Kinematic Cusps: Determining the Missing Particle Mass at Colliders

In many extensions of the SM, neutral massive stable particles (dark matter candidates) are produced at colliders in pairs due to an exact symmetry called a"parity". These particles escape detection, rendering their mass measurement difficult. In the pair production of such particles via a specific ("antler") decay topology, kinematic cusp structures are present in the invariant mass and angular distributions of the observable particles. Together with the end-points, such cusps can be used to measure the missing particle mass and the intermediate particle mass in the decay chain. Our simulation of a benchmark scenario in a Z' supersymmetric model shows that the cusp feature survives under the consideration of detector simulation and the standard model backgrounds. This technique for determining missing particle masses should be invaluable in the search for new physics at the LHC and future lepton colliders.

Introduction. Pauli's postulation of a new particle that escapes from detection and carries away energy and angular momentum in β decay not only laid out the foundation for the weak interaction, but also rightfully introduced the first dark matter (DM) particle, the neutrino. Ever since then, attempts to determine the masses and other properties of the neutrinos have led to many research efforts in nuclear physics, particle physics, astroparticle physics and cosmology. If the upcoming experiments at the CERN LHC find evidence of large missing energy events beyond the Standard Model (SM) expectations, it will be a new exciting discovery. Most significantly, this may hold the key to explain the missing mass puzzle in the Universe, the dark matter. It is thus of fundamental importance to determine the mass and properties of this missing particle in LHC experiments, to uncover its underlying dynamics and to check its consistency with dark matter expectations.
However, this task appears to be extremely difficult, even with the establishment of missing energy events at the LHC. In hadronic collisions, the undetermined longitudinal boost of the parton-level scattering leads to the ambiguity of their partonic c.m. energy in the lab frame. Furthermore, with a conserved discrete quantum number (generically called a "parity") that keeps the lightest particle in the new sector stable, the missing particles always come in pairs and thus the final state kinematics is even less constrained. Great efforts have been made to reconstruct the mass of the missing particle in SUSY-like cascade decays. The mass difference of two parity-odd particles may be measured by the maximum end-points of invariant mass distributions [1]. Attempts have been made to determine the missing particle mass by the construction of new transverse variables [2,3]. With long cascade decay chains, it may be possible to determine the masses through the on-shell mass constraints combining information from multiple events [4].
We propose a more favorable configuration in which the two missing parity-odd particles (X) come along with two visible SM particles (a), from the decay of a heav- FIG. 1: The "antler" decay diagram, a heavy particle (D) to two visible SM particles (a) and two missing particles (X), via two on-shell intermediate particles (B). ier parity-even particle (D) through intermediate parityodd particles (B), as shown in Fig. 1. We dub this decay topology as the "antler" diagram. The advantage of antler decays is that once the parent mass (m D ) is known, the masses of the missing particle (m X ) and the intermediate particle (m B ) can be determined by measuring the energy-momenta of the visible particles (a) without combinatoric complications.
The antler topology is common in many scenarios with DM particle candidates. Familiar examples include the following theoretically well-motivated models: The precondition is that the mass of particle D is known a priori. This can be achieved since its even-parity allows its decay into two observable SM particles. In this regard, the antler topology is equally applicable to a lepton collider where the c.m. energy is accurately known.
Here, we explore two kinematic distributions: (i) M aa , the invariant mass of the two visible particles a 1 and a 2 , and (ii) cos Θ, cosine of the angle between one of the two visible particles (say a 1 ) and the pair c.m. moving direction in their c.m. frame. We show these distributions in Figs. 2 and 3. The distributions have, in addition to the end points, unique geometrical structures, the cusps (denoted by vertical lines), which carry direct information on m B and m X .
A cusp is the point where two curves with different slopes meet. Cusps in the antler decay have outstanding merits for determining the missing particle mass: (i) a cusp is experimentally easy to identify due to its pointed feature; its sharpness depends on the mass spectra of the particles involved; (ii) looking for a cusp is statistically advantageous since it has large (in most cases, maximum) event rate; (iii) cusps can determine both the masses of the intermediate particle B and the missing particle X; (iv ) there is no combinatoric complication due to its simple decay topology; (v ) spin correlations of the decay processes do not change the position of the cusps.
