Spin and Chirality Effects in Antler-Topology Processes at High Energy e^{+}e^{-} Colliders

Spin and Chirality Effects in Antler-Topology Processes at High Energy Colliders

S.Y. Choi, N.D. Christensen, D. Salmon, and X. Wang

Department of Physics, Chonbuk National University, Jeonbuk 561-756, Korea
Pittsburgh Particle physics, Astrophysics, and Cosmology Center, Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, USA
Department of Physics, Illinois State University, Normal, IL 61790, USA
July 31, 2019

We perform a model-independent investigation of spin and chirality correlation effects in the antler-topology processes at high energy colliders with polarized beams. Generally the production process can occur not only through the -channel exchange of vector bosons, , including the neutral Standard Model (SM) gauge bosons, and , but also through the - and -channel exchanges of new neutral states, and , and the -channel exchange of new doubly-charged states, . The general set of (non-chiral) three-point couplings of the new particles and leptons allowed in a renormalizable quantum field theory is considered. The general spin and chirality analysis is based on the threshold behavior of the excitation curves for pair production in collisions with longitudinal and transverse polarized beams, the angular distributions in the production process and also the production-decay angular correlations. In the first step, we present the observables in the helicity formalism. Subsequently, we show how a set of observables can be designed for determining the spins and chiral structures of the new particles without any model assumptions. Finally, taking into account a typical set of approximately chiral invariant scenarios, we demonstrate how the spin and chirality effects can be probed experimentally at a high energy collider.


1 Introduction

The monumental discovery Aad:2012tfa (); Chatrchyan:2012ufa () of the Higgs boson at the CERN Large Hadron Collider (LHC) has filled in the only missing piece of the SM of electroweak and strong interactions, completing its gauge symmetry structure and electroweak symmetry breaking (EWSB) through the so-called Brout-Englert-Higgs (BEH) mechanism Englert:1964et (); Higgs:1964ia (); Higgs:1964pj (); Higgs:1966ev (); Kibble:1967sv (). Nevertheless, there are several compelling indications that the SM needs to be extended by including new particles and/or new types of interactions. Once any new particle indicating new physics beyond the SM is discovered at the LHC or high energy colliders, one of the first crucial steps is to experimentally determine its spin as well as its mass because spin is one of the canonical characteristics of all particles required for defining a new theoretical framework as a Lorentz-invariant quantum field theory Wigner:1939cj ().

Many models beyond the SM Wess:1974tw (); Nilles:1983ge (); Haber:1984rc (); Chung:2003fi (); Weinstein:1973gj (); Weinberg:1979bn (); Susskind:1978ms (); ArkaniHamed:1998nn (); Randall:1999ee (); Appelquist:2000nn (); ArkaniHamed:2001nc (); Csaki:2003dt (); Csaki:2003zu () have been proposed and studied not only to resolve several conceptual issues like the gauge hierarchy problem but also to explain the dark matter (DM) composition of the Universe with new stable weakly interacting massive particles Griest:2000kj (); Bertone:2004pz (); Ade:2013zuv (). For this purpose, a (discrete) symmetry such as paity in supersymmetric (SUSY) models and Kaluza-Klein (KK) parity in universal extra-dimension (UED) models is generally introduced to guarantee the stability of the particles and thus to explain the DM relic density quantitatively. As a consequence, the new particles can be produced only in pairs at high energy hadron or lepton colliders, leading to challenging signatures with at least two invisible final-state particles.

Figure 1: The correlated process characterized by the antler-topology diagram. Here, the invisible final-state particle might be charge self-conjugate, i.e. .

