The Determination of the Helicity of $W'$ Boson Couplings at the LHC
SLAC-PUB-12392
The Determination of the Helicity of W ′ Boson Couplings at the
LHC ∗ †
Thomas G. Rizzoa
Stanford Linear Accelerator Center, 2575 Sand Hill Rd., Menlo Park, CA, 94025
Abstract
Apart from its mass and width, the most important property of a new charged gauge boson,
W ′, is the helicity of its couplings to the SM fermions. Such particles are expected to exist in
many extensions of the Standard Model. In this paper we explore the capability of the LHC
to determine the W ′ coupling helicity at low integrated luminosities in the ℓ+ Emiss
discovery
channel. We find that measurements of the transverse mass distribution, reconstructed from
this final state in the W −W ′ interference region, provides the best determination of this quan-
tity. To make such measurements requires integrated luminosities of ∼ 10(60) fb−1 assuming
MW ′ = 1.5(2.5) TeV and provided that the W
′ couplings have Standard Model magnitude.
This helicity determination can be further strengthened by the use of various discovery channel
leptonic asymmetries, also measured in the same interference regime, but with higher integrated
luminosities.
∗Work supported in part by the Department of Energy, Contract DE-AC02-76SF00515
†e-mail: arizzo@slac.stanford.edu
http://arxiv.org/abs/0704.0235v4
1 Introduction
The ATLAS and CMS experiments at the LHC will begin taking data in a few months and it is
widely believed that new physics beyond the Standard Model(SM) will be discovered in the coming
years. There are many expectations as to what this new physics may be and in what form it will
manifest itself, but it is likely that we will be in for a surprise. Once this new physics is discovered
our primary goal will be to understand its essential nature and how the specific discoveries, such as
the production and observed properties of new particles, fit into a broader theoretical framework.
The existence of a new charged gauge boson, W ′, or a W ′-like object, is now a relatively
common prediction which results from many new physics scenarios. These possibilities include the
Little Higgs(LH) model[1], the Randall-Sundrum(RS)[2] model with bulk gauge fields[3], Universal
Extra Dimensions(UED)[4], TeV scale extra dimensions[5, 6, 7], as well as many different extended
electroweak gauge models, such as the prototypical Left-Right Symmetric Model(LRM)[8, 9]. Al-
though the physics of a new Z ′ has gotten much attention in the literature[10], the detailed study
of a possible W ′ has fared somewhat less well[11]. Perhaps the most important property of a W ′,
apart from its mass and width, is the helicity of its couplings to the fermions in the SM. For all of
the models discussed in the literature above, these couplings are either purely left- or right-handed,
apart from some possible small mixing effects. Determining the helicity of the couplings of a newly
discovered W ′ is thus the first major step in opening up the underlying physics as it is an order
one discriminator between different classes of models.‡
As will be discussed below, there have been many suggestions over the last 20-plus years
as to how to measure the helicity of W ′ couplings, all of which have their own strengths and
weaknesses. These analyses have generally relied upon the use of the narrow width approximation.
However, in employing this approximation much valuable information about the properties of the
W ′ can be lost, in particular, that obtained from W −W ′ interference. The goal of this paper will
‡This is similar in nature to determining whether the known light neutrinos are Dirac or Majorana particles.
be to explore the effects of this interference on the transverse mass dependent distributions of the
W ′. As we will see the rather straightforward measurement of the transverse mass distribution
itself will allow us obtain the necessary W ′ helicity information. Furthermore, we will demonstrate
that such measurements will require only relatively low integrated luminosities for W ′ masses which
are not too large, and will employ the traditional ℓ+EmissT W
′ discovery channel.
Section II of the paper contains some background material and a historically-oriented
overview of previous ideas that have been suggested to address the W ′ helicity issue including
a discussion of their various strengths and weaknesses. Section III will present an analysis of
the W ′ transverse mass distribution and its helicity dependence for a range of W ′ masses, cou-
pling strengths and LHC integrated luminosities. The use of various asymmetries evaluated in
the W −W ′ interference region in order to assist with the W ′ helicity determination will also be
discussed. Section IV contains a final summary and discussion of our results.
2 Background and History
Let us begin by establishing some notation; since much of this should be fairly familiar we will be
rather sketchy and refer the interested reader to Ref.[10] for details.
