Probing leptoquark production at IceCube

We emphasize the inelasticity distribution of events detected at the IceCube neutrino telescope as an important tool for revealing new physics. This is possible because the unique energy resolution at this facility allows to separately assign the energy fractions for emergent muons and taus in neutrino interactions. As a particular example, we explore the possibility of probing second and third generation leptoquark parameter space (coupling and mass). We show that production of leptoquarks with masses \agt 250 GeV and diagonal generation couplings of O(1) can be directly tested if the cosmic neutrino flux is at the Waxman-Bahcall level.


I. INTRODUCTION
Leptoquarks are SU (3)-colored particles which simultaneously carry non-zero baryon and lepton quantum numbers. They are predicted in several models (such as SU (5) [1] or Pati-Salam SU (4) [2]) addressing the unification of the lepton and quark sectors of the standard model (SM). In such models, the masses of the leptoquarks are generically superheavy, on the order of the GUT scale, which puts them out of reach of direct experimental access. Nevertheless, since leptoquarks with electroweak scale masses are not disallowed for any fundamental reason, it is of interest to conduct experimental searches to delimit their properties [3]. It is, of course, important to note that in order to avoid rapid baryon decay, the simultaneous trilinear coupling of the leptoquark to a purely hadronic channel needs to be excluded [4].
In general, the couplings of the leptoquark need not be generation-diagonal, and the problem of extracting limits on couplings and masses is complicated by the presence of a large-dimensional parameter space. Experiments at HERA have placed lower limits of O(300 GeV) on first generation leptoquark masses, for trilinear couplings of electroweak gauge strength [5]. Similar bounds, under the same assumptions, have been found at LEP from their search for anomalous 4-fermion vertices [6]. For first generation leptoquark trilinear couplings which are much smaller than gauge strength (as is the case for the Yukawas in the SM), the mass bounds are greatly weakened.
At the Tevatron, the leptoquarks could be produced in pairs, with identification made through decay topologies. In this way, the bounds are not dependent on the trilinear couplings, except for decay branching fractions. In the case of first [7] and second [8] generation leptoquarks, the final state topology consists of 2 hadronic jets + 2 charged leptons, and the resulting lower limits on the leptoquark mass are around 250 GeV. In the case of the third generation, a lower limit of 219 GeV has been recently reported by the DØ Collaboration [9], by tagging on 2 b jets + missing energy. For this value of the leptoquark mass, the decay into tτ is largely suppressed compared to the bν τ channel, so that the mass bound is nearly independent of even the branching fraction. As the explored mass region becomes larger, the tτ channel becomes more available and thus the mass limit obtained is pushed a bit lower (to ≥ 213 GeV).
In this work, we explore the possibility of probing second and third generation leptoquark parameter space (coupling and mass) with the IceCube neutrino detection facility [10]. This experiment, located below the surface of the Antarctic ice sheet at the geographic South pole, is required to be sensitive to the best estimates of potential cosmic ray neutrino fluxes. When completed, the telescope will consist of 80 kilometer-length strings, each instrumented with 60 10-inch photomultipliers spaced by 17 m. The deepest module is 2.4 km below the surface. The strings are arranged at the apexes of equilateral triangles 125 m on a side. The instrumented (not effective!) detector volume is a cubic kilometer. IceTop, a surface array of Cerenkov detectors deployed over 1 km 2 above IceCube, augments the deep-ice component by providing a tool for calibration, background rejection and airshower physics. The expected energy resolution is ±0.1 on a log 10 scale. Construction of the detector started in the Austral summer of 2004/2005 and will continue for 6 years, possibly less. At present, data collection by the first 9 strings has begun.
The event signatures are grouped as tracks, showers, or a combination of the two. Tracks include muons resulting from both cosmic ray showers and from charged current (CC) interaction of muon neutrinos. Tracks can also be produced by τ leptons arising in ultra-high energy ν τ CC interactions. Showers are generated by neutrino collisions (ν e or low energy ν τ CC interactions, and all neutral current interactions) inside or near the detector, and by muon bremsstrahlung radiation near the detector.
