#  Electroweak vacuum decay in metric-affine gravity

Ioannis D. Gialamas Hardi Veermäe Laboratory of High Energy and Computational
Physics, National Institute of Chemical Physics and Biophysics,
Rävala pst. 10, 10143, Tallinn, Estonia

###### Abstract

We investigate the stability of the electroweak vacuum in metric-affine
gravity in which the Standard Model Higgs boson can be non-minimally coupled
to both the Ricci scalar and the Holst invariant. We find that vacuum
stability is improved in this framework across a wide range of model
parameters.

##  1 Introduction

It is well known that the potential of the Higgs boson in the Standard Model
(SM) is deeper at high energies than in the electroweak vacuum permitting its
decay through quantum tunneling [1, 2, 3, 4]. Although this does not
invalidate the SM, the electroweak vacuum is predicted to be metastable in the
absence of contributions from UV physics [5, 6, 7, 8, 9, 10, 11, 12, 13].

Coleman and De Luccia [14] were the first to delve into the matter of
gravitational effects on vacuum decay. Subsequently, multiple studies of
gravitational corrections have been performed [15, 16, 17, 18, 19, 20, 21, 22,
23], along with discussions about the impact of black holes on the
amplification or reduction of the vacuum decay rate [24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40].

In this letter, we extend the calculation of gravitational corrections to
vacuum decay in the context of metric-affine gravity. There, in contrast to
general relativity, the connection is taken to be an independent variable
without the usual symmetries of the Levi-Civita one. As a result, the Riemann
tensor does not possess the symmetries it has in the metric case, and thus the
gravitational action should be extended by including an additional scalar
curvature invariant - the Holst invariant. This term commonly appears in Loop
Quantum Gravity [41] and has been studied in various branches of high energy
physics, such as black hole thermodynamics [42, 43]. Lately, its significance
in inflationary cosmology [44, 45, 46, 47, 48, 49], and high energy physics
phenomenology [50, 51, 52, 53, 54], has garnered a great deal of attention.

##  2 The gravitational action

Metric-affine theories of gravity treat the metric tensor $g_{\mu\nu}$ and the
connection $\tensor{\Gamma}{{}^{\lambda}_{\mu}{}_{\nu}}$ as independent
variables. This should be contrasted with the usual metric gravity that uses
the Levi-Civita connection
$\left\\{\tensor{}{{}^{\lambda}_{\mu}{}_{\nu}}\right\\}$, which is completely
determined by the metric. It is useful to decompose the connection
$\tensor{\Gamma}{{}^{\lambda}_{\mu}{}_{\nu}}$ as

$\tensor{\Gamma}{{}^{\lambda}_{\mu}{}_{\nu}}\equiv\left\\{\tensor{}{{}^{\lambda}_{\mu}{}_{\nu}}\right\\}+\tensor{C}{{}^{\lambda}_{\mu}{}_{\nu}}\,,$
(1)

where $\tensor{C}{{}^{\lambda}_{\mu}{}_{\nu}}$ is dubbed the distortion
tensor. The Riemann tensor
$\tensor{\mathcal{R}}{{}^{\mu}_{\nu}{}_{\rho}{}_{\sigma}}$ is constructed from
the connection $\Gamma$ in the usual way and one can form two scalars that are
linear on it. These are the Ricci scalar and the Holst invariant [55, 56]

$\mathcal{R}\equiv\tensor{\mathcal{R}}{{}^{\mu}{}^{\nu}_{\mu}{}_{\nu}}\,,\qquad\widetilde{\mathcal{R}}\equiv\frac{1}{2}\tensor{\epsilon}{{}^{\mu}{}^{\nu}{}^{\rho}{}^{\sigma}}\tensor{\mathcal{R}}{{}_{\mu}{}_{\nu}{}_{\rho}{}_{\sigma}}\,,$
(2)

where $g$ is the determinant of the metric tensor and
$\epsilon^{\mu\nu\rho\sigma}$ is the totally antisymmetric tensor. In metric
gravity, the Holst invariant vanishes identically due to the symmetries of the
Riemann tensor.

The most general action linear in the Riemann tensor and containing terms of
at most dimension 4 has the form111Natural units with $c=\hbar=1$ are used
throughout this paper.

