How to Get ${\Delta}G(x)$ from Direct Photon + Jet Measurements Chris Allgower¹
$^1\,$High Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439 Talk for RHIC Spin Discussion Group, March 17 1998

Experimental Side:

What You Measure:

Counts in Luminosity Monitors for each Beam State L_ij

Photon Angle ${\theta}_{\gamma}$

Photon Energy $E_{\gamma} = P_{\gamma}$

Jet Angle ${\theta}_{\mbox{\large jet}}$

Jet Momentum $P_{\mbox{\large jet}}$

$\sqrt{s}$ of the Proton Beams

Proton Beam Polarization P_b

(Note - P_b may differ for different beams and/or beam states.)

Steps in the Process:

Step 0.

Design and Build the Experiment

Step 1.

For each event that passes cuts (i.e. good jet, $E_{\gamma}$ above threshold, coplanarity of jet and photon, etc., etc., etc.) calculate

$$P_T = P_{\gamma} \sin {\theta}_{\gamma} = P_{\mbox{\large jet}} \sin {\theta}_{\mbox{\large jet}}$$

$${\eta}_{\gamma} = -\ln \left( \tan \frac{{\theta}_{\gamma}}{... ... -\ln \left( \tan \frac{{\theta}_{\mbox{\large jet}}}{2}\right)$$

$$X_1 = \left( \frac{2P_T}{\sqrt{s}} \right) \left( \frac{e^{{... ...{-\eta}_{\gamma}} + e^{{-\eta}_{\mbox{\small jet}}}}{2}\right).$$

Step 2.

X₁ and X₂ are the initial-state Bjorken X's of the quark and the gluon, but you don't know which is which.${}^{*}\;$ So, define X_a and X_b to be

$$X_a = \mbox{min}\left( X_1, X_2 \right)$$

$$X_b = \mbox{max}\left( X_1, X_2 \right).$$

Each event has a unique pair of X_a and X_b.

Divide the acceptance into an equal number of X_a, X_b bins. For n divisions in X, one obtains $\frac{n(n+1)}{2}$ bins in (X_a, X_b):

^*(Probabilities of which X is which are known. Gluons likely at low X.)

Step 2. (Cont.)

Using this binning system, you count the number of events in each bin of (X_a, X_b) for each combination of the four spin states of the beam $(++,\, +-,\, -+,\, --)$. After subtracting backgrounds, you obtain the raw counts for direct photon + jet events:

$$N_{++}(X_a, X_b),\;\; N_{+-}(X_a, X_b),\;\; N_{-+}(X_a, X_b),\;\; N_{--}(X_a, X_b).$$

Note:

Possible sources of background might include events with multiple direct photons or one direct photon and multiple jets. In any case, systematic errors due to background subtraction enter in at this stage.

Step 3.

In each bin, divide N_ij(X_a, X_b) by the product of the raw counts in the luminosity monitors for the two beams:

$$n_{ij}(X_a, X_b) = \frac{N_{ij}(X_a, X_b)} {L_{ij}^{\mbox{\... ...\,\cdot\, L_{ij}^{\mbox{\normalsize (counterclockwise beam)}}}$$

Note that there is only one value of L_ij for each beam, and that all the N_ij's of the various (X_a, X_b) bins are divided by a common product of L_ij values.

The above assumes that there is no transverse jitter in the beam positions or changes in beam focusing due to switching between beam spin states. Any such changes will affect the collision rate in a way that can cause false asymmetries. This issue is a critical one, as it is potentially a major source of systematic errors. For more on this topic, stay tuned for Dave Underwood's talk on luminosity monitoring at the RIKEN workshop next April.

Step 4.

Assuming P_b is the same for both spin states of both beams, A_LL(X_a, X_b) is given by

$$A_{LL}(X_a, X_b) = \frac{1}{(P_b)^{\, \scriptscriptstyle 2}}... ...{-+}} {n_{++}\; +\; n_{--}\; +\; n_{+-}\; +\; n_{-+}} \right].$$

You calculate this quantity for each bin of (X_a, X_b).

$\underline{\mbox{Note for Inclusive direct gamma measurements:}}$

Recall that knowledge of the jet angle was required for calculating X_a and X_b. Lacking this, you can try using single tracks in the acceptance to estimate a jet angle, or some other way of estimating where the jet went. Alternatively, you can simply calculate A_LL as a function of P_T, which is obtainable from just the photon energy and angle. If you do this however, you can't use A_LL(P_T) to extract a polarized gluon distribution function ${\Delta}G(X)$.

Phenomenological Side:

Step 5.

Warning: This step has not yet been worked out in detail, and what follows is a description of a method which is still in the conceptual stage. The details of the formulas used here may be wrong in some respects, but they will serve for now to illustrate how it will be possible to obtain ${\Delta}G(X)$.

Step 5. (Cont.)

