Polarized ZZ pairs in gluon fusion and vector boson fusion at the LHC

Pair production of helicity-polarized weak bosons ( V λ = W ± λ , Z λ ) from gluon fusion ( gg → V λ V ′ λ ′ ) and weak boson fusion ( V 1 V 2 → V λ V ′ λ ′ ) are powerful probes of the Standard Model, new physics, and properties of quantum systems. Measuring cross sections of polarized processes is a chief objective of the Large Hadron Collider’s (LHC) Run 3 and high luminosity programs, but is limited by the simulation tools that are presently available. We propose a method for computing polarized cross sections that works by directly modifying Feynman rules instead of (squared) amplitudes. The method is applicable to loop-induced computations and includes both interference and off-shell effects. As a demonstration, we report the prospect of observing and studying polarized Z λ Z λ ′ pairs when produced via gluon fusion and electroweak processes in final-states with four charged leptons at the LHC. Our Feynman rules are publicly available as a set of Universal FeynRules Object libraries called SM Loop VPolar .

Despite the physics needs and motivations, there is a limited supply of predictions for integrated (σ) and differential (dσ) cross sections for martina.javurkova@cern.ch(contact author), rruiz@ifj.edu.pl,rclsa@umass.edu,jsandesara@umass.eduTeV-scale proton (p) collisions with intermediate or final-state helicity-polarized particles, e.g., Here and throughout, λ denotes the helicity of V and q is its momentum.Predictions are notably scarce beyond tree level in perturbation theory.Typically, helicity-polarized cross sections are obtained by first constructing the matrix elements (or their square) for an unpolarized process, and then decomposing the numerator of an intermediate, resonant propagator V into the outer product of one or more pairs of polarization vectors (or spinors for fermions).This is known generically as the "truncated-propagator" method.
By combining discrete Monte Carlo sampling over helicities with the spin-correlated narrow width approximation (NWA), a generalization of the truncated-propagator method was developed [46] that did not require modifying intermediate momenta.This led to the automation of helicity-polarized cross sections at LO for arbitrary tree-level processes [46].More recently, the computation of polarized cross sections with the NWA was automated at (approximately) NLO in QCD for arbitrary processes by projecting unpolarized cross section onto a basis of polarizations, and hence providing polarized cross sections [45].
We attempt to advance this program by proposing another method for computing polarized cross sections that extends the truncated-propagator method to individual Feynman rules.The broad idea is that the (coherent/interfering) sum of polarizations of a single, unpolarized intermediate vector boson can be realized as the (coherent/interfering) sum of several, polarized states.In other words, the graph of a vector boson (or fermion) can be split into the sum of several propagating states.Polarized cross sections that include full off-shell effects, interference with non-resonant contributions, polarized t-channel contributions, and loop-induced contributions can then be obtained via diagram selection, which is a common technique in realistic simulations for the LHC.
As a demonstration and using publicly available Monte Carlo (MC) simulation tools, we use the new method to estimate the sensitivity of the LHC experiments to helicity-polarized ZZ pairs produced via gluon fusion (ggF).We also estimate the sensitivity to EW processes, including weak boson fusion (VBF) and annihilation channels.We focus on the data set sizes expected by the end of the LHC's Run 3 era of operations and with the full dataset of the high-luminosity (HL) LHC.This work continues as follows: In Sec. 2 we describe our proposed method for computing polarized cross sections and its practical implementation.In Sec. 3 we present LHC predictions for the production of Z λ Z λ ′ pairs from gluon fusion.In Sec. 4 we present our projections for measuring polarized ZZ cross sections.Further applications are discussed in Sec. 5, and we conclude in Sec. 6. AppendixA-AppendixC contain additional technical details to reproduce our work.