$S=\int{{\rm
d}}^{4}x\sqrt{-g}\bigg{[}\frac{\mathcal{R}}{2}f(h)+\frac{\widetilde{\mathcal{R}}}{2}\tilde{f}(h)-\frac{1}{2}(\partial
h)^{2}-V(h)\bigg{]}\,,$ (3)

where $V(h)$ is the Higgs potential,

$f(h)=1/\kappa+\xi h^{2},\qquad\tilde{f}(h)=\beta/\kappa+\tilde{\xi}h^{2}\,,$
(4)

are non-minimal couplings, $\beta$, $\xi$ and $\tilde{\xi}$ are constant
couplings, and $\kappa=1/M_{\rm Pl}^{2}$. The function $1/\tilde{f}(h)$ can be
thought of as a field-dependent Barbero-Immirzi parameter [56, 57]. We will
first work out the formalism without assuming a specific functional form of
$f$, $\tilde{f}$, and $V$ and suppress their arguments for notational brevity.

In addition to the Ricci and Holst terms, metric-affine gravity permits the
construction of 20 additional scalars with mass dimension 2 from torsion
$T_{\rho\mu\nu}\equiv 2\Gamma_{\rho[\mu\nu]}$ and non-metricity
$Q_{\rho\mu\nu}\equiv\nabla_{\rho}g_{\mu\nu}$ [58, 59]. Although such terms
are not considered in this work, our perturbative results can be
straightforwardly extended to include them as will be explained below. We will
also neglect couplings between the connection and fermions as they will only
generate Planck-suppressed four-fermion and higher-order scalar-fermion
interactions [60, 51, 53, 47] and do not affect the leading order corrections
to vacuum stability.

We remark that the action (3) also appears in Einstein-Cartan gravity. A
crucial distinction with the current case is that the Einstein-Cartan
connection is decomposed using the Levi-Civita connection and torsion and,
unlike in metric-affine gravity, the non-metricity is taken to be zero.
Nevertheless, as we will show below, the metric-affine framework contains both
the metric and Palatini formulations as limiting cases and the non-metricity
may be taken to zero without loss of generality.

In order to study bounce solutions, we construct the Euclidean action (3) by
analytically continuing the Lorentzian signature $(-,+,+,+)$ to the Euclidean
one $(+,+,+,+)$. Then, to bring the action to a more conventional form, we
will first integrate out the connection. To this aim, we will express the
Ricci scalar and the Holst invariant in terms of the metric Ricci scalar
$R[g]$ and the distortion tensor,

$\displaystyle\mathcal{R}=$ $\displaystyle
R+D_{\mu}\tensor{C}{{}^{\mu}_{\nu}{}^{\nu}}-D_{\nu}\tensor{C}{{}^{\mu}_{\mu}{}^{\nu}}+\tensor{C}{{}^{\mu}_{\mu}{}_{\lambda}}\tensor{C}{{}^{\lambda}_{\nu}{}^{\nu}}-\tensor{C}{{}^{\mu}_{\nu}{}_{\lambda}}\tensor{C}{{}^{\lambda}_{\mu}{}^{\nu}}\,,$
(5a) $\displaystyle\widetilde{\mathcal{R}}=$
$\displaystyle\epsilon^{\mu\nu\rho\sigma}\left(D_{\mu}\tensor{C}{{}_{\rho}{}_{\nu}{}_{\sigma}}+\tensor{C}{{}_{\rho}{}_{\mu}{}_{\lambda}}\tensor{C}{{}^{\lambda}_{\nu}{}_{\sigma}}\right)\,,$
(5b)

where $D$ denotes the covariant derivative of the Levi-Civita connection.
Substituting Eq. (5a) and (5b) into the action (3), yields222Although the
Holst term picks up an imaginary unit when continuing to Euclidean space,
analogously to the CP violating topological term in Yang-Mills theory (e.g.,
see Ref. [61]), its effect is negated due to the dependence of
$\epsilon^{\mu\nu\rho\sigma}\epsilon_{\mu\nu\rho\sigma}={\rm sign}(g)\,4!$ on
the sign of the metric determinant.

