Study on Anomalous Neutral Triple Gauge Boson Couplings from Dimension-eight Operators at the HL-LHC

TL;DR

This study investigates anomalous neutral triple gauge boson couplings at the HL-LHC using dimension-eight operators, analyzing specific processes and distributions to set sensitivity limits on coupling parameters.

Contribution

It provides the first detailed sensitivity analysis of dimension-eight operator-induced aNTGCs at the HL-LHC, including detector effects and different luminosity scenarios.

Findings

Sensitivity limits for C_{BW} and C_{BB} couplings are established.

Constraints are provided for integrated luminosities of 300 fb^{-1} and 3000 fb^{-1}.

Analysis includes transverse momentum and angular distributions with detector effects.

Abstract

The anomalous neutral triple gauge boson couplings (aNTGCs) for the Z gamma gamma and Z gamma Z vertices described by dimension-eight operators are examined through the process pp to l^{+}l^{-} gamma at the High-Luminosity Large Hadron Collider (HL-LHC). We performed an analysis on transverse momentum of photon and angular distribution of charged lepton in the final state including detector effects. Sensitivity limits of the C_{\widetilde{B}W}, C_{BB} couplings are obtained at 95 % C.L. to constrain for the range [-1.88: 1.88] TeV^{-4}, [-1.47: 1.47] TeV^{-4} and [-1.14:1.14] TeV^{-4}, [-0.86: 0.86] TeV^{-4} with an integrated luminosity of 300 fb^{-1} and 3000 fb^{-1}, respectively.

Tables

Table 1. Table 1: The number of signal events and χ 2 superscript 𝜒 2 \chi^{2} results for various coupling value of C B ~ ​ W / Λ 4 subscript 𝐶 ~ 𝐵 𝑊 superscript Λ 4 C_{\widetilde{B}W}/\Lambda^{4} after applied kinematic cuts presented in the text using cos ⁡ θ l ∗ superscript subscript 𝜃 𝑙 \cos\theta_{l}^{*} distributions of the p ​ p → l − ​ l + ​ γ → 𝑝 𝑝 superscript 𝑙 superscript 𝑙 𝛾 pp\to l^{-}l^{+}\gamma process with L i ​ n ​ t = 3000 subscript 𝐿 𝑖 𝑛 𝑡 3000 L_{int}=3000 fb -1 .

$C_{\tilde{B}W}/{\Lambda }^4$ (TeV-4)	Number of events	${\chi }^2\left({\delta }_{sys}=0\right)$	${\chi }^2\left({\delta }_{sys}=3\%\right)$	${\chi }^2\left({\delta }_{sys}=5\%\right)$
3.0	811	269.02	190.31	125.20
5.0	1458	2171.40	1536.12	1010.53
7.0	2464	8743.88	6185.72	4069.24

Table 2. Table 2: The number of signal events and χ 2 superscript 𝜒 2 \chi^{2} results for various coupling value of C B ​ B / Λ 4 subscript 𝐶 𝐵 𝐵 superscript Λ 4 C_{BB}/\Lambda^{4} after applied kinematic cuts presented in the text using cos ⁡ θ l ∗ superscript subscript 𝜃 𝑙 \cos\theta_{l}^{*} distributions of the p ​ p → l − ​ l + ​ γ → 𝑝 𝑝 superscript 𝑙 superscript 𝑙 𝛾 pp\to l^{-}l^{+}\gamma process with L i ​ n ​ t = 3000 subscript 𝐿 𝑖 𝑛 𝑡 3000 L_{int}=3000 fb -1 .

$C_{BB}/{\Lambda }^4$ (TeV-4)	Number of events	${\chi }^2\left({\delta }_{sys}=0\right)$	${\chi }^2\left({\delta }_{sys}=3\%\right)$	${\chi }^2\left({\delta }_{sys}=5\%\right)$
2.0	695	120.71	85.40	394.09
3.0	1037	726.09	513.66	337.91
4.0	1464	2199.84	1556.24	1023.77

Table 3. Table 3: Obtained limits on C B ~ ​ W / Λ 4 subscript 𝐶 ~ 𝐵 𝑊 superscript Λ 4 C_{\widetilde{B}W}/\Lambda^{4} and C B ​ B / Λ 4 subscript 𝐶 𝐵 𝐵 superscript Λ 4 C_{BB}/\Lambda^{4} at 95% C.L. with L i ​ n ​ t = 300 subscript 𝐿 𝑖 𝑛 𝑡 300 L_{int}=300 fb -1 and 3000 3000 3000 fb -1 by assuming a non-zero dimension-eight operator at a time for p ​ p → l − ​ l + ​ γ → 𝑝 𝑝 superscript 𝑙 superscript 𝑙 𝛾 pp\to l^{-}l^{+}\gamma process.