At hadron colliders like the LHC such a signal with invisible particles is usually insufficiently constrained for full kinematic reconstructions, rendering the unambiguous and precise determination of the masses, spins and couplings of (new) particles produced in the intermediate or final stages challenging, even if conceptually possible, as demonstrated in many previous works on mass measurements Lester:1999tx (); Barr:2003rg (); Cho:2007qv (); Barr:2007hy (); Cho:2007dh (); Tovey:2008ui (); Cheng:2008hk (); Barr:2009jv (); Matchev:2009ad (); Polesello:2009rn (); Konar:2009wn (); Cohen:2010wv (); Alwall:2009sv (); Artoisenet:2010cn (); Alwall:2010cq (); Han:2009ss (); Han:2012nm (); Han:2012nr (); Swain:2014dha () and on spin determination Barr:2004ze (); Smillie:2005ar (); Datta:2005zs (); Barr:2005dz (); Meade:2006dw (); Alves:2006df (); Athanasiou:2006ef (); Wang:2006hk (); Smillie:2006cd (); Choi:2006mt (); Kilic:2007zk (); Alves:2007xt (); Csaki:2007xm (); Wang:2008sw (); Burns:2008cp (); Cho:2008tj (); Gedalia:2009ym (); Ehrenfeld:2009rt (); Edelhauser:2010gb (); Horton:2010bg (); Cheng:2010yy (); Buckley:2010jv (); Chen:2010ek (); Chen:2011cya (); Nojiri:2011qn (); MoortgatPick:2011ix ().

In contrast to hadron colliders, an collider Behnke:2013xla (); Baer:2013cma (); Behnke:2013lya (); Koratzinos:2013chw (); Gomez-Ceballos:2013zzn (); Accomando:2004sz (); Linssen:2012hp () has a fixed center-of-mass (c.m.) energy and c.m. frame and the collider can be equipped with longitudinally and/or transversely polarized beams. These characteristic features allow us to exploit several complementary techniques at colliders for unambiguously determining the spins as well as the masses of new pairwise-produced particles, the invisible particles from the decays of the parent particles and the particles exchanged as intermediate states, with good precision. In the present work we focus on the following production-decay correlated processes


dubbed antler-topology events Han:2009ss (), which contain the production of an electrically charged pair in collisions followed by the two-body decays, and , giving rise to a charged lepton pair and an invisible pair (See Fig.1).

The invisible particle may be charge self-conjugate, i.e. . Nevertheless, it is expected to be insubstantial quantitatively whether the particle is self-conjugate or not, unless the width of the parent particle is very large and there exist large chirality mixing contributions Hagiwara:2005ym (). So, any interference effects due to the charge self-conjugateness of the invisible particle will be ignored in the present work.111An indirect but powerful way of checking the charge self-conjugateness of the particle is to study the process to which the self-conjugate particle can contribute through its -channel exchange. The mode is under consideration as a satellite mode at the ILC.

If the parent particle carries an electron number or a muon number , then the final-state leptons must be or , respectively, if electron and muon numbers are conserved individually and the invisible particles, and , carry no lepton numbers. On the other hand, if the parent particle carries no lepton number, the final-state leptons can be any of the four combinations, , and the invisible particles, and , must carry the same lepton number as , respectively.

Once the masses of new particles are determined by (pure) kinematic effects Christensen:2014yya (), a sequence of techniques increasing in complexity can be applied to determine the spins and chirality properties of particles in the correlated antler-topology process at colliders Battaglia:2005zf (); Choi:2006mr (); Buckley:2007th (); Buckley:2008eb (); Boudjema:2009fz (); Christensen:2013sea ():

  • Rise of the excitation curve near threshold with polarized electron and positron beams;

  • Angular distribution of the production process;

  • Angular distributions of the decays of polarized particles;

  • Angular correlations between decay products of two particles.

While the first and second steps (a) and (b) are already sufficient in the case with a spin-0 scalar as will be demonstrated in detail, the production-decay correlations need to be considered for the case with a spin-1/2 fermion and a spin-1 to determine the spin unambiguously; in principle a proper combination of these complementary techniques enables us to determine the spins of the invisible particles, and , and all the intermediate particles exchanged in -, - or -channel diagrams participating in the production process. For our numerical analysis we follow the standard procedure. We show through detailed simulations how the theoretically predicted distributions can be reconstructed after including initial state QED radiation (ISR), beamstrahlung and width effects as well as typical kinematic cuts.

