# Estimation of error of A_L including detector response, May 2008 (Jan)

This set of analysis was done in preparation to PAC presentation at BNL, May 2008, ( Jan)

# Section A

Definitions:

S1,S2 - "signal" yields for 2 spins states (helicity)   "1", "2" , S1+S2=S

B1,B2 - "background" yields for 2 spins states  "1", "2" , B1+B2=B

N1=S1+B1;  N2=S2+B2;  N1+N2=N=S+B - "raw" yields  measured in real experiment

Assumptions:

• background is unpolarized, so   B1=B2=B/2
• there are 2 independent experiments:
1. measure spin independent yields for signal counts 's' and  background 'b' counts, yielding the fraction w=b/s, V(w)=w2 *(1/b +1/s) ; (e.g. M-C Pythia simulation using W and QCD events)
2. measure helicity dependent raw yields N1=S1+B1 and  N2=S2+B2 ; (e.g. theory calculation with specific assumption of AL(W+) , fixed eta & pT ranges, assumed LT & W reco efficiency)  yielding raw spin asymmetry :

ALraw=(N1-N2)/(N1+N2)= (S1-S2)/(S+B);  V(ALraw)= 4*N1*N2/N3

I used capital & small letters to distinguish this two experiments.

Problem: find statistical error of 'signal' asymmetry:

ALsig= (S1-S2)/S

Solution:

1. ALsig= (1+w) * ALraw
2. V(ALsig)= (1+w)2*V(ALraw ) + V(w)*(ALraw)2  where V(x) denotes variance of x, V(N)=N.

# Section B - theory

Model predictions of A_L for W+, W- , used files:

rb800.w+pola_grsv00_2.root,  rb800.w-pola_grsv00_2.root, rb800.w+unp_ct5m.root       rb800.w-unp_ct5m.root

LT=800/pb,

Brown oval shows approximate coverage of IST+FGT

Red diamond shows region with max A_L and ... little yield. # Section C - W reco efficiency, fixed ET using fact=1.25

From Brian, e/h algo ver 2.4, LT=800/pb.

This version uses only tower seed in bin 6-11, this is main reason efficiency is of 40%.

I'll assume in further calculation the W-reco efficiency is of 70%, flat in lepton PT>20 GeV.

Left: W-yield black=input, green - after cut 15.

Right: ratio. h->Smooth() was used for some histos. ```PT=20.6  sum1=  1223
PT=25.6  sum1=  1251  sum2= 473 att=1/2.6
PT=30.6  sum1=   987  sum2= 406 att=1/2.4
PT=35.6  sum1=   771  sum2= 343 att=1/2.2
PT=40.6  sum1=   372  sum2= 166 att=1/2.2
PT=45.6  sum1=    74  sum2=  16 att=1/4.6
```

# Section D - QCD reco efficiency,fixed ET using fact=1.25

From Brian, e/h algo ver 2.4, LT=800/pb. h->Smooth() was used for some histos.

This version uses only tower seed in bin 6-11

Left: QCD-yield black=input, green - after cut 15.

Right: ratio= QCD attenuation, not the a;go gets ~3x  'weaker' at PT =[34-37] GeV , exactly where we need it the most PT averaged attenuation of QCD events

```PT=20.6  sum1=2122517
PT=25.6  sum1=528917  sum2=3992 att=1/133
PT=30.6  sum1=135252  sum2= 736 att=1/184
PT=35.6  sum1= 38226  sum2= 320 att=1/120
PT=40.6  sum1= 11292  sum2= 127 att=1/89
PT=45.6  sum1=  3153  sum2=  41 att=1/77
```

# Section E -  QCD/W  ratio after e/h cuts, algo ver 2.4,fixed ET using fact=1.25

From Brian. h->Smooth() was used for some histos.

Left: final yield of QCD events (blue) and W-events (green) after e/h algo.

Right: ratio.

I'll assume w=b/s is better than the red line, a continuous ET dependence:

w(pt=20)=10
w(pt=25)=1.
w(pt=40)=0.5 ```PT=25.6  sum1=  3992  sum2= 473 att=1/8.4
PT=30.6  sum1=   736  sum2= 406 att=1/1.8
PT=35.6  sum1=   320  sum2= 343 att=1/0.9
PT=40.6  sum1=   127  sum2= 166 att=1/0.8
PT=45.6  sum1=    41  sum2=  16 att=1/2.5
```

