# Intriguing Plasma Composition Pattern in a Solar Active Region: a Result of
Non-Resonant Alfvén Waves?

Teodora Mihailescu Mullard Space Science Laboratory, University College
London, Holmbury St Mary, Dorking, Surrey, RH5 6NT, UK  David H. Brooks
College of Science, George Mason University, 4400 University Drive, Fairfax,
VA 22030, USA  J. Martin Laming Space Science Division, Naval Research
Laboratory, Code 7684, Washington, DC 20375, USA  Deborah Baker Mullard Space
Science Laboratory, University College London, Holmbury St Mary, Dorking,
Surrey, RH5 6NT, UK  Lucie M. Green Mullard Space Science Laboratory,
University College London, Holmbury St Mary, Dorking, Surrey, RH5 6NT, UK
Alexander W. James Mullard Space Science Laboratory, University College
London, Holmbury St Mary, Dorking, Surrey, RH5 6NT, UK  David M. Long
Astrophysics Research Centre, Queen’s University Belfast, University Road,
Belfast, BT7 1NN, Northern Ireland, UK  Lidia van Driel-Gesztelyi Mullard
Space Science Laboratory, University College London, Holmbury St Mary,
Dorking, Surrey, RH5 6NT, UK  LESIA, Observatoire de Paris, Université PSL,
CNRS, Sorbonne Université, Univ. Paris Diderot, 5 place Jules Janssen, F-92195
Meudon, France  Konkoly Observatory, Research Centre for Astronomy and Earth
Sciences, Konkoly Thege út 15-17., H-1121, Budapest, Hungary  Marco Stangalini
ASI Italian Space Agency, Via del Politecnico, s.n.c I-00133—Roma, Italy

###### Abstract

The plasma composition of the solar corona is different from that of the solar
photosphere. Elements that have a low first ionisation potential (FIP) are
preferentially transported to the corona and, therefore, show enhanced
abundances in the corona compared to the photosphere. The level of enhancement
is measured using the FIP bias parameter. In this work, we use data from the
EUV Imaging Spectrometer (EIS) on Hinode to study the plasma composition in an
active region following an episode of significant new flux emergence into the
pre-existing magnetic environment of the active region. We use two FIP bias
diagnostics: Si X 258.375 Å/S X 264.233 Å (temperature of approximately 1.5
MK) and Ca XIV 193.874 Å/Ar XIV 194.396 Å (temperature of approximately 4 MK).
We observe slightly higher FIP bias values with the Ca/Ar diagnostic than Si/S
in the newly emerging loops, and this pattern is much stronger in the
preexisting loops (those that had been formed before the flux emergence). This
result can be interpreted in the context of the ponderomotive force model,
which proposes that the plasma fractionation is generally driven by Alfvén
waves. Model simulations predict this difference between diagnostics using
simple assumptions about the wave properties, particularly that the
fractionation is driven by resonant/non-resonant waves in the
emerging/preexisting loops. We propose that this results in the different
fractionation patterns observed in these two sets of loops.

FIP bias, Composition, Corona

††software: SolarSoftWare (Freeland & Handy, 1998), EISPAC (Weberg et al.,
2023)

##  1 Introduction

One of the major open questions in solar physics is why the elemental
composition of some regions in the solar corona is different to that of the
underlying photosphere. The relative abundance of different elements is
spatially homogenous in the photosphere (Asplund et al., 2009), but varies in
the corona, with a strong dependence on the first ionization potential (FIP)
of the element (Meyer, 1985). The abundances of elements with a low FIP ($<$10
eV) are often enhanced in the corona, while the abundances of high FIP
elements ($>$10 eV) appear to be unchanged. This is called the FIP effect. The
degree of enhancement of an element is calculated using the FIP bias
parameter. In the extreme ultraviolet (EUV), this is measured as the abundance
ratio of a low-FIP element to a high-FIP element, relative to the same
photospheric ratio. Observed FIP bias values typically vary between 1 and 4
(e.g. Baker et al., 2013; Del Zanna, 2013) but higher values that go up to 8
have also been reported (e.g. Widing & Feldman, 2001).

Various processes have been proposed to be responsible for this effect, such
as diffusion or inefficient Coulomb drag (e.g. von Steiger & Geiss, 1989;
Marsch et al., 1995; Pucci et al., 2010; B\o et al., 2013), thermoelectric
driving (Antiochos, 1994), chromospheric reconnection (Arge & Mullan, 1998) or
ion cyclotron wave heating (Schwadron et al., 1999) . Among the many
candidates, a collisionless wave-particle mechanism based on the ponderomotive
force (Laming, 2004, 2015) appears to be able to describe this phenomenon more
realistically than previously suggested mechanisms. The ponderomotive force
arises as the reaction of the plasma to the refraction of Alfvén waves in the
chromosphere and only acts on charged particles. In the chromosphere, low-FIP
elements are predominantly ionized (ionization fraction $>$99%; see Laming,
2015, and references therein), while high-FIP elements are mostly neutral.
Therefore, the ponderomotive force separates the low-FIP ions from the high-
FIP neutrals by preferentially transporting them upwards to the corona. The
result can then be observed in the corona as enhanced abundances of low-FIP
elements in the corona compared to the photosphere. This is interesting
because it places the fractionation closer to the beginning of the chain of
events that lead to coronal heating, rather than being at the endpoint of the
thermalization of that energy as would be the case for some of the diffusion
based mechanisms. Numerical simulations by Dahlburg et al. (2016) support the
presence of the ponderomotive acceleration in solar coronal loops, with the
appropriate magnitude and direction, and suggest it is a “by-product” of
coronal heating. Recent work by Baker et al. (2021); Stangalini et al. (2021);
Murabito et al. (2021) found magnetic fluctuations in the chromosphere being
magnetically connected to regions of high FIP bias in the corona which
supports this theoretical model. More recent numerical simulations by Réville
et al. (2021) using a shell turbulence model found that, under the assumption
that turbulence is the main driver of coronal heating and solar wind
acceleration, a ponderomotive force can appear in the chromosphere and the
transition region, and can be strong enough to create the FIP effect.
Martínez-Sykora et al. (2023) use a combination of IRIS observations and a
2.5D radiative magnetohydrodynamics (MHD) model of the solar atmosphere to
investigate the multifluid effects on the ponderomotive force associated with
Alfvén waves.

The strongest FIP bias is typically observed in active regions. There are
large variations in the measured FIP bias among active regions (Brooks et al.,
2015; Mihailescu et al., 2022) and the overall FIP bias values observed in an
active region also vary with time as the active region goes through the
different stages that make up its lifetime. Previous studies found that
emerging flux carries plasma with photospheric composition, i.e., FIP bias of
approximately 1, at first (Widing & Feldman, 2001). Then it increases with
time in the emergence phase (Widing & Feldman, 2001) and early decay phase
(Baker et al., 2018), suggesting that the higher level of magnetic activity
observed during the emergence phase is linked to the processes that drive the
FIP effect. After an active region enters its decay phase, the FIP bias starts
to decrease (Baker et al., 2015; Ko et al., 2016) until it reaches the FIP
bias of the surrounding quiet Sun (Ko et al., 2016).

In addition, FIP bias values within an active region show a broad distribution
(Mihailescu et al., 2022), indicating that processes acting on sub-active
region scales can influence the FIP bias in different ways. For example, Baker
et al. (2013) found that the FIP bias is highest at the loop footpoints and
shows a mild enhancement along some of the active region loops. The same study
also found that photospheric reconnection that manifests as photospheric flux
cancellation and subsequent formation of a flux rope leads to FIP bias values
closer to 1. This makes plasma composition a powerful tool for obtaining
insight into the magnetic configuration and formation history of solar
structures (see e.g., Fletcher et al., 2001; James et al., 2017; Baker et al.,
2022).

