Probing Neutral Triple Gauge Couplings with Z ∗ γ ( ν ¯ νγ ) Production at Hadron Colliders

We study probes of neutral triple gauge couplings (nTGCs) via Z ∗ γ production followed by off-shell decays Z ∗ → ν ¯ ν at the LHC and future pp colliders, including both CP-conserving (CPC) and CP-violating (CPV) couplings. We present the dimension-8 SMEFT operators contributing to nTGCs and derive the correct form factor formulation for the off-shell vertices Z ∗ γV ∗ ( V = Z, γ ) by matching them with the dimension-8 SMEFT operators. Our analysis includes new contributions enhanced by the large off-shell momentum of Z ∗ , beyond those of the conventional ZγV ∗ vertices with on-shell Zγ . We analyze the sensitivity reaches for probing the CPC/CPV nTGC form factors and the new physics scales of the dimension-8 nTGC operators at the LHC and future 100 TeV pp colliders. We compare our new predictions with the existing LHC measurements of CPC nTGCs in the ν ¯ νγ channel and demonstrate the importance of our new method


Introduction
Neutral triple gauge couplings (nTGCs) are attracting increased theoretical and experimental interest [1][2][3][4] [5][6].This is largely driven by the fact [7][8] that the nTGCs do not appear in the Standard Model (SM) Lagrangian, nor do they show up in the dimension-6 Lagrangian of the SM Effective Field Theory (SMEFT) [10], suggesting that they could open up a unique new window to physics beyond the SM that may first appear at the dimension-8 level.The SMEFT provides a powerful universal framework to formulate model-independently such new physics beyond the SM, by parametrizing the low-energy effects of possible high-mass new physics in terms of operators composed of the SM fields that incorporate the full SU(3)⊗SU( 2)⊗U (1) gauge symmetry of the SM.
There have been extensive studies of the dimension-6 operators [10][11][12] in the SMEFT, including the experimental constraints on their coefficients and hence on the associated new physics cutoff scale Λ.But these studies do not involve nTGCs because they first appear as a set of dimension-8 operators of the SMEFT.In general, dimension-8 operators make interference contributions to amplitudes at O(1/Λ 4 ), and thus can contribute to cross sections at the same order.But dimension-6 SMEFT operators contributing to amplitudes at O(1/Λ 2 ) also contribute to cross sections at O(1/Λ 4 ) in general, complicating efforts to isolate any dimension-8 contributions.The absence of dimension-6 contributions to nTGCs avoids this complication, making them an ideal place to probe the new physics at dimension-8 level.
Previous phenomenological studies of nTGCs in the SMEFT formalism have mainly focused on CP-conserving (CPC) operators that contribute to scattering amplitudes involving the neutral triple gauge vertex (nTGV) ZγV * (V = Z, γ) through the reaction f f → Zγ .In the case of e + e − colliders, on-shell Zγ production with Z → ℓ − ℓ + , ν ν, q q final states have been considered, but at pp colliders off-shell invisible decays Z * → ν ν cannot be separated from on-shell Z decays because of the insufficient kinematic information of Z boson.Hence, for the ν νγ channel it is important to include the off-shell production of Z * → ν ν at pp colliders.
In this work, we study the nTGVs with two off-shell bosons, Z * γV * (V = Z, γ), which include contributions from additional dimension-8 operators that were not considered in the previous nTGC studies.We include a new analysis of CP-violating (CPV) nTGCs.As the basis for this work, we first formulate the correct CPC and CPV form factors of the doubly off-shell nTGVs Z * γV * that are compatible with the full electroweak SU(2)⊗U(1) gauge symmetry of the SM [13].By matching the CPC and CPV dimension-8 nTGC operators with the corresponding nTGC form factors, we derive the correct formulations of the CPC and CPV Z * γV * form factors.We then use these formulations to study the sensitivity reaches of the LHC and the projected 100 TeV pp colliders for probing the CPC and CPV nTGCs through the reaction pp(q q) → Z * γ → ν νγ .We further compare our new predictions with the existing LHC measurements [5] of CPC nTGCs in the ν νγ channel, and demonstrate the importance of using our new nTGC form factor formulation for the correct LHC experimental analysis.

2.
Formulating Z * γV * Form Factors from Matching the SMEFT In previous works [1][2][3], we studied the dimension-8 SMEFT operators that generate nTGCs and their contributions to Zγ production at the LHC and future colliders.In particular, we studied systematically ZγV * vertices including their matching with the corresponding SMEFT operators and the correct formulation of nTGC form factors that respect the full electroweak SU( 2)⊗U (1) gauge symmetry of the SM.However, unlike the case of e + e − collisions where the on-shell constraint can be imposed on the invisible decays of Z→ ν ν , this is not possible in pp collisions, for which only the missing transverse momentum can be measured.Moreover, since Z boson is an unstable particle, the invariant-masses of ℓ + ℓ − and q q final states from Z decays are not exactly on-shell, in general.For these reasons, it is important to study the nTGVs with the final-state Z * off-shell, as well as the initial-state V * .
