Impact of three-dimensional polarization profiles on spin-dependent measurements in colliding beam experiments

The Relativistic Heavy Ion Collider (RHIC) is the only collider of spin polarized protons [1]. During beam acceleration and storage the polarization P evolves. Polarization profiles, i.e., variation of polarization value versus betatron amplitudes, develop. These lead to profiles in the polarization measured in a polarimeter, i.e., variations of the polarization value versus the horizontal, vertical, or longitudinal space amplitude [2–4], and affect the observed polarization and polarization-weighted luminosity (figure of merit, FOM) in colliding beam experiments. The development of polarization profiles is the primary reason for the reduction of the average polarization [4,5]. We calculate average polarizations and figures of merit for profiles in all three dimensions, and give examples for RHIC. Like in RHIC we call the two colliding beams Blue and Yellow. We use the overbar to designate intensity-weighted averages as measured in polarimeters (e.g. P), and angle brackets to designate luminosity-weighted averages in colliding beam experiments (e.g. hPi).


I. INTRODUCTION
The Relativistic Heavy Ion Collider (RHIC) is the only collider of spin polarized protons [1]. During beam acceleration and storage the polarization P evolves. Polarization profiles, i.e., variation of polarization value versus betatron amplitudes, develop. These lead to profiles in the polarization measured in a polarimeter, i.e., variations of the polarization value versus the horizontal, vertical, or longitudinal space amplitude [2][3][4], and affect the observed polarization and polarization-weighted luminosity (figure of merit, FOM) in colliding beam experiments. The development of polarization profiles is the primary reason for the reduction of the average polarization [4,5]. We calculate average polarizations and figures of merit for profiles in all three dimensions, and give examples for RHIC. Like in RHIC we call the two colliding beams Blue and Yellow. We use the overbar to designate intensity-weighted averages as measured in polarimeters (e.g. P), and angle brackets to designate luminosity-weighted averages in colliding beam experiments (e.g. hPi).

II. COORDINATES
We use normalized horizontal, vertical, and longitudinal phase-space coordinates [6] ðx;x 0 Þ ¼ ðx; x x þ x x 0 Þ ðy;ỹ 0 Þ ¼ ðy; y y þ y y 0 Þ ðs;s 0 Þ ¼ where ðx; x 0 Þ and ðy; y 0 Þ are the horizontal and vertical phase-space coordinates, and ð x ; x Þ and ð y ; y Þ the respective lattice functions. is the rf phase, C the circumference, h the harmonic number, the slip factor, Q s the synchrotron tune, and p=p the relative momentum deviation. In the normalized coordinates the linear motion in phase space is represented by a circle on a Poincaré surface of section, and all coordinates have the dimension length.

III. DISTRIBUTIONS
As was shown by measurements in RHIC transverse intensity profiles [4,7,8] and transverse polarization profiles [2][3][4] can both be well approximated by Gaussian distributions. Similar approximations can also be used for longitudinal profiles [9,10]. The intensity distribution in phase space can then be written as where N b is the bunch intensity, and x;y;s are the rms beam sizes. The polarization P can be approximated as Pðx;x 0 ;y;ỹ 0 ;s;s 0 Þ ¼ P 0 exp À x 2 þx 02 2 2 x;P À y 2 þỹ 02 2 2 y;P À s 2 þs 02 2 2 s;P : With this dependence the polarization is a function of the normalized horizontal ( ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi x 2 þx 02 p ), vertical ( ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi y 2 þỹ 02 p ), and longitudinal ( ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 2 þs 02 p ) betatron amplitudes. The maximum polarization P 0 is reached for zero amplitudes in all dimensions. We also introduce the quantities Without any polarization profiles we have x;P ! 1, y;P ! 1, s;P ! 1, and R x ¼ R y ¼ R s ¼ 0.
In general, the phase-space distributions I [Eq. (2)] and P [Eq. (5)] are dependent on the location in a storage ring. For small enough betatron amplitudes the betatron motion can be linearized around the closed orbit and its representation on the Poincaré surface of sections is a circle at all locations in the ring. In this case, the distribution I is independent of the ring location. The situation is more complicated for polarization profiles P although it is still a reasonable assumption that the profile parameters R x;y;s are independent of the ring location for small betatron amplitudes [6,11].

IV. POLARIZATION MEASUREMENTS
Two types of polarimeters are used in RHIC to provide proton beam polarization measurements. One type of polarimeter uses a polarized atomic hydrogen gas target [12,13] and measures the average polarization over all particles: Here and in the following, the overbar denotes the intensity-weighted average. The other type of polarimeter uses an ultrathin carbon target [14][15][16] and is capable of measuring intensity and polarization profiles in both transverse directions. In a horizontal profile measurement with a thin vertical target, we have Similarly, we have for a vertical profile measurement with a thin horizontal target and for a longitudinal profile measurement A longitudinal profile can be obtained through time binning in polarimeter measurements [10].

