PHYSICAL REVIEW C 81, 044910 (2010)
Production of exotic atoms at energies available at the CERN Large Hadron Collider
C. A. Bertulani and M. Ellermann Department of Physics and Astronomy, Texas A&M University–Commerce, Commerce, Texas 75429, USA (Received 17 March 2010; published 29 April 2010)
We study in details the space-time dependence of the production of muonic, pionic, and other exotic atoms by the coherent photon exchange between nuclei at the Large Hadron Collider at CERN. We show that a multipole expansion of the electromagnetic interaction yields an useful insight of the bound-free production mechanism which has not been explored in the literature. Predictions for the spatial, temporal, and angular distribution, as well as the total cross sections, for the production of exotic atoms are also included.
DOI: 10.1103/PhysRevC.81.044910 PACS number(s): 25.75.−q, 25.20.Lj, 13.60.Le
I. INTRODUCTION Protonium has been studied theoretically mainly by using nonrelativistic quantum mechanics, which yields predictions It is undeniable that ultraperipheral collisions (i.e., colli- for its binding energy and lifetime. The lifetimes are predicted sions dominated by the electromagnetic interaction) between in the range of 0.1–10 µs. While protonium production has relativistic heavy ions allows interesting physics to be probed. a very small cross section in p-p collisions at CERN, the As far back as 1989 it was noticed that even the Higgs boson cross section is appreciable for heavy ions, e.g., for Pb-Pb could be produced at comparable rates as in central collisions collisions. In this case, the antiproton will be captured in between relativistic heavy ions [1,2], the advantage being an orbit around one of the Pb nuclei. Such a system would that ultraperipheral collisions are cleaner in the sense that be of large interest for understanding internucleon forces in nothing else than the Higgs boson would be observed. By charge-conjugate channels. now, ultraperipheral collisions are known as an excellent tool Most of our calculations will be for the production of for several interesting phenomena and have been discussed in muonic atoms and p-p collisions. There is no qualitative numerous publications, reviews, and popular articles (see, e.g., difference for the production of other exotic atoms via the Refs. [3,4]). same mechanism, except for the obvious reduction of the A process of interest in ultraperipheral collisions with production yields due to mass differences. The production relativistic heavy ions is the production of pairs in which one cross sections for Pb-Pb collisions will be enhanced by a huge of the particles is captured in an orbit around one of the ions amount, approximately equal to 1010 − 1012, but the details in the collider (“bound-free” pair production). In particular of the production mechanism are practically the same as in the process of pair production with capture was used as a p-p collisions. In Sec. II we show how a multipole expansion tool to generate the first artificial antihydrogen atom in the allows for a clear space-time description of the bound-free laboratory [5]. The pioneer CERN experiment was followed by production mechanism. Useful approximate formulas are another experiment at Fermilab [6]. The number of antiatoms derived. In Sec. III we present our numerical results and produced was shown to be in good agreement with detailed discuss the physics properties of our results. Our conclusions theoretical predictions [7]. are presented in Sec. IV. In this article we extend these studies by considering the production of exotic atoms in p-p and heavy-ion collisions such as those being carried out at the Large Hadron Collider (LHC) at CERN. In our notation, an exotic atom is an atom II. PRODUCTION OF EXOTIC ATOMS IN ION-ION in which one electron has been replaced by other particles of COLLISIONS the same charge. In contrast to previous approaches, we show In the frame of reference of one nucleus, the time-dependent that a multipole expansion of the electromagnetic field yields electromagnetic field generated by the other nucleus is given several insights in the production mechanism which have not by the Lienard-Wiechert potential been explored in the literature. Our calculations apply to the = = production of muonic atoms, pionic atoms, protonium, and Aµ(r,t) vµφ, with vµ (1, v), and so on. The Bohr radius for a muonic atom is much closer Ze2 φ(r,t) = , (1) to the nucleus than in an ordinary atom, and corrections due |R − R (t)| to quantum electrodynamics are important. The energy levels and transition rates from excited states to the ground state where of muonic atoms also provide experimental tests of quantum r = (x,y,z), R = (x,y,z), R = (b ,b ,vt), electrodynamics. Hadronic atoms, such as pionic hydrogen and x y kaonic hydrogen, also provide interesting experimental probes and v is the relative velocity between the nuclei which we of the theory of quantum chromodynamics. A protonium is will take to lie along the z axis. is the relativistic Lorentz antiprotonic hydrogen, a composite of a negatively charged factor = 1/(1 − v2)1/2 (otherwise explicitly stated, here we antiproton paired with a positively charged proton or nucleus. use the unitsh ¯ = c = 1).
