Diboson production in the SMEFT from gluon fusion

Precision measurements of diboson production at the LHC is an important probe of the limits of the Standard Model. The gluon-fusion channel of this process offers a connection between the Higgs and top sectors. We study in a systematic way gluon-induced diboson production in the Standard Model Effective Field Theory. We compute the amplitudes of double Higgs, double Z/W and associated ZH production at one loop and with up to one insertion of a dimension-6 operator. We study their high-energy limit and identify to which operators each channel could be most sensitive. To illustrate the relevance of these processes, we perform a phenomenological study of associated ZH production. We show that for some top operators the gluon-induced channel can offer competitive sensitivity to constraints obtained from top quark production processes.


Introduction
The main goal of collider phenomenology at this time is to assimilate the lessons from the early runs of the Large Hadron Collider (LHC) and set the stage to fully exploit the data to be collected during Run 3 and the High-Luminosity LHC (HL-LHC) era.In light of the absence of new resonances, the search for Beyond the Standard Model (BSM) Physics at the LHC is moving towards the indirect precision measurement paradigm.
The first manifestation of a new particle at energies beyond our direct reach would likely be a rise of the high-energy tails in differential distributions with respect to the Standard Model (SM) prediction.Hence, the precise measurement of such distributions represents an excellent opportunity to test our current best description of Nature and understand how much space is left for New Physics (NP).Not only this could yield powerful insights into the physics that awaits us at higher energies, but also will constitute a key piece in the legacy of the (HL-)LHC program.
Assuming that all BSM particles have a mass larger than the energies we can probe directly, calls for the use of an Effective Field Theory (EFT) in order to parametrise their effects.The Standard Model Effective Field Theory (SMEFT) offers a powerful and systematic method to parametrise deviations from the Standard Model and interpret (HL-)LHC data.It is built from the SM, respecting its field content, gauge symmetries and Electroweak Symmetry Breaking (EWSB) pattern, and adding higher-dimensional operators.As with any other EFT, it is only valid up to a cutoff energy scale Λ.
Formally, the SMEFT Lagrangian is an infinite series in the energy dimension of the operators, where each operator O (d) k has energy dimension (d), c k is an adimensional coupling that we call Wilson coefficient (WC), and the series converges as long as the typical energy of the process obeys E/Λ 1. Normalising the WCs in this way makes explicit how higher-order operator contributions are further suppressed by the cutoff scale.The leading deviations with respect to the SM of relevance for collider physics are generated by operators of dimension 6, on which we will focus.
Not every process is well-suited to constrain SMEFT WCs via precision measurements.Those in which amplitudes with one dimension-6 operator insertion interfere with the SM and generate growth with energy with respect to the SM cross-section are particularly promising.If the process has also low background levels, the high-energy tails of differential distributions could give stringent constraints on the value of the WCs.Drell-Yan [1][2][3][4][5][6] and diboson production  have become the two quintessential processes in this program thanks to combining both features with an interesting breadth of interactions to probe.
Diboson, i.e.V V ( ), V H, and HH production1 , allows the study of several EW couplings strictly determined by the SM gauge symmetry.
At the LHC, V V ( ) and V H production are largely dominated by the quark-initiated channel whilst HH production is induced by top quark loops.The tree-level quark initiated contributions to V V ( ) and V H are typically studied to provide information on the light quark couplings to the gauge bosons, the gauge boson self-interaction and the couplings of the Higgs to the gauge bosons.
The gap between V V ( ) and V H production and the top sector is bridged by the gluon-fusion channel of diboson production.Though limited to neutral final states and suppressed due to its loop-induced nature, gluon fusion can still give a relevant contribution to W W , ZZ and ZH production at the LHC in the high-energy regime [33][34][35][36][37].In all cases, the cross-section is dominated by top-quark-loop diagrams, sometimes interacting directly with a virtual Higgs boson.Furthermore, the different energy behaviour of the quark-and gluon-initiated channels offers the opportunity to amplify the contribution of the latter by probing particular kinematic regions.
This link to the top sector is of particular interest since the top quark is another candidate to show signs of BSM physics at colliders.Already in the SM, the top sector is closely connected to the Higgs/EW one via the near-unity top Yukawa coupling, and the top EW gauge couplings.Furthermore, several classes of BSM models tend to relate closely the BSM fate of the top quark, the Higgs boson and the EW gauge bosons.As a reflection of this, the effort to perform combined fits of the EW, Higgs and top sectors has increased remarkably in recent years [38,39].
In light of this connection to top physics, several groups have studied gluon-initiated diboson production before and stressed its potential to constrain top couplings [40][41][42][43][44].To fully explore the potential of this class of processes we aim to systematically analyse the high-energy behaviour of the gluon-initiated diboson production amplitudes generated by dimension-6 SMEFT operators.Such a study will enable us to identify which operators are more promising, especially in the context of high-energy measurements at the LHC, HL-LHC and future colliders.Analyses with the same spirit for quark-initiated diboson and t t production are available in the literature [13,45], as well as studies within the on-shell scattering amplitudes formalism [46,47].
To then demonstrate in a realistic manner how the loop induced component of the diboson process plays a special role, we focus on ZH production.Previous studies of this process focused on one of the initial states, either gg or q q, and set the other to its SM value to play the role of a background [24,31,32,42].This methodology can be justified by the fact that quark-and gluon-initiated ZH production probe disjoint sets of SMEFT dimension-6 operators.When those sets are linked by a flavour hypothesis such as Flavour Universality, the sensitivity arises mainly from the quark-initiated process due to a higher cross-section and steeper energy growth.
However, the progress towards global fits of experimental data and more general flavour assumptions calls for a joint study of the different initial-state channels.In this work, we perform for the first time such joint phenomenological analysis of both the quark and gluon-induced production channels.
This paper is organised as follows.In Section 2, we compute analytically the one-loop amplitudes of gg → HH, gg → ZH, and gg → ZZ/W W in the SMEFT with up to one insertion of dimension-6 operators and study their energy behaviour.Then, in Section 3, we extend an analysis of q q → ZH at (HL-)LHC going beyond Flavour Universality and properly accounting for the contribution of the gluon-initiated channel.This allows us to explore the prospects of ZH in probing particular dimension-6 operators.In this way, we perform a truly comprehensive study of the (HL-)LHC reach of pp → ZH in the SMEFT framework.
2 Energy-growing helicity amplitudes We use the Warsaw basis of dimension-6 operators along with a U(2) q ×U(3 3 flavour assumption and the operator definitions from the SMEFTatNLO model [48,49].The operators relevant to gluon-initiated diboson production are presented in Table 1.We note here that we do not discuss operators which enter only through the modification of the input parameters as these will not lead to energy growth.The lefthanded quark doublets are denoted by q i for the first two generations and by Q for the third-generation, while t denotes a right-handed top-quark field and u i , d i the right-handed u, c and d, s, b fields respectively.All the quarks are considered to be massless except for the top.The Higgs doublet is given by ϕ with vacuum expectation value v/ √ 2. We define

Purely bosonic
where the covariant derivative is given by: W µ , B µ are the gauge bosons fields, g and g are the SU (2) L and U (1) Y couplings respectively and τ I are the Pauli sigma matrices.Furthermore G A µν , W µν and B µν stand for the SU (3) C , SU (2) L and U (1) Y field strength tensors.T A = 1 2 λ A are the SU (3) generators where λ A are the Gell-Mann matrices.The strong coupling constant is denoted by g s , θ W represents the weak mixing angle and In the rest of this section, we present the leading high-energy behaviour of the gg → HH, ZH, ZZ, W W amplitudes in the SM and with the insertion of one dimension-6 SMEFT operator.We analyse separately each possible helicity configuration of the initial and final state particles.The analytical expression for the full amplitude of each process was obtained in Wolfram Mathematica 12.3 using the FeynCalc [50][51][52], FeynHelpers [53], Package-X [54] and FeynArts [55] packages.
We only consider leading order diagrams, which are one-loop QCD diagrams in all cases except for the insertion of O ϕG , which introduces a contact term between the Higgs boson and the gluons and hence generates tree-level diagrams for gg → HH, ZZ, W W .All the loops considered here are finite except when O tG enters in HH, ZZ and W W production.The UV divergence can be reabsorbed in the renormalisation of O ϕG , as discussed in [56][57][58][59][60][61].In the M S scheme, c ϕG is renormalised as: where µ EF T denotes the renormalisation scale of the EFT and the counterterm is given by: In this work we consider µ = µ EF T .
The helicity amplitudes have been obtained using a modified version of HelCalc [45], where the polarisation vectors are defined in the following way: In the above, ε ± µ denotes the transverse polarisation vectors and ε 0 µ stands for the longitudinal one, where p is the 4-momentum of a particle, p, M and E are its 3-momentum, mass and energy respectively, and θ is the angle of the momentum with respect to the beam axis.Finally, as we are interested in the high-energy limit we performed a series expansion in energy to find the leading contribution to the helicity amplitudes in the high √ s limit.In this expansion we also consider the mass of the final state bosons to be significantly smaller than the centre of mass energy.
We validated our results by comparing the numerical values obtained from our analytical predictions with the numerical predictions given by the SMEFTatNLO model in Madgraph5 aMC@NLO v3.4.The analytical expressions for the full helicity amplitudes match with Madgraph.We note that for each amplitude we provide only the leading term(s) in the energy expansion.The level of agreement between this leading term and the full amplitude depends on the process and operator, with operators that rescale the SM and s-channel contributions tending to the high-energy limit fast.For these amplitudes the leading term provides an estimate to the total amplitude to about 10% at around 5 TeV.Convergence is slower for the top dipole operators and when the leading term is a constant.Often subleading terms can be numerically significant even at energies around 10 TeV depending on the kinematic configuration.In particular this is the case for O tG in ZH, ZZ and W W production and for O ϕG in gg → ZH.For conciseness, we report only the leading term(s) as this gives a reasonably reliable expectation for the tails of the distributions at the HL-LHC and FCC-hh.
When the amplitude behaves at high energies like s α log 2 ( s ), with α a semi-integer, we found that the leading term in the high-energy regime can be computed by applying the method of regions [62,63].The squared logarithm arises from phase-space regions in which ones of the loop lines becomes soft and collinear, extracted by expanding the loop traces in the corresponding region.This configuration leads to a scaleless integral which can be evaluated by neglecting the top masses in the denominator and computing the appropriate residues.The integral is regulated in the IR and UV regions by m t and √ s respectively and yields log 2 ( s ).This technique applies to both triangle and box diagrams, with either an SM-like Lorentz structure or one insertion of a dipole operator.For loops with planar topology, this procedure is analogous to the s-channel unitarity cut employed in Ref. [33,34], but it can be extended to non-planar topologies allowing to reproduce the results in a wider variety of helicity configurations.Non-planar diagrams are key in reproducing the right behaviour when one dipole operator is inserted.We note here that the simple s-channel unitarity cut argument is not enough to link the behaviour of the loop-induced gg → V V amplitude to the one of the underlying t t → V V , when there are contributions from the non-planar topologies as is the case in several of the amplitudes that we compute.We have cross-checked several of our results using this method.For clarity we show the detailed computation for an example in Appendix A.

gg → HH
The first process we consider is double Higgs production.This process is of particular interest as it constitutes the simplest process which can probe the Higgs self coupling.From our perspective it is the simplest of the processes we study as it involves two scalars in the final state.For the CP even operators we consider, flipping all ± ↔ ∓ in a helicity combination does not modify the helicity amplitude.This, combined with Bose symmetry gives two independent helicity configurations for gg → HH.These two configurations correspond to s-wave and d-wave contributions, with the boxes contributing to both s-and d-wave and triangles only to s-wave amplitudes.Relevant diagram topologies with possible insertions of SMEFT operators are shown in Fig. 1.
The leading behaviour in the high energy limit for the different helicity amplitudes is presented in Table 2 and the results are consistent with [40].For the SM, we give the schematic energy behaviour of all independent helicity configurations.For the SMEFT operators we give the full analytical expression of the leading term up to a phase.As we are interested in studying amplitudes which grow in the high energy limit, in this section only the growing SMEFT helicity amplitudes are shown.The results for the constant and decaying amplitudes of gg → HH, ZH, ZZ can be found in Appendix B. We note here that O ϕ , which enters in the triple Higgs vertex, does not lead to any growth with energy as it simply rescales the SM triangle amplitude.We start by commenting on the Yukawa operator, O tϕ , which leads to a growth in energy when the two incoming gluons have the same helicity and tends to a constant amplitude otherwise.This is because O tϕ rescales the t tH vertex and introduces a new t tHH vertex.This fact combined with the presence of both triangles and boxes which scale differently with the modification of the Yukawa coupling leads to a non-trivial impact of this operator on HH.This is in contrast with single Higgs production where the Yukawa operator acts as a simple rescaling of the SM prediction.It is instructive to note that for the SM, the box diagrams tend to a constant at high energy while the triangle ones decrease for same helicities and do not contribute for opposite helicities [64,65].The t tHH interaction leads to a new triangle diagram without a Higgs propagator shown in the bottom left of Fig. 1.The absence of the s-channel propagator and of the corresponding 1/s suppression leads to a logarithmic growth.
High energy behaviour of the gg → HH helicity amplitudes in the SM and modified by SMEFT operators.The " − " and " " denote when a helicity amplitude is not growing or is equal to 0 respectively.λ g1 , λ g2 , λ H1 , λ H2 represent the polarisation of the two incoming gluons and the two outgoing Higgs bosons and cθ stands for the cosine of the collision angle in the centre of mass frame.We also keep implicit the overall colour factor δ ab , where a, b are the colours of the incoming gluons, as well as the overall dependence on the WCs and Λ 2 .
The purely bosonic operator O dϕ shifts the kinetic term for the Higgs field and the canonical form is restored through the following Higgs field redefinition [49,66]: As a result of the field redefinition, O dϕ shifts all Higgs couplings to fermion and gauge bosons in the same way.The HHH interaction receives both a rescaling and an additional momentum dependent correction.This additional correction has two powers of momentum which, in the (+ + 0 0) helicity configuration, cancel out the 1/s dependence of the triangle diagrams, leading to a logarithmic growth of the amplitude.It should be noted that an alternative Higgs field redefinition can be performed [67] in which the Higgs self coupling does not have any momentum dependence.However in this different redefinition, O dϕ introduces a t tHH vertex as shown in the bottom-left of Fig. 1.Since this diagram does not have the 1/s dependence of the Higgs propagator, it can grow logarithmically as discussed above, reproducing the energy growth that we observe for O dϕ with the field redefinition from Eq. (2.7).
For both O tϕ and O dϕ , it is worth noting that the energy growth observed in the gg → HH process is related to the energy growth in the underlying t t → HH process.The sub-process amplitudes for the opposite top helicities grow linearly with energy due to the presence of the contact t tHH interaction, see for example Ref. [45] for the crossed tH → tH process.In this case the leading contributions come from triangle diagrams, by definition planar, and the result is consistent with the s-channel unitarity expectation, confirmed also by our calculation of the corresponding loops in the soft and collinear limits.
The chromomagnetic operator O tG , which alters the t tg vertex and introduces a t tgH vertex, induces a growth with energy for both helicity configurations.The quadratic growth in the center of mass energy is due to the box diagrams with a modified t tg vertex and the triangle diagrams with a t tgH vertex.The modified vertex, in both cases adds a power of momentum and a different Lorentz structure than in the SM.It should be noted that we have kept µ EF T explicit in our expressions.For the reasonable choice of µ EF T = √ s the logarithmic term vanishes and both helicity amplitudes grow quadratically with the energy.
A direct coupling between the Higgs boson and the gluons is introduced by O ϕG and gg → HH becomes a tree-level process with either a ggH or a ggHH vertex.In the (+ + 0 0) helicity configuration the 3-point vertices diagram goes to a constant in the high energy limit as the 1 s behaviour of the Higgs propagator is cancelled by the two powers of momentum in the ggH vertex.Thus the 4-point vertex diagram which does not have a propagator grows linearly with s.The behaviour of the O ϕG amplitudes is identical to that expected from the infinite top mass limit of the SM amplitudes.The different energy behaviour compared to the SM and the large energy growth observed demonstrate once more why such an approximation is not appropriate for the double Higgs process, in contrast with single Higgs production.
To conclude our discussion of double Higgs production, we note that as the SM amplitudes are constant for both helicity configurations, the interference between the SM and the growing SMEFT amplitudes will also grow in the high-energy limit.Using the same symmetries as in the previous subsection, we find that there are 5 independent helicity configurations for gg → ZH.The high energy behaviour of the helicity amplitudes is presented for the SM and in the presence of SMEFT operators in Tables 3 and 4. For clarity the tables only include the helicity configurations which lead to a growth for at least one of the operators considered.The amplitudes are given in terms of the vector and axial-vector parts of the SM t tZ vertex which can be expressed as:

