A search for physics beyond the Standard Model

by measuring the parity-violating asymmetry in electron scattering at intermediate energies has been identified to be within the reach of experiments at CEBAF. In particular spin-0 isospin-0 nuclei in the low-A region are appropriate for these measurements because of the simple form nuclear current takes on in this case.

Note that the dependence on the nuclear wave-function that is present in the Coulomb form factors FC0 cancel out in the ratio, leaving only the residual difference between the strangeness form factor and the electric isoscalar form factor intrinsic to the nucleon itself. The RV factors carry the corrections to the tree-level Standard Model asymmetry. The goal of the experiment would be to measure RV(T=0) as accurately as possible. The Gamma(q) factor contains corrections arising from the T>0 admixture in the nuclear ground state, and is estimated to be at or below the 1% level for the nuclei studied below.

Survey of possible targets

Below is shown the nuclear level structure of the 0 nuclei in the P-shell region. The numbers along the bottom are atomic number A.

In order to separate elastic scattering from inelastic reactions, one would prefer a nucleus with the largest separation between the ground and the first excited states. Oxygen is the best candidate from this point of view, and also with regard to the cross section, which goes as Z2. Carbon is also interesting because it makes for a simple and compact target, with good heat tolerance properties. Nuclei heavier than 16O are less desirable from the point of view of low-lying excited states (see following figure). Also the ground states of these nuclides contain increasing admixtures of T>0 which complicate the interpretation of the data.

The lightest 0+ nucleus is 4He, which is interesting in spite of its low Z because of the simplicity of its excitation spectrum. It has no excited levels below 20MeV which is where the continuum begins. A very nice compilation of the full spectrum of nuclear levels for a large number of light nuclides is found here.

In this study I concentrate on three specific targets, 16O, 12C and 4He. Water is used for the oxygen target.

Water target

In all of the following figures I plot lab scattering angle on the x-axis and use colour to distinguish different electron beam energies. First I show the Standard Model asymmetry for  16O.
Fig. 1

The strangeness radius is set to zero in the above calculation. The effect of a non-zero rho(s) will be shown in Fig. 3. In the following plot is shown the uncertainty in the measurement of the above asymmetry for an experiment performed under the following assumptions.

Fig. 2

The statistical error alone reaches a minimum at a scattering angle of zero degrees for all beam energies, because the cross section rises faster than A2 is dropping as one approaches zero degrees. However this does not take into account the inherent difficulties involved with controlling beam conditions well enough to measure very small differences in experimental rates. Even if the rates are high enough that statistical accuracy is not a problem, one must also ensure that the magnitude of the asymmetry is large enough to measure under real experimental conditions. I take this into account by introducing into the experimental error a systematic uncertainty SA as follows.

The above figure of 10-8 for SA is probably pessimistic for what is possible at CEBAF. The calculation will be repeated later with an order of magnitude lower systematic error.

To obtain an idea of what might stand in the way of comparing a measured asymmetry with the Standard Model value, I plot below the percent uncertainty in A arising from the poorly known value of the strangeness radius of the nucleon rho(s). For this figure I have taken rho(s) to be uncertain by ±1. Right now I think this number is not known even that well. Comparing with Fig. 2 shows that parameter would need to be constrained before one could reach the level of measuring EW corrections.

Fig. 3

Next Figs. 2 and 3 are shown overlayed. This shows that running at higher energy is relatively more sensitive to the strangeness form-factor. The best idea might be to divide the running between high energy to measure rho(s) and low energy where sensitivity to the residual uncertainty in rho(s) is the least.

Fig. 4

To see how these result depend on the systematic error on A I repeat the above figure for SA = 10-9. The optimum measurement angles move forward, as expected, and the experimental error on A for the conditions listed above improves somewhat. The colour coding is the same as in Figs. 1-4.

Fig. 5

Now, for example, one might fix the spectrometer at 6° in the lab and split the running between 1.5GeV (to measure rho(s)) and 600MeV to measure EW corrections.

So far we have only looked at 16O and with a water target there would also be a contribution from hydrogen. A comparison of the cross section for electron scattering from 16O and 1H (among others) is shown next. The cross sections are multiplied by q4 to facilitate the comparison at low angles.

Fig. 6

The Standard Model asymmetry for the proton is shown next. There are contributions from the strange quarks involving both rho(s) and mu(s), but these are poorly known at present and have been left out of the calculation.

