#### Higher order QCD predictions for associated Higgs production with anomalous couplings to gauge bosons

W
Higher order QCD predictions for associated Higgs
Ken Mimasu 0 1 3
Veronica Sanz 0 1 3
Ciaran Williams 0 1 2
0 239 Fronczak Hall, Bu alo , 14260 U.S.A
1 Pevensey 2 Building , Brighton, BN1 9QH U.K
2 Department of Physics, University at Bu alo, The State University of New York , USA
3 Department of Physics and Astronomy, University of Sussex
We present predictions for the associated production of a Higgs boson at NLO+PS accuracy, including the e ect of anomalous interactions between the Higgs and gauge bosons. We present our results in di erent frameworks, one in which the interaction vertex between the Higgs boson and Standard Model W and Z bosons is parameterized in terms of general Lorentz structures, and one in which Electroweak symmetry breaking is manifestly linear and the resulting operators arise through a six-dimensional e ective eld theory framework. We present analytic calculations of the Standard Model and Beyond the Standard Model contributions, and discuss the phenomenological impact of the higher order pieces. Our results are implemented in the NLO Monte Carlo program MCFM, and interfaced to shower Monte Carlos through the Powheg box framework.
Higgs; E ective eld theories; Higgs Physics; Perturbative QCD
1 Introduction 2 3
The e ective Standard Model
Calculation
Amplitudes for W H production
Amplitudes for ZH production
3.3 Implementation in Monte Carlo codes
3.1
3.2
4.1
4.2
4
Results 5
Conclusions
Fixed order results
NLO + Parton shower results
4.2.1
4.2.2
Gluon initiated contribution to HZ
EFT e ects
boson interactions
nitions and their contributions to EW parameters and gauge
1
Introduction
The LHC's discovery of a particle consistent with the predicted Standard Model (SM) Higgs
boson has opened the door to a full understanding of electroweak symmetry breaking in
nature. One of the key aims of Run II of the LHC is to study the properties and interactions
of the Higgs in as much detail as possible, with the ultimate goal of con rming, or seriously
constraining, the possibility of new physics and/or anomalous interactions.
One of the most interesting electroweak processes to study at the LHC is the interaction
of the Higgs boson with massive vector bosons (W; Z). The primary role of the Higgs is
to generate masses for these particles and ensure perturbative unitarity in vector boson
scattering and any deviation from the SM Higgs-vector boson vertex could be indicative
of new physics contributions. At the LHC the dominant Higgs production mechanism
occurs through the fusion of gluons via a top quark loop. Therefore the total inclusive
Higgs cross section at the LHC is more sensitive to the top Yukawa coupling than potential
anomalous interactions of the Higgs with vector bosons. An obvious place to study the
interaction between the Higgs and vector bosons are the decays H ! V V . However since
the Higgs is considerably lighter than the 2mV threshold the decay phase space is restricted,
forcing one of the nal state vector bosons o -shell. Consequently, anomalous interactions
that modify the high energy behaviour of the vertex, are suppressed due to the kinematic
requirements. Accordingly, the best places to constrain anomalous interactions of the Higgs
and vector bosons are those sensitive to said vertex in production, namely Vector Boson
Fusion (VBF), Higgs in association with a hard jet, and associated production (V H). Of
{ 1 {
these, associated production | which occurs through an s-channel production mechanism
| is particularly appealing, since one can directly probe the high energy behaviour of the
interaction through, for instance, the invariant mass of the Higgs Vector system, mV H .
A simple way to encode e ects of new physics in the Higgs sector is to study Higgs
anomalous couplings (HAC) [1, 2]. This parametrization does not rely on assumptions
about whether EWSB is linearly or non-linearly realized, as it only relies on the Higgs as
a scalar degree of freedom and the preservation of U(1)EM , i.e. by saturating all possible
Lorentz structures in the vertex with the lowest number of derivatives. This
parametrization was successfully used at LEP in the study of anomalous trilinear gauge couplings [3{5]
and adopted in the study of BSM e ects in the Higgs couplings.
An alternative way to describe indirect e ects of new physics is to use an E ective Field
Theory (EFT) approach. Within this approach, one could assume a linear realization of
EWSB with the Higgs as a doublet of SU(2), and write down all the relevant operators
which satisfy SU(2)L
U(1)Y [6]. This e ective Lagrangian can be written in several
equivalent ways which account for the choice of a basis. In this paper we will be using the
proposal in refs. [7, 8]. A translation into other choices of basis can be done using, e.g.
the tool Rosetta [9]. Also, one could write an EFT for a non-linear realization of EWSB
as in refs. [10{15]. In either case, there is a correspondence between the HAC and EFT
approaches, see e.g. [8].
the following reaction,
In this paper we focus on searching for BSM e ects in Higgs production in association
with a massive vector boson. The Higgs associated production process is de ned through
(1.1)
(1.2)
(1.3)
q(p1) + q(p2) ! V
! V (pV ) + H(pH )
where V represents an electroweak vector boson. In the SM V is constrained to be either a
W or a Z, whilst including the higher dimensional operators allow for potential exchange
of an o -shell virtual photon. The massive bosons are unstable and their decay products
are measured at collider experiments. Leptonic decays of the vector boson are the cleanest
experimentally, whilst the decay H ! bb corresponds to the maximal Higgs branching ratio.
Therefore, unless otherwise stated, the process we study in this paper corresponds to
q(p1) + q(p2) ! V (! `1(p3) + `2(p4)) + H(! b(p5) + b(p6))
`1 and `2 correspond to either two charged leptons (V = Z) or a charged lepton and
neutrino (V = W ).
process
Associated production of a Higgs boson and a Z includes the following production
g(p1) + g(p2) ! Z(! ` (p3) + `+(p4)) + H(! b(p5) + b(p6)) :
This process is formally O( S2) and therefore a consistent treatment in a perturbative
expansion would
rst include this piece at NNLO. However, the large gluon
ux at the
LHC, coupled with the boosted topology typical of experimental searches results in a
signi cant contribution from this initial state. The SM contribution corresponds to two
types of topologies, one in which the top loop radiates the Z, and one in which the Z is
produced at the HZZ vertex. In the SM these two terms destructively interfere.
