Systematic error analysis in the absolute hydrogen gas jet polarimeter atRHIC I

A. A. Poblaguev∗, A. Zelenski, G. Atoian, Y. Makdisi, J. Ritter

Brookhaven National Laboratory, Upton, New York 11973, USA

Abstract

The Polarized Atomic Hydrogen Gas Jet Target polarimeter (HJET) is operated at the Relativistic Heavy IonCollider (RHIC) since 2004 to measure the absolute polarization of each colliding proton beam. Polarimeterdetectors and data acquisition were upgraded in 2015 to increase the solid angle, energy range, and to improvethe energy and time resolution. These upgrades along with an improved beam intensity and polarizationallowed us to greatly reduce the statistical and systematic errors for the proton polarization measurementsin RHIC Runs 15 (Ebeam = 100 GeV) and 17 (255 GeV). For a typical 8 hour RHIC store, the measuredproton beam average polarization was about Pbeam ∼ 55 ± 2.0stat ± 0.3syst%. The elastic pp analyzingpower, AN(t)∼0.04, was determined with a precision of about |δAN(t)|∼0.0002 in the momentum transfersquared range 0.001<−t<0.020 GeV2. In this paper we present a detailed systematic error analysis of thepolarization measurements at HJET. Perspectives of using the HJET based absolute polarimeters in thefuture Electron Ion Collider (EIC) will be also discussed.

Keywords: RHIC; Absolute proton beam polarization; Polarized hydrogen jet target; Coulomb nuclearinterference; Analyzing power; Systematic uncertainties;

1. Introduction

A polarized hydrogen gas jet polarimeter is usedto measure absolute proton beam polarization atthe Relativistic Heavy Ion Collider (RHIC). Thistechnique is based on concurrent measurements ofthe beam and target (jet) spin correlated asym-metries in the elastic proton–proton scattering inthe Coulomb-Nuclear Interference (CNI) region [1,2, 3]. Due to the identity of the beam and tar-get particles, polarization of the accelerated protonbeam is directly related to the proton target polar-ization, which can be determined precisely by a con-ventional Breit–Rabi polarimeter[4]. The polarime-ter target is a polarized atomic hydrogen beam (H-jet), which crosses the RHIC beam in the verticaldirection. It is nearly a point-like target, which

IThis manuscript has been authored by employees ofBrookhaven Science Associates, LLC under Contract No.de-sc0012704 with the U.S. Department of Energy∗Corresponding authorEmail address: poblaguev@bnl.gov (A. A. Poblaguev)

does not introduce any perturbation of recoil pro-tons , due to the absence of any target walls orwindows. With the high intensity circulating pro-ton current (∼2×1013 p at a 70 kHz revolution fre-quency) in the storage rings and the jet thickness of

∼1012 atoms/cm2, the event rate (for a reasonable

detector acceptance) is sufficiently high to achieve∼ 5% statistical error accuracy in an hour of mea-surement. At this target thickness, the polarimeteris non-destructive, i.e. it can be operated continu-ously, without any effect on the beams and with nobackground generation for the other experiments.

In the RHIC Spin Program [5], this techniquewas implemented in the Polarized Atomic Hydro-gen Gas Jet Target [6] (HJET) which has been inoperation since 2004. The original purpose of HJETwas to measure the absolute proton beam polar-ization at 100–255 GeV with accuracy better thanδP/P . 5% [3, 5] and to calibrate the fast p-CarbonCNI polarimeter [7] which was considered as themain tool to monitor the proton beam polarization.

It was proven in the data analysis [8, 9], thatthe HJET performance did satisfy the design goals.

Preprint submitted to Nucl. Instrum. Methods A June 16, 2020

arX

iv:2

006.

0839

3v1

[ph

ysic

s.in

s-de

t] 1

5 Ju

n 20

20

107 ns2-3

Jet

Figure 1: The RHIC polarized proton beams crossing thejet.

The systematic error in absolute proton beam po-larization measurement was dominated by uncer-tainty of the molecular hydrogen (or water vapor)contamination of the jet atomic beam and was eval-uated to be δP/P ∼ 3%. However, statistical ac-curacy of the measurements was relatively large,δP ∼3% for an 8 hour exposition. Since the HJEToperation stability appeared to be much better thanthat of the p-Carbon polarimeter, it was also real-ized that the continuous calibration of the p-Carbonis important to maintain the systematic errors un-der control.

The HJET detectors and data acquisition wereupgraded in 2015 [10]. The upgrades included theinstallation of new Si strip detectors, which pro-vided better energy resolution and a significantlylarger solid angle. The wave form digitizers wereupgraded from CAMAC, 8 bit, 140 MHz to VME,12 bit, 250 MHz. Subsequently, the statistical er-ror of the beam polarization measurements wasgreatly reduced to about δP ∼ 2% for an 8 hstore. Large statistics of about 109 elastic eventsper beam per Run acquired during RHIC Run 15(Ebeam = 100 GeV) [11] and Run 17 (255 GeV) [12]allowed us to study in depth systematic uncertain-ties and to develop an efficient background sub-traction method. As a result, the systematic er-rors in the beam polarization measurements werereduced to σsyst

P /P . 0.5%. Also, the elastic ppsingle AN(t)∼0.04 and double ANN(t)∼0.002 spinanalyzing powers have been precisely, |δAN,NN(t)|∼2×10−4, measured [13] in the momentum transfersquared range of 0.001<−t<0.020 GeV2.

In this paper we present the detailed descriptionof the HJET data processing including the analysisof the systematic errors in the RHIC proton beam

cold head

dissociator

isolation valve

six-pole magnets

RF transitions

holding fieldmagnet

Recoil spectrometer

RHIC beam

detector

Bre

it-R

abi P

ola

rim

eter

Ato

mic

Be

am S

ou

rce

cold head

dissociator

isolation valve

six-pole magnets

RF transitions

holding fieldmagnet

Recoil spectrometer

RHIC beam

detector

Bre

it-R

abi P

ola

rim

eter

Ato

mic

Be

am S

ou

rce

Figure 2: The beam view of the Polarized Atomic HydrogenGas Jet Target polarimeter at RHIC.

polarization and elastic pp analyzing powers mea-surements. Based on the obtained results, perspec-tives of using the HJET based absolute polarimetersat future Electron Ion Collider (EIC) will be alsodiscussed.

2. Polarized proton beams at RHIC

RHIC is the world’s first and only polarizedproton–proton collider [14]. The two independentstorage rings, known as: blue and yellow, have sixintersection points (IP’s), where beam collisions arepossible. At HJET location (IP12), both verticallypolarized proton beams cross the jet as depicted inFig 1. The beam separation vertically was 3 mm inRHIC Run 15 (100 GeV) and 2 mm in Run 17 (255GeV).

Each beam consisted of 111 bunches with 106.5 nsspacing. The beam intensity was (2.0–2.3) ×1011 p/bunch. The bunch spin pattern (see Fig. 1)is produced in the source [15] fulfilling a require-ment to minimize the spin correlated systematicuncertainties in the RHIC experiments. The proton

2

Radius [cm]0 10 20 30 40 50 60 70 80

Mag

netic

Fie

ld [

kG]

1.5−

1.0−

0.5−

0.0

0.5

1.0

1.5

Figure 3: The holding magnetic field as a function of dis-tance from the chamber center on the horizontal plane.

beam polarization in RHIC Runs 15 & 17, averagedover the beam intensity profile, was about 55-60%with a low <1%/h decay rate.

Since the RHIC transverse rms beam size ofabout 0.3–0.5 mm is much smaller than the jettransverse profile, the average beam polarizationand not its profile is measured by HJET.

3. HJET polarimeter

The HJET polarimeter [6] includes three majorparts: the polarized Atomic Beam Source (ABS),a scattering chamber with a holding field magnet,and the Breit–Rabi polarimeter (BRP) (see Fig. 2).The polarimeter axis is vertical and the recoil pro-tons are detected in the horizontal plane in the di-rection perpendicular to the beam. The commonvacuum system is assembled from nine identicalvacuum chambers, which provide nine stages of dif-ferential pumping. The system building block is acylindrical vacuum chamber 50 cm in diameter and32 cm long. Each chamber is pumped by two tur-bomolecular pumps with 1000 l/s pumping speedeach.

3.1. The Atomic Beam Source

The ABS part of HJET includes five vacuumchambers and five differential vacuum stages (seeFig. 2). The distance from the nozzle to the col-lision point is 127 cm. The dissociator design isdescribed in detail in Ref. [6]. The combinationof powerful vacuum system with a carefully de-signed separating magnet system allows for the de-livery of a record atomic beam intensity of 1.2 ×

≈ 0.6 cm (FWHM)

Hydrogen Jet

L=77 cm

≈20

cm

L=77 cm

≈20

cm

≈ 0.6 cm (FWHM)

Hydrogen Jet

L=77 cm

≈20

cm

Figure 4: A schematic view of the HJET recoil spectrome-ter.

1017 atoms/s [6]. The atomic beam is focused pre-cisely at the collision point, where the jet thick-ness is about 1.2×1012 atoms/cm

2and the profile is

well approximated by a Gaussian distribution withσ≈2.6 mm.

The proton spin direction in HJET is con-trolled by switching of RF-transitions, the StrongField Transition (SFT) and Weak Field Transition(WFT). Typically, the jet spin was reversed every5 min.

The proton polarization is vertical in the scat-tering chamber and is determined by the holdingfield magnet strength. The magnetic field steeringeffect on the recoil protons is minimized by usingcompensation coils, whose fields are adjusted (seeFig. 3) to keep the total vertical field integral alongthe proton paths close to zero.

