# Low-complexity CNNs for acoustic scene classification

###### Abstract

This technical report describes the SurreyAudioTeam22’s submission for DCASE
2022 ASC Task 1, Low-Complexity Acoustic Scene Classification (ASC). The task
has two rules, (a) the ASC framework should have maximum 128K parameters, and
(b) there should be a maximum of 30 millions multiply-accumulate operations
(MACs) per inference. In this report, we present low-complexity systems for
ASC that follow the rules intended for the task.

Index Terms—  Acoustic scene classification, Low-complexity, Pruning,
Convolution neural network.

##  1 Introduction

Convolutional neural networks (CNNs) have shown state-of-the-art performance
in comparison to traditional hand-crafted methods in various domains [1].
However, CNNs are resource hungry due to their large size and memory [2, 3],
and hence it is difficult to deploy CNNs on resource constrained devices. For
example, Cortex-M4 devices (STM32L496@80MHz or Arduino Nano 33@64MHz) have a
maximum allowed limit is 128K parameters and 30M multiply-accumulate
operations (MACs) per inference corresponding to an audio of 1 second length.
Thus, the issue of reducing the size and the computational cost of CNNs has
drawn a significant amount of attention in the detection and classification of
acoustic scenes and events (DCASE) research community.

This report aims to design “low-complexity CNNs” for ASC which have a maximum
number of parameters less than 128K and a maximum number of MACs less than
30M. The baseline CNN designed for the DCASE 2022 task 1 has 46512 parameters,
29.24M MACs and the accuracy is approximately 43% with 1.575 log-loss.

We propose the following steps to obtain low-complexity CNNs.

  1. (a)

We design a “low-complexity” CNN that has a fewer parameters and MACs with
better performance than that of the baseline CNN.

  2. (b)

A filter pruning method is applied to compress the “low-complexity CNN” of (a)
further. Subsequently, we quantize each parameter from float32 to INT8 data
type, reducing networks memory by 4x.

  3. (c)

An ensemble approach is proposed which combines predictions obtained from (b)
low-complexity CNNs.

The rest of the report is organized as follows. In section 2, a procedure to
obtain low-complexity CNN and an ensemble framework is described. A brief
overview of dataset used, features used for experimentation and experimental
setup are described in Section 3. Section 4 presents experimental analysis.
The submitted entries for the DCASE 2022 task 1 challenge are included in 5.
Finally, conclusion is presented in Section 6.

##  2 Low-complexity CNNs

Proposed low-complexity architecture: We propose a CNN architecture which
consists of three convolutional layers (C1, C2 & C3), two pooling layers (P1,
P2), a dense layer (D1) and a classification layer. The details of different
layers is given in Table 1. The proposed architecture requires approximately
5M MACs to produce an output corresponding to an input of size (40 x 51), and
has 14886 parameters.

Filter Pruning: To eliminate redundant filters from the low-complexity
architecture as given in Table 1, we apply a filter pruning strategy. For each
convolutional layer, we identify filter pairs which are similar. Our
hypothesis is that similar filters produce similar output and hence,
contribute to redundancy only. Therefore, one of the similar filters can be
eliminated. The similarity between the filters is measured using a cosine
distance. We identify the closest filter pairs for each layer separately. A
filter from each pair is deemed redundant and eliminated from the network.
More information about the similarity based filter pruning method can be found
at [4].

The number of redundant filters obtained after performing similarity-based
filter pruning for C1 layer is 4 out of 16, C2 layer is 4 out of 16 and C3
layer is 10 out of 32. We obtain 6 different pruned networks that are obtained
after pruning C1 layer only, C2 layer only, C3 layer only, C1 and C2 layers,
C2 and C3 layers, C1 and C2 and C3 layers. The number of MACs and the number
of parameters for each pruned network are given in Table 2.

Quantization: To reduce the network size further, we perform quantization on
parameters of each pruned network using TFlite optimization, and quantized the
network parameters from 32-bit floating point to 8-bit integers (INT8).

An ensemble framework: Next, the predictions obtained from various pruned
networks are aggregated together in an ensemble framework. The total number of
parameters in the ensemble framework that aggregates predictions from all 6
pruned networks are 70.97K and the total number of MACs are 23.84M.

Table 1: Proposed low-complexity architecture. Here, BN stands for
batchnormalization, tanh is hyperbolic tangent activation function and ReLU is
a rectified linear unit activation function.

Layer name | Description | Number of filters/dense units | Filter/pooling size | input | output
---|---|---|---|---|---
C1 | Convolution + BN + tanh | 16 | (3 x 3) | (40 x 51) | (40 x 51)
C2 | Convolution + BN + ReLU | 16 | (3 x 3) | (40 x 51) | (40 x 51)
P1 | Average Pooling | - | (5 x 5) | (40 x 51) | (8 x 10)
C3 | Convolution + BN + tanh | 32 | (3 x 3) | (8 x 10) | (8 x 10)
P2 | Average Pooling | - | (4 x 10) | (8 x 10) | (2 x 1)
Dense | Dense + tanh | 100 | 1 | 64 | 100
Classification | Classification + softmax | 10 | 1 | 100 | 10

Table 2: Various low-complexity CNNs obtained after pruning and applying
quantization (INT8). Here, the frameworks in the bold entries are used for
evaluation on DCASE challenge dataset.

Sr No. | Network Name | Pruned layer | Architecture (C1-C2-C3-Dense) | Parameters | Size (KB) | MACs (millions)
---|---|---|---|---|---|---
1 | Unpruned low-complexity | NA | 16-16-32-100 | 14886 |  18.59 | 5.41
2 | Pruned_C1 | C1 | 12-16-32-100 | 14254 | 17.86 | 4.16
3 | Pruned_C2 | C2 | 16-12-32-100 | 13138 |  16.85 | 4.13
4 | Pruned_C3 | C3 | 16-16-22-100 | 11396 | 15.11 | 5.29
5 | Pruned_C12 | C1 + C2 | 12-12-32-100 | 12650 | 16.26 | 3.18
6 | Pruned_C23 | C2 + C3 | 16-12-22-100 | 10008 | 13.73 | 4.04
7 | Pruned_C123 | C1 + C2 + C3 | 12-12-22-100 | 9520 | 13.14 | 3.08

Table 3: Performance obtained when different pruned and quantized networks are
used in the ensemble framework. Here, the frameworks in the bold entries are
used for evaluation on DCASE challenge dataset.

Sr No. | Ensemble framework | Number of parameters | MACs (millions) | Size (KB) | Accuracy (%) | Log-loss
---|---|---|---|---|---|---
1 | All pruned networks except Pruned_C1 | 56712 | 19.72 | 75.09 | 47.14 | 1.394
2 | All pruned networks except Pruned_C2 | 57828 | 19.75 | 76.10 | 47.10 | 1.396
3 | All pruned networks except Pruned_C3 | 59570 | 18.60 | 77.84 | 47.26 | 1.392
4 | All pruned networks except Pruned_C12 | 58316 | 20.70 | 76.69 | 47.45 | 1.394
5 | All pruned networks except Pruned_C23 | 60958 | 19.84 | 79.22 | 47.52 | 1.389
6 | All pruned networks except Pruned_C123 | 61446 | 20.80 | 79.81 | 47.35 | 1.392
7 | Ensemble on all pruned networks | 70966 | 23.84 | 92.95 | 47.45 | 1.389

##  3 Experimental dataset, feature extraction and experimental setup

Dataset used: We use the TAU Urban Acoustic Scenes 2022 Mobile, development
dataset [5]. The dataset contains recordings from 12 European cities in 10
different acoustic scenes using 4 different devices. Each audio recording has
1 second length. The dataset is divided in training and validation sets. The
training dataset consists of 139620 audio examples and the validation dataset
consists of 29680 audio examples.

Feature extraction: For time-frequency representations, log-mel band energies
of size (40 $\times$ 51) corresponding to an audio signal of 1 second length
are extracted. A Hamming asymmetric window of length 40ms, and a hop length of
20ms is used to extract magnitude spectrogram. Next, log-mel spectrogram is
computed using 40 mel bands.

Experimental Setup: The unpruned low-complexity network is trained using the
training dataset with a batch size of 64 with an Adam optimizer for 1000
epochs. A categorical cross-entropy loss function is used during the training
process. We apply an early stopping criterion to yield the best network that
gives the minimum log-loss for the validation dataset.

After obtaining the trained unpruned low-complexity CNN, the filter pruning
strategy described in Section 2 is applied to eliminate redundant filters. To
regain the loss in performance due to pruning, the pruned networks are fine-
tuned with a similar process to that used for the training of the unpruned
low-complexity network.

##  4 Performance Analysis

Figure 1 shows the accuracy and the log-loss obtained for the unpruned low-
complexity network and the various pruned networks. The unpruned low-
complexity network gives 1.475 log-loss and 45.92% accuracy. Eliminating
filters from the unpruned network results in a significant reduction in
performance, but this is almost entirely restored after fine-tuning.

The performance obtained after aggregating predictions from various pruned
networks is given in Table 3. The ensemble framework improves the performance
in comparison to that of individual pruned networks.

Figure 1: (a) Accuracy and (b) log-loss obtained after pruning different
intermediate layers (C1, C2, C3, C1+C2, C2+C3, C1+C2+C3) in the unpruned low-
complexity CNN. The accuracy and the log-loss is obtained with and without
performing the fine-tuning of the pruned network.

Comparison with DCASE 2022 Task 1 baseline network: In comparison to the DCASE
2022 task 1 baseline network, the proposed unpruned low-complexity network
improves the accuracy by approximately 3 percentage points, improves log-loss
by 0.102 points with approximately 5x reduction in MACs and 3x reduction in
the parameters. Utilizing ensemble of pruned networks improves the accuracy by
approximately 4.5 percentage points, improves log-loss by 0.19 points with
1.5x reduction in MACs, however requires approximately 14K more parameters in
comparison to that of the DCASE 2022 task 1 baseline network.

