Gluonfusion Higgs production in the Standard Model Effective Field Theory
HJE
Gluonfusion Higgs production in the Standard Model
Nicolas Deutschmann 0 1 2 4 5
Claude Duhr 0 1 2 3 5
Fabio Maltoni 0 1 2 5
Eleni Vryonidou 0 1 2 5
0 CH1211 Geneva 23 , Switzerland
1 Chemin du Cyclotron 2 , 1348 LouvainLaNeuve , Belgium
2 F69622 , Villeurbanne , France
3 Theoretical Physics Department , CERN
4 Univ. Lyon , Universite Lyon 1, CNRS/IN2P3, IPNL
5 Science Park 105 , 1098 XG, Amsterdam , The Netherlands
We provide the complete set of predictions needed to achieve NLO accuracy in the Standard Model E ective Field Theory at dimension six for Higgs production in gluon fusion. In particular, we compute for the rst time the contribution of the chromomagnetic relevant operators by implementing the results into the MadGraph5 aMC@NLO framework. This allows us to compute total cross sections as well as to perform event generation at NLO that can be directly employed in experimental analyses.
Universite catholique de Louvain

E
operator QL
qRG at NLO in QCD, which entails twoloop virtual and oneloop real
contributions, as well as renormalisation and mixing with the Yukawa operator
y QL qR
and the gluonfusion operator
y
GG. Focusing on the topquarkHiggs couplings, we
consider the phenomenological impact of the NLO corrections in constraining the three
1 Introduction
2
3
3.1
3.2
3.3
3.4
4.1
4.2
4.3
4
Phenomenology
Crosssection results
Di erential distributions
Renormalisation group e ects
5
Conclusion and outlook
1
Introduction
Gluon fusion in the SM E ective Field Theory
Virtual corrections
Computation of the twoloop amplitudes UV & IR pole structure
Analytic results for the twoloop amplitudes
Renormalisation group running of the e ective couplings
Five years into its discovery at the LHC, the Higgs boson is still the centre of attention
of the highenergy physics community. A wealth of information has been collected on
its properties by the ATLAS and CMS experiments [1{5], all of which so far support the
predictions of the Standard Model (SM). In particular, the size of the couplings to the weak
vector bosons and to the electrically charged third generation fermions has been con rmed,
and the rst evidence of the coupling to second generation fermions (either charm quark
or muon) could arrive in the coming years, if SMlike.
The steady improvement in the precision of the current and forthcoming Higgs
measurements invites to explore physics beyond the SM not only via the search of new
resonances, as widely pursued at the LHC, but also via indirect e ects on the couplings of the
Higgs boson to the known SM particles. The most appealing aspect of such an approach
is that, despite being much more challenging than direct searches both experimentally and
theoretically, it has the potential to probe new physics scales that are beyond the
kinematical reach of the LHC. A powerful and predictive framework to analyse possible deviations
in the absence of resonant BSM production is provided by the SM E ective Field Theory
(SMEFT) [6{8], i.e., the SM augmented by higherdimensional operators. Among the most
interesting features of this framework is the possibility to compute radiative corrections in
the gauge couplings, thus allowing for systematic improvements of the predictions and a
strong coupling constant typically entail large e ects at the LHC both in the accuracy
{ 1 {
and the precision. They are therefore being calculated for a continuously growing set of
processes involving operators of dimension six featuring the Higgs boson, the bottom and
top quarks and the vector bosons. Currently, predictions for the most important
associated production channels for the Higgs boson are available in this framework, e.g., VH,
VBF and ttH [10{12]. For topquark production, NLO results for EW and QCD inclusive
production, i.e., tj and tt, and for topquark associated production ttZ, tt have also
appeared [13{18]. The e ect of dimensionsix operators has also become available recently
for topquark and Higgs decays [19{23].
The situation is somewhat less satisfactory for gluon fusion, which, despite being a
loopinduced process in the SM, is highly enhanced by the gluon density in the proton and
corrections are now known up to N3LO in the limit of a heavy top quark [24{26]. The
full quarkmass dependence is known up to NLO [27{30], while at NNLO only subleading
terms in the heavy topmass expansion [31{34] and leading contributions to the top/bottom
interference [35, 36] are known. Beyond inclusive production, the only available NNLO
result is the production of a Higgs boson in association with a jet in the in nite topmass
limit [37{39], while cross sections for H + njets, n = 2; 3, are known only at NLO in the
heavy topmass expansion [40, 41].
In the SMEFT, most studies have been performed at LO, typically using
approximate rescaling factors obtained from SM calculations. Higherorder results have only been
considered when existing SM calculations could be readily used within the SMEFT. The
simplest examples are the inclusion of higher orders in the strong coupling to the
contribution of two speci c dimensionsix operators, namely the Yukawa operator ( y )QL qR
and the gluonfusion operator ( y )GG. The former can be accounted for by a
straightforward modi cation of the Yukawa coupling of the corresponding heavy quark, b or t, while
the latter involves the computation of contributions identical to SM calculations in the
limit of an in nitelyheavy top quark. Results for the inclusive production cross section
including modi ed top and bottom Yukawa couplings and an additional direct Higgsgluons
interaction are available at NNLO [42] and at N3LO [43, 44]. At the di erential level,
phenomenological studies at LO have shown the relevance of the high transverse momentum
region of the Higgs boson in order to resolve degeneracies among operators present at the
inclusive level [12, 45{47]. Recently, the calculation of the Higgs spectrum at NLO+NNLL
level for the Yukawa (both b and t) and Higgsgluons operator has appeared [48, 49].