Helicity polarization as a Feynman rule
After EWSB, the propagator of a massive, onshell gauge boson V with momentum q and mass M V is related to its polarization vectors (ε) in the Unitary gauge by the completeness relationship The sum runs over the longitudinal (λ = 0) and transverse (λ = ±) states of V .Eq. ( 2) can be extended to off-shell V by extending the sum to the so-called "auxiliary/scalar" polarization (λ = A), and to the t-channel production by including λ = A and summing over (−1) [47,48].Strictly speaking, the helicity of V in a particular frame is only well-defined, and hence experimentally accessible, when V is on shell.This is because helicity labels an eigenstate in the mass basis and off-shell states are not eigenstates of this basis.However, in particular circumstances [49], Eq. ( 2) is also meaningful for t-channel exchanges [50][51][52].
The decoherence of a heavy object's helicity when off-shell is evident in its propagator, which can be identified as a coherent sum of external polarization vectors (or spinors for fermions).That is to say, using Eq. ( 2) one can express the propagator of V (with mass and width M V , Γ V ) as Here, η λ = +1, unless λ = 0 and V λ is in the tchannel; in that case η λ = −1.In Ref. [46], the quantity in the summation of Eq. ( 4) is labeled the "helicity-truncated" propagator Π V λ µν , Helicity-polarized matrix elements, and hence helicity-polarized cross sections (dσ λ ), can then be defined by using the truncated propagator of Eq. ( 5).The unpolarized (M) and polarized (M λ ) matrix elements are related by summing over λ: Here, M µ i and M ν f are the Green's functions (diagram fragments) that sandwich the full propagator V , i.e., Eq. ( 3), in the unpolarized matrix element.
After squaring the polarized matrix element and integrating over phase space one obtains dσ λ .The difference between the unpolarized cross section (dσ unpol.) and summing over all polarized cross sections is the interference between helicity states, The "interference" term here also includes the contribution from non-resonant diagrams.
The magnitude of the interference is process dependent but can be incorporated by adjusting the definition of Π V λ µν in Eq. ( 5) to include multiple polarizations [46].For example: one can define the "transverse" propagator Π V T µν as the sum over the λ = ±1 states.The interference is also phasespace dependent, the impact of which can be adjusted by restrictions on external and intermediate momenta.Finally, the accuracy of dσ λ depends on how many intermediate states are polarized.
The traditional truncated-propagator method has several advantages: (i) off-shell effects are included, (ii) it is generalizable to fermions and high-spin states, (iii) interference between polarizations can be included, and (iv) it is applicable to tree-and loop-level processes.A significant disadvantage, however, is the need to modify the propagator routines (software libraries) in existing event generators.Such work is technically challenging to implement into event generators capable of simulating arbitrary processes because the mappings between particles, their Feynman graphs, and their propagators are typically hard coded.Alterations can lead to undesired consequences for unpolarized subgraphs [45,46].
To ameliorate this difficulty, we propose treating the truncated propagator in Eq. ( 5) as a Feynman rule itself.If one considers the decomposition of a state V as a combination of polarized states V λ , (9) and propagates this throughout a Lagrangian, then graphically one can make the identification: That is, the truncated propagator Π V λ µν in Eq. ( 5) is the propagator of a vector that has a fixed polar-ization.One can then interpret Eq. ( 6) as the amplitude for a process that is mediated by a single particle V with interfering polarizations λ and interpret Eq. ( 7) as the sum of interfering diagrams for a process mediated by a collection of particles V λ , where each V λ has its own propagator.This identification is one chief result of our work.
The key to realizing this approach in realistic simulations is the fact that modern, multi-purpose event generators are sufficiently flexible to handle new Feynman rules1 , including alternative propagators, as input libraries.Passing Eq. (10) or Eq. ( 5) to event generators as an input library is an efficient alternative to modifying the built-in propagators for SM particles, which are typically hardcoded.Moreover, this method is compatible with the automated computation of so-called R 2 and ultraviolet counter-terms at NLO in QCD [57].Therefore, "polarization as a Feynman rule" is also applicable to automated loop-induced processes.
As a brief comment, treating polarization as a Feynman rule raises the prospect that new rules are not individually Lorentz invariant, and that all external-particle momenta must be written in the same frame through the whole flow of a numerical calculation.In general, individual Feynman rules are inherently not Lorentz invariant: many rules in the SM carry spinor or vector indices.As a consequence, numerical implementations of the helicity amplitude method for computing matrix elements (for example: HELAS [58] as implemented in MadGraph [59]) already require a reference frame to be chosen before phase space integration.For LO predictions in mg5amc, including those that are loop-induced, helicity amplitudes are evaluated by default in the partonic frame, though other choices of frame are possible [46].Our method does not change this operation; rather, the method relies and exploits these generator-dependent capabilities (and their numerical efficiencies).

