The second Kaluza-Klein neutral Higgs bosons in the minimal Universal Extra Dimension model

Loop-induced decay of a neutral Higgs boson into a pair of gluons or photons has great implications for the Higgs discovery at the LHC. If the Higgs boson is heavy with mass above $\sim 500\gev$, however, these radiative branching ratios are very suppressed in the standard model (SM), as the new decay channels are kinematically open. We note that these radiative decays can be sizable for the heavy CP-odd second Kaluza-Klein (KK) mode of the Higgs boson, $\chi^\tw$, in the minimal universal extra dimension model: highly degenerate mass spectrum of the theory prohibits kinematically the dominant KK-number-conserving decays into the first KK modes of the $W$, $Z$ and top quark. We find that the CP-even decay of $h^{(2)} \to g g$ is absent at one-loop level since $h^{(2)}$ couples with different mass eigenstates of $t_{1,2}^\on$ while a gluon does with the same mass eigenstates. The $h^\tw$ production at the LHC is very suppressed. On the contrary, the process $ gg \to \chi^\tw \to \gamma\gamma$ in an optimal scenario can be observed with manageable SM backgrounds at the LHC.


I. INTRODUCTION
The universal extra dimension (UED) model [1] has recently drawn a lot of interest as it suggests solutions for proton decay [2], the number of fermion generations [3], and supersymmetry breaking [4]. Based on a flat five-dimensional (5D) spacetime, this model assumes that all the standard model (SM) fields propagate in the additional extra dimension y with size R, compactified over an S 1 /Z 2 orbifold. This universal accessibility to the extra dimension protects the Kaluza-Klein (KK) number conservation at tree level and the KK parity conservation at loop level. This new parity invariance has two significant implications in the phenomenology. First, the compactification scale can come down as low as about 300 GeV since the contributions of the KK modes to electroweak precision observables arise only through loops. Second, the exact invariance of the KK parity allows the cold dark matter candidate, the lightest KK particle (LKP) [5].
The identity of the LKP depends crucially on the radiative corrections to the KK masses.
There are two types of radiative corrections to the KK mass. The first is the bulk correction from compactification over finite distances, which is well-defined and finite. The second type corrections are from the boundary kinetic terms, which are incalculable due to unknown physics at the cutoff scale Λ. The minimal version of this model, called the mUED model, is based on the assumption that the boundary kinetic terms vanish at the cut-off scale. Then radiative corrections to the KK masses are well-defined, leading to the first KK mode of the U(1) Y gauge boson B (1) as the LKP [6]. Many interesting phenomenological signatures of the mUED have been studied [7].
New particle contents and their phenomenology of the UED model resemble those of a supersymmetry model with R parity conservation: all the SM particles have their heavy partner with odd parity; the decay of each heavy partner ends up with the lightest new particle (missing energy signal) plus some SM particles. There are three distinctive features of, especially, the mUED model. First, the new heavy partner has the same spin as the corresponding SM particle. In the literature, the spin discrimination in supersymmetry and UED models have been studied extensively, although very challenging at the LHC [8,9].
The second characteristic is nearly degenerate mass spectra of new particles [9,10]. High degeneracy in the KK masses at the same KK level makes the decay products of a heavy new particle consist of very soft SM particles with missing energy. At the LHC, this is to be overwhelmed by QCD backgrounds. The third characteristic is the presence of even KK parity heavy particles, the second KK modes. The even parity allows their decay into two SM particles, which can be smoking-gun signatures of this model. In Ref. [9], it was shown that 100 fb −1 data of the LHC can discover the second KK modes of the Z boson and the photon through the decays into two leptons.
In this paper, we focus on the massive scalar particles with even KK parity, the second KK modes of the Higgs boson. The n-th KK modes of a SU(2) doublet Higgs field consist of CP-even neutral h (n) , CP-odd neutral χ (n) , and charged scalars φ ±(n) . In the literature, the Higgs sector in the mUED model has been studied, mostly focused on the effects of the first KK modes. The zero mode of the Higgs boson has O(10%) increase in its gluon fusion production and the O(10%) decrease in the h → γγ decay width, by the first KK mode effects through loops [11]. The phenomenological signature of h (1) was also discussed, concluding that the production at the LHC is suppressed because the dominant channel is through the production and subsequent decay of the first KK mode of the b quark [12]. The detection of h (1) is expected even more challenging because the decay products involve too soft SM particles. However the second KK Higgs bosons can avoid these difficulties.
In the mUED model, therefore, the loop-induced decay of the second KK Higgs boson can be substantial. Their Feynman diagrams are illustrated in Figs. 1 and 2. As shall be shown, the vertex of h (2) -g-g vanishes even at one-loop level because the CP-even h (2) couples with different mass eigenstates of t (1) 1,2 while the gluon with the same mass eigenstates. The production of the CP-even neutral Higgs boson through gluon fusion at the LHC is very suppressed. On the contrary, the decay of the CP-odd scalar χ (2) into gg is substantial, which leads to sizable gluon fusion production at the LHC. In addition, BR(χ (2) → γγ) is not extremely small as in the SM. At the LHC, the heavy CP-odd neutral Higgs boson is FIG. 1. The Feynman diagrams for the decay of the CP-even neutral Higgs bosons at one-loop level. h (2) → gg channel is prohibited in this model. V (1) are first KK modes of gauge bosons and produced through the gluon fusion, and can be detected by two hard photons. The SM backgrounds are to be shown manageable. This is our main results.
The organization of the paper is as follows. In the next section, we briefly review the model and describe the effective interactions focused on the Higgs and top quark sector.
Section III deals with the production and decay of the second KK modes of the neutral Higgs bosons. For the process of gg → χ (2) → γγ, we give details of the SM backgrounds and the kinematic cuts to see the signal in Sec. IV. We conclude in Sec. V.
In this model, all the SM fields propagate freely in the 5D bulk. The zero mode of each 5D field corresponds to the SM particle. In order to obtain chiral zero mode of a fermion from a 5D vector-like fermion field, we compactify the extra dimension on an S 1 /Z 2 orbifold. We assign odd parity under the Z 2 orbifold symmetry to the zero mode fermion with wrong chirality. This extends the fermion sector to accommodate both SU(2)-doublet and SU(2)-singlet SM fermions. For the third generation quarks, e.g., we have In addition, the fifth-dimensional gauge field V 5 (x, y) has odd Z 2 parity.
for a fermion f . Here n is called the KK number. Note that, e.g., the KK modes of SU(2)-doublet top quark have both chiralities.
At tree level, the KK mass is where M n = n/R, and m 0 is the corresponding SM particle mass. All the KK mode masses are highly degenerate. In Ref. [6], it was shown that the radiative corrections generate significant changes in the KK masses. In the minimal model based on the assumption of vanishing boundary kinetic terms at the cutoff scale Λ, the corrections are well-defined and finite.

A. The Higgs sector
The 4D effective Lagrangian in the Higgs sector is obtained by integrating out the extra dimensions y: where D M is the covariant derivative given by Lagrangian in Eq. (5) yields the following 4D potential of the Higgs boson: Positive µ 2 (or negative mass squared) generates non-zero vacuum expectation value (VEV) for the SM Higgs boson H (0) , which triggers the electroweak symmetric breaking. However the condition R −1 > µ leads to positive mass squared parameters for all the KK Higgs bosons: the KK Higgs bosons do not have non-zero VEV.
The n-th KK mode of the SU(2)-doublet Higgs boson is where h (n) and χ (n) are the CP-even and CP-odd neutral scalar fields, respectively. The mass Z is a linear combination of χ (n) and the fifth component of the n-th KK mode of the Z boson, Z 5(n) : Its orthogonal combination is the Goldstone mode G µ [11]. Note that χ (n) Z and Z (n) have the same mass at tree level. The KK masses of neutral Higgs bosons are where m 2 h = λ h v 2 and v ≈ 246 GeV is the VEV of the SM Higgs boson. The radiative mass correction to n-th KK scalar masses is where µ is the regularization scale [6]. For the second KK mode production, we put µ = 2R −1 [12].

