# Historical Inertia: An Ignored but Powerful Baseline for Long Sequence Time-
series Forecasting

Yue Cui
University of Electronic Science
and Technology of China
Chengdu, 611731
<EMAIL_ADDRESS>
&Jiandong Xie
Cloud BU
Huawei Technologies Co. Ltd.
Chengdu, 611731
<EMAIL_ADDRESS>
&Kai Zheng
University of Electronic Science
and Technology of China
Chengdu, 611731
<EMAIL_ADDRESS>

###### Abstract

Long sequence time-series forecasting (LSTF) has become increasingly popular
for its wide range of applications. Though superior models have been proposed
to enhance the prediction effectiveness and efficiency, it is reckless to
ignore or underestimate one of the most natural and basic temporal properties
of time-series, the historical inertia (HI), which refers to the most recent
data-points in the input time series. In this paper, we experimentally
evaluate the power of historical inertia on four public real-word datasets.
The results demonstrate that up to 82% relative improvement over state-of-the-
art works can be achieved even by adopting HI directly as output.

##  1 Introduction

Time series forecasting, i.e., given historical values of time series and
making prediction for future time-slots, can be deemed as one of the main
enablers of modern society. An accurate prediction model can benefit a wide-
range of applications, e.g., predicting stock prices (Xu et al.,, 2020, Box et
al.,, 2015), monitoring traffic flows and electricity consumption (Zhou et
al.,, 2021, Wu et al.,, 2020, Shih et al.,, 2019, Lai et al.,, 2018).

Rather than the typical setting of predicting values of limited number of
time-steps, i.e. 48 steps or fewer (Wu et al.,, 2020, Xu et al.,, 2020, Shih
et al.,, 2019, Lai et al.,, 2018), an emerging line of work focuses on the
problem of long sequence time-series forecasting (LSTF), where up to 720 steps
can be predicted at a time (Zhou et al.,, 2021). Such an increasing sequence
length can be troublesome to most existing works, which are designed for
relatively short prediction horizon.

To deal with the challenges of effectively modeling temporal correlations in
long sequence and efficiently operating on long inputs and outputs, the state-
of-the-art (SOTA) work Informer (Zhou et al.,, 2021) proposes a novel variant
of Transformer (Vaswani et al.,, 2017) to reduce time and space complexity
while maintaining prediction accuracy, which is indeed a breakthrough. Despite
that the extensive experiments on five real-world datasets demonstrates
Informer’s superiority to its baselines, the enhanced performance can be
limited when considering the baseline of taking the most recent values in
inputs as outputs, which can be referred to as the historical inertial (HI).

In this paper, we first address this issue by providing an experimental
evaluation of the proposed baseline HI and SOTA models and on a variety of
public rel-world datasets, and then make an comprehensive discussion on why HI
is powerful and how HI could serve as a potential trick.

##  2 Problem and The Proposed Baseline

Long Sequence Time-series Forecasting: At time $t$, given a $L_{x}$-length
time series as input, i.e., $\mathcal{X}(t)=\\{X_{1}(t),...,X_{L_{x}}(t)\\}$,
where
$X_{i}(t)=[x_{i,1}(t),..,x_{i,d_{x}}(t)]\in\mathbb{R}^{d_{x}},i\in[1,...,{L_{x}}]$,
is the observed univariate ($d_{x}=1$) or multivariate ($d_{x}>1$) variable at
the $i$-th time-stamp, the goal of long sequence time-series forecasting
(LSTF) is to predict the corresponds $L_{y}$-length sequence $\Delta$ steps
ahead, i.e., $\mathcal{Y}(t)=[Y_{1}(t),..,Y_{L_{y}}(t)]$, where
$Y_{i}(t)=[y_{i,1}(t),..,y_{i,d_{y}}(t)]\in\mathbb{R}^{d_{y}}$ and $d_{x}\geq
d_{y}\geq 1$. When $d_{x}=d_{y}$,
$\mathcal{Y}(t)=[X_{L_{x}+\Delta+1}(t),..,X_{L_{x}+\Delta+L_{y}}(t)]$.

Historical Inertia: The historical inertial (HI) baseline takes $L_{y}$-length
subsequence of $\mathcal{X}(t)$ as prediction results, i.e.,
$\hat{\mathcal{Y}(t)}=[X_{L_{x}-L_{y}-1}(t),..,X_{L_{x}}(t)]$.

Note that the HI baseline requires prediction length to be no longer than the
input length, i.e., $L_{x}\geq L_{y}$, which is not necessary for learning-
based LSTF models. Considering that in real application scenarios, the dataset
is usually orders of magnitude larger than $L_{y}$, this condition can be
easily achieved.

##  3 Experiment and Results

###  3.1 Datasets and Metrics

We compare HI with SOTA models on four real-world public datasets.

ETT (Electricity Transformer Temperature) 111https://github.com/zhouhaoyi/:
The ETT dataset from the Informer paper contains 2-years electric power
deployment collected from two Chinese counties. There are 7 features in total.
Three sub-datasets are generated, ETTh1 and ETTh2 are hourly data and ETTm1 is
in 15-minutes level. In the univariate forecasting task, the feature "oil
temperature" is chosen as prediction target.

Electricity 222https://github.com/laiguokun/multivariate-time-series-data: The
raw electricity dataset is from the UCI Machine Learning Repository
333https://archive.ics.uci.edu/ml/datasets/ElectricityLoadDiagrams20112014,
which contains electricity consumption of 370 clients every 15 minutes from
2011 to 2014. We use the pre-processed dataset from Lai et al., (2018), which
reflects hourly consumption of 321 clients from 2012 to 2014. The last client
(column) is used as prediction target in the univariate forecasting task.

Statistics of above datasets can be found in Table 1.

Table 1: Statistics of dataset. Dataset | # samples | # variables | Sample rate
---|---|---|---
ETTh1 | 17420 | 7 | 1 hour
ETTh2 | 17420 | 7 | 1 hour
ETTm1 | 69680 | 7 | 15 minutes
Electricity | 26304 | 321 | 1 hour

As a common practice, we evaluate the models by two metrics: Mean Square Error
(MSE) and Mean Absolute Error (MAE), which are computed as:

$MSE=\frac{1}{T}\sum\limits_{t=1}^{T}\frac{1}{L_{y}\times
d_{y}}\sum\limits_{i=t_{1}}^{t_{L_{y}}}\sum\limits_{j=1}^{d_{y}}(\hat{y}_{i,j}(t)-y_{i,j}(t))^{2},$
(1) $MAE=\frac{1}{T}\sum\limits_{t=1}^{T}\frac{1}{L_{y}\times
d_{y}}\sum\limits_{i=t_{1}}^{t_{L_{y}}}\sum\limits_{j=1}^{d_{y}}|\hat{y}_{i,j}(t)-y_{i,j}(t)|$
(2)

where $\hat{y}$ is the prediction output, $y$ is the ground-truth value,
$t\in[t_{1},t_{T}]$ is the time instance in test set.

All above settings are consistent with the Informer paper. Note that we
eliminate the dataset Weather that is also used in the paper. Because only raw
data is available and the preprocessing operations are unclear.

###  3.2 Competitors

####  3.2.1 Univariate LSTF SOTA Models

Eight models ranging from traditional statistical methods to recent-proposed
deep models are included as competitors for the task of univariate time series
forecasting.

  * •

Prophet (Taylor and Letham,, 2018): A modular regression model that models
common feature of time series in scale-aware way.

  * •

ARIMA (Box et al.,, 2015): An autoregressive integrated moving average-based
model for stock price prediction.

  * •

DeepAR (Salinas et al.,, 2020): An autoregressive recurrent neural network.

  * •

LSTMa (Bahdanau et al.,, 2014): A recurrent neural network-based neural
machine translation model designed for long sentences.

  * •

Reformer (Kitaev et al.,, 2020): An efficient variant of Transformer using
locality-sensitive hashing and reversible residual layers.

  * •

LogTrans (Li et al.,, 2019): An efficient variant of Transformer using
convolutional attention and sparse attention.

  * •

Informer (Zhou et al.,, 2021): An efficient variant of Transformer using
ProbSparse self-attention and self-attention distilling.

  * •

Informer- (Zhou et al.,, 2021): A variant of Informer removing the ProbSparse
self-attention mechanism.

####  3.2.2 Multivariate LSTF SOTA Models

Besides above mentioned LSTMa, Reformer, LogTrans, Informer and Informer-,

  * •

LSTNet (Lai et al.,, 2018): A deep neural network that combines convolutional
neural networks and recurrent neural networks,

is also used as a competitor in the task of multivariate time series
forecasting.

###  3.3 Implementation Details

Basically, we follow the common practice in the community as described in Zhou
et al., (2021). $\Delta$ is fixed as 1. Prediction length is set as [24, 48,
168, 366, 720] for ETTh1 and ETTh2, [24, 48, 96, 288, 672] for ETTm1 and [48,
168, 366, 720, 960] for Electricity. We split the ETT datasets into 12:4:4 and
Electricity dataset into 15:3:4 for training, validation and test. Above
implementation settings are consistent with the Informer paper. Since the
method of HI doesn’t require training, when the dataset split is fixes the
performance is fixed. Thus, only one iteration is sufficient to compute the
final results.

###  3.4 Main Results

Table 2 and Table 3 provide the main experimental results of HI and SOTA
models. The best results are highlighted in bold. The last line in each Table
calculates HI’s relative improvement over the best SOTA model, which is
calculated as $(best\\_SOTA\\_Model-HI)/best\\_SOTA\\_Model$. Numbers in green
indicate positive and in red indicate negative. All reported results are on
test set. Besides the results of HI, numbers are referenced from the updated
results on the official GitHub repository of Informer
444https://github.com/zhouhaoyi/Informer2020/tree/main/img. We also follow the
same scaling strategy as Informer does.

We observe that HI achieves state-of-the-art results in many cases, especially
for the task of multivariate forecasting, in which the relative improvement
can up to 82%. In the following, we discuss experimental results of univariate
and multivariate LSTF respectively.

####  3.4.1 Univariate LSTF Results

In the task of predicting a single variable over time, HI outperforms SOTA
models significantly on ETTh1 and ETTm1 dataset. Informer and its variant
almost dominate the ETTh2 dataset while DeepAR, Informer and HI claims part of
the best results on the Electricity dataset.

Table 2: Summary of univariate long sequence time-series forecasting comparison results. Dataset | ETTh1 | ETTh2 | ETTm1 | Electricity
---|---|---|---|---
Method | Metric | 24 | 48 | 168 | 336 | 720 | 24 | 48 | 168 | 336 | 720 | 24 | 48 | 96 | 288 | 672 | 48 | 168 | 336 | 720 | 960
Prophet | MSE | 0.115 | 0.168 | 1.224 | 1.549 | 2.735 | 0.199 | 0.304 | 2.145 | 2.096 | 3.355 | 0.120 | 0.133 | 0.194 | 0.452 | 2.747 | 0.524 | 2.725 | 2.246 | 4.243 | 6.901
MAE | 0.275 | 0.330 | 0.763 | 1.820 | 3.253 | 0.381 | 0.462 | 1.068 | 2.543 | 4.664 | 0.290 | 0.305 | 0.396 | 0.574 | 1.174 | 0.595 | 1.273 | 3.077 | 1.415 | 4.264
ARIMA | MSE | 0.108 | 0.175 | 0.396 | 0.468 | 0.659 | 3.554 | 3.190 | 2.800 | 2.753 | 2.878 | 0.090 | 0.179 | 0.272 | 0.462 | 0.639 | 0.879 | 1.032 | 1.136 | 1.251 | 1.370
MAE | 0.284 | 0.424 | 0.504 | 0.593 | 0.766 | 0.445 | 0.474 | 0.595 | 0.738 | 1.044 | 0.206 | 0.306 | 0.399 | 0.558 | 0.697 | 0.764 | 0.833 | 0.876 | 0.933 | 0.982
DeepAR | MSE | 0.107 | 0.162 | 0.239 | 0.445 | 0.658 | 0.098 | 0.163 | 0.255 | 0.604 | 0.429 | 0.091 | 0.219 | 0.364 | 0.948 | 2.437 | 0.204 | 0.315 | 0.414 | 0.563 | 0.657
MAE | 0.280 | 0.327 | 0.422 | 0.552 | 0.707 | 0.263 | 0.341 | 0.414 | 0.607 | 0.580 | 0.243 | 0.362 | 0.496 | 0.795 | 1.352 | 0.357 | 0.436 | 0.519 | 0.595 | 0.683
LSTMa | MSE | 0.114 | 0.193 | 0.236 | 0.590 | 0.683 | 0.155 | 0.190 | 0.385 | 0.558 | 0.640 | 0.121 | 0.305 | 0.287 | 0.524 | 1.064 | 0.493 | 0.723 | 1.212 | 1.511 | 1.545
MAE | 0.272 | 0.358 | 0.392 | 0.698 | 0.768 | 0.307 | 0.348 | 0.514 | 0.606 | 0.681 | 0.233 | 0.411 | 0.420 | 0.584 | 0.873 | 0.539 | 0.655 | 0.898 | 0.966 | 1.006
Reformer | MSE | 0.222 | 0.284 | 1.522 | 1.860 | 2.112 | 0.263 | 0.458 | 1.029 | 1.668 | 2.030 | 0.095 | 0.249 | 0.920 | 1.108 | 1.793 | 0.971 | 1.671 | 3.528 | 4.891 | 7.019
MAE | 0.389 | 0.445 | 1.191 | 0.124 | 1.436 | 0.437 | 0.545 | 0.879 | 1.228 | 1.721 | 0.228 | 0.390 | 0.767 | 1.245 | 1.528 | 0.884 | 1.587 | 2.196 | 4.047 | 5.105
LogTrans | MSE | 0.103 | 0.167 | 0.207 | 0.230 | 0.273 | 0.102 | 0.169 | 0.246 | 0.267 | 0.303 | 0.065 | 0.078 | 0.199 | 0.411 | 0.598 | 0.280 | 0.454 | 0.514 | 0.558 | 0.624
MAE | 0.259 | 0.328 | 0.375 | 0.398 | 0.463 | 0.255 | 0.348 | 0.422 | 0.437 | 0.493 | 0.202 | 0.220 | 0.386 | 0.572 | 0.702 | 0.429 | 0.529 | 0.563 | 0.609 | 0.645
Informer- | MSE | 0.092 | 0.161 | 0.187 | 0.215 | 0.257 | 0.099 | 0.159 | 0.235 | 0.258 | 0.285 | 0.034 | 0.066 | 0.187 | 0.409 | 0.519 | 0.238 | 0.442 | 0.501 | 0.543 | 0.594
MAE | 0.246 | 0.322 | 0.355 | 0.369 | 0.421 | 0.241 | 0.317 | 0.390 | 0.423 | 0.442 | 0.160 | 0.194 | 0.384 | 0.548 | 0.665 | 0.368 | 0.514 | 0.552 | 0.578 | 0.638
Informer | MSE | 0.098 | 0.158 | 0.183 | 0.222 | 0.269 | 0.093 | 0.155 | 0.232 | 0.263 | 0.277 | 0.030 | 0.069 | 0.194 | 0.401 | 0.512 | 0.230 | 0.447 | 0.489 | 0.540 | 0.582
MAE | 0.247 | 0.319 | 0.346 | 0.387 | 0.435 | 0.240 | 0.314 | 0.389 | 0.417 | 0.431 | 0.137 | 0.203 | 0.372 | 0.554 | 0.644 | 0.359 | 0.503 | 0.528 | 0.571 | 0.608
HI | MSE | 0.046 | 0.069 | 0.116 | 0.137 | 0.186 | 0.095 | 0.150 | 0.257 | 0.318 | 0.449 | 0.023 | 0.039 | 0.046 | 0.081 | 0.115 | 0.872 | 0.328 | 0.415 | 1.178 | 1.302
MAE | 0.166 | 0.210 | 0.271 | 0.306 | 0.351 | 0.231 | 0.300 | 0.409 | 0.465 | 0.549 | 0.115 | 0.156 | 0.167 | 0.229 | 0.270 | 0.690 | 0.393 | 0.463 | 0.836 | 0.894
Improve | MSE | 50 % | 56% | 37% | 36% | 28% | 2% | 3% | 11% | 23% | 62% | 23% | 41% | 75% | 80% | 78% | 327% | 4% | 0% | 118% | 124%
MAE | 33% | 34% | 37% | 17% | 17% | 4% | 4% | 5% | 12% | 27% | 16% | 20% | 55% | 58% | 58% | 93% | 10% | 11% | 46% | 47%

####  3.4.2 Multivariate LSTF Results

In the task of predicting multiple variables over time, HI significantly
outperforms SOTA models in almost all tests. The relative improvement of HI
ranges from 2% to 82%.

