Dilution Factor for Exclusive Channels

Transcription

1 Dilution Factor for Exclusive Channels S.E. Kuhn 1, 1 Old Dominion University, Norfolk, Virginia (Dated: May 3, 2010) In this note, we propose a unified way to calculate dilution factors for various exclusive channels like p(e,e p)π 0, p(e,e π + )n, d(e,e π )pp and d(e, e p)n on standard solid state polarized targets containing NH 3 and ND 3 (as well as 4 He coolant plus foils), as have been used in EG1, EG4 and EG1-DVCS. We will use double spin asymmetries and the experimental conditions of EG1 as an example, but similar considerations apply to the other experiments and to target single spin asymmetries for the same channels. Keywords: nucleon structure, spin I. INTRODUCTION Whenever we want to calculate asymmetries proportional to the target spin (double spin asymmetries or target single spin asymmetries), we have to divide the measured raw asymmetry by the dilution factor, i.e. the fraction of all events coming from the polarized nucleon or nuclear species of interest. This is true for both inclusive channels (measuring spin structure functions like g 1 and A 1 ), semi-inclusive channels and exclusive channels. Dilution factors for inclusive analyses have been calculated using either a sophisticated model for the cross sections of all nuclear species present in the target (NH 3 or ND 3 as well as 4 He coolant and various foils made of plastic and/or Aluminum) [1] or a combination of data taken from several different types of special runs (on empty targets, Carbon targets and even 15 N targets) [2] to approximate scattering from the non-hydrogen nuclei of Ammonia targets. Traditionally, exclusive analyses [3, 4] have relied on a simplified method using only Carbon target data. This approach was somewhat justified by the much smaller background contribution for exclusive channels as well as the larger statistical errors for these data sets that did not warrant an excessive effort in getting the background exactly right. However, there is a potential problem with the way the Carbon target background has traditionally been cross-normalized to the standard Ammonia target data, namely by using the low-mass tails of the missing mass distribution from each target. This is explained in more detail below. In addition, it may well be worth improving the background subtraction method, given the much higher statistical accuracy available with the new data sets from EG1b, EG4 and EG1-DVCS. This is the goal of this note. In the following, we will distinguish two cases: 1. We call fully exclusive final state channels those where all but one of the final state particles expected in the reaction of interest are detected, Electronic address: skuhn@odu.edu II. leading to an expectation of a (theoretically perfectly) narrow missing mass distribution for that one undetected particle, centered on its known mass. Examples for these reactions are p(e, e p)π 0, p(e, e π + )n, d(e, e π p)p, d(e, e p)n and even inclusive p(e, e )p. In these cases, one expects that the missing mass resolutions for the same reactions on nuclear background targets will be broadened and can somehow be separated. However, as it turns out (see below), the effect is not as simple as one would naively expect, and in particular once the rather finite CLAS resolution is included, one has to be careful where to look for this broadening. In particular, it is not clear a priori that the broadening leads to a long tail towards low missing mass values. 2. We call not-quite-exclusive channels those where the reaction occurs on a deuteron, but we pretend that they occur on a neutron (or proton) at rest. The most important examples for this are inclusive deuteron breakup d(e, e )pn and the channel d(e, e π )pp where one proton is assumed to be nearly at rest while the other carries most of the missing momentum. The kinematics for this case are quite different than for the previous one. KINEMATICS FOR FULLY EXCLUSIVE FINAL STATES In this section, we will explore the kinematics of reactions where all but one final state particle are detected, and how they are modified when they take place on a nuclear target (where this is necessarily no longer the case). A. p(e, e )p As a first example, we study fully exclusive scattering on hydrogen in the quasi-elastic region (which can be considered as a special case of one missing final state

