Is there new particle running in the loop-induced $H\gamma\gamma$ and $Hgg$ vertex?

Loop-induced $Hgg$ and $H\gamma\gamma$ coupling play an important role in the Higgs production and decay at the LHC. These vertices can be affected by various New Physics contributions, including new particles, anomalous Yukawa couplings, and so on. We point out some ratios of the Higgs signal strengths could help disentangle the contributions of new particles from other sources of New Physics.


I. INTRODUCTION
The first run of the LHC was very successful with the discovery of the Higgs boson [1,2].
The measured couplings of the observed Higgs boson to fermions and gauge bosons are compatible with the Standard Model (SM) predictions at a typical O(10 − 20)% precision level [3]. The next step would be precision Higgs coupling measurements, which will be performed at the Run II of the LHC and its high-luminosity upgrade [4,5], as well as planned high-energy colliders [6,7]. These precision measurements will allow more complete understanding of Electro-Weak Symmetry Breaking (EWSB) and are crucial for searching for physics Beyond the Standard Model (BSM) [8,9].
BSM physics can contribute to all the Higgs couplings, including the couplings to fermions Hf f and massive gauge bosons HV V , which control various Higgs production and decay channels at the LHC. Unlike the tree-level couplings, the Higgs couplings to photon Hγγ and gluon Hgg are generated by loop diagrams in the SM. Therefore, they can be affected by BSM physics in a variety of ways [10,11]. Generally, we can divide the various BSM sources into two scenarios. In the first scenario, the Hgg and Hγγ couplings are affected only through the loop diagrams in the SM, where the Hf f or HW W couplings are modified.
In the second scenario, new particles also enter in the loop diagrams and affect the Hgg and Hγγ couplings directly. Even if deviation of the rate for the gluon-gluon fusion production or h → γγ decay were observed in the future, it is not straightforward to determine whether the anomaly is due to contributions from modified tree-level couplings or from new particles.
One approach to probe these two different BSM sources would be to check if the SM treelevel coupling scale factors are sufficient to fit the Higgs data [12]. Alternatively, in this paper, we will build ratios from the Higgs signal strengths to disentangle the contributions of new particles from other BSM sources. All possible BSM contributions are parameterized in the framework of Effective Field Theory (EFT). Numerical analysis with the LHC Run I data is also performed. This paper is organized as follows: In Sec. II, BSM contributions to the Higgs production and decay channels at the LHC are investigated in the EFT framework. Ratios of the Higgs signal strengths are built to analyze the BSM contributions. We present our numerical results in Sec. III and conclude in Sec. IV.