gg → ZH
where ϕQ .Representative diagrams for this process are shown in Fig. 2. The weak dipole operator O tZ , which enters in the t tZ vertex, does not contribute to gg → ZH due to charge conjugation invariance [51].Interestingly O ϕW , O ϕB and O ϕW B , do not enter either despite modifying the ZZH vertex and introducing a γZH one.The ggZ loop function, combined with the momentum dependent ZZH interaction contracted with the gauge boson propagator and the external Z polarisation vector lead to a vanishing amplitude.This is in contrast with the tree-level q q → ZH amplitudes in the presence of the Higgs-gauge operators which not only do not vanish but can also lead to energy growing amplitudes for both massless and massive quarks [13,27,45].
Finally, we note here that even though massless quarks can potentially enter in the SM gg → ZH process, in particular in the triangle loops with a Z propagator, the currentcurrent operators modifying light quark couplings do not affect this process.In fact those operators enter both in the triangle diagram with a Z propagator and in the diagram with a q qZH vertex and those two diagrams cancel each other out as we will also discuss in the following.
: High energy behaviour of the gg → ZH helicity amplitudes in the SM and with modified top-gluon and Higgs-gluon interactions.The cosine and the sine of the weak angle are represented by c w and s w respectively.For readability we have defined The t tZ vertex can be modified by O ϕt and O (−) ϕQ , which rescale the SM couplings of the Z boson to the right-handed top singlet and the left-handed third generation doublet respectively.Only the axial vector coupling of the top to the Z enters in this process, and therefore the modification of this coupling is what determines the impact of these operators on the amplitudes.These two operators also introduce a t tZH vertex as well as a b bZH one from O (−)  ϕQ .The modified triangle diagrams with a Z propagator and with a t tZH vertex cancel each other exactly, invalidating naive expectations from tree-level t t → ZH [41].More precisely in q q → ZH the longitudinal part of the Z propagator vanishes for external massless quarks leading to energy-growing amplitudes, while its contraction with the oneloop form factor gives a contribution that cancels the ones of the transverse propagator and the t tZH vertex.
Given this cancellation, the behaviour of O ϕt and O (−) ϕQ amplitudes can be simply understood from the SM box diagrams with a rescaled t tZ interaction.Boxes grow log-arithmically in the (+ + 0 0) helicity configuration and decrease in all other cases.Their growth is not observed in the SM due to the logarithmic terms being exactly cancelled by the triangle diagrams.Both operators therefore lead to a logarithmic growth with energy when the two incoming gluons have the same polarisation and the Z boson is longitudinally polarised.
Another consequence of the cancellation between triangle diagrams with O ϕt and O (−) ϕQ is that they generate the same behaviour as the Yukawa operator O tϕ .The latter can only enter in box diagrams with a rescaled t tH interaction.Hence, gg → ZH is only sensitive to the linear combination c (−) ϕQ − c ϕt + ctϕ yt .We will discuss this degeneracy further in our phenomenological analysis of this process in Section 3.
The chromomagnetic dipole operator O tG leads to a growth in energy for all the helicity configurations, which is due to the new t tgH vertex and the modified Lorentz structure of the t tg one.The leading growth happens for (+ − 0 0) which grows quadratically.In (+ + 0 0) the quadratic growth of the box, triangles with a Z propagator, and t tgH vertex diagrams cancel each other out such that the helicity amplitude grows logarithmically.We note here that amplitudes for this operator are typically more complex functions of the scattering angle, due to the different possibilities of this operator entering the Feynman diagrams.
We conclude our discussion of gg → ZH by briefly mentioning O ϕG .The process remains loop induced as shown in the right-most diagram of Fig. 2 and its amplitude grows when the Z is longitudinally polarised.Finally it should be noted that only O tG and O ϕG lead to growing interferences with the SM: this is the case in the (+ + 0 0) helicity configuration for both operators, and additionally in the (+ + 0 +) and (+ − 0 +) configurations for O tG .We now turn our attention to ZZ production.There are 36 possible helicity combinations for gg → ZZ, but using the Bose symmetry of the initial state gluons and final state Zs and the fact that all operators considered are CP-even leads to 10 independent helicity combinations. 2.
Table 5: High energy behaviour of the gg → ZZ helicity amplitudes in the SM and in the presence of top dipole operators.λ g1 , λ g2 , λ Z1 , λ Z2 represent the polarisations of the two incoming gluons and the two outgoing Z bosons respectively.For readability we define  ϕQ also modifies the Zb L bL vertex.All three operators induce a growth with energy in the (+ + 0 0) helicity configuration.This is because in the SM, the top boxes and top triangles each either tend to a constant or decrease with energy except in the (+ + 0 0) configuration where they each grow logarithmically [33,34,41,44].Those growths cancel each other out such that the full SM (+ + 0 0) amplitude tends to a constant.However O ϕt , O (−) ϕQ only enter in the box diagrams and O tϕ only enters in the triangle ones, thus the logarithmic growths are not cancelled by any other diagrams.In this case our results obtained by expanding the loop calculation in the soft and collinear region confirm that only planar topologies end up contributing in the high-energy region (in particular the region where the top propagator between the initial state gluons becomes soft) and the s-channel unitarity cut argument applies.
From Table 6 we notice that there is a degeneracy in the high-energy behaviour of these three operators.We note though that this degeneracy is only present in the leading term, and subleading terms differ for these operators.In passing we also comment on the corresponding light quark operator amplitudes (O (3)  ϕQ , O (3) ϕq i , O ϕu , O ϕd ) that tend to at most a constant in the high energy limit.As such, their contributions are suppressed compared to those of the top operators in the high-energy region that we are interested in.
The top chromomagnetic operator, O tG , generates helicity amplitudes with the largest growths in (+ + 0 0) and (+ − 0 0), which rise quadratically with the energy.As expected, the amplitudes which involve a Higgs propagator depend on µ EF T after the renormalisation of the corresponding UV divergence.Those diagrams enter in the (+ + ++), (+ + −−) and (+ + 0 0) amplitudes, however in the (+ + −−) configuration the leading term (log 2 s m 2 t ) comes from the box diagrams and thus does not depend on µ EF T .While most helicity configurations grow with energy, the dominant growths happen when the two Z bosons are longitudinal.
The weak dipole operator O tZ modifies the t tZ vertex by adding a power of momentum and changing its Lorentz structure compared to the SM boxes.The different gamma matrix structure of the loop leads to a logarithmic growth in the (+ + +−), (+ + −−) and (+ − −−) helicity configurations.For this operator the amplitude is proportional to the vector coupling of the top to the Z and interestingly the leading growth arises for transverse Z bosons rather than longitudinal ones.This is consistent also with observations made in the tree level t t → ZZ amplitudes, where no energy growth is observed in purely longitudinal boson states [45].The analytic computation with the method of regions, introduced in Subsection 2.1, shows that the growth for the (+++−) and (++−−) helicity configurations is generated exclusively by box diagrams with planar topology, while the (+ − −−) amplitude receives logarithmic contributions only from non-planar boxes.Both planar and non-planar diagrams generate logarithmically-growing (+ + 00) amplitudes, however, each set of diagrams cancels exactly the other one.Furthermore, this computation explains the relative factor of three in front of the log 2 between the (+++−) and (++−−) configurations: in the first one, the squared logarithm arises only when the line between the gluons is soft meanwhile, in the second growth is generated when any of the lines between gluons or between one gluon and one Z become soft.
The gauge operators O ϕB and O ϕW modify the ZZH vertex and enter in the same three helicity configurations as the SM s-channel diagrams.In the SM, the ZZH vertex contracted with the Z bosons polarisation vectors goes to a constant when the Z are transverse and grows ∝ s when they are longitudinal.However O ϕB and O ϕW change the Lorentz structure of the ZZH vertex such that its contraction with the Z polarisation vectors grows ∝ s when the Z are transverse and tends to a constant when they are longitudinal.Adding the 1 s and log 2 s m 2 t dependence of the Higgs propagator and the top loop respectively, leads to the observed logarithmic growth of the triangle diagrams in the (+ + ++) and (+ + −−) configurations, similarly to what is observed in the underlying t t → ZZ process.
A direct coupling between the gluons and the Higgs boson is induced by O ϕG and gg → ZZ becomes a tree level process which, like the SM triangle diagrams, is only nonzero in the (+ + ++), (+ + −−) and (+ + 0 0) helicity configurations.The ggH vertex has two powers of momentum which cancel the 1/s coming from the Higgs propagator such that the energy behaviour of the diagram is determined by the contraction of the ZZH vertex with the Z polarisation vectors.This contraction tends to a constant when the Z bosons are transverse and to a quadratic growth when the Z bosons are longitudinal, leading to the observed amplitude growth [41].
For this process, almost all the SMEFT growing helicity amplitudes lead to a growing interference with the SM.The only exceptions are for O tG in the (+ + − 0), (+ − −+) and (+ + ++) configurations where the growth of the SMEFT amplitude is not enough to overcome the suppression of the SM amplitude.As in gg → ZZ there are 36 possible helicity combinations for gg → W + W − and the Bose symmetry of the initial state and CP invariance lead to 15 independent helicity configurations.For this process we only focus on the operators which lead to growing amplitudes.This is the only process considered which probes the tbW interaction and thus is sensitive to the O (3)  ϕQ and O tW operators.Example diagrams are shown in Fig. 4 and the schematic high energy behaviour of the SM amplitudes and the SMEFT growing helicity amplitudes are presented in Tables 8 and 9. Some of the O tG results are given up to a function of cθ, g 1/2 (cθ), which we have extracted numerically.
All the operators considered in this subsection lead to similar energy behaviours in gg → W W and gg → ZZ.Similarly to ZZ production, all the SMEFT growing helicity amplitudes lead to a growing interference with the SM except for some O tG amplitudes.Most W W helicity amplitudes modified by O tG grow with energy and the leading growths are the quadratic ones of the (+ + 0 0) and (+ − 0 0) configurations, for which both the box diagrams and the diagrams with a H propagator grow ∝ s.As in ZZ production, the diagrams with a Higgs propagator only enter in the (++++), (++−−) and (++0 0) helicity configurations and are UV divergent.Hence those three amplitudes depend on µ EF T , but the renormalisation scale is not part of the leading energy behaviour for (+ + −−).Next, O tW affects the tbW vertex by adding a power of momentum and modifying its Lorentz structure compared to the SM boxes and it is therefore similar to O tZ .Hence O tW has a similar energy behaviour as O tZ and it grows logarithmically for the (+ + −−), (+ + −+) and (+ − ++) helicity configurations.
O (3)  ϕQ and O tϕ rescale the SM tbW and t tH and vertices respectively and both lead to a logarithmic growth when the two W bosons are longitudinally polarised.As in the previous processes this can be understood from the SM diagrams: in the (+ + 0 0) helicity configuration the box and triangle diagrams each grow logarithmically such that they cancel each other out and the overall (+ + 0 0) amplitude tends to a constant [34,71].As O tϕ only enters in the triangle diagrams, there is no other diagram to cancel the logarithmic growth.The same applies for O (3)  ϕQ which only enters in the box diagrams.Finally O ϕG and O ϕW , which induce a ggH vertex and modify the W W H one respectively, only enter when both gluons and both W bosons have the same helicity.The behaviour of the amplitudes is determined by the contraction of the W W H vertex with the W polarisation vectors and the discussion from Subsection 2.4 applies here as well.