We denote the couplings of the SM fermions to the Wi = (W = WSM ,W
′) as
Vff ′C
i f̄γµ(1− hiγ5)f
i + h.c. , (1)
where for the case of Wi = WSM , the coupling strength(for leptons and quarks, respectively)
and helicity factors are given by C
i , hi = 1 and Vff ′ is the CKM(unit) matrix when f, f
′ are
quarks(leptons); note that the helicity structure for both leptons and quarks is assumed to be the
same as in all the model cases above.§ Following the notation given in Ref.[10], with some obvious
§For simplicity in what follows we will further assume that the corresponding RH and LH CKM matrices are
identical up to phases and we will generally neglect any possible small effects arising from W − W ′ mixing. In the
case of RH couplings, we will further assume that the SM neutrinos are Dirac fields.
modifications, the inclusive pp → W+i → ℓ+ν +X differential cross section can be written as
dτ dy dz
|Vqq′ |2
(1 + z2) + 2AG−
, (2)
where K is a kinematic/numerical factor that accounts for NLO and NNLO QCD corrections[12]
as well as leading electroweak corrections[13] and is roughly of order ≃ 1.3 for suitably defined
couplings, τ = M2/s (
s = 14 TeV at the LHC) with M2 being the lepton pair invariant mass.
Furthermore,
Pij(CiCj)
ℓ(CiCj)
q(1 + hihj)
2 (3)
Pij(CiCj)
ℓ(CiCj)
q(hi + hj)
where the sums extend over all of the exchanged particles in the s-channel. Here
Pij = ŝ
(ŝ−M2i )(ŝ−M2j ) + ΓiΓjMiMj
[(ŝ −M2i )2 + Γ2iM2i ][i → j]
, (4)
with ŝ = M2 being the square of the total collision energy and Γi the total widths of the exchanged
Wi particles. Note that we have employed z = cos θ, the scattering angle in the CM frame defined
as that between the incoming u-type quark and the outgoing neutrino (both being fermions as
opposed to being one fermion and one anti-fermion). Furthermore, the following combinations of
parton distribution functions appear:
q(xa,M
2)q̄′(xb,M
2)± q(xb,M2)q̄′(xa,M2)] , (5)
where q(q′) is a u(d)−type quark and xa,b =
τe±y are the corresponding parton momentum
fractions. Analogous expressions can also be written in the case of W−i exchange by taking z → −z
and interchanging initial state quarks and anti-quarks.
In most cases of interest one usually converts the distribution over z above into one over
the transverse mass, MT , formed from the final state lepton and the missing transverse energy
associated with the neutrino; at fixed M , one has z = (1 −M2T /M2)1/2. The resulting transverse
mass distribution can then be written as
dy J(z → MT )
dτ dy dz
, (6)
where Y = min(ycut,−1/2 log τ) allows for a rapidity cut on the outgoing leptons and J(z → MT )
is the appropriate Jacobian factor[15]. In practice, ycut ≃ 2.5 for the two LHC detectors. Note
that dσ
will only pick out the z-even part of dσ
dτ dy dz
as well as the even combination of terms in
the product of the parton densities, G+
. In the usual analogous fashion to the Z ′ case[10], as we
will see in our discussion below, one can also define the forward-backward asymmetry as a function
of the transverse mass, in principle prior to integration over the rapidity y, AFB(MT , y), whose
numerator now picks out the z-odd terms in dσ
dτ dy dz
as well as the odd combination of terms in
the parton densities G−
To be complete, we note that historically when discussing new gauge boson production,
particularly when dealing with states which are weakly coupled as will be the case in what fol-
lows, use is often made of the narrow width approximation(NWA). In the W ′ case of relevance
here, the NWA essentially replaces the integration over dτ ∼ dM by a δ function, i.e., the W ′ is
assumed to be produced on-shell. Thus, for any smooth function f(M), essentially,
dM f(M)
dM f(M) π
ΓW ′δ(M −MW ′) → π2ΓW ′f(MW ′), apart from some overall factors. Note that
use of the NWA implies that we evaluate quantities on the ‘peak’ of the W ′ mass distribution, i.e.,
at M = MW ′ . This approximation is usually claimed to be valid up to O(ΓW ′/MW ′) corrections(at
worst), but there are occasions, e.g., when W −W ′ interference is important, when its use can lead
to a loss of valuable information and may even lead to wrong conclusions[16]. Unfortunately, in the
W ′ case, the quantity M itself is not a true observable due to the missing longitudinal momentum
of the neutrino.
Given this background, let us now turn to an historical discussion of the determination of
the W ′ coupling helicity. To be concrete, we will consider two different W ′ models; we will assume
for simplicity that C
= 1 in both cases and that only the value of hW ′ = ±1 distinguishes them.
In this situation, employing the NWA, the cross section for on-shell W ′ production (followed by its
leptonic decay) is proportional to ∼ (1+h2W ′) and is trivially seen to be independent of the helicity
of the couplings. We would thus conclude that cross section measurements are not useful helicity
discriminants. More interestingly, as was noted long ago[17], we find that the rapidity integrated
value of AFB, given in the NWA by
AFB ∼
h2W ′
(1 + h2W ′)
, (7)
also has the same value for either purely LH or RH couplings¶. Thus, in the NWA, AFB provides
no help in determining the W ′ coupling helicity structure for the cases we consider here. However,
we note that if the quark and leptonic coupling helicities of the W ′ are opposite, then the value of
AFB will flip sign in comparison to the above expectation.
It is apparent from this result that some other observable(s) must be used to distinguish
these two cases. Keeping the NWA assumption, the first suggestion[18] along these lines was to
examine the polarization of τ ’s originating in the decay W ′ → τν. In that paper it was explicitly
shown that the the energy spectrum of the final state particle in the decay τ → ℓ, π or ρ (in the τ
rest frame) was reasonably sensitive to the original W ′ helicity since the τ itself effectively decays
only through the SM LH couplings of the W (provided the W ′ is sufficiently massive as we will
assume here). The difficulty with this method is that the observation of this decay mode at the
LHC is not all that straightforward and even the corresponding Z ′ → ττ mode, which is somewhat
easier to observe, is just beginning to be studied by the LHC experimental collaborations[19].
¶This follows immediately from the fact that we have assumed that both the hadronic and leptonic couplings of
the W ′ have to have the same helicity.
Clearly, measuring the polarization of the τ ’s in W ′ → τν will be reasonably difficult in the LHC
detector environment and may, at the very least, require large integrated luminosities even for a
relatively light W ′. The results of detailed studies by the LHC collaborations to address this issue
are anxiously awaited.
In the early 90’s, two important NWA-based methods for probing the helicity of the W ′
were suggested[20]. The first of these is an examination of the rare decay mode W ′ → ℓ+ℓ−W
(with the W decaying into jets); in particular, one makes a measurement of the ratio of branching
fractions
B(W ′ → ℓ+ℓ−W )
B(W ′ → ℓν)
, (8)
obtained by employing the NWA. RW is expected to be roughly ∼O(0.01) or so after suitable cuts.
One of the main SM backgrounds, i.e., WZ production, can essentially be removed by demanding
that the dileptons do not form a Z, demanding that the mass of the jjℓℓ system be not far from
the (already known) value of MW ′ and that of the dijets reconstructs to the W mass. Even after
there requirements, however, some background from the continuum would remain. Furthermore,
as the energy of the final state W increases it is more likely that the resulting dijets will coalesce
into a single jet depending on the jet cone definition which is employed. In this case, at the very
least, a very large additional background from single jets may appear; it is also possible that the
events with a final state W would be completely lost without the dijet mass reconstruction. The
3ℓ+EmissT final state, with suitable cuts, would be obviously cleaner and would avoid some of these
issues but at the price of an overall suppression due to ratio of branching fractions of ≃ 1/3 thus
reducing the mass range over which this process would be useful.
In a general gauge model, the amplitude for this process is the sum of two graphs. In
the first graph, W
′− → ℓ−ν̄∗, i.e., the production of a virtual neutrino followed by the ‘decay’
ν̄∗ → ℓ+W−. Clearly, if the W ′ couples in a purely RH manner to the SM leptons then this graph
will vanish in the limit of massless neutrinos due to the presence of two opposite helicity projection
operators. This graph will, of course, be non-zero only if the W ′ couples in an at least partially LH
manner. The second graph involves the presence of the trilinear couplings W ′ZW and W ′Z ′W ;
recall that in any model with a W ′, a Z ′ will also appear just based on gauge invariance. In this
case, the decay proceeds as W ′ → WZ/Z ′∗ → Wℓ+ℓ−, noting that the on-shell SM Z contribution
can be removed by a suitable cut on the dilepton invariant mass. The main issue is the size of
the W ′Z ′W (and W ′ZW ) couplings and this can involve such things such as, e.g., the detailed
electroweak symmetry breaking patterns of the given model under study. Generically in extra
dimensional models[3, 4, 5, 6, 7], these couplings are absent in the limit of small mixing due the
orthogonality of the Kaluza-Klein wavefunctions of the states. In models where the SM SU(2)L
arises from a diagonal breaking of the form G1 ⊗ G2 → SU(2)Diag , such as in LH models[1], the
W ′Z ′W coupling is of order the SM weak coupling, g, while the W ′ZW coupling is either of order g
or can be mixing angle suppressed. In other cases, such as in the LRM[8], where SU(2)L⊗SU(2)R
just breaks to SU(2)L, the W
′ZW,WZ ′W couplings are only generated by mixings and for the
diagrams of interest are not longitudinally enhanced. Since the amplitude associated with the pure
leptonic graphs are absent in this case, the entire amplitude is mixing angle suppressed so that
this process has an unobservably small rate. In fact, there are no known models where the W ′
helicity is RH and the W ′ZW,WZ ′W couplings are not mixing angle suppressed‖. Thus, based
on known models, it appears that the observation of the rare decay W ′ → ℓ+ℓ−W would be a
compelling indication that the W ′ is at least partially coupled in a LH manner with apparently
most of the serious SM backgrounds being removable by conventional cuts. However, in making
a truly model-independent analysis one must exercise care in the use of this result. A detailed
analysis of the signal and backgrounds, including that for the jjℓ+ℓ− final state, for such decays
including realistic detector effects would be very useful in addressing all these issues and should be
performed. However, it also seems clear that is unlikely that a reliable measurement of RW can be
made with relatively low integrated luminosities.
‖In a fundamental UV complete theory, this may follow directly from arguments based solely on gauge invariance
and the requirement of high energy unitarity.
A second, imaginative possibility is to observeWW ′ associated production[20] withW → jj
for the same reasons as above. Many of the arguments made in the previous paragraph will
also apply in this case as well since the diagrams responsible for this process are quite similar to
previously discussed. Essentially these graphs are obtained by crossing, with the final state leptons
now replaced by an initial state qq̄. In this case one looks for the jjℓEmissT final state with the
ℓEmissT transverse mass peaking near MW ′ . One would anticipate this cross section to be of order
∼ 0.01 of that of the W ′ discovery channel. The main issues here are, as above, the SM backgrounds
and the nature of the triple gauge vertices. It is not likely that a reliable measurement of this cross
section will be performed with low luminosities that could be interpreted in a model-independent
way until all of the background and detector issues are dealt with. Again, a detailed analysis
including detector effects should be performed.
3 W −W ′ Interference as a Function of MT
What we have learned from the previous discussion is that tools which employ the NWA are not
particularly useful when we are trying to determine the W ′ coupling helicity with relatively low
luminosities in an easily examined final state. One of the key reasons for this is that the use of NWA
does not allow us to examine the influence of W − W ′ interference to which we now turn[21]∗∗.
To be specific, in the analysis that follows, we will employ the CTEQ6M parton densities[25]
and will restrict our attention only to the ℓ = e final state since it is better measured at these
energies[23] yielding a better MT resolution. Furthermore, we will assume that only SM particles
are accessible in the decay of theW ′ so that the total width can be straightforwardly calculated from
the assumptions described above and its assumed mass value; for example, we obtain Γ(W ′) = 51.9
GeV assuming a W ′ mass of 1.5 TeV including QCD corrections. NLO QCD modifications to the
distributions we discuss below have been ignored but those distributions we consider are rather
∗∗We note in passing that the usual experimental analyses at LHC[23] performed by both the ATLAS and CMS
collaborations (as well as those at the Tevatron by CDF and D0[22]) ignore the effects of W −W ′ interference since
these contributions are absent from default versions of stand-alone PYTHIA[24].
Figure 1: Transverse mass distribution for the production of a 1.5 TeV W ′ including interference
effects at the LHC displayed on both log and linear scales assuming an integrated luminosity of
300 fb−1. The lowest histogram is the SM continuum background. The upper blue(middle red)
histogram at MT = 600 GeV corresponds to the case of hW ′ = −1(1).
robust against large corrections.
Figure 2: Same as in the previous figure but now on a linear scale with lower luminosities and
smeared by the detector resolution. In the top(bottom) panel an integrated luminosity of 30(10)
fb−1 has been assumed. Detector smearing has now been included assuming δMT /MT = 2%.
The most obvious distribution to examine first is dσ
itself; for the moment let us restrict
ourselves to the two cases where C
= 1 and hW ′ = ±1. Fig. 1 shows this distribution for a large
integrated luminosity, assuming MW ′ = 1.5 TeV[22], as well as the SM continuum background
††. In
††Note that we would expect to see many excess events for such W ′ masses as only ≃ 25 pb−1 of luminosity would
obtaining these and other MT -deprndent distributions below, a cut on the lepton rapidity, |ηℓ ≤ 2.5,
has been applied. Several things are immediately clear: (i) In the region near the Jacobian peak
both distributions are quite similar; this is not surprising as this is the region where the NWA is
most applicable since now MT ≃ M and W −W ′ interference is minimal. In this limit we would
indeed recover our earlier result that the cross section is helicity independent. (ii) In the lower MT
region where interference effects are important the two models lead to quite different distributions.
