# Tevatron run ii LuminositY, Emittance and Collision Point Size

J. Slaughter, J. Estrada, K. Genser, A. Jansson, P. Lebrun, J. C. Yun,

Fermi National Accelerator Laboratory, Batavia, IL 60510,

S. Lai,

University of Toronto, Toronto, Canada M5S 1A7

Abstract

We compare the Tevatron luminosity as measured by the CDF and D0
experiments with that computed from machine characteristics. We also
compare the CDF measurements of the size of the interaction region with
that predicted by machine parameters. Although these results are still
preliminary, they show promise as a useful crosscheck of the
instrumentation and our understanding of the Tevatron machine
characteristics.

## Introduction

Understanding the behavior of the Tevatron depends on understanding the
instrumentation and the lattice parameters. The experiments D0 and CDF
provide independent measurements of the luminosity\[[^1]\] and of the
size of the luminous region. The luminosity (10^30^cm^‑2^sec^‑1^) can
also be calculated from lattice parameters and beam measurements in the
Tevatron, as

$L = \frac{\text{10}^{- 5}\text{fBN}_{p}N_{\overline{p}}\left( 6\beta_{r}\gamma_{r} \right)}{2\text{πβ}^{}\sqrt{\left( \varepsilon_{p} + \varepsilon_{\overline{p}} \right)_{x}\left( \varepsilon_{p} + \varepsilon_{\overline{p}} \right)_{y}}}H\left( \sigma_{l}/\beta^{} \right)$.
(1)

Here, *N~p\ ~*and $N_{\overline{p}}$ are the numbers of protons and
anti-protons per bunch (\~10^9^), *B* is the number of bunches (36), *f*
is the revolution frequency (47.7 KHz) and $\beta_{r}\gamma_{r}$=1045 is
the relativistic factor, * β^∗ ^*is ^ ^the beta function at the
interaction point (measured in cm, and assumed equal in x and y). *H* is
the hourglass factor, a function of the bunch length σ*~t~* and
*β^∗^*^ ^, and finally *ε~p~* and $\varepsilon_{\overline{p}}$ are the
proton and anti-proton 95% normalized emittances in π-mm-mr. Comparing
the calculated luminosity to the measured one provides a crosscheck
between the Tevatron beam instrumentation and the detectors.

The experiments very accurately measure the size of the luminous regions
at the interaction points as a function of z (distance along the beam
axis) from the distributions of primary vertices as measured by the
silicon vertex detectors. In the simplest model,

$\sigma(z) = \sqrt{\varepsilon\left( \beta^{} + (z - z0)^{2}\beta^{} \right)}$,
(2)

where *σ * is the rms beam size, *ε* =
(*ε~p~*$\varepsilon_{\overline{p}}$)/((*ε~p~*+$\varepsilon_{\overline{p}}$)
is a pseudo-emittance and *z~0~* is the z location where * β*  is a
minimum. Note that if the p and pbar emittances are equal, the
pseudo-emittance is half the equivalent beam emittance.

Although not discussed in this paper, CDF can measure the bunch lengths
of the p and pbar bunches separately by using the Central Outer Tracker
(COT) information in conjunction with the time of flight (TOF)
system\[[^2]\].

## Size of the Luminous Regions

CDF has fit for *ε, β^∗ ^* and *z~0\ ~*in Equation 2 for 20 runs in 14
stores from the last 6 months of 2002. Figure 1 shows typical
distributions for one run with fits to Equation 2. The average values of
the parameters are given in Table 1. The average *β*\* is 38 cm, 9%
larger than the nominal 35 cm and the minimums are displaced with
respect to the center of CDF. It is expected that *β*\* and *z~0~* are
the same store to store, but not *ε*. Fixing *β*\* to the average *β*\*,
38.6 cm in x and 38.0 in y, and then refitting improves the fits, while
fixing *β*\* to the nominal 35 cm gives slightly worse fits. The average
over 20 runs of the run-by-run ratio of x and y pseudo-emittances is
1.01±.04. If the proton and anti-proton emittances are assumed to be the
same[^3], then the pseudo-emittances imply that their values are
15.7±1.5 π-mm-mr in the horizontal plane and 15.8±1.5 π-mm-mr in the
vertical plane. The largest uncertainties in the vertex measurements
are, first, that they integrate over a significant time period, ranging
from 4 to 15 hours with an average of 8 hours, in which the emittances
are changing, and, second, they rely on proper subtraction of the
measurement resolution. In this preliminary analysis the experimental
resolution that is subtracted in quadrature from the uncorrected beam
width is about equal to the resulting beam width.

