Rare Z-decay into light pseudoscalar bosons in the simplest little Higgs model

Nuclear Physics B 853 (2011) 625–634 www.elsevier.com/locate/nuclphysb

Rare Z-decay into light pseudoscalar bosons in the simplest little Higgs model Lei Wang, Xiao-Fang Han ∗ Department of Physics, Yantai University, Yantai 264005, China Received 25 April 2011; received in revised form 21 June 2011; accepted 9 August 2011 Available online 17 August 2011

Abstract The simplest little Higgs model predicts a light pseudoscalar boson η and opens up some new decay modes for Z-boson, such as Z → f¯f η, Z → ηηη, Z → ηγ and Z → ηgg. We examine these decay modes in the parameter space allowed by current experiments, and find that the branching ratios can reach 10−7 ¯ for Z → bbη, 10−8 for Z → τ¯ τ η, and 10−8 for Z → ηγ , which should be accessible at the GigaZ option of the ILC. However, the branching ratios can reach 10−12 for Z → ηηη, and 10−11 for Z → ηgg, which are hardly accessible at the GigaZ option. © 2011 Elsevier B.V. All rights reserved. Keywords: Simplest little Higgs model; Z-decay; Pseudoscalar; GigaZ

1. Introduction Little Higgs theory [1] has been proposed as an interesting solution to the hierarchy problem. So far various realizations of the little Higgs symmetry structure have been proposed [2–5], which can be categorized generally into two classes [6]. One class use the product group, represented by the littlest Higgs model [3], in which the SM SU(2)L gauge group is from the diagonal breaking of two (or more) gauge groups. The other class use the simple group, represented by the simplest little Higgs model (SLHM) [4], in which a single larger gauge group is broken down to the SM SU(2)L . Since these little Higgs models mainly alter the properties of the Higgs boson, hints of these models may be unraveled from various Higgs boson processes. The phenomenology of Higgs bo* Corresponding author.

E-mail address: [email protected] (X.-F. Han). 0550-3213/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2011.08.009

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son in these little Higgs models has been widely studied [7–10]. In addition to the SM-like Higgs boson h, the SLHM predicts a pseudoscalar boson η, whose mass can be as low as O(10 GeV). The constraint from the non-observation in the decay Υ → γ η excludes η with mass below 5–7 GeV [11]. Thus, the Z-decays into η can be open for a light η, such as Z → f¯f η (f = b, τ ), Z → ηηη, Z → ηγ and Z → ηgg. The next generation Z factory can be realized in the GigaZ option of the International Linear Collider (ILC) [12]. The ILC is a proposed electron–positron collider with tunable energy ranging from 400 GeV to 500 GeV and polarized beams in its first phase, and the GigaZ option corresponds to its operation on top of the resonance of Z-boson by adding a bypass to its main beam line. About 2 × 109 Z events can be generated in an operational year of 107 s of GigaZ, which implies that the expected sensitivity to the branching ratio of Z-decay can be improved from 10−5 at the LEP to 10−8 at the GigaZ [12]. Therefore, it will offer an important opportunity to probe the new physics via the exotic or rare decays of Z-boson. The Z-boson flavor-changing neutral-current (FCNC) decays have been studied in many new physics models. For the lepton (quark) flavor violation decays, the branching ratios can be enhanced sizeably compared with the SM predictions [13,14]. In addition to the FCNC processes, the branching ratios for the processes of the Z-decay into light Higgs boson(s) in some new physics models can be accessible at the GigaZ option of the ILC [15–18]. In this paper, we will focus on the processes of the Z-decays into η in the SLHM, namely Z → f¯f η (f = b, τ ), Z → ηηη, Z → ηγ and Z → ηgg. This work is organized as follows. In Section 2 we recapitulate the SLHM. In Section 3 we study the five decay modes, respectively. Finally, we give our conclusion in Section 4. 2. Simplest little Higgs model The SLHM is based on [SU(3) × U (1)X ]2 global symmetry. The gauge symmetry SU(3) × U (1)X is broken down to the SM electroweak gauge group by two copies of scalar fields Φ1 and Φ2 , which are triplets under the SU(3) with aligned VEVs f1 and f2 . The uneaten five pseudo-Goldstone bosons can be parameterized as ⎛ ⎞ ⎛ ⎞ 0 0 i − Θ itβ Θ ⎝ tβ ⎠ ⎝ Φ2 = e (1) Φ1 = e 0 , 0 ⎠, f1 where

