# S2-MLPv2: Improved Spatial-Shift MLP Architecture for Vision

Tan Yu, Xu Li, Yunfeng Cai, Mingming Sun, Ping Li

Cognitive Computing Lab
Baidu Research
10900 NE 8th St. Bellevue, Washington 98004, USA
No.10 Xibeiwang East Road, Beijing 100193, China
{tanyu01, lixu13, caiyunfeng, sunmingming01<EMAIL_ADDRESS>

###### Abstract

Recently, MLP-based vision backbones emerge. MLP-based vision architectures
with less inductive bias achieve competitive performance in image recognition
compared with CNNs and vision Transformers. Among them, spatial-shift MLP
(S2-MLP), adopting the straightforward spatial-shift operation, achieves
better performance than the pioneering works including MLP-mixer and ResMLP.
More recently, using smaller patches with a pyramid structure, Vision
Permutator (ViP) and Global Filter Network (GFNet) achieve better performance
than S2-MLP. In this paper, we improve the S2-MLP vision backbone. We expand
the feature map along the channel dimension and split the expanded feature map
into several parts. We conduct different spatial-shift operations on split
parts. Meanwhile, we exploit the split-attention operation to fuse these split
parts. Moreover, like the counterparts, we adopt smaller-scale patches and use
a pyramid structure for boosting the image recognition accuracy. We term the
improved spatial-shift MLP vision backbone as S2-MLPv2. Using 55M parameters,
our medium-scale model, S2-MLPv2-Medium achieves an $83.6\%$ top-1 accuracy on
the ImageNet-1K benchmark using $224\times 224$ images without self-attention
and external training data.

##  1 Introduction

Recently, extensive studies on computer vision are conducted to achieve high
performance with less inductive bias. Two types of architectures emerge
including vision Transformers (Dosovitskiy et al., 2021; Touvron et al., 2020)
and MLP-based backbones (Tolstikhin et al., 2021; Touvron et al., 2021a).
Compared with _de facto_ vision backbone CNN (He et al., 2016) with delicately
devised convolution kernels, both vision Transformers and MLP-based backbones
have achieved competitive performance in image recognition without expensive
hand-crafted design. Specifically, vision Transformer models stack a series of
Transformer blocks, achieving the global reception field.

MLP-based methods such as MLP-Mixer (Tolstikhin et al., 2021) and ResMLP
(Touvron et al., 2021a) achieve the communication between patches through
projections along different patches implemented by MLP. Different from MLP-
Mixer and ResMLP, spatial-shift MLP (S2-MLP) (Yu et al., 2021b) adopts a very
straightforward operation, spatial shifting, for communications between
patches, achieving higher image recognition accuracy on ImageNet1K dataset
without external training data. In parallel, Vision Permutator (ViP) (Hou et
al., 2021) encodes the feature representation along the height and width
dimensions and meanwhile exploits the finer patch size with a two-level
pyramid structure, achieving better performance than S2-MLP. CCS-MLP (Yu et
al., 2021a) devises a circulant token-mixing MLP for achieving the
translation-invariance property. Global Filter Networks (GFNet) (Rao et al.,
2021b) exploits 2D Fourier Transform to map the spatial patch features into
the frequency domain and conducts the cross-patch communications in the
frequency domain. As pointed by Rao et al. (2021b), the token-mixing operation
in the frequency domain is equivalent to depthwise convolution with circulant
weights. To achieve a high recognition accuracy, GFNet also utilizes patches
of smaller size with a pyramid structure. More recently, AS-MLP (Lian et al.,
2021) axially shifts channels of the feature map and devises a four-level
pyramid, achieving excellent performance. In parallel, Cycle-MLP (Chen et al.,
2021a) devises several pseudo-kernels for spatial projection and also achieves
outstanding performance. It is worth noting that, both AS-MLP (Lian et al.,
2021) and Cycle-MLP (Chen et al., 2021a) are based on the well-devised four-
level pyramid.

(a) Spatial-shift operations in S2-MLP (b) Spatial-shift operations in
S2-MLPv2

Figure 1: Comparisons between the spatial-shift operations in S2-MLP (Yu et
al., 2021b) and the proposed S2-MLPv2. In S2-MLP, the channels are equally
divided into four parts, and each part shifts along different directions. An
MLP is conducted on the shifted channels. In contrast, in S2-MLPv2, the
$c$-channel feature map is expanded into the $3c$-channel feature map. Then
the expanded map is equally split into three parts along the channel
dimension. For each part, we conduct different spatial-shift operations. Then
the shifted parts are merged through the split-attention operation (Zhang et
al., 2020) to generate the $c$-channel feature map.

In this work, we rethink the design of spatial-shift MLP (S2-MLP) (Yu et al.,
2021b) and propose an improved spatial-shift MLP (S2-MLPv2). Compared with the
original S2-MLP, the modifications are mainly conducted on two aspects:

  * •

As visualized in Figure 1 (b), we expand the feature map along the channel
dimension and split the expanded feature map into multiple parts. For
different parts, we conduct different spatial-shift operations. We exploit the
split-attention operation (Zhang et al., 2020) to fuse these split parts.

  * •

We adopt smaller-scale patches and the hierarchical pyramid structure like
existing MLP-based architectures such as ViP (Hou et al., 2021), GFNet (Rao et
al., 2021b), AS-MLP (Lian et al., 2021) and Cycle-MLP (Chen et al., 2021a).

We term the improved spatial-shift MLP architecture as S2-MLPv2. We visualize
the difference between the original spatial-shift MLP (S2-MLP) and the
improved verision, S2-MLPv2, in Figure 1. Our experiments conduct on the
public benchmark, ImageNet-1K, demonstrates the state-of-the-art image
recognition accuracy of the proposed S2-MLPv2. Specifically, using 55M
parameters, our medium-scale model, S2-MLPv2-Medium achieves $83.6\%$ top-1
accuracy using $224\times 224$ images without self-attention and external
training data.

##  2 Related Work

vision Transformer. vision Transformer (ViT) (Dosovitskiy et al., 2021) crops
an image into $16\times 16$ patches, and treat each patch as a token in the
input of Transformer. These patches/tokens are processed by a stack of
Transformer layers for communications with each other. It has achieved
competitive image recognition accuracy as CNNs using huge-scale pre-training
datasets. DeiT (Touvron et al., 2020) adopts more advanced optimizer as well
as data augmentation methods, achieving promising results using medium-scale
pre-training datasets. Pyramid vision transformer (PvT) (Wang et al., 2021b)
and PiT (Heo et al., 2021) exploit a pyramid structure which gradually shrinks
the spatial dimension and expands the hidden size, achieving better
performance. Tokens-to-Token (T2T) (Yuan et al., 2021) and Transformer-in-
Transformer (TNT) (Han et al., 2021) improve the effectiveness of modeling the
local structure of each patch/token. To overcome the inefficiency of the
global self-attention, Swin (Liu et al., 2021b) conducts the self-attention
within local windows but achieves the global reception field through shifting
the window settings. Shuffle Transformer (Huang et al., 2021) also exploits
the local self-attention windows and achieves the cross-window communications
through switching the spatial dimension and the feature dimension. Twins (Chu
et al., 2021a) enhances the self-attention within local windows by the global
sub-sampled attention. DynamicViT (Rao et al., 2021a) and SViTE (Chen et al.,
2021b) exploit the sparsity for achieving high efficiency. CaiT (Touvron et
al., 2021b) explores the extremely deep architecture by stacking tens of
layers. Recently, PVTv2 (Wang et al., 2021a) improves PvT using overlapping
patch embedding, convolutional feedforward networks, and linear-complexity
attention layers. CSwin Transformer (Dong et al., 2021) improves Swin through
cross-shaped windows computing self-attention in the horizontal and vertical
stripes in parallel. Focal Transformer (Yang et al., 2021) also develops more
advanced local windows which attend fine-grain tokens only locally, but the
summarized ones globally.

MLP-based architectures. MLP-mixer (Tolstikhin et al., 2021) is the pioneering
work for MLP-based vision backbone. It proposes a token-mixing MLP consisting
of two fully-connected layers for communications between patches. Res-MLP
(Touvron et al., 2021a) simplifies the token-mixing MLP to a single fully-
connected layer and explores the deeper architecture with more layers.
Spatial-shift MLP backbone (S2-MLP) (Yu et al., 2021b) adopts the spatial-
shift operation for cross-patch communications. Vision Permutator (ViP) (Hou
et al., 2021) mixes tokens along the height dimension and the width dimension,
separately. Meanwhile, ViP adopts a pyramid structure as PvT (Wang et al.,
2021b) and achieves considerably better performance than MLP-mixer, Res-MLP
and S2-MLP. CCS-MLP (Yu et al., 2021a) rethinks the design of token-mixing MLP
in MLP-mixer and Res-MLP, and propose a circulant channel-specific MLP.
Specifically, they devise the weight matrix of token-mixing MLP as a circulant
matrix, taking fewer parameters. Meanwhile, the multiplication between vector
and circulant matrix can be efficiently computed through Fast Fourier
Transform (FFT). Global Filter Network (GFNet) (Rao et al., 2021b) maps the
patch features to the frequency domain through 2D FFT and mixes the tokens in
the frequency domain. As proved by Rao et al. (2021b), the global filter in
GFNet is equivalent to a depthwise global circular convolution with the filter
size H $\times$ W. Meanwhile, GFNet also exploits pyramid structure for
boosting the recognition accuracy. In this work, we rethink the design of
S2-MLP and considerably improves its performance in image recognition.

##  3 Preliminary

###  3.1 Spatial-shift MLP (S2-MLP)

In this section, we briefly review the structure of S2-MLP (Yu et al., 2021b)
architecture. It consists of the patch embedding layer, a stack of S2-MLP
blocks and the classification head.

Patch embedding layer. It first crops an image of $W\times H\times 3$ size
into $w\times h$ patches. Each patch is of $p\times p\times 3$ size and
$p=\frac{W}{w}=\frac{H}{h}$. It then maps each patch into a $d$-dimensional
vector through a fully-connected layer.

Figure 2: The structure of an S2-MLP block.

Spatial-shift MLP block. As visualized in Figure 2, it consists of four MLP
layers for mixing channels and a spatial shift layer for mixing patches. Below
we only introduce the spatial-shift module. Given an input tensor
$\mathcal{X}\in\mathbb{R}^{w\times h\times c}$, it first equally splits
$\bm{\mathcal{X}}$ into four parts $\\{\bm{\mathcal{X}}_{i}\\}_{i=1}^{4}$
along the channel dimension and shifts them along four directions:

$\begin{split}&\bm{\mathcal{X}}[2:h,:,1:c/4]\leftarrow\bm{\mathcal{X}}[1:h-1,:,1:c/4],\\\
&\bm{\mathcal{X}}[1:h-1,:,c/4+1:c/2]\leftarrow\bm{\mathcal{X}}[2:h,:,c/4+1:c/2],\\\
&\bm{\mathcal{X}}[:,2:w,c/2:3c/4]\leftarrow\bm{\mathcal{X}}[:,1:w-1,c/2:3c/4],\\\
&\bm{\mathcal{X}}[:,1:w-1,3c/4:c]\leftarrow\bm{\mathcal{X}}[:,2:w,3c/4:c].\end{split}$
(1)

It is worth noting that, S2-MLP (Yu et al., 2021b) stacks $N$ Spatial-shift
MLP blocks with the same settings and does not exploit pyramid structure as
its MLP-backone counterparts such as Vision Permutator (Hou et al., 2021) and
Global Filter Network (GFNet) (Rao et al., 2021b).

###  3.2 Split Attention

Vision Permutator (Hou et al., 2021) adopts split attention proposed in
ResNeSt (Zhang et al., 2020) for enhancing multiple feature maps from
different operations. Specifically, we denote $K$ features maps of the same
size $n\times c$ by $[\mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{K}]$
where $n$ is the number of patches and $c$ is the number of channels, the
split-attention operation first averages them and obtains

$\mathbf{a}=\sum_{k=1}^{K}\mathbf{1}\mathbf{X}_{k},$ (2)

where $\mathbf{1}\in\mathbb{R}^{n}$ is the $n$-dimensional row vector with all
$1$s. Then $\mathbf{a}\in\mathbb{R}^{c}$ goes through a stack of MLPs and
generates

$\hat{\mathbf{a}}=\sigma(\mathbf{a}\mathbf{W}_{1})\mathbf{W}_{2},$ (3)

where $\sigma$ is the activation function implemented by GELU,
$\mathbf{W}_{1}\in\mathbb{R}^{c\times\bar{c}}$ and
$\mathbf{W}_{2}\in\mathbb{R}^{\bar{c}\times Kc}$ are weights of MLPs and the
output $\hat{\mathbf{a}}\in\mathbb{R}^{Kc}$. Then $\hat{\mathbf{A}}$ is
reshaped into a matrix $\hat{\mathbf{{A}}}\in\mathbb{R}^{K\times c}$, which is
further processed by a softmax function along the first dimension and
generates
$\bar{\mathbf{{A}}}=\mathrm{softmax}(\hat{\mathbf{A}})\in\mathbb{R}^{K\times
c}$. Then it generates the attended feature map $\hat{\mathbf{X}}$ where each
row of $\hat{\mathbf{X}}$, $\hat{\mathbf{X}}[i,:]$, is computed by

$\hat{\mathbf{X}}[i,:]=\sum_{k=1}^{K}\mathbf{X}_{k}[i,:]\odot\bar{\mathbf{A}}[k,:],$
(4)

where $\odot$ denotes the element-wise multiplication between two vectors.

##  4  S2-MLPv2

In this section, we introduce the proposed S2-MLPv2 architecture. Similar to
S2-MLP backbone, S2-MLPv2 backbone consists of the patch embedding layer, a
stack of S2-MLPv2 blocks and the classification head. Since we have introduced
the patch embedding layer in the previous section, we only introduce the
proposed S2-MLPv2 block below.

###  4.1 S2v2 block

Figure 3: The structure of an S2-MLPv2 block.

The S2-MLPv2 block consists of two parts, the S2-MLPv2 component and the
channel-mixing MLP (CM-MLP) component. Given an input feature map
$\bm{\mathcal{X}}\in\mathbb{R}^{w\times h\times c}$, it conducts

$\begin{split}\bm{\mathcal{Y}}&=\mathrm{S}^{2}\textit{-}\mathrm{MLPv2}(\mathrm{LN}(\bm{\mathcal{X}}))+\bm{\mathcal{X}},\\\
\bm{\mathcal{Z}}&=\mathrm{CM}\textit{-}\mathrm{MLP}(\mathrm{LN}(\bm{\mathcal{Y}}))+\bm{\mathcal{Y}}.\\\
\end{split}$ (5)

The channel-mixing MLP (CM-MLP) adopts the same structure as MLP-mixer
(Tolstikhin et al., 2021) and ResMLP (Touvron et al., 2021a) and thus we skip
their details here. Below we only introduce the proposed S2-MLPv2 component in
detail.

Given an input feature map $\bm{\mathcal{X}}\in\mathbb{R}^{w\times h\times
c}$, the proposed S2-MLPv2 component first expands the channels of
$\bm{\mathcal{X}}$ from $c$ to $3c$ by an MLP:

$\hat{\bm{\mathcal{X}}}=\mathrm{MLP}_{1}(\bm{\mathcal{X}})\in\mathbb{R}^{w\times
h\times 3c}.$ (6)

Then it equally splits the expanded feature map $\hat{\bm{\mathcal{X}}}$ along
the channel dimension into three parts:

${\bm{\mathcal{X}}}_{1}=\hat{\bm{\mathcal{X}}}[:,:,1:c],{\bm{\mathcal{X}}}_{2}=\hat{\bm{\mathcal{X}}}[:,:,c+1:2c],{\bm{\mathcal{X}}}_{3}=\hat{\bm{\mathcal{X}}}[:,:,2c+1:3c].$
(7)

It shifts ${\bm{\mathcal{X}}}_{1}$ and ${\bm{\mathcal{X}}}_{2}$ through two
two spatial-shift layers $\mathrm{SS}_{1}(\cdot)$ and
$\mathrm{SS}_{2}(\cdot)$. $\mathrm{SS}_{1}(\cdot)$ conducts the same spatial-
shift operation as equation 1. In contrast, $\mathrm{SS}_{2}(\cdot)$ conducts
an asymmetric spatial-shift operation with respect to
$\mathrm{SS}_{1}(\cdot)$. To be specific, given the feature map
$\bm{\mathcal{X}}_{2}$, $\mathrm{SS}_{2}(\bm{\mathcal{X}}_{2})$ conducts:

$\begin{split}&\bm{\mathcal{X}}_{2}[:,2:w,1:c/4]\leftarrow\bm{\mathcal{X}}_{2}[:,1:w-1,1:c/4],\\\
&\bm{\mathcal{X}}_{2}[:,1:w-1,c/4+1:c/2]\leftarrow\bm{\mathcal{X}}_{2}[:,2:w,c/4+1:c/2],\\\
&\bm{\mathcal{X}}_{2}[2:h,:,c/2:3c/4]\leftarrow\bm{\mathcal{X}}_{2}[1:h-1,:,c/2:3c/4],\\\
&\bm{\mathcal{X}}_{2}[1:h-1,:,3c/4:c]\leftarrow\bm{\mathcal{X}}_{2}[2:h,:,3c/4:c].\end{split}$
(8)

Algorithm 1 Pseudocode of our S2-MLPv2 module.

⬇

def spatial_shift1(x):

b,w,h,c = x.size()

x[:,1:,:,:c/4] = x[:,:w-1,:,:c/4]

x[:,:w-1,:,c/4:c/2] = x[:,1:,:,c/4:c/2]

x[:,:,1:,c/2:c*3/4] = x[:,:,:h-1,c/2:c*3/4]

x[:,:,:h-1,3*c/4:] = x[:,:,1:,3*c/4:]

return x

def spatial_shift2(x):

b,w,h,c = x.size()

x[:,:,1:,:c/4] = x[:,:,:h-1,:c/4]

x[:,:,:h-1,c/4:c/2] = x[:,:,1:,c/4:c/2]

x[:,1:,:,c/2:c*3/4] = x[:,:w-1,:,c/2:c*3/4]

x[:,:w-1,:,3*c/4:] = x[:,1:,:,3*c/4:]

return x

class S2-MLPv2(nn.Module):

def __init__(self, channels):

super().__init__()

self.mlp1 = nn.Linear(channels,channels*3)

self.mlp2 = nn.Linear(channels,channels)

self.split_attention = SplitAttention()

def forward(self, x):

b,w,h,c = x.size()

x = self.mlp1(x)

x1 = spatial_shift1(x[:,:,:,:c/3])

x2 = spatial_shift2(x[:,:,:,c/3:c/3*2])

x3 = x[:,:,:,c/3*2:]

a = self.split_attention(x1,x2,x3)

x = self.mlp2(a)

return x

Note that we intentionally devise $\mathrm{SS}_{1}(\cdot)$ and
$\mathrm{SS}_{2}(\cdot)$ in an asymmetric structure so that they are
complementary to each other. Meanwhile, we do not shift
${\bm{\mathcal{X}}}_{3}$ and just keep it as well.

After that, $\\{\bm{\mathcal{X}}_{k}\\}_{k=1}^{3}$ are reshape into matrices
$\\{\mathbf{X}_{k}\\}_{k=1}^{3}$ where $\mathbf{X}_{k}\in\mathbb{R}^{wh\times
c}$, which are fed into a split-attention (SA) module as equation 2, equation
3, and equation 4, to generate

$\hat{\mathbf{X}}=\mathrm{SA}(\\{{\mathbf{X}}_{k}\\}_{k=1}^{3}).$ (9)

Then the attended feature map $\mathbf{A}$ is further fed into another MLP
layer for generating the output

$\bar{\mathbf{X}}=\mathrm{MLP}_{2}(\hat{\mathbf{X}}).$ (10)

The structure of the proposed S2-MLP module is visualized in Figure 3 and the
details are listed in Algorithm 1.

###  4.2 Pyramid structure

Following vision Permutator (Hou et al., 2021), we also exploit the two-level
pyramid structure to enhance the performance. To make a fair comparison with
Vision Permutator (Hou et al., 2021), we adopt the exact same pyramid
structure. The details are in Table 1. We notice that counterpart works such
as PVTv2 (Wang et al., 2021a), AS-MLP (Lian et al., 2021), and Cycle-MLP (Chen
et al., 2021a) adopt more advanced pyramid structure with smaller patches in
the early blocks. The smaller patches might be better at capturing the fine-
grained visual details and lead to higher recognition accuracy. Nevertheless,
due to the limited computing resources, it is unfeasible for us to re-
implement all these pyramid structures. Moreover, we also notice that Vision
Permutator devises a large model with considerably more parameters and FLOPs.
Nevertheless, due to the limited computing resources, the large model is not
feasible for us, either.

3 Settings |  Patch Size |  $\\#$ of Tokens |  Hidden Size |  $\\#$ of Blocks |  Patch Size |  $\\#$ of Tokens |  Hidden Size |  $\\#$ of Blocks |  Expa. Ratio
---|---|---|---|---|---|---|---|---|---
3 Small/7 | $7\times 7$ | $32^{2}$ | $192$ | $4$ | $2\times 2$ | $16^{2}$ | $384$ | $14$ | $3$
Medium/7 | $7\times 7$ | $32^{2}$ | $256$ | $7$ | $2\times 2$ | $16^{2}$ | $512$ | $17$ | $3$
3 |  |  |  |  |  |  |  |  |
Table 1: The configurations of the two-level pyramid structure used in our
S2-MLPv2. We exploit both small and medium settings, which are the exactly
same as Vision Permutator (Hou et al., 2021) fo a fair comparison. AS-MLP
(Lian et al., 2021) and Cycle-MLP (Chen et al., 2021a) adopt more advanced
four-level pyramid structure with patches of the smaller scale.

##  5 Experiments

We testify the proposed S2-MLPv2 on ImageNet-1K dataset (Deng et al., 2009).
We do not use external data for training. The implementation is based on the
PaddlePaddle deep learning platform.

Implementation details. Following DeiT (Touvron et al., 2020), we adopt AdamW
(Loshchilov & Hutter, 2019) as optimizer. We train both the small model and
the medium model using four NVIDIA A100 GPU cards. For the small model, we set
the batch size as 1024. In contrast, for the medium model, we only set the
batch size as 744 due to the GPU memory limitation of four NVIDIA A100 GPU
cards. We set the initial learning rate as 2e-3 and decay it to 1e-5 in 300
epochs using a cosine function. The weight decay rate is set to be 5e-2
following previous works (Touvron et al., 2020; Hou et al., 2021). We also
conduct warming up in the first 10 epochs following Hou et al. (2021).
Moreover, as adopted by Touvron et al. (2020); Hou et al. (2021), we conduct
multiple data augmentation methods including Rand-Augment (Cubuk et al.,
2020), Mixup (Zhang et al., 2018) and CutMix (Yun et al., 2019) and CutOut
(Zhong et al., 2020). Like DeiT (Touvron et al., 2020) and Vision Permutator
(Hou et al., 2021), we adopt exponential moving average (EMA) model (Laine &
Aila, 2016). Meanwhile, we also use label smoothing (Szegedy et al., 2016)
with a smooth ratio of $0.1$ and DropPath (Huang et al., 2016) with a drop
ratio of $0.1$ for both small and medium settings.