The general dimension-8 SMEFT Lagrangian takes the following form: where the dimensionless coefficients cj may take either sign, sign(c j ) = ± . CPC: CPV: CPV: CPV: where H denotes the Higgs doublet of the SM, and the covariant derivative term D σ B µν = ∂ σ B µν holds for the Abelian field strength.The operators (2.2a) and (2.2c)-(2.2e)were given in [8], to which we have further added an independent CPC operator (2.2b).
For the dimension-8 CPC and CPV nTGC operators containing pure gauge fields only, we have the following: where the operators (O G+ , O G− ) are CPC [2], and the two new CPV operators ( O G+ , O G− ) are constructed.We note that the operators (2.2)-( 2.3) belong to the two classes, F 2 ϕ 2 D 2 and F 3 D 2 .From the classification of [9] the relevant dimension-8 operators contains F 2 H 2 D 2 , F 2 ψ 2 D, and F 4 , where F 2 ψ 2 D and F 4 do not explicitly contain any nTGC vertices and thus are not included here.Using integration by parts and equations of motion (EOM), we find that F 3 D 2 type can be converted into three types of operators , where F 2 H 2 D 2 corresponds to our Eq.(2.2).Moreover, the F 2 H 2 D 2 type operators contribute to the form factors , as shown by Eqs.(2.6)(2.8)below.Hence, the operator types F 2 ϕ 2 D 2 and F 3 D 2 provide the optimal basis for the current nTGC study.These are further explained in Sec.S1 of the Supplemental Material [21].
The conventional formalism for nTGC form factors was proposed over 20 years ago [7] and respects only the residual gauge symmetry U (1) em .However, as we stressed in Ref. [1], it does not respect the full electroweak SU( 2)⊗U( 1) gauge symmetry of the SM, and leads to large unphysical high-energy behaviors of certain scattering amplitudes [1].We thus proposed [1] a new formulation of the CPC form factors of the nTGVs ZγV * that is compatible with the full SM gauge group with spontaneous electroweak symmetry breaking.We will construct an extended formulation to include the CPV nTGVs for the present study.
The doubly off-shell nTGC form factors for Z * γV * vertices are more complicated than the ZγV * form factors. Matching with the dimension-8 nTGC operators of the SMEFT, we can parametrize the Z * γV * vertices in terms of the following form factors: where expressions for X βµ 1V and X αβ 3V are given in Appendix S1 and we find that they make vanishing contributions to the reaction f f→ Z ( * ) γ with Z ( * ) →f ′ f ′ .Hence, the present analysis only involves the vertices Γ αβµ Z * γV * .We present first the CPC parts of the Γ αβµ Z * γV * vertices: In the above, we use the hat symbol to distinguish off-shell form factors ( ĥZ 3 , ĥγ 3 , ĥ4 ) from their on-shell counterparts (h γ 3 , h Z 3 , h 4 ) as studied in [1].The CPC form-factor parameters (h γ 31 , ĥγ 3 , ĥZ 3 , ĥ4 ) can be mapped precisely to the cutoff scales , (2.6a) where [Λ 4 j ] ≡ sign(c j )Λ 4 j .The above relations hold for any momentum q 1 of Z * , and for q 2  1 =M 2 Z the off-shell form factors ( ĥZ 3 , ĥγ 3 , ĥ4 ) reduce to the on-shell cases, ĥ4 = h 4 , ĥZ 3 = h Z 3 , and ĥγ 3 +h γ 31 = h γ 3 , where h γ 31 is part of the on-shell form factor, h γ 31 = h γ 3 − ĥγ 3 , for q 2 1 =M 2 Z .We note that the formulations of the off-shell nTGC form factors Z * γV * in Eq.(2.5) (the CPC case) and in Eq.(2.7) (the CPV case) were not given explicitly in the previous literature, including Refs.[7]- [8].As we discuss in Section 3, CMS [4] and ATLAS [5] measured the CPC nTGC form factors h γ 3 and h V 4 via the reaction pp(q q) → Z * γ → ν νγ , but used the conventional CPC nTGC form factor formulation of ZγV * in which both the final-state bosons Zγ are assumed to be on-shell [7].This means that their measurement of h γ 3 is simply equivalent to measuring the form factor h γ 31 in Eq.(2.5a) above [14].Hence the CMS and ATLAS analyses [4][5] missed the new form factor ĥγ 3 in Eq.(2.5a), whose contribution dominates over that of h γ 31 in the ν νγ channel for both the LHC and 100TeV pp colliders, as we will demonstrate in Section 3.