A probe of top couplings at (HL-)LHC
Some of the growths discussed in the previous section can be observed in differential distributions of the relevant diboson processes.Typically the energy growing amplitudes lead to harder tails of differential distributions, dominated by the dimension-6 squared contributions.The impact of these is a larger deviation from the SM in the higher energy regions compared to the threshold, even when considering energies as low as √ s = 400 − 1000 GeV as studied for example for gg → ZZ, W W in [72].In some cases top operators lead to harder distributions also at the interference level, one such example being O tG which induces a growth for several helicity configurations of all processes considered here.
The energy growths at the amplitude level are thus particularly interesting as they could, under certain circumstances, offer a handle to improve our sensitivity and probe operators otherwise poorly constrained, as hinted in previous studies [41][42][43].Whilst similar sensitivity studies can be performed for all processes discussed above, we focus on gg → ZH to explore the potential impact of these growing amplitudes in probing top couplings in a realistic analysis.
This section is devoted to revisiting the possibility of using gg → ZH at HL-LHC to improve the constraints on dimension-6 SMEFT WCs, in particular those related to the top quark.Under the flavour symmetry U(2) q ×U(3) d ×U(2) u × (U(1) × U(1) e )3 , this process is sensitive to the operators O (−) ϕQ , O ϕt , O tϕ , O tG , and O ϕG , as shown before.We neglect the effect of the last 2 since they are stringently constrained by current measurements 3  For convenience, during the rest of this work, we will employ dimensionful WCs obtained by absorbing the 1/Λ 2 factor into their definition.All the results presented in this section include the cross-section terms that are quadratic on the WCs unless stated otherwise.
To assess the sensitivity of this process to the aforementioned operators, we add the gluon-initiated signal contribution to an analysis originally focused on quark-initiated diboson production [32].This addition is meaningful only after relaxing the flavour assumption in said analysis from its original Flavour Universality to the one mentioned in the previous paragraph.Thus, our results are fully compatible with the SMEFiT fit [38,49].Moreover, this allows us to study the flavour-assumption dependence of the light-quark operator bounds.
We discuss the general analysis strategy and how we simulated the collider events in Subsection 3.1.Then, in Subsection 3.2, we ponder the relevance of higher-order QCD corrections for gluon-initiated ZH production and explain how we accounted for them.The latter is crucial since it determines the interplay between the different initial state channels, which we discuss in Subsection 3.3.There, we also explain how the quark-initial state process is affected by the change in flavour assumptions.The projected bounds on the WCs from this analysis at HL-LHC are presented in Subsection 3.4.