###  5.1 Comparisons with state-of-the-art methods

3 Model | Pyramid | Para. | FLOPs |  Train Size |  Test Size |  Top-1 Acc. ($\%$)
---|---|---|---|---|---|---
3 Small models
3 EAMLP-14 (Guo et al., 2021) |  | 30M | $-$ | 224 | 224 | $78.9$
ResMLP-S24 (Touvron et al., 2021a) |  | 30M | 6.0B | 224 | 224 | $79.4$
gMLP-S (Liu et al., 2021a) |  | 20M | 4.5B | 224 | 224 | $79.6$
GFNet-S (Rao et al., 2021b) |  | 25M | 4.5B | 224 | 224 | $80.0$
GFNet-H-S (Rao et al., 2021b) | ✓ | 32M | 4.5B | 224 | 224 | $81.5$
AS-MLP-T (Lian et al., 2021) | ✓ | 28M | 4.4B | 224 | 224 | $81.3$
CycleMLP-B2 (Chen et al., 2021a) | ✓ | 27M | 3.9B | 224 | 224 | $81.6$
ViP-Small/7 (Hou et al., 2021) | ✓ | 25M | 6.9B | 224 | 224 | $81.5$
S2-MLPv2-Small/7 (ours) | ✓ | 25M | 6.9B | 224 | 224 | $\mathbf{82.0}$
3 Medium models
3 MLP-mixer (Tolstikhin et al., 2021) |  | 59M | 11.6B | 224 | 224 | $76.4$
EAMLP-19 (Guo et al., 2021) |  | 55M | $-$ | 224 | 224 | $79.4$
S2-MLP-deep (Yu et al., 2021b) |  | 51M | 10.5B | 224 | 224 | $80.7$
CCS-MLP-36 (Yu et al., 2021a) |  | 43M | 8.9B | 224 | 224 | $80.6$
GFNet-B (Rao et al., 2021b) |  | 43M | 7.9G | 224 | 224 | $80.7$
GFNet-H-B (Rao et al., 2021b) | ✓ | 54M | 8.4B | 224 | 224 | $82.9$
AS-MLP-S (Lian et al., 2021) | ✓ | 50M | 8.5B | 224 | 224 | $83.1$
CycleMLP-B4 (Chen et al., 2021a) | ✓ | 52M | 10.1B | 224 | 224 | $83.0$
ViP-Medium/7 (Hou et al., 2021) | ✓ | 55M | 16.3B | 224 | 224 | $82.7$
S2-MLPv2-Medium/7 (ours) | ✓ | 55M | 16.3B | 224 | 224 | $\mathbf{83.6}$
3 Large models
3 CycleMLP-B5 (Chen et al., 2021a) | ✓ | 76M | 12.3B | 224 | 224 | $83.2$
AS-MLP-B (Lian et al., 2021) | ✓ | 88M | 15.2B | 224 | 224 | $83.3$
ViP-Large/7 (Hou et al., 2021) | ✓ | 88M | 24.3B | 224 | 224 | $83.2$
3 |  |  |  |  |  |

Table 2: Comparisons with MLP-like backbones on ImageNet-1K benchmark without
extra data. Our S2-MLPv2-Medium/7 achieves the state-of-the-art performance on
the benchmark among medium-scale MLP models and even outperforms the existing
large-scale MLP models. M denotes million and B denotes billion.

Comparisons with existing MLP-like methods. In Table 5.1, we compare our
S2-MLPv2 with existing MLP-like backbones including MLP-Mixer (Tolstikhin et
al., 2021), EAMLP (Guo et al., 2021), ResMLP (Touvron et al., 2021a), gMLP
(Liu et al., 2021a), S2-MLP-deep (Yu et al., 2021b), CCS-MLP (Yu et al.,
2021a), GFNet (Rao et al., 2021b), AS-MLP (Lian et al., 2021), CycleMLP (Chen
et al., 2021a) and ViP (Hou et al., 2021) on both small and base settings.
Among them, MLP-Mixer, ResMLP, gMLP, S2-MLP, CCS-MLP do not exploit the
pyramid structure, and thus they cannot achieve competitive recognition
accuracy compared with GFNet, AS-MLP, CycleMLP, and ViP, which are with the
pyramid structure as shown in Table 5.1. Meanwhile, as shown in the table, our
S2-MLPv2 consistently outperforms its counterparts in both small and medium
settings using a comparable number of parameters. Meanwhile, our medium model
performs even better than the large models of AS-MLP (Lian et al., 2021),
CycleMLP (Chen et al., 2021a) and ViP (Hou et al., 2021) with considerably
more parameters. On the other hand, we notice that both S2-MLPv2 and ViP take
more FLOPs compared with GFNet, AS-MLP, CycleMLP. This is due to that ours and
ViP use a very coarse pyramid structure, whereas GFNet, AS-MLP, CycleMLP use a
more advanced pyramid. Both S2-MLPv2 and ViP might potentially reduce FLOPs by
using a well-devised pyramid structure like GFNet, AS-MLP, CycleMLP.

Comparisons with CNNs and vision Transformers. We compare the proposed
S2-MLPv2 with CNN models including ResNet50 (He et al., 2016), RegNet
(Radosavovic et al., 2020) and EfficientNet (Tan & Le, 2019). Meanwhile, we
also compare with vision Transformer models including ViT (Dosovitskiy et al.,
2021), DeiT (Touvron et al., 2020), PVT (Wang et al., 2021b), T2T (Yuan et
al., 2021), TNT (Han et al., 2021), PiT (Heo et al., 2021), MViT (Fan et al.,
2021) CaiT (Touvron et al., 2021b), Swin (Liu et al., 2021b), Shuffle
Transformer (Huang et al., 2021), Nest Transformer (Zhang et al., 2021), Focal
Transformer (Yang et al., 2021), and CSWin (Dong et al., 2021).

As shown in Table 5.1, our S2-MLPv2-Medium achieves comparable accuracy as its
vision Transformer counterparts using fewer parameters but more FLOPs. Using a
more advanced pyramid as PvTv2, the FLOPs of our S2-MLPv2-Medium might be
reduced. Noting that, compared with vision Transformer requiring complex self-
attention operations, ours is much simpler in formulation and takes
consideraly fewer parameters, making it a competitive choice in practical
deployment.

3 Model | Scale | Top-1 ($\%$) | Params (M) | FLOPs (B)
---|---|---|---|---
3 CNN-based models
3 ResNet50 (He et al., 2016) | $224$ | $76.2$ | $25.6$ | $4.1$
RegNetY-16GF (Radosavovic et al., 2020) | $224$ | $80.4$ | $83.6$ | $15.9$
EfficientNet-B3 (Tan & Le, 2019) | $300$ | $81.6$ | $12$ | $1.8$
EfficientNet-B5 (Tan & Le, 2019) | $456$ | $84.0$ | $30$ | $9.9$
3 Transformer-based models
3 ViT-B/16∗ (Dosovitskiy et al., 2021) | $224$ | $79.7$ | $86.4$ | $17.6$
DeiT-B/16 (Touvron et al., 2020) | $224$ | $81.8$ | $86.4$ | $17.6$
PVT-L (Wang et al., 2021b) | $224$ | $82.3$ | $61.4$ | $9.8$
TNT-B (Han et al., 2021) | $224$ | $82.8$ | $65.6$ | $14.1$
T2T-24 (Yuan et al., 2021) | $224$ | $82.6$ | $65.1$ | $15.0$
CPVT-B (Chu et al., 2021b) | $224$ | $82.3$ | $88$ | $17.6$
PiT-B/16 (Heo et al., 2021) | $224$ | $82.0$ | $73.8$ | $12.5$
MViT-B-24 (Fan et al., 2021) | $224$ | $83.1$ | $53.5$ | $10.9$
CaiT-S32 (Touvron et al., 2021b) | $224$ | $83.3$ | $68$ | $13.9$
Swin-B (Liu et al., 2021b) | $224$ | $83.3$ | $88$ | $15.4$
Shuffle-B (Huang et al., 2021) | $224$ | $84.0$ | $88$ | $15.6$
Nest-B (Zhang et al., 2021) | $224$ | $83.8$ | $68$ | $17.9$
PvTv2-B4 (Wang et al., 2021a) | $224$ | $83.6$ | $62.6$ | $10.1$
Focal-Base (Yang et al., 2021) | $224$ | $83.8$ | $89.8$ | $16.0$
CSWin-B (Dong et al., 2021) | $224$ | $84.2$ | $78$ | $15.0$
3 Our models
3 S2-MLPv2-Small/7 | $224$ | $82.0$ | $25$ | $6.9$
S2-MLPv2-Medium/7 | $224$ | $83.6$ | $55$ | $16.3$
3 |  |  |  |

Table 3: Comparisons with CNN and Transformer models on ImageNet-1K without
extra data. ViT-B/16∗ denotes the result of ViT-B/16 reported by Tolstikhin et
al. (2021) with extra regularization. Compared with Transformer-based models,
our S2-MLPv2-Medium/7 model achieves comparable recognition accuracy on the
benchmark without self-attention and considerably fewer parameters.

###  5.2 Ablation studies

Influence of the pyramid structure. To evaluate the influence of the pyramid
structure on the proposed S2-MLPv2, we compare the Small/7 settings and the
Small/14 settings. The details of Small/7 settings and the Small/14 settings
are in Table 4. Both of them are the same as that in Vision Permutator (Hou et
al., 2021). Specifically, the initial patch size in Small/7 is $7\times 7$,
which is smaller than the $14\times 14$ patches in the Small/14 settings.
Intuitively, the smaller patches are beneficial to modeling fine-grained
details in the images and tend to achieve higher recognition accuracy. Table 5
compares the performance of these two settings. As shown in the table, by
utilizing the pyramid, S2-MLPv2-Small/7 achieves a considerably better
performance than S2-MLPv2-Small/14.

3 Settings |  Patch Size |  $\\#$ of Tokens |  Hidden Size |  $\\#$ of Blocks |  Patch Size |  $\\#$ of Tokens |  Hidden Size |  $\\#$ of Blocks |  Expa. Ratio
---|---|---|---|---|---|---|---|---|---
3 Small/7 | $7\times 7$ | $32^{2}$ | $192$ | $4$ | $2\times 2$ | $16^{2}$ | $384$ | $14$ | $3$
Small/14 | $14\times 14$ | $16^{2}$ | $384$ | $4$ | $2\times 2$ | $16^{2}$ | $384$ | $14$ | $3$
3 |  |  |  |  |  |  |  |  |
Table 4: The configurations of the Small/7 settings with the pyramid structure and Small/14 without the pyramid structure. 3 Settings | Pyramid |  Top-1 ($\%$) |  $\\#$ of parameters |  FLOPs
---|---|---|---|---
3 S2-MLPv2-Small/7 | ✓ | $82.0$ |  $25$M |  $6.9$B
S2-MLPv2-Small/14 |  | $80.9$ |  $30$M |  $5.7$B
3 |  |  |  |
Table 5: Comparisons between Small/7 and Small/14 settings.

Influence of the split attention. Recall from equation 9 that, we use the
split-attention (SA) for fusing the feature maps
$\mathbf{A}=\mathrm{SA}(\\{{\mathbf{X}}_{k}\\}_{k=1}^{3})$ . An alternating
fusing manner is sum-pooling them implemented by
$\mathbf{A}={\sum_{k=1}^{3}{\mathbf{X}}_{k}}/3$. We compare these two manners
in Table 6. The experiments are conducted in Small/7 settings. As shown in
Table 6, the split attention significantly outperforms sum pooling with a
slight increase in the number of parameters and FLOPs.

3 Settings |  Top-1 ($\%$) |  $\\#$ of parameters |  FLOPs
---|---|---|---
3 Sum-pooling | $79.8$ |  $22$M |  $6.9$B
Split-attention | $82.0$ |  $25$M |  $6.9$B
3 |  |  |
Table 6: Performance comparisons between split attention and sum pooling. The
experiments are conducted in Small/7 settings.

Influence of each split. As equation 9, we fuse three splits
$\\{{\mathbf{X}}_{k}\\}_{k=1}^{3}$ through the split attention. In this
section, we evaluate the influence of removing one of them. The experiments
are conducted in Small/7 settings. As shown in Table 7, when using only
${\mathbf{X}}_{1}$ and ${\mathbf{X}}_{3}$, the top-1 accuracy drops from
$82.0\%$ to $81.6\%$. Meanwhile, when removing ${\mathbf{X}}_{3}$, the top-1
accuracy decreases to $81.6\%$.