Next, using the Lagrangian (S5) for nTGVs, we construct the following off-shell CPV nTGVs Γ αβµ Z * γV * : which have important differences from the conventional CPV nTGC form factors [7] as we explain in Appendix S1.We note that when the final-state Z boson is on-shell (q 2 1 = M 2 Z ), the above off-shell form factors should reduce to the on-shell ones, h γ 11 + ĥγ , and ĥ2 = h 2 .In the on-shell limit, the longitudinally polarized on-shell external state Z L (q 1 ) should satisfy the equivalence theorem (ET) [15], which puts nontrivial constraints on the structure of form factors, as shown in Appendix S1.
Then, we can match these CPV nTGC form factors to the dimension-8 gauge-invariant CPV operators (2.2)- (2.3) in the broken phase and derive the following correspondence relations: , (2.8a) ) ) ) form factors arise as a consequence of spontaneous electroweak symmetry breaking, and would vanish if ⟨H⟩ = 0 .We also note that ĥγ 1,3 and ĥ2,4 arise from the electroweak rotation of the BW 3 W 3 vertex, so the ĥγ 1 term in Eq.( 2.7a) and ĥγ 3 term in Eq.( 2.5a) would vanish if s W = 0 .
In passing, we have also derived the perturbative unitarity bounds on the cutoff scales Λ j and the form factors h V j in Section S3 of the Supplemental Material.We verify that these bounds are much weaker than our current collider limits (presented in Section 3) and thus do not affect our collider analyses.

Probing nTGCs with Z * γ (ν νγ) Production at Hadron Colliders
The invariant mass of Z ( * ) →ν ν decays cannot be measured in the reaction pp(q q)→ν νγ at hadron colliders, so it becomes impossible to determine whether the invisible Z-decay is on- shell or not.Hence analyzing the ν νγ production with off-shell Z * decays is important for hadron colliders.
The cross section for q q → Z ( * ) γ at a pp collider can be expressed as where the functions F q/p and F q/p are the parton distribution functions (PDFs) of the quark and antiquark in the proton beams, and ŝ = x 1 x 2 s .The PDFs depend on the factorization scale µ, which we set to µ= √ ŝ /2 in our leading-order analysis.We use the PDFs of the quarks q = u, d, s, c, b and their antiquarks determined by the CTEQ collaboration [16].In principle, ŝ can be determined by measuring the invariant-mass of observable final-state particles.ATLAS measurements of M ℓℓγ during the LHC Run-2 reached around 3 TeV.Accordingly, in our analysis of Z * γ (ν νγ) production we consider the relevant range ŝ < 3 TeV for the LHC and ŝ < 23 TeV for a 100TeV pp collider.
We compute the partonic cross-section in three parts, where σ 0 is the SM contribution, σ 1 is the contribution of the nTGCs interfering with the SM amplitude, and σ 2 denotes the squared nTGC contribution.We present explicit formulae for The CPC and CPV amplitudes do not interfere for q q → Z ( * ) γ, hence the contributions of the CPV nTGCs to σ 1 vanish.Moreover, we find that σ 1 ≪ σ 2 holds for the CPC nTGCs at the LHC and future pp colliders, making σ 1 negligible in this analysis [17].We further observe that each CPV nTGC gives the same contribution to σ 2 as that of the corresponding CPC nTGC.
The final state γ is the only visible particle when Z ( * ) has invisible decays, and the longitudinal momentum of the ν ν pair cannot be measured at hadron colliders.Hence, the photon's transverse momentum P γ T is the main variable that can be used to distinguish the new physics signals from the SM backgrounds.We present in Fig. 1 |×10 4 2.5 2.0 1.0 |h γ  31,11 |×10 7 8.3 5.7 4.0   Table 1: Sensitivity reaches on probing the CPC and CPV nTGC form factors at the 2σ level, as obtained by analyzing the reaction pp(q q) → Z * γ → ν νγ at the LHC (13 TeV) and the 100 TeV pp collider, for the indicated integrated luminosities.In the last two rows, the ĥγ 3,1 sensitivities (red color) include the Z * -momentum-square (q 21 ) enhanced off-shell effects, whereas the h γ 31,11 sensitivities (blue color) do not.
