The systematic error on the selection e

ciency was obtained by summing

in quadrature all uncertainties.

Luminosity

The uncertainty on the integrated luminosity for the dataset studied in this note has been determined [13]

to be 1.8%. This systematic uncertainty does not apply to the measurement of the non-prompt production fraction.

Total Uncertainties

Figures 9–11 summarise the total systematic and statistical uncertainties on the measurement of the non- prompt production fraction and the prompt/non-prompt cross-sections.

[GeV]

(2S) pT

ψ

10 20 30 40 50 60 70 80 100

Fractional Uncertainty [%]

-30 -20 -10 0 10 20

30 ATLASPreliminary ^s=7TeV,

∫

Non-prompt fraction |y| < 0.75

Total Uncertainty

Total Systematics Statistical

Muon Reconstruction Pion Reconstruction Trigger

Inner Detector tracking Fit Model

Selection Criteria

[GeV]

(2S) pT

10 20 30 40 50 60 70 80ψ 10²

Fractional Uncertainty [%]

-30 -20 -10 0 10 20

30 ATLASPreliminary ^s=7TeV,

∫

[GeV]

(2S) pT

ψ

10 20 30 40 50 60 70 80 100

Fractional Uncertainty [%]

-30 -20 -10 0 10 20

30 ATLASPreliminary ^s=7TeV,

∫

Figure 9: Summary of the positive and negative uncertainties for the non-prompt fraction measurement.

This plot does not include the spin-alignment uncertainty.

[GeV]

(2S) pT

ψ

10 20 30 40 50 60 70 80 100

Fractional Uncertainty [%]

-60 -40 -20 0 20 40 60

ATLASPreliminary ^s=7TeV,

∫

Total Uncertainty Total Systematics Statistical

Muon Reconstruction Pion Reconstruction Trigger

Inner Detector tracking Fit Model

Selection Criteria

[GeV]

(2S) pT

ψ

10 20 30 40 50 60 70 80 100

Fractional Uncertainty [%]

-60 -40 -20 0 20 40 60

ATLASPreliminary ^s=7TeV,

∫

[GeV]

(2S) pT

ψ

10 20 30 40 50 60 70 80 100

Fractional Uncertainty [%]

-60 -40 -20 0 20 40 60

ATLASPreliminary ^s=7TeV,

∫

Figure 10: Summary of the positive and negative uncertainties for the prompt cross-section measurement.

This plot does not include the constant 1.8% luminosity uncertainty or the spin-alignment uncertainty.

[GeV]

(2S) pT

ψ

10 20 30 40 50 60 70 80 100

Fractional Uncertainty [%]

-60 -40 -20 0 20 40 60

ATLASPreliminary ^s=7TeV,

∫

Total Uncertainty Total Systematics Statistical

Muon Reconstruction Pion Reconstruction Trigger

Inner Detector tracking Fit Model

Selection Criteria

[GeV]

(2S) pT

10 20 30 40 50 60 70 80ψ 10²

Fractional Uncertainty [%]

-60 -40 -20 0 20 40 60

ATLASPreliminary ^s=7TeV,

∫

[GeV]

(2S) pT

ψ

10 20 30 40 50 60 70 80 100

Fractional Uncertainty [%]

-60 -40 -20 0 20 40 60

ATLASPreliminary ^s=7TeV,

∫

Figure 11: Summary of the positive and negative uncertainties for the non-prompt cross-section mea-

surement. This plot does not include the constant 1.8% luminosity uncertainty or the spin-alignment

uncertainty.

6 Results

The corrected non-prompt ψ(2S) production fraction, and the prompt and non-prompt ψ(2S) production cross-sections are measured in intervals of ψ(2S) transverse momentum and three ranges of ψ(2S) rapid- ity. All measurements are presented assuming the ψ(2S) is produced isotropically. Figure 12 shows the fully-corrected measured non-prompt production fraction as a function of

. A rise in the relative non- prompt production rate is observed as transverse momentum is increased. No significant dependence on rapidity is observed. Results [10] from the CMS experiment are superposed for comparison, and appear consistent with the ATLAS results, taking into consideration the slightly di

erent rapidity intervals used in the two experiments.

Fully-corrected measurements of the differential prompt and non-prompt cross-sections are presented in Figures 13(a) and 13(b), and compared to earlier results from LHCb [9] and CMS [10] in similar or neighbouring rapidity intervals, which are also presented under the isotropic assumption. The ATLAS data points are placed at the mean of the weighted p

distribution in each interval of p

(indicated by the horizontal error bars).

The prompt and non-prompt cross-section measurements are compared with a variety of theoretical models in Figures 14–15.

Prompt Cross-Section versus Theory

The Non-Relativistic QCD (NRQCD) [22] model (also known as the Colour Octet Model) is an effec- tive field theory for the heavy quark sector, which in addition to colour-singlet production modes allows for the possibility of formation of the heavy quark pair in a coloured state, which subsequently evolves into a physical singlet quarkonium bound state through the non-perturbative emission of soft gluons.

These non-perturbative long distance dynamics are quantified within NRQCD matrix elements (LDME)

that describe the probability for a heavy quark-antiquark pair in a particular state

to evolve into a heavy quarkonium

a strength of the approach is that for a particular partonic process these matrix elements should be uni- versal, the individual matrix elements are often poorly constrained by data, and the resultant predictions for experimental observables can be significantly modified by small changes in the fits. This can result in notably di

erent theoretical predictions even for the same order in the perturbative expansion. The NRQCD predictions presented here are derived using HELAC-ONIA [23, 24, 25, 26], an automatic ma- trix element generator for the calculation of the heavy quarkonium helicity amplitudes in the framework of NRQCD factorisation.

Both LO and NLO predictions from NRQCD are shown in comparison with the experimental data in Figure 14(a). Uncertainties on the predictions come from the uncertainties on the choice of scale, charm quark mass and LDME as discussed in Ref. [24]. In the NLO results, NLO Colour Octet LDME from Ref. [27] are used. As Colour Octet LDME from NLO fits to data cannot reasonably be used for an LO calculation, the LO results are derived using the values in Table 1 of Ref. [28], which are obtained by fitting the LO calculation to Tevatron Run-1 data. Leading order predictions show agreement with the data, albeit with large uncertainties. Next-to-leading order predictions do well at describing the shape and normalisation of prompt production data over a large range of transverse momenta. However, at the highest transverse momenta probed, NRQCD tends to predict a harder spectrum than observed. The ratio of theory to data is shown in Figure 14(b) where a deviation from the data at high

is clearly seen.

The

-factorisation [29] approach takes a parton-level cross-section prediction from the colour sin-

glet model [30, 31, 32] and incorporates initial-state radiation e

ects through parton transverse momen-

tum (k

) dependent parton distributions. These

-dependent parton distributions are not well-constrained

phenomenologically, and there are possible unresolved theoretical issues, but study of quarkonium pro-

duction o

ers an important testing ground for these alternative QCD calculation approaches. Parameter