The paper is organized as follows. In Sect.2 we describe a general theoretical framework for the spin and chiral effects in antler-topology processes at high energy colliders. In Sect.3 we present the complete amplitudes and polarized cross sections for the production process in the center-of-mass (c.m.) frame with the general set of couplings listed in Appendix A. The technical framework we have employed is the helicity formalism Jacob:1959at (). Then, we present in Sect.4 the complete helicity amplitudes of the two-body decays and with general couplings given in Appendix A. Sect.5 describes how to obtain the fully-correlated six-dimensional production-decay angular distributions by combining the production helicity amplitudes and the two two-body decay helicity amplitudes and by implementing arbitrary electron and positron polarizations Hikasa:1985qi (); Hagiwara:1985yu (); MoortgatPick:2005cw (); Choi:2006vh (); Ananthanarayan:2008dr (). Sect.6, the main part of the present work, is devoted to various observables: the threshold-excitation patterns, the production angle distributions equipped with polarized beams, the lepton decay polar-angle distributions and the lepton angular-correlations of the two two-decay modes. They provide us with powerful tests of the spin and chirality effects in the production-decay correlated process. While all the analytic results are maintained to be general, the numerical analyses are given for the theories with (approximate) electron chirality conservation such as SUSY and UED models and a subsection will be devoted to a brief discussion of the possible influence from electron chirality violation effects. Finally, we summarize our findings and conclude in Sect.7. For completeness, we include three appendices in addition to Appendix A. In Appendix B, we list all of the Wigner -functions used in the main text d functions:1957rs (). In Appendix C, we describe how to obtain the expression of the production matrix element-squared for arbitrary polarized electron and positron beams. Finally, in Appendix D we give an analytic proof of the presence of a twofold discrete ambiguity in determining the momenta in the process , even if the masses of the particles, and (), are a priori known.

2 Setup for model-independent spin determinations

Generally, the production part of the antler-topology process (1.1) can occur through -, - and/or -channel diagrams in renormalizable field theories, as shown in Fig.2. Which types of diagrams are present and/or significant depend crucially on the nature of the new particles, , and as well as the SM leptons and on the constraints from the discrete symmetries conserved in the theory.

Figure 2: New -channel -exchange diagrams (including the standard - and -exchange diagrams), new -channel -exchange diagrams and new -channel -exchange diagrams to the pair-production process .

We assume that the new particles, , and , are produced on-shell in the antler-topology process (1.1), and they are uncolored under the SM strong-interaction group so that they are not strongly interacting.222In addition, assuming the widths of the new particles to be much smaller than their corresponding masses, we neglect their width effects for any analytic expressions, although we consider them in numerical simulations in the present work. Motivated mainly by the DM problem, the new particles are assumed to be odd under a conserved discrete -parity symmetry. Therefore, they can only be produced in pairs at high energy hadron and lepton colliders with an initial -parity even environment such as LHC, ILC, TLEP and CLIC, etc. Furthermore, the invisible particle participating in the two-body decay , if the decay mode is present, is included among the particles exchanged in the -channel diagram of the production process . This implies that unavoidably at least one of the particles is lighter than the particle in the antler-topology process with .

As the as well as the electron is singly electrically-charged, the - and -channel processes are mediated by (potentially several) neutral particles, and , but any -channel processes must be mediated by (potentially several) doubly-charged particles, . In passing, we note that most of the popular extensions of the SM such as supersymmetry (SUSY) and universal extra-dimension (UED) models contain no doubly-charged particles so that there exist only -channel and/or -channel exchange diagrams but no -channel exchange diagrams contributing to the production process . The -channel scalar-exchange contributions may be practically negligible as well because the electron-chirality violating couplings of any scalar to the electron line are strongly suppressed in proportion to the tiny electron mass in those SUSY and UED models.