# Section A) Theoretical calculations:

Assumed LT=800/pb

fpol=new TFile("rb800.w+pola_grsv00_2.root");   <--GRSV-VAL (maximal W polarization)
funp=new TFile("rb800.w+unp_ct5m.root");

histo 215

Total W+ yield for lepton ET[20,45] GeV and eta [1,2] is of 7101 for unpolarized cross section and of -2556 for the helicity dependent part.

Assuming 70% beam polarization measured spin dependent asymmetry:

eps_L= P* del/sum= -0.25 +/-0.012

(assuming err(eps)=1/sqrt(sum) )

Fig 1 , W+ : top row - unpol & pol cross section GRSV-VAL (maximal W polarization),

bottom left: integrated over eta, black=unpol, red=pol

bottom right: asy=P *pol/unpol vs. lepton PT, green=fit Total W- yield for lepton ET[20,45] GeV and eta [1,2] is of 5574 for unpolarized cross section and of +2588 for the helicity dependent part.

fpol=new TFile("rb800.w-pola_grsv00_2.root");
funp=new TFile("rb800.w-unp_ct5m.root");

histo 215

Assuming 70% beam polarization measured spin dependent asymmetry:

eps_L= P* del/sum= +0.325 +/-0.013

Fig 2.  W- GRSV-VAL (maximal W polarization) # Section  B) Folding in e+,e- reconstruction and QCD background

Assumptions:

1. LT=800/pb
2. beam pol P=0.7
3. e+,e- reco efficiency is 70%, no PT dependence
4. QCD background to W contamination, after e/h algo no spin dependece
5.  lepton PT range w=backg/signal 20-25 GeV 5.0 +/- 10% 25-30 GeV 1.0  +/- 10% 30-35 GeV 0.8 +/- 10% 35-40 GeV 0.7 +/- 10% 40-45 GeV 0.6 +/- 10%

Formulas:

• Theory yields : unpol=sig0(PT) & pol=del(pt) , S1=(sig0+del)/2, S2=(sig0-del)/2
• Measured yields N1, N2 , for 2 helicity states
• N1(PT)= eff*[ sig0+del + sig0 * w) /2
• N2(PT)= eff*[ sig0-del + sig0 * w) /2
• ALraw(PT)= P* (N1-N2)/ (N1+N2)
• V(ALraw(PT))= 1/(N1+N2) <-- variance
• ALsig(PT)= (1+w(PT))* ALraw(PT)
• V(ALsig)= (1+w)2*V(ALraw ) + V(w)*(ALraw)2
• dil=1+w
• QA= |(ALsig)/err(ALsig)| - must be above 3 for meaningful result

# Fig 3,  Results for W + GRSV-VAL (maximal W polarization)

Left : N1(PT)=red, N2(PT) blue. Right: reconstructed signal AL ```ipt=0  y-bins=[41,50] unpol=1826.5 pol=-327.3  AL=-0.125 +/- 0.0234   QA=1.8
B2S=5.0 , N1=3721 N2=3950 ALraw=-0.021 +/- 0.011, dil=6.00  ALsig=-0.125 +/- 0.069

ipt=1  y-bins=[51,60] unpol=1403.0 pol=-265.8  AL=-0.133 +/- 0.0267   QA=2.9
B2S=1.0 , N1=889 N2=1075 ALraw=-0.066 +/- 0.023, dil=2.00  ALsig=-0.133 +/- 0.046

ipt=2  y-bins=[61,70] unpol=1233.7 pol=-384.7  AL=-0.218 +/- 0.0285   QA=4.7
B2S=0.8 , N1=643 N2=912 ALraw=-0.121 +/- 0.025, dil=1.80  ALsig=-0.218 +/- 0.047