Analysis strategy
The final-state configuration of ZH/W H production that is most useful for precision differential measurements is that in which the weak gauge boson decays to charged leptons or neutrinos and the Higgs boson decays to a b-quark pair.Several studies have highlighted this process as a powerful BSM probe, in particular in the boosted Higgs regime [15,18,23,24,32,73].However, the limited cross-section at high energies allows for a noticeable gain of sensitivity when the boosted and resolved regimes are combined, as shown in Ref. [32].We adopt the analysis strategy developed in the latter reference, which exploits the high-energy tails of p T distributions and reproduces published ATLAS ZH/W H analyses [74,75].
We take the quark-initiated signal and background simulations from Ref. [32], adapting the former to our flavour assumption as discussed in detail in Subsection 3.3.Those simulations were performed at NLO in QCD, except for the t t process, which is the subleading background process in the 0-lepton channel and was simulated at LO with one additional hard jet.We add to the signal the contribution of the gg → ZH process, which we simulate at LO in QCD and for a centre-of-mass energy of 13 TeV to ensure compatibility with the q q → ZH results.The difference in cross sections between 13 and 14 TeV is negligible in comparison with the impact of the increased luminosity from LHC Run 3 to HL-LHC.This also allows us to obtain results for both LHC Run 3 and HL-LHC by a simple luminosity rescaling.For more details on the simulations, see App.C and Ref. [32].
The collider events are classified into two categories, boosted and resolved, according to the presence of a boosted Higgs candidate or two resolved b-jets respectively.This classification is done following an adapted version of the scale-invariant tagging procedure [76,77].Furthermore, the events are split in three channels according to the number of charged leptons in the final state, ranging from 0 to 2. Selection cuts and bins are optimised independently for each of these 6 categories.
Here, we only consider the 4 categories comprised by the 0-and 2-lepton channels since the 1-lepton channel concerns only W H production.Both 0-and 2-lepton channels use a binning in p T,min = min{p Z T , p H T }, but the bin limits are tailored to each category as seen in Table 10.This reflects their different cross-sections and energy distributions.Among the different selection cuts, the most effective to reduce the background is the cut on the invariant mass of the Higgs candidate.Additionally, a veto on untagged jets helps to control the background in the 0-lepton channel, while in the 2-lepton channel a cut on the p T imbalance of the charged leptons has a similar effect.More details on the selection cuts can be found in App.D and Ref. [32].

Higher order QCD corrections
The full computation at NLO in QCD of gg → ZH in the SM shows significant corrections in the tails of the p T distributions with respect to the LO result [78].This could have a significant impact on precision measurements that seize on them, such as the one considered here.This process has not been computed at QCD NLO in SMEFT yet.However, it is known that a sizeable part of the corrections introduced at NLO come from the real emission, in particular in the high-energy tails of p T distributions [78,79].This allows us to partially account for the NLO corrections in the presence of dimension-6 operators by resorting to the simulation of the process with 0 and 1 jet merged.Details on the simulations can be found in Appendix C.2.We show in the left panel of Fig. 5 the p Z T distributions in the SM at LO, NLO and from the 0 + 1 jet merged samples.We show the NLO results from Ref. [78] for 2 different choices of the renormalization scale µ, µ = m ZH and µ = H T .In the case of LO, we include the result from Ref. [78] and our parton-level results obtained with MadGraph5.NLO corrections give a harder p T distribution and that effect is well reproduced by the 0 + 1 jet merged samples even at very high energies.However, the latter underestimates the cross-section w.r.t.LO at energies below ∼ 200 GeV.
The result of turning on the dimension-6 operator O The interference effects are however reduced in the merged samples.This is clearly visible in Fig. 6, where we show the ratio between different simulation orders for the SM and the two c In Fig. 6, we also include the SM ratio and the NLO/LO k-factors from Ref. [78].From this figure, it is clear that the merged 0 + 1 jet samples in the SM underestimate the NLO corrections for p Z T < 500 GeV but constitute a very good approximation for higher p Z T .The right panel of Fig. 6 shows a zoom-in on the mid-energies region that is most relevant for (HL-)LHC analyses.The NLO/LO k-factor is roughly constant in that region, while the rate 0 + 1 jet merged over LO is close to it only for p T 350 GeV.Higher values of the WC can yield starker differences between the LO and merged samples.Such values would be excluded for c (−) ϕQ already from this ZH analysis, but not for c ϕt or c tϕ .Hence, we show in Fig. 7, the (0 + 1 jet)/LO ratio for several positive values of c ϕt around the expected HL-LHC bounds, as will be shown in Subsection 3.4.These ratios show a sharp peak at mid-low energies, which decreases in height and moves towards lower energies for increasing values of the WC.
The peak is caused by the negative interference between O ϕt and the SM and the partial deletion of its effect by the hard emission and shower.At higher energies, the interference becomes less relevant and is overtaken by the O(Λ −4 ) piece of the cross section.The energies at which the interference effect becomes noticeable and at which is surpassed by the quadratic piece move downwards as c ϕt increases, shifting the peak position accordingly.This non-trivial behaviour of the ratios hints at a possible strong dependence of the NLO/LO k-factors on the value of the WCs, in particular when the interference is relevant.Its detailed study is beyond the scope of this work.Thus, higher-order corrections due to real emissions might have a sizeable effect on the bounds that can be put on WCs.Whilst different simulations can lead to ratios which can be as large as 10, one has to explore the impact of these corrections in a realistic analysis setup.
In the left panel of Fig. 8, we show the number of events expected at HL-LHC for gg → ZH → ν νb b as a function of c (−) ϕQ obtained with LO and 0 + 1 jet merged simulations and the analysis strategy presented in Section 3.1.The dots represent the WC values for which we simulated and analyzed the signal, and the lines are the quadratic functions fitted to them.We show only the 0-lepton channel since it drives the sensitivity of the analysis and checked that using a different WC would yield qualitatively similar results.The differences between the LO and 0+1 jet results are negligible in most bins, except near the minimum, where the difference is relatively bigger but irrelevant for HL-LHC luminosities since both cases would yield 0 events.This is due in part to the presence of a jet-veto in the analysis and the binning in p T,min , as can be deduced from the right panel of Fig. 8, in which we show the number of events after removing the jet veto and binning in p Z T instead of p T,min .The difference between LO and 0 + 1 jet merged remains small but the gap widens in two situations.First, when c (−) ϕQ is negative, due to the reduced impact of the destructive SM interference in the merged samples.Second and most notably, in the highest-energy resolved bin, where the resolved regime favours a low p H T while the bin definition requires a high p Z T , giving place to a p T imbalance that can be compensated by a hard jet.Another factor that reduces the impact of the real emission is that the (HL-)LHC analysis is mostly sensitive to the [250, 500] GeV energy region, where the 0 + 1 jet merged samples underestimate the NLO corrections, as can be seen in Fig. 5 and 6.As a check of this phenomenon, we obtained the post-analysis cross-sections but using only events generated with a parton-level cut p Z T > 600 GeV.The results are presented in Fig. 9, where the left (right) panel shows the result of using the analysis with (without) jet veto and binning in p T,min (p Z T ).Only the highest-energy bins of both the boosted and resolved modes receive significant contributions from the p Z T > 600 GeV region.The analysis with a jet veto and the binning in p T,min shows negligible differences between the simulation orders.
For the case of no jet veto and binning in p Z T , shown on the right panel of Fig. 9, both simulation orders differ by between ∼ 50% and ∼ 300%, depending on the value of c (−) ϕQ and whether it is the resolved or boosted mode.The biggest differences are found in the resolved mode for the same reasons as before and near the minimum of the cross-section, where the interference is more relevant.Furthermore, we checked that for this analysis version and in this energy regime the degeneracy among c (−) ϕQ and c ϕt present at LO is broken by the additional jet as the ZHj amplitudes do not simply depend on the axial vector coupling of the top.However, such effect becomes negligible in the full analysis with a jet veto, binning in p T,min and including lower energies.
Thus, the number of events at HL-LHC obtained with our analysis strategy is insensitive to the use of 0 + 1 jet merged samples.Instead, a simple rescaling of the LO cross-section by a constant SM k-factor offers a better approximation of the impact of NLO corrections for bounds at (HL-)LHC.Due to the sensitivity of our (HL-)LHC analysis to the p T ∈ [250, 500] GeV region, we average the k-factor computed in Ref. [78] over those energies.This procedure yields a k-factor of 2.0 − 2.1 (3.0 − 3.4) when averaging over p Z T (p H T ) depending on the renormalization scale.We pick the conservative value of 2.0 for our analysis and briefly comment on the results obtained with other choices.From the previous discussion, we expect this simple rescaling to be inaccurate at higher energies, and hence the simulation of 0+1 jet merged samples could be preferable for future colliders such as FCC-hh.
0.001 0.010 0.100 3.3 Flavour assumptions and interplay with the q q channel.
The gluon-initiated channel is subdominant in ZH production at hadron colliders where it formally enters at NNLO.This can be seen explicitly in Fig. 10, where we show the number of events at HL-LHC per bin in each channel and category generated by each of the signal and background processes in the SM.In all cases, the gluon-initiated contribution to the signal is one order of magnitude smaller than the one from the quark-initiated channel.Additionally, the background overwhelms the signal in most bins.Therefore, the contribution from gluon-initiated ZH production would not modify ostensibly the flavouruniversal results in Ref. [32].Which dimension-6 operators are constrained and how much each production channel contributes to the total sensitivity is a flavour-symmetry dependent statement.Here, ϕQ .While additional operators modify the inclusive qq → ZH cross section, they do not generate growth with energy and consequently, our sensitivity to them is negligible [32,80].Since only the b b initial state is sensitive to the heavy-quark operators, qq → ZH can only probe the combination 2c ϕQ .When combined with the gluon-initiated channel, this leaves two unconstrained directions in the space of the heavy-quark operators.There are no flat directions in the space of the lightquark operators and the discussion on the sensitivity to each of their Flavour-Universal versions in Ref. [31,32] applies straightforwardly.
The abandonment of Flavour Universality implies that the cross-section of the q q channel as a function of the WCs could map onto the new flavour scenario in a non-trivial way.Parton-level analysis shows that the contribution of the b b initial state at (HL-)LHC is 1% in the energy regime that matters for sensitivity to NP, p T ∈ [200, 600] GeV,