Practical Implementation
To implement our proposal, we start from the default implementation of the full SM Lagrangian into FeynRules [53][54][55][56].We add four new "particles" to the model: V T , V 0 , V A and V X for V = W ± and Z, which carry the same properties as the SM W ± and Z bosons.We then make a field redefinition V = V T +V 0 +V A +V X .The V λ inherit all interactions of the W and Z; redundant operators, e.g., ∆L = M2 Z Z µ T Z Aµ , are removed.We extract QCD renormalization and R 2 counter terms (CTs) up to the first order in α s using FeynRules (v2.3.36) with NLOCT (1.02) [57] and FeynArts (v3.11) [60].At this point, the V λ still have the same propagators as V and subsequently inherit all the CTs associated with V due to the V = V T + V 0 + V A + V X redefinitions.Importantly, the CTs here are purely QCD CTs at O(α s ), not mixed QCD⊗EW CTs.This means that our CTs do not depend on the helicity of V .While the gg → e + e − µ + µ − process at LO contains no UV divergencies, R 2 terms are present.
We then package all Feynman rules into a single set of software libraries 2 .It is at this point that we modify the propagator library so that the "new" particles have propagators given by Eq. ( 5).Explicitly, V T carries summed λ = ±1 polarizations and V 0 (V A ) carries the λ = 0 (A) polarization.W X and Z X , which are not used in the study, are copies of the unpolarized SM W and Z that we included for diagnostic purposes / closure tests.
We use the new UFO (SM Loop VPolar) in conjunction with MadGraph5 aMC@NLO (mg5amc) (v3.4.0) [59,61,62] to build the polarized matrix element, and hence compute the polarized cross section, for the full, loop-induced 2 → 4 process gg → e + e − µ + µ − at LO, i.e., at O(α s α 2 ).We include the full dependence of the top quark [62].We do not impose the NWA but remove all diagrams with photons; we justify removing photons in the next section.This level of diagram removal is a built-in ability mg5amc [61,62].For unpolarized propagators, we obtain 16 diagrams mediated by top triangles and top boxes, including diagrams with an internal Higgs, and obtain 14 diagrams each for massless u-and d-quark loops.
According to our method, all intermediate Z subgraphs, including non-resonant subgraphs, are split into "polarized" subgraphs.(For multiboson processes, this can generate a large diagram multiplicity.)For the doubly polarized matrix elements (λ 1 , λ 2 ) = (0, 0) and (T, T ), we remove all diagrams containing undesired V λ .Again, this type of diagram removal is a built-in option in mg5amc.
In both cases, we obtain 16 (14) diagrams when the top quark (a light quark) runs in the loop.
For the mixed polarized channel (λ 1 , λ 2 ) = (0, T )+(T, 0), we exploit the topology, coupling order, absence of photons.These conditions dictate that two exactly Z λ propagators must appear in the full 2 → 6 process.We employ a type diagram filtering accept/reject diagrams3 .For diagrams in-duced by triangle loops, we require that exactly one Z 0 and one Z T appear in any multi-vertex, tree subgraph attached to the loop.For diagrams induced by boxes, we require that one of the tree subgraphs attached to the loop contains Z 0 and another contains Z T .We generate the expected 32 (28) diagrams for top (light) quark loops.
The matrix element for our closure test is obtained by removing all diagrams with photons and the auxiliary "Z X " state.We otherwise keep all diagrams with Z λ ∈ {Z T , Z 0 , Z A }.For top quark diagrams, this generates 16 × 3 2 = 144 individual diagrams while 14 × 3 2 = 126 diagrams are generated for loops with light quarks.