The measured FIP bias values can also vary depending on the diagnostic used to
measure it. While elements are broadly categorised into low-FIP and high-FIP,
different low-FIP elements can show different levels of enhancement, and high-
FIP elements can in some instances show enhancement too. The clearest example
is S (FIP = 10.36 eV) which sits close to the boundary between low-FIP and
high-FIP elements. In some instances it shows no or little enhancement (i.e.,
it behaves like a high-FIP element), while in others it shows significant
enhancement (i.e., it behaves like a low-FIP element, see e.g., the coronal
hole measurements of Brooks & Warren, 2011). In addition, in the EUV, FIP bias
diagnostics also have an associated temperature given by the contribution
functions of the lines used. For example, the Si X 258.38 Å/S X 264.22 Å
diagnostic captures cooler coronal plasma as the formation temperature of the
lines involved is around 1.5 MK, while the Ca XIV 193.87 Å/Ar XIV 194.40 Å
diagnostic captures hotter coronal plasma since the lines involved form at
around 4 MK. Therefore, differences in FIP bias values measured with different
diagnostics can be due to either the elements themselves behaving differently
or the diagnostic probing plasma at different temperatures. Ko et al. (2016)
found a high correlation (correlation coefficient varying between 0.76 and
0.91 depending on the region selected) between the FIP bias measured with Si X
258.38 Å/S X 264.22 Å and Fe XII 195.12 Å/S X 264.22 Å
($\text{log}(\text{T}_{\text{MAX}})=6.2$) in a decaying active region. This
indicates that Fe and Si show similar FIP enhancement relative to S (note that
S is the high-FIP element in both diagnostics). To et al. (2021) found more
significant differences between FIP bias diagnostics when analysing the Si X
258.38 Å/S X 264.22 Å compared to Ca XIV 193.87 Å/Ar XIV 194.40 Å FIP bias
values in a small flare. This study suggests that the elements considered to
be high-FIP could be behaving differently, i.e., S could be acting like a low-
FIP element, as previously suggested by Laming et al. (2019). However, the
conditions under which this phenomenon can happen are not yet fully
understood.

This work investigates strikingly different fractionation patterns in
different parts of an active region, again involving variations in the
behavior of S. We use a combination of two FIP bias diagnostics characterising
the two fractionation patterns and ponderomotive force model simulations to
further investigate mechanisms that could be responsible for the observed
differences. Section 2 presents the photospheric and coronal evolution of the
active region in the time leading up to the plasma composition observations.
Section 3 describes the composition observations and the method for obtaining
the FIP bias measurements. Section 4 presents the ponderomotive force model
(Laming, 2004, 2015) simulation results and how they can contribute to
understanding the plasma composition observations. Finally, Section 5 provides
a summary and discussion of the results in this study.

##  2 Evolution of AR 12665

###  2.1 White Light Continuum and Photospheric Magnetic Field Evolution

Figure 1: SDO AIA 193 Å context image of target AR 12665 on 2017 July 9 at
03:59 UT. Figure 2: SDO HMI Continuum emission and photospheric magnetic field
evolution of AR 12665, every twelve hours, prior to and during the Hinode/EIS
scans. Red (blue) contours represent areas of HMI line of sight photospheric
magnetic field strength above (below) 200 G (-200 G). Yellow contours
represent values below 25,000 ct/s in the continuum emission, indicating the
location of the sunspot umbrae.

NOAA AR 12665 (see Figure 1) was first observed by the Solar Dynamics
Observatory (SDO; Pesnell et al., 2012) Helioseismic and Magnetic Imager (HMI;
Scherrer et al., 2012; Schou et al., 2012) at the eastern limb on 2017 July 5,
with sunspots already present in both polarities. A new episode of significant
flux emergence had begun between the leading and following polarities just as
the AR region rotated into Earth view. By July 8, the emerging flux formed a
new positive sunspot, which started moving towards (see top panels in Figure
2) and then orbiting the preexisting leading sunspot in a counterclockwise
direction (see bottom panels in Figure 2). They eventually collided around
12:00 UT on July 9 and became one sunspot consisting of two umbrae separated
by a light bridge within one common penumbra. Approximately one day later the
light bridge disappeared (not pictured). This orbiting motion lasted for
multiple days and was studied in detail by James et al. (2020). For the first
approximately 1.5 days, the light bridge constantly separated the new from the
preexisting positive sunspot umbrae. This enabled us to track the evolution of
the newer and older parts of the active region separately.

###  2.2 Coronal Evolution

In the EUV, images from the SDO Atmospheric Imaging Assembly (AIA; Lemen et
al., 2012) showed two main loop populations, namely the new loops and the
preexisting loops. The new loops had been formed recently by the flux
emergence and were rooted in the newly formed positive sunspot (see Figure 2).
They were bright, relatively small hot loops in the core of the active region
(see Figure 3). The preexisting loops had been part of the active region since
before the flux emergence and were rooted in the preexisting positive sunspot
(see Figure 2). They were fainter, high-lying warm loops located in the
southern part of the active region (see Figure 3).

In the few days running up to the EIS scans on July 9 at 01:08 UT and 14:15
UT, the active region showed modest activity, with only one C1.0 flare on July
7 at 13:37 UT and one C3.4 flare on July 8 at 23:50 UT. On July 9, however,
the flaring activity becomes more intense, with one M1.3 flare at 03:09 UT and
four C-class flares at 06:15 UT, 07:28 UT, 08:55 UT and 11:44 UT being
observed at the boundary between the new and the preexisting loop populations.

Figure 3: SDO AIA 193 Å passband images at the times matching the middle time
of the EIS raster scans. Yellow contours represent values below 25,000 ct/s in
the continuum emission, indicating the location of the sunspots. Black dotted
(dashed) lines indicate representative examples of loops belonging to the new
(preexisting) loop populations.

##  3 Plasma Composition

###  3.1 Hinode EIS Observations

The FIP bias was calculated using observations from the EUV Imaging
Spectrometer (EIS; Culhane et al., 2007) on Hinode (Kosugi et al., 2007). The
EIS dataset contains 6 scans of the active region over a period of
approximately 13 hours. In this work, we analyse the first and the last scans
in this series. Details of the EIS scans used are provided in Table 1.

Table 1: Summary of Hinode/EIS study details and emission lines used for creating the FIP bias measurements. EIS Study Details |
---|---
Raster middle times |  09/07/2017 01:08 09/07/2017 14:15
Study acronym | DHB_007
Study number | 544
Field of view | 492” $\times$ 512”
Rastering | 2” slit, 123 positions, 4” coarse step
Exposure time | 30s
Total raster time | 1h 1m 30s
Reference spectral window | Fe XII 195.12 Å
DEM lines |  Fe VIII 185.213 Å, Fe VIII 186.601 Å, Fe IX 188.497 Å, Fe IX 197.862 Å, Fe X 184.536 Å, Fe XI 188.216 Å, Fe XI 188\. 299 Å, Fe XII 192.394 Å, Fe XII 195.119 Å, Fe XIII 202.044 Å, Fe XIII 203.826 Å, Fe XIV 264.787 Å, Fe XIV 270.519 Å, Fe XV 284.16 Å, Fe XVI 262.984 Å, Ca XIV 193.874 Å, Ca XV 200.972 Å.
Density diagnostic lines |  Fe XIII 202.04 Å, Fe XIII 203.82 Å
Line ratio lines |  Si X 258.38 Å, S X 264.22 Å, Ca XIV 193.87 Å, Ar XIV 194.40 Å

###  3.2 Method

We used two line pair diagnostics: Si X 258.38 Å (low FIP, FIP = 8.25 eV) & S
X 264.22 Å (high FIP, FIP = 10.36 eV) and Ca XIV 193.87 Å (low FIP, FIP = 6.11
eV) & Ar XIV 194.40 Å (high FIP, FIP = 15.76 eV). According to the CHIANTI
database (Dere et al., 1997) Version 10 (Del Zanna et al., 2021), the
theoretical formation temperatures for the two diagnostics are different: the
Si X 258.38 Å and S X 264.22 Å lines have a formation temperature of
$\text{log}(\text{T}_{\text{MAX}})=6.2$, while the Ca XIV 193.87 Å and Ar XIV
194.40 Å form at a temperature of $\text{log}(\text{T}_{\text{MAX}})=6.7$ and
$\text{log}(\text{T}_{\text{MAX}})=6.6$ respectively. The lines involved in
the two diagnostics were fitted using the Python EISPAC software (Weberg et
al., 2023) and line ratios for each diagnostic were calculated in every pixel
in the EIS rasters to obtain an approximation of the FIP bias (see second
column in Figures 5 and 5).

Line ratios, however, are sensitive to temperature and density effects, so
they only provide context. Corrected FIP bias measurements were also
calculated in a few key locations. In these locations, spectra were averaged
over multiple pixels (creating a macropixel) for all the lines included in the
calculation. The method used for the FIP bias calculation in each of these
macropixels uses a differential emission measure (DEM) to correct the
temperature effects and a density analysis to correct for density effects. The
DEM was derived using a series of Fe lines supplemented with a couple of Ca
lines for additional high temperature constraints (see Table 1) and the
density was calculated using the Fe XIII 202.04 Å/Fe XIII 203.82 Å diagnostic.
A Markov-Chain Monte Carlo (MCMC) algorithm in the PintOfAle (Kashyap & Drake,
1998, 2000) SolarSoft (Freeland & Handy, 1998) package was used to compute the
DEM distribution, and the CHIANTI database (Dere et al., 1997) version 10 (Del
Zanna et al., 2021) to compute the contribution functions (G(T,n)) for each of
the spectral lines involved in the diagnostic. We used the photospheric
abundances of Scott et al. (2015a) and Scott et al. (2015b). Note that using
different sets of photospheric abundances could result in slightly different
FIP bias measurements. This method is described in detail by Brooks & Warren
(2011); Brooks et al. (2015).

