# Modeling the seasonal variability and the governing factors of Ocean
Acidification over the Bay of Bengal region

A.P Joshi1* and H.V Warrior1 1Department of Ocean Engineering and Naval
Architecture, IIT Kharagpur, Kharagpur-721302, West Bengal, India

###### Abstract

The Bay of Bengal (BoB) is a high recipient of freshwater flux from rivers and
precipitation, making the region strongly stratified. The strong
stratification results in a thick barrier layer formation, which inhibits
vertical mixing making this region a low-productive zone. In the present
study, we attempt to model the pH of the BoB region and understand the role of
different governing factors such as sea-surface temperature (SST), sea-surface
salinity (SSS), dissolved inorganic carbon (DIC), and total alkalinity (TALK)
on the seasonality of sea-surface pH. We run a set of sensitivity experiments
to understand the role of each of the governing factors. The results show that
the SST, SSS, and DIC are the principal drivers affecting the sea-surface pH,
while TALK plays a buffering role. The SST and DIC are consistently found to
be opposite to each other. The pre-monsoon season (MAM) has shown to have an
almost equal contribution from all the drivers. In the pre-monsoon season, the
SST and DIC are balanced by TALK and SSS. The role of SSS is significantly
dominant in the second half of the year. Both SST and SSS counter the role of
DIC in the southwest monsoon season. The strong stratification plays an
essential role in modulating the pH of the BoB region. The thickness of the
barrier layer formed in the sub-surface layers positively affects the sea-
surface pH. The northern BoB is found to be more alkaline than the southern
BoB. Our study highlights the complexity of ocean acidification in the BoB
region compared to the other part of the world ocean.

###### keywords:

Bay of Bengal (BoB) , pH , sea-surface Salinity (SSS) , Stratification ,
Barrier Layer Thickness (BLT)

††journal: Continental Shelf Research

##  1 Introduction

The pronounced absorption of the anthropogenic atmospheric carbon dioxide by
oceans ($2.5\pm 0.6$ GtC yr-1, [11]) causes a drop in oceanic pH referred to
as Ocean Acidification. A few studies indicate a drop of 0.1 units in the
upper ocean pH, and it is predicted to drop by almost 0.3 to 0.5 units by the
end of this century [32, 10, 21]. This reduction in sea-surface pH has a major
effect on the biological process and production in oceans [12].These long-term
variations in oceanic pH values include short-term fluctuations (diurnal,
seasonal, and interannual). The extent of vulnerability of these short-term
fluctuations is equal or sometimes greater than the long-term changes [29, 54,
46]. Hence to improve the predictions of ocean pH variability, identification
of drivers affecting it is of utmost importance.

The Bay of Bengal (BoB) region chosen in this study is peculiar by reference
to its geographical settings (enveloped by land from the three sides and open
in the south), the substantial freshwater influx from rivers and precipitation
[49], and the seasonal reversal of coastal currents [27, 40]. The abundance of
freshwater in the BoB region makes this region one of the challenging zones
for ocean acidification studies. The ocean acidification scenario for the BoB
region, as reported by [10], suggests the pH to be below 8.0 by 2050 and may
go below 7.8 in 2095. Since the BoB is a reservoir of an abundance of marine
species, especially shells and coral reefs, the ocean acidification scenario
presented by [10] is of great worry to the scientist, environmentalists, and
policymakers.

[26] noted the lowest pH in Mahanadi estuaries ($7.02\pm 0.1$) and highest in
the Ganges estuaries ($8.13\pm 0.24$). [35], based on ship observations in the
western coast of BoB, showed that the northwestern coast has higher pH
($8.45\pm 0.003$) than the southwestern coast ($8.12\pm 0.03$). The lowest pH
was observed in the Godavari estuaries ($8.07\pm 0.05$), while the Mahanadi
estuaries demonstrated the highest pH values ($8.41\pm 0.005$). The sharp
gradient between the northwestern and southwestern pH values are associated
with the difference in salinity ($31.96\pm 0.88$ for Godavari estuaries and
$23.24\pm 1.84$ for Mahanadi estuaries).

[34] compares the measured hydrographic properties and inorganic carbon
components of the BoB between March-April of 1991 and 2011, this work also
includes a time series observation of 8 years from 2005 to 2013. All these
observations were of the western coast of BoB. The study suggests that the
rate of decrease in pH in the southwestern coast was consistent with the
global trend, but the northwestern coast experienced a higher decrement rate
of pH (3 to 5 times higher). The time series observation at Vishakhapatnam
(located on the southwestern coast of BoB) and Paradip (located on the
northwestern coast of BoB) demonstrates a decline in the pH at -0.0015 units
year-1 and -0.005 units year-1, respectively. The higher declining rate in at
the Paradip station is attributed to enhanced aerosol depositions in the BoB.

The seasonal reversing coastal currents are known as the East India Coastal
Currents (EICC) or Western Boundary Currents (WBC), play a significant role in
controlling the sea-surface salinity (SSS) of the BoB region. From February to
May, the northward moving EICC increases the salinity, which weakens the
stratification, and enhances coastal upwelling, resulting in a decrease of
sea-surface pH [37]. The EICC flows southwards during October to December,
which brings in low saline, more basic waters from north to the western coast
of BoB, that increases the pH of this zone [37]. Since the circulation pattern
of BoB consists of many eddies, they play a significant role in modulating the
sea-surface pH. [36], based on the observation in the western BoB, reports the
presence of low pH values in the cyclonic eddy and no-eddy regions (upper
200m) of BoB, whereas the high pH waters extends up to 150 m to 175 m deep in
the anticyclonic eddy regions.

Using Multiple Linear Regression (MLR) [44] shows an increase in pH in the BoB
(1998-2015), the only exception being the head bay region. The near proximity
of the head bay to the land makes it vulnerable to the atmospheric and river
pollutants. The acidification of the head bay is associated with the rise in
sea-surface temperature and atmospheric depositions, as well as decrease in
freshwater discharge. The SSS plays a major role in controlling the
seasonality of the northern BoB (seasonal amplitude of $\approx$ 0.18 units)
[4]. [4] using interannual model for the North Indian Ocean (NIO) shows the
contribution of the various drivers on the seasonality of total pH and pCO2
for the Arabian as well as the BoB region.