B. The top quark sector
Due to large top quark mass, there is non-negligible mixing between the KK modes of SU(2)-doublet and SU(2)-singlet top quarks in the same KK level. Their 4D effective Lagrangian is whereH = iσ 2 H * . As the zero mode of the Higgs boson develops non-zero VEV of v, the mass matrix of the KK top quark becomes non-diagonal. Including the radiative corrections to the mass, the n-th KK mass term for the top quark is where δm T (n) and δm t (n) are, respectively, the radiative corrections to the SU(2)-doublet and SU(2)-singlet top quarks, given by [6] δm where y t = m t /v. The KK mass of the SU(2)-doublet top quark T (n) has larger radiative correstions than that of the SU(2)-singlet top quark t (n) .
Two mass eigenstates of the n-th KK mode are denoted by t and T (n) through the mixing angle θ The mixing angle θ and the physical masses are, to a good approximation,

III. THE LHC REACH FOR THE SECOND KK HIGGS BOSONS
The 4D interaction Lagrangian for h (2) is and that for χ Similar expressions for other KK fermions such as the KK tau lepton can be inferred with the replacement of θ τ ∼ m τ R ≪ 1.
The tree level decay rates of CP-even h (2) are where V µ , S is the symmetric factor (S = 1/2 for V Note that g H 2 Z 1 Z 1 and g H 2 W 1 W 1 are the same to leading order because of very small KK Weinberg angle [6]. Compared to the SM coupling of h-W -W , g H 2 W 1 W 1 has additional factor of 1/ √ 2. At one-loop level, h (2) decays into a pair of top quarks, and a pair of photons. Their decay rates are The normalized amplitude for spin-1 particles is given bŷ where the universal scalar function f (τ ) is The CP-odd χ (2) has only radiative decays because of its small mass both in the light and heavy m h cases. The decay rates are The amplitude for spin-1/2 particles is where f (τ ) is given in Eq. (28). For more complicated expressions of Γ(h (2) /χ (2) → W W, ZZ, Zγ) we refer the reader to Ref. [15,16].
Brief comments on the lower bounds on R −1 are in order here. Indirect observables put rather strong constraint on R −1 . Electroweak precision data with the subleading new physics contributions and two-loop corrections to the SM ρ parameter leads to R −1 > ∼ 600 (300) GeV for m h = 115 (600) GeV at 90% confidence level [17]. In addition, the B → X s γ branching ratio constrains this model more seriously since the mUED KK modes interfere destructively with the SM amplitude [18]. At the 95% (99%) confidence level, the bound is R −1 > ∼ 600 (300) GeV. Since we are focused on the direct probe of this model, we take flexible parameter space of R −1 ∈ [350, 600] GeV as marginally allowed by indirect constraints, which is commonly searched in the literature [19]. For the Higgs boson mass, we take two cases, the light Higgs boson case of m h = 120 GeV and the heavy Higgs boson case of m h = 600 GeV.
At tree level, only the KK-number-conserving interactions are possible: a second KK mode mainly decays into two first KK mode particles. The mass spectra of the first and second KK modes determine the kinematic permission of each decay channel. In Table   I (2) . An interesting observation is that large m h or large Higgs quartic coupling λ h makes negative contributions through the radiative corrections as in Eq. (12). This negative δm 2 H (n) contribution applies to the CP-odd χ (n) identically, while the tree level KK mass of χ (n) has the SM Z boson mass, not the SM Higgs boson mass. Therefore, the KK modes of the CP-odd Higgs boson become lighter as m h increases.
Another tree level decay mode, h (2) /χ (2) → ℓ (2) R ℓ, is also very suppressed because of the same reasons. For definiteness, we present the masses of the second KK tau leptons and Γ(h (2) → τ (2) R τ ) with the fixed RΛ = 20 and m h = 120 GeV:  (2) and χ (2) can decay into a pair of top quarks through the triangle diagram (see Figs. 1 and 2). As shall be shown, this tt decay mode is dominant for both h (2) and χ (2) . Unfortunately the huge SM tt backgrounds obstruct the observation of the signal. Second types of radiative decays are into a SM gauge boson pair of W W , ZZ, Zγ, γγ and gg. For the decay of h (2) and χ (2) these decays are through the first KK gauge boson and the first KK top quarks respectively.
