What Do We Get With 10⁷ Tagged Photons per Second?

Curtis A. Meyer

October 28, 1998

Abstract:

This paper is suppsed to be a bottom up analysis on the number of events we can expect in one year of HALL D running using 10⁷ tagged photons per seconds. Based on several more-or-less reasonable assumptions, we find a significant amount of data in the 1.0 to 2.3 GeV/c² mass range using an 8 GeV primary beam electron.

The Photon Beam

  Based on the rate discussions that took place at the F.S.U. Eight+ workshop, I have decided to compile this document to examine the physics we can potentially reach with R=10⁷ tagged photons per second. I have assumed that we have an 8GeV electron beam being used to generate tagged photons from a normal bremsstrahlung source. The tagger is able to tag a 3.0 GeV energy bite topping out at 95% of the beam energy -- 4.6 GeV up to 7.6 GeV. The Bremsstrahlung spectrum is given as: N(E) = N₀/E, where $N_{0} = R \cdot \left [ \ln (E_{\mathrm high}) - \ln ( E_{\mathrm low} ) \right]$.This spectrum is shown in figure 1. For 10⁷ tagged photons, there are $3\times 10^{7}$ untagged photons in the range from $E_{\gamma}=1GeV$up to $E_{\gamma}=4.6 GeV$. Extending this down to the single-pion threshold at 13#1 yields a total $5.4\times 10^{7}$ untagged photons. Finally to be complete the number of photons between energies E₁ and E₂ is given as:

$$N(E) = N_{0} \ln \left( \frac{E_{2}}{E_{1}} \right ) .$$

 

Figure:   The Photon Energy Spectrum from an 8GeV electron beam.

In addition to the beam rate, we have assumed a 30 cm long liquid hydrogen target. We use the standard values for the density of hydrogen of $\rho_{LH2}= 0.070 gm/cm^{3}$, or $4.2\times 10^{22} {\mathrm protons}/cm^{3}$.This leads to

$$N_{\mathrm target} = 1.2 \times 10^{24} {\mathrm protons}/cm^{2} .$$

With this, we can now compute various production rates:

$$N_{\mathrm Events} = N_{\gamma} \cdot N_{\mathrm target} \cdot \sigma .$$

Known Photoproduction Cross Sections

Table 1 summarizes some of the known cross sections from photoproduction experiments. This is in no way meant to be complete, but rather indicate that many of the more common cross sections are on the order of $0.5\mu b$, and that typical numbers of events are only a few thousand.

 

$$E_{\gamma}\,\, \left[ GeV\right ] \sigma \,\, \left [ \mu b \right ]$$ Reaction Events Reference

$$\gamma p\rightarrow \Delta^{++}\pi^{-} 3.7\pm 0.4$$ 2.8 [1]

$$1.0\pm 0.1$$ 4.7 [1]

$$\gamma p\rightarrow \pi^{+}\pi^{-}\pi^{+}\pi^{-}p 4.0\pm 0.5$$ 4-6 [2]

$$4.8\pm 0.5$$ 6-8 [2]

$$4.5\pm 0.6$$ 8-12 [2]

$$\gamma p\rightarrow \pi^{+}\pi^{-}\pi^{-}\Delta^{++} 1.65\pm 0.2$$ 4-6 [2]

$$1.8\pm 0.2$$ 6-8 [2]

$$1.1\pm 0.2$$ 8-12 [2]

$$\gamma p \rightarrow n\pi^{+}\pi^{+}\pi^{-}$$ 16.5-21.0 3781 [4],[5]

$$\gamma p \rightarrow p\pi^{+}\pi^{+}\pi^{-}\pi^{-}\pi^{\circ}$$ 16.5-21.0 2553 [6]

$$\gamma p \rightarrow \Delta^{++}\pi^{+}\pi^{-}\pi^{-}\pi^{\circ} 0.6\pm 0.1$$ 16.5-21.0 2553 [6]

$$\gamma p \rightarrow \Delta^{++}b_{1}(1235) 0.6\pm 0.1$$ 16.5-21.0 2553 [6]

$$\gamma p \rightarrow a_{2}^{-}\Delta^{++} 0.45\pm 0.05$$ 19 200 [3]

Rate Estimates

In table 2 is shown a spread sheet calculation for the number of events expected for a uniform cross section of $\sigma=1.00\mu b$.We assume a tagged rate of 10⁷ photons per second distributed according to the bremsstrahlung spectrum as shown in figure 1. We have cut the spectrum into approximately 0.250 GeV/c wide bins in $E_{\gamma}$. The dependence of the cross sections on t will be discussed in section 4. In computing the number of events per year, we have assumed that there are 100 days of running per year, and that the reconstruction efficiency is 10%.