There are 8 such locations in total. For each of the 2 rasters, 4 macropixels
were selected: one for each loop population and each diagnostic (see first
panel of Figures 5 and 5). This was done to obtain representative FIP bias
values for each of the loop populations in both diagnostics. Slightly
different macropixels were chosen for the two diagnostics. This is because the
formation temperatures for the lines involved in the two diagnostics are
different. For each diagnostic, the macropixels were chosen such that emission
in the lines involved in the diagnostic is maximised.

Figure 4: Si X/S X results. From left to right: Hinode EIS Si X 258.38 Å
intensity, Hinode EIS Si X 258.38 Å/S X 264.22 Å line ratio, SDO AIA DEM in
the log(T)=6.0 - 6.2 and the log(T)=6.2 - 6.4 temperature bins (computed using
the method developed by Hannah & Kontar, 2012, 2013). The boxes indicate the
locations of the macropixels for this diagnostic.

Figure 5: Ca XIV/Ar XIV results. From left to right: Hinode EIS Ca XIV 193.87
Å intensity, Hinode EIS Ca XIV 193.87 Å/Ar XIV 194.40 Å line ratio, SDO AIA
DEM in the log(T)=6.4 - 6.6 and the log(T)=6.6 - 6.8 temperature bins
(computed using the method developped by Hannah & Kontar, 2012, 2013). The
boxes indicate the locations of the macropixels for this diagnostic.

###  3.3 Results

The FIP bias values are summarised in Table 2. In the emerging loops, the Si/S
FIP bias increases with time from 2.3 to 3.0 and the Ca/Ar FIP bias from 2.6
to 3.3. This result agrees with previous studies which found increasing FIP
bias in the emergence phase of an active region (Widing & Feldman, 2001; Baker
et al., 2018). While the increase is observed in both diagnostics, the Ca/Ar
values are slightly higher than the Si/S ones (albeit close to the 0.3
uncertainty in the measurements previously estimated by Brooks et al., 2017).

In the preexisting loops, the FIP bias behaviour is different. The Si/S FIP
bias changes slightly from 1.8 to 2.0, but this is within the 0.3 error limit.
The Ca/Ar FIP bias shows an increase from 4.3 to 7.8 and, more importantly,
shows consistently high values. This is very interesting because these values
are significantly higher than the Si/S FIP bias values in the same loop
population. To check that these high values are indeed representative of the
entire preexisting loops population, we examine the line ratio maps (see
second column of Figure 5). The line ratio maps show high values everywhere in
the preexisting loops population, suggesting that the high FIP bias values are
not isolated to the location of the macropixels that were chosen for the FIP
bias calculation. Of course, we must be cautious when analysing line ratio
maps as they are sensitive to temperature and density effects. In particular,
for the Ca/Ar line ratio, significant temperature effects due to plasma above
$\text{log}(T)=6.6$ must be considered (see e.g. Feldman et al., 2009; Doschek
& Warren, 2017; To et al., 2021). The DEM analysis (see third and fourth panel
in Figures 5, 5) shows that the emission in these preexisting loops is mostly
cooler, with most of the emission coming from the $\text{log}(T)=6.2-6.4$ and
$\text{log}(T)=6.4-6.6$ temperature bins, making it unlikely that the high
values we see is a temperature effect.

The large Ca/Ar FIP bias values observed in the preexisting loops are not
common, but similarly high values have been observed before, for example, in
the Ca/Ar ratio in post flare loops (Doschek et al., 2018), in the Mg/Ne ratio
in an emerging flux region (Young & Mason, 1997; Widing & Feldman, 2001), in
the Mg/O ratio in coronal mass ejection cores (Landi et al., 2010), and in
various diagnostics in post coronal mass ejection current sheets (Ko et al.,
2003; Ciaravella et al., 2002). The newest aspect of the present results is
the different behaviour of the two diagnostics used to probe the plasma
composition in the two loop populations. While in the emerging loops, the Si/S
and Ca/Ar diagnostics indicate similar FIP bias values, in the preexisting
loops there is a significant difference between the lower Si/S FIP bias values
and higher Ca/Ar FIP bias values. This raises the question of whether the
mechanism driving the FIP effect has different characteristics in the two loop
populations, so this possibility was explored further using simulations from
the ponderomotive force model.

Table 2: Hinode/EIS FIP bias results summary Raster Time | FIP Bias
---|---
| Emerging Loops | Preexisting Loops
| Si/S | Ca/Ar | Si/S | Ca/Ar
2017 July 9 01:08 UT | 2.3 | 2.6 | 1.8 | 4.3
2017 July 9 14:15 UT | 3.0 | 3.3 | 2.0 | 7.8

##  4 The Ponderomotive Force Model

The ponderomotive force model (Laming, 2004, 2015) is a 1D static model which
proposes that the FIP effect is generated in the chromosphere by Alfvén wave
activity originating in the corona. At coronal loop footpoints, refraction of
these Alfvén waves in the high density gradient of the chromosphere (see
Figure 6a) generates a ponderomotive force. This ponderomotive force acts on
the ionized material (i.e. mostly low-FIP elements since they have a much
higher ionization fraction than the high-FIP elements, see Figure 6b,c),
preferentially bringing them upwards to the top of the transition region. Once
plasma reaches the transition region, two things happen: 1) the temperature
increases enough to ionize all elements, so low-FIP and high-FIP elements are
no longer separated into ions and neutrals and 2) there is no significant
density gradient anymore, so the ponderomotive force disappears. This means
that the fractionation process stops at the top of the transition region, and
the fractionation pattern is locked in. From here, the fractionated plasma is
transported up into the corona through other mechanisms. Note that the current
implementation of the ponderomotive force model uses a 1D static model
chromosphere, and so it does not cover this last part of the chain.

The pattern and strength of the fractionation process depend on the height in
the chromosphere at which the ponderomotive force is generated. This, in turn,
is dictated by whether the Alfvén wave driver is in resonance with the loop or
not. Resonance here means that the wave travel time from one loop footpoint to
the other is an integral number of wave half-periods. Resonant waves
accumulate much more wave energy in the corona than in the chromospheric
footpoints, and drive the ponderomotive force close to the top of the
chromosphere (see Figure 6e). This results in mild fractionation levels (see
Figure 6f) because of the ionized background gas. Non-resonant waves drive the
ponderomotive force at lower heights (see Figure 6h), which results in
stronger fractionation levels (see Figure 6i) because the background gas is
neutral.

###  4.1 Model Simulations

We use the ponderomotive force model to make predictions of the fractionation
patterns in the two loop populations shown in Figures 5 and 5. We find that
the simulation predictions match best with the Hinode EIS observations when
assuming the driver is resonant waves in the emerging loops and non-resonant
waves in the preexisting loops. This scenario is described in detail below.

We first estimate the resonant frequency of the emerging loops using the
following parameters: loop length, plasma density along the loop and magnetic
field strength. The loop lengths were estimated using a Potential Field Source
Surface (PFSS) model (IDL SolarSoft package provided by Schrijver & DeRosa,
2003) of the active region. In the PFSS model, a representative loop was
selected for each loop population and its length was calculated. The plasma
density along the loop was measured in the macropixel boxes shown in Figures
5, 5 using the Fe XIII 202.04 Å/Fe XIII 203.82 Å diagnostic from Hinode EIS.
The photospheric magnetic field strength was measured using SDO HMI. All loop
parameters are summarised in Table 3. The resonant frequency is given
approximately by

$f_{\text{resonance}}=\frac{v_{A}}{2L},$ (1)

where L is the loop length and $v_{A}$ is the Alfvén speed calculated as:

$v_{A}=\frac{B}{\sqrt{4\pi\rho}},$ (2)

where B is the coronal magnetic field strength, and $\rho$ is the loop
density. We construct loop models matching the parameters in Table 3, where
the 0 to 2500 km portions at each end of the loop are taken to be the
chromospheric part of the loop, with the chromospheric model given by Avrett &
Loeser (2008). The loop resonant frequencies are taken from calculations of
the Alfvén wave propagation and identifying the frequency at which wave
transmission from the chromosphere into the corona is maximized.

