Charm production by cosmic muons

Narrow muon bundles in underground detectors permit to study muoproduction reactions that take place in the surrounding rock. We analyze the relevance of a QED+QCD reaction, muoproduction of “open charm”. The contribution to double muon events is estimated to be 4−8% of the one due to QED “trident” process, for an ideal detector located under a rock depth of 3 km water equivalent, and an observation threshold of 1 GeV. In recent years, there has been a certain experimental [1, 2] and theoretical interest [3, 4, 5, 6] on “narrow muon bundles” (multiple muons with a lateral separation less than a few meters) in underground detectors. These events have been observed as a peak at small lateral separation, and interpreted as an induced flux. In fact, the energetic muons that propagate underground in roughly ∼ 1% of the cases interact and produce other muons. Thence, an analysis of these events requires to consider high energy muoproduction processes, in the rock surrounding the detector. Up to now, the process considered was the muon “trident” reaction [7] μZ → μZ μμ, where a muon pair is formed in the field of the nucleus. For muons propagating in high Z materials an amplification factor Z/A (= 5.5 for standard rock, A=22 and Z=11) is present, due to coherent character of the reaction. This interpretation has been pursued since the first evidences obtained in cosmic ray experiments [9]. The trident reaction leads mostly to narrow bundles of three or two muons in an underground detector (one produced muon may stop before reaching the detector); a reference ratio in an ideal detector is of 3 double e-mail: [email protected] It is assumed that an effective rejection of muoproduced π, γ, e ... can be achieved. muons per triple muon, for a threshold of observation Eh = 1 GeV, and a depth h = 3 km w.e. of standard rock. Recent studies [5, 6], however, suggest that existing interpretations are insufficient to quantitatively account for the whole “narrow muon” data set. In this work we analyze the role of another high energy process as source of prompt muons: production of charmed states due to cosmic (atmospheric) muons, whose relevant energies range from E ∼ 100 GeV up to tenths of TeV‘s (for studies in laboratory, see [10]). More specifically, we are concerned with the “open charm” reaction of muoproduction: μN → μ cc̄X (X denotes a byproduct which does not concern us). This process is stipulated by QED and QCD interactions, while weak interactions provide the instability of charmed states: c → Xc → μX. In order to obtain a simple estimate of the flux of double muons due to this process, we adopted the collinear approximation, considering only how the initial muon with energy E branches into those of the final muons (E ′ and E ) and proceeded in the following way: 1) First, we calculated the cross section of muoproduction dσμN→μcc̄X/dE ′dEc at leading order (LO) in αs, double differential in the energies of the scattered muon, E ′ and of the charm, Ec (see appendix). This can be done with a limited effort by following the calculations documented in [11], where the cross section integrated over the hadronic phase space dΦhadr was obtained: In fact, neglecting the gluon mass, the differential expression is simply dΦhadr = dEc/(8πEγ), where Eγ = E − E ′ is the energy of the virtual photon emitted by the muon. In the actual calculation, that requires integrating over the photon virtuality Q and the gluon momentum fraction x, we use the GRV98 gluon distribution [12], and set: mc = 1.5, 1.35 or 1.2 GeV. We multiplied the cross section by the factor K(E) = σNLO/σLO (where σ is the total cross section) to describe next-to leading order QCD effects [13, 14, 15]. The differential cross section increases with E ′ with a “1/v behavior” and than rapidly decreases to zero in the range of energies of interest; instead, it is rather mildly distributed in Ec. The total cross section σ increases with E, due to the smaller values of x that are probed by the virtual photon, and to well known characteristics of the gluon distribution. Its value is 4× 10 cm when E = 1 TeV for mc = 1.35 GeV (almost equal to the trident cross section at the same energy); LO cross section increases by 50% if mc is lowered to 1.2 GeV, while decreases by 30% if mc is 1.5 GeV. We neglect the energy loss in the rock of the charmed hadrons Xc, for a 200 GeV D ± meson travels on average just 3 cm in the rock before decaying. Also, we found convenient to relate Ec to the zenith angle and velocity of emission in the gluon-gamma c.m.s. as follows: Ec/Eγ = (1 + β ∗ c cos θ ∗ c )/2, where β ∗ c = √ 1− 4m2c/(p+ q) 2 (p and q are the momenta of the gluon and of the virtual photon) 2) We estimate a “scaling” probability dPc→μ/dw ≡ BRc→μ × ρc→μ(w) that a charm yields a muon with a certain energy fraction w = E ′′/Ec, by first hadronizing the charm into a D meson (using the normalized distribution of [16] with ǫD = 0.135) and then letting it decay with a Kμ3 distribution (that is, retaining only the D mass, and neglecting the Q 2 dependence of the form factors). The resulting normalized probability ρc→μ(w) falls strongly with the energy fraction w; the median of the distribution is in fact 〈E 〉 = 0.15 × Ec. We took as an effective branching ratio of charm into muons the value BRc→μ = 8% [17], and multiplied the result by two, to account for the fact that a charm or an anticharm can yield a muon. Notice, incidentally, that the corresponding yield of triple muon is negligible, due to an a priori 4% suppression factor. 3) At this stage, we can calculate the cross section dσμN→μμ/dE dE , where E ′′ is the energy of the produced muon, and, with that, the cross section σμN→μμ(E, f) for production of two muons, each with a fraction of the initial muon energy greater than f. Due to the behaviors of dσμN→μcc̄X and dPc→μ with E ′ and E ′′ mentioned above, this cross section diminishes dramatically with f ; when E = 1 TeV, it drops down by one order of magnitude already when f ≈ 0.07. This cross section enters the elementary yield of double muons in the detector, which depends linearly on the infinitesimal depth crossed dh (in gr/cm): dYμ→μμ(E, h ) = dh ×NA × σμN→μμ(E, f) where f = Eh′ E (1) NA = 6.023 × 10 23 is the number of nucleons in 1 mole (multiplying by dh, we obtain the density of targets per cm). The energy losses are evaluated in continuous energy loss approximation, Eh′ = (Eh+ǫ) exp[ (h−h ) / h0 ]−ǫ, where ǫ ≈ 600 GeV and h0 ≈ 2.5 km w.e. are phenomenological parameters, and Eh = 1 GeV is the (typical) threshold for detection. Multiplying this by the single muon flux differential in dE, dFμ, we get the differential double muon flux induced by “open charm” reaction at the depth h. The total flux is then: