The Higgs boson decay into decaying to identical fermion pairs
In order to investigate various decay channels of the Higgs boson or the hypothetical dilaton, we consider a neutral particle with zero spin and arbitrary parity. This particle can decay into two off-mass-shell bosons ( and ) decaying to identical fermion-antifermion pairs (): . We derive analytical formulas for the fully differential width of this decay and for the fully differential width of ( stands for , , or ). Integration of these formulas yields some Standard Model histogram distributions of the decay which are compared with corresponding Monte Carlo simulated distributions obtained by ATLAS and with ATLAS experimental data.
Keywords: Higgs boson; decay to fermion-antifermion pairs; identical fermions.
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PACS numbers: 12.15.Ji, 12.60.Fr, 14.80.Bn, 14.80.Ec
The boson discovered in 2012 by the CMS and ATLAS collaborations was reported to have a mass about 125 GeV and some decay modes predicted for the Standard Model (SM) Higgs boson. Since that time, the observed particle, called the Higgs boson, has been intensively studied (see, for example, Refs. 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27). A main goal of experiments on the Higgs boson physics has been to prove or disprove the hypothesis that is the SM Higgs boson. Apart from the decay channels, the SM predicts that has . The followed thorough analysis has fine-tuned the mass of , which is GeV according to Ref. 28, and has yielded some information on its spin and its parity.
In particular, the observation of the and modes (see, for example, Ref. 29) means that the Higgs boson spin is zero, one, or two while the fact that decays to and the Landau-Yang theorem exclude the spin-one variant. Further, the analyses presented in Ref. 30, 31 rule out many spin-two hypotheses at a 99% confidence level (CL) or higher. Therefore, we conclude that the spin of the Higgs boson is zero with a probability of about 99%.
To clarify the properties of , in Ref. 32 we study the decay of a spin-zero particle into two off-mass-shell bosons and . Since is defined as an elementary neutral particle with zero spin, our study applies to the Higgs boson. Moreover, it can apply to the dilaton if this boson actually exists.
The amplitude of the decay depends (see Eq. (4) in Ref. 32) on 3 complex-valued functions of the invariant masses of and . These functions determine the properties of the boson and are called the couplings. Using the CMS and ATLAS experimental data on the decay (where stands for , , or ), these collaborations in Refs. 30, 31, 29 and we in Ref. 32 have obtained some constraints on the couplings. These constraints demonstrate that is not a -odd state and it may be the SM Higgs boson, another -even state, or a boson with indefinite parity. Besides, as shown in Ref. 32, a non-zero imaginary part of the couplings is not excluded, which can be related to small loop corrections and possibly to a non-Hermiticity of the interaction.
Thus, the parity of the Higgs boson is not yet fully ascertained. Moreover, in some supersymmetric extensions of the SM there are neutral bosons with negative or indefinite parity. That is why it is now important to establish the properties of the Higgs boson.
Aiming at that, we consider the decay of the particle into and which then decay to fermion-antifermion pairs and respectively. While in Ref. 32 we study in detail the decays with the non-identical fermions, , in the present paper the case is under investigation. The masses of the fermions and are neglected in both papers.
We are motivated to consider the decay into identical fermions by the following. In Refs. 31, 30 the CMS and ATLAS collaborations analyze 95 events . 53 of them are the decays to identical leptons, namely to or . In spite of the fact that the decays to the identical leptons make up about 55% of the measured decays , the distributions of the former decays have not been properly analytically studied.
The SM total widths of the decays into identical fermions are studied in Refs. 36, 37 and are calculated in Ref. 38. Some distributions of the decay are plotted in Ref. 30, 31 for the SM Higgs boson and some spin-zero states beyond the SM. In the present paper we perform a more general study and consider the decay with allowance for all the possible properties of the particle .
In Sec. 2 we derive an analytical formula for the fully differential width of the decay to identical fermions. Section 3 shows a comparison of some distributions of the decay to identical leptons with those for the decay into non-identical ones. For this comparison we obtain an exact analytical formula for a certain differential width of the decay to non-identical fermions (see Appendix B). We analyze the usefulness of all the compared distributions for obtaining constraints on the couplings. In Sec. 4 we derive some SM histogram distributions of the decay by Monte Carlo (MC) integration and compare them with the corresponding simulations presented in Ref. 30 and with the experimental distributions from Ref. 30.
