Large invisible decay of a Higgs boson to neutrinos

Osamu Seto seto@physics.umn.edu Department of Life Science and Technology, Hokkai-Gakuen University, Sapporo 062-8605, Japan

Abstract

We show that the standard model (SM)-like Higgs boson may decay into neutrinos with a sizable decay branching ratio in one well-known two Higgs doublet model, so-called neutrinophilic Higgs model. This could happen if the mass of the lighter extra neutral Higgs boson is smaller than one half of the SM-like Higgs boson mass. The definite prediction of this scenario is that the rate of the SM-like Higgs boson decay into diphoton normalized by the SM value is about $0.9$ . In the case that a neutrino is Majorana particle, a displaced vertex of right-handed neutrino decay would be additionally observed. This example indicates that a large invisible Higgs boson decay could be irrelevant to dark matter.

^†^†preprint: HGU-CAP-038

I Introduction

The newly discovered particle at the Large Hadron Collider (LHC) is now identified as a Higgs boson Aad:2012tfa (); Chatrchyan:2012ufa (). Its measured properties such as spin, parity, and couplings are consistent with the Higgs boson $h$ in the standard model (SM) of particle physics Aad:2013wqa (); Aad:2013xqa (); Chatrchyan:2013iaa (); Chatrchyan:2013mxa () within uncertainties, which are not very small yet. Possible deviations from the SM prediction on the Higgs boson also have been examined.

One of those is an invisible Higgs boson decay. Actually, an invisible Higgs boson decay occurs even in the SM through an off-shell $Z$ boson $Z$ pair into four neutrinos $ν$ , as $h \to Z^{*} Z^{*} \to 2 ν 2 ¯ ν$ . Its branching ratio in the SM is of the order of $10^{- 3}$ . If once it is found with a larger branching ratio than that due to SM processes, this must be a sign of a beyond the SM (BSM). Such BSM models include, for instance, a light neutralino in supersymmetric models Griest:1987qv (), a Majoron Joshipura:1992ua (), graviscalars Giudice:2000av (), fourth generation neutrino Belotsky:2002ym () and Higgs portal dark matter InvisibleDecayByDM (). Searches of invisible decays of the Higgs boson $h$ have been carried out and to date only an upper bound on the branching ratio of the invisible decay has been obtained Aad:2014iia (); Chatrchyan:2014tja ().

In this paper, we show that the Higgs boson $h$ would decay into four neutrinos through an extra Higgs boson, which can be seen as the invisible decay, in a class of the two Higgs doublet model (THDM). The remarkable feature in this scenario is that the invisible final states are a SM particle, neutrinos, compared with other BSM models mentioned above where final invisible states are new hypothetical particles, such as a supersymmetric particle or dark matter. We consider the so-called neutrinophilic THDM Ma (); Wang (); Nandi (), where one Higgs doublet provides the mass of the SM fermions, while the other generates neutrino Dirac masses with its small vacuum expectation value (VEV). Phenomenology of the charged Higgs boson was studied in Refs. Davidson:2009ha (); Haba:2011nb (). In this paper, we will study a possible phenomenology of neutral Higgs bosons in those models. Because of these Yukawa couplings, both extra $C P$ -even and extra $C P$ -odd neutral Higgs bosons, $H$ and $A$ , respectively, couple mostly with neutrinos. Thus, through interactions between the SM-like Higgs boson $h$ and the extra Higgs bosons $H (A)$ , as the $Z$ boson makes new decay processes

h \to H^{(*)} H^{(*)} o r A^{(*)} A^{(*)} \to 2 ν 2 ¯ ν

(1)

arise. ¹¹1 This possibility was briefly mentioned in Ref. Davidson:2009ha () for a very heavy SM-like Higgs boson in a different type of neutrinophilic Higgs model. If the intermediate $H$ or $A$ is off shell, the resultant contribution is comparable to the SM contribution by the $Z$ boson and is not so large. However, if either $H$ or $A$ is on shell, the resultant invisible decay width is large.

Ii Neutrinophilic two Higgs doublet model

The Higgs sector is of the so-called neutrinophilic THDM, where one Higgs doublet $Φ_{1}$ with its VEV $v_{1}$ generates the mass of the SM fermions, while the other $Φ_{2}$ generates neutrino Dirac masses through its VEV $v_{2} ≪ v_{1}$ . Such a Yukawa coupling is realized by introducing the softly broken $Z_{2}$ -parity charge assigned as in Table 1.