The purpose of this work is to provide the contribution of the chromomagnetic operator
QL
qRG to inclusive Higgs production at NLO in QCD, thereby completing the set of
predictions (involving only CP even interactions) needed to achieve NLO accuracy in the
SMEFT for this process. The rst correct computation at oneloop of the contribution
of chromomagnetic operator of the top quark to gg ! H has appeared in the erratum of
ref. [50] and later con rmed in refs. [12, 49]. The LO contribution of the chromomagnetic
operator of the topquark to H+jet was computed in ref. [12]. An important conclusion
drawn in ref. [12] was that even when the most stringent (and still approximate) constraints
from tt production are considered [14], this operator sizably a ects Higgs production, both
in gluon fusion (single and double Higgs) and ttH production.
{ 2 {
At LO the chromomagnetic operator enters Higgs production in gluon fusion at one
loop. Therefore NLO corrections in QCD entail twoloop virtual and oneloop real
contributions. The latter can nowadays easily be computed using an automated approach. The
former, however, involve a nontrivial twoloop computation that requires analytic
multiloop techniques and a careful treatment of the renormalisation and mixing in the SMEFT,
both of which are presented in this work for the rst time. In particular, while the full
mixing pattern of the SMEFT at one loop is known [51{53], a new twoloop counterterm
enters our computation, and we provide its value for the rst time here. Moreover, we
present very compact analytic results for all the relevant amplitudes up to two loop order.
Focusing on possibly anomalous contributions in topquarkHiggs interactions, we then
leading logarithmic renormalisation group running of the Wilson coe cients. In section 4
we perform a phenomenological study at NLO, in particular of the behaviour of the QCD
and EFT expansion at the total inclusive level and provide predictions for the pT spectrum
of the Higgs via a NLO+PS approach.
2
Gluon fusion in the SM E ective Field Theory
The goal of this paper is to study the production of a Higgs boson in hadron collisions in
the SMEFT, i.e., the SM supplemented by a complete set of operators of dimension six,
O1 =
O2 = gs2
y
y
v
2
2
v
2
2
QL ~ tR ;
G
a Ga ;
O3 = gs QL ~ T a
tR G
a ;
{ 3 {
LEFT = LSM + X
Cb
i
2 Oi + h.c. :
i
The sum in eq. (2.1) runs over a basis of operators Oi of dimension six, is the scale of new
physics and Cib are the (bare) Wilson coe cients, multiplying the e ective operators. A
complete and independent set of operators of dimension six is known [7, 55]. In this paper,
we are only interested in those operators that modify the contribution of the heavy quarks,
bottom and top quarks, to Higgs production in gluon fusion. Focusing on the top quark,
there are three operators of dimension six that contribute to the gluonfusion process,
(2.1)
(2.2)
(2.3)
(2.4)
g
H
(Φ† Φ) Q¯LΦ qR
(Φ† Φ) GG
Q¯LΦ σqRG
where gs is the (bare) strong coupling constant and v denotes the vacuum expectation
value (vev) of the Higgs eld
( ~ = i 2 ). QL is the lefthanded quark SU(2)doublet
containing the top quark, tR is the righthanded SU(2)singlet top quark, and Ga is the
gluon eld strength tensor. Finally, T a is the generator of the fundamental representation
of SU(3) (with [T a; T b] = 12 ab) and
= 2i
can be obtained by simply making the substitutions f
, tR ! bRg. Second, while
O2 is hermitian O1 and O3 are not.1 In this work, we focus on the CP even contributions
of O1 and O3. For this reason, all the Wilson coe cients Ci with i = 1; 2; 3 are real.
Representative Feynman diagrams contributing at LO are shown in gure 1. ~
!
In the SM and at leading order (LO) in the strong coupling the gluonfusion process
is mediated only by quark loops. This contribution is proportional to the mass of the
corresponding quark and therefore heavy quarks dominate.
While we comment on the b (and possibly c) contributions later, let us focus on the leading contributions coming from the top quark, i.e., the contributions from the operators of dimension six shown in eqs. (2.2){(2.4). The (unrenormalised) amplitude can be cast in the form
i S
C1b v2
2 b
Ab(g g ! H) =
s [(p1 p2) ( 1 2) (p1 2) (p2 1)]
1
v Ab;0(mtb; mH )
(2.5)
+ p
2 2 Ab;1(mtb; mH )+
C2b v
2 Ab;2(mtb; mH )+ p
Cb
3
2 2 Ab;3(mtb; mH ) +O(1= 4) ;
where sb = gs2=(4 ) denotes the bare QCD coupling constant and mH and mtb are the bare
masses of the Higgs boson and the top quark. The factor S = e E (4 ) is the usual MS
factor, with
E =
0(1) the EulerMascheroni constant and
is the scale introduced by
dimensional regularisation. For i = 0, the form factor Ab;i denotes the unrenormalised SM
contribution to gluon fusion [56], while for i > 0 it denotes the form factor with a single2
1Note that in eq. (2.1) we adopt the convention to include the hermitian conjugate for all operators, be
they hermitian or not. This means that the overall contribution from O2 in LEFT is actually 2C2O2= 2
2According to our power counting rules, multiple insertions of an operator of dimension six correspond
.
to contributions of O(1= 4) in the EFT, and so they are neglected.
{ 4 {
operator Oi inserted [48, 50, 57]. The normalisation of the amplitudes is chosen such that
all coupling constants, as well as all powers of the vev v, are explicitly factored out. Each
form factor admits a perturbative expansion in the strong coupling,
Ab;i(mtb; mH ) =
1
X
k=0
S
2
b k
s
A(bk;i)(mtb; mH ) :
(2.6)
Some comments about these amplitudes are in order. First, after electroweak symmetry
breaking, the operator O1 only amounts to a rescaling of the Yukawa coupling, i.e., Ab;1 is
simply proportional to the bare SM amplitude. Second, at LO the operator O2 contributes
at tree level, while the SM amplitude and the contributions from O1 and O3 are
loopinduced. Finally, this process has the unusual feature that the amplitude involving the
chromomagnetic operator O3 is ultraviolet (UV) divergent, and thus requires
renormalisation, already at LO [12, 49, 50]. The UV divergence is absorbed into the e ective coupling
that multiplies the operator O2, which only enters at tree level at LO. The renormalisation
at NLO will be discussed in detail in section 3.
