Leptoquark toolbox for precision collider studies
HJE
Leptoquark toolbox for precision collider studies
Ilja Dorsner 0 1 2 4
Admir Greljo 0 1 2 3
0 avours (u , d, s, c, and b). Vector LQ imple
1 Johannes GutenbergUniversitat Mainz , 55099 Mainz , Germany
2 Ruđera Boskovica 32 , 21000 Split , Croatia
3 Faculty of Science, University of Sarajevo
4 University of Split, Faculty of Electrical Engineering
We implement scalar and vector leptoquark (LQ) models in the universal FeynRules output (UFO) format assuming the Standard Model fermion content and conservation of baryon and lepton numbers. Scalar LQ implementations include nexttoleading order (NLO) QCD corrections. We report the NLO QCD inclusive cross sections in protonproton collisions at 13 TeV, 14 TeV, and 27 TeV for all onshell LQ production processes. These comprise (i) LQ pair production (pp ! production (pp ! `) for all initial quark mentation includes adjustable nonminimal QCD coupling. We discuss several aspects of LQ searches at a hadron collider, emphasising the implications of SU(2) gauge invariance, electroweak and avour constraints, on the possible signatures. Finally, we outline the highpT search strategy for LQs recently proposed in the literature to resolve experimental anomalies in Bmeson decays. In this context, we stress the importance of complementarity of the three LQ related processes, namely, pp !
Beyond Standard Model; Heavy Quark Physics

`, and pp ! ``.
1 Introduction
2 Implementation and validation
of uni cation of the quarks and leptons of the SM [1]. There exists a number of indepth
reviews of various aspects of the LQ physics one can consult [2{5]. These aspects are
related to the avour physics e ects, collider physics signatures, and proton decay signals,
to name a few. In this note we revisit the production mechanisms of LQs in the
protonproton collisions in view of the need for an uptodate Monte Carlo event generator output
that can be used for the current and future experimental searches and search recasts [6{
9]. We especially address the single LQ production in association with a lepton and the
LQ pair production including important nexttoleading order (NLO) QCD corrections. A
sample of leading order (LO) Feynman diagrams for these processes involving scalar LQs
is shown in gure 1.
{ 1 {
We stress from the onset that there already exist explicit calculations of the pair
production of scalar LQs at the NLO level [10, 11] as well as several studies of the NLO
e ects on the single LQ production [7, 12, 13] at the LHC. One of our aims is to ll
in the missing pieces with respect to the latter process, especially in the context of the
sea quark initiated production. In fact, it is very important to entertain a possibility
of an LQ dominantly coupled to heavy fermions as motivated by the pattern of fermion
masses and mixing parameters, and as recently suggested by the hints on lepton
avour
universality violation in Bmeson decays. (See, for example, ref. [14] for more details.)
With this possibility in mind we also address single production of vector LQs through the
bottomgluon fusion processes.
Since the number of LQs is nite one can easily classify them [15]. We provide, as an
integral part of this analysis, readytouse universal FeynRules (UFO) [16] model les for
all scalar LQs as well as one vector LQ that are particularly suited for the avour
dependent studies of the LQ signatures at colliders within the MadGraph5 aMC@NLO [17]
framework. We validate our numerical results with the existing NLO calculations for the
pair production and present novel results for the single production of scalar (vector) LQs
at the NLO (LO) level. These results, in our view, can be particularly useful for the
current and future LHC data analyses and accurate search recasts. The UFO model les are
publicly available at http://lqnlo.hepforge.org.
The outline of the manuscript is as follows. We present the setup for our LQ signature
studies in section 2. This is followed by section 3 on numerical analysis that is subdivided
into the LQ pair production subsection and the single LQ + lepton production subsection.
The strategy for LQ searches inferred from Bphysics anomalies is described in section 4.
We present our conclusions in section 5. Most of our numerical results are summarised in
HJEP05(218)6
Implementation and validation
Scalar LQ setup
appendix A.
2
2.1
(3; 2; 1=6), S~1
The scalar LQ models we implement comprise S3
(3; 3; 1=3), R2
(3; 2; 7=6), R~2
(3; 1; 4=3), and S1
(3; 1; 1=3), where we specify transformation
properties under the SM gauge group. First (second) integer in the brackets corresponds to
the dimension of the irreducible representation of SU(3) (SU(2)) that the LQ belongs to
whereas the rational number is the LQ U(1) hypercharge. Our hypercharge normalisation
is such that the electric charge of S1 is 1=3 in units of the absolute value of the electron
charge. We assume that lepton number and baryon number are conserved quantities.
We use FeynRules 2.0 [16] to prepare the model les for each LQ representation. The
inclusion of NLO QCD corrections is possible in modern Monte Carlo frameworks that are
capable of the automated generation of the corresponding born, oneloop and real matrix
elements, and subtraction of infrared singularities. To this purpose, we use the NLOCT
package (version 1.02) [18] together with FeynArts (version 3.9) [19] to generate the
relevant UV and R2 counterterms at oneloop level in QCD. The resulting models are
exported in the UFO format, which can be directly used within MadGraph5 aMC@NLO
{ 2 {
row) and single scalar LQ plus lepton production (bottom row).
framework, where all required oneloop amplitudes are automatically generated with
MadLoop [20] and Ninja [21, 22]. The corresponding real amplitudes are generated from the
underlying UFO model, while the infrared subtraction of the real contributions is
automatically performed a la FKS [23] in MadFKS [24].
The kinetic and mass terms are implemented in the same manner for all the scalar
LQs and are given by
Lkinetic = (D
)y (D
)
m2LQ y ;
(2.1)
where D is the appropriate covariant derivative and
the second term in eq. (2.1) all the components within the given LQ multiplet , when
transforms nontrivially under SU(2), are assumed to have the same mass. This assumption
is driven by the electroweak precision measurements. (See, for example, section 4.2 in ref. [5]
for more details.) The mass splitting which generates small enough oneloop correction to
oblique Zpole observable can be completely neglected in view of the current direct limits
on LQ masses. In other words, one expects correlated signal in searches for the samemass
LQ states with di erent electric charge (by one unit). A combination of such searches can
improve the overall sensitivity with respect to the parameter space of the nontrivial SU(2)
= S3; R2; R~2; S~1; S1. As implied by
LQ multiplet(s).
For the avour dependent part of the lagrangian we closely follow notation of ref. [5]
and implement the most general form of Yukawa couplings. The fermion content is taken
to be purely that of the SM. We explicitly assume that the unitary transformations of
the rightchiral fermions are not physical. In our convention the
CabbiboKobayashiMaskawa (CKM) rotations reside in the uptype quark sector whereas the
PontecorvoMakiNakagawaSakata (PMNS) rotations originate from the neutrino sector. These
rota{ 3 {
tions provide connection between Yukawa couplings in the case when LQ interacts with
the SU(2) doublet(s) of the SM fermions. For example, one set of the R2 Yukawa couplings
that features the CKM matrix V is
LYRu2kawa
+y2LiRj eiRR2a QjL;a = +(y2LRV y)ij eiRujLR25=3
+ y2LiRj eiRdjLR22=3 ;
(2.2)
3 matrix in the avour space, QL is a leftchiral quark doublet, eR is a
rightchiral charged lepton, a = 1; 2 is an SU(2) index, and i; j = 1; 2; 3 are avour indices.
The couplings of R~2, on the other hand, feature the PMNS matrix U since the relevant
where y2LR is the 3
lagrangian reads
where y~2RL is the 3
~
LYRu2kawa
y~2RiLj diRR~2a abLjL;b =
y~2RiLj diRejLR~2=3 + (y~2RLU )ij diR LjR~2 1=3;
2
(2.3)
HJEP05(218)6
3 matrix in the avour space, LL is a leftchiral lepton doublet, dR
is a rightchiral downtype quark, and ab is LeviCivita symbol. The hermitian conjugate
parts are omitted from eqs. (2.2) and (2.3) for brevity. Note that our convention allows one
to completely neglect the PMNS rotations as the neutrino
avour is not relevant for the
processes we are interested in. In the actual model le implementations the PMNS matrix
is thus set to be an identity matrix whereas the only relevant angle in the CKM matrix
is taken to be the Cabbibo one. These assumptions can be modi ed using the parameter
restriction les that are provided with each LQ model.
One could also entertain a possibility of introducing one or more rightchiral neutrinos
thus extending the SM fermion sector. This would allow one to study one additional scalar
LQ state  S1
(3; 1; 2=3)  and to consider additional sets of Yukawa couplings for
R~2 and S1 [5]. These three scalar multiplets have the same transformation properties under
the SM gauge group as the squarks, where the role of the rightchiral neutrino(s) could be
played by neutralino(s). The rightchiral neutrino introduction would, in principle, yield
the same phenomenological signatures that one has for those LQs that couple to the
leftchiral neutrinos as long as the rightchiral neutrinos are light enough. This fact and the
close correspondence between the LQ and squark properties is often used to reinterpret
dedicated experimental searches for supersymmetric particles in terms of limits on the
allowed LQ parameter space. See, for example, ref. [8] for a recent recast along these
lines. Be that as it may, the model les we provide can be modi ed to incorporate these
hypothetical fermionic elds and associated interactions.
We always consider a scenario where the SM is extended with a single scalar LQ
multiplet. From these single LQ model les one can easily generate more complicated
scenarios of new physics (NP) when two or more scalar LQs are simultaneously present at
the energies relevant for collider physics. Since the LQ electric charge eigenstates coincide
with the mass eigenstates in the single LQ extensions we use this property to uniquely
denote a given LQ component. For example, R5=3 (R~2 1=3) is denoted as R2p53 (R2tm13)
2
in model les. The fact that FeynRules 2.0 [16] does not accommodate antifundamental
representation of SU(3) has prompted us to implement all the LQs as triplets of colour in
model les.
In the MadGraph5 aMC@NLO model parameter card of a given LQ scenario one
can modify the LQ mass mLQ and its Yukawa couplings. For example, the 13 element
{ 4 {
of the Yukawa coupling matrix y~RL from eq. (2.3) is denoted as yRL1x3 in the associated
2
model le. For the total decay width of a given LQ we assume that all the quark masses
except the top quark can be neglected and provide relevant expressions. Note that mass
eigenstates that originate from the same LQ multiplet do not need to have the same decay
width. To that end we denote the associated decay widths di erently. For example, the
decay width of R5=3 (R22=3) is denoted as WR253 (WR223).
2
In order to validate the NLO QCD implementation we generate the LQ decay process
with MadGraph5 aMC@NLO. This calculation consists of one born, one virtual and
two realradiation diagrams. The analytic formula for the partial decay width for massless
fermions and lagrangian de ned as L
( yq` qPL;R`
+ h.c.) is [25]
(
! q`) = jyq`j2mLQ
16
1 +
9
2
s = gs2=(4 ) is the strong coupling constant, PL and PR are the standard
leftand rightchiral projection operators, and yq` is the Yukawa coupling strength. Our
numerical result for the NLO QCD correction factor (KF
1
0:0043), obtained using
MadGraph5 aMC@NLO, agrees perfectly with the analytic formula in eq. (2.4). That
is, by reproducing nite oneloop corrections, we have validated the implementation of the
corresponding QCD counterterms in the UFO model(s).
2.2
Vector LQ setup
The phenomenology of vector LQ states is sensitive to their UV origin. The only vector
LQ UFO implementation we opt to provide here is the one that involves U1
(3; 1; 2=3)
eld. This vector boson has attracted a lot of attention recently [14, 26{28] and its model
le can be appropriately modi ed to represent other vector LQ states, if needed, for avour
dependent studies.
The kinetic and mass terms of U1 are
LkUi1netic =
1
2
U y U
igs U1y T aU1 G
a + m2U1 U1y U1 ;
(2.5)
where U
= D U1
D U1 is a eld strength tensor and
is a dimensionless coupling
that depends on the UV origin of the vector. For the YangMills case
= 1, while for the
minimal coupling case
= 0. Precision calculations with vector LQ require UV completion
and this ambiguity is only in part captured by the
dependence that we study in section 3.3.
In fact, unitarization of the highpT scattering amplitudes requires additional dynamics not
far beyond the LQ mass scale, which can impact the production processes in a nontrivial
way. For example, an extra colour octet vector might exist in a complete model and give an
schannel contribution to the LQ pair production. Therefore, in this paper, we concentrate
on the LO e ects in QCD. Implementation of the benchmark UV realisations together
with the NLO QCD corrections is left for future work. For a complementary study of NLO
QCD e ects in vector LQ processes see ref. [13].
The Yukawa part of the U1 lagrangian is de ned in eq. (4.3) in section 4. For simplicity,
here we implement the following lagrangian L
(gbL bL
LU1 + gtL tL
LU1 + h.c.)
{ 5 {
where gbL (gtL ) is the coupling strength of U1 with the bottomtau (topneutrino) pair.
(SU(2) gauge invariance predicts gbL = gtL .) The model les can easily be modi ed to
include other interactions. The relevant parameters that one can vary are mU1 (mLQ),
(kappa), gtL (gtL), and gbL (gbL). The U1 particle (antiparticle) name in the model le
is vlq (vlq ). LQ total decay width is denoted with wLQ, and should be correspondingly
adjusted for a given set of input parameters. For example, the LO partial decay width
(U1 ! b +) = jgbL j2mU1 =(24 ) if one neglects b and
this implementation are presented in section 3.3.
3
Numerical analysis
masses. Numerical results using
The UFO implementation at the NLO in QCD allows us to calculate the total inclusive
cross section for either LQ pair production or single LQ production within the
MadGraph5 aMC@NLO framework for a given LQ mass. We have also prepared a simpli ed
scalar LQ model le, named Leptokvark NLO, that can be used to e ciently determine
inclusive cross sections at the tree level and NLO level. This simpli ed model uses the fact
that the pair production of any scalar LQ is, for all practical purposes, solely QCD driven
whereas the single scalar LQ production in association with lepton depends only on the
particular quark avour that the LQ couples to and the associated coupling strength, as
discussed below.
3.1
Scalar LQ pair production
LQ pair production at hadron collider(s) is a QCD driven process that is, at this point,
completely determined by the LQ mass and strong coupling constant due to the existing
experimental measurements on atomic parity nonconservation [30, 31] and the current
direct search limits on LQ masses at the LHC. The atomic parity nonconservation
measurements limit the allowed strength of the LQ couplings to the rst generation of quarks
and leptons [
5, 6
]. These need to be small and, as such, cannot a ect LQ pair production.
One might think that it could be possible to a ect pair production with the large enough
Yukawa couplings to the second and/or third generation of quarks thereby avoiding atomic
parity violation constraints. Another option is to have couplings between the
rst
generation quarks and second and/or third generation leptons. These particular possibilities
are of limited interest due to the fact that for such large couplings (and masses) single
LQ process is expected to be dominant. (See, for example, gure 3 in ref. [
6
].) We thus
completely neglect tchannel contribution towards pair production of LQs in our numerical
studies. The Feynman diagram that depicts the contribution that we neglect is shown in
the third panel of the rst row of gure 1.
The dominant pair production mechanism at the LHC is a gluongluon scattering
followed by the quarkantiquark annihilation with the representative Feynman diagrams
shown in the rst and second panel of the rst row of gure 1, respectively. The latter
process grows in importance as the LQ mass increases. Di erential and integral cross
sections for these processes at the tree level [32] and NLO level [10] are wellknown. We
{ 6 {
HJEP05(218)6
mass energy as a function of the LQ mass mLQ. The central values are obtained using
xed
factorisation and renormalisation scales
F =
R = mLQ. The total uncertainty (shown with
bands) is obtained by adding the PDF and perturbative uncertainties in quadrature, where the
former one is given by the 68% C.L. ranges when averaging over the PDF replicas while the latter
one is estimated varying factorisation and renormalisation scales within
Prediction for the single LQ production (pp !
` plus pp !
`) initiated from up, down, strange,
charm, and bottom initialstate
avours is marked with u, d, s, c, and b, respectively, while the
F ; R 2 [0:5; 2] mLQ.
LQ pair production (pp !
) is denoted with LQ pair. All single LQ production cross sections
correspond to the case when the coupling strength of the LQ to the quarklepton pair is set to
one (yq` = 1). (Right panel) yq` = p
pair= single(yq` = 1) as a function of the LQ mass for all
initialstate quarks at 13 TeV. The three lines for each quark
avour are obtained using central,
plus, and minus predictions for the total cross sections, and the shaded area indicates the size of
prediction uncertainty. We have checked that the contribution of the Yukawa dependent diagram
with tchannel lepton to LQ pair production is negligible in determining these lines.
nd perfect agreement between our results and analytic expressions that are available in
the literature for the same choice of PDFs.
We use our simpli ed model le to evaluate total inclusive cross section
pair at the
NLO level as a function of LQ mass mLQ for 13 TeV, 14 TeV, and 27 TeV centerofmass
energies for the protonproton collisions for the PDF4LHC15 PDF sets [29] using
MadGraph5 aMC@NLO. Again, all the model les we provide for scalar LQs yield exactly
the same result. The cross section dependancy on the renormalisation and factorisation
scale variations is also taken into account in our evaluation, as well as uncertainty due
to the PDF determination. The following scale variations are used in this determination,
R; F = mLQ=2; mLQ; 2mLQ, using the method of ref. [17]. The relevant NLO results for
pair are summarised in appendix A in tables 1, 2, and 3 for 13 TeV, 14 TeV, and 27 TeV,
respectively, where we also present the PDF uncertainty. The uncertainties are quoted in
per cent units with respect to the cross section central value.
Finally, we present in gure 2 total inclusive cross section at NLO in QCD for scalar LQ
pair production (black band) in protonproton collisions at 13 TeV centerofmass energy
as a function of the LQ mass mLQ for the PDF4LHC15 nlo mc [29] PDF sets.
{ 7 {
protonproton collisions for the NNPDF23NLO [33] and CTEQ6M [34] sets at 13 TeV centerofmass
energy as a function of the LQ mass mLQ. The LQ coupling strength to bottom quark and lepton
is set to one (yb` = 1).
