# Combined Analysis of the CP Properties of The Higgs Boson in Effective Higgs
Lagrangian

Rong<EMAIL_ADDRESS>Ying
<EMAIL_ADDRESS>Corresponding author.  School of
Science, Xi’an Jiaotong University, Xi’an, 710049, China

###### Abstract

CP violation effects of the Higgs stem from not only CP-violation interactions
but also an ambiguous defined CP state. The two CPV sources are coherently
studied based on an effective Higgs Lagrangian. The constraints from unitarity
limits for $WW$ and $ZZ$ scatterings are proposed to restrict Higgs couplings
to weak gauge bosons. Five interesting cases are classified in terms of CPV
sources to investigate the Higgs CP properties. The allowed ranges are shown
from fitting results to the signal strengths of the Higgs measured by ATLAS
and CMS.

PACS numbers: 12.60.Fr, 14.80.Bn, 11.30.Na

Key words: Higgs, effective Lagrangian, CP violation, signal strength

##  I Motivation

Although much work has been done on The Boson , found in 2012 at the LHC with
a mass of about $125$ GeV HiggsFound , there is no definite answer yet whether
it is the Standard Model (SM) Higgs boson or not. CP property and self-
interaction are two important check points. Compared with the latter, the
former is more easy to examine in present and coming experiments. Differing
from the CP-even state in SM, The Boson can be explained as an ambiguous CP
defined state, like a mixing of a CP-even state and a CP-odd state in the two
Higgs doublet model (2HDM) 2HDM . It suggests a kind of CP violation (CPV)
effect. The CP mixing angle has been constrained by the Higgs experiment
signal in much of the literature HiggsCP ; reRatio . Additionally, the CPV
effect can also arise from another source, namely, CPV interactions beyond the
SM. In Ref.HVVexp , CPV couplings in $HVV$ vertices (with $V=Z,W$) are
required to describe the general tensor structures appearing in high energy
experiments. Other CPV interactions with defined CP parity Higgs are often be
investigated in terms of high dimension operators in ChangPRD ; GavelaJHEP .
Nevertheless, two kinds of CPV effect, ambiguous CP defined state and CPV
interactions, can correct Higgs production and decay channels measured by
ATLAS and CMS, individually and coherently. The two must also be analyzed
together. Indeed, there is no reason to neglect one or other. Differing from
articles addressing a single CPV source, the purpose of this paper to present
a combined analysis of the two kinds of CPV effect in observations of The
Boson , thereby providing a better means to discriminate between the SM Higgs
and beyond.

To start a general investigation, we first review a Higgs electroweak chiral
effective theory in Sec.II. The proposal in Ref.LMWangHiggs  is a development
with a scalar Higgs from the electroweak chiral Lagrangian in
Ref.AppelquistEWCL . As electroweak symmetry can be broken according to many
models that go beyond the SM, such as the strong coupled Higgs model, where
the Lagrangian adopts a nonlinear-realized Goldstone boson to describe
electroweak symmetry breaking. The physical Higgs after electroweak symmetry
broken is only treated as a singlet scalar. We add CPV Higgs interactions to
the Lagrangian to complete it in $p^{4}$ order. The effective Higgs Lagrangian
provides a larger parameter space to investigate CP properties of The Boson .
In Sec.III, we investigate the Higgs interactions with weak gauge bosons.
These couplings provide not only CP conserved vertexes but also CPV form. On
the other hand, they are consistently associated with Higgs productions and
decay at the LHC. The relation between the $HVV$ tensor structure and low-
energy coefficients of the Lagrangian is studied. Theoretical constraints are
also proposed from unitarity limits in $WW/ZZ$ scatterings. In Sec.IV, we
classify five interesting cases according to the CPV source. Five classes are
identified and we calculate the Higgs decay width into the different channels
of each class. To avoid complexity in the calculations of Higgs production, we
choose the sum of signal strengths $\mu_{ref}=\sum_{X}\mu_{X}$ as a benchmark
to define $H\rightarrow XX$ relative ratios of branching ratios
$D_{x}\equiv\mu_{x}/\mu_{ref}$. Based on this study, we fit the parameters in
Higgs effective theory to the experimental signal strength at $\sqrt{s}=8$ TeV
and in Sec.V give the optimal parameter values and ranges for the different
cases. Finally, a short summary is given in Sec.VI.

##  II Nonlinear-realized effective Higgs Lagrangian

In the section, we review the construction of a nonlinear-realized effective
Higgs Lagrangian presented in LMWangHiggs  and generalize it to include CPV
terms.

The Goldstone bosons is expressed as unitary unimodular matrices $U$ that are
associated with the cosets of $SU(2)_{L}\otimes U(1)_{Y}/U(1)_{em}$. A
physical Higgs after EW symmetry broken is labeled by a singlet scalar $h$.
The covariant derivatives of the Goldstone fields are written

$\displaystyle
D_{\mu}U=\partial_{\mu}U+ig\frac{\tau^{a}}{2}W^{a}_{\mu}U-igU\frac{\tau_{3}}{2}B_{\mu}.$
(1)

Here, $W_{\mu}^{a}$ ($a=1,2,3$) and $B_{\mu}$ are $SU(2)_{L}$ and $U(1)_{Y}$
gauge fields, respectively. We adopt $SU(2)_{L}$ covariant building blocks
AppelquistEWCL

$\displaystyle
T_{3}=U^{\dagger}\tau_{3}U,~{}~{}~{}V_{\mu}=(D_{\mu}U)U^{\dagger},$ (2)

to express the effective Higgs Lagrangian. As $h$ is a singlet scalar, it
couples to an effective field operator $\mathcal{O}$ in any way

$\displaystyle\left(1+\sum_{i=1}^{N}c^{i}\frac{h^{i}}{\Lambda^{i}}\right)\cdot\mathcal{O}.$
(3)

The high power of the Higgs field implies that a strongly coupled case
appearing at energy scale $\Lambda$ has been recovered. The factor including
the Higgs field in Eq.3 can be integrated into a random function for $h$,
i.e., the Higgs field $h$ is counted as order $p^{0}$. The lowest order of the
chiral Lagrangian is just that of the Higgs potential

$\displaystyle\mathcal{L}_{0}$ $\displaystyle=$ $\displaystyle-V(h)$ (4)

The details of $V(h)$ depend on the nature of electroweak symmetry breaking.
In general, the function can be expressed as a power series in the Higgs field
$V(h)=\sum_{i=2}^{\infty}\lambda_{i}h^{i}$ that includes the mass term (for
$i=2$) and all possible self-interactions. The next-to-leading order is
$p^{2}$ for this Lagrangian,

$\displaystyle\mathcal{L}_{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}f^{2}\langle{{V}_{\mu}{V}^{\mu}}\rangle+\frac{1}{4}\beta
f^{2}\langle{T{V}_{\mu}}\rangle^{2}+\frac{1}{2}(\partial_{\mu}h)^{2}$ (5)

in which $f$ takes the scale of spontaneous breaking for electroweak symmetry
and $\langle~{}\rangle$ denotes trace in the weak isospin space. The first
term is the Lagrangian of the nonlinear sigma model. The second corresponds to
leading-order custodial symmetry-violating interactions. Because $\beta$
depends on the Higgs field $h$, it involves interactions with the EW gauge
bosons. When neglecting the non-standard EW rotations between $W^{3}_{\mu}$
and $B_{\mu}$, the term yields the $H^{N}ZZ$ vertex with $N$ Higgs legs. The
last term in Eq.5 is the Higgs kinetic term.

The Lagrangian at $p^{4}$ order can be divided into three contributions,

$\displaystyle\mathcal{L}_{4}=\mathcal{L}_{K}+\mathcal{L}_{B}+\mathcal{L}_{H}$
(6)

with kinetic term $\mathcal{L}_{K}$, interaction terms without derivatives of
the Higgs $\mathcal{L}_{B}$, and those with derivatives of the Higgs
$\mathcal{L}_{H}$. The kinetic term can be expressed as

$\displaystyle\mathcal{L}_{K}=-\frac{1}{2}\langle{W_{\mu\nu}W^{\mu\nu}}\rangle-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}.$
(7)

$\mathcal{L}_{B}$ has a similar form to the electroweak chiral Lagrangian in
AppelquistEWCL

