PHYSICAL REVIEW D 74, 115015 (2006) Light Higgs production in hyperon decay
Xiao-Gang He* Department of Physics and Center for Theoretical Sciences, National Taiwan University, Taipei, Taiwan
Jusak Tandean† Department of Mathematics/Physics/Computer Science, University of La Verne, La Verne, California 91750, USA
G. Valencia‡ Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA (Received 30 October 2006; published 22 December 2006) A recent HyperCP observation of three events in the decay ! p is suggestive of a new particle with mass 214.3 MeV. In order to confront models that contain a light Higgs boson with this observation, it is necessary to know the Higgs production rate in hyperon decay. The contribution to this rate from penguinlike two-quark operators has been considered before and found to be too large. We point out that there are additional four-quark contributions to this rate that could be comparable in size to the two-quark contributions, and that could bring the total rate to the observed level in some models. To this effect we implement the low-energy theorems that dictate the couplings of light Higgs bosons to hyperons at leading order in chiral perturbation theory. We consider the cases of scalar and pseudoscalar Higgs bosons in the standard model and in its two-Higgs-doublet extensions to illustrate the challenges posed by existing experimental constraints and suggest possible avenues for models to satisfy them.
DOI: 10.1103/PhysRevD.74.115015 PACS numbers: 14.80. j, 11.30.Rd, 12.15.Ji, 13.30.Ce
do)scalars to hadrons. In this paper we generalize existing I. INTRODUCTION studies appropriate for kaon decay to the case of hyperon Three events for the decay mode ! p with a decay. This allows us to discuss the production of light dimuon invariant mass of 214:3 0:5 MeV have been (pseudo)scalars in hyperon decay consistently, including recently observed by the HyperCP Collaboration [1]. It is the effects of both the two- and four-quark operators with possible to account for these events within the standard the aid of PT. We consider the cases of scalar and model (SM) when long-distance contributions are properly pseudoscalar Higgs bosons in the SM and in the two- included [2]. However, the probability of having all three Higgs-doublet model (2HDM), expressing our results in events at the same dimuon mass in the SM is less than 1%. a form that can be easily applied to more complicated This suggests a new-particle interpretation for the events, Higgs models. 2:4 8 for which the branching ratio is 3:1 1:9 1:5 10 [1]. This paper is organized as follows. We begin by collect- This possibility has been explored to some extent in the ing in Sec. II the existing constraints on light Higgs bosons literature, where it has been shown that kaon decays place from kaon, B-meson, and hyperon decays if we interpret severe constraints on the couplings of the hypothetical new the HyperCP events as being mediated by a light Higgs particle [3–5]. In particular, it was found that the flavor- boson. In Secs. III and IV we compute the production rates changing coupling of the new state, X,tods has to be of a in both kaon and hyperon decays for a light scalar and pseudoscalar or axial-vector nature to explain why the state pseudoscalar Higgs boson, respectively. Finally, in Sec. V has not been seen in K ! . At least one model we summarize our results and state our conclusions. containing a particle with these properties has appeared in the literature [6]. II. SUMMARY OF EXISTING CONSTRAINTS All these previous analyses of X considered only the effects of two-quark operators for dsX . However, it is well In Ref. [3] we parametrized the possible couplings of the known in the case of light Higgs production in kaon decay new particle, X,tods and assuming that it had definite that there are also four-quark operators that can contribute parity. Whereas this is a reasonable assumption for the at the same level as the two-quark ones [7–10]. These four- diagonal couplings of X to fermions, it is not for its quark contributions are most conveniently described in flavor-changing neutral couplings (FCNCs). Two-quark chiral perturbation theory ( PT) which implements low- FCNCs are predominantly induced by Higgs-penguin dia- energy theorems governing the couplings of light (pseu- grams, which result in left- and right-handed couplings, implying that the scalar and pseudoscalar ones are present *Electronic address: [email protected] simultaneously. For this reason, we revisit the existing †Electronic address: [email protected] constraints for X being a scalar particle, H , or a pseudo- ‡Electronic address: [email protected] scalar particle, A, assuming them to have two-fermion
1550-7998=2006=74(11)=115015(12) 115015-1 © 2006 The American Physical Society XIAO-GANG HE, JUSAK TANDEAN, AND G. VALENCIA PHYSICAL REVIEW D 74, 115015 (2006) FCNCs described by struct a model that can satisfy all the existing constraints.