Couplings	$L_{int}=300$ fb-1			$L_{int}=3000$ fb-1
(TeV-4)	${\delta }_{sys}=0$	${\delta }_{sys}=3\%$	${\delta }_{sys}=5\%$	${\delta }_{sys}=0$	${\delta }_{sys}=3\%$	${\delta }_{sys}=5\%$
$C_{\tilde{B}W}/{\Lambda }^4$	$\left[-1.88:1.88\right]$	$\left[-1.89:1.89\right]$	$\left[-1.92:1.92\right]$	$\left[-1.14:1.14\right]$	$\left[-1.22:1.22\right]$	$\left[-1.34:1.34\right]$
$C_{BB}/{\Lambda }^4$	$\left[-1.47:1.47\right]$	$\left[-1.49:1.49\right]$	$\left[-1.51:1.51\right]$	$\left[-0.86:0.86\right]$	$\left[-0.93:0.93\right]$	$\left[-1.02:1.02\right]$

Equations

$${\mathcal{L}}^{nTGC}={\mathcal{L}}^{SM}+\sum _i\frac{C_i}{{\Lambda }^4}({\mathcal{O}}_i+{\mathcal{O}}_i^{†})$$

$${\mathcal{O}}_{BW}$$

$${\mathcal{O}}_{WW}$$

$${\mathcal{O}}_{BB}$$

$${\mathcal{O}}_{\tilde{B}W}$$

$$|M{|}^2=|M_{SM}{|}^2+2Re(M_{SM}{M}_{D8}^{\ast })+|M_{D8}{|}^2.$$

$$\Delta R(l,\gamma )={\left[(\Delta {\varphi }_{l,\gamma }{)}^2+(\Delta {\eta }_{l,\gamma }{)}^2\right]}^{1/2}.$$

$${\chi }^2=\sum _i^{n_{bins}}{\left(\frac{N_i^{NP}-N_i^B}{N_i^B{\Delta }_i}\right)}^2$$

Full text

Gauge boson interactions, dimension-eight operators, high-luminosity LHC

I Introduction

The Standard Model (SM) through the non-Abelian $SU(2)_{L}\times U(1)_{Y}$ gauge group of the electroweak sector predicts the gauge boson self-interactions. The tree-level vertices of three neutral gauge bosons are not allowed since it violates the underlying $SU(2)_{L}\times U(1)_{Y}$ symmetry. Deviations of triple gauge couplings from SM expectations can provide clues about the new physics beyond the SM. The effect of new physics can be parametirized in a model independent way by an effective Lagrangian at TeV energy scale. Using effective field theory (EFT), the Lagrangian for neutral triple gauge boson interactions can be written as (Degrande:2013kka, )

[TABLE]

where $\Lambda$ is the new physics scale, $i$ runs over the label of the four operators expressed as

[TABLE]

where $\tilde{B}_{\mu\nu}$ is the $B$-field strength tensor.

The coefficients $C_{\widetilde{B}W}$ (CP conserving) and $C_{BB}$, $C_{BW}$ , $C_{WW}$ (CP violating) of dimension-eight operators describe anomalous Neutral Triple Gauge Couplings (aNTGC). The contributions from new physics is expected to be suppressed by the inverse powers of the scale of new physics. When the new physics appears at high energy scale, the largest contribution to the subprocess $q\bar{q}\to Z\gamma$ is expected from the interference between the SM and the dimension-eight operators. The resulting matrix-element squared for the process $pp\to Z\gamma$ (where $Z\to l^{+}l^{-}$ with $l^{\pm}=e^{\pm},\mu^{\pm}$) is given by

[TABLE]

Here, the last term could be small due to the factor $\sim C_{i}^{2}\Lambda^{-8}$ when the $\Lambda$ kept as high energy scale. However, second term may contribute importantly since the interference takes contribution proportional to $\sim C_{i}\Lambda^{-4}$. The total cross section for $pp\to l^{+}l^{-}\gamma$ process can be written as $\sigma_{tot}=\sigma_{SM}+\sigma_{D8}+\sigma_{Int}$ with the leading order SM cross section $\sigma_{SM}$, the dimension-eight term cross section $\sigma_{D8}$ and the interference term cross section $\sigma_{Int}$.

Although the dimension-six operators do not induce aNTGC at the tree-level, they can have an effect at the one-loop level. The one-loop contributions from dimension-six operators would be of the order $(\alpha/4\pi)(\hat{s}/\Lambda^{2})$ while the tree-level contribution from the dimension-eight operators are of the order $(\hat{s}v^{2}/\Lambda^{4})$ (Degrande:2013kka, ). As a result, the contribution of the dimension-eight operators dominates the one-loop contribution of the dimension-six operators for $\Lambda\lesssim 2v\sqrt{\pi/\alpha}\approx 10$ TeV.

