# High-Speed (7,2) Compressor Using A Fast Carry-Generation Logic based on
Sorting Network

Wenbo Guo  School of Integrated Circuits
Tsinghua University
Beijing, China
<EMAIL_ADDRESS>
###### Abstract

Fast binary compressors are the main components of many basic digital
calculation units. In this paper, a high-speed (7,2) compressor with a fast
carry-generation logic is proposed. The carry-generation logic is based on the
sorting network, and it can generate a carry bit within 2 logical stages other
than 3 stages as in previous school book full adders. Collaborating with the
adjusted full adder logic, the proposed (7,2) compressor achieves using only
11 basic logical stages. Testing this new design in a binary arry with 7 rows
and 8 columns, and the result shows that this design have higher proformance
than previous designs. This method is suitable for high proformance cases in
multiplication design or other cryptography hardware blocks.

###### Index Terms:

(7,2) compressor, multiplier, full adder, sorting network

##  I Introduction

Multiplication is a very common operation in digital devices. And the
performance of multiplication is the bottleneck of DSP. Fast multiplier
consists of 3 parts: partial product generation, partial product reduction,
and vector merge addition. Wallace Tree [1] [2] is proposed to parallelly
compress the partial product with full adders, and the full adders are now
known as the (3,2) compressors. Thereafter, various methods are proposed to
construct a more efficient compressor to further speed-up the reduction of
partical products. Larger compressors have been widely used, such as (4,2),
(5,2), (7,2) [3] [4] [5] compressors, but this reduction step still spend the
most time in a multiplication operation. Besides, in many cryptography
hardware blocks, such as modular multiplication, also require high-speed
compressors to speed them up.

(7,2) compressor is proved to be a hign efficient method [6], and many papers
have discussed on this. Booth [6] and [7] are using special mechods to reduce
the number of basic logical stages. The method in [7] reduced the logical
layer to 12, but still using too many XOR gate which is slow and hard to
optimize in logical level. The method in [6] also implemente a 12-logical-
layer design with the help of (7,3) counter in [8]. But this design did not
optimize for it.

Further reduce the basic logical layers of (7,2) compressors is difficult.
However, it is still possible by using a special carry-generation logical. The
contributions of this paper are listed below:

  * 1.

We proposed a carry-generation logic that generates a carry bit within 2 basic
logical stages while traditional full adders require 3 basic logical stages.

  * 2.

The adjusted full adder is introduced. It is designed to collaborate with the
carry-generation unit.

  * 3.

We propose the new (7,2) compressor that only consumes 11 basic ligical
stages. According to the synthesis result, which is synthsised by Synopsys DC,
our design has less time delay.

This paper will present our design firstly, and then present the comparison
between our design and [6] or [7].

##  II Method and implementation

###  II-A Sorting Network of 1-bit Numbers

For two 1-bit numbers, sorting is a simple operation. The circuit in Fig. 1
can easily sort the two input 1-bit numbers. $Out_{1}$ is always the larger
one and $Out_{2}$ is always the minor one. This circuit only consumes an AND
gate and an OR gate.

Figure 1: Sorter of two 1-bit numbers.

Fig. 2 shows a 4-input sorting network [9]. Each of the vertical lines
represents a sorter in Fig. 1. After three stages of sorting, the inputs are
sorted. We know that the delay of a sorter is one stage of basic logical gate,
so the sorting network consumes three stages of basic logical gates.

Figure 2: 4-bit sorting network

###  II-B Fast Carry-Generation

In a basic full adder, suppose that the input bits are A, B, and C, the carry
bit is generated by equation (1). This will consume three stages of basic
logic. However, carry generation and propagation are the bottleneck of the
performance in a multiplier. $AB$ represents the AND of $A$ and $B$. $A+B$
represents the OR of them.

$Carry=AB+AC+BC$ (1)

As we can see, the last stage in Fig. 2 only sort the second and the third
bits. So it is clear that the first bit is the largest one and the fourth bit
is the smallest one after two stages of sorting. Then if we choose one number
randomly from the second and the third bits, this one is no lager than the
first bit and no less than the fourth bit. That means the first bit, the bit
randomly selected and the fourth bit have been in order, and we named them as
X, Y and Z. The first two stages in Fig. 1 is represented as Half Sorter as
shown in Fig. 2. Add X, Y and Z up as binary numbers to generate carry and
sum, the result is shown in Table I. Note that X, Y, Z are in order.

TABLE I: Truth table of X, Y, Z. X | Y | Z | Carry | Sum
---|---|---|---|---
0 | 0 | 0 | 0 | 0
1 | 0 | 0 | 0 | 1
1 | 1 | 0 | 1 | 0
1 | 1 | 1 | 1 | 1

Because they are ordered, so there are only four possible combinations. Based
on the truth table in Table I, the bool expression is simplified as equation
(2) and (3). With this special full adder structure (SFA), carry bit can be
generated with two basic logic stages, faster than the traditional full adder
logic which will comsume 3 logic stages.