Polarized Z λ Z λ ′ from gluon fusion
As a first demonstration of our method we present total and differential predictions for polarized Z λ Z λ ′ production from ggF.We construct our matrix elements and simulate our signal processes at LO with mg5amc as described in Sec.2.1.We use the NNPDF31+LUXqed NLO parton distribution function (PDF) (lhaid=324900) [63][64][65].DGLAP evolution and PDF uncertainty extraction are handled by LHAPDF (v6.3.0)[66].Further simulation details are given in AppendixC.

Polarized cross sections in proton collisions
In Fig. 1 (top panel) we show the hadron-level cross section [fb] as a function of collider energy √ s [TeV] for the full loop-induced, 2 → 4 process at LO for unpolarized ZZ pairs (black band) as well as polarized Z T Z T (blue band), Z T Z 0 (purple band), and Z 0 Z 0 (green band) pairs.The Z T Z 0 rate includes both (T, 0) and (0, T ) helicity configurations.Helicity is defined in the (gg) frame.The unpolarized prediction does not employ the  12), along with the analogous Z T Z T (blue), Z T Z 0 (purple), and Z 0 Z 0 (green) rates.Band thickness denotes the scale uncertainty.Lower panel: The ratio of the Z T Z T (blue) and Z X Z X (red) cross sections and their uncertainties relative to the unpolarized rate.(Z X is the sum of the Z T , Z 0 , and Z A channels at the amplitude level.)Larger (transparent) bands are the scale uncertainties; smaller (solid) bands are the PDF uncertainties.methodology of Sec. 2 and serves as a check.Band thickness corresponds to the scale uncertainty.
We remove photon diagrams, which are less important around the Z resonance, to make the relative importance of individual polarization configurations clearer.We also impose the following phase space cuts on the charged leptons: where SF denotes same-flavor lepton pairs and charged leptons are ranked by p T , i.e., p i T > p i+1 T .These generator cuts reflect selection cuts that would be imposed in a real analysis in ATLAS or CMS.For all polarization channels and collider energies, we estimate4 missing QCD corrections by applying a constant K-factor of [67,68] K NLO = 1.86 . ( Over the range √ s = 1 − 100 TeV, the fiducial cross sections and relative scale uncertainties for various polarization configurations roughly span We do not show rates with the auxiliary polarization λ = A since their contraction with currents containing massless leptons vanishes by the Dirac equation.For lower collider energies, we observe that the (0, 0) rate is comparable but still lower than the mixed (0, T ) + (T, 0) rate.At higher collider energies, however, this difference disappears.
For concreteness, the QCD-improved cross sections (first column), scale uncertainties [%] (second), and PDF uncertainties [%] (third) at the LHC's current energy of √ s = 13 TeV are Here and below, Z X denotes the sum over Z T , Z 0 , and Z A channels at the amplitude level and shows that the unpolarized rate is recovered.By closure, the interference is δσ = σ ZZ − λλ ′ σ λλ ′ ∼ O(+1) ab, which is below our MC statistical uncertainty for 400k events.In terms of polarization fractions the LO predictions for various channels are: This is consistent with the results of Ref. [34].
In the lower panel of Fig. 1 we show the ratio of polarized (fiducial) cross sections σ λλ , along with their uncertainties relative to the central unpolarized ZZ rate.The larger (transparent) bands are the scale uncertainties at LO and reach approximately δσ scale λλ /σ ZZ ∼ ±20%−±40%.The smaller   12) and ( 19), with and without spin correlation.The ratios are with respect to the unpolarized result with spin correlation.
(solid) bands are the PDF uncertainties and reach about δσ PDF λλ /σ ZZ ≲ ±1.5% for √ s ≳ 5 TeV.Specifically, we show the Z T Z T ratio (blue bands), which sits uniformly at around (σ T T /σ ZZ ) ∼ 90% − 95%, and the rate of Z X Z X (red bands) sits uniformly at unity.The Z X Z X ratio serves as a closure test and check of our methodology.

