The anomalous $Zb\bar{b}$ couplings: From LEP to LHC

The bottom quark forward-backward asymmetry ($A_{FB}^b$) data at LEP exhibits a long-standing discrepancy with the standard model prediction. We propose a novel method to probe the $Zb\bar{b}$ interactions through $gg\to Zh$ production at the LHC, which is sensitive to the axial-vector component of the $Zb\bar{b}$ couplings. We demonstrate that the $Zh$ data collected at the 13 TeV LHC can already resolve the apparent degeneracy of the anomalous $Zb\bar{b}$ couplings implied by the LEP precision electroweak measurements, with a strong dependence on the observed distribution of the $Z$ boson transverse momentum. We also show the potential of the HL-LHC to either verify or exclude the anomalous $Zb\bar{b}$ couplings observed at LEP through measuring the $Zh$ production rate at the HL-LHC, and this conclusion is not sensitive to possible new physics contribution induced by top quark or Higgs boson anomalous couplings in the loop.

The bottom quark forward-backward asymmetry (A b F B ) data at LEP exhibits a long-standing discrepancy with the standard model prediction. We propose a novel method to probe the Zbb interactions through gg → Zh production at the LHC, which is sensitive to the axial-vector component of the Zbb couplings. The apparent degeneracy of the anomalous Zbb couplings implied by the LEP precision electroweak measurements seems to be resolved by the current 13 TeV LHC Zh data, which is however dominated by the two data points with high transverse momentum of Z boson whose central values are in conflict with the standard model prediction. We also show the potential of the HL-LHC to either verify or exclude the anomalous Zbb couplings observed at LEP through measuring the Zh production rate at the HL-LHC, and this conclusion is not sensitive to possible new physics contribution induced by top quark or Higgs boson anomalous couplings in the loop.
Introduction: The LEP and SLC experiments have measured the Z boson couplings and found most of the electroweak data are consistent with the standard model (SM) predictions with a remarkable precision [1]. However, there are still some experimental results which cannot be explained within the SM framework. A notorious example is that the bottom quark forward-backward asymmetry (A b F B ) measured at the LEP presents a 2.5σ deviation with respect to the SM prediction [1]. As a result, it requires some degree of tuning of the left and right-handed Zbb couplings. One class of intriguing models proposed in the literature to explain the puzzling A b F B data is to allow a sizable right-handed Zbb coupling, while keeping the left-handed Zbb coupling about the same as the SM value [2][3][4][5]. Although such a large discrepancy in A b F B could be an evidence of new physics (NP) beyond the SM, it is also important to exclude the possibility that it was caused by statistical fluctuation or some subtle systematic errors in experiments. Resolving this puzzle has became one of the core tasks of the next generation lepton colliders, e.g. CEPC, ILC, CLIC and FCC-ee, which has received much attention by the high energy physics community [6][7][8][9]. It has been shown that the Zbb anomalous couplings could be well constrained at the future lepton colliders [8]. However, a direct measurement of the Zbb couplings at the Large Hadron Collider (LHC) is often ignored in the literature due to the huge backgrounds for detecting the Z boson decaying into a bottom quark and antiquark pair, i.e., Z → bb or Zb associated production [10,11].
In this Letter, we propose a novel method to probe the Zbb couplings through the associated production of Z and Higgs boson (h) via gg → Zh at the LHC. The Zbb couplings contribute to the Zh associated production through bottom quark loop effects in gluon fusion channel, cf. Fig. 1. This process has been widely used to constrain the top quark anomalous couplings, e.g. Ztt, htt, and it has been shown to be sensitive to many NP ef- fects [12][13][14][15][16][17][18][19][20][21][22]. For the first time, we demonstrate that this process can also be used to constrain the bottom quark anomalous couplings and to resolve the A b F B puzzle. Owing to charge conjugation invariance, the Z-boson couples only axially to the internal quarks in the loop of diagrams shown in Fig. 1, so that the contribution from a mass-degenerate weak doublet of quarks vanishes. It is worthwhile noting that this conclusion will not be influenced by higher order QCD corrections, because QCD theory preserves vector current conservation due to the symmetry of parity [23][24][25][26]. Such property leads to the conclusion that the gg → Zh production in the SM would only be sensitive to physics of the third generation quarks, i.e, the bottom and top quarks. Furthermore, as to be shown below, the contribution from the bottom quark is comparable to the top quark in gg → Zh production. Therefore, such process could be used to probe the axial-vector component of the Zbb interaction at hadron colliders.