Fields	$Z_{2}$ parity	Lepton number
First Higgs doublet, $Φ_{1}$	$+$	0
Second Higgs doublet, $Φ_{2}$	$-$	0
Lepton doublet, $L$	$+$	1
Right-handed neutrino, $ν_{R}$	$-$	$1$
Right-handed charged lepton, $ℓ_{R}$	$+$	$1$
Others	$+$	0

Table 1: The assignment of

Z_{2}

parity and lepton number.

The Yukawa interaction is given by

L_{Y} = - y_{ℓ_{α}} {¯ ¯¯ ¯ L}_{α} Φ_{1} ℓ_{R_{α}} - y_{u_{α}} {¯ ¯¯ ¯ Q}_{α} {~ Φ}_{1} u_{R_{α}} - y_{d_{α}} {¯ ¯¯ ¯ Q}_{α} Φ_{1} d_{R_{α}} - y_{α i}_{α} {~ Φ}_{2} ν_{R_{i}} + H . c .,

(2)

where $~ Φ = i σ_{2} Φ^{*}$ , $Q$ ( $L$ ) is the left-handed $S U (2)$ doublet quark (lepton), and $u_{R}$ , $d_{R}$ , $e_{R}$ , and $ν_{R}$ are the right-handed (RH) $S U (2)$ singlet fermions, respectively. $α$ denotes flavor where we neglect mixing in quarks and $i$ represents the generation index of RH neutrinos. If we admit lepton number violation in theory, the lepton number violating Majorana mass term

L_{M} = - \frac{1}{2} ¯ ¯¯¯¯¯ ¯ ν_{R_{i}}^{c} M_{i} ν_{R_{i}}

(3)

also can be introduced Ma (). The scalar potential is given by

	$V$	$=$	$μ_{1}^{2} \| Φ_{1} \|^{2} + μ_{2}^{2} \| Φ_{2} \|^{2} - (μ_{12}^{2} Φ_{1}^{†} Φ_{2} + H . c .)$		(4)
			$+ λ_{1} \| Φ_{1} \|^{4} + λ_{2} \| Φ_{2} \|^{4} + λ_{3} \| Φ_{1} \|^{2} \| Φ_{2} \|^{2} + λ_{4} \| Φ_{1}^{†} Φ_{2} \|^{2} + {\frac{λ_{5}}{2} (Φ_{1}^{†} Φ_{2})^{2} + H . c .},$		(4)

where $μ_{12}$ is the soft breaking parameter of the $Z_{2}$ parity, as introduced above. Conditions that the potential(4) is bounded from below and a stable vacuum are given by Kanemura:2000bq ()

λ_{1} > 0, λ_{2} > 0, 2 \sqrt{λ_{1} λ_{2}} + λ_{3} + min [0, λ_{4} - | λ_{5} |] > 0.

(5)

Components in two Higgs doublets, each with a VEV, are parameterized as

Φ_{1} = ⎛ ⎝ \begin{matrix} ϕ_{1}^{+} \frac{v_{1} + h_{1} + i a_{1}}{\sqrt{2}} \end{matrix} ⎞ ⎠, Φ_{2} = ⎛ ⎝ \begin{matrix} ϕ_{2}^{+} \frac{v_{2} + h_{2} + i a_{2}}{\sqrt{2}} \end{matrix} ⎞ ⎠ .

(6)

Following the concept of neutrinophilic Higgs model, we take $v_{1} ≃ v ≃ 246$ GeV and $v_{2} ≪ v_{1}$ . The smallness of $v_{2}$ is due to the small $μ_{12}^{2}$ Ma (). We define $tan β = v_{1} / v_{2}$ as usual; this corresponds to $tan β ≫ 1$ . The states $h_{1}$ and $h_{2}$ are diagonalized to the mass eigenstates ( $h$ and $H$ ) as

(\begin{matrix} h_{1} h_{2} \end{matrix}) = (\begin{matrix} cos α & - sin α sin α & cos α \end{matrix}) (\begin{matrix} h H \end{matrix}) .