Constraints at (HL-)LHC
Here, we present the projected 95% C.L. bounds on dimension-6 WCs at the HL-LHC from the process pp → ZH for the flavour assumption U(2) q ×U(3) d ×U(2) u × (U(1) × U(1) e ) 3 .We assume SM-like measurements, uncorrelated observables and three different systematic uncertainties: 1%, 5%, and 10%.Our HL-LHC projections for 1-operator fits are summarised in Table 11 (see App. E for the projections corresponding to LHC Run 3 luminosity).At the HL-LHC, 5% syst.uncertainties start overshadowing the statistical ones.This value is the most realistic assumption for HL-LHC according to current measurements [82].The bounds at LHC Run 3 are still statistically limited, hence HL-LHC can improve them by 30 − 50%.
A comparison of the 1-op.bounds on c (3) ϕq,ii , c ϕu and c ϕd against the bound on their flavour-universal counterparts from Ref. [32] reveals that the new flavour assumption and the addition of gg → ZH has had no relevant effect on c     Among the heavy-quark operators, c ϕQ is the best constrained one, despite modifying only the b b channel.The same process is expected to give a worse bound on c (−) ϕQ by a factor of 2, but this should be partially compensated by the contribution of the gluonfusion process.In practice, the latter contribution is small and its main effect is a shift of the bounds toward negative values.The dominance of the quark-initiated channel can also be seen in the great difference between the bounds on c ϕQ is led by the piece of the cross-section that is quadratic on the WCs, while for c tϕ and c ϕt , the cross-section is in an intermediate regime between interference-and quadratic-domination, as evidenced by the high asymmetry of the bounds around 0.
For the results presented here, we have rescaled the gg → ZH cross section by a constant k-factor of 2 to account for NLO QCD corrections, see Subsection 3.2 for details.interaction between gluons and Higgses.Two additional operators, O tϕ and O dϕ , modify SM vertices, even changing the Lorentz structure, and induce a milder logarithmic growth.
In the case of ZH production, we find five operators which generate energy growth.
ϕQ and O tG generate energy-growing amplitudes in W W production.The first four of them generate logarithmic growths, with O tϕ and O (3) ϕQ doing it only for longitudinally polarised W bosons with same-helicity gluons, whilst O ϕW and O tW require transversely polarised W bosons.The amplitudes with O tG behave ∼ s in the high-energy limit when both W s are longitudinal, while one longitudinal and one transverse W makes them ∝ √ s and the behaviour is logarithmic when all bosons are transverse.Motivated by the analytical results and the absence of combined phenomenological analysis of the quark and gluon-initiated ZH production, we perform a complete analysis of ZH production at (HL-)LHC.We assessed how the flavour assumptions must be relaxed to make the addition of the gluon-fusion process meaningful and its effects on the reach of the quark-initiated channel.We also discussed in detail the possible effects of NLO QCD corrections to gg → ZH and approximately included them in our analysis.
Our results show that the flavour-universal bounds on 2-fermion operators constrained by V H production can be safely translated to bounds on light-quark operators thanks to the small contribution of b b → ZH.Nevertheless, the b b initial-state contribution is large enough to provide competitive bounds on c (3) ϕQ and dominate the sensitivity to c (−) ϕQ .The latter WC is the only one probed by both quark and gluon initial states, albeit the inclusion of the gluon channel does not improve the bound sizeably.Gluon-fusion ZH production offers sensitivity to c ϕt and c tϕ , otherwise inaccessible in diboson processes, with the former being competitive against current bounds.
The interplay of the two initial states is not enough to constrain all directions in the space spanned by the heavy-quark operators, since b b → ZH is sensitive to 2c Several research directions remain open and deserve further exploration.The use of a differential distribution to distinguish better between quark-and gluon-initiated processes is an interesting possibility to explore.Going beyond ZH, the combination of various loop-induced channels, several of which exhibit energy-growing amplitudes for the same operators is an obvious next step.One can envision that a combination of the ZH, ZZ and W W processes could improve constraints on several of the coefficients discussed here and break the degeneracies which plague the ZH channel.Tailored analyses would then be designed for the other processes.Eventually, this class of processes will be included fully in global EFT fits, where a large number of processes is considered to extract maximal information on the coefficients.
Finally, the study of dimension-8 contributions to diboson processes is a lightly trodden path that might be of special relevance to gluon-initiated processes.We have seen that only one dimension-6 operator can promote these processes from loop induced to tree-level and hence offer a steeper growth with energy.Several dimension-8 operators can generate gg → V V ( ), V H, HH at tree-level with a distinctive energy growth, as found recently via on-shell amplitude techniques [47].We leave a detailed study of those operators for future work.
operators, the expressions for amplitudes that decrease with energy were obtained by numerical fits of the amplitudes and for those operators we only give the energy dependence of the helicity amplitudes.In the case of top quark operators, numerical fits were used for amplitudes that decrease faster than 1/ √ s.All the other helicity amplitudes were obtained analytically with Mathematica following the procedure described in the main text.Finally, the dependence on the WCs and Λ 2 is implicit.