$\displaystyle\mathcal{L}_{B}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\alpha^{(0)}_{1}gg^{\prime}B_{\mu\nu}\langle{TW^{\mu\nu}}\rangle+\frac{i}{2}\alpha^{(0)}_{2}g^{\prime}B_{\mu\nu}\langle{T[{V}^{\mu},{V}^{\nu}]}\rangle+i\alpha^{(0)}_{3}g\langle{W^{\mu\nu}[{V}^{\mu},{V}^{\nu}]}\rangle$
(8)
$\displaystyle+\alpha^{(0)}_{4}\langle{{V}_{\mu}{V}_{\nu}}\rangle^{2}+\alpha^{(0)}_{5}\langle{{V}_{\mu}{V}^{\mu}}\rangle^{2}+\alpha^{(0)}_{6}\langle{{V}_{\mu}{V}_{\nu}}\rangle\langle{T{V}^{\mu}}\rangle\langle{T{V}^{\nu}}\rangle$
$\displaystyle+\alpha^{(0)}_{7}\langle{{V}_{\mu}{V}^{\mu}}\rangle\langle{T{V}_{\nu}}\rangle^{2}+\frac{1}{4}\alpha^{(0)}_{8}g^{2}\langle{TW_{\mu\nu}}\rangle^{2}+\frac{i}{2}\alpha^{(0)}_{9}g\langle{TW^{\mu\nu}}\rangle\langle{T[{V}_{\mu},{V}_{\nu}]}\rangle$
$\displaystyle+\frac{1}{2}\alpha^{(0)}_{10}\langle{T{V}^{\mu}}\rangle^{2}\langle{T{V}^{\nu}}\rangle^{2}+\alpha^{(0)}_{11}g\epsilon^{\mu\nu\rho\lambda}\langle{T{V}_{\mu}}\rangle\langle{{V}_{\nu}W_{\rho\lambda}}\rangle$
$\displaystyle+\alpha^{(0)}_{12}g\langle{T{V}^{\mu}}\rangle\langle{{V}^{\nu}W_{\mu\nu}}\rangle+\alpha^{(0)}_{13}gg^{\prime}\epsilon^{\mu\nu\rho\lambda}B_{\mu\nu}\langle{TW_{\rho\lambda}}\rangle$
$\displaystyle+\alpha^{(0)}_{14}g^{2}\epsilon^{\mu\nu\rho\lambda}\langle{TW_{\mu\nu}}\rangle\langle{TW_{\rho\lambda}}\rangle$

Here, a superscript $(i)$ attached to an $\alpha$ coefficients indicates the
number of obvious partial derivatives of the Higgs field. As we have
mentioned, the coefficients depend on the Higgs field $h$, i.e.,

$\displaystyle\alpha^{(0)}_{i}(h)=\sum_{j=0}^{\infty}\alpha^{(0),j}_{i}\frac{h^{j}}{\Lambda^{j}}.$
(9)

For convenience, we omit the functional symbol and denote this as
$\alpha^{(0)}_{i}$. The contribution $\mathcal{L}_{H}$ with the derivatives of
the Higgs field can be written as

$\displaystyle\mathcal{L}_{H}$ $\displaystyle=$
$\displaystyle(\partial_{\mu}h)\Big{\\{}\alpha^{(1)}_{1}\langle{T{V}^{\mu}}\rangle\langle{{V}_{\nu}{V}^{\nu}}\rangle+\alpha^{(1)}_{2}\langle{T{V}_{\nu}}\rangle\langle{{V}^{\mu}{V}^{\nu}}\rangle+\alpha^{(1)}_{3}\langle{T{V}_{\nu}}\rangle\langle{T[{V}^{\mu},{V}^{\nu}]}\rangle$
(10)
$\displaystyle+i\alpha^{(1)}_{4}g\langle{T{V}_{\nu}}\rangle\langle{TW^{\mu\nu}}\rangle+i\alpha^{(1)}_{5}g^{\prime}\langle{T{V}_{\nu}}\rangle
B^{\mu\nu}+i\alpha^{(1)}_{6}g\langle{T{V}_{\nu}W^{\mu\nu}}\rangle+i\alpha^{(1)}_{7}g\langle{{V}_{\nu}W^{\mu\nu}}\rangle\Big{\\}}$
$\displaystyle+\epsilon^{\mu\nu\rho\sigma}(\partial_{\mu}h)\Big{\\{}\frac{\alpha^{(1)}_{8}}{2}\langle{T{V}_{\nu}}\rangle\langle{T[{V}_{\rho},{V}_{\sigma}]}\rangle+i\alpha^{(1)}_{9}g\langle{T{V}_{\nu}}\rangle\langle{TW_{\rho\sigma}}\rangle+i\alpha^{(1)}_{10}g^{\prime}\langle{T{V}_{\nu}}\rangle
B_{\rho\sigma}$
$\displaystyle+i\alpha^{(1)}_{11}g\langle{T{V}_{\nu}W_{\rho\sigma}}\rangle+i\alpha^{(1)}_{12}g\langle{{V}_{\nu}W_{\rho\sigma}}\rangle\Big{\\}}$
$\displaystyle+(\partial_{\mu}h)(\partial_{\nu}h)\Big{\\{}\alpha^{(2)}_{1}\langle{T{V}^{\mu}}\rangle\langle{T{V}^{\nu}}\rangle+\alpha^{(2)}_{2}\langle{{V}^{\mu}{V}^{\nu}}\rangle\Big{\\}}$
$\displaystyle+(\partial_{\mu}h)^{2}\Big{\\{}\alpha^{(2)}_{3}\langle{T{V}_{\nu}}\rangle^{2}+\alpha^{(2)}_{4}\langle{{V}_{\nu}{V}^{\nu}}\rangle\Big{\\}}$
$\displaystyle+\alpha^{(3)}_{1}(\partial_{\mu}h)^{2}\partial_{\nu}h\langle{T{V}^{\nu}}\rangle+\alpha^{(4)}_{1}(\partial_{\mu}h)^{4}$

Note that five new terms corresponding to $\alpha^{(1)}_{j}$ for $j=8..12$
that go beyond the effective Higgs Lagrangian in ref.LMWangHiggs  have been
added, and thus $\mathcal{L}_{H}$ breaks CP.

Aside from the above bosonic contribution, the Lagrangian involving the
fermionic interactions can be generalized as

$\displaystyle\mathcal{L}_{F}=-\sum_{f=u,d,l}Y^{f}\bar{f}_{i}(g_{E}^{f}+g_{O}^{f}\gamma_{5})fh$
(11)

with the (family universal) SM Yukawa coupling $Y^{f}$. Here, $g_{E}^{f}$
($g_{O}^{f}$) parameterizes the scalar type (pseudo-scalar type) couplings
beyond the SM. It is inspired from the ambiguity in the CP property of the
light Higgs that comes from the mixing of a CP-even state and a CP-odd state
in 2HDM. The SM with a CP-even scalar corresponds to having $g_{E}^{f}=1$ and
$g_{O}^{f}=0$.

The effective Higgs Lagrangian describes nature in which, irrespective of the
CP property, there is no new particles with masses between that of the EW
gauge bosons and of the Higgs. It is valid for The Boson found at the LHC.

##  III $HVV$ effective vertices and unitarity constraints

Starting from the effective Higgs Lagrangian, we can read off the full
interaction vertexes. In this section, we focus on the Higgs interaction with
EW gauge bosons. From the point of view of experiment, the interactions should
include all possible tensor structures which can be measured at colliders. The
vertex function $V^{\mu\nu}_{HVV}$ ($V=W,Z$) for
$HV_{\mu}(p_{1})V_{\nu}(p_{2})$ is taken

$\displaystyle
V^{\mu\nu}_{HVV}=g_{HVV}^{(SM)}\left(a_{V}g_{\mu\nu}+b_{V}\frac{p_{\mu}p_{\nu}}{m_{V}^{2}}+c_{V}\epsilon_{\mu\nu\rho\lambda}\frac{k_{\rho}p_{\lambda}}{m_{V}^{2}}\right)$
(12)

Here, $a_{V}$ rescales the SM coupling and $b_{V}$ and $c_{V}$ are the CP
conserving and CP violating couplings, respectively, that depend on momenta.
They are related to the effective Lagrangian coefficients by

$\displaystyle g_{HZZ}^{SM}(a_{Z}-1)$ $\displaystyle=$ $\displaystyle
i\left(g^{2}\alpha^{(1),0}_{4}-g^{\prime
2}\alpha^{(1),0}_{5}+\frac{g^{2}}{2}\alpha^{(1),0}_{7}\right)p^{2}$ (13)
$\displaystyle-i\frac{s_{W}^{2}}{\Lambda}\left(g^{2}\alpha_{8}^{(0),1}-2g^{\prime
2}\alpha_{1}^{(0),1}\right)q_{1}\cdot q_{2}$
$\displaystyle\frac{g_{HZZ}^{SM}}{m_{Z}^{2}}b_{Z}$ $\displaystyle=$
$\displaystyle-i(2g^{2}\alpha^{(1),0}_{4}-g^{\prime
2}\alpha^{(1),0}_{5}+g^{2}\alpha^{(1),0}_{7})$ (14)
$\displaystyle+i\frac{s_{W}^{2}}{\Lambda}\left(g^{2}\alpha_{8}^{(0),1}-2g^{\prime
2}\alpha_{1}^{(0),1}\right)$
$\displaystyle\frac{g_{HZZ}^{SM}}{m_{Z}^{2}}c_{Z}$ $\displaystyle=$
$\displaystyle-i(-2g^{2}\alpha^{(1),0}_{9}+2{g^{\prime}}^{2}\alpha^{(1),0}_{10}-g^{2}\alpha^{(1),0}_{12})$
(15)
$\displaystyle+4i\frac{s_{W}^{2}}{\Lambda}\left(g^{2}\alpha_{14}^{(0),1}-g^{\prime
2}\alpha_{13}^{(0),1}\right)$ $\displaystyle g_{HWW}^{SM}(a_{W}-1)$
$\displaystyle=$ $\displaystyle-\frac{i}{4}g^{2}\left(\alpha^{(1),0}_{6}p\cdot
k-\alpha^{(1),0}_{7}p^{2}\right)$ (16)
$\displaystyle\frac{g_{HWW}^{SM}}{m_{W}^{2}}b_{W}$ $\displaystyle=$
$\displaystyle-\frac{i}{2}g^{2}\left(\alpha^{(1),0}_{6}+\alpha^{(1),0}_{7}\right)$
(17) $\displaystyle\frac{g_{HWW}^{SM}}{m_{W}^{2}}c_{W}$ $\displaystyle=$
$\displaystyle\frac{i}{2}g^{2}\alpha^{(1),0}_{12}$ (18)

with $p=q_{1}+q_{2}$ and $k=q_{1}-q_{2}$. We find that $a_{Z}$, $b_{Z}$, and
$c_{Z}$ stem from not only $\mathcal{L}_{H}$ but also $\mathcal{L}_{B}$. The
latter is suppressed by the strong coupling scalar $\Lambda$.