3.2. The Breit–Rabi polarimeter

The RF-transitions are shielded from the longi-tudinal holding magnetic field by additional mag-netic screens. The polarimeter magnet system wasdesigned for high efficiency beam transport to theatomic beam detector. This allowed a very accu-rate measurement of the RF transition efficiencies,which appeared to be 99.9 ± 0.1% both for WFTand SFT. In this case the proton polarization inthe jet hydrogen atoms is defined by the strength(∼1.2 kG) of the holding field magnet and, thus, isknown with a high accuracy, e.g. in Run 17 it was

Pjet = 0.957± 0.001. (1)

3.3. The recoil proton spectrometer

The recoil protons from the RHIC beam scatter-ing on the jet in the scattering chamber (Cham-ber 6) are counted in the recoil spectrometer de-picted in Fig. 4.

3

Figure 5: A schematic view of the p↑p↑ spin correlatedasymmetry measurements at HJET. The recoil protons arecounted in left/right symmetric detectors. The beam movesalong the z-axis. The transverse polarization direction isalong the y-axis.

For the recoil proton measurements, we use 8pairs of the Si strip detectors. Each detector con-sists of 12 vertically oriented strips of 3.75×45 mm2

area, 470µm thickness, and a ∼ 0.37 mg/cm2

uni-form dead-layer.

For elastic scattering, the spectrometer geometryallows us to detect CNI recoil protons with kineticenergy up to TR ≈ 10-11 MeV. Protons with an en-ergy above 7.8 MeV punch through the Si detector(only part of the proton kinetic energy is detected).

For signal readout we use a 12-bit 250 MHzFADC250 wave-form digitizers [16]. A full wave-form (80 samples) is recorded for every triggered(∼0.5 MeV threshold) event.

The recoil proton energy, time, and the signalshape parametrization determined in the waveformfit as well as the recoil angle tagged by the Si striplocation provided an ability to reliably isolate theelastic pp events.

3.4. Special test measurements

For the detailed study of the HJET performanceincluding evaluation of the specific backgrounds, itwas helpful to operate the HJET in special modes.For example, injecting H2 (or O2) gas (while the jetoff) to Chamber 7 allowed us to evaluate how therecoil proton tracking in the magnetic field affectsthe results of the measurements. This diffused H2

to the scattering chamber emulated beam scatter-ing off the beamline gas.

To evaluate the atomic beam contamination byH2, we carried out the measurement with the dis-sociator RF-power off.

[MeV] RT0 2 4 6 8 10

CN

IN

A

0.01

0.02

0.03

0.04

Figure 6: A basic theoretical prediction [1] for the elastic ppsingle spin analyzing power at RHIC energies.

Special tests also included the empty target (jetoff) and single RHIC beam measurements.

HJET was routinely operated in parasitic modeduring Heavy Ion RHIC Runs. This not onlyallowed us to study the analyzing power of thep↑A scattering in the CNI region for many nucleusspecies but to also conduct various tests of HJET.

4. General analysis of the spin dependentmeasurements at HJET

4.1. Spin dependent correlations

For the vertically polarized proton beam and tar-get (the jet), the recoil proton azimuthal angle ϕdistribution can be written [17] by

d2σ

dtdϕ=

1

2π

dσ

dt×[1 +

(AjNPj +AbNPb

)sinϕ

+(ANN sin2 ϕ+ASS cos2 ϕ

)PbPj

].

(2)

The positive signs of the beam Pb and target Pjpolarizations correspond with the spin up direction.In Eq. (2), ϕ is defined in accordance with Fig. 5.Since cosϕ ≈ 0 at HJET, the measurements are notsensitive to ASS. The analyzing powers Ab,jN (t, s)and ANN(t, s) are, generally, functions of the centralmass energy squared s and 4-momentum transfersquared t, which for the fixed target proton–protonscattering can be expressed as

s = 2mp(mp + Ebeam), t = −2mpTR. (3)

where Ebeam is the beam energy, TR is the recoilproton kinetic energy, and mp is a proton mass.

For elastic p↑p↑ scattering, the target and beamsingle spin analyzing powers are the same, AjN =

4

AbN =AN. At the HJET, i.e. for small angle scat-tering at high energies, the analyzing power can bewell approximated [1] by the CNI term

ACNIN (s, t) ≈

√2TRmp

κ Tc/TR(Tc/TR)2 + 1

(4)

where κ = µp− 1 = 1.793 is proton’s anomalousmagnetic moment, Tc = 4πα/mpσtot ≈ 1 MeV, andσtot(s)∼40 mb is proton–proton total cross section.ACNI

N (TR), shown in Fig. 6, has a maximum of about0.047 at TR=

√3Tc≈1.7 MeV.

The HJET measured [13] single AN(TR) and dou-ble ANN(TR) spin analyzing powers are shown inFigs. 36 and 37, respectively.

4.2. Single spin analyzing power

Using the HJET, the beam polarization can bedetermined with no prior knowledge of AN(TR).However, since the analyzing power can be preciselymeasured at HJET, a theoretical understanding ofAN(TR) is important for the interpretation of theresults of this study. A detailed theoretical analysisof AN(t, s) and ANN(t, s) was given in Ref. [18]. Inthis approach, the single spin analyzing power canbe written as

AN(TR) = A(0)N (TR)× αN (1 + βNTR/Tc) (5)

where A(0)N (TR) is the analyzing power if the

hadronic spin–flip amplitude is disregarded. For

the elastic pp scattering, A(0)N (TR) can be calcu-

lated [18] using QED helicity amplitudes in onephoton approximation [19] and the experimental re-sults [20] for the pp differential cross-section and for

the proton rms charge radius. Such defined A(0)N is

consistent with ACNIN with the relative accuracy of

about 5%. Possible corrections to A(0)N , e.g. due

to the hadronic spin flip amplitudes [18], absorptivecorrections [21], etc. can be approximated by a lin-ear function of TR, parametrized by two factors [13],

αN = 1 +O(10−2) and βN = O(10−2). (6)

In Ref. [18], the expression for AN(t, s) was givenwith some simplifications. However, for the exper-imental accuracy achieved at HJET, some of theomitted corrections to AN(t, s) [21, 22] should betaken into account.

4.3. Experimental determination of the spin corre-lated asymmetries

At the HJET, three spin-correlated asymmetries

ajN =AjN|Pj |, abN =AbN|Pb|, aNN =ANN|PjPb| (7)

can be determined (see Appendix A) from the elas-tic events discriminated by the left/right (LR) de-tector location and the beam (↑↓) and jet (+−) spindirections:

N(↑↓)(+−)RL ∝

(1+ηεηja

jN+ηεηba

bN+ηjηbaNN

)× (1 + ηjλj) (1 + ηbλb) (1 + ηεε)

(8)

where λj,b are the jet/beam intensity asymmetriesand ε is the right/left detector acceptance asymme-try. The sign of the ηj(+−) = ±1 is defined by thejet spin (+−) and similarly for ηb(↑↓) and ηε(RL).

4.4. General description of the systematic errors’sources

In a basic approximation, i.e. if all asymmetries,aj,bn , aNN, λj,b, and ε, used in Eqs. (8) are mutuallyuncorrelated, the solution given by (A.11)–(A.16) isfree of systematic errors. However, more commonly,the possible correlations between the asymmetriesmight result in the systematic uncertainties in themeasurements.

Generally, the effective analyzing power

A(eff)N (TR) in a measurement differs from the

actual one, AN(TR), due to the background and/orthe errors in the energy calibration:

δAN =A(eff)N −AN

=b

1 + b

(A

(bgr)N −AN

)− dAN (TR)

dTRδTR

(9)

where b is the background to the signal ratio and

A(bgr)N is the effective analyzing power for back-

ground events. δAN may be interpreted as asystematic error in the definition of the analyz-ing power. Commonly, it is not the same forthe left/right detectors and for the beam/jet spinasymmetries. For example, for the proton beamscattering on the beam gas unpolarized hydrogen,

A(bgr)N = 0 for the jet spin, while A

(bgr)N = AN for

the beam spin.A possible spin correlated asymmetry of the de-

tector acceptance ω,

δω =ω↑ − ω↓

ω↑ + ω↓, (10)

5

can lead to a systematic error in the measured aN.In Run 15, a significant electronic noise correla-

tion with the WFT (i.e. with the jet spin) was ob-served in two Si detectors. Subsequently, the eventselection efficiency in these detectors appeared tobe the jet spin dependent giving non-zero δω

It should also be kept in mind, that the absolutevalues of the polarization for the up and down spinsare not necessarily the same. In this paper, exceptEq. (2), we assume

P = |P | = |P↑|+|P ↓|

2, δP =

|P ↑|−|P ↓||P ↑|+|P ↓|

. (11)

For the HJET data analysis, δPjet =δPbeam =0 wasfound to be an acceptable approach.

In summary, the leading order systematic errorsin the measured jet (beam) asymmetries can begiven by

δasystN = P

δA(R)N +δA

(L)N

2+δωR−δωL

2(12)

δλsyst = PδA

(R)N −δA

(L)N

2+δωR+δωL

2(13)

δεsyst = δP AN (14)

where δA(L,R)N and δωL,R are the average correc-

tions in left/right detectors for the jet (beam) ana-lyzing power and for the acceptance dependence onthe jet (beam) spin direction, respectively.

4.4.1. Routine control of systematic errors

For elastic pp scattering and a systematic errorfree measurement, the measured ratio abN/a

jN as well

as the measured beam and jet intensity asymme-tries λb,j must not depend on the recoil proton en-ergy TR:

d(abN/a

jN

)dTR

=dλb

dTR=dλj

dTR= bNN(TR) = 0. (15)

Also, both the jet and beam spin normalized asym-metries

aj,bn = aj,bN /A(0)N = Pj,b αN (1 + βNTR/Tc) (16)

[see Eq. 5)] are expected to be linear functions ofTR with the same factors αN and βN.