g For this purpose, we consider a pseudoscalar A with two- L H m d 1 s m d 1 s H H:c:; quark couplings as in Eq. (1b) supplemented with simple H sd v s 5 d 5 parametrizations for the four-quark amplitudes for both (1a) kaon and hyperon decays. For B-meson decay, we assume igA that the two-quark contribution completely dominates. L m d 1 s m d 1 s A H:c:; Asd v s 5 d 5 (1b) A. K ! A where the g’s are coupling constants, m is a quark mass, Introducing the dimensionless quantity M4K for the four- q quark contribution, we express the amplitude for K ! and v 2 1=4G 1=2 246 GeV. In addition, the diagonal F and its branching ratio, respectively, as couplings to charged leptons are assumed to have definite A 2 2 2 parity and be proportional to the lepton mass, mK m mK iM K ! A g M ; A v 4K v (6) g‘m‘ ig‘m‘ LH ‘ ‘‘H ; LA‘ ‘ 5‘A: (2) 8 2 v v B K ! A 4:43 10 jgA 1:08M4Kj : For a (pseudo)scalar of mass 214.3 MeV, it is then This mode is constrained by its nonobservation in the BNL natural to assume that the decay X ! will domi- E865 [11] or FNAL HyperCP [12] measurements of K ! nate over the other kinematically allowed modes: X ! . It is also constrained by its nonobservation in 0 e e , , . We will restrict ourselves to this case, the isospin-related mode KS ! by CERN NA48 assuming that B X ! 1. This is true, for ex- [13]. Of these three experiments, E865 had the best statis- ample, for a light SM Higgs boson where g‘ 1, or for tics, collecting 430 events in K ! . A new light pseudoscalars in the 2HDM types I and II, where particle A of mass 214.3 MeV would have contributed g‘ cot and tan , respectively. In all these cases, only in their first dimuon-mass bin, where 0:21 GeV < 2 5 X ! e e are suppressed at least by me=m 10 . m < 0:224 GeV and approximately 30 events were ob- To be consistent with the HyperCP observation, X must served. To obtain a conservative bound, we assume that all be short lived and decay inside the detector. This is com- the events in the first bin are statistically Gaussian and can patible with the estimate for the total width A be attributed to the new particle (either a scalar or a 10 7 MeV [5] of a pseudoscalar particle, A. It was shown pseudoscalar). Further assuming uniform acceptance, we in Ref. [3] that the muon anomalous magnetic moment obtain at 95% C.L. imposes the constraint 9 B K ! X & 8:7 10 : (7)
jg‘j & 1:2: (3) The NA48 Collaboration collected 6 events for KS ! A coupling satisfying this constraint implies a width 0 [13], and none of them have the 214.3 MeV 7 invariant mass required if they originate from the new A & 3:7 10 MeV; (4) particle A. Using the KS flux and the acceptance at low consistent with the observation. In contrast, the corre- m in Ref. [13], we estimate a single event sensitivity of 0:6 10 sponding constraint for a scalar particle is jg‘j & 0:98, 5:3 0:4 10 . With no events observed and Poisson leading to a longer lifetime, statistics, this translates into the 95%-C.L. bound
9 0 9
H & 6:9 10 MeV: (5) B KS ! X & 1:8 10 : (8) The estimated lifetime for the HyperCP particle is there- We employ these bounds when we discuss specific models, fore consistent with that of a pseudoscalar or scalar that but for now we use the E865 result in Eq. (7), combined decays predominantly into muons. with Eq. (6), to find In addition to the two-quark contributions to the ampli- 9 jg 1:08M j & 4:4 10 : (9) tudes for K ! H A and ! pH A induced by A 4K the interactions in Eq. (1), we will also include contribu- tions arising from the usual SM four-quark j Sj 1 op- B. ! pA erators, along with flavor-conserving couplings of H A . We will adopt the chiral-Lagrangian approach to evaluate In this case, we need two new dimensionless quantities the hadron-level interactions. A4 and B4 to parametrize the effect of the four-quark Later on we will discuss specific models and consider operators, writing the amplitude as the bounds appropriate for them, including all the relevant M ! pA ip A B ; (10a) two- and four-quark contributions. It is useful to start with pA pA 5 one example to illustrate the ingredients needed to con- where
115015-2 LIGHT HIGGS PRODUCTION IN HYPERON DECAY PHYSICAL REVIEW D 74, 115015 (2006) 0 m mN f g mb
Ap g A4 ; L s 1 bX H:c:; (14) A A v v Xbs v 5 2 m mN mK f where g0 g0 ig0 for X H A . This leads to the BpA gA D F 2 2 B4 ; H A v mK mA v partial decay rate
(10b) 3 0 2 mb b ! sX ’jg j : (15) the parameters D and F coming from a chiral Lagrangian 8 v2 to be discussed in a later section and f 92:4 MeV Using for illustration m 4:3 GeV and the B lifetime being the pion-decay constant. The resulting branching b [14] results in ratio is 8 0 2
b ! sX 1:3 10 jg j : (16) 6 2 B B ! pA 1:91 10 jgA 0:36A4j One could obtain a similar number for b ! dX. 4:84 104jg 0:14B j2 (11) A 4 The latest experimental average B b ! s 1:23 6 4:27 1:22 10 [14] covers the full kinematic range with the choice D F 0:25. Combining the statistical 0 and systematic errors of the HyperCP measurement [1]in for m . To constrain g , it is better to limit the comparison quadrature, we require to the measured rate at the lowest measured m invariant- mass bin. BABAR quotes in Table 2 of Ref. [15] 2:8 8
! B pA 3:1 2:4 10 ; (12) B b ! s‘ ‘ m‘ ‘ 2 0:2 GeV;1:0 GeV and therefore 0:07 6 0:08 0:36 0:04 10 : (17) 7 jgA 0:36A4j 1:3 0:6 10 ; (13) This is an average for electrons and muons, but no notice- able difference between them was found. Belle quotes on where we have used the larger of the errors in Eq. (12) and Table 4 of Ref. [16] the corresponding number ignored the contribution from the P-wave term in Eq. (11),
assuming that B & A . This assumption is satisfied by all 4 4 B b ! s‘ ‘ m 2 0:2 GeV;1:0 GeV the models we discuss, but when checking a specific ‘ ‘ 11:3 4:8 4:6 10 7: (18) model, we do so without neglecting B4. 2:7 A comparison of Eqs. (9) and (13) shows why it is not To be conservative, we constrain the Higgs coupling by possible to have a (pseudo)scalar with penguinlike flavor- requiring that the induced rate be below the 95%-C.L. changing neutral couplings, as in Eq. (1), as an explanation upper range of the measured b ! s‘ ‘ rate in the lowest for the HyperCP result given the constraints from kaon measured m bin. Thus, combining errors in quadrature decay. It also shows how this is no longer true if there are four-quark contributions to the amplitudes that are compa- for the more restrictive BABAR result gives rable to the penguin amplitudes. In particular, if we assume 7 B b ! s‘ ‘ & 8:0 10 BABAR m‘ ‘ <1 GeV that in a given model gA, M4K, and A4 have comparable magnitudes, we see that in order to satisfy both Eqs. (9) and (19) (13) we need a cancellation between the two- and four- and, correspondingly, quark contributions to the kaon amplitude that reduces 0 8 them by a factor of about 20. As we will show in later jg j & 7:8 10 : (20) sections, this is possible in many models. For this cancel- The exclusive B ! K; K? modes have been mea- lation to work, however, gA and M4K must also have similar phases. As we will see, this is a requirement that sured, but the resulting constraints are not better than Eq. (20). This constraint, Eq. (20), is difficult to satisfy is much harder to satisfy. In the simple models we consider 0 in this paper, the phase of g is much larger than the phase in models where gA and g are related by top-quark A Cabibbo-Kobayashi-Maskawa (CKM) angles, as happens of M4K so that the cancellation does not happen for the imaginary part. in the simple models we consider here.
C. b ! sX III. SCALAR HIGGS BOSON Finally, we consider the constraints on the new particle In this section we discuss in detail the case of a light from its nonobservation in B-meson decay. In this case, the Higgs boson in the standard model and in the two-Higgs- four-quark contributions are negligible, and we can neglect doublet model. We will use known low-energy theorems to ms compared to mb. The Lagrangian for b ! sX can then implement the four-quark contributions to kaon and hy- be expressed as peron amplitudes.
115015-3 XIAO-GANG HE, JUSAK TANDEAN, AND G. VALENCIA PHYSICAL REVIEW D 74, 115015 (2006) A. Two-quark j Sj 1 interactions which selects out s ! d transitions, M 2 2 2 2 The effective Lagrangian for the sdH coupling, where diag m;^ m;^ ms diag m ;m ; 2mK m = 2B0 the H is either the standard-model Higgs boson H0 or the quark-mass matrix in the isospin-symmetric limit mu lightest scalar Higgs boson h0 in the 2HDM, has been md m^ , and the baryon and meson fields represented by i’=f much discussed in the literature [7,8,17–19] and can be the usual 3 3 matrices B and e , respectively. written as LH sd in Eq. (1a), where To derive amplitudes, we also need the chiral G X F 2 Lagrangian for the strong interactions of the hadrons g p m V V F q ; (21) H 2 q qd qs 4 2 q u;c;t [22,23]. At leading order in the derivative and ms expan- sions, it can be written as with V being the elements of the CKM matrix and F q
kl depending on the model. In the SM, for a Higgs mass much Ls hBi @ B V ;B i m0hBBi smaller than the W mass, DhB 5fA ;Bgi FhB 5 A ;B i
F q 3=4; (22) bDhB fM ;Bgi bFhB M ;B i b0hM ihBB i whereas in the 2HDM the expression for F q is much 1f2h@ y@ i 1f2B hM i; (28) lengthier [18,19]. 4 2 0 Using CKM and mass parameters from Ref. [20], we 1 y y where V 2 @ @ , m0 is the baryon mass find in the SM i y y in the chiral limit, A 2 @ @ , and M 6 yM y My , with further details being given in g 1:3 0:6i 10 ; (23) H Ref. [3]. to be compared with Eqs. (9) and (13) above. Employing From LH and Ls, we derive the leading-order diagrams the expression for F q derived in Ref. [19], we obtain a shown in Fig. 1 for ! pH , yielding the amplitude
similar number in the 2HDM type II, for instance, m m m2 7 N K
! gH 5:0 1:9i 10 (24) M2q pH gH 2 2 p v mK m for the parameters 2 2 m mN mK m gH D F 2 2 tan ’ 2:57; sin ’0:149; v mK m (25) H mH 250 GeV; p 5 ; (29) where tan is the ratio of vacuum expectation values of the where the two terms correspond to the two diagrams, two Higgs doublets, the mixing angle in the neutral- respectively, m ;N are isospin-symmetric masses, and we Higgs-boson mass matrix, and mH the mass of the have used the relations m mN 2 bD bF ms m^ , 1 2 2 charged Higgs bosons. We note that the and values mK B0 m^ ms , and m 2B0m^ derived from Ls. above satisfy the constraint sin2 < 0:06 from Numerically, we will allow D and F to have the ranges CERN LEP [21]. We see right away that gH can be in 0:6 D 0:8 and 0:4 F 0:5 [23], leading to the right ball park to explain the HyperCP observation,
Eq. (13), but conflicts with the kaon bound, Eq. (9). 0:1 D F 0:4; (30) To evaluate the hadronic amplitudes from this two-quark which is their combination occurring in our amplitudes. contribution, we employ chiral perturbation theory. Using It follows that the contribution of L to the branching the operator matching of Ref. [3], we write the lowest- H sd ratio of ! pH for m 214:3 MeV and the middle order chiral realization of as H LH sd value D F 0:25 is in the SM LH bDhBfhH ;Bgi bFhB hH ;B i b0hhH ihBBi 1 2 2f B0hhH i H:c:; (26) where h i Tr in flavor-SU(3) space, f f 92:4 MeV, and K¯ 0 y y H Σ + p Σ + p
h 2g hM Mh ; (27) H H v (a) (b) with h being a 3 3 matrix having elements hkl k2 3l FIG. 1. Diagrams contributing to ! pH arising from 1 2 2 at leading order in PT. The square vertices come from We have also set mH =mW in F q , where is defined in LH sd Ref. [19]. LH in Eq. (26), and the solid vertex from Ls in Eq. (28).