2 The fully differential width
We consider a neutral particle with zero spin and arbitrary parity. It can decay into two fermion-antifermion pairs, and , through the two off-mass-shell bosons ( and ):
If ( is the mass of the particle , is the mass of the quark, is the mass of the quark), which holds for , then . If , which is possible if is the dilaton, then can be the top quark as well.
In Ref. 32 we considered decays
at the tree level.
The present paper shows our analysis of decay (2) in the case of the identical fermions, :
The matrix element of decay (2) is
where the matrix elements and correspond to the diagrams (a) and (b) in Fig. 1 respectively. Namely,
and ( and ) are the 4-momenta of the particles and ( and ) in the rest frame of ;
and are the 4-momenta of and respectively in the rest frame of in diagram Fig. 1 (a);
and are respectively the pole mass and the total width of the boson;
is the amplitude of the decay where and are respectively the momentum and the helicity of the boson in the rest frame of ;
is the amplitude of the decay where and ( and ) are respectively the momentum and the polarization of () in the rest frame of , is the helicity of decaying ;
and are the 4-momenta of and respectively in the rest frame of in diagram Fig. 1 (b);
From the conservation of the energy-momentum 4-vectors we find all the possible values of and :
where is the mass of the fermion .
The amplitude is
where , is the Fermi constant, , , and are some complex-valued dimensionless functions of and , with being the polarization 4-vector of the boson with a momentum and a helicity , is the 4-momentum of the boson in its own rest frame, is the Levi-Civita symbol ().
The values of the couplings , , and reflect the properties of the particle . Specifically, at the tree level the correspondence shown in Table 1 takes place.
For the SM Higgs boson the loop corrections change slightly the tree-level values , , (see, for example, Refs. 40, 41, 31, 42). In particular, the SM electroweak radiative diagrams tune the value of the coupling , beginning from the next-to-leading order, while a contribution to appears at the three-loop level, so that and (see Ref. 43). Physics beyond the SM is the additional source of a possible deviation from the values , , .
Calculating Lorentz-invariant amplitude (2) in the rest frame of , we derive that
where , .
We take the amplitude from the SM (see, for example, Ref. 44).
Further, to describe decay (2), let us introduce the following angles (see Fig. 2): () is the angle between the momentum of () in the rest frame of and the momentum of () in the rest frame of () (in other words, () is the polar angle of the fermion ()) and is the azimuthal angle between the planes of the decays and . For decay (2), we can arbitrarily choose the boson which we will call , and then we will refer to the other boson as .
As for and , an explicit calculation yields
The expression for the amplitude is analogous to Eq. (2):
where . Calculating in the rest frame of , we get
3 Invariant mass and angular distributions
Integrating Eq. (Appendix A) numerically, we can obtain some distributions of decay (2). Moreover, numerical integration of Eq. (5) in Ref. 32 yields distributions for decay (2). In Figs. 3 and 4 we compare certain distributions of (2) with those of (2). We define the weak mixing angle as , where is the mass of the boson, and use the values of the constants in Table 2 neglecting their experimental uncertainties.
First, we show the SM distribution for any decay with different from (see Fig. 3a) and that for any decay where stands for , , or (see Fig. 3b). We see peaks at or and a flat surface outside the peaks for either dependence. For the decay into non-identical fermions the SM values of on the peaks are about 120 times greater than the values on the “plateau” (the square GeV). However, for the decay into identical leptons this ratio varies from 3 to 55 if we take , as the indicative point on the peak and on the plateau we consider the points on the line from GeV to GeV. Moreover, the SM probability that in a decay either boson has an invariant mass less than 50 GeV is
while the corresponding probability for the decay is much higher, of about 21%.
Figure 4 shows the distributions , , and for the decay to non-identical leptons and the decay to identical ones. The definitions and explicit formulas for the differential widths and are given in Appendix C (see Eqs. (C.1), (C.9), (C.10), and (C.15)).
The distributions in Fig. 4 are presented at the following four sets of values of the couplings , , and :
The dependences in the upper plot of Fig. 4a are calculated using Eq. (A.2) from Ref. 32 and Eq. (Appendix B) from this paper. To obtain the lines shown in the two other plots of Fig. 4a, we first integrate Eq. (Appendix A) with a MC method and obtain four sets of dots. Then we fit each set by means of the method of least squares. In order not to clutter the plots, we show only the fitting lines and do not present the dots.
