#### Jet azimuthal angle correlations in the production of a Higgs boson pair plus two jets at hadron colliders

Eur. Phys. J. C
Jet azimuthal angle correlations in the production of a Higgs boson pair plus two jets at hadron colliders
Junya Nakamura 0
Julien Baglio 0
0 Institute for Theoretical Physics, University of Tübingen , Auf der Morgenstelle 14, 72076 Tübingen , Germany
Azimuthal angle correlations of two jets in the process pp → H H j j are studied. The loop induced O(αs4α2) gluon fusion (GF) sub-process and the O(α4) weak boson fusion (WBF) sub-process are considered. The GF sub-process exhibits strong correlations in the azimuthal angles φ1,2 of the two jets measured from the production plane of the Higgs boson pair and the difference between these two angles φ1 − φ2, and a very small correlation in their sum φ1 + φ2. Using a finite value for the mass of the loop running top quark in the amplitude is crucial for the correlations. The impact of a non-standard value for the triple Higgs self-coupling on the correlations is found small. The peak shifts of the azimuthal angle distributions reflect the magnitude of parity violation in the gg → H H amplitude and the dependence of the distributions on parity violating phases is analytically clarified. The normalised distributions and the peak positions of the correlations are stable against the variation of factorisation and renormalisation scales. The WBF sub-process also produces correlated distributions and it is found that they are not induced by the quantum effect of the intermediate weak bosons but mainly by a kinematic effect. This kinematic effect is a characteristic feature of the WBF sub-process and is not observed in the GF sub-process. It is found that the correlations are different in the GF and in the WBF sub-processes. As part of the process dependent information, they will be helpful in the analyses of the process pp → H H j j at the LHC.
1 Introduction . . . . . . . . . . . . . . . . . . . . . .
2 Helicity amplitudes for the process pp → V ∗ V ∗ j j
→ H H j j . . . . . . . . . . . . . . . . . . . . . . .
2.1 VBF amplitudes . . . . . . . . . . . . . . . . . .
2.2 Helicity amplitudes for the splitting processes . .
2.3 Helicity amplitudes for the processes
V V → H H . . . . . . . . . . . . . . . . . . . .
3 Azimuthal angle correlations . . . . . . . . . . . . . .
3.1 The gluon fusion process . . . . . . . . . . . . .
3.2 The weak boson fusion process . . . . . . . . . .
4 Summary and discussion . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction
The discovery of a Higgs boson with a mass around 125
GeV in 2012 is the main discovery of Run I of the Large
Hadron Collider (LHC) [1,2]. The study of its properties has
started and until now they are compatible with the standard
model (SM) hypothesis [3–5]. To probe the mechanism of
electroweak symmetry breaking [6–9] directly one would
want to measure the triple Higgs self-coupling that is one of
the key parameters of the scalar potential. This is one of the
main goals of the future high-luminosity LHC and the future
circular collider (FCC) in hadron–hadron mode, a potential
100 TeV proton–proton collider following the LHC at CERN
(for reviews of the FCC physics potential, see Refs. [10,11]).
In this view the production of a pair of Higgs bosons needs
to be observed and has been extensively studied over the
last years [12–32] (see also Refs. [33–35] for studies at the
FCC). The gluon fusion (GF) sub-process [36–39] and the
weak boson fusion (WBF) sub-process [36,40–42] are the
two main sub-processes; see e.g. Ref. [43] for a review of
SM studies. Both of the two sub-processes are sensitive to
the triple Higgs self-coupling. In the WBF sub-process, we
have access to the coupling between the two Higgs bosons
and the two weak bosons, too. In Refs. [23,30] studies using
the production of a pair of Higgs bosons plus two hadronic
jets has been conducted, using both the GF and the WBF
sub-processes. The main advantage of the latter sub-process
is the fact that the theoretical uncertainties are under
control [19,27], but the phenomenological studies suffer from
the difficulty to separate the GF contributions from that of
the WBF.
Azimuthal angle correlations of two jets produced together
with heavy particles have been actively studied as a provider
of important information as regards the heavy particles [44–
50]. The correlations are induced by only a certain type
of sub-processes, called vector boson fusion (VBF)
subprocesses,1 in which a heavy object is produced by a fusion
of two vector bosons emitted from incoming two coloured
particles. The correlations arise from the quantum effect of
the two fusing intermediate vector bosons [42,47].2 A set of
cuts on the rapidity y1,2 of the two jets, y2 < 0 < y1 and
y1 − y2 3 and an upper cut on the transverse momentum
pT of the two jets are therefore crucial for the two jets to show
strong correlations if any, since they enhance contributions
from VBF sub-processes [47]. These rapidity cuts are often
called VBF cuts.
The azimuthal angle correlations of the two jets both in
the GF sub-process and in the WBF sub-process of the
single Higgs boson plus two jets production process pp →
H j j [42,44–47,52] are nowadays a common knowledge,
have been studied in detail [53–65] and applied in many
phenomenological studies. However, the azimuthal angle
correlations of the two jets in the Higgs boson pair plus two
jets production process pp → H H j j have not been
studied thoroughly. To our knowledge, the correlations in the GF
sub-process, which is an one-loop induced O(αs4α2) process
at leading order (LO), have not been studied in the literature.
One of the reasons may be that event generations are still
challenging even with an advanced calculation technique. For the
GF sub-process in the process pp → H j j , the approach of
using the effective interactions between the Higgs boson and
gluons is known to work quite well as long as the pT of the
jets are small enough [66,67]. The azimuthal angle
correlation after the VBF cuts is also described correctly [67].
Therefore event generations can be easily performed with
a tree-level event generator which implements the effective
interactions. In contrast, for the GF sub-process in the process
pp → H H j j , the effective interaction approach is known
not to work well in describing the distributions of several
observables [23,30]. It is naively expected that this
observation is the same for the azimuthal angle correlations. The
process pp → H H j j at LO with the exact one-loop amplitude
has been calculated for the first time in Ref. [23] and
subsequently phenomenology is studied in Ref. [30]. The fully
automated event generation for one-loop induced processes
1 In this paper, the term “vector boson fusion” is used to refer to both
gluon fusion and weak boson fusion.
2 The azimuthal angle correlations of two outgoing electrons in e+e−
collisions, induced by the quantum effect of two intermediate virtual
photons, have been discussed a long time ago; see e.g. [51].
is now available in MadGraph5_aMC@NLO [68,69]. This
achievement will activate further phenomenological studies
of the process pp → H H j j including studies which use the
azimuthal angle correlations. The azimuthal angle
correlations of the two jets in the WBF sub-process, which is an
O(α4) process at LO, have been studied in [42]. There, only
the azimuthal angle difference of the two jets is studied as an
azimuthal angle observable.
In this paper, we study the azimuthal angle correlations
of the two jets in the process pp → H H j j . Instead of
considering all of the sub-processes contributing to the process
pp → H H j j , we consider only the VBF sub-processes:
qq → qq V ∗V ∗ → qq H H (V = W, Z , g),
qg → qgV ∗V ∗ → qg H H (V = g),
gg → ggV ∗V ∗ → gg H H (V = g),
where q denotes a light quark or a light antiquark, g denotes a
gluon and V ∗ denotes an intermediate off-shell vector boson.
The two Higgs bosons are produced by a fusion of two vector
bosons emitted from incoming two coloured particles. More
precisely, we calculate the amplitudes contributed from only
t -channel Feynman diagrams shown in Fig. 13 and obtain the
approximate analytic cross section formula. We call them
VBF amplitudes and VBF diagrams, respectively, in this
paper. The circles denote all the Feynman diagrams
contributing to the process V ∗V ∗ → H H at LO. When the two
virtual vector bosons are weak bosons (WBF sub-process),
there are four tree-level Feynman diagrams. When the two
virtual vector bosons are gluons (GF sub-process), there are
eight one-loop Feynman diagrams in which heavy quarks
run. We calculate these LO diagrams exactly.
Our calculations give not the exact LO results but
approximate ones, since diagrams other than the VBF diagrams are
not considered. In a kinematic region where the virtualities
of the intermediate vector bosons are small (this is achieved
by the VBF cuts and the pT upper cut), the VBF amplitudes
dominate and thus our approximate analytic cross section
gives a valid approximation to the exact LO result. We call
it the VBF kinematic region. For the WBF sub-process, we
explicitly compare our approximate result with the exact LO
result and find good agreement between the two results. For
the GF sub-process, in order to make our discussion clearer,
we list the calculation methods which we discuss:
Calc.1 Our approximate analytic cross section, to which
only the VBF diagrams contribute. The mass of the
loop running top quark mt in the gg → H H
amplitude is set finite.
3 All pictures in this paper are drawn by using the program
Fig. 1 Feynman diagrams calculated in this paper. Here V ∗ stands for either an off-shell weak boson W/Z or an off-shell gluon g. The second and
third diagrams only contribute in the case V = g. The circles denote all the Feynman diagrams contributing to the process V ∗V ∗ → H H at LO
Calc.2 Our approximate analytic cross section, to which
only the VBF diagrams contribute. The mass of the
loop running top quark mt in the gg → H H
amplitude is set extremely large.
Calc.3 The LO cross section, to which all LO diagrams
contribute. The mass of the loop running top quark mt in
the full amplitude is set extremely large.
Calc.4 The LO cross section, to which all LO diagrams
contribute. The mass of the loop running top quark mt in
the full amplitude is set finite.