We have studied anomalous neutral triple gauge boson couplings from dimension-eight operators via the $pp\to l^{+}l^{-}\gamma$ process at High Luminosity Large Hadron Collider (HL-LHC) with 14 TeV center of mass energy, and we expect an enhancement due to the existence of aNTGCs with high $p_{T}$ photon in the final state (Barger:1984yn, ; Baur:1992cd, ; Baur:1997kz, ). The HL-LHC may provide a portal to complete opportunities at the LHC for discovery of new physics beyond the SM. The HL-LHC program as a top priority in particle physics in the context of developing the strategy for particle physics (ATLAS-Collaboration:2012jwa, ; ATLAS-collaboration:2012iza, ) is planned at two benchmark values of integrated luminosity: the 300 fb*-1* expected by the end of Run 3, and the 3000 fb*-1* expected to be delivered by the HL-LHC (ATLAS:2013hta, ).

II Cross Sections

The leading order Feynman diagrams corresponding to the process $pp\to l^{+}l^{-}\gamma$ are given in Fig. 1. In this figure, the first diagram contains the anomalous $Z\gamma\gamma$ and $ZZ\gamma$ couplings, and second diagram contains only the anomalous $Z\gamma\gamma$ couplings, while the others come from SM electroweak processes. The operators in Eqs. (2)-(5) are implemented through FeynRules package (Alloul:2013bka, ) as a Universal FeynRules Output (UFO) (Degrande:2011ua, ). After implementation of this UFO model file the cross section of $pp\to l^{+}l^{-}\gamma$ process at the center of mass energy of 14 TeV is calculated with MadGraph5_aMC@NLO (Alwall:2014hca, ). In the study, we focus on CP-even $C_{\widetilde{B}W}$ coupling and CP-odd $C_{BB}$ coupling because the deviation in cross section from its SM value for these couplings is larger than that for $C_{BW}$, $C_{WW}$ as mentioned in Ref. (Senol:2018cks, ). The Fig. 2 shows the cross sections of the $pp\to l^{+}l^{-}\gamma$ process as a function of two dimension-eight couplings $C_{\widetilde{B}W}$, $C_{BB}$. The cross sections are calculated via generator level cuts as follows:

• •

photon transverse momentum $p_{T}^{\gamma}>100$ GeV

• •

photon pseudorapidity $|\eta^{\gamma}|<2.5$

• •

charged lepton transverse momentum $p_{T}^{l}>10$ GeV and pseudorapidity $|\eta^{l}|<2.5$

• •

the invariant mass of final state charged leptons $m_{ll}>50$ GeV

• •

charged lepton - photon separation $\Delta R(l,\gamma)>0.7$, the separation between the charged lepton and photon in the pseudorapidity-azimuthal angle plane is defined by

[TABLE]

For the calculation of cross sections, only one coupling at a time is varied from its SM value.

III analysis framework

The study on effective dimension-eight aNTG couplings $C_{\tilde{B}W}$, $C_{BB}$ and SM contribution as well as interference between effective couplings and SM contributions have been performed via $pp\to l^{+}l^{-}\gamma$ process where $l^{\pm}=e^{\pm},\mu^{\pm}$. For the detailed analysis we follow steps as shown in Fig 3. The signal and background events are generated with MadGraph5_aMC@NLO applying generator-level cuts for pseudo-rapidity $|\eta^{l,\gamma}|<2.5$, and transverse momentum $p_{T}^{l,\gamma}>20$ GeV and passed through the Pythia 6 (Sjostrand:2006za, ) for parton showering and hadronization. The detector effects are included by ATLAS detector card in Delphes 3.3.3 (deFavereau:2013fsa, ) package. Then, all events are analyzed with with ROOT (Brun:1997pa, ) by using the ExRootAnalysis utility (exroot, ).