The main difficulty of measuring the Zbb couplings at the LHC via gg → Zh process comes from the contamination of the top quark contribution in the loop. One can combine the other measurements at the LHC to constrain the top quark anomalous couplings, e.g. Ztt and htt couplings [27][28][29][30][31][32]. In this letter, we demonstrate that one could determine the Zbb couplings through detecting Zh associated production at the LHC, and the results are not sensitive to the top quark nor the Higgs boson anomalous couplings. Furthermore, we show that the implication of the A b F B data at LEP can either be verified or excluded if the central value of the signal strength is found to be less arXiv:2101.06261v2 [hep-ph] 23 Jul 2021 than what SM predicts at the High luminosity LHC (HL-LHC), a proton-proton collider to operate at a center-ofmass energy of 14 TeV with an integrated luminosity of 3 ab −1 . In that case, our method can also resolve the degeneracy of the Zbb couplings, presently allowed by the precision electroweak data at LEP and SLC.
Zh production via gluon fusion: We consider the following effective Lagrangian related to Zh associated production, where g W is the weak gauge coupling, c W is the cosine of the weak mixing angle θ W and v = 246 GeV is the Higgs vacuum expectation value. The gauge coupling strength modifiers κ b,t v,a and κ t,Z are introduced to include possible NP effects. The vector and axial-vector couplings of Z boson to bottom (b) and top (t) quarks in the SM We calculate the helicity amplitudes M λ1,λ2,λ3 of the channel g(λ 1 )g(λ 2 ) → Z(λ 3 )h using Fey-nArts [33] and FeynCalc [34], where λ i = +, −, 0 labels the helicity of particle i. Below, we show the explicit expression of the dominant helicity amplitudes, which can be written, separately for triangle ( ) and box ( ) diagrams, as where and s, t, u are the usual Mandelstam variables for describing the scattering of gg → Zh. The other couplings are defined as g hZZ = 2m 2 , while the other helicity configurations could be ignored due to their small numerical contributions, about 0.1% at the 14 TeV LHC, to the inclusive production cross section. The definition of the scalar functions F and F 0 ++ in Eq. (2) could be found in Ref. [35]. We have compared our analytical results to MadGraph5 [36] and found perfect agreement.
Sensitivity at the LHC: Below, we consider the impact of the non-standard Zbb, Ztt, htt and hZZ couplings to the inclusive cross section of gg → Zh at the 14 TeV The contours of R Zh in the plane of anomalous couplings at the 14 TeV LHC. The cyan region denotes the constraints, at 1σ level, provided by the measurements of pp → Zh at the 13 TeV LHC, while the yellow shaded region denotes the impact after removing the two high p Z T data, depicted in Fig. 4 of Ref. [37]. The orange bands in (b), (c) and (d) come from the constraints of Ztt, htt and hZZ coupling measurements at the 13 TeV LHC, respectively. LHC. In order to compare σ(gg → Zh) with the SM prediction, we define a ratio R Zh ≡ σ(gg → Zh)/σ(gg → Zh) SM . Figure 2 displays the contours of R Zh in the plane of anomalous couplings. The cyan region denotes the constraints from the measurements of inclusive cross section and transverse momentum distribution of Z boson (p Z T ) in the pp → Zh production at the 13 TeV LHC [37][38][39][40][41]. The dominant constraint on the parameter space comes from the ATLAS Zh data with high p Z T [37], which shows a large deviation from the central value of the signal strength (µ Zh ≡ σ(pp → Zh)/σ(pp → Zh) SM )   Fig.4 of Ref. [37]. If we exclude those two high p Z T data points, whose central values happen to be much smaller than the SM predictions, the allowed parameter space is depicted as the yellow band in Fig. 2(a), which shows a much wider uncertainty in κ b a as compared to the cyan band (with 0.84 < κ b a < 1.17 at 1σ level). Hence, it is important to improve the measurement of the p Z T distribution of Zh production at the HL-LHC. The higher order QCD correction effects are taken into account by introducing a constant k-factor for both qq → Zh and gg → Zh production processes in the analysis, with k qq = 1.3 and k gg = 2.7 [23,42]. The orange bands in Fig. 2(b)-(d) show the constraints imposed by the measurements of Ztt, htt and hZZ couplings at the 13 TeV LHC [43][44][45][46][47][48][49], respectively. They were obtained by analyzing the production of Ztt, Ztj, htt and gg → h → ZZ * , etc. The cyran region shown in Fig. 2(b) is constrained by the 13 TeV Zh data while letting both κ t a and κ t v freely vary within the allowed range imposed by the measurement of Ztt couplings. This yields only a slightly larger uncertainty band as 0.66 < κ b a < 1.23. Similar analyses, but separately for the anomalous couplings κ t and κ Z , are shown in Fig. 2(c) and (d). Within the constraints from the current data, the cross section σ(gg → Zh) could differ from the SM prediction by about 20% ∼ 30% (or 100%, if without the inclusion of two above-mentioned high p Z T data points) which is a large enough deviation that can be detected at the HL-LHC [50].