$Carry=Y$ (2)

$Sum=X(\overline{Y}+Z)$ (3)

###  II-C Adjusted Full Adder

Then let us talk about full adders. Usually, full adder is implemented by (4)
and (5). The symbol $\oplus$ means XOR. A, B and C are inputs, Carry and Sum
are outputs. Sum is on the critical path. Formula (5) can be changed to:
$Sum=C\oplus(A\overline{A}+A\overline{B}+\overline{A}B+B\overline{B})$. Note
that
$A\overline{A}+A\overline{B}+\overline{A}B+B\overline{B}=(A+B)(\overline{A}+\overline{B})=(A+B)\overline{(AB)}$.
Suppose that $h_{1}=A+B$ and $h_{2}=AB$, then the formula (5) can be rewritten
to formula (6). Rewrite the formula (5) to formula (6) does not reduce the
logical stages, but formula (6) is more convenient with the subsequent
analysis.

$Carry=C(A+B)+AB$ (4)

$Sum=A\oplus B\oplus C$ (5)

$Sum=C\overline{(h_{1}\overline{h_{2}})}+\overline{C}(h_{1}\overline{h_{2}})$
(6)

Fig. 3 is the logical implementation for formula (4) and (6). As can be seen
in Fig 3, if the MUX requires 2 logical stages, the sum comsumes 4 logical
stages. The input signal C is only used by the MUX, and there are two logical
stages from C to Sum, two logical stages from C to Carry. That means C is used
after A and B, and it does not matter that if C is late to 2 logical stages of
delay. The rest of this chapter will discuss how to use this feature to
optimize latency.

Figure 3: Adjusted full adder lgic

###  II-D Implementation of (7,2) Compressor

Fig. 4 shows the overall design of (7,2) compressor. Numbers in parentheses
means how many logical stages are used from input. Because of the special
logic of full adder discusseed above which can generate a Carry bit with only
2 logical stages, the input signal $C_{i2}$ can be input to the full adder.
This helps the whole design reduced a logical stage. Each full adder in Fig. 4
have a circular marker which means the ”$C$” input in Fig. 3. This input bit
can be later than $A$ and $B$ for 2 logical stages.

Figure 4: Overall design of (7,2) compressor

By this structure, the latency of a (7,2) compressor is reduced to 11 logical
stages’s delay.

TABLE II: Performance comparison Method | TSMC 90nm | TSMC 65nm | TSMC 28nm
---|---|---|---
| Delay($ps$) | Area($um^{2}$) | Delay($ps$) | Area($um^{2}$) | Delay($ps$) | Area($um^{2}$)
[1] | 970 | 1975.0 | 550 | 1157.8 | 154 | 583.5
[2] | 973 | 1822.6 | 562 | 956.2 | 155 | 559.6
proposed | 937 | 2348.9 | 544 | 1055.5 | 150 | 748.6

##  III Performance Comparison

In this chapter, we will make a comparison with previous methods proposed in
[6] and [7]. To make a clear comparison, all these designs will be utilized in
a binary array of 7 rows and 8 columns. They are used to compress the binary
array into 2 rows, just like the second step of a multiplier. Then a vector
merge adder, which is implemented with Kogge-Stone algorithm, will plus them
to one row. The only defference among them is the structure of (7,2)
compressor. All the Verilog HDL codes are synthesised with Synopsys Design
Compiler, with TSMC 90nm, 65nm and 28nm process, to find the minimum delay of
each design. All the results are shown in Table II. Methods in [6] and [7]
consume 12 logical stages, so their delays are close. The proposed method in
this paper comsumes 11 logical stages, so it has lower delay comprared to [6]
or [7].

As shown in Table II, the delay of this module reduced when the process nodes
are reduced. But the implementations with method in [6] and [7] have almost
the same delay whichever the process is used. The implementation with the
proposed method have less delay than them, the saving delay is approximately
one logical stage delay. Througn the data in table 2, we can sum up that with
the reduction of process nodes, the influence of logic optimization decreases
gradually. And because of the special full adder logic, which will comsume
more ligical gates compared with tranditional full adder logic, is used, the
area is larger than the methods in [6] and [7]. That means under this extreme
high performance design condition, a small delay reduce requires a significant
amount of area to be consumed as a cost.