Fig. 7

The precision with which the proton asymmetry can be measured under the above experimental conditions is shown in Fig. 8. A systematic error SA = 10-9 was assumed. This is not really relevant here because a water target would produce many more quasi-elastic than free proton events, so it is a different experiment. Fig. 8 also shows that the proton asymmetry is best measured at high energy, instead of around or below 1GeV, where one wants to run when measuring on a T=0 nucleus.

Fig. 8

Carbon target

I now move on from water to look at a carbon target. For these studies the luminosity, and hence target thickness in nuclei/cm2, are kept the same. In Fig. 9 is shown the experimental error on the asymmetry measurement with a the systematic error kept at SA = 10-9. It appears that the error for carbon is about 25% larger than that obtained with an oxygen (water) target of the same thickness in nuclei/cm2.
Fig. 9

Helium target

Fig. 10 shows the same quantity calculated using a helium target, all other things kept the same. It appears that one does not really win in any respect by going down from 16O to lighter nuclei. The cross section is dropping fast enough that one is still confined to very forward angles to reach the desired statistical precision.
Fig. 10

Of all 0+ targets, helium has the largest cross section from Q2=0.1GeV2 and greater. This makes it the best target for measuring rho(s). This measurement might be made at an intermediate value of Q2, for example with a forward-scattering experiment at a beam energy 2-3GeV, and then the value used in the interpretation of data taken below 1GeV where the contribution from the strangeness form-factor to the total asymmetry is smaller.

Comparison of three targets

These results show that in general one would like to measure the asymmetries at angles in the interval 5°-8° in the lab. In the conditions above we also assumed a solid-angle of 0.01sr. If I assume the spectrometer acceptance is symmetric in x and y this translates to an area roughly 6°x6° on the unit sphere. To maintain this acceptance if we have to accept a 6° bite then it should be over the range 5°-11°, or centered at 8°. Since this is more or less independent of energy and target, let us now switch to fixing the scattering angle and plotting measurement error versus beam energy. All three 0+ T=0 targets are shown together in Fig. 11. The lower curves are the same data from Figs. 5,9-10 but now plotted vs energy instead of angle. The green curve indicates the contribution to the error on the asymmetry from an error in the strangeness radius of ±1. The upper curves are the result of throwing away the part of the radiative tail of the elastic peak that overlaps with the first excited level (or continuum in the case of helium) of the recoil nucleus. Such a cut would be necessary in order not to contaminate the measurement with T>0 transitions.
Fig. 11

In addition to nuclear excited states, with a water target one must also be concerned at some level with separating out the contribution of scattering from the free protons. The following simplified picture of the focal plane serves to illustrate the kinematic separation between the two elastic peaks as a function of angle.

Fig. 12

At a beam energy of 1GeV the proton lies beyond the excited nuclear levels for most of the accessible angular range, but at 600MeV the separation is less clear.

Fig. 13

However at these low values of Q2 the cross section for 1H is a factor of 50 less than 16O (see Fig. 6) and the asymmetry of the proton is an order of magnitude less than for oxygen, so the effect is small and easily corrected for at low Q2. The same is not true at higher values of Q2, as one approaches the the diffraction minimum for 16O, where hydrogen would dominate the asymmetry. But there the kinematic separation is large. So the presence of hydrogen in water nowhere diminishes its effectiveness as an oxygen target.

The outcome of this preliminary comparison is that 16O in the form of water provides the best target for a precision measurement of the weak isoscalar charge of the nucleon. As was already shown by Musolf et.al. it is possible, within parameters reasonable for an experiment at CEBAF, to make a 1% measurement of this charge which would be competitive with the limits currently established by atomic parity-violation experiments. However there are still some experimental questions that have to be addressed for an experiment that would like to run at a luminosity of 1038cm-2s-1.

  1. How does the use of a thick target affect the ability to resolve elastic from inelastic scattering?
  2. What are the practical limits on luminosity that arise from the power dissipation requirements for each type of target?
  3. Can any of the existing spectrometers at CEBAF operate at angles forward of 10° with an acceptance of 0.01sr and still obtain the necessary separation between the elastic and inelastic channels? What limits does this place on target length?
The first of these considerations is taken up in detail in the remaining body of this report. The second is covered in a preliminary way in Appendix A. It requires further study once the desirability of one of the possible targets has been identified. The third is independent of target type and requires the selection of a particular spectrometer to be adapted to the needs of this experiment. Appendix B outlines some work underway in this regard.

