# 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
```