3.2
Single scalar LQ production
The single LQ production in association with a lepton at tree level is induced via partonic
process gq ! `
and involves two diagrams that are shown in the second row of gure 1.
The NLO QCD corrections to this process involve virtual oneloop and realradiation
diagrams, which all have the same linear dependence on the Yukawa coupling yq` de ned
through L
(yq` qPL;R`
+ h.c.). The interference e ects that might be relevant when
LQ simultaneously couples to two fermion pairs of the same avour but di erent chirality
structure are suppressed by the
nalstate lepton mass and can thus be safely neglected.
Therefore, the inclusive NLO QCD Kfactor is rather model independent, i.e., it does not
depend on the speci c LQ representation, nor the
nalstate lepton avour, chirality, and
charge. It only depends on the avour of the initialstate quark in the treelevel diagram,
the LQ mass mLQ, and trivially on the coupling since single
jyq`j2.
We use our simpli ed NLO model le to evaluate total inclusive single LQ
production cross sections
single
u;d;s;c;b for the protonproton collisions using the PDF4LHC15 PDF
sets [29]. These cross sections are due to production through the corresponding quark
avour, as indicated in the subscript, where we set the associated Yukawa coupling strength
{ 8 {
to one (yq` = 1), and vary the LQ mass mLQ. Again, the single LQ production cross section
is proportional to a square of the coupling strength and can thus be trivially rescaled. We
furthermore evaluate single
u;d;s;c;b for 13 TeV, 14 TeV, and 27 TeV centerofmass energies. The
cross section dependancy on the renormalisation and factorisation scale variations is also
taken into account in this evaluation following the method of ref. [17]. The relevant NLO
results are summarised in appendix A in tables 1, 2, and 3 for 13 TeV, 14 TeV, and 27 TeV,
respectively, where we also present the PDF uncertainty. The uncertainties are quoted in
per cent units with respect to the cross section central value.
We present total inclusive single LQ production cross sections in the left panel of
gure 2 in protonproton collisions at 13 TeV centerofmass energy as a function of the
LQ mass mLQ for the PDF4LHC15 nlo mc [29] PDF sets. We, again, take that the relevant
Yukawa coupling strength between a leptoquark and a quarklepton pair equal to one
(yq` = 1). In the right panel of gure 2 we show what values of Yukawa couplings one needs
to use to have equality between the total inclusive single LQ production cross section and
the total inclusive LQ pair production cross section for a given initial quark
avour as a
function of the LQ mass. This plot clearly shows the importance of single LQ production
in the heavy LQ regime.
We furthermore present in gure 3 total inclusive single LQ production cross sections
at the NLO level (upper panel) and a ratio of NLO and LO cross sections (lower panel) for
the NNPDF23NLO [33] and CTEQ6M [34] sets at 13 TeV protonproton centerofmass energy
as a function of the LQ mass mLQ. The lower panel of gure 3 shows the Kfactor for these
PDF sets.
3.3
Single vector LQ production from b quark
In this numerical exercise we study the dependence of the single vector LQ plus lepton
production on the (adjustable) nonminimal QCD coupling
de ned in eq. (2.5).
We
present in
gure 4 ratio of total inclusive cross sections at the LO in QCD for the single
production of a vector LQ and a scalar LQ through a fusion of b and b quarks with gluons
in protonproton collisions at 13 TeV and 27 TeV as a function of the LQ mass mLQ. We
explicitly take that both scalar and vector LQs couple to the bottomtau pair with the
same Yukawa coupling strength (gbL = yb ). Since
with regard to the Yukawa coupling the ratio b
vector and b
b
scalar scale in the same way
vector= bscalar we present in gure 4 is Yukawa
coupling independent. This simply means that the knowledge of the total inclusive cross
section for the single production of a scalar LQ, for a given strength of Yukawa coupling,
allows one to obtain corresponding cross section for vector LQ. We consider both the
Yang
Mills case
= 1 and the minimal coupling case
= 0 to capture
dependance. We nd
that for a xed mLQ the ratio
Note that the ratio
vector= bscalar is parton distribution function (PDF) insensitive and its
b
vector= bscalar grows with the increase in value of
b
parameter.
value decreases as the LQ mass increases.
We, for de niteness, use nn23lo1 PDFs to
perform the numerical calculation. The cross sections are evaluated for
R; F = mLQ,
where R ( F ) is the renormalisation (factorisation) scale.
{ 9 {
HJEP05(218)6
20
r
a vector LQ ( bvector) and a scalar LQ ( bscalar) through a fusion of b and b quarks with gluons for
nn23lo1 PDFs in protonproton collisions at 13 TeV and 27 TeV as a function of the LQ mass mLQ.