(7)

Because of $tan β ≫ 1$ , $ϕ_{1}^{+}$ and $a_{1}$ are mostly eaten by the $W$ and $Z$ bosons, while we can identify the physical states as $H^{+} ≃ ϕ_{2}^{+}$ , $A ≃ a_{2}$ , $H ≃ h_{2}$ , and $h ≃ h_{1}$ . Then, the mixing angle is found to be

sin α ≃ \frac{v_{2}}{v_{1}} .

(8)

Automatically almost, the so-called SM limit $sin (β - α) = 1$ is realized. Because of $tan β ≫ 1$ , $α ≃ 0$ is realized. The Higgs boson $h$ with the mass of $125$ GeV is also composed as $h ≃ h_{1}$ . From Eq. (2), the Yukawa interactions of extra neutral Higgs bosons are written as

	$L_{Y}$	$\supset$	$- \sum f = u_{i}, d_{i}, ℓ_{i} \frac{m_{f}}{v} \frac{sin α}{sin β} ¯ ¯ ¯ f H f - i \frac{m_{u_{i}}}{v} (- cot β) {¯ ¯ ¯ u}_{i} A γ_{5} u_{i} - i \sum f = d_{i}, ℓ_{i} \frac{m_{f}}{v} cot β ¯ ¯ ¯ f A γ_{5} f$		(9)
			$- \frac{y_{α i}}{\sqrt{2}} cos α_{α} H P_{R} ν_{i} + i \frac{y_{α i}}{\sqrt{2}} sin β_{α} A P_{R} ν_{i} + H . c .$		(9)

We find that $H$ or $A$ decays into mostly neutrinos for

y_{α i} ≫ \frac{\sqrt{2} m_{f}}{v tan β} .

(10)

Masses of extra Higgs bosons are given by

$m_{H}^{2}$	$=$	$μ_{2}^{2} + \frac{λ_{3} + λ_{4} + λ_{5}}{2} v^{2},$	(11)
$m_{A}^{2}$	$=$	$μ_{2}^{2} + \frac{λ_{3} + λ_{4} - λ_{5}}{2} v^{2},$	(12)
$m_{H^{\pm}}^{2}$	$=$	$μ_{2}^{2} + \frac{λ_{3}}{2} v^{2} .$	(13)

To be consistent with the electroweak precision test, one neutral Higgs boson mass should be close to the charged Higgs boson mass as

m_{H^{+}} ≃ m_{A} o r m_{H^{+}} ≃ m_{H} .

(14)

Interactions of extra Higgs bosons with $h$ is

L \supset - v (\frac{1}{2} (λ_{3} + λ_{4} - λ_{5}) A^{2} + \frac{1}{2} (λ_{3} + λ_{4} + λ_{5}) H^{2} + λ_{3} | H^{+} |^{2}) h .

(15)

Iii Exotic SM-like Higgs boson decay

Now we consider a case where either $H$ or $A$ is light enough to be produced on shell by the $h$ decay. There are two mass spectra of Higgs bosons that are as consistent with the electroweak precision test:

m_{H} < m_{h} / 2 ≪ m_{H^{+}} ≃ m_{A}

(16)

and

m_{A} < m_{h} / 2 ≪ m_{H^{+}} ≃ m_{H} .

(17)

From the mass formulas(11), (12), and (13), we find that mass spectra (16) and (17) can be realized for

0 > λ_{4} ≃ λ_{5} a n d 0 > λ_{4} ≃ - λ_{5},

(18)

respectively.

With couplings (9) and (15), if $h$ decays into $H$ or $A$ , which decays into neutrinos, then this fraction is measured as its invisible decay.

Figure 1: Contours of the decay branching ratio of $h$ into $H H$ in the $λ_{H} - m_{H}$ plane.