0556-2813/2010/81(4)/044910(6) 044910-1 ©2010 The American Physical Society C. A. BERTULANI AND M. ELLERMANN PHYSICAL REVIEW C 81, 044910 (2010)
In first-order perturbation theory, the pair-production am- For the negative particle we use the distorted hydrogenic plitude at time t for a collision with impact parameter b = wave function [7] for capture to the ground state, 2 + 2 bx by is given in terms of the transition density ρ(r) and 1 Z 3/2 i the current density j(r), as = √ − 0γ · ∇ − −(r) 1 γ u(0)exp( Zr/a) , t π a 2 1 iωt 3 µ a1st(p,b,t) = dt e d rjµ(r)A (r,t ), (2) (7) i −∞ where ω = + m − I, with equal to the energy of the where u is the negative particle spinor, 0 is the energy of = 2 free positive particle and I being the ionization energy of the exotic atom bound-state, and a 1/me is the hydrogen the negative captured particle; m is the rest mass of either Bohr radius. As noted in Ref. [7], the reason why we need particle. The transition current is given in terms of the Dirac to keep the corrections in the wave functions to first order in Ze2 is because for m these corrections yield a term of the matrices γµ, and the particle and antiparticle wave functions, same order of magnitude as the plane-wave to the total cross i.e., j (r) = e−γ +, where − is the wave function for µ µ section. The reason for this are the small distances which enter the captured negative particle and + that of the free positive in the calculation of the integral in Eq. (2). The wave function particle. corrections are essential to properly account for the short The multipole expansion of j Aµ has been extensively µ distance effects. Such effects are not as important in the discussed in details in Refs. [8–10]. Replacing the Schrodinger¨ case of bound-free production of heavy particle-antiparticle by the Dirac currents in their results, one finds that pairs, but we keep them for a more accurate description of the j (r)Aµ(r,t) = V (r,b,t), where π = E,M and l = µ πlκ πlκ process. 0, 1, 2,..., and κ =−l,...,l denote the multipolarities. For To gain insight into the pair production probability with electric E1 (electric dipole) and E2 (electric quadrupole) capture and its distribution in space-time, we will compare our multipolarities, numerical results with simplified calculations which neglect
VE1κ (r,b,t) the above mentioned short-distance corrections of the wave function of the captured negative particle as well as of the free = − (r) (1 − γ0γz)rY1κ (rˆ) + (r) √ positive particle. This is a reasonable approximation for small Ze2 2π/3 ∓b, (if κ =±1) values of the free particle energy, i.e., for ∼ m,asweshow × √ (3) (b2 + 2v2t2)3/2 2vt (if κ = 0), later. This approximation allows us to obtain the coordinate integral in a1st analytically by using V (r,b,t) E2κ √ 2 3/2 2 Ze 3π/10 1 Z 3 −ip·r λ −Zr/a = − (r) (1 − γ0γz)r Y2κ (rˆ) + (r) F (p) = √ d re r Y (rˆ)e (b2 + 2v2t2)5/2 λκ a λκ ⎧ π ⎧ ⎪ 2 =± xp = ⎨ b , (if κ 2) ⎨ + 2 3 Y1κ (pˆ)forλ 1, √ a 5/2 (1 xp ) 2 = × ∓( + 1)bvt, (if κ =±1) (4) 32 π 2 ⎪ √ Z ⎩ − 6(a/Z)xp = ⎩ + 2 4 Y2κ (pˆ)forλ 2, 2/3(22v2t2 − b2)(ifκ = 0). (1 xp ) (8) In addition, for magnetic dipole (M1) [10], where xp = ap/Z and pˆ is the unit vector for the positive VM1κ ( r,b,t ) particle momentum direction with respect to a z axis pointing pz along the beam direction. To calculate total cross sections we = − (r) (1 − γ0γz)γzrY1κ (rˆ) + (r) m √ keep the wave function corrections, Eqs. (6) and (7). In this Ze2 2π/3 ±b, (if κ =±1) case, some of the coordinate integrals need to be performed × (5) (b2 + 2v2t2)3/2 0(ifκ = 0). numerically. The exotic atom production probability density, at time t For the positive particle wave function we use a plane wave and for a collision with impact parameter b, is obtained by the and a correction term to account for the distortion due to square of expression (2), i.e., the nuclear charge [7]. The wave function is given by + = 2 N [v(p)exp(ip · r) + +], where N () = exp(πa+/2)(1 + P(p,b,t) = |a1st(p,b,t)| , (9) 2 ia+), with a+ = Ze m/p, with p being the proton momentum, spins 2 v(p) the particle spinor, and |N ()| = 2πa+/[exp(2πa+) + where the sum over spins is performed with standard trace 1]. The correction term + is given by Eq. (B5) of Ref. [7], which for our purposes can be written as techniques. The production probability, at time t and for a collision 2 0 + γ · − Ze 3 −iq·r 2γ i (q p) with impact parameter b, is obtained by integration over +(r) = v(p) d qe , (6) 2π 2 (p − q)2(q2 − p2) momentum, 3 = 2 + 2 d p with p m . Parts of this integral can be done P (b, t) = P(p,b,t). (10) analytically. (2π)3