Since the on-shell particles, , and as well as the virtual intermediate particles, and , are not directly measured, their spins and couplings as well as masses are not a priori known. The neutral state can be a spin-0 scalar, , or a spin-1 vector boson, , including the standard gauge bosons as well. Each of the other intermediate particles can be a spin-0 scalar, a spin-1/2 fermion or a spin-1 vector boson, assigned in relation to the spin of the particle . In any Lorentz-invariant theories, there exist in total twenty () different spin assignments for the production-decay correlated antler-topology process (1.1) as


with spins up to and couplings consistent with renormalizable interactions. The symbols used for the particles in our analysis are listed in Tab.1 along with their charges, spins and parities. Generically, the intermediate states, , and may stand for several different states, although typically the on-shell particle or stands for a single state. Note that, if the parent particle turns out to be a spin-0 or spin-1 particle, then the daughter particles, and , and the - and -channel intermediate particles and are guaranteed to be spin-1/2 particles.

  Particle Spin   Charge    Parity
Table 1: List of symbols used for the particles in our analysis with their electric charges, spins and parities. The symbol denotes an electron or a muon . The last three lines are for the new particles exchanged in the -, - and -channel diagrams including the neutral electroweak gauge bosons, and , exchanged in the -channel diagram in the production process, .

Among the elementary particles discovered so far, the electron is the lightest electrically-charged particle in the SM. Its mass is much smaller than the vacuum expectation value (vev) GeV of the SM Higgs field, the weak scale for setting the masses of leptons and quarks, as well as the c.m. energies of future high-energy colliders. Any kinematic effects due to the electron mass are negligible so that the electron will be regarded as a massless particle from the kinematic point of view in the present work. The near masslessness of the electron is related to the approximate chiral symmetry of the SM. Any new theory beyond the SM should guarantee the experimentally-established smallness of the electron mass. This is a challenge in new theories beyond the SM since they usually involve larger mass scale(s) than the weak scale. One simple and natural protection mechanism is chiral symmetry.333Other possible solutions for getting a massless fermion naturally is that the fermion is a Nambu-Goldstone fermion, the super-partner of an unbroken gauge boson or the super-partner of a Goldstone boson.

Nevertheless, we do not impose any type of chiral symmetry so as to maintain full generality in our model-independent analysis of spin and chirality effects, emphasising the importance of checking experimentally to what extent the underlying theory possesses chiral symmetry. In each three-point vertex involving a fermion line, i.e. two spin-1/2 fermion states, we allow for an arbitrary linear combination of right-handed and left-handed couplings. Only in our numerical examples will every interaction vertex involving the initial line and the final-state lepton () be set to be purely chiral, as is nearly valid in typical SUSY and UED models, apart from tiny contaminations proportional to the electron or muon masses generated through the BEH mechanism of EWSB Englert:1964et (); Higgs:1964ia (); Higgs:1964pj (); Higgs:1966ev (); Kibble:1967sv ().

3 Pair Production Processes

In this section we present the analytic form of helicity amplitudes for the production process


with the -, - and -channel contributions as depicted in Fig.2 with the general three-point couplings listed in Appendix A. Here, we discuss only the amplitudes for on-shell pair production. The technical framework for our analytic results is the standard helicity formalism Jacob:1959at ().

The helicity of a massive particle is not a relativistically invariant quantity. It is invariant only for rotations or boosts along the particle’s momentum, as long as the momentum does not change its sign. In the present work, we define the helicities of the in the c.m. frame. Helicity amplitudes contain full information on the production process and enable us to take into account polarization of the initial beams in a straightforward way as described in Appendix C.

Generically, ignoring the electron mass, we can cast the helicity amplitude into a compact form composed of two parts - an electron-chirality conserving (ECC) part and an electron-chirality violating (ECV) part - as


where with the difference of the helicities and that of the helicities . Here, and are the spin of the electron and the particle , respectively. No helicity indices are needed when the spin of the particle is zero, i.e. . After extracting the spin value of the electron and , takes two values of while takes two values of or three values for or , respectively. Frequently, in the present work we adopt the conventions, and , will be used to denote the sign of the re-scaled helicity values for the sake of notational convenience. The angle in Eq.(3.2) denotes the scattering angle of with respect to the direction in the c.m. frame. The explicit form of the functions needed here is reproduced in Appendix B.