We present bvector= bscalar for both the YangMills case
= 1 and the minimal coupling case
= 0.
4
Banomalies inspired LQ search strategy
Semileptonic Bmeson decays have recently received a lot of attention in view of an
increasing set of experimental measurements that contradict the SM predictions. Despite the
fact that a convincing evidence for NP is still missing, the case for it looks very promising
as the coherent picture of deviations seems to be solidifying. (See, for example, ref. [14]
for more details.) While the experimental and theoretical endeavour in Bphysics slowly
keeps moving forward, it is important and timely to provide consistent NP scenarios or,
better still, NP models that are able to predict smoking gun signatures in other (ongoing)
searches, in particular, at the highpT frontier experiments, such as ATLAS and CMS.
Anomalies in Bmeson decays consistently point to a violation of lepton avour
universality (LFU) and can be grouped into two di erent classes. These are (i) deviations from
=` (where ` = e; ) universality in semitauonic decays as de ned by R(D( )) observables
(b ! c` charged currents) [35{37] and (ii) deviations from
as de ned by R(K( )) observables (b ! s`` neutral currents) [38, 39]. Further evidence of
=e universality in rare decays
coherent deviations in rare b ! s
angular distributions of B ! K
+
transitions has been observed in the measurements of
[40, 41]. The overall statistical signi cance of the
discrepancies in the clean LFU observables alone is at the level of 4
for both charged and
neutral current processes. See, for example, refs. [42{46].
R(D( )) anomaly: the enhancement of O(20%) on top of the SM treelevel
CKMfavoured contribution to b ! c
transition requires large NP e ect that is, presumably,
treelevel generated. A careful consideration based on the perturbative unitarity implies
that the scale of NP is rather low [47], i.e., in the TeV ballpark, making it an ideal physics
case for the LHC. Analysis of the lowenergy process in the SM e ective
eld theory
(SM EFT) requires NP in (at least) one of the d = 6 semileptonic fourfermion operators
mass spectrum, the strongest bound comes from the DrellYan production of a dilepton pair via a
t channel LQ exchange since amplitude scales with yq2`. Finally, in the intermediate mass range,
production of a single LQ in association with a lepton is expected to be the most sensitive probe
as the associated amplitude scales linearly with yq`.
OVL
OT
(QL
(QL
kQL)(LL
uR)i 2(LL
kLL), OSR
(dRQL)(LLeR), OSL
(QLuR)i 2(LLeR), and
eR), where k, k = 1; 2; 3, are Pauli matrices, and uR are the
rightchiral uptype quark
elds. For example, a very good
t to data is obtained with
a shift in OVL operator only, giving a universal enhancement in all b ! c
Nonetheless, several other scenarios are tting data well [48].
processes.
In the simplest case, these e ective operators can be generated by a treelevel exchange
of a single mediator, de ning simpli ed benchmark models for the LHC studies. These
(3; 3; 2=3) (S3
(3; 3; 1=3)), doublet vector (scalar)
include triplet vector (scalar) U
3
V
2
(3; 2; 5=6) (R2
(3; 2; 7=6)), and singlet vector (scalar) U
1
(3; 1; 2=3) (S1
(3; 1; 1=3)) [5]. Triplets induce OVL operator only, when integrated out, while, for example,
U1 induces OVL and OSR , and S1 yields OVL and OSL
1=4 OT .
Since the scale required to t R(D( )) is rather low the main challenge is not only
to reconcile it with the nonobservation of other related signals such as avour changing
neutral currents (FCNC) in the downtype quark sector (e.g. [49{52]) or other treelevel
avour changing processes (e.g. [53]) but also with
decays and electroweak precision
observables [54, 55] as well as highpT production of
leptons [56]. These constraints
suggest that the LQ couples dominantly (but not entirely) to the third generation fermions.
A typical coupling tting the anomaly is given by yb
mLQ=1 TeV.
There are two important implications of this discussion. First, the value of the
{q{`
coupling yq` emerging from the lowenergy
t suggests that three types of LQ processes
are relevant at the LHC. In addition to the widely studied LQ pair production, single LQ
plus
lepton production (from initial b quark) turns out to be crucial as indicated in the
right panel of gure 2. Moreover, as shown in ref. [56], a virtual LQ exchange in tchannel
can give a sizeable contribution to
lepton pair production in the limit of complete
alignment with the downtype quarks. These three processes scale di erently with the
coupling yq` and can thus provide complementary information. Possible exclusion plot one
could potentially generate by using this feature is sketched in
gure 5. It is, in our view,
crucial to perform such a combined analysis to scrutinise the available parameter space as
much as possible.
interactions
Second, the decay channels of the LQ resonances are predicted. Let us illustrate this
point on a few examples. We, in particular, consider U1, S1, and S3, with the corresponding
LS3
LS1
LU1
y3LiLj QLC i;a ab( kSk)bcLjL;c + h.c. ;
3
y1LiLj QL
C i;aS1 abLjL;b + y1RiRj uCR iS1ejR + h.c. ;
x1LiLj QiL;a
U1 LjL;a + x1RiRj diR
U1 ejR + h.c. :
(4.1)
(4.2)
(4.3)
HJEP05(218)6
For instance, U
1 decays dominantly to b and t
nal states. If no rightchiral couplings
(x1RR) are present, the branching ratios are predicted to be B(U1 ! b ) = B(U1 ! t ) =
0:5 , motivating a search for t b
nal state in addition to conventional bb
and tt
searches [57, 58]. This changes with the inclusion of sizeable x1R3R3 in favour of U1 ! b
decay. On the contrary, S1 resonance decays to b and t
nal states. As in the previous
example, the exact branching ratios depend on the relative strengths of the left and
rightcouplings. Another very instructive example is that of the scalar triplet S3, which has three
degenerate resonances of a di erent charge. The decay modes and branching ratios are xed
assuming the dominant coupling to be y3L3L3, in particular, B(S31=3
! b ) = B(S31=3
! t ) =
0:5 , while B(S32=3
! t ) = 1:0 , and B(S34=3
! b ) = 1:0 . As illustrated by these examples,
the highpT searches might also prove useful to distinguish the underlying LQ model.
The above discussion implicitly assumed the dominant LQ couplings to be with the
third family, as predicted in most models with conventional avour structure, and as
(usually) required by the FCNC constraints. However, a viable possibility in some models is to
have sizeable coupling to q2{`3 fermion current. Here, the LQ tends to decay to light jets
(as opposed to b and t) and the production from initial s (c) avours (as opposed to b) is
preferred. Such example has been studied in ref. [59] utilising large ys coupling.
R(K( )) anomaly: rare Bdecays are generated at oneloop level in the SM, and su er
from additional CKM and GIM suppression. Therefore, the e ective scale indicated by the
anomaly in rare b ! s`` transitions is m =pys yb
30 TeV, where yq is the relevant
{q{` coupling. If the anomalies are in muons, as suggested by the angular observables,
then V
A operator structure has to be generated. At tree level, this can be achieved by
an exchange of a single mediator such as S3, U1 or U3 .
The main implication of such a large e ective NP scale is that the LQ pair production
might be the only relevant process at the LHC (unless one of the two couplings, i.e., ys
or yb , is extremely large, or the LQ couples to the valance quarks [60]). This, however,
is not the case at the future circular hadron collider (FCChh), where a much heavier LQ
could potentially be probed. (See gure 10 of ref. [61].) On the other hand, the width of
an LQ at the TeV scale could be dominated by other decay channels (other than b or
s ). For instance, the interesting option is the decay to third family. Indeed given the
same e ective V
A structure, a coherent picture of Banomalies is emerging [14] when
requiring (i) a new dynamics in (dominantly) leftchiral semileptonic transitions, and (ii) a
avour structure implying dominant (but not exclusive) couplings to the third family. The
highpT phenomenology of the combined solution is very similar to the R(D( )) discussion
above. An exceptional working model is U
1 vector LQ. For UV completion as a massive
gauge boson see refs. [26, 27].
5
Conclusions
We address the need for an uptodate Monte Carlo event generator output that can be
used for the current and future experimental searches and search recasts concerning scalar
and vector LQs.
We implement and provide readytouse LQ models in the universal
FeynRules output format assuming the SM fermion content and conservation of baryon
and lepton numbers for all scalar LQs as well as one vector LQ. Scalar LQ implementations
include NLO QCD corrections. We validate our numerical results with the existing NLO
calculations for the pair production and present novel results for the single LQ production
for scalar (vector) LQs at the NLO (LO) level. The numerical output comprises the NLO
QCD inclusive cross sections in protonproton collisions at 13 TeV, 14 TeV, and 27 TeV
centerofmass energies as a function of the LQ mass. These results can be particularly
useful for the current and future LHC data analyses, accurate search recasts, and the avour
dependent studies of the LQ signatures at colliders within the MadGraph5 aMC@NLO
framework. We also discuss aspects of the LQ searches at a hadron collider and outline the
highpT search strategy for LQs recently proposed in the literature to resolve experimental
anomalies in Bmeson decays.
Acknowledgments
This work has been supported in part by Croatian Science Foundation under the project
7118. I.D. would like to thank the CERN Theoretical Physics Department on hospitality,
where part of this work was done. We would like to thank Darius A. Faroughy for useful
discussions.
A
LQ cross sections in protonproton collisions
We present in tables 1, 2, and 3 total inclusive cross sections in pb for the PDF4LHC15 PDF
sets [29] as a function of the LQ mass mLQ at 13 TeV, 14 TeV, and 27 TeV centerofmass
energies for the protonproton collisions, respectively. These results are valid in the narrow
width approximation. For the discussion on the e ects beyond this approximation see
ref. [13].
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
LQ mass at 13 TeV centerofmass energy for the protonproton collisions.
u;d;s;c;b describe single LQ production cross sections through corresponding
quark
avour when the associated Yukawa coupling strength is set to one.
The cross section
dependancy on the change in the renormalisation ( R) and factorisation ( F ) scales is taken into
account through the following scale variations:
F = mLQ=2; mLQ; 2mLQ. First (second)
uncertainty is due to the renormalisation
R and factorisation
F scale (PDF) variations and is
given in per cent units.
HJEP05(218)6
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
75:1+12:3%+2:6%
12:1%
2:6%
2:25+11:5%+4:0%
12:5%
4:0%
0:222+11:1%+5:2%
12:7%
0:037+10:9%+6:4%
12:9%
5:2%
6:4%
0:00787+11:5%+7:7%
13:4%
7:7%
0:00204+11:5%+9:1%
13:6%
9:1%
0:000574+12:1%+10:8%
14:0%
10:8%
0:000177+12:1%+12:5%
14:2%
12:5%
0:0000585+13:2%+14:7%
14:9%
14:7%
0:0000193+13:3%+17:0%
15:1%
17:0%
8:68+7:6%+1:4%
LQ mass at 14 TeV centerofmass energy for the protonproton collisions.
pair corresponds to the
u;d;s;c;b describe single LQ productions through corresponding quark
avour
when the associated Yukawa coupling strength is set to one. The cross section dependancy on the
change in the renormalisation ( R) and factorisation ( F ) scales is taken into account through the
following scale variations:
F = mLQ=2; mLQ; 2mLQ. First (second) uncertainty is due to the
R and factorisation F scale (PDF) variations and is given in per cent units.
HJEP05(218)6
0:106+9:6%+4:7%
11:3%
4:7%
0:0138+9:8%+5:9%
11:7%
5:9%
0:00257+9:8%+7:1%
11:9%
7:1%
0:00059+10:4%+8:5%
12:4%
8:5%
0:000159+10:5%+10:0%
12:6%
10:0%
0:0000453+10:8%+11:6%
12:9%
11:6%
0:0000139+11:0%+13:6%
13:2%
13:6%
0:676+6:6%+1:5%
6:5%
1:5%
0:149+6:6%+1:9%
6:7%
1:9%
0:0442+6:9%+2:2%
7:1%
2:2%
0:0158+7:3%+2:6%
7:5%
2:6%
0:00635+7:4%+2:9%
7:7%
2:9%
0:00279+7:6%+3:3%
7:9%
3:3%
0:00131+7:8%+3:7%
8:2%
3:7%
0:000643+8:0%+4:1%
8:4%
4:1%
0:000332+8:3%+4:6%
8:7%
4:6%
0:000175+8:6%+5:1%
8:9%
5:1%
0:0000947+8:6%+5:6%
9:1%
5:6%
0:388+6:6%+2:2%
6:6%
2:2%
0:0797+6:9%+2:4%
7:0%
2:4%
0:0224+7:2%+2:6%
7:4%
2:6%
0:0076+7:4%+2:9%
7:7%
2:9%
0:00289+7:7%+3:2%
8:0%
3:2%
0:00122+7:9%+3:6%
8:3%
3:6%
0:000548+8:1%+4:0%
8:5%
4:0%
0:000261+8:4%+4:5%
8:8%
4:5%
0:000129+8:7%+5:1%
9:1%
5:1%
0:0000657+8:8%+5:6%
9:2%
5:6%
0:0000343+9:2%+6:3%
9:5%
6:3%
LQ mass at 27 TeV centerofmass energy for the protonproton collisions.
pair corresponds to the
u;d;s;c;b describe single LQ productions through corresponding quark
avour
when the associated Yukawa coupling strength is set to one. The cross section dependancy on the
change in the renormalisation ( R) and factorisation ( F ) scales is taken into account through the
following scale variations:
F = mLQ=2; mLQ; 2mLQ. First (second) uncertainty is due to the
R and factorisation F scale (PDF) variations and is given in per cent units.
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