In particular, for the LH case with hW ′ = 1, we observe a destructive interference with the SM
amplitude producing a distribution that lies below that of the pure SM continuum background.
(This is not surprising as the overall signs of the W and W ′ contributions are the same but we are
at values of
ŝ that are above MW yet below MW ′ so that the relevant propagators have opposite
signs.) However, for the RH case with hW ′ = −1, there is no such interference and therefore the
resulting distribution always lies above the SM background. It is fairly obvious that these two
distributions are trivially distinguishable at these large integrated luminosities. Note that other
contributions to the SM background, e.g., those from the decay of top quarks as well as guage
boson pairs, have been shown to be rather small at these masses at the detector level [23], at the
level of a few percent, and will be ignored in the analysis that follows.
Fig. 2 shows the same dσ
distribution on a linear scale but now for far smaller integrated
luminosities that may be obtained during early LHC running; here we include the effects of detector
smearing, with δMT /MT ≃ 2%, which is somewhat less important in the very large statistics sample
cases shown above. It is immediately apparent that even with only ∼ 10 fb−1 of luminosity the
two cases remain quite distinct; however, it also appears unlikely that much smaller luminosities
would be very useful in this regard. This result is a significant improvement over previous attempts
to determine the W ′ coupling helicity with low luminosities in clean channels.
At this point there are several important questions one might ask: (i) What happens for
a more massive W ′, i.e., how much luminosity will be needed in such cases to distinguish W ′
be needed to discover(5σ) such as state at the LHC.
Figure 3: High luminosity plot of the transverse mass distribution assuming MW ′ = 2.5(3.5) for
the upper(lower) pair of histograms along with the SM continuum background. In the interference
region near ≃ 0.5MW ′ the upper(lower) member of the pair corresponds to the case of hW ′ = −1(1).
Detector smearing has now been included assuming δMT /MT = 2%.
couplings of opposite helicities? (ii) What if the the W ′ couplings are weaker than our canonical
choice above? (iii) Do other observables, e.g., AFB, measured in the interference region below the
Jacobian peak assist us in model separation? (iv) In the case where the W ′ is a KK excitation,
does the presents of the additional W KK tower members alter these results? (v) In the discussion
above we have assumed that CℓW ′ = C
W ′ ; what would happen, e.g., if their signs were opposite
thus modifying the interference bewteen the W and W ′? (vi) What if the W ′ couplings are not
purely chiral and are an admixture of LH and RH helicities? It is to these issues that we now turn.
Fig. 3 provides us with a high luminosity overview for the more massive cases where MW ′ =
2.5 or 3.5 TeV. In the MW ′ = 2.5 TeV case, Fig. 4 demonstrates that the full 300 fb
−1 luminosity
is not required to distinguish the two possibly helicities; ∼ 60fb−1 seems to be the approximate
minimum luminosity that appears to be necessary. For higher masses, distinguishing the two cases
becomes far more difficult due to the smaller production cross section as we see from Fig. 5 for
the case of MW ′ = 3.5 TeV assuming a luminosity of 300 fb
−1; essentially the full luminosity is
Figure 4: Transverse mass distribution assuming a mass of 2.5 TeV for the W ′ along with the
SM continuum background; the upper(lower) panel corresponds to a luminosity of 300(75) fb−1.
In the interference region near ≃ 0.5MW ′ the upper(lower) histogram corresponds to the case of
hW ′ = −1(1).
required for model distinction in this case.
Figure 5: High luminosity plot of the transverse mass distribution assuming MW ′ = 3.5 TeV along
with the SM continuum background. In the interference region near ≃ 0.5MW ′ the upper(lower)
histogram corresponds to the case of hW ′ = −1(1).
What if the W ′ couplings are weaker? Clearly if they are too weak there will be insufficient
statistics to discriminate the two possible coupling helicity assignments for any fixed value of MW ′ .