For the event selection, we require one photon and at least two charged leptons ($l^{\pm}=e^{\pm},\mu^{\pm}$); same flavor but opposite sign and the angular separation between lepton and photon $\Delta R(l,\gamma)>0.7$. As seen from Fig. 4(left pad), the transverse momentum distribution of photon (in the final state for $pp\to l^{+}l^{-}\gamma$) for the signal deviates significantly from that of SM background for all couplings starting from $200$ GeV. The invariant mass distributions of the $l^{+}l^{-}\gamma$ system for signal are shown in Fig. 4(right pad). The deviations at large values of $m_{ll\gamma}>500$ GeV become more pronounced. Therefore, we impose the following cuts in addition to above mentioned initial cuts: (a) $p_{T}^{\gamma}>400$ $(300)$ GeV, (b) $m_{ll\gamma}>500$ GeV and (c) $m_{ll}>50$ GeV in order to separate signal from the SM background efficiently. Fig. 5 shows $\cos\theta_{l}^{*}$ distributions of signal for $C_{\widetilde{B}W}/\Lambda^{4}$ (left pad), $C_{BB}/\Lambda^{4}$ (right pad) couplings and SM background. Here, $\cos\theta_{l}^{*}$ is the polar angle in the $l^{+}l^{-}$ rest frame with respect to the $l^{+}l^{-}$ direction in the $l^{+}l^{-}\gamma$ rest frame. Since the angular distribution of final state particles from signal and background processes are different, we have used this distribution as tool to obtain attainable limits on effective dimension-eight aNTG couplings. Distributions given in Fig. 4 and Fig. 5 are normalized to the number of expected events which is defined to be the cross section of each processes times integrated luminosity of $L_{int}=3000$ fb*-1*.

In order to obtain $95\%$ C.L. limits on the aNTG couplings, a $\chi^{2}$ criterion with and without systematic error is applied. Here $\chi^{2}$ function is defined as follows

[TABLE]

where $N_{i}^{NP}$ is the total number of events in the existence of effective couplings, $N_{i}^{B}$ is total number of events of the corresponding SM backgrounds in $i$th bin of the $\cos\theta_{l}^{*}$ distributions, $\Delta_{i}=\sqrt{\delta_{sys}^{2}+1/N_{i}^{B}}$ is the combined systematic ($\delta_{sys}$) and statistical errors in each bin. The one-parameter $\chi^{2}$ results of signal events obtained from $cos\theta_{l}^{*}$ distributions are $C_{\widetilde{B}W}/\Lambda^{4}=3.0,5.0,7.0$ TeV*-4* and $C_{BB}/\Lambda^{4}=2.0,3.0,4.0$ TeV*-4* as given in Tables  1 and 2, respectively. In these tables, a cut on the photon transverse momentum $p_{T}^{\gamma}>400$ GeV and integrated luminosity of 3000 fb*-1* are considered while only one coupling at a time is varied from its SM value. The two-dimensional $95\%$ C.L. intervals in planes of $C_{\widetilde{B}W}/\Lambda^{4}$ and $C_{BB}/\Lambda^{4}$ are presented in Fig. 6. One can also find one-dimensional confidence intervals on the relevant parameter axes.

IV Discussion and Conclusion

The effects of dimension-eight operators in $Z\gamma\gamma$ and $Z\gamma Z$ vertices are investigated via the $pp\rightarrow l^{-}l^{+}\gamma$ process. Both the final state photon transverse momentum ($p_{T}^{\gamma}$) and polar angle ($\cos\theta_{l}^{\star}$) are considered as a discriminant to extract bounds for $C_{\widetilde{B}W}/\Lambda^{4}$ and $C_{BB}/\Lambda^{4}$ couplings. Our expected limits on dimension-eight aNTG couplings at $95\%$ C.L. for HL-LHC with $L_{int}=300$ fb*-1* and $3000$ fb*-1* are given in Table 3 as $[-1.88:1.88]$ TeV*-4* and $[-1.14;1.14]$ TeV*-4* for $C_{\widetilde{B}W}/\Lambda^{4}$, and the limits are $[-1.47:1.47]$ TeV*-4* and $[-0.86;0.86]$ TeV*-4* for $C_{BB}/\Lambda^{4}$ (where $p_{T}^{\gamma}>400$ GeV applied), respectively. The $95\%$ C.L. current limits on dimension-eight operators converted from coefficients of dimension-six operators for the process $pp\to ZZ\to l^{+}l^{-}l^{\prime+}l^{\prime-}$ at $\sqrt{s}=13$ TeV and $L_{int}=36.1$ fb*-1* from ATLAS Collaboration are reported as $-5.9~{}\textrm{TeV}^{-4}<C_{\widetilde{B}W}/\Lambda^{4}<5.9~{}\textrm{TeV}^{-4}$ and $-2.7~{}\textrm{TeV}^{-4}<C_{BB}/\Lambda^{4}<2.8~{}\textrm{TeV}^{-4}$ (Aaboud:2017rwm, ). Comparing with the current experimental results, we obtain $5$ and $3$ times better sensitivity for dimension-eight couplings $C_{\widetilde{B}W}/\Lambda^{4}$ and $C_{BB}/\Lambda^{4}$ , respectively. We conclude that the limits on aNTG couplings would be improved from the HL-LHC results once the systematic uncertainties are kept under control.

Acknowledgements.

Authors’ work was partially supported by Turkish Atomic Energy Authority (TAEK) under the project grant no. 2018TAEK(CERN)A5.H6.F2-20.