ipt=3  y-bins=[71,80] unpol=1811.9 pol=-1041.9  AL=-0.403 +/- 0.0235   QA=10.0
B2S=0.7 , N1=713 N2=1443 ALraw=-0.237 +/- 0.022, dil=1.70  ALsig=-0.403 +/- 0.040

ipt=4  y-bins=[81,90] unpol=808.8 pol=-525.2  AL=-0.455 +/- 0.0352   QA=8.1
B2S=0.6 , N1=269 N2=637 ALraw=-0.284 +/- 0.033, dil=1.60  ALsig=-0.455 +/- 0.056

sum2=7084.000000 sum3=-2544.850098 asy=-0.251
```

# Fig 4,  Results for W - GRSV-VAL (maximal W polarization) ```ipt=0  y-bins=[41,50] unpol=1239.1 pol=490.8  AL=0.277 +/- 0.0284   QA=3.2
B2S=5.0 , N1=2774 N2=2430 ALraw=0.046 +/- 0.014, dil=6.00  ALsig=0.277 +/- 0.086

ipt=1  y-bins=[51,60] unpol=1452.7 pol=641.2  AL=0.309 +/- 0.0262   QA=6.6
B2S=1.0 , N1=1241 N2=792 ALraw=0.154 +/- 0.022, dil=2.00  ALsig=0.309 +/- 0.047

ipt=2  y-bins=[61,70] unpol=1426.5 pol=689.9  AL=0.339 +/- 0.0265   QA=7.5
B2S=0.8 , N1=1140 N2=657 ALraw=0.188 +/- 0.024, dil=1.80  ALsig=0.339 +/- 0.045

ipt=3  y-bins=[71,80] unpol=1135.3 pol=596.1  AL=0.368 +/- 0.0297   QA=7.6
B2S=0.7 , N1=884 N2=467 ALraw=0.216 +/- 0.027, dil=1.70  ALsig=0.368 +/- 0.049

ipt=4  y-bins=[81,90] unpol=313.8 pol=166.4  AL=0.371 +/- 0.0565   QA=4.3
B2S=0.6 , N1=234 N2=117 ALraw=0.232 +/- 0.053, dil=1.60  ALsig=0.371 +/- 0.086

sum2=5567.280273 sum3=2584.398926 asy=0.325
```

# Another set of results for W+, W- with 2 x worse B/S (the same PT dependence).

Fig 5.  W+ GRSV-VAL (maximal W polarization) Fig 6.  W- GRSV-VAL (maximal W polarization) # Cross check of my code vs. A_L from FGT proposal

Input from RHICBOS , GRSV-VAL model:

```  if(Wsign==1) {             fpol=new TFile("rb800.w+pola_grsv00_2.root");              funp=new TFile("rb800.w+unp_ct5m.root");              WPM="W+ ";     } else {           fpol=new TFile("rb800.w-pola_grsv00_2.root");           funp=new TFile("rb800.w-unp_ct5m.root");             WPM="W- ";     }
```

The sign of pol cross section from RHICBOS has reversed convention, I have changed  it to Medison convention.

hpol->Scale(-1.);

# Fig 1 W-

from my macro, compare bottom right to blue from fig 2a # Fig 3 W+,

from my macro, compare bottom right to blue from fig 2b # Calculation of error of A_L for LT=300/pb,   for W± , eta=±[1,2],  PT>20 GeV/c • Input from RHICBOS , GRSV-STD model:
• The sign of pol cross section from RHICBOS has reversed convention, I have changed  it to Medison convention.
hpol->Scale(-1.);
• beam Pol=70%
• e+,e- reco efficiency is 70%, no PT dependence
• QCD background to W signal (B/S) contamination, after e/h algo no spin dependece
A B C
ET_range
(EEMC 3x3_cluster)
assumed w=backg/signal QCD eve
suppression needed for(B)
20-25 GeV  5.0 +/- 20%  - (for W+ or W-)
25-30 GeV  1.0  +/- 20% 1/539 or 1/520
30-35 GeV  0.8 +/- 20% 1/196 or 1/169
35-40 GeV  0.7 +/- 20% 1/43  or 1/ 69
40-45 GeV  0.6 +/- 20% 1/33 or 1/86
45-50 GeV  0.5 +/- 20% 1/119 or 1/289
*) based on full Pythia+GSTAR+BFC simulations of QCD events,
(study 1 of S/B, A_L, LT=800/pb (Jan) section E)
after 3x3 EEMC cluster is found
• eta of the lepton [1,2] (polarized beam is heading toward Endcap)