$\displaystyle S_{E}=$ $\displaystyle\int{{\rm
d}}^{4}x\sqrt{g}\bigg{[}-\frac{R}{2}f+\frac{1}{2}(\partial h)^{2}+V$
$\displaystyle-\frac{f}{2}\left(D_{\mu}\tensor{C}{{}^{\mu}_{\nu}{}^{\nu}}-D_{\nu}\tensor{C}{{}^{\mu}_{\mu}{}^{\nu}}+\tensor{C}{{}^{\mu}_{\mu}{}_{\lambda}}\tensor{C}{{}^{\lambda}_{\nu}{}^{\nu}}-\tensor{C}{{}^{\mu}_{\nu}{}_{\lambda}}\tensor{C}{{}^{\lambda}_{\mu}{}^{\nu}}\right)$
$\displaystyle-\frac{i\tilde{f}}{2}\epsilon^{\mu\nu\rho\sigma}\left(D_{\mu}\tensor{C}{{}_{\rho}{}_{\nu}{}_{\sigma}}+\tensor{C}{{}_{\rho}{}_{\mu}{}_{\lambda}}\tensor{C}{{}^{\lambda}_{\nu}{}_{\sigma}}\right)\bigg{]}\,.$
(6)

The distortion tensor obeys an algebraic non-homogeneous linear equation of
motion. Thus, in order to integrate it out in full generality, it is
sufficient to find a particular solution to this equation [47]. Such a
solution is given by [49]

$C_{\mu\nu\rho}\,=\frac{1}{2}\left(\,g_{\nu\mu}\partial_{\rho}X-g_{\nu\rho}\partial_{\mu}X-i\epsilon_{\mu\nu\rho\sigma}\partial^{\sigma}Y\right)\,,$
(7)

where

$f=e^{X}\cos(Y),\qquad\tilde{f}=e^{X}\sin(Y)\,.$ (8)

This solution is metric compatible, i.e.,
$Q_{\rho\mu\nu}=-2C_{(\nu|\rho|\mu)}=0$, and has torsion
$T_{\rho\nu\mu}=2C_{\rho[\nu\mu]}$. So, the theory is dynamically equivalent
to the Einstein-Cartan theory. On the other hand, the particular solution (7)
is not the general one because of the projective symmetry of the action,
$C_{\rho\nu\mu}\to C_{\rho\nu\mu}+g_{\rho\mu}A_{\nu}$, which can be used to
induce the non-metricity $Q_{\rho\mu\nu}=-2g_{\mu\nu}A_{\rho}$. In particular,
the Palatini limit with $\tilde{f}=0$ is obtained by choosing
$A_{\nu}=\partial_{\nu}X/2$.

Substituting (7) in the action (2) gives

$S_{E}=\int{{\rm d}}^{4}x\sqrt{g}\left[-\frac{R}{2}f+\frac{1}{2}K(\partial
h)^{2}+V\right]\,,$ (9)

where the contribution of the independent connection is now fully captured by
the kinetic function333The primes denote differentiation with respect to the
argument of the function, that is, depending on the context, with respect to
$h$ or $r$.

$K=1+\frac{3}{2}\frac{f\tilde{f}^{\prime
2}-2f^{\prime}\tilde{f}\tilde{f}^{\prime}-ff^{\prime
2}}{f^{2}+\tilde{f}^{2}}\,.$ (10)

Note that the action (9) does not depend on the sign of $\tilde{f}$.

For specific combinations of the involved functions, the general metric-affine
theory interpolates continuously between the metric ($K=1$) and the Palatini
($K=1-(3/2)f^{\prime 2}/f$) theories. In accordance with Ref. [44], we find
that these scenarios correspond to

$\displaystyle\tilde{f}=$ $\displaystyle
cf^{2}-\frac{1}{4c}\,,\qquad\text{(metric)}$ (11a) $\displaystyle\tilde{f}=$
$\displaystyle cf\,,\,\qquad\qquad\text{(Palatini)}$ (11b)

with $c$ a constant. As an important case, the metric formulation can be
obtained in the limit in which the constant part of $\tilde{f}$ is large. More
specifically, without loss of generality we can consider
$\tilde{f}(h)=\beta/\kappa+\tilde{f}_{1}(h)$, where $\tilde{f}_{1}$, $f$ are
arbitrary functions of $h$. If $|\beta|/\kappa\gg\tilde{f}_{1},f$, then

$K=1-\frac{3\kappa}{\beta}\,f^{\prime}\tilde{f}^{\prime}+\mathcal{O}\left(\frac{\kappa}{\beta}\right)^{2}\,,$
(12)

and thus the metric-affine theory approaches the purely metric theory when
$|\beta|\to\infty$. With $f$, $\tilde{f}$ given by (4), the Palatini
formulation corresponds to $\beta=\tilde{\xi}/\xi$, of which
$\beta=\tilde{\xi}=0$ is only a special case. Consequently, as $\beta$ ranges
from $-\infty\to\tilde{\xi}/\xi\to\infty$, the metric formulation is
continuously deformed to the Palatini one and back.