Table 4: Performance obtained when different pruned networks are used in the
ensemble framework.

Framework | Accuracy | log-loss | MACs (million) | Parameters
---|---|---|---|---
DCASE 2022 Task 1 baseline | 43% | 1.575 | 29.23 | 46512
Unpruned low-complexity CNN | 45.92% | 1.475 | 5.44 | 14886
Ensemble on all pruned networks except Pruned_C23 | 47.52% | 1.389 | 19.84 | 60958

##  5 Submitted Entries

In this section, a detail of various submitted models is described.

  1. 1.

Singh_Surrey_task1_1 (SurreyAudioTeam22_1, Surrey_4M): This challenge entry
includes predictions from the Pruned_C2 network. Please see Table 2, Sr No. 3,
for more detail of the network.

  2. 2.

Singh_Surrey_task1_2 (SurreyAudioTeam22_2, Surrey_5M): This challenge entry
includes predictions from the unpruned low-complexity network. Please see
Table 2, Sr No. 1, for more detail of the network.

  3. 3.

Singh_Surrey_task1_3 (SurreyAudioTeam22_3, Surrey_19M): This challenge entry
includes predictions from the ensemble framework which combines all pruned
networks except Pruned_C3. Please see Table 3, Sr No. 3, for more detail of
the network.

  4. 4.

Singh_Surrey_task1_4 (SurreyAudioTeam22_4, Surrey_20M): This challenge entry
includes predictions from the ensemble framework which combines all pruned
networks except Pruned_C23. Please see Table 3, Sr No. 5, for more detail of
the network.

The pruned quantized networks can be found at this link 111Link: Pruned
Quantized models, evaluation scripts.. The network size, the number of MACs
are computed using this link222Link: Model size and complexity calculation..

##  6 Conclusion

This report focuses on designing low-complexity system for acoustic scene
classification. A filter pruning, quantization and ensemble procedure is
applied to obtain compressed, accelerated, and low-size CNN. The proposed
framework shows promising results in terms of reduction in parameters and
performance.

##  7 Acknowledgements

This work was partly supported by Engineering and Physical Sciences Research
Council (EPSRC) Grant EP/T019751/1 “AI for Sound (AI4S)”, a PhD scholarship
from the Centre for Vision, Speech and Signal Processing (CVSSP), Faculty of
Engineering and Physical Science (FEPS), University of Surrey, a Research
Scholarship from the China Scholarship Council (CSC) No. 202006470010 and a
Newton Institutional Links Award from the British Council (Grant number
623805725). For the purpose of open access, the authors have applied a
Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript
version arising. For more information about the AI4S project, please follow
the link: https://ai4s.surrey.ac.uk/index.

## References

  * [1] J. Gu, Z. Wang, J. Kuen, L. Ma, A. Shahroudy, B. Shuai, T. Liu, X. Wang, G. Wang, J. Cai, _et al._ , “Recent advances in convolutional neural networks,” _Pattern Recognition_ , vol. 77, pp. 354–377, 2018.
  * [2] K. Simonyan and A. Zisserman, “Very deep convolutional networks for large-scale image recognition,” _International Conference on Learning Representations, ICLR (arXiv preprint arXiv:1409.1556)_ , 2015.
  * [3] A. Krizhevsky, I. Sutskever, and G. E. Hinton, “ImageNet classification with deep convolutional neural networks,” in _Advances in Neural Information Processing Systems_ , 2012, pp. 1097–1105.
  * [4] A. Singh and M. D. Plumbley, “A passive similarity based cnn filter pruning for efficient acoustic scene classification,” _arXiv preprint arXiv:2203.15751_ , 2022.
  * [5] T. Heittola, A. Mesaros, and T. Virtanen, “Acoustic scene classification in DCASE 2020 Challenge: Generalization across devices and low complexity solutions,” in _Proceedings of the Detection and Classification of Acoustic Scenes and Events (DCASE) Workshop_ , 2020. [Online]. Available: https://arxiv.org/abs/2005.14623


# Tutorial on algebraic deletion correction codes

Kedar Tatwawadi, Shubham Chandak
Stanford University Corresponding author:
kedart<EMAIL_ADDRESS>
(April 2019)

###### Abstract

The deletion channel is known to be a notoriously difficult channel to design
error-correction codes for. In spite of this difficulty, there are some
beautiful code constructions which give some intuition about the channel and
about what good deletion codes look like. In this tutorial we will take a look
at some of them. This document is a transcript of my talk at the coding theory
reading group on some interesting works on deletion channel. It is not
intended to be an exhaustive survey of works on deletion channel, but more as
a tutorial to some of the important and cute ideas in this area. For a
comprehensive survey, we refer the reader to the cited sources and the
following papers and surveys [8, 10, 12].

We also provide implementation of VT codes that correct single
insertion/deletion errors for general alphabets at
https://github.com/shubhamchandak94/VT_codes/.

##  1 Introduction

Deletion channel is one of the most fundamental of the channels, and is still
not well understood. Most of you would know what a deletion channel/error
looks like. But, to give an example, this is what a single deletion looks
like:

$10110\rightarrow 1010$

Here the decoder receives the message $1010$ and needs to recover the original
message. This corresponds to a single deletion, either in position 3 or 4. We
cannot say what position the deletion occurred.

In relation to our favourite erasure channel, where such a single error might
look like:

$10110\rightarrow 10e10$

Here, we know what positions the erasure happened. The deletion channel is in
fact a strictly worse channel, as we can convert the erasure channel output
into a deletion channel output, by simply removing all the ‘$e$’ symbols from
the output.

We will also see that it has connections to our second favourite channel the
Binary Symmetric Channel (BSC) channel.

###  1.1 Capacity of Deletion Channel

We define a binary deletion channel, BDC with deletion rate $\alpha$, where
each symbol in input $x^{n}$ can get deleted with probability $\alpha$. This
definition has similarities with the BEC and BSC, but that’s where the
similarity ends. Surprisingly unlike BEC and BSC, we know quite less about
BDC.

  1. 1.

We still do not know the capacity of the BDC channel. One reason is that BDC
is not really a discrete memoryless channel (DMC). For a DMC, you should be
able to write:

$p(y^{n}|x^{n})=\prod_{i=1}^{n}p(y_{i}|x_{i})$

We cannot write the same for deletion channels, one reason: as the output does
not have “length” $n$, and is in fact a random variable. Shannon theory gives
us a nice characterization for the capacity of DMCs:

$C=\max_{p(x)}I(X;Y)$ (1)

But, without this nice expression, finding the capacity is a much more
difficult task.

  2. 2.

We can of course get some bounds on the deletion channel capacity. For
example: we know that binary erasure channel is strictly better than the BDC.
Thus:

$C_{BDC}(\alpha)\leq 1-\alpha$ (2)

This has been an open problem since quite some time now, but there has been
some recent progress on this, which I will briefly talk about.

Most of the results are a part of the survey by Mitzmacher [10]. Kalai,
Mitzenmacher, and Sudan [7] showed that when $\alpha\rightarrow 0$ the
capacity is almost equal to the BSC. Best known capacity lower bounds I am
aware of are [5]: $0.11(1-\alpha)$ . Recently, improved capacity upper bounds
were obtained by Cheraghchi [4] which state that:

$C_{BDC}\leq(1-\alpha)\log\Phi$ (3)

for $\alpha\geq 0.5$, where $\Phi$ is the golden ratio. For those of you who
are interested, they essentially find capacity bounds for a channel known as
Poisson repeat channel (the number of times each symbol is repeated is a
Poisson random variable), which are then ported over for deletion channel as a
special case.

This is mainly to give a sense of how difficult the deletion channel analysis
is, and how little we really know about it. Typically, not knowing the
capacity directly translates to not knowing good codes. But, surprisingly we
know some nice code constructions in specific cases, which we will discuss
next.

Deletion channel is also related to the insertion channel, or the indel
channel, where both insertions and deletions happen, or general edit distance
channels: indels + substitutions/reversals. A better understanding of the
deletion channel is useful not just for communication, but for the problem of
denoising, which is quite common (say when we type things and miss a
character).

##  2 Adversarial deletion error

We spoke about “Shannon-type” random deletion error capacity. But, for
majority of the talk, we will mainly talk about the adversarial error, where
when we say $e$ errors, we mean at max $e$ symbols are deleted. In this
context let us define an $e$-deletion error correction code.

###### Definition 1

For any vector $u\in\mathrm{F}_{2}^{n}$ we define $D_{e}(u)$ as the
descendants; i.e. all the vectors in $\mathbb{F}_{2}^{n-e}$ obtained after $e$
deletions.

For example:
For $u=01101$, $D_{1}(u)=\\{0110,0111,0101,1101\\}$
For $u=00000$, $D_{1}(u)=\\{0000\\}$

Thus the descendant balls need not be of the same sizes unlike the Hamming
balls we are familiar with. And this will lead to some interesting scenarios
as we will see. In fact we can analyze the size of 1-deletion balls:

###### Lemma 1

The size of 1-deletion Descendant ball $D_{1}(u^{n})$ of a vector $u^{n}$ is
equal to the number of “runs” in $u^{n}$.

For example:
$u^{7}=0010111$ has $4$ runs in total $00,1,0,111$. Thus, $|D_{1}(u^{7})|=4$.
It is easy to see why this is true: Any deletion in a run essentially leads to
the same $n-1$ length sequence in the descendant ball.

###### Definition 2

We call a subset $C\subset\mathbb{F}_{2}^{n}$ to be a $e$-error correction
code, if for any $u,v\in C,u\neq v$, $D_{e}(u)\cap D_{e}(v)=\phi$

We will mainly deal with $e=1$ during the talk.

###  2.1 Repetition coding

Lets start with the simplest of the $1$-deletion correction codes you can
think of: “repetition codes”. We repeat every symbol $2$ times. Is that
sufficient? Let us say, we observe some symbol an odd number of times, then we
know that a symbol was deleted in that runs of $0$’s or $1$’s. Note that, we
still cannot figure out the location of the deletion, but can figure out what
was deleted.

This idea can in fact be extended to correct $e$ deletions by repeating every
symbol $e+1$ times. But, this is quite bad: For correcting $1$ error, our
communication rate is $0.5$. Still better that the BSC case, where we had to
repeat things 3 times.

Cool! Can we do better? We will next look at a cool class of codes known as
the VT-codes; or the Varshamov-Tenengolts code. But before doing let’s look at
a puzzle.

###  2.2  A puzzle

I believe the puzzle has some connection with VT-codes, might help with the
understanding; but if not, it is a cool simple puzzle! So lets say, Mary is
the Queen of the seven kingdoms, and she had ordered 100 big barrels filled
with gold coins, where each coin has a weight of 10gm. But, she knows from her
secret agency that one of the barrels contains counterfeit coins weighing only
9gm. She has an electric weighing scale which she can use; so the question is:

_How can she determine which barrel contains the counterfeit coins with a
single measurement_

The solution is simple: she takes $j$ coins from the $j^{th}$ barrel and
places them on the electronic weighing scale. Now if the weight is less that
expected by $r$ grams, then it is the $r^{th}$ barrel which is counterfeit! We
will come back to this puzzle :)

##  3 VT-codes

Allright! We are all set to define VT-codes.

###### Definition 3

Varshamov-Tenengolts code $VT_{a}(n)\subset\mathbb{F}^{n}_{2}$ is defined as:

$VT_{a}(n)=\left\\{x^{n}|\sum_{i=1}^{n}ix_{i}\equiv a\mod(n+1)\right\\}$ (4)

Some historical context on these codes: These codes were first proposed as
error correction codes for 1-bit Z-channel error! Which means essentially
$1\rightarrow 0$, but not the other way round. Z-channel errors are known as
asymmetric bit-flips. Varshamov and Tenengolts proposed these codes in 1965
[14], and then Levenshtein discovered that these codes in fact work well also
for the deletion channel as well!

#### Z-channel correction

So before we look into 1 deletion correction, let us see how they can correct
one Z-channel error, i.e. one of the $1$’s can flip to a $0$.

Let $\hat{x}^{n}$ be the received symbol: then we can still compute:

$S=a-\sum_{i=1}^{n}i\hat{x}_{i}$

In case, there is a $1\rightarrow 0$ flip at position $j$, then $S=j$. Thus,
we can correct the Z-channel error! Here is where the similarity with the
puzzle can be seen.

###  3.1  VT-codes decoding

We are all set to discuss the decoding for deletion channel:

  1. 1.

First of all, note that if it is only a deletion channel, then “error
detection” comes for free from the length of the code, unlike the bit-flipping
error

  2. 2.

Let $y^{n-1}$ be the received erroneous codeword after $1$ deletion. Define:

$\displaystyle w$ $\displaystyle=\sum_{i=1}^{n-1}y_{i}$ (5) $\displaystyle S$
$\displaystyle=a-\sum_{i=1}^{n-1}iy_{i}$ (6)

  3. 3.

Let $p$ be the position at which deletion occurred, as in $x_{p}$ was deleted.
Let $L_{0},L_{1}$ be the counts of 1,0 to the left of position $p$. and
$R_{0},R_{1}$ be the counts to the right. In that case: $S=R_{1}$ if $x_{p}=0$
and $S=p+R_{1}$ if $x_{p}=1$.

$\displaystyle S$ $\displaystyle=p+R_{1}$ $\displaystyle=L_{0}+L_{1}+1+R_{1}$
$\displaystyle=w+L_{0}+1$

Thus, as $R_{1}<w+1$, if $S\leq w$, $0$ is deleted at a position with $R_{1}$
1’s to its right, otherwise, $1$ is deleted such that there are $L_{0}=S-w-1$
zeros to its left.

  4. 4.

Note that we can thus uniquely determine the $x^{n}$ sequence, but we cannot
determine the exact location of the deletion here, as it can be any $0$ or $1$
in the run we identified.

This beautiful decoding algorithm was given by Levenshtein in his 1965 work
[8]. Also most of the things which I will talk are from the survey by Sloane
[12].

###  3.2 VT-codes rate

As the VT-codes are combinatorial, we can get exact formulae for their sizes.
Exact combinatorial formulate can be found in [12]. Here are few interesting
things:

  1. 1.

###### Lemma 2

For some $a\in[0,n]$:

$|VT_{a}(n)|\geq\frac{2^{n}}{n+1}$

As every $x^{n}\in\mathbb{F}_{1}^{n}$ lies in exactly one of $VT_{a}(n)$ sets:

$\sum_{a=0}^{n}|VT_{a}(n)|=2^{n}$

This leads to the lemma.

  2. 2.

It can be in fact shown that $a=0$ leads to the largest code size, and $a=1$
the smallest size.

$|VT_{0}(n)|\leq|VT_{a}(n)|\leq|VT_{1}(a)|$

  3. 3.

For $n=2^{q}-1$, all the VT-code sizes are in fact equal to
$\frac{2^{n}}{n+1}$.

###  3.3 Optimality of VT-codes

We say a $e$-error correction code is “optimal” if it has the smallest size
amongst all $e$-error correction codes. Let us analyze the “optimality” of VT-
codes.

  1. 1.

Levenshtein [8] showed that, optimal $1$-deletion correction codes have
asymptotic sizes $\approx\frac{2^{n}}{n}$. This makes VT-codes asymptotically
optimal.

  2. 2.

People have not been able to prove that VT-codes are optimal non-
asymptotically. Finding the “optimal” 1-deletion code is a NP-hard problem, as
it involves finding the independent set on the graph where vertices are
connected if they lead to the same deletion descendants [12]. But for $n\leq
8$ it is known that they are optimal, using computer programs. Sizes of these
codes are:

$1,2,2,4,6,10,16,30$

For higher $n$ due to the exponential nature of the algorithms, we cannot say
anything yet.

  3. 3.

VT-codes also have the property that they are “perfect codes” [9] which
implies that their descendent sets which are disjoint cover the entire
$2^{n-1}$ sized. For example:

$\displaystyle VT_{0}(3)$ $\displaystyle=\\{000,101\\}$ $\displaystyle
Descendants$ $\displaystyle=\\{00,10,01,11\\}$

Levenshtein showed that surprisingly, this is true for all $a$, which is quite
cool in itself! The perfect codes analogy comes from Hamming codes being
Perfect. But, unlike the Hamming distortion case, here perfect codes does not
imply optimal codes? Why? For example, code $\\{000,111\\}$ is not perfect but
is perhaps a better code that $VT_{0}(3)$, potentially there might be a larger
code for larger $n$. Why does this happen? Because, the number of descendants
are not fixed, some have $1$ and some have more.

  4. 4.

Linearity: VT codes are linear until $n=4$ but never after! [12]. Variants of
VT-code (restrictions on VT-codes) can be made linear by considering
redundancy to be $\sqrt{n}$ as against $\log n$. Althought I am not sure how
is linearity of codes useful, if the decoding is still non-linear (linear time
complexity, but non-linear in nature).

###  3.4 Systematic Encoding

Now that we have taken a look at the linear-time decoding of VT-codes, it is a
natural question to ask if there exist a nice way to encode data. This problem
surprisingly remained open for more than 30 years until 1998 when Abdel-Gaffar
et al. [1] provided a very convenient way of in fact “systematic encoding” of
data.

  1. 1.

For $n=2^{q}-1$, let $m=2^{q}-q-1$ be the number of data bits, and $q$ are the
“parity” bits. Let the data bits be $b_{1},b_{2},\ldots,b_{m}$, and the code-
word to be formed $x^{n}$ in $VT_{a}(n)$.

  2. 2.

Fill in the data except in positions $1,2,2^{2},2^{3},\ldots,2^{q-1}$. Thus
$x^{n}$ codeword looks like:

$x_{1},x_{2},b_{1},x_{4},b_{2},b_{3},b_{4},x_{8},\ldots,b_{m}$

We can compute: $S=a-\sum_{i=1}^{2^{q}-1}ix_{i}\mod(2^{q})$. As most of the
positions of $x^{n}$ (except $x_{2^{j}},j\in[0,q-1]$) are decided, to obtain
$S=0$, we need:

$\sum_{j=0}^{q-1}2^{j}x_{2^{j}}\mod(2^{q})=\hat{a}$

We can now conveniently choose $x_{2^{j}},j\in[0,q-1]$ as the $q$-bit binary
expansion of $\hat{a}$.

  3. 3.

Note that $m=2^{q}-q-1,n=2^{q}-1$ for 1-deletion correction is exactly same as
the rate of Hamming code. Not sure if this is a coincidence or something more!

##  4 Insertion + Deletion + Substitution codes

We looked at $1$-deletion correction channels in depth. In the next part of
the tutorial, we will extend this understanding to more general scenarios. The
first scenario is $1$ insertion instead of $1$ deletion error.

###  4.1 General indel error codes

Levenshtein [8] showed this general lemma:

###### Lemma 3

Any $s$-deletion correction code can also correct $s_{1}$ deletions and
$s_{2}$ insertion errors where $s_{1}+s_{2}\leq s$.

Note that here we do not need to know $s_{1},s_{2}$ beforehand. The general
proof is quite simple. Here, we will prove the simpler version of
$1$-insertion error, as that is sufficient to get an intuitive understanding.

  1. 1.

Let us assume that $C$ is a $1$-deletion correction code. We want to show that
$C$ can correct $1$-insertion errors as well.

  2. 2.

Let us assume that on the contrary, there exist codewords $x^{n},y^{n}\in C$
such that after one insertion error in them the resulting noisy codewords
$\hat{x}^{n+1},\hat{y}^{n+1}$ are equal. Let the insertion occured at position
$i$ in $\hat{x}^{n+1}$ and at position $j$ in $\hat{y}^{n+1}$.

  3. 3.

