Nonstandard Higgs couplings from angular distributions in \(h\rightarrow Z\ell ^+\ell ^\)
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Abstract
We compute the fully differential rate for the Higgsboson decay \(h\rightarrow Z\ell ^+\ell ^\), with \(Z\rightarrow \ell ^{'+}\ell ^{'}\). For these processes we assume the most general matrix elements within an effective Lagrangian framework. The electroweak chiral Lagrangian we employ assumes minimal particle content and Standard Model gauge symmetries, but it is otherwise completely general. We discuss how information on new physics in the decay form factors may be obtained that is inaccessible in the dileptonmass spectrum integrated over angular variables. The form factors are related to the coefficients of the effective Lagrangian, which are used to estimate the potential size of newphysics effects.
Keywords
Form Factor Electroweak Symmetry Breaking Effective Field Theory Total Decay Rate Angular Asymmetry1 Introduction
The recent discovery of a light scalar \(h\) by ATLAS [1] and CMS [2] has been a major step forward in our understanding of electroweak symmetry breaking. The first run of the LHC has established its mass with an accuracy of better than \(1\,\%\) and has provided evidence for its scalar nature with spinparity \(0^+\) [3]. Furthermore, decay rates to gaugeboson pairs show no significant deviations from their Standard Model (SM) values [4, 5] within the present accuracy of around 20–30 % [6, 7]. The overall agreement with the Standard Model is so far impressive.
However, theoretical arguments suggest that deviations should be expected. Their absence would actually be rather puzzling and would point to a finetuned solution for electroweak symmetry breaking, where the lightness of the Higgs would remain unexplained. Deviations from the Standard Model parameters open the gate to new physics, expected to lie at the Terascale in the form of weakly or strongly coupled new interactions. So far the LHC has been able to test total decay rates of \(h\) into gaugeboson pairs. However, LHC run 2, with a substantial increase in luminosity, will provide enough statistics to probe also differential distributions, thereby testing the Standard Model in much greater detail.
In this paper we will study in a modelindependent way the impact of new physics in the full angular distribution of \(h\rightarrow Z{\ell }^+{\ell }^\) decay, with the \(Z\) onshell and eventually decaying into a lepton pair. We will argue that \(h\rightarrow Z{\ell }^+{\ell }^\) is a useful channel not only for spin identification [8, 9, 10, 11, 12], but also to test nonstandard couplings: it provides a rich 4body angular distribution with a clean 4lepton finalstate signature. For earlier work see [13, 14].
Our results can be parametrized in terms of six independent dynamical form factors, which include the effects of virtual electroweak bosons (\(\gamma \) and \(Z\)) as well as heavier states, whose effects at the electroweak scale are encoded in contact interactions. Since we aim at model independence, we will study the new physics contributions using the effective field theory (EFT) scheme developed in [15, 16], which is the most general EFT of the electroweak interactions. As opposed to particular models, the resulting set of newphysics coefficients will remain undetermined. However, their natural sizes can still be estimated with the aid of powercounting arguments.
Certain aspects of this decay mode have already been discussed recently [17, 18, 19], with a focus on the dileptonmass distribution. The observation there is that mass distributions can unveil newphysics structures in an otherwise SMcompatible integrated decay rate. This, however, comes at the expense of some finetuning in the newphysics parameters. In contrast, by exploiting angular distributions one can identify structures that do not contribute to the integrated decay rate. Thus, one can still be compatible with the SM decay rates without tuning the newphysics parameters.
As opposed to loopinduced processes, such as \(h\rightarrow \gamma Z\), \(h\rightarrow Z\ell ^+\ell ^\) does not look a priori like a promising testing ground for newphysics effects. As we will show below, they are expected, at most, at the few \(\%\) level in certain observables. \(h\rightarrow Z\ell ^+\ell ^\) is, however, an exceptionally clean decay mode and the natural suppression of new physics can be compensated with statistics. In fact, the LHC running at 14 TeV with an integrated luminosity of 3000 fb\(^{1}\) will potentially be sensitive to newphysics effects in \(h\rightarrow Z\ell ^+\ell ^\). Our analysis also shows that CPodd effects in \(h\rightarrow Z\ell ^+\ell ^\) are expected only at the permille level.
The remainder of this paper will be organized as follows: in Sect. 2 we will derive the full angular distribution for \(h\rightarrow Z{\ell }^+{\ell }^\). Expressions for the dynamical form factors in terms of EFT coefficients will be given in Sect. 3, with a discussion of their expected sizes in both weakly and strongly coupled scenarios. In Sect. 4 we will discuss some selected angular observables. Conclusions are given in Sect. 5, while an appendix with kinematical details is provided for reference.
2 Angular distribution for \(h\rightarrow Z{\ell }^+{\ell }^\)
The angular distribution in \(h\rightarrow Z\ell ^+\ell ^\) is similar to the one in the rare \(B\)meson decay \(B\rightarrow K^*\ell ^+\ell ^\), which has been discussed for instance in [20, 21, 22, 23]. However, in the present case the angles \(\alpha \) and \(\beta \) are on an equal footing, and accordingly the angular dependence in (6) is symmetric under the interchange of \(\alpha \) and \(\beta \). Note in particular that the forward–backward asymmetry term \(J_3\) is proportional to the product \(\cos \alpha \, \cos \beta \), thus representing a kind of correlated double asymmetry in \(\alpha \) and \(\beta \). It vanishes when either \(\alpha \) or \(\beta \) are integrated over their full range. This is in contrast to \(B\rightarrow K^*\ell ^+\ell ^\), where a forward–backward asymmetry in the single angle \(\alpha \) exists due to the more complicated structure of the hadronic transition \(B\rightarrow K^*\).
3 Form factors from effective Lagrangian