II. THEORETICAL FRAMEWORK
Considering current Higgs measurements at the LHC, a general theoretical framework can be provided by the effective Lagrangian [13,14], with v = 246 GeV. In the SM, the coupling c f (f = c, b, t, τ ) and c V (V = W, Z) are generated by tree-level diagrams and equal to unity, while c γ and c g arise at loop level and vanish here by definition. Note that c γ and c g are reserved only for new particle contributions.
The contributions from the Hf f or HV V are added separately. Generally, all these couplings could deviate from their SM values due to BSM physics. It is convenient to define the following effective couplings where x f = 4m 2 f /m 2 h , and the normalized ones The C SM i 's can be obtained from (2) by setting c f = c V = 1, c γ = c g = 0, and it turns out that C SM where These effective couplings describe the interactions of the Higgs boson to the SM fermions and gauge bosons, and can be straightforwardly used to investigate the Higgs observables, such as Γ(h → γγ) ∝ |C γ | 2 and σ ggF ∝ |C g | 2 . Generally, numerical results of these effective couplings are where the contributions from the light fermions c and τ are less than 1% and have been neglected. Fitting of C i to various experimental data has been performed by several groups [16][17][18][19][20] to test the SM and also to search for the BSM.
In the BSM, the effective couplings C i in eq. (2) may deviate from the SM values. Generally there are two scenarios: Scenario I: c f , c V = 1 and c γ , c g = 0 Among the couplings c i in eq. (1), only the Higgs couplings to the fermions and massive gauge bosons c f and c V are directly modified by the BSM contributions. Therefore, the BSM effects on the effective couplings to gluon and photon are implemented through the loop diagrams in the SM with the modified c f and c V , whose numerical results read This scenario usually corresponds to the BSM models with large extra dimension [21] or BSM candidates without new electromagnetically charged particles coupled to Higgs such as gauged L µ − L τ extensions of the SM [22,23].
Scenario II: c f , c V = 1 and c γ , c g = 0 In some BSM candidates, new electrically charged or colored particles couple to the Higgs boson and generate loop-induced Hγγ or Hgg couplings, which give rise to nonzero c γ or c g . In this scenario, the effective Higgs couplings to gluon and photon can be generally written as This scenario is the most general case, since all the couplings c i could deviate from their SM values [24][25][26]. It is noted that nonzero c γ can also be generated by the anomalous triple gauge coupling W + W − γ, which, however, is strongly constrained at high-energy colliders [27][28][29][30][31][32][33] or from the B and K meson decays [34,35].
In the following, we will build some observables to discriminate these two scenarios.
At the LHC, Higgs boson production mainly occurs through the following channels, where the effective couplings relevant for each process are also listed. It is noted that, associated production with a Z boson also includes a process gg → Zh induced by top quark loops, which are affected by the effective coupling C t [3]. For simplicity, we only consider the associated production with a W boson in the following. In this work, the most relevant decay modes of the Higgs boson are listed below: The SM predictions on the Higgs production cross sections and decay branching ratios have been summarized in refs. [12,36,37].
To characterise the Higgs boson property, a signal strength is usually defined for a particular Higgs production and decay process i → h → j at the LHC [3], where σ i (i = ggF, V BF, W H, ZH, ttH) and B j (j = ZZ * , W W * , γγ, τ τ, bb) are the production cross section for i → h and the branching ratio of the decay h → j, respectively.
Observables with the superscript "SM" indicate their SM predictions. Therefore, µ i = 1, µ j = 1 and µ j i = 1 in the SM. It is straightforward to represent the signal strengths with the normalized couplings defined in eq. (3), which read and with r Γ = Γ H /Γ SM H . Here, Γ H denotes the total decay width of the Higgs boson and depends on all the couplings in the effective Lagrangian eq. (1).
Instead of the signal strengths themselves, it is interesting to consider their ratios, which can get rid of the total Higgs decay width [38][39][40][41][42][43][44][45][46]. Among all possible combinations, ratios of the signal strengths with same production or decay mode are the simplest, such as the ratios defined below where V denotes the V BF or W H production mode. Since either the decay or production signal strength is canceled in each ratio, we can reduce them to the following basic ones With eq. (10) and (11), it's easy to obtain In scenarios I and II, their numerical results read Scenario I Scenario II where the contribution from b quark has been neglected. In scenario I, the ratios r 1 and r 2 only depend on one parameter c t /c V , and the ratio r 3 is a constant. So these three ratios are strongly correlated with each other. In scenario II, however, the ratios receive additional contributions from c γ or c g , and the correlation between them no longer exists. Therefore, it is possible to distinguish between scenario I and II by investigating the correlations between the ratios r 1 , r 2 and r 3 .
Before presenting the numerical analysis, a few comments are given here: • When obtaining the numerical results of r 1 , r 2 and r 3 in eq. (15), the contribution from b quark is neglected. Since it only accounts for around 10% of the effective coupling of the Higgs boson to gluon C g , the one-parameter correlation between r 1 , r 2 and r 3 in scenario I is not polluted so much even after including this contribution. Furthermore, in the case that the Yukawa couplings c f are universal, the correlations in scenario I hold exactly.
• As advocated in refs. [38,39], theoretical uncertainties in the ratios r i are largely eliminated, since either same production or decay channels are taken. Furthermore, some systematic errors, such as the one related with luminosity measurements, also cancel out in the ratios. h → γγ [56][57][58][59][60][61][62][63], and the NLO EW corrections are also available [64][65][66][67][68][69]. Although they modify the effective couplings C g and C γ by around 10% level, their effects appear in both the denominator and numerator and are expected to be largely canceled in the signal strengths µ ggF and µ γγ . We leave the high-order QCD and EW corrections to the ratios r 1−3 for future work.
• It is possible to define other ratios which present similar features as r 1−3 , such as For simplicity, these ratios are not included in this paper.