Formulas:

• Theory yields : unpol=sig0(PT) & pol=delL(pt) , S1=(sig0+delL)/2, S2=(sig0-delL)/2
• Measured yields N1, N2 , for 2 helicity states
• N1(PT)= eff*[ sig0 +P*delL + sig0 * w) /2
• N2(PT)= eff*[ sig0 -P*delL + sig0 * w) /2
• ALraw(PT)= 1/P (N1-N2)/ (N1+N2)
• V(ALraw(PT))= 1/P2 1/(N1+N2) <-- variance
• ALsig(PT)= (1+w(PT))* ALraw(PT)
• V(ALsig)=1/P2 (1+w)2*V(ALraw ) + V(w)*(ALraw)2
• dil=1+w
• QA= |(ALsig)/err(ALsig)| - must be above 3 for meaningful result

# Table 1,  W+ , eta=[1,2] , LT=300/pb

1 reco EMC reco  20-25 25-30 30-35 35-40 40-45 45-50 ET GeV AL/err 2 3 4 5 6 7 8 9 10 reco W+ yield helicity: S1, S2 reco W+   unpol yield QCD Pythia accepted yield assumed B/S reco signal  AL+err AL dilution: 1+B/S QCD yield  w/ EMC cluster needed QCD  suppression *) 242 ,236 479 2397 5.0 0.019 +/-0.160 0.1 6.00 192 ,176 368 368 1.0 0.062 +/-0.105 0.6 2.00 198343 1/539 190 ,133 323 259 0.8 0.249 +/-0.109 2.3 1.80 50719 1/196 333 ,141 475 332 0.7 0.576 +/-0.098 5.9 1.70 14334 1/43 151,61 212 127 0.6 0.606 +/-0.132 4.6 1.60 4234 1/33 13,6 19 9 0.5 0.457 +/-0.393 1.2 1.50 1182 1/119

# Fig 3, W-, eta=[1,2] , ideal detector # ## Table 2,  W- , eta=[1,2] , LT=300/pb

1 reco EMC  reco  20-25 2.0   25-30 3.8 520 ET GeV AL/err 2 3 4 5 6 7 8 9 10 reco W+ yield  helicity: S1, S2 reco W-   unpol yield QCD Pythia accepted yield assumed  B/S reco signal  AL+err AL dilution:  1+B/S QCD yield  w/ EMC cluster needed QCD  suppression *) 116,208 325 1626 5.0 -0.403 +/-0.205 6.00 133,248 381 381 1.0 -0.431 +/-0.112 2.00 198343 126,247 374 299 0.8 -0.461 +/-0.107 1.80 50719 97,200 298 208 0.7 -0.495 +/-0.115 1.70 14334 26,55 82 49 0.6 -0.506 +/-0.203 1.60 4234 2,5 8 4 0.5 -0.521 +/-0.617 1.50 1182

# Fig 5, W+, eta=[-2,-1] , ideal detector # ## Table 3,  W+ , eta=[-2,-1] , LT=300/pb

1 reco EMC  reco  20-25 2.5   25-30 3.4 531 ET GeV AL/err 2 3 4 5 6 7 8 9 10 reco W+ yield  helicity: S1, S2 reco W-   unpol yield QCD Pythia accepted yield assumed  B/S reco signal  AL+err AL dilution:  1+B/S QCD yield  w/ EMC cluster needed QCD  suppression *) 316,168 484 2424 5.0 0.436 +/-0.175 6.00 235,137 373 373 1.0 0.375 +/-0.111 2.00 198343 189,132 322 258 0.8 0.252 +/-0.109 1.80 50719 255,223 479 335 0.7 0.094 +/-0.085 1.70 14334 109,102 212 127 0.6 0.048 +/-0.124 1.60 4234 10,9 19 9 0.5 0.027 +/-0.393 1.50 1182