The peak features of the cusps arise from the different origins for the M aa and cos Θ distributions. Roughly speaking, the triangular shape of the M aa distribution comes from folding diagonally the flat distribution of d 2 Γ/d cos θ 1 d cos θ 2 , where θ 1 and θ 2 are the scattering angles of two visible particles. Reorganized along M aa , the two apexes of the triangle at both ends correspond to M min aa and M max aa , while the middle apex corresponds to the cusp. The largest overlapping area below the middle apex explains the largest phase space density. In the distribution about cos Θ, the on-shell mass condition allows only a limited range for cos Θ. Since the Jacobean factor leads to a steep increase at the end points, we have a cusp feature in the cos Θ distribution. We will leave the details to a future work [9].
Cusp and Edge in M aa Distribution. We first only show the phase space distributions for on-shell particles. It is convenient to use the rapidities: the rapidity η of particle B and the rapidity ζ of particle a in the rest frames of their parents D and B, respectively. In terms of the mass  Table I. parameters, the rapidities η and ζ are given by Here and henceforth we use a shorthanded notation of c x ≡ cosh x. Obtaining the rapidities would be equivalent to measuring the masses m B and m X .
(1) m a = 0 case: Consider a to be massless first for simplicity. One would naively expect the invariant mass to have an end-point M max aa = m D − 2m X . However, due to the on-shell constraint for the particle B, we find a different end-point: In addition, the M aa distribution has a cusp at This is remarkable since the ratio M max aa /M cusp aa = e 2η is governed by the initial decay D → BB and thus gives m B . The product M max aa M cusp aa depends on the secondary decay B → aX and gives m X . Furthermore, dΓ/dM aa is of the form (4) Figure 2 shows dΓ/dM aa for four sets of representative masses specified in Table I. The choice of the parameters for Mass I is motivated by the Z (2) decay in the UED model [7]. Since the two subsequent decays Z (2) → L (1) L (1) and L (1) → ℓγ (1) occur near the mass threshold, Mass I is to be called the "near threshold case". For comparison, we consider the mass parameters with sizable gap in Mass III, the "large mass gap mD (GeV) mB (GeV) ma (GeV) mX (GeV )  Mass I  1250  600  0  550  Mass II  1000  440  0  300  Mass III  1000  350  0  200  Mass IV  600  250 mZ 100  (2) m a = 0 case: If the SM particle a is massive (a Z boson or a top quark), we call it the "massive case", as for the parameter choice in Mass IV, that are motivated by the MSSM heavy Higgs boson decay associated with two SM Z bosons. In this case, the analytic form of dΓ/dM aa is given by three pieces (the explicit forms are not very illuminating and thus not given here). The maximum of M aa is The positions of M min aa and M cusp aa are as follows, depending on the relations of the two rapidities η and ζ: A few remarks are in order. First, the results of Eqs. (5) and (6) for η < ζ/2 include the massless case, which corresponds to ζ → ∞. Second, for all three regions in Eq. (6), the M aa distribution shows a sharp cusp, as illustrated by the Mass IV case in Fig. 2. Third, the case ζ < η has different M min aa , rather than 2m a as naively expected. It is due to the enhanced boost of the two fast-moving parent B's. This shift helps to resolve the ambiguity among the three regions. Still a two-fold ambiguity in the η < ζ/2 and ζ/2 < η < ζ regions remains since we do not know whether the measured M cusp aa is 2m a c η−ζ or 2m a c η .