Table 3: Summary of multivariate long sequence time-series forecasting comparison results. Dataset | ETTh1 | ETTh2 | ETTm1 | Electricity
---|---|---|---|---
Method | Metric | 24 | 48 | 168 | 336 | 720 | 24 | 48 | 168 | 336 | 720 | 24 | 48 | 96 | 288 | 672 | 48 | 168 | 336 | 720 | 960
LSTMa | MSE | 0.650 | 0.702 | 1.212 | 1.424 | 1.960 | 1.143 | 1.671 | 4.117 | 3.434 | 3.963 | 0.621 | 1.392 | 1.339 | 1.740 | 2.736 | 0.486 | 0.574 | 0.886 | 1.676 | 1.591
MAE | 0.624 | 0.675 | 0.867 | 0.994 | 1.322 | 0.813 | 0.221 | 1.674 | 1.549 | 1.788 | 0.629 | 0.939 | 0.913 | 1.124 | 1.555 | 0.572 | 0.602 | 0.795 | 1.095 | 1.128
Reformer | MSE | 0.991 | 1.313 | 1.824 | 2.117 | 2.415 | 1.531 | 1.871 | 4.660 | 4.028 | 5.381 | 0.724 | 1.098 | 1.433 | 1.820 | 2.187 | 1.404 | 1.515 | 1.601 | 2.009 | 2.141
MAE | 0.754 | 0.906 | 1.138 | 1.280 | 1.520 | 1.613 | 1.735 | 1.846 | 1.688 | 2.015 | 0.607 | 0.777 | 0.945 | 1.094 | 1.232 | 0.999 | 1.069 | 1.104 | 1.170 | 1.387
LogTrans | MSE | 0.686 | 0.766 | 1.002 | 1.362 | 1.397 | 0.828 | 1.806 | 4.070 | 3.875 | 3.913 | 0.419 | 0.507 | 0.768 | 1.462 | 1.669 | 0.355 | 0.368 | 0.373 | 0.409 | 0.477
MAE | 0.604 | 0.757 | 0.846 | 0.952 | 1.291 | 0.750 | 1.034 | 1.681 | 1.763 | 1.552 | 0.412 | 0.583 | 0.792 | 1.320 | 1.461 | 0.418 | 0.432 | 0.439 | 0.454 | 0.589
LSTNet | MSE | 1.293 | 1.456 | 1.997 | 2.655 | 2.143 | 2.742 | 3.567 | 3.242 | 2.544 | 4.625 | 1.968 | 1.999 | 2.762 | 1.257 | 1.917 | 0.369 | 0.394 | 0.419 | 0.556 | 0.605
MAE | 0.901 | 0.960 | 1.214 | 1.369 | 1.380 | 1.457 | 1.687 | 2.513 | 2.591 | 3.709 | 1.170 | 1.215 | 1.542 | 2.076 | 2.941 | 0.445 | 0.476 | 0.477 | 0.565 | 0.599
Informer- | MSE | 0.620 | 0.692 | 0.947 | 1.094 | 1.241 | 0.753 | 1.461 | 3.485 | 2.626 | 3.548 | 0.306 | 0.465 | 0.681 | 1.162 | 1.231 | 0.334 | 0.353 | 0.381 | 0.391 | 0.492
MAE | 0.577 | 0.671 | 0.797 | 0.813 | 0.917 | 0.727 | 1.077 | 1.612 | 1.285 | 1.495 | 0.371 | 0.470 | 0.612 | 0.879 | 1.103 | 0.399 | 0.420 | 0.439 | 0.438 | 0.550
Informer | MSE | 0.577 | 0.685 | 0.931 | 1.128 | 1.215 | 0.720 | 1.457 | 3.489 | 2.723 | 3.467 | 0.323 | 0.494 | 0.678 | 1.056 | 1.192 | 0.344 | 0.368 | 0.381 | 0.406 | 0.460
MAE | 0.549 | 0.625 | 0.752 | 0.873 | 0.896 | 0.665 | 1.001 | 1.515 | 1.340 | 1.473 | 0.369 | 0.503 | 0.614 | 0.786 | 0.926 | 0.393 | 0.424 | 0.431 | 0.443 | 0.548
HI | MSE | 0.426 | 0.498 | 0.653 | 0.690 | 0.714 | 0.266 | 0.379 | 0.572 | 0.567 | 0.635 | 1.395 | 1.668 | 0.423 | 0.526 | 0.655 | 0.328 | 0.212 | 0.247 | 0.469 | 0.518
MAE | 0.390 | 0.423 | 0.509 | 0.527 | 0.563 | 0.304 | 0.374 | 0.481 | 0.500 | 0.530 | 0.720 | 0.821 | 0.387 | 0.444 | 0.508 | 0.329 | 0.279 | 0.312 | 0.439 | 0.471
Improve | MSE | 26% | 27% | 30% | 37% | 41% | 63% | 74% | 82% | 78% | 82% | 356% | 259% | 38% | 50% | 45% | 2% | 40% | 34% | 20% | 13%
MAE | 29% | 32% | 32% | 39% | 37% | 54% | 63% | 68% | 61% | 64% | 95% | 75% | 37% | 44% | 45% | 16% | 34% | 28% | 0% | 14%

##  4 Discussion

Given above results, it could be concluded that as a naive method, HI is
powerful but unfortunately ignored as a baseline in LSTF tasks. However, it is
more important why it is powerful and how we could benefit from it.

###  4.1 Why Historical Inertial Is Powerful

A common belief is that predictable time series should have tractable patterns
in phase and magnitude. The very first reason HI is powerful is that it
guarantees the outputs are in similar magnitude of the inputs. This is
especially true in the scenario of long sequence time-series forecasting,
because the temporal patterns of a time-series can be more steady if viewed in
the long run.

However, phase is much more tricky. On the one hand, longer time series
provide more evident periodic patterns that can not be reflected in short
horizons. This increases the chance that the HI be of similar phase as the
prediction target, especially in the case that the prediction length is an
exact integer multiple of the time series’ period when their is any. On the
other hand, HI could also badly hurt the prediction results when 1) there is
no periodic pattern; 2) the periodic pattern is not included in historical
data; 3) or the historical data is in opposite phase as the prediction target.
The multivariate prediction results on ETTm1 dataset serves a good evidence of
above statements. Since the data was sampled by 15-minute, a prediction length
of 24 or 48 is too short to reflect periodic patterns. Therefore, HI performs
much worse than SOTA models. However, for predicting length of 96, 288 and
672, where the 1-day (4$\times$24 data-points) period is well covered, the
relative improvement surges.

###  4.2 Benefit from Historical Inertia

Being so powerful, HI has the potential to serve as an effective trick. We now
discuss possible ways of implementations from the perspectives of post-process
and pre-process.

####  4.2.1 Hybrid Model

A model may benefit from combining the basic model’s and HI’s results in a
post-process fashion. For example, the simplest implementation would be making
weighted summation of the two prediction sequences.

####  4.2.2 AutoML

The modeling capacity of complex model is definitely valuable, but just in
some cases the answer to the question is too simple, which might not be
answered well when it is complicated by the model. It is desirable that a
model’s structure or complexity can be adaptable to the input, which is also
known as automated machine learning (AutoML) (e.g. Yao et al., 2018). A simple
implementation could be when a specific dataset is given, the model may first
analyze its temporal pattens in a pre-processed way, and then score whether
the basic model, HI or some median variants should be used for prediction.