# Fig 7, W-, eta=[-2,-1] , ideal detector # ## Table 3,  W+ , eta=[-2,-1] , LT=300/pb

1 reco EMC  reco  20-25 0.1 515 25-30 0.3 540 ET GeV AL/err 2 3 4 5 6 7 8 9 10 reco W+ yield  helicity: S1, S2 reco W-   unpol yield QCD Pythia accepted yield assumed  B/S reco signal  AL+err AL dilution:  1+B/S QCD yield  w/ EMC cluster needed QCD  suppression *) 157,151 308 1544 5.0 0.024 +/-0.199 6.00 795943 187,180 367 367 1.0 0.029 +/-0.105 2.00 198343 187,179 367 293 0.8 0.033 +/-0.100 1.80 50719 151,144 295 206 0.7 0.033 +/-0.108 1.70 14334 41,40 81 49 0.6 0.026 +/-0.200 1.60 4234 3,3 7 3 0.5 0.012 +/-0.624 1.50 1182

# study 5 charge sign discrimination # Fig 1 charge reco misidentification (details), M-C simulations

FGT 6 identical disk have active area Rin=11.5 cm, Rout=37.6 cm;
Z location: 70,80,90, 100,110,120 cm with respect to STAR ref frame. # Fig 2

Unpolarized yield for W+, RHICBOS # Fig 3

Unpolarized yield for W-, RHICBOS # Fig 4

Unpolarized & pol yield for W+, RHICBOS, ideal detector # Fig 5

Unpolarized & pol yield for W-, RHICBOS, ideal detector # Estimated statistical uncertainty for AL for charge leptons from W decay reconstructed in the Endcap

Fig 1 - Ideal detector Fig 2 - Realistic detector efficiency & hadronic background, ideal charge reco

 LT=300/pb LT=100/pb kinematics W+, forward 8.6 5.3 W-, forward 6.7 3.9 W+, backward 5.1 3.0 W-, backward 0.3 0.2

 LT=300/pb LT=100/pb kinematics W+, forward 8.6 5.3 W-, forward 8.2 4.7 W+, backward 5.9 3.4 W-, backward 3.9 2.3 Fig 3 - Realistic detector efficiency, hadronic background, and charge reco missing

# study 7 revised for White Paper, AL(eta), AL(ET) , (Jan)

Plots show AL for W+, W- as function of ET (fig1) and eta (fig2,3)

I assumed beam pol=70%, electron/positron reco off 70%, QCD background included, no vertex cut (as for all earlier analysis).

For AL(ET)  I integrated over eta [-2,-1] or [1,2] and assumed the following  B/S(ET) = 5.0 for ET>20 GEV, 1.0 for ET>25, 0.9 for ET>30,....

For AL(Eta)  I integrated over ET>25 GeV and assumed a constant in eta & ET B/S=0.8.

Fig 1. AL(ET). Only Endcap coverage is shown. ( EPS.zip ) Fig 2. AL(Eta) . Only Endcap coverage is shown. ( EPS.zip ) Fig 3. AL(Eta) has continuous eta-axis, binning is exactly the same as in Fig 2.  It includes Endcap & Barrel coverage

(PS.gz) generated by take2/do5.C, doAll21() # study 8 no QCD backg AL(eta), AL(ET), also rapidity (Jan)

Plots show AL for W+, W- as function of ET (fig1,2) and eta (fig3)

I assumed beam pol=70%, electron/positron reco off 70%, no vertex cut (as for all earlier analysis).

NO QCD background dilution.