The goal of this paper is to compute the NLO corrections to the gluonfusion process
with an insertion of one of the dimension six operators in eqs. (2.2){(2.4). We emphasise
that a complete NLO computation requires one to consider the set of all three operators
in eq. (2.2){(2.4), because they mix under renormalisation [51{53]. At NLO, we need to
consider both virtual corrections to the LO process g g ! H as well as real corrections
due to the emission of an additional parton in the
nal state. Starting from NLO, also
partonic channels with a quark in the initial state contribute. Since the contribution
from O1 is proportional to the SM amplitude, the corresponding NLO corrections can be
obtained from the NLO corrections to gluonfusion in the SM including the full topmass
dependence [27, 28, 30, 58]. The NLO contributions from O2 are also known, because they
are proportional to the NLO corrections to gluonfusion in the SM in the limit where the
top quark is in nitely heavy [59] (without the higherorder corrections to the matching
coe cient). In particular, the virtual corrections to the insertion of O2 are related to
the QCD form factor, which is known through three loops in the strong coupling [60{69].
Hence, the only missing ingredient is the NLO contributions to the process where the
chromomagnetic operator O3 is inserted. The computation of this ingredient, which is one
of the main results of this paper, will be presented in detail in the next section.
As a
nal comment, we note that starting at two loops other operators of EW and
QCD nature will a ect gg ! H. In the case of EW interactions, by just looking at the SM
EW contributions [70, 71], it is easy to see that many operators featuring the Higgs eld
will enter, which in a few cases could also lead to constraints, see, e.g., the trilinear Higgs
self coupling [72, 73]. In the case of QCD interactions, operators not featuring the Higgs
eld will enter, which, in general, can be more e ciently bounded from other observables.
For example, the operator gsf abcGa G
b G
c contributes at two loops in gg ! H and at
one loop in gg ! Hg. The latter process has been considered in ref. [74], where e ects on
the transverse momentum of the Higgs were studied. For the sake of completeness, we have
reproduced these results in our framework, and by considering the recent constraints on
this operator from multijet observables [75], we have con rmed that the Higgs pT cannot
{ 5 {
be signi cantly a ected. For this reason we do not discuss further this operator in this
paper. Fourfermion operators also contribute starting at two loops to gluon fusion but as
these modify observables related to top quark physics at leading order [76, 77] we expect
them to be independently constrained and work under the assumption that they cannot
signi cantly a ect gluon fusion.
Virtual corrections
Computation of the twoloop amplitudes
In this section we describe the virtual corrections to the LO amplitudes in eq. (2.5). For
HJEP12(07)63
the sake of the presentation we focus here on the calculation involving a top quark and
discuss later on how to obtain the corresponding results for the bottom quark. With the
exception of the contributions from O2, all processes are loopinduced, and so the virtual
corrections require the computation of twoloop form factor integrals with a closed
heavyquark loop and two external gluons. We have implemented the operators in eqs. (2.2){(2.4)
into QGraf [78], and we use the latter to generate all the relevant Feynman diagrams. The
QGraf output is translated into FORM [79, 80] and Mathematica using a custommade code.
The tensor structure of the amplitude is xed by gaugeinvariance to all loop orders, cf.
eq. (2.5), and we can simply project each Feynman diagram onto the transverse polarisation
tensor. The resulting scalar amplitudes are then classi ed into distinct integral topologies,
which are reduced to master integrals using FIRE and LiteRed [81{85]. After reduction,
we can express all LO and NLO amplitudes as a linear combination of one and twoloop
master integrals.
The complete set of one and twoloop master integrals is available in the literature [58,
86{88] in terms of harmonic polylogarithms (HPLs) [89],
Z z
0
H(a1; : : : ; aw; z) =
dt f (a1; t) H(a2; : : : ; aw; z) ;
mm2Ht2 =
p1
p1
(1
x
x)2
;
4=
4= + 1
1
:
{ 6 {
1
t
;
1
w!
H(0; : : : ; 0; z) =
w t{izmes}
logw z :
f (1; t) =
1
1
t
;
In the case where all the ai's are zero, we de ne,
f (0; t) =
f ( 1; t) =
1
1 + t
:
The number of integrations w is called the weight of the HPL. The only nontrivial
functional dependence of the master integrals is through the ratio of the Higgs and the top
masses, and it is useful to introduce the following variable,
(3.1)
(3.2)
(3.3)
(3.4)
(3.5)
The change of variables in eq. (3.4) has the advantage that the master integrals can be
written as a linear combination of HPLs in x. In the kinematic range that we are interested
in, 0 < m2H < 4mt2, the variable x is a unimodular complex number, jxj = 1, and so it can
be conveniently parametrised in this kinematics range by an angle ,
x = ei ;
In terms of this angle, the master integrals can be expressed in terms of (generalisations
of) Clausen functions (cf. refs. [58, 90{93] and references therein),
Clm1;:::;mk ( ) =
( Re Hm1;:::;mk e
Im Hm1;:::;mk e
i ; if k + w even ;
i ; if k + w odd ;
(3.6)
(3.7)
HJEP12(07)63
where we used the notation
(jm1j {1z) ti}mes
(jmkj {1z) ti}mes
Hm1;:::;mk (z) = H( 0; : : : ; 0 ; 1; : : : ; 0; : : : ; 0 ; k; z) ;
i
sign(mi) :
(3.8)
The number k of nonzero indices is called the depth of the HPL.
