application/xmlRare decay [formula omitted] constraints on the light CP-odd Higgs in NMSSMQin ChangYa-Dong YangPhysics Letters B 676 (2009) 88-93. doi:10.1016/j.physletb.2009.04.081journalPhysics Letters BCopyright © 2009 Elsevier B.V. All rights reserved.Elsevier B.V.0370-26936761-31 June 20092009-06-0188-93889310.1016/j.physletb.2009.04.081http://dx.doi.org/10.1016/j.physletb.2009.04.081doi:10.1016/j.physletb.2009.04.081http://vtw.elsevier.com/data/voc/oa/OpenAccessStatus#Full2014-01-01T00:14:32ZSCOAP3 - Sponsoring Consortium for Open Access Publishing in Particle Physicshttp://vtw.elsevier.com/data/voc/oa/SponsorType#FundingBodyhttp://creativecommons.org/licenses/by/3.0/

Journals

S300.3

PLB25835S0370-2693(09)00531-010.1016/j.physletb.2009.04.081Elsevier B.V.Phenomenology

Fig. 1

Relevant Feynman diagram within NMSSM.

Fig. 2

Triangle diagram for π0→e+e− process.

Fig. 3

The dependence of B(π0→e+e−) on the parameter |Xd| with mA10=mπ0/2, 214.3 MeV and 3 GeV, respectively. The horizontal lines are the KTeV data, where the solid line is the central value and the dashed ones are the error bars (1σ).

Fig. 4

Constraints on the NMSSM parameter space through B(π0→e+e−), B(ϒ(1S)→γA10), B(ϒ(3S)→γA10) and aμ, respectively. The shaded regions are allowed by the labeled processes.

Fig. 5

sin2θ versus the mass difference of the unmixed states with |Xd|=0.05 and 0.18. The solid and the dashed lines denote the real and the imaginary parts of sin2θ, respectively.

Rare decay π0→e+e− constraints on the light CP-odd Higgs in NMSSM

Qin

Chang

Ya-Dong

Yang

⁎

yangyd@iopp.ccnu.edu.cn

aInstitute of Particle Physics, Huazhong Normal University, Wuhan, Hubei 430079, PR ChinabDepartment of Physics, Henan Normal University, Xinxiang, Henan 453007, PR ChinacKey Laboratory of Quark & Lepton Physics, Ministry of Education, PR China⁎Corresponding author at: Institute of Particle Physics, Huazhong Normal University, Wuhan, Hubei 430079, PR China.

Editor: B. Grinstein

Abstract

We constrain the light CP-odd Higgs A10 in NMSSM via the rare decay π0→e+e−. It is shown that the possible 3σ discrepancy between theoretical predictions and the recent KTeV measurement of B(π0→e+e−) cannot be resolved when the constraints from ϒ→γA10, aμ and π0→γγ are combined. Furthermore, the combined constraints also exclude the scenario involving mA10=214.3 MeV, which is invoked to explain the anomaly in the Σ+→pμ+μ− decay found by the HyperCP Collaboration.

PACS

13.25.Cq14.80.Cp

Introduction

Theoretically, the rare decay π0→e+e− starts at the one loop level in the Standard Model (SM), which has been extensively studied [1–10] since the first investigation in QED by Drell [1]. It is nontrivial to make precise predictions of the branching ratio BSM(π0→e+e−) because its sub-process involves the π0→γ∗γ∗ transition form factor. In Refs. [2–5], the decay was studied via the Vector-Meson Dominance (VMD) approach, where the results are in good agreement with each other and converge in B(π0→e+e−)∼(6.2–6.4)×10−8. By using the measured value of B(η→μ+μ−) to fix the counterterms of the chiral amplitude in Chiral Perturbation Theory (ChPT), Savage et al. predicted B(π0→e+e−)=(7±1)×10−8[6]. Using a procedure similar to that used in Ref. [6] (although with an updated measurement of B(η→μ+μ−)), Dumm and Pich predicted (8.3±0.4)×10−8[7]. Alternatively, using the lowest meson dominance (LMD) approximation to the large-Nc spectrum of vector meson resonances to fix the counterterms, Knecht et al. predicted (6.2±0.3)×10−8[8], which is about 4σ lower than the value predicted by Ref. [7] but which agrees with the others. Most recently, using a dispersive approach to the amplitude and the experimental results of the CELLO [11] and CLEO [12] Collaborations for the pion transition form factor, Dorokhov and Ivanov [9] have found that(1)BSM(π0→e+e−)=(6.23±0.09)×10−8, which is consistent with most theoretical predictions of BSM(π0→e+e−) in the literature. Moreover, their prediction that B(η→μ+μ−)=(5.11±0.2)×10−6 agrees with the experimental data (which gives a value of (5.8±0.8)×10−6[13]).Experimentally, the accuracy of the measurements of the decay has increased significantly since the first π0→e+e− evidence was observed by the Geneva-Saclay group [14] in 1978 with BSM(π0→e+e−)=(22−11+24)×10−8. A detailed summary of the experimental situation can be found in Ref. [15]. Recently, using the complete data set from KTeV E799-II at Fermilab, the KTeV Collaboration has made a precise measurement of the π0→e+e− branching ratio [16](2)BKTeVno-rad(π0→e+e−)=(7.48±0.29±0.25)×10−8, after extrapolating the full radiative tail beyond (me+e−/mπ0)2>0.95 and scaling their result back up by the overall radiative correction of 3.4%.As was already noted in Ref. [9], the SM prediction given in Eq. (1) is 3.3σ lower than the KTeV data. The authors have also compared their result with estimations made by various approaches in the literature and found good agreements. Further analyses have found that QED radiative contributions [17] and mass corrections [18] are at the level of a few percent and are therefore unable to reduce the discrepancy. Although the discrepancy might be due to hadronic dynamics that are as of yet unknown, it is equally possible that this discrepancy is caused by the effects of new physics (NP). In this Letter we will study the latter possibility.As is known that leptonic decays of pseudoscalar mesons are sensitive to pseudoscalar weak interactions beyond the SM. Precise measurements and calculations of these decays will offer sensitive probes for NP effects at the low energy scale. Of particular interest to us is the rare decay π0→e+e−, which could proceed at tree level via a flavor-conserving process induced by a light pseudoscalar Higgs boson A10 in the next-to-minimal supersymmetric standard model (NMSSM) [19]. We will look for a region of the parameter space of NMSSM that could resolve the aforementioned discrepancy of B(π0→e+e−) at 1σ. Then, we combine constraints from aμ and the recent searches for ϒ(1S),(3S)→γA10 by CLEO [20] and BaBar [21], respectively.2

The amplitude of π0→e+e− in the SM and the NMSSM

The NMSSM has generated considerable interest in the literature, which extends the minimal supersymmetric SM (MSSM) by introducing a new Higgs singlet chiral superfield Sˆ to solve the known μ problem in MSSM. The superpotential in the model is [19](3)WNMSSM=QˆHˆuhuUˆC+HˆdQˆhdDˆC+HˆdLˆheEˆC+λSˆHˆuHˆd+13κSˆ3, where κ is a dimensionless constant and measures the size of Peccei–Quinn (PQ) symmetry breaking.In addition to the two charged Higgs bosons, H±, the physical NMSSM Higgs sector consists of three scalars h0, H1,20 and two pseudoscalars A1,20. As in the MSSM, tanβ=vu/vd is the ratio of the Higgs doublet vacuum expectation values vu=〈Hu0〉=vsinβ and vd=〈Hd0〉=vcosβ, where v=vd2+vu2=2mW/g≃174 GeV. Generally, the masses and singlet contents of the physical fields depend strongly on the parameters of the model (such as, in particular, how well the PQ symmetry is broken). If the PQ symmetry is slightly broken, then A10 can be rather light, and its mass is given by(4)mA102=3κxAk+O(1tanβ) with the vacuum expectation value of the singlet x=〈S〉; meanwhile, another pseudoscalar A20 has a mass of order of mH±.For π0→e+e− decay, the NMSSM contributions are dominated by A10. The couplings of A10 to fermions are [22](5)LAi0ff¯=−ig2mW(Xdmdd¯γ5d+Xumuu¯γ5u+Xℓmℓℓ¯γ5ℓ)A10 where Xd=Xℓ=vxδ− and Xu=Xd/tan2β; thus, the contribution of the u¯γ5uA10 term in π0→e+e− could be neglected in the large tanβ approximation.To the leading order, the relevant Feynman diagram within NMSSM is shown in Fig. 1

. We obtain its amplitude as(6)MA10=−GF2memπ03fπ01mπ02−mA102Xd2, which is independent of md, since md in the coupling of A10d¯γ5d is canceled by the md term of the hadronic matrix(7)〈0|d¯γ5d|π0〉=−i2fπ0mπ022md.