The experimental situation is greatly simplified for neutrino energy E ν 10 6 GeV. A cut at this energy is sufficient to reduce the great majority of background from muon bremsstrahlung and tracks arising from muons produced in cosmic ray showers. Moreover, the flux of atmospheric neutrinos is low above this en-ergy [11], so this cut generates a very pure sample of extraterrestrial neutrinos. Of particular interest here, for E ν > 10 6 GeV, there is sufficient energy resolution (±0.2 on a log 10 scale) to separately assign the energy fractions in the muon track and the hadronic shower, allowing the determination of the inelasticity distribution. Similarly, in the energy decade 10 6.5 < E ν /GeV < 10 7.5 one can expect good resolution (less than 5%) in "double bang" events generated by incoming ν τ 's. Again, this will allow a reasonable measurement of the inelasticity distribution.
In this study, we emphasize the inelasticity (y) distribution of events as an important tool for detection of new physics. In particular, we will find that the y distribution of events generated through resonant leptoquark production differs substantially from the SM prediction. If the event rate for the new physics turns out to be comparable to SM expectations, then the y profile of the measured data can be used to probe the coupling-mass leptoquark parameter space. The outline of the paper is as follows: In Sec. II we derive the relevant y-distribution of events generated through production and decay of a scalar leptoquark under the assumption of diagonal generation coupling. Armed with this distribution, in Sec. III we present a statistical method for assessing the significance of discovery criteria. Our conclusions are collected in Sec. IV.

II. LEPTOQUARK PHENOMENOLOGY
leptoquark couplings has been presented in [12]. To illustrate our proposal, we consider the simple case of SU(2)-singlet scalar leptoquarks S i which interact with quarks and leptons through the Lagrangian Here Q i = (u i d i ) T and L i = (ν i l i ) T stand for quark and lepton SU(2) left-handed doublets, u iR and l iR are righthanded singlets, and g L and g R are the corresponding coupling constants. Subindices i, running from 1 to 3, label the quark or fermion family. For simplicity we will assume that the interaction conserves leptoquark family quantum numbers separately (i.e., there is no mixing between different families). Thus in the following subindices i will be dropped, and up-and down-like quarks will be denoted generically with U and D respectively. We will be considering the inclusive ν P scattering schematically shown in Fig 1. The cross-hatched circle includes both resonant leptoquark production and decay, as well as u-channel exchange of a leptoquark leading to the same final state. We will not be considering couplings g L(R) 2, since such a coupling could lead to Landau singularities at low energies. Hence, we will assume that the resonant cross section largely dominates the process, and the narrow width δ-function approximation will be valid.
FIG. 1: Schematic diagram of a neutrino-parton collision, in which a neutrino with momentum k hits a quark with momentum p = xP giving rise to a secondary charged lepton and quark with momenta k ′ and p ′ , respectively. Here x is the fractional energy of the struck parton in the nucleon having momentum P.
Let us assume that an incoming neutrino collides with a proton target with center-of-mass energy s. If the neutrino hits a down-like parton D, the inclusive cross section for the process shown in Fig. 1 in the parton model is given by where D(x) is the corresponding parton distribution function (pdf). The neutrino-parton cross section reads where F = 2xs = 2ŝ is the invariant flux, m U is the mass of the outgoing up-like quark, and E ′ is the lab energy of the outgoing charged lepton. In the resonant approximation, the amplitude for the production of a left-handed charged lepton is given by where M and Γ are the mass and width of the leptoquark.