##  3 Corrections to vacuum decay in metric-affine gravity

Figure 1: The minimal bounce action (16) for $\beta=0,\pm 6$ and various
values of $\xi$ and $\tilde{\xi}$. The horizontal line in brown indicates the
bounce action in the absence of gravity, $S_{0}=8\pi^{2}/(3|\lambda(\mu)|)$.
The metric and Palatini cases are represented by the black-dashed and gray-
dot-dashed curves, respectively.

To compute the minimal bounce action, we will look for $O(4)$ symmetric
solutions [62], with the line element ${\rm d}s^{2}={\rm d}r^{2}+\rho^{2}{\rm
d}\Omega_{3}^{2}$, where ${\rm d}\Omega_{3}^{2}$ denotes the line element of
the unit 3-sphere and the Higgs field depends only on the radial coordinate
$r$, i.e. $h=h(r)$. In this background, the metric Ricci scalar reads
$R=6(1-\rho\rho^{\prime\prime}-\rho^{\prime 2})/\rho^{2}$ and the Euclidean
action can be recast as

$S_{E}=2\pi^{2}\int{\rm
d}r\,\rho^{3}\Bigg{[}3f\,\frac{\rho\rho^{\prime\prime}+\rho^{\prime
2}-1}{\rho^{2}}+\frac{1}{2}Kh^{\prime 2}+V\Bigg{]}\,.$ (13)

The bounce solution is determined by the following equations of motion444The
$\rho$ equations of motion follow from the $rr$ component of the Einstein
equations, but can also be derived by varying the action (13).

$\displaystyle\rho^{\prime 2}$
$\displaystyle=1+\frac{\rho^{2}}{3f}\left(\frac{1}{2}Kh^{\prime
2}-V-3\frac{\rho^{\prime}}{\rho}f^{\prime}h^{\prime}\right)\,,$ (14a)
$\displaystyle h^{\prime\prime}$
$\displaystyle=-3\frac{\rho^{\prime}}{\rho}h^{\prime}+\frac{1}{K}\left(-\frac{1}{2}K^{\prime}h^{\prime
2}+V^{\prime}-\frac{R}{2}f^{\prime}\right)\,.$ (14b)

To obtain the bounce action, we will adopt the perturbative method proposed in
Ref. [15] and look for solutions as a series in $\kappa$, i.e.

$\displaystyle h(r)=$ $\displaystyle h_{0}(r)+\kappa
h_{1}(r)+\mathcal{O}(\kappa^{2})\,,\qquad$ (15a) $\displaystyle\rho(r)=$
$\displaystyle
r+\kappa\rho_{1}(r)+\kappa^{2}\rho_{2}(r)+\mathcal{O}(\kappa^{3})\,,$ (15b)

This approach is suitable when the gravitational corrections are relatively
small, and, as we will demonstrate, this technique is adequate for elucidating
the differences that emerge due to the inclusion of the Holst invariant. In a
similar vein, the bounce action (13) can be expanded as

$S_{E}=S_{0}+\kappa S_{1}+\mathcal{O}(\kappa^{2})\,.$ (16)

The leading order solution555See also [63] for exact solutions for vacuum
decay in Higgs-like unbounded potentials. $h_{0}(r)$ of (14) is the so-called
Fubini instanton [64, 65]

$h_{0}(r)=\sqrt{\frac{2}{|\lambda|}}\frac{2\mu}{1+\mu^{2}r^{2}}\,,$ (17)

with $\mu$ being an arbitrary scale of the bounce. It solves the equation of
motion in the absence of gravity ($\rho_{0}=r$) and with $V(h)=\lambda
h^{4}/4$ assuming that the Higgs quartic coupling $\lambda$ is constant and
negative. The leading order contribution to the action (13) is

$S_{0}=2\pi^{2}\int{{\rm d}}r\,r^{3}\left(\frac{h_{0}^{\prime
2}}{2}+V(h_{0})\right)=\frac{8\pi^{2}}{3|\lambda|}\,,$ (18)

and gives the bounce action in the absence of gravity. To obtain the
gravitationally corrected action one must account for the running of $\lambda$
[19, 17]. We will evaluate $\lambda$ at the scale of the bounce $\mu$ and then
minimize the action with respect to $\mu$. The running of $\lambda$ is
computed at a 3-loop level [11] with the relevant parameters taken from [66].