In this study, we aim to model the seasonality of pH (based on the
climatological modeling) for the whole Bay of Bengal region and explore the
physical factors driving the seasonality of pH in this region. As the BoB is
recipient of large freshwater influx from precipitation and rivers, it affects
the stratification dynamics of the BoB. Through this study we attempt to
analyze the role of different drivers influencing the pH variability and the
role of stratification in monitoring the pH variability. The remaining paper
is structured as follows: Section 2 describes the Data and Methodology used in
this study; Results and Discussion are presented in section 3; Conclusion in
section 4.

##  2 Data and Methodology

###  2.1 Model

In our previous study [19], we exhibit a comparison between two types of model
configurations (the first one using the [9] bulk formulation to calculate the
wind stress and evaporation. In the second configuration, we externally
provide the Wind Stress and evaporation minus precipitation (E-P) data). We
concluded that the method of externally providing the wind stress and E-P data
emulated the carbonate chemistry satisfactorily. We provide an exhaustive
evaluation of the different physical (current, sea-surface temperature, sea-
surface salinity, barrier layer thickness, mixed layer depth) and carbonate
parameters (dissolved inorganic carbon, total alkalinity, pH, and pCO2) in
[19]. Using the same model, we further explore the influence of the freshwater
plume spreading and the barrier layer thickness on the sea-surface pCO2 [18].
Using this model we also explore the different mechanisms and drivers
affecting the BoB [20]. Since this study is an extension of our previous work,
to avoid recapitulation we provide a summarized model configuration
information. We encourage readers to refer to our previous works [19, 18, 20]
for a comprehensive model information.

Figure 1:  The contours represent the model bathymetry. The yellow box marks
the study region, while the red star shows the Rama buoy location. The broken
black line divides the study region into the Northern Bay of Bengal (N-BoB)
and the Southern Bay of Bengal (S-BoB). The rivers included in this study have
been demonstrated using green dots.

The present study couples the Regional Oceanic Modeling System (ROMS)
(physical model) [38, 39] to the Pelagic Interaction Scheme for Carbon and
Ecosystem Studies (PISCES) (biogeochemical model) [3, 2]. Fig.1 reveals the
study region, which is considered in this paper. The contours show the
bathymetry (extracted from ETOPO2 [42]), and the yellow box indicates the
domain of analysis. The rivers with high discharge volume are only included in
this study (refer to Fig.1). The climatological river discharge data is
included in the coupled model from [6, 7]. The horizontal resolution is
$1/7^{o}$ with 32 sigma layers as vertical resolution.

Near the surface, refinement is achieved by assigning 12 vertical layers in
the top 100 m at the maximum depth position. The K-profile parameterization
turbulence scheme (KPP model) [22] is adopted in the present study. We choose
the physical tracers for initial and lateral forcing from the World Ocean
Atlas, 2009 (WOA09) [1, 15]. The Comprehensive Ocean-Atmospheric Data Set
(COADS) [55] is exploited to provide the heat fluxes, temperature, humidity
(both relative and specific), and density of air. The E-P and shortwave
radiations are also provided from the COADS data set. The monthly climatology
of winds and wind stresses are used from the QuikSCAT satellite scatterometer
from 1999-2009 [24, 31].

We provide the nutrients and oxygen for the upper ocean from the WOA09
dataset, while for the interior oceans, oxygen and nutrients from the World
Ocean Atlas PISCES (WOAPisces) [3] are used. The carbonate parameters
(Dissolved Inorganic Carbon (DIC), Total Alkalinity (TALK), Dissolved Organic
Carbon (DOC), etc.) are taken from the WOAPisces dataset. The recent
parameterization scheme of [8] is used for the PISCES model.

Figure 2: The atmospheric pCO2 from the Rama buoy for the year 2014 (panel a),
2015 (panel b), and 2016 (panel c). The reconstructed atmospheric pCO2 of the
year 2015 is provided in panel d. The black broken line shows the mean
atmospheric pCO2.

For the atmospheric pCO2, we analyze the open ocean Rama buoy values (Fig.2).
As the observations are available for the 2014 full year, whereas January-June
for 2015 and March-December for 2016, we reconstruct the atmospheric pCO2 for
2015 using the available data. The reconstructed 2015 atmospheric pCO2 data
contains the original values for the January to June period, while the second
half of the year is constructed by averaging the observations from 2014 and
2016. Fig.2 d shows that the fluctuations in the atmospheric pCO2 are
relatively small, and the mean is 379.79 $\mu atm$. Since the previous study
[19] shows that the carbonate parameters are reasonably emulated by the
coupled model using 377 $\mu atm$ ($<1\%$ from the mean atmospheric pCO2 of
the reconstructed 2015 year), we continue using 377 $\mu atm$ for the present
study.

We run the ROMS model for 30 years before adding the PISCES model. Then both
the models coupled together are further run for another 30 years. Finally, we
average the last three years to create the climatology of the BoB region.

###  2.2 pH calculation and sensitivity of pH seasonality to its governing
factors

The sea-surface pH can be factorized as the effect of SST, SSS, DIC, and TALK
[48, 50, 43]. The effect of nutrients and other minor ions (like borate,
sulfate, and fluoride) on the sea-surface pH can be neglected and treated as
residuals [14]. Hence the sea-surface pH in its linearized form can be written
as:

$\frac{dpH}{dt}=\frac{\partial pH}{\partial DIC}\frac{dDIC}{dt}+\frac{\partial
pH}{\partial TALK}\frac{dTALK}{dt}+\frac{\partial pH}{\partial
SST}\frac{dSST}{dt}+\frac{\partial pH}{\partial SSS}\frac{dSSS}{dt}$ (1)

Based on the above equation, we construct the sea-surface pH for the BoB
region using the CO2SYS [23, 51] subroutine. The CO2SYS subroutine abides by
the guidelines set by Ocean Carbon-cycle Intercomparison Project (OCIMP). The
sea-surface pH calculated using CO2SYS is referred to as control pH (pHCTRL).

We perform four sensitivity experiments to understand the effect of each of
the governing factors (SST, SSS, DIC, and TALK) on the seasonal variability of
seas-surface pH. The annual climatological mean is provided to the variable of
interest, while other factors are allowed to evolve throughout the year. We
refer to the sensitivity cases by pHX, where X represents each governing
factor. The difference between pHCTRL and the pHX divulge the effect of the
variable or governing factor X.

###  2.3 Observational data used for model evaluation

We use two observational datasets to evaluate the modeled pH. The description
of these datasets are as follows:

####  2.3.1 Rama Buoy Data

The only buoy data available in the BoB region is located at 15° N, 90° E (as
shown in Fig.1). This buoy measures the ocean surface pH, pCO2, salinity, and
temperature. The data from this Rama buoy is available from 24 November 2013
to 20 November 2018. Though the buoy is reactivated from January 2020, the
data is not available for the public. [19] provide a table indicating the
three deployments of this buoy within the aforementioned period. The method
used for calculating the carbonate and physical variables is described
meticulously by [46]. Unfortunately, pH is the least continuous data measured
by the Rama buoy.