For AL(ET)  I integrated over eta [-2,-1] , [-1,+1], or [1,2] and assumed no background

For AL(Eta)  I integrated over ET>25 GeV and assumed no background

Fig 1 ( PS.zip ) Fig 2 ( PS.zip )

Accounted for 2 beams at mid rapidity. Fig 3 # The goal is to provide overall STAR sensitivity for LT=100/pb & 300/pb.

Common assumptions:

• beam pol=70%
• W reco efficiency 70%
• no losses due to the vertex cut
• integrated over ET>20 GeV

This page is tricky, different assumptions/definitions are used for different eta ranges.

# A) Forward rapidity: eta  range [+1,+2],  (shown on fig 1a+b in study 7 revised for White Paper, AL(eta), AL(ET) , (Jan))

determine the degree to which we can measure an asymmetry different from zero.

• QCD background added, B/S changes with ET.
• sensitivity  is defined as 3 x stat_error of reco AL
• method: fit constant to 'data', use 3x error of the fit
 LT=100/pb LT=300/pb W+, forward 0.27 0.15 W-, forward 0.30 0.18

I fit constant to the black points what is equivalent to taking the weighted average.

The value of the average is zero but the std dev of the average tells us sigma(measured AL).
In the table I'm reporting 3 x this sigma.

E.g.   for W+ forward we could distinguish on 3 sigma level  between 2 models of AL  if the values of AL differ by at least of 0.27 if we are given LT=100/pb.

# B) Mid-rapidity: eta  range [-1,+1],  (shown on fig 2a+b in study 8 no QCD backg AL(eta), AL(ET), also rapidity (Jan) )

determine the ratio of  difference of DNS-MIN and DNS-MAX to sigma(measured AL)

• account for 2x larger yield due to 2 polariazed beams
• QCD background NOT added
• sensitivity  is defined as ratio
• avr difference of DNS-MIN and DNS-MAX is used
 avr(ALMIN-ALMAX) LT=100/pb LT=300/pb W+, mid 0.15 13 21 W-, mid 0.34 13 22

C) Backward rapidity: eta  range [-2,-1],  (shown on fig 1c+d in study 7 revised for White Paper, AL(eta), AL(ET) , (Jan))

determine the ratio of  difference of DNS-MIN and DNS-MAX to sigma(measured AL)

• QCD background added, B/S changes with ET.
• sensitivity  is defined as ratio
• avr difference of DNS-MIN and DNS-MAX is used
 avr(ALMIN-ALMAX) LT=100/pb LT=300/pb W+, backward 0.10 1 2 W-, backward 0.5 5 9

# D) Mid-rapidity: eta  range [-1,+1], QCD background added (fig shown below, PS.zip )

determine the ratio of  difference of DNS-MIN and DNS-MAX to sigma(measured AL)

• account for 2x larger yield due to 2 polariazed beams
• QCD background added, using B/S(ET) from M-C study at forward rapidity- it is the best what we can do today
• sensitivity  is defined as ratio
• avr difference of DNS-MIN and DNS-MAX is used
 avr(ALMIN-ALMAX) LT=100/pb LT=300/pb W+, mid 0.15 7 11 W-, mid 0.34 10 15 # Projection of STAR sensitivity for AL for W+,W-  at mid rapidity for LT=10/pb and pol=60%

Common assumptions:

# Fig 1,  no QCD background ( PS.zip )

Dashed area denotes pt-averaged statistical error of STAR measurement. # study 9b sensitivity at mid rapidity LT=10/pb Pol=50% or 60%

Projection of STAR sensitivity for AL for W+,W-  at mid rapidity for LT=10/pb and pol=60% or 50%

Common assumptions:

• LT=10 pb -1
• beam pol=60% or 50%
• W reco efficiency 70%
• no losses due to the vertex cut
• integrated over eta [-1,+1]
• see remaining details in
•
 beam pol  avr AL(W+)THEORY=0.35  avr AL(W-)THEORY=0.15 . sig ALSTAR STAR signifcance sig ALSTAR STAR signifcane 50% 0.092 3.8 sigma 0.18 0.8 sigma 60% 0.077 4.6 sigma 0.15 1 sigma