At the HL-LHC, many experimental measurements can be further improved as compared to the present data. The error of the signal strengths of Zh and htt productions can be reduced to about 4.2% and 4.3%, respectively, while the uncertainty of the branching ratio of h → ZZ * would be at 2.9% [50]. The limit on κ t a will also be highly improved through measuring the tth, thj and gg → ZZ productions [17,32]. We summarize the expected constraints from the above production processes at the HL-LHC in Figs. 3 and 4. Fig. 3 shows that the expected precision for measuring κ b a at the HL-LHC is sensitive to the central value of the signal strength µ 0 Zh . Taking µ 0 Zh = 0.93 as an illustration, κ b a will be constrained to be [0. 40,0.78] at the 1σ level when we consider κ b a alone. As shown in Fig. 1, the Zh associated production cross section also depends on other parameters in Eq. (1). Fig. 4 shows the results when we consider only two parameters at a time [50]. The blue band represents the 1σ uncertainty of the Zh associated production cross section measurement with µ 0 Zh = 0.93. In the same figure, the orange shaded regions show the expected 1σ constraints provided by the Ztt, htt and hZZ coupling measurements at the HL-LHC, respectively. It shows that the HL-LHC measurements of the other processes, such as Ztt, htt and hZZ, will mainly constrain all the anomalous couplings in Eq. (1) except for κ b a . Hence, any substantial deviation observed in µ Zh would be ascribed to κ b a , whose allowed range will not be noticeably modified by a combined fit with the inclusion of the above-mentioned processes.
Break the Zbb coupling degeneracy: The Zbb couplings are also well constrained by the LEP and SLC electroweak data and mainly determined by the Z-pole measurements at the LEP: where the sum in the denominator includes all quarks except the top quark. The R b measurement agrees with the SM prediction very well, while the A b FB at the Z-pole exhibits a large deviation from the SM prediction with a significance around 2.5σ [1,51]. The SLC with polarized beam can directly probe the bottom quark asymmetry A b which was found to be consistent with the SM within 1σ [1]. As pointed out in Ref. [2], the Z-pole data alone can not fully determine the Zbb interactions due to the sign ambiguities of the couplings. With the help of A b F B measurements conducted in the off-shell Z boson region, the part of parameter space with κ b a,v < 0 has been excluded. However, it remains to be difficult to resolve the apparent degeneracy in the parameter space with κ b a,v > 0 due to the limited statistics for data away from the Z-pole. As shown in Fig. 5, both sets of  a . The cyan and gray regions come from the R b , A b F B , and A b measurements at LEP and SLC, respectively. The orange band in (a) comes from the measurements of inclusive cross section and p Z T distribution of Zh associated production at the 13 TeV LHC, while (b) is the result after removing the two high p Z T data in Fig. 4 of Ref. [37]. consistent with the precision electroweak data at LEP and SLC.