We estimated the resonance angular frequency of the emerging loops to be
$\Omega_{\text{ELresonance}}$ to be 0.351 $\text{rad~{}s}^{-1}$. We then run
the model simulations assuming the fractionation process is driven by Alfvén
waves at this frequency in both the emerging and the preexisting loops. This
means at resonance with the emerging loops, and off resonance with the
preexisting ones. A discussion on why this is likely to be an appropriate
choice is provided in Section 4.2.

Results for the resonant case are shown in Figure 6d,e,f. In this case, the
ponderomotive acceleration starts to increase in the middle of the
chromosphere and is highest at the top of the chromosphere and the transition
region (Figure 6e). As a result, the abundances of Si and Ca (relative to H,
i.e. absolute abundances) start increasing slightly from the middle of the
chromosphere, and show the highest enhancement around the top of the
chromosphere and the transition region (Figure 6f). We focus on the relative
abundance predictions at the top of the transition region (i.e. 2500 km above
the photosphere in these simulations) since the model suggests that abundance
ratios are locked in once the plasma leaves this layer and is transported into
the corona. Hence these are the ones to be compared with coronal observations.
At the top of the transition region, Ar and S are essentially not
fractionated. Ca and Si are both significantly fractionated. Ca shows a
stronger fractionation than Si (Figure 6f) which is likely because the
ionization fraction of Ca is higher than the one of Si (Figure 6b) which means
a larger fraction of the Ca atoms are affected by the ponderomotive force.
This could explain why the Hinode EIS observations show slightly higher Ca/Ar
FIP bias values than Si/S FIP bias values in the emerging loops.

Figure 6: Ponderomotive force model predictions. First row shows the variation
with height of the following parameters: a) electron temperature and density,
b) ionization fraction variation for low-FIP elements, c) ionization fraction
for high-FIP elements. Second row shows the variation with height of d)
downward and upward Alfvén wave energy fluxes for waves of assumed coronal
origin e) ponderomotive acceleration f) FIP bias relative to H in the resonant
fractionation case. Third row shows the variation with height of g) Alfvén
wave energy fluxes; the downward and upward fluxes are identical, h)
ponderomotive acceleration i) FIP bias relative to H in the non-resonant
fractionation case.

Results for the non-resonant case are shown in Figure 6g,h,i. In this case,
the fractionation happens lower down in the chromosphere and is overall
stronger: the ponderomotive acceleration starts increasing at the bottom of
the chromosphere, reaches a maximum at the middle of the chromosphere and then
decreases (Figure 6h). The abundances of Si and Ca (relative to H) start
increasing from the bottom of the chromosphere, which results in stronger
enhancements at the top of the transition region compared to the resonant
case. Interestingly, in this case S behaves similarly to Si and Ca rather than
Ar: it becomes enhanced as well, but to a lower degree than Si and Ca. As in
the resonant case, Ar shows no fractionation. This could explain the strong
difference between the Ca/Ar FIP bias and the Si/S FIP bias in the Hinode EIS
measurements of the preexisting loops. As in the resonant case, Ca is more
strongly enhanced than Si due to its higher ionization fraction. However, in
the non-resonant case, the largest discrepancy between diagnostics comes from
the fact that S experiences significant fractionation as well, i.e. does not
behave like a high-FIP element anymore. Therefore, the enhancement of Si is
underestimated when measured relative to S.

Table 3: Parameters used for Ponderomotive Force Model Predictions

Parameter | Emerging Loops | Preexisting Loops
---|---|---
L | 100 Mm | 510 Mm
$\rho$ | $10^{9.5}\text{cm}^{-3}$ | $10^{8.5}\text{cm}^{-3}$
B | 250 G | 200G
$v_{A}$ |  $10^{4}$ km s-1 |  $2.5\times 10^{4}$ km s-1
$f_{\text{resonance}}$ | 0.05 s-1 | 0.025 s-1
$\Omega_{\text{resonance}}$ | 0.351 rad s-1 | 0.157 rad s-1
$A_{\text{chromo}}$ | 0.22 km/s | 0.03 km/s
$A_{\text{coronal}}$ | 44 km/s | 14 km/s

Note. — Parameters listed here follow the same notation in Equations 1 and 2.
For the wave amplitudes, $A_{\text{chromo}}$ is the chromospheric wave
amplitude at the $\beta=1$ layer used as input for the model simulations and
$A_{\text{coronal}}$ is the coronal wave amplitude predicted by the model.

It is important to note that the main free parameter when running the model
and making FIP bias predictions is the chromospheric amplitude
$A_{\text{chromo}}$ of the wave that drives the fractionation process. In the
absence of observations that can help constrain the amplitudes of these waves
at the chromospheric level, we need to make a guess for the amplitudes to be
able to make a FIP bias prediction. Small changes in the wave amplitudes
result in large changes in the estimated fractionation strength at the top of
the transition region. However, while the strength of the fractionation is
strongly dependent on the amplitude, the fractionation patterns for different
elements mainly depend on the frequency of the wave driver (and, more
specifically, on how close the frequency of the driver is to the resonant
frequency of the loop) rather than the amplitude, so we can use the model
predictions to obtain a qualitative understanding of the relative enhancement
of different elements.

In the example described in this section and shown in Figure 6, we initiate
the calculations with the wave amplitudes $A_{\text{chromo}}$ given in Table
3. These are the input chromospheric wave amplitudes which resulted in
fractionation patterns that best matched the Hinode EIS observations at 01:08
UT. The model predicts the waves to develop amplitudes $A_{\text{coronal}}$ in
the coronal portions of the loops. In the chromosphere, where the FIP
fractionation is calculated, the resonant wave amplitude returns to
$A_{\text{chromo}}=0.22$ km s-1. The non-resonant wave amplitude is much
larger, of order 1 km s-1. This is characteristic of the non-resonant waves,
in that more wave energy accumulates in the chromosphere than in the corona.

Using the inputs listed in Table 3, the model predicts the following results:
in the emerging loops (resonant wave driver case), a Si/S FIP bias of 1.8 and
Ca/Ar FIP bias of 2.7 and, in the preexisting loops (non-resonant wave driver
case), a Si/S FIP bias of 1.5 and Ca/Ar FIP bias of 5.1. These values are
directly calculated from the simulation composition patterns at transition
region level shown in Figure 6f,i. Following the same approach,
$A_{\text{chromo}}$ can be changed slightly to obtain FIP bias values that
match the Hinode EIS observations at 14:15 UT as well (see Table 4). The key
result, however, is that, while the predicted FIP bias values depend on the
selected $A_{\text{chromo}}$, the significant differences between the two
diagnostics depend on whether the wave is resonant or not.

Table 4: Ponderomotive Force Model Predictions with Variable
$A_{\text{chromo}}$

Emerging Loops | Preexisting Loops
---|---
$A_{\rm chromo}$ | Si/S | Ca/Ar | $A_{\rm chromo}$ | Si/S | Ca/Ar
0.20 | 1.61 | 2.31 | 0.025 | 1.32 | 3.13
0.22 | 1.80 | 2.70 | 0.03 | 1.48 | 5.1
0.26 | 2.22 | 4.00 | 0.035 | 1.68 | 7.0
0.30 | 2.87 | 6.1 | 0.04 | 1.93 | 14.7

Note. — For the wave amplitudes, $A_{\text{chromo}}$ is the chromospheric wave
amplitude at the $\beta=1$ layer. $A_{\rm corona}$ changes in proportion.
Parameters are chosen to span the range of Si/S and Ca/Ar in Table 2.

While one could model Si/S and Ca/Ar coming from different strands with
different wave populations within each loop, we consider it a success that the
same fractionation process (i.e. one wave in each loop, of the same frequency)
for both Si/S and Ca/Ar reduces the former and increases the latter in going
from emerging to preexisting loops. One might achieve better agreement between
model and observation with more waves, but this would be at the expense of
more model parameters.

###  4.2 Alfvén Waves Origin Discussion

The ponderomotive force model simulations suggest that the fractionation
pattern observed in the two loop populations could be explained if the driver
is resonant waves in the emerging loops and non-resonant waves in the
preexisting loops. This naturally raises a question regarding the origin of
these waves. We propose that the waves giving rise to the fractionation seen
in the emerging loops have a coronal driver, and consider both a coronal and a
photospheric driver for the for the waves giving rise to the fractionation
observed in the preexisting loops. The emerging loops make up the very active
core of the active region, where nanoflares, for example, can give rise to
coronal Alfvén waves that are naturally at resonance with the loop. The
preexisting loops are less active, which means an external driver is more
likely to generate the waves needed for the fractionation.