Inserting the analytic expressions for the master integrals into the amplitudes, we can
express each amplitude as a Laurent expansion in
whose coe cients are linear
combinations of the special functions we have just described. The amplitudes have poles in
which
are of both ultraviolet (UV) and infrared (IR) nature, whose structure is discussed in the
next section.
3.2
UV & IR pole structure
In this section we discuss the UV renormalisation and the IR pole structure of the LO and
NLO amplitudes. We start by discussing the UV singularities. We work in the MS scheme,
and we write the bare amplitudes as a function of the renormalised amplitudes as,
Ab( sb; Cib; mtb; mH ) = Zg 1 A( s( 2); Ci( 2); mt( 2); mH ; ) ;
(3.9)
where Zg is the eld renormalisation constant of the gluon eld and
mt( 2) are the renormalised strong coupling constant, Wilson coe cients and top mass in
the MS scheme, and
denotes the renormalisation scale. The renormalised parameters are
s( 2), Ci( 2) and
related to their bare analogues through
(3.10)
S
sb =
Cib =
2 Z s s( 2) ;
ai ZC;ij Cj ( 2) ;
mtb = mt( 2) +
mt ;
{ 7 {
with (a1; a2; a3) = (3; 0; 1). Unless stated otherwise, all renormalised quantities are
assumed to be evaluated at the arbitrary scale 2 throughout this section. We can decompose
the renormalised amplitude into the contributions from the SM and the e ective operators,
similar to the decomposition of the bare amplitude in eq. (2.5)
A(g g ! H) =
and each renormalised amplitude admits a perturbative expansion in the renormalised
strong coupling constant,
Ai(mt; mH ) =
s k
Ai(k)(mt; mH ) :
The presence of the e ective operators alters the renormalisation of the SM parameters.
Throughout this section we closely follow the approach of ref. [12], where the
renormalisation of the operators at one loop was described. The oneloop UV counterterms for the
strong coupling constant and the gluon eld are given by
1
X
k=0
s C3 1
2
s C3 1
2
2
mt2
Zg = 1 + Zg;SM +
Z s = 1 + Z s;SM
2
mt2
2
mt2
p
p
2 v mt + O( s2) ;
2 v mt + O( s2) ;
Zg;SM =
Z s;SM =
s 1
6
4
s 0
+ O( s2) ;
s 1
6
2
mt2
+ O( s2) ;
where Zg;SM and Z s;SM denote the oneloop UV counterterms in the SM,
(3.11)
(3.12)
where Nc = 3 is the number of colours and Nf = 5 is the number of massless avours.
We work in a decoupling scheme and we include a factor
2=mt2
into the counterterm.
As a result only massless avours contribute to the running of the strong coupling, while
the top quark e ectively decouples [59]. The renormalisation of the strong coupling and
the gluon eld are modi ed by the presence of the dimension six operators, but the e ects
cancel each other out [50]. Similarly, the renormalisation of the top mass is modi ed by
the presence of the e ective operators,
mt =
2 2 v mt2 + O( s2) ;
where the SM contribution is
mtSM =
s mt + O( s2) :
{ 8 {
In eq. (3.16) we again include the factor into the counterterm in order to decouple the e ects from operators of dimension six from the running of the top mass in the
2=mt2
MS scheme.
written in the form
The renormalisation of the e ective couplings Cib is more involved, because the
operators in eqs. (2.2){(2.4) mix under renormalisation. The matrix ZC of counterterms can be
ZC = 1 + Z(0) +
C
s
Z(1) + O( s2) :
C
We have already mentioned that the amplitude Ab;3 requires renormalisation at LO
in the strong coupling, and the UV divergence is proportional to the LO amplitude
A(b0;2) [12, 49, 50]. As a consequence, ZC(0) is nontrivial at LO in the strong coupling,
At NLO, we also need the contribution ZC(1) to eq. (3.18). We have
Z(0) = BB 0 0
C
8mt2 1
v2 C
z23
1
6
C ;
C
A
where, apart from z23, all the entries are known [51{53]. z23 corresponds to the counterterm
that absorbs the twoloop UV divergence of the operator O3, which is proportional to the
treelevel amplitude A(b0;2) in our case. This counterterm is not available in the literature, yet
we can extract it from our computation. NLO amplitudes have both UV and IR poles, and
so we need to disentangle the two types of divergences if we want to isolate the counterterm
z23. We therefore rst review the structure of the IR divergences of NLO amplitudes, and
we will return to the determination of the counterterm z23 at the end of this section.
A oneloop amplitude with massless gauge bosons has IR divergences, arising from
regions in the loop integration where the loop momentum is soft or collinear to an external
massless leg. The structure of the IR divergences is universal in the sense that it factorises
from the underlying hard scattering process. More precisely, if A
oneloop amplitude describing the production of a colourless state from the scattering of
(1) denotes a renormalised
two massless gauge bosons, then we can write [94]
A
(1) = I(1)( ) A(0) + R ;
where A(0) is the treelevel amplitude for the process and R is a processdependent
remainder that is nite in the limit
! 0. The quantity I(1)( ) is universal (in the sense that it
does not depend on the details of the hard scattering) and is given by
I(1)( ) =
e E
(1
)
s12
i0
2
3
2 +
2
0
;
where s12 = 2p1p2 denotes the centerofmass energy squared of the incoming gluons.