3 ${\mathbf{X}}_{1}$ | ${\mathbf{X}}_{2}$ | ${\mathbf{X}}_{3}$ |  Top-1 ($\%$) |  $\\#$ of parameters |  FLOPs
---|---|---|---|---|---
3 ✓ | ✓ | ✓ | $82.0$ |  $25$M |  $6.9$B
✓ |  | ✓ | $81.6$ |  $22$M |  $6.2$B
✓ | ✓ |  | $81.7$ |  $22$M |  $6.2$B
3 |  |  |  |  |
Table 7: The influence of each split. The experiments are conducted in Small/7
settings.

##  6 Conclusion

In this paper, we improve the spatial-shift MLP (S2-MLP) and propose an
S2-MLPv2 model. It expands the feature map and splits the expanded feature map
into three splits. It shifts each split individually and then fuses the split
feature maps through split-attention. Meanwhile, we exploit the hierarchical
pyramid to improve its capability of modeling fine-grained details for higher
recognition accuracy. Using $55$M parameters, our S2-MLPv2-Medium model
achieves $83.6\%$ top-1 accuracy on ImageNet1K dataset using $224\times 224$
images without external training datasets, which is the state-of-the-art
performance among MLP-based methods. Meanwhile, compared with Transformer-
based methods, our S2-MLPv2 model has achieved comparable accuracy without
self-attention and fewer parameters.

Compared with the pioneering MLP-based works such as MLP-mixer, ResMLP as well
as recent MLP-like models including Vision Permutator and GFNet, another
important advantage of the spatial-shift MLP is that the shapes of spatial-
shift MLPs are invariant to the input scale of images. Thus, the spatial-shift
MLP model pre-trained by images of a specific scale can be well adopted for
down-stream tasks with various-sized input images.

The future work will be devoted to continuously improving the image
recognition accuracy of the spatial-shift MLP architecture. A promising and
straightforward direction is to attempt smaller-size patches and the more
advanced four-level pyramid as CycleMLP and AS-MLP for further reducing the
FLOPs and shortening the recognition gap between the Transformer-based models.

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# Revisiting pseudo-Dirac neutrino scenario after recent solar neutrino data

Saeed Ansarifard<EMAIL_ADDRESS>Yasaman Farzan<EMAIL_ADDRESS>School of physics, Institute for Research in Fundamental Sciences (IPM)
P.O.Box 19395-5531, Tehran, Iran

###### Abstract

It is still unknown whether the mass terms for neutrinos are of Majorana type
or of Dirac type. An interesting possibility, known as pseudo-Dirac scheme
combines these two with a dominant Dirac mass term and a subdominant Majorana
one. As a result, the mass eigenstates come in pairs with a maximal mixing and
a small splitting determined by the Majorana mass. This will affect the
neutrino oscillation pattern for long baselines. We revisit this scenario
employing recent solar neutrino data, including the seasonal variation of the
7Be flux recently reported by BOREXINO. We constrain the splitting using these
data and find that both the time integrated solar neutrino data and the
seasonal variation independently point towards a new pseudo-Dirac solution
with nonzero splitting for $\nu_{2}$ of $\Delta m_{2}^{2}\simeq 1.5\times
10^{-11}$ eV2. We propose alternative methods to test this new solution. In
particular, we point out the importance of measuring the solar neutrino flux
at the intermediate energies $1.5~{}{\rm MeV}<E_{\nu}<3.5~{}{\rm MeV}$ (below
the Super-Kamiokande detection threshold) as well as a more precise
measurement of the $pep$ flux. The code is available on Github

##  I Introduction

Lepton flavor violation is the cornerstone of the modern neutrino physics,
having been observed in various neutrino experiments such as solar,
atmospheric, reactor and long baseline neutrino experiments. The three
neutrino mass and mixing scheme has been established as the standard solution
to the observed lepton flavor violation in evolution of neutrino states. It is
not however known whether the neutrino mass term also violates lepton number
or not. In other words, we do not know if the mass terms for neutrinos are of
Majorana type or of Dirac type. In general, we can simultaneously write
Majorana ($\mu$) and Dirac mass ($m$) terms for neutrinos. At the limit where
Majorana term is much smaller than the Dirac term (i.e., in the limit $\mu\ll
m$), the scheme is called pseudo-Dirac. This limit is of interest from both
model building and phenomenological point of view. It is straightforward to
show that the mass eigenstates composing the active states will split to pairs
of Majorana states with maximal mixing and tiny mass squared differences given
by $\Delta m^{2}=\mu m$. For baselines much smaller than $E_{\nu}/(2\mu m)$,
the neutrino oscillation pattern will be similar to what expected for the
standard three neutrino mass and mixing scheme. For baselines comparable to
$E_{\nu}/(2\mu m)$ or larger, the active to active neutrino oscillation
probability will be smaller than that expected within the standard scheme as a
part of the active flux can oscillate to sterile neutrinos. There is already
rich literature on the potential of various neutrino observations to test this
scenario. Upcoming terrestrial experiments such as DUNE and JUNO can test
$\Delta m^{2}\sim 10^{-5}$ eV2 [1]. The galactic supernova neutrinos can probe
$\Delta m^{2}$ down to $10^{-20}$ eV2 [2]. Ultrahigh energy cosmic neutrinos
can be sensitive to $\Delta m^{2}>{\rm few}\times 10^{-18}$ eV2 [3, 4, 5, 6,
7, 8]. Finally, the solar neutrinos can be sensitive to $\Delta m^{2}\raise
1.29167pt\hbox{$\;>$\kern-7.5pt\raise-4.73611pt\hbox{$\sim\;$}}10^{-13}$ eV2
[9].

The possible effects of pseudo-Dirac neutrino scheme on solar neutrinos has
been already discussed in the literature [9, 10]. Ref [9] constrains the
splittings of $\nu_{1}$ and $\nu_{2}$ and finds a solution at $\sim 10^{-11}$
eV2 for the neutrino data. Since the publication of Ref. [9], BOREXINO has
released more data, with a relatively precise measurement of the $pep$ flux as
well as the measurement of seasonal flux variation. Moreover, the Super-
Kamiokande data has been updated. We revisit the pseudo-Dirac scheme with the
latest available BOREXINO and Super-Kamiokande solar data, also taking into
account the precise measurement of $\Delta m_{21}^{2}$ by KamLAND. Similarly
to Ref. [9], we find a solution with nonzero pseudo-Dirac splitting. We
discuss the importance of the precise measurement of 8B flux at energies
between 1.5 MeV to 3 MeV (that is below the detection threshold of Super-
Kamiokande and above the $pep$ line) to test this solution.

In the range that the oscillation length due to the pseudo-Dirac mass
splitting, $\Delta m^{2}$, is comparable to the variation of the Earth-Sun
distance during a year (resulting from the eccentricity of the Earth orbit),
we expect a signature in the seasonal variation. We examine the recently
reported seasonal variation of the 7Be flux to search for such a variation.
Independently of the time integrated analysis, this data also points towards a
pseudo-Dirac solution with the same range of $\Delta m^{2}$. We propose a few
alternative methods to test this new non-trivial solution.

The paper is organized as follows: In sect. II, we review the oscillation of
pseudo-Dirac neutrinos. This discussion is complemented in the appendix with a
focus on matter effects as well as on the eccentricity of the Earth orbit. In
sect. III, we summarize the basis of our analysis and define the relevant
$\chi^{2}$ tests. In sect. IV, we show the implications of the solar neutrino
data for the pseudo-Dirac scheme. The concluding remarks and suggestions for
further study are given in sect. V.

##  II Oscillation of pseudo-Dirac solar neutrinos

Within the pseudo-Dirac scheme, the neutrino states $\Psi^{T}=(\nu_{L}\
\nu_{R})$ have both Dirac mass ($m$) and Majorana mass ($\mu$) terms of form

$\bar{\Psi}m\Psi+\bar{\Psi}^{c}\mu\Psi\ \ \ \ \ {\rm with}\ \mu\ll m$ (1)

where $\Psi^{c}=-i\gamma^{2}\Psi^{*}$. In general, both $m$ and $\mu$ are
$3\times 3$ matrices in the flavor space. For simplicity, we assume that $m$
and $\mu$ can be simultaneously diagonalized. Then, as shown in the appendix,
each Dirac mass eigenstate, $\nu_{i}$ splits to two Majorana states with a
maximal mixing and a splitting of $\Delta m_{i}^{2}=2\mu_{i}m_{i}$. Thus, the
$\nu_{e}$ survival probability and the probability of the conversion of
$\nu_{e}$ into sterile neutrinos can be written as

$\displaystyle P_{ee}(E_{\nu},L,r)\equiv P(\nu_{e}\to\nu_{e})=$ (2)
$\displaystyle\cos^{4}\theta_{13}\Biggl{(}\cos^{2}\theta_{12}\cos^{2}\theta_{M}$
$\displaystyle\cos^{2}(\frac{\Delta m^{2}_{1}}{4E_{\nu}}L)$
$\displaystyle+\sin^{2}\theta_{12}\sin^{2}\theta_{M}$
$\displaystyle\cos^{2}(\frac{\Delta m^{2}_{2}}{4E_{\nu}}L)\Biggl{)}$
$\displaystyle+\sin^{4}\theta_{13}\cos^{2}(\frac{\Delta
m^{2}_{3}}{4E_{\nu}}L)$

and

$\displaystyle P_{es}(E_{\nu},L,r)\equiv\sum_{i}P(\nu_{e}\to\nu_{s_{i}})$
$\displaystyle=$ (3)
$\displaystyle\cos^{2}\theta_{13}\Biggl{(}\cos^{2}\theta_{M}$
$\displaystyle\sin^{2}(\frac{\Delta m^{2}_{1}}{4E_{\nu}}L)$
$\displaystyle+\sin^{2}\theta_{M}$ $\displaystyle\sin^{2}(\frac{\Delta
m^{2}_{2}}{4E_{\nu}}L)\Biggl{)}$
$\displaystyle+\sin^{2}\theta_{13}\sin^{2}(\frac{\Delta
m^{2}_{3}}{4E_{\nu}}L)$

where $\theta_{M}$ is the effective mixing at the production point of the
$\nu_{e}$ inside the sun (at a distance of $r$ from the Sun center) given by
Eq. (22) in the appendix. $L$ is the distance between Sun and Earth and
$E_{\nu}$ is the neutrino energy.

Figure 1:  Average survival probability of solar neutrinos,
$P(\nu_{e}\to\nu_{e}$) versus neutrino energy for various values of the
splitting. Averaging is taken over the production point assuming the
distribution of 8B production. The $L$ dependency is marginalized by taking
temporal averaging on a year. (For more clarification see the bulk of the
paper and the appendix.) The black curve corresponds to the pure standard MSW
effect. The standard neutrino parameters are set equal to the best fit values
as in [11]222NuFIT 5.1 (2021), www.nu-fit.org. In the upper (lower) panel, we
have set $\Delta m_{2}^{2}=0$ ($\Delta m_{1}^{2}=0$) and only one $\Delta
m_{i}^{2}$ is set nonzero. The blue, green and red lines respectively
correspond to $\Delta m_{i}^{2}=1\times 10^{-11}$ eV2, $\Delta
m_{i}^{2}=1.5\times 10^{-11}$ eV2 and $\Delta m_{i}^{2}=3\times 10^{-11}$ eV2.