As $\hat{x}^{n+1}=\hat{y}^{n+1}$, even after deleting symbols in position
$i,j$ in both $\hat{x},\hat{y}$, results in $n-1$ vectors which are equal.
However, the $n-1$ length codewords are in fact $1$-deletion descendants of
the codewords $x^{n},y^{n}\in C$. This is contradictory to the definition of
deletion correction codes, as no descendants can be equal. Thus, $C$ has to be
$1$-insertion correction as well.

Note that, this is more of an existential result, and efficient deletion error
decoding algorithms might not translate into efficient insertion detection
algorithms.

###  4.2 VT-codes for 1-insertion, deletion, substitution correction

Levenshtein showed the surprising fact that with a simple modification
standard VT-codes can be converted into 1-insertion, deletion and substitution
correction codes. The modification is as follows:

$\hat{VT}_{a}(n)\subset\mathbb{F}^{n}_{2}$ is defined as:

$\hat{VT}_{a}(n)=\left\\{x^{n}|\sum_{i=1}^{n}ix_{i}\equiv
a\mod(2n+1)\right\\}$ (7)

Let us try to understand why $\hat{VT}_{a}(n)$ work:

  1. 1.

First of all, from the length of the code, we know whether there is an
insertion, deletion or a substitution.

  2. 2.

Recollect that deletion error correction in $VT_{a}(n)$ codes only depends on
distinct remainders modulus $(n+1)$. This should still hold true if the
modulus is taken $m\geq n+1$. Thus, with $m=2n+1$, 1-deletion correction still
holds.

  3. 3.

1-insertion code ability was already shown from the general lemma earlier.
However, using a similar remainder trick $\mod 2n+1$, insertions can in fact
be corrected efficiently using the $\hat{VT}_{a}(n)$ codes.

  4. 4.

The only case remaining to analyze is the 1-substitution or 1-bitflip case.
Let $0\rightarrow 1$ at position $p\in[1,n]$ in the codeword $x^{n}$ resulting
in the noisy codework $\hat{x}^{n}$. Clearly:

$\displaystyle S$ $\displaystyle=a-\sum_{i=1}^{n}i\hat{x}_{i}\mod(2n+1)$
$\displaystyle=2n-p$

If $1\rightarrow 0$, then:

$\displaystyle S$ $\displaystyle=a-\sum_{i=1}^{n}i\hat{x}_{i}\mod(2n+1)$
$\displaystyle=p$

As all these $S$ values are distinct, we can correct for 1-bitflip. Note that,
this construction is not optimal just for 1-bitflip, as it essentially encodes
1 bit less than Hamming codes.

##  5 VT-codes for larger alphabet

Creating deletion codes for larger alphabets becomes a bit tricky. Of course,
repetition coding still works on non-binary alphabets.

#### Code which does not work

When I started thinking about this problem, I came up with this code, which
has a bug! Let us still take a look at it, as it gives some understanding as
to the intricacies of code-design:

Define $\beta_{i}$ for $i\in[2,n]$ as:

$\displaystyle\beta_{i}$ $\displaystyle=1,\ x_{i}\neq 0$ $\displaystyle=0,\
\mathrm{otherwise}$

Then we consider code as:

$\displaystyle C=\bigg{\\{}x^{n}|$
$\displaystyle\sum_{i=1}^{n}i\beta_{i}=a\mod(n+1),$
$\displaystyle\sum_{j=1}^{n}x_{i}=b\mod(q)\bigg{\\}}$

Let us look at the argument as to why the code works, and try to find the bug!
_Argument: The first equation, similar to the binary VT code will tell us the
position of the deletion, and the second equation tells us the value. _

Why does the above argument not work. The reason is that binary VT-codes not
not actually tell us the position of the deletion correctly. They can tell us
in which “run” the deletion happened, and hence obtain the codeword correctly,
but not the exact position.

How do we solve for that?

#### Code which works

This code appears in the work of Tenengolts in 1984. [13]
Define $\alpha_{i}$ for $i\in[2,n]$ as:

$\displaystyle\alpha_{i}$ $\displaystyle=1,\ x_{i}\geq x_{i-1}$
$\displaystyle=0,\ x_{i}<x_{i-1}$

Then we consider the code:

$\displaystyle C=\bigg{\\{}x^{n}|$
$\displaystyle\sum_{i=2}^{n}(i-1)\alpha_{i}=a\mod(n),$
$\displaystyle\sum_{j=1}^{n}x_{i}=b\mod(q)\bigg{\\}}$

Let us try to analyze the decoding for this code:

  1. 1.

First of all, as in the previous code, from the second constraint
$\sum_{j=1}^{n}x_{i}=b\mod(q)$, we figure out what is the value of the deleted
symbol. What remains to determine is its position.

  2. 2.

The $\alpha$ sequence is essentially capturing the monotonous regions of the
$x^{n}$ sequence. $\alpha_{i}=1$, when the sequence is increasing (non-
decreasing), and $0$ when it is decreasing. Thus, a deletion in $x^{n}$
sequence will in fact lead to exactly $1$ deletion in the $\alpha$ sequence
(the position of deletion might be shifted by 1, but it does not matter as VT-
codes do not correct for position).

  3. 3.

As the first equation is a VT-code, it can determine the run in which the
deletion occurred. As every run in $\alpha$ sequence corresponds to monotonic
increasing/decreasing subsequence in $x^{n}$, from the value of the deleted
symbol, we can correctly place it and complete the decoding!

Efficient systematic encoding for this code was discovered recently in a work
by Abroshan et al. [2] and is included in the implementation at
https://github.com/shubhamchandak94/VT_codes/.

##  6 Bursty deletion codes

In this section we will look at Bursty deletions. By a single bursty deletion
of size $s$, we mean that some consecutive $s$ symbols were deleted. Note
that, $s$-bursty deletion correction codes can correct for exactly 1 burst of
size $s$, but surprisingly they need not correct for a burst of size $s-1$.
For example: $C=\\{0101,1010\\}$ can correct 2-bursty errors, but not single
deletion errors!

###  6.1 VT-code based construction

We consider a construction based on single-deletion correction VT-codes to
correct a single burst of $s$ deletions. How should one do that? One simple
trick is to distribute these $s$ deletions across the $n$ length sequence, so
that each $n/s$ length subsequence has exactly 1 deletion. For simplicity, let
$n/s=k$.

Then the codeword $x^{n}$ has the property that each of the $s$ rows below,
belong to a $VT_{a}(n)$ code.

$\displaystyle x_{1},x_{s+1},x_{2s+1},\ldots,x_{n-s+1}$ $\displaystyle
x_{2},x_{s+2},x_{2s+2},\ldots,x_{n-s+2}$ $\displaystyle\;\;\;\;\vdots$
$\displaystyle x_{s},x_{2s},x_{2s+2},\ldots,x_{n}$

  1. 1.

Let us analyze the scenario of 1 bursty error of size $s$ from position
$x_{j},\ldots,x_{j+s-1}$ are deleted. This corresponds to exactly 1 deletion
in each of the rows.

  2. 2.

We still need to figure out which symbols of the deleted codeword belong to
which rows, as the alignment might no longer be true. The cool thing is that,
if there are _exactly_ $s$ consecutive deletions, then every $s^{th}$ symbol
will still be correctly aligned to the rows.

  3. 3.

Thus, our code can in fact correct a bursty error of $s$ deletions, but need
not correct lower number of bursts, which is quite unusual!

One important caveat to observe here is that the position of deletion in each
of the rows is the same, or shifted by 1. Thus, if one of the rows is a VT-
code, and the other rows just tell whether the error position is odd or even,
that is enough to resolve the bursty error. This observation is the basis of
further improvements to bursty error correction. For more details take a look
at this paper: [11].

##  7 Multiple Deletions

One would imagine that it should be possible to extend the elegant
construction of VT-codes from single deletion correction to multiple
deletions. However, this problem has proved to be much more difficult.

There have been some recent works which extend the single deletion errors to
multiple errors. Gabrys et al [6] provide an extension of the VT-codes idea to
correct two deletions. There have been other recent works which provide
multiple deletion correcting codes using different (non VT-code based) ideas
[3].

## Acknowledgement

I would like to thank Jay Mardia and Mary Wootters for interesting discussions
on deletion codes.

## References

  * [1] Khaled AS Abdel-Ghaffar and Hendrik C Ferreira.  Systematic encoding of the Varshamov-Tenengol’ts codes and the Constantin-Rao codes.  IEEE Transactions on Information Theory, 44(1):340–345, 1998.
  * [2] Mahed Abroshan, Ramji Venkataramanan, and Albert Guillen I Fabregas.  Efficient systematic encoding of non-binary vt codes.  In 2018 IEEE International Symposium on Information Theory (ISIT), pages 91–95. IEEE, 2018.
  * [3] Joshua Brakensiek, Venkatesan Guruswami, and Samuel Zbarsky.  Efficient low-redundancy codes for correcting multiple deletions.  IEEE Transactions on Information Theory, 64(5):3403–3410, 2017\.
  * [4] Mahdi Cheraghchi.  Capacity upper bounds for deletion-type channels.  Journal of the ACM (JACM), 66(2):9, 2019.
  * [5] Eleni Drinea and Michael Mitzenmacher.  Improved lower bounds for the capacity of iid deletion and duplication channels.  IEEE Transactions on Information Theory, 53(8):2693–2714, 2007\.
  * [6] Ryan Gabrys and Frederic Sala.  Codes correcting two deletions.  IEEE Transactions on Information Theory, 65(2):965–974, 2018.
  * [7] Adam Kalai, Michael Mitzenmacher, and Madhu Sudan.  Tight asymptotic bounds for the deletion channel with small deletion probabilities.  In 2010 IEEE International Symposium on Information Theory, pages 997–1001. IEEE, 2010.
  * [8] Vladimir I Levenshtein.  Binary codes capable of correcting deletions, insertions, and reversals.  In Soviet physics doklady, pages 707–710, 1966.
  * [9] Vladimir I Levenshtein.  On perfect codes in deletion and insertion metric.  Discrete Mathematics and Applications, 2(3):241–258, 1992.
  * [10] Michael Mitzenmacher et al.  A survey of results for deletion channels and related synchronization channels.  Probability Surveys, 6:1–33, 2009.
  * [11] Clayton Schoeny, Antonia Wachter-Zeh, Ryan Gabrys, and Eitan Yaakobi.  Codes correcting a burst of deletions or insertions.  IEEE Transactions on Information Theory, 63(4):1971–1985, 2017\.
  * [12] Neil JA Sloane.  On single-deletion-correcting codes.  Codes and designs, 10:273–291, 2000.
  * [13] Grigory Tenengolts.  Nonbinary codes, correcting single deletion or insertion (Corresp.).  IEEE Transactions on Information Theory, 30(5):766–769, 1984.
  * [14] RR Varshamov and GM Tenengolts.  Codes which correct single asymmetric errors (in Russian).  Automatika i Telemkhanika, 161(3):288–292, 1965.