2 2 particle only ). In this case, the missing mass is W = (M 2 + 2Mν Q 2 ) 1/2 ; (1) (here and in the following M = GeV stands for the average nucleon mass). For an ideal detector (and ignoring energy loss, radiative effects etc.), one expects a distribution for W that is a delta-function centered at W = M for a hydrogen target at rest (at least for W < 1.07 GeV). In contrast, if the same reaction (proton knockout) occurs on a nucleus, the measured W (calculated the same way) shifts by an amount W = E + p2 m 2M p m M q cos(θ pq), (2) where E is the one-nucleon removal energy plus the recoil energy of the A 1 final state nucleus, and p m is its recoil momentum. As one can see from Eq. 2. binding effects shift the measured W to higher masses on average, but with a large smearing out due to the second term which can be both positive and negative. This Doppler broadening of the W distribution can lead to significant tails at low values of W, in particular for relative large momentum transfer q Q Q 2 /4M 2 and for larger nuclei with higher Fermi momentum. Under the assumption that events on free protons do not contribute to these tails, one can cross normalize the W distribution for Ammonia, Carbon and Empty targets in this region to get the correct parameters for a description of the nonhydrogen background in the former in terms of the two latter, see below [2]. It is appropriate to mention at this point that a simulation of the non-hydrogen background in Ammonia targets (nitrogen, 4 He, and foils) with 12 C counts alone is not quite correct. Since the 12 C slab in a typical Carbon target is significantly thinner than the effective length ( packing fraction ) occupied by ammonia (NH 3 or ND 3 ) in Ammonia targets, one has to correct for the difference in the amount of 4 He present. While the average binding energy in 4 He is not too different from heavier nuclear targets, the momentum distribution is noticeably narrower, leading both to a different behavior in the tails and to a more pronounced peak around the quasi-elastic value W = M. On the other hand, all indications are that 12 C and 14 N or 15 N are very similar in every aspect, except for the extra neutron in 15 N. B. p(e, e π + )n Naively, one would expect that the same general pattern from the discussion of elastic (inclusive) proton scattering translates to cases with more final state particles, as long as all but one final state particles are detected and the missing mass of that particle is calculated. However, the kinematics are subtly different in this case. Let s consider next the process p(e, e π + )n. In principle, this is rather similar to inclusive elastic scattering, in that the only unobserved final state particle is an intact proton. In fact, all of the equations used above can be used if one replaces W with the missing mass, m miss = ( (M + ν E π +) 2 ( q p π +) 2) 1/2, (3) and observes that the role played previously by Q 2 is now played by t, the 4-momentum transfer squared to the unobserved nucleon; the angle in Eq. 2 must be replaced by the angle between the momentum q p π + and the recoil momentum of the nuclear remnant. With these modifications, the general observations made above apply; however, the Doppler-broadening of the missing mass distribution from nuclear targets in this case depends not only on the electron kinematics, but also directly on t and therefore on the variable cosθ, where θ is the CMS angle between the outgoing pion and the virtual photon. This makes the cross normalization between Ammonia and Carbon targets more difficult, in particular if one tries to do this separately for each cosθ bin to compensate for the imperfect approximation Carbon = Ammonia - proton. For this reason, it is much better to follow the data driven method described below. C. p(e, e p)π 0 This reaction differs from the previous ones in that the unobserved final state particle is a pion. This channel has been usually selected by detecting the recoil proton and calculating the missing mass of the neutral pion: m miss = ( (M + ν E p ) 2 ( q p p ) 2) 1/2, (4) which should yield a narrow peak centered on m miss = m π 0 for a free proton target (there could be additional peaks at m miss = 0 for elastic scattering and photon production and additional strength at higher missing masses for 2-pion, rho, omega etc. production). In the case of production on a nucleus, there are complications because there could be contributions from π production on a neutron, plus quasi-elastic scattering, photon production, and multi-particle final states even underneath the π 0 peak. In principal, one could once again assume that all strength on either side of the free proton peak comes from nuclear contributions, and one could normalize to the corresponding spectra from Carbon (and Empty) targets. However, because of the large number of confounding reactions, which might not be all equivalent on 12 C vs. nitrogen, it is again much better to use the data-driven method below. D. Deuterium channels Here, we consider the two fully exclusive channels d(e, e p)n and d(e, e π p)p, which in some sense are the analogues to the channels discussed in subsections II A

