Longitudinal and transverse spin transfer to $Λ$ and $\barΛ$ hyperons in p+p collisions at STAR
Qing-Hua Xu




§ INTRODUCTION


The polarizations of Λ and Λ̅ hyperons have been studied extensively in various aspects of spin effects in high energy reactions due to the self spin-analyzing parity violating decay.


In particular, Λ polarization transferred from polarized lepton or hadron beams (usually referred to as “spin transfer") in different reactions provides a natural connection to polarized fragmentation functions and the polarized parton densities of the nucleon. 
A number of measurements have been made in polarized lepton-nucleon deep inelastic scattering (DIS), and in polarized hadron-hadron collisions in the STAR experiment at RHIC.

As the s(s̅) quark plays a dominant role in Λ (Λ̅) hyperon's spin content, the measurements of spin transfer to Λ (Λ̅) hyperon provide a natural connection to the polarized parton distribution of (anti-)strange quarks in nucleon, which are not well constrained yet in experiment.  



In particular, the transverse spin transfer to Λ (Λ̅) in hadron-hadron collisions provides a natural connection to transversity of (anti-)strange quarks through polarized fragmentation functions <cit.>. 
Similarily, the longitudinal spin transfer to Λ (Λ̅) hyperons in hadron-hadron collisions provides sensitivity to the helicity distribution of (anti-)strange quarks <cit.>.


In this contribution, we report the first measurement of transverse spin transfer to Λ and Λ̅ hyperons in transversely polarized proton-proton collisions at √(s)=200 GeV with the STAR experiment at RHIC <cit.>. 
Then we report an improved STAR measurement of the longitudinal spin transfer D_LL to Λ and Λ̅ hyperons in longitudinally polarized proton–proton collisions at √(s) = 200 GeV <cit.>.



§ FIRST MEASUREMENT OF TRANSVERSE SPIN TRANSFER TO Λ AND Λ̅ HYPERONS AT RHIC


The transverse spin transfer, D_TT, to the Λ in proton-proton collisions is defined as:

    D_TT≡dσ^(p^↑p →Λ^↑X)-dσ^(p^↑p →Λ^↓X)/dσ^(p^↑p →Λ^↑X)+dσ^(p^↑p →Λ^↓X)
      =
      dδσ^Λ/dσ^Λ,

where ↑ (↓) denotes the positive (negative) transverse polarization direction of the particles and δσ^Λ is the transversely polarized cross section.
Within the factorization framework,  δσ^Λ can be written as the convolution of parton transversity, polarized partonic cross section and the polarized fragmentation function <cit.>.



The polarization of Λ (Λ̅) hyperons, P_Λ (Λ̅), can be measured from the angular distribution of the final state particles via their weak decay channel Λ→ pπ^- (Λ̅→p̅π^+),

    dN/ dcosθ^* ∝
      A
      (1+α_Λ (Λ̅)P_Λ (Λ̅)cosθ^*),

where A is the detector acceptance varying with θ^* as well as other observables, α_Λ (Λ̅) is the weak decay parameter, and θ^* is the angle between the Λ (Λ̅) polarization direction and the (anti-) proton momentum in the Λ (Λ̅) rest frame. 
For the D_TT measurements, the transverse polarization direction of the outgoing fragmenting parton is used to obtain θ^*.
Since there is a rotation along the normal direction of the scattering plane between the transverse polarization directions of incoming and outgoing quarks <cit.>, the momentum direction of the outgoing parton is required and the reconstructed jet axis adjacent to the Λ (Λ̅) is used as the substitute for the direction of the outgoing fragmenting quark.
Previously, only the spin transfer along the normal direction of the Λ production plane, D_NN, was measured with fixed target proton-proton collisions by the E704 Collaboration at Fermilab <cit.>.

The data for the D_TT measurement here were collected at RHIC with the STAR experiment in the year 2012 with transversely polarized proton beams, which amount to an integrated luminosity of 18 pb^-1. 

The average transverse polarizations for the two beams were 58% and 64%.