The first candidate is of coronal origin. We speculate that resonant waves in
the emerging loops could be communicated to the pre-existing loops where they
will be non-resonant. To see this we write an equation of motion for waves on
the pre-existing loop (2) forced by kink oscillations of the emerging loop (1)
with displacement $x_{2}$

$\rho_{2}\left(\ddot{x_{2}}+\Omega_{2}^{2}x_{2}\right)=-{\partial\over\partial
r}\left({\delta B_{\perp 1}^{2}\over 8\pi}+{B_{0}\delta B_{||1}\over
4\pi}\right).$ (3)

Here we assume $\delta B_{1}\propto 1/r$, where $\delta B_{1}$ is the magnetic
field perturbation in emerging loop 1, with components perpendicular and
parallel to the ambient magnetic field $B_{0}$ indicated. The first Alfvénic
term on the right hand side $\delta B_{\perp 1}^{2}/8\pi=\rho_{1}\delta
v_{\perp 1}^{2}/2$ and and oscillates at $2\Omega_{1}$ to give

${x_{2}\over
x_{1}}={\rho_{1}\over\rho_{2}}{2\Omega_{1}^{2}\over\Omega_{2}^{2}-4\Omega_{1}^{2}}{x_{1}\over
R}.$ (4)

where $R$ is the separation between the flux tubes. With $x_{1}=44/0.351=125$
km, equation 4 indicates that $x_{2}/x_{1}=0.3$ requires a separation between
flux tubes of 2” - 4”, much smaller than the observed separation of order
100”.

The effect of the second, compressive, term on the right hand side is more
model dependent (e.g. Mikhalyaev & Solov’ev, 2005; Verwichte et al., 2006).
The $r$-component of the wavevector exterior to the emerging loop can be
written when the plasma beta $\beta<<1$

$k_{r}^{2}=\Omega^{2}-k_{z}^{2}v_{Ae}^{2}={v_{A1}^{2}-v_{Ae}^{2}\over
1+\rho_{e}/\rho_{1}}$ (5)

where
$\Omega^{2}=\left(B_{e}^{2}+B_{0}^{2}\right)/\left(\rho_{e}+\rho_{1}\right)/4\pi$
is the tube oscillation frequency in terms of exterior and interior ambient
magnetic fields $B_{e}$, $B_{0}$, and densities $\rho_{e}$, $\rho_{1}$
(Mikhalyaev & Solov’ev, 2005). When $v_{Ae}>v_{A1}$ (the usual case), $k_{r}$
is imaginary and the exterior wave is evanescent, meaning that the kink
oscillations are trapped inside the loop, making transfer of wave energy from
one loop to the other unlikely. Such a situation has been studied in detail
for magnetosonic waves escaping from reconnection current sheets by
Provornikova et al. (2018).The forgoing is doubtless oversimplified, and many
possibilities must exist for the excitation of wave modes as magnetic flux
emerges. Waves generated by nanoflares are, however, highly localised to
single loops rather than loop populations which raises questions about whether
this type of wave transfer could realistically take place.

A second candidate could be of photospheric origin, and include p- and g-mode
oscillations or other perturbations of the photospheric plasma flows generated
by the flux emergence happening in the close vicinity of the preexisting loops
footpoints (see Figure 2). These can act as a driver for both resonant and
non-resonant waves. Among the wide range of possible photospheric
perturbations, those with long wavelengths could naturally couple and perturb
the neighbouring preexisting loops population at the same time. In this
regard, it is worth noting that the global rotational motion seen in the
active region suggests a photospheric or sub-photospheric driver on large
spatial scales. Interestingly, Grant et al. (2022) have detected coherent
waves across multiple pores in the photosphere suggesting a coupled wave
excitation mechanism and a driver acting on scales of several tens of Mm. The
rotational motion observed in the active region studied here could also drive
torsional Alfvén waves. The associated spatial scales of the driver may
explain coupled behaviour of different loops. The S enhancement in the
preexisting loops (as suggested by the Hinode EIS observations) and the fact
that, as suggested by the model, the fractionation process takes place lower
down in the chromosphere in the non-resonant case support the scenario of a
photospheric origin for the waves driving fractionation in the preexisting
loops. It is, however, important to note that the low frequencies and long
wavelengths implied for 3 or 5 minutes p-modes which make them reflect more
easily in the chromosphere mean that high chromospheric wave amplitudes, $>10$
km s-1 are required. Higher frequency non-resonant waves, such as modeled in
Figure 6, reduce this amplitude to $\sim 1$ km s-1.

These are a couple of options that could explain the presence of non-resonant
waves in the preexisting loop. However, understanding the exact origin of
these waves is beyond the scope of this work and should be investigated in
depth in a separate study.

##  5 Summary and Discussion

Spectral analysis of Hinode EIS observations of NOAA AR 12665 show very
different FIP bias values in two parts of the active region. The emerging
loops, i.e. the new part of the active region, show enhanced Si/S FIP bias
(2.3 to 3.0) and slightly higher Ca/Ar FIP bias (2.6 to 3.3). The preexisting
loops, i.e. the old part of the active region, show more modest Si/S FIP bias
(1.8 to 2.0) but very strong Ca/Ar FIP bias (4.3 to 7.8). The Ca/Ar FIP bias
is slightly higher than the Si/S FIP bias in the emerging loops, but much
higher than the Si/S FIP in the preexisting loops. We find that the
ponderomotive force model is able to predict this effect using simple
assumptions about the properties of the waves driving the fractionation
process in the two loops.

We propose that the fractionation pattern observed in the emerging loops can
be given by resonant Alfvén waves of coronal origin. In this case, Ar and S
show no fractionation, while Ca and Si show significant fractionation (Ca
slightly higher than Si which could explain the slightly stronger Ca/Ar FIP
bias compared to Si/S FIP bias). This can be explained by fractionation
occurring at the top of the chromosphere (as was previously suggested for the
hot core loops from measurements of significantly higher FIP bias values by
Brooks & Yardley, 2021).

The fractionation pattern observed in the preexisting loops can be given by
non-resonant waves. In this case, Ca and Si show stronger fractionation than
in the previous case (Ca again slightly higher than Si) and Ar again shows no
fractionation. The key difference is that, in these conditions, S shows
significant fractionation as well, resulting in a much lower Si/S FIP bias
than Ca/Ar FIP bias. This can be explained by fractionation occurring lower
down in the chromosphere (as was previously suggested by Laming et al., 2019).
Note that although the two diagnostics measure different FIP bias levels, they
are both detecting coronal abundances in both the hot core loops and pre-
existing loops. It is the combination of Si/S and Ca/Ar FIP bias measurements
that allow further probing of the model predictions, and the development of
the resonant/non-resonant wave explanation of the wider loop environment.

It is important to note that the model is only analysing the chromospheric and
transition region environment, predicting abundances at the top of the
transition region. Transport mechanisms from the top of the transition region
to the corona must be considered to be able to make a prediction of the FIP
bias in the corona. The model suggests that the abundance change at the top of
the transition region can be reached within minutes (Laming, 2015). However,
previous studies (Widing & Feldman, 2001; Baker et al., 2018), found that the
FIP bias increases with time in the emergence phase over hours to days. This
indicates that transport processes that bring the fractionated plasma into the
corona are much slower than the processes that drive the fractionation, so
predicted chromospheric abundances will not be reflected in the corona right
away. Nevertheless, assuming that these coronal transport timescales are
similar in the emerging and preexisting loops, we can compare qualitative
trends predicted by the ponderomotive force model to our Hinode EIS
observations.

We observe an increase in both the Si/S FIP bias and the Ca/Ar FIP bias of the
emerging loops over the 13 hour period between the two Hinode EIS scans. This
could indicate that the fractionated plasma is slowly being transported to the
corona, i.e. the fractionation pattern at the top of the transition region is
slowly being reflected in the corona. The strong increase of the Ca/Ar FIP
bias in the preexisting loops over the 13 hour period could be explained
following the same reasoning, with the exception that the final Ca/Ar FIP bias
is higher so the increase appears to be more drastic. The steady increase
assumption requires reasonably quiet coronal conditions. However, we observe 1
M-class and 4 C-class flares happening at the boundary between these two loop
populations in between the first and the second Hinode EIS scans. In the EUV,
previous studies showed that flaring can either temporarily reduce the FIP
bias to photospheric values (Warren, 2014) or increase it (To et al., 2021).
While the boxes selected for our FIP bias measurements are located further
away from the flaring sites, we cannot exclude the possibility that the
flaring activity influences the coronal transport mechanisms in the loops we
are studying.

