Correlated forward-backward dissociation and neutron spectra as a luminosity monitor in heavy ion colliders ∗ Anthony J. Baltz, Chellis Chasman, and Sebastian N. White Brookhaven National Laboratory, Upton, New York 11973 (January 12, 1998) 8 9 9 1 n Abstract a J 3 1 Detection in zero degree calorimeters of the correlated forward-backward 1 Coulomb or nuclear dissociation of two colliding nuclei is presented as a prac- v 2 tical luminosity monitor in heavy ion colliders. Complementary predictions 0 0 1 are given for total correlated Coulomb plus nuclear dissociation and for cor- 0 8 related forward-backward single neutrons from the giant dipole peak. 9 / x PACS: 25.75.-q; 29.40.Vj; 29.27.-a e - l keywords: Heavy ions; Calorimeter; Luminosity; Collider; Dissociation c u n : v i I. INTRODUCTION X r a In a heavy ion collider such as RHIC the cross section for Coulomb dissociation of one of the ions in a 100 GeV × 100 GeV Au + Au collision will be many times the geometric cross section, about 95 barns [1]. With such a large cross section available, it has been suggested that the correlated forward-backward Coulomb dissociation would provide a clean monitor of beam luminosity with the very forward (zero degree) calorimeters proposed for RHIC [2]. In fact the calculated cross section for mutual Coulomb dissociation was found to be quite large, about 4 barns. But there is an intrinsic limitation in the precision of ∗ Corresponding author. Tel.: 1-516-344-5488; fax: 1-516-344-3253; e-mail: [email protected]. 1 the calculated mutual Coulomb dissociation: a lower cutoff in impact parameter must be chosen, and the calculated mutual Coulomb dissociation varies approximately as the inverse square of this cutoff. However, it was found that if one adds nuclear dissociation to the mutual Coulomb dissociation calculation, the cutoff merges into the nuclear surface and the calculated cross section of 11 barns is relatively insenstive to parameter variation [3]. Details of these calculations are presented in Section III below. The proposed zero degree calorimeter [4] detects the total neutral energy in the forward direction. Forneutronsofthebeammomentumthedetectedsignalvariesthenasthenumber of neutrons. While the dissociation cross section includes excitations of the nucleus up to many GeV, the largest contribution is for excitations of less than about 25 MeV, the region of the giant dipole resonance, amounting to about 65 barns. Photonuclear data [5] indicate that the excited state decays predominantly by single neutron emission in this region; we will show that the Coulomb dissociation to a single neutron final state is large (50 barns) and contained in a single energy peak within about 10% of the beam energy. In Section IV we present details and consider the detection of correlated single neutron peaks. Based on the above we will present several complementary methods of using calculations of mutual Coulomb plus nuclear dissociation along with zero degree calorimeter measure- ments as an absolute luminosity calibration at heavy ion colliders such as RHIC and LHC. But first we will begin with some experimental considerations. II. EXPERIMENTAL CONSIDERATIONS The primary quantities of interest are: (1) the luminosity, (2) the distribution and cen- troid of the interaction region along the beam direction, and (3) the distribution in time of the interactions relative to the nominal bunch crossing time. A basic requirement on the reaction chosen for these measurements (particularly of luminosity) is that it be relatively background free and that it have a straightforward acceptance, implying simple correction of measured rates for experimental details. Finally, if this reaction is to be used as an absolute 2 luminosity determination, it will be necessary to reduce uncertainties in the calculated cross section to ≤ 5% or better. We deal below with the issues pertaining to luminosity. The other principal measurements rely on the timing resolution of the neutron detectors. What is the rate from which lumnosity is calculated? We propose to measure the coinci- dence rate requiring forward and backward energy of ≥ 60 GeV within 2 mrad of each beam direction. These requirements are largely chosen to minimize uncertainties in the calculated cross section. The coincidence requirement is also necessary to reduce contamination from background sources. The60GeVthresholdchoice(assuming100GeV/ubeamenergy)issimplyarequirement of at least 1 fragmentation neutron emitted into the detector and allows for linewidth and experimental resolution. (A resolution of ≤ 20% at 100 GeV is expected for a practical detector). Since at a collider the forward direction is only accessible downstream of beam steering magnets, the contribution from protons and charged