for the LHC (13TeV) in plot (a) and for the projected 100 TeV pp collider in plot (b).In each plot, the SM cross section is given by the black curve, and the contributions of the nTGC form factors ( ĥ4 , ĥZ 3 , ĥγ 3 , h γ 31 ) (taking a reference value 10 −4 ) are shown by the blue, green, red and red-dashed curves.We observe from Eq.(2.5a) that the ĥγ 3 contribution is strongly enhanced by the off-shell Z * momentum factor q 2 1 /M 2 Z , whereas the ĥγ 31 contribution is not.Unlike ĥγ 3 , this does not happen to ĥZ 3 because Eq.( 2.5b) shows that the numerator factor q 2 3 − q 2 1 exhibits a strong cancellation between q 2 3 (= ŝ) and q 2 1 when q 2 1 goes far off-shell, which is particularly relevant for the high-energy 100 TeV pp collider.We find that the ĥγ 3 contribution (red solid curve) is larger than that of ĥγ 31 (red dashed curve) by about a factor of (2−3) for the LHC, as seen in Fig. 1(a), and by a large factor of (43−77) in the case of the 100 TeV pp collider, as seen in Fig. 1(b).We note that the ĥ4 contribution is much larger than that of ĥZ 3 , ĥγ 3 , and h γ 31 , because the ĥ4 terms are enhanced by an extra large momentum factor of q 2 q 3 or q 2 3 , as shown in Eq. (2.5).
In order to optimize the detection sensitivity, we divide events into bins of the P T (γ) distribution, whose widths we take as ∆P T = 100 GeV for the LHC and ∆P T = 500 GeV for the 100 TeV pp collider.Then, we compute the significance Z bin for each bin, and construct the following total significance measure: Since the SM contribution σ 0 becomes small when the photon P T is high, we determine the statistical significance by using the following formula for the background-with-signal hypothesis [20]: where L is the integrated luminosity and ϵ denotes the detection efficiency.For our analysis we choose an ideal detection efficiency ϵ = 100% unless specified otherwise.We present sensitivities for probing the CPC and CPV nTGC form factors and the corresponding new physics scales of the dimension-8 nTGC operators in Tables 1 and 2, respectively.For the LHC, we find that the sensitivities to ĥ2 and ĥ4 can reach the level of O(10 −5 −10 −6 ), whereas the sensitivities to ĥZ 3,1 , ĥγ nTGC form factors at the projected 100 TeV pp collider are generally much higher than those at the LHC, by a factor of O(10 2 −10 3 ).At the LHC the sensitivities to ĥγ 3,1 are stronger than those to h γ 31,11 and ĥZ 3,1 by about (50−60)%, whereas at the 100TeV pp collider the sensitivities to ĥγ 3,1 are much higher than those to h γ 31,11 and ĥZ 3,1 , by factors of O( 10).We recall that both CMS (using 19.6/fb of Run-1 data) [4] and ATLAS (using 36.9/fb of Run-2 data) [5] measured the CPC nTGC form factors (h V 3 , h V 4 ) via the reaction pp(q q) → Z * γ → ν νγ ; but they used the conventional CPC nTGC form factor formulation of the ZγV * vertex with both Zγ being on-shell [7].This differs from our off-shell formulation of the Z * γV * vertex in Eq. (2.5), which gives the correct nTGC form factor formulation for analyzing the reaction pp(q q) → Z * γ → ν νγ at hadron colliders.For instance, using 36.1/fb of Run-2 data at the LHC(13TeV), ATLAS obtained the following bounds at 95% C.L. [5]: For a quantitative comparison, we impose the same cut P γ T > 600 GeV as that of the ATLAS analysis [5] and choose a detection efficiency ϵ = 70% [5][19].We then derive the following LHC bounds on the CPC nTGCs at 95% C.L.: Comparing our results (3.6) with the ATLAS result (3.5), we find that our bounds on h γ 31 and ĥZ 3 agree well with the ATLAS bounds on h γ 3 and h Z 3 (to within a few percent), whereas our bound on ĥγ 3 is significantly stronger than the ATLAS bound on h γ 3 by about 60%.This is because in our off-shell formulation of Eq.(2.5) the form factor ĥγ 3 is enhanced by the Z * off-shell factor q 2 1 /M 2 Z , but h γ 31 and ĥZ 3 are not and thus their bounds are quite similar to the case of assuming on-shell invisible Z decays.On the other hand, our ĥ4 bound in Eq. (3.6) is much weaker than the ATLAS bounds on (h γ 4 , h Z 4 ), by a large factor of ∼ 32, because the conventional nTGC form factor formulae are incompatible with the gauge-invariant SMEFT formulation of dimension-8 (including spontaneous electroweak gauge symmetry breaking), as we explain in Appendix S1 and Table S3.We emphasize that the existing ATLAS result [5] used the conventional on-shell nTGC formula of the vertex ZγV * [7] for analyzing the ν νγ channel, which caused a significant underestimate of the sensitivity to ĥγ 3 by about 60%.This underlines the importance for the on-going LHC experimental analyses to use the correct theoretical formalism to analyze the ν νγ channel for probing the nTGCs, as proposed in this work.
In Table 2 we demonstrate that the sensitivities to new physics scales in the coefficients of O G+ and O G+ can reach (2.7− 4.1) TeV at the LHC, and (19 − 28) TeV at the 100 TeV pp collider.The sensitivities to probing new physics scales of other nTGC operators are around (1−1.8) TeV at the LHC and (3.9−9.3)TeV at the 100TeV pp collider.We find that the sensitivities to the coefficients of O G− , O G− and ĥγ 1,3 are significantly higher than that of the case by assuming the on-shell Zγ final states [1].In addition, we have studied the perturbative unitarity constraints [22] on the nTGC contributions, as presented in Section S3 of the Supplemental Material [21].We demonstrate [21] that the unitarity bounds on Λ j and h V j are much weaker than our current collider bounds in Tables 1-2, hence they do not affect our collider analyses.
Finally, we analyze the correlation contours (95% C.L.) between the sensitivities to the form factors of the off-shell nTGC vertex Z * γV * , as shown in Fig. 2, where we have chosen an ideal detection efficiency ϵ = 100%.We find that the behavior of the ĥ4 − ĥZ 3 correlation is similar to that of their on-shell counterparts h 4 and h Z 3 [1].Specifically, the ĥ4 − ĥV 3 correlation is rather small, while the sensitivity to ĥγ 3 is much greater than that to h γ 31 , especially at a 100 TeV pp collider.On the other hand, the correlation between ĥZ 3 − ĥγ 3 is large at the LHC, but almost invisible at the 100 TeV pp collider.This is because the ĥγ 3 contribution is enhanced by the off-shell momentum-square q 2 1 /M 2 Z of Z * and ĥZ 3 is not, which make the ĥZ 3 − ĥγ 3 correlation suppressed by 1/ q 2 1 .At the LHC the on-shell and off-shell contributions to h γ 3 are comparable and thus the ĥZ 3 − ĥγ 3 correlation can be significant, whereas their correlation is much suppressed by 1/ q 2 1 at the 100 TeV pp collider and thus becomes nearly invisible.We note that no correlation exists between the CPC and CPV nTGCs because their amplitudes only differ by ±i.Finally, correlations among ĥ2 , ĥZ 1 , ĥγ 1 , and h γ 11 are similar to those of the corresponding CPC nTGCs, because the squared term σ 2 dominates the signal cross section at the LHC and at the 100TeV pp collider.

Conclusions
In this work, we have demonstrated that the reaction pp(q q)→ Z * γ → ν νγ can probe sensitively both the CPC and CPV nTGCs at the LHC and at the projected 100TeV pp colliders.It has comparable sensitivities to the on-shell production channels pp(q q)→Zγ with Z → ℓ + ℓ − [1].Because the on-shell constraint cannot be imposed on the final state Z boson with invisible decays at hadron colliders, we studied the Z * γ production with off-shell decays Z * →ν ν .For this, we presented a new general formulation of the CPC/CPV doubly off-shell Z * γ V * form factors and their nontrivial matching with the full SU( 2)⊗U(1) gauge-invariant SMEFT operators of dimension-8.In consequence, we found that the conventional on-shell form factors for ZγV * vertices [7] are inadequate, and the conventional form factors Eqs.(2.5) and (2.7).
In the ν νγ production channel at pp colliders, only the photon P γ T is measurable, and we have studied how to optimize the sensitivities to nTGCs by proper choices of P γ T bins at the LHC and the projected 100 TeV pp colliders.Since the interference cross section is always negligible relative to the squared term at these colliders [17], the sensitivities to the CPV nTGCs are similar to those to the corresponding CPC nTGCs.
We have presented the prospective sensitivity reaches on the nTGC form factors ĥV j (compatible with the full electroweak gauge symmtry with spontaneous breaking) at the LHC and the 100 TeV pp collider in Table 1, and the sensitivity reaches on probing the new physics scales Λ j of the corresponding dimension-8 nTGC operators in Table 2.We found in Table 1 that including the off-shell decays Z * → ν ν can increase the sensitivity reaches for ( ĥγ 3 , ĥγ 1 ) by (50−60)% at the LHC and by a factor of O( 10) at the 100 TeV pp collider.We present in Fig. 2 the correlations between the sensitivities to each pair of nTGC form factors.
We further made quantitative comparisons between the existing ATLAS measurements [5] of the CPC nTGC form factors and our new predictions, as shown in Eqs.(3.5)- (3.6).We found that for the ν ν γ channel the LHC sensitivity reaches on probing the nTGC form factors ĥγ 3 and ĥ4 differ significantly from the ATLAS results (using the conventional ZγV * formulae which are inconsistent with spontaneous breaking of the electroweak gauge symmetry).
This work establishes a new perspective for on-going experimental probes of the new physics in nTGCs at the LHC and the projected 100 TeV pp colliders.We look forward to continuing the fruitful cooperation with the LHC experimental groups, extending their on-going nTGC analyses by using our new nTGC formulation, which consistently incorporates the nTGC form factors with the corresponding gauge-invariant dimension-8 operators of the SMEFT.
[13] We emphasize that the conventional electroweak form factor formulation imposes only the U(1) gauge symmetry of QED, and is in general incompatible with the SMEFT framework, which takes into account the full SU( 2)⊗U( 1) electroweak gauge symmetry of the SM.We stress that it is important to match precisely the form factors with the corresponding SMEFT operators in the broken phase, which can place additional nontrivial constraints on the structure of the form factors as a result of the spontaneous electroweak symmetry breaking of the SM.We demonstrate this for our correct formulation of the nTGC form factors in Section 3.
[14] We note that the CMS [4] and ATLAS [5] collaborations also measured the CPC nTGC form factor h V 4 using the conventional formula, which gives rise to unphysically large highenergy behavior [1].
[15] For a comprehensive review, see H. J.In this Supplemental Material, we provide further details to support the analyses in the main text1 .These include three parts.In Section S1, we first derive systematically the neutral triple gauge boson vertices (nTGVs) at the Lagrangian level from the corresponding SU( 2)⊗U( 1) gauge-invariant dimension-8 operators of the SM Effective Field Theory (SMEFT) in its electroweak broken phase.Then, we present a systematic matching between the doubly off-shell Z * γV * form factors and the SMEFT operators, which puts nontrivial constraints on the neutral triple gauge coupling (nTGC) form factors and ensures the correct high energy behaviors.In Section S2, we provide cross section formulae for CP-conserving (CPC) and CP-violating (CPV) nTGC contributions to the reaction q q → Z * γ .For comparison with the Tables I-II of the main text, we further present the combined sensitivitiy reaches on probing the nTGC form factors h V j and new physics scales Λ j for both the ν νγ channel 1 and ℓ − ℓ + γ channel [1].Finally, in Section S3, we present systematically the inelastic unitarity bounds on the nTGC form factors h V j and new physics scales Λ j for both the CPC and CPV cases.These perturbative unitarity bounds are shown to be much weaker than our collider sensitivity reaches for the relevant energy scales, so they do not affect our present collider analyses.

S1. Matching Z * γV * Form Factors to the SMEFT Operators
For the present nTGC study, we have considered two classes of dimension-8 operators, F 2 H 2 D 2 and F 3 D 2 , where the symbols (F, H, D) denote the gauge field strength, Higgs doublet, and covariant derivative, respectively.Using integration by parts and equation of motion, F 3 D 2 can be converted to operators of the forms F 4 , F 2 ψ 2 D, and F 2 H 2 D 2 , where ψ denotes fermion field, and the classes of F 4 and F 2 ψ 2 D have no direct contribution to nTGCs.The operator type F 3 D 2 was not chosen as the basis in Ref. [9] since its general classification [9] was not optimized for studying any specific type of new physics operators such as the nTGCs.We note that the operator class F 2 H 2 D 2 can contribute only to the form factors h V 1 and h V 3 , but not to the form factors h 2 and h 4 .Thus, we find that without including the class F 3 D 2 , there would be no corresponding operators contributing to the form factors h 2 and h 4 .Hence, choosing the operator basis of F 2 H 2 D 2 and F 3 D 2 is necessary for a complete formulation of nTGCs and their phenomenological studies.As we have verified [1][2], for the reaction f f → Zγ, the contributions of h 2,4 from the F 3 D 2 type operators are equivalent to that of certain combinations of F 2 H 2 D 2 and F 2 ψ 2 D operators.We studied [1][2] the fermionic operators of type F 2 ψ 2 D : They are connected to nTGC operators by equation of motions, but they do not contribute directly to the nTGCs.Working in the electroweak broken phase of the SM, SU(2) W ⊗U(1) Y →U( 1) em , we expand the CPC operators and derive the relevant neutral triple gauge vertices (nTGVs) as follows: where we adopt the notations We then expand the CPV operators in the electroweak broken phase respecting only the residual SM gauge symmetry SU(3) C ⊗U(1) em .In this way we derive the relevant CPV nTGVs as follows: where we have defined (s 2W , c 2W ) ≡ (sin 2θ W , cos 2θ W ). We note that in Eq.( S4b) the factor ∂ •Z vanishes for an on-shell Z boson, while for an off-shell Z * it also leads to vanishing results as long as the vector boson couples to a pair of fermions or gauge bosons with equal mass.The same is true for the factor ∂ •A.Thus, we can drop terms with either ∂ •Z or ∂ •A for the present study.Our next step is to classify the complete structure of the nTGVs in the broken electroweak phase that are generated by the above SMEFT operators of dimension-8.The Lagrangian of these nTGVs in the electroweak broken phase contains the following CPC and CPV nTGC form factors, which generally hold for all fields being off-shell: Matching the above form factor Lagrangian with the corresponding dimension-8 nTGC operators in Eq.(S4), we derive the following relations for the CPC nTGC form factors: and for the CPV form factors: where [Λ 4 j ] ≡ sign(c j )Λ 4 j .We stress that the nTGC form factor formulas in Eq.( S5) generally hold at the Lagrangian level with all the fields being off-shell, and they are consistently built upon the SMEFT broken phase formulation of the nTGCs.This provides a fully general and consistent off-shell formulation of nTGCs at the Lagrangian level.