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

Qin Changa,b, Ya-Dong Yanga,c
aInstitute of Particle Physics, Huazhong Normal University, Wuhan, Hubei
430079, P. R. China
bDepartment of Physics, Henan Normal University, Xinxiang, Henan 453007, P. R.
China
cKey Laboratory of Quark $\&$ Lepton Physics, Ministry of Education, P.R.
China

###### Abstract

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.

PACS Numbers: 13.25.Cq, 14.80.Cp

##  1 Introduction

Theoretically, the rare decay $\pi^{0}\to e^{+}e^{-}$ starts at the one loop
level in the standard model (SM), which has been extensively studied [1, 2, 3,
4, 5, 6, 7, 8, 9, 10] since the first investigation in QED by Drell [1]. It is
nontrivial to make precise predictions of the branching ratio ${\cal
B}_{SM}(\pi^{0}\to e^{+}e^{-})$ because its sub-process involves the
$\pi^{0}\to\gamma^{*}\gamma^{*}$ transition form factor. In Refs.[2, 3, 4, 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 ${\cal
B}(\pi^{0}\to e^{+}e^{-})\sim 6.2-6.4\times 10^{-8}$. By using the measured
value of ${\cal B}(\eta\to\mu^{+}\mu^{-})$ to fix the counterterms of the
chiral amplitude in Chiral Perturbation Theory (ChPT), Savage et al. predicted
${\cal B}(\pi^{0}\to e^{+}e^{-})=(7\pm 1)\times 10^{-8}$ [6]. Using a
procedure similar to that used in Ref.[6] (although with an updated
measurement of ${\cal B}(\eta\to\mu^{+}\mu^{-})$), Dumm and Pich predicted
$(8.3\pm 0.4)\times 10^{-8}$ [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 $(6.2\pm
0.3)\times 10^{-8}$ [8], which is about $4\sigma$ 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

${\cal B}_{SM}(\pi^{0}\to e^{+}e^{-})\,=\,(6.23\,\pm\,0.09)\times 10^{-8},$
(1)

which is consistent with most theoretical predictions of ${\cal
B}_{SM}(\pi^{0}\to e^{+}e^{-})$ in the literature. Moreover, their prediction
that ${\cal B}(\eta\to\mu^{+}\mu^{-})=(5.11\pm 0.2)\times 10^{-6}$ agrees with
the experimental data (which gives a value of $(5.8\pm 0.8)\times 10^{-6}$
[13]).

Experimentally, the accuracy of the measurements of the decay has increased
significantly since the first $\pi^{0}\to e^{+}e^{-}$ evidence was observed by
the Geneva-Saclay group [14] in 1978 with ${\cal B}_{SM}(\pi^{0}\to
e^{+}e^{-})=(22^{+24}_{-11})\times 10^{-8}$. 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 $\pi^{0}\to e^{+}e^{-}$ branching ratio [16]

${\cal B}^{no-rad}_{KTeV}(\pi^{0}\to
e^{+}e^{-})\,=\,(7.48\,\pm\,0.29\,\pm\,0.25)\times 10^{-8},$ (2)

after extrapolating the full radiative tail beyond
$(m_{e^{+}e^{-}}/m_{\pi^{0}})^{2}>0.95$ 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
$3.3\sigma$ 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 $\pi^{0}\to
e^{+}e^{-}$, 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
${\cal B}(\pi^{0}\to e^{+}e^{-})$ at $1\sigma$. Then, we combine constraints
from $a_{\mu}$ and the recent searches for $\Upsilon(1S),(3S)\to\gamma
A^{0}_{1}$ by CLEO [20] and BaBar [21], respectively.

##  2 The amplitude of $\pi^{0}\to e^{+}e^{-}$ 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 $\hat{S}$ to solve the known $\mu$ problem in MSSM. The
superpotential in the model is [19]

$W_{NMSSM}=\hat{Q}\hat{H}_{u}h_{u}\hat{U}^{C}+\hat{H}_{d}\hat{Q}h_{d}\hat{D}^{C}+\hat{H}_{d}\hat{L}h_{e}\hat{E}^{C}+\lambda\hat{S}\hat{H}_{u}\hat{H}_{d}+\frac{1}{3}\kappa\hat{S}^{3},$
(3)

where $\kappa$ is a dimensionless constant and measures the size of Peccei-
Quinn (PQ) symmetry breaking.

In addition to the two charged Higgs bosons, $H^{\pm}$, the physical NMSSM
Higgs sector consists of three scalars $h^{0},\,H_{1,2}^{0}$ and two
pseudoscalars $A_{1,2}^{0}$. As in the MSSM, $\tan\beta=v_{u}/v_{d}$ is the
ratio of the Higgs doublet vacuum expectation values $v_{u}=\langle
H_{u}^{0}\rangle=v\sin\beta$ and $v_{d}=\langle H_{d}^{0}\rangle=v\cos\beta$,
where $v=\sqrt{v_{d}^{2}+v_{u}^{2}}=\sqrt{2}m_{W}/g\simeq 174GeV$. 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 $A_{1}^{0}$ can be rather
light, and its mass is given by

$m^{2}_{A^{0}_{1}}=3\kappa xA_{k}+{\cal O}(\frac{1}{\tan\beta})$ (4)

with the vacuum expectation value of the singlet $x=\langle S\rangle$;
meanwhile, another pseudoscalar $A_{2}^{0}$ has a mass of order of
$m_{H^{\pm}}$.

For $\pi^{0}\to e^{+}e^{-}$ decay, the NMSSM contributions are dominated by
$A_{1}^{0}$. The couplings of $A_{1}^{0}$ to fermions are [22]

$\mathcal{L}_{A_{i}^{0}f\bar{f}}=-i\frac{g}{2m_{W}}\Big{(}X_{d}m_{d}\bar{d}\gamma_{5}d+X_{u}m_{u}\bar{u}\gamma_{5}u+X_{\ell}m_{\ell}\bar{\ell}\gamma_{5}\ell\Big{)}A_{1}^{0}$
(5)

where $X_{d}=X_{\ell}=\frac{v}{x}\delta_{-}$ and $X_{u}=X_{d}/\tan^{2}\beta$;
thus, the contribution of the $\bar{u}\gamma_{5}uA_{1}^{0}$ term in
$\pi^{0}\to e^{+}e^{-}$ could be neglected in the large $\tan\beta$
approximation.

Figure 1:  Relevant Feynman diagram within NMSSM.

To the leading order, the relevant Feynman diagram within NMSSM is shown in
Fig. 1. We obtain its amplitude as

$\mathcal{M}_{A_{1}^{0}}=-\frac{G_{F}}{\sqrt{2}}m_{e}m_{\pi^{0}}^{3}f_{\pi^{0}}\frac{1}{m_{\pi^{0}}^{2}-m_{A_{1}^{0}}^{2}}X_{d}^{2},$
(6)

which is independent of $m_{d}$, since $m_{d}$ in the coupling of
$A^{0}_{1}\bar{d}\gamma_{5}d$ is canceled by the $m_{d}$ term of the hadronic
matrix

$\langle\,0\,|\,\bar{d}\gamma_{5}d\,|\,\pi^{0}\rangle\,=-\frac{i}{\sqrt{2}}f_{\pi^{0}}\frac{m_{\pi^{0}}^{2}}{2m_{d}}\,.$
(7)

In the SM, the normalized branching ratio of $\pi^{0}\to e^{+}e^{-}$ is given
by [9]

$R(\pi^{0}\to e^{+}e^{-})=\frac{{\cal B}(\pi^{0}\to e^{+}e^{-})}{{\cal
B}(\pi^{0}\to\gamma\gamma)}=2\left(\frac{\alpha_{e}}{\pi}\frac{m_{e}}{m_{\pi^{0}}}\right)^{2}\beta_{e}(m^{2}_{\pi^{0}})|{\cal
A}(m^{2}_{\pi^{0}})|^{2}$ (8)

where $\beta_{e}(m^{2}_{\pi^{0}})=\sqrt{1-4\frac{m_{e}^{2}}{m_{\pi^{0}}^{2}}}$
and ${\cal A}(m^{2}_{\pi^{0}})$ 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].

Figure 2:  Triangle diagram for $\pi^{0}\to e^{+}e^{-}$ process.

The Feynman diagram that proceeds via two photon intermediate states is shown
in Fig. 2. We start with the $\pi^{0}\gamma^{*}\gamma^{*}$ vertex

$H_{\mu\nu}=-i\,e^{2}\,\epsilon_{\mu\nu\alpha\beta}\,k^{\alpha}\,(q-k)^{\beta}f_{\gamma^{\ast}\gamma^{\ast}}\,F_{\pi^{0}\gamma^{\ast}\gamma^{\ast}}(k^{2},(q-k)^{2})$
(9)

where $k$ and $q-k$ are the momenta of the two photons,
$f_{\gamma^{\ast}\gamma^{\ast}}=\frac{\sqrt{2}}{4\pi^{2}\,andf_{\pi^{0}}}$ is
the coupling constant of $\pi^{0}$ to two real photons.
$F_{\pi^{0}\gamma^{\ast}\gamma^{\ast}}(k^{2},(q-k)^{2})$ is the transition
form factor $\pi^{0}\to\gamma^{\ast}\gamma^{\ast}$, which is normalized to
$F_{\pi^{0}\gamma^{\ast}\gamma^{\ast}}(0,0)=1$. The amplitude of Fig. 2 is
written as

$\mathcal{M}_{SM}(\pi^{0}\to
e^{+}e^{-})=ie^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{L^{\mu\nu}H_{\mu\nu}}{(k^{2}+i\varepsilon)\big{(}(k-q)^{2}+i\varepsilon\big{)}\big{(}(k-p)^{2}-m_{e}+i\varepsilon\big{)}},$
(10)

with

$L^{\mu\nu}\,=\,\bar{u}(p,s)\gamma^{\mu}(\not\\!p\,-\not\\!k\,+m_{e})\gamma^{\nu}v(q-p,s^{\prime}).$
(11)

There is a known, convenient way to calculate $L^{\mu\nu}$ with the projection
operator for the outgoing $e^{+}e^{-}$ pair system[23]

$\displaystyle\mathcal{P}(q-p,p)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\big{[}v(q-p,+)\otimes\bar{u}(p,-)+v(q-p,-)\otimes\bar{u}(p,+)\big{]}$
(12) $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{2t}}\big{[}-2m_{e}q_{\mu}\gamma^{\mu}\gamma^{5}+\frac{1}{2}\epsilon_{\mu\nu\sigma\tau}\big{(}p^{\sigma}(q-p)^{\tau}-(q-p)^{\sigma}p^{\tau}\big{)}\sigma^{\mu\nu}+t\gamma^{5}\big{]}$

where $t=q^{2}=m_{\pi^{0}}^{2}$. After some calculations, we get

$\mathcal{M}_{SM}(\pi^{0}\to
e^{+}e^{-})=2\sqrt{2}\,\alpha^{2}\,m_{e}\,m_{\pi^{0}}\,f_{\gamma^{\ast}\gamma^{\ast}}\,A(m^{2}_{\pi})$
(13)

where the reduced amplitude $A(q^{2})$ is

${\cal
A}(q^{2})=\frac{2\,i}{q^{2}}\int\frac{d^{4}k}{\pi^{2}}\,\frac{k^{2}\,q^{2}\,-\,(q\cdot
k)^{2}}{(k^{2}+i\varepsilon)\big{(}(k-q)^{2}+i\varepsilon\big{)}\big{(}(k-p)^{2}-m_{e}+i\varepsilon\big{)}}F_{\pi^{0}\gamma^{\ast}\gamma^{\ast}}(k^{2},(q-k)^{2}).$
(14)

We note that the ${\cal A}(q^{2})$ derived here is in agreement with Ref. [9].
Further evaluation of the integrals of ${\cal A}(q^{2})$ is quite subtle and
lengthy [2, 24], and only the imaginary part of ${\cal A}(m_{\pi^{0}}^{2})$
can be obtained model-independently[1, 2]. In the following calculations, we
quote the result of Ref. [9],

${\cal A}(m_{\pi}^{2})=(10.0\pm 0.3)-i17.5.$ (15)

With Eq. (6) and Eq. (13), we get the total amplitude

$\mathcal{M}=2\sqrt{2}\,\alpha^{2}\,m_{e}\,m_{\pi^{0}}\,f_{\gamma^{\ast}\gamma^{\ast}}\,A(m^{2}_{\pi})-\frac{G_{F}}{\sqrt{2}}m_{e}m_{\pi^{0}}^{3}f_{\pi^{0}}\frac{1}{m_{\pi^{0}}^{2}-m_{A_{1}^{0}}^{2}}X_{d}^{2}.$
(16)

##  3 Numerical analysis and discussion

Now, we are ready to discuss the effects of $A_{1}^{0}$ numerically, with a
focus on the $m_{A^{0}_{1}}<2m_{b}$ scenarios. The dependence of ${\cal
B}(\pi^{0}\to e^{+}e^{-})$ on the parameter $|X_{d}|$ is shown in Fig. 3 with
$m_{A_{1}^{0}}=m_{\pi}/2,\,214.3{\rm MeV},\,3{\rm GeV}$ as benchmarks. We have
used the input parameters ${\cal B}(\pi^{0}\to\gamma\gamma)=0.988$ and
$f_{\pi^{0}}=(130.7\pm 0.4)~{}{\rm MeV}$ [13]. As shown in Fig. 3, ${\cal
B}(\pi^{0}\to e^{+}e^{-})$ is very sensitive to the parameter $|X_{d}|$ and
$m_{A_{1}^{0}}$. For $m_{A_{1}^{0}}<m_{\pi^{0}}$, the NMSSM contribution is
deconstructive and reduces ${\cal B}(\pi^{0}\to e^{+}e^{-})$ at small
$|X_{d}|$ region. For $m_{A_{1}^{0}}>m_{\pi^{0}}$, the NMSSM contribution is
constructive and could enhance ${\cal B}(\pi^{0}\to e^{+}e^{-})$ to be
consistent with the KTeV measurement ${\cal B}^{no-rad}_{KTeV}(\pi^{0}\to
e^{+}e^{-})=(7.48\pm 0.38)\times 10^{-8}$ (where $|X_{d}|$ strongly depends on
$m_{A^{0}_{1}}$).

Figure 3:  The dependence of ${\cal B}(\pi^{0}\,{\to}\,e^{+}e^{-})$ on the
parameter $|X_{d}|$ with $m_{A_{1}^{0}}=m_{\pi^{0}}/2$, $214.3{\rm~{}MeV}$ and
$\rm 3~{}GeV$, respectively. The horizontal lines are the KTeV data, where the
solid line is the central value and the dashed ones are the error bars
($1\sigma$).

I. Constraint on the scenario of $m_{A^{0}_{1}}=214.3{\rm MeV}$
It is interesting to note that the HyperCP Collaboration [25] has observed
three events for the decay $\Sigma^{+}\to p\mu^{+}\mu^{-}$ with a narrow range
of dimuon masses. This may indicate that the decay proceeds via a neutral
intermediate state, $\Sigma^{+}\to pP^{0},P^{0}\to\mu^{+}\mu^{-}$, with a
$P^{0}$ mass of $214.3\pm 0.5{\rm MeV}$. The possibility of $P^{0}$ has been
explored in the literature [26, 28, 29, 30]. The authors have proposed
$A_{1}^{0}$ 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 $\pi^{0}\to e^{+}e^{-}$ decay.

Taking $m_{A_{1}^{0}}=214.3{\rm~{}MeV}$, we find that ${\cal B}(\pi^{0}\to
e^{+}e^{-})$ is enhanced rapidly and could be consistent with the KTeV data
within $1\sigma$ for

$|X_{d}|=14.0\pm 2.4.$ (17)

However, the upper bound $|X_{d}|<1.2$ from the $a_{\mu}$ constraint has been
derived and used in the calculations of Ref. [26, 29]. So, with the assumption
that $m_{A_{1}^{0}}=214.3{\rm~{}MeV}$, our result of $|X_{d}|$ violates the
upper bound with a significance of $5\sigma$.

Recently, CLEO [20] and BaBar [21] have searched for the CP-odd Higgs boson in
radiative decays of $\Upsilon(1S)\to\gamma A^{0}_{1}$ and
$\Upsilon(3S)\to\gamma A^{0}_{1}$, respectively. For $m_{A_{1}^{0}}=214{\rm
MeV}$, CLEO gives the upper limit

${\cal B}(\Upsilon(1S)\to\gamma A_{1}^{0})<2.3\times
10^{-6}~{}~{}~{}~{}(90\%~{}C.L.)$ (18)

which constrains $|X_{d}|<0.16$.

The BaBar Collaboration has searched for $A_{1}^{0}$ through
$\Upsilon(3S)\to\gamma A_{1}^{0}$, $A_{1}^{0}\to\textrm{invisible}$ in the
mass range $m_{A_{1}^{0}}\leq 7.8{\rm~{}GeV}$ [21]. From Fig. 5 of Ref. [21],
we read

${\cal B}(\Upsilon(3S)\to\gamma A_{1}^{0})\times{\cal
B}(A_{1}^{0}\to\textrm{invisible})\lesssim 3.5\times
10^{-6}~{}~{}~{}~{}(90\%~{}C.L.)$ (19)

for $m_{A_{1}^{0}}=214{\rm~{}MeV}$. Assuming ${\cal
B}(A_{1}^{0}\to\textrm{invisible})\sim 1$, we get the conservative upper limit
$|X_{d}|<0.19$

All of these upper limits are much lower than the limit of Eq.17 set by
$\pi^{0}\to e^{+}e^{-}$; therefore, the scenario where $m_{A_{1}^{0}}\simeq
214{\rm~{}MeV}$ in NMSSM could be excluded by combining the constraints from
$\pi^{0}\to e^{+}e^{-}$ and the direct searches for $\Upsilon$ radiative
decays.
II. Constraints on the parameter space of $m_{A^{0}_{1}}-|X_{d}|$

To show the constraints on NMSSM parameter space from $\pi^{0}\to e^{+}e^{-}$,
we present a scan of $m_{A^{0}_{1}}-|X_{d}|$ space, as shown in Fig. 4. In
order to scan the region of $m_{A_{1}^{0}}\sim m_{\pi^{0}}$, the amplitude of
the $A_{1}^{0}$ contribution in Eq. (6) is replaced by the Breit-Wigner
formula

$\mathcal{M}_{A_{1}^{0}}\,=-\,\frac{G_{F}}{\sqrt{2}}m_{e}m_{\pi^{0}}^{3}f_{\pi^{0}}\frac{1}{m_{\pi^{0}}^{2}-m_{A_{1}^{0}}^{2}+i\Gamma(A_{1}^{0})m_{A_{1}^{0}}}X_{d}^{2}\,.$
(20)