{ 9 {
(3.18)
(3.19)
(3.20)
(3.21)
(3.22)
Since in our case most amplitudes are at one loop already at LO, we have to deal
with twoloop amplitudes at NLO. However, since the structure of the IR singularities is
independent of the details of the underlying hard scattering, eq. (3.21) remains valid for
twoloop amplitudes describing loopinduced processes, and we can write
Ai
(1) = I(1)( ) Ai
(0) + Ri ;
0
i
We have checked that our results for amplitudes which do not involve the operator O3
have the correct IR pole structure at NLO. For A(31), instead, we can use eq. (3.23) as a
constraint on the singularities of the amplitude. This allows us to extract the twoloop UV
Note that the coe cient of the double pole is in fact xed by requiring the anomalous
dimension of the e ective couplings to be
nite. We have checked that eq. (3.24) satis es
this criterion, which is a strong consistency check on our computation.
Let us conclude our discussion of the renormalisation with a comment on the
relationship between the renormalised amplitudes in the SM and the insertion of the operator
O1. We know that the corresponding unrenormalised amplitudes are related by a simple
rescaling, and the constant of proportionality is proportional to the ratio Cb=mtb. There is a
1
priori no reason why such a simple relationship should be preserved by the renormalisation
procedure. In (the variant of) the MSscheme that we use, the renormalised amplitudes
are still related by this simple scaling. This can be traced back to the fact that the MS
counterterms are related by
mtSM =
s Z(1)
C 11 + O( s2) :
If the top mass and the Wilson coe cient C1b are renormalised using a di erent scheme
which breaks this relation between the counterterms, the simple relation between the
amplitudes A0
(1) and A1
(1) will in general not hold after renormalisation.
3.3
Analytic results for the twoloop amplitudes
In this section we present the analytic results for the renormalised amplitudes that enter
the computation of the gluonfusion cross section at NLO with the operators in eqs. (2.2){
(2.4) included. We show explicitly the oneloop amplitudes up to O( 2) in dimensional
regularisation, as well as the
nite twoloop remainders Ri de ned in eq. (3.21). The
amplitudes have been renormalised using the scheme described in the previous section and
all scales are xed to the mass of the Higgs boson, 2 = m2H .
The operator O2 only contributes at one loop at NLO, and agrees (up to normalisation)
with the oneloop corrections to Higgs production via gluonfusion [59]. The amplitude is
independent of the top mass through one loop, and so it evaluates to a pure number,
p
A2
(0) =
32 2 2 and R2 = 16 i 3 0 ;
(3.26)
where 0 is de ned in eq. (3.15). The remaining amplitudes have a nontrivial functional
dependence on the top mass through the variables
and
de ned in eq. (3.4) and (3.6).
We have argued in the previous section that in the MSscheme the renormalised amplitudes
A(01) and A(11) are related by a simple rescaling,
We therefore only present results for the SM contribution and the contribution from O3. We
have checked that our result for the twoloop amplitude in the SM agrees with the results
of refs. [27, 28, 30, 58]. The twoloop amplitude A(31) is genuinely new and is presented
The oneloop amplitude in the SM can be cast in the form (3.27) (3.28) (3.29)
(3.30)
(3.31)
A(00) = a0 + (a1 + log a0) + 2 a2 + log a1 + log2 a0 + O( 3) ;
where we have de ned the function
1
2
where the coe cients ai are given by
a0 =
a1 =
a2 =
The nite remainder of the twoloop SM amplitude is
R0 =
0
2
28 Cl2( ) 28 Cl3( )+28 3 +5 2 log +28 2 48 log
252
R( )+
(4
12 p(4
)
)
13 2 +24 log
16 ;
R( ) =
16 2 Cl 2( ) +
28 2 Cl2( )
+ 96 Cl 4( ) 72 Cl4( )
128
3
+ p(4
2
)
where the coe cients bi are given by
2
2
p(4
1
4
;
4 Cl 2( ) + 8 Cl 3( ) + 2
The nite remainder of the twoloop amplitude A(31) is
R3 = i mt 0 1
+ 2 log
+
2 m2t (16 log + 35)
The oneloop amplitude involving the operator O3 is
[ log(4
) 2 Cl 2( )
its inverse, e.g.,
Although the main focus of this paper is to include e ects from dimension six operators
that a ect the gluonfusion cross section through the top quark, let us conclude this section
by making a comment about e ects from the bottom, and to a lesser extent, the charm
quark. The amplitudes presented in this section are only valid if the Higgs boson is lighter
than the quarkpair threshold,
< 4. It is, however, not di cult to analytically continue
our results to the region above threshold where
> 4. Above threshold, the variable x
de ned in eq. (3.5) is no longer a phase, but instead we have
1 < x < 0. As a consequence,
the Clausen functions may develop an imaginary part. In the following we describe how
one can extract the correct imaginary part of the amplitudes in the region above threshold
We start from eq. (3.7) and express all Clausen functions in terms of HPLs in x and
1
2i
Cl2( ) = Im H(0; 1; x) =
HPLs evaluated at 1=x can always be expressed in terms of HPLs in x. For example,
one nds
2
3
;
H(0; 1; 1=x) =
H(0; 1; 1=x) + i H(0; x)
H(0; 0; x) +
jxj = 1 and Im x > 0 :
Similar relations can be derived for all other HPLs in an algorithmic way [89, 95, 96].
The previous equation, however, is not yet valid above threshold, because the logarithms
H(0; x) = log x may develop an imaginary part. Indeed, when crossing the threshold x
approaches the negative real axis from above, x ! x + i0, and so the correct analytic
continuation of the logarithms is
H(0; x) = log x ! H(0; x) + i :
The previous rule is su cient to perform the analytic continuation of all HPLs appearing
in our results. Indeed, it is known that an HPL of the form H(a1; : : : ; ak; x) has a branch
point at x = 0 only if ak = 0, and, using the shu e algebra properties of HPLs [89], any
HPL of the form H(a1; : : : ; ak; 0; x) can be expressed as a linear combination of products of
HPLs such that if their last entry is zero, then all of its entries are zero. The amplitudes can
therefore be expressed in terms of two categories of HPLs: those whose last entry is nonzero
and so do not have a branch point at x = 0, and those of the form H(0; : : : ; 0; x) =
which are continued according to eq. (3.37).