A similar expression, replacing g 2 L → g L g R , holds for the decay through the right-handed channel. It is easy to see that there is no interference between M L and M R . Now in the narrow resonance approximation one has Then, after summing over spins of outgoing fermions one arrives to where we have neglected both the mass of the down-like quark and the mass of the outgoing charged lepton. We have kept instead the m U dependence, since it is relevant in the case of the third family, where the outgoing quark would be a top. As a further assumption, we will consider that the leptoquark width is dominated by the U l and Dν quark-lepton channels. This leads to where λ U = m 2 U /M 2 . Substituting Eqs. (3), (6) and (7) into Eq. (2) yields where Q 2 = −(k − k ′ ) 2 , and the inelasticity y is defined as y = (E ν − E ′ )/E ν , E ν being the lab energy of the incoming neutrino. Indices α = 1, 2, 3 correspond to the up-like quarks U = u, c, t, respectively. After adequate changes of variables and integrations [13] we find The inelasticity y lies in the range λ U ≤ y 1. Note that the y distribution of the resonant process is approximately flat (at the energies of interest, the Q 2 dependence of the pdf can be neglected), in contrast to that characterizing SM charged current (CC) processes in which where G F = 1.16632×10 −5 GeV −2 is the Fermi constant, M W is the mass of the W gauge boson, and q(x) and q(x) stand for combinations of quark and anti-quark proton pdf's, respectively [14]. The y dependence of the SM cross section is shown in Fig. 6 of Ref. [15]. In the next section we exploit the differing y dependences of the leptoquark and SM interactions to constrain the parameter space of the new physics.

III. SENSITIVITY REACH AT ICECUBE
To evaluate the prospects for probing leptoquark production at IceCube, one has to estimate the "beam luminosity", i.e. the magnitude of the (yet to be detected) neutrino flux. We know that cosmic accelerators produce particles with energies in excess of 10 11 GeV (we do not know where or how [16]), and a neutrino beam is expected to come in association with these cosmic rays [17]. However, given our ignorance of the opacity of the sources, it is difficult to calculate the magnitude of the neutrino flux. The usual benchmark here is the so-called Waxman-Bahcall (WB) flux (all flavours), which is derived assuming that neutrinos come from transparent cosmic ray sources [18], and that there is adequate transfer of energy to pions following pp collisions. Here we will rely on this expression to estimate the event rates needed to quantify the IceCube sensitivity to leptoquark production. However, one should keep in mind that if there are in fact "hidden" sources which are opaque to ultra-high energy cosmic rays, then the expected neutrino flux will be higher [19]. Moreover, if the extragalactic cosmic rays begin to dominate over the Galactic component at energies as low as ∼ 10 9 GeV, as suggested recently [20], then the required power of the extragalactic sources will increase by a factor of ∼ 2, implying a concommitantly larger neutrino flux [21]. IceCube is sensitive to both downward and upward coming cosmic neutrinos. However, to remain conservative with our statistical sample, here we select only downward going events. To a good approximation, the expected number of such events at IceCube is given by where n T is the number of target nucleons in the effective volume, T is the running time, and σ tot (E ν ) is the total neutrino-nucleon cross section. In our analysis we are interested only in CC contained events, for which an accurate measurement of the inelasticty can be obtained. The IceCube's effective volume for (background rejected) contained events is roughly 1 km 3 [22], which corresponds to n T ≃ 6 × 10 38 .
Hereafter we focus on neutrino energies in the range 10 7 < E ν /GeV < 10 7.5 , where the background from atmospheric neutrinos is negligible, but the extraterrestrial FIG. 2: Sensitivity reach of IceCube (90% CL) to probe second generation SU(2)-singlet scalar leptoquark parameter space (coupling and mass). For comparison, the existing limit (95% CL) reported by DØCollaboration [8] is also shown.