044910-2 PRODUCTION OF EXOTIC ATOMS AT ENERGIES... PHYSICAL REVIEW C 81, 044910 (2010)
If one neglects the derivative corrections in the particle and of the potential in terms of a momentum transform, as done in antiparticle wave function, the sum over spins in Eq. (9) Ref. [7], i.e., yields Ze eiq·[R−R (t)] φ(r,t) = d3q . (13) 2 2 3 √ 2 2π q d p P (b, t) = 4m + mN ()Fλκ (p) (2π)3 This introduces extra integrations over the virtual momentum λκ q increasing considerably the numerical effort [7]. 2 × |Gλκ (b, ω, t)| , (11) Recently, it was shown that one can treat the effects of close collisions (R t 1 1 π iωt close = − ∗ 2 { + } dt e VE1κ (r,b,t ) δκ,0n (r) (1 γ0γz) Y10 (rˆ) p (r) 4 Ze v g0 () c0g2 () i −∞ r2 3 ⎧ ⎪ Rω Rω Rω ⎨⎪ sin − cos if R vt 1 v v v × , (16) ω2 ⎪ Rω ⎩⎪ e−iRω/v i − + ieitω(itω − 1) if R>vt v For 1, g0() + c0g2() = 2ln(2) − 15/2. When t = Note that we keep the relative velocity v between the ∞ only the upper term in the last part of this equation colliding nuclei in all formulas, although v ∼ c.Thisis contributes (R<∞) to the production probability. necessary because sometimes important combinations of 1 and We use the above separation of close and distant collisions v conspire and combine into Lorentz factors = (1 − v2)−1/2 to make an estimate of their relative contributions to the in subsequent steps of the calculations. total cross section. To obtain the total cross sections we use the formalism developed in Ref. [7] which allows a better reduction of the integrations in momentum space. Adapting III. RESULTS their results for ion-ion collisions, we get At the Large Hadron Collider (LHC) at CERN the Lorentz γ factor in the laboratory frame, lab, is 7000 for p-p and 3000 for Pb-Pb collisions. The relationship between the Lorentz 256πZ7e4 ∞ p ∞ q(q2 + ω2/2v2) contraction factor associated with the relative velocity between σ = d |N ()|2 dq the colliding nuclei, and the collider energy per nucleon, 5 2 − 2 2 2 2 3a m−I ω 0 (q ω / v ) = 2 − 2 E/A, in GeV, is given by 2(lab 1) 2(1.0735E/A) . (p2 + 3q2 + 3ω2/v2)(3p2 + q2 + ω2/v2) This means that for the production of exotic atoms we have × . (17) (q2 + ω2/v2 − p2)6 effectively 108 for p-p and 107 for Pb-Pb collisions. 044910-3 C. A. BERTULANI AND M. ELLERMANN PHYSICAL REVIEW C 81, 044910 (2010) 1.2 ) ∞ 0.8 E1 (b, E1 0.4 P(b,t)/P E2 0 -2024 ct [10-6 fm] FIG. 1. (Color online) Relative probability PEλ(b, t)/PE1(b, t = FIG. 2. (Color online) Coordinate distribution in the plane along ∞), for production of a muonic atom in p-p collisions at the LHC the beam axis of the probability for production of muonic atoms with with an impact parameter b = amuon = 255 fm, where amuon is the capture in the K shell with p-p collisions at the LHC. The impact = = Bohr radius for a muonic atom. parameter is chosen as b amuon 255 fm and is along the vertical axis. In Fig. 1 we plot the relative probability =∞ The above discussion implies that in order to calculate pair PEλ(b, t)/PE1(b, t ), for production of a muonic production with capture in ultraperipheral collisions at the atom in p-p collisions at the LHC with an impact parameter LHC one needs to consider only the E1 field, with κ =±1, b = amuon = 255 fm, where amuon is the Bohr radius for −8 in Eq. (3). The asymptotic production probability amplitude, a muonic atom. The time scale is in units of 10 fm/c. Eq. (2), for a given impact parameter b, is then given by The probability is normalized so that it is equal to the asymptotic value of the E1 multipolarity. In absolute Ze2 2π values the probabilities are very small, justifying the use of aκ=±1(p,b) = 2κi ξK1(ξ) perturbation theory. The E2 probability tends to its asymptotic vb 3 value faster than E1. This can be understood from Eqs. (3) 3 × d rn (r) (1 − γ0γz)rY1κ (rˆ) p (r) , and (4) as the time dependence of the E2 field is determined by a higher inverse power of time. The asymptotic