Finally, this result is particularly relevant for connection science studies.
Typically the FIP bias diagnostics used in remote sensing studies (e.g., Si/S,
Fe/S, Ca/Ar, Mg/Ne) are different from the ones used for in-situ measurements
(e.g., Fe/O), so understanding when these diagnostics behave differently is
important for connnecting the two types of measurements. According to the
ponderomotive force model, resonant waves drive little fractionation in larger
loops (e.g., the ones in this study) and no fractionation in open loops
(Laming, 2015). However, as the present work suggests, fractionation can still
be driven in these loops by non-resonant waves if an external driver is
present. In this case the Si/S FIP bias would be very low (and this can be
extended to other low-FIP elements relative to S) so the area under study
could mistakenly be believed to show no/weak fractionation unless a second
diagnostic (not including S) is used. Therefore, understanding what
differences to expect between different FIP bias diagnostics and being able to
predict in what conditions S starts experiencing significant fractionation is
important for connecting in-situ plasma parcels to their origin on the Sun.
This is especially important given that S is commonly used in both remote
sensing and in-situ FIP bias diagnostics.

We would like to thank Tom Van Doorsselaere for insightful comments and
discussions that helped develop some of the ideas presented in this work. The
EIS Chief Observer for July 9 was H. P. Warren. T.M. acknowledges funding the
Science and Technology Funding Council (STFC) PhD studentship ST/V507155/1 and
the support of a Royal Astronomical Society Field Trip grant to carry out part
of this work. The work of D.H.B. was performed under contract to the Naval
Research Laboratory and was funded by the NASA Hinode program. J.M.L. is
supported by NASA HSR Grant NNH22OB102 and by Basic Research Funds of the
Office of Naval Research. D.B. is funded under Solar Orbiter EUI Operations
grant number ST/X002012/1 and Hinode Ops Continuation 2022-25 grant number
ST/X002063/1. A.W.J. acknowledges funding from the STFC Consolidated Grant
ST/W001004/1. D.M.L. is grateful to the STFC for the award of an Ernest
Rutherford Fellowship (ST/R003246/1). L.v.D.G. acknowledges the Hungarian
National Research, Development and Innovation Office grant OTKA K-131508.
Hinode is a Japanese mission developed and launched by ISAS/JAXA,
collaborating with NAOJ as a domestic partner, and NASA and STFC (UK) as
international partners. Scientific operation of Hinode is performed by the
Hinode science team organized at ISAS/JAXA. Support for the post-launch
operation is provided by JAXA and NAOJ (Japan), STFC (UK), NASA, ESA, and NSC
(Norway). CHIANTI is a collaborative project involving George Mason
University, the University of Michigan (USA), University of Cambridge (UK) and
NASA Goddard Space Flight Center (USA).

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Physics Department, University of Roma “La Sapienza” and INFN, Sezione di
Roma,
Piazzale Aldo Moro 5, 00185 Roma, Italy Dipartimento di Fisica dell’Università
di Pisa and INFN, Sezione di Pisa,
Largo Bruno Pontecorvo 3, I-56127 Pisa, Italy Istituto Nazionale di Fisica
Nucleare, Sezione di Roma Tre,
Via della Vasca Navale 84, I-00146 Rome, Italy Scuola Normale Superiore,
Piazza dei Cavalieri 7, 56126 Pisa, Italy Istituto Nazionale di Fisica
Nucleare, Sezione di Pisa,
Largo Bruno Pontecorvo 3, I-56127 Pisa, Italy

# LFU ratios in B decays using Lattice QCD and Unitarity

G. Martinelliins:x  M. Naviglioins:y  S. Simulains:w L. Vittorioins:zins:zz
Speakerins:xins:xins:yins:yins:wins:wins:zins:zins:zzins:zz

###### Abstract

We present the results of the application of the unitarity-based Dispersion
Matrix approach to semileptonic charged-current $B$ decays. This method allows
to achieve a non-perturbative and completely model-independent determination
of the hadronic form factors. Starting from lattice results available at large
values of the momentum transfer, the behaviour of the form factors in their
whole kinematical range is obtained without introducing any explicit
parameterization of their momentum dependence. We will focus on the analysis
of Lepton Flavour Universality by computing the $\tau/\mu$ ratios of the
branching fractions of the $B\to D^{(*)}\ell\nu$ and $B\to\pi\ell\nu$ decays.
The most important result is that, for the first time, the discrepancies
between the SM expectation values and the measurements of the Lepton Flavour
Universality ratios for the $B\to D^{(*)}\ell\nu$ decays are reduced at the
1.3$\sigma$ level for each of the two channels, separately.

##  1 State-of-the-art of exclusive $B\to D^{(*)}\ell\nu$ decays

$B\to D^{(*)}\ell\nu$ decays are among the most challenging processes in the
phenomenology of flavor physics, since they are affected by two unsolved
problems.

On the one hand, we have the so-called $|V_{cb}|$ _puzzle_ , $i.e.$ the
discrepancy between the inclusive and the exclusive determinations of the CKM
matrix element $|V_{cb}|$. According to the FLAG Review 2021 [1], there is a
$\sim 2.8\sigma$ tension between the exclusive estimate (that depends on the
form factors parametrization) and the inclusive one, namely

$|V_{cb}|_{\rm excl}\times
10^{3}=39.36(68),\,\,\,\,\,\,\,\,\,\,\,\,|V_{cb}|_{\rm incl}\times
10^{3}=42.00(65).$ (1)

A new more precise estimate of the inclusive value has also recently appeared
[2], namely $|V_{cb}|_{\rm incl}=42.16(50)$, which is compatible with the
inclusive FLAG value in Eq. (1).

On the other hand, a strong tension exists between the theoretical value and
the measurements of $R(D^{(*)})$, which are a fundamental test of Lepton
Flavour Universality (LFU) and are defined as

$R(D^{(*)})\equiv\frac{\Gamma(B\to D^{(*)}\tau\nu_{\tau})}{\Gamma(B\to
D^{(*)}\ell\nu_{\ell})},$ (2)

where $\ell$ denotes a light lepton. The HFLAV Collaboration [3] computed the
world averages of the available measurements of the $R(D^{(*)})$ ratio and of
their SM theoretical predictions. From the numerical point of view, we have

$R(D)_{\rm SM}=0.299\pm 0.003,\,\,\,\,\,R(D)_{\rm exp}=0.339\pm 0.026\pm
0.014$ (3)

for the $B\to D$ case and

$R(D^{*})_{\rm SM}=0.254\pm 0.005,\,\,\,\,\,R(D)_{\rm exp}=0.295\pm 0.010\pm
0.010$ (4)

for the $B\to D^{*}$ one. As clearly stated by HFLAV Collaboration, the
averages of the measurements of $R(D)$ and $R(D^{*})$ exceed the corresponding
SM predictions by 1.4$\sigma$ and 2.8$\sigma$, respectively. If we also take
into consideration the experimental correlation between these two quantities,
namely $\rho=-0.38$, the resulting difference with the SM predictions is
increased at the 3.3$\sigma$ level.

##  2 The Dispersion Matrix approach to Form Factors in $B\to D^{(*)}\ell\nu$
decays

Let us focus on semileptonic $B\to D^{(*)}$ decays. In case of production of a
_pseudoscalar_ meson, $i.e.$ the $B\to D\ell\nu$ case, the differential decay
width reads

$\frac{d\Gamma}{dq^{2}}=\frac{G_{F}^{2}|V_{cb}|^{2}\eta_{EW}^{2}}{24\pi^{3}}\left(1-\frac{m_{\ell}^{2}}{q^{2}}\right)^{2}\times\left[C_{+}(q^{2})+C_{0}(q^{2})\right],$
(5)

where

$\displaystyle C_{+}(q^{2})$ $\displaystyle\equiv$
$\displaystyle|\vec{p}_{D}|^{3}\left(1+\frac{m_{\ell}^{2}}{2q^{2}}\right)|f^{+}(q^{2})|^{2},$
(6) $\displaystyle C_{0}(q^{2})$ $\displaystyle\equiv$ $\displaystyle
m_{B}^{2}|\vec{p}_{D}|\left(1-\frac{m_{D}^{2}}{m_{B}^{2}}\right)^{2}\frac{3m_{\ell}^{2}}{8q^{2}}|f^{0}(q^{2})|^{2}.$
(7)

In this case only two Form Factors (FFs) are present, namely
$f^{+}(q^{2}),\,f^{0}(q^{2})$. Moreover, $\vec{p}_{D}$ represents the
3-momentum of the produced $D$ meson and $m_{\ell}$ is the mass of the
produced lepton.