Since the possible δAN and δω are expected tobe significantly dependent on TR, the experimentaltests of Eqs. (15) and (16) are critically importantfor detecting systematic errors in the HJET mea-surements.

5. Recoil proton kinematics

5.1. Elastic pp scattering

For elastic proton–proton scattering, there is astrict correlation between the z coordinate of therecoil proton in the detectors and its kinetic energy

z−zjet = L

√TR

2mp

Ebeam+mp

Ebeam−mp+TR= κp

√TR.

(17)Here, zjet is the coordinate of the scattering point(in the jet). At HJET, κp has only a weak depen-

dence on the beam energy, κp= 17.92 mm/MeV1/2

(100 GeV) and κp=17.82 mm/MeV1/2

(255 GeV).For the vertically oriented Si strips with fixed

3.75 mm z coordinate spacing, it is convenient todefine the strip-corresponding energy

√Tstrip =

zstrip − 〈zjet〉κp

. (18)

The average recoil proton energy shift between twoconsecutive strips is

∆√Tstrip ≈ 0.210 MeV1/2 (19)

Eq. (17) allows one to relate the recoil proton en-ergy distribution in a Si strip

dNstrip

d√TR∝ dσ

dt

√TR × f

(κp√TR − κp

√Tstrip

).

(20)to the jet target density profile f(z − 〈zjet〉). Suchan evaluated jet profile must be the same for all Sistrips. However, due to the actual strip width of3.75 mm, the measured profile rms will be overesti-mated by about 9%.

5.2. Inelastic pp scattering

For inelastic scattering, p+p→ X+p, we shouldreplace in Eq.(17) with

κp → κ′p = κp×(

1 +mp∆

TREbeam

),

∆ = MX −mp ≥ mπ.

(21)

For the 100 GeV proton beam, the inelastic eventsare mostly beyond the detector’s acceptance. How-ever, for the 255 GeV beam, the acquired pp datacontamination by the inelastic events should beproperly treated in the data analysis.

6

[cm] jetz10− 0 10

[cm

]

det

z

5−

0

5

1 MeV=RT

9 MeV=RT

(L)

(R)

Figure 7: The correlation between the z-coordinates in thedetectors (filled areas) zdet and in a vertex zjet for 1 MeV and9 MeV recoil protons. The results for left and right detectorsare labeled by (L) and (R), respectively. Dashed lines showthe dependencies if there is no magnetic field.

5.3. Beam ion scattering on the jet proton

We also operated HJET in the Heavy Ion RHICRuns. The main goal was to measure the p↑A an-alyzing powers for several nuclei, Au, Ru, Zr, Al,and d. Since the beam energy Ebeam is reportedat RHIC in GeV/nucleon units, the equivalent jetproton energy in the ion frame is equal to

Ep = EbeammpA

M≈ Ebeam (22)

where A is the ion’s mass number and M is its mass.For the elastic scattering of the ion beam on the jetproton, one substitutes in Eq. (17) with

κA ≈L√2mp

√Ep+m2

p/M

Ep−m2p/M

. (23)

For heavy ions, e.g. for Au, κA ≈ L/√

2mp =

17.76 mm/MeV1/2

is beam energy independent. Incase of an inelastic scattering of the ion, MX =M+∆, the correction factor to κA is the same asgiven in Eq. (21), if we substitute Ebeam→Ep.

5.4. Tracking in the holding field magnet

The magnetic field (see Fig. 3) may significantlyalter the correlation (17) between the recoil pro-ton’s energy and coordinate in the Si detector

z → z + bMF/√TR. (24)

The displacement factor bMF is defined by the fieldintegral

|bMF| =qL

c×∫ L

0

(1− r

L

) H(r)dr√2mp

. (25)

where qL/c=2.3×10−2 MeV/G andH(r) is verticalcomponent of the holding magnetic field shown inFig. 3. The sign of bMF is different for the left andright detectors. To minimize the effect, the magnetcoils currents are adjusted to keep the displacementfactor close to zero.

A horizontal beam shift ∆x can also contributeto (24)

|b′MF| =qL

c×H(0)∆x√

2mp

. (26)

Factors bMF and b′MF effectively alter ∆√Tstrip

(19) depending on the strip location and, thus,can be determined in the data analysis. In RHICRun 15, it was |bMF|∼0.7 mm·MeV1/2 and |b′MF|∼2 mm·MeV1/2.

The magnetic field correction (24) was evaluatedassuming beam scattering at z = 〈zjet〉. It willbe shown below that the beam scattering on thebeam gas hydrogen in a wide range of z is im-portant for evaluation of the background contribu-tion. For such events, the dependence (17) can alsobe strongly violated. The calculated dependencieszdet = f(zjet, TR) for two recoil proton energies,1 MeV and 9 MeV are shown in Fig. 7.

According to (17), for any given recoil proton en-ergy TR, this background event rate is expected tobe the same in all Si strips (neglecting the smallvariation of the strips acceptance). For small TR,the strip-to-strip rate alteration may be substantialdue to the magnetic field.

6. Data Processing

6.1. Signal waveform parametrization

In the data analysis, the signal shape (Fig. 8) wasparametrized by the following function

W (t) = p+A

(t− tinτs

)nexp

(− t− ti

τs+ n

)(27)

if t > ti and W (t) = 0 if (t < ti). Here, p is thepedestal, A is signal amplitude, ti is signal starttime, n and τs are signal shape parameters. Thewaveform has maximum W (tm) = p+A at tm =ti+nτs.

7

Sample Number0 20 40 60 80

WFD

cou

nts

500

1000

1500

2000 419.9p = 1377.5A = 30.8 =it 3.39n = 3.95 =sτ

Figure 8: A signal waveform in HJET (the filled histogram).The solid red line indicates the time interval which is usedin the waveform fit. The dashed blue line is the waveformfunction W (t) beyond this interval. The sampling time isabout 4.1 ns.

Kinetic Energy [MeV]2−10 1−10 1 10 210 310

/g]

2St

oppi

ng P

ower

[M

eV c

m

1

10

210

310

α

p

Si

Figure 9: Stopping power in silicon material for α-particleand proton as a function of the injection energy of each par-ticle.

Amplitude [WFD counts]0 500 1000 1500 2000

Eve

nts

/ bin

0

200

400

600

800Am (5.486 MeV)241

Gd (3.183 MeV)148

Amplitude [WFD counts]1000 1100 1200

Eve

nts

/ bin

1

10

210Gd148 = 1155.1⟩A⟨

= 8.09σ

= 20.2 keVEσ = 297 keVDLE

Amplitude [WFD counts]2000 2100 2200

Eve

nts

/ bin

1

10

210

310Am241 = 2112.4⟩A⟨

= 8.06σ

= 20.1 keVEσ = 208 keVDLE

Figure 10: Signal amplitude distribution in the α-source calibration. For 241Am, multiple peaks around 5.486 MeV are takeninto account in the fit.

Due to energy losses in the Si detector entrancewindow, the reconstructed kinetic energy of the re-coil proton is expressed via the measured signal am-plitude as

TR = gA+ Eloss(gA, xDL) (28)

where g is the ADC gain and xDL is the dead layerthickness. Energy losses Eloss were evaluated us-ing stopping power dependence on TR [23] shown inFig. 9.

6.2. Energy calibration of the Si detectors

For energy calibration, all Si strips are exposedto α-particles from two alpha-sources, 148Gd (3.183MeV) and 241Am (5.486 MeV). A typical signalamplitude distribution in a Si strip is shown inFig. 10. The 241Am spectrum was approximatedby five Gaussian distributions

dN/dA ∝5∑i=1

wi exp− (A− 〈A〉Ei/E3)

2

2σ2(29)

where Ei = (5.388, 5.443, 5.486, 5.511, 5.544) MeVare the 241Am alpha energies and wi =(1.66, 13.1, 84.8, 0.225, 0.37) are the respective rel-ative intensities [24]. For 148Gd, a simple Gaussianfunction was used.

The two different energies of α-particles allow usto determine both the gain g ∼ 2.5 keV/cnt and

dead-layer thickness xDL ∼ 0.37 mg/cm2

in everySi strip. The energy resolution, σE ∼ 20 keV, isdominated by electronic noise.

The described method appeared to be a reliableinstrument to calibrate the HJET detectors, whichgives well reproducible results. However, it doesnot contain an intrinsic mechanism to control forsystematic errors, e.g. due to possible uncertaintiesof the dead-layer description. Extrapolation of theα-particle measurements to the proton detection isalso model dependent. Thus, we cannot exclude thecalibration related systematic error

∆calibT = δ0T + δ1

TTR. (30)

8

Si strip number0 2 4 6 8 10 12

Eve

nts

/ str

ip

310

410MeV = 2RT

Figure 11: Elastic event rates in a Si detector strips for fixedrecoil proton energy. The coordinate z0(TR) correspondingto the jet center can be found in a Gaussian (plus flat back-ground) fit of the distribution.

]1/2 [MeVRT1.0 1.5 2.0 2.5

[m

m]

⟩z⟨

0

10

20

30

40

Figure 12: The average (all HJET detectors) jet center coor-dinate 〈z0〉 dependence on the recoil proton energy TR. Thesolid line indicates the result of linear fit in the 1.1 – 6.5 GeVenergy range.

in the recoil proton energy TR measurement. Eventhough such an uncertainty must not affect the re-sult of the beam polarization measurement, it maybe essential for the experimental determination ofAN(t).