The radiative decays of Φ (2) (= h (2) , χ (2) ) into a pair of gluons or photons require more If the SM Higgs boson is heavy, e.g., m h = 600 GeV, the second KK mode of CP-even Higgs boson becomes also heavy as in Table I 2,1 is also kinematically allowed. In Fig. 3, we present the branching ratio of h (2) as a function of its mass in the light SM Higgs boson case (m h = 120 GeV). The deacy intott is dominant because of the large Yukawa coupling and strong coupling as in Eq. (21). The next dominant decay mode is KKnumber conserving decay of h (2) → B (1) B (1) . This invisible branching ratio is about 1%.
Narrow kinematic phase space from the degenerate mass spectrum suppresses this decay.  The produced (1) . At the LHC, this signal is overwhelmed by QCD backgrounds. The next dominant decay mode is into a top quark pair, which becomes more significant as m h (2) increases. Kaluza-Klein number conserving modes into W (1) W (1) , Z (1) Z (1) , B (1) B (1) , and t (1)t(1) follow. Note that for R −1 > 420 GeV and m h = 600 GeV, h (2) → t (1)t(1) is not kinematically allowed. Radiative decays into a pair of SM gauge bosons are very suppressed.
The CP-odd Higgs boson χ (2) does not have large enough mass for KK-number-conserving decays. Only radiative decays are allowed. This pattern remains the same for the heavy Higgs boson case because the χ (2) mass decreases with increasing m h as discussed before. In   (2) is into a pair of top quarks. The next dominent one is into a gluon pair with BR(χ (2) → gg) ≈ 20 − 40%. We expect quite efficient production of χ (2) through the gluon fusion at the LHC. Decays into a pair of the SM gauge bosons follow, in the order of ZZ, γγ, Zγ, and W W . For the detection of χ (2) at the LHC, the γγ mode is expected to be most efficient. The dominant decay mode into tt suffers from large SM background with the cross section of ∼ 900 pb [21]. Other channels into W 's or Z's have additional suppression from their small branching ratios of leptonic decay. The decay into γγ has the branching ratio of ∼ 0.1%. As shall be seen below, gg → χ (2) → γγ in an optimal scenario has a good chance to be observed at the LHC.
In all three cases of Figs. 3−5, tt decay mode is dominant, even though it is generated at one-loop level. Suppressed KK-number-conserving decays are attributed to degenrate masses and thus small phase space. Much smaller branching ratios of the decays into the SM gauge bosons than that into tt can be understood by two factors. First, the coupling strength of h (2) -t-t, of which the dominant part is proportional to y t g 2 s , is much larger than that of h (2) -V µ -V ν which is proportional to g 3 . Second the decay amplitude of h (2) → tt is characterized by m h (2) while that of h (2) → V µ V µ by m W .
In Fig. 6 we compare the total decay widths of χ (2) and h (2) with that of the SM Higgs boson. We set ΛR = 20, and m h = 120, 600 GeV for h (2) . The kinematic closure of many KK-number-conserving decays suppresses their total decay widths quite a lot. The second KK Higgs bosons, even though very heavy, are not obese like the SM one. At a collider, they are expected to appear as resonances.
At the LHC, the most promising production is that of χ (2) through gluon fusion process, pp → gg → χ (2) . The production cross section at the parton level is given bŷ (32) Figure 7 shows the production cross section σ(pp → gg → χ (2) ) as a function of R −1 at the LHC with √ s = 14 TeV. We take two cases of m h = 120, 600 GeV. For the parton distribution function, we have used the MRST 99 [22]. In the heavy SM Higgs boson case, FIG. 6. The total decay widths of χ (2) , h (2) and the SM Higgs boson with respect to their masses.
the production cross of pp → gg → χ (2) is larger than that in the light Higgs case with the given R −1 . It is mainly because of lighter χ (2) mass with large m h , as shown in Table I. In addition, large m h case is much less constrained by indirect observables such as electroweak precision data and B → X s γ: for m h = 600 GeV, R −1 > 300 GeV and m χ (2) > 560 GeV.