The polarization-weighted polar-angle differential cross sections of the production process can be cast into the form


with the relative opening angle of the electron and positron transverse polarizations and the speed of pair-produced particles, where is the degrees of longitudinal and transverse polarizations and is the relative opening angle of the transverse polarizations. The ECC and ECV production tensors and are defined in terms of the reduced production helicity amplitudes by


with or simply for notational convenience. (For more detailed derivation of the polarized cross sections, see Appendix C.) The polarized total cross section can then be obtained by integrating the differential cross section over the full range of .

If all of the coupling coefficients are real and all the particle widths are neglected, the following relations must hold for both the ECC and ECV parts of the production helicity amplitudes:


as a consequence of invariance in the absence of any absorptive parts. Therefore, violation of this relation indicates the presence of re-scattering effects. On the other hand, invariance leads to the relation:


independently of the absorptive parts so that the relation can be directly used as a test of CP conservation. Similarly, it is easy to see that P invariance leads to the relation for both the ECC and ECV amplitudes:


which is violated usually through chiral interactions such as weak interactions in the SM.

Applying the and symmetry relations to the ECC and ECV production tensors, (3.4) and (3.5), we can classify the six polar-angle distributions in Eq.(3.3) according to their and properties as shown in Tab.2. We find that the two combinations, and , contributing to the unpolarized part are both - and -even whereas the terms, and , linear in the degrees of longitudinal polarization are -odd and -even. One of the two transverse-polarization dependent parts, , is both - and -even and the other one, , is both - and -odd. Unlike the other five distributions, the distribution vanishes due to CPT invariance if all the couplings are real.

  Polar-angle distributions
  even   even
  odd   even
  even   even
  odd   even
  even   even
  odd   odd
Table 2: and properties of the production polar-angle distributions separable with initial beam polarizations.

As can be checked with the expression of the last line in Eq.(3.3), the transverse-polarization dependent parts can be non-zero only in the presence of some non-trivial ECV contributions so that they serve as a useful indicator for the ECV parts. If both the electron and positron longitudinal polarizations are available, then we can obtain the ECC and ECV parts of the unpolarized cross section separately. For the degrees of longitudinal polarization the ECC and ECV parts of the cross section are given by the relations:


where the upper arrow () or down arrow () indicates that the direction of longitudinal polarization is parallel or anti-parallel to the particle momentum with the first and second one for the electron and positron, respectively. Furthermore, we can construct two -odd -asymmetric quantities, of which one is ECC and the other is ECV, as


These observables, and , are expected to play a crucial role in diagnosing the chiral structure of the ECC and ECV parts of the production process, respectively. Furthermore, Eq.(3.9) and Eq.(3.11) are powerful even when electron chirality invariance is violated. As we will see, they enable us to extract the ECC parts separately so that the analysis of observables discussed in Sect.6 can be adopted without any further elaboration.

3.1 Charged spin-0 scalar pair production

The production of an electrically charged spin-0 scalar pair in collisions


is generally mediated by the -channel exchange of neutral spin-0 and spin-1 (including the standard and bosons), by the -channel exchange of neutral spin-1/2 fermions , and also by the -channel exchange of doubly-charged spin-1/2 fermions . The - or -channel diagrams can contribute to the process only when the produced scalar has the same electron number as the electron or positron in theories with conserved electron number. (Again, are twice the electron and positron helicities and the convention is used.)

The amplitude of the scalar-pair production process in Eq.(3.13) can be expressed in terms of four generalized ECC and ECV bilinear charges, and , in the c.m. frame as


where with and is the scattering polar angle between with respect to the direction in the c.m. frame. Explicitly, the ECC and ECV reduced helicity amplitudes are given in terms of all the relevant 3-point couplings listed in Appendix A by


in terms of the boost factor and the re-scaled angle-independent -channel propagator and the re-scaled angle-dependent -channel and -channel propagators, and defined as


with and in the c.m. frame. All of the propagators are constant, i.e. independent of the polar angle at threshold with , i.e. when the scalar pair are produced at rest. (The width appearing in the -channel propagator is supposed to be much smaller than and the c.m. energy so that their effects will be ignored in our later numerical analyses.)