The primary detector sub-system is the Time Projection Chamber (TPC) <cit.>, which provides tracking for charged particles in the pseudo-rapidity range of |η|<1.3 with full azimuthal coverage.
The Barrel Electromagnetic Calorimeter (BEMC) <cit.> and Endcap Electromagnetic Calorimeter (EEMC) <cit.> were used in generating the pimary jet trigger information at STAR.
The data sample for this analysis were recorded with jet-patch (JP) trigger conditions which required a transverse energy deposit E_T in BEMC or EEMC patches (each covering Δη×Δϕ = 1×1) exceeding certain thresholds.

The Λ and Λ̅ candidates were reconstructed from their dominant weak decay channels, Λ→ pπ^- and Λ̅→p̅π^+, with a branching ratio of 63.9% <cit.>.
The selection procedure of Λ and Λ̅ candidates is based on the topology of the weak decay in a similar way as the previous longitudinal spin transfer measurement  in Ref. <cit.>.

In addition, reconstructed jets were employed to obtain the momentum direction of the fragmenting quark and thus the transverse polarization direction of hyperons for the D_TT measurement.
Jets were reconstructed using the anti-k_T algorithm <cit.> and
then the association between Λ (Λ̅) candidates and the adjacent reconstructed jet was made by constraining the radius,
Δ R=√((Δη)^2+(Δϕ)^2)<0.6,
between the Λ (Λ̅) momentum direction and the jet axis in η-ϕ space.

The bin counting method was used to obtain the raw yields of Λ and Λ̅ candidates. 
The signal mass windows were chosen to be about 3σ of the mass width.

The residual background fractions r were estimated by the side-band method and found to be 7%∼10%.
The background subtraction was done at the spin transfer level as D_TT =
 
  ( D_TT^raw-rD_TT^bkg )/(1-r), with D_TT^raw and D_TT^bkg defined as the spin transfer for signal and side-band regions.



To minimize the systematics associated with acceptance and relative luminosity, D_TT for hyperon candidates has been extracted from the asymmetry in small cosθ^* intervals using<cit.>


    D_TT = 
      1/ α P_beam<cosθ^*>  √(N^↑(cosθ^*)N^↓(-cosθ^*) )
      -√(N^↓(cosθ^*)N^↑(-cosθ^*) ) / √(N^↑(cosθ^*)N^↓(-cosθ^*) )
      +√(N^↓(cosθ^*)N^↑(-cosθ^*) ) ,

where α_Λ=0.642±0.013 <cit.>, α_Λ̅ = -α_Λ, P_beam is the beam polarization and ⟨cosθ^*⟩ denotes the average value in the cosθ^* interval.
N^↑(↓) is the Λ yield in the corresponding cosθ^* bin when the proton beam is upward (downward) polarized.
The relative luminosity between N^↑ and N^↓ is cancelled in this cross-ratio asymmetry. 
The acceptance is also cancelled as the acceptance in a small cosθ^* interval is expected to remain the same when flipping the beam polarization <cit.>.





The STAR results for D_TT versus hyperon p_T are shown in Fig. <ref> for Λ and Λ̅ at both positive and negative η regions relative to the polarized beam in proton-proton collisions at √(s)=200 GeV. 
These results provide the first measurements on transverse spin transfer of hyperons in proton-proton collisions at a high energy of √(s)=200 GeV.
The D_TT results for Λ and Λ̅ are consistent with zero within uncertainties. 
The data cover p_T up to 7 GeV/c, where D_TT = 0.031 ± 0.033(stat.) ± 0.008(sys.) for Λ and D_TT = -0.034 ± 0.040(stat.) ± 0.009(sys.) for Λ̅ at <η> = 0.5 and <p_T> = 6.7 GeV/c. 
The data are comparable to a model estimation calculated at <η>=± 0.5, which is available for RHIC energy, with simple assumptions of the transversity (using the DSSV helicity distribution as input) and the SU6 picture of fragmentation functions <cit.>.
Since strange and anti-strange quarks are expected to carry a significant part of the spins of Λ and Λ̅, the measurements of transverse spin transfer to them can provide sensitivity to the transversity distribution of strange quarks.
Knowledge on transversity of valence quarks has been learned mostly from DIS experiments,
but the transversity distribution of (anti-)strange quarks is still poorly known in experiment <cit.>. 