fragments is expected to be negligible [6,7]. Further specifics are detailed in the following. A. Energy Threshold In order to include a real energy threshold in the coincidence requirement we consider it essential that some fraction of triggers populate an energy peak which can be clearly resolved from the continuum. The peak will be used to adjust trigger energy scale and to compute inefficiencies after the fact. We examine below the energy spectrum of neutrons emitted inCoulomb dissociation. Singleneutron emission turns outto beprominent because of the importance of the giant dipole resonance; the natural energy spread is smaller than the expected detector resolution. B. Experimental Backgrounds There are two principal types of experimental background that concern us, those from beam interactions (which are therefore proportional to luminosity) and those which arise 3 from single beam interactions with residual gas in the machine, for example, (and do not depend directly onluminosity). The coincidence requirement is designed tolargely eliminate the latter. Single beam rates can be estimated from the nominal conditions and vacuum require- ments called for in the experimental areas. Beam gas rates are expected to be lower than the minimum bias interaction rate at full luminosity (i. e. the signal to be measured, roughly 7 barns cross section for Au on Au). Since the actual minimum bias rate can be signifi- cantly lower than the nominal design value during commissioning the beam gas rate could become significant. Similarly, rates due to interactions of the beam halo, which are difficult to calculate a priori, could also be a large background source. The requirement of forward-backward coincidences greatly reduces the full counting rate due to these sources except for accidentals. It also makes possible the secondary measure- ments described above. The spatial distribution of interactions is calculated from forward- backward time differences. Assuming a time resolution of order 200 psec, we calculate a spatial resolution of a few cms. Backgrounds due to accidental coincidences between the two beam directions, including contributions from beam-beam interactions such as single beam Coulomb dissociation (95 barns at RHIC), are found to be negligible. C. Luminosity Calibration and Cross section Uncertainties Aswasindicated intheIntroduction, mutualCoulombdissociationcalculations areprob- ably limited to an uncertainty of not less than about 5%. There is a lower impact parameter cutoff in the Weizsacker-Williams formalism that has no clear experimental counterpart. In fact, it is difficult to define an experimental cut which would clearly distinguish between strong interaction final states and those that arise from the high energy component of the Weizsacker-Williams spectrum. For this reason in Section III we examine the possibility of calculating a cross section which isactually thesum of Coulomb andnuclear processes leading to theforward-backward 4 coincidence of neutral beam energy clusters. We find that this almost completely eliminates the above uncertainties. D. Beam Fragmentation in Nuclear Collisions Before we perform this calculation of Coulomb plus nuclear dissociation we must address the question of the fraction of nuclear interactions, as a function of impact parameter, in which heavy ion collisions lead to a neutral beam energy cluster within 2 mr of the beam direction. Fortunately this measurement has been recently performed in a dedicated test at the CERN SPS [8]. For our purposes the probability of producing less than one neutral forward cluster, even in a central collision, is negligible. In what follows, we take the probability of a neutral beam energy cluster within 2 mr of the beam direction to be unity. III. MUTUAL COULOMB PLUS NUCLEAR DISSOCIATION Thecrosssection forheavy-iondissociationmaybeaccuratelyexpressed intermsof(gen- erally experimentally known) photo-dissociation cross sections σ (ω) of the same nucleus ph over an appropriate energy range of photon energies ω. This is the so-called Weizsacker- Williams expression 2αZ2 ∞ bω σ = p dωωσ (ω) bdbK2( ), (1) dis πγ2 Z ph Z 1 γ b0 where K is the usual modified Bessel function. The lower cutoff of the impact parameter 1 integral b is normally fixed at a value to separate pure Coulomb excitation from nuclear 0 processes. b provides a somewhat arbitrary parameter dependence which we deal with 0 below. We may define a probability of dissociation, P(b) as a function of impact parameter b ∞ σ = 2π P(b)b db. (2) dis Z b0 Then inverting the order of integration in Eq. (1) we have 5 αZ2 bω P(b) = p dωωσ (ω)K2( ). (3) π2γ2 Z ph 1 γ If we assume independence of the various modes of dissociation, then the probability of at least one dissociation