Single spin asymmetry using cross formula

Single Spin asymmetry using cross formula

Equation one shows the cross-formula used to calculate the single spin asymmetry.

AP=[sqrt(N(L,up)N(R,down))-sqrt(N(L,down)N(R,up))]/[sqrt(N(L,up)N(R,down))+sqrt(N(L,down)N(R,up))]

Equation 1

where N is a particle yield, L(eft) and R(ight) indicate the side of the polarised beam to which the particle is produced and arrows indicate the polarisation direction of the beam. Equation one cancels acceptance and beam luminosity and allows simply the raw yields to be used for the calculation. The asymmetry can be calculated twice; once for each beam, summing over the polarisation states of the other beam to leave it "unpolarised". I previously used only particles produced at forward η when calculating the blue beam asymmetry, and backward η for yellow, but I now sum over the full η range for each. Equations two and three give the numbers for up/down polarisation for blue (westward at STAR) and yellow (eastward) beams respectively in terms of the contributions from the four different beam polarisation permutations, and these permutations are related to spin bits numbers in table one.

N(blue,up)=N(upUp)+N(downUp),N(blue,down)=N(downDown)+N(upDown)

Equation 2

N(yellow,up)=N(upUp)+N(upDown),N(yellow,down)=N(downDown+N(downUp)

Equation 3

(in e.g. N(upUp), The first arrow refers to yellow beam polarisation, the second to blue beam.)

Beam polarisation		4-bit spin bits
Yellow	Blue	4-bit spin bits
Up	Up	5
Down	Up	6
Up	Down	9
Down	Down	10
Table 1

The raw asymmetry is calculated for each RHIC fill, then divided by the polarisation for that fill to give the physics asymmetry. Final polarisation numbers (released December 2007) are used. The error on the raw asymmetry is calculated by propagation of the √(N) errors calculated for each particle yield. The final asymmetry error incorporates the polarisation error (statistical and systematic errors summed in quadrature). The fill-by-fill asymmetries for each K⁰_S and Λ for each beam are shown in figures one and two. Anti-Λ results shall be forthcoming. An average asymmetry is calculated by performing a straight line χ² fit through the fill-by-fill values with ROOT. Table one summarises the asymmetry results. The asymmetry error is the error from the ROOT fit and is statistical only. All fits give a good χ² per degree of freedom and are consistent with zero within errors.

$Blue beam K0s A_{N}$

Figure 1a: K⁰_S blue beam asymmetry

$Yellow beam K0s A_{N}$

Figure 1b: K⁰_S yellow beam asymmetry

$Blue beam Λ A_{N}$

Figure 2a: Λ blue beam asymmetry

$Yellow beam Λ A_{N}$

Figure 2b: Λ yellow beam asymmetry

The above are summed over the entire p_T range available. I also divide the data into different transverse momentum bins and calculate the asymmetry as a function of p_T. Figures three and four show the p_T-dependent asymmetries. No p_T dependence is discernible.

$K^{0}_{S} blue beam A_{N} as a function of p_{T}$

Figure 3a: K⁰_S p_T-dependent blue beam A_N

$K^{0}_{S} yellow beam A_{N} as a function of p_{T}$

Figure 3b: K⁰_S p_T-dependent yellow beam A_N

$Λ blue beam A_{N} as a function of p_{T}$

Figure 4a: Λ p_T-dependent blue beam A_N

$Λ yellow beam A_{N} as a function of p_{T}$

Figure 4b: Λ p_T-dependent yellow beam A_N