A simple way to evaluate the calibration qual-ity is to test how it satisfies Eq. (17). For that,TR dependence of the recoil proton mean z(TR)coordinate in a Si detector was being determinedas shown in Fig. 11. On average (over all Si de-tectors) the value 〈z〉, the magnetic field correc-tions (25) and (26) are canceled and the value ofκp can be determined in a linear fit of 〈z(TR)〉 (seeFig. 12). For the energy range 1.1 – 6.5 MeV, the

values of κp found, 17.94(2) mm/MeV1/2

(100 GeV)

and 17.80(2) mm/MeV1/2

(255 GeV), are in goodagreement with those calculated in section 5.1.

However, it should be noted that such evaluatedvalues of κp are fit range dependent. A more de-tail study of systematic uncertainties (30) shouldinclude a combined analysis of the measured jet pro-files (20) and the measured time dependence on thekinetic energy [26]. Also, calibration systematic er-

rors may lead to the non-linearity of AN/A(0)N (5),

which also may be used to constrain δ0T and δ1

T . Ad-mitting, that the result of this study appeared tobe unstable due to the large number of free param-eters used, we established the following upper limiton the systematic errors in the energy calibration

δ0T = 0± 15 keV, δ1

T = 0± 0.01, (31)

where δ0T and δ1

T [see Eq. (30)] variations are uncor-related.

10

210

310

[MeV] Detected Energy2 4 6 8 10 12

[ns]

T

ime

0

20

40

60

80

100

elastic protons

- particlesα

A protonsp

nuclei

prompts

Figure 13: A typical time-amplitude distribution in a Sistrip. The number of events per bin is limited to 3000 toclearly isolate elastic protons in the plot.

6.3. Overview of the signal time-amplitude distri-bution at HJET

A typical time-amplitude distribution of theevents detected in a Si strip is displayed in Fig. 13.The detected particles can be readily identified bythe correlation between measured energy and timeof flight. It should be noted that a detected parti-cle with energy above some threshold (7.8 MeV for aproton) punch through the Si strip and, thus, onlypart of the kinetic energy is measured. For suchevents, the larger kinetic energy the smaller signalamplitude is determined.

Elastic pp events can be readily recognized bythe measured energy peak around the strip locationdependent value of Tstrip (18). In addition to theproton line containing the elastic events, one can

9

[MeV]kinE0 2 4 6 8 10 12 14

Am

plitu

de [

WFD

cnt

s]

0

1000

2000

3000

[MeV]kinE0 2 4 6 8 10 12 14

[ns

]mtδ

0

2

4

6

8

[MeV]kinE0 2 4 6 8 10 12 14

)α(n

/ n

1.0

1.1

1.2

Figure 14: Simulation of waveform shape parameters dependence on recoil proton energy. n(α) denotes mean waveform shapeparameter n determined in the α-calibration. Vertical dashed lines indicate the threshold energy of 7.8 MeV for punch-throughprotons.

10

210

310

Amplitude [WFD cnts]1000 2000 3000

)α(n

/ n

0.8

1.0

1.2

1.4

3 MeV 6 MeV

8 MeV

13 MeV

Figure 15: Event selection cut (solid magenta line) to sep-arate punch-through and stopped protons for the 0.5<TR<13 MeV energy range. No other event selection cuts had beenapplied in this plot.

also identify α-particles and heavier nuclei. Thesespecies which cannot be produced in the pp scatter-ing, indicate a noticeable contribution from inelas-tic pA scattering where A is a nucleus (Nitrogen,Oxygen, . . . ) in the beam gas or in the jet.

The event statistics is strongly dominated by fastparticles, so-called prompts, which are supposed tobe π, p, α, . . . , emitted in the inelastic pp and/orpA scattering. Switching off the jet reduces promptsrate by factor 3–4.

6.4. Reconstruction of the punch-through protons

The recoil spectrometer geometry allows detec-tion of the recoil protons with kinetic energy up to10–11 MeV. For the detected energy above 5 MeVthe stopped and punch-through protons cannot beefficiently separated by comparing the measuredsignal time and the deposited energy only.

It was found in the data analysis that the signalwaveform shape (27) is significantly dependent on

10

210

310

Kinetic Energy [MeV]2 4 6 8 10 12

Tim

e [

ns]

0

20

40

60

80

100

Figure 16: The same data as in Fig. 13, but after recon-struction of the recoil proton kinetic energy and time tm.

the kinetic energy Ekin of an incident recoil pro-

ton. With parameter τs = τ(α)s fixed to the value

determined in the α-calibration, the simulated de-pendence of A, δtm (the correction to the measuredtime), and n/n(α) on Ekin is shown in Fig. 14. Thesimulation [25] is based on the evaluation of chargecollection in the Si strip and it was adjusted usingthe α-calibration data.

To separate stopped and punch-through protons,for every pair of measured A and n within the eventselection cut shown in Fig. 15 the corresponding re-coil proton kinetic energy was determined in thesimulation. The resulting time-amplitude distribu-tion (including the signal time correction) is dis-played in Fig. 16.

6.5. Event selection

To study the spin correlated asymmetries, wehave to isolate the elastic events.

First, we should verify that the detected particleis a proton. For that, we compare the measured

10

[ns]tδ10− 0 10

Eve

nts

/ bin

0

10

20

310×

]1/2 [MeVRT1.5 2.0 2.5 3.0

Eve

nts

/ bin

0

5

10

310×

Figure 17: Elastic pp events isolation. Event selection cuts are shown by the dashed red lines. The δ√T cut is applied for

events in the δt histogram, and the δt cut is applied in the TR histogram. Filled histograms show the distributions before thebackground subtraction.

]1/2 [MeVRT1.0 1.5 2.0 2.5 3.0

RTdN

/ d

410

510

610

]1/2[MeV RT1.0 1.5 2.0 2.5 3.0

Si s

trip

num

ber

0

2

4

6

8

10

12

Si strip number0 2 4 6 8 10 12

RT

dN/ d

410

510

610]1/2[MeV = 1.8RT

Figure 18: The superposition of dN/s√TR distributions for all Si strips (left). The markers colors depending on the strip

location in a detector and recoil proton energy are explained in center histogram. Green color is used for elastic events, red andblue for backgrounds, and violet specifies the area contaminated by the inelastic events. An example of background subtractionfor fixed, TR= 3.6 MeV recoil proton energy is shown in the right histogram. The background determined in the red (and/orblue, depending on TR) area is extrapolated to the signal (green) area.

signal time t with the time corresponding to themeasured recoil proton kinetic energy TR(A,n)

δt = t−t0−ToF = t−t0−L

c

√mp

2TR≈ 0 (32)

Here, L = 769 mm is the distance to detector, cis speed of light, and t0 is the time offset. Sincethe δt distribution is dominated by the beam bunchlongitudinal profile, it must be the same for all Sistrips.

Second, we should check that the missing massMX , defined as

M2X = (pbeam + pjet − precoil)

2, (33)

is equal to a proton mass, MX =mp. This conditionleads to Eq. (17). Therefore, using definition (18),it may be presented as

δ√T =

√TR −

√Tstrip ≈ 0 (34)

Since δ√T distribution is dominated by the jet den-

sity profile; it has to be the same in all of the Sistrips.

An elastic pp event isolation based on the δt andδ√T cuts is illustrated in Fig. 17.

6.6. Background subtraction

The molecular hydrogen (H2) in the beam gas(including water vapor and H2 from the jet atomichydrogen recombination on the walls) effectively di-lute the target polarization. The H2 contamina-tion in the polarized jet was measured by using aquadrupole mass-analyzer [27]. Only an upper limitof ∼ 3% on possible polarization dilution was ob-tained in these earlier studies and the molecularhydrogen was considered to be the main source ofsystematic errors of about σsyst

P /P ∼ 2–3% [8, 9].

To improve, a background subtraction methodwas developed and implemented. The method is

11

1

10

210

310

410

Amplitude [WF cnts]500 1000 1500 2000 2500

Tim

e [n

s]

0

50

100

[ns]tδ30− 20− 10− 0 10 20 30

Eve

nts

/ bin

0

2

4

6

8

310×

]1/2 [MeVRT1.0 1.5 2.0

Eve

nts

/ Bin

0

2

4

6310×

Figure 19: An example of the background subtraction in case of very high background rate. Gray filled and blue solid linehistograms show event distributions respectively before and after background subtraction. Dashed red lines indicate eventselection cuts used.

based on an assumption, that, for any TR, the back-ground rate is the same in all strips of a Si de-tector (see, for example, Fig. 11). Thus, the mea-sured event rates for TR outside the elastic peak|δ√T |>0.6 MeV can be used to evaluate (and sub-

tract) the background under the elastic peak in theother Si strips. To illustrate, in Run 17 (255 GeV)the measured dN/d

√TR distributions in all 96 Si

strips are displayed in Fig. 18. The 255 GeV datais noticeably contaminated by inelastic pp eventsand complicates the basic assumption of the back-ground subtraction. This is why the backgroundselection criteria were modified to cut off the in-elastic events.

In the data analysis, the background rate was de-termined, separately for each Si detector, as a func-tion of δt (32) and

√TR. To properly account for

the possible spin correlated effects associated withthe background, every combination of the beam andjet spins was analyzed independently. A typical re-sult of the background subtraction, seen in Fig. 17,suggests that the residual background contributionto the elastic events is well below 1%.

The efficiency of background subtraction isdemonstrated in Fig. 19. For the illustration,we used uncommon data, obtained in RHIC 16deuteron–Gold Run, with very high backgroundlevel. Even in this case, the method works suffi-ciently well.

The background subtraction is routinely used inthe data analysis which allows us to dilute the back-ground to a sub-percent (on average) level.