We propose another independent observable to break this ambiguity, (∆|p a T |) max , which is the maximum of the difference between the magnitudes of the two transverse momenta of a 1 and a 2 . We find the expression to be which is invariant under longitudinal boost. Note that the two-fold ambiguity happens when η < ζ leading to (∆|p a T |) max = 2m a cosh ζ sinh η. This sinh η-dependence can provide independent information from M cusp aa . Cusp in Angular Distribution. Another useful observable is the distribution about cos Θ, where Θ is the angle of a visible particle, say a 1 , in the c.m. frame of a 1 and a 2 , with respect to their c.m. moving direction. We find the expression of dΓ/d cos Θ in the D-rest frame for m a = 0 to be remarkably simple: The distribution has a sharp end-point, another cusp, with the highest event rate at the boundary: While this variable is unambiguous in the lab frame at a lepton collider, we cannot determine the longitudinal motion of the particle D in hadronic collisions. After convoluting with the parton distribution functions to convert it to the lab frame, dΓ/d cos Θ in Eq. (8) is smeared. In Fig. 3, we compare the cos Θ distribution in the rest frame of D (thin curves) with that in the lab frame at the LHC (thick curves) for the near threshold case (Mass I) and the large mass gap case (Mass III). We have assumed that D is produced by direct s-channel gg or qq annihilation so that the longitudinal momentum distribution of D is obtained from the parton distribution of the incident protons. The convolution effects with the partons smear out the sharp cos Θ cusp in the lab frame. For the near threshold case (Mass I), the two sharp rises at both ends get much less pronounced, although it is still possible to read the edge point off in the distribution. For the large mass gap case (Mass III), as the end point position approaches towards cos Θ = ±1, the sharpness of two cusps maintains better. It is interesting to note that the cusp in the M aa distribution and that in the cos Θ distribution provide complementary information for determining m B . The invariant mass distribution yields a better resolution for the near threshold case, while the angular distribution provides better one for the large mass gap case.
To some extent, the cos Θ distribution in the D-rest frame may be obtained even in hadron collisions. The smearing effects can be effectively modeled by the wellknown parton distribution functions in the large x region. Or the direct resonant decay of D into SM particles allows to extract the velocity distribution of D, which can be used to recover dΓ/d cos Θ in the D-rest frame. Discussion. Until now, we have ignored the effects of the matrix elements regarding the spin correlations between the initial state and final state particles. We have confirmed that, for the four processes in Eq. (1), including the full matrix elements does not change the shape of the distributions. The deviations from the phase space predictions become appreciable when the fermions and vector bosons (like the Z (2) in UED) have chiral couplings for both D → BB and B → aX decays. Even in this extreme case, the cusped peak remains at the same location and its height is changed by about 2%. However, non-vanishing decay widths of parent particles can alter the shape of the distributions. In Fig. 4, we show the M aa distributions for different finite decay widths Γ B with the Mass I spectrum. With Γ B ∼ m B /100, such distortions do not have a big effect. With Γ B ∼ m B /10, the finite decay width effect can change the cusp shape as well as the positions of M cusp aa and M max aa . In the limit Γ B ∼ m B , the momenta of the visible particles a 1 and a 2 span all of the allowed phase space given by m D and m X , regardless of m B , and thus the maximum of M aa approaches m D − 2m X , denoted by the vertical dashed arrow in Fig. 4. The end point measurement with the known m D leads to the missing particle mass m X , just like the direct neutrino mass determination in tritium decays. In the scenarios listed in Eq. (1), the intermediate particles (χ 0 2 ,l ± , L (1) and t − ) typically have decay widths that are smaller than one percent of their masses. The cusp shape in the M aa distribution remains intact.
Our cusp distributions of the signals are quite distinctive from the SM background kinematics. For instance, the leading SM backgrounds for ℓ + ℓ − + / E T are mainly from W + W − /ZZ → ℓ + ℓ − νν, while those for ZZ + / E T from ZZZ/W + W − Z, which should be well predicted in the SM. The optimal strategy to suppress the SM backgrounds is different case by case. We leave the detailed analyses to [9]. Conclusions. New techniques to measure the missing particle mass are proposed based on an "antler decay topology" (D → B + B → aX + aX), with a final state of two visible particles (a) and two missing particles (X) via intermediate particles (B). This antler decay has no combinatoric complications. We found cusp structures in the invariant mass distribution of two visible particles and an angular distribution. The cusp structure in the antler decay has large (in most cases largest) event rates unlike the end points. With its unique geometric feature, the cusp is easy to identify. Along with the end points of the distributions, the missing particle mass as well as the intermediate particle mass can be determined. A possible two-fold ambiguity can be resolved by another new observable, the difference between the magnitudes of transverse momenta of two observable SM particles. We believe that this technique will be invaluable for searches for new physics at the LHC.