excitation of the nucleus is given by the usual Poisson distribution. If the first order probability of Coulomb dissociation at a given inpact parameter b is P (b), C then the probability of at least one mutual Coulomb dissociation Pm(b) is given by C Pm(b) = (1−e−PC(b))2. (4) C The corresponding probability of at least one mutual nuclear dissociation Pm(b) is given in N terms of a first order probability P (b) by N Pm(b) = (1−e−PN(b)). (5) N Note that there is no square here; all nuclear dissociation is mutual. P (b) is evaluated from N the Glauber model: P (b) = dxdydz ρ (x−b,y,z )(1−eσNN dztρt(x,y,zt)). (6) N Z p p p R The densities ρ ,ρ are parameterized with a Fermi function. For Au the charge distribution p t parameters are R = 6.38,a = .535 fm [9]. From Hartree-Fock calculations one finds that C the neutron radius should be a little larger [10]. We take the neutron radius of Au to be 6.6 fm and then average for a best overall radius of 6.5. At RHIC energies the total nucleon-nucleon cross section σ is 50 mb [11]. NN Consider now the probability of survival without mutual Coulomb or nuclear dissociation P (b). It istheproduct ofthe separateprobabilities. Since allnuclear dissociationis mutual, S the nuclear-Coulomb mutual dissociation contribution is redundant. We have P (b) = (1−Pm(b))(1−Pm(b)) S C N = e−PN(b)(2e−PC(b) −e−2PC(b)). (7) The probability of mutual excitation is then 6 P(b) = 1−e−PN(b)(2e−PC(b) −e−2PC(b)). (8) Let us take the standard design case of 100 GeV + 100 Gev Au + Au at RHIC. Seen in the frame of the nucleus being dissociated, the equivalent γ of the other ion providing the equivalent photons is 23,000. The experimental photo-dissociation cross section σ (ω) that ph we utilized is shown in Fig. 1, which we have taken from Ref. [1]. In Table I we present calculated first order and unitarity corrected mutual Coulomb dissociation cross sections for two values of b2, 2.25 barns and 3 barns, corresponding to 0 r values of 1.29 and 1.49 respectively when one expresses b in term of 2×r A1/3. Cross 0 0 0 sections are tabulated asa function of a cutoff in ω, which should correspond in some fashion to an experimental acceptance. For comparison, the corresponding cross sections for 2.76 TeV + 2.76 TeV Pb + Pb collisions at LHC are shown in Table II. Fig. 2 shows the calculated probabilities of correlated forward-backward dissociation as a function of impact parameter. At impact parameters of 13 fm and below the mutual dissociation probability becomes unity from the nuclear dissociation alone. Table III shows the parameter dependence of correlated dissociation cross sections. The first four rows show how the calculated mutual dissociation cross section becomes independent of the cutoff parameter for values less than about 15 fm. The fifth row compared to the second row shows how a small increase in the nuclear size parameter causes a small increase in the computed cross section. The sixth row shows how a 20% reduction in the nucleon-nucleon cross section leads to a less than 1% reduction in the mutual dissociation cross section. The overall conclusion is that we can predict a mutual Coulomb plus nuclear dissociation cross section of 11 barns with an error of less than about 5%. IV. FORWARD-BACKWARD ENERGY PEAK From Eq.(1) we can write down the differential Weizsacker-Williams cross section for heavy-ion dissociation 7 dσ 2αZ2ω ∞ bω dis = p σ (ω) bdbK2( ). (9) dω πγ2 ph Z 1 γ b0 σ (ω) is the photon cross section; in the present calculation we take it from the photo- ph dissociation data leading to only a single neutron in the final state [5]. The square of the Bessel function gives the the energy and impact parameter dependent number of equivalent photons from the heavy ion. The impact parameter integral may be approximated very accurately for b ω << γ to 0 yield ∞ bω γ2 2γ γ2 .681γ bdbK2( ) = [ln( )−γ −.5] = ln( ). (10) Z 1 γ ω2 b ω euler ω2 b ω b0 0 0 Putting in the factor of h¯c explicitly we obtain the familiar form dσ 2αZ2 .681h¯cγ dis p = σ (ω)ln( ). (11) ph dω πω b ω 0 The energy dependence of the (γ, 1n) cross section on Au [5], σ (ω), is shown as the ph dashed line in Fig. 3. The Au + Au cross section for single neutrons in either forward beam direction, obtained fromEq. (11), is shown as the solid line in Fig. 3. Notes the difference in scale on the plot of the two cross sections. Setting the impact parameter cut off b to 15 fm 0 and integrating from the threshold at 8.1 MeV up to 24.0 MeV, we find that the Coulomb excitation to a single neutron final state, σ , is 50.6 barns. dis The angular distribution of neutrons emitted from the giant dipole resonance has been parameterized in the form A+Bsin2(θ), and for Au A/B has been measured to be .58/.38 [12]. If one relativistically boosts the