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# Inverse Potentials for $\alpha-\alpha$ Scattering using Gaussian Interaction

Anil Khachi1, Shikha Awasthi2, Lalit Kumar2 and O.S.K.S Sastri2

1 St. Bede’s College, Navbahar, Shimla, India
2Department of Physics and Astronomical Sciences,
Central University of Himachal Pradesh,
Dharamshala, India

###### Abstract

Alpha-alpha scattering phase shifts (SPS) have been determined using phase
function method (PFM) with local nuclear interaction between two alpha
clusters modeled as double Gaussian potential and Coulomb repulsion in the
form of $erf()$ function. The obtained SPS for various $\ell$-channels have
been compared with experimental ones with mean absolute percentage error
(MAPE) as a measure. The model parameters have been optimised by suitable
optimisation technique by looking for minimum value of MAPE. The results for
$\ell$=0+, 2+ and 4+ partial waves have been obtained, to match with
experimental SPS, with MAPE values of $7.6$, $2.7$ and $6.5$ respectively for
data up to $120$ MeV, while for higher states 6+, 8+ and 10+ has MAPE of
$1.7,7.4$ and $10.4$ respectively for data from $53-120$ MeV. On extrapolation
for data in range Eℓab. = $23-120$ MeV, using the optimised parameters, the
SPS are found to be in close agreement with experimental ones for first three
channels.

keywords: alpha-alpha scattering, phase function method (PFM), scattering
phase shifts, Double Gaussian potential

##  1 Introduction

To understand the structure of nucleus one of the essential methods involved
is scattering phenomena. The core purpose of this paper is to investigate
scattering of alpha particles which are in relative motion for all partial
waves. Rutherford and Chadwick were the first who experimentally studied
$\alpha-\alpha$ scattering in the year 1927 and since then a large amount of
experimental data is available given by (i) Afzal et al. [1] (ii) S. Chien and
Ronald E brown [2] (iii) Igo [3] (iv) Darriulat, Igo,Pug, Holm [4](v) Nilson
[5] and others. Alpha-alpha problem has been extensively studied both
experimentally and theoretically with alpha particle having some sole
properties like (i) zero spin and isospin (ii) tight binding energy of $28.3$
MeV having property to form cluster like states for lighter nuclei (6,7Li,
9Be, 12C and 16O are $\alpha$-structured) with alpha particle being the core
nuclei in the cluster (iii) small root mean square radius of $1.44$ fm.
In 1940’s for $\alpha-\alpha$ scattering experiments only naturally occurring
$\alpha$-sources like polonium, thorium and radium were used, which did not
result in very accurate results from experiments. Later on, with the
advancement of technology, accelerators were used in scattering processes and
highly accurate phase shifts were observed. The importance of $\alpha-\alpha$
scattering is that the study provides information regarding force field in
nearest surrounding of He-nuclei and also provide information regarding energy
levels of 8Be nucleus.
Haéfner was one of the first who studied 8Be properties in 1951 by using
phenomenological potential [6]. Later on Nilson, Briggs, Jentschke and others
used Haéfner potential for $\ell=$2 state from the ground state of $96$ KeV.
Later on, Nilson et al. [5] extended Hafner model [6] to include $\ell$= 0, 2
and 4 and found that best agreement of phase shift with experiments require
small value of $R=3.49$ fm. Also, Nilson found that testing the efficiency of
the potential required data above $22.9$ MeV, which was unfortunately not
available at that time. Later in the year 1958 Spuy and Pienaar [7] made
phenomenological analysis up to $6$ MeV where they concluded that for $E<6$
MeV one needed velocity dependent interaction for fitting S and D waves. Later
on, Wittern [8] in the year 1959 derived same semi-phenomenological potential
and reached to same conclusion as given by Spuy and Pienaar. Later from
1960-1965 more phenomenological study was done by Igo who made optical model
analysis for $\alpha-\alpha$ in range $E=23.1-47.1$ MeV and Darriulat, Igo and
Pugh [3] who used energy independent but strongly $\ell$ dependent complex
Wood Saxon potential for range $E=53-120$ MeV where they failed to fit the
phase shifts using one single common potential for all the partial waves.
Thus, it has been concluded, that single potential common to all $\ell$ do not
exist phenomenologically. In short, $\alpha-\alpha$ potentials are found to be
strongly $\ell$ dependent.
Buck et al. [9] used Gaussian potential of the form

$V(r)=-V_{a}exp{(-\alpha r^{2})}+z_{1}z_{2}\frac{e^{2}}{r}erf(\beta r)$ (1)

having single term with only two free parameters Va ($-122.6225$ MeV) and
$\alpha$ ($0.22$ $fm^{-2}$). The erf() function was included to take into
account for the Coulomb interaction and model parameters were obtained to fit
all even $\ell$ channels upto $E_{\ell ab.}<$ $80$ MeV. Ali and Bodmer [10]
proposed two term phenomenological potential with 4 parameters and was able to
fit the phase shift for $\ell$ = 0, 2 and 4 partial waves with good accuracy
to the experimental data. Later Darriulat et al., [4] used the following
Woods-Saxon potential, which has more than 6 parameters:

$V_{\alpha\alpha}(r)=\frac{U_{1}}{[1+exp(r-r_{1})/a_{1}]}-\frac{U_{2}}{[1+exp(r-r_{2})/a_{2}]}+i\frac{W}{[1+exp(r-r_{3})/a_{3}]}+V_{C}(r)$
(2)

with inelastic effects duly included in it for laboratory energy $53-120$ MeV
and $\ell$= 0, 2, 4, 6, 8 and 10 partial waves phase shift was fitted for
$E>53$ MeV.
Phase function method (PFM) was used by Jana et al. [11] for $\alpha-\alpha$
scattering by using angular momentum-dependent complex Saxon-Woods potential
as was suggested by Darriulat et al for $\ell$=0,2 and 4 partial wave only.
Odsuren et al. [12] calculated partial scattering cross section using Gaussian
potential for $\alpha-\alpha$, $n-\alpha$ and $p-\alpha$ using complex scaling
method (CSM). Scattering phase-shifts are commonly obtained analytically using
S-matrix [13] and Jost function methods [14]. Recently, there has been a
renewed interest in the use of Phase function method (PFM) or Variable phase
approach (VPA) by Laha and group and they have applied the method for studying
various light nuclei scattering problems which include study of nucleon-
nucleon (N-N), nucleon-nucleus (N-n) and nucleus-nucleus (n-n) [15, 16, 17,
18] scattering using a variety of two term potentials such as modified Hulthen
and Manning-Rosen. While traditional S-matrix approaches depend on wave-
functions obtained by solving time independent Schr$\ddot{o}$dinger equation
(TISE), PFM requires only potential function to obtain the scattering phase
shifts.
In this paper, our main objective is to obtain the scattering phase-shifts for
alpha-alpha system using two term Gaussian potential and Coulomb term included
as an $erf()$ function in elastic region only i.e ($E_{\ell}$=$0-23$ MeV) by
employing PFM for all even partial waves including $\ell$ = 6, 8 and 10. We
have included above partial waves because although their contribution to total
cross section is small, but it can not be neglected. Above elastic region we
have extrapolated the results upto $120$ MeV energies. Instead of following
Ali procedure [10] we have given free run to the parameters for all partial
waves. We have extended the calculations for higher $\ell$ channels like
$\ell$=6,8 and 10 which is missing in Ali et al. work. PFM has been employed
in tandem with model parameter optimising technique to obtain the results.

##  2 Methodology:

###  2.1 Model of Interaction:

The interaction between the two alpha particles is written as a combination of
nuclear and Coulomb part as

$V_{\alpha\alpha}=V_{\alpha\alpha}^{(N)}+V_{C}$ (3)

where the nuclear part is

$V_{\alpha\alpha}^{(N)}(r)=V_{r}exp(-\mu_{r}^{2}r^{2})-V_{a}exp(-\mu_{a}^{2}r^{2})$
(4)

Here, Vr is attraction strength and Va is repulsion in MeV. $\mu_{r}$ and
$\mu_{a}$ are the inverse ranges in $fm^{-2}$. The Coulomb potential $V_{c}$
is having form

$V_{c}=\frac{4e^{2}}{r}erf(\beta
r)~{}~{}\text{and}~{}~{}\beta=\frac{\sqrt{3}}{2R_{\alpha}}$ (5)

Interaction from this expression is called as improved Coulomb interaction. It
is due to finite size of the $\alpha$-particles, which is given by RMS value
of its radius. That is, Rα = 1.44 fm.