The density profile of the Sun is exponentially suppressed with the distance
from the Sun center so $P_{ee}$ strongly depends on the production point, $r$.
As explained in the appendix for each component of the solar flux component
(i.e., $j\in\\{pp,\ ^{7}{\rm Be},\ pep,\ ^{8}B\\}$) we should average
$P_{ee}(E_{\nu},L,r)$ and $P_{es}(E_{\nu},L,r)$ over the production point,
using the production point spatial distribution inside the Sun associated with
each flux component. More details can be found in the appendix. Hereafter, we
show the averaged survival probability over the production point of component
$j$ by $\bar{P}^{j}_{ee}(E_{\nu},L)$. Fig. 2 illustrates the oscillation
probability averaged with the distribution of 8B production at several values
of $\Delta m_{1}^{2}$ and $\Delta m_{2}^{2}$. The black lines correspond to
the standard MSW solution. We expect for relatively large $\Delta m_{i}^{2}$,
the deviation to be more significant for large energies and relatively
suppressed at low energies because the effect is given by the ratio $\Delta
m_{i}^{2}/E_{\nu}$. This behavior is demonstrated by the red lines which
correspond to $\Delta m_{1}^{2}$ or $\Delta m_{2}^{2}=3\times 10^{-11}$ eV2.
For $\Delta m_{i}^{2}\sim 10^{-11}$ eV2, the deviation is specially
significant for intermediate values of energies $1.5~{}{\rm
MeV}<E_{\nu}<3~{}{\rm MeV}$, lying below the detection threshold of Super-
Kamiokande where the solar neutrino data is lacking.

Figure 2:
$\Delta\bar{P}_{ee}=\bar{P}^{Be7}_{ee}(E_{\nu},L_{max})-\bar{P}^{Be7}_{ee}(E_{\nu},L_{min})$
versus $\Delta m_{1}^{2}$ and $\Delta m_{2}^{2}$ at $E_{\nu}=0.862$ MeV (i.e.,
for 7Be line). $L_{max}=1.52\times 10^{8}$ km and $L_{min}=1.47\times 10^{8}$
km are the Earth-Sun distance at aphelion and perihelion, respectively.

The average Sun Earth distance is $\sim$ 150 million km but due to the
eccentricity of the Earth orbit around the Sun, $L$ varies during a year with
$L_{max}-L_{min}=5$ million km. Fig. 2 shows

$\Delta\bar{P}_{ee}\equiv\bar{P}^{{}^{7}{\rm
Be}}_{ee}(E_{\nu},L_{max})-\bar{P}^{{}^{7}{\rm Be}}_{ee}(E_{\nu},L_{min})$

versus $\Delta m_{1}^{2}$ and $\Delta m_{2}^{2}$ at $E_{\nu}=0.862$ MeV which
is the energy of the 7Be line. As seen in these figures, for $\Delta
m^{2}>10^{-11}$ eV2, the seasonal variation can be sizable. BOREXINO has
recently published seasonal variation of the 7Be flux reaching the Earth. We
shall examine whether this piece of information can help to constrain the
parameter space of the pseudo-Dirac scheme.

##  III Analysis of the solar neutrino data

As seen in Eqs. (2,3) and Eq. (22), the solar neutrino flux on the Earth
depends on $\theta_{12},\ \Delta m_{21}^{2}$ and $\theta_{13}$ as well as on
$\Delta m_{i}^{2}$. Because of the smallness of $\sin^{2}\theta_{13}$, the
sensitivity of the solar data to $\theta_{13}$ is negligible so we fix
$\theta_{31}=8.57^{\circ}$ throughout our analysis [11]. Historically, the
solar neutrino data have provided the first measurement of $\theta_{12}$ and
$\Delta m_{21}^{2}$ within the standard neutrino mass and mixing paradigm
(i.e., setting $\Delta m_{i}^{2}=0$). These measurements were confirmed by the
KamLAND reactor experiment which measured the $\bar{\nu}_{e}$ flux from
reactors active throughout Japan. The baseline of KamLAND, $L_{Kam}$ was less
than 200 km so for the values of $\Delta m_{i}^{2}$ of our interest
($10^{-12}$ eV${}^{2}\leq\Delta m_{i}^{2}\leq 10^{-9}$ eV2), we can write
$\Delta m_{i}^{2}L_{Kam}/(2E_{\nu})\ll 1$. Thus, the neutrino oscillation in
the KamLAND experiment was sensitive only to $\theta_{12}$ and $\Delta
m_{21}^{2}$. As a result, the determination of these parameters by KamLAND is
also valid for our pseudo-Dirac scenario with $\Delta m_{i}^{2}\neq 0\ll
10^{-9}$ eV2. Indeed, the determination of $\Delta m_{21}^{2}$ by KamLAND
suffers from much smaller uncertainty than that by the solar neutrino data. We
therefore treat $\Delta m_{21}^{2}$ by a nuisance parameter with a mean value
of $\Delta\bar{m}_{21}^{2}=7.54\times 10^{-5}$ eV2 and an error
$\sigma_{\Delta m^{2}_{21}}=0.5\times 10^{-5}$ eV2 as measured by KamLAND
[12]. The effects of $\Delta m_{3}^{2}$ on $P_{ee}$ and on $P_{es}$ are
respectively suppressed by $\sin^{4}\theta_{13}$ and $\sin^{2}\theta_{13}$ so
the sensitivity to $\Delta m_{3}^{2}$ is low. We therefore set $\Delta
m_{3}^{2}=0$ and focus on the effects of nonzero $\Delta m_{1}^{2}$ and
$\Delta m_{2}^{2}$ on the solar neutrinos. We employ the latest solar data
both from Super-Kamiokande and from BOREXINO to extract information on $\Delta
m_{1}^{2}$ and $\Delta m_{2}^{2}$. We also set $\theta_{12}$ as a free
parameter to “re”-measured from the solar neutrino data in the presence of
nonzero $\Delta m_{1}^{2}$ or $\Delta m_{2}^{2}$.

We use the $\chi^{2}$ analysis in order to constrain the allowed regions for
the free parameters of the theory, separately defining $\chi^{2}$ for each
$\Delta m_{i}^{2}$ setting the rest equal to zero:

$\displaystyle\chi^{2}_{\rm min}(\Delta m_{i}^{2},\theta_{21})=$ (4)
$\displaystyle{\rm\min\limits_{nusiance}}$
$\displaystyle\left\\{\chi^{2}_{Su}+\chi^{2}_{Bo}+\left(\frac{\Delta
m_{21}^{2}-{\Delta\bar{m}_{21}^{2}}}{\sigma_{\Delta
m^{2}_{21}}}\right)^{2}\right\\}$

where $Su$ ($Bo$) indicates the Super-Kamiokande (BOREXINO) experiment.
$\chi^{2}_{Su}$ is defined as follows [13]:

$\displaystyle\chi^{2}_{Su}=\sum_{k}\left[\left(\frac{D_{k}-f_{k}(\delta_{B},\delta_{S},\delta_{R})\alpha^{B8}\mathcal{T}^{B8}_{k}}{\sigma_{k}}\right)^{2}\right]$
(5)
$\displaystyle+\left(\frac{\alpha^{B8}-1}{\sigma_{\alpha^{B8}}}\right)^{2}+\delta_{B}^{2}+$
$\displaystyle\delta_{S}^{2}+\delta_{R}^{2}$

Subscript $k$ runs over the 13 bins of 8B solar neutrino energy spectrum
starting from $3.5\ \rm MeV$ to $10\ \rm MeV$. Super-Kamiokande covers the
energy range $3.5\ \rm MeV$ to $19.5\ \rm MeV$. However, the effect of $\Delta
m_{i}^{2}$ in the range under study in this paper will be significant only at
energies lower than $10\ \rm MeV$. As a result, we do not consider the higher
energy bins and we do not therefore need to worry about the $hep$ data events.
333Regarding to day/night effect see the Appendix. Similarly for the BOREXINO
experiment we have

$\chi^{2}_{Bo}=\sum_{j}\left[\left(\frac{D^{j}-\alpha^{j}\mathcal{T}^{j}}{\sigma^{j}}\right)^{2}+\left(\frac{\alpha^{j}-1}{\sigma_{\alpha^{j}}}\right)^{2}\right]$
(6)

Superscript $j$ runs over $pp$, 7Be and $pep$ solar neutrino event rate
(counts per day per 100 t).

Figure 3: The annually averaged BOREXINO (left) and Super-K (right) data
points. Predictions for the splitting values of $\Delta m_{2}^{2}=1\times
10^{-11}$ eV2 (blue), $\Delta m_{2}^{2}=1.5\times 10^{-11}$ eV2 (green) and
$\Delta m_{2}^{2}=3\times 10^{-11}$ eV2 (red) are illustrated. The standard
MSW prediction (black) is added for comparison. We have taken $\Delta
m_{21}^{2}=7.5\times 10^{-5}$ eV2 and $\theta_{12}=33.4^{\circ}$ [11].

$D$ represent the background-subtracted measured data. $\sigma$ include both
statistical and systematic errors. Their values are taken from ref [15] and
table 1 of ref [16] for Super-kamiokande and BOREXINO respectively. While the
Super-Kamiokande data covers the solar neutrino spectrum with energies above
3.5 MeV, the BOREXINO data provides precision measurement of the low energy
part of the spectrum. The data used in Eqs. (5,6) is averaged over a year.

$\mathcal{T}$ is the prediction which will be discussed below. $\alpha^{j}$
are added as nuisance parameters to account for the flux normalization
uncertainty in the predictions of various solar neutrino components. The
uncertainty values for $pp$ and $pep$ are taken equal to $1\%$ and those for
7Be and 8B are taken equal to $6\%$ and $12\%$, respectively [17]. We also
consider the energy correlated systematic uncertainties in spectral shape,
energy scale and energy resolution of 8B by adding nuisance parameters
$\delta_{B}$,$\delta_{S}$ and $\delta_{R}$, respectively. The dependence of
$f_{k}(\delta_{B},\delta_{S},\delta_{R})$ on the nuisance parameters and
energy bins are explained in ref [18]. The prediction of the theory is derived
by

$\displaystyle\mathcal{T}^{j}_{k}(t_{0},t_{1},\Delta t)$
$\displaystyle=\frac{t_{0}}{\Delta t}\mathcal{N}_{\rm
det}\int_{t_{1}}^{t_{1}+\Delta t}dt\int dE_{\nu}\int_{0}^{T_{e}^{max}}dT_{e}$
(7) $\displaystyle R^{j}_{k}(T_{e})\phi^{j}(L)$
$\displaystyle\frac{d\lambda^{j}}{dE_{\nu}}(E_{\nu})\Biggl{[}\frac{d\sigma_{e}}{dT_{e}}(E_{\nu},T_{e})\bar{P}^{j}_{ee}(E_{\nu},L)$
$\displaystyle+\frac{d\sigma_{\mu,\tau}}{dT_{e}}$
$\displaystyle(E_{\nu},T_{e})\biggl{(}1-\bar{P}^{j}_{ee}(E_{\nu},L)-\bar{P}^{j}_{es}(E_{\nu},L)\biggl{)}\Biggl{]}.$

where $\Delta t$ is the time period over which the temporal average is taken.
For annually averaged data, $\Delta t$ should be of course taken to be a year;
then, $\mathcal{T}^{j}_{k}$ will be independent of $t_{1}$, i.e., independent
of the start of the data taking period. $t_{0}$ determines the temporal unit
of data which for the BOREXINO and Super-Kamiokande experiments are
respectively taken to be a day and a year. For the BOREXINO experiment,
$\mathcal{N}_{\rm det}=3.307\times 31\ \rm per\ 100\ ton$. For the Super-
Kamiokande experiment, $\mathcal{N}_{\rm det}=({10}/{18})({1}/{m_{p}})$ is the
number of electron at each kilo ton of the Super-Kamiokande detector and
$m_{p}$ is the mass of the proton in kilotons. $R^{j}(T_{e})$ is the detector
performance function to measure the $j$th component. $T_{e}$ is the recoil
energy of the scattered electron. The $R^{j}(T_{e})\sim 1$ for all three
components $(pp,pep,^{7}$Be) because we have used the total event rate for the
case of BOREXINO. $R_{k}^{B8}(T_{e})$ for $k$th energy bin of ${}^{8}B$
follows a Gaussian function computed in Ref. [13].