In order to examine a realistic example of this situation, we consider the case of the second W
KK excitation in the UED model[4, 26] with a conserved KK-parity. In such a scenario the LH
couplings of this field to SM fermions vanish at tree level but are induced by one loop effects. In
this case one finds that the effective values of Cℓ,q are distinct but are qualitatively of order ∼ 0.05
though we employ the specific values obtained in Ref.[4, 26] below in the actual calculations. Fig. 6
shows the transverse mass distributions in this case assuming that MW ′ =1 TeV for the second level
KK state. The signal for this W KK state is clearly visible above the SM background. However,
we also see that for even for these high luminosities and low masses the two helicity choices are
not distinguishable. Clearly, one cannot determine the W ′ coupling helicity for such very weak
interaction strengths. Semi-quantitatively, we find that that this breakdown in the discriminating
power occurs when (CℓCq)1/2 ∼ 0.1 at these luminosities and masses.
Figure 6: High luminosity plot of the transverse mass distribution assuming MW ′ = 1 TeV for
the second W KK level in the UED model smeared by detector resolution as above. As usual the
lower histogram is the SM background while the other two correspond to the signal cases with
hW ′ = −1(1) and are essentially indistinguishable.
We now turn to the next question we need to address: can asymmetries be useful in strength-
ening our ability to determine the W ′ coupling helicity? We know from the discussion above that
the answer is apparently ‘no’ in the NWA limit, i.e., when MT ≃ M . Thus we must focus our at-
tention on the MT region below the peak where W−W ′ interference is strongest or, more generally,
examine the asymmetries’ MT -dependence directly. The most obvious quantity to begin with is the
y-integrated value of AFB for both W
′± channels. To make such a measurement, we need to know
several things in addition to the sign of the lepton (which we assume can be done with ≃ 100%
efficiency). At the parton level, in the case of W ′− for example, the relevant angle used to define
AFB lies between the incoming d-type quark and the outgoing ℓ
−. Reconstructing this direction
presents us with two problems: first, since the longitudinal momentum of the ν is unknown there
is an, in principle, two-fold ambiguity in the motion of the center of mass in the lab frame; this
can cause a serious dilution of the observed asymmetry but can be corrected for statistically using
Monte Carlo once the W ′ mass is known. Second, even when it is determined, the direction of
motion of the center of mass is not necessarily that of the d-type quark though it is likely to be
so when the boost of the center of mass frame is large. The later problem also arises for the case
of a Z ′ and has also been shown to be mostly correctable in detailed Monte Carlo studies[27]. For
the moment, let us forget these issues and ask what the y-integrated AFB(MT ) looks like in both
ℓ± channels; the results are shown in Fig. 7 assuming high luminosities and MW ′ = 1.5 TeV. Here
we see that these integrated quantities, even for luminosities of 300 fb−1, are essentially useless
in distinguishing the two coupling helicity cases. Furthermore, we also see that the two coupling
helicities lead to essentially identical results when MT ≃ MW ′ as would be expected based on the
NWA. A short analysis indicates that approximately ten times more integrated luminosity would
be required before some separation in the two cases becomes possible[28]. Clearly this situation
would only become worse if we were to raise the mass of the W ′ or reduce its coupling strength.
It is perhaps possible that some information is lost by only using the integrated quantity
AFB and we need to consider instead AFB(yW ), where yW is the rapidity of the center of mass frame.
This distribution is odd under the interchange yW → −yW at the LHC so we can simply fold this
distribution over the yW = 0 boundary to double the statistics. Furthermore, by integrating over
a wide MT range in the interference region below the W
′ peak, e.g., 0.4 ≤ MT ≤ 1 TeV in the case
of a 1.5 TeV W ′, further statistics can be gained. Fig 8 shows the resulting AFB(yW ) distributions
for a W ′± with mass of 1.5 TeV assuming a luminosity of 300 fb−1 for hW ′ = ±1. At these large
luminosities, the AFB(yW ) distributions for the two helicity choices are clearly distinguishable but
this will certainly become more difficult for lower luminosities or for larger masses. We find that we
essentially loose all coupling helicity information when the luminosity falls much below ≃ 100fb−1
for this W ′ mass.
The next observable we consider is the charge asymmetry, AWQ(yW ):
AWQ(yW ) =
N+(yW )−N−(yW )
N+(yW ) +N−(yW )
, (9)
where N±(yW ) are the number of events with charged leptons of sign ± in a given bin of rapidity.
Note that at the LHC, AWQ(yW ) is symmetric under yW → −yW so that we can again fold the
Figure 7: The y-integrated value of AFB, as a function of the transverse mass, assuming a mass of
1.5 TeV for the W ′+(W ′−) in the top(bottom) panel. Here an integrated luminosity of 300 fb−1
has been assumed. The two essentially indistinguishable histograms correspond to the two possible
choices of the helicity, hW ′ = ±1.