§ IMPROVED MEASUREMENT OF LONGITUDINAL SPIN TRANSFER TO Λ (Λ̅) HYPERONS AT STAR


The longitudinal spin transfer, D_LL, to Λ  hyperon in polarized proton-proton collisions is defined as:

    D_LL≡dσ_p^+p →Λ ^+ X-dσ_p^+p →Λ ^-X/dσ_p^+p →Λ ^+ X+dσ_p^+p →Λ ^-X,

where the superscripts + or - denote the helicity of the beam proton or the Λ hyperon. 


The data sample for this analysis was recorded with the STAR experiment in 2009 and corresponds to an integrated luminosity of about 19 pb^-1 with an average longitudinal beam polarization, P_beam, of 57%± 4.7%.
The figure-of-merit of this data sample, P_beam^2ℒ, is about twelve times higher than that of the previous STAR measurement of D_LL <cit.>.
Similar to the 2012 data sample, the events here were recorded also with a jet-patch trigger condition in the BEMC or EEMC.


The reconstruction of Λ (Λ̅) is similar to that done for the D_TT analysis. 
The detailed selection cuts were tuned in p_T intervals so as to keep as much signal as possible, while keeping the residual background at an acceptable level of about 10%.
In addition, the Λ and Λ̅ baryons were required to be associated with a reconstructed jet that satisfied the trigger conditions.
Here, the jet sample was reconstructed using the mid-point cone algorithm, as in several previous STAR jet analyses <cit.>.

For this association, no significant difference is expected either for anti-kt or mid-point cone jet algorithm.
The association required that the reconstructed Λ or Λ̅ candidate was within the jet cone of radius ΔR =√((Δη)^2+(Δϕ)^2) <0.7.  



Following our previous D_LL measurement <cit.>, the longitudinal spin transfer D_LL was extracted from the asymmetry:

    D_LL=1/α_Λ(Λ̅) P_beam<cosθ^*>N^+ -R N^-/N^+ + R N^-,


where N^+ (N^-) is the number of Λ or Λ̅ candidates in the cosθ^* interval when the beam is positively (negatively) polarized.
<cosθ^*> is the average value of cosθ^* in this interval, but different from the case for D_TT, here θ^* is defined as the angle between Λ (Λ̅) momentum direction and the (anti-) proton momentum in the Λ (Λ̅) rest frame.
R denotes the relative luminosity ratio for these two beam polarization states and has been measured with Beam-Beam Counters <cit.>.













Figure <ref> shows our new STAR results on the spin transfer D_LL to the Λ and to the Λ̅ as functions of p_T for positive η.

The longitudinal spin transfer is found to be D_LL = -0.036 ± 0.048 (stat) ± 0.013 (sys) for Λ and D_LL = 0.032 ± 0.043 (stat) ± 0.013 (sys) for Λ̅ at <η> = 0.5 and <p_T> = 5.9 GeV/c.
The data are compared with the theory expectations from Ref. <cit.>, with D_LL for Λ and Λ̅ combined, and from Ref. <cit.>, with D_LL separately for Λ and for Λ̅.
The data are still consistent with zero within uncertainty, and also provide no evidence for a difference between  Λ and Λ̅ D_LL.
However, the data are below the DSV scenario 3 theory expectation assuming that the quark polarized fragmentation functions are flavor-independent <cit.>.





§ SUMMARY AND OUTLOOK


In summary, we report the first measurement of the transverse spin transfer, D_TT, to Λ and Λ̅ in transversely polarized proton-proton collisions at √(s) = 200 GeV at RHIC-STAR, and an improved measurement of the longitudinal spin transfer, D_LL, to Λ hyperons and Λ̅ anti-hyperons in longitudinally polarized proton–proton collisions at √(s) = 200 GeV from STAR.

The results for D_TT are found to be consistent with zero for Λ and Λ̅ within uncertainties, and are also consistent with model predictions.

The D_LL data do not provide evidence for a non-vanishing spin transfer within uncertainties, but tend to be below a model expectation based on the extreme assumption on polarized fragmentation functions.

STAR is expanding its acceptance by series of detector upgrades including an ongoing upgrade to the inner sectors of the TPC and the proposed forward detector upgrades <cit.>,  which would enable a number of interesting hyperon polarization measurements, including spin transfers D_TT, D_LL and other hyperon polarization measurements such as the induced transverse polarization in unpolarized proton-proton collisions, at very forward rapidities in proton-proton collisions.

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