Fermionic tensor operators are in principle also present, but they turn out to be negligible: first, they have a chiral suppression and second, they do not interfere with the Standard Model and thus can only appear at NNLO.

For simplicity, the list above includes only fermions of the first family. The extension to include the second family is, however, trivial.

The \(f_i(h/v)\) above are generic functions with modeldependent coefficients [16]. As a result, the previous operators contain all the possible powers of \(h\). In the following, \(a_i\) and \(b_i\) will denote, respectively, the dimensionless Wilson coefficients for the pieces without \(h/v\) and linear in \(h/v\), which are the relevant ones for the process under study.
4 Observables and form factor determination
In Sect. 2 we pointed out that at NLO there are six independent form factors entering the dynamical functions \(J_i\). With high enough statistics one can fit the full distribution \(J\) to experimental data. However, at least in the first stages of the run 2 at the LHC, where statistics will be rather limited, it is more efficient to devise a set of observables that can project out the different form factor combinations through angular asymmetries.
In order to assess the experimental relevance of these asymmetries, we will rely on numerical estimates of newphysics effects based on general powercounting arguments. Accordingly, one would naively expect the NLO coefficients given in the previous section to be generically of \({\mathcal {O}}(v^2/\Lambda ^2)\), with \(\Lambda \sim 4\pi v\). Therefore, keeping track of the gauge couplings, we will assume \(F_1=a+{\mathcal {O}}(v^2/\Lambda ^2)\), \(g_{V,A}=g_{V,A}^{(0)}+g{\mathcal {O}}(v^2/\Lambda ^2)\), \(b_{2,3}^{(\gamma )}\sim e^2{\mathcal {O}}(v^2/\Lambda ^2)\), and \(h_{V,A}\sim g {\mathcal {O}}(v^2/\Lambda ^2)\).
The main source of deviations from the SM comes from \(a\) in \(F_1\). This parameter measures the signal strength of \(h\rightarrow ZZ^*\), and it is currently constrained to deviate less than \(20\,\%\) from the SM. Since our conclusions will be independent of it, we will set \(a=1\) and \(F_1=1\) for simplicity. Newphysics corrections are then naturally dominated by \(\delta g_{V,A}\) and \(h_{V,A}\). \(\delta g_{V,A}\) are constrained by the \(Z\) partial width and LEP data sets bounds on them at the \(10^{3}\) level [33, 39], which is within the EFT expectation. \(h_{V,A}\) are instead unconstrained, and might in principle attain values larger than the naive EFT dimensional estimate because of numerical enhancements. Consider, for instance, the local \(h\rightarrow Z\ell ^+\ell ^\) couplings \(h_{V,A}\) to be induced by the treelevel exchange of a composite heavy vector resonance \(R\), mediating \(h\rightarrow Z R^*\), \(R^*\rightarrow \ell ^+\ell ^\). Then \(h_{V,A}\sim v^2/M^2_R\sim v^2/\Lambda ^2\). If \(M_R\) is numerically smaller than \(\Lambda \approx 3\,\mathrm{TeV}\) by a factor of three, say, the resulting value of \(h_{V,A}\) might be 5–10 times bigger than the naive EFT estimate. This assumes consistency with other phenomenological constraints, which is plausible in view of the free parameters in this scenario.
For simplicity we will consider a scenario where \(h_{V,A}\ne 0\), with all other corrections set to zero. Due to the smallness of \(g_V\) in the SM, the most sensitive probes of new physics are those linear in \(G_V\), namely \(A_{\alpha \beta }\) and \(B_{\phi }\), with corrections that can easily reach 50–100 %. Incidentally, notice that neither \(A_{\alpha \beta }\) nor \(B_{\phi }\) are constrained by the angular distributions collected for the spinparity analysis [3]. This has to be compared with the mass distribution, with typical corrections of a few \(\%\). However, both corrections are uncorrelated. Qualitatively, \(h_V\) controls \(A_{\alpha \beta }\) and \(B_{\phi }\) while \(h_A\) affects the mass distribution. Thus, one can get large corrections on the former while barely affecting the latter.
With the LHC running at 14 TeV and with an integrated luminosity of 3000 fb\(^{1}\), one expects around 6400 reconstructed events for \(h\rightarrow Z\ell ^+\ell ^\) [40]. With such statistics one could in principle reach a 1–2 % sensitivity in the observables that we are discussing. Since the overall effects for \(A_{\alpha \beta }\) and \(B_{\phi }\) lie around the \(\%\) level, as illustrated in Fig. 2, they could be accessible at the LHC, at least in its final stage. Regarding the CPodd sector, within the range of validity of our EFT, the asymmetries \(C_{\phi }\) and \(D_{\phi }\) are expected to be below the permille level and thus clearly out of reach for detection at the LHC.
These estimates could be made more precise by analysing the size of the backgrounds associated to the specific angular dependences. Such an analysis goes beyond the scope of the present paper, but naively they should be substantially reduced as compared to the total decay rate [11, 41, 42, 43]. In this case, \(A_{\phi }\) might turn out to be especially suited to extract \((G_V^2+G_A^2)\) with higher precision than through the total decay rate.
5 Conclusions
We have studied, in a general and systematic way, how the decay \(h\rightarrow Z\ell ^+\ell ^\) can be used to probe for physics beyond the Standard Model in the Higgs sector. For this purpose we have employed a general parametrization of the amplitude in terms of form factors, neglecting lepton masses. In view of the large gap between the electroweak scale and the expected scale of new physics, an effective field theory approach appears to be the most efficient tool. We have computed the form factors in terms of the coefficients of an effective Lagrangian, which is defined by the SM gauge symmetries, a light scalar singlet \(h\), and the remaining SM particles, but is otherwise completely general.