Given its phenomenological relevance signi cant theoretical e ort has been invested in
the V H processes. The NNLO corrections for the inclusive (i.e. on-shell V H) cross section
were presented in ref. [16] based on the previous calculation of corrections to the Drell-Yan
process [17]. The results of ref. [16] included the on-shell contributions from the gg pieces
described in the proceeding paragraphs (a study of the gg pieces at NLO in the heavy top
EFT was presented in ref [18]). A second type of heavy quark initiated contribution arises
at NNLO and contains a qq pair. The leading contributions were calculated terms of an
asymptotic expansion in mt2 in ref. [19]. A fully di erential calculation for the Drell-Yan
type W H process was presented in [20], and was extended to include NLO H ! bb decays
in [21, 22]. A similar calculation for ZH, including the gg diagrams was presented in
ref [23] while a complete, fully di erential calculation of the NNLO production (Drell-Yan
and heavy quark) with decays for bb at NLO can be found in ref. [24]. The calculation
of the NLO EW corrections for W H were presented in [25]. Fully di erential predictions
to the H ! bb decay were computed at NNLO in [26] (the total inclusive H ! bb rate is
known to O( s4) [27]). In addition to the parton level calculations discussed above there
has been signi cant progress matching parton level predictions to parton showers, allowing
for full simulation of the LHC collisions. An NLO matched prediction for V H using the
Powheg [28] framework was
rst presented in ref. [29] and using the MC@NLO [30]
setup [31]. Studies using merged NLO samples of V H + 0 and 1 jet were in Powheg [32]
and SHERPA [33]. A study for W H including anomalous couplings was presented at parton
level in VBFNLO in ref. [34]. EW corrections have been implemented in the HAWK Monte
Carlo code [35], including a study of anomalous HV V interactions at NLO in QCD.
The aim of this paper is to provide a framework to combine the precision predictions
described above with the anomalous coupling prediction in a general HAC or EFT
framework. We will do this be by calculating the V H processes at NLO including a general
parameterization of the HV V vertex. We then interface our results to the
POHWEGBOX [28, 29, 36] allowing for full event simulation. This paper proceeds as follows, in
section 2 we discuss the implementation of the anomalous couplings through the EFT
Lagrangian. Section 3 presents our calculation in detail and provides the amplitudes for V H
production at NLO including the anomalous couplings. Section 4 includes predictions at
xed order and NLO+PS accuracy and we present phenomenological results for LHC Run
II. We draw our conclusions in section 5.
2
The e ective Standard Model
In this paper we focus on the e ects of heavy New Physics in production of a Higgs boson
in association with a vector boson.
We are interested, hence, in three-point functions
involving the Higgs and two vector bosons [1, 2, 8]
LHAC =
41 gh(1z)zZ
21 gh(1w)wW
21 gh(1a)zZ
Z
h+ 21 gh(3z)zZ Z h
1
4 g~hzzZ
~
Z
W y h
hgh(2w)wW @ W y h+h:c:i+gh(3w)wW W y h
F
h
1
2 g~hazZ
~
F
h
h
1
2 g~hwwW
W~ y h
(2.1)
{ 3 {
2v2
c
u
v2 yu y
ig cW
m2W
2ig cHW
m2W
ig c~HW D
m2W
+ gs2 c~g y G
m2W
B
2v2
c
d
v2 yd y
y QLuR +
yT2k D
!
D W k +
D
yT2kD
W k +
a Ga
y D!
ig0 cB
2m2W
y D!
QLdR +
y D!
ig0 cHB D
m2W
f
W k +
ig0 c~HB D
m2W
yT2kD
a Gea +
where
is the Higgs doublet,
yD
1
c
l
v2 y`
yD
B
e
c
6
v2
y
B
+
y
3
LLeR + h:c:
g02 c~
m2W
y B
a G b G
e
B
e
c
;
g3 c~3W
m2W
0
iG+
ijkW i W j W
f
k + gs3 c~3G fabcG
m2W
(2.2)
(2.3)
(2.4)
as well as, possibly, couplings of the Higgs to a vector boson and two fermions. These
Higgs anomalous couplings (HAC) are a model-independent parametrization which respects
the fundamental symmetries of the SM at energies below electroweak symmetry breaking
(EWSB), namely Lorentz and U(1)EM invariance, assuming that the Higgs is a neutral,
scalar particle.
The HAC can be related to an E ective Field Theory approach (EFT), where new
resonances participating in EWSB are integrated out. The relation between HAC and
EFT depends on assumptions of how EWSB occurs, e.g. whether the symmetry is linearly
or non-linearly realized. In this paper we will match results in terms of HAC with a linearly
realized EFT in which the Higgs h is part of a doublet of SU(2)L. We follow the conventions
for EFT operators in [7, 8], which are based on the work in ref. [37]. The relevant part of
the Lagrangian is as follows,
and the dual eld strength tensors are de ned by
B
e
=
1
2
B
;
f
W k =
1
2
W
k
;
G
e
a =
1
2
G
a :
In table 1 we show the relation between HAC and coe cients of the EFT. The basis we
have chosen in this paper is not a unique choice and one can use, for instance, Rosetta [9]
as a tool to translate among di erent basis.
The e ect of HAC/EFT on the Higgs coupling to vector bosons is to introduce a
nontrivial momentum dependence in the vertex, as one can see by inspecting the Feynman
rules of the Higgs with W W , ZZ and Z
vector bosons, which are presented in gure 1.