Using the explicit form of functions (see Appendix B), we obtain the polarization-weighted differential cross sections of the production of scalar particles as


where and are the degrees of longitudinal and transverse polarizations and the relative opening angle of the transverse polarizations. The polarized total cross section can be then obtained by integrating the differential cross section over the full range of . One noteworthy point is that the transverse-polarization dependent parts on the last line in Eq.(3.19) survive even after the integration if there exist any non-trivial ECV amplitudes.

Inspecting the polarization-weighted differential cross sections in Eq.(3.19), we find the following aspects of the scalar pair production:

  • As previously demonstrated in detail for the production of scalar smuon or selectron pairs in SUSY models, the ECC part of the production cross section of an electrically-charged scalar pair in collisions, originated from the system, has two characteristic features. Firstly, the cross section rises slowly in -waves near the threshold, i.e. as the ECC amplitudes are proportional to . Secondly, as the total spin angular momentum of the final system of two spinless scalar particles is zero, angular momentum conservation generates the dependence of the ECC part of the differential cross section, leading to the angular distribution near the threshold.

  • However, the two salient features of the ECC parts are spoiled by any non-trivial ECV contributions originated from -channel scalar exchanges or - and -channel spin-1/2 fermion exchanges with both left-handed and right-handed couplings. Near the threshold the ECV amplitudes become constant. Therefore, in contrast to the ECC part the ECV part of the total cross section rises sharply in -waves and the ECV part of the differential cross section is isotropic.

  • As mentioned before, even in the presence of both the ECC and ECV contributions, the electron and positron beam polarizations can provide powerful diagnostic handles for differentiating the ECC and ECV parts. On one hand, the presence of the ECV contributions, if not suppressed, can be confirmed by transverse polarizations.444As is well known, transversely-polarized electron and positron beams can be produced at circular colliders by the guiding magnetic field of storage rings through its coupling to the magnetic moment of electrons and positrons. On the other hand, longitudinal electron and positron polarizations enable us to extract out the ECC parts and to check the chiral structure of the three-point , and couplings.

  • Then, the polar-angle distribution can be used for confirming the presence of - or -channel exchanges, as the distribution is peaked near the forward and/or backward directions for the - and/or -channel contributions.

  • If there exist only -channel contributions, then the ECC and ECV part of the angular distribution is proportional to and to a constant in the scalar-pair production in collisions, respectively.

To find which of the these aspects are unique to the spin-0 case we need to compare them with the spin-1/2 and spin-1 case.

Asymptotically the ECV amplitudes become vanishing and the ECC ones remain finite as can be checked with Eqs.(3.15) and (3.16). As the c.m. energy increases, the ECV contributions diminish and the ECC part of the unpolarized cross section of a scalar-pair production scales as


in the absence of both - and -channel contributions, following the simple scaling law , and the cross section scales in the presence of the -channel and -channel contributions as


as expected from the near-forward and near-backward enhancements of the - and -channel exchanges. (The expression on the last line in Eq.(3.21) is obtained by replacing all the intermediate masses by the scalar mass as a typical mass scale.) As the ECC part of the -pair production cross section is zero in strict forward and backward direction due to angular momentum conservation, the cross section remains scale-invariant apart from the logarithmic coefficients.

3.2 Charged spin-1/2 fermion pair production

The analysis presented in Subsect.3.1 for the scalar pair production repeats itself rather closely for new spin-1/2 fermion states, . In addition to the standard and exchanges, there may exist the -, - and -channel exchanges of new spin-0 scalar states, and , and new spin-1 vector states, and . Despite the complicated superposition of scalar and vector interactions, the helicity amplitudes of the production of an electrically-charged fermion pair, , can be decomposed into the ECC and ECV parts as in Eq.(3.2) with , , and . Explicitly, employing the general couplings listed in Appendix A, we obtain for the ECC helicity amplitudes for which :


for the same helicities, , and


for the opposite helicities, with the boost factors, and