We note that the full amplitude in Calc.3 and Calc.4
contains contributions from diagrams with a triangle, box,
pentagon or hexagon loop. Calc.3 is equivalent to the
calculation that is performed by a tree-level event generator which
implements the effective interactions between gluons and the
Higgs bosons (we use MadGraph5_aMC@NLO [68] to
generate the Calc.3 result). Calc.4 is the one that was performed
in Ref. [23]. In the VBF kinematic region, we expect that
Calc.1 reproduces the Calc.4 result and Calc.2 reproduces
the Calc.3 result. In this paper, we compare the Calc.2 result
with the Calc.3 result and find good agreement between the
two results in the inclusive cross section and the azimuthal
angle correlations. We do not perform Calc.4 and thus do not
compare the Calc.1 result with the Calc.4 result. We stress
that, if it is confirmed that Calc.2 reproduces the Calc.3 result,
then we can be confident that Calc.1 is able to reproduce the
Calc.4 result. This is because the dominance of the VBF
amplitudes in the VBF kinematic region is solely due to the
propagator factors of the intermediate gluons and the details
of the core process gg → H H in Calc.1 and Calc.2, which
are represented by the circles in Fig. 1, have nothing to do
with the validity of the approximation. We mention that the
validity of the approximation has been demonstrated in other
processes, too. Ref. [47] studies the processes pp → j j ,
where = H, A, G denotes the Higgs boson, a parity-odd
Higgs boson and a spin-2 massive graviton, respectively. In
Ref. [50], the processes pp → Q Q¯ j j , where Q denotes the
top quark or the bottom quark, are studied. In the both
studies, it has been shown that the approximate cross sections
reproduce the exact LO results in the VBF kinematic region.
In order to measure the azimuthal angle correlations, we
study four observables: φ1, φ2, φ = φ1 − φ2 and φ+ =
φ1 + φ2, where φ1,2 are azimuthal angles of the two jets
measured from the production plane of the Higgs boson pair
( φ is irrelevant to the production plane as is clear from its
definition). φ is the observable which is sensitive to the
property of the Higgs boson in the process pp → H j j [44–
47,49] and thus has been the subject of many studies. The
processes pp → G j j [47] and pp → Q Q¯ j j [50] exhibit
strong correlations in φ+. To our knowledge, correlations
in φ1,2 have not been addressed in any hadronic process in
the literature. Our explicit findings can be summarised as
follows. The GF sub-process has strong correlations in φ
and φ1,2, and the pT of the Higgs boson can be an useful
measure to enhance or suppress these correlations. Using
the finite mt value is important to produce the correlations
correctly. Violation of the parity invariance of the gg →
H H amplitude appears as the peak shifts of the correlations.
The impact of a non-standard value for the triple Higgs
selfcoupling on the correlations is smaller than that on other
observables, such as the invariant mass of the two Higgs
bosons, of the inclusive process pp → H H . The correlation
in φ+ is negligibly small in most every case. The normalised
distributions of and the peak positions of the correlations
are stable against scale variations in the parton distribution
functions and the strong couplings. The WBF sub-process
produces correlated distributions in all of the azimuthal angle
observables and they are not induced by the quantum effect
of the intermediate weak bosons but mainly by a kinematic
effect. This kinematic effect is a characteristic feature of the
WBF sub-process and is not observed in the GF sub-process.
The impact of a non-standard value for the triple Higgs
selfcoupling on the correlations is not significant in the WBF
sub-process, too. The correlations in the GF and WBF
subprocesses are found to be different.
The paper is organised as follows. In Sect. 2, we perform a
calculation of the VBF amplitudes. Since it can be shown that
the azimuthal angle correlations arise from the interference
of amplitudes with various helicities of the two intermediate
vector bosons [42,47], we employ a helicity amplitude
technique. Our calculation is performed based on the method
presented in Ref. [47]. A full analytic set of helicity amplitudes
is presented. Since the material in Sect. 2 is rather technical,
the reader who is interested in only the results may skip this
section. In Sect. 3, a detailed study of the azimuthal angle
correlations is presented. First, we discuss the GF sub-process
in Sect. 3.1. The squared VBF amplitude for the four
subprocesses Eq. (1.1) is given in a compact form. This analytic
formula is found to be quite useful in making expectations
of the correlations. The correlations in different kinematic
regions of the two Higgs bosons and those in non-standard
values for the Higgs triple self-coupling are studied. The
impact of parity violation in the gg → H H amplitude on
the correlations is also studied. Next, we discuss the WBF
sub-process in Sect. 3.2. The squared VBF amplitude is given
in a simple form by keeping only the dominant terms. The
correlations in non-standard values for the Higgs triple
selfcoupling are studied. In Sect. 4 we summarise our findings
and give some comments.
2 Helicity amplitudes for the process pp → V ∗ V ∗ j j
→ H H j j
In this section we present a full analytic set of helicity
amplitudes contributed from the vector boson fusion (VBF)
diagrams. Our calculation is based on the method presented
in Ref. [47]. We present a more complete discussion on the
treatment of the intermediate off-shell gluons in the VBF
diagrams. In addition, we discuss the importance of an
appropriate choice of gauge-fixing vectors for the polarisation vectors
of the external gluons. We believe that the above two remarks
provide sufficient justification for repeating some of the
calculations of Ref. [47].
2.1 VBF amplitudes
We first introduce a common set of kinematic variables
→ a3 k3, σ3 + a4 k4, σ4 + V1 q1, λ1 + V2 q2, λ2
(2.1)
where a1,2,3,4 can be quarks, antiquarks or gluons, V1,2 are
intermediate off-shell vector bosons, H denotes the Higgs
boson and the four-momentum ki and helicity σi of each
particle are shown. This assignment for the VBF sub-processes
is more apparent in Fig. 2.
The external particles take helicities σi = ±14 and the
intermediate vector bosons take helicities λi = ±1, 0. Colour
indices are suppressed. The VBF helicity amplitude can be
expressed as follows:
4 We define the helicity operator for a two-component spinor by p ·
σ /| p| with the Pauli matrices σ , so a quark also takes σ = ±1 in our
notation. Sometimes we simply write σ = ± instead of σ = ±1, and
do the same for λ.
The λ = s component μ(q, λ = s) = qμ/ −q2 is the
scalar part of a virtual weak boson and vanishes when it
Fig. 2 The assignment of the four-momenta and helicities of each
particle for the VBF sub-processes
σ3σ4 μ μ
Mσ1σ2 = JV11a1a3 k1, k3; σ1, σ3 JV22a2a4 k2, k4; σ2, σ4
× DμV11μ1 q1 DμV22μ2 q2 μH1HμV21V2 q1, q2, q3, q4 ,
μ
where JViiai ai+2 (ki , ki+2; σi , σi+2) is a current involving the
off-shell vector boson Vi and the two external quarks,
antiquarks or gluons, DVii i (qi ) is the Vi propagator and
μ μ
μ μ
1 2
H H V1V2 q1, q2, q3, q4 is the tensor amplitude for the pro
cess V V
1 2 → H H . When we denote a helicity amplitude in
this paper, we show only helicity indices.
For the weak boson fusion (WBF) sub-process, we have
(V1, V2) = (W +, W −), (W −, W +) and (Z , Z ), and only
the VBF diagram shown in Fig. 1 (left) contributes to the
VBF amplitude. The VBF amplitude is gauge invariant on
its own. This is apparent from the fact that only the VBF
diagram exists in some of the WBF sub-processes, for instance
there are no other diagrams describing the sub-process ud →
ud H H . We choose the unitary gauge for the weak boson
propagator,
= q2 − m2V −1.
We express the projector part of the above propagator in terms
of polarisation vectors in the helicity basis (λ = ±1, 0, s):
q2
− 1 − m2
V
q2
− 1 − m2
V
−q2
−q2
= −1 λi +1 J μi
Vi ai ai+2 ki , ki+2; σi , σi+2 μi qi , λi ∗,
(2.7a)
H H V1V2 q1, q2, q3, q4 .
Equation (2.7a) represents a helicity amplitude for the
splitting process ai → ai+2Vi , where Vi is off-shell. This will
be derived in Sect. 2.2. Equation (2.7b) represents a helicity
amplitude for the process V1V2 → H H , where V1 and V2
are off-shell. This will be presented in Sect. 2.3.
The gluon fusion (GF) sub-process (V1, V2) = (g, g)
is more complicated than the WBF sub-process. The VBF
amplitude for the qq initiated sub-process (a1, a2) = (q, q)
is gauge invariant on its own, the reason being the same as
for the WBF amplitude. If we choose the Feynman–’t Hooft
gauge for a gluon propagator, the projector part of the
propagator is:
−q2
−q2
− μ q, λ = s ∗ μ q, λ = s .