Figure 3: The observation pH data for the years 2014, 2015, and 2017 from Rama
buoy.

Fig.3 shows the available pH data from Rama buoy. The paucity of real time
observation is clearly depicted through Fig.3. January to May is a common
period between these three years, the pH data for December is available for
2013 and 2014, whereas the pH data for June is only present in 2014. However,
in-spite of the limitations recent studies have utilized the RAMA mooring data
for validation purpose [19, 44, 4]. To validate our climatology model, we
average the available observation data (Fig.5).

####  2.3.2 Takahashi Data

Due to the absence of observational data in the BoB region, we choose the
monthly pH climatology of [47] to evaluate the model performance. This data
will be referred to as the Takahashi Data in this manuscript and has been used
in previous studies for validating modeled pH [19, 44, 4]. The pH monthly
climatology is calculated using the observed pCO2 and calculated potential
alkalinity (PALK) data. The PALK is calculated using a PALK-salinity
relationship, which has shown an inability to capture the effect of the high
freshwater influx in the BoB region [47]. Hence the pH climatology from
Takahashi data showed a significant deviation from the observed pH of the BoB
region. The horizontal resolution of 4° $\times$ 5° does not allow the effect
of small-scale activities to be adequately captured. Despite the limitations
of the Takahashi data, the paucity of observed pH data justifies its use for
evaluating the model performance.

###  2.4 Statistics used to aid in model evaluation

Statistical Parameters |  Description
---|---
Correlation Coefficient (r) |  Compares the variability between the model and observation data. It ranges from -1 to +1, where the negative value indicates an inverse relationship while the positive values represent a positive relationship.
Average Error (AE) |
Absolute Average Error (AAE) |  These parameters represent the bias between observation and model data.
Root Mean Square Error (RMSE) |
Reliability Index (RI) |  The mean deviation between the model and observed value is given by RI. A value near zero indicates the model to be reliable [45].
Cost Function (CF) |  It is a measure of ”goodness of fit.” It rates the model in the following manner [5] :
|  CF $<$ 1 = excellent ; 1 $\leq$ CF $\leq$ 2 = good
|  2 $\leq$ CF $\leq$ 3 = average ; CF $>$ 3 = poor
Percentage Bias (PB) |  Similar to CF it is a performance indicator which rates the model in the following manner [5] :
|  PB $<$ 10 = excellent ; 10 $\leq$ PB $\leq$ 20 = very good
|  20 $\leq$ PB $\leq$ 30 = good ; PB $>$ 30 = poor
Model Efficiency Factor (MEF) |  It determines the predictability of the model concerning mean observed values. MEF quantifies the model efficiency in the following manner [25] :
|  MEF $\geq$ 0.65 = excellent ; 0.5 $\leq$ MEF $<$ 0.65 = very good
|  0.2 $\leq$ MEF $<$ 0.5 = good ; MEF $<$ 0.2 = poor
Table 1: Statistical parameters used in this study.

Table 1 provides a list of statistical indices employed in the present study
to evaluate the model performance. The formulation of each of these parameters
is provided in Appendix A of [19]; hence it is not repeated here.

##  3 Results and discussion

###  3.1 Model Evaluation

The present modeled pH performance is evaluated against the two observational
data described in sec2.3. When comparing the model and observation data, we
interpolate the model data to the grid resolution of the observation data
using the nearest-neighbor interpolation method.

Figure 4: In panel (a), the model annual mean pH (color contour) is overlaid
with the annual mean pH from the Takahashi data (black lines), and the bias
(Model - Observation) is shown in panel (b).

Fig.4a shows the modeled annual mean pH for the BoB overlaid with the
observations from Takahashi data. We observe that the model data shows high
spatial variability of the sea-surface pH, while the spatial variability in
the Takahashi data is low. The high pH value ($\approx$ 8.08 units) observed
from the model in the northern region indicates the alkaline nature of the
ocean. The nature of the southern region is comparatively acidic ($\approx$
8.04 units). The low salinity in the northern region due to the high discharge
from rivers and the precipitation could contribute to the high pH values in
the north [4]. As mentioned in sec2.3.2, the Takahashi data fails to account
for the freshwater, which may be attributed to the absence of high pH values
from the Takahashi data. The model tends to have a positive bias in the north
($\approx$ 0.01 units), as shown in Fig.4b. The pH gradually decreases
southwards (Fig.4a), this could be due to the high acidic and less discharge
from the peninsular rivers. The high saline waters of the Arabian Sea enters
the simulation domain decreasing the pH ($\approx$ 8.02 units) in the
southwestern region.

Figure 5: Monthly variation in the sea-surface pH from the model and the RAMA
buoy.

We compare The monthly variability of the modeled sea-surface pH for BoB
against the Rama buoy data in Fig.5. The paucity of data in the BoB region
adheres as a significant hindrance while evaluating the model. Despite the
observational data limitations, the model emulates the monthly cycle
variability satisfactorily while comparing with buoy data. The model shows a
overestimation in May and June, which could be due to the unavailability of
recent years data (since the pH is increasing in the central BoB [44]) and
relatively higher vertical mixing in the model [19, 18]. The high peak pH
value from model pH data noted in the December month could be due to the
maximum thickness of the barrier layer [18] inhibits mixing of acidic sub-
surface waters with the sea-surface. The FPS distributes the freshwater over
the sea-surface of the BoB region, which may affect the solubility and
dissociation constants (as they are functions of temperature and salinity).
The FPS begins in June, peaking over the post-monsoon season. Thus this
freshwater spread may have a significant role in determining the seasonal
variability of the pH in BoB. To quantify the model performance, we use
different statistical indices as mentioned in sec 2.4 (Table 2).

Model Vs. RAMA buoy data
---
AAE | AE | CF | r | RMSE | PB | RI | MEF
0.006 | 0.005 | 0.316 | 0.94 | 0.0063 | 0.056 | 1.0002 | 0.87
Table 2: Statistical comparison of model-simulated pH with the RAMA buoy data.

