Rare decay $\pi^{0}\to e^+ e^-$ constraints on the light CP-odd Higgs in NMSSM

We constrain the light CP-odd Higgs $A_{1}^{0}$ in NMSSM via the rare decay $\pi^{0}\to e^{+}e^{-}$. It is shown that the possible $3\sigma$ discrepancy between theoretical predictions and the recent KTeV measurement of ${\cal B}({\pi}^{0}\to e^+ e^-)$ cannot be resolved when the constraints from $\Upsilon\to\gamma A_1^0$, $a_{\mu}$ and $\pi^{0}\to \gamma \gamma$ are combined. Furthermore, the combined constraints also exclude the scenario involving $m_{A_1^0}=214.3$ MeV, which is invoked to explain the anomaly in the $\Sigma^{+}\to p \mu^{+}\mu^{-}$ decay found by the HyperCP Collaboration.

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-N c 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 B SM (π 0 → e + e − ) = (6.23 ± 0.09) × 10 −8 , which is consistent with most theoretical predictions of B SM (π 0 → e + e − ) in the literature.
Experimentally, the accuracy of the measurements of the decay has increased significantly since the first π 0 → e + e − evidence was observed by the Geneva-Saclay group [14] in 1978 with B SM (π 0 → e + e − ) = (22 +24 −11 ) × 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 π 0 → e + e − branching ratio [16] B no−rad KT eV (π 0 → e + e − ) = (7.48 ± 0.29 ± 0.25) × 10 −8 , after extrapolating the full radiative tail beyond (m e + e − /m π 0 ) 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 π 0 → e + e − , which could proceed at tree level via a flavor-conserving process induced by a light pseudoscalar Higgs boson A 0 1 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(π 0 → e + e − ) at 1σ. Then, we combine constraints from a µ and the recent searches for Υ(1S), (3S) → γA 0 1 by CLEO [20] and BaBar [21], respectively.
2 The amplitude of π 0 → 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 superfieldŜ to solve the known µ problem in MSSM. The superpotential in the model is [19] 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 h 0 , H 0 1,2 and two pseudoscalars A 0 1,2 . As in the MSSM, tan β = v u /v d is the ratio of the Higgs doublet vacuum expectation values 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 A 0 1 can be rather light, and its mass is given by with the vacuum expectation value of the singlet x = S ; meanwhile, another pseudoscalar A 0 2 has a mass of order of m H ± .
For π 0 → e + e − decay, the NMSSM contributions are dominated by A 0 1 . The couplings of A 0 1 to fermions are [22] where X d = X ℓ = v x δ − and X u = X d / tan 2 β; thus, the contribution of theūγ 5 uA 0 1 term in π 0 → e + e − could be neglected in the large tan β approximation.
To the leading order, the relevant Feynman diagram within NMSSM is shown in Fig. 1. We obtain its amplitude as which is independent of m d , since m d in the coupling of A 0 1d γ 5 d is canceled by the m d term of the hadronic matrix In the SM, the normalized branching ratio of π 0 → e + e − is given by [9] where β e (m 2 and A(m 2 π 0 ) 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 π 0 γ * γ * vertex where k and q−k are the momenta of the two photons, f γ * γ * = √ 2 4π 2 andf π 0 is the coupling constant of π 0 to two real photons. with There is a known, convenient way to calculate L µν with the projection operator for the outgoing e + e − pair system [23] where t = q 2 = m 2 π 0 . After some calculations, we get where the reduced amplitude A(q 2 ) is We note that the A(q 2 ) derived here is in agreement with Ref. [9]. Further evaluation of the integrals of A(q 2 ) is quite subtle and lengthy [2,24], and only the imaginary part of A(m 2 π 0 ) can be obtained model-independently [1,2]. In the following calculations, we quote the result of Ref. [9], With Eq. (6) and Eq. (13), we get the total amplitude 3 Numerical analysis and discussion Now, we are ready to discuss the effects of A 0 1 numerically, with a focus on the m A 0 1 < 2m b scenarios. The dependence of B(π 0 → e + e − ) on the parameter |X d | is shown in Fig. 3 with 3MeV, 3GeV as benchmarks. We have used the input parameters B(π 0 → γγ) = 0.988 and f π 0 = (130.7 ± 0.4) MeV [13]. As shown in Fig. 3 I. Constraint on the scenario of m A 0 1 = 214.3MeV 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 + → pP 0 , P 0 → µ + µ − , with a P 0 mass of 214.3 ± 0.5MeV. The possibility of P 0 has been explored in the literature [26,28,29,30]. The authors have proposed A 0 1 as a candidate for the P 0 , 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 π 0 → e + e − decay.
Taking m A 0 1 = 214.3 MeV, we find that B(π 0 → e + e − ) is enhanced rapidly and could be consistent with the KTeV data within 1σ for However, the upper bound |X d | < 1.2 from the a µ constraint has been derived and used in the calculations of Ref. [26,29]. So, with the assumption that m for All of these upper limits are much lower than the limit of Eq.17 set by π 0 → e + e − ; therefore, the scenario where m A 0 1 ≃ 214 MeV in NMSSM could be excluded by combining the constraints from π 0 → e + e − and the direct searches for Υ radiative decays.
II. Constraints on the parameter space of m A 0 1 − |X d | To show the constraints on NMSSM parameter space from π 0 → e + e − , we present a scan of Fig. 4. In order to scan the region of m A 0 1 ∼ m π 0 , the amplitude of the A 0 1 contribution in Eq. (6) is replaced by the Breit-Wigner formula With the assumption that A 0 1 just decays to electron and photon pairs for m A 0 1 ∼ m π 0 , the decay width of A 0 1 could be written as with where r = 1 for leptons and r = N c for quarks, k i = m 2 i /m 2 As shown in Fig. 4, only two narrow connected bands of the |X d | − m A 0 1 space survive after the KTeV measurement of B(π 0 →e + e − ), which show that π 0 →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 a µ simultaneously.
The contributions of A 0 1 to a µ are given by [34] δa µ (A 0 It has been found that the A 0 1 contribution is always negative at the one loop level and worsens the discrepancy in a µ ; however, it could be positive and dominated by the two loop contribution for A 0 1 > 3GeV [34]. One should note that there are other contributions to a µ in NMSSM; for instance, the chargino/sneutino and neutralino/smuon loops. Moreover, the discrepancy △a µ could be resolved without pseudoscalars [34]. So, putting a constraint on |X d | via a µ is a rather model-dependent process. There are two approximations with different emphases on the role of A 0 1 ; namely, (i) assuming that △a µ is resolved by other contributions and requiring that A 0 1 contributions are smaller than the 1σ error-bar of the experimental measurement, and (ii) assuming that the A 0 1 contributions are solely responsible for △a µ . In Ref. [26], approximation (i) has been used to derive an upper bound of |X d | < 1.2. We present the a µ 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 a µ and B(π 0 →e + e − ): one is for m A 0 1 ∼ 3 GeV with |X d | > 150 and another one is for m A 0 1 ∼ 135 MeV with |X d | < 1.
Of particular interest, as shown in Fig. 4(a), is the parameter space around m A 0 1 ∼ 135 MeV with |X d | < 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 where the constraint on m A 0 1 is dominated by B(π 0 →e + e − ) and the limit of |X d | is dominated by B(Υ(3S) → γA 0 1 ). At first sight, the uncertainties in the abovementioned two parameters are too different. We find that the difference arises from our assumption Γ(A 0 1 ) ≃ Γ(A 0 1 → e + e − ) + Γ(A 0 1 → γγ). From Eqs. (20) and (21), one can see that the X 2 d factor in M A 0 1 could be canceled out by the one in Γ(A 0 1 ) when m A 0 1 approaches m π 0 , which results in a very sharp peak for position of m A 0 1 . Thus, with the well measured quantities given in Eq. (20) and the sensitivity of the peak, m A 0 1 turns out to be well-constrained. Furthermore, if we take m A 0 1 = m π 0 , we find that X 2 d is canceled out exactly, so there is no parameter to tune; however, we have B(π 0 → e + e − ) ≫ 1, which violates the unitary bound and is thus excluded.
Obviously, we obtain the SM result when θ is small.
With |X d | = 0.05 and 0.18, Fig. 5 shows sin 2θ as a function of the difference between m A 0 1 and m π 0 . We note that the imaginary part of sin 2θ is negligibly small, since Γ A 0 1 m A 0 1 +Γ π 0 m π 0 ≪ δm 2 . So, the normalization parameter N of the mixing states is nearly unity. Combining the constraints from B(Υ(3S) → γA 0 1 ) and B(π ′0 → e + e − ), we get This confirms the results of our straightforward calculation from Eq. (26), but gives a somewhat stronger constraint on |X d |. With this constraint, we get which is also in agreement with Eq. (27). Furthermore, we get | sin θ| 2 = 0.31 ± 0.19.
It is well known that the decay width of π 0 → γγ agrees perfectly with the SM prediction, so it is doubtful that that π 0 → γγ would be compatible with Higgs with a degenerate mass m π 0 . Using the fitted result | sin θ| 2 = 0.31 ± 0.19 and one can easily observe that is needed to give Γ(π ′ → γγ) ≃ Γ(π 0 → γγ). However, it would require a too large value of |X d | ≃ 10 3 ; therefore, the degenerate case is excluded.