The decay width of $h \to H H$ is given by

Γ (h \to H H) = λ_{H}^{2} \frac{v^{2}}{32 π m_{h}} {(1 - \frac{4 m_{H}^{2}}{m_{h}^{2}})}^{1 / 2},

(19)

with $λ_{H} = λ_{3} + λ_{4} + λ_{5}$ . For the case of $h \to A A$ , we obtain the same result just by replacing $λ_{H}$ and $m_{H}$ with $λ_{A} = λ_{3} + λ_{4} - λ_{5}$ and $m_{A}$ , respectively. The LHC constraints on exotic decay modes, $h \to H H$ or $h \to A A$ , indicate that $λ_{H} (λ_{A}) ≲ O (10^{- 2})$ is allowed. We define

B r (h \to H H) = \frac{Γ (h \to H H)}{Γ (h \to a l l)},

(20)

which is shown in Fig. 1. By combining the constraint on $λ_{H} (λ_{A})$ and Eq. (18), we find that, for the light $H$ ( $A$ ),

λ_{3} ≃ - λ_{4} - (+) λ_{5}

(21)

should be positive and of $O (1)$ , which leads to the deviation in the diphoton decay rate of $h$ from the SM value HHG (), as shown in Fig. 2. Here, the brown and magenta shaded regions correspond to the region where the extra light Higgs boson is tachyonic and its on shell production is kinematically forbidden, respectively. One can find that this scenario predicts $R_{γ γ} ≃ 0.9$ . The signal strength of $h \to γ γ$ has been reported as $1.17 \pm 0.27$ by ATLAS Aad:2014eha () and $1.14_{- 0.23}^{+ 0.26}$ by CMS Khachatryan:2014ira ().

Figure 2: Contours of $h \to γ γ$ rate normalized by the SM value on the $λ_{3} - m_{H^{\pm}}$ plane. In the magenta shaded region, the lighter extra Higgs boson is too heavy to be produced by the $h$ decay. In the brown shaded region, the extra light Higgs boson is tachyonic. Here, the extra Higgs boson mass is evaluated with $λ_{4} = - λ_{5}$ .

In the following sections, we discuss more detailed phenomenology which depends on neutrino mass nature.

Iv Dirac neutrino case

The condition (10) is satisfied by a large $tan β$ for the Dirac neutrino case. Thus, the light $H$ is, in practice, invisible and we have $B r (h \to i n v) = B r (h \to H H)$ (shown in Fig. 1). The same is true for a light $A$ as well.

The constraint on the charged Higgs boson, which decays into a lepton and a neutrino, is, in fact, stringent. This decay mode is similar to that of a slepton in supersymmetric models Davidson:2009ha () and masses of the first and the second generation slepton is constrained as $m_{~ l} ≳ 300$ GeV by ATLAS Aad:2014vma () or $m_{~ l} ≳ 260$ GeV by CMS Khachatryan:2014qwa (). Although some differences due to its decay branching ratio exist Davidson:2009ha (), roughly speaking, there is a similar bound on $H^{+}$ . Referring to Fig. 2, we find that a rather large coupling $λ_{3} ≳ 3 (2)$ for the ATLAS (CMS) bound is required.

V Majorana neutrino case

With the presence of the Majorana mass term (3), the neutrino mass matrix is given as

M = ⎛ ⎜ ⎝ \begin{matrix} m_{ν}^{(l o o p)} & \frac{y}{\sqrt{2}} v_{2} \frac{y}{\sqrt{2}} v_{2} & M_{k} \end{matrix} ⎞ ⎟ ⎠,

(22)

with the radiative generated mass $m_{ν}^{(l o o p)}$ Ma:2006km (),

m_{ν}^{(l o o p)} = \sum k \frac{y_{α k} y_{k β}^{T} M_{k}}{16 π^{2}} (\frac{m_{H}^{2}}{m_{H}^{2} - M_{k}^{2}} ln \frac{m_{H}^{2}}{M_{k}^{2}} - \frac{m_{A}^{2}}{m_{A}^{2} - M_{k}^{2}} ln \frac{m_{A}^{2}}{M_{k}^{2}}) .

(23)

We obtain a light neutrino mass

(m_{ν})_{α β} = m_{ν}^{(l o o p)} - \sum k \frac{y_{α k} y_{k β}^{T} v_{2}^{2} / 2}{M_{k}},

(24)

the mass of a heavier RH-like neutrino $m_{N_{R}} ≃ M_{k}$ and the left-right mixing angle $θ$ Type1seesaw ()

sin θ ≃ \frac{y v_{2}}{\sqrt{2} M_{k}} .