A reasonably low bias (0.006) is found when comparing the modeled pH with the
Rama buoy data. A high correlation with buoy data indicates that the model
well captures the monthly variability. A low reliability index indicate the
model to be reliable for modeling the sea-surface pH. The model is rated
excellent by all the other statics indices. Especially high MEF shows the
model can replicate the sea-surface pH. The good performance of the model
against the RAMA buoy data gives us the confidence to continue further
analysis of the sea-surface pH of the BoB region. Though we agree that the
data is less but this approach to validate modeled pH is popularly adopted in
many past studies [19, 44, 4].

Thus this section highlights the need of more pH observation data in the BoB.
We also observe that the SSS may modulate the seasonality of sea-surface pH in
the BoB. As the model performance seems fair, we attempt to evaluate and
analyze various factors driving the sea-surface pH in the BoB region using
this model.

###  3.2 Effect of individual drivers on the seasonality of pH

Figure 6: Comparison between Control pH (pHCTRL) and sensitivity of each
drivers (pHX). ’X’ denotes individual driver (SST, SSS, DIC, and TALK).

Fig.6 presents the effect of individual drivers, viz, SST, DIC, TALK, and SSS
on the domain (yellow box, shown in Fig.1) averaged pH. Each panel in Fig.6
shows the effect of these individual drivers on pH, respectively. The pHCTRL
is shown with a solid black line in each of the Fig.6 panels. The seasonality
in pHCTRL seems to be driven by 6 and 12 months signal. We observe a maximum
peak of pHCTRL in October (8.0942) and a minimum crest in May (8.04). It
implies that the sea-surface of BoB is most acidic in May and most alkaline in
October. The annual mean pH of the BoB is 8.0620.

In Fig.6a, we show the effect of DIC on the pH by replacing it with the annual
mean (as described in sec 2.2). The pHDIC is significantly higher than the
pHCTRL during the period between December and February. This high value
indicates that the role of DIC is to make the BoB acidic. From March-May, the
pHDIC is lower than pHCTRL, suggesting that the DIC tends to make the surface
of BoB alkaline. Interestingly, from June to November, the effect of DIC is
negligible; in fact, the primary variation to the ocean acidification of BoB
by DIC is found in the first half of the year. The second half of the year
shows an almost negligible effect of DIC on pHCTRL.

Fig.6b exhibits the effect of temperature (Temp) on the pH. We replace the
monthly variation of Temp by annual mean and reproduce pHSST. From November to
February the pHSST is lower than pHCTRL, which implies that during this
period, the effect of SST is to reduce the acidic nature of BoB. It should be
noted that SST tends to drop from November to February (winter months); this
may be the reason that during this period, the effect of SST is to make the
sea-surface water alkaline. From March to October, the pHSST is observed to be
higher than pHCTRL; hence it reveals that the effect of SST is to make the
sea-surface acidic in this period. The higher SST during these months may
enhance the acidic nature of the BoB.

The BoB region experiences high freshwater influx, and the effect of salinity
(SSS) is revealed in Fig.6c. It is clear that for the BoB region, the Sal
plays a significant role in modulating the sea-surface pH. From January to
August, the pHSSS is higher than pHCTRL, which suggests that the role of Sal
is to increase the acidic levels during this period. Whereas from August-
December, the lower values of pHSSS with respect to pHCTRL reveal that Sal
makes BoB alkaline during this period. These results suggest that
stratification and the spreading of freshwater plume could possibly heavily
modulate the values and dynamics of the sea-surface pH.

Fig.6d presents the effect of TALK on the sea-surface pH. We observe that the
TALK effect is only significant in February and March. The effect of TALK is
almost negligible for most of the climatological year. So it is evident from
Fig.6 that in the BoB region, DIC, SST, and SSS are the major drivers of the
seasonality of pH, while TALK has a negligible effect. Further, Fig.7
summarizes the results of Fig.6 to quantify the effect of each driver.

Figure 7: Difference between pHCTRL and pHX, where ’X’ represents each driver.
SENSX defines the sensitivity of the drivers SST, SSS, DIC, and TALK.

Fig.7 reveals a summary of the effect of each driver on the sea-surface pH of
the BoB. Unlike the studies related to the Arabian Sea [43] and the
observational studies from the Pacific ocean, Mediterranean Sea, and North-
Atlantic [14], the BoB pH is significantly affected by SSS, SST and DIC.
Whereas SST and DIC were the major drivers of pH in the Arabian Sea, Pacific
Ocean, North Atlantic Ocean, and Mediterranean sea [14, 43]. From Fig.7, we
observe that except for February to March period, the effect of TALK is
negligible. In February to March period, the effect of DIC and SST is balanced
by SSS and TALK. During December and January, the effect of SST is compensated
primarily by DIC, whereas both TALK and SSS have negligible effect. The period
of August to November is strictly dominated by SSS, making the sea-surface
alkaline. The peak in October coincides with the maximum freshwater plume
spread [16, 17, 18], which indicates that the freshwater which lowers the SSS
increases the pH of the BoB region. The acidic peak in May can be attributed
to the highest SSS in the BoB region and lowest freshwater plume spread [16,
17, 18]. This domination of SSS in controlling pH in the BoB region makes this
region unique from the rest of the world ocean.

Figure 8: Seasonal effect of the drivers of pH.

In Fig.8, we demonstrate the seasonal effect of each of the governing factors
on the sea-surface pH of the BoB region. The climatological year is divided
based on the monsoon season. The December to February (DJF) is the winter
monsoon season, the June to September (JJAS) is the southwest monsoon season,
while the phases March to May (MAM) and October-November (ON) are pre-monsoon
and post-monsoon seasons, respectively. In the winter monsoon season (DJF),
the SST tends to increase the pH, while the DIC averse the rise in pH, but the
minor effect of TALK further increments the pH. The effect of SSS is almost
zero in the winter monsoon season. The pre-monsoon season (MAM) experiences
the combined effect of all the drivers. The DIC tends to increase the pH
during MAM, but the combined effect of SSS, SST, and TALK reduces the pH. Thus
the pre-monsoon season is the most acidic season among all other seasons.

In the southwest monsoon season (JJAS), pH is dominated by SST. The SST tends
to reduce the pH in the southwest monsoon season. The effect of SSS, DIC, and
TALK is minimal during the southwest monsoon season. The SSS dominates the
rise in pH during the post-monsoon season. This may be attributed to the
freshwater plume spread during this period [16, 17, 18]. The DIC effect is
countered by the TALK in the post-monsoon season, while the SST has a
negligible effect on pH during this season.