3 3 and II B. However, there is a big difference: No longer do we assume that the struck nucleon is initially at rest. This means that we treat deuterium as a nucleus, and this leads to a somewhat different relationship between the reactions of interest and those occurring on other target nuclei. Let us focus on the quasi-elastic channel first. Here, we calculate the missing mass of the unobserved neutron using m miss = ( (m D + ν E p ) 2 ( q p p ) 2) 1/2, (5) where m D is the deuteron mass, and we expect a sharp peak at m miss = M. The same reaction on a nucleus is now very similar (single-proton knockout), and the kinematics look as follows: ν = E p + m A 1 + ( q p p) 2 m A, (6) 2m A 1 where m A is the initial nuclear mass and m A 1 is the mass of the (possibly excited) final state nucleus. We assume that this mass is large enough, and the missing momentum q p p small enough, to justify the nonrelativiistic approximation for the recoiling final-state nucleus. Inserting Eq. 6 into Eq. 5 yields for the measured missing mass in case the reaction occurred on a nucleus: m miss = ((m D + m A 1 + ( q p p) 2 2m A 1 m A ) 2 ( q p p ) 2 ) 1/2 = ((M + E + ( q p p) 2 ) 1/2 ) 2 ( q p p ) 2 2m A 1 ( M 2 + 2M E m A 1 M 1/2 m p miss) 2 A 1 M + E m A 1 M p 2 miss m A 1 2M where E is the sum of excitation and proton removal energy for proton knockout from nucleus A, minus the binding energy (2.2 MeV) for deuterium, and p miss = q p p is the missing momentum. There is an important difference with equation Eq. 2: Here, there is no broadening of the peak proportional to q and to p miss ; rather, the nuclear peak is mostly shifted up, but only becomes significantly smeared out if the nucleon recoil energy p 2 miss /2M becomes comparable to the intrinsic resolution of CLAS. Note that the average value of this energy is about 30 MeV for heavy nuclei, while the upward shift E can be MeV. Therefore, the low-missing-mass tail in this case may be non-existent or at least hard to properly define. I did not go through the full kinematics for the case of the fully exclusive channel d(e, e π p)p, but I suppose it would be very similar to Eq. 7 above, modified analog to the way Eq. 2 was modified in Section II B. The (7) conclusion will be very similar: There likely is no reliable tail at low missing mass that could be used for successful cross normalization. III. KINEMATICS FOR NOT-QUITE-EXCLUSIVE DEUTERON CHANNELS For the inclusive reaction d(e, e )pn, we have in the past used a similar approach as in Section II A, simply pretending that the struck nucleon (p or n) is initially at rest. This method is clearly somewhat problematic, since even in deuterium, the nucleons have an initial momentum distribution, albeit with a smaller width (lower Fermi momentum), leading to a shift and broadening akin to Eq. 2 even for the signal. It is non-trivial to find the right W region for which the measured counts are not due to some uncontrolled background, yet little to no deuterium counts contribute. Much care went into the analysis of EG1 inclusive deuterium data to get reliable values for packing fraction and other parameters so that the data driven method could be applied. Again, for the not-quite-exclusive channel d(e, e π )pp, similar concerns apply, with modifications as in Section II B. Yet, in our analyses so far, we have tried to define a missing mass range below the nucleon mass where we can cross-normalize carbon counts to ND 3 counts in the hopes that D contributes little and that 12 C is a good enough proxy for 15 N. Both of these assumptions are suspect. Again, the method introduced below is less prone to be overly sensitive to these details. IV. DATA DRIVEN METHOD FOR INCLUSIVE DOUBLE SPIN ASYMMETRIES In the following, we reproduce the relevant sections from the original write-up Data Analysis for EG2000. This can be used as reference in the remainder of this note. We have updated the text somewhat to reflect the accumulated information over the years, but it is mostly written from the perspective of inclusive analyses. A. Definitions Assume some n-dimensional bin in the relevant kinematic variables (W and Q 2 for inclusive analysis). We define as N +/ A,C,MT the number of counts with +/ beam helicity on all Ammonia, Carbon and empty target runs (all targets assume the presence of 4 He coolant) for a given running condition, and FC +/ the corresponding Faraday cup counts. Then we define the following