emitted soft neutron energy to the lab frame, then one obtains the spectra of Fig. 4. The solid line is the predicted spectrum from the experimental ratio of A/B. For comparison, a purely isotropic distribution (dot-dashed line) and a sin2(θ) distribution (dashed line) are also shown. From Fig. 4 it is clear that no matter what the the angular distribution of the emit- ted neutrons is, there will be a huge one neutron peak of about 50 barns. Because of its size, this peak will stand out with very little background: other single neutron contribu- tions from higher energy dissociation will be relatively negligible [13] [14]; soft two neutron 8 spectrometer contributions will be centered at about twice the single neutron energy of 100 GeV; higher numbers of neutral particles will be correspondingly higher in energy. Thus the large, well characterized single neutron peak provides an ideal component for a Au + Au beam luminosity monitor in the forward calorimeter at RHIC. Now let us consider mutual Coulomb dissociation in which one or both of the reaction products is a single neutron at the beam momentum. The previous analysis made use of the lowest order expression Eq. (11) for the single neutron cross section which is not properly unitarized and contains a logarithmic dependence on b . A more accurate expression may 0 be obtained in analogy with Section III by again assuming a Poisson distribution in the number of excitations. Then the probability of one and only one neutron excitation P (b) 1n may be expressed terms of the first order probability of one neutron excitation P1 (b) and 1n the normalization factor e−PN(b)−PC(b) which involves all excitations P (b) = P1 (b)e−PN(b)−PC(b). (12) 1n 1n Note that the normalization factor provides a natural impact parameter cutoff; we will not need a dependence on b in our cross section expression Eq. (2). Multiple neutron final 0 states completely dominate for impact parameters smaller than grazing. The expression for mutual Coulomb dissociation in which both of the neutral reaction products are single neutrons is the square of the Eq. (12) expression. The analagous ex- pression for mutual Coulomb dissociation in which one reaction products is a single neutron and the other is any excitation is P (b) = P1 (b)e−PN(b)−PC(b)(1−e−PN(b)−PC(b)). (13) 1n,xn 1n Table IV shows the dependence of computed cross sections on the radius parameter of the nuclear density. For Au + Au 6.38 corresponds to the the radius of the proton density while 6.50 is probably a more realistic value, an average between a proton density and an expected neutron density. In any case the dependences are not large. The mutual cross sections vary by about 1% over this range while the uncorrelated cross sections vary by less than a tenth of one per cent. 9 The correlated single neutron cross section is predicted tobe .45 barns and the correlated cross section fora single neutronin onespecified detector alongwith any neutral in theother (including possibly a single neutron) is three times that, 1.35 barns. For completeness the 11 barn cross section of Table III for any mutual excitation is repeated. Corresponding cross sections for Pb + Pb at LHC have also been computed and are also displayed in Table IV. For Pb the charge distribution parameters are R = 6.624,a = .549 C fm [9] and a more realistic nucleon matter radius is taken to be 6.65. At LHC energies the total nucleon-nucleon cross section σ is taken to be 85 mb [11]. Again radius dependences NN are not large. One might consider using the predicted 49 barn single neutron peak as a luminosity monitor. The limitation ofthis method comes from theluminosity independent backgrounds due to beam gas and beam halo interactions discussed in Section II. The 102 barn nuclear plusCoulombdissociationcrosssectionsuffersfromthesamelimitationswithoutthepositive energy signal of the neutron peak. Requiring a forward-backward single coincidence should largely eliminate these backgrounds with the experimental advantage of having at least one the neutron peaks. V. Z DEPENDENCE We have performed calculations involving mutual Coulomb dissociation for Au + Au and Pb + Pb reactions to exploit multi-barn cross sections. Unfortunately these large cross sections do not persist to collisions of lighter ion species: mutual dissociation cross sections scale approximately as Z6. This is because single Coulomb dissociation scales as Z2 times the number of nucleons for the highest excitations or as Z2 times (NZ)/A for the giant dipole resonance. Table V presents some representative comparisons. Clearly for ions as light as Ca + Ca and O + O correlated forward-backward energy deposit will be dominated by purely nuclear collisions, and will lead to a simple geometric mutual cross section. 10