1
n!
logn x,
Using the procedure outlined above, it is possible to easily perform the analytic
continuation of our amplitudes above threshold. The resulting amplitudes contribute to the
gluonfusion process when light quarks, e.g., massive bottom and/or charm quarks, are
taken into account. Hence, although we focus primarily on the e ects from the top quark
in this paper, our results can be easily extended to include e ects from bottom and charm
quarks as well.
(3.36)
(3.37)
3.4
After renormalisation, our amplitudes depend explicitly on the scale , which in the
following we identify with the factorisation scale
F . It can, however, be desirable to choose
di erent scales for the strong coupling constant, the top mass and the e ective couplings.
In this section we derive and solve the renormalisation group equations (RGEs) for these
parameters.
Since we are working in a decoupling scheme for the top mass, the RGEs for the
strong coupling constant and the top mass are identical to the SM with Nf = 5 massless
avours. We have checked that we correctly reproduce the evolution of s and mt in the
MS scheme, and we do not discuss them here any further. For the RGEs satis ed by the
e ective couplings, we nd
dC
d log 2 =
C ;
where C = (C1; C2; C3)T , and the anomalous dimension matrix is given by
= BB0 0
0
B
B
B
B
0 0
0 0
0
0
1
mt( 2) CCCC +
v
1
C
C
A
0
s( 2) BBBB 0 0
C
C
A
23
As already mentioned in the previous section, the double pole from the twoloop
counterterm in eq. (3.24) cancels. We can solve the RGEs in eq. (3.38) to one loop, and we nd
C1( 2) = C1(Q2)
C2( 2) = C2(Q2) + p2 C3(Q2)
16 2 log
Q2
s(Q2)
log
2
Q2
C1(Q2) + 8 C3(Q2) mt2(Q2)
v2
+ O( s(Q2)2) ;
p
2 s(Q2)
192 3 C3(Q2) log
Q2
5 log
2
Q2
69 + O( s(Q2)2) ;
C3( 2) = C3(Q2) + C3(Q2) s(Q2)
6
log
Q2 + O( s(Q2)2) :
We show in gure 2 the quantitative impact of running and mixing by varying the
renormalisation scale from 10 TeV to mH =2 in two scenarios: one where all Wilson coe cients are
equal at 10 TeV and another where only C3 is nonzero. This latter example serves as a
reminder of the need to always consider the e ect of all the relevant operators in
phenomenological analyses as choosing a single operator to be nonzero is a scaledependent choice.
4
4.1
Phenomenology
Crosssection results
In this section we perform a phenomenological study of Higgs production in the SMEFT,
focusing on anomalous contributions coming from the top quark. Results are obtained
2 mt(Q2)
2 mt(Q2)
v
v
2
(3.38)
(3.39)
(3.40)
C
C
C
1
2
3
within the MadGraph5 aMC@NLO framework [54]. The computation builds on the
implementation of the dimensionsix operators presented in ref. [12]. Starting from the
SMEFT Lagrangian, all treelevel and oneloop amplitudes can be obtained automatically
using a series of packages [97{102]. The twoloop amplitudes for the virtual corrections
are implemented in the code through a reweighting method [103, 104]. Within the
MadGraph5 aMC@NLO framework NLO results can be matched to parton shower programs,
such as PYTHIA8 [105] and HERWIG++ [106], through the MC@NLO [107] formalism.
Results are obtained for the LHC at 13 TeV with MMHT2014 LO/NLO PDFs [108],
for LO and NLO results respectively. The values of the input parameters are
mt = 173 GeV ;
mH = 125 GeV ;
mZ = 91:1876 GeV ;
E1W = 127:9 ;
GF = 1:16637
10 5 GeV 2
:
The values for the central scales for
R; F and
EFT are chosen as mH =2, and we work
with the top mass in the onshell scheme.
We parametrise the contribution to the cross section from dimensionsix operators as
= SM + X 1TeV2
Ci i + X 1TeV4
2
4
CiCj ij :
i
i j
Within our setup we can obtain results for SM, i, and ij . We note here that results for
single Higgs and H + j production in the SMEFT were presented at LO in QCD in ref. [12].
The normalisation of the operators used here di ers from the one in ref. [12], but we have
found full agreement between the LO results presented here and those of ref. [12] when
this di erence is taken into account. Furthermore, the SM topquark results obtained here
have been crosschecked with the NLO+PS implementation of aMCSusHi [109].
Our results for the total cross section at the LHC at 13 TeV at LO and NLO are
shown in table 1. We include e ects from bottomquark loops (topbottom interference
and pure bottom contributions) into the SM prediction by using aMCSusHi. However, in
this rst study, we neglect bottomquark e ects from dimensionsix operators in i and ij
C
C
1
2
3
(10 TeV)
HJEP12(07)63
(4.1)
(4.2)
(4.3)
13 TeV
SM
1
2
3
11
22
33
12
13
23
Ki = KU + s i(0) ;
(1)
i
as we assume them to be subleading. As mentioned above, our analytic results and MC
implementation can be extended to also include these e ects. We see that the contributions
from e ective operators have Kfactors that are slightly smaller then their SM counterpart,
with a residual scale dependence that is almost identical to the SM. In the following we
present an argument which explains this observation.