$\phi^{j}(L)$ is the solar neutrino flux normalization:

$\phi^{j}(L)\equiv\bar{\phi}^{j}\frac{\bar{L}^{2}}{L^{2}}$ (8)

$\bar{\phi}^{j}$ is solar standard model prediction [17] calculated at
$\bar{L}$, temporal average of sun to earth distance. $L$ is the Sun to Earth
distance which varies during a year due to the Earth orbit eccentricity.
$\frac{d\lambda^{j}}{dE_{\nu}}(E_{\nu})$ is the solar neutrino spectrum. For
the $pp$ and ${}^{8}B$ components which have continuous spectrum,
$\phi^{pp}(L)({d\lambda^{pp}}/{dE_{\nu}})$ and
$\phi^{B8}(L)({d\lambda^{B8}}/{dE_{\nu}})$ are in unit of $[\rm
cm^{-2}s^{-1}MeV^{-1}]$. The normalization of the monoenergetic 7Be and $pep$
fluxes are in the unit of $[\rm cm^{-2}s^{-1}]$.

The differential cross sections of the electron scattered by neutrinos of
different flavors ($e,\mu,\tau$) are [19]:

$\displaystyle\frac{d\sigma_{e(\mu,\tau)}}{dT_{e}}=$ (9)
$\displaystyle\frac{G_{F}^{2}m_{e}}{2\pi}$
$\displaystyle\Biggl{[}\biggl{(}2\sin^{2}(\theta_{W})\pm
1\biggl{)}^{2}+\biggl{(}2\sin^{2}(\theta_{W})(1-\frac{T_{e}}{E_{\nu}})\biggl{)}^{2}$
$\displaystyle-2\sin^{2}(\theta_{W})\biggl{(}2\sin^{2}(\theta_{W})\pm
1\biggl{)}\frac{m_{e}T_{e}}{E_{\nu}^{2}}\Biggl{]}$

where for $\nu_{e}$ (for $\nu_{\mu},\nu_{\tau}$) we should take the plus
(minus) sign. We take the Weinberg angle as
$\sin^{2}(\theta_{W})=0.22342$444https://pdg.lbl.gov/. The maximum recoil
energy of the electron is given by

$T_{e}^{max}=\frac{E^{2}_{\nu}}{E_{\nu}+m_{e}/2}.$

##  IV Results

In sect. IV.1, we first study how by combining the BOREXINO data on the 7Be,
$pep$ and $pp$ event rate with the Super-Kamiokande solar neutrino spectrum
data, we can constrain $\Delta m_{1}^{2}$ and $\Delta m_{2}^{2}$.
Surprisingly, we find that there is a new solution in the range of $\Delta
m_{2}^{2}=(1-2)\times 10^{-11}$ eV2. We discuss whether the measurement of the
total active neutrino fluxes by SNO or by current and future direct dark
matter search experiments can test this new solution with nonzero $\Delta
m_{2}^{2}$. In sect. IV.2, we study the effects of $\Delta m_{1}^{2}$ and
$\Delta m_{2}^{2}$ on the seasonal variation of the 7Be flux and contrast it
with the recent BOREXINO data release on the seasonal variation of the 7Be
flux on the Earth. Surprisingly, this data independently points towards the
same solution. We show that our new solution favors the GNO radio chemical
experiment and is compatible with combined experiments Gallex and GNO [21]. We
then discuss the prospect of testing this solution by a more precise
measurement of the seasonal variation of the 7Be flux.

###  IV.1 Total solar flux integrated over year(s)

In this sub-section, we analyze the time-integrated BOREXINO and Super-
Kamiokande solar neutrino data. The data points are shown in Fig. 3. The
vertical axis in the left (right) panel is the number of counts per day per
100 tons (the number of counts over MC (unoscillated) per year per kilo ton).
The prediction of the pseudo-Dirac scenario with $\Delta
m_{2}^{2}=(1,1.5,3)\times 10^{-11}$ eV2 are also shown. To obtain these
predictions, we have set $\Delta m_{1}^{2}=0$ and have used Eq. (7). The
standard MSW scheme (i.e, $\Delta m_{1}^{2}=\Delta m_{2}^{2}=0$) is added for
comparison. As seen from the figure, large values of the splitting such as
$\Delta m_{2}^{2}\sim 3\times 10^{-11}$ eV2 can be ruled out by the ${}^{8}B$
data points. Although the range few$\times 10^{-12}$ eV${}^{2}<\Delta
m_{2}^{2}<10^{-11}$ eV2 is consistent with the Super-Kamiokande data, it is
located out of one sigma error of the precise 7Be line measurement by
BOREXINO. As demonstrated by the green curve and triangle, The $\Delta
m_{2}^{2}=1.5\times 10^{-11}$ eV2 also gives a good fit to 8B data points as
well as to the BOREXINO data points. Ref. [9] had also found this solution
with $\Delta m_{2}^{2}\sim 10^{-11}$ eV2. Our results with updated solar data
[16] which includes the relatively precise pep line measurement confirms their
finding. Notice that the prediction with $\Delta m_{2}^{2}=1.5\times 10^{-11}$
eV2 for the $pep$ line is smaller than that with $\Delta m_{2}^{2}=0$.
Improving the precision of the $pep$ line can therefore test this non-trivial
solution.

Figure 4: $\Delta\chi^{2}$ as a function of $\Delta m_{i}^{2}$ minimized over
$\theta_{12}$. The dashed and dotted horizontal lines respectively correspond
to the $68\%$ and $95.45\%$ confidence levels.

Fig. 4 shows $\Delta\chi^{2}$ versus $\Delta m_{1}^{2}$ and $\Delta
m_{2}^{2}$. $\chi^{2}$ is defined by Eq. (4). To compute $\Delta\chi^{2}$, we
have minimized over $\theta_{12}$ and have subtracted the minimum $\chi^{2}$
with respect to $\Delta m_{i}^{2}$ . As seen from the figure, the values of
$\Delta m_{2}^{2}$ ($\Delta m_{1}^{2}$) larger than $2\times 10^{-11}$ eV2
($1.5\times 10^{-12}$ eV2) is ruled out at 2$\sigma$. This figure also
demonstrates that $1\times 10^{-11}~{}eV^{2}<\Delta m_{2}^{2}<2\times
10^{-11}~{}eV^{2}$ provides a fit comparable with SM when $\Delta m_{2}^{2}\to
0$.555We have also performed a similar analysis with the official Super-
Kamiokande data release in 2016 [13] which confirmed the current results,
except that those data tended to have lower values and thus the region found
in the interval of $\Delta m_{2}^{2}=(1,2)\times 10^{-11}$ compared to the
$\Delta m_{2}^{2}\to 0$ have lower $\chi^{2}$ value; i.e., providing a
slightly better fit than the standard MSW with $\Delta m_{2}^{2}=0$. $\Delta
m_{2}^{2}<1.5\times 10^{-12}$ eV2 is allowed within 1$\sigma$ C.L. The
1$\sigma$ and 2$\sigma$ contours of $\Delta m_{2}^{2}$ versus $\theta_{12}$
are shown in Fig. 5 . As seen from the figure, the values of $\theta_{12}$ at
the solutions that we have found are consistent with the $\theta_{12}$
measurement by the global neutrino data analysis.

Figure 5: Allowed $1\sigma$ and $2\sigma$ regions of joint $\chi^{2}$ analysis
for the $(\Delta m^{2}_{2},\theta_{21})$ space, assuming $\Delta m^{2}_{1}$ is
zero. The black and grey band show the $1\sigma$ and $2\sigma$ allowed range
of $\theta_{12}$ by global neutrino data analysis [11].

Let us now discuss the implication of the SNO measurement of the total active
solar neutrino flux. The SNO experiment has extracted the total flux by
measuring the Deuteron dissociation rate, $\nu+D\to\nu+n+p$ with a precision
of $8\%$ [23, 24]. This measurement is well-consistent with the standard solar
model prediction for the total neutrino flux within the uncertainties. In our
model, the measured total active flux is suppressed by $(1-P_{es}(E_{\nu}))$.
The SNO detection threshold is practically above 5 MeV.666 The natural energy
threshold, which is set by the binding energy of the Deuteron nucleus, is 2.2
MeV. For $E_{\nu}>5$ MeV and $\Delta m_{2}^{2}<2\times 10^{-11}$ eV2, $P_{es}$
is below 10% and as a result, the suppression of the total active flux
measurement relative to the SM prediction will be within the flux prediction
uncertainty of 12 % [17] and cannot therefore be resolved. For lower energies
(below the SNO threshold), the deviation should be more significant. The total
flux with lower energy threshold can also be measured by the coherent elastic
neutrino nucleus scattering at large scale direct dark matter experiments such
as the ongoing XENONnT and LZ experiments and future DARWIN experiment,
promising to test this model.

###  IV.2 Seasonal variation

Figure 6: Seasonal variation of event rate (per day per 100t) for 7Be. The
data points are taken from [26]. The predictions for various values of $\Delta
m_{i}^{2}$ are shown by curves. The upper (lower) panel corresponds to nonzero
$\Delta m_{1}^{2}$ (nonzero $\Delta m_{2}^{2}$). The values of the standard
mixing parameters are fixed to $\theta_{12}=33.4^{\circ}$ and $\Delta
m_{21}^{2}=7.5\times 10^{-5}$ eV2.

Recently, the BOREXINO experiment has released data on the residual of the 7Be
neutrino event rate, showing modulation due to the seasonal variation [26].
The selected events include an electron with a recoil energy larger than $0.3$
MeV. The rate is given in terms of per day per 100t. The data points, covering
a period of almost 10 years, are a time series binned in time intervals of 30
days. The annual trend of the data is subtracted.777For the exact definition
of tend and residue, the readers may consult [26].  We use this new data to
independently examine the validity of the new solution ($\Delta
m_{2}^{2}\simeq(1-2)\times 10^{-11}$ eV2) found in sect. IV.1. Furthermore,
studying the seasonal variation is an alternative approach to probe the
pseudo-Dirac mass splitting. The data points along with the predictions with
various values of $\Delta m_{1}^{2}$ and $\Delta m_{2}^{2}$ are shown in Fig.
6.

As seen in Fig. 2, depending on the exact value of $\Delta m_{1}^{2}$ and/or
$\Delta m_{2}^{2}$ in the range $\sim(1-2)\times 10^{-11}$ eV2, the pseudo-
Dirac scheme can lead to enhancement or suppression of the seasonal
modulation. This behavior is also confirmed in Fig. 6. In the following, we
focus on $\Delta m^{2}_{2}$ and set $\Delta m^{2}_{1}=0$. To constrain $\Delta
m^{2}_{2}$, we define $\chi^{2}$ as

$\chi^{2}=\sum_{t}\frac{\left[D^{Be}_{t}-\biggl{(}\mathcal{T}^{Be}({\rm
day},t,{\rm m})-\mathcal{T}^{Be}({\rm
day,t,y})\biggl{)}\right]^{2}}{\sigma^{2}_{t}}$ (10)

where $t$ runs over the 120 bins, each bin corresponding to a one month data
taking periods. $\rm m$ and $\rm y$ stand for month and year respectively.
$D^{Be}_{t}$ is residual of the events per day per 100 ton which are modeled
as a time series trend [26]. $\sigma^{2}_{t}$ is the corresponding error at
each bin $t$, as shown in Fig. 6. $\mathcal{T}^{Be}({\rm day},t,{\rm month})$
is computed using Eq. (7) by replacing the lower limit of the electron recoil
energy with $T_{e}=0.3$ MeV and integrating over monthly periods.
$\mathcal{T}^{Be}({\rm day},t,{\rm year})$ is computed using the same formula
with an averaging priod of a year. As discussed before, $\mathcal{T}^{Be}({\rm
day},t,{\rm year})$ should be independent of $t$. We fix
$\theta_{12}=33.4^{\circ}$, $\Delta m_{21}^{2}=7.54\times 10^{-5}$ eV2 and
$\Delta m_{1}^{2}=0$ but vary $\Delta m_{2}^{2}$. Similarly to the previous
section, we invoke the standard solar model for the flux normalization, with
negligible uncertainty. As seen from Fig. 7, the $\chi^{2}$ analysis using
this new data set independently supports the enhancement of the modulation
which occur in the $1.4\times 10^{-11}~{}{\rm eV}^{2}<\Delta m_{2}^{2}<2\times
10^{-11}~{}{\rm eV}^{2}$. This non-trivial solution falls in the 2$\sigma$
region of the annually averaged data that we have found in sect. IV.1. The
non-trivial solution that we have found provides a better fit to the seasonal
variation (see Fig. 7) than the standard model with $\Delta m_{2}^{2}\to 0$.
This is because the data shows about $10\%$ more enhanced modulation than the
$1/L^{2}$ modulation expected in the standard MSW scenario. On the other hand,
a cancellation on the modulation takes place in the range $0.6\times
10^{-11}~{}{\rm eV}^{2}<\Delta m_{2}^{2}<1.3\times 10^{-11}~{}{\rm eV}^{2}$
which is clearly ruled out with current data. The standard solution ($\Delta
m_{2}^{2}\to 0$) is allowed at just 80 % confidence level. However, the fact
that the two independent measurements, namely the Super-Kamiokande time
integrated solar data with $E_{\nu}>3$ MeV and the seasonal variation of the
7Be flux measured by BOREXINO, as well as the time integrated BOREXINO solar
neutrino data simultaneously point towards the same nonzero value of $\Delta
m_{2}^{2}$ makes it imperative to look for ways to test this new solution.