In case of production of a _vector_ meson, $i.e.$ the $B\to D^{*}\ell\nu$
case, the expression of the differential decay width is

$\frac{d\Gamma_{\tau}}{dw}=\frac{d\Gamma_{\tau,1}}{dw}+\frac{d\Gamma_{\tau,2}}{dw},$
(8)

where

$\displaystyle\frac{d\Gamma_{\tau,1}}{dw}$ $\displaystyle=$
$\displaystyle\frac{\eta_{EW}^{2}|V_{cb}|^{2}G_{F}^{2}m_{D^{*}}^{2}}{48\pi^{3}m_{B}}\sqrt{w^{2}-1}\left(1-\frac{m_{\tau}^{2}}{q(w)^{2}}\right)^{2}\left(1+\frac{m_{\tau}^{2}}{2q(w)^{2}}\right)\,K(w),$
(9) $\displaystyle K(w)$ $\displaystyle\equiv$ $\displaystyle
2\,q^{2}(w)\left(f(w)^{2}+m_{B}^{2}m_{D^{*}}^{2}\left(w^{2}-1\right)g(w)^{2}\right)+\mathcal{F}_{1}(w)^{2},$
(10) $\displaystyle\frac{d\Gamma_{\tau,2}}{dw}$ $\displaystyle=$
$\displaystyle\frac{\eta_{EW}^{2}|V_{cb}|^{2}G_{F}^{2}m_{B}^{5}}{32\pi^{3}}\frac{m_{\tau}^{2}(m_{\tau}^{2}-q(w)^{2})^{2}r^{3}(1+r)^{2}(w^{2}-1)^{3/2}P_{1}(w)^{2}}{q(w)^{6}}.$
(11)

In this case we have four FFs to deal with, $i.e.$
$f(w),g(w),\mathcal{F}_{1}(w),P_{1}(w)$. Note that in Eqs. (5) and (8) we
refer equivalently to the momentum transfer $q^{2}$ or to the recoil $w$,
since they are related by the following 1-to-1 correspondence

$q^{2}(w)=m_{B}^{2}+m_{D^{(*)}}^{2}-2m_{B}m_{D^{(*)}}w.$ (12)

Now, our goal is to describe the FFs entering in $B\to D^{(*)}\ell\nu$ decays
by using the novel Dispersion Matrix (DM) method [4], which was originally
proposed in [5]. The DM method allows us to study the FFs in a non-
perturbative and model-independent way, since, starting from the available
LQCD computations of the FFs at high momentum transfer (or, equivalently, at
low recoil), we can extrapolate their behaviour in the opposite kinematical
region. To this end, we do not assume any functional dependence of the FFs on
$q^{2}$ (or, equivalently, on $w$) and we use only non-perturbative inputs.
Moreover, the resulting bands of the FFs will be independent of the
experimental determinations of the differential decay widths.

From the mathematical point of view, the starting point is to focus on one of
the six FFs defined above, for instance $f$, and then consider the matrix

$\mathbf{M}=\left(\begin{tabular}[]{ccccc}$\chi$&$\phi
f$&$\phi_{1}f_{1}$&$...$&$\phi_{N}f_{N}$\\\\[5.69054pt] $\phi
f$&$\frac{1}{1-z^{2}}$&$\frac{1}{1-zz_{1}}$&$...$&$\frac{1}{1-zz_{N}}$\\\\[5.69054pt]
$\phi_{1}f_{1}$&$\frac{1}{1-z_{1}z}$&$\frac{1}{1-z_{1}^{2}}$&$...$&$\frac{1}{1-z_{1}z_{N}}$\\\\[5.69054pt]
$...$&$...$&$...$&$...$&$...$\\\\[5.69054pt]
$\phi_{N}f_{N}$&$\frac{1}{1-z_{N}z}$&$\frac{1}{1-z_{N}z_{1}}$&$...$&$\frac{1}{1-z_{N}^{2}}$\end{tabular}\right),$
(13)

where we have introduced the conformal variable $z$ defined as

$z(t)=\frac{\sqrt{t_{+}-t}-\sqrt{t_{+}-t_{-}}}{\sqrt{t_{+}-t}+\sqrt{t_{+}-t_{-}}},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,t_{\pm}=(m_{B}\pm
m_{D^{(*)}})^{2}$ (14)

or, equivalently, as

$z=\frac{\sqrt{w+1}-\sqrt{2}}{\sqrt{w+1}+\sqrt{2}}.$ (15)

In this expression, $\phi_{i}f_{i}\equiv\phi(z_{i})f(z_{i})$ (with
$i=1,2,...N$) represent the known values of the quantity $\phi(z)f(z)$
corresponding to the values $z_{i}$ at which the FFs have been computed on the
lattice. The kinematical function $\phi(z)$ has a specific expression for each
of the aforementioned FFs. The general forms for each case can be found in
[6]. Finally, the susceptibility $\chi(q^{2})$ is related to the derivative
with respect to $q^{2}$ of the Fourier transform of suitable Green functions
of bilinear quark operators and follows from the dispersion relation
associated to a particular spin-parity quantum channel. Note that they have
been computed for the first time on the lattice in [7] for $b\to c$ quark
transitions. At this point, one can demonstrate from first principles that
$\det\mathbf{M}\geq 0$. Then, the positivity of the determinant, which we will
refer to as _unitarity filter_ hereafter, allows to compute the lower and the
upper bounds of the FF of interest for each generic value of $z$, $i.e.$

$f_{\rm lo}(z)\leq f(z)\leq f_{\rm up}(z).$ (16)

The explicit definitions of $f_{\rm lo}(z),\,f_{\rm up}(z)$ can be found in
[4].

##  3 An instructive example: the semileptonic $B\to D^{*}$ channel

Let us discuss in detail the semileptonic $B\to D^{*}$ decay, which is very
challenging due to the high number of FFs involved. In [8] we have computed
the unitarity bands of the FFs, starting from the final results of the
computations on the lattice performed by the FNAL/MILC Collaborations [9].
There, in the ancillary files, the authors give the synthetic values of the
FFs $g(w),f(w),\mathcal{F}_{1}(w)$ and $\mathcal{F}_{2}(w)$ at three non-zero
values of the recoil variable ($w-1$), namely $w=\\{1.03,1.10,1.17\\}$,
together with their correlations. Note that the FF $\mathcal{F}_{2}(w)$ is
directly related to the $P_{1}(w)$ one, in fact
$P_{1}(w)=\mathcal{F}_{2}(w)\sqrt{r}/(1+r)$, where $r\equiv
m_{D^{*}}/m_{B}\simeq 0.38$.

In Fig. 1 we show the results of our DM analysis as red bands. The DM
unitarity bands are built up through bootstrap events that satisfy exactly
both the unitarity filter of the DM method and the Kinematical Constraints
(KCs)

$\displaystyle\mathcal{F}_{1}(1)$ $\displaystyle=$ $\displaystyle
m_{B}(1-r)f(1),$ (17) $\displaystyle P_{1}(w_{max})$ $\displaystyle=$
$\displaystyle\frac{\mathcal{F}_{1}(w_{max})}{m_{B}^{2}(1+w_{max})(1-r)\sqrt{r}},$
(18)

where $w_{\rm max}\simeq 1.5$. These unitarity bands allow us to compute new
_fully-theoretical_ values of the LFU ratio $R(D^{*})$, thus obtaining

$R(D^{*})=0.275\pm 0.008.$ (19)

For the first time, the compatibility between this theoretical determination
of $R(D^{*})$ and the corresponding HFLAV world average of the measurements is
at the $\sim 1.3\sigma$.