Hence, our analysis shows that SSS is a major driver of pH along with SST and
DIC in the BoB region. The role of SSS in modulating the pH may be due to the
high freshwater flux, strong stratification, and thick barrier layer,
inhibiting vertical mixing. The seasonality in SSS and SST also affects the
sea-surface pH of BoB. As SSS has a significant role in modulating the
seasonality of pH in the BoB region, further analysis on the effect of
stratification and barrier layer over the pH becomes highly important.

###  3.3 Role of stratification on seasonality of pH

Due to the low-density freshwater influx from rivers and precipitation, a
shallow layer of freshwater forms in the sub-surface waters, leading to the
strong stratification in the BoB region [40, 53, 30]. The strong
stratification leads to a barrier layer formation. [41, 40] reported observing
a low saline strip of freshwater along the western coast in July-August 1989
and December 1991, respectively. The formation and spreading of the freshwater
plume in the summer monsoon are meticulously described through observation
[52]. [16, 17, 33] explain the role of winds, circulation, and rivers from a
modeling perspective, in the formation and spreading of the freshwater plume.
The freshwater spread over the surface of BoB, and the barrier layer thickness
(BLT) thus formed restricts vertical mixing. This inhibition of vertical
mixing does not allow the sub-surface nutrients and chemicals from reaching
the surface, which makes the BoB region a low-productive zone [28, 13]. Our
previous work [18] shows the extent by which BLT and the freshwater plume
influences the sea-surface pCO2. The method to calculate the BLT is same for
this study, hence we encourage readers to refer to our previous work for
detailed explanation of BLT calculation and its influence on the sea-surface
pCO2.

Figure 9: Spatial correlation between BLT and sea-surface pH.

To examine the relationship between the BLT and the sea-surface pH in the BoB
region, we make a spatial correlation between the BLT and pH, as shown in
Fig.9. The correlation is used as a proxy to highlight the relation between
BLT and pH. As we observe from Fig.9, the open oceans of the BoB region shows
a high positive correlation between pH and BLT, suggesting that the thicker
the barrier layer higher the sea-surface pH. This may suggest that the thick
barrier layer impedes the mixing of sub-surface acidic waters with the surface
waters of BoB, resulting in higher pH at the sea-surface. The high positive
correlation also suggests that the months or seasons having low BLT would
result in acidic sea-surface waters.

Figure 10: The seasonal spatial plot showing pH (contours) overlaid with BLT
in black broken lines.

In Fig.10, we emulate the spatial seasonal pH overlaid with the seasonal BLT.
We observe that indeed the areas having thicker barrier layer seem to have
higher pH values. The barrier layer is known to be thickest during the winter
monsoon period (DJF), and the area having thicker barrier layers is spread the
most during this season. Hence the higher pH value area is seen to be more
widely spread during the winter monsoon season. The pre-monsoon season (MAM)
is observed to have the shallowest barrier layer, and hence the interaction
between sub-surface and surface waters must be predominant. Thus the pre-
monsoon season seems to be the season having the most acidic sea-surface. In
Fig.8, we see that the pH in the pre-monsoon season has a significant
contribution from all the governing factors, the interaction between sub-
surface and surface waters may contribute to the equal roles of all drivers.

Figure 11: Comparison of the depth-wise pH profile between the northern
(N-BoB) and southern (S-BoB) BoB.

The southwest monsoon season (JJAS) in Fig.10 exhibits a strip of low acidic
waters slanting from the north BoB along the western coast. The beginning of
freshwater plume spread in the southwest monsoon season seems to draw the low-
saline waters through the west coast [41], which increases the pH in this
region. The post-monsoon season (ON) marks the largest freshwater plume spread
and the lowest SSS, resulting in the highest pH, especially in the northern
region. It is observed from Fig.10 that the northern region of BoB (as shown
in Fig.1) has higher pH values except for the pre-monsoon season. The
freshwater influx from high discharge rivers like Ganga, Brahmaputra, and
Irrawaddy, along with the high precipitation, maybe why the high pH and high
BLT values in the north. The southern sea-surface is comparatively acidic in
nature. Especially the southwestern region seems to be relatively acidic
throughout the climatological year. To further visualize the effect of
stratification, we plot the depth-wise annual mean pH (up to 100m) (Fig.11)
for the N-BoB and S-BoB demarcated by the broken black line in Fig.1.

We observe in Fig.11 that the pH in the N-BoB is lower than the S-BoB below
20m depth. On the surface, the N-BoB has a pH higher than the S-BoB. We know
that the N-BoB is more influenced by the freshwater influx due to the BLT and
consequently stronger stratification; hence the acidic lower pH waters are
restricted at around 20 to 22m depth. The pH in N-BoB is constant till 10m
depth, whereas the S-BoB pH is constant almost till 20m depth, which indicates
that the inhibition of vertical mixing in N-BoB is lower than the S-BoB. Hence
the effect of stratification in the N-BoB plays a vital role in making the
N-BoB sea-surface alkaline compared to the S-BoB.

##  4 Conclusion

In the present study, we attempt to explore the climatological ocean
acidification state of the BoB region. We use the outputs of a coupled
physical-biogeochemical model to reconstruct the pH for the BoB region. The
paucity of data from the chemical parameters has been a challenge to establish
the model worldwide. The high freshwater influx in the BoB region makes this
region even more challenging as mathematical relations of chemical parameters
based on SSS do not stand valid in this region. Nevertheless, considering
these challenges, we try to validate our modeled pH with the limited data
available. The modeled pH performs reasonably well when compared with RAMA
buoy and Takahashi data. The correlation of 0.94 with RAMA buoy implies that
the model well captures the monthly variability. The biases have been
reasonably low when comparing with both the observational data. The comparison
also highlights the necessity of more real-time chemical parameter
observations in the BoB region.

Further, we analyze the role of each governing factor, viz, SST, SSS, DIC, and
TALK, in modulating the seasonality of pH. The SSS has emerged as one of the
major drivers of pH along with SST and DIC. The first six months of the
climatological year have a combined effect of SST, DIC, and SSS on pH, but SSS
significantly dominates the last six months. The SST reduces the sea-surface
acidity in the winter monsoon season and post-monsoon season, but it increases
the acidity in pre-monsoon and southwest monsoon seasons. The winter monsoons
experience the maximum effect of SST on pH, and the post-monsoon season has a
minimum SST effect on pH. SSS increases the pH significantly in the post-
monsoon season. The SSS reduces the pH in the pre-monsoon season. The effect
of SSS is minor during the southwest monsoon and winter monsoon seasons. DIC
increases acidity in the sea-surface during the winter monsoon season but
reduces acidity during the pre-monsoon season. The effect of TALK is lowest
among all the governing factors.