###  2.2 Phase Function Method:

The Schr$\ddot{o}$dinger wave equation for a spinless particle with energy E
and orbital angular momentum $\ell$ undergoing scattering with interaction
potential V(r) is given by

$\frac{\hbar^{2}}{2\mu}\bigg{[}\frac{d^{2}}{dr^{2}}+\bigg{(}k^{2}-\frac{\ell(\ell+1)}{r^{2}}\bigg{)}\bigg{]}u_{\ell}(k,r)=V(r)u_{\ell}(k,r)$
(6)

Where

$k_{c.m}=\sqrt{\frac{2\mu E_{c.m}}{{\hbar^{2}}}}~{}fm^{-1}$ (7)

with $\frac{\hbar^{2}}{2\mu}=10.44217$ MeV $fm^{2}$. For $\alpha-\alpha$
system, centre of mass energy $E_{c.m.}$ is related to laboratory energy by
following relation for non-relativistic kinematics

$E_{c.m}=\frac{M_{\alpha}}{M_{\alpha}+M_{\alpha}}E_{\ell ab}=0.5E_{\ell ab}$
(8)

PFM or VPA is one of the important tools in scattering studies for both local
[19] and non-local interactions [11]. The mathematical foundation of PFM
method is well known in theory of differential equations, that a linear
homogeneous equation of second order, such as Schr$\ddot{o}$dinger equation ,
can be reduced to a nonlinear differential equation (NDE) of first order-the
Riccati equation [20]. The phase equation which was independently worked out
by Calogero [21] and Babikov [22] is written in the following form.

$\delta_{\ell}^{\prime}(k,r)=-\frac{V(r)}{k(\hbar^{2}/2\mu)}\big{[}\cos(\delta_{\ell}(r))\hat{j}_{\ell}(kr)-\sin(\delta_{\ell}(r))\hat{\eta}_{\ell}(kr)\big{]}^{2}$
(9)

This NDE is numerically integrated from origin to the asymptotic region using
suitable numerical techniques, thereby obtaining directly the values of
scattering phase shift for different values of projectile energy in laboratory
frame. The central idea of VPA is to obtain the phase shift $\delta$ directly
from physical quantities such as interaction potential V(r), instead of
solving TISE for wave functions u(r), which in turn are used to determine
$\delta_{\ell}(k,r)$. With initial condition $\delta(0)=0$. The phase shift
$\delta_{\ell}$ can be seen as real function of $k$ and characterizes the
strength of scattering of any partial wave i.e. say $\ell^{th}$ partial wave
of the potential V(r). In the above equation and are the Bessel functions.
Since we are only focusing on obtaining scattering phase shifts for $\ell$= 0
partial wave, the Riccati-Bessel function [21] is given by
$\hat{j_{0}}=\sin(kr)$ and similarly the Riccati-Neumann function is given by
$\hat{\eta_{0}}=-\cos(kr)$, thus reducing eq. 9 to

$\displaystyle\delta_{0}^{\prime}(k,r)$ $\displaystyle=\begin{aligned}
&-\frac{V(r)}{k(\hbar^{2}/2\mu)}\bigg{[}\sin(kr+\delta_{0})\bigg{]}^{2}\\\
\end{aligned}$ (10) PFM equation for D-wave takes following form
$\displaystyle\delta_{2}^{\prime}(k,r)$ $\displaystyle=\begin{aligned}
&-\frac{V(r)}{k(\hbar^{2}/2\mu)}\bigg{[}-\sin{\left(kr+\delta_{2}\right)}-\frac{3\cos{\left(\delta_{2}+kr\right)}}{kr}+\frac{3\sin{\left(\delta_{2}+kr\right)}}{k^{2}r^{2}}\bigg{]}^{2}\\\
&\end{aligned}$ (11) PFM equation for G-wave takes following form
$\displaystyle\delta_{4}^{\prime}(k,r)$ $\displaystyle=\begin{aligned}
\begin{cases}&-\frac{V(r)}{k(\hbar^{2}/2\mu)}\bigg{[}\sin{\left(\delta_{4}+kr\right)}+\frac{10\cos{\left(\delta_{4}+kr\right)}}{kr}-\frac{45\sin{\left(\delta_{4}+kr\right)}}{k^{2}r^{2}}-\frac{105\cos{\left(\delta_{4}+kr\right)}}{k^{3}r^{3}}\\\
&+\frac{105\sin{\left(\delta_{4}+kr\right)}}{k^{4}r^{4}}\bigg{]}^{2}\end{cases}\end{aligned}$
(12) PFM equation for $\ell$=6 is $\displaystyle\delta_{6}^{\prime}(k,r)$
$\displaystyle=\begin{aligned}
&-\frac{V(r)}{k(\hbar^{2}/2\mu)}\bigg{[}\cos(\delta_{6}(k,r))\hat{\tau}(kr)-\sin(\delta_{6}(r))\hat{\xi}(kr)\bigg{]}^{2}\end{aligned}$
(13) where $\hat{\tau}(kr)$ is calculated as $\displaystyle\hat{\tau}(kr)$
$\displaystyle=\begin{aligned}
\begin{cases}&(kr)^{6}\bigg{\\{}\bigg{(}-21(kr)\bigg{[}495-60(kr)^{2}+(kr)^{4}\bigg{]}\cos(kr)\\\
&+\bigg{[}-10395+4725(kr)^{2}-210(kr)^{4}+(kr)^{6}\bigg{]}\sin(kr)\bigg{)}\bigg{\\}}\end{cases}\end{aligned}$
(14) and $\hat{\xi}(kr)$ is calculated as $\displaystyle\hat{\xi}(kr)$
$\displaystyle=\begin{aligned}
\begin{cases}&(kr)^{6}\bigg{\\{}\bigg{(}-21(kr)\bigg{[}495-60(kr)^{2}+(kr)^{4}\bigg{]}\sin(kr)\\\
&+\bigg{[}-10395+4725(kr)^{2}-210(kr)^{4}+(kr)^{6}\bigg{]}\cos(kr)\bigg{)}\bigg{\\}}\end{cases}\end{aligned}$
(15) PFM equation for $\ell$=8 is $\displaystyle\delta_{8}^{\prime}(k,r)$
$\displaystyle=\begin{aligned}
&-\frac{V(r)}{k(\hbar^{2}/2\mu)}\bigg{[}\cos(\delta_{8}(k,r))\hat{\zeta}(kr)-\sin(\delta_{8}(k,r))\hat{\chi}(kr)\bigg{]}^{2}\end{aligned}$
(16) where we have $\displaystyle\hat{\zeta}(kr)$
$\displaystyle=\begin{aligned}
\begin{cases}&\frac{1}{(kr)^{8}}\bigg{\\{}9(kr)\bigg{[}-225225+30030(kr)^{2}-770(kr)^{4}+4(kr)^{6}\bigg{]}\cos(kr)\\\
&+\bigg{[}2027025-945945(kr)^{2}+51975(kr)^{4}-630(kr)^{6}+(kr)^{8}\bigg{]}\sin(kr)\bigg{\\}}\end{cases}\end{aligned}$
(17) and $\hat{\chi}(kr)$ is calculated to be $\displaystyle\hat{\chi}(kr)$
$\displaystyle=\begin{aligned}
\begin{cases}&\frac{1}{(kr)^{8}}\bigg{\\{}9(kr)\bigg{[}-225225+30030(kr)^{2}-770(kr)^{4}+4(kr)^{6}\bigg{]}\sin(kr)\\\
&-\bigg{[}2027025-945945(kr)^{2}+51975(kr)^{4}-630(kr)^{6}+(kr)^{8}\bigg{]}\cos(kr)\bigg{\\}}\end{cases}\end{aligned}$
(18) PFM equation for $\ell$=10 takes form
$\displaystyle\delta_{10}^{\prime}(k,r)$ $\displaystyle=\begin{aligned}
&-\frac{V(r)}{k(\hbar^{2}/2\mu)}\bigg{[}\cos(\delta_{10}(k,r))\hat{\kappa}(kr)-\sin(\delta_{10}(k,r))\hat{\gamma}(kr)\bigg{]}^{2}\end{aligned}$
(19) where $\hat{\kappa}(kr)$ is calculated to be
$\displaystyle\hat{\kappa}(kr)$ $\displaystyle=\begin{aligned}
\begin{cases}&-(55(kr)\bigg{[}11904165-1670760(kr)^{2}+51597(kr)^{4}-4684(kr)^{6}\\\
&+(kr)^{8}\bigg{]}\cos(kr)+\bigg{[}-654729075+310134825(kr)^{2}-18918900(kr)^{4}\\\
&+315315(kr)^{6}-1485(kr)^{8}+(kr)^{10}\bigg{]}\sin(kr))(kr)^{-10}\end{cases}\end{aligned}$
(20) and $\hat{\gamma}(kr)$ comes out to be $\displaystyle\hat{\gamma}(kr)$
$\displaystyle=\begin{aligned}
\begin{cases}&-(55(kr)\bigg{[}11904165-1670760(kr)^{2}+51597(kr)^{4}-4684(kr)^{6}\\\
&+(kr)^{8}\bigg{]}\sin(kr)+\bigg{[}-654729075+310134825(kr)^{2}-18918900(kr)^{4}\\\
&+315315(kr)^{6}-1485(kr)^{8}+(kr)^{10}\bigg{]}\cos(kr))(kr)^{-10}\end{cases}\end{aligned}$
(21)