⎡⎛

f2

⎛ ⎞⎤ ⎞ 00 100 H⎟ η ⎜ 1 ⎢⎜ ⎟⎥ (2) Θ = ⎣⎝ 0 0 ⎠ + √ ⎝ 0 1 0 ⎠⎦ , f 2 001 H† 0  f = f12 + f22 and tβ ≡ tan β = f2 /f1 . Under the SU(2)L SM gauge group, η is a real scalar, while H transforms as a doublet and can be identified as the SM Higgs doublet. The kinetic term in the non-linear sigma model is  2     ∂μ + igAa T a − i gx B x Φj  , (3) LΦ = μ μ   3 j =1,2  where gx = g tan θW / 1 − tan2 θW /3 with θW being the electroweak mixing angle. As Φ1 and Φ2 develop their VEVs, the new heavy gauge bosons Z  , Y 0 , and X ± get their masses proportional to f . A novel coupling of the SM-like Higgs boson h can be derived from Eq. (3),

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   mZ 1  μ LZhη = √ η∂ h − h∂ μ η . tβ − tβ 2f

(4)

The gauged SU(3) symmetry promotes the SM fermion doublets into SU(3) triplets. There are two possible gauge charge assignments for the fermions: the ‘universal’ embedding and the ‘anomaly-free’ embedding. The first choice is not favored by the electroweak precision data [4], so we focus on the second way of embedding. The quark Yukawa interactions for the third generation and the first two generations can be written respectively as λm j d c (5) d ij k Φ1i Φ2 Qk3 + h.c., Λ m mn λ ∗j c c L1,2 = iλd1n d1n QTn Φ1 + iλd2n d2n QTn Φ2 + i u ucm ij k Φ1∗i Φ2 Qkn + h.c., (6) Λ where n = 1, 2 are the first two generation indices; i, j, k = 1, 2, 3; Q3 = {tL , bL , iTL } and Qn = c runs over (d c , s c , bc , D c , S c ); d c and d c are linear combinations of d c {dnL , −unL , iDnL }; dm 1n 2n c and D for n = 1 and of s c and S c for n = 2; ucm runs over (uc , cc , t c , T c ). For simplicity, we assume the quark flavor mixing are small and neglect the mixing effects. Eqs. (5) and (6) contain the Higgs boson interactions and the mass terms for the three generations of quarks:   Lt  −f λt2 xλt cβ t1c (−s1 tL + c1 TL )G1 (η) + sβ t2c (s2 tL + c2 TL )G2 (η) + h.c., (7) dn  dn c ∗ Ldn  −f λ2 xλ cβ d1 (s1 dnL + c1 DnL )G1 (η)  (8) + sβ d2c (−s2 dnL + c2 DnL )G∗2 (η) + h.c., λb (9) Lb  − f 2 sβ cβ s3 bc bL G3 (η) + h.c., Λ λq (10) Lq  − f 2 sβ cβ s3 q c qL G∗3 (η) + h.c. (q = u, c), Λ where L3 = iλt1 t1c Φ1† Q3 + iλt2 t2c Φ2† Q3 + i

xλt ≡

λt1 , λt2

xλdn ≡

tβ (h + v) , s1 ≡ sin √ 2f

λd1n λd2n

,

f2 sβ ≡  , f12 + f22

(h + v) s2 ≡ sin √ , 2tβ f

cβ ≡ 

f1

,

f12 + f22

(h + v)(tβ2 + 1) s3 ≡ sin , √ 2tβ f

tβ2 2 tβ 1 1 G1 (η) ≡ 1 − i √ η − η , G2 (η) ≡ 1 + i √ η − 2 η2 , 2 4f 4tβ f 2 2f 2tβ f     1 1 1 2 2 1 − η , η− t G3 (η) ≡ 1 + i √ tβ − β tβ tβ 4f 2 2f

(11)

with h and v being the SM-like Higgs boson field and its VEV, respectively. The mass eigenstates are obtained by mixing the corresponding interaction eigenstates, e.g., the mass eigenstates c , T c ) are respectively the mixtures of (t , T ) and (t c , T c ). The diagonaliza(tmL , TmL ) and (tm L L m tion of the mass matrix in Eqs. (7) and (8) is performed numerically in our analysis, and the top quark (T , D, d, S, s) couplings of h and η bosons can also be obtained without resort to any m expansion of v/f . From Eqs. (9) and (10), we can get directly the couplings −i vf xf hf¯f and mf ¯ v yf ηf γ5 f (f = b, u, c) with

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  v 1 v 1 xf = √ (tβ + ) cot √ , tβ + tβ tβ 2f 2f

  v 1 yu = yc = −yb = √ . tβ − tβ 2f

(12)

The charged lepton couplings with h and η are the same as those of b quark, but replacing mb with the corresponding lepton mass. Hereafter we denote the mass eigenstates without the subscript ‘m’ for simplicity. The Yukawa and gauge interactions break the global symmetry and then provide a potential for the Higgs boson. However, the Coleman–Weinberg potential alone is not sufficient since the generated h mass is too heavy and the new pseudoscalar η is massless. Therefore, one can introduce a tree-level μ term which can partially cancel the h mass [4,9]:   √ †  H H η † 2 2 2 . (13) cos −μ (Φ1 Φ2 + h.c.) = −2μ f sβ cβ cos √ f cβ sβ 2sβ cβ f The Higgs potential becomes 1 V = −m2 H † H + λ(H † H )2 − m2η η2 + λ H † H η2 + · · · , 2 where m2 = m20 −

μ2 , sβ cβ

λ = λ0 −

μ2 12sβ3 cβ3 f 2

,

λ = −

(14)

μ2 4f 2 sβ3 cβ3

(15)

with m0 and λ0 being respectively the one-loop contributions to the H mass and the quartic couplings from the contributions of fermion loops and gauge boson loops [4]. After EWSB, the coupling hηη can be obtained from the term λ H † H η2 in Eq. (14). The Higgs VEV and the masses of h and η are given by   m2 μ2 v m2η = cos √ , m2h = 2m2 , . (16) v2 = λ sβ cβ 2f sβ cβ The Coleman–Weinberg potential involves the following parameters: f, xλt , tβ , μ, mη , mh , v. Due to the modification of the observed W gauge boson mass, v is defined as [9]   8 6 v04 tβ − tβ + tβ4 − tβ2 + 1 v02 tβ4 − tβ2 + 1 − v  v0 1 + , 12f 2 180f 4 tβ2 tβ4

(17)

(18)

where v0 = 246 GeV is the SM Higgs VEV. Assuming that there are no large direct contributions to the potential from physics at the cutoff, we can determine other parameters in Eq. (17) from f , tβ and mη with the definition of v in Eq. (18). 3. The rare Z-decay into η In the SLHM, the rare Z-decays Z → f¯f η (f = b, τ ), Z → ηηη, Z → ηγ and Z → ηgg can be depicted by the Feynman diagrams shown in Fig. 1, Fig. 2, Fig. 3 and Fig. 4, respectively. For the decay Z → ηηη, Ref. [15] shows the contributions of the scalar-loop diagrams can also be important for the large hηη coupling. Here we do not consider the contributions of the loop diagram since the hηη coupling does not have the large factor of enhancement in the SLHM (see λ in Eq. (15)).

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Fig. 1. Feynman diagrams for Z → f¯f η (f = b, τ ) in the SLHM.

Fig. 2. Feynman diagrams for Z → ηηη in the SLHM.

Fig. 3. Feynman diagrams for Z → ηγ in the SLHM. f Denotes the charged fermions in SM, the new quarks T , D, and S.

Fig. 4. Feynman diagrams for Z → ηgg in the SLHM. f Denotes the SM quarks, the new quarks T , D, and S. The diagrams by exchanging the two gluons are not shown here.

The calculations of the loop diagrams in Fig. 3 and Fig. 4 are straightforward. Each loop diagram is composed of some scalar loop functions [19] which are calculated by using LoopTools [20]. In Appendix A, we list the amplitudes for Z → f¯f η, Z → ηηη and Z → ηγ , respectively. The expressions for the amplitude of Z → ηgg are very lengthy, which are not presented here.

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Fig. 5. For f = 5.6 TeV, the branching ratios of Z → f¯f η (f = b, τ ), Z → ηηη, Z → ηγ and Z → ηgg versus the η boson mass.

In our calculations, we take mt = 173.3 GeV [21] and the other SM input parameters as Ref. [22]. The free SLHM parameters are f, tβ , mη , xλd and xλs . As shown above, the parameters xλt , μ, mh can be determined by f , tβ , mη and v. To satisfy the bound of LEP2, we require that mh is larger than 114.4 GeV [23]. Certainly, due to the suppression of hZZ coupling [9], the LEP2 bound on mh should be loosened to some extent. The recent studies about Z leptonic decay and e+ e− → τ + τ − γ process at the Z-pole show that the scale f should be respectively larger than 5.6 TeV and 5.4 TeV [24]. Here, we assume the new flavor mixing matrices in lepton and quark sectors are diagonal [6,25,26], so that f and tβ are free from the experimental constraints of the lepton and quark flavor violating processes. The large values of f can suppress the SLHM predictions sizeably. However, the factor tβ in the couplings of h and η can be taken as a large value to cancel the suppression of f partially. For the perturbation to be valid, tβ cannot be too large for a fixed f . If we require O(v04 /f 4 )/O(v02 /f 2 ) < 0.1 in the expansion of v, tβ should be below 28 for f = 5.6 TeV. Besides, the large f and tβ are favored by a light η boson. For example, mη should be no less than 45 GeV (170 GeV) for tβ = 15 (8) and f = 3 TeV, in order to generate a Higgs boson whose mass is larger than 114.4 GeV. Therefore, in our calculations, we take f = 5.6 TeV and tβ = 28, 26, 24, respectively. The small mass of the d (s) quark requires one of the couplings λd1 and λd2 (λs1 and λs2 ) to be very small, so there is almost no mixing between the SM down-type quarks and their heavy partners. We assume λd1 (λs1 ) is small, and take xλd = 1.1 × 10−4 (xλs = 2.1 × 10−3 ), which can make the masses of D and S be in the range of 1 TeV and 2 TeV for other parameters taken in our calculations. In fact, our results show that different choices of xλd and xλs cannot have sizable effects on the result. In Fig. 5, we plot the decay branching ratios of Z → f¯f η (f = b, τ ), Z → ηηη, Z → ηγ and Z → ηgg versus the η mass for tβ = 28, 26, 24, respectively. Because of the constraint of mh > 114.4 GeV, the lower bound of mη is enhanced for the small tβ . Fig. 5 shows that the ratios can reach 7 × 10−12 for Z → ηηη with mη = 15 GeV and tβ = 28, and 5 × 10−11 for Z → ηgg with mη = 5 GeV and tβ = 28, which are too small to be detectable at the GigaZ option of the ILC. The reason should be partly from the suppression of three-body phase space. Besides, for the decay Z → ηgg, we find there is a strong cancellation between the contributions of different diagrams shown in Fig. 4, which can reduce the branching ratio sizeably. As the increasing of mη , the phase space of Z → ηηη is suppressed, and the hηη coupling which is proportional to m2η is enhanced, so the ratio reaches its peak for mη = 15 GeV and tβ = 28. In addition to the