As SSS is a major contributor in modulating the seasonality of pH, the role of
stratification on sea-surface ocean acidification becomes crucial in the BoB
region. The sea-surface pH shows a strong positive correlation with the BLT,
which indicates that thicker barrier layer areas have higher sea-surface pH.
The N-BoB is found to be more alkaline than the S-BoB. The spreading of the
freshwater plume that reduces the SSS may be why the higher pH values in the
N-BoB. Below 20 m depth, the N-BoB has more acidic waters than the S-BoB. The
stronger stratification and thick barrier layer inhibit these high acidic sub-
surface waters from reaching the surface resulting in a more alkaline sea-
surface in N-BoB than the S-BoB.

Our study is one of the first climatological modeling approaches for the pH in
the BoB region. The study becomes more important as it provides a significant
idea of the governing factors of pH in the open ocean part of BoB. With global
warming, these roles may change in the future. The relationship between BLT
and sea-surface pH gives a different perspective which may be used to
artificially (using a neural network approach) generate pH data. However,
interannual modeling has to be done, as cyclones and higher local winds may
change the role of stratification on pH. In the next part of this study, we
will evaluate the effect of other mechanisms like biological production,
solubility, and carbon dioxide flux on the sea-surface pH.

## Acknowledgment

The present work is carried out at the Indian Institute of Technology,
Kharagpur. The authors extend gratitude towards the institute for providing
support for this work. We also thank Prof. Arun Chakraborty for introducing us
to the Regional Ocean Model (ROMS). The first author acknowledges the Ministry
of Education, India, for providing the scholarship assistance. The RAMA buoy
pH data is downloaded from
https://www.nodc.noaa.gov/ocads/oceans/Moorings/BOBOA.html. The Takahashi data
is downloaded from https://www.nodc.noaa.gov/ocads/oceans/ndp_094/ndp094.html.
There is no conflict of interest between the authors to declare.

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[a]B.F.L. Ward

# New Results from IR-Improved Amplitude-Based Resummation in Quantum Field
Theory

S. Jadach222Deceased. W. Placzek  M. Skrzypek  Z.A. Was  S.A. Yost

###### Abstract

There is a continuing effort to support and prepare the precision physics
programs for the present and planned future colliders such as HL-LHC, FCC,
CLIC, CEPC, and CPPC. We discuss new results from IR-improved amplitude-based
resummation in quantum field theory relevant to such support and preparation
with some emphasis on the interplay between soft and collinear resummation
algebras.

BU-HEPP-23-02, Oct., 2023

##  1 In Memoriam

Sadly, my (BFLW) close friend and collaborator, Prof. Stanislaw Jadach, passed
away suddenly on February 26, 2023. His CERN Courier***CERN Courier, May/June
2023 issue, p.59. obituary is reproduced here in Fig. 1.

Figure 1: CERN Courier obituary for the late Prof. Stanislaw Jadach.

He was a pioneering member of the RADCOR International Advisory Board and he
was the Chair of the Local Organizing Committee when the Institute of Nuclear
Physics hosted the 1996 RADCOR in Krakow, Poland. His many outstanding
contributions to our field helped to keep our field alive. We all miss him
dearly. This contribution is dedicated in memoriam to him.

##  2 Introduction

The future of precision quantum field theory is dictated by the planned future
colliders: FCC [1], CLIC [2], ILC [3] CEPC [4], CPPC [5], $\ldots$. For
example, in the case of the FCC-ee, factors of improvement from $\sim 5$ to
$\sim 100$ are needed from theory. Resummation is key to such improvements in
many cases. In this context, we discuss here the amplitude-based resummation
following from the methodology of Yennie, Frautschi, and Suura (YFS) in Ref.
[6] as it has been realized Refs. [7, 8, 9, 10, 11, 12, 13]. This approach to
resummation treats the resummation of infrared (IR) singularities to all
orders in the loop expansion. One of us (BFLW) has extended [14] this
methodology to non-Abelian gauge theories as well.

It is encouraging that the need for precision theory for the future collider
physics objectives is appreciated at world-leading laboratories such as CERN,
as we illustrate in Fig. 2 with excerpts from Ref. [15]. The future options
for CERN are shown featuring the FCC and the important role of higher-order
calculations for its background processes is a theory highlight. We can hope
that the funding agencies take note of the implied connection.

(a) (b) Figure 2: Excerpts from Ref. [15] on the state of CERN: (a), Future
Options and R& D; (b), Theory highlights.

The fundamental attraction of the YFS approach is that there is no limit in
principle to the precision one may achieve as long as one calculates the
corresponding hard radiation residuals to the desired order in the respective
coupling constant. This should be contrasted with other methods of resummation
in which various degrees of freedom are integrated out to allow the
resummation thereby engendering an intrinsic uncertainty in observables which
require the reinstatement of those degrees of freedom; for, the reinstatement
can only be done approximately.

More specifically, YFS methods in QED are exact to all orders in the infrared
limit but treat collinear big logs perturbatively in the hard photon
residuals. DGLAP-based collinear factorization as recently done at the next-
leading log level in Refs. [16, 17, 18, 19], for example, treats collinear
logs to all orders but has a non-exact infrared limit. In what follows, after
briefly reviewing in the next Section the elements of exact amplitude
resummation theory which we employ, we first present (in Section 4) new
results for precision collider physics and pioneering results in quantum
gravity using the usual YFS methods. We then investigate in Section 5
improving the collinear limit of YFS theory. A key point is the following:
Exact amplitude-based resummation realized on an event-by-event basis gives
enhanced precision for a given level of exactness and this is essential for
future precision physics as exemplified by CERN.