The following speculations were used to justifythe method. For the pA background, which is dom-inant in the HJET measurements, the rate shouldbe strip independent due to a relatively small sizeof the detectors and absence of a strict correlation

between the recoil angle and kinetic energy.

For the beam gas H2 background, one can expectalmost uniform hydrogen density in the scatteringchamber. Furthermore, according to Eq. (17), thebackground rate dNH2

/dTR should be the same inall Si strips. However, the recoil proton trackingin the magnetic field (see Fig. 7) results in strip-to-strip alteration of this background rate. Thateffect requires special data processing which will bediscussed in section 7.2.

Other possible errors in the background subtrac-tion will be analyzed in sections 7.1 (the atomichydrogen jet contamination by H2) and 7.3 (the in-elastic pp background).

7. Absolute proton beam polarization mea-surements at HJET

To determine the RHIC proton beam polariza-tion, we compare the measured beam and jet singlespin asymmetries

Pbeam =(abeam

N /ajetN

)× Pjet. (35)

Here, abeamN and ajet

N are the measurement averagedvalues, depending on the event selection cuts used.Since both asymmetries are determined using thesame events, the result must be independent of theanalyzing power AN(t) details.

To discuss experimental uncertainties of the mea-surements, we will use 255 GeV polarized protonbeam data acquired during a 14-week RHIC Run 17.

Two main sets of event selection cuts were used.The first one accepts as many elastic events as rea-sonably possible to minimize the statistical uncer-

12

RHIC Fill Number20600 20700 20800 20900

⟩N

A⟨je

tP

=

jet

Na

0.030

0.035

0.040

= 0.03622(5)⟩a⟨ = 263.4 / 1802χ

Blue beam = 0.03618(5)⟩a⟨

= 171.2 / 1802χ

Yellow beam

March April May

Figure 20: The RHIC store average jet spin asymmetriesmeasured with blue and yellow beams in Run 17.

Recoil Proton Energy [MeV]1 2 3 4 5 6 7 8 9 10

Bea

m P

olar

izat

ion

0.4

0.5

0.6

0.7

= 0.5461(16)⟩P⟨ = 93.4 / 322χ

Blue beam = 0.5527(15)⟩P⟨

= 214.4 / 322χ

Yellow beam

Figure 21: The RHIC Run 17 averaged blue and yellowbeam polarization dependence on the recoil proton energy.

tainty of the measurements:

Cuts I: 0.6 < TR < 10.6 MeV,

|δt| < 7 ns,∣∣∣δ√T ∣∣∣ < 0.40 MeV1/2.

(36)

The second set minimizes uncontrollable systematiccorrections:

Cuts II: 2.0 < TR < 9.5 MeV,

|δt| < 7 ns,

− 0.18 < δ√T < 0.30 MeV1/2.

(37)

Explanation of these cuts will be given below.For Cuts I, the RHIC fill number dependence of

ajetN is displayed Fig. 20. One can see good agree-

ment between asymmetries measured with blue andyellow beams. Some excess of the χ2 over NDF,observed for blue beam only, can be attributed topossible indication of the fill-by-fill fluctuations of

]1/2 [MeVRT1.5 2.0 2.5

[a.

u.]

R d

N/d

T-1

/dt

σd

0.00

0.01

)10× (2Molecular H

]1/2 [MeVRT1.5 2.0 2.5

[a.

u.]

R d

N/d

T-1

/dt

σd 0.0

0.5

1.0

↑Atomic H

Figure 22: The jet density profile evaluation using themeasured dN/dTR distribution in a Si strip. Upper: themolecular hydrogen component isolated in a run with thedissociator RF-discharge off. Lower: the polarized atomichydrogen profile in a regular measurement. The graphs havethe same normalization per integral beam intensity in themeasurements.

the systematic corrections. However, these fluctu-ations are smaller than the statistical errors of themeasurements. For the RHIC Run 17 average esti-mates, the long-term instability of the jet asymme-try measurement must not exceed∣∣∣∣δaNaN

∣∣∣∣long-term

. 0.1% (38)

This value includes the possible fluctuations of thejet polarization, instability of the energy calibra-tion, fluctuations of the signal contamination bybackgrounds, etc.

In a short term beam polarization measurement,ajetN in Eq. (35) can be substituted by the Run av-

erage⟨ajetN

⟩value

Pbeam =abeamN

〈ajetN 〉

Pjet (1 + δcorr) =abeamN

AeffN

(39)

where δcorr denotes correction, which has to be ap-plied to compensate the systematic error. The sig-nificance of this correction for Cuts I is followed

13

6Beam

Jet

9 mm

15 mm

Scattering Chamber

30 mm

7

5

Figure 23: The molecular hydrogen flow in HJET. The pres-sures in the chambers shown are given for a regular HJETrun.

3 mm

Figure 24: HJET construction element which partiallyshadows recoil protons from the beam scattering on the beamgas H2.

from the measured beam polarization dependenceon the recoil proton energy (Fig. 21).

Excellent stability of the measured jet spin asym-metry during the Run allows us to use the RHICRun average effective analyzing power Aeff

N . Thisdilutes statistical uncertainties in short-term mea-surements. To incorporate the Run average system-atic corrections to Aeff

N , we may process the datausing a special set of event selection cuts (Cuts II)for which δcorr can be precisely evaluated:

AeffN =

⟨ajetN

⟩Pjet

(1 + δcorr

) ⟨abeamN

⟩⟨abeamN

⟩ . (40)

To distinguish between Cuts I and Cuts II values,the latter are marked by tilde.

7.1. Molecular hydrogen in the jet

Typically, mass fraction of the molecular hydro-gen flow from the dissociator is about 10%. Since

]1/2 [MeVRT1.4 1.6 1.8 2.0 2.2

RT

dN/ d

8

10

12

14

310×

Figure 25: Equidistant dips in the dN/d√TR distributions

in the consecutive Si strips used to calibrate the beam gasH2 background.

the H2 component is not focused by the sextupolemagnets, only a small fraction of these moleculescan directly reach the collision point, 127 cm awayfrom the dissociator nozzle. By turning off the dis-sociator RF-discharge, the atomic component in thejet is eliminated while the H2 density is increased byfactor ∼ 10. In a such measurement (see Fig. 22),we have evaluated the molecular hydrogen fractionin the polarized atomic jet and, subsequently, thecorresponding systematic correction to the mea-sured beam polarization(

δP

P

)syst

jet H2

= +0.06± 0.06 %, (41)

where the specified error reflects the possible (upperlimit) uncertainty in the dissociation degree.

7.2. Molecular hydrogen in the beam gas

As shown in Fig. 23, the beam gas hydrogen ex-cess in the scattering Chamber 6 is caused by H2

diffusion from Chamber 7 (the scattered and recom-bined jet atomic hydrogen) and Chamber 5 (unfo-cused hydrogen atoms recombined to molecules).The diffusion H2 density profile (seen by the beam)is much wider than that of the atomic hydrogen pro-file in the jet. It was confirmed in the special testsby injecting H2 gas to Chamber 7 or to Chamber 5.

To evaluate the beam gas H2 density, we no-ticed that the recoil protons from the beam scat-tering on H2 at z ≈ ±12 mm are shadowed byd= 3 mm wide wall in the HJET construction (de-picted in Fig. 24). The efficiency of such shadow-ing is up to d/dstr = 80% where dstr = 3.7 mm isthe Si strip active width. In the dN/d

√TR distri-

butions (see Fig. 25), the effect manifests itself by

14

]1/2 [MeVRT1.5 2.0 2.5

Bac

kgro

und

[a.

u.]

10

15

20

25

30

Left

Right

Figure 26: Comparison of the background dN/d√TR rate

in left and right side detectors.

the equidistant dips, relative to the strips commonbackground. Comparing the event rates in the dipswith the jet center (see, for example, Fig. 18) onecan estimate the atomic mass fraction of the beamgas H2 in the jet center

b(0)H2

= 0.41± 0.04 %. (42)

which leads to the effective, Cuts II average,

bH2= 0.56± 0.06 %. (43)

background rate in the Run 17 measurements. Suchevaluated bH2

is provided by one (forward) beamonly and gives about 20% to the total backgroundrate for the 2–5 MeV recoil protons.

The recoil proton tracking in the holding fieldmagnet (see Fig. 7) strongly violates the depen-dence (17) and, thus, affects the beam gas H2 back-ground subtraction. The effect can be readily seenby comparing the experimentally evaluated back-ground in left and right detectors (Fig. 26).

Residual, i.e. after subtraction, backgrounds cal-culated separately for the forward (the beam whichpolarization is measured by the considered detec-tor) and backward (the opposite direction beam)beams and for the left and right detectors are shownin Fig. 27.

For the spin pattern used in RHIC, the polar-izations of forward and backward beams are un-correlated on average. Therefore, for the back-ward beam, the residual H2 background does notalter the measured beam polarization (the ratio

abeamN /ajet

N ). Subsequently, to evaluate respectivesystematic errors in the beam polarization mea-surement, one should only use the forward beamresidual background. According to Fig. 27, for the

[MeV]RT0 2 4 6 8 10

MH

b / b

2−

0

2 beamForward

0.10− Left:

0.11 Right:

[MeV]RT0 2 4 6 8 10

MH

b / b

0

5 beamBackward

0.10− Left:

1.49− Right:

Figure 27: Simulation of the residual H2 backgrounds forforward and backward beams. Backgrounds in left and right(relative to the forward beam) detectors are shown sepa-rately. The average values are given for the 2<TR<10 MeVenergy range.

Cuts II recoil proton energy cut TR > 2 MeV theresulting systematic corrections to the measuredbeam polarization is negligible. Possible variationsin the simulation model do not change this conclu-sion.