+---------+------+--------+---------+-------+-------+--------+--------+
|         | β~x~ | β~y~   | ε~x~    | ε~y~  | Z~0x~ | Z~0y~  | equi   |
|         |      |        |         |       |       |        | valent |
|         | \(   | \(cm\) | π-mm-mr | π-    | \     | \(cm\) | emi    |
|         | cm\) |        |         | mm-mr | (cm\) |        | ttance |
|         |      |        |         |       |       |        | π      |
|         |      |        |         |       |       |        | -mm-mr |
+---------+------+--------+---------+-------+-------+--------+--------+
| β~x     | 38.6 | 38     | 7.9±1.1 | 7.    | 14.   | -9     | 15     |
| ,~ β~y~ | ±2.5 | .0±3.0 |         | 8±1.0 | 2±1.6 | .2±1.7 | .7±1.5 |
| free    |      |        |         |       |       |        |        |
+---------+------+--------+---------+-------+-------+--------+--------+
| β~x     | 38.6 | 38.0   | 8.0±1.1 | 7.    | 14.   | -9     | 15     |
| ,~ β~y~ |      |        |         | 9±1.1 | 3±3.1 | .2±1.4 | .8±1.5 |
| fixed   |      |        |         |       |       |        |        |
+---------+------+--------+---------+-------+-------+--------+--------+
| β~x     | 35.0 | 35.0   | 8.3±1.2 | 8.    | 13.   | -8     |        |
| ,~ β~y~ |      |        |         | 1±1.2 | 3±1.1 | .5±1.4 |        |
| fixed   |      |        |         |       |       |        |        |
+---------+------+--------+---------+-------+-------+--------+--------+

Table 1 Luminosity weighted averages over 20 runs from fits to Equation
2. The errors are the weighted standard deviations. The emittances ε~x~
and ε~y~ are pseudo-emittances as defined in the text. The column
labeled equivalent emittance is obtained by inverting *ε* =
(*ε~p~*$\varepsilon_{\overline{p}}$)/2(*ε~p~*+$\varepsilon_{\overline{p}}$),
assuming the proton and anti-proton emittances are the same.

## Luminosity Determinations

Using Equation 1, we compare the luminosities as measured by CDF and D0
with the calculated luminosities. The intensities are measured using the
FBIs (Fast Bunch Integrator)\[[^4]\], The hourglass factor is calculated
using the bunch length *σ~t~* from the SBD (Sampled Bunch
Display)\[[^5]\] and the emittances are measured by either the flying
wire system (FW)\[[^6]\] or the synchrotron light (SL) system\[[^7]\].
Since the flying wires cause background for the experiments, they are
only flown once at the beginning of the store and then again at the end,
just before terminating the store. The SL measurements are made
continuously throughout the HEP store.

![](media/image1.png){width="3.261111111111111in"
height="2.6909722222222223in"}

Figure 1 Example plot for one run of 20 runs showing luminous region
size versus z and the fits to Equation 2. Typical correlation
coefficients of the fits are about -0.8 for (*β*\*,* ε*), 0.5 for
(*β*\*, *z~0\ ~*), and -0.6 for (*ε, z~0~*)

Figure 2 plots the ratio of the bunch-by-bunch calculated luminosity to
the CDF luminosity at the beginning of the store as a function of CDF
luminosity for a sample of 52 recent stores. The calculated luminosity
uses the FW emittances. Figure 3 shows the same ratio but for D0, using
5 stores. Each store has approximately 100 measurements taken every 10
minutes. The calculated luminosity uses the SL emittances and assumes
that the horizontal emittance equals the vertical one for p and pbar
respectively. The main characteristic of the plots is a systematic scale
factor that is a function of luminosity, although with different
intercepts and slopes. In Figure 2 the slope is -0.22±0.03 and the
intercept is 0.92±.02(Fit 1). In Figure 3, the slope is --0.0634±0.0002
and the intercept is 0.7702±0.0003(Fit 2). The equivalent analysis but
for CDF, gives --0.0826±0.0012 and 0.7455±0.0008(Fit 3).

We now investigate possible explanations for both the dependences on
measured luminosity and the differences in the fit parameters. Note that
multiplying the calculated luminosity by a constant factor changes both
the intercept and the slope of the fits to the ratio.

![](media/image2.wmf){width="3.1125in" height="2.339583333333333in"}

Figure 2 Ratio of calculated to measured CDF per bunch luminosity at the
beginning of the store for a sample of 52 stores versus CDF luminosity.

We first consider the measured luminosities. Both CDF and D0 quote
statistical errors of less than 1% and systematic errors of about ±5%.
The ratio of the total D0 measured luminosity to that of CDF versus CDF
luminosity is linear, with an intercept of 0.984±.001 and a slope of
--0.00194±.00005. (At CDF luminosities of 20 and 40 10^-30^ cm^-2^
sec^-1^, the ratio is 0.95 and 0.91 respectively.) This difference is
enough to account for the difference between the D0 and CDF versions of
the fits to the data in Figure 3 quoted above. Although there are
significant store-to-store variations, within errors the measured
luminosities for CDF and D0 are linear with proton and anti-proton
intensities, with no offset.