115015-4 LIGHT HIGGS PRODUCTION IN HYPERON DECAY PHYSICAL REVIEW D 74, 115015 (2006)
7 y y B2q ! pH 40 1 10 ; (31) Lw hDhB f h ; Bgi hFhB h ; B i f2hh@ @ yi 2 ~ f2B hh M yi where we have ignored the imaginary (CP-violating) part 8 8 0 of the amplitude, and the two numbers correspond to the H:c:; (36) contributions from the scalar and pseudoscalar flavor- changing couplings, respectively. Evidently, the scalar where hD;F can be extracted from hyperon nonleptonic 8 contribution is much larger than what HyperCP saw, but decays, 8 7:8 10 from K ! , the sign fol- the pseudoscalar contribution is within the range. This, lowing from various predictions [7–9,26], and ~8 is un- however, is only part of the story, as there are in addition known as it does not contribute to any process with only four-quark contributions to be discussed in the next kaons and pions. subsection. The four-quark j Sj 1 interactions of a light Higgs Also from L , we derive the leading-order diagram for boson H arise from its tree-level couplings to quarks and H K ! H , which is that in Fig. 1(a) with (p) replaced W bosons, as well as from its coupling to gluons induced by a triangle diagram with heavy quarks in the loop. To by K ( ) and arises from the scalar coupling in LH sd. The resulting amplitude is obtain the relevant chiral Lagrangians, one starts with Ls;w above and follows the prescription given in Refs. [7–9,19].
p 0 0 The results are M2q K ! H 2M2q K ! H
(32) g m2 1 1 H K ; LH c f2h@ y@ i c f2B hM i v s 4 1 2 2 0 1 H H and so M K ! 0H ReM K ! H . f2B hM^ M i k m hBB i 2q L 2q 0 v 1 0 v Dropping again the imaginary parts of the amplitudes, 2 ^ ^ we obtain in the SM the branching ratios k2 bDhBfM ;Bgi bFhB M ;B i H 4 ^ B2q K ! H 9:3 10 ; b0hM ihBBi ; (37) (33) v 0 3 B2q KL ! H 3:9 10 :
LH c f2hh@ @ yi 2 ~ c f2B hh M yi These numbers would easily be incompatible with that in w 8 3 8 4 0 Eq. (7) and the 95%-C.L. bound2 H 2 ~ f2B hh M^ M yi 8 0 v 0 10 B KL ! < 4:9 10 ; (34) H k h hB f yh ; Bgi h hB yh ; B i 3 D F v but, as in the case, there are four-quark contributions H:c:; (38) that have to be considered as well. The situation is similar in the 2HDM. Adopting the real where part of the coupling in Eq. (24), for example, we find
c 2k ;c 3k 1;c 4k 2k ; 8 1 G 2 G 3 G W B2q ! pH 56 1 10 ; c4 5kG 2kW 1;k1 kG;k2 1; (39) K ! : 4; (35) B2q H 1 3 10 y y y k3 3kG 2kW; M^ M ^ M^ ; 0 4 B2q KL ! H 5:4 10 : with
2 2ku kd
B. Four-quark j Sj 1 interactions k ; M^ diag k m;^ k m;^ k m ; G 27 u d d s The hadronic interactions of a light Higgs boson due to (40) four-quark j Sj 1 operators are best accounted for in the chiral-Lagrangian approach. The dominant contribution is the expression for kG corresponding to 3 heavy and 3 light generated by the j Ij 1 2 component of the effective quarks. The parameters ku;d, kW, and kG come from the Hamiltonian transforming as 8L; 1R . The corresponding couplings of H to light quarks, W , and the gluons, Lagrangian at leading order is given by [23,25] respectively, and depend on the model of the Higgs sector. Thus 2We have inferred this number from Ref. [24] which reported
0 10 k k k 1 in the SM; (41) B KL ! < 3:8 10 at 90% C.L. u d W
115015-5 XIAO-GANG HE, JUSAK TANDEAN, AND G. VALENCIA PHYSICAL REVIEW D 74, 115015 (2006) cos 7 7 A 0 3:25 10 ;B0 26:67 10 ; (47) k k ;k sin in the 2HDM I; p p u d sin W (42) up to an overall sign, in our numerical evaluation of the four-quark contributions to ! pH we will explore cos sin different hD–hF values accordingly. k ;k ; H u d We can also derive from L ; the corresponding sin cos (43) s w leading-order diagrams for K ! H , which are the upper kW sin in the 2HDM II: three in Fig. 2 with (p) replaced by K ( ) and yield
The parameters c for the meson terms have already 1;2;3;4 M K ! H 8 2 k k m2 m2 m2 been obtained in the literature [7–9,19,27], whereas the 4q v W G K H new ones k follow from how the baryon parameters 2 1;2;3 kd ku m depend on masses: m0 , bD;F;0 1, mq, and 3 2 ~8 2 hD;F =mW, where is a QCD mass scale. Note that 4 kG kW mK; (48) we work in that basis in which the mass terms in the v Lagrangians are not diagonal and must therefore include
the corresponding tadpole diagrams in our calculation. K0 ! 0 p 8 2 k k m2 m2 m2 H M4q H G W K H For ! pH , we derive from L ; the diagrams 2v s w shown in Fig. 2, finding 2 2 m mK
k k h h u d m2 m2 M ! pH k 3k 2k D F p K 4q d G W v ~ 8 2 p 4 kW kG mK 4 kG kW D F ~8 2v m m m2 m2 m2 m2 N K p ; k k K : (49) v m2 m2 5 d u m2 m2 K H K (44) Since ~8 is unknown, in evaluating its effect on K ! where the first term comes from the upper three diagrams, H we will allow it to vary from 10 to 10 times 8. which are at leading order, and the second term results Naively we would expect 8 and ~8 to be of the same order. from the lower two diagrams, which are at next-to-leading order. Now, the combination hD–hF also occurs in the C. Total contributions 0 amplitude for ! p , which we write as The total amplitude for K ! H comes from the 0 sum of the contributions in Eqs. (32) and (48). If the M ! p ip A 0 B 0 ; (45) p p 5 CP-violating terms in the amplitudes are ignored, it is possible for the two-quark and four-quark contributions where from Ls;w to cancel. We show this possibility in Fig. 3, where we hD hF plot the resulting branching ratio in the SM as a function of Ap 0 ; 2f the ratio r8 ~8= 8 for mH 214:3 MeV. We find that (46) K ! 0 when r ’ 5:1 and that, as the hD hF m mN B H 8 B 0 D F : p 2f m m figure indicates, for only a very narrow range of r8 around N this value does the branching ratio ever fall below the Since from experiment [28] upper limit in Eq. (7). In Fig. 3, we also plot the corre-
Σ + p Σ + p Σ + p Σ + p
K¯ 0 K¯ 0 K¯ 0 Σ + p Σ + p
H FIG. 2. Diagrams contributing to ! pH arising from the four-quark operators. The square vertices come from Lw in H Eqs. (36) and (38), whereas the dots are from Ls in Eqs. (28) and (37).
115015-6 LIGHT HIGGS PRODUCTION IN HYPERON DECAY PHYSICAL REVIEW D 74, 115015 (2006)
10 -3 10 − 5
) − 10 -5 10 6
π p 10 − 7 → 10 -7
K
( 10 − 8 10 -9 10 − 9
-8 -6 -4 -2 0 0.15 0.2 0.25 0.3 0.35 0.4 r8 D F
FIG. 3 (color online). Contributions of real parts of total FIG. 4 (color online). Contribution of the real part of the total 0 amplitude for ! p to its branching ratio in the SM as a amplitudes for K ! H (solid curve) and KL ! H H (dashed curve) in the SM to their branching ratios as functions function of D F for mH 214:3 MeV and r8 ’ 5:1. The of r ~ = for m 214:3 MeV. The horizontal lines are solid (dotted) curve corresponds to hD hF extracted from the 8 8 8 H 0 the corresponding upper bounds in Eqs. (7) and (34). P-wave (S-wave) fit to the ! p data using Eqs. (46) and (47). The dashed lines correspond to the upper and lower bounds in the HyperCP result. sponding branching ratio of the isospin-related mode 0 KL ! H . − For ! pH , the total amplitude results from adding 10 3 the contributions in Eqs. (29) and (44). Including only the − 5 real part of amplitudes again, and using r8 ’ 5:1 deter- 10 mined above, we plot in Fig. 4 the branching ratio in the SM as a function of D F for the range in Eq. (30). This 10 − 7 K figure shows that the curve resulting from the P-wave fit using Eqs. (46) and (47) satisfies the HyperCP constraints 10 − 9 for certain D F values. In Figs. 5 and 6, we display the corresponding branching 246810 ratios in the 2HDM II obtained using the parameters in r8 Eqs. (24) and (25). In contrast to the SM case, here B K ! H 0 when r8 ’ 6:7, but the vanishing FIG. 5 (color online). Contributions of real parts of total 0 of the KL rate occurs at a different r8 value due to amplitudes for K ! H (solid curve) and KL ! H 0 (dashed curve) in the 2HDM II to their branching ratios as M4q KL ! H and ReM4q K ! H being 3 functions of r8 ~8= 8 for m 214:3 MeV and the parame- unequal with ku kd in Eq. (43). As a consequence, the H two kaon constraints cannot be satisfied simultaneously. ters in Eq. (25). The horizontal lines indicate the upper bounds in Eqs. (7) and (34). Furthermore, the ! pH curve that falls within the HyperCP limits is the one resulting from the S-wave fit using Eqs. (46) and (47). To summarize this section, a light Higgs boson in the − SM can be made compatible with the empirical bounds for 10 5 ! pH , while satisfying the constraints from K ! 10 − 6
, if the real part of the two-quark (penguin) contribu- p H − tion to the respective amplitudes is combined with the four- 10 7
quark contribution. Moreover, in the 2HDM such a particle 10 − 8 can satisfy all these constraints if its diagonal couplings to − 9 the up- and down-type quarks are the same. For this to 10 happen in either model, it is necessary for the two ampli- 0.15 0.2 0.25 0.3 0.35 0.4 tudes to cancel precisely, and we have shown that this is D F possible for certain values of the hadronic constants ~8, hD hF, and D F. Although ~8 is not known, unlike FIG. 6 (color online). Contribution of the real part of the total amplitude for ! pH to its branching ratio in the 2HDM as a function of D F for mH 214:3 MeV and the parameters 3 in Eq. (25). The solid (dotted) curve corresponds to h h We note that, although ~8 is not known from experiment, D F P S ! p 0 there are model calculations [8,26] of it yielding j ~8= 8j 0:2. extracted from the -wave ( -wave) fit to the data This would make the kaon rates greatly exceed their bounds, as using Eqs. (46) and (47). The dashed lines correspond to the can be seen from Figs. 3 and 5. upper and lower bounds from the HyperCP result.
115015-7 XIAO-GANG HE, JUSAK TANDEAN, AND G. VALENCIA PHYSICAL REVIEW D 74, 115015 (2006) hD hF and D F which are extractable from hyperon types I and II of the model and can be written as LAsd nonleptonic and semileptonic decays [23], it has a definite in Eq. (1b), where value in the SM and cannot be fine-tuned. We note that in G X A q A q F 2 1 2 all the ! pH cases discussed above the p 5 term g p mqV Vqs ; (52) A 2 qd 3 in the amplitude is small compared to the p term and 16 2 q u;c;t tan tan that, therefore, the ~8 contributions are important only in with A q being functions of m , m , and m , whose the kaon cases. We also note that flipping the signs of Ap 0 1;2 q W H expressions can be found in Ref. [31]. The leading-order and Bp 0 in Eq. (47), whose overall sign is not fixed by experiment, would prevent the cancellation in the hyperon chiral realization of LAsd is then case from occurring and thus result in rates much above the LA bDhBfhA;Bgi bFhB hA;B i b0hhAihBBi bounds. 1 2 It turns out that the imaginary part of the penguin 2f B0hhAi H:c:; (53) amplitude is sufficient to eliminate these scalar particles where as candidates for the HyperCP events, as it cannot be canceled by the four-quark amplitudes [29], having a size y y A hA 2igA hM Mh : (54) of v 7 The leading-order diagrams for K ! and ! A jImgH j 5:8 10 ; (50) pA arising from LA, plus Ls, are similar to those in the much larger than allowed by Eq. (9) with ImM4K 0. The case of standard-model Higgs bosons, displayed in Fig. 1. scaling of the penguin amplitude to the B-meson system is The resulting amplitudes are also incompatible with the b ! sX bound. In the SM p G X K ! 2 K0 ! 0 0 3 F 2 4 M2q A M2q A g p mqVqsVqb 1:7 10 ; (51) H 2 2 2 16 2 q u;c;t mK m igA ; (55) which is much larger than allowed by Eq. (20). In the v 2HDM, the relative size is also too large: 0 4 m m jg =g j jVtb=Vtdj. N H H M2q ! pA igA p igA D F Both of these problems are associated with a structure in v which the Higgs-penguin amplitude is dominated by dia- m m m2 N K p : grams with up-type quarks and W bosons in the loops. It 2 2 5 v mK mA may be possible to remedy these problems in models with additional contributions to the penguin, for example, from (56) supersymmetric (SUSY) partners. If the penguin can be sufficiently suppressed, Eqs. (44) and (48) suggest that B. Four-quark j Sj 1 interactions models in which kW kG could satisfy the kaon bounds while being able to account for the HyperCP result. The diagonal couplings of A to light quarks in the 2HDM are described by [32] IV. PSEUDOSCALAR HIGGS BOSON iA iA
L q M ~ q q Mq~ H:c:; (57) We now consider the possibility that the new particle is a Aqq 5 v L R v light CP-odd pseudoscalar, A, in the two-Higgs-doublet where model. Specifically, we do so in types I and II of the model. T q uds ; M~ diag lum;^ ldm;^ ldms ; (58) A. Two-quark j Sj 1 interactions with The two-quark flavor-changing couplings of A in the
2HDM are induced at one loop and have been evaluated in lu ld cot in the 2HDM I; (59) Refs. [31,32]. The effective Lagrangian is the same in
lu cot ; ld tan in the 2HDM II: 4One could arrive at a similar conclusion about H in the (60) 2HDM II by analyzing the decay ! 0H , whose amplitude depends on the four-quark parameters kd ku [27]. Thus, from Since the Lagrangian for the quark masses is Lq the 90%-C.L. bound B ! 0H < 5 10 6 [30], one ex- q LMqR H:c:, the effect of LqqA on interactions de- tracts jkd kuj < 0:45 for mH 214:3 MeV, which is incom- patible with the limit derived from Eq. (43) plus the LEP scribed by Ls;w can be taken into account using Ls;w and 2 ~ constraint sin < 0:06 [21], namely jkd kuj substituting M with MiA=v [10]. The resulting j2 cos = sin 2 j > 1:9. Lagrangians are
115015-8 LIGHT HIGGS PRODUCTION IN HYPERON DECAY PHYSICAL REVIEW D 74, 115015 (2006)
1 2 1 LA b hB fM~ ;Bgi b hB M~ ;B i b hM~ ihBB i L m2 m2 m2 s D F 0 1A 2 1 3 K 3 1 i f 2 2 ~ A A f B0hM i ; (61) 1 p 2lu ld ; (64) 2 v 6v i A 2 y A which modifies the - mixing generated by A.
~ L w 2 ~8f B0hh M i H:c:; (62) 1 A Ls v A From Ls;w , we derive the leading-order diagrams where shown in Fig. 7 for K ! A, where y y y
~ ~ ~ M M M : (63) 0 8 cos 1 sin ; 8 sin 1 cos : A In addition, if the SU(3)p singlet 1 is included in Ls;w by (65) replacing with exp i 2=3 1=f , the coupling of A to two gluons via the axial anomaly gives rise to [10] The resulting amplitudes are
2 p i 8 lu ld m 2 2 2 2 2 M4q K ! A i 8 2mK m 3mA c 8 mK m s 2v p 2 2 2 2 2 4ldmK 3ld lu m c 2 2ldmK lum ld 2lu m~0 s 2 2 2 2 2 i 8 2mK m 3mA s 6 m m v A p p 2 2 2 2 2 2 2 4ldmK 3ld lu m s 2 2ldmK lum ld 2lu m~0 c 8 mK m c 2 2 ; (66a) 6 m 0 mA v i l l 2m2 m2 m2 m2 p 0 0 8 u d K A 2 2 2 2 2 M4q K ! A p i 8 2m m 3m c 8 m m s 8 m2 m2 v K A K A p 4l m2 3l l m2 c 2 2l m2 l m2 l 2l m~2 s d K d u p d K u d u 0 2 2 6 2 m m v pA 2 2 2 2 2 i 8 2m m 3m s 8 m m c K A p K 4l m2 3l l m2 s 2 2l m2 l m2 l 2l m~2 c d K d u p d K u d u 0 ; (66b) 2 2 6 2 mA m 0 v where 2 2 2 2 1 2 c cos ; s sin ; m~ m m m : (67) 0 1 3 K 3 2 The ~8 contributions to this amplitude cancel completely, as already noted in Ref. [10]. Numerically, m~0 ’ 819 MeV from 0 fitting to the mass after diagonalizing the 8;1 masses derived from the Lagrangians in Eqs. (28) and (64), and consequently ’ 19:7 . The leading-order four-quark contributions to ! pA arise from the diagrams in Fig. 8 and can be expressed as
π 0,η,η K 0 K π K π Σ + p
π 0,η,η π 0,η,η π 0,η,η π 0,η,η K π π Σ + p Σ + p K K Σ + p
FIG. 7. Diagrams contributing to K ! A arising from the FIG. 8. Diagrams contributing to ! pA arising from the A four-quark operators. The dots come from Ls in Eqs. (28) and four-quark operators. The square vertices come from Lw in A (61), whereas the square vertices are from Lw in Eqs. (36) and Eq. (36), whereas the dots are from the Lagrangians in (62). Eqs. (28), (61), and (64).
115015-9 XIAO-GANG HE, JUSAK TANDEAN, AND G. VALENCIA PHYSICAL REVIEW D 74, 115015 (2006)
M 4q ! pA ip ApA BpA 5 ; (68) where p 2 2 2 2 2 2 2 fAp 0 ld lu m fAp 0 f 4ldmK 3ld lu m c 2 2ldmK lum ld 2lu m~0 c s g ApA 2 2 2 2 2 mA m v 2 m mA v p 2 2 2 2 2 2 fAp 0 f 4ldmK 3ld lu m s 2 2ldmK lum ld 2lu m~0 c s g 2 2 ; (69) 2 m 0 mA v p 2 2 2 2 2 2 2 fBp 0 ld lu m fBp 0 f 4ldmK 3ld lu m c 2 2ldmK lum ld 2lu m~0 c s g BpA 2 2 2 2 2 mA m v 2 m mA v p 2 2 2 2 2 2 fBp 0 f 4ldmK ld lu m s 2 2ldmK lum ld 2lu m~0 c s g 2 2 ; (70) 2 m 0 mA v
0 where Ap 0 and Bp 0 are given in Eq. (46). We note that M4q KS ! H and ReM4q K ! H being contributions with 8 or ~8 appear only at next-to-leading unequal with lu ld in Eqs. (59) and (60). We note that order. the situation is not much different if the signs of Ap 0 and Bp 0 in Eq. (47) are both flipped. C. Total contributions To summarize this section, we have found that it is The total amplitudes for K ! A result from adding possible for the real part of the penguin amplitude to cancel the contributions in Eqs. (55) and (66). If the CP-violating against the four-quark amplitude to approximately satisfy terms in the amplitudes are ignored, it is possible for the the kaon bounds while explaining the HyperCP observa- two-quark and four-quark contributions to cancel. We tion with a 2HDM pseudoscalar. Unlike the scalar case, show this possibility in Fig. 9, where we plot the resulting there is no free hadronic parameter at leading order in PT branching ratios as functions of the charged-Higgs-boson in this case. The cancellation must happen as a function of mass for mA 214:3 MeV and different tan values in the short-distance parameters that determine the size of the the 2 versions of the 2HDM. The total amplitude for ! amplitudes. pA is the sum of the contributions in Eqs. (56) and (68). If A feature shared by scalars and pseudoscalars in the the experimental values of Ap 0 and Bp 0 in Eq. (47), as 2HDM is that the imaginary part of the penguin amplitude well as the middle value D F 0:25, are used in the is incompatible with the kaon bounds in Eq. (9) and has no total amplitude, the resulting branching ratios in the 2HDM counterpart that could cancel it in the four-quark ampli- are displayed in Fig. 10. tude. A related problem is that the scaling of the penguin We have found that only one of the kaon bounds can be amplitude to the B system is also incompatible with satisfied if the HyperCP result is assumed to be mediated observation. by A in the 2HDM. However, for certain tan and mH In view of these flaws, it is tempting to search for a values near the ones indicated in Figs. 9 and 10, all the model in which the penguin amplitudes are completely kaon and hyperon constraints can be nearly satisfied si- suppressed, and the 2HDM II seems to allow us to do multaneously. Part of the difficulty in satisfying all of the that. In the 2HDM II the penguin amplitudes are propor- constraints lies with the vanishing of the K and KS rates tional to lu, whereas the four-quark amplitudes receive occurring at different mH values, which is due to contributions from both lu and ld in Eq. (60). Thus the
2HDM I withtanβ = 4 2HDM II with tanβ = 0.9
− − 10 7 10 7
π − π − 10 8 10 8
K K − − 10 9 10 9
265 270 275 280 285 290 290 300 310 320 330 mH GeV mH GeV
0 FIG. 9 (color online). Contributions of real parts of total amplitudes for K ! A (solid curve) and KS ! A (dashed curve) in the 2HDM to their branching ratios as functions of charged-Higgs-boson mass for mA 214:3 MeV and tan 4 0:9 in type I(II) of the model. The horizontal lines indicate the upper bounds in Eqs. (7) and (8).