The λ = s component μ(q, λ = s) = qμ/ −q2 again
vanishes when it couples with a quark current. As a result,
the following replacement is possible without any
approximation:
couples with a light quark current. As a result, the following
replacement is possible when the external quarks are assumed
to be massless:
The polarisation vectors μ(qi , λi = ±, 0) will be defined
later once the kinematics of the weak bosons is fixed. After
that, one can confirm Eq. (2.4) explicitly. By inserting the
identity Eq. (2.5) into the VBF helicity amplitude in Eq. (2.2),
it can be expressed as a product of three helicity amplitudes:
Therefore, for the qq initiated sub-process, we can arrive at
the same expression as in Eq. (2.6). In Ref. [47], the above
procedure and thus the expression Eq. (2.6) is used not only
for the qq initiated sub-process but also for the qg and gg
initiated sub-processes (a1, a2) = (q, g), (g, g).5 We point
out that this approach does not necessarily calculate the
offshell effects of the intermediate gluons correctly for the qg
and gg initiated sub-processes and only introduces
unnecessary complications in the amplitude calculation. Since the
off-shell gluon amplitude (MgHgH )λ λ is not enhanced in
the on-shell limit of the gluons, we 1ca2n always expand the
off-shell gluon amplitude around the on-shell limit. To make
our discussion simpler, let us consider an amplitude which
involves only one off-shell gluon. The off-shell gluon
amplitude can be expanded as
where Q is the virtuality of the gluon and the first term in
the right hand side (RHS) of the first equation is the
onshell amplitude, which is gauge invariant. As we will see in
Sect. 2.2, the amplitude for the splitting process in Eq. (2.7a),
where an off-shell gluon with virtuality Q is emitted, has an
overall factor of Q. By considering this factor and Q−2 in the
propagator factor of the off-shell gluon, we find the following
term in a VBF amplitude:
While the first term in the RHS of the above equation is
gauge invariant, the rest terms are generally dependent on
a gauge-fixing choice for the off-shell gluon. As we have
mentioned above, for the qq initiated sub-process, the VBF
amplitude is gauge invariant on its own and hence not only
the first term in the RHS of Eq. (2.11) but also the other terms
as a whole are gauge invariant. This is not the case for the
qg and gg initiated sub-processes. For the qg and gg
initiated sub-processes, the second and higher terms in the RHS
of Eq. (2.11), which are not enhanced in the on-shell limit,
become gauge invariant only after contributions from other
diagrams are also included. Therefore, in our method where
only the VBF diagrams are calculated, we can calculate the
off-shell effect of the intermediate gluons correctly only for
5 The λ = s component μ(q, λ = s) = qμ/ −q2 vanishes when it
couples to a gluon current, too.
the qq initiated sub-process. For the qg and gg initiated
subprocesses, therefore, we take the following approach:
Completely ignore the off-shell effect of the intermediate gluons
and look at only a kinematic region where the virtualities
of the gluons are small (Q → 0). This is possible because
the off-shell effect in the amplitude will not be essential as
long as we look at the small virtuality region, as is clear from
Eq. (2.11). The unitarity condition gives the following
equation for a gluon propagator in its on-shell limit q2 → 0:
= Dg q
λ=±
= q2 −1.
By using this, we can express the VBF amplitude for the GF
sub-process as a product of three helicity amplitudes in the
same way that we do for the WBF sub-process and the qq
initiated GF sub-process in Eq. (2.6). The differences from
the qq initiated GF sub-process are (1) the λ = 0
component μ(q, λ = 0) in Eq. (2.6) is neglected, (2) the off-shell
gluon amplitude (MVH1HV2 )λ1λ2 is replaced by the on-shell
gluon amplitude in which q12 = 0 and q22 = 0. Note that
these two approximations are nothing less than the two
fundamental ingredients of the equivalent photon approximation
(EPA) [71,72]. As a rule of using the EPA, we should clarify
the kinematic region where the approximated VBF amplitude
is a good approximation to the exact VBF amplitude. For the
process pp → H H j j , it should be −q12 < sˆ and −q22 < sˆ,
where sˆ = M H2 H . We use the EPA not only for the qg and
gg initiated sub-processes but also for the qq initiated
subprocess, so that we can consistently use the on-shell gluon
amplitude (MVH1HV2 )λ λ . It should be noted that, while we
1 2
use the on-shell gluon amplitude for the process gg → H H ,
the amplitude for an off-shell gluon emission in Eq. (2.7a)
is still calculated without ignoring the off-shell effect of the
gluon (q12,2 = 0), as in other applications of the EPA.
2.2 Helicity amplitudes for the splitting processes
We derive the helicity amplitude (2.7a) for the splitting
processes. We use the chiral representation for Dirac matrices.
Since we neglect the mass of the external quarks, the
helicity of the quark is equal to its chirality and the helicity of
the antiquark is opposite to its chirality. As a result, the
helicity of ai is always equal to that of ai+2 (σi = σi+2)
for the quark and antiquark splitting processes. Otherwise,
the amplitude vanishes. By introducing one common
current Jˆiμ, we can express the quark and antiquark currents
μ
JVi ai ai+2 ki , ki+2; σi , σi+2 in a compact manner:
JVi ai ai+2 ki , ki+2; σi , σi+2 = gσVi ai ai+2 Jˆiμ ki , ki+2; σi , σi+2 ,
μ
i
(2.13a)
where u(k, σ )α represents the two-component Weyl u-spinor
with its four-momentum k, helicity σ and chirality α (= ±1),
and σ±μ = (1, ±σ ) with the Pauli matrices σ . Note that we
can use the above Jˆiμ for the antiquark current, too, since
The couplings between quarks and vector bosons relevant to
our study are summarised as follows:
g−W −du
= g+W −u¯d¯
where gs is the QCD coupling, t a is the generator of the SU(3)
group, gw = e/ sin θw, gz = e/(sin θw cos θw) with θw being
the Weinberg mixing angle and e being the proton charge,
Vud is an element of the CKM matrix, Qq is the electric
charge of the quark in units of e, Tu3 = 1/2 and Td3 = −1/2.
The gluon current involving three gluons is also expressed
by Eq. (2.13a) with
ki , σi · ki+2, σi+2 ∗ −ki+2 − ki μ,
where f abc is the structure constant of the SU(3) group.
Generally speaking, helicities are frame dependent and so
are helicity amplitudes. When we calculate the VBF helicity
amplitude Mσσ31σσ42 in Eq. (2.6), we must choose one frame at
first and then define the four-momenta and the helicities of all
the external particles in this particular frame. For our
calculation we choose the centre-of-mass (c.m.) frame of the two
intermediate vector bosons moving along the z-axis, which
is shown in the middle of Fig. 3. We call it the VBF frame.
Note that the production plane of the two Higgs bosons
coincides with the plane of the x –z axes. All of the three helicity
amplitudes in the VBF helicity amplitude must be evaluated
in the VBF frame. However, by using a property of
helicities, we can justify a calculation of the helicity amplitude
(JaVi iai+2 )σλii σi+2 for the splitting processes in a different frame.
Fig. 3 The coordinate systems in the q1 Breit frame (left), the c.m.
frame of the two colliding vector bosons (middle) and the q2 Breit
frame (right). The y-axis points to us. The directions of the z-axis in
the three coordinate systems are chosen common, so that the coordinate
system in one of the frames can coincide with that in the other of the
frames by a single boost along the z-axis
Because the helicity of a massless quark is frame
independent and furthermore the helicity of a massive vector boson
is invariant under Lorentz boosts along its momentum
direction as long as the boosts do not change the sign of its
threemomentum,6 the helicity amplitude (JaVi iai+2 )σλii σi+2 for the
quark splitting processes is invariant under Lorentz boosts
along the z-axis from the VBF frame, as long as the boosts
do not change the sign of the three-momentum of Vi . By
this property, it is justified to calculate the helicity amplitude
(JaV11a3 )σλ1σ for the quark splitting process in the q1 Breit
1 3
frame, to which k1,3 and q1 in the VBF frame can move
by a single Lorentz boost along the negative direction of the
z-axis. Similarly, it is justified to calculate the helicity
amplitude (JaV22a4 )σλ22σ4 for the quark splitting process in the q2 Breit
frame, to which k2,4 and q2 in the VBF frame can move by a
single Lorentz boost along the positive direction of the z-axis.
The q1 and q2 Breit frames are illustrated in the left and right
of Fig. 3, respectively. A calculation of the helicity
amplitude (JaVi iai+2 )σλii σi+2 for the gluon splitting process in the qi
Breit frame is also justified by appropriately choosing
gaugefixing vectors for the polarisation vectors of the external
gluons. Although the helicity of a gluon is frame independent,
the polarisation vectors of the gluon are dependent on their
gauge-fixing vectors and hence the helicity amplitude for the
gluon splitting process is in general frame dependent.
However, if we choose the gauge-fixing vectors in a way that their
directions are invariant under Lorentz boosts along the z-axis,
the helicity amplitude for the gluon splitting process also
becomes invariant under the same boosts, again as long as
the boosts do not change the sign of the three-momentum of
the intermediate off-shell gluon. This can be simply achieved
by choosing all the gauge-fixing vectors along the z-axis.
We parametrise the four-momenta k1, k3 and q1 in the q1
Breit frame as
6 This is also the case for an intermediate off-shell gluon.
q1μ = k1μ − k3μ = 0, 0, 0, Q1 ,
k1μ = 2 cQos1θ1 1, sin θ1 cos φ1, sin θ1 sin φ1, cos θ1 ,
k3μ = 2 cQos1θ1 1, sin θ1 cos φ1, sin θ1 sin φ1, − cos θ1 ,
q2μ = k2μ − k4μ = 0, 0, 0, −Q2 ,
k2μ = − 2 cQos2θ2 1, sin θ2 cos φ2, sin θ2 sin φ2, cos θ2 ,
k4μ = − 2 cQos2θ2 1, sin θ2 cos φ2, sin θ2 sin φ2, − cos θ2 ,
where Q2 = −(q2)2 > 0, π/2 < θ2 < π and 0 < φ2 <
2π . We define polarisation vectors for the intermediate vector
boson V1 in the q1 Breit frame by
It is easy to explicitly confirm that the above sets of the
polarisation vectors satisfy the identity for the propagator
of a weak boson in Eq. (2.4). We use the same polarisation
vectors μ(qi , λi = ±) for the intermediate weak bosons
(Vi = W, Z ) and gluons (Vi = g). As a result, the λi = ±
helicity amplitudes (JaVi iai+2 )σ±σ for a weak boson emission
i i+2
from a quark and those for a gluon emission from a quark are
the same, except for couplings. (Let us recall that the λ = 0
components of the intermediate gluons are neglected.)
As we have discussed above, the gauge-fixing vectors for
polarisation vectors of the external gluons should be chosen
along the z-axis. For the polarisation vectors of the two
external gluons in the q1 Breit frame, we choose a common vector
n1μ = (1, 0, 0, −1), i.e. the light-cone axial gauge. With this
choice, the polarisation vectors in the q1 Breit frame are
Similarly, a vector n2μ = (1, 0, 0, 1) is commonly chosen
for the polarisation vectors of the two external gluons in the
q2 Breit frame, then the polarisation vectors in the q2 Breit
frame are
It is easy to confirm that the above polarisation vectors with
their gauge-fixing vectors satisfy the unitarity condition for
an on-shell gluon:
With our preparations up to now, we can easily derive
the helicity amplitude (JaVi iai+2 )σλii σi+2 for the quark splitting
process and that for the gluon splitting process in the Breit
frames. Following Ref. [47], we write the amplitudes as
The coupling factor is already defined in Eq. (2.13a). The
common amplitude Jˆi λσii σi+2 is summarised in Table 1. Note
that we adopt the phase convention for the two-component
Weyl spinors developed in refs. [73,74]. Since we use the
same phase convention for the spinors and the same
gaugefixing vectors for the external gluons with Ref. [47], the
amplitudes in Table 1 are consistent with those in Tables
1 and 2 of Ref. [47] including the common overall phase.
2.3 Helicity amplitudes for the processes
V V → H H
Finally we present the helicity amplitudes for the processes
V1V2 → H H , (MVH1HV2 )λ1λ2 defined in Eq. (2.7b). As we
have mentioned in Sect. 2.2, we evaluate the amplitudes in
the VBF frame. We parametrise the four-momenta q1, q2, q3
and q4 in the VBF frame as
where sˆ is the c.m. energy squared sˆ = (q1 + q2)2 and
β = 1 − 4m2H/sˆ with mH being the Higgs boson mass.
The polarisation vectors for the vector bosons V1,2 in the
VBF frame can be simply obtained by boosting those in the
q1,2 Breit frames [Eqs. (2.19), (2.20)] to the VBF frame along
the z-axis:
−q23, 0, 0, −q20 .
Needless to say, the λ1,2 = ±1 components remain the same
after the boost. The helicity amplitude for the WBF process
Table 1 The helicity
λi defined in
amplitudes Jˆi σi σi+12Breit frame
Eq. (2.24) in the q
(left row) and the q2 Breit frame
(right row). The amplitudes of
an off-shell vector boson (weak
boson or gluon) emission from a
quark are given in the upper part
and those of an off-shell gluon
emission from a gluon are given
in the lower part
Table 2 Functions Fi [a1a2] in
Eq. (3.1) for the qq initiated
sub-process (a1, a2) = (q, q)
(upper left), the qg initiated
sub-process (a1, a2) = (q, g)
(upper right), the gq initiated
sub-process (a1, a2) = (g, q)
(lower left) and the gg initiated
sub-process (a1, a2) = (g, g)
(lower right). θ1,2 are defined in
the q1,2 Breit frames Eqs. (2.17)
and (2.18)
1 2 → H H , where (V1, V2) = (W +, W −), (W −, W +) or
V V
(Z , Z ), is given by [42]
H H 4
MgHgH λ1λ2 = G2√Fα2sπsˆ δb1b2 μ1 q1, λ1
V V
− 2m2V1 Dμ11μ2 q1 − q3 + Dμ11μ2 q1 − q4
where GF is the Fermi constant and λh is the factor
rescaling the triple Higgs self-coupling√:λHHH = λh λSHMHH,
where λSHMHH = 3m2H/v with v−2 = 2GF. The standard
model predicts λh = 1. The notation for the propagators
is defined in Eq. (2.3). By using the four-momenta and the
polarisation vectors defined above, we can obtain the
offshell weak boson amplitude.
As we have discussed in Sect. 2.1, we use the on-shell
gluon amplitude for the GF process gg → H H . Using the
notation given in the appendix of Ref. [39], we write the
where b1,2 are the colour indices of the gluons, and F , F
and G are form factors [39] consisting of the scalar loop
functions after tensor reduction and Aiμ1μ2 are the tensor
structures of the process. Not only we neglect the λ1,2 = 0
components of the gluons, but we also set Q1 = Q2 = 0
in the four-momenta in Eq. (2.25). Our definitions of the
polarisation vectors μ(qi , λi = ±) in Eqs. (2.26) and (2.27)
actually correspond to an axial gauge nμ = (1, 0, 0, −1) for
an on-shell gluon q1μ ∝ (1, 0, 0, 1) and to an axial gauge
μ
nμ = (1, 0, 0, 1) for an on-shell gluon q2 ∝ (1, 0, 0, −1),
respectively.
In order to simplify further analyses, we introduce the
following amplitude:
Then the amplitude Eq. (2.29) has a simpler form,
H H GFαssˆ δb1b2
After a simple manipulation in the VBF frame, we find
The parity invariance of the amplitude is apparent, (MgHgH )++
= (MgHgH )−− and (MgHgH )+− = (MgHgH )−+. Up to now we
have assumed the Higgs sector of the standard model. In this
paper, we study the impact of parity symmetry violation in
the GF process gg → H H .7 We can read off parity violating
phases from a parity-odd process gg → H A, where A is a
parity-odd Higgs boson. By using the tensor Aiμ1μ2 for the
process gg → H A [39], we find the amplitude in the VBF
frame:
In order to evaluate the impact of parity violation, we
introduce two angles (or phases) ξ1,2 (−π/2 ≤ ξ1,2 ≤ π/2) and
write the amplitude as
The two phases ξ1,2 parametrise the magnitude of parity
violation in the process gg → H H and independently affect
the λ = λ1 − λ2 = 0 helicity states and λ = ±2 helicity
states, respectively. We believe that this simplified
introduction of parity violating phases is enough for studying the
impact of parity violation on the azimuthal angle
correlations.
7 The parity symmetry violation in the GF process indicates charge
conjugation and parity (CP) symmetry violation in the Higgs sector.
3 Azimuthal angle correlations
In this section we present a detailed study of the azimuthal
angle correlations of the two jets by using the helicity
amplitudes presented in Sect. 2.
3.1 The gluon fusion process
The azimuthal angle correlations of the two jets can be
analytically apparent, once we obtain the squared VBF amplitude.
There are four gluon fusion (GF) sub-processes (V1, V2) =
(g, g), namely the qq initiated sub-process (a1, a2) = (q, q),
the qg initiated sub-processes (a1, a2) = (q, g), (g, q) and
the gg initiated sub-process (a1, a2) = (g, g). The squared
VBF amplitude for the four GF sub-processes has the
following compact form, after averaging over the initial state
colours and helicities and summing over the final state
colours and helicities:
2 2 2
+ |Mˆ +−| + |Mˆ −+| + |Mˆ −−|
− 2F1[a1a2] Re Mˆ ++Mˆ ∗−+ + Re Mˆ +−Mˆ ∗−−
− 2F2[a1a2] Re Mˆ +−Mˆ ∗++ + Re Mˆ −−Mˆ ∗−+
+ 2F3[a1a2]Re Mˆ ++Mˆ ∗−− cos 2(φ1 − φ2)
+ 2F3[a1a2]Re Mˆ +−Mˆ ∗−+ cos 2(φ1 + φ2)
In order to simplify our writing, we introduce the notation
Mˆ λ1λ2 , which denotes the helicity amplitude (MgHgH )λ λ .
Ca1a2 are the colour factors from the splitting proces1se2s
a1 → a3 + g1∗ and a2 → a4 + g2∗, and they take values
Cqq = 16/9, Cqg = Cgq = 4 and Cgg = 9. Fi [a1a2]
are functions of the kinematic variables θ1,2 defined in the
q1,2 Breit frames Eqs. (2.17) and (2.18), and summarised in
Table 2. The azimuthal angles φ1,2 are also defined in the
Breit frames. b1,2 are the colour indices of the intermediate
gluons and the average over b1,2 gives 1/8; see Eq. (2.29).
The colour factors Ca1a2 and the functions Fi [a1a2] for
antiquarks are the same as those for quarks, since the QCD
interaction does not distinguish quarks and antiquarks.