*N~p\ ~*and $N_{\overline{p}}$ have been calibrated to ±2%. *H* is not
very sensitive to errors in σ~t~ and is quite constant for the data in
Figures 2 and 3. For the data in Figure 2, the average *H* is .48±.02,
and the average *σ~t~* is 2.4±.4 ns for anti-protons and 2.5±.5 ns for
protons. By definition, the* β∗* dependence is independent of
luminosity. A larger than nominal * β∗* as indicated by the luminous
region measurements reported above makes the disagreement slightly
worse. If *β∗ *were different at D0 than at CDF, it could help explain
the differences between Fits 2 and 3*.* Most Tevatron experts believe
the lattice parameters are only know to ±10%. These two factors could
explain an intercept and slope factor of the order of ±10%.

The final factor in Equation 1 is the combination of emittances
*F~emit~*=$\sqrt{\left( \varepsilon_{p} + \varepsilon_{\overline{p}} \right)_{x}\left( \varepsilon_{p} + \varepsilon_{\overline{p}} \right)_{y}}$.
A major difference between the data sets in Figures 2 and 3 is that the
calculated luminosity uses the flying wire emittances in the first case
and the synchrotron light emittances in the second case. There is direct
evidence of a difference between the flying wire emittances and those
measured by synchrotron light. Both measurements of emittances are
recorded simultaneously at the beginning of each store. Comparing the
ratios of emittances bunch by bunch over a large number of stores gives
the relationships between the two methods shown in Table 2. However,
this study gives no information as to whether either of them is correct.
Preliminary beam scraping results give good agreement with proton
vertical emittances and show significant discrepancies for the
anti-proton vertical emittance measurements, in agreement with Table 2,
indicating that the flying wire emittances are somewhat more reliable
than the synchrotron light measurements. However, there is also evidence
that the lattice parameters at the synchrotron light and flying wire
locations are different from what is assumed in these calculations and
could explain some of the inconsistencies.

The effective emittance is defined by solving Equation 1 for the factor
*F~emit~*. This can then be directly compared to *F~emit,~* evaluated
with FW emittance measurements. For the same data as in Figure 2
averaged over all 36 bunches in the machine, the ratio of the FW
*F~emit~*. to the effective emittance is linear in measured luminosity
with an intercept of 0.97±0.03 and a slope of --0.004±.001 For
luminosities of 20 and 40 10^30^cm^--2^sec^--1^, this translates into
ratios of effective emittance to *F~emit~* of .84-.94 and .74-.88
respectively. This result is very consistent with the ratio (.85) of the
average effective emittance 19.6±2.9 and the average flying wires
emittance factor 23.0±3.2 π mm mr evaluated for the same data.

## Discussion and Conclusions

In this paper we have summarized current, as yet preliminary,
understanding of the agreement between accelerator performance numbers
as measured by Tevatron instrumentation compared to what is seen by the
experiments. While there is qualitative agreement, much remains to be
done. The different aspects of the comparisons need to be done on the
same data sets. D0 is doing a luminous region study similar to that of
CDF and both experiments are working on another method\[[^8]\] of
measuring the size of the luminous region in which the measurement
resolution factors out, providing an important crosscheck. More scraping
studies and better knowledge of the lattice parameters will help
calibrate the flying wires and the synchrotron light. *We thank all
members of the Sequenced Data Acquisition team for their dedicated
support.*

![](media/image3.wmf){width="2.673611111111111in"
height="2.613888888888889in"}

Figure 3 Ratio of calculated to measured D0 bunch-by-bunch luminosity
during 5 stores versus D0 measured luminosity.

  -------------------------------- -------------------- ------------------
  Quantity                         Slope                Intercept

  *ε~x~* for protons               1.12±0.14            8.9±2.0

  *ε~y\ ~*for protons              1.28±0.17            -0.5±4.7

  *ε~x\ ~*for anti-protons         0.65±0.21            34.2±4.3

  *ε~y\ for~* anti-protons         0.64±0.21            14.1±6.5
  -------------------------------- -------------------- ------------------

Table 2. Coefficients for the best linear fit formula for synchrotron
light emittances (SL) in terms of the flying wire (FW) emittances (SL =
Intercept + Slope\*FW).

References

[^1]: D0 Collaboration, Fermilab_Conf-98/395-E, and for CDF, D. Acosta
    *et al.*, Nucl. Instrum. Meth. A **494**, 57 (2002).

[^2]: M. Jones *et al,* Fermilab TM-2172.

[^3]: The flying wires give a ratio of anti-proton to proton emittance
    of 1.00±.22 in the horizontal plane and 1.03±.13 in vertical plane,
    based on data from 95 stores.

    This works was supported by Department of Energy of the U.S. under
    contract no DE-AC002-76CH03000 and by the Natural Sciences and
    Engineering Research Council of Canada.

[^4]: G. Vogel *et al*, A Multi-Batch Fast Bunch Integrator for the
    Fermilab Main Ring

[^5]: S. Pordes *et al*, Measurements of Proton and Antiproton beam
    intensities in the Tevatron\' these proceedings

[^6]: W. Blokland *et al*, A new Flying Wire System for the Tevatron,
    PAC\`97.

[^7]: H. Cheung, Performance of a Beam Monitor in the Fermilab Tevatron
    Using Synchrotron Light, these proceedings

[^8]: L. Ristori *et al,* CDF Note 4189.