##  3 Brief Review of Exact Amplitude-Based Resummation Theory

As it is still not generally familiar, we include here a synopsis of exact
amplitude-based resummation theory. The master formula that exhibits the
theory is as follows:

$\displaystyle d\bar{\sigma}_{\rm res}=e^{\rm
SUM_{IR}(QCED)}\sum_{{n,m}=0}^{\infty}\frac{1}{n!m!}\int\prod_{j_{1}=1}^{n}\frac{d^{3}k_{j_{1}}}{k_{j_{1}}}$
(1)
$\displaystyle\prod_{j_{2}=1}^{m}\frac{d^{3}{k^{\prime}}_{j_{2}}}{{k^{\prime}}_{j_{2}}}\int\frac{d^{4}y}{(2\pi)^{4}}e^{iy\cdot(p_{1}+q_{1}-p_{2}-q_{2}-\sum
k_{j_{1}}-\sum{k^{\prime}}_{j_{2}})+D_{\rm QCED}}$ (2)
$\displaystyle{\tilde{\bar{\beta}}_{n,m}(k_{1},\ldots,k_{n};k^{\prime}_{1},\ldots,k^{\prime}_{m})}\frac{d^{3}p_{2}}{p_{2}^{\,0}}\frac{d^{3}q_{2}}{q_{2}^{\,0}},$
(3)

where the new†††The non-Abelian nature of QCD requires a new treatment of the
corresponding part of the IR limit [20] so that we usually include in ${\rm
SUM_{IR}(QCED)}$ only the leading term from the QCD exponent in Ref. [20] –
the remainder is included in the residuals $\tilde{\bar{\beta}}_{n,m}$ .(YFS-
style) residuals
$\tilde{\bar{\beta}}_{n,m}(k_{1},\ldots,k_{n};k^{\prime}_{1},\ldots,k^{\prime}_{m})$
have $n$ hard gluons and $m$ hard photons. The new residuals and the infrared
functions ${\rm SUM_{IR}(QCED)}$ and ${D_{\rm QCED}}$ are defined in Ref. [21,
22]. As explained in Ref. [21, 22], parton shower/ME matching engenders the
replacements
$\tilde{\bar{\beta}}_{n,m}\rightarrow\hat{\tilde{\bar{\beta}}}_{n,m}$, which
allow us to connect with MC@NLO [23, 24], via the basic formula

${d\sigma}=\sum_{i,j}\int
dx_{1}dx_{2}{F_{i}(x_{1})F_{j}(x_{2})}d\hat{\sigma}_{\rm res}(x_{1}x_{2}s).$
(4)

New results in precision LHC and FCC physics have been obtained using Eq.(3).
One of us (BFLW) has extended the latter equation to general relativity as an
approach to quantum gravity. In each respective application, our new results
are accompanied with new perspectives. We discuss such new results and
perspectives in the next Section.

##  4 New Perspectives for Precision Physics: High Energy Colliders and
Quantum Gravity

The realization of eq.(3) in the MC event generator ${\cal KK}$MC-hh [12] by
four of us (SJ, BFLW, ZAW, SAY) allows a new perspective on the expectations
for precision physics for the Standard Theory EW interactions at HL-LHC. This
is illustrated by the plots in Fig. 3 in the ATLAS analysis [25] of $Z\gamma$
production at 8 TeV.

Figure 3: ATLAS analysis of $Z/\gamma$ production at $8$ TeV.

The data are compared to the Powheg-Pythia8-Photos [26, 27, 28, 29, 30, 31],
Sherpa2.2.4(YFS) [32, 33], and ${\cal KK}$MC-hh predictions for the $\gamma
p_{T}$ spectrum. At this level of the uncertainties in the data at this point,
all three predictions are in reasonable agreement with the data. At HL-LHC,
with 10 times the current statistics, a precision test against the theories
will obtain.

An important issue is the effect of QED contamination in non-QED PDFs. In a
new perspective toward this issue [34, 35, 36] we use Negative ISR (NISR)
evolution to address the size of this contamination directly. We have, using a
standard notation for PDFs and cross sections, the cross section
representation

$\begin{split}\sigma(s)&=\frac{3}{4}\pi\sigma_{0}(s)\\!\\!\\!\sum_{q=u,d,s,c,b}\int
d\hat{x}\;dzdrdt\;\int
dx_{q}dx_{\bar{q}}\;\delta(\hat{x}-x_{q}x_{\bar{q}}zt)\\\ &\times
f^{h_{1}}_{q}(s\hat{x},x_{q})f^{h_{2}}_{\bar{q}}(s\hat{x},x_{\bar{q}})\;\rho_{I}^{(0)}\big{(}\gamma_{Iq}(s\hat{x}/m_{q}^{2}),z\big{)}\;\rho_{I}^{(2)}\big{(}-\gamma_{Iq}(Q_{0}^{2}/m_{q}^{2}),t\big{)}\;\\\
&\times\sigma^{Born}_{q{\bar{q}}}(s\hat{x}z)\;\langle
W_{MC}\rangle,\end{split}$ (5)

which includes an extra convolution with the well known second order
exponentiated ISR “radiator function” $\rho_{I}^{(2)}$ with the negative
evolution time argument $-\gamma_{Iq}(Q_{0}^{2}/m_{q}^{2})$ defined in Ref.
[34]. The QED below $Q_{0}$ is thus removed. We illustrate this in Fig. 4 from
Ref. [35]

Figure 4: The distribution for $P_{T_{\gamma}}$ of the photon for which it is
greatest for events with at least one photon and each lepton having
$p_{T\ell}>25GeV,\eta_{\ell}<2.5$ calculated with (0) FSR only (black). (1)
FSR + ISR (blue). and (2) FSR + ISR with NISR (red) for NNPDF3.1-LuxQED NLO
PDFs. For comparison, (3) shows FSR + ISR with ordinary NNPDF3.1 NLO PDFs
(green). The center graph shows ISR on/off ratios (1)/(0) (blue),(2)/(0) (red)
and (3)/(0) (green). The right-hand graph shows the fractional differences
((1)- (2))/(0) in red and ((2)- (3))/(0) in green.

for the $P_{T_{\gamma}}$ for the photon for which it is the largest in
$Z\gamma^{*}$ production and decay to lepton pairs at the LHC at $8~{}TeV$ for
cuts as described in the figure. As we see in the figure, the results show
that the effect of QED contamination in non-QED PDFs is below the errors on
the PDFs in agreement with arguments in Ref. [37].

For the planned EW/Higgs factories, five of us (SJ, WP, MS, BFLW, SAY) have
discussed in Refs. [38, 39, 40] the new perspectives for the BHLUMI [8]
luminosity theory error. This new perspective is illustrated in Fig. 5 [40]
wherein we show the current purview for the FCC-ee at $M_{Z}$ and that for the
proposed higher energy colliders.

(a) (b) Figure 5: Current purview on luminosity theory errors: (a), FCC-ee at
$M_{Z}$; (b), proposed higher energy colliders

In addition to the improvements at $M_{Z}$ shown in Fig. 5(a) to 0.007%, the
use of the results in Ref. [41] together with lattice methods [42, 43] opens
the possibility that item (c) in Fig 5(a) could be reduced by a factor of
6‡‡‡The formula to be studied is
$\Delta\alpha_{had}(t)=\Delta\alpha_{had}(-Q^{2}_{0})|_{lat}+[\Delta\alpha_{had}(t)-\Delta\alpha_{had}(-Q^{2}_{0})]|_{pQCDAdler}$
with lat denoting the methods of Refs. [42, 43] and pQCDAdler denoting the
methods of Ref. [41]..