We discuss the most general observables arising from the full angular distribution of the 4lepton final state in \(h\rightarrow Z\ell ^+\ell ^\), \(Z\rightarrow \ell ^{'+}\ell ^{'}\). The nine coefficients \(J_i\) describing the angular distribution are expressed through the six form factors \(G_{V,A}\), \(H_{V,A}\), and \(K_{V,A}\).
 Interesting observables, besides the dileptonmass spectrum \(d\Gamma /ds\), can be constructed from the angular distribution. Examples are:

The forward–backward asymmetry \(A_{\alpha \beta }\) measuring \(J_3\) and \(B_{\phi }\) measuring \(J_6\). These quantities are strongly suppressed in the SM because of the smallness of the vectorial coupling \(g_V\). On the other hand, this implies an enhanced relative sensitivity to new physics. The required precision of a few \(\%\) might be within reach of the LHC.

\(J_7\) or \(J_9\) give similar information as \(d\Gamma /ds\), but should have different experimental systematics because of the characteristic angular dependence associated with them.

CP violation in the coupling of \(h\) to electroweak bosons is probed by \(J_4\), \(J_5\), \(J_8\), which enter the terms in the decay distribution odd in the angle between the dilepton planes \(\phi \). Their effects are, however, expected at the permille level and thus out of reach of the LHC.


The form factors are expressed in terms of the coefficients of the complete effective Lagrangian at nexttoleading order, \({\mathcal {O}}( v^2/\Lambda ^2\sim 1/(16\pi ^2))\). We use the electroweak chiral Lagrangian, extended to include a light Higgs singlet \(h\), and take into account all NLO newphysics effects at tree level, including the renormalization of SM fields and parameters. The effective Lagrangian for a linearly realized Higgs is also considered with operators up to dimension 6.

Based on effectivetheory power counting, the potentially dominant impact of new physics arises from the leadingorder \(hZZ\) coupling \(a\), which only affects the overall decay rate, but not the angular and dileptonmass distributions. The latter can only be modified by the NLO coefficients in the Lagrangian.