We can describe the total cross
section for Higgs boson production to a good accuracy by taking the limit of an in nitely
heavy top quark, because most of the production happens near threshold. In this e ective
theory where the top quark is integrated out, all contributions from SMEFT operators can
be described by the same contact interaction
Ga Ga H. The Wilson coe cient
can be
written as
= 0 + X Ci i ;
as a perturbative series i =
Kfactor Ki can be decomposed as
where 0 denotes the SM contribution and i those corresponding to each operator Oi
in the SMEFT. As a result each i is generated by the same Feynman diagrams both at
LO and NLO in the in nite topmass EFT. The e ect of radiative corrections is, however,
not entirely universal as NLO corrections to the in nite topmass EFT amplitudes come
both from diagrammatic corrections and corrections to the Wilson coe cients i, which
can be obtained by matching the SMEFT amplitude to the in nite top mass amplitude,
as illustrated in gure 3. Indeed, each i can be expressed in terms of SMEFT parameters
(0) + s i
i
(1) + O( s2). In the in nite top mass EFT, each
where KU is the universal part of the Kfactor, which is exactly equal to K2. By
subtracting K2 to each Ki in the in nite top mass limit numerically (setting mt = 10TeV),
(4.4)
(4.5)
mass EFT at LO (left) and at NLO (right). The NLO amplitude in the in nite topmass EFT
contains two elements: diagrammatic corrections, which contribute universally to the Kfactors
and Wilson coe cient corrections, which are nonuniversal.
we could extract the ratios s ii and check explicitly that these nonuniversal corrections
are subdominant compared to the universal diagrammatic corrections, which explains the
similarity of the e ects of radiative corrections for each contribution.
Our results can be used to put bounds on the Wilson coe cients from measurements
of the gluonfusion signal strength
ggF at the LHC. Whilst here we do not attempt to
perform a rigorous t of the Wilson coe cients, useful information can be extracted by a
simple t. For illustration purposes, we use the recent measurement of the gluonfusion
signal strength in the diphoton channel by the CMS experiment [110]
which we compare to our predictions for this signal strength under the assumption that
the experimental selection e ciency is not changed by BSM e ects
ggF = 1:1
0:19;
ggF = 1 +
C1 1 + C2 2 + C3 3
SM
;
where we set
= 1TeV and kept only the O(1= 2) terms. We therefore nd that we can
put the following constraint on the Wilson coe cients with 95% con dence level:
0:28 <
0:128C1 + 114C2 + 2:28C3 < 0:48:
While the correct method for putting bounds on the parameter space of the SMEFT is
to consider the combined contribution of all relevant operators to a given observable, the
presence of unconstrained linear combinations makes it interesting to consider how each
operator would be bounded if the others were absent in order to obtain an estimate of
the size of each individual Wilson coe cient. Of course such estimates must not be taken
as actual bounds on the Wilson coe cients and should only be considered of illustrative
(4.6)
(4.7)
(4.8)
value. We obtain
most 10%.
3:8 < C1 < 2:2; 0:0025 < C2 < 0:0043; 0:12 < C3 < 0:21 :
(4.9)
For these individual operator constraints, the impact of the ii terms on the limits is at
For reference we note that if one includes the O(1= 4) contributions the linear
combination in the bound becomes a quadratic one:
0:28 <
0:128C1 + 114C2 + 2:28C3 + 0:0038C12 + 3000C22
+ 1:13C32
6:78C1C2
0:138C1C3 + 122C2C3 < 0:48 :
(4.10)
101
e
p
0
0
O2
O3
O
L
N
M
a
_
5
h
p
a
r
G
d
a
M
O
L
N
p
a
r
G
d
a
M
0.12 pμRp=→μF=HμiEnFTth=e62E.F5TGLeHVC13
NLO+PS, MMHT2014NLO
0.10 Interference with the SM
transverse momentum. Right: Higgs rapidity. SM contributions and individual operator
contributions are displayed. Lower panels give the ratio over the SM.
Di erential distributions
In the light of di erential Higgs measurements at the LHC, it is important to examine
the impact of the dimensionsix operators on the Higgs pT spectrum. It is known that
measurements of the Higgs pT spectrum can be used to lift the degeneracy between O1 and
O2 [12, 45, 111]. For a realistic description of the pT spectrum, we match our NLO
predictions to the parton shower with the MC@NLO method [107], and we use PYTHIA8 [105]
for the parton shower. Note that we have kept the shower scale at its default value in
MC@NLO, which gives results that are in good agreement with the optimised scale choice
of ref. [112], as discussed in ref. [109].
The normalised distributions for the transverse momentum and rapidity of the Higgs
boson are shown in gures 4 for the interference contributions. The impact of the O(1= 4
)
terms is demonstrated in gure 5 for the transverse momentum distribution. We nd that
the operators O3 and O2 give rise to harder transverse momentum tails, while for O1 the
shape is identical to the SM. The dimensionsix operators have no impact on the shape of
the rapidity distribution. The O(1= 4) contributions involving O3 and O2 are harder than
those involving O1.
Finally we show the transverse momentum distributions for several benchmark points
which respect the total crosssection bounds in
gure 6. The operator coe cients are
chosen such that eq. (4.10) is satis ed. We
nd that large deviations can be seen in the
tails of the distributions for coe cient values which respect the total crosssection bounds.
4.3
Renormalisation group e ects
The impact of running and mixing between the operators is demonstrated in gure 7,
where we show the individual (O(1= 2)) contributions from the three operators in
gluonfusion Higgs production at LO and NLO, as a function of EFT, assuming that C3 = 1,
C1 = C2 = 0 at EFT = mH =2 and
= 1 TeV. While at
= mH =2 the only contribution
101
e
p
σ
M 2
S
O2O3
O
L
L
N
N
_
5
h
h
p
p
a
a
r
r
G
G
d
d
a
a
M
M
101
ibn102
/
σ
d
σ
/
1
103
102
101
] 100
b
p
[
n
i
b
r
e
pσ101
102
103
2
M
tehS 1.5
r
veo 1
o
i
taR 0.5
0
0
50
ii. Right: interference between operators, ij. SM contributions and operator contributions are
displayed. Lower panels give the ratio over the SM.