These NDE’s equations (10-19) are numerically integrated from origin to the
asymptotic region using RK-4/5 method, thereby obtaining directly the values
of scattering phase shift for different values of projectile energy in lab
frame. The central idea of VPA is to obtain the phase shift $\delta$ directly
from physical quantities such as interaction potential V(r), instead of
solving TISE for wave functions $u(r)$, which in turn are used to determine
$\delta$.

##  3 Simulation of Results and Discussion

Here, we have used double Gaussian potential and calculated the SPS using
phase function method (PFM). Our obtained potentials are real and are obtained
at elastic energies, $E_{\ell ab.}$= 0-23 MeV. The extrapolated curves are
just an indication of how phase shift varies at higher energies (E $>$ 23 MeV)
and are observed to be in good match with the experimental data upto 120 MeV
on extrapolation. Some distortion from the experimental data above 23 MeV may
be due to the absence of inelastic potential in our calculations. We have
extended our work by taking additional $\ell$ = 6, 8 and 10 partial wave data
which have not been investigated using PFM in any recent literature.
Potentials associated with partial waves are shown in figures 3. Standard
procedures were used to numerically calculate the phase shifts using equations
(10-19). Our approach has been to vary all the 4 parameters for $\ell$ = 0-10,
but we observed that for $\ell$ = 0, 2 and 4 all parameters are required to
get their respective phase shifts but for $\ell$ = 6, 8 and 10, it was
observed that keeping or removing the repulsive core did not effect the phase
shifts i.e. MAPE almost remained same, this must be due to the centrifugal
barrier concealing the inner repulsive core. So, only two parameters were
adjusted by to get phase shifts for $\ell$ = 6, 8 and 10 partial waves.
Similar observation was found in work done recently by Darriulat [4] and Laha
et al. [15] where it was observed that a single static potential can not give
phase shift for all the partial waves. Double term Gaussian potential is
giving similar nature potential as was given by Darriulat et al. with
repulsive core and an attractive outer region. At $E_{\ell ab.}$ = 0.6 MeV the
value of our phase shift for $\ell$ = 0 is 182.2 degree while experimental
value is 178 $\pm$ 1 degree. Phase shift for $\ell$ = 0 changes its sign at
$E_{\ell ab.}$ = 20 MeV, which is in close agreement to the work done by Ali
and Bodmer [10], Tombrello and Senhouse [23] and Afzal [1] and continues to be
in negative phase up to highest beam energy used ($120$ MeV). Our S-wave phase
shift are consistent with experimental data (with MAPE=0.86$\%$) above $23$
MeV even on extrapolation and can be clearly observed in figure 2. D-wave
phase shift become appreciable at $E_{\alpha}$ $>2.5$ MeV and at $E_{\ell
ab}$=11.8 MeV maximum value of our phase shift for $\ell$ = 2 is 115.56 degree
while experimental value is 114.9$\pm$2 degree, while the G-wave shift rises
from $E_{\alpha}$ $>11$ MeV and at $E_{\ell ab}$=77.5 MeV maximum value of our
phase shift for G-wave is $140.52$ degree while experimental value is 137
$\pm$ 1.8 degree. The $\ell$=6 and $\ell$=8 phase shifts can be seen to become
active above $E_{\alpha}$ $\approx$ $30$ and $50$ MeV respectively.
The repulsive core for $\ell$=0 is more strong than D, G, I, K and M partial
waves which can be observed form figure 3 and thus our potential is consistent
with the results of Darriulat and Laha et al. Here it should be noted that the
attractive interaction between two alpha particles is an important element
which is responsible for keeping the nuclei against Coulombic repulsion. Also
here it should be noted that the potential for $\ell$=6,8 and 10 in figure 3
should not be taken too seriously since experimental data above $54$ MeV was
only considered to get the nature of interaction so it must be dealt
cautiously. By utilising PFM method we have been able to construct appropriate
phenomenological potential for entire range of energy for $\ell$=0-10 partial
waves without incorporating the effects of inelastic processes. Including
inelastic process may further bring down the MAPE and may even give
approximate realistic potentials for the entire energy range.

States |  | $V_{a}$
---
(MeV)
| $\mu_{a}$
---
$(fm^{-2})$
| $V_{r}$
---
(MeV)
| $\mu_{r}$
---
$(fm^{-2})$
| MAPE
---
(%)
$S^{+}$ | 15.2767 | 0.1851 | 693.912 | 0.942 | 7.6
$D^{+}$ | 73.9102 | 0.4741 | 334.852 | 1.0441 | 2.7
$G^{+}$ | 246.917 | 0.47458 | 235.731 | 3.9989 | 6.5
$I^{+}$ | 184.747 | 0.63178 | – | – | 1.7
$K^{+}$ | 6.28974 | 0.2271 | – | – | 7.4
$M^{+}$ | 12.2422 | 0.25907 | – | – | 10.4
Figure 1: Phase shift for S,D,G (Left) and I,K, M (Right) states. Experimental
data is taken from Heydenberg and Temmer, Russel, Phillips and Reich and
Jones, Phillips and Miller E=0.15-9 MeV (1956-60), Nilson, Jentschke, Briggs,
Kerman, Snyder E=12.3-22.9 MeV (1958). Figure 2: Phase shift for S,D,G,I,K and
M-states. From 23 MeV on wards we have extrapolated data all the way up to 120
MeV only for first three channels. Experimental data is taken from Heydenberg
and Temmer, Russel, Phillips and Reich and Jones, Phillips and Miller E=0.15-9
MeV (1956-60), Nilson, Jentschke, Briggs, Kerman, Snyder E=12.3-22.9 MeV
(1958), Igo E=23.1-47.1 MeV (1960), Tombrello and Senhouse E=3.84-11.88 MeV
(1963), Darriulat, Igo and Pugh E=53-120 MeV (1965) and Chen and Ronald
E=18-29.5 MeV (1974). Figure 3: Potential plots for all channels without
centrifugal potential a function of distance $r$(fm).

###  3.1 Phase shift, amplitude and wavefunction vs. distance $r$(fm)

One can use equations provided by Zhaba [25] to obtain scattering amplitudes
and wavefunctions for various partial waves .The phase shift, amplitude and
wavefunction plots for $\ell$=0, 2 and 4 states are shown in figure 4.

Figure 4: Phase shift vs. $r$ and amplitude vs. $r$ plots for $\ell$=0, 2 and
4 states.

Amplitude function equation for $\ell=0$ is given as $\displaystyle
A_{0}^{\prime}=$ $\displaystyle\begin{aligned}
&-\frac{A_{0}V(r)}{k\left(\frac{\hbar^{2}}{2\mu}\right)}\left[\cos\delta_{0}\sin(kr)-\sin\delta_{0}(-\cos(kr))\right]\\\
&\times\left[\sin\delta_{0}\sin(kr)+\cos\delta_{0}(-\cos(kr))\right]\end{aligned}$
(23) Amplitude function equation for $\ell=2$ is given as $\displaystyle
A_{2}^{\prime}=$ $\displaystyle\begin{aligned}
&-\frac{A_{2}V(r)}{k\left(\frac{\hbar^{2}}{2\mu}\right)}\left[\cos\delta_{2}\left(\left(\frac{3}{(kr)^{2}}-1\right)\sin(kr)-\frac{3}{(kr)}\cos(kr)\right)\right.\\\
&\times\left(\sin\delta_{2}\left(\left(\frac{3}{(kr)^{2}}-1\right)\sin(kr)-\frac{3}{(kr)}\cos(kr)\right)+\cos\delta_{2}\left(\left(-\frac{3}{(kr)^{2}}+1\right)\cos(kr)\right.\right.\\\
&\left.\left.-\frac{3}{(kr)}\sin(kr)\right)\right]\end{aligned}$ (24)
Similarly one can obtain amplitude equation for $\ell=4$ wave.