value of (18) the E2 probability is about a factor 4 smaller than the E1 case. As a function of the impact parameter, the E2 probability where ξ = ωb/ v and K1 is the modified Bessel function 2 decreases with an additional 1/b dependence as compared to of first order. Note that ξK1(ξ) 1forξ 1. For ξ>1, E1. This yields cross sections for pair production with capture ξK1(ξ) drops to zero exponentially. This means that the 2 due to the E2 field being much smaller than that with E1. production probability drops as 1/b up to bmax ≈ hc/m¯ ≈ The probabilities and cross sections are also much smaller 1010(MeV fm)/mc2 for p-p collisions at the LHC. For pro- in the case of the M1 multipolarity, due to the factor (pz/m) duction of muons and pions with capture this means bmax ≈ in Eq. (5). This is also substantiated√ by the approximation√ in 108 fm, whereas for proton-antiproton with capture, this means 7 Eq. (11), which has a factor − m instead of the + m bmax ≈ 10 fm. which appears in the electric multipole cases. For low positive In Fig. 2 we show the coordinate space distribution for particle momentum this leads to a further suppression of the production of muons with capture in the K shell in p-p M1 multipolarity. collisions at the LHC. The distribution is shown in a plane We thus conclude that the E1 multipolarity alone is containing the beam direction and the impact parameter vector. responsible for most part of the bound-free pair-production The calculation is done for the E1 multipolarity and for an probability. This comes as no surprise because of the very large impact parameter b = amuon. The positive muon energy is factor in the frame of reference of the exotic atom. The spatial taken as = 1.1mµ and its direction of emission is chosen distribution of the time-dependent field is compressed as a as 100◦ when measured along the direction of motion of the pancakelike object with a spatial width z = c t = b/ .For muonic atom. One notes that the production mechanism is b = 200 fm this is equal to z 10−6 fm for the LHC. Even more efficient in regions perpendicular to the beam axis. The for very large impact parameters, e.g., 1A˚ = 105 fm, z is darker areas are representative of larger production rates in small compared to the nuclear sizes. The E2 field is a measure coordinate space. Figure 3 shows the same result but as seen of the “tidal” force, proportional to the spatial spreading of the in the plane perpendicular to the beam axis. In this case, the electric field [3]. This “tidal” effect becomes larger at smaller impact parameter vector lies along the x direction. We observe velocities, when the field lines are not as compressed. The that the probability density is largest within a toruslike region large value of also leads to a complete dominance of the with a radius r ≈ amuon from the origin of the muonic atom, κ =±1 component of the E1 field in Eq. (3). with the torus axis along the beam direction. 044910-4 PRODUCTION OF EXOTIC ATOMS AT ENERGIES... PHYSICAL REVIEW C 81, 044910 (2010) 30 ] -11 20 P(b) [10 10 0 0 100 200 300 400 500 FIG. 3. (Color online) Same as the previous figure, but in the b [fm] plane perpendicular to the beam axis. The impact parameter vector lies along the x direction. FIG. 4. (Color online) Probability of muonic atom production in p-p collisions at the LHC as a function of the impact parameter. We now use Eq. (16) to obtain the production probability of muonic atoms at the LHC at zero impact parameter. We find that the probability is five orders of magnitude smaller direction of motion of the nucleus capturing the muon. The higher the positive muon energy is, the more backward peaked than that with b = amuon. We easily understand this result by inspection of Eq. (16). When t =∞ only the upper the distribution becomes. The most probable energy of the term in the last part contributes (R<∞) to the production free muon is nonrelativistic, i.e., m. For such low-energy probability. This term oscillates harmonically as a function of muons the opening angle for emission of the free muon is ωR/v ≈ 2mR/ v.ForR along the z direction this variable larger. In a collider, a Lorentz transformation of these results varies as 2mc2z/hc¯ . As the largest contribution to the