{ 4 {
gh(1z)z
gh(2z)z
gh(3z)z
g~hzz
gh(1a)z
gh(2a)z
g~haz
gh(1w)w
gh(2w)w
g~hww
2g h
c2W mW
g
c2W mW
gmW h1
2
c
W
2g h
c2W mW
gsW h
cW mW
gsW h
cW mW
gsW h
cW mW
m2Wg cHW
g h
mW
m2Wg c~HW
cHBs2W
4c s4W + c2W cHW
h
(cHW + cW )c2W + (cB + cHB)s2W i
12 cH
2cT + 8c sc24W i
W
c~HBs2W
4c~ s4W + c2W c~HW
i
i
cHW
cHW
c~HW
cHB + 8c s2 i
W
cHB
are particularly relevant in the high-pT region.
We observe that in possible models which may generate these anomalous couplings,
i.e. UV completions, not all Lorentz structures may be simultaneously generated. Indeed,
in a large class of models, HAC of the type gh(2v)v do not occur, e.g. in 2HDMs [38, 39],
radion/dilaton exchange [38, 39] or supersymmetric loops involving sfermions or
gauginos [40{42]. This makes the study of phenomenology in which gh(2v)v = 0 and gh(2v)v 6= 0
particularly interesting.
Finally, we comment on the current bounds for these operators from a global analysis
including LEP and LHC Run 1 performed in refs. [43, 44], see refs. [45{47] for other
examples of ts in this context. This analysis took into account all the CP-even operators
including pure gauge and operators involving fermions, but not the CP-odd couplings g~.
In table 2 we show the result of a global t. When gh(2v)v = 0, i.e. cW =
from a global t of Run1 data are more stringent [43, 44]
cHW , the limits
cHW =
cW = (0:004; 0:02) :
(2.5)
{ 5 {
Zν(q)
W+μ(p)
W−ν (q)
Aμ(p)
Zν(q)
H
H
H
:
:
:
"
i ⌘ μν⇣cosgθW MZ + gh(1z)z p · q + gh(2z)z (p2 + q2)⌘−
gh(1z)z qμpν − g˜hzz✏μνρσ qρpσ − gh(2z)z (pμpν + qμqν)
"
i ⌘ μν⇣gMW + gh(1w)w p · q + gh(2w)w (p2 + q2)⌘−
gh(1w)w qμpν − g˜hww✏μνρσ qρpσ − gh(2w)w (pμpν + qμqν)
"
i ⌘ μν ⇣gh(1a)z p · q + gh(2a)z p2⌘ − gh(1a)z qμpν −
g˜haz✏μνρσ qρpσ − gh(2a)z pμpν
#
#
#
and Higgs- -Z vertices in the Higgs
Anomalous Coupling description of eq. (2.1). All momenta are owing into the vertex.
OW = i2g
Operator
Hy aD$ H
OB = i2g0
HyD$ H
OHW = ig(D H)y a(D H)W a
OHB = ig0(D H)y(D H)B
Coe cient
Constraints
D W a
m2W2 (cW
cB)
( 0:035; 0:005)
m2W2 (cW + cB) ( 0:0033; 0:0018)
m2W2 cHW
m2W2 cHB
( 0:035; 0:015)
( 0:045; 0:075)
computed in terms of the modi ed Feynman rules presented in
gure 1 such that the
anomalous couplings are a function of gh(iV) V . At NLO accuracy the production and decay for
pp ! V H ! leptons+bb completely factorize due to SU(3) color structure. This is because
gluon radiation linking the initial state to the nal state has no contribution at NLO since
its interference with the LO amplitude results in a contribution proportional to Tr(T a),
{ 6 {
HJEP08(216)39
where Ta is an SU(3) generator. We therefore present amplitudes for pp ! V H !leptons
H, and allow the subsequent MC code to decay the Higgs boson (PYTHIA, or MCFM). In
this paper the MCFM prediction corresponds to a LO decay, whilst the PYTHIA prediction
includes e ects from the parton shower.
where we have de ned the full amplitude A(0) in terms of a color stripped primitive
amplitude A(50); at LO the color factor is simply ji12 . In addition to the extraction of the
overall color factor in eq. (3.1) we have also extracted the electroweak pre-factors from the
fermionic W vertices. We note however that we have not extracted the g from the HW W
vertex since this will be modi ed during our calculation. Although we will consider decays
of the Higgs to bb, for simplicity we suppress the decay of the Higgs in this section.
We will also require the amplitudes needed to construct the NLO corrections. This
consists of two new amplitudes, the one-loop virtual amplitude A(1) and the real emission
amplitude containing an additional parton A(60)(in this case an additional gluon). The
5
virtual primitive amplitude is de ned as follows,
A(51)(1q; 2q; 3`; 4 ; H) =
g
p
2
2
i1
The real emission amplitude including the parton 7g,1 is thus
A(60)(1q; 2q; 3`; 4 ; H; 7g) = gs
(T a7 )ij12 A(60)(1q; 2q; 3`; 4 ; H; 7g) :
The color stripping for the real emission amplitude is slightly more complicated than the
LO and depends on the color matrix T a7 , however there is still only one unique color
ordering which simpli es the calculation signi cantly.
Since we are interested in associated production we are able to factorize the QCD
corrections which a ect the initial state, from the modi cations to the HV V vertex. The
factorization proceeds as follows for all of the primitive amplitudes we have considered,
A(ji)(p1 + p2 ! V1(! H + V2) + X) = JSM(p1; p2; V1; X)P V1 (P12X )V (V2; H) : (3.4)
In the above equation JSM(p1; p2; V1; X) represents the production of a (chiral) current in
the SM, if j = 5 then X = 0, whilst for the real emission amplitude j = 6 and X corresponds
to the emission of an additional gluon. The second current V (V2; H) corresponds to the
1Our naming convention follows the implementation in MCFM such that p5 and p6 are reserved for the
decay of the Higgs boson to bb:
(3.2)
(3.3)
splitting of the initial vector boson V1 into V2 and H, with the subsequent decays of V2
to leptons included. Finally the two currents are connected by vector boson propagator,
which in the unitary gauge is de ned as follows,
P W (k) =
k2
i
m2W
g
k k
M W2
Since the W bosons decay we work in the complex mass scheme. However, in this
calculation the longitudinal k k pieces do not contribute since JSM(p1; p2; V1; X)P 12X = 0
for massless initial and nal states. We also frequently use the following function in our
calculations
HJEP08(216)39
JVirt(1u ; 2d+; P12) = 2
1
2
3
2
2
s
+
2
2
7
2
JLO(1u ; 2d+; P12) : (3.9)
The virtual current corresponding to a vertex correction, is also very simple,
Finally the current corresponding to the emission of an additional gluon, necessary in the
real calculation has two possible helicity con gurations,
PV (s) =
s
s
MV2 + iMV V
and notation of the spinor helicity formalism in the following de nitions. We refer readers
unfamiliar with the spinor-helicity formalism to one of the many reviews. For instance a
detailed introduction can be found in ref. [48].
