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

Photon Energy

Jet Angle

Jet Momentum

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, above threshold, coplanarity of jet and photon, etc., etc., etc.) calculate

**Step 2.**

*X _{1}* and

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 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:

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:

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

You calculate this quantity for each bin of (*X*_{a}, *X*_{b}).

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 .

**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 .

**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

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
:

which written more explicitly is something like

The pieces in the above will be described shortly, but in principle
everything is known except for and . If you binned *X*_{a} and
*X*_{b} into n bins each, then you can form an overdetermined system of equations in the n
unknowns .

**Step 5. (Cont.)**

In the previous equation, the pieces are:

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

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

Tree-level spin-average Born cross section for direct photon + jet production, which is a function of only, and is calculable from PQCD. This will be a combination of terms from both and processes. Presumably they are also flavor-dependent.

Tree-level Born asymmetry for direct photon + jet production, which is a function of only, and is calculable from PQCD. This would also be a flavor-dependent combination of terms from both and .

(Note: It is known that tends to dominate at most angles.)

**Step 5. (Cont.)**

The angles and are given by

where is given by

where is the 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)

Once you have values of , do a fit to the points and calculate

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

where is the quantity known from the EMC/NMC experiments:

where

is the fraction of
the proton spin carried by things other than quarks and gluons. (i.e. *L*_{z}, sea quarks, etc.)