Thick target effects

The benchmark luminosity used by Musolf et.al. of 1038cm-2s-1 translates into the target thickness shown in the following table for each type of target. A beam current of 100uA is assumed.
target length (cm) radiation length (cm) target length (%rad.len.)
water 4.9 36.1 13.6
carbon 1.4 18.8 7.4
helium (liquid) 8.6 756 1.1

The two effects related to resolution that arise when considering targets as thick as 0.1X0 are external bremsstrahlung and multiple scattering in the target.

internal bremsstrahlung

I use the term internal bremsstrahlung to refer to the Q.E.D. radiative corrections that are associated with the primary scattering event. This is not a thick-target effect, and was already taken into account in Fig. 11, but is described here to make clear its distinction from external bremsstrahlung discussed in the next section. The correction to the elastic scattering cross section due to internal bremsstrahlung is contained in the logarithmic term of the following n.l.o. elastic scattering cross section.

The magnitude of the correction depends on Ecut where one cuts off the radiative tail of the scattered electron energy distribution. The requirement the the experiment discriminate between elastic and inelastic scattering sets a natural scale for Ecut for each type of target, which I take to be 6MeV for water, 4MeV for carbon, and 20MeV for helium. The exact values do not matter very much because they only enter the cross section through a logarithm. Under these conditions there is a reduction in the cross section of about 30% for all three targets due to internal bremsstrahlung. This is represented by the difference between the two curves in each colour in Fig. 11. This correction is independent of target thickness.

external bremsstrahlung

There is another effect, also classed as a radiative correction, which does depend on target thickness, which I call external bremsstrahlung. This has to do with radiation that occurs in a separate scattering event from the primary one. The incoming electron may radiate some energy in a prior interaction in the target before it undergoes the primary collision, or the radiation may occur after the primary scattering event; the effect is the same. The problem is actually more complicated than simply considering just these two cases. The energy-loss distribution due to bremsstrahlung rises like 1/k for k going to zero, with the result that for any finite target thickness there is a certain energy scale below which the multiple-radiation effects become important and one can no longer consider just single-photon radiative corrections. This is true for all target thicknesses; however, for very thin targets the scale can be pushed far into the infrared, leaving single-photon corrections dominant at energy scales that are relevant to the problem. Here we are interested in effects that are important at the MeV scale.

For the bremsstrahlung spectrum I take the following simplified formula, which has the correct infrared behaviour, the right average radiation rate (if x is measured in radiation lengths) and cuts off at the endpoint.

I now divide the target up into n slices along the incident beam direction and apply this formula for the energy loss K distribution of the electron as it traverses each layer. Since we are considering targets much less than X0 thick and are only interested in the distribution near the peak, I use the same value of 1GeV for E in all of the layers. In this case the distribution g(K) for electrons emerging from the target is given by

where the exponent (n) refers to auto-convolution n times. I take the incident beam lineshape g0(K) to be a delta function for our purposes. This function is normalized to unity and has the correct mean energy loss in the limit of large n. This convolution is best carried out numerically. I carried it out for this study with n=1000, although for these target thicknesses convergence was obtained already with n=100. In the next figure is shown the energy profile for a 1GeV incident electron exiting from a target of 1% radiation length, in the vicinity of the peak. Zero in this plot corresponds to a 1GeV transmitted electron, 0.01 to a 0.990GeV electron, and so on. The vertical axis is probability normalized to 1 over the whole spectrum out to 1GeV. For this case most of the beam remains in the first bin, that is, within 1MeV of the incident beam energy.
Fig. 14a

For a target of 0.1 radiation lengths or larger, as shown in the next figure, the first bin no longer contains the majority of the scattered beam.

Fig. 14b

To apply these results to the problem at hand, I assume that a measurement at low angles involves a target geometry equivalent to the slices described above. This means that no matter where inside the target the primary scattering occurs, the electron must traverse the same amount of target material on the way to the detector. An exception to this assumption will be considered later in the case of an extended helium target. Integrating the probabilities in Figs. 14a-b up to Ecut gives the fraction of the elastic scattering events that enter into the measured elastic peak at the detector. This can be multiplied by the target thickness to yield an effective target thickness, that is, the equivalent amount of target that contributes to the measurable elastic cross section. This is shown for three different values of Ecut in the next figure. Note that these curves apply to any target material because thickness is measured in radiation lengths.