C Simulation details
In this work, we only consider the 0-and 2-lepton channels of (W/Z)H production, which correspond to the Z → ν ν and Z → + − decay channels respectively.The signal process in each of the channels is mostly composed of ZH production, with the Z decaying to the appropriate leptons.The 0-lepton channel also receives a significant signal contribution from W H production with W → τ ν τ and missing products of the tau decay [32,74,75] The production of Zb b is the main background process in both channels and, in fact, is the only relevant one in the 2-lepton channel.The 0-lepton channel background is also composed of t t and W b b production.The event generation was performed using MadGraph5_aMC@NLO v.2.7.3 [83] and the parton shower was performed with Pythia 8.2 [84].We used the SMEFTatNLO v.1.0.3 UFO model for all processes [49].The simulation of quark-initiated signal and background processes was performed as described in Ref. [32].This means that, since we limit ourselves  account for most of the NLO QCD corrections.Additionally, the quark-initiated part of the signal process includes NLO EW corrections via a k-factor.In the following, we detail the generation of the gluon-initiated processes that were added in this work.

C.1 LO generation
We generated the signal processes gg → ννH and gg → + − H at LO in QCD.The efficiency with which the generated events pass the selection cuts was improved via the application of generation-level cuts and a binning in p Z T .These cuts and bins are detailed in Table 17.and we implemented a loop filter in MadGraph such that only the diagrams in Fig. 13 were included in the case of the SM.The dimension-6 operators we study can modify the Zt t, Zq q, and t tH couplings and introduce the additional topologies shown in Fig.The (0+1) jet process was simulated only in the Z → νν decay channel since the comparison between QCD orders is independent of the EW decay channel.The generationlevel cuts used can be found in Table 18.We checked that the k ⊥ − cutoff [GeV] setting leads to a successful merging by looking at the p j T and ∆R jj distributions and ensuring they were smooth.
Table 19: Projected bounds at 95% C.L. from one-dimensional fits on the dimension-6 SMEFT WCs probed by pp → ZH at LHC Run 3 with integrated luminosity of 300 fb −1 .The WCs are in units of TeV −2 .The gluon-initiated channels were rescaled with a constant k-factor of 2.0 to account for NLO QCD corrections.Left: Light-quark WCs.Right column: Heavy-quark WCs.
simple rescaling since the change in energy has a negligible impact.The number of signal events is given as a quadratic function of the studied WCs.The background contributions include the main background processes, as described in Ref. [32].

F.1 The 0-lepton channel
The number of signal and background events in the 0-lepton channel at HL-LHC is reported in tables 20 and 21 for the resolved and boosted channels respectively.In this channel, the number of signal events includes the contributions from both ZH → ν νb b and W H → ν b b with a missing lepton.The latter process only contributes to the SM, c ϕq,ii and (c ϕq,ii ) 2 coefficients.
Table 20: Number of expected signal events as a function of the WCs (in units of TeV −2 ) and of total background events in the pp → ZH → ν νb b channel, resolved category, at HL-LHC.The Monte Carlo errors on the fitted coefficients, when not explicitly specified, are 5 %.
Table 22: Number of expected signal and background events in the pp → ZH → + − b b channel, resolved category, at HL-LHC.The Monte Carlo errors on the fitted coefficients, when not explicitly specified, are 5 %.

F.2 The 2-lepton channel
In this subsection, we present the number of signal and background events in the 2-lepton channel at HL-LHC.Tables 22 and 23 show the results in the resolved and boosted categories respectively.The signal is composed of the processes q q → ZH → + − b b and gḡ → ZH → + − b b.

Figure 1 :
Figure 1: Diagram topologies that enter in the computation of gg → HH in SMEFT at oneloop.The empty dots represent couplings that could be either SM-like or modified by dimension-6 operators.The filled dots represent vertices generated only by dimension-6 operators.Only one insertion of dimension-6 operators is allowed per diagram.

Figure 2 :
Figure 2: Diagram topologies that enter in the computation of gg → ZH in SMEFT at oneloop.The empty dots represent couplings that could be either SM-like or modified by dimension-6 operators.The filled dots represent vertices generated only by dimension-6 operators.Only one insertion of dimension-6 operators is allowed per diagram.

Figure 3 :
Figure 3: Diagram topologies that enter in the computation of gg → ZZ in SMEFT at oneloop.The empty dots represent couplings that could be either SM-like or modified by dimension-6 operators.The filled dots represent vertices generated only by dimension-6 operators.Only one insertion of dimension-6 operators is allowed per diagram.

Figure 4 :
Figure 4: Diagram topologies that enter in the computation of gg → W W in SMEFT at oneloop.The empty dots represent couplings that could be either SM-like or modified by dimension-6 operators.The filled dots represent vertices generated only by dimension-6 operators.Only one insertion of dimension-6 operators is allowed per diagram.

Table 10 :
p T,min bins used in the (HL-)LHC analysis of the 0-and 2-lepton channels.
ϕQ can be seen in the right panel of Fig. 5.The chosen values of c (−) ϕQ are similar to the HL-LHC bounds to be presented in Subsection 3.4.The positive interference between O (−) ϕQ and the SM generates a concavity in the LO p Z T distribution for negative c (−) ϕQ .As the energy increases, the contribution of the squared EFT amplitude becomes more relevant and the total O (−) ϕQ cross-section exceeds the SM curve for energies at the edge of our plot.All the merged samples produce harder tails than their LO counterparts.Notice that the merged cross section with negative c (−) ϕQ also shows the interference effects, since it is below the merged SM curve for p Z T ∈ [200 − 900] GeV.

Figure 5 :
Figure 5: Differential cross-section of gg → ZH with respect to p Z T at different orders.Left panel: SM p Z T distribution at LO and NLO from Ref. [78], the latter for 2 different renormalization scales µ.We also show SM distributions at LO and 0 + 1 jet merged obtained with MadGraph5 (MG5) and Pythia8 (PY8).Right panel: p Z T distribution at LO and 0 + 1 jet merged in the SM and when turning on the operator O (−) ϕQ for 2 different values of its WC, while setting all other WCs to zero.Results obtained with MadGraph5 and SMEFTatNLO.

Figure 6 :
Figure 6: Ratio of differential cross sections for gg → ZH for different computation orders.The blue curves show the Standard Model results, with the different shades representing different computation orders.The NLO/LO ratios were extracted from Ref. [78].The red and green curves show the result in SMEFT for two representative values of c (−) ϕQ , while setting all other WCs to zero.These representative values are near the 95% C.L. bound on this operator.The shaded regions indicate the statistical uncertainty.The right panel shows a zoomed-in version of the figure on the left.

Figure 7 :
Figure 7: Ratio of differential cross sections for gg → ZH between 0 + 1 jet merged and LO.The light blue curve shows the Standard Model result.The remaining curves show the result in SMEFT for several positive values of c ϕt with all other WCs set to zero.The shaded regions indicate the statistical uncertainty.The right panel shows a zoom-in of the plot on the left.