In SM, $WW$ and $ZZ$ scattering amplitudes are quadratic divergence if only
the weak boson and photon exchange process are considered. The unitarity limit
is at $1.4$TeV. If our effective theory is required to be valid at a higher
energy scale $\Lambda$ than $1.4$TeV, The Boson field must be responsible for
unitarity recovery. At present, the fact that no other new particle except
125GeV Higgs has been found just corresponds to the case. In general,
unitarity constraint refers to set unitarity limit to a variational $\Lambda$.
Couplings $a_{V},b_{V},c_{V}$ depend on a chosen $\Lambda$. It is complex
situation. Now, one consider a simply case: a strict unitarity constraint that
improve unitarity limit to arbitrary energy. Hence, the $HVV$ interaction
should be constrained by unitarity limits in the $WW$/$ZZ$ scatterings
amplitudes.

The Higgs exchange $WW$ scattering amplitude under effective vertex
$V^{\mu\nu}_{HWW}$ in Eq. (12) is

$\displaystyle i\mathcal{M}^{WW}_{H}/(g_{HWW}^{(SM)})^{2}$ $\displaystyle=$
$\displaystyle\left[a_{W}(\epsilon_{a}\cdot\epsilon_{b})+b_{W}\frac{(\epsilon_{a}\cdot
p_{s})(\epsilon_{b}\cdot
p_{s})}{m_{W}^{2}}\right]\frac{i}{p_{s}^{2}-m_{H}^{2}}$ (19)
$\displaystyle\times\left[a_{W}(\epsilon_{c}\cdot\epsilon_{d})+b_{W}\frac{(\epsilon_{c}\cdot
p_{s})(\epsilon_{d}\cdot p_{s})}{m_{W}^{2}}\right]$
$\displaystyle+\Big{[}a_{W}(\epsilon_{a}\cdot\epsilon_{c})+b_{W}\frac{(\epsilon_{a}\cdot
p_{t})(\epsilon_{c}\cdot
p_{t})}{m_{W}^{2}}\Big{]}\frac{i}{p_{t}^{2}-m_{H}^{2}}$
$\displaystyle\times\Big{[}a_{W}(\epsilon_{b}\cdot\epsilon_{d})+b_{W}\frac{(\epsilon_{b}\cdot
p_{t})(\epsilon_{d}\cdot p_{t})}{m_{W}^{2}}\Big{]}$
$\displaystyle+\Big{[}a_{W}(\epsilon_{a}\cdot\epsilon_{d})+b_{W}\frac{(\epsilon_{a}\cdot
p_{u})(\epsilon_{d}\cdot
p_{u})}{m_{W}^{2}}\Big{]}\frac{i}{p_{u}^{2}-m_{H}^{2}}$
$\displaystyle\times\Big{[}a_{W}(\epsilon_{b}\cdot\epsilon_{c})+b_{W}\frac{(\epsilon_{b}\cdot
p_{u})(\epsilon_{c}\cdot p_{u})}{m_{W}^{2}}\Big{]}$

Here, $p_{s}=p_{a}+p_{b}=p_{c}+p_{d}$, $p_{t}=p_{a}-k_{c}=k_{d}-p_{b}$, and
$p_{u}=p_{a}-k_{d}=k_{c}-p_{b}$. Notice that the CPV term corresponding to
$c_{W}$ in the vertex (12) does not contribute to the amplitude because of the
fourth order totally anti-symmetric tensor $\epsilon^{\mu\nu\rho\sigma}$.
Expanding $i\mathcal{M}^{WW}_{H}$ in terms of the invariant momentum $s$ ($s$
is equal to $p_{s}^{2}$ for s-channel, $p_{t}^{2}$ for t-channel and
$p_{u}^{2}$ for u-channel), we find the leading-order term is of order
$s^{3}$,

$\displaystyle{i\mathcal{M}^{WW}_{H}=\frac{ib_{W}^{2}s^{3}}{128m_{W}^{8}}(g_{HWW}^{(SM)})^{2}(3+\cos^{2}\theta)^{2}+\mathcal{O}(s^{2}).}$
(20)

That is, the $p_{\mu}p_{\nu}$-type coupling in Eq. (12) involves a worse case
than one in the standard EW theory that yields only $s^{2}$-order divergence
when removing the SM Higgs exchange contribution. To avoid the catastrophe, we
must limit the vanishing of $b_{W}$. The left divergence at order $s^{2}$ is
similar to the SM case. We must set $a_{W}=1$ to cancel the $s^{2}$-order
divergence to recover unitarity of the $WW$ scattering amplitude.

Similarly, we find the same results in $ZZ$ scattering:

  * •

$c_{Z}$ gives no contribution to the $ZZ$ scattering amplitude;

  * •

$b_{Z}$ is restricted to zero in order to cancel the $s^{3}$-order divergence;

  * •

$a_{Z}$ must take unity to cancel the remaining divergence in the $ZZ$
scattering amplitude without Higgs exchange.

##  IV CPV effects and LHC phenomenology

From Sec.II, recall the two kinds of CPV effects: one from CPV interactions
and the other from the ambiguity in the CP property of the Higgs field. The
first does not depend on a CP-even or CP-odd Higgs, but the second does. The
main CPV interactions that are related to LHC phenomenology are $HZZ$ and
$HWW$ measured in $H\rightarrow WW^{*}$ and $H\rightarrow ZZ^{*}$ decays,
respectively. They are controlled by $c_{Z}$ and $c_{W}$ in Eq.12. The
parameters related to the Higgs CP property are fermion couplings: $g_{E}^{f}$
and $g_{O}^{f}$ for $f=u,d,l$. A popular idea is that The Boson is regarded as
a mixed state of the light CP-even state $H$ and CP-odd state $A$ in the 2HDM:

$\displaystyle h=\cos\theta H+\sin\theta A$ (21)

with mixing angle $\theta$ CPmixingHiggs . The scalar couplings then take the
form $g_{E}^{f}=\cos\theta$, but the pseudo-scalar couplings still need to be
determined by the models. More generally, the fermion coupling of CP-even $H$
is not required, similar to those in SM, which relaxes the limit for
$g_{E}^{f}$. Thus, there are eight parameters describing CPV effects:
$c_{Z},~{}c_{W}$, $g_{E}^{f}$, and $g_{O}^{f}$ ($f=u,d,l$).

To investigate CPV effects in detail, we classify the five classes in terms of
two CPV sources: ambiguous CP state and CPV couplings. The former can induce
CPV interactions, and the later do not vanish even if $h$ reduces to CP-even
eigenstate. Five classes are listed as

  * •

case 0: SM Higgs with $c_{Z}=c_{W}=0$, $g_{E}^{f}=1$ and $g_{O}^{f}=0$—there
is no free parameter and no CPV effect;

  * •

case 1: pure CP-even Higgs with $g_{O}^{f}=0,g^{f}_{E}=1$ and $c_{Z},c_{W}$
free—this is an extended SM Higgs with anomalous $HVV$ coupling. CPV effects
only come from CP-even Higgs interactions;

  * •

case 2: 2HDM Higgs with $c_{Z}=c_{W}=0$, $g_{E}^{f}=\cos\theta$ and
$g_{O}^{f}$ free—the dependent parameters are the CP mixing angle $\theta$ and
three pseudo-scalar couplings $g^{f}_{O}$. The CPV effects complete stem from
CP-mixing state. CPV in the case can be transferred to all Higgs interactions,
which controlled by CP-mixing angle;

  * •

case 3: multi-CPV Higgs only with $g_{E}^{f}=\cos\theta$ and other parameters
free—this is a simple combination between ambiguous CP defined state and CPV
couplings to EW bosons. CPV interactions are not vanishing even if CP mixing
angle tends to zero;

  * •

case 4: the most general Higgs with all parameters free—this is an extension
of case 3. Ambiguous CP defination is not required to come from CP eigenstate
mixing.