However, there is some uncertainty in determin-ing the H2 background caused by non-uniform shad-owing of a Si detector. This uncertainty may beaccounted as a related systematic error:(σP

P

)syst

beam H2

. 0.1% (44)

In the analyzing power measurements, we cor-rected the residual H2 background using thesimulation for the left/right detectors and for-ward/backward beams.

7.3. Inelastic pp scattering

For inelastic proton beam scattering on a pro-ton target (21), the recoil proton z-coordinate de-pendences on TR and ∆ [see Eq. (21)] is shown inFig. 28. For the 255 GeV beam, about half of theSi strips are exposed to inelastic protons. It was

15

]1/2[MeV RT0 1 2 3

[mm

] ⟩

jet

z⟨ −z

0

20

40

60

123456789

101112

XMissing Mass M

(elastic)pm

π+mpm

(1232)∆m

N(1440)m

Figure 28: The recoil proton z-coordinate dependence onkinetic energy TR and missing mass MX . The shown linesshould be smeared with σz ∼ 2.6 mm due to the jet densityprofile.

]1/2 [MeVRTδ0.4− 0.2− 0.0 0.2 0.4

Bea

m P

olar

izat

ion

0

1

2

= 0.5392(17)⟩P⟨ = 16.7 / 232χ

Blue beam = 0.5445(17)⟩P⟨

= 27.3 / 232χ

Yellow beam

Cuts I

Cuts II

Figure 29: The measured beam polarization dependenceon δ√T event selection cuts. The fit results are given for

Cuts II.

observed (see Fig. 27 in Ref. [28]) that the inelas-tic beam spin asymmetry is larger than the elas-tic one while the jet spin asymmetry is smaller.Therefore, the inelastic background increases themeasured beam polarization in HJET. To separateelastic and inelastic pp-scattering events we alteredthe δ

√T cut as shown in Fig. 29. By varying the

cut, the systematic error, for Cuts II, was estimatedas (

δP

P

)syst

pp→Xp= +0.15± 0.15 %. (45)

7.4. pA scattering

The pA scattering gives the dominant contribu-tion to the recoil proton background which is as-sumed to be Si strip zstr coordinate independent.

]1/2 [MeVRT1.0 1.5 2.0

[a.

u.]

RT

dN /

d

2−10

1−10

11

10

6

12

Figure 30: Comparison of the dN/d√TR (background) dis-

tributions for Si strips 1 and 10. Elastic peak (strip 6) isshown for normalization. Solid line (strip 12) gives an esti-mate of the inelastic pp background in this strip.

In units of kstr, the strip number in a detector, thebackground rate can be approximated by

η(TR, zstr) = dN(pA)bgr /d

√TR

∝ 1 + c1(TR)kstr + c2(TR)k2str

(46)

where c1,2 are expected to be small. Since bothRHIC beams, blue and yellow, contribute to thebackground in each Si strip, c1 must be additionallysuppressed.

To evaluate a possible non-uniformity of η(zstr)we compare dN/d

√TR distributions in two strips,

kstr=1 (zstr ≈ 11 mm) and 10 (45 mm). The dis-tributions, shown in Fig. 30 were averaged over all8 Si detectors to eliminate discrepancy due to theleft/right dependent backgrounds. To avoid cor-ruption by the inelastic pp events, we have used100 GeV data. For that beam energy, the inelas-tic background can be detected only in strip 12.For the

√TR range 1.2–1.7 MeV1/2, where the con-

tribution from the beam scattering of the jet pro-tons is strongly suppressed, the η1(TR) and η10(TR)backgrounds are consistent at the 0.01% level rela-tive to elastic peak rate. Thus, systematic errors inpA background subtraction can be neglected. How-ever, since the obtained result is slightly dependenton the cuts used and because the entire Cuts II TRrange was not tested, we used a more conservativeestimate (σP

P

)syst

pA< 0.2% (47)

based on the observed fluctuations of the back-ground lines. The specified systematic error is more

16

Si strip number0 10 20 30 40 50 60 70 80 90

[W

FD u

nits

]no

ise

σ

0

10

20

ONWFT

OFFWFT

10×) OFFσ−ONσ(

[MeV]RT2 4 6 8

RTje

tna

1.0

1.2

1.4

1.6

1.8

[MeV]RT2 4 6 8

10

× RT

jet

λ

0.3−

0.2−

0.1−

0.0

Figure 31: Left: Effective electronic noise for the jet spin up and down in Run 15. The difference scaled by factor 10 is shownby the histogram. For the measured jet spin (center) and intensity (right) asymmetries, � points are for the all blue Si detectorsand N are for the analysis with no strips 72–83 data used.

due to the background subtraction method uncer-tainties rather than due to the pA background itself.

7.5. Noise correlation with the jet spin

It was found that the WFT induces a 14 MHzringing in some detectors. In Run 15, this effectivenoise (see Fig. 31) in blue upper left detector (strips72–83) resulted in a dependence of the event selec-tion efficiency on the jet spin state. Therefore, forblue beam, the asymmetry δωL (10) appeared tobe significantly non-zero and, in accordance withEqs. (12), (13), the correlated changes of the mea-sured ajet

n (TR) and λjet(TR) were observed. Remov-ing strips 72–83 from the analysis fixed the distri-butions. For the beam spin asymmetries, the issuewas not detected.

In RHIC Run 17, no evidence of the WFT re-lated systematic error was found. The upper limit,mostly defined by the statistical uncertainties in thestudied distributions, was established as(σP

P

)syst

WFT< 0.2%. (48)

7.6. Systematic uncertainties summary

The systematic uncertainties in the beam polar-ization measurements are summarized in Table 1.For event selection Cuts II, the total correction tocompensate a systematic error bias was evaluatedas

δcorr = −0.21± 0.37 % (49)

According to Eq. (40) and using 〈Pjet〉 = 0.957, theeffective Run 17 blue and yellow beam spin analyz-ing powers are equal to(

AeffN

)B

= 3.749± 0.013stat ± 0.014syst %, (50)(AeffN

)Y

= 3.739± 0.012stat ± 0.014syst %, (51)

Source δP/P [%] σP /P [%]

A. Long term stability 0.1B. Jet polarization 0.1C. Jet H2 +0.06 0.06D. Beam gas H2 . 0.1E. pA . 0.2F. pp→ Xp +0.15 0.15G. WFT . 0.2

Total +0.21 . 0.37

Table 1: Systematic uncertainties summary for the eventselection Cuts II. Some systematic uncertainties were can-celed in the beam/jet asymmetry ratio and, therefore, wereomitted here.

respectively. The effective systematic error in theHJET beam polarization measurements can be ap-proximated by

σsystP /P . 0.5% (52)

The provided value includes statistical uncertain-ties in evaluation of the Run 17 average analyzingpowers (50), (51).

8. Single spin analyzing power measurement

Low systematic uncertainties in the HJET spincorrelation measurements gave a possibility of highprecision determination [13] of the elastic pp singlespin analyzing power

AN(t) = ajetN (TR)/Pjet (53)

at two proton beam energies, 100 and 255 GeV.To fit AN(t), it is convenient to use αN and βN

based parametrization (5). However, since the main

17

[MeV] RT2 4 6 8 10

RT

jet

na

0.8

1.0

1.2

= 0.9643(52)Nα∼ = 0.0164(14)

Nβ

= 76.1 / 862χBlue beam

= 0.9548(51)Nα∼ = 0.0201(13)

Nβ

= 74.2 / 862χYellow beam

[MeV] RT2 4 6 8 10

RT

beam

na0.4

0.5

0.6

0.7

0.8Run15 (100 GeV)

= 0.5470(42)Nα∼ = 0.0193(20)

Nβ

= 88.1 / 862χBlue beam

= 0.5768(42)Nα∼ = 0.0184(18)

Nβ

= 72.1 / 862χYellow beam

[MeV] RT2 4 6 8 10

RT

jet

λ

5−

0

53−10×

=-0.00045(9)⟩j

λ⟨ = 75.2 / 872χ

Blue beam

=-0.00022(9)⟩j

λ⟨ = 89.7 / 872χ

Yellow beam

[MeV] RT2 4 6 8 10

RT

beam

λ

5−

0

53−10×

= 0.00004(7)⟩b

λ⟨ = 110.7 / 872χ

Blue beam

=-0.00121(7)⟩b

λ⟨ = 82.0 / 872χ

Yellow beam

[MeV] RT2 4 6 8 10

RT

jet

na

0.8

1.0

1.2

= 0.9386(46)Nα∼ = 0.0069(10)

Nβ

= 57.4 / 742χBlue beam

= 0.9319(45)Nα∼ = 0.0086(10)

Nβ

= 96.7 / 742χYellow beam

[MeV] RT2 4 6 8 10

RT

beam

na

0.4

0.5

0.6

0.7Run17 (255 GeV)

= 0.5336(29)Nα∼ = 0.0092(13)

Nβ

= 87.6 / 862χBlue beam

= 0.5428(28)Nα∼ = 0.0099(13)

Nβ

= 84.3 / 862χYellow beam

[MeV] RT2 4 6 8 10

RT

jet

λ

4−

2−

0

2

43−10×

= 0.00010(5)⟩j

λ⟨ = 103.8 / 872χ

Blue beam

= 0.00030(5)⟩j

λ⟨ = 83.0 / 872χ

Yellow beam

[MeV] RT2 4 6 8 10

RT

beam

λ

4−

2−

0

2

43−10×

= 0.00012(5)⟩b

λ⟨ = 96.5 / 872χ

Blue beam

=-0.00016(5)⟩b

λ⟨ = 105.8 / 872χ

Yellow beam

Figure 32: Measured asymmetries in RHIC Run15 (100 GeV) and Run 17 (255 GeV). Blue/red polymarker colors are used for

blue / yellow RHIC beams. The fit energy range is 1.9 < TR < 9.9 MeV for the 255 GeV ajn(TR) and 0.7 < TR < 9.9 MeV forall other graphs. The fit parameter αN is defined as αN = 〈P 〉αN.

goal of this analysis was experimental isolation ofthe hadronic spin flip amplitude r5 [18], the fit inRef. [13] was done in terms of Re r5 and Im r5 usingthe theoretically expected expressions for αN(r5)and βN(r5) [22].