Power counting gives a typical size of the NLO coefficients of \(\sim v^2/\Lambda ^2\sim 1\,\%\), up to coupling constants and numerical factors. With this estimate the newphysics effects are typically small. In particular, the contributions of the virtual \(Z\) and \(\gamma \), which could in principle be inferred from the profiles of the different mass distributions turn out to be at the permille level and therefore too small to be detected. Somewhat larger effects (up to 5 %) may be possible in specific scenarios, for instance from enhanced \(hZ\bar{l}l\) local couplings \(h_{V,A}\) in a strongly interacting Higgs sector. Quantities such as \(A_{\alpha \beta }\) and \(B_{\phi }\), with their large sensitivity to NP corrections, could be especially interesting in this respect.

For the quantitative extraction of newphysics coefficients from data, radiative corrections have to be taken into account. To NLO (one loop) in the Standard Model they have been computed in [44, 45].
Notes
Acknowledgments
We thank Elvira Rossi, Mario Antonelli and Wolfgang Hollik for useful discussions. O. C. wants to thank the University of Naples for very pleasant stays during the different stages of this work. O. C. is supported in part by the DFG cluster of excellence ’Origin and Structure of the Universe’ and the ERC Advanced Grant project ’FLAVOUR’ (267104). G. D’A. is grateful to the Dipartimento di Fisica of Federico II University, Naples, for hospitality and support and acknowledges partial support by MIUR under project 2010YJ2NYW.
References
 1.G. Aad et al., ATLAS Collaboration. Phys. Lett. B 716, 1 (2012). [arXiv:1207.7214 [hepex]]
 2.S. Chatrchyan et al., CMS Collaboration. Phys. Lett. B 716, 30 (2012). [arXiv:1207.7235 [hepex]]CrossRefADSGoogle Scholar
 3.G. Aad et al. [ATLAS Collaboration]. [arXiv:1307.1432 [hepex]]
 4.A. Djouadi, Phys. Rept. 457, 1 (2008). [ hepph/0503172]CrossRefADSGoogle Scholar
 5.S. Heinemeyer et al. LHC Higgs Cross Section Working Group Collaboration. [arXiv:1307.1347 [hepph]]
 6.ATLAS Collaboration, ATLASCONF2013034Google Scholar
 7.CMS Collaboration, CMSPASHIG13005Google Scholar
 8.S.Y. Choi, D.J. Miller, M.M. Mühlleitner, P.M. Zerwas, Phys. Lett. B 553, 61 (2003). [hepph/0210077]CrossRefADSGoogle Scholar
 9.A. De Rujula, J. Lykken, M. Pierini, C. Rogan, M. Spiropulu, Phys. Rev. D 82, 013003 (2010). [arXiv:1001.5300 [hepph]]CrossRefADSGoogle Scholar
 10.Y. Gao, A.V. Gritsan, Z. Guo, K. Melnikov, M. Schulze, N.V. Tran, Phys. Rev. D 81, 075022 (2010). [arXiv:1001.3396 [hepph]]CrossRefADSGoogle Scholar
 11.S. Bolognesi, Y. Gao, A.V. Gritsan, K. Melnikov, M. Schulze, N.V. Tran, A. Whitbeck, Phys. Rev. D 86, 095031 (2012). [arXiv:1208.4018 [hepph]]CrossRefADSGoogle Scholar
 12.T. Modak, D. Sahoo, R. Sinha, H.Y. Cheng. [arXiv:1301.5404 [hepph]]
 13.V.D. Barger, K.m. Cheung, A. Djouadi, B.A. Kniehl, P.M. Zerwas. Phys. Rev. D 49, 79 (1994) [hepph/9306270]
 14.D. Stolarski, R. VegaMorales, Phys. Rev. D 86, 117504 (2012). [arXiv:1208.4840 [hepph]]CrossRefADSGoogle Scholar
 15.G. Buchalla, O. Catà, JHEP 1207, 101 (2012). [arXiv:1203.6510 [hepph]]CrossRefADSGoogle Scholar
 16.G. Buchalla, O. Catà, C. Krause. [arXiv:1307.5017] [hepph]
 17.G. Isidori, A.V. Manohar, M. Trott. [arXiv:1305.0663 [hepph]]
 18.B. Grinstein, C.W. Murphy, D. Pirtskhalava. [arXiv:1305.6938 [hepph]]
 19.G. Isidori, M. Trott . [arXiv:1307.4051 [hepph]]
 20.U. Egede, T. Hurth, J. Matias, M. Ramon, W. Reece, JHEP 0811, 032 (2008). [arXiv:0807.2589 [hepph]]CrossRefADSGoogle Scholar
 21.W. Altmannshofer, P. Ball, A. Bharucha, A.J. Buras, D.M. Straub, M. Wick, JHEP 0901, 019 (2009). [arXiv:0811.1214 [hepph]]CrossRefADSGoogle Scholar
 22.F. Kruger, L.M. Sehgal, N. Sinha, R. Sinha, Phys. Rev. D 61, 114028 (2000) [Erratumibid. D 63, 019901 (2001)] [hepph/9907386]
 23.C. Bobeth, G. Hiller, G. Piranishvili, JHEP 0807, 106 (2008). [arXiv:0805.2525 [hepph]]CrossRefADSGoogle Scholar
 24.R. Alonso, M.B. Gavela, L. Merlo, S. Rigolin, J. Yepes, Phys. Lett. B 722, 330 (2013). [arXiv:1212.3305 [hepph]]CrossRefADSGoogle Scholar
 25.R. Alonso, M.B. Gavela, L. Merlo, S. Rigolin, J. Yepes, Phys. Rev. D 87, 055019 (2013). [arXiv:1212.3307 [hepph]]CrossRefADSGoogle Scholar
 26.R. Contino, C. Grojean, M. Moretti, F. Piccinini, R. Rattazzi, JHEP 1005, 089 (2010). [arXiv:1002.1011 [hepph]]CrossRefADSGoogle Scholar
 27.R. Contino. [arXiv:1005.4269 [hepph]]
 28.B. Holdom, Phys. Lett. B 258, 156 (1991)CrossRefADSGoogle Scholar
 29.G. Buchalla, O. Catà, R. Rahn, M. Schlaffer. [arXiv:1302.6481 [hepph]]
 30.W. Buchmüller, D. Wyler, Nucl. Phys. B 268, 621 (1986)CrossRefADSGoogle Scholar
 31.B. Grzadkowski, M. Iskrzynski, M. Misiak, J. Rosiek, JHEP 1010, 085 (2010). [arXiv:1008.4884 [hepph]]
 32.G. Passarino, Nucl. Phys. B 868, 416 (2013). [arXiv:1209.5538 [hepph]]
 33.A. Pomarol, F. Riva. [arXiv:1308.2803 [hepph]]
 34.G.F. Giudice, C. Grojean, A. Pomarol, R. Rattazzi, JHEP 0706, 045 (2007). [hepph/0703164]CrossRefADSGoogle Scholar
 35.A. Soni, R.M. Xu, Phys. Rev. D 48, 5259 (1993). [hepph/9301225]CrossRefADSGoogle Scholar
 36.D. Chang, W.Y. Keung, I. Phillips, Phys. Rev. D 48, 3225 (1993). [hepph/9303226]CrossRefADSGoogle Scholar
 37.R.M. Godbole, D.J. Miller, M.M. Mühlleitner, JHEP 0712, 031 (2007). [arXiv:0708.0458 [hepph]]CrossRefADSGoogle Scholar
 38.Y. Sun, X.F. Wang, D.N. Gao. [arXiv:1309.4171 [hepph]]
 39.Z. Han, W. Skiba, Phys. Rev. D 71, 075009 (2005). [hepph/0412166]CrossRefADSGoogle Scholar
 40.I. Anderson, S. Bolognesi, F. Caola, Y. Gao, A.V. Gritsan, C.B. Martin, K. Melnikov, M. Schulze et al. [arXiv:1309.4819 [hepph]]
 41.J.S. Gainer, K. Kumar, I. Low, R. VegaMorales, JHEP 1111, 027 (2011). [arXiv:1108.2274 [hepph]]CrossRefADSGoogle Scholar
 42.P. Avery, D. Bourilkov, M. Chen, T. Cheng, A. Drozdetskiy, J.S. Gainer, A. Korytov, K.T. Matchev et al., Phys. Rev. D 87(5), 055006 (2013) [arXiv:1210.0896 [hepph]]
 43.Y. Chen, N. Tran, R. VegaMorales, JHEP 1301, 182 (2013). [arXiv:1211.1959 [hepph]]CrossRefADSGoogle Scholar
 44.A. Bredenstein, A. Denner, S. Dittmaier, M.M. Weber, Phys. Rev. D 74, 013004 (2006). [hepph/0604011]CrossRefADSGoogle Scholar
 45.B.A. Kniehl, O.L. Veretin, Phys. Rev. D 86, 053007 (2012). [arXiv:1206.7110 [hepph]]CrossRefADSGoogle Scholar
 46.N. Cabibbo, A. Maksymowicz, Phys. Rev. 137, B438 (1965) [Erratumibid. 168, 1926 (1968)]Google Scholar
 47.A. Pais, S.B. Treiman, Phys. Rev. 168, 1858 (1968)CrossRefADSGoogle Scholar
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