III. NUMERICAL ANALYSIS
As discussed in the previous section, the correlations between the ratios r i in scenario I are slightly polluted by the contributions from light fermions. To account for such contributions, we perform numerical analysis in two cases: the scale factors for the Yukawa couplings are universal or non-universal, which correspond to the following parameter sets: universal coupling non-universal coupling We perform a global fit for each parameter set. The package Lilith-1.1.3 with database DB 15.09 [70] is used to take into account the Higgs signal strengths measured by LHC Run I [71,72] and Tevatron [73]. Our parameter scan doesn't include the LHC Run II data, since the corresponding Higgs signal strengths haven't been fully available yet [74,75].
Our global fit shows that O(20%) deviations from the SM values are allowed in the fit of (c V , c F ), while O(50%) in the other three fits. The fit results for the parameter set (c V , c F ) and (c V , c b , c t , c τ , c γ , c g ) can also been found in ref. [3] and [76], respectively.
We can obtain the experimental values of r 1 , r 2 and r 3 from the following ratios. [12] 0.082 ± 0.009 0.0854 ± 0.0010 0.0067 ± 0.0010 EXP [3] 0.109 ± 0.034 0.069 ± 0.018 0.022 ± 0.007 To be conservative, experimental errors have been symmetrized. Compared to the SM predictions, the current experimental uncertainties are rather large, which are mainly due to the limited statistics available [3] and expected to be improved significantly at the future LHC.
In the case of universal scale factors for the Yukawa couplings, the ratios r i in scenario I and II are shown in Fig. 1. In scenario I, as expected, r 1 and r 2 is strongly correlated with each other and r 3 is kept a constant. However, the three ratios can vary independently in a wide range in scenario II. The numerical results in the case of the non-universal scale factors are shown in Fig. 2. Although the contributions from light fermions are included generally, the correlation between r 1 and r 2 in scenario I is still very strong and r 3 can depart from its SM value by at most 5%. Considering the LHC Run I data, the measured ratios prefer scenario II, but with large uncertainty. At the future LHC, any observed deviation from the correlations in scenario I would indicate non-vanishing coupling c γ or c g , which can provide a hint of new particles running in the loop of the Hγγ or Hgg coupling.

IV. CONCLUSIONS
In this paper, we have investigated how to use the Higgs observables at the LHC to determine if new particles enter the loops of the Hγγ and Hgg effective couplings. Accordingly, we considered two general BSM scenarios. In scenario I, only tree-level Higgs couplings to fermions and massive gauge bosons are affected. They can still change the Hγγ and Hgg couplings through the corresponding SM loop diagrams. In scenario II, besides the modified tree-level couplings, there are new particles which can enter in the loop-induced vertices and they modify the Hγγ and Hgg couplings directly.
Effects of these two BSM scenarios on the Higgs observables at the LHC have been investigated in the EFT framework. We have constructed ratios of the Higgs signal strengths to distinguish between the two scenarios. We find that, in scenario I, the ratios r 1 = µ ggF /µ V and r 2 = µ γγ /µ V V strongly correlate with each other and r 3 = µ ggF /µ ttH can deviate from its SM value by at most 5%. However, the ratios in scenario II vary independently in a wide range. Due to these different features, the ratios can provide information on whether there are new particles running in the loops of Hgg and Hγγ vertices. With large statistics expected at the future LHC, these ratios can be determined with a high precision, which makes them powerful tools to analysis possible anomalous Higgs couplings.