Figure 7: $\Delta\chi^{2}$ as a function of $\Delta m_{2}^{2}$ using just the
seasonal variation data set. The values of the standard mixing parameters are
fixed to $\theta_{12}=33.4^{\circ}$ and $\Delta m_{21}^{2}=7.54\times 10^{-5}$
eV2. The dashed curve illustrates the forecast for similar experiment with
error $\sigma_{t}\sim 0.5$ per day per 100t; i.e., with errors reduced to
almost half of the current values. The right panel of Fig. 4 is added as
orange curve for comparison in the zoomed box.

We also examine this non-trivial solution with the data previously released by
the radio chemical gallium experiments Gallex and GNO. We use the combined
result for the variation of the total $\nu_{e}$ during a year. The data is
taken from fig 7 of [21]. The results are illustrated in Fig. 8. These two
experiments had capability to record the low energy $\nu_{e}$ above a
threshold of $0.24\ \rm MeV$. They might therefore be used to examine the low
energy effect of pseudo-Dirac scheme for both annually averaged along with
seasonal variation data. Although, the region of our interest, $\Delta
m_{2}^{2}\sim 1.5\times 10^{-11}\rm eV^{2}$, is compatible with the annually
averaged data, due to large uncertainty the effect of seasonal variation is
not apparent. The main source of uncertainties comes from the deviation of
Gallex results from the GNO data. We have highlighted the 1$\sigma$ averaged
result of the Gallex in the Fig. 8 with light brown. It is clear that the
combined result is affected by the tendency of Gallex data to higher values
which is even at odds to the standard MSW scheme. We have also highlighted the
1$\sigma$ averaged result of GNO with light blue. There is no preferences
between the non-trivial solution and standard MSW when we just consider the
GNO result at annually averaged level.

Figure 8: Combined Gallex and GNO in solar neutrino unit (SNU) versus $L$. The
color code for different $\Delta m_{2}^{2}$ is same as previous plots. The
brown (blue) band is the 1$\sigma$ allowed region derived by Gallex (GNO),
averaging over time.

In the previous sections, we proposed four alternative methods to test this
solution. In the following, we investigate how by improving the seasonal
variation measurements, this new solution can be tested.

Let us suppose the true value of $\Delta m_{2}^{2}$ is close to the best fit
that we have found and then study what level of precision is required to rule
out the standard solution with $\Delta m_{2}^{2}=0$. The dashed line in Fig. 7
shows the value of $\Delta\chi^{2}$ versus $\Delta m_{2}^{2}$ setting
$\sigma_{t}\sim 0.5\ (\rm count/day\ per\ 100t)$. Notice that such
$\sigma_{t}$ requires a factor of 2 reduction in the current uncertainty. As
seen from the figure, with such an improvement, the standard solution can be
ruled out at better than 2 $\sigma$ C.L.

Let us now discuss how small $\sigma_{t}$ should be in order to obtain a
desired precision on $\Delta m_{2}^{2}$. To answer this question, we assume
that the error value $\sigma_{t}\sim\sigma_{F}$ is equal for all bins and
utilize the Fisher forecast formalism [28] with

$\sigma_{F}=\sigma_{\Delta
m^{2}_{i}}\sqrt{\sum_{t}\left(\frac{\partial\mathcal{T}^{Be}({\rm day},t,{\rm
m})}{\partial\Delta m^{2}_{i}}\right)^{2}}$ (11)

$\sigma_{F}$ is the ideal measurement error in order to have
$2\times\sigma_{\Delta m^{2}_{i}}$, the $1\sigma$ allowed region for parameter
$\Delta m^{2}_{i}$. The sum is over one year data points binned in 30 days and
$\mathcal{T}^{Be}$ is the prediction for 7Be neutrino event rate with
$T_{e}>0.3\ {\rm MeV}$, similarly to the current measurement. We assume the
true value $\Delta m_{2}^{2}=1.5\times 10^{-11}$ eV2 which is in the range of
$\sim(1-2)\times 10^{-11}$ eV2. The result is shown in Fig. 9. Error values of
order of $\sigma_{F}\sim 1\ (\rm count/day100t)$ lead to $1\sigma$ range
$2\times\sigma_{\Delta m^{2}_{i}}\sim 10^{-12}$ eV2. In particular, we obtain
$2\times\sigma_{\Delta m^{2}_{i}}\sim 0.5\times 10^{-12}$ eV2 reducing them to
$\sigma_{F}\sim 0.5\ (\rm count/day100t)$.

Figure 9: The precision required in the time variation measurement to
determine $\Delta m_{2}^{2}$ with a precision of $\sigma_{\Delta m_{2}^{2}}$.
We have assumed that the true value is $\Delta m_{2}^{2}=1.5\times 10^{-11}$
eV2.

##  V Discussion and conclusions

We have studied the oscillation of the solar neutrinos within the pseudo-Dirac
scheme. Our focus has been on the splittings of $\nu_{1}$ and $\nu_{2}$
states, $\Delta m_{1}^{2}$ and $\Delta m_{2}^{2}$ of order of $10^{-13}$
eV${}^{2}-10^{-10}$ eV2 which are relevant for solar neutrinos. Since the
contribution of $\nu_{3}$ to the solar neutrinos ($\nu_{e}$ at the production)
is suppressed by $\theta_{13}$, a splitting in $\nu_{3}$ will not affect the
solar neutrino data. To derive bounds on the splitting, we have used the
latest BOREXINO and Super-Kamiokande solar neutrino data and have employed the
$\Delta m_{21}^{2}$ measurement by KamLAND. We have found that these data rule
out $\Delta m_{1}^{2}$ and $\Delta m_{2}^{2}$ above $2\times 10^{-11}$ eV2.
However, we find a new solution in the range of $\Delta
m_{2}^{2}\sim(1-2)\times 10^{-11}$ eV2 and $\Delta m_{1}^{2}=0$ which fits the
solar neutrino data (especially the 8B data measured by Super-Kamiokande) in
addition to the standard three neutrino scenario with $\Delta m_{1}^{2}=\Delta
m_{2}^{2}=0$. We have discussed the possibility of ruling out this solution
with the total active neutrino flux measurement by SNO. We found that the
deviation due to $P(\nu_{e}\to\nu_{s})$ at this solution for neutrinos with
energy above the SNO detection threshold can hide within the flux prediction.
We have examined the robustness of this new solution against the accumulation
of more solar data. The data available by 2016 slightly prefers this solution
to the standard MSW. Ref. [9] also confirms this solution.

We have examined the possibility of testing this non-trivial pseudo-Dirac
solution with the recent data release by BOREXINO on the seasonal variation of
7Be [26]. Surprisingly, the seasonal variation also points towards a solution
with $1.4\times 10^{-11}{\rm eV}^{2}<\Delta m_{2}^{2}<2\times 10^{-11}{\rm
eV}^{2}$, independently. Indeed, this solution fits the seasonal variation
better than the standard three neutrino scheme but, the $\Delta
m_{1}^{2}=\Delta m_{2}^{2}=0$ solution is still allowed at 80% C.L. We have
discussed how reducing the uncertainty in the measurement of seasonal
variation can help to measure $\Delta m_{2}^{2}$ with better precision or set
a bound on it.

We also examine our new solution with radio chemical gallium experiments
Gallex and GNO. The solution is in agreement with the combined Gallex and GNO
averaged results, favoring GNO over Gallex. However, due to large
uncertainties it is not possible to test the effect of seasonal variation with
this old data.

We have proposed four independent approaches to test the non-trivial solution
that we have found: (1) Measurement of the 8B flux in the energy range below 4
MeV with a moderate precision of 20 % (or better) can test the solution. The
proposed THEIA detector [29], with a relatively low detection energy threshold
will be able to perform such a measurement. (2) By improving the precision of
the measurement of the $pep$ line, the solution can be tested. (3) Reducing
the uncertainty in the seasonal variation of the 7Be line by half can test
this solution. (4) Finally, the measurement of the total active solar flux via
coherent elastic $\nu$ nucleus scattering by direct dark matter search
experiments can provide an alternative method for testing the solution.

We have found that both time integrated solar neutrino data and the 7Be time
variation, independently from each other, constrain $\Delta
m_{1}^{2}<1.5\times 10^{-12}$ eV2 and $\Delta m_{2}^{2}<2\times 10^{-11}$ eV2
at $2\sigma$ C.L.

All material and code of this article are publicly available on
https://github.com/SaeedAnsarifard/SolarNeutrinos-pseudoDirac.git *

##  Appendix A Pseudo-Dirac scheme in the presence of matter

In this appendix, we derive dispersion relation and the energy momentum
eigenstates for the pseudo-Dirac scheme in the presence of matter effects. We
compute $P(\nu_{e}\to\nu_{e})$ and $P(\nu_{e}\to\nu_{s})$ for solar neutrinos
in the pseudo-Dirac neutrino scheme. We then formulate the time dependence
(seasonal variation) of the flux arriving to the Earth, considering the
eccentricity of the Earth orbit around the Sun.