Figure 8 :
Figure 8: Number of events at HL-LHC from gg → ZH → ν νb b in the different categories and bins as function of c (−) ϕQ , with all other WCs to zero.The different colours identify the different categories and bins.The squares (circles) represent the results obtained with simulations of 0+1 jet merged (at LO+PS) and the continuous (dashed) line shows the quadratic fit to them.Left panel: Analysis with jet veto and binning in p T,min , as used in the rest of this work.Right panel: Analysis without jet veto and binning on p Z T .

Figure 9 :
Figure 9: Number of events at HL-LHC from gg → ZH → ν νb b from the high-energy regime as function of c (−) ϕQ , with all other WCs set to zero.The high-energy condition was enforced by applying a generation-level cut p Z T > 600 GeV.The different colours identify the different categories and bins.Only the highest energy bins receive relevant contributions in this regime.The squares (circles) represent the results obtained with simulations of 0+1 jet merged (at LO) and the continuous (dashed) line shows the quadratic fit to them.Left panel: Analysis with jet veto and binning in p T,min , as used in the rest of this work.Right panel: Analysis without jet veto and binning on p Z T .

Figure 10 :
Figure 10: Number of SM events at HL-LHC for each of the processes contributing to the signal and background in our analysis of pp → ZH.The processes in shades of brown and orange are background processes, while the ones in blue are considered part of the signal.The x-axis represents the analysis bins used in each channel and category.Top left: 0-lepton channel, resolved category.Top right: 0-lepton channel, boosted category.Bottom left: 2-lepton channel, resolved category.Bottom right: 2-lepton channel, boosted category.

( 1 ) 3 ]
ϕq,ii , c ϕu and c ϕd , as expected.The worsening of the bounds on c(3) ϕq,ii is due to the exclusion from this work of the 1-lepton channel, which probes only this coefficient.Heavy-quark operators are less stringently constrained than their light counterparts.Those that affect the b b → ZH process, c(3)ϕQ and c (−) ϕQ , have bounds between one and two orders of magnitude worse than the ones for light-quark operators.c ϕt and c tϕ , probed only via gluon fusion, are constrained even more loosely and their bounds are up to two orders of magnitude worse than the best ones.× 10 −2 1% syst.[−2.1, 1.8] × 10 −2 5% syst.[−3.8,2.7] × 10 −2 10% syst. c

Figure 11 :
Figure 11: Projected 95% C.L. bounds on c ϕt (upper row) and c (−) ϕQ (lower row) at LHC and HL-LHC from a one-operator fit as a function of the maximal-invariant-mass cut M .We also show the bound from the global fit to LHC data performed by the SMEFiT collaboration [38] with a dark green line.The dashed grey line shows the expected value of the WC in strongly-coupled NP scenarios, c ∼ 4π 2 /M 2 .Left column: Projection for HL-LHC with different levels of systematic uncertainty.Right column: Projection for LHC and HL-LHC with a fixed 5% syst.uncertainty and different integrated luminosities, L.
ϕQ and gg → ZH probes c (−) ϕQ − c ϕt + ctϕ yt .The remaining flat directions could be lifted with the full inclusion of NLO QCD corrections to gg → ZH, although with limited sensitivity.

Figure 13 :
Figure 13: Representative diagrams included in the computation of gg → ZHj in SMEFT with SM topology.The empty dots represent couplings that could be either SM-like or modified by dimension-6 operators.Only one insertion of dimension-6 operators is allowed per diagram.

Figure 14 :
Figure 14: Feynman diagrams of gg → ZH with new topologies introduced by the dimension-6 operator O (−) ϕQ .The dimension-6 insertion is marked with a black dot.

Table 1 :
Dimension-6 operators O i and their associated Wilson Coefficients c i entering in gg → HH, ZH, ZZ, W W .

Table 4 :
High energy behaviour of the gg → ZH helicity amplitudes with modified top-Z and top-Higgs interactions.

Table 6 :
High energy behaviour of the gg → ZZ helicity amplitudes modified by top operators.

Table 7 :
High energy behaviour of the gg → ZZ helicity amplitudes modified by the purely bosonic operators.The operators probed by gg → ZZ can be divided into three categories.First, O tG , O tZ , O tϕ , O ϕt and O (−) ϕQ enter in the top quark couplings with the Higgs, Z bosons and gluons.Then O ϕB , O ϕW and O ϕG modify the bosonic Higgs couplings.Finally, O ϕu , O ϕd , O (−) ϕq i , O (3) ϕq i and O (3) ϕQ all modify the light quark couplings with the Z boson.Example diagrams are shown in Fig. 3.The high energy behaviour of the SM helicity amplitudes, first discussed in [69, 70], and of the SMEFT growing amplitudes are presented in Tables 5-7.
The operators O ϕt , O (−) ϕQ and O tϕ enter gg → ZZ by rescaling the Zt R tR , the Zt L tL and the t tH interactions respectively.In addition O (−) λ g1 , λ g2 , λ W + , λ W −

Table 8 :
High energy behaviour of the gg → W + W − helicity amplitudes in the SM and modified by top dipole operators.For readability we define f 3

Table 9 :
High energy behaviour of the gg → W + W − helicity amplitudes modified by top and purely bosonic operators.

Table 11 :
Projected bounds at 95% C.L. from one-dimensional fits on the dimension-6 SMEFT WCs probed by pp → ZH at HL-LHC with integrated luminosity of 3 ab −1 .The WCs are in units of TeV −2 .The gluon-initiated channels were rescaled with a constant k-factor of 2.0 to account for NLO QCD corrections.All the results include the quadratic EFT corrections.Left: Light-quark WCs.Right column: Heavy-quark WCs.
Only one of them, O tG , induces amplitudes that are linear in either √ s or s.The √ s behaviour arises for three different helicity configurations with a transversely polarised Z boson, while the fastest growth appears when the Z is longitudinally polarised and the incoming gluons have opposite helicities.The other 4 operators, O ϕG , O ϕt , O ϕQ , and O tϕ , induce a logarithmic growth for a longitudinal Z.The first of them does it regardless of the gluon helicities, while the rest require same-helicity gluons.An exact cancellation of the triangle diagrams renders ZH sensitive to only the linear combination c production is the process with the largest number of growth-inducing dimension-6 operators.O tG generates amplitudes that behave ∝ s at high energies whenever both Z bosons are longitudinal.If only one is longitudinally polarised, this same operator induces a linear growth with energy, and if both Z bosons have transverse polarisations, their amplitudes grow logarithmically.The only other operator capable of producing amplitudes that grow quadratically with energy is O ϕG , but it enters only for the (+, +, 0, 0) helicity configuration.Three operators, O ϕB , O ϕW , and O tZ , generate a logarithmic growth for amplitudes with only transversely polarised Z bosons and the last two further require samehelicity gluons.The configuration with same-helicity gluons and longitudinally polarised Z bosons also allows the operators O tϕ , O ϕt and O ϕW , O tW , O tϕ , O

Table 13 :
λ H 1 , λ H 2 High energy behaviour of the gg → ZH helicity amplitudes modified by SMEFT operators.

Table 15 :
High energy behaviour of the gg → ZZ helicity amplitudes in the presence of purely bosonic operators.to(HL-)LHC, all the relevant processes were simulated at NLO in QCD, except for the t t background process.The latter was simulated at LO with one additional hard jet to λ g1 , λ g2 , λ Z1 , λ Z2

Table 16 :
High energy behaviour of the gg → ZZ helicity amplitudes modified by light quark operators.

Table 17 :
Parton level generation cuts for gg → ZH at 13 TeV at LO.C.2 0 + 1 jet merged generationThe 0 + 1 jet merged samples were generated including all the following processes: