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arXiv:hep-ph/0610362v1 27 Oct 2006 ‡ † ∗ eit Σ mediate epee sanwpril fms 1. e.Hwvr exist However, MeV. 214.3 mass of new a as terpreted nti etrw hwta h HprPpril”cnb iden be can mode particle” standard “HyperCP supersymmetric the next-to-minimal the that in show we letter nontrivi this is it In and interpretation, this constrain severely aaee pc hr the where space parameter lcrncades [email protected] address: Electronic [email protected] address: Electronic lcrncades [email protected] address: Electronic eateto hsc n etrfrTertclSciences, Theoretical for Center and Physics of Department h yeC olbrto a bevdtreeet o the for events three observed has collaboration HyperCP The + → pµ + a yeC bevdaLgtHgsBoson? Higgs Light a Observed HyperCP Has µ − eateto ahmtc/hsc/optrScience, Mathematics/Physics/Computer of Department talvlcnitn ihteHprPobservation. HyperCP the with consistent level a at nvriyo aVre aVre A970 USA 91750, CA Verne, La Verne, La of University oaSaeUiest,Ae,I 01,USA 50011, IA Ames, University, State Iowa A 1 0 eateto hsc n Astronomy, and Physics of Department a aif l h xsigcntansfo anand from constraints existing the all satisfy can Dtd etme 0 2018) 20, September (Dated: ua Tandean Jusak ioGn He Xiao-Gang .Valencia G. Abstract 1 lt osrc oe ossetwt l h data. the all with consistent model a construct to al ‡ ∗ † ,the l, n aafo anand kaon from data ing ie ihtelgtpedsaa Higgs pseudoscalar light the with tified ea Σ decay ainlTia nvriy Taipei University, Taiwan National A 1 0 nti oe hr r ein of regions are there model this In . + → pµ + µ B − msndcy and decays - hc a ein- be may which B hep-ph/0610362 msndecays -meson Three events for the decay mode Σ+ pµ+µ− with a dimuon invariant mass of 214.3 MeV have → been recently observed by the HyperCP Collaboration [1]. It is possible to account for these events within the (SM) when long-distance contributions are properly included [2]. How- ever, the probability of having all three events at the same dimuon mass, given the SM predictions, is less than one percent. This suggests a new-particle interpretation for these events, for which the branching ratio is 3.1+2.4 1.5 10−8 [1]. −1.9 ± × This possibility has been explored to some extent in the literature [3, 4, 5], where it has been shown that kaon decays place severe constraints on the flavor-changing two- couplings of the hypothetical new particle, X. It has also been claimed that a light sgoldstino is a viable candidate [6]. It is well known in the case of light Higgs production in kaon decay that, in addition to the two-quark flavor-changing couplings, there are comparable four-quark contributions [7]. They arise from the combined effects of the usual SM four-quark ∆S = 1 operators and the flavor- | | conserving couplings of X. We have recently computed the analogous four-quark contributions to light Higgs production in decay [8] and found that they can also be comparable to the two-quark contributions previously discussed in the literature. The interplay between the two- and four-quark contributions makes it possible to find models with a light responsible for the HyperCP events that has not been observed in kaon or B-meson decay. However, it is not easy to devise such models respecting all the experimental constraints. In most models that can generate dsX¯ couplings, the two-quark operators have the structure d¯(1 γ )sX. Since the part without γ contributes significantly to K πµ+µ−, their ± 5 5 → data imply that these couplings are too small to account for the HyperCP events [3, 4, 5]. In some models, there may be parameter space where the four-quark contributions mentioned above and the two-quark ones are comparable and cancel sufficiently to lead to suppressed K πµ+µ− rates + + → while yielding Σ pµ µ− rates within the required bounds. However, since in many models the → flavor-changing two-quark couplingsqq ¯ ′X are related for different (q, q′) sets, experimental data on + − B-meson decays, in particular B Xsµ µ , also provide stringent constraints. For these reasons, → the light (pseudo)scalars in many well-known models, such as the SM and the two-Higgs-doublet model, are ruled out as candidates to explain the HyperCP events [8]. 0 In this paper we show that a light pseudoscalar Higgs boson, A1, in the next-to-minimal su- persymmetric standard model (NMSSM) [9] can be identified with the X particle satisfying the constraints

± ± 0 < −9 0 0 < −9 (K π A1) 8.7 10 , (KS π A1) 1.8 10 , B → ∼ × B → ∼ × 0 < −7 (B XsA1) 8.0 10 , (1) B → ∼ × ± ± + − 0 + − obtained in Ref. [8] from the measurements of K π µ µ [10, 11], KS π µ µ [12], and + − → → + B Xsµ µ [13, 14]. We include both two- and four-quark contributions to the K and Σ decays, → neglecting the latter contributions to the B decay.

In the NMSSM there is a gauge-singlet Higgs field N in addition to the two Higgs fields Hu and Hd responsible for the up- and down-type quark masses in the MSSM. We follow the specific model described in Ref. [15], with suitable modifications. The superpotential of the model is given by

W = QY H U + QY H D + LY H E + λH H N 1 kN 3 , (2) u u d d e d d u − 3 2 where Q, U, D, L, and E represent the usual quark and fields, Yu,d,e are the Yukawa couplings, and λ and k are dimensionless parameters. The soft--breaking term in the Higgs potential is

V = m2 H 2 + m2 H 2 + m2 N 2 λA H H N + 1 kA N 3 + H.c. , (3) soft Hu | u| Hd | d| N | | − λ d u 3 k and the resulting Higgs potential has a global U(1)R symmetry in the limit that the parameters Aλ, Ak 0 [16]. → There are two physical pseudoscalar in the model which are linear combinations of the 0 pseudoscalar components in Hu,d and N. The lighter one, A1, has a mass given by m2 = 3k x A + (1/ tan β) (4) A k O in the large-tan β limit, where x = N is the vacuum expectation value of N and tan β is the ratio h i of vacuum expectation values of the two Higgs doublets. If the U(1)R symmetry is broken slightly, the mass of A0 becomes naturally small, with values as low as 100 MeV phenomenologically 1 ∼ allowed [15, 16]. We will now show that this CP -odd scalar can play the role of the X particle. 0 At tree level, the A1 couplings to up- and down-type and to in this model are diagonal and described by iA0 ig m = l m uγ¯ u + l m dγ¯ d 1 , = ℓ ℓ ℓγ¯ ℓ A0 (5) LAqq − u u 5 d d 5 v LAℓ v 5 1 in the general notation of Ref. [8] used to compute the effect of four-quark operators, where v δ l l = g = − , l = d . (6) d − ℓ √2 x u tan2 β with v = 246 GeV being the electroweak scale and δ = (Aλ 2kx)/(Aλ + kx). Thus, lu can be − − neglected in the large-tan β limit. The interactions in qq combined with the operators due to W LA exchange between quarks induce the four-quark contributions discussed earlier. 0 For an A1 of mass 214.3 MeV as we propose, the decay into a -antimuon pair completely 0 + − dominates over the other kinematically allowed modes: A1 e e , νν,¯ γγ. Accordingly, we + → assume (X µ µ−) 1. In addition, the muon anomalous magnetic moment imposes the B → ∼ constraint [3] g < 1.2 . (7) | ℓ| 0 ∼−7 This bound results in an A1 width ΓA < 3.7 10 MeV, which is consistent with the estimate of −7 × 10 MeV for the width of the HyperCP∼ particle [5] if gℓ = ld = (1). An ld of order one is also | | | | O consistent with the general estimates [15] for the size of vδ−/ √2 x . At one-loop level, the two-quark flavor-changing Lagrangian for s dA0 can be written as  → 1 ig = A m d¯(1 + γ )s m d¯(1 γ )s A0 + H.c., (8) LAsd v s 5 − d − 5 1 where   G g = F V ∗ V C , (9) A √ 2 qd qs Aq 8 2 π q=u,c,t X 3 with Vqr being the elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix and CAq depending ′ 0 on the in the loops [15]. The corresponding coupling constant gA in b sA1 has a similar ∗ → expression in terms of VqsVqb CAq. 0 In the large-tan β limit, the pseudoscalar A1 in this model does not couple to up-type quarks and Eq. (9) is completely determined by parameters in the supersymmetry (SUSY) sector. In particular,

CAq can be zero if a super-Glashow-Iliopoulos-Maiani mechanism is operative [15]. In this limit, + 0 0 the decays Σ pA1 and K πA1 are dominated by four-quark contributions [8], and the mode 0 → → b sA is negligible. However, the results of Ref. [8] imply that a vanishing g due to C 0 → 1 A Aq → would not satisfy all the existing constraints. This leads us to some details of the model. Assuming that there is only stop mixing, and that sup and scharm are degenerate, we have for s dA0 or b sA0 [15], → 1 → 1

CAq = δqt tan β ld Xi + λ mtv sin θt˜ cos θt˜ Yuw , (10) "− i=1,2 u,w=1,2 # X X where Xi and Yuw are loop-integral functions of parameters in the chargino and squark mixing matrices given in Ref. [15], and θt˜ is the stop mixing angle. Consequently, in this scenario the CKM factors involving the top-quark are the relevant ones. It then follows from Eq. (9) that CAq of the + 0 −8 0 right magnitude for (Σ pA ) 3 10 will yield (B XsA ) that violates its limit in B → 1 ∼ × B → 1 Eq. (1) by about three orders of magnitude. Now, the penguin amplitude in this model is dominated by squark and chargino intermediate states, and the particular scaling with the top-quark CKM factor in Eq. (10) follows from the assumption of stop mixing only. If we assume instead that there is only sup mixing, CAq will be 0 proportional to δqu. In this case, it is possible for A1 to be the HyperCP particle without violating the bounds in Eq. (1). To be specific, assuming that sup mixing is dominant, and that stop and scharm are degenerate, we obtain

CAq = δqu tan β ld Xi + λ muv sin θu˜ cos θu˜ Yuw . (11) "− i=1,2 u,w=1,2 # X X In addition, Eq. (11) has a vanishing CP -violating phase if all the SUSY parameters are real. This is needed if the contributions from two- and four-quark operators are to partially cancel each other [8], as needed to satisfy the kaon bounds in Eq. (1). To simplify the discussion, we now consider the case where θu˜ = 0. In this case G g = F V ∗ V √2 m tan β m U V D y D y l , (12) A √ 2 ud us W χi i2 i1 3 c,i − 3 u,i d 8 2 π i X    where Uij and Vij are the unitary matrices diagonalizing the chargino mass matrix, D3(y) = 2 2 y ln y/(1 y) is a loop-integral function, and y = m /m , with mχi being a chargino mass [15]. − r,i r˜ χi Therefore, depending on whether the sup mass is larger or smaller than the scharm mass, gA can take either sign. The expressions for the four-quark contributions to the kaon and hyperon decays are lengthy 0 and in general depend on the effective coupling of A1 to a pair [8]. For the simple case above,

4 0 the contributions to A1gg are from quarks in the loop in the large-tan β limit, and the squarks do not contribute since there is no left- and right-squark mixing. Since lu 0, the four-quark → contributions can be written purely in terms of ld. Using the results in Ref. [8] (with D F =0.25 ), − the full amplitudes are

m2 m2 m2 i (K± π±A0) = g K − π 1.08 10−7 l K , (13) M → 1 A v − × d v 

mΣ m f (Σ+ pA0) = g − p 6.96 10−7 l π ip¯Σ+ M → 1 A v − × d v  2  mΣ + m m  f (0.25) g p K + 5.72 10−6 l π ipγ¯ Σ+. (14) − A v m2 m2 × d v 5  K − A   0 0 The corresponding amplitude for KS π A , which we have not displayed, can be derived from → 1 Ref. [8].

Choosing, for example, ld =0.35, we show the resulting branching ratios in Fig. 1. Even though 0 it is possible for A1 to explain the HyperCP observation for a wide range of parameters, as the figure indicates, we have found that there are only rather narrow ranges of gA and ld for which the −7 −7 kaon bounds are also satisfied, namely 1.0 10 < g /ld < 1.3 10 . The B-meson bound is × A × automatically satisfied with our assumptions above.∼ ∼ ∗ We remark that gA in Eq. (12) is proportional to GF VudVus tan β mW mχi , which is of (1) −7 O for tan β = (30) and mχi = (100 GeV). Thus, in order to have gA = (10 ) with these O O −O7 parameter choices, one needs Ui2Vi1 D y D y to be of order 10 as well. Since as 3 c,i − 3 u,i mentioned above there is a super-GIM mechanism that can be at work, as long as m˜ and m˜ are    u c

10-7

10-8 B

10-9

0.2 0.4 0.6 0.8 1 7 10 gA

+ 0 + + 0 0 0 FIG. 1: Branching ratios of Σ pA (solid curve), K π A (dashed curve), and KS π A → 1 → 1 → 1 (dotted curve) as functions of gA for ld = 0.35. The horizontal lines indicate the bounds from the HyperCP < 7 < result and the kaon bounds from Eq. (1). The allowed range in this case is 0.38 10 gA 0.43. ∼ ∼

5 degenerate enough, one can get an arbitrarily small gA to satisfy the required value. It turns out that there is also SUSY parameter space where the desired value of gA can be produced without very fine-tuned degeneracy. We have carried out a detailed exploration of the parameter space. In general, if the chargino masses are larger than the squark masses, fine-tuned squark masses, with splitting of less than one GeV between sup and scharm masses, are required. However, if the chargino masses [of (100 GeV)] O are smaller than the squark masses [of (1 TeV)], there is some parameter space satisfying the O constraints with squark-mass splitting of a few GeV to tens of GeV. Taking λx = 150 GeV, − mc˜ =2.5 TeV, and tan β = 30, we display in Fig. 2 the allowed ranges of mu˜ mc˜ and m2/( λx) −7 −7 − − corresponding to the required values 1.0 10 < g /ld < 1.3 10 , where m is the first element × A × 2 of the chargino undiagonalized mass-matrix [15].∼ In this∼ case, the two chargino eigenmasses are, < < < < respectively, 100 GeV mχ1 150GeV and 200GeV mχ2 1500 GeV. These chargino masses, especially the lighter one,∼ are∼ in the interesting range which∼ may∼ be probed at the LHC and ILC. 0 In this figure, we also display the much larger (gray) regions where A1 can explain the HyperCP observation. In conclusion, we have shown that a light pseudoscalar Higgs in the NMSSM can be identified as the possible new particle responsible for the HyperCP events while satisfying all constraints from kaon and B-meson decays. We have found that there are large regions in the parameter space where the particle can explain the HyperCP observation. This by itself is not trivial, considering that the SM, for example, yields rates that are too large by more than a factor of ten [8]. Only after the kaon bounds are imposed does the allowed parameter space become much more restricted. We believe that this is sufficiently intriguing to warrant a revisiting of these kaon bounds, perhaps by the NA48 experiment.

15

10

5 L GeV H 0 Ž c m - Ž u

m -5

-10

-15 2 4 6 8 10 m2H-ΛxL

0 FIG. 2: Parameter space for mu˜ mc˜ and m /( λx) where A can explain the HyperCP events (gray − 2 − 1 regions) and simultaneously satisfy the kaon bounds (black regions), with the input described in the text.

6 Acknowledgments

The work of X.G.H. was supported in part by NSC and NCTS. The work of G.V. was supported in part by DOE under contract number DE-FG02-01ER41155.

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