integral to the laboratory frame implies that all particles are seen along the beam direction, within an opening angle of order of 1/. arises for z ≈ amuon, this variable is much larger than the unity, causing the harmonic functions to oscillate wildly, and That is, all positive particles are seen along the same direction leading to a small value of the integral over coordinates (matrix as the beam. element). Using Eq. (17) we calculate the cross sections for produc- We now perform an estimate of the contribution of tion of exotic atoms in p-p and Pb-Pb collisions at the LHC. collisions with small impact parameters by using Eq. (16) Our results are shown in Table I. Whereas the cross sections and integrating it numerically over time, space, and impact are small for p-p collisions, they are by no means negligible 7 parameters for several energies and angles of the emitted for Pb-Pb collisions. This is due to the factor Z in Eq. (17), positive muon. To obtain these estimates, we have used the functions − and + without the derivative corrections. The integrals are performed for b 044910-5 C. A. BERTULANI AND M. ELLERMANN PHYSICAL REVIEW C 81, 044910 (2010) TABLE I. Cross sections for production of exotic atoms in p-p large impact parameters. Whereas most of our discussion and and Pb-Pb collisions at the CERN Large Hadron Collider (LHC). conclusions have been based on calculations of muonic atoms, the qualitative aspects will not change for pionic, protonic, Exotic atom p-p Pb-Pb or other exotic atoms, except for the magnitude of the cross Hydrogen 63.4 pb 132 b sections. The space-time and impact parameter dependencies Muonic 44.8 × 10−4 pb 0.16 mb will also be similar for Pb-Pb collisions. Pionic 21.3 × 10−4 pb 0.09 mb We considered the capture of negative particles in the K Kaonic 1.3 × 10−4 pb 4.3 µb shell of either the proton of the Pb nucleus. This is the largest ρ atom 0.51 × 10−4 pb 1.3 µb contribution to the capture cross section. Inclusion of capture to Protonium 0.09 × 10−4 pb 0.3 µb all other shells will increase by about 20% of the our calculated cross sections, according to early estimates for electron-pair production with capture [3]. Depending on the relevance of this although a reduction of this Z dependence arises from the process to peripheral collisions with relativistic heavy ions and distortion factor N . on the accuracy attained by the experiments, further theoretical studies on the capture to higher orbits could be necessary. The distortion of the wave functions due to the nuclear size should IV. CONCLUSIONS also be considered. Due to their small charges, p-p collisions will not yield a measurable amount of exotic atoms, but in Because of the very strong electromagnetic fields of short Pb-Pb collisions we expect an abundant number of events of duration, new and interesting physics arise at the LHC. We production of exotic atoms. This will open opportunities to have studied the space-time dependence of the cross sections study the properties of exotic atoms and their decay widths. for production of exotic atoms in ultraperipheral collisions at the LHC. We have considered the case of muonic, pionic and antiprotonic (or protonium) atoms. A very transparent ACKNOWLEDGMENTS and simple formulation is obtained using the lowest-order corrections for the positron and electron wave functions and a This work was partially supported by the US DOE grants multipole expansion of the electromagnetic field separated in DE-FG02-08ER41533 and DE-FC02-07ER41457 (UNEDF, the regions of small (i.e., b smaller than the Bohr radius) and SciDAC-2) and the Research Corporation. [1] E. Papageorgiu, Phys. Rev. D 40, 92 (1989). [7]C.A.BertulaniandG.Baur,Phys.Rev.D58, 034005 (1998). [2] M. Grabiak, B. Muller,¨ W. Greiner, G. Soff, and P. Koch, J. Phys. [8] C. A. Bertulani, C. M. Campbell, and T. Glasmacher, Comput. G 15, L25 (1989). Phys. Commun. 152, 317 (2003). [3]C.A.BertulaniandG.Baur,Phys. Rep. 163, 299 (1988). [9] H. Esbensen and C. A. Bertulani, Phys.Rev.C65, 024605 [4] C. A. Bertulani and G. Baur, Phys. Today, March 47, 22 (1994). (2002). [5] G. Baur et al., Phys. Lett. B 368, 251 (1996). [10] K. Ogata and C. A. Bertulani, Prog. Theor. 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