# Explanation of Jet 4_0

The fundamental algorithms are explained in the published references. There are many definitions that one might think of using for a triply differential jet cross section. The definition that you use makes a difference. Our recommendation is to use the cross section d sigma / d xA d xB dy* . Our paper hep-ph/9412342 explains why. The program presented here calculates d sigma / d xA d xB dy* .

Most of the methods used in Jet version 4_0 are the same used in the earlier version 3.4. Therefore, one should see the explanation of Jet version 3.4.

The major difference is that jet4_0 makes a three dimensional table for the triply differential jet cross section. The entries in this table are the K factor minus 1, where the K factor is the ratio of the NLO cross section to the LO cross section. Then readjet4_0 can read this table and produce cross sections by interpolation in the table, multiplying the K factor by the LO cross section (which is easy to calculate).

The entries in the table at each point in a lattice of jet variables (xA, xB, y*) are calculated as integrals of (K - 1) over a region near to the lattice point, using a gaussian weight.

### What to do

Compile jet4_0.f together with jetsubs4_0.f. Make sure you have a parton distribution table such as cteq3m.ptn in the same directory as the program. (See A Potpourri of Partons.) Run the program. This produces a table of K - 1 values. Compile readjet4_0.f and run it in order to display the information in the table in whatever way you like.

### readjet4_0

readjet4_0 gives results for each point in (xA, xB, y*) requested by the user. These results are interpolated from the table. The calculated cross section is in the column labelled ``Xsect.'' The estimate of the statistical error on the K factor is given under ``Kerror.'' This is thus an estimate of the fractional statistical error on the cross section. There is also an error on the cross section due to using K - 1 smeared over a region near each lattice point in place of K - 1 exactly at the lattice point. An estimated correction to K - 1 is given under ``Ksmear.'' The size of ``Ksmear'' can serve as an estimate of the error from this source. (The program does not add this correction to K - 1 in calculating ``Xsect.'')

For each point (xA, xB, y*) reported, readjet4_0 also gives the values of (ET, y1, y2) for an ideal two jet system with these values of (xA, xB, y*). This transformation is based on 2 parton --> 2 parton kinematics. Thus this is a ``Born level'' relation. This information is intended to convey an idea of what values of jet transverse energies and rapidities are explored by the cross section at this (xA, xB, y*) value.

### Parameters

The program produces a table of data for K - 1 and its statistical error at lattice points that are evenly spaced in log(xA/(1-xA)), log(xB/(1-xB)), and y*. To fix the the lattice in xA and xB the program asks

Give X grid: Xmin, Xmax, Number of points

One could choose, for example,

0.01,0.8,11

To fix the lattice in y*, the program asks

Give YSTAR grid: YSTARMAX, Number of points

One could choose, for example,

2,7

To fix smearing distances in log(xA/(1-xA)), log(xB/(1-xB)), and y*, the program asks

Give smearing distances in ln(x/(1-x)) and ystar

One could choose, for example,

1,1

For each lattice point, the program will calculate an integral of K - 1 times a gaussian weight factor that effectively limits the integral to a region near to the lattice point. Big smearing distances make the region bigger and hence improve the statistical error. However, if the distanes get too big, so that K - 1 varies substantially over one smearing length, then the smearing error (reported by readjet) will go up.

To fix the renormalization and factorization scales, the program asks

Give scale choices: A_uv, A_co

One could choose, for example,

1,1

(Please see our published paper for the meaning of these parameters.) The program asks how many thousands of Monte Carlo points (or rather ``Reno'' points) you want. The more the better if you want small statistical errors. But of course, ``more'' takes longer. Probably about 1000 thousand is enough.

### Warning

Producing a three dimensional array of ``theory'' integrals all at once inevitably limits the number of Monte Carlo points that contribute to each integral. This limits the statistical accuracy of the integrations. There are parameters built into the program to determine where the Monte Carlo points go in the two and three parton momentum space. These parameters are tuned to put most points in the middle of the region of most physical interest. The result is that the statistical errors near the edges of a typical (xA, xB, y*) lattice are large, and the results are not reliable there.

Davison E. Soper, Institute of Theoretical Science, University of Oregon, Eugene OR 97403 USA soper@bovine.uoregon.edu