The first term in the right hand side (RHS) of Eq. (3.1)
contributes to the inclusive cross section after a phase space
integration, while the other terms give the azimuthal angle
distributions of the two jets. The azimuthal angles φ1,2 of
the two jets are defined in the q1,2 Breit frames, respectively,
and they remain the same in the VBF frame. In the limit
that each of the two jets in the proton–proton (pp) frame
is collinear to the incoming parton that emits it (collinear
limit), the emitted two intermediate vector bosons also move
on the z-axis. After rotating the two jets and the two Higgs
bosons around the z-axis in such a way that the two Higgs
bosons have zero azimuthal angle [Let us recall that the two
Higgs bosons have zero azimuthal angle in the VBF frame;
see Eq. (2.25)] and after a single boost along the z-axis, all of
these particles can be studied in the VBF frame. Therefore, in
the collinear limit, the azimuthal angles of the two jets after
the single rotation around the z-axis are identical to φ1 and φ2.
We apply the VBF cuts and an upper transverse momentum
pT cut on the jets in the pp frame and these cuts reproduce the
collinear limit to some extent. Hence the azimuthal angles of
the two jets in the pp frame after the rotation around the
zaxis should not be very different from φ1,2 defined in the q1,2
Breit frames. We perform the rotation of the two jets and the
two Higgs bosons around the z-axis in the following way:
1. Go to the centre-of-mass (c.m.) frame of the two Higgs
bosons and then rotate the two Higgs bosons around the
z-axis by φ˜ in a way that the two Higgs bosons have zero
azimuthal angle.
2. Rotate the two jets in the pp frame around the z-axis by
φ˜ .
3. Measure the azimuthal angles of the two jets.
In the collinear limit, it is clear that the azimuthal angles of the
two jets measured after this rotation coincide with φ1,2. Note
that this rotation is necessary for the azimuthal angles and the
sum of them to show meaningful distributions, because the
process pp → H H j j in the pp frame is completely
symmetric around the z-axis. This rotation is, however, not needed
for the difference of the two azimuthal angles. Before we
show numerical results, we point out characteristic features
of the GF sub-process in the standard model (SM) already
expected from the analytic formula Eq. (3.1):
• The azimuthal angles of the two jets show the same
distribution due to the parity invariance of the amplitude
Mˆ λ λ ; see Eq. (2.32).
• All o1f2the azimuthal angle observables show cosine
distributions, again due to the parity invariance of the
amplitude.
Violation of the parity invariance of the amplitude should
appear as a deviation from the above expectations. This case
will be studied at the end of this subsection.
We show numerical results for the 14 TeV LHC. We do not
study decays of the two Higgs bosons and assume that they
can be reconstructed. An outgoing quark, antiquark or gluon
is identified as a jet. The following set of parameters are
chosen: mH = 125.5 GeV, mt = 173.5 GeV and αs(mZ) = 0.13.
We use the CTEQ6L1 [75] set for the parton distribution
functions (PDFs) and have chosen the input value for the
strong coupling constant accordingly. For the factorisation
scales in the PDFs, we choose a fixed value of 25 GeV, which
corresponds to the lower cutoff on the transverse momentum
pT of the jets (see below). The renormalisation scales in the
√
strong couplings are chosen as αs( s)2αs(150GeV)2, where
√sˆ is the invariant mass of the two Hˆ iggs bosons. Using the
two different scales in the strong couplings can be considered
as a better choice, since we look at only a kinematic region
where the virtualities of the gluons are small (Q1,2 → 0)
and this separates the two splitting processes from the
process gg → H H in time-scale. The renormalisation scales in
the strong couplings of the splitting processes correspond to
the upper cutoff on the pT of the jets. Different choices for
the factorisation and renormalisation scales are also studied,
when we discuss the uncertainties about scale choices. The
following cuts are applied on the rapidity y and pT of the
two jets in the pp frame:
The above rapidity cuts are the VBF cuts. As we have already
mentioned in Sect. 1, the VBF cuts and the upper pT cut
enhance the contributions from the VBF sub-processes. This
can be understood as follows. The virtuality Q1 of the
intermediate vector boson V1 is
Q21 = − k1 − k3
where the momentum assignment given in Eq. (2.1) is used
and θ13 is the angle between k1 and k3. The VBF cuts make
k3 collinear to k1 (θ13 → 0), then Q1 is decreased and the
VBF amplitude is enhanced. The upper pT cut additionally
enhances the VBF amplitude. In a collinear case (θ13 → 0),
the pT of k3 is
so the upper pT cut reasonably implies the upper cut on Q1.
Only the VBF cuts may be enough to enhance contributions
from the VBF sub-processes. However, we need to impose
the upper pT cut, too, because we perform the further two
approximations in calculating the VBF amplitude for the GF
sub-process and these approximations are justified only when
Q21 < sˆ and Q22 < sˆ (see Sect. 2.1). Throughout our analyses,
the four-momentum of the jet which has a positive
rapidity y1 in the pp frame is used for calculating its azimuthal
Fig. 4 The normalised differential cross section of the GF process as
a function of φ1 (upper left), φ2 (upper right), φ (lower left) and
φ+ (lower right). The correspondence between curves and simulation
angle labelled φ1 and that of the other jet which has a
negative rapidity y2 in the pp frame is used for calculating its
azimuthal angle labelled φ2. Azimuthal angles labelled φ1,2
in our numerical results shown below are not those defined
in the q1,2 Breit frames anymore. The phase space
integration and event generations are performed with the programs
BASES and SPRING [76]. The scalar loop functions are
calculated with the program FF [77].
We show the normalised differential cross section as a
function of φ1 (upper left), φ2 (upper right), φ = φ1 − φ2
(lower left) and φ+ = φ1 + φ2 (lower right) in Fig. 4 and
the inclusive cross sections of qq H H , gg H H , qg H H final
states and the sum of these final states in pp collisions in
Table 3. The calculation methods are the following:
Calc.1 Our approximate analytic cross section, to which
only the VBF diagrams contribute. The mass of the
loop running top quark in the gg → H H amplitude
[Mˆ λ1λ2 in Eq. (3.1)] is set finite (173.5 GeV).
methods is shown inside the upper left panel. The simulation methods
are explained in text
Table 3 The inclusive cross sections in units of femtobarn for various
final states in pp collisions, produced by the three different methods
which are the same as those used in Fig. 4 and explained in text. The
statistical uncertainty for the last digit is shown in the parentheses
Calc.2 Our approximate analytic cross section, to which
only the VBF diagrams contribute. The mass of the
loop running top quark in the gg → H H amplitude
[Mˆ λ λ in Eq. (3.1)] is set extremely large (14 TeV).
Calc.3 The L1 O2 cross section, to which all LO diagrams
contribute. The mass of the loop running top quark in
the full amplitude is set extremely large.
Calc.4 The LO cross section, to which all LO diagrams
contribute. The mass of the loop running top quark in
the full amplitude is set finite.
The above methods are already explained in Sect. 1. Although
we do not perform Calc.4, it is listed above for the sake
of our discussion. In Fig. 4, the blue solid curve, the red
solid curve and the black dashed curve represent the Calc.1
result, the Calc.2 result and the Calc.3 result, respectively.
Calc.3 is performed by implementing the following effective
Lagrangian density [78] into an UFO file [79] with the help of
FeynRule [80] version 1.6.18 and subsequently using the
UFO file in MadGraph5_aMC@NLO[68] version 5.2.2.1:
2GF 1/2 Fμaν F a,μν H − 2α4sπ √2GF Fμaν F a,μν H H,
where Fμaν is the gluon field strength tensor and H is the
Higgs boson field. The good agreement between the Calc.2
result and the Calc.3 result in all of the panels of Fig. 4
and in the inclusive cross sections of Table 3 (within 5%)
shows that Calc.2 successfully reproduces the Calc.3 result.
We stress that this observation indicates that Calc.1 is able
to reproduce the Calc.4 result, for the following reason. Our
approximate analytic cross section in Calc.1 and Calc.2 relies
only on the fact that the VBF diagrams dominate in a
kinematic region where the virtualities of the intermediate
gluons are small [this kinematic region is achieved by the cuts
in Eq. (3.2)]. This fact is solely due to the propagator
factors of the intermediate gluons and the details of the core
process gg → H H have nothing to do with the validity of
the approximation. Therefore, once it is observed that Calc.2
reproduces the Calc.3 result, then we can be confident that
Calc.1 is also able to reproduce the Calc.4 result. The large
discrepancies between the Calc.1 result and the Calc.2 (or
Calc.3) result in Fig. 4 and Table 3 show the importance of
using a finite value for the mass mt of the loop running top
quark in the amplitude. In Calc.4, the diagrams other than
the VBF diagrams should also include some effects of finite
mt. However, only the VBF diagrams dominate, after all, in
the kinematic region at which we are looking. Therefore, we
are able to deny the possibility that the effects of finite mt
we have found are washed out by some effects of finite mt
existing in the diagrams other than the VBF diagrams.
The reason why we do not find correlated distributions in
φ1,2 in the Calc.2 (or Calc.3) result can be understood from
the squared VBF amplitude in Eq. (3.1). The second and third
terms in the RHS shows that the correlations in φ1,2 arise from
theλ i=nte±rf2ersetantcees,owf hthereeamλpl=ituλd1es− Mλˆ2λ.1Iλn2 t hfoerlargλe =mt 0li manitd,
the amplitude for λ = ±2 states vanishes and hence the
correlations also vanish. The GF sub-process exhibits the largest
correlation in φ and an almost zero correlation in φ+. The
squared VBF amplitude in Eq. (3.1) tells us that the
correlation in φ arises from the interference of the amplitudes
Mˆ λ λ for λ = 0 states (the fourth term in the RHS) and the
corre1la2tion in φ+ arises from the interference of the
amplitudes for λ = ±2 states (the fifth term in the RHS). The
reason of the large correlation in φ and the small correlation
in φ+ is because the amplitudes for λ = 0 states are much
larger than those for λ = ±2 states in a large part of the
phase space (this will be confirmed explicitly below soon).
We briefly mention differences in the correlations between
the processes pp → H H j j and pp → H j j . The process
gg → H has non-zero amplitudes Mˆ λ1λ2 only for λ = 0
states. Therefore, the process pp → H j j exhibits a large
correlation in φ [44–47] but no correlations in φ1,2 and
φ+.
In Fig. 5 we show the differential cross section as a
function of φ1,2 (left), φ = φ1 − φ2 (middle) and φ+ = φ1 + φ2
(right), contributed by all of the sub-processes (black curve,
labelled (1) pp), by the qg and gq initiated sub-processes
(blue curve, labelled (2) qg + gq), by the gg initiated
subprocess (red curve, labelled (3) gg) and by the qq initiated
sub-process (green curve, labelled (4) qq). Calc.1 is used to
produce all of the numerical results hereafter in this
subsection. As we have already discussed before showing the
numerical results and have actually confirmed in Fig. 4, φ1
and φ2 show the same distribution due to the parity
invariance of the amplitude in the SM. Therefore, we show only
the φ1 distribution and label it φ1,2 instead of showing the
two distributions, until we study parity violation.
We now study the uncertainties stemming from the choice
of the factorisation scales in the PDFs and the renormalisation
scales in the strong couplings. We set the notations for the
factorisation scales as follows: μF1 denotes the factorisation
scale in the PDF of the proton that moves along the positive
direction of the z-axis, μF2 denotes that in the PDF of the
proton that moves along the negative direction of the z-axis.
Let us remind that our default setting is μF1 = μF2 = 25
GeV. We write μF1,2 as
μF1 = α × pT 1, μF2 = α × pT 2,
where pT1 is the pT of the jet with a positive
rapidity, pT2 is the pT of the jet with a negative rapidity,
and α is a constant factor. We use the following
expression for the renormalisation scales in the strong couplings,
αs(μgg→H H )2αs(μR1)αs(μR2):
where β is a constant factor. Let us remind that out default
√
sˆ, μR1 = μR2 = 150 GeV. In the
Fig. 5 The differential cross section in units of femtobarn as a function
of φ1,2 (left), φ (middle) and φ+ (right), contributed by different GF
sub-processes. All of the sub-processes contribute to the black curve,
the qg and gq initiated sub-processes to the blue curve, the gg initiated
sub-process to the red curve and the qq initiated sub-process to the
green curve
Fig. 6 Left: the differential cross sections of the GF sub-process in
units of femtobarn as a function of φ, with different factorisation
scales. The renormalisation scales are fixed to our default choice. The
solid curve shows the result for our default choice and , and × points
give the α = 1, α = 2 and α = 1/2 results in Eq. (3.6), respectively.
The ratios with respect to the solid curve result are shown in the lower
part. Right: the differential cross sections of the GF sub-process in units
of femtobarn as a function of φ, with different renormalisation scales.
The factorisation scales are fixed to our default value. The solid curve
shows the result for our default choice and , and × points give the
β = 1, β = 2 and β = 1/2 results in Eq. (3.7), respectively. The ratios
with respect to the solid curve result are shown in the lower part
left panel of Fig. 6, we show the differential cross sections
as a function of φ, with different α values. The
renormalisation scales are fixed to our default choice. The solid curve
shows the result for our default choice and , and × points
give the α = 1, α = 2, α = 1/2 results, respectively. In the
right panel of Fig. 6, we show the differential cross sections
as a function of φ, with different β values. The
factorisation scales μF1,2 are fixed to our default choice. The solid
curve shows the result for our default choice and , and
× points give the β = 1, β = 2, β = 1/2 results,
respectively. The ratios with respect to the solid curve result are
shown in the lower part of each panel. The panels show the
large differences between the different scale choice results
in the differential cross section. This is due to the fact that
Fig. 7 List plots of C R2φ1,2 (left), C R2 φ (middle) and C R2φ+ (right) defined in Eq. (3.8) with the pT of the Higgs boson (in GeV) in the c.m.
frame of the two Higgs bosons
the scale dependence in the inclusive cross section is large.
Nevertheless, the important observation is that the ratios with
respect to the calculation with our default choice of scales
are almost flat, as shown in the lower part of each panel. This
means that the normalised differential cross section is robust
against the scale variations and the correlation in φ is not
washed out by this theoretical uncertainty. We note that the
same is confirmed for the other observables φ1,2 and φ+, too.
In order to study how the azimuthal angle correlations
depend on the kinematics of the two Higgs bosons, we
introduce the following three quantities:
Re Mˆ ++Mˆ ∗−+ + Re Mˆ +−Mˆ ∗−− ,
|Mˆ ++|2 + |Mˆ +−|2 + |Mˆ −+|2 + |Mˆ −−|2
(3.8a)
These are simply the coefficients of the azimuthal angle
dependent terms divided by the coefficient of the azimuthal
angle independent term in Eq. (3.1). Although the functions
Fi [a1a2] are omitted, these quantities are useful in that an
azimuthal angle observable should show a strong
correlation in a kinematic region where the corresponding quantity
C Ri is enhanced. We find that all of the quantities C Ri are
well correlated with the pT of the Higgs boson in the c.m.
frame of the two Higgs bosons, as shown in Fig. 7.
Figure 7 presents list plots of C R2φ1,2 (left), C R2 φ (middle)
and the small value of C R2φ in the most part of the phase
space explain the large correl+ation in φ and the small
correlation in φ+ observed in Fig. 4. The plots indicate that the
correlation in φ is enhanced as the pT of the Higgs boson is
decreased, while the correlations in φ1,2 and φ+ are enhanced
as the pT of the Higgs boson is increased. This is confirmed
in Fig. 8. In Fig. 8, we show the normalised differential cross
section as a function of φ1,2 (left), φ = φ1 − φ2 (middle)
and φ+ = φ1 + φ2 (right) with different values for the lower
cutoff on the pT of the Higgs boson in the c.m. frame of the
two Higgs bosons, blue solid curve: pT > 0 GeV (no cut),
black dashed curve: pT > 150 GeV, and red solid curve:
pT > 250 GeV.
Next, we study how the azimuthal angle correlations
depend on the triple Higgs self-coupling. Equations (2.31)
and (2.32) tell us that λh , which is the factor re-scaling the
triple Higgs self-coupling, affects only the amplitude Mˆ λ λ
for λ = 0 states. However, as is clear from the quantit1ie2s
in Eq. (3.8), λh affects all of the correlations. For an extreme
example, if we could make the amplitude for λ = 0 states
to be identically zero (this is actually not possible because the
amplitude depend not only on λh but also on the kinematic
of the process gg → H H ), we would observe a large
correlation only in φ+. In Fig. 9, we study the correlations with
three different values for λh , the blue solid curve: λh = 0,
the black dashed curve: λh = 1 (the SM prediction), and the
red solid curve: λh = 2. The impact of a non-standard value
for λh (λh = 1) in the distributions is visible but not large.
Actually this is much smaller than that in other observables,
such as the pT of the Higgs boson [18,19] or the invariant
mass of the two Higgs bosons [19,22], of the inclusive
process pp → H H . The azimuthal angle correlations may not
be useful to probe λh .
Finally, we study the impact of parity symmetry violation
on the azimuthal angle correlations. Let us remind that we
have introduced two phases ξ1,2 in Eq. (2.34) and these phases
Fig. 8 The normalised differential cross section of the GF process as a
function of φ1,2 (left), φ (middle) and φ+ (right), with three different
values for lower cutoff on pT of the Higgs boson in the c.m. frame of
the two Higgs bosons. The correspondence between the curves and the
cutoff values is shown inside the left panel
Fig. 9 The normalised differential cross section of the GF process as a
function of φ1,2 (left), φ (middle) and φ+ (right), with three different
values for the triple Higgs self-coupling re-scaling factor λh . The
corparametrise the magnitude of parity violation in the process
gg → H H . If they are non-zero, the gg → H H amplitude
Mˆ λ1λ2 is not parity invariant anymore: Mˆ ++ = Mˆ −− if
ξ1 = 0, Mˆ +− = Mˆ −+ if ξ2 = 0. The squared VBF
amplitude Eq. (3.1) tells us that violation of the parity invariance
of the amplitude appears as the following deviations from the
standard model predictions: (1) the φ1 and φ2 distributions
are not necessarily equal to each other, (2) the azimuthal angle
observables do not necessarily show cosine distributions. In
order to make these expectations more explicit, we write the
amplitude Mˆ λ1λ2 with the phases ξ1,2 in the following way:
These are simply obtained by putting Eq. (2.34) into the
amplitude in Eq. (2.31) and substituting the terms including
the form factors with A and B. By inserting the above
amplitudes into the azimuthal angle dependent terms in Eq. (3.1),
each term becomes
−2 Re Mˆ ++Mˆ ∗−+ + Re Mˆ +−Mˆ ∗−−
+ (Re → Im, cos → sin )
−2 Re Mˆ +−Mˆ ∗++ + Re Mˆ −−Mˆ ∗−+
+ (Re → Im, cos → sin )
2Re Mˆ ++Mˆ ∗−−
2Re Mˆ +−Mˆ ∗−+
Fig. 10 The normalised differential cross section of the GF process
as a function of φ1 (left column), φ2 (middle column) and φ (right
column), with different values for the parity violating phases ξ1,2. In all
of the panels, the SM prediction is shown by the black dashed curve.
The correspondence between the curves and values for ξ1,2 is shown
inside each panel. A cutoff on the pT of the Higgs boson pT > 200
GeV is imposed in the φ1 and φ2 panels. Each row has the same set of
the parity violating phases
These results show that parity violation appears as peak shifts
of the azimuthal angle distributions. The results also tell us
an important fact that the peak shifts of the distributions
reflect only the magnitude of parity violation (recall that ξ1,2
parametrise the magnitude of parity violation in our study).
It is an easy exercise to confirm that this is true no matter how
parity violating phases are introduced. From the above
analytic results, it is also apparent how each observable depends
on the phases ξ1,2. φ is sensitive to ξ1 and φ is sensitive to
+
ξ2. φ1 and φ2 are sensitive to both ξ1 and ξ2. Although we are
far less likely to be able to measure ξ2 in the φ distribution
+
because of the very small correlation in φ+, we may use φ1
Fig. 11 The differential cross sections of the GF sub-process in units
of femtobarn as a function of φ (left panel) and φ1 (right panel), with
different factorisation and renormalisation scale choices. The set of the
parity violating phases is chosen as (ξ1, ξ2) = (30◦, 60◦). By using the
notations in Eqs. (3.6) and (3.7), the solid curves show the α = β = 1
results and and × points give the α = β = 2 and α = β = 1/2
results, respectively. The ratios with respect to the solid curve result are
shown in the lower part of each panel
We note in passing the impact of parity violation in the
process pp → H j j , which can be considered as a simpler
case. Since the process gg → H has non-zero amplitudes
Mˆ λ1λ2 only for λ = λ1 − λ2 = 0 states, the process
pp → H j j has only Eq. (3.11c) in its cross section formula.
Therefore, parity violation in the process gg → H appears
as a peak shift only in the φ distribution [45,46].
While the phase dependence on the correlations is
apparent from Eq. (3.11), we present numerical results, too. In
Fig. 10 we show the normalised differential cross section
as a function of φ1 (left column), φ2 (middle column) and
φ = φ1 − φ2 (right column) with different values for ξ1,2.
The correspondence between the curves and values for ξ1,2 is
shown inside each panel. In all of the panels, the SM
prediction is shown by the black dashed curve. A cutoff, pT > 200
GeV, is imposed on the pT of the Higgs boson in the c.m.
frame of the two Higgs bosons, when we produce the φ1
and φ2 plots, in order to enhance the correlations in φ1,2; see
Fig. 8. The distributions of φ+ = φ1 + φ2 are not shown
anymore, since we have found that the correlation in φ+ is
very small in most every case; see Figs. 8 and 9.
In Fig. 11, we study the uncertainties about the scale
choices in the case of parity violation. The differential cross
sections as a function of φ are shown in the left panel and
that as a function of φ1 is shown in the right panel. The phase
(ξ1, ξ2) = (30◦, 60◦) is chosen. A cutoff, pT > 200 GeV, is
imposed on the pT of the Higgs boson in the c.m. frame of
the two Higgs bosons, for the results in the right panel. The
notations in Eqs. (3.6) and (3.7) are used. The solid curves
show the α = β = 1 results and and × points give the
α = β = 2 and α = β = 1/2 results, respectively. The ratios
with respect to the solid curve result are shown in the lower
part of each panel. An important observation is that the peak
positions which are shifted by the parity violation, namely
the φ or φ1,2 values that give the highest cross sections, do
not depend on the scale choices. This fact is important from
the point of view of parity violation measurements.
3.2 The weak boson fusion process
The weak boson fusion (WBF) sub-process (V1, V2) =
(W +, W −), (W −, W +) and (Z , Z ) consists of only the
qq initiated sub-process (a1, a2) = (q, q). However,
the squared VBF amplitude for the WBF sub-process
takes a more complicated form than that for the GF
subprocess in Eq. (3.1) because of the following two
reasons: (1) the helicity λ1,2 = 0 components of the
intermediate weak bosons additionally induce 16 azimuthal
angle dependent terms, such as cos (φ1 − φ2), (2) we
cannot simply take averages for the initial helicities and
take summations for the final helicities, since the
electroweak interactions distinguish different helicity states. The
squared VBF amplitude for four sets of the helicities of
the external quarks must be prepared: (σ1, σ3, σ2, σ4) =
(+, +, +, +), (+, +, −, −), (−, −, +, +), (−, −, −, −).
The amplitude (MVH1HV2 )λ λ for helicity λ1,2 = 0 weak
1 2
bosons gives a dominant contribution in the WBF
subprocess and the amplitudes for other helicities are much
smaller than the above amplitude. Thus we can expect that
all of the azimuthal angle correlations are small, because the
correlations arise from the interference of the amplitudes for
various helicities; see Eq. (3.1). If we keep only terms which
contain at least one (MVH1HV2 )00, the squared VBF amplitude
for (σ1, σ3, σ2, σ4) = (+, +, +, +) has the following form:
M++++ 2 =
+ 2 s1t1Re Mˆ ++Mˆ 0∗0 + s2t2Re Mˆ 00Mˆ ∗−−
− 2 s1t2Re Mˆ +−Mˆ 0∗0 + s2t1Re Mˆ 00Mˆ ∗−+
+ Re → Im, cos → sin
amplitude (MVH1HV2 )λ1λ2 . s0,1,2 and t0,1,2 are functions of θ1,2
defined in the q1,2 Breit frames Eqs. (2.17) and (2.18) and
given by
sin2 θ
1 , s1 =
s0 = 2 cos2 θ1
√
s2 =
t2 =
sin2 θ
2 , t1 =
t0 = 2 cos2 θ2
√
The squared VBF amplitude for the other three helicity
states can be simply obtained by exchanging s1,2 and t1,2
in |M++++|2 in the following ways:
|M++++|2 → |M−−++|2 by s1 ↔ s2,
|M++++|2 → |M++−−|2 by t1 ↔ t2,
|M++++|2 → |M−−−−|2 by s1 ↔ s2 and t1 ↔ t2.
The couplings should also be changed accordingly. The
coefficients of cos φ1,2 terms actually contain one Mˆ 00, too.
However, the cos φ1,2 terms cannot give correlated
distributions in the process pp → H H j j , because we cannot
distinguish the two Higgs bosons. More practically, an azimuthal
angle dependent term which changes its overall sign under the
transformations φ1 → φ1 + π and φ2 → φ2 + π gives only
a flat distribution after the phase space integration. The first
term in the right hand side (RHS) of Eq. (3.12) contributes to
the inclusive cross section after the phase space integration
and the other terms give the correlations in φ = φ1 − φ2
and φ+ = φ1 − φ2. An interesting difference from the
GF sub-process is that the sine terms do not vanish even
when the amplitude is parity invariant (Mˆ ++ = Mˆ −− and
Mˆ +− = Mˆ −+), if the amplitude contains an imaginary part.
This is because the interaction between the external quarks
and the intermediate weak boson already violates the
parity symmetry. In our tree-level calculation, the amplitude is
purely real and we will observe only cosine distributions.
We show numerical results for the 14 TeV LHC. The setup
and phase space cuts are the same as in the GF study in
Sect. 3.1. Therefore the numerical results in this Section can
be directly compared with those in Sect. 3.1. The only
difference is the scale choice in the PDFs. The pT of the jet
with a positive rapidity is used for the scale in the PDF of the
incoming parton which moves along the positive direction of
the z-axis, and the pT of the jet with a negative rapidity is
used for the scale in the PDF of the other incoming parton. In
Fig. 12, we show the normalised differential cross section as
a function of φ1 (upper left), φ2 (upper right), φ = φ1 − φ2
(lower left) and φ+ = φ1 + φ2 (lower right). The blue solid
curve, labelled (1) VBF, represents the result according to
our approximate analytic cross section, to which only the
VBF diagrams contribute. The black dashed curve, labelled
(2) MG, represents the result according to the exact LO cross
section, to which not only the VBF diagrams but also the
s-channel and u-channel diagrams contribute, generated by
MadGraph5_aMC@NLO [68] version 5.2.2.1. The red solid
curve, labelled (3) VBF w/o QE, represents the result
according to our approximate analytic cross section from which the
azimuthal angle dependent terms are removed on purpose,
that is, the quantum effects of the intermediate weak bosons
expressed as the interference of the amplitudes for
different helicities are killed. Therefore, the differences between
the result (1) and the result (3) in the distributions visualise
the contribution from the azimuthal angle dependent terms.
The good agreement between the result (1) and the result
(2) confirms the validity of our approximate analytic cross
section. The φ1 and φ2 plots show correlated distributions.
However, the agreement between the result (1) and the result
(3) reveals that the correlated distributions are not induced by
the quantum effects of the intermediate weak bosons but by a
kinematic effect. The small discrepancies between the result
(1) and the result (3) in the φ and φ+ plots come from the
cos φ and the cos φ+ terms in Eq. (3.12), respectively. The
smallness of the discrepancies is as expected [see the
discusFig. 12 The normalised differential cross section of the WBF process
as a function of φ1 (upper left), φ2 (upper right), φ (lower left) and
φ+ (lower right). The correspondence between curves and simulation
methods is shown inside the upper left panel: (1) The blue solid curve
represents the result according to our approximate analytic cross
secsion above Eq. (3.12)]. In the φ and φ+ plots, the result (3)
again shows correlated distributions, which must be induced
by a kinematic effect. Therefore, the WBF sub-process
produces the correlated distributions in all of the azimuthal angle
observables; however, they are mainly induced by a
kinematic effect. We note that the GF sub-process produces only
flat distributions in all of the azimuthal angle observables,
when the quantum effects of the intermediate gluons are
killed in the same way as above. It can be concluded that
the non-flat distributions induced by a kinematic effect are a
characteristic feature of the WBF sub-process.
We study how the azimuthal angle distributions depend
on the triple Higgs self-coupling. We have observed the
kinematic effect on the distributions in non-standard cases
λh = 1, too, where λh is the factor re-scaling the triple Higgs
self-coupling. The three cases λh = 0, 1, 2 produce the
simition, (2) The black dashed curve represents the exact LO result, (3) The
red solid curve represents the result according to our approximate
analytic cross section from which the quantum effects of the intermediate
weak bosons are removed on purpose
lar distributions in the azimuthal angle observables, when the
quantum effects of the intermediate weak bosons are killed.
In Fig. 13 we show the normalised differential cross section
as a function of φ1,2 (left), φ = φ1 − φ2 (middle) and
φ+ = φ1 + φ2 (right) with three different values of λh , the
blue solid curve: λh = 0, the black dashed curve: λh = 1 (the
SM prediction), and the red solid curve: λh = 2. The
correlated distributions in the φ1,2 plot are completely induced
by a kinematic effect. We find that, when λh = 2, the
coefficient of the cos (φ1 − φ2) term [the second term in the RHS
of Eq. (3.12)] is large enough to flip the φ distribution. The
impact of a non-standard value for λh (λh = 1) is again not
so significant. However, differently from the GF sub-process
which is actually the O(αs2) correction to the inclusive GF
sub-process gg → H H , the WBF sub-process is a LO
treelevel process and so the correlations should be used together
with other observables, such as the invariant mass of the
Higgs boson pair, to probe λh .
4 Summary and discussion
In this paper, we have studied the azimuthal angle
correlations of two jets in the production of a Higgs boson pair
plus two jets pp → H H j j . Based on the known fact that
the azimuthal angle correlations are induced by the
quantum effects of the two intermediate vector bosons in vector
boson fusion (VBF) sub-processes, we have calculated the
amplitudes contributed from only VBF Feynman diagrams.
As VBF sub-processes, we have considered the gluon fusion
(GF) sub-process, which is an one-loop O(αs4α2) process at
leading order (LO), and the weak boson fusion (WBF)
subprocess, which is an O(α4) process at LO. We have used a
helicity amplitude technique for evaluating the VBF
amplitudes. Based on the method presented in Ref. [47], we have
divided a VBF amplitude into two amplitudes for off-shell
vector boson emissions (q → q V ∗ or g → gV ∗) and one
amplitude for the Higgs boson pair production by a fusion
of the two off-shell vector bosons (V ∗V ∗ → H H ), and
presented each of the three amplitudes in the helicity basis. The
quantum effects of the intermediate vector bosons are still
included correctly and are expressed as the interference of the
V ∗V ∗ → H H amplitudes with various helicities of the
vector bosons. With this method, we have obtained the squared
VBF amplitude in a compact form, from which we can
easily make an expectation on the azimuthal angle correlations,
both for the GF sub-process and for the WBF sub-process. We
have numerically compared our approximate analytic cross
sections with the exact LO results and have observed the
good agreement between the two results both in the
inclusive cross sections and in azimuthal angle distributions, after
the VBF cuts and the upper transverse momentum pT cut on
the jets are imposed (for the GF sub-process, the
comparison is performed only in the large mt limit). As azimuthal
angle observables, we have studied four observables: φ1, φ2,
φ = φ1 − φ2 and φ+ = φ1 + φ2, where φ1,2 are the
azimuthal angles of the two jets measured from the
production plane of the Higgs boson pair.
In the GF sub-process, using a finite mt value is found
to be important to produce the azimuthal angle correlations
correctly. The GF sub-process exhibits large correlations in
φ1,2 and φ. The pT of the Higgs boson is found to be
useful in controlling these correlations. The correlation in φ
is enhanced when the pT of the Higgs boson is decreased
and the correlations in φ1,2 are enhanced when the pT of the
Higgs boson is increased. We have found that the
correlation in φ+ is very small in most every case. The impact of a
non-standard value for the triple Higgs self-coupling on the
correlations is found to be much smaller than that in other
observables, such as the invariant mass of the Higgs boson
pair, of the inclusive process pp → H H . In order to study
the impact of parity violation on the correlations, we have
introduced two independent phases ξ1,2 which parametrise
the magnitude of parity violation in the process gg → H H .
They are introduced in a way that ξ1 affects the gg → H H
amplitude for λ = λ1 − λ2 = 0 helicity states, where
λ1,2 are helicities of the gluons, and ξ2 affects the amplitude
for λ = ±2 helicity states. We have analytically shown
that parity violation appears as peak shifts of the correlations
and that the peak shifts reflect only the magnitude of
parity violation. We have also shown that φ is sensitive to ξ1,
φ+ is sensitive to ξ2, and φ1,2 are sensitive to both ξ1 and ξ2.
Although we are far less likely to be able to measure ξ2 in the
φ+ distribution because of the very small correlation in φ+,
we may use φ1 and φ2 to probe ξ2 instead. The scale
uncertainties in the GF sub-process are large in the inclusive cross
section, as expected from the study of the GF sub-process
in the process pp → H j j [67] or pp → H H . However,
the uncertainties in the normalised distributions of the
correlations are found very small. Therefore, the prediction of
the correlations is stable and thus useful, as long as the
normalised distributions are discussed. The peak positions of
the correlations, namely the φ or φ1,2 values that give the
highest cross sections, are also found robust against the scale
choices. This fact further motivates us to measure parity
violation by using the correlations. While we can naively expect
that the azimuthal angle distributions in the WBF sub-process
are almost flat, we have actually observed correlated
(nonflat) distributions. We have found that they are not induced
by the quantum effects of the intermediate weak bosons but
by a kinematic effect. Since we do not find a similar
kinematic effect in the GF sub-process, we conclude that this is
a characteristic feature of the WBF sub-process. The impact
of a non-standard value for the triple Higgs self-coupling is
not so significant in the WBF sub-process, too.
The parton level event samples of the process pp →
H H j j are exclusively generated and each of the two
outgoing partons is identified as a jet in our numerical
studies. When more realistic event generations are intended to
be performed, merging the parton level event samples with
the leading logarithmic parton shower [81–83] and
subsequently proceeding to a hadronisation procedure will be a
promising approach. However, a careful merging procedure
is required for correctly reproducing the azimuthal angle
correlations after the merging procedure, because the
correlations studied in this paper are completely process dependent
( pp → H H j j ) and those process dependent angular
correlations are not described correctly by the parton shower.
Contamination from the parton shower may lead to a wrong
prediction [84].
The azimuthal angle correlations revealed in this paper
will help the analyses of the process pp → H H j j at
Fig. 13 The normalised differential cross section of the WBF process
as a function of φ1,2 (left), φ (middle) and φ+ (right), with three
different values for the triple Higgs self-coupling re-scaling factor λh . The
Fig. 14 The normalised differential cross section as a function of φ1,2 (left), φ (middle) and φ+ (right) for the GF process (red solid curve) and
the WBF process (black dashed curve)
the LHC. Since the production cross section of the process
pp → H H j j is small, we should use as much process
dependent information as possible to extract the events of the
process. The azimuthal angle correlations are obviously a part
of the process dependent information. It is a known issue that
separating the contributions coming from the GF sub-process
and those coming from the WBF sub-process is difficult in
the production of a Higgs boson pair plus two jets [23]. One
possible application of the correlations is to help
disentangling the GF sub-process and the WBF sub-process by using
the fact that these two sub-processes exhibit different
correlations, as shown in Fig. 14. We have studied only the signal
processes in this paper and so the impact of the correlations
in a realistic situation is not so clear yet. The fully automated
event generation for loop induced processes is now available
in MadGraph5_aMC@NLO [68, 69]. This achievement will
activate phenomenological studies of the process. We hope
that further phenomenological studies including the uses of
the azimuthal angle correlations will be performed both by
theorists and by experimentalists.
Acknowledgements J.N. is grateful to Kentarou Mawatari for
answering several questions on physics of jet angular correlations. J.N. would
also like to thank Kaoru Hagiwara for valuable discussions, and Junichi
Kanzaki for his help on using programs BASES and SPRING. The
authors would also like to thank Barbara Jäger for useful discussions.
The work of the authors (J.N. and J.B.) is supported by the Institutional
Strategy of the University of Tübingen (DFG, ZUK 63) and in addition
J.B. is supported by the DFG Grant JA 1954/1.
Open Access This article is distributed under the terms of the Creative
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ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
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