In this study, the quadratic sum of uncertaintiesB, C, and F in Table 1 of 0.25% was attributed tothe effective systematic error in the value of the jetpolarization, and the mean value of Pjet was cor-rected by 0.06%. In the combined Regge fit of the100 and 255 GeV data [9], this systematic uncer-tainty was assumed to be strictly correlated for theboth energies. Similarly, the quadratic sum of un-certainties A and G was interpreted as systematicerror in Pjet, uncorrelated for the 100 and 255 GeV.

The systematic corrections due to the data con-tamination by the inelastic pp events were evalu-ated using the same method as in section 7.3. Thedependence of measured r5 on the δ

√TR event se-

lection cut was studied.In the beam polarization measurement [28], some

systematic uncertainties common to ajN and ab

N

were effectively canceled. However, these uncer-tainties should be thoroughly re-assessed in the ex-perimental evaluation of the single spin analyzingpower:– Compared to section 7.2, a more accurate analy-sis of the magnetic field corrections to the evaluatedbeam gas H2 background is needed. The residualbackground, both for forward and backward beams,

was simulated and corrections to the backgroundsubtractions were applied. The simulation includedpossible non-uniformity of the beam gas density dueto pumping and used a more accurate descriptionof the effective collimator shown in Fig. 24. Foradjustment, we compared the calculated and mea-sured difference in the left/right background rates(see Fig. 26).– The vertical size of the detectors effectively alterthe jet polarization P eff

jet = Pjet〈sinϕ〉 ≈ 0.997Pjet.– Systematic uncertainties in the energy calibra-tion of the detectors, can alter the result of themeasurement of r5 or, equivalently, αN and βN. Toestimate these systematic errors, we used the cali-bration uncertainties given in Eq. (31) consideringthem strictly correlated for 100 and 255 GeV mea-surements.

The effective, i.e. after systematic corrections,jet polarization used in data analysis was P eff

jet =

0.954 (100 GeV) and P efftext = 0.953 (255 GeV). The

small difference is due to change of the holding fieldmagnet currents.

The event selection cuts

100 GeV: |δt| < 5.9 ns,

|δ√T | < 0.36 MeV1/2,

255 GeV: |δt| < 7.1 ns,

− 0.18 < δ√T < 0.36 MeV1/2,

(54)

were optimized using procedures similar to that il-

18

[GeV]LabE100 150 200 250 300

+φIm

/+φ

Re

=

ρ

0.2−

0.1−

0.0

Figure 33: Comparison of the elastic pp forward imaginaryto real ratio ρ determined in (�) the HJET analyzing powerfit [13] and (•) the unpolarized measurements [20].

[MeV] p× + 10RT0 10 20 30 40

NA

= P

Na

0.02

0.03

0.04

(Jet Blue) p=0

(Jet Yellow) p=1

(Beam Blue) p=2

(Beam Yellow) p=3

Figure 34: The combined single spin analyzing power fit inRun 17 (255 GeV). All the displayed data was concurrentlyfit using four free parameters, αN, βN, and Run averagevalues of the blue and yellow beam polarizations.

lustrated in Fig. 29. The normalized single spin,aN(TR)/AN(TR) (16), and intensity, λ(TR) (8),asymmetries determined with these cuts are shownin Fig. 32.

For the 100 GeV beam, all dependencies agreewith a systematic error free expectation. For exam-ple, four statistically independent measurements ofthe slope βN are well consistent 〈βN〉 = 0.0185 ±0.0008 (χ2/ndf = 3.93/3).

For 255 GeV, our assumption that ajn(TR) is a

linear function of TR is strongly violated at low en-ergies, TR < 1.9 MeV, but there are no issues withother distributions. Such a signature attributes theproblem to an overestimated H2 background sub-traction. For low TR, the accuracy of the H2 back-ground evaluation may be affected by the recoilproton tracking in the magnetic field and, for the255 GeV beam, by the data contamination from

100× N

β1.7 1.8 1.9

100

× )1 − Nα(

0.0

0.5

1.0

100 GeV = beamE

100× N

β0.75 0.80 0.85 0.90

100

× )1 − Nα(

2.6−

2.4−

2.2−

2.0−

1.8−255 GeV = beamE

Figure 35: 1-σ correlation contours, in terms of αN and βN,for the HJET measured single spin–flip hadronic amplitudesr5. Orange and violet ellipses are for systematic and total(stat.+syst.) uncertainties, respectively.

inelastic pp → Xp events. However, as we cur-rently have no proven model to describe the phe-nomenon, we have eliminated the 255 GeV low en-ergy (TR < 1.9 MeV) jet asymmetry aj

N results fromthe analysis. With such a cut, the βN test gives〈βN〉 = 0.0084± 0.0006 (χ2/ndf = 3.99/3).

An incorrect value of the forward elastic real-to-imaginary ratio ρ used in the calculation of

A(0)N (t) [Eq. (5)] may result in a false non-linearity

of Eq. (5). In the fits with ρ being a free parame-ter, we obtained ρ = −0.050±0.025 (100 GeV) andρ = −0.028 ± 0.018 (255 GeV), the values whichagree (Fig. 33) with unpolarized pp data to withinone standard deviation. So, this test does not in-dicate any statistically significant discrepancy withthe theoretical expectation (5).

The described fit of ρ provided an important con-tribution to the upper limit (31) for the systematicuncertainties in the energy calibration.

To determine the hadronic spin–flip amplituder5 (separately for 100 and 255 GeV), we concur-rently fit all four (the jet and beam spin for both

RHIC beams) measured asymmetries, aj,bN (t) =

Pj,bAN(t, r5) as illustrated in Fig. 34.

The detailed description of the fit results is givenin Ref. [13]. The correlation contours for αN ≈1− (2/κ) Im r5 and βN ≈ −(2/κ) Re r5 are shownin Fig. 35. The measured analyzing power depen-dence on TR, including the experimental uncertain-ties, are shown in Fig. 36. It should be noted thatthe drown analyzing powers can be interpreted [13]as model independent, i.e. insensitive to the actualtheoretical parametrization of AN(t) used in the fit.Instructions to calculate AN(t) including extrapo-lations to other energies can be found in Ref. [13].

19

[MeV] RT2 4 6 8

0.00

0.01

0.02

0.03

0.04

)R

(TNA = 100 GeVbeamE

50×

[MeV] RT2 4 6 8

0.00

0.01

0.02

0.03

0.04

)R

(TNA = 255 GeVbeamE

50×

Figure 36: Elastic pp single spin analyzing powers measured at HJET [13]. Filledareas are 50-fold systematic and total (stat+syst) errors in the measurements.Dashed lines specifies the theoretical predictions given by Eq. (4).

GeV RT2 4 6 8 10

0.000

0.001

0.002

0.003

0.004 )R

(TNNA

GeV =100beamE

GeV =255beamE

Figure 37: Elastic pp double spin an-alyzing powers measured at HJET [13].Filled areas indicate ±1σ statistical er-rors.

9. Double spin analyzing power measure-ment

The jet spin correlated systematic uncertaintiesare canceled in the ratio, aNN(TR)/aj

N(TR), of themeasured asymmetries. This statement was veri-fied by comparing the ratio for data with and with-out background subtraction. Therefore, for experi-mental determination of the double spin analyzingpower ANN(t), it is convenient to use the followingrelation:

ANN(t=−2mpTR) =A2

N(TR, r5)

abN(TR)

× aNN(TR)

ajN(TR)

(55)

Since AN(TR, r5) is well known from the single spinasymmetry study and since there are no issues withsystematic error in the experimental determinationof ab

N(TR), the experimental uncertainties of suchdefined ANN(t) are strongly dominated by the sta-tistical errors of the measured aNN(TR).

Double spin analyzing power measured atHJET [13] is shown in Fig. 37.

10. HJET in Heavy Ion beam

The HJET performance in a heavy ion RHICbeam is very similar to that of in the proton beam.The observed correlation between the recoil angleand recoil energy is well consistent with Eq. (23)suggesting a strong dominance of the elastic and/ornucleus excitation pp → pA∗ scattering. Therefore,the p↑A single spin analyzing power can be studiedin the same way as it was done to measure elas-tic p↑p AN even though we cannot experimentallyseparate the elastic and A∗ events at the low t.Some preliminary results (statistical errors only),

]2 [GeVt−0.000 0.005 0.010 0.015 0.020

(t)

NA

0.05−

0.00

0.05Preliminary

Figure 38: Preliminary results for p↑p (• and • for blueand yellow beams, respectively), p↑Al (H), and p↑Au (�)analyzing powers measured with 100 GeV/n beams in RHICRun15.

obtained in 2015 are displayed in Fig. 38. Thedata obtained with 100 GeV/nucleon d, Al, Ru, Zr,and Au beams as well as energy scans for Deu-terium (9.8, 19.5, 31.2, and 101 GeV/n) and Gold(3.85, 4.59, 5.76, 9.8, 13.2, 19.5, 27.2, 31.2, and100 GeV/n) are being analyzed.