Assuming the LHC integrated luminosity of 100 fb −1 , about 10,000 events of χ (2) production are expected for R −1 = 500 GeV. The most of χ (2) 's decay into a pair of top quark or gluon jets, which suffers from huge QCD backgrounds. For heavier χ (2) with mass above 1 TeV, top tagging becomes efficient and it can be a good channel to test the model. For R −1 < 500 GeV, however, top tagging efficient drops too much [23,24]. The next dominant decay modes are into ZZ, Zγ and γγ. Considering small leptonic branching ratio of Z, detection efficiency for the Z boson is low. Thus χ (2) → γγ is most promising decay channel to test the mUED model for 300 GeV ≤ R −1 ≤ 600 GeV. Since the branching ratio of BR(χ (2) → γγ) is about 0.1%, we will have dozens of events of γγ pair from the χ (2) decays. For the optimal case of the detection of pp → gg → χ (2) → γγ, we take m h = 600 GeV, R −1 = 300 GeV, m χ (2) = 560 GeV.
We adopt the K-factor of 1.3, which represents the enhancement from higher order QCD processes [25]. Then the χ (2) production cross section at the LHC is about 0.61 pb. With Br(χ (2) → γγ) ≈ 10 −3 and the integrated luminosity of 100 fb −1 , σ χ (2) Br(χ (2) → γγ) has about 60 events. And the invariant mass distributions of two photons will show a resonant peak at the χ (2) mass. This special decay of χ (2) → γγ can be a smoking gun signature to discriminate the mUED from SM or minimal supersymmetric standard model (MSSM) type heavy Higgs decay [15].

IV. BACKGROUND STUDY
The photon events suffer from huge backgrounds from QCD processes. In order to suppress the QCD background, we first select the photons of E γ > 50 GeV in the simulated events and take the most energetic two photons as the photon candidates. Having applied CUT I, we reduce the background events by five order of magnitude.
For the next step, we use the longitudinal boost invariance at the LHC. The two photons in our signal have back-to-back momenta in the transverse plane. We apply the following CUT II: CUT II (1): The magnitudes of the transverse momenta of two photons are same, CUT II (2): The opening angle of the transverse momenta of two photons are in the opposite direction, −1 < cos θ T < −0.985.
With the CUT II applied, the background events are reduced by three order of magnitude, leaving 5.4 × 10 6 events as the SM backgrounds.
Finally we apply the kinematic cut on the invariant mass distribution of two photons.
Most QCD background photons have their invariant mass distribution in the low mass region, less than 300 GeV. With both CUT I and CUT II, there are less than 50 events per 10 GeV. On the contrary, our signal χ (2) → γγ in the optimal scenario has about 61 events.
Therefore we have a very sharp peak over the SM backgrounds.

V. CONCLUSIONS
The probe of massive Higgs bosons with mass above 500 GeV beyond the observed SM particles is an interesting possibility at the LHC. Within the SM, the Higgs boson can be that heavy. In the MSSM, additional heavy CP-even and CP-odd neutral Higgs bosons can be good candidates. At the LHC, however, their detection is very challenging. The production of this heavy Higgs boson, mainly through the gluon fusion, is reduced by the kinematic suppression. And the detection is not clean: the SM heavy Higgs boson is too obese (Γ h SM ≃ m h SM ) to clearly declare the observation from the golden ZZ → 4ℓ mode; in the decoupling limit the MSSM heavy neutral Higgs bosons mainly decay into tt and bb for the small and large tan β case, respectively, which suffer from the QCD backgrounds.
We found that the second KK modes of the Higgs boson in the mUED model are also very interesting candidates for massive Higgs bosons. And they have very distinctive features from the heavy Higgs bosons in the SM and MSSM. Highly degenerate mass spectrum within the given KK level closes kinematically most of the KK-number-conserving decays into the first KK modes. This kinematic closure leads to quite distinctive phenomenology compared with the heavy Higgs boson(s) in the SM and MSSM. First their total decay width is much small, which leads to a sharp resonance at the LHC. The second characteristic is the large branching ratio of CP-odd χ (2) decay into two photons or two gluons.
It is also remarkable that h (2) → gg, γγ through the KK fermion (mainly top) loops is prohibited since the coupling of the CP-even second KK Higgs boson with the first KK fermions are off-diagonal. The h (2) production through the gluon fusion is not feasible at the LHC. On the contrary, the CP-odd χ (2) has diagonal Yukawa couplings, though suppressed