Figure 8: The value of AFB as a function the center of mass rapidity, yW , integrated over the
transverse mass bin 400-1000 GeV assuming a mass of 1.5 TeV for theW
′−) in the top(bottom)
panel. An integrated luminosity of 300 fb−1 has been assumed and the distribution has been folded
around yW = 0. The upper(lower) set of data points in the top(lower) panel for small values of yW
corresponds to the choice of hW ′ = −1. Note that we have chosen signs to make the ranges of AFB
comparable in both cases.
distribution around yW = 0. Fig. 9 shows this distribution, integrated over the interference region
0.4 ≤ MT ≤ 1 TeV, assuming MW ′ = 1.5 TeV and a luminosity of 300 fb−1. It is clear that at this
level of integrated luminosity the two distributions are reasonably distinguishable. However, as we
lower the luminosity or raise the mass of the W ′ the quality of the separation degrades significantly.
Certainly for luminosities less that ≃ 100 fb−1, this asymmetry measurement would not be very
helpful. Thus AWQ(yW ) is not a very useful tool for coupling helicity determination until high
luminosities become available.
Figure 9: The W − W ′ induced charge asymmetry, assuming MW ′ = 1.5 TeV, as a function the
center of mass rapidity, yW , integrated over the transverse mass bin 400-1000 GeV. An integrated
luminosity of 300 fb−1 has been assumed and the distribution has been folded around yW = 0.
The upper set of data points at low values of yW corresponds to the choice of hW ′ = 1.
A last asymmetry possibility to consider is the rapidity asymmetry for the final state charged
leptons themselves, Aℓ(yℓ):
Aℓ(yℓ) =
N+(yℓ)−N−(yℓ)
N+(yℓ) +N−(yℓ)
, (10)
which is also an even function of yℓ so the distribution can again be folded around yℓ = 0. The
resulting distribution can be seen in Fig. 10 for large integrated luminosities. Here we again see
reasonable model differentiation at low values of yℓ <∼ 1 but this fades in utility as integrated
luminosities drop much below ≃ 100 fb−1 as the two curves are generally rather close.
From this general discussion of possibly asymmetries that one can form employing this final
state we can thus conclude that their usefulness in coupling helicity determination will require
≃ 100fb−1.
Figure 10: The W − W ′ induced lepton asymmetry, assuming MW ′ = 1.5 TeV, as a function
the lepton’s rapidity, yℓ, integrated over the transverse mass bin 400-1100 GeV. An integrated
luminosity of 300 fb−1 has been assumed and the distribution has been folded around yℓ = 0. The
upper set of data points at low values of yℓ corresponds to the choice of hW ′ = 1.
In the case of extra dimensions we know that an entire tower of W ′-like KK states is
expected to exist. Do the presence of these additional states modify the results we have obtained
above for an ordinary W ′? To address this, consider the simplified case of a second W ′-like KK
state which have the same coupling strength as the SM W and is twice as heavy as the W ′ discussed
above, i.e.,3 TeV. Now imagine that the coupling helicity of this second state is uncorrelated with
that of the W ′; in the MT distribution in the W −W ′ interference region influenced by this state?
The upper panel in Fig. 11 addresses this issue for modest luminosities including the effects of
smearing. The upper(lower) set of three histograms corresponds to the case where hW ′ = −1(1)
and either there is no W ′′, as above, or hW ′′ = ±1. This demonstrates that the existence of the
extra KK states has little influence on the results we obtained above independent of their coupling
helicities.
Up to now we have assumed that CℓW ′ = C
W ′ ; what if this was no longer true? How would
the MT distribution and our ability to determine coupling helicity be modified? The simplest case
to examine is CℓW ′ = −C
= 1 with hW ′ = ±1. (Note that interchanging the signs of these two
couplings, i.e., which one of these two couplings we choose to be negative, has no physical effect on
the MT distribution or on any of the asymmetries discussed earlier.) The result of this investigation
is shown in the lower panel of Fig. 11. Here the red(green) histograms correspond to the cases
analyzed above where CℓW ′ = C
W ′ = 1 and hW ′ = 1(−1) whereas the blue(magenta) histograms
corresponds to the cases where CℓW ′ = −C
= 1 with hW ′ = 1(−1). It is clear from this Figure that
the MT distribution distinguishes only three of these cases with the C
W ′ = ±C
= 1, hW ′ = −1
possibilities being degenerate. The reason for this is that in both these cases there is no interference
with the SMW ′ exchange and in the pureW ′ term in the cross section this sign change is irrelevant;
these two degenerate cases are, of course, separable using the information obtained from AFB as
they produce values with opposite sign.