With the assumption that $A_{1}^{0}$ just decays to electron and photon pairs
for $m_{A_{1}^{0}}\sim m_{\pi^{0}}$, the decay width of $A_{1}^{0}$ could be
written as

$\Gamma(A_{1}^{0})=\Gamma(A_{1}^{0}\to
e^{+}e^{-})+\Gamma(A_{1}^{0}\to\gamma\gamma)$ (21)

with

$\displaystyle\Gamma(A_{1}^{0}\to e^{+}e^{-})$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}G_{F}}{8\pi}m_{e}^{2}m_{A_{1}^{0}}X_{d}^{2}\sqrt{1-4\frac{m_{e}^{2}}{m_{A_{1}^{0}}^{2}}}\,,$
$\displaystyle\Gamma(A_{1}^{0}\to\gamma\gamma)$ $\displaystyle=$
$\displaystyle\frac{G_{F}\alpha^{2}}{8\sqrt{2}\pi^{3}}m_{A_{1}^{0}}^{3}X_{d}^{2}|\sum_{i}rQ_{i}^{2}k_{i}F(k_{i})|^{2},$
(22)

where $r=1$ for leptons and $r=N_{c}$ for quarks,
$k_{i}=m_{i}^{2}/m_{A_{1}^{0}}^{2}$ and $Q_{i}$ is the charge of the fermion
in the loop. The loop function $F(k_{i})$ reads [31]

$F(k_{i})=\left\\{\begin{array}[]{ll}-2\big{(}\arcsin\frac{1}{2\sqrt{k_{i}}}\big{)}^{2}\,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\textrm{for}\,k_{i}\geq\frac{1}{4}\,,\\\
\frac{1}{2}\Big{[}\ln\big{(}\frac{1+\sqrt{1-4k_{i}}}{1-\sqrt{1-4k_{i}}}\big{)}+i\pi\Big{]}^{2}\,~{}~{}~{}~{}~{}~{}\textrm{for}\,k_{i}<\frac{1}{4}\,.\end{array}\right.$

Figure 4:  Constraints on the NMSSM parameter space through ${\cal
B}(\pi^{0}{\to}e^{+}e^{-})$, ${\cal B}(\Upsilon(1S)\to\gamma A_{1}^{0})$,
${\cal B}(\Upsilon(3S)\to\gamma A_{1}^{0})$ and $a_{\mu}$ respectively. The
shaded regions are allowed by the labeled processes.

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 ${\cal
B}(\pi^{0}{\to}e^{+}e^{-})$, which show that $\pi^{0}{\to}e^{+}e^{-}$ 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 $\Upsilon$ decays
and $a_{\mu}$ simultaneously.

To include the $a_{\mu}$ constraint, we use the experimental result that [32]
$a_{\mu}(Exp)=(11659208.0\pm 6.3)\times 10^{-10}$ and the SM prediction [33]
$a_{\mu}(SM)=(11659177.8\pm 6.1)\times 10^{-10}$. The discrepancy is

$\triangle\,a_{\mu}=a_{\mu}(Exp)-a_{\mu}(SM)=(30.2\pm 8.8)\times
10^{-10}(3.4\sigma)$ (23)

which is established at a $3.4\sigma$ level of significance.

The contributions of $A_{1}^{0}$ to $a_{\mu}$ are given by [34]

$\displaystyle{\delta}a_{\mu}({A_{1}^{0}})$ $\displaystyle=$
$\displaystyle{\delta}a_{\mu}^{1-loop}({A_{1}^{0}})+{\delta}a_{\mu}^{2-loop}({A_{1}^{0}})\,,$
(24) $\displaystyle{\delta}a_{\mu}^{1-loop}({A_{1}^{0}})\,$ $\displaystyle=$
$\displaystyle\,-\sqrt{2}G_{F}\frac{m_{\mu}^{2}}{8\pi^{2}}|X_{d}|^{2}f_{1}\big{(}\frac{m_{A_{1}^{0}}^{2}}{m_{\mu}^{2}}\big{)}\,,$
$\displaystyle{\delta}a_{\mu}^{2-loop}({A_{1}^{0}})\,$ $\displaystyle=$
$\displaystyle\,\sqrt{2}G_{F}\alpha\frac{m_{\mu}^{2}}{8\pi^{3}}|X_{d}|^{2}\Big{[}\frac{4}{3}\frac{1}{\tan^{2}\beta}f_{2}\big{(}\frac{m_{t}^{2}}{m_{A_{1}^{0}}^{2}}\big{)}+\frac{1}{3}f_{2}\big{(}\frac{m_{b}^{2}}{m_{A_{1}^{0}}^{2}}\big{)}+f_{2}\big{(}\frac{m_{\tau}^{2}}{m_{A_{1}^{0}}^{2}}\big{)}\Big{]}$

with

$\displaystyle f_{1}(z)$ $\displaystyle=$
$\displaystyle\int_{0}^{1}dx\frac{x^{3}}{z(1-x)+x^{2}}\,,$ (25) $\displaystyle
f_{2}(z)$ $\displaystyle=$ $\displaystyle
z\int_{0}^{1}dx\frac{1}{x(1-x)-z}\ln\frac{x(1-x)}{z}.$

It has been found that the $A^{0}_{1}$ contribution is always negative at the
one loop level and worsens the discrepancy in $a_{\mu}$; however, it could be
positive and dominated by the two loop contribution for $A^{0}_{1}>3{\rm
GeV}$[34]. One should note that there are other contributions to $a_{\mu}$ in
NMSSM; for instance, the chargino/sneutino and neutralino/smuon loops.
Moreover, the discrepancy $\triangle a_{\mu}$ could be resolved without
pseudoscalars [34]. So, putting a constraint on $|X_{d}|$ via $a_{\mu}$ is a
rather model-dependent process. There are two approximations with different
emphases on the role of $A^{0}_{1}$; namely, (i) assuming that $\triangle
a_{\mu}$ is resolved by other contributions and requiring that $A_{1}^{0}$
contributions are smaller than the $1\sigma$ error-bar of the experimental
measurement, and (ii) assuming that the $A_{1}^{0}$ contributions are solely
responsible for $\triangle a_{\mu}$. In Ref. [26], approximation (i) has been
used to derive an upper bound of $|X_{d}|<1.2$. We present the $a_{\mu}$
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 $a_{\mu}$ and ${\cal B}(\pi^{0}{\to}e^{+}e^{-})$: one
is for $m_{A_{1}^{0}}\sim 3{\rm~{}GeV}$ with $|X_{d}|>150$ and another one is
for $m_{A_{1}^{0}}\sim 135{\rm~{}MeV}$ with $|X_{d}|<1$.

In the searches for $\Upsilon\to\gamma A^{0}_{1}$ decays, CLEO [20] obtains
the upper limits for the product of ${\cal B}(\Upsilon(1S)\to\gamma
A^{0}_{1})$ and ${\cal B}(A^{0}_{1}\to\tau^{+}\tau^{-})$ or ${\cal
B}(A^{0}_{1}\to\mu^{+}\mu^{-})$, while BaBar presents upper limits on ${\cal
B}(\Upsilon(3S)\to\gamma A^{0}_{1})\times{\cal B}(A^{0}_{1}\to invisible)$.
All these limits fluctuate with the mass of $A^{0}_{1}$ frequently. For
simplicity, we take the loosest upper limit ${\cal B}(\Upsilon(1S)\to\gamma
A^{0}_{1})\times{\cal B}(A^{0}_{1}\to\tau^{+}\tau^{-})<6\times 10^{-5}$ of
CLEO and assume ${\cal B}(A^{0}_{1}\to\tau^{+}\tau^{-})=1$. Similarly, we also
use the loosest upper limits on ${\cal B}(\Upsilon(3S)\to\gamma
A^{0}_{1})\times{\cal B}(A^{0}_{1}\to invisible)<3.1\times 10^{-5}$ of BaBar
[21] and assume ${\cal B}(A^{0}_{1}\to invisible)=1$. With the loosest upper
limits, we get their bounds on the $|X_{d}|-m_{A^{0}_{1}}$ space, which are
shown in Fig. 4. From the figure, we can see the bounds (excluding the
parameter space $X_{d}>1$) for $0<m_{A^{0}_{1}}<7.8{\rm~{}GeV}$. Fig. 4(b)
shows that there is no region of parameter space satisfying all the
aforementioned constraints if the contribution of $A^{0}_{1}$ is required to
solely resolve the $a_{\mu}$ discrepancy.

Of particular interest, as shown in Fig. 4(a), is the parameter space around
$m_{A_{1}^{0}}\sim 135{\rm~{}MeV}$ 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 Fig. 5 for the value
$m_{A_{1}^{0}}\sim 135{\rm~{}MeV}$: ${\cal B}(\Upsilon(3S)\to\gamma
A_{1}^{0})\times{\cal B}(A_{1}^{0}\to invisible)\lesssim 3.3\times 10^{-6}$.
With the assumption that ${\cal B}(A_{1}^{0}\to invisible)\simeq 1$ and the
constraints from ${\cal B}(\pi^{0}{\to}e^{+}e^{-})$, we get

$|X_{d}|=0.10\pm 0.08,~{}~{}~{}~{}m_{A_{1}^{0}}=134.99\pm 0.01\,{\rm MeV},$
(26)

where the constraint on $m_{A_{1}^{0}}$ is dominated by ${\cal
B}(\pi^{0}{\to}e^{+}e^{-})$ and the limit of $|X_{d}|$ is dominated by ${\cal
B}(\Upsilon(3S)\to\gamma A_{1}^{0})$. At first sight, the uncertainties in the
above mentioned two parameters are too different. We find that the difference
arises from our assumption $\Gamma(A_{1}^{0})\simeq\Gamma(A_{1}^{0}\to
e^{+}e^{-})+\Gamma(A_{1}^{0}\to\gamma\gamma)$. From Eqs. (20) and (21), one
can see that the $X_{d}^{2}$ factor in ${\cal M}_{A_{1}^{0}}$ could be
canceled out by the one in $\Gamma(A_{1}^{0})$ when $m_{A_{1}^{0}}$ approaches
$m_{\pi^{0}}$, which results in a very sharp peak for position of
$m_{A_{1}^{0}}$. Thus, with the well measured quantities given in Eq. (20) and
the sensitivity of the peak, $m_{A_{1}^{0}}$ turns out to be well-constrained.
Furthermore, if we take $m_{A_{1}^{0}}=m_{\pi^{0}}$, we find that $X_{d}^{2}$
is canceled out exactly, so there is no parameter to tune; however, we have
${\cal B}(\pi^{0}\to e^{+}e^{-})\gg 1$, which violates the unitary bound and
is thus excluded.