4 4 quantities: n C,MT = N+ C,MT + N C,MT FC + + FC (8) n A = 1 ( N + A 2 + N A /FC ) (9) Note that in some cases (especially for lower beam energies), the Faraday Cup counts have to be corrected for the fraction of beam that misses the Faraday cup entirely because of multiple scattering in the target and overfocussing by the target magnet. Additional information on this issue can be found at Ralph Minehart s page minehart/monitors/summary.txt on the EG2000 website and at Rob Fersch s CUE directory /home/fersch/eg1b/runinfo/runinfo.txt. [5] Item Value Comments F l F Al: µm = g/cm 2 Kapton: µm = g/cm 2 before 27997; +80 µm => g/cm 2 after. Tot: g/cm 2 / g/cm 2 (I ve tried to account for all material within 5 cm of the target center) This is from measurements by Chris and Raffaella. 50 µm Kapton could be up to 85 µm (or less because of perforation). Extra 80 µm Kapton foil was added after run (Assume F is proportional to mass number A, so mass density is needed here). C l C g/cm 2 = mol/cm 2 Needed to calculate f f (0.200 after run 27997) Ratio of previous two numbers He g/cm 3 = mol/cm 3 Triple-checked L 1.90 cm From analysis of n MT /n C ; measurement by Chris Keith and Stephen Bueltmann, gave 1.80 cm. Drawing says 2.26 cm; post mortem measurement 1.66 cm (windows bulging in not likely for run) C 2.17 g/cm 3 = mol/cm 3 Could be 2.16 g/cm 3 (measurement) or g/cm 3 (standard literature) or 2.2 g/cm 3 (SLAC number) l C 0.23 cm Other numbers quoted include cm A (NH3) A (ND3) and 0.24 cm g/cm 3 = mol/cm 3 These numbers disagree a bit they g/cm 3 = mol/cm 3 should be the same in mol/cm 3. l A 0.6 cm Must be extracted from data (packing fraction); this is just a wild guess FIG. 1: Table of approximate values for various parameters reproduced from EG1b. For the following, we define several useful quantities: Densities (in mol/cm 3 = g/cm 3 divided by molecular weight): ρ F,C,He,A for foils, 12 C, 4 He and Ammonia Target lengths (in cm): l F,C,A (Note: l A is also known as packing fraction ) Total length of the target cell ( banjo ) from entrance to exit foil: L (this value changed during EG1 because of the use of a minicup insert inside the banjo that was supposed to contain the 4 He coolant but sometimes overflowed) Cross sections (nuclear, NOT per nucleon, where applicable): σ F,C,He,N,D,H To simplify the expressions, we define the ratio of counts from foils to those from the 12 C slab in the Carbon target f = ρ Fl F σ F ρ C l C σ C. (10) In Fig. 1, we reproduce a table from the original write up which further defines these quantities and gives typical values for EG1b. The most precise numerical values for the quantity L for EG1b were extracted by Rob Fersch and are listed here (see Footnote[5]): ± ± ± ± ± ± ± ± ± ± ± ± Similarly, the best EG1b values for the packing fraction l A are given for NH3: and for ND3: ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

5 ± ± ± ± ± B. Ansatz We can now write the Faraday Cup normalized counts defined in the previous Section for all four targets as sums of contributions from entrance and exit foils (F), liquid Helium-4 coolant (He), Carbon-12 (C), Nitrogen-15 (N) and Hydrogen or Deuterium (H/D) as follows: n MT = (ρ F l F σ F + ρ He Lσ He )F = (fρ C l C σ C + ρ He Lσ He ) F (11) n C = ((1 + f)ρ C l C σ C + ρ He (L l C )σ He ) F (12) n A = (fρ C l C σ C + ρ He (L l A )σ He + ρ A l A (σ N + 3σ H/D ))F (13) Here, F is a common factor that includes the conversion from mol/cm 2 and Faraday cup clicks to luminosity, and the acceptance and efficiency of CLAS for the given kinematic bin. We define two new spectra which can be calculated from the empty and Carbon target spectra: n 12 C = L n C L l C n MT = ρ C l C σ C F(14) L + fl C L + fl C n 4 He = 1 + f f n MT n C = ρ He σ He F(15) L + fl C L + fl C The second of these can be used to account for the difference in the amount of helium in ammonia targets versus the carbon target; however, one has to make a correction to n MT before using it in this (and the following) application(s) due to the fact that external radiative effects are significantly more important in carbon and ammonia targets than in empty targets. This has been taken into account for the case of inclusive EG1b analysis by calculating radiative correction factors for each bin in W and Q 2, based on Peter Bosted s radiated cross section models. These correction factors can be found in a set of tables, once again in Rob Fersch s CUE directory /home/fersch/eg1b/analysis/asymmetry/ under the filenames MTcorr1.606GeV.txt MTcorr1.723GeV.txt MTcorr2.286GeV.txt MTcorr2.561GeV.txt MTcorr4.238GeV.txt MTcorr5.615GeV.txt MTcorr5.725GeV.txt MTcorr5.743GeV.txt These arrays represent numbers that should multiply the number of counts in the empty target before using them in the equations above and below. It is somewhat difficult to know how to apply the corresponding corrections for more exclusive channels (see next Section). For the final step, we need to come up with a scale factor that allows us to predict the cross section from the 15 N in ammonia in terms of the cross section for 12 C, per nucleus. There is an elaborate discussion on this topic in the