####  3.1.1 Wavefunction vs. distance $r$(fm)

The equation and figures for wavefunction vs. $r$(fm) for $\ell$=0, 2 and 4
states are given in figures 5.

Wavefunction equation for $\ell=0$ is given as $\displaystyle u_{0}(r)=$
$\displaystyle\begin{aligned}
A_{0}(r)\left[\cos\delta_{0}(r)\cdot\sin(kr)-\sin\delta_{0}(r)\cdot\cos(kr)\right]\\\
\end{aligned}$ (27) Wavefunction equation for $\ell=2$ is given as
$\displaystyle u_{2}(r)=$ $\displaystyle\begin{aligned}
&A_{2}(r)\left[\cos\delta_{2}(r)\left(\left(\frac{3}{(kr)^{2}}-1\right)\sin(kr)-\frac{3}{(kr)}\cos(kr)\right)\right.\\\
&\left.-\sin\delta_{2}(r)\left(-\left(-\frac{3}{(kr)^{2}}+1\right)\cos(kr)-\frac{3}{(kr)}\sin(kr)\right)\right]\end{aligned}$
(28) Similarly one can obtain $\ell=4$ wavefuncton

Figure 5: Wavefunction vs $r$(fm) plots for $\ell$=0, 2 and 4 states.

###  3.2 Cross-section:

Partial cross-section plot is shown in Fig. 2 for different states calculated
using the obatained SPS values. Partial cross-section has been calculated by
expression [24]

$\sigma_{\ell}=\frac{4\pi}{k^{2}}(2\ell+1)\sin^{2}{\delta_{\ell}(k)}$ (30)

Figure 6: Cross section for D and G channels as function of center of mass
energy $E_{C.M}$.

##  4 Conclusion

The scattering phase shifts for $\ell$ = 0 (S-channel), $\ell$ = 2 (D-channel)
and $\ell$ = 4 (G-channel) have been computed upto 23 MeV and the best fitted
parameters are found to give good match with the experimental data when
extrapolated to the inelastic region of E $>$ 40 MeV up to $120$ MeV. For
$\ell$ = 6 (I-channel), $\ell$ = 8 (K-channel) and $\ell$ = 10 (M-channel)
also, a good match has be seen with the experimental data. Including inelastic
process may further bring down the MAPE and may even give approximate
realistic potentials for the entire energy range and shall be taken up in
future. At last, the computed cross section results are compared with that of
Odsuren et al., and are found to be in good agreement. In addition to earlier
works done by Laha et al., using PFM for $\ell$ = 0, 2 and 4, we have explored
the applicability of PFM to even higher states upto $\ell$ = 6, 8 and 10.
Thus, PFM stands as an efficient tool for phase shift calculations in quantum
mechanical scattering problems for local as well as non-local potentials. We
may summarise our entire work in 3 main points:
(i) Double Gaussian potential results in effective inverse interaction
potentials for alpha-alpha scattering for all $\ell$-channels.
(ii) Phase Function Method has been shown to be an effective tool for phase
shift calculations.
(iii) Phases shifts calculated for elastic region have been found to give good
results for inelastic energy data upto $120$ MeV for all partial waves.

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Parton distribution functions and intrinsic
charm at LHCb

Cristina Sánchez Gras

on behalf of the LHCb collaboration

Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands

Presented at the Low-$x$ Workshop, Elba Island, Italy, September 27–October 1
2021

###### Abstract

AAt LHCb, proton parton distribution functions (PDFs) can be studied in a
unique phase space complementary to that accessible by ATLAS and CMS,
corresponding to low and high values of Bjorken-$x$. The measurements of
vector boson production in the forward region, with and without an associated
jet, are presented. These measurements can be used to constrain the proton
PDFs, and in particular, the production of a $Z$ boson in association with a
$c$-jet can be studied to measure the intrinsic charm content of the proton.

##  1 Introduction

Initially designed for the study of forward beauty and charm physics, the LHCb
detector has the pseudorapidity ($\eta$) coverage $2<\eta<4.5$ lhcbdetector .
Its remarkable vertex reconstruction and particle identification performance,
together with its high momentum resolution, have now established it as a
general purpose detector. The forward coverage allows LHCb to reach
Bjorken-$x$ values (where $x$ is the fraction of momentum carried away by a
parton) complementary to those accessible by other general purpose detectors,
such as CMS cms  and ATLAS atlas . This allows to probe proton parton
distribution functions (PDFs) at very low- and high-$x$ values.

On the low-$x$ side, central exclusive production (CEP) of $J/\psi$ and
$\psi(2S)$ mesons in proton-proton ($pp$) collisions can probe the gluon PDF
down to $x\sim 10^{-6}$ lhcbCEP . For high values of $x>0.1$, $Z$ boson
production in association with charm jets can be used to determine the
intrinsic charm content of the proton lhcbIC . The most recent LHCb results
for these two cases are presented in this note.

##  2 Central exclusive production of $J/\psi$ and $\psi(2S)$ mesons in $pp$
collisions

The diffractive process $pp\rightarrow p+X+p$ in which two protons stay
unscathed following their interaction is known as central exclusive production
(CEP). The proton interaction takes place via the exchange of colourless
objects. In the case of vector meson production, the exchange of a photon and
a pomeron receives the name of photoproduction. The cross-section for
photoproduction is proportional to the square of the gluon PDF (at leading
order) in perturbative quantumchromodynamics (QCD). In $pp$ collisions with
the LHCb coverage, the gluon PDF can then be proved at very low-$x$ values of
$x\sim 10^{-6}$.

This process has a very distinctive signature: low final state multiplicity,
as only the muons that follow the meson decay are present in the detector, and
large rapidity gaps (regions of no activity) around the dimuon system. The
latter can be spoiled when one of the protons dissociates after the
interaction, making it an inelastic CEP process. The proton remnants in that
case are produced outside the $2<\eta<5$ coverage and escape detection. To
veto these processes, three (two) scintillator stations are in place around
the beam pipe upstream (downstream) from the interaction point, conforming
HeRSCheL. The HeRSCheL (High Rapidity Shower Counters for LHCb) detector
consists on five 60 cm x 60 cm stations equipped with four scintillating pads
each herschel . It increases the LHCb coverage to $1.5<\eta<10$ and
$-10<\eta<5,\ -3.5<\eta<-1.5$ in the forward and backward regions,
respectively. Charged particles produced when a proton dissociates trigger
detection by the scintillating pads. A $\chi^{2}$-like figure of merit built
with the activity registered at the HeRSCheL stations is used to veto these
events.

CEP events are selected by requiring exactly two muon tracks and imposing a
veto on the activity in HeRSCheL. The number of signal events is obtained by
fitting the dimuon squared transverse momentum ($p_{\rm T}^{2}$) distribution.
Following Regge theory, the cross-section for $J/\psi$ and $\psi(2S)$ CEP
events follows ${\rm d\sigma}/{\rm d}y\sim\exp(-bp_{\rm T}^{2})$, with $b\sim
6$ (GeV$/c)^{-2}$. The background arising from the dissociation of one of the
protons is parametrised in a sample with the HeRSCheL veto inverted. Aside
from this, two more backgrounds need to be accounted for: nonresonant dimuon
production and feed-down to $J/\psi$ from $\psi(2S)$ and $\chi_{c_{J}}\
(J=0,1,2)$. Nonresonant dimuon production takes place when both protons
interact electromagnetically via photon-photon exchange. This background is
measured by fitting the dimuon mass spectrum, and its $p_{\rm T}^{2}$ shape
modelled from simulated samples. The $J/\psi$ feed-down background concerns
$\chi_{c_{J}}\rightarrow J/\psi\gamma$ and $\psi(2S)\rightarrow J\psi X$
decays where only the $J/\psi$ is fully reconstructed. Its contribution is
estimated combining data and simulation. The $J/\psi$ and $\psi(2S)$ $p_{\rm
T}^{2}$ distributions are shown in Fig. 1. A fit to the background-subtracted
shapes is used to determine the number of signal events.