These cases completely describe all possible CP properties of The Boson .

Now, let us consider the Higgs decay widths at the LHC. ATLAS and CMS have
measured signal strengths at $ZZ^{*},WW^{*},\gamma\gamma,\tau\tau,b\bar{b}$
decay channels. They are sensitive to different parameters.

Under the narrow width approximation, we get the width of
$\Gamma_{H\rightarrow VV^{(*)}\rightarrow 4l}$(V=Z or W)

$\displaystyle\Gamma_{H\rightarrow VV^{*}\rightarrow f_{1}f_{2}f_{3}f_{4}}$
$\displaystyle=\frac{1}{\pi^{2}}\int
dm_{1}^{2}dm_{2}^{2}\frac{m_{V}\Gamma_{V\rightarrow
f_{1}f_{2}}m_{V^{*}}\Gamma_{V^{(*)}\rightarrow f_{3}f_{4}}\Gamma_{H\rightarrow
VV^{*}}}{[(m_{1}^{2}-m_{V}^{2})^{2}+m_{V}^{2}\Gamma_{V}^{2}][(m_{2}^{2}-m_{V^{*}}^{2})^{2}+m_{V^{*}}^{2}\Gamma_{V^{*}}^{2}]}.$
(22)

Here $m_{1}$ and $m_{2}$ are the virtualities for the intermediate vector
bosons. $\Gamma_{V}$($\Gamma_{V^{*}}$) and $m_{V}$($m_{V^{*}}$) are the total
width and mass of V(V∗), respectively. The Higgs decay width to $ZZ^{*}$
($WW^{*}$) with $HZZ$ ($HWW$) couplings given by ref.HVVexp  is written

$\displaystyle\Gamma_{H\rightarrow VV^{*}}$ $\displaystyle=$
$\displaystyle\frac{G_{F}\beta}{D\sqrt{2}m_{H}^{3}m_{V}^{4}\pi}(4a_{V}^{2}(\beta^{2}+12m_{1}^{2}m_{2}^{2})m_{V}^{4}+b_{V}^{2}\beta^{4}$
(23)
$\displaystyle+32c_{V}^{2}m_{1}^{2}m_{2}^{2}\beta^{2}+4a_{V}b_{V}m_{V}^{2}\beta^{2}\sqrt{\beta^{2}+4m_{1}^{2}m_{2}^{2}}),$

where D is 64(32) for V=Z(W) and the $\beta$ is defined as

$\displaystyle\beta=\sqrt{(m_{H}^{2}-(m_{1}-m_{2})^{2})(m_{H}^{2}-(m_{1}+m_{2})^{2})}.$
(24)

The sequential decay widths of Z or W into fermions are

$\displaystyle\Gamma_{Z\rightarrow f\bar{f}}$ $\displaystyle=$
$\displaystyle\frac{G_{F}m_{Z}^{3}}{6\sqrt{2}\pi}(v_{f}^{2}+a_{f}^{2}),$ (25)
$\displaystyle\Gamma_{W\rightarrow f\bar{\nu_{f}}}$ $\displaystyle=$
$\displaystyle\frac{G_{F}m_{W}^{3}}{6\sqrt{2}\pi}.$ (26)

Higgs di-photon decay is an important channel appearing at one-loop level. The
main contributions come from the $W$ loop and the top loop in the SM. Because
there are more tensor structures beyond the SM, the effective $HWW$ coupling
may correct the $W$ loop contribution. Its result however shows that the CP
violating term with $\epsilon^{\mu\nu\rho\sigma}$ makes no contribution
because the Higgs di-photon decay conserves CP. Thus, the
$H\rightarrow\gamma\gamma$ amplitude is a sum of SM-like terms with
$g_{E}^{f}$ rescaling and fermionic pseudo-scalar terms with $g_{O}^{f}$,

$\displaystyle\Gamma_{\gamma\gamma}$ $\displaystyle=$
$\displaystyle\frac{\alpha^{2}g^{2}}{4^{5}\pi^{3}}\Bigg{(}\left|\frac{4}{3}g^{u}_{E}F_{1/2}(\tau_{t})+\frac{1}{3}g^{d}_{E}F_{1/2}(\tau_{b})+F_{1}(\tau_{W})\right|^{2}$
(28)
$\displaystyle+\left|\frac{4}{3}g^{u}_{O}A_{1/2}(\tau_{t})+\frac{1}{3}g^{d}_{O}A_{1/2}(\tau_{b})\right|^{2}\Bigg{)}$

where

$\displaystyle F_{1}$ $\displaystyle=$ $\displaystyle
2+3\tau+3\tau(2-\tau)f(\tau)$ (29) $\displaystyle F_{1/2}$ $\displaystyle=$
$\displaystyle-2\tau[1+(1-\tau)f(\tau)]$ (30) $\displaystyle A_{1/2}$
$\displaystyle=$ $\displaystyle-2\tau f(\tau)$ (31)

and $\tau_{i}=4m_{i}^{2}/m_{H}^{2}$ HiggsHunter .

The Higgs decay to fermion pairs can be expressed as

$\displaystyle\Gamma_{f\bar{f}}=\frac{N_{C}G_{F}}{4\sqrt{2}\pi}m_{f}^{2}m_{H}(1-\frac{4m_{f}^{2}}{m_{H}^{2}})^{\frac{3}{2}}\left((g_{E}^{f})^{2}+R^{f}(g_{O}^{f})^{2}\right)$
(32)

with QCD correction factor $R^{f}$ from the scalar and pseudo-scalar decays
decayQCDcorr .

##  V Fitted results

The signal strength of $pp\rightarrow H\rightarrow XX$ at the LHC has been
measured by ATLAS and CMS. In theory, the signal strength can be calculated
using the narrow-width approximation

$\displaystyle\mu_{X}^{th}=\frac{\sigma^{th}(pp\rightarrow
H)}{\sigma^{SM}(pp\rightarrow H)}\frac{\Gamma^{th}(H\rightarrow
XX)}{\Gamma^{SM}(H\rightarrow XX)}.$ (33)

However, to calculate Higgs production channels is very complicated. To avoid
the problem, we adopt relative ratios of the branching ratios reRatio

$\displaystyle D_{X}\equiv\frac{\mu_{X}}{\mu_{ref}}.$ (34)

with reference $\mu_{ref}$. Thus it only depends on relative decay widths

$\displaystyle D^{th}_{X}={\frac{\Gamma^{th}(H\rightarrow
XX)}{\Gamma^{SM}(H\rightarrow
XX)}}\Bigg{/}{\frac{\Gamma^{th}_{ref}}{\Gamma^{SM}_{ref}}}.$ (35)

However, the signal strength of $b\bar{b}$ channel, $\mu_{b}$, is measured
from the associated production of the higgs boson and the electroweak gauge
boson(W or Z)(VH), which differs from $\mu_{Z}$ and $\mu_{W}$ that are
dominated by the gluon fusion channel (ggF). The production channel will not
drop out in the retio of $\mu_{b}/\mu_{ref}$ if the production modes are
different between them. An alternative scheme conveniently adopted by
experimental collaborations is to group five production modes, gluon
fusion(ggF), vector boson fusion(VBF), associated production with vector
bosons(VH), and associated production with top pair(ttH), into two effective
modes, ggF$+$ttH and VBF$+$VH. These combined signal strengths are calculated
in paper combinedsingalstrength  and listed in Tab. 1.

Table 1: Combined signal strength: $\mu^{ggF+ttH}$ and $\mu^{VBF+VH}$ for different decay modes. $VV$ takes $WW,ZZ$. Channel | $\mu^{ggF+ttH}$ | $\mu^{VBF+VH}$
---|---|---
$VV$ | $0.91\pm 0.16$ | $1.01\pm 0.49$
$\gamma\gamma$ | $0.98\pm 0.28$ | $1.72\pm 0.59$
$b\bar{b}$ | $-0.23\pm 2.86$ | $0.97\pm 0.38$
$\tau\tau$ | $1.07\pm 0.71$ | $0.94\pm 0.65$

Thus, two choices of reference signal strengths can be made independently. One
is the sum of all channels from $\mu^{ggF+VF}$, i.e.
$\mu_{ref1}=\sum_{X=Z,W,\gamma,b,\tau}\mu^{ggF+VF}_{X}$; the other is
$\mu_{ref2}=\sum_{X=Z,W,\gamma,b,\tau}\mu^{VBF+VH}_{X}$.

With the above formula, we can constrain the parameters by minimizing the
$\chi^{2}$ function

$\displaystyle\chi^{2}=\sum_{X=Z,W,\gamma,b,\tau}\left(\frac{D^{th}_{X}-D^{exp}_{X}}{\delta
D^{exp}_{X}}\right)^{2}.$ (36)

Figures 1–4 show the allowed area for cases 1–4, respectively.

Figure 1: Fitted result in $c_{Z}-c_{W}$ plane for case 1 at $1\sigma$ (green)
and $2\sigma$ (yellow). Black dot represent the best fitted point.