Spin-correlation in polarized gg → ZZ → 4ℓ
Measurements of ZZ polarization in ggF processes require public MC generators matched to parton showers that can simultaneously model loop-induced processes, helicity-polarized intermediate states, and still preserve spin correlation.
Spin correlations are essential for the observables used in a typical 4ℓ-analyses.For example: matrix-element discriminants rely on spin correlations between final-state leptons to separate signal processes from irreducible background [69][70][71].
As another demonstration of our method, Fig. 2 shows a comparison between simulations of the gg → ZZ → 4ℓ process with and without spin correlation.The variable θ 1 is the angle between the three-momentum of V 2 and the three-momentum of the hardest fermion in the decay of V 1 , ℓ 11 , in the rest-frame of V 1 , where V 1 is the same-flavor ℓ + ℓ − pair whose mass is closest to M Z .
The cos θ 1 distribution captures the spin correlation in the process that enters the matrixelement discriminants used in Higgs analyses.A recent measurement by the ATLAS collaboration of polarization states in q q → ZZ → 4ℓ uses a reweighting based on this variable to approximate the effect of polarization in gg → ZZ → 4ℓ [72].The comparison shows how acceptance estimates would be incorrect if spin correlations were ignored.The largest acceptance difference is observed in Z 0 Z 0 production (green), with smaller effects in Z T Z T (red).We note the importance of spin correlation in ggF already at lowest order.

Sensitivity to longitudinally polarized diboson pairs from gluon fusion
We now present our projections for extracting polarization cross sections from LHC collisions.
We employ the so-called template method, where signal processes are treated as linear combinations of sub-processes, or templates, with unknown weights and fit these weights to (simulated) data.By using individual Z λ Z λ ′ helicity configurations as our templates, we can identify the unknown weights as the polarization fractions f λλ ′ .
We choose the following polarization configurations as our basis of templates: (i) Z 0 Z 0 , (ii) Z T Z T , and (iii) Z T Z 0 .This last one is defined by subtracting the Z 0 Z 0 and Z T Z T channels from the unpolarized process.Formally, the Z T Z 0 template includes a contribution from interference but this is estimated to be small by our closure test.Formally, the Z T Z 0 template includes the Z A component and interference.For resonant diagrams, the Z A contribution is zero because we assume finalstate charged leptons are massless.For simplicity,  22).Templates are stacked and normalized to L = 500 fb −1 of integrated luminosity in √ s = 13 TeV LHC collisions.Contributions from longitudinally polarized Z 0 Z 0 pairs in ggF (blue dash) and EW (red dash) production modes are overlaid.The black solid line indicates the total expected (Total Exp.) number of events with acceptance effects from selection criteria but without taking into account detection efficiency.The hatched band is an estimate of modeling and experimental uncertainties.
we will keep the notation Z T Z 0 for all these contributions hereafter.To maximize the sensitivity to longitudinally polarized Z 0 Z 0 pairs, events are divided into two categories, called "ggF" and "EW", according to the selection criteria below.

Event selection and observable definitions
Simulated events are selected using criteria similar to the H ( * ) → ZZ → 4ℓ analyses performed by ATLAS and CMS [69,71,74,75].Specifically, we require signal and background processes to satisfy the selection cuts given in Eq. ( 12) but tighten the m SF ℓℓ requirement such that where (ℓ 1 ℓ 2 ) is the same-flavor pair with mass closest to M Z .Jets are reconstructed by clustering all final-state hadrons and photons using the anti-k t algorithm [76] with a radius parameter of R = 0.4.Jet candidates are required to satisfy Jets that overlap with any signal lepton are discarded.To increase sensitivity to the EW mode, EW events are required to additionally satisfy All other events are assigned to the ggF region.For both regions, templates are built using the matrix-element (ME) kinematic discriminants The templates for the ggF and EW regions are shown in Figs.3a and 3b, respectively.