Fig. 14c

These results show

  1. that there is a limit to the useful luminosity that one can obtain for a given beam current and target material, that turns out to be on the order of 5% rad.len., and
  2. that there is an optimum target thickness that yields the maximum usable luminosity for elastic scattering, which depends somewhat on the elastic cutoff, but is on the order of 10% rad.len.
To explore what ramifications this might have for the experiment in question, I now re-evaluate the usable luminosity for each type of target under optimal conditions.

cutoff
(MeV) 		max. eff. target
thickness
(% rad.len.) 		phys. target
thickness
(cm) 		eff. luminosity
(1038cm-2s-1)
16O (water) 6 4.5 4.3 0.33
12C (graphite) 4 4.2 2.1 0.57
4He (liquid) 20 5.9 113 5.4
4He (short) 20 1.6 12.5 1.35

These are reasonable thicknesses for physical targets, except for the case of helium, which is highlighted in red in the table. There are several reasons why a target more than 1m long is a difficulty, not the least of which is the considerable amount of power that would have to be dissipated by the crygenics. This matter is taken up again in the next section. For the present purposes, I consider a 12.5cm (300W) target to be within the limits of what is practical, and proceed with the comparison on this basis. The parameters for such a target are shown in the last line of the above table.

I now repeat the calculation of the experimental error on the measured asymmetry using the corrected values for the effective luminosity for each target. As before, the upper curve in each colour is the more important one, as it contains the cross section that one would actually measure.

Fig. 15

Now all three targets are quite close to one another in the ultimate limits of their precision, although they differ somewhat in the kinematics where the optimum precision is reached. Carbon has edged out water by a small margin as the best target from the point of view of absolute precision.

Just as an exercise, I set aside questions of technical feasibility and spectrometer optics, and consider what the ultimate limits would be with a 1m helium target. This is shown in the next figure. Both this and the preceding figure are evaluated assuming a systematic error on A of 10-9.

Fig. 16

In the case of an extended target like this, it is no longer valid to consider that all of the electrons traverse the full thickness of the target. If the target were in the form of a tube of transverse dimensions only a few cm then the electrons would exit the target not far downstream of the primary vertex, giving the effect of a thinner target. They would, however, have to penetrate the walls of the target vessel at rather low angles, which would restore a part of the losses. For this study I assumed an aluminum target vessel with walls 300um thick, and recalculated the elastic yield taking into account the fact that a greater number of the detected elastic events come from the upstream region of the target, before the beam has begun to degrade due to radiation. The result is the lower pair of curves in Fig. 16, the upper (final) one of which largely overlaps with the lower curve of the "fat target" calculation.

multiple scattering

The two previous sections dealt with the scattering of the electrons from their nominal trajectory with respect to their energy. Along with this there will also be an accompanying scatter from the nominal trajectory with respect to direction. Multiple scattering arises mainly from Coulomb scattering from nuclei, both with and without accompanying bremsstrahlung. Besides the primary scattering event which provides the signal being measured, the electron suffers many small-angle deflections which add up to a total deflection angle distributed as a Gaussian with an r.m.s. given by

where lambda is the target thickness in radiation lengths. From Fig. 1 it is clear that it is not sufficient to simply measure an asymmetry accurately, but one must also know with sufficient accuracy at what value of Q2 it applies. For finite acceptance this involves an average over angles. In order to compare this average with a calculation one must know the shape of the envelope around the accepted angular range, which involves both the optics of the spectrometer and the smearing due to multiple scattering.

At 1GeV with a target thickness of 0.1 rad.len. the r.m.s. angular spread from multiple scattering is 3.9mr, or about ¼°. At 5° Q2 changes by 9% over 4mr, which implies the same rate of change for the asymmetry, and double for the cross section. At 8° the figure is 6% for the asymmetry. Multiple scattering in the interior of the accepted angular region does affect the result, which involves only the sum of all counts within the acceptance. Hence the only thing of concern with regard to multiple scattering is its effect on the acceptance near the edges, with the low-angle extreme being the most sensitive.

The implications for alignment would be that the overall spectrometer angle would have to be known to within ±0.1mr to ensure that geography would not contribute a significant amount to the total experimental error. With the proper placement and alignment of focal plane counters, it should not be a problem to reach the necessary precision with respect to angle, for a 1% measurement of the asymmetry.

Summary

The net effect of the corrections considered above is to decrease the effective luminosity, and hence statistical accuracy, of the measurement. The question then arises whether a systematic precision at the level of 10-9 is still needed, or whether 10-8 is sufficient. The next figure shows the combination of the preceding two, but calculated for the case SA = 10-8.