Lastly, and to be more general, we must at least consider possible scenarios where the
couplings of the W ′ to SM fermions are a substantial admixture of both LH and RH helicities,
though obvious examples of such kinds of models are apparently absent from the existing literature.
To get a feel for such a possibility, we perform two analyses: first, we set Cℓ,q = 1 as before and
vary the values of hW ′ between pairs of positive and negative values. As we do this, the helicity
of the couplings of the W ′ will vary as will its total decay width which behaves as ∼ 1 + h2W . In
a second analysis, we can rescale the values of the Cℓ,q so that the W ′ width is held fixed. In this
case, as we will see, the resulting histograms for the transverse mass distribution lie especially close
to one another. The results of these two sets of calculations are shown in Fig. 12 in the case of
large integrated luminosities assuming the default value of MW ′ = 1.5 TeV. In the first analysis
shown in the top panel, we see that at these assumed luminosities all of the different histograms
Figure 11: SmearedMT distributions for several scenarios; the top panel, the lower(upper) compares
the single W ′ case discussed above to that where a second KK state, W ′′, exists with coupling
helicities uncorrelated to that of the W ′. Details are given in the text. In the lower panel, we
compare the cases for hW ′ = ±1 allowing for the possibility that CℓW ′ = ±C
with the signs
uncorrelated with the coupling helicity; the details are discussed in the text.
are distinguishable and not just the two pairs of cases with opposite helicities. This result generally
remains true down to luminosities ∼ 75fb−1 or so. If we are only interested in separating opposite
helicity pairs then we find that the cases hW ′ = ±0.8(0.6, 0.4, 0.2) can be distinguished down to
luminosities of order ∼ 10(25, 50, 75)fb−1 , respectively.
In the second analysis, as seen in the lower panel of the figure, the histograms for hW ′ =
0.8, 0.6 and 0.4 (as well as for their corresponding opposite helicity partners) are very close to one
another and are essentially inseparable even at these high luminosities. However, the two sets of
opposite helicity histograms remain distinguishable and this will remains true down to luminosities
of order 30 − 75 fb−1. It would seem from these analyses that the transverse mass distribution
will play the dominant role in W ′ coupling helicity determination in all possible cases although
somewhat higher integrated luminosities may be required in some scenarios.
4 Summary and Conclusions
Apart from its mass and width, the most important property of a new charged gauge boson, W ′, is
the helicity of its couplings to the SM fermions. Such particles are predicted to exist in the TeV mass
range in many new physics models and this coupling helicity is an order one discriminator between
the various classes of models. The main difficulties with the existing techniques for determining
this helicity are potentially threefold: (i) they require rather high integrated luminosities even for
a relatively light W ′, and/or (ii) they are sufficiently intricate as to require a detailed background
and detector study to determine their feasibility, and/or (iii) they make use of more complex final
states other than the standard ℓ + EmissT discovery channel. Some of these techniques also suffer
from employing the narrow width approximation which can result in loss of valuable information
regarding the effects of W − W ′ interference. In this paper we propose a simple technique for
making this helicity determination at the LHC. In order to attempt to circumvent all of these
difficulties, we have examined the W −W ′ interference region of the transverse mass distribution
Figure 12: Same as the linear plot shown in Fig.1, but now for other values of the coupling
helicities. From top to bottom the pairs of histograms in the upper panel correspond to h(W ′) =
±0.8,±0.6,±0.4and±0.2, respectively. The next lowest single histogram corresponds to the case of
pure vector couplings, i.e., h(W ′) = 0. In producing these results we have assumed that the values
of the Cℓ,q=1. In the lower panel, we show the same result now but with the overall couplings
rescaled so as to keep the W ′ width a constant.
for the ℓ + EmissT discovery mode. We have found that this distribution is particularly sensitive
to the helicity of the W ′ couplings. In particular, using this technique we have shown that such
helicity differentiation requires only ∼ 10(60, 300) fb−1 assuming MW ′ = 1.5(2.5, 3.5) TeV and
provided that the W ′ has Standard Model strength couplings. This helicity determination can be
further strengthened by the use of various discovery channel leptonic asymmetries also measured
in the same interference regime once higher integrated luminosities are available as well as by the
more traditional approaches. Hopefully the LHC will observe a W ′ so that this approach can be
employed.
Acknowledgments
The author would like to thank A. De Roeck, S.Godfrey and J. Hewett for input and
discussions related to this paper.
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	Introduction
	Background and History
	W-W' Interference as a Function of MT
	Summary and Conclusions