Statistical model and expected sensitivity
Expected sensitivity is assessed for two integrated luminosity scenarios at √ s = 13 TeV: L = 500 and L = 3000 fb −1 .The first scenario provides a good approximation for an analysis using the full Run 2 and Run 3 data sets of the LHC.The latter corresponds to the full HL-LHC program.
The expected sensitivity is probed by a profile log-likelihood ratio test statistic [77].The likelihood is built from the product of Poisson probability densities over the bins shown in Fig. 3a and 3b, and Gaussian constraint terms to represent systematic uncertainties with nuisance pa- rameters (NP) α.This profile is given by λ(µ ggF 00 , µ ggF 00 , α) where σ α is the uncertainty associated with α.
We consider three uncorrelated sources of systematic uncertainties: a 10% experimental uncertainty on all processes, a 25% modeling uncertainty on ggF, and a 5% modeling uncertainty on all other processes.We also consider the optimistic scenario where ggF modeling uncertainties are reduced to 15% [29].The function N (µ ggF 00 , µ EW 00 ) is the number of ZZ → 4ℓ events and is given by the sum of individual channels Here, µ ggF 00 and µ EW 00 are signal strengths and multiply the SM predictions for gg → Z 0 Z 0 → 4ℓ and EW qq → Z 0 Z 0 qq → 4ℓqq, respectively.The µ 00 are varied to maximize the log-likelihood function; other polarization components remain unchanged.We further take the natural logarithm of the likelihood and multiply by −2 (2NLL) to obtain a χ 2 asymptotic distribution.The test statistic 2NLL is obtained by profiling the NPs and any signal strength not being inferred.
With the HL-LHC, ATLAS and CMS can constrain the production of gg → Z 0 Z 0 → 4ℓ (qq → Z 0 Z 0 qq → 4ℓqq) to within 2.5× (4.5×) the SM prediction.The expected significance for observing gg → Z 0 Z 0 → 4ℓ is 1.3σ with L = 3000 fb −1 , which is comparable to the expected sensitivity for qq → Z 0 Z 0 jj → 4ℓjj assessed in Ref. [78].The corresponding reach at Run 3 is 4× (8×) the SM prediction.By multiplying by the ratio σ 00 /σ unpol.with respected to the unpolarized cross-section, limits on µ 00 can be translated into limits on the polarization fraction f 00 .
To extract the helicity fractions of Z λ Z λ ′ pairs f λλ ′ (relative to the SM prediction f SM λλ ′ ) we again use the template method.We impose the normalization f 00 + f T T + f T 0+0T +int.= 1, where f T 0+0T +int.accounts for both the (0, T ) + (0, T ) polarization configurations and interference.This reduces the number of independent helicity parameters in the fit to only two: one for the longitudinal fraction (f 00 /f SM 00 ) and the other for the transverse fraction (f T T /f SM T T ).Two statistical models are built to probe the ggF and EW channels individually.
The model for the ggF process is given by Here, s, s 00 , s T T and s T 0 are the integrals over the templates of the fiducial, longitudinal and transverse polarizations in both the ggF and EW regions.The expected sensitivity for measuring the values of f 00 /f SM 00 and f T T /f SM T T in ggF and EW events is shown in Figs.4a and 4b, respectively.We anticipate that projected sensitivities could be improved by employing more sophisticated analysis techniques, e.g., deep neural networks, that are common in ATLAS and CMS frameworks.