Conclusion
We have studied the decay π 0 → e + e − in the NMSSM and shown that it is sensitive to the light CP-odd Higgs boson A 0 1 predicted in the model. The possible discrepancy between the KTeV Collaboration measurement [16] and the theoretical prediction of B(π 0 → e + e − ) could be resolved in NMSSM by the effects of A 0 1 at the tree level. However, it excludes a large fraction of the parameter space of m A 0 1 − |X d |. To further constrain the parameter space, we have included bounds from muon g − 2 and the recent searches for A 0 1 from radiative Υ decays performed at CLEO [20] and BaBar [21]. Combining all these constraints, we have found that • B(π 0 → e + e − ) and B(Υ → γA 0 1 ) put strong constraints on the NMSSM parameter X d and m A 0 1 . Due to their different dependences on the two parameters, the interesting scenario where m A 0 1 = 214.3 MeV is excluded, which would invalidate the A 0 1 hypothesis for the three HyperCP events [25].
In this Letter, we have worked in the limit of X d ≫ X u , i.e., the large tan β limit. If we relax the limit and take Eq.5 as a general parameterization of the couplings between a pseudoscalar and fermions, theū−u−A 0 1 coupling should be included. However, its contribution is deconstructive to the contributions from X d , since the π 0 flavor structure is (uū − dd). To give a result in agreement with the KTeV Collaboration measurement [16], X u ≫ X d 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 m A 0 1 < 7.8GeV in the large tan β limit that is consistent with the experimental constraints. The HyperCP 214.3MeV resonance and the possible 3.3σ discrepancy in π 0 → 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 π 0 → e + e − decay. If the discrepancy still persists, it would be an important testing ground for NP scenarios with a light pseudoscalar boson.