In the SM, the normalized branching ratio of π0→e+e− is given by [9](8)R(π0→e+e−)=B(π0→e+e−)B(π0→γγ)=2(αeπmemπ0)2βe(mπ02)|A(mπ02)|2 where βe(mπ02)=1−4me2mπ02 and A(mπ02) is the reduced amplitude.To add the NMSSM amplitude to the above amplitudes consistently, we rederive the SM amplitude to look into possible differences between the conventions used in our Letter and the ones used in Ref. [9]. The Feynman diagram that proceeds via two photon intermediate states is shown in Fig. 2

. We start with the π0γ∗γ∗ vertex(9)Hμν=−ie2ϵμναβkα(q−k)βfγ*γ*Fπ0γ*γ*(k2,(q−k)2) where k and q−k are the momenta of the two photons, fγ*γ*=24π2andfπ0 is the coupling constant of π0 to two real photons. Fπ0γ*γ*(k2,(q−k)2) is the transition form factor π0→γ*γ*, which is normalized to Fπ0γ*γ*(0,0)=1. The amplitude of Fig. 2 is written as(10)MSM(π0→e+e−)=ie2∫d4k(2π)4LμνHμν(k2+iε)((k−q)2+iε)((k−p)2−me+iε), with(11)Lμν=u¯(p,s)γμ(p̸−k̸+me)γνv(q−p,s′). There is a known, convenient way to calculate Lμν with the projection operator for the outgoing e+e− pair system [23](12)P(q−p,p)=12[v(q−p,+)⊗u¯(p,−)+v(q−p,−)⊗u¯(p,+)]=122t[−2meqμγμγ5+12ϵμνστ(pσ(q−p)τ−(q−p)σpτ)σμν+tγ5] where t=q2=mπ02. After some calculations, we get(13)MSM(π0→e+e−)=22α2memπ0fγ*γ*A(mπ2) where the reduced amplitude A(q2) is(14)A(q2)=2iq2∫d4kπ2k2q2−(q⋅k)2(k2+iε)((k−q)2+iε)((k−p)2−me+iε)Fπ0γ*γ*(k2,(q−k)2). We note that the A(q2) derived here is in agreement with Ref. [9]. Further evaluation of the integrals of A(q2) is quite subtle and lengthy [2,24], and only the imaginary part of A(mπ02) can be obtained model-independently [1,2]. In the following calculations, we quote the result of Ref. [9],(15)A(mπ2)=(10.0±0.3)−i17.5. With Eqs. (6) and (13), we get the total amplitude(16)M=22α2memπ0fγ*γ*A(mπ2)−GF2memπ03fπ01mπ02−mA102Xd2.

Numerical analysis and discussion

Now, we are ready to discuss the effects of A10 numerically, with a focus on the mA10<2mb scenarios. The dependence of B(π0→e+e−) on the parameter |Xd| is shown in Fig. 3

with mA10=mπ/2, 214.3 MeV, 3 GeV as benchmarks. We have used the input parameters B(π0→γγ)=0.988 and fπ0=(130.7±0.4) MeV[13]. As shown in Fig. 3, B(π0→e+e−) is very sensitive to the parameter |Xd| and mA10. For mA10<mπ0, the NMSSM contribution is deconstructive and reduces B(π0→e+e−) at small |Xd| region. For mA10>mπ0, the NMSSM contribution is constructive and could enhance B(π0→e+e−) to be consistent with the KTeV measurement BKTeVno-rad(π0→e+e−)=(7.48±0.38)×10−8 (where |Xd| strongly depends on mA10).