Let us start with one flavor state with the effective Lagrangian

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\bar{\Psi}i\partial\cdot\gamma\Psi-m\bar{\Psi}\Psi-V\bar{\Psi}\gamma^{0}P_{L}\Psi-\mu\bar{\Psi}^{c}P_{R}\Psi-\mu\bar{\Psi}P_{L}\Psi^{c}$
(12) $\displaystyle=$
$\displaystyle\bar{\Psi}^{c}i\partial\cdot\gamma\Psi^{c}-m\bar{\Psi}^{c}\Psi^{c}+V\bar{\Psi}^{c}\gamma^{0}P_{R}\Psi^{c}-\mu\bar{\Psi}^{c}P_{R}\Psi-\mu\bar{\Psi}P_{L}\Psi^{c}$
(13)

where $\Psi$ is a general Dirac spinor and $\Psi^{c}=-i\gamma^{2}\Psi^{*}$.
Taking the derivative of the Lagrangian in the first line of Eq. (13) with
respect to $\bar{\Psi}$, we arrive at the Euler-Lagrange equation,

$i\partial\cdot\gamma\Psi-m\Psi-V\gamma^{0}P_{L}\Psi-\mu P_{L}\Psi^{c}=0.$
(14)

Similarly taking derivative of the Lagrangian in the second line of Eq. (13)
with respect to $\bar{\Psi}^{c}$, we find

$i\partial\cdot\gamma\Psi^{c}-m\Psi^{c}+V\gamma^{0}P_{R}\Psi^{c}-\mu
P_{R}\Psi=0.$ (15)

Applying $(i\gamma\cdot\partial-V\gamma^{0}P_{L})$ and
$(i\gamma\cdot\partial+V\gamma^{0}P_{R})$ respectly to Eqs. (14,15), we obtain
the following relations

$\displaystyle-\partial^{2}\Psi-2iVP_{L}\partial_{3}\Psi$
$\displaystyle=m^{2}\Psi+m\mu\Psi^{c}-\mu^{2}P_{R}\Psi$ (16)
$\displaystyle-\partial^{2}\Psi^{c}+2iVP_{R}\partial_{3}\Psi^{c}$
$\displaystyle=m^{2}\Psi^{c}+m\mu\Psi+\mu^{2}P_{L}\Psi^{c}$

where we have taken the third ($z$) direction along the momentum ($p$) of the
particle. Remembering $P_{L}\Psi=\nu_{L}$, $P_{L}\Psi^{c}=\nu_{R}^{c}$,
$P_{R}\Psi=\nu_{R}$, $P_{R}\Psi^{c}=-\nu_{L}^{c}$ and using $P_{L}P_{R}=0$, we
obtain

$\displaystyle(E^{2}-p^{2})\left(\begin{matrix}\nu_{L}\cr\nu_{R}^{c}\end{matrix}\right)=$
$\displaystyle\left(\begin{matrix}2pV+m^{2}&m\mu\cr
m\mu&m^{2}+\mu^{2}\end{matrix}\right)\left(\begin{matrix}\nu_{L}\cr\nu_{R}^{c}\end{matrix}\right)$
(17) $\displaystyle=$ $\displaystyle\left[\left(\begin{matrix}2pV&0\cr
0&0\end{matrix}\right)+\left(\begin{matrix}0&m\cr
m&\mu\end{matrix}\right)^{2}\right]\left(\begin{matrix}\nu_{L}\cr\nu_{R}^{c}\end{matrix}\right)$

and

$(E^{2}-p^{2})\left(\begin{matrix}\nu_{R}\cr\nu_{L}^{c}\end{matrix}\right)=\left(\begin{matrix}m^{2}-\mu^{2}&-m\mu\cr-m\mu&m^{2}-2pV\end{matrix}\right)\left(\begin{matrix}\nu_{R}\cr\nu_{L}^{c}\end{matrix}\right)$
(18)

Let us focus on Eq. (17). We should of course take the ultra relativistic
limit, $p\gg V,m,\mu$ so the energy eigenvectors correspond to the
eigenvectors of

$\left(\begin{matrix}V+\frac{m^{2}}{2p}&\frac{m\mu}{2p}\cr\frac{m\mu}{2p}&\frac{m^{2}+\mu^{2}}{2p}\end{matrix}\right).$
(19)

In the limit $2Vp\ll m\mu$, we recover the famous pseudo-Dirac scheme with
maximal mixing. That is the eigenstates will be the following Majorana states
with energy eigenvalues as

$\chi_{1}=\frac{\nu_{L}+\nu^{c}_{R}}{2}~{}{\rm with}~{}E^{2}=p^{2}+m^{2}+\mu
m$ (20)

and

$\chi_{2}=\frac{\nu_{L}-\nu^{c}_{R}}{2}~{}{\rm with}~{}E^{2}=p^{2}+m^{2}-\mu
m$ (21)

As a result, active $\nu_{L}$ can oscillate to sterile neutrino with
oscillation length determined by splitting $\Delta m^{2}=2m\mu$ and maximal
mixing

$P(\nu_{L}\to\nu_{R}^{c})=\sin^{2}(\frac{\mu m}{2p}L)$

and

$P(\nu_{L}\to\nu_{L})=1-P(\nu_{L}\to\nu_{R}^{c})=\cos^{2}(\frac{\mu m}{2p}L)$

Notice that the active and sterile neutrinos respectively correspond to
$\nu_{L}$ and $\nu_{R}$ (or $\nu_{R}^{c}$). We therefore use $\nu_{R}^{c}$ and
$\nu_{s}$ interchangeably: $P(\nu_{L}\to\nu_{R}^{c})=P(\nu_{a}\to\nu_{s})$.

For $2pV\gg\mu m$, the mixing between $\nu_{L}$ and $\nu_{R}^{c}$ will be
suppressed by $\mu m/(2pV)$ so the oscillation to sterile neutrino will be
negligible. For the sake of simplicity, for the three neutrino flavors, we
strict ourselves to the case that the $m$ and $\mu$ matrices can
simultaneously be diagonalized. Thus, in the mass basis all terms will be
diagonal except for the effective potential
$V\sum_{i,j}U_{ei}U_{ej}^{*}\bar{\Psi}_{j}\gamma^{0}\Psi_{i}$.

Now, let us consider solar neutrinos with $\mu m/(2p)\sim 1/L$ where $L$ is
the Earth Sun distance. Within the Sun, the matter effects will dominate and
will suppress the $\nu_{Li}$ and $\nu^{c}_{Ri}$ mixing. That is within the
Sun, we shall have the standard MSW effect and the $\nu_{e}$ state after
crossing the Sun will emerge at the Sun surface as an incoherent combination
of $\nu_{L1}$, $\nu_{L2}$ and $\nu_{L3}$ with pobabilities
$\cos^{2}\theta_{M}\cos^{2}\theta_{13}$,
$\sin^{2}\theta_{M}\cos^{2}\theta_{13}$ and $\sin^{2}\theta_{13}$. Here,
$\theta_{M}$ is the effective 12 mixing at the production point of $\nu_{e}$:

$\displaystyle\cos 2\theta_{M}$ $\displaystyle=$ (22)
$\displaystyle\frac{\Delta m_{21}^{2}\cos 2\theta_{12}-VE_{\nu}}{\left[(\Delta
m_{21}^{2}\cos 2\theta_{12}-VE_{\nu})^{2}+(\Delta m_{21}^{2}\sin
2\theta_{12})^{2}\right]^{1/2}}$

in which $V=2\sqrt{2}G_{F}N_{e}(r)|_{\rm at~{}production}$. Notice that we
have taken into account two facts: (i) conversion in the Sun is adiabatic;
(ii) the matter effect on $\theta_{13}$ is negligible due to suppression with
$2Vp/(\Delta m_{31}^{2}\sin\theta_{13})\ll 1.$ The mass eigenstates on their
way to Earth can oscillate into their sterile counter-part ($\nu^{c}_{Ri}$)
with maximal mixing so the survival probability up to the Earth surface can be
written as

$\displaystyle P_{ee}=$
$\displaystyle\cos^{4}\theta_{13}\Biggl{(}\cos^{2}\theta_{12}\cos^{2}\theta_{M}\cos^{2}(\frac{\mu_{1}m_{1}}{2p}L)$
(23)
$\displaystyle\qquad\qquad+\sin^{2}\theta_{12}\sin^{2}\theta_{M}\cos^{2}(\frac{\mu_{2}m_{2}}{2p}L)\Biggl{)}$
$\displaystyle+\sin^{4}\theta_{13}\cos^{2}(\frac{\mu_{3}m_{3}}{2p}L)$

and

$\displaystyle P_{es}=$
$\displaystyle\cos^{2}\theta_{13}\Biggl{(}\cos^{2}\theta_{M}\sin^{2}(\frac{\mu_{1}m_{1}}{2p}L)$
(24)
$\displaystyle\qquad\qquad+\sin^{2}\theta_{M}\sin^{2}(\frac{\mu_{2}m_{2}}{2p}L)\Biggl{)}$
$\displaystyle+\sin^{2}\theta_{13}\sin^{2}(\frac{\mu_{3}m_{3}}{2p}L)$

For simplicity, we denote $P_{ee}\equiv P(\nu_{e}\to\nu_{e})$ and
$P_{es}\equiv\sum_{i}P(\nu_{e}\to\nu_{s_{i}})$. This formula corresponds to
that in Ref. [10] in the limit $\mu_{i}m_{i}\ll VE_{\nu}$. For relatively high
energy solar neutrinos, the oscillation in Earth due to matter effects (i.e.,
Day/Night effect) can also be important but our focus is on the intermediate
energy solar neutrinos for which the matter effects are negligible.

Through $\theta_{M}$, $P_{ee}$ depends on the location of $\nu_{e}$ production
inside the Sun. We define

$\bar{P}^{j}_{ee}(E_{\nu},L)=\int_{0}^{R_{\odot}}P_{ee}(E_{\nu},L,r)\Phi_{j}(r)dr$
(25)

where $j$ can be any of the flux components $pp$, 7Be, $pep$ and 8B.
$\Phi_{j}(r)$ is the flux from radius $r$ taken from
888http://www.sns.ias.edu/ jnb/. Notice that $\Phi_{j}(r)dr$ includes the
volume factor ($r^{2}dr$) in its definition and vanishes at $r=0$. The
dependence of $\Phi_{j}(r)$ on $r$ is different for the $j$ modes. For
example, while for $j=pp$ the flux peaks at $r\simeq 0.1R_{\odot}$, for
$j=^{8}$B, the peak is at $r\simeq 0.05R_{\odot}$, The dependence of
$P_{ee}(E_{\nu},L,r)$ on $r$ is through the dependence of $\theta_{M}$ on
$N_{e}(r)$. Let us define

$\Delta m_{i}^{2}=2\mu_{i}m_{i}.$

Let us now discuss the time dependence of the flux throughout a year. The sun
Earth distance during a year varies between $L_{max}=152.1\times 10^{6}$ km
(aphelion ocurring around July 4th) and $L_{min}=147.1\times 10^{6}$ km
(perihelion ocurring around January 4th). That is the orbit of the Earth
around the Sun can be written as

$L(\theta)=\frac{a(1-e^{2})}{1+e\cos\theta}$ (26)

in which $a=(L_{min}+L_{max})/2$ and the eccentricity is
$e=(L_{max}-L_{min})/(L_{max}+L_{min})=0.0167$. The conservation of the
angular momentum imples $dt=(L^{2}/H)d\theta$ in which
$H=|\vec{r}\times\dot{\vec{r}}|$. The number of events during a time interval
$(T_{1},T_{2})$ is proportional to

$\displaystyle\int_{t_{1}}^{t_{1}+\Delta
t}\frac{P_{ee}\sigma_{e}+(1-P_{ee}-P_{es})\sigma_{\mu}}{L^{2}}dt=$ (27)
$\displaystyle\int_{\theta_{1}}^{\theta_{2}}\frac{P_{ee}\sigma_{e}+(1-P_{ee}-P_{es})\sigma_{\mu}}{H}d\theta$

where $\sigma_{e}$ and $\sigma_{\mu}$ are respectively the scattering cross
sections of $\nu_{e}$ and $\nu_{\mu}$ (or $\nu_{\tau}$) at the detector. To
compute the number of events during a time interval we should know the
relation between $\theta$ and time. Replacing $L(\theta)$ given in Eq. (26) we
find

$\int_{t_{1}}^{t_{1}+\Delta
t}\frac{H}{a^{2}(1-e^{2})^{2}}dt=\int_{\theta_{1}}^{\theta_{2}}\frac{d\theta}{(1+e\cos\theta)^{2}}$
(28)

which yields

$H=\frac{6.55a^{2}(1-e^{2})^{2}}{1~{}{\rm year}}.$

We have used these formulas to study the seasonal variation of the 7Be solar
flux. As shown in [31], the widths of 7Be and $pep$ lines are of order of
kinetic energy in the sun center $\Delta E_{\nu}\sim 0.6$ keV. Thus, as long
as $\Delta m^{2}L/E\ll 1000$, we have $(\Delta E_{\nu}/E_{\nu})(\Delta
m^{2}L/E)\ll 1$, the finite width of these lines will not smear the
oscillatory behavior.

###### Acknowledgements.

The authors are grateful to P. Zakeri for the collaboration in the early
stages of this work. This project has received funding/support from the
European Union’s Horizon 2020 research and innovation programme under the
Marie Skłodowska -Curie grant agreement No 860881-HIDDeN. YF has received
financial support from Saramadan under contract No. ISEF/M/401439. She would
like to acknowledge support from the ICTP through the Associates Programme and
from the Simons Foundation through grant number 284558FY19.

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