In using the binning scheme already described, we have added the contributions from both of the two choices of assigning X_a and X_b to the quark and the gluon. To disentangle this, we need to reflect this in the formalism. For the moment, suppose that in addition to having measured A_LL(X_a, X_b) with a polarized beam, we also measured the unpolarized (spin-average) absolute differential cross section for direct photon + jet production using unpolarized proton beams. In this case, the measured cross section would be given by

$${\sigma}^{\mbox{\normalsize measured}}_{{\gamma}+\mbox{\norm... ...a, X_b) + {\sigma}_{{\gamma}+\mbox{\normalsize jet}}(X_b, X_a),$$

where the terms on the right-hand side are true theoretical cross sections where you know which X is which.

Step 5. (Cont.)

Now consider the product of A_LL(X_a, X_b) with ${\sigma}^{\mbox{\normalsize measured}}_{{\gamma}+\mbox{\normalsize jet}}(X_a, X_b)$:

which written more explicitly is something like

The pieces in the above will be described shortly, but in principle everything is known except for ${\Delta}G(X_a)$ and ${\Delta}G(X_b)$. If you binned X_a and X_b into n bins each, then you can form an overdetermined system of $\frac{n(n+1)}{2}$ equations in the n unknowns ${\Delta}G(X_i)$.

Step 5. (Cont.)

In the previous equation, the pieces are:

$\parbox{1.5in}{$G(x), q(x)$}$

Spin-average parton distribution functions obtained from fits to data of unpolarized deep inelastic scattering experiments and others such as jet-jet, Drell-Yan.

$\parbox{1.5in}{${\Delta}q(x)$}$

Polarized quark distribution functions obtained from fits to data of polarized deep inelastic scattering experiments. (EMC, SLAC, HERA, etc.)

$\parbox{1.5in}{$\hat{\sigma}^{{\gamma}+\mbox{\normalsize jet}}_{f}({\theta})$}$

Tree-level spin-average Born cross section for direct photon + jet production, which is a function of ${\theta}_{CM}$ only, and is calculable from PQCD. This will be a combination of terms from both $qg \rightarrow q\gamma$ and $q\bar{q} \rightarrow g\gamma$ processes. Presumably they are also flavor-dependent.

$\parbox{1.5in}{$\hat{a}^{{\gamma}+\mbox{\normalsize jet}}_{\scriptscriptstyle LL,f}({\theta})$}$

Tree-level Born asymmetry for direct photon + jet production, which is a function of ${\theta}_{CM}$ only, and is calculable from PQCD. This would also be a flavor-dependent combination of terms from both $qg \rightarrow q\gamma$ and $q\bar{q} \rightarrow g\gamma$.

(Note: It is known that $qg \rightarrow q\gamma$ tends to dominate at most angles.)

Step 5. (Cont.)

The angles ${\theta}_{ab}$ and ${\theta}_{ba}$ are given by

$${\theta}_{ab} = \frac{X_a\, e^{\,-y_{\mbox{\small jet}}} - X... ...{\,-y_{\mbox{\small jet}}} + X_b\, e^{\,y_{\mbox{\small jet}}}}$$

$${\theta}_{ba} = \frac{X_b\, e^{\,-y_{\mbox{\small jet}}} - X... ...\,-y_{\mbox{\small jet}}} + X_a\, e^{\,y_{\mbox{\small jet}}}},$$

where $y_{\mbox{\normalsize jet}}$ is given by

$$y_{\mbox{\normalsize jet}} = \ln \left( \cot \left( \frac{{\theta}_{\mbox{\small jet}}}{2} \right) \right),$$

where ${\theta}_{\mbox{\normalsize jet}}$ is the ${\theta}_{CM}$ of the jet.

(See section 4.1 of C. Bourrely et. al., Phys. Rep. 177, 319 (1989) for more information. Also listed there are various expressions for tree-level cross sections and asymmetries.)

Step 6. (If desired)

$\underline{\mbox{Spin Content of the Proton Carried by Gluons:}}$

Once you have values of ${\Delta}G(X_i)$, do a fit to the points and calculate

$${\Delta}G = \int_{0}^{1} {\Delta}G(x) dx,$$

which is half the fraction of the proton spin carried by gluons. The formula for the proton spin is

$$\mbox{Proton Spin} = \frac{1}{2} = {\Delta}\Sigma + {\Delta}G + \mbox{other},$$

where ${\Delta}\Sigma$ is the quantity known from the EMC/NMC experiments:

$${\Delta}\Sigma = {\Delta}u + {\Delta}d + {\Delta}s,$$

where

$${\Delta}q = \int_{0}^{1} \left[ {\Delta}q(x) + {\Delta}\overline{q}(x) \right]dx.$$

$\Rightarrow 2\left[ \frac{1}{2} - {\Delta}\Sigma - {\Delta}G \right]$ is the fraction of the proton spin carried by things other than quarks and gluons. (i.e. L_z, sea quarks, etc.)