V. LUMINOSITY
For the following we recall the luminosity formula [6,17,18] where f c is the bunch collision frequency, and the subscripts B and Y describe quantities of the Blue and Yellow beams, respectively. Note that the distributionsÎ are only 3-dimensional and also time dependent: v is the common velocity of particles in the bunch, and c the speed of light. The rms beam sizes x;y;s are functions of the time t. With neither transverse offset nor crossing angle, the luminosity can be written as [17,19] where the superscript * denotes quantities at the interaction point, and the function hðt x ; t y Þ is the hourglass factor A similar expression holds for t 2 y .

VI. AVERAGE POLARIZATIONS AND FIGURES OF MERIT IN COLLIDING BEAM EXPERIMENTS
Let us introduce polarization moments for two colliding beams as where m and n are non-negative integers and the angle brackets indicate the luminosity-weighted average over the polarization distributions. P B and P Y denote the polarizations of the Blue and Yellow beam, respectively. Important quantities for a collider experiment are average polarizations and figures of merit. For single spin measurements with the Blue and Yellow beams, respectively, these can be expressed through the polarization moments M mn as For double spin experiments we have Polarization profiles dilute the measured spin asymmetries and rescaling is needed to get the physics asymmetries. Statistical uncertainties in the measurement scale as 1= ffiffiffiffiffiffiffiffiffiffiffi FOM p , so figures of merit describe the experimental sensitivity or resolution.
We now calculate the moments M mn using the luminosity formulas in the previous section. With Eq. (15) we have where the time-dependent polarization functions in 3 spacial dimensionsP k (k ¼ m; n) are given bŷ x À kR y y 2 2 2 y À kR s s 2 2 2 s : (20) Equation (19) can be expressed as The last 4 lines of the above expression have the same form as Eq. (10).
and a similar expression for t 2 y;mn . Note that due to polarization profiles, the polarizations observed by colliding beam experiments [Eqs. (16) and (17) with the solution of Eq. (24)] generally differ from the average polarization measured by a polarimeter [Eq. (6)].

VII. SIMPLIFIED CASE
To simplify the general solution of Eq. (24) considerably, we make the following assumptions: (i) short bunches, i.e., no hourglass effects;B , s; (24) can then be written as and the cases of Eqs. (16)-(18) become The ratio between the polarization P measured by a polarimeter and the polarization hPi observed in a single spin colliding beam experiment is for the simplified case for R x ; R y ( 1: (31)
We consider the two cases of R x ¼ 0 and R x ¼ R y , both with equal polarization profiles in both beams. The former case is expected if all machines in the acceleration chain are perfectly flat, the spin direction is always vertical, and the horizontal and vertical planes decoupled. The latter case is expected for fully coupled machines. Profile parameters of R x % R y % 0:2 were observed with an ultrathin carbon target at 250 GeV in 2011 [4]. The measurements of longitudinal profile parameter R s have not been yet finalized but were found to be small [10]. Figure 1(a) shows the relative reduction (i.e. relative to the case without polarization profiles) of the average polarization P measured by a polarimeter as a function of the vertical profile parameter R y . The two cases R x ¼ 0 and R x ¼ R y are shown for R s ¼ 0; 0:01; 0:05; 0:1 each. Figure 1(b) displays the ratio of the polarization measured by a polarimeter to the polarization seen in a single spin colliding beam experiment. Figure 2 exhibits the effect of the polarization profiles on the average polarization and figure of merit in single spin colliding beam experiments. Figure 3 shows the effect in double spin experiments.
The evaluations above assume that the beam polarization profiles (namely the profile parameters R x;y;s ) do not change from the polarimeter location in the ring to the experimental collision points, even if they are separated by  ''Siberian snakes'' (magnets that flip the spin direction by 180 degrees), and by spin rotators (magnets that rotate the spin from vertical to longitudinal or horizontal, and back) [1]. This is a separate topic for careful investigation and confirmation. A dedicated simulation study is under way to better understand proton spin dynamics in RHIC, and in particular to test these assumptions [21].

IX. SUMMARY
The development of polarization profiles have an impact on both the average beam polarization and the polarization-weighted luminosity (figures of merit) in colliding beam experiments. An example is RHIC with typical profile parameter R values of 0.2 in both transverse planes. Because of the profiles, the polarization measured by polarimeters is different from the polarization observed by the experiments, and corrections depend on profiles in all three planes. For precision spin experiments polarimetry must provide measurements and monitoring of the polarization profiles in all three dimensions. We analytically derived formulas to quantify these effects for 3-dimensional Gaussian profiles, which is a valid assumption for RHIC.