What can change in this table? Clearly there are detector advantages to running with a shorter target. Trigger hardware around the target needs fewer channels, so shortening the target by a factor of 2 is not unreasonable. The beam rate could also be raised. CLAS is currently running at about $5\cdot 10^{7}$ tagged photons per second over the energy range of 0.8 GeV up to 3.8 GeV. Doubling the current rate would lead to a total situation not too far off from the current situation in Hall B. Taken together, these of course cancel each other.

 

$$E_{\gamma} \gamma$$ 2c|Event Rates Reconstructed

$$0.640\times 10^{6}/s 66\, 350/{\mathrm day} 660\, 000/{\mathrm year}$$ 4.60 to 4.75 0.77/s

$$1.000\times 10^{6}/s 103\, 700/{\mathrm day} 1\, 040\, 000/{\mathrm year}$$ 4.75 to 5.00 1.20/s

$$0.970\times 10^{6}/s 100\, 500/{\mathrm day} 1\, 010\, 000/{\mathrm year}$$ 5.00 to 5.25 1.16/s

$$0.930\times 10^{6}/s 96\, 400/{\mathrm day} 960\, 000/{\mathrm year}$$ 5.25 to 5.50 1.12/s

$$0.880\times 10^{6}/s 91\, 200/{\mathrm day} 910\, 000/{\mathrm year}$$ 5.50 to 5.75 1.06/s

$$0.850\times 10^{6}/s 88\, 100/{\mathrm day} 880\, 000/{\mathrm year}$$ 5.75 to 6.00 1.02/s

$$0.810\times 10^{6}/s 84\, 000/{\mathrm day} 840\, 000/{\mathrm year}$$ 6.00 to 6.25 0.97/s

$$0.780\times 10^{6}/s 80\, 900/{\mathrm day} 810\, 000/{\mathrm year}$$ 6.25 to 6.50 0.94/s

$$0.750\times 10^{6}/s 77\, 800/{\mathrm day} 780\, 000/{\mathrm year}$$ 6.50 to 6.75 0.90/s

$$0.720\times 10^{6}/s 74\, 600/{\mathrm day} 750\, 000/{\mathrm year}$$ 6.75 to 7.00 0.86/s

$$0.700\times 10^{6}/s 72\, 600/{\mathrm day} 730\, 000/{\mathrm year}$$ 7.00 to 7.25 0.84/s

$$0.670\times 10^{6}/s 69\, 500/{\mathrm day} 700\, 000/{\mathrm year}$$ 7.25 to 7.50 0.80/s

$$0.260\times 10^{6}/s 27\, 000/{\mathrm day} 270\, 000/{\mathrm year}$$ 7.50 to 7.60 0.31/s

$$10.0\times 10^{6}/s 1\, 037\, 000/{\mathrm day} 10\, 400\, 000/{\mathrm year}$$ 4.60 to 7.60 12.0/s

Rate Estimates as a Funtion of t  

We now want to flip the rates given in table 2 into corresponding bins of mass and t. We consider the idealized reaction in 1 where the mass of X, m_x can vary, but the nucleon N is restricted to either a proton or a neutron, both of which we shall assume have the same mass.  

$$\gamma p \rightarrow X N$$ (1)

We are now interested in the minimum value of t which can be reached for a particular photon energy, $E_{\gamma}$ and m_x. We can obtain the smallest and largest values from the center-of-mass quantities as in equation 2 with $s=2E_{\gamma}m_{N}+m^{2}_{N}$. 

$$t_{0} (t_{1}) = \left [ \frac{m^{2}_{\gamma} - m^{2}_{x} - m... ...erbrace{\left( p_{\gamma} \pm p_{x} \right)^{2}}_{\mathrm com}$$ (2)

In the center of mass, we find that the $p_{\gamma}$ is given in equation 3.  

$$p_{\gamma} = \frac{m_{N}E_{\gamma}} {\sqrt{m^{2}_{N} + 2E_{\gamma}m_{N}}}$$ (3)

 

$$p_{x} = \left [ \frac{ (s-m^{2}_{x}+m^{2}_{N})^{2}}{4s} - m^{2}_{N}\right]^{\frac{1}{2}}$$ (4)

We can now simplify equation 2 to equation 5.  

$$t_{0} (t_{1}) = \frac{m^{4}_{x}}{4s} - \underbrace{(p_{\gamma}\pm p_{x})^{2}}_{\mathrm com}$$ (5)

In figure 2 we now plot the limits on $\mid t\mid$ as a function of m_x for photon energies within the tagging range. Figure 3 shows the same plot, but restricted to values of t smaller than 1GeV/c².

 

Figure:  The allowed values of t as a function of m_x for several photon energies. The minimum and maximum photon energies correspond to the ends of the tagging range for an 8GeV electron beam.

 

Figure:  The minimum values of t as a function of m_x for several photon energies. The minimum and maximum photon energies correspond to the ends of the tagging range for an 8GeV electron beam.