Figure 2: Fitted results for case 2 at $1\sigma$ (green) and $2\sigma$
(yellow) in the planes of (a) $\alpha-g^{u}_{O}$, (b) $\alpha-g^{d}_{O}$ and
(c) $\alpha-g^{l}_{O}$. Other parameters are taken from the best fit values.
Black dots represent the best fitted points.

In Fig. 1, $c_{Z}$ is left more space than $c_{W}$. The reason is that apart
from the $H\rightarrow VV$ decay, $c_{W}$ also takes part in di-photon decay
to constraint it more, but $c_{Z}$ does not. The same case appears in Fig.
3(a). Case 2 in Fig. 2 shows that the pseudo-scalar couplings $g^{d}_{O}$ and
$g^{l}_{O}$ have more deviation from SM that $g^{u}_{O}$. However, $g^{u}_{O}$
has more error. More details will follow using updated LHC data.

Figure 3: Fitted results for case 3 at $1\sigma$ (green) and $2\sigma$
(yellow) in the planes of (a) $c_{Z}-c_{W}$, (b) $\alpha-g^{u}_{O}$, (c)
$\alpha-g^{d}_{O}$ and (d) $\alpha-g^{l}_{O}$. Other parameters are taken from
best-fit values. Black dots represent the best fitted points.

In case 4, Fig. 4(a) shows that $c_{Z}$ and $c_{W}$ almost have the same
allowed areas as that in case 3. It is not enlarged despite more free
parameters. The scalar coupling of the up quark, $g^{u}_{E}$, is not sensitive
to the minimum of chi-squared function, and is not plotted. The pseudo-scalar
couplings of the d-quark and leptons, $g^{d}_{O}$ and $g^{l}_{O}$, exhibit an
observable deviation to SM in Fig. 4(b,c).

Figure 4: Fitted results for case 4 at $1\sigma$ (green) and $2\sigma$
(yellow) in the planes of (a) $c_{Z}-c_{W}$, (b) $g^{d}_{E}-g^{d}_{O}$ and (c)
$g^{l}_{E}-g^{l}_{O}$. Other parameters are taken from best-fit values. Black
dots present the best fitted points.

##  VI Summary

An Higgs effective Lagrangian is completed, which provides a complete CPV
sources and a general parameter space to describe CP properties of The Boson
and helps to improve the theoretical calculation of the Higgs signal strength.
Using the effective Higgs Lagrangian, we investigated two possible sources of
CPV, corresponding to ambiguous CP defined state and CPV interactions. The
unitarity limit of $WW$ and $ZZ$ scatterings only requires $b_{V}$ to vanish
and keeps $c_{V}$ independent. By fitting signal strengths of the Higgs, the
pseudo-scalar coupling $g^{f}_{O}$ and anomalous coupling $c_{Z}$ obviously
exhibit deviations to the SM results. The allowed area of $c_{Z}$ is not
noticeably larger when more free parameters are included from cases 1 through
4. All cases have shown that there are a abundant parameter space, even if
unitarity limit is severely set to arbitrary large energy. So, the two kinds
of CPV from ambiguous CP definition and anomalous couplings can coexist. Even
under These results come from an imprecise signal strength, which will be
improved at the updated LHC in the near future.

## acknowledgement

We thank Prof. Qing Wang for helpful discussions. This work is supported by
the National Natural Science Foundation of China (No. 11105152).

## References

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# Approximate Wireless Communication for Federated Learning

Xiang Ma1, Haijian Sun2, Rose Qingyang Hu1, and Yi Qian3
1Department of Electrical and Computer Engineering, Utah State University,
Logan, UT
2School of Electrical and Computer Engineering, University of Georgia, Athens,
GA
3Department of Electrical and Computer Engineering, University of Nebraska-
Lincoln, Lincoln, NE
Emails<EMAIL_ADDRESS><EMAIL_ADDRESS>2hun@uga,edu, 3<EMAIL_ADDRESS>
###### Abstract

This paper presents an approximate wireless communication scheme for federated
learning (FL) model aggregation in the uplink transmission. We consider a
realistic channel that reveals bit errors during FL model exchange in wireless
networks. Our study demonstrates that random bit errors during model
transmission can significantly affect FL performance. To overcome this
challenge, we propose an approximate communication scheme based on the
mathematical and statistical proof that machine learning (ML) model gradients
are bounded under certain constraints. This bound enables us to introduce a
novel encoding scheme for float-to-binary representation of gradient values
and their QAM constellation mapping. Besides, since FL gradients are error-
resilient, the proposed scheme simply delivers gradients with errors when the
channel quality is satisfactory, eliminating extensive error-correcting codes
and/or retransmission. The direct benefits include less overhead and lower
latency. The proposed scheme is well-suited for resource-constrained devices
in wireless networks. Through simulations, we show that the proposed scheme is
effective in reducing the impact of bit errors on FL performance and saves at
least half the time than transmission with error correction and retransmission
to achieve the same learning performance. In addition, we investigated the
effectiveness of bit protection mechanisms in high-order modulation when gray
coding is employed and found that this approach considerably enhances learning
performance.

###### Index Terms:

approximate communication, federated learning, wireless networks, bit error
rate, modulation

##  I Introduction

Federated learning (FL) [1] enables local devices to perform machine learning
(ML) tasks while still benefiting from the learning generalization ability
provided by model parameter sharing. It does not require sharing locally
collected data among devices and the server. Instead, only model parameters
are shared, thereby effectively protecting data privacy. The FL system is
composed of a central parameter server (PS) and a large number of smart local
clients (LCs). LCs gather data from onboard sensors and execute a predefined
ML task based on the global model broadcast from the PS. After computation,
each LC sends local models to the PS for aggregation, and then the updated
global model is redistributed back to LCs. This process is repeated until the
global model converges.

For edge devices such as UAVs serving as LCs, wireless networks are usually
employed to connect them to the PS. However, the nature of wireless channels
often results in erroneous information transmission. To address this issue,
modern wireless communications utilize forward error correction (FEC) methods,
such as convolutional code and low-density parity check code (LDPC), to detect
and correct received bit errors. The basic principle of FEC is to encode the
message with redundant information in the form of an error correction code
(ECC). The receiver can correct the error bits without knowing the actual bits
sent by the transmitter. Packet retransmission can be employed when the number
of errors exceeds the correction capability of ECC. Although FEC and packet
retransmission are powerful, they increase computation and communication
overhead, leading to extra power consumption and transmission delays during FL
model aggregation. In [2], the authors focused on transmission bit errors in
FL but only in a packet erasure channel.

Stochastic gradient descent (SGD) is a widely used optimization method in
distributed ML. In FL, each client performs SGD on ML tasks, then a single-
step gradient is calculated and sent to the central PS in every communication
round. This method, called FedSGD [3], serves as a baseline algorithm for FL.
However, for large-scale distributed ML models with millions of parameters,
transmitting gradients can cause high delay. Advanced transmission schemes
such as non-orthogonal multiple access (NOMA) [4] are good options, but they
need to equip with complex decoding methods. Gradient compression is a
promising approach to addressing this challenge, where extensive research has
shown the effectiveness of gradient sparsification and quantization with
little performance loss. For instance, 1-bit SGD was applied in [5] to reduce
gradient transmission size, and in [6], it was shown that 99% of gradients
could be dropped. Therefore, we are motivated to apply _approximate wireless
communication_ to transmit those gradients, i.e., allowing lossy transmission
(with errors) in exchange for low latency, low overhead, and less FEC
computation in this paper. We would like to emphasize that gradient
compression is different from and runs parallel to our proposed approximate
wireless communication method, even though both can result in information
errors. The tolerance of gradient quantization errors is based on the
assumption that the gradient magnitude is small enough. Empirical studies in
[7, 8, 9] have shown that the gradients are close to Gaussian distribution,
and most gradient values fall in the range of $(-1,1)$ or even $(-0.01,0.01)$.
Approximate wireless communication for media data transmission has been
proposed in [10, 11], and similar ideas can be applied to FL gradient
transmission.

In this study, we present a theoretical analysis of bounded ML gradients in
commonly used ML settings. Specifically, we prove that gradients in fully
connected neural network models and convolutional neural network models are
bounded under commonly used conditions. Based on this analysis, we set a limit
on the erroneous gradient, together with the approximate transmission in
practical wireless networks. Simulation results demonstrate that our proposed
method is effective in reducing the impact of bit errors on FL performance,
and saves half the time than transmission with Error Correction and
ReTransmission (ECRT). The rest of this paper is organized as follows. Section
2 introduces the FL model in wireless networks, Section 3 presents the
theoretical analysis of bounded gradients, Section 4 describes the proposed
method, Section 5 presents simulation results, and Section 6 concludes the
paper.

##  II System Model

The FL model is considered as follows. The FL system consists of $M$ LCs,
which are connected to the PS through wireless channels. The overall data
amount $D$ is distributed among $M$ devices, with each device $m$ containing
$D_{m}$ data.

###  II-A FL System Model

FL is an iterative ML algorithm that performs local computation and global
aggregation in each round. Local computation is followed by the LC model
uploading, and after global aggregation, the global model is downloaded to
each LC. This process is repeated until the model converges. The objective
function of FL can be defined as:

$\min_{\bm{w}\in R^{d}}f(\bm{w})\quad\text{where}\quad
f(\bm{w})\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{1}{|D|}\sum_{i=1}^{|D|}f_{i}(\bm{w}),$
(1)

where $|D|$ is the size of dataset $D$.
$f_{i}(\bm{w})=C(\bm{x_{i}},\bm{y_{i}};\bm{w})$ is the cost or loss function
used to measure the inference error between the data sample
$(\bm{x_{i}},\bm{y_{i}})$ and the inference made by model parameters $\bm{w}$.
For classification problems in ML, the cross-entropy function is commonly used
as the loss function $C$, particularly in neural network models. In multiclass
classification, the label $\bm{y_{i}}$ is typically one-hot encoded to ensure
each label carries equal weight.

As the data is distributed among $M$ LCs and not on the same device, the
objective function (1) needs to be rewritten as follows:

$f(\bm{w})=\sum_{m=1}^{M}\frac{|D_{m}|}{|D|}F_{m}(\bm{w}),$ (2)

where $F_{m}(\bm{w})=\frac{1}{|D_{m}|}\sum_{i\in D_{m}}f_{i}(\bm{w})$. By
distributing the data and computation across multiple devices, a conventional
centralized ML problem can be transformed into a distributed FL problem.

Since the cost function for neural networks is typically non-convex, it is
challenging to solve directly and find the global minimum. Therefore, the
gradient descent method is an iterative optimization algorithm commonly used
in ML to find a local minimum point. Stochastic gradient descent (SGD) is a
variant of the gradient descent method that can be helpful in escaping local
minimums by selecting data samples randomly. As a result, gradients play a
central role in the learning process. The gradient is defined as:

$g=\nabla_{\bm{w}}C(\bm{x_{i}},\bm{y_{i}};\bm{w}).$ (3)

The local gradient at each LC in each round can be written as

$g_{t}^{m}=\nabla F_{m}(w_{t}).$ (4)

And the global gradient after aggregation is

$g_{t}=\sum_{m=1}^{M}\frac{|D_{m}|}{|D|}g_{t}^{m}.$ (5)

The PS stores the model weights from the last round $w_{t}$, and then updates
the global model as follows

$w_{t+1}=w_{t}-\eta g_{t}.$ (6)

Here, $\eta$ is the learning rate, which typically falls within the range of
$(0,1)$.

###  II-B Wireless Channel Model

Federated Learning is an upper-layer algorithm that does not have knowledge of
the lower-layer gradient transmission details. Typically, transmission takes
place over wireless channels when LCs are smart sensors or UAVs. For the
uplink channel from LCs to PS, we consider the fading channel, which can lead
to random bit errors. For the downlink channel, we assume the PS can deliver
global gradients to LCs with negligible errors, this can be justified by
higher PS transmit power (hence higher SNR) [2].

In the uplink, a time division scheme can be used where each user is assigned
to a specific time slot while sharing the same channel. The received signal at
the PS can be expressed as follows

$r_{t}^{m}=\sqrt{p_{t}^{m}(d^{m})^{-\alpha}}h_{t}^{m}g_{t}^{m}+n_{t}^{m},$ (7)

where $r_{t}^{m}$ represents the signal received at the PS from client $m$.
The transmission power is denoted as $p_{t}^{m}$, and the small scale fading
is denoted as $h_{t}^{m}$, which is assumed to be complex normal Gaussian
distributed, i.e., $h_{t}^{m}\sim\mathcal{CN}(0,1)$. The distance between the
PS and client $m$ is represented by $d^{m}$, and the path-loss exponent is
denoted as $\alpha$. The additive noise is given as
$n^{t}\sim\mathcal{CN}(0,\sigma^{2})$. PS has the knowledge of the channel
gain, i.e., $c_{t}^{m}=\sqrt{p_{t}^{m}(d^{m})^{-\alpha}}h_{t}^{m}$, and only
the noise serves as an error source.

The entire transmission process can be described as follows. First, the
gradients are converted from decimal format to binary format. The bits are
then mapped to symbols using a QAM modulation scheme. The symbols are then
transmitted through the wireless fading channel. At the receiver end, the
signal is decoded with maximum likelihood estimation and then demodulated to
the closest point in the constellation,

$\hat{g}_{t}^{m}=\arg_{\bar{g}_{t}^{m}\in\mathcal{G}}\min||r_{t}^{m}-\sqrt{p_{t}^{m}(d^{m})^{-\alpha}}h_{t}^{m}\bar{g}_{t}^{m}||^{2},$
(8)

where $\mathcal{G}$ is the symbol points set of the constellation diagram.

##  III Bounded Gradients under Constraints - A Sketch of Proof

###  III-A Gradient Backpropagation

In machine learning, particularly in deep neural networks, backpropagation is
widely used for calculating gradients in each layer. In a fully connected
neural network, the feed-forward equation at each neuron can be expressed as

$\displaystyle z_{j}^{l}=b_{j}^{l}+\sum_{k}w_{jk}^{l}a_{k}^{l-1},$ (9)
$\displaystyle a_{j}^{l}=\sigma(z_{j}^{l}).$

Here, $b$ is the bias, $w$ is the weights, $z$ is the intermediate output, and
$a$ is the final output after the activation function $\sigma(\cdot)$. $l$
represents $l$-th layer and $j,k$ are indices. The corresponding four
fundamental equations in back-propagation for a fully connected network is

$\displaystyle\delta_{j}^{L}=\frac{\partial C}{\partial
z_{j}^{L}}=\frac{\partial C}{\partial a_{j}^{L}}\frac{\partial
a_{j}^{L}}{\partial z_{j}^{L}}=\frac{\partial C}{\partial
a_{j}^{L}}\sigma^{\prime}(z_{j}^{L}),$ (10a)
$\displaystyle\delta_{j}^{l}=\frac{\partial C}{\partial
z_{j}^{l}}=\sum_{k}\frac{\partial C}{\partial z_{k}^{l+1}}\frac{\partial
z_{k}^{l+1}}{\partial a_{j}^{l}}\frac{\partial a_{j}^{l}}{\partial
z_{j}^{l}}=\sum_{k}\delta_{k}^{l+1}w_{kj}^{l+1}\sigma^{\prime}(z_{j}^{l}),$
(10b) $\displaystyle\frac{\partial C}{\partial b_{j}^{l}}=\frac{\partial
C}{\partial z_{j}^{l}}\frac{\partial z_{j}^{l}}{\partial
b_{j}^{l}}=\delta_{j}^{l},$ (10c) $\displaystyle\frac{\partial C}{\partial
w_{jk}^{l}}=\frac{\partial C}{\partial z_{j}^{l}}\frac{\partial
z_{j}^{l}}{\partial w_{jk}^{l}}=\delta_{j}^{l}a_{k}^{l-1}.$ (10d)

Here, $L$ is the final layer index in the neural network, and $\delta_{j}^{l}$
is the defined “error” in $l$-th layer at node $j$.

To ensure that the gradient $\nabla C=\frac{\partial C}{\partial w_{jk}^{l}}$
is bounded, it is necessary to limit $\delta_{j}^{l}$ and $a_{k}^{l-1}$ based
on equation (10d). These two terms are discussed separately. $a_{k}^{l-1}$ is
the activation function output of neuron $k$ at the $(l-1)$-th layer. It
depends on the activation function being used. For example, the Sigmoid
function ensures that $a_{k}^{l-1}$ is in the range $(0,1)$ regardless of the
input $z_{k}^{l-1}$, while the ReLU activation function requires the input to
be bounded. Further discussion and mathematical expressions on activation
functions can be found in [12].

The calculation of $\delta_{j}^{l}$ is described in equation (10b), which
involves a summation of products of next layer errors $\delta_{k}^{l+1}$,
weights from node $j$ in the $l$-th layer to next layer $w_{kj}^{l+1}$, and
the derivative of activation function $\sigma^{\prime}(z_{j}^{l})$. There are
three terms in this equation, and the summation requires the number of neurons
in each layer to be finite. The derivative of the activation function
$\sigma^{\prime}(z_{j}^{l})$ also depends on the activation function used,
with the derivative being in the range $(0,0.25)$ for the Sigmoid function and
$\\{0,1\\}$ for ReLU. The weight $w_{kj}^{l+1}$ depends on model
initialization, learning rate $\eta$, and last round gradient based on
equation (6). Weight initialization methods typically generate random weight
values in the range $(-1,1)$ or even smaller, and there are newer
initialization methods such as [13] and [14]. Without loss of generality, we
assume that the weight value $w_{kj}^{l+1}$ is bounded. The error
$\delta_{k}^{l+1}$ can be written in the same way as in equation (10b) with
elements in the $(l+2)$-th layer, and this process continues all the way back
to the final layer. In classification problems, the softmax function is
commonly used as the activation function in the final layer to normalize the
output class probabilities. When the cross-entropy loss function is used, it
can be combined with the softmax function. For cross-entropy loss function,

$C=-\sum_{i}y_{i}log(p_{i}),$ (11)

where $y_{i}$ is the input truth label, $p_{i}$ is the softmax probability for
the $i$-th class

$p_{i}=\sigma(z_{i})=\frac{e^{z_{i}}}{\sum_{k}e^{z_{k}}},$ (12)

and the derivative is

$\frac{\partial p_{i}}{\partial z_{j}}=\left\\{\begin{aligned}
&p_{i}(1-p_{j}),&&\text{if}\ i=j;\\\ &-p_{j}\cdot p_{i},&&\text{if}\ i\neq
j.\end{aligned}\right.$ (13)

Equation (10a) can be written as

$\displaystyle\delta_{j}^{L}=\frac{\partial C}{\partial
p_{i}^{L}}\frac{\partial p_{i}^{L}}{\partial z_{j}^{L}}$
$\displaystyle=-\sum_{i}y_{i}\frac{\partial log(p_{i})}{\partial
p_{i}}\frac{\partial p_{i}}{\partial z_{j}},$ (14)
$\displaystyle=-\sum_{i}y_{i}\frac{1}{p_{i}}\frac{\partial p_{i}}{\partial
z_{j}},$ $\displaystyle=-y_{j}(1-p_{j})-\sum_{i\neq
j}y_{i}\frac{1}{p_{i}}(-p_{j}\cdot p_{i}),$
$\displaystyle=p_{j}\cdot\sum_{i}y_{i}-y_{j}.$

Since $y$ is a one-hot encoded label vector, so $\sum_{i}y_{i}=1$, that is

$\delta_{j}^{L}=p_{j}-y_{j}.$ (15)

As $p_{j}$ takes values between 0 and 1 and $y_{j}$ is either 0 or 1,
$\delta_{j}^{L}$ lies in the interval $(-1,1)$.

To summarize, in a fully connected neural network with cross-entropy as the
cost function and softmax function as the activation function in the final
layer, the final layer error $\delta_{j}^{L}$ is in the range $(-1,1)$. In
addition, if the weights are assumed in the range $(-1,1)$ and Sigmoid
functions are used as activation functions in other layers, the gradient
$\frac{\partial C}{\partial w_{jk}^{l}}$ is bounded by the sum of the number
of neurons after $l$-th layer, denoted as $B^{l}$.

###  III-B Gradient in Convolutional Neural Network

Modern image recognition tasks often use convolutional neural networks (CNNs)
as an advanced technique. CNNs are a special variant of feedforward networks
that consist of three types of layers: convolutional layers, pooling layers,
and fully connected layers. The feedforward process of a CNN can be written
as:

$\displaystyle
z_{j,k}^{1}=b_{j,k}^{1}+\sum_{p}\sum_{q}w_{p,q}^{1}x_{j+p,k+q}^{0},$ (16a)
$\displaystyle a_{j,k}^{1}=\sigma(z_{j,k}^{1}),$ (16b) $\displaystyle
a_{j,k}^{2}=\max(a_{2j,2k}^{1},a_{2j+1,2k}^{1},a_{2j,2k+1}^{1},a_{2j+1,2k+1}^{1}),$
(16c) $\displaystyle z_{i}^{3}=b_{i}^{3}+\sum_{j,k}w_{i;j,k}^{3}a_{j,k}^{2},$
(16d) $\displaystyle a_{i}^{3}=\sigma(z_{i}^{3}).$ (16e)

For the sake of simplicity, we assume that this CNN network comprises only
three layers. Equation (16a) and (16b) represent the convolutional layer,
equation (16c) represents the max pooling layer with a $2\times 2$ kernel, and
equations (16d) and (16e) represent the fully connected layer. Here, $p$ and
$q$ denote the indices of convolutional kernels.

Now the backpropagation for the CNN network becomes

$\displaystyle\delta_{i}^{3}$ $\displaystyle=\frac{\partial C}{\partial
z_{i}^{3}}=\frac{\partial C}{\partial a_{i}^{3}}\frac{\partial
a_{i}^{3}}{\partial z_{i}^{3}}=\frac{\partial C}{\partial
a_{i}^{3}}\sigma^{\prime}(z_{i}^{3}),$ (17a) $\displaystyle\delta_{j,k}^{1}$
$\displaystyle=\frac{\partial C}{\partial z_{j,k}^{1}}=\sum_{i}\frac{\partial
C}{\partial z_{i}^{3}}\frac{\partial z_{i}^{3}}{\partial
a_{s,t}^{2}}\frac{\partial a_{s,t}^{2}}{\partial z_{j,k}^{1}},$ (17b)
$\displaystyle=\sum_{i}\delta_{i}^{3}w_{i;s,t}^{3}\frac{\partial
a_{s,t}^{2}}{\partial a_{j,k}^{1}}\frac{\partial a_{j,k}^{1}}{\partial
z_{j,k}^{1}},$
$\displaystyle=\sum_{i}\delta_{i}^{3}w_{i;s,t}^{3}\frac{\partial
a_{s,t}^{2}}{\partial a_{j,k}^{1}}\sigma^{\prime}(z_{j,k}^{1}),$
$\displaystyle=\begin{cases}\sum_{i}\delta_{i}^{3}w_{i;s,t}^{3}\sigma^{\prime}(z_{j,k}^{1}),&\text{if
case 1;}\\\ 0,&\text{otherwise;}\end{cases}$ $\displaystyle\frac{\partial
C}{\partial w_{i;j,k}^{3}}$ $\displaystyle=\frac{\partial C}{\partial
z_{i}^{3}}\frac{\partial z_{i}^{3}}{\partial
w_{i;j,k}^{3}}=\delta_{i}^{3}a_{j,k}^{2},$ (17c) $\displaystyle\frac{\partial
C}{\partial w_{p,q}^{1}}$ $\displaystyle=\frac{\partial C}{\partial
z_{j,k}^{1}}\frac{\partial z_{j,k}^{1}}{\partial
w_{p,q}^{1}}=\delta_{j,k}^{1}x_{j+p,\ k+q}^{0}.$ (17d)

Here, case 1 is $a_{j,k}^{2}=\text{max}(a_{2s,2t}^{1},\ a_{2s+1,2t}^{1},\
a_{2s,2t+1}^{1},\ a_{2s+1,2t+1}^{1})$ in equation (17b), and
$s=\frac{j}{2},n=\frac{k}{2}$.

Similarly, when the cross entropy serves as the loss and the activation
function in the last layer is the softmax function, $\delta_{i}^{3}$ lies in
the range $(-1,1)$. In order to bound the gradient $\frac{\partial C}{\partial
w_{i;j,k}^{3}}$ in the fully connected layer, it is necessary to ensure that
$a_{j,k}^{2}$ or
$\max(a_{2j,2k}^{1},a_{2j+1,2k}^{1},a_{2j,2k+1}^{1},a_{2j+1,2k+1}^{1})$ is
also bounded. If the Sigmoid function is used as the activation function in
the first layer, then $a_{j,k}^{2}$ is bounded within the range of $(0,1)$,
which results in $\frac{\partial C}{\partial w_{i;j,k}^{3}}$ being bounded
within the range of $(-1,1)$. The gradient $\frac{\partial C}{\partial
w_{p,q}^{1}}$ in equation (17d) can be bounded by considering the two terms
involved. Firstly, $x_{j+p,k+q}^{0}$ is the input and is bounded. Secondly,
$\delta_{j,k}^{1}$ can be expressed as either 0 or
$\sum_{i}\delta_{i}^{3}w_{i;s,t}^{3}\sigma^{\prime}(z_{j,k}^{1})$. We know
that $\delta_{i}^{3}$ is in the range $(-1,1)$ and
$\sigma^{\prime}(z_{j,k}^{1})$ is in the range $(0,0.25)$ if the activation
function is Sigmoid. For $w_{i;s,t}^{3}$, its value depends on the model
initialization, learning rate $\eta$, and the last round gradient based on
equation (6), as we discussed above. If we assume $w_{i;s,t}^{3}$ is bounded
in $(-1,1)$, then $\frac{\partial C}{\partial w_{p,q}^{1}}$ can be bounded by
the number of neurons in the last layer, also denoted as $B^{l}$.

##  IV Proposed Method

In section 3, we have presented mathematical proofs that under certain
conditions, the gradients are bounded by $B^{l}$. Empirically, it has been
shown in [7, 8, 9] that the gradients are not only bounded but also bounded
within the range of $(-1,1)$ or even a smaller range. This allows us to have
an expected value (statistically) for the received gradient at the PS.
Correspondingly, we first design a QAM encoding scheme for the bounded
gradients,

###  IV-A QAM Encoding

In ML, gradients are commonly expressed using 32-bit floating-point numbers.
These numbers follow the format defined by the IEEE-754 standard, which
assigns the first bit to the sign and the next 8 bits to the exponent, leaving
the final 23 bits for the fraction. Bits in different locations have varying
importance. The sign bit controls the sign of the gradient value, while the
exponent part defines the integer and decimal values. The fraction part only
controls the decimal value, and thus, the exponent bits are more important
than the fraction bits. Furthermore, the bits located on the left side of the
exponent are more important than those on the right.