Measuring Quantum Properties
Spin, and hence spin-correlation in production and decay chains of heavy particles, is a fundamental quantum property of elementary particles.With our methodology, we can additionally propose two further tests of quantum properties in multiboson systems at high energies: the existence of spin correlation and quantum entanglement.
To show the importance of spin correlation in Z λ Z λ ′ production, we build a statistical model that assesses the sensitivity of the LHC data to spin-correlation following Ref.[79].The model describes the number of ZZ → 4ℓ events (N ) as a combination of the gg → ZZ → 4ℓ process with and without spin correlation, parameterized by f .
Inference on the parameter f , which represents the degree of observed spin correlation, is performed with the same test statistic from Eq. (23).Templates in cos θ 1 , as shown in Fig. 2, are built for all processes.Fig. 5a shows that with Run 2 and 3 data alone, a 5σ exclusion of the null hypothesis (no correlation) may already be possible.
Recently, several groups have discussed the possibility of measuring quantum entanglement with H → V V [19][20][21][22][80][81][82] and t t processes [18,83] at the LHC.The ATLAS collaboration also recently published the first experimental result using t t events [84].In Ref. [20], the authors showed that the absence of quantum entanglement in H → V V decays is equivalent to testing the hypothesis that only the H → V 0 V 0 polarization configuration occurs in nature.As models testing quantum entanglement are similar to those testing polarization and spin correlation, our methodology is applicable.Specifically, our Feynman rules can be used to compute off-shell Higgs production with four final-state fermions when one or both intermediate weak bosons are polarized.
Here we show the LHC is able to falsify the (null) hypothesis that there is no entanglement in H → V V splittings and hence no gg → H * → Z T Z T → 4ℓ subprocess, in the high-mass region where the Higgs boson is produced off-shell.We extend the statistical model of Eq. ( 24) to describe separately the contribution of diagrams with an s-channel Higgs boson (S), the contribution from non-Higgs, "box" diagrams (B), and the interference (I) between them.The method described in Sec. 2 is also employed for these additional templates.The interference template is defined by subtracting the signal and background processes from the full process.This is given by The signal strength µ Higgs T T is the cross section for Z T Z T pairs normalized to the SM prediction.
The expected sensitivity is assessed by using templates of the observable O ggF ME defined in Eq. (22a).For Runs 2 and 3, Fig. 5b shows that the no entanglement hypothesis (µ Higgs T T = 0) can be rejected with approximately 2σ significance.An even stronger rejection can be obtained by optimizing the observable for Higgs processes.

Summary and Conclusion
In this work we proposed a method for computing polarized cross sections that works at the level of Feynman rules.Previous methods typically work at the amplitude or squared-amplitude levels and require specialized event generators.As illustrated in Eqs. ( 6)- (7), the basic idea is to identify the sum over polarizations in a single (unpolarized) propagator subgraph as the sum over multiple (polarized) propagator subgraphs.Our method's efficient implementation into existing, public simulation tools enables realistic simulations of tree-and loop-induced processes, including gg → V λ V λ ′ → 4ℓ, 2ℓ2ν while maintaining spin correlation, off-shell effects, as well as interference between resonant and non-resonant diagrams.After exploring some phenomenological predictions for polarized Z λ Z λ ′ pairs, we presented sensitivity projections for measuring polarized Z λ Z λ ′ pairs when produced via gluon fusion and EW processes during Run 2 and Run 3 of the LHC, and for the larger HL-LHC dataset.We also presented prospects for measuring quantum properties, e.g., entanglement in multiboson systems.
We believe that the techniques describe in this work can be used by experimental collaborations to further explore the production of Higgs bosons at high virtuality and by the theory community to explore ramifications for new physics.While we have focused only on a SM case study, the basic idea can be applied to resonances in new physics scenarios as well as to weak bosons and the top quark in the Standard Model Effective Field Theory.Finally, there are field-theoretic aspects of "polarization as a Feynman rule" that merit further investigation and we encourage such studies.
for internal/external parameters are inherited.We then make the field (re)definition and remove new but unphysical two-point vertices.
We also (re)assign the following PIDs: The fields Z(239) and W ± (±249) are designated Unphysical->True, meaning that they do not appear in the final UFO library.Importantly, a feature in mg5amc is to override particles labels when PIDs match those of SM particles.This means that in mg5amc, the unpolarized fields Z X (23) and W ± X (±24) are automatically relabeled z, w+/w-.