3.1

Constraint on the scenario of mA10=214.3 MeV

It is interesting to note that the HyperCP Collaboration [25] has observed three events for the decay Σ+→pμ+μ− with a narrow range of dimuon masses. This may indicate that the decay proceeds via a neutral intermediate state, Σ+→pP0, P0→μ+μ−, with a P0 mass of 214.3±0.5 MeV. The possibility of P0 has been explored in the literature [26,28–30]. The authors have proposed A10 as a candidate for the P0, and have also shown that their explanation could be consistent with the constraints provided by K and B meson decays [26,27]. It would be worthwhile to check on whether the explanation could be consistent with the π0→e+e− decay.Taking mA10=214.3 MeV, we find that B(π0→e+e−) is enhanced rapidly and could be consistent with the KTeV data within 1σ for(17)|Xd|=14.0±2.4. However, the upper bound |Xd|<1.2 from the aμ constraint has been derived and used in the calculations of Refs. [26,29]. So, with the assumption that mA10=214.3 MeV, our result of |Xd| violates the upper bound with a significance of 5σ.Recently, CLEO [20] and BaBar [21] have searched for the CP-odd Higgs boson in radiative decays of ϒ(1S)→γA10 and ϒ(3S)→γA10, respectively. For mA10=214 MeV, CLEO gives the upper limit(18)B(ϒ(1S)→γA10)<2.3×10−6(90% C.L.) which constrains |Xd|<0.16.The BaBar Collaboration has searched for A10 through ϒ(3S)→γA10, A10→invisible in the mass range mA10⩽7.8 GeV[21]. From Fig. 5 of Ref. [21], we read(19)B(ϒ(3S)→γA10)×B(A10→invisible)≲3.5×10−6(90% C.L.) for mA10=214 MeV. Assuming B(A10→invisible)∼1, we get the conservative upper limit |Xd|<0.19.All of these upper limits are much lower than the limit of Eq. (17) set by π0→e+e−; therefore, the scenario where mA10≃214 MeV in NMSSM could be excluded by combining the constraints from π0→e+e− and the direct searches for ϒ radiative decays.3.2

Constraints on the parameter space of mA10−|Xd|

To show the constraints on NMSSM parameter space from π0→e+e−, we present a scan of mA10−|Xd| space, as shown in Fig. 4

. In order to scan the region of mA10∼mπ0, the amplitude of the A10 contribution in Eq. (6) is replaced by the Breit–Wigner formula(20)MA10=−GF2memπ03fπ01mπ02−mA102+iΓ(A10)mA10Xd2. With the assumption that A10 just decays to electron and photon pairs for mA10∼mπ0, the decay width of A10 could be written as(21)Γ(A10)=Γ(A10→e+e−)+Γ(A10→γγ) with(22)Γ(A10→e+e−)=2GF8πme2mA10Xd21−4me2mA102,Γ(A10→γγ)=GFα282π3mA103Xd2|∑irQi2kiF(ki)|2, where r=1 for leptons and r=Nc for quarks, ki=mi2/mA102 and Qi is the charge of the fermion in the loop. The loop function F(ki) reads [31]F(ki)={−2(arcsin12ki)2forki⩾14,12[ln(1+1−4ki1−1−4ki)+iπ]2forki<14.