To break this degeneracy, one needs to analyze the Zh data collected at LHC. In Fig. 5, we compare the precision on the determination of the axial-vector component of the Zbb anomalous coupling via the measurements of inclusive cross section and transverse momentum distribution of Z boson in the Zh production at the 13 TeV LHC to that implied by the precision electroweak data at LEP and SLC. Here, we focus on the parameter space with κ b a,v > 0. The cyan and gray shaded regions denote the constraint from R b and (A b F B , A b ) measurements with 1σ accuracy, respectively. The orange region in Fig. 5(a) is consistent with the current Zh production measurements, while the orange band in Fig. 5(b) shows the allowed parameter space region after we re-move the above-mentioned two high p Z T data points [37]. It appears that the current measurement of the Zh inclusive cross section at the LHC has broken the degeneracy in the allowed κ b a and κ b v values implied by the precision electroweak data, resulting the preferred values of κ b a and κ b v to be close to 1, the SM values. However, the degeneracy would remain after removing the two high p Z T data points whose central values are in conflict with the SM predictions. Hence, it is important to have a more precise measurement of the p Z T distribution in the Zh events at the LHC.
Next, we discuss the potential of the HL-LHC to break the above-mentioned degeneracy and determine the value of κ b a . The expected constraint on κ b a derived from the Zh inclusive cross section measurement at the HL-LHC, assuming an projected uncertainty ∆µ = 4.2% [50], is shown as the orange band in Fig. 6(a), in which the two horizontal lines indicate the two values (1.03 and 0.67) of κ b a consistent with the precision electroweak data at LEP and SLC. The degeneracy of κ b a (and κ b v ) found in interpreting the precision electroweak data at LEP and SLC can be broken by the measurement of Zh production cross section at HL-LHC if µ 0 Zh is measured to be less than 0.964 (indicated by the black perpendicular line), which excludes the solution of κ b a = 1.03 allowed by the precision electroweak data and implies new physics beyond the SM must exist (for µ 0 Zh = 1). In case that µ 0 Zh is measured to be less than 0.923 (indicated by the purple perpendicular line), the Zh cross section measurement at HL-LHC would exclude the interpretation of the precision electroweak data at LEP and SLC by introducing merely the anomalous κ b a and κ b v couplings. Moreover, in that case, κ b a can be well determined. On the other hand, when µ 0 Zh = 1, it becomes challenging to test against the A b F B measurement at LEP by measuring the Zh cross section at HL-LHC, due to the large uncertainty in κ b a , cf. Fig. 3. To achieve that goal, a much higher precision of the Zh cross section measurement is needed. Fig. 6(b) shows the required precision ∆µ to break the apparent degeneracy in the Zbb couplings, as implied by the LEP and SLC electroweak data, is 3.5% (black perpendicular line). To exclude the interpretation of the A b F B data at LEP by introducing merely the anomalous κ b a and κ b v couplings would require ∆µ = 0.5% (indicated by the purple perpendicular line). However, current study on the Zh cross section measurement at HL-LHC [50] projects a 2.6% statistical error, 1.3% experimental systematic error and 3.1% theoretical uncertainty. Hence, ∆µ = 3.5% might be hopeful to reach at HL-LHC, with an improvement in the theoretical calculation accuracy. But, it would be very challenging to reach ∆µ = 0.5%. Nevertheless, measuring the cross section of Zh production can clearly break the apparent degeneracy of the Zbb couplings, as implied by the precision electroweak data at LEP and SLC.
Conclusions: In this Letter, we propose a novel signature to probe the anomalous Zbb couplings through measuring the cross section of the Zh associated production via gluon-gluon fusion at the LHC. Our method could be used to break the apparent degeneracy in the Zbb couplings, as implied by the LEP and SLC precision electroweak data, including the long-standing discrepancy of the A b F B data from LEP. We show that the Zh cross section at the one-loop level depends on the axial-vector component of the Zbb coupling. The determination of κ b a is sensitive to the central value of the signal strength (µ 0 Zh ) of Zh production, and is not sensitive to possible new physics contribution induced by top quark or Higgs boson anomalous couplings in the loop. The HL-LHC measurements of the other processes, such as Ztt, htt and hZZ, will mainly constrain all the anomalous couplings in Eq. (1) except for κ b a . Hence, any substantial deviation observed in µ Zh would be ascribed to κ b a . If µ 0 Zh is found to be noticeably less than 1, the degeneracy of the Zbb couplings found interpreting the precision electroweak data at LEP and SLC can be broken and new physics beyond the SM must exist.