AppendixC. Computational Setup
We assume n f = 5 massless quarks, with a Cabibbo-Kobayashi-Maskawa matrix equal to the identity matrix, i.e., no quark flavor mixing.Other SM inputs are set to the following values: (C.1) The central (ζ = 1) collinear factorization and renormalization scales are set to be half the sum of the transverse energy of final-state particles (f ): Scale uncertainties are estimated by varying ζ discretely over the range {0.5, 1.0, 2.0}.For select processes, we include up to one additional parton in the matrix element using the multi-leg matching (MLM) [85,86].For MLM in the gluon fusion channel, diagrams in which neither a Z nor a H is attached to the quark loop are filtered out since they are part of the qq annihilation channel.Simulated events are showered and hadronized using Pythia8 [87] (tune=pp:14) and matched to the NNPDF23qed LO PDF (PDF:pSet=13) [88].Hadrons are clustered using FastJet [89,90].
Using SM Loop VPolar in MadGraph5 aMC@NLO (mg5amc), loop-induced processes such as Eq. ( 11) can be simulated at lowest order and be passed to other programs for further processes, e.g., parton showering.For gg → Z T Z T → e + e − µ + µ − , the relevant mg5amc syntax to build matrix elements that includes spin-correlation, off-shell effects, and interference is: import model SM_Loop_ZPolar generate g g > e+ e-mu+ mu-QED=4 QCD=2 [noborn = QCD] / a z z0 za output ZPolar_gg_eemm_TT_QED4_QCD2_XLO The g g > e+ e-mu+ mu-syntax will interfere all instances of Z λ Z λ ′ .To filter out unwanted diagrams, including photon contributions, the "/ a z z0 za" syntax is added.The matrix elements for the analogous (i) Z 0 Z 0 , (ii) Z A Z A , and (iii) Z X Z X channels can be generated using the filters: (i) /a z za zt, (ii) / a z z0 zt, (iii) / a z.For the mixed-polarization configuration Z 0 Z T +Z T Z 0 , we apply the diagram selection described in Sec.2.1 and available from the repository given in Footnote 3. The relevant mg5amc syntax is then The cross sections in Eq. ( 15) can be obtained using the following (and similar) commands: launch ZPolar_gg_eemm_00_QED4_QCD2_XLO analysis=off set pdlabel lhapdf set lhaid 324900 set lhc 13 set nevents 4k set dynamical_scale_choice 3 set me_frame [3,4,5,6]

Figure 1 :
Figure 1: Upper panel: As a function of √ s, the unpolarized gg → ZZ → e + e − µ + µ − cross section (black band) with phase space cuts of Eq. (12), along with the analogous Z T Z T (blue), Z T Z 0 (purple), and Z 0 Z 0 (green) rates.Band thickness denotes the scale uncertainty.Lower panel: The ratio of the Z T Z T (blue) and Z X Z X (red) cross sections and their uncertainties relative to the unpolarized rate.(Z X is the sum of the Z T , Z 0 , and Z A channels at the amplitude level.)Larger (transparent) bands are the scale uncertainties; smaller (solid) bands are the PDF uncertainties.

Figure 2 :
Figure 2: Comparison between simulated polarized gg → Z λ Z λ ′ → e + e − µ + µ − events with phase space cuts of Eqs.(12) and (19), with and without spin correlation.The ratios are with respect to the unpolarized result with spin correlation.

Figure 3 :
Figure 3: Templates for the observables used in the (a) ggF and (b) EW regions as defined in Eq. (22).Templates are stacked and normalized to L = 500 fb −1 of integrated luminosity in √ s = 13 TeV LHC collisions.Contributions from longitudinally polarized Z 0 Z 0 pairs in ggF (blue dash) and EW (red dash) production modes are overlaid.The black solid line indicates the total expected (Total Exp.) number of events with acceptance effects from selection criteria but without taking into account detection efficiency.The hatched band is an estimate of modeling and experimental uncertainties.

Figure 5 :
Figure 5: (a) Test statistics scan with respect to the parameter f which represents the degree of spincorrelation in gg → ZZ → 4ℓ process, assuming L = 500 fb −1 of LHC data.(b) Same as (a) but with respect to signal strength multiplying the SM production of polarized gg → H * → Z T Z T → 4ℓ.

Figure B. 6
Figure B.6 shows the expected sensitivity for the observation of longitudinally polarized Z 0 Z 0 pairs in (a,c) gg → ZZ → 4ℓ and (b,d) qq → ZZjj → 4ℓjj production in the high-mass regime (a,b) after Runs 2 and 3 of LHC data taking, i.e., L = 500 fb −1 at √ s = 13 TeV, as well as (c,d) with the full HL-LHC data set, i.e., L = 3000 fb −1 .