The
rst current we de ne is the modi ed decay current including the e ect of the
dimension-6 operators W (P2; 3 ; 4e++ ; H234), where P2 is the in owing momenta, H234 is
the outgoing Higgs boson, and p3 and p4 represent the momenta of the nal state leptons.
The explicit form for this current is as follows,
W (P2;3 ;4e++ ;H234) =
h3j j4](gmW + gh(2W) W (P22 + s34))
gh(2W) W P2 h3jP2j4]
Next we consider the chiral currents in the SM. The current needed for the construction
of the LO amplitude is rather simple,
JLO(1u ; 2d+; P12) =
i h1j j2] :
ig~hW W ) (h3j jP2j3i[43] + h34i[4jP2j j4]) :
(3.5)
(3.6)
(3.7)
(3.8)
(3.10)
(3.11)
iPW (s34)
s34
(gh(1W) W
2
2
s
JReal(1u ; 2d+; 3g+; P123) =
JReal(1u ; 2d+; 3g ; P123) =
i h1jP123j j1i
h2 3i h3 1i
i [2jP123j j2] :
Contracting these various currents together results in the amplitudes for the production of
W H at NLO in QCD including the e ects of the dimension-6 operators. For example the
LO contraction is,
Next we consider the production of a Z boson in association with a Higgs boson. The
situation is slightly more complicated than the W H example considered in the previous
section as the internal boson can be either a Z or a virtual photon
. In the SM the latter
case does not occur, but the full general anomalous coupling parametrization allows for
this contribution. We therefore parametrize the LO amplitude as follows,
to the selection of L and R helicities for particles 1 and 3. The left and right handed
couplings (in the SM) are de ned as follows,
vL` =
vLq =
1
1
2Q` sin2 W
sin 2 W
2Qq sin2 W
sin 2 W
;
;
v
R` =
v
Rq =
2Q` sin2 W
sin 2 W
2Qq sin2 W
sin 2 W
:
(3.14)
(3.15)
g ! cos 2
g
W
and g(i)
hW W
! gh(iZ) Z in eq. (3.7).
The sign in the vLq distinguishes between up (+) and down ( ) type quarks. The amplitudes
involving Z exchange can be obtained from the results presented in the previous section.
The results for the analogous case in which W ! Z can be obtained by simply swapping
The results then correspond to the LL
con guration, other con gurations can be obtained from fermion line reversal symmetries.
The current for a virtual photon exchange is given by,
J (P2; 3` ; 4`+; H234) =
iPZ (s34) ( (gh(1a)Z
s34
2
ig~haZ ) (h3j j4] (h3j2j3] + h4j2j4])
2(p3 + p4 ) h3j2j4]) + gh(2a)Z h3j j4] p22
(p2 ) h3j2j4]
: (3.16)
)
The SM currents are related to those described in the previous section.
3.3
Implementation in Monte Carlo codes
The amplitudes calculated above have been implemented into the parton level Monte Carlo
code MCFM. Infrared divergences are regulated using the Catani Seymour Dipole
subtraction method [49]. We use the default MCFM electroweak parameters, which correspond to
{ 9 {
the following choices,
MZ = 91:1876 GeV;
The remaining EW parameters are de ned in terms of the above input parameters. Since
we make the choice of de ning the input of our program in terms of the dimension-6
Wilson coe cients of eq. (2.2), some additional e ects are taken into account to fully map the
physical e ects of the EFT description into our HAC Lagrangian. Of the operators that we
consider in our implementation | OW ; OB and O
| lead to non-canonical SU(2)
U(1)
gauge boson kinetic terms after electroweak symmetry breaking.
The eld and gauge
coupling rede nitions necessary to canonically normalise the theory lead to O(
cations of both the EW parameters in terms of the inputs as well as the couplings of gauge
bosons to fermions as compared to the usual SM expectations. There is some freedom in
how these rede nitions are performed and therefore the places in which these corrections
appear although physical observables are naturally independent of such choices at this
order in the EFT expansion. Appendix A describes the choices we make and therefore the
2)
modiorigin of the relations and corrections that follow.
For the EW parameters, we work in the mW ; mZ ; GF scheme, and de ne the SM values
for the Weinberg angle, electric charge and Higgs v.e.v as
while the left and right handed couplings of the Z to a fermion, f , with weak isospin T
and electric charge Qf are shifted as follows,
c~W =
mW
mZ
;
e~ = 2
mW
v
s
1
mm2W2Z ;
v2 =
1
2GF
:
These are corrected due to a relative shift in the Z-boson mass
which rede nes the mixing angle and (mZ ) as follows,
mZ =
e~
2
v
2
8c~2W m2W
c^2W = c~2W (1 + 2 mZ )
c~
s~
2
W
2
W
e = e~ 1
mZ
= p4
(mZ );
2cB + c~2W cW ;
while the de nition of the Higgs v.e.v in terms of GF is unchanged. All other EW
parameters are derived from the modi ed values c^2W and e using SM relations.
