Spin and Chirality Effects in Antler-Topology Processes at High Energy Colliders
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.
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.
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
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
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
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
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.
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.
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
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
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
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
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
to the unpolarized part are both - and -even whereas the terms,
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.
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
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
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