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Table 1 ¯ Maximal branching ratios of Z → bba, Z → τ¯ τ a, Z → γ a, Z → aaa and Z → agg in several new physics models, namely the SLH, the left–right twin Higgs model (LRTH), the nearly minimal supersymetric standard model (nMSSM), the next-to-minimal supersymmetric standard model (NMSSM), the lepton-specific two Higgs doublet model (L2HDM) and the type-II two Higgs doublet model (Type-II 2HDM). The results of the last four models are from Ref. [15], and the results of the LRTH are from Ref. [18]. ¯ BR(Z → bba) BR(Z → τ¯ τ a) BR(Z → γ a) BR(Z → aaa) BR(Z → agg)

SLHM

LRTH

nMSSM

NMSSM

L2HDM

Type-II 2HDM

O(10−7 )

O(10−8 )

O(10−7 )

O(10−6 )

O(10−8 )

O(10−8 ) O(10−12 ) O(10−11 )

O(10−9 ) – –

O(10−10 ) O(10−3 ) –

O(10−10 ) O(10−6 ) –

O(10−10 ) O(10−3 ) –

O(10−5 ) O(10−6 ) O(10−9 ) O(10−3 ) –

O(10−8 )

O(10−9 )

O(10−8 )

O(10−7 )

O(10−4 )

couplings Zηh and hηη which are suppressed by the scale f , the factor

1 sˆ −m2h

contributed by the

intermediate state h shown in Fig. 2 (see Eq. (A.4)) can suppress the branching ratio of Z → ηηη sizeably. ¯ 10−8 for Z → ητ¯ τ , and 10−8 Fig. 5 shows that the ratios can reach 10−7 for Z → ηbb, for Z → ηγ for a light η boson with tβ = 28, which should be accessible at the GigaZ option. ¯ and Z → The three ratios decrease with increasing of mη , especially for the decays Z → ηbb ητ¯ τ . For tβ = 26, the lower bound of mη is enhanced to 24 GeV to satisfy the constraint of mh > 114.4 GeV, and the ratios of Z → ητ¯ τ is below 10−8 , which is hardly accessible at the GigaZ option. For tβ = 24, the lower bound of mη is enhanced to 43 GeV, and the ratios of the ¯ three decays are too small to detectable at the GigaZ option. The branching ratio of Z → ηbb ¯ always dominates over Z → ητ¯ τ since the coupling ηbb is larger than ητ¯ τ , and the former can be enhanced by the color factor. Although the decay Z → ηγ occurs at one-loop level, the branching ratio can be comparable with those of Z → ηf¯f because of the large couplings ηt¯t and ηT¯ T , and the suppression of the three-body phase space for Z → ηf¯f . ¯ In Table 1, we list the maximal branching ratios of Z → bba, Z → τ¯ τ a, Z → γ a, Z → aaa and Z → agg in several new physics models, where the a denotes the light pseudoscalar boson in each model. Ref. [18] does not give the numerical results of the decay Z → aaa in LRTH, but expects the branching ratio to be much below the sensitivity of the GigaZ (10−8 ). Since different models predict different patterns of the branching ratios and the differences are sizable, the measurement of these rare decays at the GigaZ may be utilized to distinguish the models. For example, if the decay Z → τ¯ τ a is found at the GigaZ, the LRTH will be disfavored; If the decay Z → aaa is found, the SLH and LRTH will be disfavored; If the decay Z → γ a is found, the SLH will be favored, and other five models will be disfavored. 4. Conclusion In the framework of the simplest little Higgs model, we studied the rare Z-decays Z → f¯f η, Z → ηηη, Z → ηγ and Z → ηgg. In the parameter space allowed by current experiments, the ¯ 10−8 for Z → τ¯ τ η, and 10−8 for Z → ηγ , which branching ratios can reach 10−7 for Z → bbη, should be accessible at the GigaZ option of the ILC. However, the branching ratios can reach 10−12 for Z → ηηη, and 10−11 for Z → ηgg, which are too small to be detectable at the GigaZ.