(25)

$m_{ν}^{(l o o p)}$ could be indeed dominant —– or at least comparable with tree level seesaw mass —– because of $λ_{5} = \pm O (1)$ from Eqs. (18) and (21).

The charged Higgs boson decays into $c b$ or $t b$ , depending on the mass in the neutrinophilic Higgs model Haba:2011nb (). The results for the $H^{\pm} \to t b$ mode with a normalizing production cross section of $1$ pb can be found in Ref. CMS:2014pea (). However, this constraint is not so stringent because the actual production cross section is not so large. The LHC data constrains the mass of $H^{\pm}$ decaying into $b c$ between $90$ and $150$ GeV Aad:2013hla () and $H^{\pm}$ decaying into $τ ν$ Aad:2014kga ().

We note here one cosmological argument on the Majorana neutrino case. The lepton number violation by the Majorana nature of neutrino plays an important role in cosmology. Several cosmological discussions on neutrinophilic Higgs model were held in Refs. HabaSeto (); Haba:2013pca (); Choi:2012ap (). One of them is an enhancement $Δ L = 2$ washout process by large Yukawa couplings and relatively light RH neutrinos in a neutrinophilic Higgs model HabaSeto (). Although a discussion of baryogenesis is beyond the scope and purpose of this paper, as a necessary condition to have nonvanishing baryon asymmetry in our Universe, we roughly evaluate the condition of no strong washout of lepton asymmetry, ²²2 This is because lepton asymmetry is a potential source of the baryon asymmetry in our Universe in the large class of baryogenesis scenario FukugitaYanagida (). provided a nonvanishing lepton asymmetry has been generated by any means of a higher energy physics process. If this condition were violated, it would be difficult to explain nonvanishing baryon asymmetry in our Universe, because any generated lepton asymmetry is washed out. The washout rate is given by $Γ_{Δ L = 2} ≃ y^{4} T$ . The condition $Γ_{Δ L = 2} < H (T)$ at $T ≃ 100$ GeV, with $H$ being the cosmic expansion rate, is rewritten as

y ≲ 10^{- 4},

(26)

which would be regarded as a cosmologically favored region. ³³3 One known mechanism of baryogenesis which works without any lepton asymmetry is “baryogenesis via neutrino oscillation” Akhmedov:1998qx (); Asaka:2005pn ().

Now we discuss the decay of $H$ or $A$ . For $m_{N_{R}} < m_{H / A}$ , an extra neutral Higgs boson $H (A)$ decay produces one light left-handed-like neutrino $ν$ and the other heavy RH-like neutrino $N_{R}$ . The amplitude is calculated as

¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ | M (H / A \to ν N_{R}) |^{2} = 2 | y |^{2} (p_{1} \cdot p_{2}) = | y |^{2} ((p_{1} + p_{2})^{2} - m_{N_{R}}^{2}),

(27)

where $p_{1}$ and $p_{2}$ are outgoing momentum of $ν$ and $N_{R}$ , respectively. In addition, $m_{ν}$ is neglected and indexes of $y$ are omitted. The decay width is given by

Γ (H / A \to ν N_{R})

=

\frac{1}{16 π m_{H / A}^{3}} \sum | y |^{2} (m_{H / A}^{2} - m_{N_{R}}^{2})^{2} .

(28)

Here, the summation $\sum$ is taken for all kinematically possible modes. An extra Higgs boson decays into SM fermions, mostly the bottom quark, through a tiny mixing $α$ . Thus, its decay width is strongly suppressed by a large $tan β$ as

Γ (H / A \to b ¯ b) ≃ \frac{3}{8 π} {(\frac{m_{b}}{v tan β})}^{2} m_{H / A} .

(29)

Here, we define

B r (H / A \to i n v) = \frac{Γ (H / A \to ν N_{R})}{Γ (H / A \to b ¯ b) + Γ (H / A \to ν N_{R})},

(30)

and

B r (h \to 2 H / 2 A \to 2 ν 2 N_{R}) = B r (h \to H H / A A) B r (H / A \to i n v) .