In Refs. [44, 45] one of us (BFLW) has shown that amplitude-based resummation
applied to quantum gravity tames its UV divergences. One of the many
consequences is, using a standard notation,

$\begin{split}\rho_{\Lambda}(t_{0})&\cong\frac{-M_{Pl}^{4}(1+c_{2,eff}k_{tr}^{2}/(360\pi
M_{Pl}^{2}))^{2}}{64}\sum_{j}\frac{(-1)^{F}n_{j}}{\rho_{j}^{2}}\cr&\qquad\quad\times\frac{t_{tr}^{2}}{t_{eq}^{2}}\times(\frac{t_{eq}^{2/3}}{t_{0}^{2/3}})^{3}\cr&\cong\frac{-M_{Pl}^{2}(1.0362)^{2}(-9.194\times
10^{-3})}{64}\frac{(25)^{2}}{t_{0}^{2}}\cr&\cong(2.4\times
10^{-3}eV)^{4}.\end{split}$ (6)

Here, $t_{0}$ is the age of the universe and we take it to be $t_{0}\cong
13.7\times 10^{9}$ yrs and $t_{tr}\sim 25t_{Pl}$ [46, 44, 45] is the
transition time between the Planck regime and the classical Friedmann-
Robertson-Walker(FRW) regime. In the estimate in (6), the first factor in the
second line comes from the radiation dominated period from $t_{tr}$ to
$t_{eq}$ and the second factor comes from the matter dominated period from
$t_{eq}$ to $t_{0}$. The estimate in (6) is close to the experimental result
[47]§§§See also Ref. [48, 49] for analyses that suggest a value for
$\rho_{\Lambda}(t_{0})$ that is qualitatively similar to this experimental
result. $\rho_{\Lambda}(t_{0})|_{\text{expt}}\cong((2.37\pm 0.05)\times
10^{-3}eV)^{4}$.

##  5 Improving the Collinear Limit in YFS Theory

It is known [50] that in the usual YFS theory the virtual infrared function
$B$ in the s-channel resums (exponentiates) the non-infrared term
$\frac{1}{2}Q_{e}^{2}{\alpha\over\pi}L$ in
$e^{+}(p_{2})\;e^{-}(p_{1})\rightarrow\bar{f}(p_{4})\;e^{-}(p_{3})$ using an
obvious notation where the respective big log is $L=\ln(s/m_{e}^{2})$ when
$s=(p_{1}+p_{2})^{2}$ is the center-of-mass energy squared. It is also known
from Ref. [51] that the term $\frac{3}{2}Q_{e}^{2}{\alpha\over\pi}L$
exponentiates – see also Refs. [16, 17, 18, 19] for recent developments in
attendant collinear factorization approach. Does the YFS theory allow an
extension that would also exponentiate the latter term? In Ref. [50], three of
us (SJ, BFLW, ZAW) have answered this question in the affirmative.

Specifically, we find that the virtual infrared function $B$ in the s-channel
can be extended to

$\begin{split}B_{CL}&\equiv B+{\bf\Delta B}\\\ &=\int{d^{4}k\over
k^{2}}{i\over(2\pi)^{3}}\bigg{[}\bigg{(}{2p-k\over 2kp-k^{2}}-{2q+k\over
2kq+k^{2}}\bigg{)}^{2}{\bf-\frac{4pk-4qk}{(2pk-k^{2})(2qk+k^{2})}}\bigg{]},\end{split}$
(7)

while the real infrared function $\tilde{B}$ can be extended to

$\begin{split}\tilde{B}_{CL}&\equiv\tilde{B}+{\bf\Delta\tilde{B}}\\\
&=\frac{-1}{8\pi^{2}}\int\frac{d^{3}k}{k_{0}}\bigg{\\{}\large(\frac{p_{1}}{kp_{1}}-\frac{p_{2}}{kp_{2}}\large)^{2}+{\bf\frac{1}{kp_{1}}\large(2-\frac{kp_{2}}{p_{1}p_{2}}\large)}\\\
&\qquad\qquad+{\bf\frac{1}{kp_{2}}\large(2-\frac{kp_{1}}{p_{1}p_{2}}\large)}\bigg{\\}},\end{split}$
(8)

where the extensions are indicated in boldface in an obvious notation. The YFS
infrared algebra is unaffected by these extensions while the $B_{CL}$ does
exponentiate the entire $\frac{3}{2}Q_{e}^{2}{\alpha\over\pi}L$ term and the
$\tilde{B}_{CL}$ does carry the respective collinear big log of the exact
result in Ref. [52] in the soft regime.

The corresponding collinear extension of the CEEX soft eikonal amplitude
factor defined in Ref. [53] for the photon polarization $\sigma$ and $e^{-}$
helicity $\sigma^{\prime}$ is given by

$\begin{split}\mathfrak{s}_{CL,\sigma}(k)=\sqrt{2}Q_{e}e\bigg{[}-\sqrt{\frac{p_{1}\zeta}{k\zeta}}\frac{<k\sigma|\hat{p}_{1}-\sigma>}{2p_{1}k}+{\bf\delta_{\sigma^{\prime}\;-\sigma}\sqrt{\frac{k\zeta}{p_{1}\zeta}}\frac{<k\sigma|\hat{p}_{1}\sigma^{\prime}>}{2p_{1}k}}\\\
+\sqrt{\frac{p_{2}\zeta}{k\zeta}}\frac{<k\sigma|\hat{p}_{2}-\sigma>}{2p_{2}k}+{\bf\delta_{\sigma^{\prime}\sigma}\sqrt{\frac{k\zeta}{p_{2}\zeta}}\frac{<\hat{p}_{2}\sigma^{\prime}|k-\sigma>}{2p_{2}k}}\bigg{]},\end{split}$
(9)

where from Ref. [53] $\zeta\equiv(1,1,0,0)$ for our choice for the respective
auxiliary vector in our Global Positioning of Spin (GPS) [54] spinor
conventions with the consequent definition $\hat{p}=p-\zeta m^{2}/(2\zeta p)$
for any four vector $p$ with $p^{2}=m^{2}.$ The collinear extension terms are
again indicated in boldface.

We expect these extended infrared functions to give in general a higher
precision for a given level of exactness [55].

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