As shown in Fig. 4, only two narrow connected bands of the |Xd|−mA10 space survive after the KTeV measurement of B(π0→e+e−), which show that π0→e+e− is very sensitive to NP scenarios with a light pseudoscalar neutral boson.In the following, we will determine which part of the remaining parameter space could satisfy the constraints enforced by radiative ϒ decays and aμ simultaneously.To include the aμ constraint, we use the experimental result that [32]aμ(Exp)=(11659208.0±6.3)×10−10 and the SM prediction [33]aμ(SM)=(11659177.8±6.1)×10−10. The discrepancy is(23)Δaμ=aμ(Exp)−aμ(SM)=(30.2±8.8)×10−10(3.4σ) which is established at a 3.4σ level of significance.The contributions of A10 to aμ are given by [34](24)δaμ(A10)=δaμ1-loop(A10)+δaμ2-loop(A10),δaμ1-loop(A10)=−2GFmμ28π2|Xd|2f1(mA102mμ2),δaμ2-loop(A10)=2GFαmμ28π3|Xd|2[431tan2βf2(mt2mA102)+13f2(mb2mA102)+f2(mτ2mA102)] with(25)f1(z)=∫01dxx3z(1−x)+x2,f2(z)=z∫01dx1x(1−x)−zlnx(1−x)z. It has been found that the A10 contribution is always negative at the one loop level and worsens the discrepancy in aμ; however, it could be positive and dominated by the two loop contribution for A10>3 GeV[34]. One should note that there are other contributions to aμ in NMSSM; for instance, the chargino/sneutino and neutralino/smuon loops. Moreover, the discrepancy Δaμ could be resolved without pseudoscalars [34]. So, putting a constraint on |Xd| via aμ is a rather model-dependent process. There are two approximations with different emphases on the role of A10; namely, (i) assuming that Δaμ is resolved by other contributions and requiring that A10 contributions are smaller than the 1σ error-bar of the experimental measurement, and (ii) assuming that the A10 contributions are solely responsible for Δaμ. In Ref. [26], approximation (i) has been used to derive an upper bound of |Xd|<1.2. We present the aμ constraints with the two approximations which are shown in Figs. 4(a) and (b), respectively.From Fig. 4(a), we can find that there are two narrow overlaps between the constraints provided by aμ and B(π0→e+e−): one is for mA10∼3 GeV with |Xd|>150 and another one is for mA10∼135 MeV with |Xd|<1.In the searches for ϒ→γA10 decays, CLEO [20] obtains the upper limits for the product of B(ϒ(1S)→γA10) and B(A10→τ+τ−) or B(A10→μ+μ−), while BaBar presents upper limits on B(ϒ(3S)→γA10)×B(A10→invisible). All these limits fluctuate with the mass of A10 frequently. For simplicity, we take the loosest upper limit B(ϒ(1S)→γA10)×B(A10→τ+τ−)<6×10−5 of CLEO and assume B(A10→τ+τ−)=1. Similarly, we also use the loosest upper limits on B(ϒ(3S)→γA10)×B(A10→invisible)<3.1×10−5 of BaBar [21] and assume B(A10→invisible)=1. With the loosest upper limits, we get their bounds on the |Xd|−mA10 space, which are shown in Fig. 4. From the figure, we can see the bounds (excluding the parameter space Xd>1) for 0<mA10<7.8 GeV. Fig. 4(b) shows that there is no region of parameter space satisfying all the aforementioned constraints if the contribution of A10 is required to solely resolve the aμ discrepancy.Of particular interest, as shown in Fig. 4(a), is the parameter space around mA10∼135 MeV with |Xd|<1 (which is still allowed with approximation (i)). To make a thorough investigation of the space, we read off the upper limits of BaBar [21] from Fig. 5

for the value mA10∼135 MeV: B(ϒ(3S)→γA10)×B(A10→invisible)≲3.3×10−6. With the assumption that B(A10→invisible)≃1 and the constraints from B(π0→e+e−), we get(26)|Xd|=0.10±0.08,mA10=134.99±0.01 MeV, where the constraint on mA10 is dominated by B(π0→e+e−) and the limit of |Xd| is dominated by B(ϒ(3S)→γA10). At first sight, the uncertainties in the above mentioned two parameters are too different. We find that the difference arises from our assumption Γ(A10)≃Γ(A10→e+e−)+Γ(A10→γγ). From Eqs. (20) and (21), one can see that the Xd2 factor in MA10 could be canceled out by the one in Γ(A10) when mA10 approaches mπ0, which results in a very sharp peak for position of mA10. Thus, with the well measured quantities given in Eq. (20) and the sensitivity of the peak, mA10 turns out to be well-constrained. Furthermore, if we take mA10=mπ0, we find that Xd2 is canceled out exactly, so there is no parameter to tune; however, we have B(π0→e+e−)≫1, which violates the unitary bound and is thus excluded.

From the results of Eq. (26), we obtain δaμ(A10)=(−9.2±8.9)×10−12 with tanβ=30 as a benchmark, which is small enough to be smeared by the chargino/sneutrino and neutralino/smuon contributions. Moreover, we have(27)Γ(A10)=(5.7±5.5)×10−13 MeV, which corresponds to τ(A10)∼1.2×10−9 s (cτ∼36 cm).For the case where A10 decays mostly to invisible particles, we take the width of A10 as a free parameter and get Γ(A10)⩽8.24×10−6 GeV, mA10=134.99±0.02 MeV and |Xd|⩽0.18. In this case, mA10 can equal mπ0, and it is found that Γ(A10)⩽3.3×10−6 GeV and |Xd|⩽0.18.3.3