flux is expected to be significant. Thus, we will consider a medium energy E ν = 10 7.25 GeV. At production, the WB flux has flavor ratios ν µ : ν e : ν τ = 2 : 1 : 0, but this quickly transforms to 1 : 1 : 1 through neutrino oscillations [23]. One has then φ να WB ( E ν ) ≃ 6 × 10 −23 GeV −1 cm −2 s −1 sr −1 (13) for each flavor α. Now it is possible to increase the ratio signal/background events by performing a cut in the inelasticity y. Given the dependence on y of the σ CC SM cross section, the flat behavior of the σ LQ cross section, and the available phase space for quark production, it is convenient to consider events with relatively large values of y. We choose here events in the range y ≥ 0.5. With this cut, the integration of Eq. (10) leads to One has to take into account that this result carries a systematic error of about 20% [14,24] due to uncertainties in the extrapolation of the pdf's [25]. For illustrative comparison the second generation leptoquark cross section in the high y region is calculated from Eq. (9) to be for fiducial values g L = g R = 1 and leptoquark mass M = 300 GeV. From Eqs. (12) and (13) we can now easily estimate the number of expected SM background events during the lifetime of the experiment. Taking T = 15 yr, for each neutrino flavor α one has where ∆E ν = 10 7.5 GeV − 10 7 GeV ≃ 2.2 × 10 7 GeV. (Note that the background is cosmic neutrinos! Todays' signal, tomorrow's background.) In the same way, the number of signal events N S will be approximately given by which for the above proposed leptoquark interaction is just a function of the couplings g R,L and the leptoquark mass M . It should be noted that the neutrino induced events do not constitute the sole background. As mentioned in the introduction there are muons (produce in the atmosphere) which traverse the detector and may deposit energy through bremsstrahlung radiation. In our energy bin, one may expect 10 muon traversals in 15 yr. However, our inelasticity cut will completely eliminate this source of background, because of the negligible probability for muons to radiate 50% of their energy.
To determine the bounds for leptoquark production, let us assume that 2 ν α -events are in fact observed with y ≥ 0.5. Then, at 90% CL, there will be an upper bound on signal events given by N (α) S ≤ 3.91 [26]. For simplicity, we consider the left-right symmetric case in which g L = g R . Then, after numerical evaluation of the leptoquark cross sections, the upper bounds on N (α) S can be translated into contours of constant likelihood in the M -g L plane. Our results are displayed in Figs. 2 and 3, where we show the sensitivity reach of IceCube together with the existing limits from DØCollaboration [8,9].
In the case of the third family, it can be seen that the sensitivity is maximal for leptoquarks of M ≃ 245 GeV. For lower leptoquark masses (in the narrow resonance limit) the allowed inelasticity range -and thus the leptoquark cross section-becomes reduced due to phase space suppression, owing to the large mass of the top quark.
In order to estimate the significance of the assumption g L = g R , we have also considered the case of purely lefthanded leptoquark currents, i.e. g R = 0. By looking at Eq. (9), it can be seen that this implies an average reduction in the leptoquark cross section by a factor of about 0.75 and 0.65 for the second and third families, respectively.

IV. CONCLUSIONS
In this paper we have introduced the measurement of inelasticity as a powerful tool for probing new physics in cosmic neutrino interactions. As an illustrative example, we have discussed the possibility of detecting leptoquark production at the IceCube neutrino telescope [27]. We estimated the expected event rate at IceCube to be comparable to the one predicted for cosmic ray facilities that make use of the atmosphere as the detector calorimeter [28]. However, the ability of IceCube to accurately measure the inelasticity distribution of events provides a unique method for SM background rejection, allowing powerful discrimination of resonant processes: we have shown that production of leptoquarks with masses 250 GeV and diagonal generation couplings of O(1) can be directly tested at the Antarctic ice-cap.
In closing, some comments are in order. First, we have not taken account of any systematic considerations concerning the detector -these are beyond the scope of the present work. Second, for reasons of simplicity, we have not included upcoming events close to the horizon. Although these Earth-skimming neutrinos have the potential of nearly doubling our signal event background (and thus nearly halving the required observation time scale), their proper consideration will require a full Monte Carlo simulation.