Next, using the general Lagrangian formulation of nTGVs in Eq.(S5), we further derive the neutral triple gauge vertices Z * γV * by requiring one photon field A µ be on-shell, where the on-shell photon field satisfies the conditions ∂ 2 A µ = 0 and ∂ µ A µ = 0 .Thus, we can derive the complete form factor formulation of the doubly off-shell nTGVs in momentum space: where the expressions of X βµ 1V and X αβ 3V are given as follows: In the above the additional form factor coefficients χ V ij are defined as follows: ) where Using the on-shell conditions for the initial/final state fermions, we can readily prove that the contributions of X βµ 1V and X αβ 3V to the physical processes of the present study vanish, so they are not needed for this work, but we list them here for completeness.
Finally, we clarify the difference between our new nTGC form factor formulation and the conventional nTGC form factor formulation [7].Since there is no conventional form factor formulation for the doubly off-shell nTGVs Z * γV * (with only γ on-shell) given in the literature [7][8], we compare our new formulation and the conventional formulation [7] for the nTGVs ZγV * when both Z and γ are on-shell.The conventional form factor parametrization of the CPC and CPV neutral triple gauge vertices Γ αβµ ZγV * yield the following formulae [7]: We found in [1] that the conventional CPC form factor h V 4 of Eq.( S11a) is incompatible with the spontaneous breaking of the full electroweak gauge group SU(2) W ⊗U(1) Y →U(1) em , and leads to unphysically large high-energy behaviors.Hence we must introduce a new form factor h V 5 as in Eq.(S12a) for matching consistently with the broken phase results originating from the CPC dimension-8 operators (2.2a)-(2.2b)and (2.3a)-(2.3b) of the SMEFT that respect the underlying electroweak SU(2) W ⊗U(1) Y gauge symmetry.Furthermore, we find that the conventional CPV form factor h V 2 of Eq.( S11b) is incompatible with the spontaneous breaking of the full electroweak gauge group SU(2) W ⊗U(1) Y →U (1) em and also causes unphysically large high-energy behavior.To construct a consistent formulation, we introduce another new form factor h V 6 as in Eq.( S12b) to match with the broken phase results originating from the CPV dimension-8 operators (2.2c)-(2.2e)and (2.3c)-(2.3d) of the SMEFT that respect the full underlying electroweak gauge symmetry.We then present the extended nTGC form factor formulations for both the CPC case [1] and the CPV case: where the form factor coefficients h V 1,2,3,4 are the conventional form factors [7] and the form factor terms ∝ h V 5 and h V 6 belong to our extension.We observe that the form factor coefficients (h V 4 , h V 5 ) (CPC) and (h V 2 , h V 6 ) (CPV) are subject to further constraints due to the matching with the broken phase results of the corresponding dimension-8 nTGC operators that respect the full SU(2) W ⊗U(1) Y electroweak gauge symmetry.These constraints ensure that for each scattering amplitude the nTGC form factor contributions have the same high-energy behaviors as those of the corresponding dimension-8 nTGC operators.For this, we make a precise matching between the above nTGC form factor formulae (S12a)-(S12b) and the dimension-8 nTGC operators (2.2)- (2.3).From these we derive the following nontrivial relations between the nTGC form factor coefficients: Imposing these relations, we can further express the formulae (S12a) and (S12b) as follows: Table S3: Comparisons of the 2σ sensitivity reaches for the form factors ( ĥ4 , ĥ2 ) formulated in the SMEFT (in red color) and for the conventional form factors (h V 4 , h V 2 ) respecting only U( 1) em (in blue color), derived from the reaction pp(q q)→ Zγ→ ν νγ at the LHC (13 TeV) and a 100 TeV pp collider, with the indicated integrated luminosities.As discussed in the text, the form-factor limits in blue color are included for illustration only, as they are incompatible with the spontaneous breaking of the SM electroweak gauge symmetry, and hence are invalid.