Evaluating the gravitational correction $S_{1}$ relies on the specific form of
$f$ and $\tilde{f}$, for which we will assume the form (4) when needed. We
will assume that the leading order gravitational corrections to the kinetic
function (10) can be expressed as

$K=1+\kappa K_{1}h^{2}+\mathcal{O}(\kappa^{2})\,,$ (19)

where $K_{1}$ is a dimensionless constant. Indeed, with $f$ and $\tilde{f}$
given by (4), we have that

$K_{1}\equiv-6\frac{\xi^{2}+2\beta\xi\tilde{\xi}-\tilde{\xi}^{2}}{1+\beta^{2}}\,.$
(20)

Note that for large $\beta$ we recover Eq. (12). At order
$\mathcal{O}(\kappa)$, the equation of motion (14a) is

$\rho_{1}^{\prime}=\frac{1}{6}r^{2}\left(\frac{1}{2}h_{0}^{\prime
2}-V(h_{0})-3f^{\prime}(h_{0})h_{0}^{\prime}\right)\,,$ (21)

independently of the shape of $\tilde{f}$. For the Fubini bounce (17), it is
solved by

$\rho_{1}=\frac{1+6\xi}{3|\lambda|/\mu^{2}}\left(r\frac{\mu^{2}r^{2}-1}{(\mu^{2}r^{2}+1)^{2}}+\mu^{-1}{\rm\arctan}(\mu
r)\right)\,.$ (22)

Knowing $\rho_{1}$ is sufficient to compute the $\mathcal{O}(\kappa)$
correction $S_{1}$ to the action. We checked explicitly that the dependence on
$h_{1}$ can be eliminated by the $h_{0}$ equations of motion and partial
integration. This is a general result, however, because $h_{0}$ minimizes the
action in the absence of gravity and thus $S_{0}[h_{0}+\kappa
h_{1}]=S_{0}[h_{0}]+\mathcal{O}(\kappa^{2})$ [19]. In all, we obtain that

$\displaystyle S_{1}$
$\displaystyle=\frac{32\pi^{2}\mu^{2}}{45\lambda^{2}(\mu)}\left((1+6\xi)^{2}+6K_{1}\right)\,$
(23)
$\displaystyle=\frac{32\pi^{2}\mu^{2}}{45\lambda^{2}(\mu)}\left((1+6\xi)^{2}-36\frac{\xi^{2}+2\beta\xi\tilde{\xi}-\tilde{\xi}^{2}}{1+\beta^{2}}\right)\,,$

where $K_{1}$ encodes the modifications resulting from an independent
connection, i.e., setting $K_{1}=0$ recovers the metric case.

The gravitational correction (23) can be negative for certain values of model
parameters. If this happens, then the action cannot be minimized with respect
to $\mu$ and the adopted perturbative approach is not applicable. However, by
minimizing $S_{1}$ with respect to $\xi$, it is straightforward to show that
$S_{1}$ is always positive when

$\beta\tilde{\xi}\geq 1/12\,.$ (24)

Otherwise, the positivity of the gravitational correction $S_{1}$ can be
achieved only in certain regions of the parameter space. Two special cases
warrant being considered more closely:

  1. 1.

$\tilde{\xi}=0$: A minimally coupled Holst term $\tilde{f}(h)=\beta/\kappa$ is
probably the simplest scenario. The region allowed by the positivity of
$S_{1}$ is

$\left|\xi+\frac{1}{6}+\frac{1}{6\beta^{2}}\right|\geq\frac{\sqrt{1+\beta^{2}}}{6\beta^{2}}\,.$
(25)

For $\beta\ll 1$, this gives $\xi\leq-1/(3\beta^{2})$ or $\xi\geq-1/12$,
while, when $\beta\gg 1$, only a narrow region around the conformal coupling
$\xi=-1/6$ is forbidden, that is, $|\xi+1/6|\geq 1/(6\beta)$.

  2. 2.

$\beta=0$: In this case, the contribution from the Holst term

$S_{1}=\frac{32\pi^{2}\mu^{2}}{45\lambda^{2}(\mu)}\left(1+12\xi+36\tilde{\xi}^{2}\right)\,,$
(26)

is always positive and will thus always improve the stability of the SM vacuum
when compared to the Palatini case. This special case is depicted in the
middle panel of Fig. 1 However, as in the Palatini limit, the positivity of
$S_{1}$ implies a strict lower bound

$\xi\geq-1/12-3\tilde{\xi}^{2}\,.$ (27)

The minimal bounce action is shown in Fig. 1 with the running of $\lambda$
computed at the 3-loop level [11]. It shows that the metric limit
$\beta\to\infty$ is realized quite well already for $|\beta|=6$ when
$\tilde{\xi}\approx 0$. For $\tilde{\xi}\gg 1$, we see that the bounce action
is typically enhanced when $\beta$ and $\xi$ have the opposite signs, thus
improving the stability of the vacuum. Additionally, in comparison to the
metric case, the stability is improved when $|\xi|<|\tilde{\xi}|$. In all
depicted cases, the regions in which $S_{1}<0$ can be observed: When
$\beta=6$, this region exists only for the $\tilde{\xi}=0$ line and is
contained in a narrow range around $\xi=-37/216$. In the $\beta=-6$ case, a
parameter region with $S_{1}<0$ can be observed for every $\tilde{\xi}$. The
disallowed $\xi$ range varies with $\tilde{\xi}$. Since the theory is
independent of the sign of $\tilde{f}$, then the $\beta=6$ panel covers the
$\beta=-6$ case with $\tilde{\xi}\in[-100,0]$ and vice versa.

Finally, it is important to point out that, as in the action (9), the
contributions from mass dimension 2 terms constructed from torsion and non-
metricity that can appear in metric-affine gravity can be reduced to a non-
canonical kinetic term in a metric theory [59]. This implies that our results
can be straightforwardly extended to include corrections to vacuum stability
from such terms by computing their contribution to the small $\kappa$
expansion (19).

##  4 Conclusions

We analyzed the stability of the electroweak vacuum in metric-affine gravity,
where the Higgs boson is expected to have an additional non-minimal coupling
to the Holst invariant. This scenario can be reformulated in terms of an
equivalent metric theory with a non-canonical kinetic term, where the
gravitational corrections to the bounce action can be studied with established
perturbative methods. Our results show that the stability of the electroweak
vacuum in metric-affine gravity is improved across a wide range of model
parameters.

A non-minimally coupled Holst term provides a class of models that
continuously connects metric and Palatini gravity. We find that the limiting
case of Palatini gravity displays the mildest improvement to vacuum stability.

## Acknowledgments

We thank Tomi Koivisto for useful comments. This work was supported by the
Estonian Research Council grants SJD18 and PSG869.

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# An Embedded Auto-Calibrated Offset Current Compensation Technique for
PPG/fNIRS System

Sadan Saquib Khan, Sumit Kumar, Benish Jan, Laxmeesha Somappa, and Shahid
Malik Sadan Saquib Khan, Sumit Kumar, Benish Jan, and Shahid Malik are with
the Centre for Sensors, Instrumentation and Cyber Physical System Engineering
(SeNSE), Indian Institute of Technology Delhi (IIT Delhi).Laxmeesha Somappa is
with the Department of Electrical Engineering, Indian Institute of Technology
Bombay (IIT Bombay). Manuscript received Month Day, Year; revised Month Day,
Year.

###### Abstract

Usually, the current generated by the photodiode proportional to the
oxygenated blood in the photoplethysmography (PPG) and functional infrared
spectroscopy (fNIRS) based recording systems is small as compared to the
offset-current. The offset current is the combination of the dark current of
the photodiode, the current due to ambient light, and the current due to the
reflected light from fat and skull [1, 2]. The relatively large value of the
offset current limits the amplification of the signal current and affects the
overall performance of the PPG/fNIRS recording systems. In this paper, we
present a mixed-signal auto-calibrated offset current compensation technique
for PPG and fNIRS recording systems. The system auto-calibrates the offset
current, compensates using a dual discrete loop technique, and amplifies the
signal current. Thanks to the amplification, the system provides better
sensitivity. A prototype of the system is built and tested for PPG signal
recording. The prototype is developed for a 3.3 V single supply. The results
show that the proposed system is able to effectively compensate for the offset
current.

###### Index Terms:

Article submission, IEEE, IEEEtran, journal, LaTeX, paper, template,
typesetting.

##  I Introduction

The Optical sensors are used in many applications thanks to their unique
features such as immunity to electromagnetic interference, small size,
lightweight and high sensitivity [3, 4, 5]. They are particularly useful in
biomedical applications since they non-invasively provide information about
the biomarkers. Various biomedical diagnostic devices such as oximeters,
cerebral oximeters, and fNIRS-based brain-imaging systems utilize optical
sensors [6, 7, 8]. In addition, they are also being explored for the non-
invasive detection of breast cancer [9, 10, 11]. However, the optical sensors
suffer from the offset current. Mostly a photodiode is used to convert the
photon into the current. The offset current may be because of the ambient
light, dark current, and also because of the light reflected from the fat,
bones, and skulls [12]. Usually, the dark current is fixed and can be
compensated. However, the offset current due to ambient light and reflection
varies with time and can be significantly higher than the current due to the
actual signal [13, 14, 15, 16].

Figure 1: Overview of PPG sensor [8].

The current from the photodiode is converted into equivalent voltage using a
trans-impedance amplifier. The value of the feedback resistor is selected
appropriately to convert the current into an output voltage. A high-value of
feedback resistor is desirable to amplify the signal and keep the current
noise at a low level. However, the large value of offset current limits the
amplification. To compensate for the effect of offset current on the output
voltage of the trans-impedance amplifier, various approaches are explored. The
current source using the digital-to-analog converter (DAC) for offset current
compensation is integrated with many system-on-chip (SoC) based PPG/fNRIS
systems. The current-DAC-based SoC systems are able to compensate wide-range
offset current from 1$\mu A$ to 128 $\mu A$ with a 7-bit resolution. However,
the
HPF for offset compensation

Figure 2: Block diagram of a four-channel impedance measurement unit for
impedance sensors Figure 3: Block diagram of a four-channel impedance
measurement unit for impedance sensors

The offset current compensation techniques, based on a negative feedback loop
(analog or digital), by nullifying the DC-voltage at the output of the
amplifier, are reported in [17, 18, 19, 20]. This continuous nullification of
the DC signal affects the shape of the signal which is misinterpreted as the
measurand. The problem is especially critical for biomedical signals such as a
PPG and functional near-infrared spectroscopy (fNIRS) for brain-computer
interfacing applications. In addition, the implementation of the analog low-
pass filter in the feedback path to nullify the offset current introduces the
delay in the sensor signal[21, 22]. Since the cut-off frequency of the filter
is near dc, it introduces a delay causing a phase error in the PPF and fNIRS
signals [23, 24, 25, 26, 27]. Typically, the cutoff frequency varies from 0.1
Hz to 5 Hz for PPG and fNIRS. Due to the extremely low cutoff frequency, the
analog low pass filter may influence the frequencies present in our signal of
interest [28, 29, 30, 31].

In this paper, we present a dual-loop auto-compensation technique for wide-
range offset current compensation. The circuit utilizes voltage feedback to
compensate for the offset current smaller than 1$\mu A$. For a wide range of
currents, the circuit utilizes a digitally controlled current source [32, 33,
34, 35]. The circuit compensates the offset current.

##  II The Proposed System

The block diagram of the proposed discrete offset compensation loop-based
system is shown in Fig. 3. The system consists of a trans-impedance amplifier,
with a digital potentiometer $R_{F}$ for programmable gain. A programmable
current source $i_{DAC}$ is used to compensate for the offset current from the
photodiode. The proposed system utilizes a dual-cancellation technique to
effectively compensate the offset voltage from the output voltage $V_{out}$ of
the trans-impedance amplifier. A low-pass filter is used to filter out the DC
offset from the voltage $V_{out}$. The DC-offset voltage $V_{dc}$ is acquired
using an analog-to-digital converter. The offset-free output is further
amplified by a variable gain non-inverting amplifier $OA_{2}$. The output
voltage $V_{s}ig$ of the amplifier is proportional to the AC signal.

The frequency of the PPG and fNIRS is usually between 0.05 Hz to 5 Hz. The
selection of an appropriate cut-off frequency for the compensation of DC-
offset is important for the effective design of the overall embedded sensing
system. For instance, ideally, a cut-off frequency of less than 0.05 Hz is
preferred for DC-offset compensation. However, the compensation time will
increase accordingly. The continuous compensation technique will also affect
the shape morphology of the PPG/fNIRS signals. In this paper, the cut-off
frequency of the low-pass filter is chosen to be around 1 Hz. The proposed
system utilizes a discrete offset compensation approach by monitoring the
offset voltage. The loop will automatically compensate the offset voltage if
the DC voltage crosses a threshold voltage limit.

Figure 4: Block diagram of a four-channel impedance measurement unit for
impedance sensors

The flow of operation of the proposed system is shown in Fig. 2. The auto-
calibration phase start with switching the opening of the switch $S_{1}$ and
connecting the switch $S_{2}$ to $V_{cm}$. During this phase, both ac and dc
current through the photo-diode is amplified by the trans-impedance amplifier.
The average dc value of $V_{out}$ is extracted using the low-pass filter. The
voltage $V_{dc}$ is acquired using an analog-to-digital converter (ADC 1), as
shown in Fig. 3. The DC current is estimated from the voltage $V_{dc}$ and
resistor $R_{F}$. Next, the magnitude of the digitally controlled current
source ($i_{DAC}$) is tuned to compensate for the DC current. Once the current
value is set, the switch $S_{1}$ is closed. Consequently, the DC offset
current from the photo-diode has sunk by the $i_{DAC}$. The output voltage of
the trans-impedance amplifier is proportional to the AC signal.

The DC offset is compensated by using the digitally controlled current source.
The magnitude of the current source is controlled by using a digital
potentiometer. However, the digital potentiometer is suffered from limited
resolution and tolerance. Once the value of $i_{DAC}$ is set to compensate for
the DC offset, the gain of the trans-impedance amplifier is increased by using
digital-potentiometer $R_{F}$. However, due to the finite resolution of the
current source, the uncompensated DC current is also amplified, which results
in a DC offset at the output of the trans-impedance amplifier. To compensate
for that, in this letter, we incorporated a second offset-compensation loop by
controlling the voltage $V_{REF}$ of the trans-impedance amplifier.

Figure 5: Experimental PPG data acquired with the traditional continuous offset cancellation technique. TABLE I: Components used for Prototyping Component | Details
---|---
LED and Photo-Diode wavelengths | 940nm
Operational Amplifier | OPA333
$i_{DAC}$ | LM334
Digital-potentiometer | AD8252
Microcontroller |  | ATSAMD21G18
---
(12-bit ADC)
(10-bit DAC)
$R_{F}$ | 3.9 k$\Omega$ \- 1 M$\Omega$
$R_{L}$ | 20 k$\Omega$
$C_{L}$ | 10 $\mu$F

##  III Experimental Setup and Results

A prototype of the proposed embedded auto-calibrated system is fabricated and
tested. The components used for the prototyping are tabulated in Table I. The
low-voltage low-power operational amplifier (OPA333) is used to implement the
trans-impedance amplifier, voltage-follower, and non-inverting amplifier. A
microcontroller with integrated ADC and DAC is used. The resolution of the ADC
can be tuned from 12 to 16 bits with oversampling and decimation. A digital
potentiometer with 8-bit resolution and 1 M$\Omega$ full-scale value is used.
The amplitude of the current source can be tuned from 1 $\mu$A to 10 $m$A.

Figure 6: Experimental PPG results obtained with (a) dc offset and with (b)
Compensated dc offset.

This study’s primary goal was to look into the DC Offset in the PPG signal. In
order to get the PPG signal, infrared light is shone on the fingertip and
reflected to a photodiode that has data on the blood and oxygen flow. From
Fig. 6a, we can clearly visualize the DC offset present in the signal which
limits the gain in the succeeding stage. Fig. 6 (b) show the dc offset
compensated PPG signals by acquiring a stable DC value using the proposed
mixed-signal system. The dc offset in the PPG signal has been successfully
eliminated up to a very high order with no effect of the delay and shape
morphing of the PPG signal.

##  IV Conclusion

A mixed-signal loop based approach for acquiring high fidelity PPG/fNIRS
signals has been demonstrated. The proposed architecture overcomes the issues
associated with large delay and shape morphing of the signals with the
traditional continuous offset cancellation technique. Finally, PPG measurement
results acquired with the proposed system are shown and the offset
cancellation with high fidelity data acquisition is verified.

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