For the Gold beam, the low energy (TR∼1 MeV)recoil proton rate is about the prompts rate. Thismakes the Gold beam a convenient tool to studysome systematic uncertainties. For example, inRHIC Run 20, we studied the magnetic field cor-rection to the low energy recoil dN/dz distribution.For that, H2 was injected to Chamber 7 and, usingthe single (yellow) Gold beam, we measured the re-coil proton rate in the Si strips with the holdingfield magnet switched on and off. The results forTR = 1.0± 0.1 MeV are shown in Fig. 39. Thesemeasurements provided experimental confirmationof the assumptions discussed in sections 5.4 and 7.2.

20

[cm]stripz4− 2− 0 2 4

Eve

nts

/ str

ip [

a.u]

0

1

2

Yellow beamleft detectorsright detectors

Magnetic Field OFF

[cm]stripz4− 2− 0 2 4

Eve

nts

/ str

ip [

a.u]

0

1

2

Magnetic Field ON

Figure 39: Recoil proton z-coordinate distribution for TR=1.0 ± 0.1 MeV with the holding field magnet On and Off.The measurements were done with yellow Gold beam andH2 injected to Chamber 7.

11. The HJET based polarimeters at EIC

The proton beam energies and intensities pro-posed for EIC [30] are similar to those at RHIC,which allows for a straightforward transfer of ourexperience with HJET to the hadronic polarime-try at EIC. However, high bunch frequency (10 nsbunch spacing) at EIC may be a challenge due tomixing signals from the differently polarized beambunches.

To evaluate the HJET performance in these con-ditions, we emulated the beam polarization mea-surement data at EIC using the real data ac-quired in RHIC Run 17 [31]. These generated eventswere processed using the regular HJET analysis de-scribed in this paper. The proton beam polariza-tion systematic error is not changed much if re-coil protons with energy cut TR > 2 MeV are se-lected [31]. A possibility to lower this threshold to. 1 MeV requires more studies including the opti-mization of the waveform shape event selection cutsshown in Fig. 15.

Employing double layer Si strip detectors to vetohigh intensity prompts is also being investigated.

12. Summary and outlook

Since it was commissioned in 2004 the HJEThas exhibited stable and reliable performance. Thesources of systematic errors are well controlled,which allowed us to measure the high energy pro-ton beam polarization with little systematic uncer-tainty σsyst

P /P . 0.5%.The accuracy achieved in determining AN(t) [13]

allows one to use a higher density unpolarized hy-drogen jet target in a high precision absolute po-larimeter, e.g., at EIC.

The unpolarized H-jet thickness of a 1014 H2/cm2

(with the jet profile similar to the HJET one) canbe produced by the cluster-jet technique, which wasdeveloped for the PANDA experiment at GSI [29].Considering the higher EIC beam intensity and apossibility to increase the recoil proton detectorssolid angle, one can expect 1000 times higher elasticpp event rate compared to that in the RHIC HJETmeasurements.

For such a polarimeter, only a 3 min. expo-sure is needed to determine the beam polarizationwith a statistical accuracy of σstat

P ∼ 1%, whilethe systematic uncertainty can be kept as low asδsystP/P . 1%. Therefore, an unpolarized jet canalso provide accurate measurement of the polariza-tion decay time and, with additional detectors atdifferent azimuthal angles, the beam’s spin tilt.

13. Acknowledgments

Many results of this work were presentedat RHIC Spin Group meetings. The au-thors are thankful to I. Alekseev, E. As-chenauer, A. Bazilevsky, K. O. Eyser, H. Huang,W. B. Schmidke, and D. Svirida for numeroususeful discussions and stimulating comments. Weacknowledge support from the Office of NuclearPhysics in the Office of Science of the US Depart-ment of Energy and the RIKEN BNL ResearchCenter.

Appendix A. Experimental determinationof the spin correlated asym-metries

For evaluation of the spin-correlated asymme-tries, an experiment schematically shown in Fig. 4 isequivalent to counting of recoil protons in right/left(RL) symmetric detectors depending on the beam

21

(↑↓) and target (+−) spins. Following Eq. (2) onecan expect

N↑+R = N0

(1 + aj

N + abN + aNN

)×

(1+λj) (1+λb) (1+ε) (1+bNN) , (A.1)

N↓+R = N0

(1 + aj

N − abN − aNN

)×

(1+λj) (1−λb) (1+ε) (1−bNN) , (A.2)

N↑−R = N0

(1− aj

N + abN − aNN

)×

(1−λj) (1+λb) (1+ε) (1−bNN) , (A.3)

N↓−R = N0

(1− aj

N − abN + aNN

)×

(1−λj) (1−λb) (1+ε) (1+bNN) , (A.4)

N↑+L = N0

(1− aj

N − abN + aNN

)×

(1+λj) (1+λb) (1−ε) (1−bNN) , (A.5)

N↓+L = N0

(1− aj

N + abN − aNN

)(1+λj) (1−λb) (1−ε) (1+bNN) , (A.6)

N↑−L = N0

(1 + aj

N − abN − aNN

)×

(1−λj) (1+λb) (1−ε) (1+bNN) , (A.7)

N↓−L = N0

(1 + aj

N + abN + aNN

)×

(1−λj) (1−λb) (1−ε) (1−bNN) (A.8)

where ajN, ab

N, and aNN are the spin correlatedasymmetries defined in Eq. (7), λj,b are the targetand beam intensity asymmetries, ε is the RL detec-tor acceptance asymmetry, and N0 is a normaliza-tion factor.

We deliberately added the variable bNN to thissystem of eight equations. The parameter bNN canbe considered as right/left asymmetric analogue ofthe double spin asymmetry aNN. According to (2),bNN = 0. However, the experimental evaluation ofbNN may serve as an indicator of some systematicuncertainties. For example, a non-zero value of bNN

can be found if both, ANN 6= ASS and cosϕL 6=cosϕR.

If all of the asymmetries are small, Eqs. (A.1)–(A.8) can be readily linearized, indicating that theabove defined asymmetries are orthogonal to eachother and are similarly normalized. Consequently,the statistical uncertainties of the experimental de-termination of the asymmetries, commonly denot-ing as ai, are mutually uncorrelated⟨

δstatai δstataj

⟩= δij/Ntot (A.9)

where δi,j is the Kronecker delta and Ntot is thetotal statistics in the measurement.

Denoting

F (A,B) = (A−B)/(A+B) (A.10)

the exact solution of Eqs. (A.1)–(A.8) can be givenas

ajN = F(√

N↑+R N↓−L +

√N↓+R N↑−L ,√

N↑−R N↓+L +

√N↓−R N↑+L

), (A.11)

abN = F(√

N↑+R N↓−L +

√N↑−R N↓+L ,√

N↓+R N↑−L +

√N↓−R N↑+L

), (A.12)

aNN = F(√

N↑+R N↓−L +

√N↓−R N↑+L ,√

N↓+R N↑−L +

√N↑−R N↓+L

), (A.13)

λj = F(

4

√N↑+R N↓+R N↑+L N↓+L ,

4

√N↑−R N↓−R N↑−L N↓−L

), (A.14)

λb = F(

4

√N↑+R N↑−R N↑+L N↑−L ,

4

√N↓+R N↓−R N↓+L N↓−L

), (A.15)

ε = F(

4

√N↑+R N↓+R N↑−R N↓−R ,

4

√N↑−L N↓−L N↑−L N↓−L

), (A.16)

bNN = F(

4

√N↑+R N↓−R N↑+L N↓−L ,

4

√N↑−R N↓+R N↑−L N↓+L

). (A.17)

Such experimentally determined asymmetries aresystematic error free as long as Eqs. (A.1)–(A.8) arevalid. The second order effects, which may resultin the possible systematic uncertainties in the mea-sured asymmetries are discussed in Section 4.4.

Eqs. (A.1)–(A.8) were derived assuming theHJET scheme of measurements in which both spinup and down beam bunches see the same target(the jet) density. More commonly, e.g. for scatter-ing of the polarized proton beams at colliders, twoasymmetries λj and λb should be replaced by fourluminosity corrections λ(↑↓)(+−) constrained by theequality

λ↑+ + λ↑− + λ↓+ + λ↓− = 0. (A.18)

In terms of the linearized equations, λ(↑↓)(+−) can,generally, be expanded via linear combinations of

22

λj , λb, and aNN. Thus, to avoid the consequent sys-tematic error in measurement of double spin asym-metry aNN, the effective dependence of luminosityon the beam spins, λ(↑↓)(+−), should be externallyevaluated [32].

For the unpolarized target (similarly for the un-polarized beam), the above solutions can be re-duced to well-known and widely used expressions

aN = F(√

N↑RN↓L,

√N↓RN

↑L

), (A.19)

λ = F(√

N↑RN↑L,

√N↓RN

↓L

), (A.20)

ε = F(√

N↑RN↓R,

√N↓LN

↑L

)(A.21)

by omitting the target polarization indexes +− inEqs. (A.12), (A.15), and (A.16), respectively. AtHJET, an unpolarized target measurements can bereasonably approximated by combining data withboth jet asymmetries, e.g. N↑+R +N↑−R →N↑R. Thisis why, the earlier analysis of HJET data [8, 9] wasbased on Eqs. (A.19)–(A.21).

If the intensity asymmetry λ is somehow known,the single spin asymmetry aN can be measured in-dependently in left or right detectors

aN = F

(N↑R1+λ

,N↓R1−λ

)= F

(N↓L

1−λ,N↑L

1+λ

)(A.22)

For example, λ can be determined, followingEqs. (A.19)–(A.21), in the part of the recoil pro-ton energy range and, than, the obtained value canbe used to determine aN for energies at which leftand right detector acceptances are significantly dif-ferent. Obviously, the method can be straightfor-wardly expanded for a more general case (A.1)–(A.8)

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