115015-10 LIGHT HIGGS PRODUCTION IN HYPERON DECAY PHYSICAL REVIEW D 74, 115015 (2006) We first discussed the case of a scalar Higgs boson. We found that the leading-order amplitudes in both kaon and 10 − 7 hyperon decays depend on an unknown low-energy con-
p stant ~ , as well as known constants from the hyperon − 8 8 10 sector. This constant is connected to a weak-mass term in the chiral Lagrangian that can be rotated away for pro- 10 − 9 cesses that involve only pseudo-Goldstone bosons and is, therefore, unknown. We applied our results to the process ! pX ! 225 250 275 300 325 350 375 400 relevant to the HyperCP observation of mH GeV p . We showed that the two-quark contributions in the SM and its 2HDM extensions are too large to explain FIG. 10 (color online). Contribution of the real part of the total the HyperCP observation. However, we also showed that amplitude for ! pA to its branching ratio in the 2HDM I there can be cancellations between the CP-conserving two- (solid curve) and II (dotted curve) as a function of charged- and four-quark contributions to this process that lead to a Higgs-boson mass for mA 214:3 MeV and tan 4 0:9 in rate comparable in size to the HyperCP observation for type I(II). The dashed lines indicate the bounds from the HyperCP result. both the SM and the 2HDM. Such cancellations occur for a certain range of known constants from the hyperon sector, the effect of ~ being small. In both cases, however, the model in the large- tan limit has lu ! 0. Unfortunately, in 8 two-quark penguin contribution has an imaginary this limit ld induces four-quark amplitudes resulting in (CP-violating) part that is too large to be compatible B ! pA with the HyperCP result. In the SM and in the 2HDM, ! 0:025; (71) B K ! X the four-quark contributions have a CP-violating part that which is inconsistent with Eqs. (9) and (13). In the 2HDM is much smaller than that of the penguin amplitude and hence these models are ruled out as explanations for the I, which has lu and ld given in Eq. (59), the four-quark amplitudes alone yield HyperCP observation. More general models with addi- tional CP-violating phases may be able to address this B4q ! pA issue. In addition, in these models the scaling of the two- 0:53 (72) B4q K ! X quark operator to the B system is incompatible with the nonobservation of a light scalar in B decay. for all values of tan , which is consistent with Eqs. (9) and We then discussed the case of a pseudoscalar Higgs (13). However, in this case it is the penguin amplitude that boson in the 2HDM. In this case we computed the eliminates the pseudoscalar as a possible HyperCP leading-order amplitudes in PT and included, as well, candidate. 0 These results suggest the ingredients of a model that can certain higher-order terms mediated by the state. The satisfy all constraints. It is necessary for the penguin resulting amplitudes for both kaon and hyperon decays do amplitudes to be dominated by additional particles, such not depend on any unknown hadronic parameters. In par- ticular, they do not depend on ~8, as observed in Ref. [10]. as SUSY partners, in such a way that gA is not propor- tional to top-quark CKM angles. We have sketched a We then applied our results to the ! pA process. scenario where this happens in Ref. [33]. Once again we found that the real part of the amplitude can be consistent with the HyperCP observation for a certain range of parameters in the 2HDM ( tan and V. SUMMARY AND CONCLUSIONS mH ), but that the imaginary part of the penguin amplitude We have summarized the existing constraints on the is too large. The scaling of the two-quark operator to the B production of a light Higgs boson in kaon and B-meson system also produces a B ! XsA rate that is too large. decays, as well as the implication of attributing the Both of these problems can be solved in more general HyperCP events to the production of a light Higgs boson models that modify the phase and scaling with CKM in hyperon decay. angles of the two-quark operator. Production rates for such a particle in kaon and hyperon In conclusion, we have shown that it is possible to inter- decays receive contributions from two- and four-quark pret the HyperCP observation as evidence for a light Higgs operators that can be comparable in some cases. We have boson, although it is not easy to arrange this in a model. investigated the interplay of both production mechanisms Typical Higgs-penguin operators have three problems: with the aid of leading-order chiral perturbation theory. To (a) if they have the right size to fit the HyperCP obser- this effect, we have implemented the low-energy theorems vation, they induce K ! X at rates larger than the governing the couplings of light (pseudo)scalars to hadrons existing bounds; at leading order in baryon PT, generalizing existing (b) if they are dominated by loop diagrams involving studies for kaon decay. up-type quarks and W bosons, they have a CP phase
115015-11 XIAO-GANG HE, JUSAK TANDEAN, AND G. VALENCIA PHYSICAL REVIEW D 74, 115015 (2006) that is too large; revisiting of the kaon and B-decay results. In particular, the (c) if they are dominated by loop diagrams involving B factories are still operational and could reanalyze the up-type quarks and W bosons, their scaling to the B very low m invariant-mass region in their measurements system is incompatible with the nonobservation of of B ! Xs modes. The NA48 experiment might B ! XsX. also be able to revisit the kaon modes. We have found in this paper that (a) can be solved in some cases by the addition of the effects of four-quark operators. ACKNOWLEDGMENTS We have suggested that more general models may be constructed to solve (b) and (c). To show that this is pos- The work of X. G. H. was supported in part by NSC and sible, we have constructed a specific example in Ref. [33]. NCTS. The work of G. V. was supported in part by DOE Disregarding existing bounds from kaon and B-meson under Contract No. DE-FG02-01ER41155. We thank decays, we have shown that many light Higgs bosons have Laurence Littenberg and Rainer Wanke for useful discus- couplings of the right size to explain the HyperCP obser- sions on the kaon bounds and Soeren Prell for useful vation. We think this is sufficiently intriguing to warrant a discussions on the B bounds.
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