original EG1b analysis note that is by now largely irrelevant (especially for the present purpose) and will not be reproduced here. For all practical purposes, we can assume that σ15 N 7 ( ) σ n = 6 + σ n, (16) where σ n is the cross section on a bound neutron in 15 N. Unfortunately, it is not always easy to figure out what σ n/ for a given reaction might be, although this may actually be a lesser problem for exclusive channels than for inclusive ones! Using this scale factor, we can now write down the number of counts coming from the non-hydrogen (deuterium) part of the full ammonia target, i.e. the counts that dilute our asymmetry signal: [ ( ) ρa l A 7 n A D = ρ C l C 6 + σ n + f + (L l A )n 4 He = n MT + + l A [ ρa ρ C l C ( σ n The dilution factor then becomes simply ] n 12 C + ) n 12 C n 4 He ] (17) DF = n A n A D n A. (18) We should add that for the inelastic part of the inclusive spectra, we used a different approach in EG1b (employing Peter Bosted s cross section models); however, for the (quasi-)elastic part (relevant for determinations of P B P T ), the method outlined above was deemed more appropriate. Since exclusive channels (with cuts on missing mass) are more akin to inclusive elastic scattering, we believe that this approach (modified as explained below) should also work for this case. V. APPLICATION OF THIS METHOD TO NON-INCLUSIVE CHANNELS The method outlined in the previous section should be directly applicable to all exclusive channels discussed earlier in this note. In particular, the relevant parameters L, l A etc. should be determined from the best and most precise method available; usually this would be inclusive scattering and direct measurements. Once all parameters are fixed, the equations in the previous section can be

6 6 simply evaluated using counts within multi-dimensional bins as needed for the exclusive analysis. There is (or at least should be) no need to cross-normalize, and in particular, all parameters should be independent of the bin kinematics. There are two problems that need to be solved, however, to make this method applicable. First, for each of the exclusive channels considered, a reasonable estimate for σ n/ is needed. (This problem is only present for experiments using 15 N in their ammonia targets; EG1- DVCS which used 14 N can simply ignore this term). Fortunately, to first order, one can make the simplifying assumption that the channels p(e, e π + )n and d(e, e p)n can only occur on protons inside the target, so in this approximation σ n/ = 0 for these channels. Vice versa, for d(e, e π p)p and d(e, e π )pp, one can reasonably assume that the reaction occurs only on neutrons, and therefore one can approximate σ n / = 1/6 as there are 6 bound neutrons in 12 C. However, one should at least be aware of possible complications - for instance, in nuclei, charge exchange reactions can occur in the final state, where the outgoing particle has a different charge than the one first produced. Therefore, some uncertainty has to be assigned to the value used for σ n / in each case. Unfortunately, for the channel p(e, e p)π 0, things might be more complicated. Ideally one would use a reasonably realistic prediction for the ratio σ n /σ p for this cross section, and then approximate σ n 1 σ n /σ p. (19) σ n /σ p The second problem is the correction for empty target runs necessary to account for the larger external radiation effects in carbon and ammonia targets. It turns out that this correction is relatively small for (quasi-)elastic scattering, where there is no background to subtract. For similar reasons, one would expect a relatively small correction for most exclusive channels; one very crude approach would be to use the average value of the correction for the region 0.9 < W < 1.0 for inclusive scattering as a first approximation. A more careful treatment of this correction is beyond the scope of this note; one should note, though, that this correction is of higher order since the whole empty target contribution to the final dilution factor itself is already a relatively small correction. This research is supported by the US Department of Energy under grant DE-FG02-96ER [1] P. E. Bosted et al. [CLAS Collaboration], Phys. Rev. C 78 (2008) [arxiv: [nucl-ex]]. [2] R. Fersch, Ph.D. thesis (William and Mary, 2008). [3] R. De Vita et al.: Phys. Rev. Lett. 88 (2002) [4] A. Biselli et al.: Phys. Rev. C 68 (2003) ; A. S. Biselli et al. [CLAS Collaboration], Phys. Rev. C 78, (2008) [arxiv: [nucl-ex]]. [5] In all cases, one must divide the Faraday Cup counts by the list number under the AVG column. The indices 1 through 12 correspond to the following EG1 settings: 1= 1.6in, 2= 1.6out, 3= 5.76out, 4= 5.73out, 5= 5.7in, 6= 2.3in, 7= 5.6in, 8= 1.7out, 9= 2.5out, 10= 2.5in, 11= 4.2in, 12= 4.2out.