The SM cross section term (σ 0 ) and the interference term (σ 1 ) are given by ) while the squared contribution (σ 2 ) includes the following terms: where the nTGC parameters ) In the above, the squared-mass M 2 Z * = q 2 1 with q 1 being the Z * momentum, and the coefficients (q L , q R ) = (T 3 −Qs Table S4: Combined sensitivity reaches on probing the CPC and CPV nTGC form factors at the 2σ level, as obtained by analyzing the reactions pp(q q) → Z * γ → ν νγ and pp(q q) → Zγ → ℓ + ℓ − γ at the LHC (13 TeV) and the 100 TeV pp collider, for the indicated integrated luminosities.In the last two rows, the ĥγ 3,1 sensitivities (red color) are significantly higher than those of h γ 31,11 (blue color) because the former contains the off-shell contributions from ν νγ channel as enhanced by Z * -momentum-square (q 2 1 ) and the latter does not.
Finally, we combine the sensitivity reaches on probing the CPC and CPV nTGC form factors by using the bounds derived from the reaction pp(q q)→Z * γ →ν νγ (given in Table I of the main text) and the reaction pp(q q) → Zγ → ℓ + ℓ − γ.We analyzed the ν νγ channel in [1] only for CPC nTGCs, and for the current combined analysis we further include the CPV nTGCs.We present the combined sensitivity reaches on probing the CPC and CPV nTGC form factors at the 2σ level, for both the on-going LHC and the projected 100 TeV pp collider, as shown in Table S4.We find that the sensitivity reaches on the form factors ĥ4,2 , ĥZ 3,1 , and h γ 31,11 are comparable for both ν νγ and ℓ + ℓ − γ channels, where the ν νγ channel has higher sensitivities than the ℓ + ℓ − γ channel by about (20−30)% at the LHC and the 100 TeV pp collider, and thus their combined sensitivities have visible improvements.On the other hand, the sensitivity reaches on the form factors ĥγ 3,1 via the ν νγ channel are significantly higher than those via the ℓ + ℓ − γ channel by about 93% at the LHC and by a large factor of ∼ 11 at the 100 TeV pp collider, because of the major off-shell enhancement on the ν νγ signals as we demonstrated in the main text.This explains why the combined sensitivity reaches on ĥγ 3,1 (shown in Table S4 and marked in red color) are dominated by the ν νγ channel and remain nearly the same as the sensitivities of the ν νγ channel alone (shown in Table I of the main text).We also analyzed the combined sensitivity reaches on the new physics cutoff scale Λ j for each given nTGC operator O j .Since the nTGC contribution to the signal cross section is dominated by the σ 2 term which is proportional to 1/Λ 8 j (as compared to σ 2 ∝ ĥ2 j for the form factor contributions), we would expect a rather minor enhancement by 2 1/16 − 1 ≃ 4.4% even if the two channels of ν νγ and ℓ + ℓ − γ would contribute equal statistic significance.We present in Table S5 the combined sensitivity reaches on the new physics scales Λ j (in TeV) of the dimension-8 nTGC operators at 2σ level.As expected, it shows that the combination of both ν νγ and ℓ + ℓ − γ channels leads to fairly minor improvements on the sensitivity reaches.Especially, there are essentially no visible improvements beyond the sensitivity bounds of the ν νγ channel (shown in Table II of the main text) for new physics scales (Λ G− , Λ G− ), which correspond to the nTGC form factors ( ĥγ 3 , ĥγ 1 ).This is because the ν νγ channel has large off-shell enhancements for the contributions of ( ĥγ 3 , ĥγ 1 ) and thus the corresponding new physics scale (Λ G− , Λ G− ), which lead to significantly higher sensitivity reaches than those of the ℓ + ℓ − γ channel.Table S5: Combined sensitivity reaches on the new physics scales Λ j (in TeV) of the dimension-

Figure 1 :
Figure 1: Differential cross sections for ν νγ production as functions of the photon transverse momentum P γ T , shown in plot (a) for the LHC(13TeV) and in plot (b) for the projected 100TeV pp collider.In each plot, the SM cross section is given by a black curve, and the contributions of the different nTGC form factors (taking a reference value 10 −4 ) are shown by the colored curves.
For each dimension-8 operator O j , we define in Eq.(2.1) the corresponding effective cutoff scale for new physics, Λ j ≡ Λ/|c j | 1/4 , and introduce the notation [Λ 4 j ] ≡ sign(c j )Λ 4 j .The following CPC and CPV nTGC operators include Higgs doublets: CPC: distributions of the Z * γ cross section

Table 2 :
Sensitivity reaches on probing the new physics scales Λ 3,1 and h γ 31,11 are of O(10 −4).The sensitivities for probing the j (TeV) of the dimension-8 nTGC operators at 2σ level, as derived by analyzing the reaction pp(q q)→Z * γ→ν νγ at the LHC (13 TeV) and at a 100 TeV pp collider, with the integrated luminosities L as shown in this table.