The coupling between a photon and fermion is corrected by the term,
e =
v
2 e
2
m2W 8
cW
fLZ =
fRZ =
e~
c~W s~W
e~s~W Q
c~W
T
3f T Z
3
f QZ ;
Qf s~2W QZ ;
(3.20)
(3.21)
(3.22)
(3.23)
(3.24)
f
3
(3.25)
(3.26)
with
T3Z =
mZ ;
QZ =
v
2
e
2
m2W 8s~2W c~2W (2cB + c~4W cW ):
(3.27)
Phenomenologically speaking, the e ect of these re-de nitions is minor, and typically
results in changes which are sub-precent compared to predictions which do not alter the EW
parameters by the EFT operators. For the choice of parameters simulated in this paper,
only cW a ects the EW parameters and neutral gauge boson couplings.
The Higgs width is also modi ed as a result of the anomalous interactions, and we
use the eHDECAY implementation described in ref. [7] to de ne the modi cations to the
The Powheg-Box [28, 36] provides a mechanism to match xed order results to parton
shower level Monte Carlo codes. In our case the implementation is rather straightforward,
in particular since associated production in the SM has already been considered in the
literature [29, 32] . Therefore to include our results in the Powheg-Box we have simply
updated the existing matrix elements with our own calculations described earlier in this
section. The only technical task is to ensure that all of the variables required in the
MCFM matrix element routines, are appropriately initialized by to the values assigned in
Powheg. This is achieved through an interface to the MCFM routines which is called
4
4.1
once at runtime.
Results
Fixed order results
In this section we present results obtained at xed order in perturbation theory. Speci cally
we study the dependence on the total rate at NLO as a function of the anomalous couplings.
In order to simplify our results we do not vary all parameters continuously. Instead we
focus on parameters which are representative of the phenomenology at the LHC. Ignoring
for now the CP-violating operators we note that the in our basis the variables cW and cHW
are su cient to probe the Lorentz structures of the Feynman rule associated with gh(1w)w and
gh(2w)w. In particular cHW = 0 and cW 6= 0 probes the regime in which ghww modi es the
(2)
vertex, and if cW =
cHW then the regime in which gh(1w)w modi es the vertex is selected.
In order to study the impact of the anomalous couplings we calculate cross sections
for the LHC at the 13 TeV in which the nal state particles have to satisfy the following
phase space selection criteria,
W H : p
ZH : p
`T > 25 GeV; j `j < 2:5;
MET > 45 GeV; 2 bjets : p
jT > 25 GeV; j bj < 2:5
`T > 25 GeV
j `j < 2:5; 2 bjets : p
jT > 25 GeV; j bj < 2:5:
We refer to this selection as our basic-cuts. Since the e ects of the EFT operators are more
apparent at higher energies we also de ne a high-pVT selection cut in which we impose an
additional cut on pVT > 200 GeV. We note that pVT is a well de ned experimental observable
for both W and Z processes.
prediction for W +H production at the 13 TeV LHC. The plots on the left hand side correspond to
the basic cut selection, whilst those on the right include an additional cut on pTZ > 200 as described
Our results for the basic and high-pVT cuts are presented in gures 2 and 3 for W H
and ZH processes respectively. The results have been obtained using MCFM, with the
parameters described in the previous section. We use the CT10 PDF set [50] for NLO
predictions and CTEQ6L1 for LO predictions. The renormalization and factorization scale
have been set to
= mV H . In each of gures 2 and 3 the plot on the left side represents the
cross section obtained using the basic cuts, whilst those on the right hand side correspond
to the cross sections obtained with the additional high-pVT selection requirement applied.
In both gures we plot the cross section as a function of cW and cHW . We present contours
which correspond to the values of cW and cHW needed to obtain a 10, 20 or 30 % deviation
from the SM prediction. For reference using our cuts described above the SM predictions
are: 9:7 fb and 1.8 fb for W +H with the basic and high-pVT cuts respectively, and 5:1 and
0:54 for ZH. Our results contain terms up to order c2X as can be clearly seen from the
gures, since the results for constant cross section form ellipses. Including these terms is
somewhat dangerous, since in general they correspond to regions in which the EFT may
be breaking down. This is because 8 dimensional operators also contribute rst at this
level, and therefore should not be ignored in the calculation. Therefore in an attempt to
quantify the range of validity of our results we present a contour which corresponds to the
regime in which the linear part of the cross section corresponds to 95% (solid) and 90%
(dashed) of the total, or in other words the higher order pieces in the EFT should be small
(both from 8 dimensional operators and loop corrections to the 6 dimensional operators
which go like quadratic pieces). Experimental results can then be used to set reliable EFT
bounds inside of this contour. We stress that values which lie outside of this contour (i.e.
larger absolute values of cX ) cannot be reliably excluded given our theoretical accuracy,
and given the form of our results, it is clear that there are regions outside of the EFT
validity which conspire to produce small changes in the total cross section.
( %
the basic cut selection, whilst those on the right include an additional cut on pTZ > 200 as described
in the text.
The hallmark of EFT operators is a lack of suppression at high energies due to poor
high-energy behaviour. Therefore, a natural place to search for the impact of the higher
dimensional operators is the region which is sensitive to larger values of s^. Since mV H
cannot be directly cut upon in the experiment for leptonic W H
nal states, we use pVT as
a proxy for s^. The plots on the right hand side of gures 2 and 3 present these results.
As expected we see a signi cant increase in sensitivity in the (cHW ; cW ) plane compared
to the more inclusive analysis. By looking at high p
around jcW + cHW j . 0:005 to around jcW + cHW j . 0:002.
VT one improves the sensitivity from
To demonstrate the exibility of our code, we present results in the HAC basis, rather
than the EFT approach, in gure 4. In this setup the anomalous couplings are
parameterized in terms of general Lorentz invariant operators in the Lagrangian. In this approach
corrections from higher dimensional operators are not a concern so we present the full
ellipses, and do not present EFT validity contours for these plots. We note that, since the
HAC basis does not necessitate deviations in the EW parameters due to kinetic mixing of
operators, we use the SM EW parameter choices (corresponding to the MCFM defaults)
for these plots.
In
gure 5 we investigate the impact of the NLO corrections to the anomalous
couplings. We de ne the following ratio,
R
NLO(cW ; cHW ) =
NLO(cW ; cHW )
NLO(0; 0) + LO(cW ; cHW )
LO(0; 0)
:
(4.1)
Here R
NLO(cW ; cHW ) is de ned as the full NLO result, divided by the NLO SM piece
plus the LO anomalous coupling pieces. The results for W +H and ZH are presented in
the
gure. As might be expected from the inclusive K-factor the deviations are around
20% depending on the position in the (cW ; cHW ) plane. Around ( 0:015; 0:01) the NLO
1.2
1.1
1.0
0.9
RNLO(cW ; cHW ) as de ned in the text, for both predictions the SM piece is included at NLO.
corrections suppress the result one would obtain if a LO prediction were used, and previous
limits in this region of phase space (using the LO prediction) may be too optimistic. On
the other hand, away from this region the corrections tend to be positive and will improve
existing limits.
We note that the region which corresponds to that in which our EFT
calculation is valid intersects the region in which the impact of the NLO corrections is
most rapidly changing. This suggests that using a at K factor to re-weight the anomalous
coupling part of the calculation is not advisable.
Finally in gure 6 we present contours of constant cross section in the (~cHW ; cHW )
plane, i.e. we study the impact of including CP odd operators. The CP odd operators
do not interfere with the SM amplitude, so they rst enter the cross section at O(c~2HW ).
This can be seen in the gures via the c~HW $
c~HW symmetry in the gure. In order to
prediction for W +H (left) and ZH (right) production at the 13 TeV LHC. The plots correspond to
the basic cut selection.
have a relatively small deviation from the SM therefore requires a negative value of cHW
which can compensate for the positive de nite correction arising from the CP odd operator.
In
gure 6 we present results for the basic cuts only, however improved results could be
obtained by optimizing the analysis to look at high p
VT observables. In addition angular
distributions between
nal state particles are particularly sensitive to the CP structure of
operators and represent a promising avenue of study.
4.2
NLO + Parton shower results
We now turn to the presentation of the xed order plus parton shower results making use
of the MCFM/Powheg-Box interface. The two processes considered are the production
of the Higgs in association with a Z or W boson, where the Higgs decays to bb and the weak
boson decays leptonically to e or . Events were generated for some characteristic values
of the EFT Wilson coe cients and showered/hadronised with Pythia8. The decay of the
Higgs was also performed by Pythia8 [51], with the total rate normalised to the NLO
production cross section times the branching fraction as calculated by eHDECAY [
52
]
to NLO accuracy in both
S and
EW. Event reconstruction and the implementation
of basic selection cuts, summarised in table 3, was performed using MadAnalysis5 [53]
which makes use of Fastjet [
54
]. Benchmark EFT scenarios are selected to be within
the allowed regions of recently performed global ts. In each case, an estimate of scale
uncertainty is evaluated by varying the renormalisation and factorisation scales between
half and twice the central scale, which is the invariant mass of the V H system. This is
combined in quadrature with the usual Monte Carlo statistics uncertainty arising from the
nite number of events generated. NLO and LO samples were generated with the CTEQ10
and CTEQ6L1 PDF sets respectively and PDF uncertainties were not estimated.
Process
Jets
Cuts
H Z ! bb `+`
H W ! bb `
kT algorithm:
R = 0:4, pT > 25 GeV & b < 2:5
2 b-jets, pT > 25 GeV, b < 2:5
1 lepton, ` (e or )
2 leptons, `+; ` (e or )
p`T < 25 GeV, j `j < 2:5
associated production modes.
Gluon initiated contribution to HZ
In order to highlight the importance of the gg-initiated contribution to ZH production,
a sample was generated separately and compared to the pure qq-initiated piece at NLO.
In general, due to the 2mt thresholds, the kinematics of the box con guration will prefer
a signi cantly harder region of phase space than the Drell-Yan like topology and, if it is
not adequately taken into account, could show up as a ctitious EFT-like signal. Figure 7
overlays the two contributions in several di erential distributions, showing the relative size
of the would-be `signal' that one may observe if the gg piece were not factored into the SM
prediction. The contribution of the sub-process to the inclusive cross section is minor, of
order 3{4%, but as it populates a high pT region, where the SM cross section is also quite
small, the gg contribution can show up as an O(10{15%) e ect in the tails of di erential
distributions, mimicking a potential EFT-like signature. The e ect of this contribution on
the Nj distribution is even more striking, given the increased emission probabilities of the
initial state, reaching around 40%. In sections 4.2.2, this contribution is taken into account
as part of the SM prediction for ZH associated production. The nal panel in gure 7 (and
similar panels in future gures) encapsulates the emission of radiation in the form of the
number of jets in the process. These jets can arise from either the matrix element or the
subsequent parton shower. Since the matrix elements can provide at most one additional
jet at NLO, the parton shower provides the additional jet multiplies beyond those of the
NLO matching, which for these processes corresponds to Nj > 3 (qq), and Nj > 2 (gg)
(and hence these rates are subject to larger theoretical uncertainties.)
LO + PS vs. NLO + PS.
To asses the impact of taking higher order e ects into account,
we now compare the new implementation at NLO to a LO one in MCFM, post showering
and hadronisation. Figure 8 depicts a selection of di erential distributions in the SM and
for one of our benchmark points of cW = 0:004 (see discussion in section 4.2.2) for HZ and
HW production respectively. Here the predominant e ect is that of a relatively at
Kfactor that is not sensitive to the presence of the new EFT interactions which, in any case,
are colour neutral. The rightmost distributions in the upper half of gure 8 show the pT
of the ZH system and are therefore sensitive to the `kick' that it receives from additional
V
e
/
b
5
2
/
dσf[bZdpT10−3
10−4
20
qq¯
qggq¯ + gg
1
dσ dNj
qq¯
qggq¯ + gg
0
di erential distributions in, from left to right, the invariant mass of the HZ system, the pT of the
Z boson and the Njets distribution normalised to the 0-jet bin after a cut on pTZ of 200 GeV. Lower
panels show the ratio of the gg- and qq-initiated contributions.
radiation. We see a mild rise in the tail between the NLO case that we would expect given
that it captures the full phase space of one additional emission compared to the LO case
which remains within the soft and collinear approximation of the parton shower.
4.2.2
EFT e ects
We now turn to examining the e ect of switching on one or more of the previously de ned
Wilson coe cients that a ect associated production in both the ZH and W H channels.
We limit ourselves to the cW and cHW coe cients as they are su cient to capture the
additional non SM-like momentum dependence brought about by dimension-6 operators.
Figure 7 displays a number of characteristic di erential distributions evaluated using the
values of cW =
0:02 and cHW
= 0:015, which saturate the bounds set by the most
recent global ts [43, 44]. In general, very large e ects are expected for such sizable values
of the coe cients and considering the discussion in section 4.1, the validity of such an
EFT description is called into question in the phase space regions where the BSM e ects
are important. Considering
gure 3, it is clear that the values of the coe cients lie well
outside of the regions in which the quadratic piece of the EFT contribution makes up
less than 10% of the overall contribution. The pbT distribution in
gure 9, for example,
highlights very clearly the onset of a breakdown of the EFT in the high pT tail, where the
relative contribution of the cW =
0:02 point changes sign, suggesting the dominance of
the (cW = )2 term. Therefore, although these values of Wilson coe cients are technically
`allowed', the evidence in this section as well as in section 4.1, suggests that we are not
yet at a point in which the sensitivity of experiments can provide meaningful information
about the coe cients a ecting this Higgs production process.
Since we have not included e ects from dimension 8 operators, which may be as large
as the aforementioned squared EFT contributions, we prefer to present results using more
conservative values of the coe cients, where the EFT interpretation is better motivated.
These values are chosen from the criteria delineated in section 4.1, i.e., the requirement
that the squared terms do not make up more than 10% of the overall contribution. This
leads us to choose jcW j; jcHW j = 0:004. Our two benchmark points derived from this
V
b
f[
dσ dMVH10−3
V
e
b
f[
5G10−2
2
dσbdpT10−3
and cHW = 0:015. From left to right, the di erential cross sections with respect to the Higgs-Z
invariant mass, mV H ; Z-boson transverse momentum, pTZ ; and the leading b-jet transverse
momentum, pbT1, are shown in the upper panels, with the percentage deviation of the EFT benchmarks
from the SM prediction, BSM shown in the lower panels.
are cW = 0:004 and cW =
cHW =
0:004. The relationship imposed in the latter
choice is motivated by the results of previous works that calculated the low energy EFT
coe cients predicted by a number of UV scenarios [38, 39]. In the Two-Higgs Doublet
Model, for example, this relationship is always satis ed at the matching scale. From a
phenomenological perspective, this relation is also special because it corresponds to the
elimination of one of the two momentum structures, gh(2v)v, present in the extended
Higgsgauge vertices (see table 1 and discussion in section 2).
For ZH production, gure 10 shows di erential distributions with respect to the
HiggsZ invariant mass, Z-boson pT and the number of jets (Nj ) normalised to the 0-jet bin
comparing the SM to the two EFT benchmarks. For the Nj distribution, an additional
cut on the Higgs pT of 200 GeV is applied in order to isolate the region where the EFT
contributions are most important. In the case of W H production, the leptonic decay of the
W + includes a neutrino which contributes to real missing energy on the event, preventing
the construction of some of the kinematic variables available to the Z-boson associated
production process, namely mV H and pVT , the total invariant mass and the vector boson
transverse momentum. We trade these two observables for the total transverse mass of the
system and the transverse momentum of the Higgs boson. Here, the total transverse mass
of the HW system is de ned including the two b-jets, the lepton and the missing transverse
energy,
m2T =
X ETi + E= T
i
!2
X p~Ti + p=~T
i
!2
;
i = b; b; l+:
(4.2)
The observable is the analogue of MV H in the ZH case and is an approximation of the
momentum
owing through the W H vertex. These variables are shown in gure 10 along
with the normalised Nj distributions, as in the ZH case.
We see that these more conservative choices for the Wilson coe cients still permit
O(20{50%) deviations in the tails of the various distributions for the cW = 0:004 benchmark
with a clear preference for large momentum
ow through the vertex. We also observe that
σ VH
d M
d
f[
dσZdpT
10−3
0.01
)% 2400
(
M
δBS −20
0
−400
d1σ dNσjj0 0.1
cHW =
0:004 using Powheg + Pythia8. Lower panels show the
percentage deviation of the EFT benchmarks from the SM prediction, BSM.
Upper row : pp ! ZH ! `+` bb. From left to right | the Higgs-Z invariant mass, MV H ;
Z-boson transverse momentum, pTZ ; and the number of jets normalised to the 0-jet bin, Nj.
Lower row : pp ! W +H ! `+ bb. From left to right | the transverse mass of the system,
mT (de ned in text); Higgs transverse momentum, pTH ; and the number of jets normalised to the
0-jet bin, Nj.
the size of the deviation in the MV H distribution for ZH correspond roughly to the size of
the deviation in p
TZ at the corresponding energy scale, i.e., MV H
pTZ =2, demonstrating
the expected correlation between the two observables. The second benchmark of cW =
cHW =
0:004 does not exhibit such large deviations, instead contributing a relatively
atter depletion of the di erential rate. This can be traced to the di erent Lorentz structure
governing the e ective vertex. The di erence between the two benchmarks shows that gh(1v)v
leads to much more striking `EFT-like' deviations than gh(1v)v. Looking more closely at the
Feynman rules of gure 1, we see that gh(2v)v goes as the square of the individual momenta of
the Z bosons, while gh(1v)v goes as the product of the two Z boson momenta. As a consequence,
in high centre of mass energy limit of the ZH production matrix element, gh(2v)v leads to a
richer energy dependence, containing terms proportional to higher powers of Mandelstam
variables / st=MZ2 ; t2=MZ2 that are not present when only considering gh(1v)v contributions.
The Nj distributions | although su ering from somewhat low MC statistics due to the
p
VT > 200 GeV requirement | appear to follow a similar trend.
5
Physics Beyond the Standard Model is likely to be connected to the Higgs sector, generically
leading to deviations in the Higgs behaviour with respect to SM predictions. These indirect
probes of new physics require a precise understanding of the SM contributions as well as the
interplay between the SM and New Physics in observables. Among the di erent LHC Higgs
observables, the production in association with a vector boson is specially sensitive to e ects
of new heavy particles in kinematic distributions and ratios of cross sections [43, 44, 55, 56].
In this paper we have presented predictions for the associated production of a Higgs
boson in association with a W or Z vector boson, including anomalous couplings
between the Higgs and vector boson, not present in the Standard Model. Our predictions
include e ects in QCD beyond the Leading Order in perturbation theory. We presented
predictions at xed order (NLO) and matched to parton showers using the Powheg
formalism (NLO+PS).
Anomalous couplings in the HV V vertex (HAC) can arise in many extensions of the
SM. A general model independent parameterization can be obtained by saturating the
Lorentz structures of the four-dimensional HV V interactions in the Lagrangian. Particular
models then correspond to some (or all) of the new couplings acquiring non-zero values.
An interesting class of models arise when the scale of new physics is large and can be
integrated out of the Standard Model. In these scenarios the SM is treated as an e ective
eld theory (EFT). We matched our results from the EFT to a linearly realized breaking
of the EW symmetry, in which the Higgs is a doublet of SU(2)L. Transitioning between
the two calculations setups is straightforward, and we presented results in both the HAC
and EFT frameworks.
In order to maximize the physics potential of the LHC it is essential that precise
theoretical predictions are used to compare theoretical predictions and experimental data.
Matched parton showers, which combine the normalization and matrix elements of a
Nextto-Leading Order calculation, and a leading logarithmic resummation of soft collinear
logarithms provide a good framework for comparing theoretical predictions to data. The
Powheg-Box provides a public format to match results obtained at xed order to parton
showers, allowing for full event simulation.
We calculated the NLO corrections to V H
production including the e ects of anomalous couplings using analytic amplitudes and the
spinor helicity formalism. We implemented this calculation into MCFM and modi ed the
existing V H processes in Powheg to incorporate our new matrix elements.
We used our results to study the phenomenological impact of our calculation at the
Run II of the LHC operating at p
s = 13 TeV. We demonstrated the capabilities of our code
both at xed order and NLO+PS accuracy generating events and showering them with the
PYTHIA parton shower. We focused on parameter selections which are consistent with
limits obtained during Run I. In this region NLO e ects change the di erential distributions
by around O(20%). Our results will be made publicly available in the released versions of
the MCFM and Powheg codes.
Acknowledgments
The work of KM and VS is supported by the Science Technology and Facilities Council
(STFC) under grant number ST/J000477/1.
Fields rede nitions and their contributions to EW parameters and
gauge boson interactions
After electroweak symmetry breaking, the SM supplemented by the dimension-6 operators
in eq. (2.2) leads to the following, non-canonical kinetic terms for the weak gauge bosons:
Lkin: =
W+ W
1
2
1
4
B
B
"
v2 g0 2
eld rede nitions canonically normalise these terms and remove the
T3
Hypercharge mixing.
W
B
! W [1 + W ]
! B [1 + B] + yW3
W3 ! W3 [1 + W ] + zB
W =
y + z
v2 g2
2 8 cW ;
W B =
v2 gg0
One may also rede ne the weak and hypercharge gauge couplings in order to absorb the
e ects of some of these shifts.
In general, the W mass is modi ed and the neutral mass matrix has one zero eigenvalue
for the photon and a modi ed mass for the Z boson. The following choice for the gauge
coupling rede nitions preserves the SM expressions for W -boson mass and interactions as
well as the de nition of the Weinberg angle in terms of the gauge couplings while shifting
the Z-mass:
with
g =
W ;
g0 =
B + W + g;
y +
g0
g
g
g0
z
) mW =
ev
2s^W
;
mZ =
ev
2s^W c^W
[1 + mZ ] ;
g
c^W = pg2 + g0 2
;
e = gs^W ;
mZ =
2 W B):
s^W (z
c^W
We choose to set the parameter z to:
z =
e2v2 s^W cW :
8 2 c^W
(A.1)
(A.2)
(A.3)
(A.4)
(A.5)
(A.6)
(A.7)
(A.8)
The shifts to the fermionic photon and Z couplings parametrised as:
Q0 = eQ [1 + e] ;
Q0Z =
e
c^W s^W
T3 1 + T Z
3
Qs^2W 1 + QZ
;
(A.9)
(A.10)
are given in eqs. (3.24) and (3.25). We note that the di erence between using the SM and
EFT de nitions of parameters multiplying a dimension-6
order in the EFT expansion.
O(
2
) contribution is higher
The extraction of the EW parameters from the fmZ ; mW ; GF g set of inputs discussed
in section 3.3 follows from these de nitions. It is important to stress that these de
HJEP08(216)39
nitions are valid for the subset of operators that are implemented in our code, namely
OW ; OHW ; OB; OHB and O . In general, the presence of a complete dimension-6 basis of
operators will lead to more modi cations, such as with the OH and OT operators modifying
the canonical normalisation of the Higgs eld and therefore its couplings. A consequence
of this can be seen in table 1, where these two Wilson coe cients appear in the SM-like
gh(3z)z structure.
Open Access.
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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