(31)

Figure 3 shows the contour plot of the invisible decay branching ratio with thick black lines as well as the contour of Eq. (25) with thin blue thin lines of $H$ and $A$ , respectively. In both cases, the invisible decay branching ratio is large for $v_{2} < 0.1$ GeV. The dashed green (thick) lines are contours of the typical size of Yukawa coupling $y = 10^{- 5} (10^{- 4})$ estimated from Eq. (24) with the atmospheric neutrino mass. As discussed above, $y ≃ 10^{- 4}$ would be critical when we consider nonvanishing baryon asymmetry in our Universe. In both panels, neutrino masses dominantly come from the tree level seesaw at the upper left region, and do from $m_{ν}^{l o o p}$ at the lower right region. In the light $A$ case shown in the right panel of Fig. 3, the destructive cancellation between $m_{ν}^{l o o p}$ and the seesaw term takes place. This cancellation makes cuspy curves of the invisible branching ratio and Yukawa couplings in contours. For the $H$ decay, $m_{H} = 60$ GeV and $m_{A} = 200$ GeV are taken. For the $A$ decay, $m_{H} = 200$ GeV and $m_{A} = 60$ GeV are taken.

Figure 3: Contours of the invisible decay branching ratio of the lighter extra neutral Higgs boson, $H$ or $A$ , for $m_{ν} = 0.05$ eV on the ${log}_{10} (m_{N_{R}} / G e V) - {log}_{10} (v_{2} / G e V)$ plane. The thin blue lines are contours of $sin θ$ . For the dashed lines, see the text. Here we take $m_{H}, m_{A} = 60$ and $200$ GeV (left panel) and $200$ and $60$ GeV (right panel).

Figure 4 is the contour plot of the invisible decay branching ratio of $h$ . Here, $m_{H} = 60$ GeV and $m_{N_{R}} = 10$ GeV are taken.

Figure 4: Contours of the decay branching ratio of the SM-like Higgs boson, $B r (h \to 2 H \to 2 ν 2 N_{R})$ , in the $λ_{H} - v_{2}$ plane. Here, we take $m_{N_{R}} = 10$ GeV.

The produced RH neutrino $N_{R}$ decays as $N_{R} \to Z^{*} ν, h^{*} ν$ or $W^{*} ℓ$ through a tiny left-right mixing of $sin θ ≲ O (10^{- 6})$ . Here, a sign of inequality becomes more appropriate as $m_{ν}^{(l o o p)}$ becomes sizable. For such a left-right mixing of the order of $10^{- 6}$ or less, the displaced vertex of $N_{R}$ decay could be generated, and the decay length of the RH neutrino becomes $c τ_{N_{R}} ≳ 1$ cm for $m_{N_{R}} = O (10)$ GeV Displaced (). For a further lighter $m_{N_{R}}$ or a much smaller $sin θ$ , $N_{R}$ would not decay inside the detector. One can see that such a small mixing is realized in Fig. 3.

On the other hand, for $m_{N_{R}} > m_{H / A}$ , $H$ or $A$ decays into SM fermions through a tiny mixing of a Higgs boson (8) as in the usual type-I THDM.

Vi Summary

We have shown that the SM-like Higgs boson could have a sizable invisible decay branching ratio such as $O (10) %$ with four neutrinos final states, $h \to 2 ν 2 ¯ ν$ for a Dirac neutrino and $h \to 2 ν 2 N_{R}$ for a Majorana neutrino, in neutrinophilic Higgs doublet models, if one of the extra Higgs bosons is light enough to be produced by the SM-like Higgs boson decay. For the Majorana neutrino, this becomes a case in the parameter region $v_{2} ≲ 0.1$ GeV. Because of this mass spectrum of Higgs bosons, the SM normalized decay rate of $h \to γ γ$ is predicted to be $0.9$ . In the Majorana neutrino case, the displaced vertex of a $N_{R}$ decay also would be observed. Although the invisible decay of the Higgs boson was recently widely discussed in InvisibleDecayByDM () or applied to dark matter physics Aad:2014iia (); Chatrchyan:2014tja (), we emphasis that such a size of invisible decay can be realized just within simple THDM without a dark matter candidate.

Acknowledgments

We would like to thank Koji Tsumura and Shigeki Matsumoto for the valuable comments. This work is supported in part by Grant-in-Aid for Scientific Research No. 2610551401 from the Ministry of Education, Culture, Sports, Science and Technology in Japan.

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