During transmission, each bit is susceptible to noise, which can cause
corruption. To avoid block corruption, we employ interleaving at the
transmitter and de-interleaving at the receiver, reducing the likelihood of
multiple error bits taking place together. In the bit representation, when the
second bit in the 32-bit representation, i.e., the first bit in the exponent
part, is 1 and all other 31 bits are 0s, the decimal value is 2. Conversely,
when the second bit in the 32-bit representation is 0, and all other 31 bits
are 1s, the magnitude is less than 2. When assuming a magnitude threshold of 1
for the gradient value, the first bit in the exponent part is always 0. This
motivated us, on the receiver side, regardless of the value decoded in the
second-bit location of the gradient, it will be set to 0, as shown in Figure
1.

Figure 1: Received Gradient Bit Representation

Moreover, we have also observed that different modulation schemes have varying
effects on bits located at different positions [11]. This is important not
only in media message transmission but also in ML model parameter
transmission. In wireless transmission, the transmission system is not aware
of the relative importance of data bits and treats all the bits equally. When
using QPSK as the modulation scheme, each symbol consists of 2 bits, with the
bit combinations being ${00,01,11,10}$. The error probability for the first
and second bits in QPSK is the same. In contrast, 16-QAM has 4 bits per
symbol, and the constellation map with gray code is shown below.

Figure 2: 16-QAM with Gray Coding Constellation Map

The bits with underlines in Figure 2 correspond to the first bit in each
symbol, which are the most significant bits (MSB), while the fourth or last
bits are the least significant bits (LSB). When the transmission probability
for each symbol is the same, the error probability for the MSB is higher than
for the LSB. For example, if the symbol $s_{0}$ is decoded with an error, it
is most likely to be decoded as $s_{1}$, $s_{4}$, or $s_{5}$. The MSB bit
remains the same, while the LSB changes twice. This is summarized in Table I.
The symbols in the other quadrants are symmetric to the first quadrant, so the
results are identical. High-order modulation schemes with gray coding provide
built-in protection for the MSB bits of gradient values in bit representation.

TABLE I: 16-QAM MSB/LSB Error Count Symbol |  Potential Error Symbol |  MSB Error Count |  LSB Error Count
---|---|---|---
$s_{0}$ |  $s_{1},s_{4},s_{5}$ |  0 |  2
$s_{1}$ |  $s_{0},s_{2},s_{4},s_{5},s_{6}$ |  2 |  3
$s_{4}$ |  $s_{0},s_{1},s_{5},s_{8},s_{9}$ |  0 |  2
$s_{5}$ |  $s_{0},s_{1},s_{2},s_{4},s_{6},s_{8},s_{9},s_{10}$ |  3 |  3

###  IV-B Approximate Wireless Transmission

To further improve gradient exchange efficiency over wireless networks, we
propose an approximate wireless transmission scheme. Essentially, since the
gradient is resilient to errors, as witnessed in the existing gradient
compression methods, the delivered messages (gradient) do not have to be
accurate. Hence we eliminate FEC and re-transmission when channel SNR is
satisfactory. While the exact SNR value is to be determined, our empirical
results have shown that at around 10-20 dB, the BER is acceptable for FL.
Notice that our approach is different from user datagram protocol (UDP), where
retransmission is not required either. The difference is that UDP works at a
higher level, and the CRC is used only to check the UDP payload. When the
error happens at the physical or MAC layer, retransmission is still issued.
Our approach eliminates both FEC and re-transmission at lower layers,
including physical and MAC layers. The benefit is three-fold: 1) it reduces
the communication overhead, so more data bits can be transmitted; 2) it
reduces the computation overhead for FEC, and this is very appealing for edge
devices; 3) it improves latency performance since no re-transmission is
required.

##  V Simulation Results

In this section, we first present the parameter settings for the simulation.
We then provide the empirical probability of bit error versus signal-to-noise
ratio (SNR) under the wireless channel mentioned earlier. Among the modulation
schemes tested, QPSK achieves a better bit error rate (BER) than 16-QAM and
256-QAM at the same SNR level. Next, we compare the FL performance under three
scenarios: ECRT transmission with error correction and retransmission, naive
erroneous transmission, and erroneous transmission with our proposed scheme.
The naive error transmission is the transmission in wireless networks with
errors without extra operation. Compared to naive erroneous transmission, our
proposed scheme achieves a high testing accuracy. Furthermore, compared to
ECRT transmission, our proposed scheme for erroneous transmission saves much
more time. Finally, we discuss different modulation schemes with gray coding
to show the built-in bit protection for MSB in the bit representation.

We consider a typical FL setting in our simulation, where $M=100$ LCs are
connected to the PS, and all LCs participate in the learning process in each
communication round. The LCs perform image classification tasks using the
MNIST dataset, which consists of handwritten digits 0-9. The training set
contains $60,000$ images, and the test set contains $10,000$ images, with each
digit having approximately $6,000$ images in the training set and $1,000$
images in the test set. To simulate a realistic scenario where data is
collected from the environment, we distribute the data in a non-iid way, with
each LC having $2$ digits and each digit having around $300$ images for
training. We use a convolutional neural network (CNN) as the ML model, with 2
convolutional layers, each having a kernel size of 5, 2 max-pooling layers
with size 2, and 2 fully connected layers. ReLU is used as the activation
function in all layers except the last one, which uses the log softmax
function. The learning rate is set to $\eta=0.01$.

We set the path loss exponent for the wireless channel as $\alpha=3$, and
consider a distance of 10m between the PS and LCs. The transmission power at
the LCs is normalized to 1. We use QPSK as the modulation scheme, and the
receiver SNR is set at $\gamma=10$ dB unless otherwise specified.

Under the specific fading channel, QPSK achieves a lower BER compared to
16-QAM and 256-QAM at the same SNR level. For QPSK, at SNR=10 dB, the BER is
approximately $4\times 10^{-2}$ while the BER is $5\times 10^{-3}$ when SNR is
20 dB.

In the ECRT scheme with error correction and retransmission, all the bits are
received correctly by the PS, which incurs a cost for forward error correction
(FEC) and possible retransmission when the error exceeds the FEC capability.
In contrast, the naive error transmission scheme involves transmitting bits
with errors without prior knowledge of the gradients, where the test accuracy
remains flat at around 10%, similar to random guessing as shown in Figure 3.
This occurs because the model cannot learn anything due to transmission
errors. Our proposed method, however, takes into account prior knowledge of
the gradient values, which are expected to be in the range of (-1, 1). This
makes the proposed scheme achieves much better results than naive error
transmission.

Figure 3: Test Accuracy v.s. Communication Time

To quantify the transmission time saved by our proposed method compared to
ECRT transmission, we employ a practical IEEE 802.11 protocol with LDPC error
correction coding. LDPC is a promising ECC that can approach the Shannon
limit. For different coding rates, there exists a trade-off between error
correction capability and transmission overhead. Lower coding rate results in
high transmission overhead but comes with high error correction capability.
Here, we use a coding rate of $1/2$ to enhance error correction. According to
[15], the minimum Hamming distance is 15 for a code rate of $1/2$ when the
code length is 648, and we search using the parity check matrix. This results
in an error correction capability of 7 bits. In Figure 3, the transmission
with LDPC coding with retransmission takes $2\times$ time than the proposed
scheme to achieve 80% accuracy at SNR=20 dB while it takes more than $3\times$
for SNR=10 dB for the LDPC coding with retransmission scheme to achieve that
performance.

(a)

(b)

Figure 4: Test Accuracy with the Same SNR/BER

To demonstrate the effectiveness of built-in MSB bit protection of high-order
modulation with gray coding, we begin by presenting the test accuracy of
different modulations at the same SNR in Figure 4(4(a)). At an SNR of 10 dB,
the BER for QPSK, 16-QAM, and 256-QAM is roughly $4\times 10^{-2}$, $10^{-1}$,
and $3\times 10^{-1}$, respectively. Because QPSK results in fewer errors, its
learning performance is better than in 16-QAM and 256-QAM.

In Figure 4(4(b)), we present a scenario where the BER is made the same for
different modulations. To accomplish this, we increase the SNR for 16-QAM to
16 dB and the SNR for 256-QAM to 26 dB. Consequently, the BER for all three
modulation schemes is $4\times 10^{-2}$. In this scenario, 256-QAM achieves
significantly better learning performance than QPSK, with smaller transmission
errors in 256-QAM than QPSK.

##  VI Conclusions

In this paper, we proposed a federated learning parameter transmission scheme
in wireless networks. Unlike existing transmission methods that rely on
forward error correction and retransmission, we proposed gradient transmission
with errors based on prior knowledge of gradient values. The gradient value is
mathematically proven to be within a small range under certain constraints, so
the received gradient value is expected to be within that range. This approach
achieves learning performance with errors much better than naive error
transmission and saves at least half time to achieve the same learning
performance as ECRT transmission. Additionally, we explored high-order
modulation and demonstrated improved learning results. In the future, our plan
is to quantify the impact of communication errors on FL performance and
examine how the number of clients influences it.

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