The resonant effects of mA10∼mπ0

So far we have included only the width effects of A10 with the Breit–Wigner formula for the propagator of A10. When the masses of A10 and π0 are very close, the mixing between the two states could modify the parton level π0–A10 coupling. In a manner analogous to Ref. [35], the mixing can be described by introducing off-diagonal elements in the A10–π0 mass matrix(28)M2=(mA102−imA10ΓA10δm2δm2mπ02−imπ0Γπ0) with δm2=GF/42fπ0mπ02Xd. The complex mixing angle θ between the states is given by(29)sin22θ=(δm2)214(mA102−mπ02−imA10ΓA10+imπ0Γπ0)2+(δm2)2. The mass eigenstates A1′0 and π′0 are obtained as(30)A1′0=1N(A10cosθ+π0sinθ),(31)π′0=1N(−A10sinθ+π0cosθ), where N=|sinθ|2+|cosθ|2. Then, we can write the decay amplitude of the “physical” state π′0 as(32)|M(π′0→e+e−)|2=1N2(|cosθ|2|M(π0→e+e−)|2+|sinθ|2|M(A10→e+e−)|2). Obviously, we obtain the SM result when θ is small.With |Xd|=0.05 and 0.18, Fig. 5 shows sin2θ as a function of the difference between mA10 and mπ0. We note that the imaginary part of sin2θ is negligibly small, since ΓA10mA10+Γπ0mπ0≪δm2. So, the normalization parameter N of the mixing states is nearly unity. Combining the constraints from B(ϒ(3S)→γA10) and B(π′0→e+e−), we get(33)|Xd|=0.17±0.01,mA10≃mπ0. This confirms the results of our straightforward calculation from Eq. (26), but gives a somewhat stronger constraint on |Xd|. With this constraint, we get(34)Γ(A10)=(9.8±1.1)×10−13 MeV, which is also in agreement with Eq. (27). Furthermore, we get |sinθ|2=0.31±0.19.It is well known that the decay width of π0→γγ agrees perfectly with the SM prediction, so it is doubtful that that π0→γγ would be compatible with Higgs with a degenerate mass mπ0. Using the fitted result |sinθ|2=0.31±0.19 and(35)|M(π′0→γγ)|2=1N2(|cosθ|2|M(π0→γγ)|2+|sinθ|2|M(A10→γγ)|2), one can easily observe that(36)|M(A10→γγ)|2≃|M(π0→γγ)|2 is needed to give Γ(π′→γγ)≃Γ(π0→γγ). However, it would require a too large value of |Xd|≃103; therefore, the degenerate case is excluded.4

Conclusion

We have studied the decay π0→e+e− in the NMSSM and shown that it is sensitive to the light CP-odd Higgs boson A10 predicted in the model. The possible discrepancy between the KTeV Collaboration measurement [16] and the theoretical prediction of B(π0→e+e−) could be resolved in NMSSM by the effects of A10 at the tree level. However, it excludes a large fraction of the parameter space of mA10−|Xd|. To further constrain the parameter space, we have included bounds from muon g−2 and the recent searches for A10 from radiative ϒ decays performed at CLEO [20] and BaBar [21]. Combining all these constraints, we have found that

•B(π0→e+e−) and B(ϒ→γA10) put strong constraints on the NMSSM parameter Xd and mA10. Due to their different dependences on the two parameters, the interesting scenario where mA10=214.3 MeV is excluded, which would invalidate the A10 hypothesis for the three HyperCP events [25].

•Although these constraints point to a pseudoscalar with mA10∼mπ0 and |Xd|=0.10±0.08 (0.17±0.01, π0−A10 mixing included) in the NMSSM, such an mA10 is excluded by π0→γγ decay.

In this Letter, we have worked in the limit of Xd≫Xu, i.e., the large tanβ limit. If we relax the limit and take Eq. (5) as a general parameterization of the couplings between a pseudoscalar and fermions, the u¯–u–A10 coupling should be included. However, its contribution is deconstructive to the contributions from Xd, since the π0 flavor structure is (uu¯−dd¯). To give a result in agreement with the KTeV Collaboration measurement [16], Xu≫Xd would be needed, which would imply possible large effects in Ψ(1S) radiative decays. Detailed discussion of this issue would be beyond the main scope of our present study. In summary, we could not find a region of parameter space of NMSSM with mA10<7.8 GeV in the large tanβ limit that is consistent with the experimental constraints. The HyperCP 214.3 MeV resonance and the possible 3.3σ discrepancy in π0→e+e− decay are still unsolved. Finally, further theoretical investigation is also needed to confirm the discrepancy between the KTeV measurements and SM predications of π0→e+e− decay. If the discrepancy still persists, it would be an important testing ground for NP scenarios with a light pseudoscalar boson.

Acknowledgements

The work is supported by the National Science Foundation under contract Nos. 10675039 and 10735080.

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