At this point, a natural question arises: why is the SM HFLAV theoretical
average in Eq. (4) so different from the DM estimate in Eq. (19)? The answer
has to be found in the shape of the FFs. In Fig. 1 the red DM unitarity bands
are compared with new green ones, taken directly from Fig. 2 of [10]. In this
article, the authors show the plot of the FFs $g(w)$, $f(w)$ and
$\mathcal{F}_{1}(w)$ obtained by fitting the Belle experimental data [11] for
the differential decay widths through the BGL parametrization [12, 13, 14].
This comparison is particularly instructive since from the green bands the
authors of [10] obtain the result $R(D^{*})=0.251^{+0.004}_{-0.005}$, which is
very similar to the SM HFLAV result (4). To achieve this goal, the authors of
[10] had also constrained the pseudoscalar FF $P_{1}(w)$ through appropriate
Heavy Quark Effective Theory relations, as described in detail in [15]. Figure
1 clearly shows that the different shapes of the FF $\mathcal{F}_{1}(w)$
induced by the LQCD or the experimental data have a fundamental impact on the
final theoretical value of $R(D^{*})$. To avoid the bias induce by the
experimental data on $R(D^{*})$, we have then adopted the DM method by using
_only_ the LQCD computations as inputs for the description of the unitarity
bands of the FFs.

Figure 1: The red bands of the FFs $g(w)$, $f(w)$, $\mathcal{F}_{1}(w)$ and
$P_{1}(w)$ are computed through the DM method after imposing both the
unitarity filter and the two KCs (17)-(18). The FNAL/MILC values [9] used as
inputs for the DM method are represented by the black diamonds. The green
bands are, instead, the FFs obtained by fitting the Belle experimental data
[11] for the differential decay widths. These bands have been taken from Fig.
2 of [10].

##  4 A summary of the DM applications to all the $b\to c$ quark transitions

The application of the Dispersive Matrix method, that we have explicitly
described in the case of semileptonic $B\to D^{*}$ decays, can be repeated to
all the other relevant $b\to c$ quark transitions in the mesonic sector,
namely the $B\to D\ell\nu$, the $B_{s}\to D_{s}\ell\nu$ and the $B_{s}\to
D_{s}^{*}\ell\nu$ processes. In the former case, in [6] we have obtained the
unitarity bands of the FFs $f_{+},\,f_{0}$, defined in Eq. (5), by using the
FNAL/MILC synthetic data in [16]. In the latter two cases, in [17] we have
extracted three values of the relevant FFs from the fits published by the
HPQCD Collaboration in [18, 19] by using the BCL parametrization [20] and we
have then implemented the DM method. In Fig. 2 we show the DM estimates of all
the LFU observables characterizing these decays, $i.e.$ the $R(D_{(s)})$ and
$R(D_{(s)}^{*})$ ratios. The black area represents the average of all the
experimental measurements of $R(D)$ and $R(D^{*})$, computed by HFLAV and
reported in Eqs. (3)-(4). The red and the green regions are, instead, the DM
predictions for $R(D_{(s)})$ and $R(D_{(s)}^{*})$ obtained in Refs. [6, 8,
17].

Our first message is that the anomalies in the $B$ sector have been very
lightened through the DM approach to the hadronic FFs. This achievement is
mainly due to the absence of the mixing between theoretical and experimental
data in the description of the FFs. Our second message is that the comparison
between $R(D^{*})$ and $R(D_{s}^{*})$, which differ by $\approx 10\%$,
highlights the possible presence of $SU(3)_{F}$ symmetry breaking effects in
semileptonic charged-current $B$ decays. In this sense, further studies of the
spectator-quark dependence of the hadronic FFs of the semileptonic $B\to
D^{(*)}$ and $B_{s}\to D_{s}^{(*)}$ transitions are called for.

Figure 2: The correlation plot for $R(D_{(s)})$ and $R(D_{(s)}^{*})$. The
black area represents the average of all the experimental measurements of
$R(D)$ and $R(D^{*})$ performed by HFLAV Collaboration. The red and the green
regions are the DM predictions for $R(D_{(s)})$ and $R(D_{(s)}^{*})$.

##  5 Lepton Flavour Universality in semileptonic $B\to\pi$ decays

The DM method can be applied to whatever semileptonic charged-current decays
of mesons and baryons. It is very instructive, thus, to investigate its
potential in the analysis of the $b\to u$ quark transitions.

Let us discuss, for instance, the case of the $B\to\pi\ell\nu$ decays [21].
They are characterized by two FFs, which are analogous to the ones defined in
Eq. (5), given the pseudoscalar nature of the $\pi$ meson. Let us call them
$f_{+}^{\pi},\,f_{0}^{\pi}$ to distinguish them from the $B\to D$ case. These
FFs have been studied by the RBC/UKQCD [22] and the FNAL/MILC [23]
Collaborations. For both channels the lattice computations of the FFs are
available in the large-$q^{2}$ region. To be more specific, the authors of
Ref. [22] provide synthetic LQCD values of the FFs (together with their
statistical and systematic correlations) at $q^{2}=\\{19.0,22.6,25.1\\}$ GeV2.
In [23], instead, only the results of BCL fits of the FFs extrapolated to the
continuum limit and to the physical pion point are available. Thus, from the
marginalized BCL coefficients we evaluate the mean values, uncertainties and
correlations of the FFs at the same three values of $q^{2}$ given in Ref.
[22].

In Fig. 3 we show the red (blue) DM bands which are obtained by using as
inputs the RBC/UKQCD (FNAL/MILC) data, respectively. In principle, when one
implements the BCL fits the mean value and the uncertainty of the FFs value
extrapolated at zero momentum transfer are not stable under variation of the
truncation order of a series expansion of the FFs. On the contrary, the DM
approach is completely independent of this issue, since no approximation due
to the truncation of a series expansion is present. In other words, we argue
that the DM method is equivalent to the results of all possible (BCL) fits
which satisfy unitarity and, at the same time, reproduce exactly the input
data. Note that this property is particularly useful in $B\to\pi\ell\nu$
decays, since here we have a long extrapolation in $q^{2}$.

Figure 3: The scalar $f_{0}^{\pi}(q^{2})$ (left panel) and vector
$f_{+}^{\pi}(q^{2})$ (right panel) FFs entering the semileptonic
$B\to\pi\ell\nu_{\ell}$ decays computed by the DM method as a function of the
4-momentum transfer $q^{2}$ using the LQCD inputs from RBC/UKQCD (red points)
and FNAL/MILC (blue squares) Collaborations. In the right panel, the vector FF
is multiplied by the factor $(1-q^{2}/m_{B^{*}}^{2})$ with $m_{B^{*}}=5.325$
GeV.

These unitarity bands can be used to compute fully-theoretical values of the
LFU ratio
$R_{\pi}^{\tau/\mu}\equiv\Gamma(B\to\pi\tau\nu_{\tau})/\Gamma(B\to\pi\mu\nu_{\mu})$,
which is equivalent to the $R(D^{(*)})$ ratios defined in Eq. (2). The DM
results are shown in Table 1 for each LQCD input. The _combined_ case in the
last column corresponds to the combination of the RBC/UKQCD and the FNAL/MILC
data, as described in [21]. They are all compatible with the only available
measurement by Belle [24]

$R_{\pi}^{\tau/\mu}|_{exp}=1.05\pm 0.51,$ (20)

which has a large uncertainty compared to theoretical predictions. However,
the expected uncertainty on the above ratio by Belle II at 50 ab-1 of
luminosity [25] is $\delta R_{\pi}^{\tau/\mu}\simeq 0.09$, comparable to our
present theoretical uncertainties. Further LQCD computations of the FFs, as
well as more precise ones, will thus be of capital importance in order to test
possible New Physics effects affecting semileptonic $B\to\pi$ decays.

Input | RBC/UKQCD | FNAL/MILC | combined
---|---|---|---
$R^{\tau/\mu}_{\pi}$ | 0.767(145) | 0.838(75) | 0.793(118)
Table 1: Theoretical values of $R_{\pi}^{\tau/\mu}$ in the case of
semileptonic $B\to\pi$ decays adopting the RBC/UKQCD, the FNAL/MILC and the
combined LQCD data as inputs for our DM method.

##  6 Conclusions

We have reviewed the main properties of the Dispersion Matrix approach, which
is an attractive tool to implement unitarity and LQCD calculations in the
analysis of exclusive semileptonic decays of mesons and baryons. It has
several interesting properties. In particular, it does not rely on any
assumption about the momentum dependence of the hadronic form factors and it
can be based entirely on first principles ($i.e.$ unitarity and analiticity)
and on non-perturbative inputs.

We have discussed the application of the DM method to the $b\to c$ and the
$b\to u$ quark transitions and the resulting theoretical determinations of the
LFU observables. In this sense, our main result is that the anomalies in
semileptonic charged-current $B$ decays have been strongly lightened. In fact,
we have consistency between theory and experiment at the $1.3\sigma$ level for
both $R(D)$ and $R(D^{*})$, separately. To achieve this goal, it is
fundamental to avoid the mixing among theoretical calculations and
experimental data to describe the shape of the FFs of interest.

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