The result for the differential cross-section in rapidity bins is shown in
Fig. 3, as well as the leading order (LO) and next-to-leading order (NLO)
Jones-Martin-Ryskin-Teubner (JMRT) theory descriptions JMRTLO ; JMRTNLO . In
the $J/\psi$ case, the data is more in agreement with the NLO description,
especially in the higher rapidity bins. For the $\psi(2S)$ the same trend is
observed, but higher statistics are needed.

---
Figure 1: Squared transverse momentum ($p_{\rm T}^{2}$) distribution for CEP
$J/\psi\rightarrow\mu^{+}\mu^{-}$ (left) and
$\psi(2S)\rightarrow\mu^{+}\mu^{-}$ (right). The different backgrounds
described in the text are indicated.

---

Figure 2: Differential cross-section results for
$J/\psi\rightarrow\mu^{+}\mu^{-}$ (left) and
$\psi(2S)\rightarrow\mu^{+}\mu^{-}$ (right) compared to LO and NLO JMRT theory
descriptions JMRTLO ; JMRTNLO .

---

Figure 3: Photoproduction cross-section results for $J/\psi$ (left) and
$\psi(2S)$. The LHCb result at $\sqrt{s}=13\ \mathrm{TeV}$ is shown together
with the LHCb results at $\sqrt{s}=7\ \mathrm{TeV}$ lhcb7tev and those from
the H1 H1param ; H1psi2S ZEUS zeusCEP and ALICE aliceCEP collaborations, and
the results from the fixed target experiments E401 e401 , E516 e516 and E687
e687 .

The measured cross-section per rapidity bins allow for the calculation of the
photoproduction cross-section, $\sigma_{pp\rightarrow p\psi p}$, as:

$\sigma_{pp\rightarrow p\psi
p}=r(W_{+})k_{+}\frac{\mathrm{d}n}{\mathrm{d}k_{+}}\sigma_{\gamma
p\rightarrow\psi p}(W_{+})\ +\
r(W_{-})k_{-}\frac{\mathrm{d}n}{\mathrm{d}k_{-}}\sigma_{\gamma
p\rightarrow\psi p}(W_{-})\,,$ (1)

where $r(W_{\pm}$ is the gap survival factor, $k_{\pm}\equiv
M_{\psi}/2e^{\pm|y|}$ is the photon energy,
$\frac{\mathrm{d}n}{\mathrm{d}k_{\pm}}$ is the photon flux and
$W_{\pm}=2k_{\pm}\sqrt{s}$ is the photon-proton system invariant mass. The
positive (negative) signs in Eq. 1 refer to the situation where the photon is
emitted by the proton travelling parallel (antiparallel) to the LHCb beam
axis. In LHCb, $W_{+}$ and $W_{-}$ contribute to the same rapidity bin and
cannot be disentangled. However, given that only about a third of the data
corresponds to $W_{-}$ and this low-energy contribution has been parametrised
for the $J/\psi$ meson by the H1 collaboration H1param , their power-law
parametrisation is used to fix it. This power-law is scaled by the ratio of
the $\psi(2S)$ and $J/\psi$ cross-sections measured by H1 H1psi2S . The
estimated photoproduction cross-section is presented in Fig. 3 and compared to
the power-law H1 fit and their results H1param ; H1psi2S , as well as to
different results from other experiments. Also shown is the JMRT NLO theory
description. In the case of the $J/\psi$, where more data is available, it is
observed that the LHCb photoproduction cross-section values at $\sqrt{s}=13\
\mathrm{TeV}$ deviate from the power-law fit at higher rapidities and are in
more agreement with the JMRT NLO description. More data is necessary to
discern the behaviour of the $\psi(2S)$ photoproduction cross-section at high
rapidities.

##  3 Intrinsic charm with $Z$ bosons produced in association with charm jets

While the extrinsic charm ($c$) content of the proton (due to perturbative
gluon radiation) has been widely established, several theory predictions
suggest that the proton also contains charm intrinsically. This could take
place in a sea-quark-like manner or as a valence-like $c$ quark, transforming
the proton wave-function into $|uudc\bar{c}\rangle$, as predicted by Light
Front QCD (LFQCD). Previous measurements have been performed at low-$Q^{2}$
measIC1 ; measIC2 . At such low energy, the theoretical treatment of hadronic
nuclear effects is challenging, and it is difficult to understand the results
as evidence or not of the proton intrinsic charm (IC) content. Nevertheless,
global PDF analysis do not exclude it at the percent level percIC1 ; percIC2 .

A proposal was made to measure the ratio of $Z+c$ jets events to that of
$Z+$jets, $\sigma(Zc)/\sigma(Zj)$ in the forward region proposalIC .
Performing this measurement at high-$Q^{2}$ with $Z$ bosons at forward
rapidities allows to access high Bjorken-$x$ values with $x>0.1$, where the
hadronic and nuclear effects are negligible. Figure 4 illustrates how the
ratio $\sigma(Zc)/\sigma(Zj)$ at high $Z$ boson rapidities would allow to
discriminate the intrinsic charm content of the proton.

Figure 4: Theory predictions including and excluding intrinsic charm content
in the proton for $\sigma(Zc)/\sigma(Zj)$ proposalIC . The range $2<y(Z)<4.5$
corresponds to the LHCb forward region.

An integrated luminosity of 6 fb-1 corresponding to the full proton-proton
collision LHCb dataset at $\sqrt{s}=13\ \mathrm{TeV}$ is used lhcbIC . Events
with $Z\rightarrow\mu^{+}\mu^{-}$ and at least one jet with transverse
momentum $p_{\mathrm{T}}>20\ \mathrm{GeV/}c$ are selected. Charm jets are
identified by using a displaced-vertex (DV) tagger in bins of
$p_{\mathrm{T}}(\mathrm{jet}),y(Z)$. A two-dimensional fit to the corrected DV
mass and the number of tracks in the DV is performed to identify the flavour
of each jet in the selected events. The result of the fit is shown in Fig. 5.
The efficiency of tagging a jet as a charm jet is estimated in simulation and
calibrated in data. The $Z_{c}$ and $Z_{j}$ yields in each $y(Z)$ bin are
corrected for their selection efficiency and resolution effects at detection.

---
Figure 5: Result of the two-dimensional fit for the corrected mass
($m_{\mathrm{cor}}$) and number of tracks ($N_{\mathrm{trk}}$) in the
dispaced-vertex (DV). The contributions for charm, beauty and light jets are
shown.

The measurement of a ratio results in most of the systematic uncertainties
cancelling out. The dominant systematic uncertainty is related to the
efficiency of identifying charm jets. A comparison of the measured
$\sigma(Zc)/\sigma(Zj)$ values to different theory predictions is shown in
Fig. 6. The first two bins are compatible with both no IC and IC allowed
content. The bin at higher $Z$ boson rapidity is consistent with proton IC
models, and is about three standard deviations away from the prediction of no
intrinsic charm content. The measurement is statistically limited and more
data is needed to draw further conclusions. Moreover, these results need to be
added to global PDF analyses for interpretation.

Figure 6: Results for $\sigma(Zc)/\sigma(Zj)$ compared to theory predictions
allowing and excluding intrinsic charm content in the proton.

##  4 Conclusions

The LHCb detector can be used to perform precision QCD measurements, both in
the low- and high-$x$ regions. The central exclusive production cross-section
of $J/\psi$ allows to probe the region $x\sim 10^{-6}$. This measurement at
$\sqrt{s}=13\ \mathrm{TeV}$ is in agreement with the JMRT NLO description, and
further data is needed to observe if the same behaviour is present for
$\psi(2S)$. The high-$x$ region provides access to large values $x>0.1$, where
the intrinsic charm of the content can be probed. While statistically limited,
the first measurement of the proton intrinsic charm content in the forward
region in proton-proton collisions has been performed.

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