To derive the distributions , , and for the decay into identical leptons, we integrate Eq. (Appendix A) with a MC method and obtain sets of dots. The lines in the upper plot of Fig. 4b consist of cubic parabolas joining the neighboring dots, since we have not been able to properly fit the dots of this plot with the method of least squares. The lines in the two other plots of Fig. 4b are least-squares fits to the corresponding dots. As in Fig. 4a, the dots are not shown to avoid cluttering of the plots.
The relative uncertainties of the dots used for plotting the dependences in Fig. 4 are estimated during the MC integration. For any of the plotted distributions, these uncertainties turned out to be virtually the same for each dot and each set (3). Thus, they depend only on what distribution we consider. One standard deviation of a fitting line has been estimated using Eq. (10) from Ref. 46. The uncertainties and one standard deviations for the distributions of the decays into non-identical or identical leptons are presented in Table 3. The estimates shown in Table 3 do not account for the uncertainties of the constants listed in Table 2.
We note that according to Fig. 3 in Ref. 47, the distinctions between the SM distributions and for the decay into non-identical leptons and those for the decay into identical ones are not as significant as these distinctions according to Fig. 4 in the present article. There can be a few sources of the differences with Fig. 3 in Ref. 47:
i) we consider the tree-level decays while the dependences in Fig. 3 of Ref. 47 are calculated at next-to-leading order (NLO) accuracy;
iii) our definitions of the boson couplings to fermions and and the asymmetry parameter are given in Appendix A. These definitions yield , , and (). However, experimental values of these parameters are different. For instance, for the electron , , and (see Ref. 45). The difference in , , and causes a certain distinction in the shapes of the distributions and ;
iv) in the present article non-histrogram distributions are plotted.
The dependences plotted in Fig. 4 almost coincide at all four sets (3). For this reason, we can get significant constraints on , , and via measurement of the distributions , , and only if these distributions are measured at very high precision. That is why in order to constrain the couplings, we should try to define observables sensitive to these couplings, like it is done in Ref. 32 for decay (2).
The distinctions between the distributions for the decay into non-identical leptons (Fig. 4a) and those for identical leptons (Fig. 4b) are due to greater values of the SM distribution on the plateau for the decay and smaller values of this distribution at the peaks and (see Fig. 3). However, these distinctions are insubstantial.
4 Comparison with experimental data
4.1 ATLAS and CMS results
In Ref. 30 the ATLAS collaboration presents experimental distributions of the decay and corresponding distributions derived with MC simulations in the SM. We take the same kinematic limitations and the bin widths as ATLAS and use Eqs. (Appendix A) and (Appendix A) to derive the SM histogram distributions of the decay which appear in Ref. 30. Comparison of our distributions with the ATLAS experimental and theoretical ones will determine the usefulness of Eq. (Appendix A).
CMS has shown experimental distributions for the decay (, , ) and corresponding MC simulations in the SM in Ref. 31. Taking the same kinematic limitations and the same bin widths as CMS, we integrate Eqs. (Appendix A) and (Appendix A) in the SM to obtain distributions for the decay .
We introduce the four following variables: () is the invariant mass of the boson which is produced in a decay and whose mass is closest to (most distant from) , () is the polar angle of the fermion whose parent boson has the invariant mass closest to (most distant from) . From the definitions of and it follows that
However, since , the quantity () can be equivalently defined as the invariant mass of the heaviest (lightest) boson produced in a decay ().
In Ref. 30 ATLAS shows distributions of , , , and (a distribution of is not presented). ATLAS selects events wherein
Here () is the pseudorapidity of the electron (muon):
where () is the polar angle of the electron (muon).
CMS paper presents distributions of , , , , and for the decay with
4.2 A discussion of plots
Integrating Eq. (Appendix D) with a MC method, we derive some SM histogram distributions of the decay (see the blue lines in Figs. 5 and 6). The bin widths in Fig. 5 are taken from Ref. 30 while those in Fig. 6 are taken from Ref. 31.
ATLAS reports about 45 events with ( is the invariant mass of the 4 final leptons) in Ref. 30 (see Table 3 there). For this reason, we have calculated our distributions shown in Fig. 5, setting in Eq. (Appendix D).