We will now make an assumption that we have a total cross section of $1\mu b$,which is constant as a function of $E_{\gamma}$ over our range of photon energies. Additionally, we will assume that the cross section has a t dependence given as $e^{-\alpha\mid t\mid}$. The only dependence on m_x will come in via the $t_{\mathrm min}$ of the given reaction. We next will bin the t versus m_x plane into 10 MeV/c² wide bins in m_x, (starting at 1GeV/c²), and in five 0.2 (GeV/c²)² wide bins in t, (starting at 0).

For each average photon energy as given in table 2, we examine each bin in the t-m plane. If the value of $\mid t_{min}\mid$ is less than or equal to the average t of the bin, we will assume 100% acceptance. Otherwise, the acceptance is taken as 0%. The number of events in this bin is then proportional to $e^{-\alpha\mid t\mid}$ multiplied by the acceptance. For each individual energy, this is normalized so that the total number of events from one year of running are correctly distributed over the plane, (numbers taken from table 2).

Finally, the summed distribution from all photon energies is formed. This leads to an expected number of events in each bin from one year of running. Figure 4 shows this distribution for slope parameters of $\alpha = 5$, (upper left), and $\alpha =9$ (lower left). For the lowest t-bin, ( 0.0 to 0.2), the data extend up to nearly 2 GeV/c² in mass. Nearly all of these bins have over 50,000 events per bin -- this is a very large number for Partial Wave Analysis!. Examination of the next two bins, (0.2 to 0.4 and 0.4 to 0.6), we extend our mass reach up to about 2.3 GeV/c². Here the statistics have dropped a bit, but we still have over 10,000 events per bin in most bins -- again a large number.

In order to try and put these numbers in perspective, we have rebinned the data in a fashion similar to that used by E852. In their $3\pi$ analysis, the bin width is 5 MeV/c² rather than our 10 MeV/c², and all data have been put into a single bin in t. At the peak of the $\pi_{2}(1670)$ they have about 4000 events per bin. Our data are shown in the upper right-hand plot of figure 4 in the same form, and the $\pi_{2}(1670)$has been marked on our figure. We have about 100,000 events per bin at the same place.

 

Figure:  The distribution of events from the full photon beam based on the one year of running data from table 2. The data are assumed to be binned in 10 MeV/c² wide bins, and there are five bins in t between 0.0 and 1.0. The upper left-hand figure assumes a slope parameter of $\alpha = 5$, while the lower left-hand figure uses $\alpha =9$.The lower right-hand figure is an expanded view of the lower left-hand one. Finally, the upper right hand figure shows the data binned in E852 style bins.

Conclusions

This report has presented a bottom up analysis of the expected event rate in one year of running HALL D with an 8 GeV electron beam. Most of the results are easily scaled to other assumptions. We reiterate that our assumptions are as follows:

• We have a tagged bremsstrahlung beam produced from an 8 GeV primary electron beam. We are tagging a 3 GeV wide bite from 104#2 up to 105#3. The total tagged rate is 10⁷/s.
• The photon beam is incident on a 30 cm long liquid hydrogen target.
• The total cross section is independent of $E_{\gamma}$ and is set to 106#4.
• We have assumed 100 days of running per year with 100% efficiency.
• We have assumed a 10% reconstruction efficiency for good events.
• The cross section is assumed to have a t dependence of $e^{-\alpha\mid t\mid}$. We have examined two values of $\alpha$.

Under these assumptions, we can collect statistics which appear to be about 25 times larger than the current best E852 statistics in one year of running.

References

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H. H. Bingham, et al., Phys. Rev. Lett. 25, 1223, (1970).$\Delta$ Production Via $\gamma p\rightarrow \Delta \pi$ by Linearly Polarized Photons at 2.8 and 4.7 GeV.

2

M. Davier, et al., Nucl. Phys. B58, 31, (1973). The Reaction $\gamma p\rightarrow \pi^{+}\pi^{-}\pi^{+}\pi^{-}p$ at High Energy and Photon Dissociation into Four Pions.

3

G. T. Condo, et al., Phys. Rev. D41, 3317, (1990).

4

G. T. Condo, T. Handler, W. M. Bugg, G. R. Blackett, M. Pisharody and K. A. Danyo, Phys. Rev. D48, 3045, (1993). Further results from charge-exchange photoproduction.

5

G. T. Condo, T. Handler, W. M. Bugg, G. R. Blackett, M. Pisharody and K. A. Danyo, Phys. Rev. D49, 2164, (1994). Onelastic $\rho^{\circ}$ photoproduction in the reaction $\gamma p\rightarrow \rho^{\circ}\pi^{+}n$.

6

G. R. Blackett, K. Danyo, T. Handler, M. Pisharody and G. .T. Condo, hep-ex/9708032. The Photoproduction of the $b_{1}(1235)\pi$ System.