From the results of Eq.26, we obtain ${\delta}a_{\mu}({A_{1}^{0}})=(-9.2\pm
8.9)\times 10^{-12}$ with $\tan\beta=30$ as a benchmark, which is small enough
to be smeared by the chargino/sneutrino and neutralino/smuon contributions.
Moreover, we have

$\Gamma(A_{1}^{0})=(5.7\,\pm\,5.5)\times 10^{-13}\,{\rm MeV},$ (27)

which corresponds to $\tau(A_{1}^{0})\sim 1.2\times 10^{-9}~{}s$ ($c\tau\sim
36{\rm~{}cm}$).

Figure 5: $\sin 2\theta$ versus the mass difference of the unmixed states with
$|X_{d}|=0.05$ and $0.18$. The solid and the dashed lines denote the real and
the imaginary parts of $\sin 2\theta$, respectively.

For the case where $A_{1}^{0}$ decays mostly to invisible particles, we take
the width of $A_{1}^{0}$ as a free parameter and get $\Gamma(A_{1}^{0})\leq
8.24\times\,10^{-6}{\rm GeV}$, $m_{A_{1}^{0}}=134.99\pm 0.02{\rm~{}MeV}$ and
$|X_{d}|\leq 0.18$. In this case, $m_{A_{1}^{0}}$ can equal $m_{\pi^{0}}$, and
it is found that $\Gamma(A_{1}^{0})\leq 3.3\times 10^{-6}{\rm~{}GeV}$ and
$|X_{d}|\leq 0.18$.
III. The resonant effects of $m_{A^{0}_{1}}\sim m_{\pi^{0}}$

So far we have included only the width effects of $A^{0}_{1}$ with the Breit-
Wigner formula for the propagator of $A^{0}_{1}$. When the masses of
$A_{1}^{0}$ and $\pi^{0}$ are very close, the mixing between the two states
could modify the parton level $\pi^{0}-A^{0}_{1}$ coupling. In a manner
analogous to Ref.[35], the mixing can be described by introducing off-diagonal
elements in the $A_{1}^{0}-\pi^{0}$ mass matrix

$\displaystyle{\cal
M}^{2}=\left(\begin{array}[]{cc}m_{A_{1}^{0}}^{2}-im_{A_{1}^{0}}\Gamma_{A_{1}^{0}}&\delta{m^{2}}\\\
\delta{m^{2}}&m_{\pi^{0}}^{2}-im_{\pi^{0}}\Gamma_{\pi^{0}}\end{array}\right)$
(30)

with $\delta{m^{2}}=\sqrt{G_{F}/4\sqrt{2}}f_{\pi^{0}}m_{\pi^{0}}^{2}X_{d}$.
The complex mixing angle $\theta$ between the states is given by

$\sin^{2}2\theta=\frac{(\delta
m^{2})^{2}}{\frac{1}{4}(m_{A_{1}^{0}}^{2}-m_{\pi^{0}}^{2}-im_{A_{1}^{0}}\Gamma_{A_{1}^{0}}+im_{\pi^{0}}\Gamma_{\pi^{0}})^{2}+(\delta
m^{2})^{2}}.$ (31)

The mass eigenstates $A_{1}^{\prime 0}$ and $\pi^{\prime 0}$ are obtained as

$\displaystyle A_{1}^{\prime 0}$ $\displaystyle=$
$\displaystyle\frac{1}{N}(A_{1}^{0}\cos\theta+\pi^{0}\sin\theta),$ (32)
$\displaystyle\pi^{\prime 0}$ $\displaystyle=$
$\displaystyle\frac{1}{N}(-A_{1}^{0}\sin\theta+\pi^{0}\cos\theta),$ (33)

where $N=\sqrt{|\sin\theta|^{2}+|\cos\theta|^{2}}$. Then, we can write the
decay amplitude of the “physical” state $\pi^{\prime 0}$ as

$|{\cal M}(\pi^{\prime 0}\to
e^{+}e^{-})|^{2}\,=\,\frac{1}{N^{2}}\Big{(}|\cos\theta|^{2}|{\cal
M}(\pi^{0}\to e^{+}e^{-})|^{2}+|\sin\theta|^{2}|{\cal M}(A_{1}^{0}\to
e^{+}e^{-})|^{2}\Big{)}.$ (34)

Obviously, we obtain the SM result when $\theta$ is small.

With $|X_{d}|=0.05$ and $0.18$, Fig. 5 shows $\sin 2\theta$ as a function of
the difference between $m_{A_{1}^{0}}$ and $m_{\pi^{0}}$. We note that the
imaginary part of $\sin 2\theta$ is negligibly small, since
$\Gamma_{A^{0}_{1}}m_{A^{0}_{1}}+\Gamma_{\pi^{0}}m_{\pi^{0}}\ll\delta m^{2}$.
So, the normalization parameter $N$ of the mixing states is nearly unity.
Combining the constraints from ${\cal B}(\Upsilon(3S)\to\gamma A_{1}^{0})$ and
${\cal B}(\pi^{\prime 0}\to e^{+}e^{-})$, we get

$|X_{d}|=0.17\pm 0.01,~{}~{}~{}~{}m_{A_{1}^{0}}\simeq m_{\pi^{0}}\,.$ (35)

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

$\Gamma(A_{1}^{0})=(9.8\,\pm\,1.1)\times 10^{-13}\,{\rm MeV},$ (36)

which is also in agreement with Eq. (27). Furthermore, we get
$|\sin\theta|^{2}=0.31\pm 0.19$.

It is well known that the decay width of $\pi^{0}\to\gamma\gamma$ agrees
perfectly with the SM prediction, so it is doubtful that that
$\pi^{0}\to\gamma\gamma$ would be compatible with Higgs with a degenerate mass
$m_{\pi^{0}}$. Using the fitted result $|\sin\theta|^{2}=0.31\pm 0.19$ and

$|{\cal M}(\pi^{\prime
0}\to\gamma\gamma)|^{2}\,=\,\frac{1}{N^{2}}\Big{(}|\cos\theta|^{2}|{\cal
M}(\pi^{0}\to\gamma\gamma)|^{2}+|\sin\theta|^{2}|{\cal
M}(A_{1}^{0}\to\gamma\gamma)|^{2}\Big{)},$ (37)

one can easily observe that

$|{\cal M}(A_{1}^{0}\to\gamma\gamma)|^{2}\simeq|{\cal
M}(\pi^{0}\to\gamma\gamma)|^{2}$ (38)

is needed to give
$\Gamma(\pi^{\prime}\to\gamma\gamma)\simeq\Gamma(\pi^{0}\to\gamma\gamma)$.
However, it would require a too large value of $|X_{d}|\simeq 10^{3}$;
therefore, the degenerate case is excluded.

##  4 Conclusion

We have studied the decay $\pi^{0}\to e^{+}e^{-}$ 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 ${\cal B}(\pi^{0}\,{\to}\,e^{+}e^{-})$
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 $\Upsilon$ decays performed at CLEO [20] and BaBar [21]. Combining
all these constraints, we have found that

  * •

${\cal B}(\pi^{0}\,{\to}\,e^{+}e^{-})$ and ${\cal B}(\Upsilon\to\gamma
A_{1}^{0})$ put strong constraints on the NMSSM parameter $X_{d}$ and
$m_{A_{1}^{0}}$. Due to their different dependences on the two parameters, the
interesting scenario where $m_{A^{0}_{1}}=214.3{\rm~{}MeV}$ is excluded, which
would invalidate the $A_{1}^{0}$ hypothesis for the three HyperCP events [25].

  * •

Although these constraints point to a pseudoscalar with $m_{A_{1}^{0}}\sim
m_{\pi^{0}}$ and $|X_{d}|=0.10\pm 0.08$ ($0.17\pm 0.01$, $\pi^{0}-A_{1}^{0}$
mixing included) in the NMSSM, such an $m_{A_{1}^{0}}$ is excluded by
$\pi^{0}\to\gamma\gamma$ decay.

In this Letter, we have worked in the limit of $X_{d}\gg X_{u}$, i.e., the
large $\tan\beta$ limit. If we relax the limit and take Eq.5 as a general
parameterization of the couplings between a pseudoscalar and fermions, the
$\bar{u}-u-A_{1}^{0}$ coupling should be included. However, its contribution
is deconstructive to the contributions from $X_{d}$, since the $\pi^{0}$
flavor structure is $(u{\bar{u}}-d{\bar{d}})$. To give a result in agreement
with the KTeV Collaboration measurement [16], $X_{u}\gg X_{d}$ would be
needed, which would imply possible large effects in $\Psi(1S)$ 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 $m_{A_{1}^{0}}<7.8{\rm GeV}$ in the large $\tan\beta$ limit that
is consistent with the experimental constraints. The HyperCP $214.3{\rm MeV}$
resonance and the possible $3.3\sigma$ discrepancy in $\pi^{0}\to e^{+}e^{-}$
decay are still unsolved. Finally, further theoretical investigation is also
needed to confirm the discrepancy between the KTeV measurements and SM
predications of $\pi^{0}\to e^{+}e^{-}$ decay. If the discrepancy still
persists, it would be an important testing ground for NP scenarios with a
light pseudoscalar boson.

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

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