Fig. 17

There is small but not insignificant decrement in precision that comes with this additional systematic error. But the major issue is that in order to address the important physics issues with such an experiment we need to reach the 1% level. Besides hoping to surmount the difficulties involved in operating a 2.5kW liquid helium target, there is another way to increase the statistical precision of the measurement, which is to run for a longer period of time. Since the combination of 100uA and 70% polarization is already a challenge to achieve over the duration of a 1000hr experiment, it is probably not wise to count on signficant improvements in these values. In the next figure I show the uncertainty for a 2000hr experiment under the same conditions as have been studied so far and a systematic error of 10-8.

Fig. 18

Finally, I repeat the calculation once again, substituting back a systematic error of 10-9. This probably comes close to what could reasonably be done in the near future at CEBAF.

Fig. 19

I would like to highlight the potential of an experiment with a water target. It is a 2000hr experiment with a 100uA beam at 70% polarization with systematics on the asymmetry controlled at the level of 10-9. It does not sound like an easy job. The result would be a 1% measurement, that meets the criteria put forward by Musolf et.al. for a significant test of physics beyond the Standard Model. I think that the importance of the issues involved warrant the kind of effort that would be necessary to make an experiment of this type a reality.

Appendix A: target heat dissipation

The ionization heating per unit length of a target is given by the product IS in W/cm where current I is measured in uA and stopping power S is measured in MeV/cm. The total power for the optimized targets discussed in the text of this report are shown in the following table.

beam heating (W/cm) target length (cm) total power (W)
16O (water) 200 3.6 720
12C (graphite) 400 1.9 760
4He (liquid) 24 113 2700
4He (short) 24 12.5 300

A water target capable of dissipating 700W would have to be of the "waterfall" type. With the beam on the order of 1mm in diameter, the waterfall could be oriented so that it presents an edge to the beam. With this geometry the target could be quite thin in the transverse horizontal direction, which would reduce the effective amount of target seen by the average scattering electron, and hence reduce radiation losses somewhat. We would need to consult people with experience with waterfall targets to see whether such a scheme could be made to work.

The graphite target is the simplest of all. In this case most of the heat dissipation takes place by blackbody radiation. If 700W is too much for a stationary target one could construct a rotating target. There is no apparent difficulty with carbon.

With helium the amount of heating that could be handled by an affordable cyrogenic system would be the limiting factor in what could be done. If a 15cm target can be made to operate successfully then several such targets could be arranged in series up to the desired length, limited by the total cooling capacity that could be put in place. Eventually there is also a limit imposed by the resolution of the spectrometer viewing an extended target, but at these forward angles I suspect that the cryogenic limits will be reached first. In Appendix B is described work that is presently underway to determine the experimental limits imposed by the spectrometer optics at forward angles.

Appendix B: spectrometer resolution

The need for energy resolution at the level of a few MeV restricts the choices to one or two spectrometers among what is on the floor at CEBAF today. The following figure, clipped from the APS news bulletin, shows the energy spectrum for electron scattering from carbon at kinematics not far from what we are presently considering.
This is an actual spectrum taken during one of the early Hall A experiments. The resolution is more than sufficient for our purposes. The question remains whether this resolution can be maintained at angles forward of 10° without sacrificing the solid-angle bite of 0.01sr.

At UConn we are in the process of setting up a description of the HMS spectrometer in the program TRANSPORT which calculates the trajectories of charged particles through a sequence of magnetic elements. In this study we aim to determine what the trade-offs are between solid-angle, resolution, and far-forward acceptance for this spectrometer. Results from this study are expected by the end of the summer.

Appendix C: nuclear level diagrams

Richard Jones

University of Connecticut / Jefferson Lab
INTERMEDIATE-ENERGY SEMILEPTONIC PROBES OF THE HADRONIC NEUTRAL CURRENT.
By M.J. Musolf (MIT, LNS & Old Dominion U. & CEBAF), T.W. Donnelly (MIT, LNS), J. Dubach (Massachusetts U., Amherst), S.J.. Pollock (Washington U., Seattle & NIKHEF, Amsterdam), S. Kowalski (MIT, LNS), E.J. Beise (Cal Tech, Kellogg Lab & Maryland U.). CEBAF-TH-93-11, Jun 1993. 233pp.
Published in Phys.Rept.239:1-178,1994
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