Higgs pT
SM
B5 (5,0.004,0.2)
B4 (2,0.006,0.3)
B3 (1,0.004,0.25)
B2 (4,0.005,0.5)
B1 (4,0.004,0.21)
p p → H in the EFT LHC13
μR=μF=μEFT=62.5 GeV
NLO+PS, MMHT2014NLO
dashed: only 1/Λ2 terms, solid: including 1/Λ4 terms
O
L
5
h
p
a
r
G
d
a
M
200pTH [GeV]
250
50
100
150
300
350
400
450
coe cients. The lower panel shows the ratio over the SM prediction for the various benchmarks
and the SM scale variation band.
is coming from the chromomagnetic operator, this contribution changes rapidly with the
scale. While the e ect of the running of C3 is only at the percent level, 3 has a strong
dependence on the scale. At the same time nonzero values of C1 and C2 are induced
through renormalisation group running, which gives rise to large contributions from O2. We
nd that the dependence on the EFT scale is tamed when the sum of the three contributions
200
150
100
50
0
−50
−100
−150
pp → H LHC13
solid: NLO, dashed: LO
100
μ EF T [GeV]
1000
.
at the LHC at 13 TeV as a function of the EFT scale. Starting from one nonzero coe cient
at
EFT = mH =2 we compute the EFT contributions at di erent scales, taking into account the
running and mixing of the operators. LO and NLO predictions are shown in dashed and solid lines
respectively.
is considered. This is the physical cross section coming from C3(mH =2) = 1 which has a
weaker dependence on the EFT scale. The dependence of this quantity on the scale gives an
estimation of the higher order corrections to the e ective operators and should be reported
as an additional uncertainty of the predictions. By comparing the total contributions at
LO and NLO we nd that the relative uncertainty is reduced at NLO.
5
Conclusion and outlook
A precise determination of the properties of the Higgs boson and, in particular, of its
couplings to the other SM particles is one of the main goals of the LHC programme of the
coming years. The interpretation of such measurements, and of possible deviations in the
context of an EFT, allows one to put constraints on the type and strength of hypothetical
new interactions, and therefore on the scale of new physics, in a modelindependent way.
The success of this endeavour will critically depend on having theoretical predictions that
at least match the precision of the experimental measurements, both in the SM and in
the SMEFT.
In this work we have computed for the rst time the contribution of the (CP even
part of the) QL
qRG operator to the inclusive Higgs production at NLO in QCD. Since
the NLO corrections for the other two (CP even) operators entering the same process
are available in the literature, this calculation completes the SMEFT predictions for this
process at the NLO accuracy. Even though our results can be easily extended to include
anomalous couplings of the bottom quark, we have considered in the detail the case where
new physics mostly a ects the topquark couplings. Our results con rm the expectations
based on previous calculations and on the general features of gluonfusion Higgs production:
at the inclusive level the Kfactor is of the same order as that of the SM and of the other
two operators. The residual uncertainties estimated by renormalisation and factorisation
scale dependence also match extremely well. The result of the NLO calculation con rms
that the chromomagnetic operator cannot be neglected for at least two reasons. The rst
is of purely theoretical nature: the individual e ects of QL ~ tRG and
y GG are very
much dependent on the EFT scale, while their sum is stable and only mildly a ected by the
scale choice. The second draws from the present status of the constraints. Considering the
uncertainties in inclusive Higgs production cross section measurements and the constraints
from tt production, the impact of the chromomagnetic operator cannot be neglected in
global ts of the Higgs couplings. As a result, a twofold degeneracy is left unresolved by
a threeoperator t using the total Higgs cross section and one is forced to look for other
observables or processes to constrain all three of the operators.
The implementation of the nite part of the twoloop virtual corrections into
MadGraph5 aMC@NLO has also allowed us to study the process at a fully di erential level,
including the e ects of the parton shower resummation and in particular to compare the
transverse momentum distributions of the SM and the three operators in the region of the
parameter space where the total cross section bound is respected. Once again, we have
found that the contributions from QL ~ tRG and
y GG are similar and produce a shape
with a harder tail substantially di erent from that of the SM and the Yukawa operator
(which are the same).
While QL ~ tRG and
y GG cannot really be distinguished in
gluonfusion Higgs production, they do contribute in a very di erent way to ttH where the
e ect of y GG is extremely weak. Therefore, we expect that H; H+jet, and ttH (and
possibly tt) can e ectively constrain the set of the three operators.
In this work we have mostly focused our attention on the topquarkHiggs boson
interactions and only considered CP even operators. As mentioned above and explained
in section 3, extending it to include anomalous couplings for lighter quarks, the bottom
and possibly the charm, is straightforward. On the other hand, extending it to include
CP odd operators requires a new independent calculation. We reckon both developments
worth pursuing.
Acknowledgments
This work was supported in part by the ERC grant \MathAm", by the FNRSIISN
convention \Fundamental Interactions" FNRSIISN 4.4517.08, and by the European Union
Marie Curie Innovative Training Network MCnetITN3 722104. E.V. is supported by the
research programme of the Foundation for Fundamental Research on Matter (FOM), which
is part of the Netherlands Organisation for Scienti c Research (NWO). ND acknowledges
the hospitality of the CERN TH department while this work was carried out.
Open Access.
This article is distributed under the terms of the Creative Commons Attribution License (CCBY 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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