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Acknowledgement We thank Junjie Cao, Zhaoxia Heng, Wenlong Sang, and Jin Min Yang for discussions. This work was supported in part by the National Natural Science Foundation of China (NNSFC) under Grant Nos. 11005089 and 11105116, and by the Foundation of Yantai University under Grant No. WL10B24. Appendix A. Amplitudes of Z → f¯f η, Z → ηηη and Z → ηγ Here we give the amplitude of the process Z(p1 , 1 ) → f¯(p4 )f (p3 )η(p2 ) (f = b, τ ). The expressions are given by M = Ma + Mb + M c ,

(A.1)

where Ma = −

    η 1 η p1 − p u(p ¯ 3 ) gL PL + gR PR (/ / 4 + mf )γ μ gLZ PL + gRZ PR 2 2 (p1 − p4 ) − mf

× v(p4 ) 1 (p1 ),     η 1 η p3 − p u(p ¯ 3 )γ μ gLZ PL + gRZ PR (/ / 1 + mf ) gL PL + gR PR Mb = − 2 2 (p1 − p3 ) − mf Mc =

× v(p4 ) 1 (p1 ), ghf f¯ gZhη (p3 + p4 )2 − m2h

u(p ¯ 3 )v(p4 )(p1 − 2p3 − 2p4 )μ μ (p1 ).

(A.2)

The amplitude of Z(p1 , 1 ) → η(p2 )η(p3 )η(p4 ) is as follows: M = M1 (p1 , p2 , p3 , p4 ) + M1 (p1 , p3 , p2 , p4 ) + M1 (p1 , p4 , p3 , p2 ),

(A.3)

where M1 (p1 , p2 , p3 , p4 ) =

ghηη gZhη sˆ − m2h

(p3 + p4 − p2 )μ μ (p1 )

with sˆ = (p3 + p4 )2 .

(A.4)

The expressions for the amplitude of Z(p1 , 1 ) → η(p3 )γ (p2 , 2 ) can be given by   mf Ncf Qf ∗ − mf Cμν g μν (A + B) 1 · 2 −2mf Cν p2ν + M= 2 2 4π  3 + mf p2 · p3 C0 + mf C0     μ + p2 · 2∗ 2mf Cμ 1 − mf p3 · 1 C0 + p2 · 1 2mf Cν 2∗ν − mf p3 · 2∗ C0  μ − 2mf p3 · 1 Cν 2∗ν + 4mf Cμν 1 2∗ν Ncf Qf β μ (C − D)mf εναμβ 1 2∗ν p2α p3 C0 , (A.5) 4π 2 where Ncf and Qf denote the color factor and the electric charge for the fermion f , reη η η η η η spectively; A = 12 (gLZ gR + gRZ gL ), B = 12 (gLZ gL + gRZ gR ), C = 12 (gLZ gR − gRZ gL ), and D = η η 1 Z η Z η Z Z 2 (gL gL − gR gR ). The parameters gL , gR , gL and gR are respectively from the couplings +i

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η η i f¯γ μ (gLZ PL + gRZ PR )f Z and f¯(gL PL + gR PR )f η with PL,R = 1∓γ 2 . C0 , Cμ and Cμν are respectively the 3-point loop functions, and their dependence is given by C0,μ,μν ≡ C0,μ,μν (p2 , −p1 , mf , mf , mf ).

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