##  IV Conclusions

In this paper, we proposed a special full adder logic. With its help, we
proposed a new (7,2) compressor structure. This method reduces the logical
stages of a (7,2) compressor to 11, less than the tranditional design methods.
The special full adder logic runs with the consider of 4 bits not 3 bits, but
get a quick carry out bit. Also therefore, consider 4 bits at the same time
consumes more areas. So this method is suitable for high performance
conditions.

## References

  * [1] C. S. Wallace, “A Suggestion for a Fast Multiplier,” in IEEE Transactions on Electronic Computers, vol. EC-13, no. 1, pp. 14-17, Feb. 1964, doi:10.1109/PGEC.1964.263830.
  * [2] R. S. Waters and E. E. Swartzlander, “A Reduced Complexity Wallace Multiplier Reduction,” in IEEE Transactions on Computers, vol. 59, no.8, pp. 1134-1137, Aug. 2010, doi: 10.1109/TC.2010.103.
  * [3] A. Najaﬁ, B. Mazloom-nezhad and A. Najaﬁ, “Low-power and high-speed 4-2 compressor,” 2013 36th International Convention on Information and Communication Technology, Electronics and Microelectronics (MIPRO), Opatija, 2013, pp. 66-69.
  * [4] A. Fathi, B. Mashouﬁ and S. Azizian, “Very Fast, High-Performance 5-2 and 7-2 Compressors in CMOS Process for Rapid Parallel Accumulations,” in IEEE Transactions on Very Large Scale Integration (VLSI) Systems, vol. 28, no. 6, pp. 1403-1412, June 2020, doi:10.1109/TVLSI.2020.2983458.
  * [5] W. Guo and S. Li, ”Fast Binary Counters and Compressors Generated by Sorting Network,” in IEEE Transactions on Very Large Scale Integration (VLSI) Systems, vol. 29, no. 6, pp. 1220-1230, June 2021, doi: 10.1109/TVLSI.2021.3067010.
  * [6] Weinan Ma and Shuguo Li, ”A new high compression compressor for large multiplier,” 2008 9th International Conference on Solid-State and Integrated-Circuit Technology, Beijing, 2008, pp. 1877-1880, doi: 10.1109/ICSICT.2008.4734925.
  * [7] M. Rouholamini, O. Kavehie, A. -P. Mirbaha, S. J. Jasbi and K. Navi, ”A New Design for 7:2 Compressors,” 2007 IEEE/ACS International Conference on Computer Systems and Applications, Amman, Jordan, 2007, pp. 474-478, doi: 10.1109/AICCSA.2007.370924.
  * [8] M. Mehta, V. Parmar and E. Swartzlander, ”High-speed multiplier design using multi-input counter and compressor circuits,” [1991] Proceedings 10th IEEE Symposium on Computer Arithmetic, Grenoble, France, 1991, pp. 43-50, doi: 10.1109/ARITH.1991.145532.
  * [9] M. H. Najaﬁ, D. J. Lilja, M. D. Riedel and K. Bazargan, “Low-Cost Sorting Network Circuits Using Unary Processing,” in IEEE Transactions on Very Large Scale Integration (VLSI) Systems, vol. 26, no. 8, pp. 1471- 1480, Aug. 2018, doi: 10.1109/TVLSI.2018.2822300.


# Scotoseesaw model implied by dark right-handed neutrinos

Phung Van Dong<EMAIL_ADDRESS>(corresponding author) Duong
Van Loi<EMAIL_ADDRESS>Phenikaa Institute for Advanced
Study and Faculty of Basic Science, Phenikaa University, Yen Nghia, Ha Dong,
Hanoi 100000, Vietnam

###### Abstract

We find a dark gauge symmetry $U(1)_{D}$ that transforms nontrivially only for
three right-handed neutrinos $\nu_{1,2,3R}$. The anomaly cancellation demands
that they have a dark charge $D=0,-1,+1$ assigned to $\nu_{1,2,3R}$,
respectively. The dark charge is broken by two units down to a dark parity,
i.e. $U(1)_{D}\to P_{D}=(-1)^{D}$, which stabilizes a dark matter candidate.
Interestingly, the model manifestly supplies neutrino masses via joint seesaw
and scotogenic mechanisms, called scotoseesaw.

##  I Motivation

The standard model arranges left-handed fermions in doublets,
$l_{aL}\equiv(\nu_{aL},e_{aL})$ and $q_{aL}\equiv(u_{aL},d_{aL})$, while it
puts right-handed fermion partners in singlets, $e_{aR}$, $u_{aR}$, and
$d_{aR}$, where $a=1,2,3$ is a family index. Since the standard model contains
neither sterile right-handed neutrino singlets nor lepton number violating
interactions, it implies massless neutrinos in contrast with experiment [1,
2].

If right-handed neutrinos $\nu_{aR}$ are included according to respective
families, possessing large Majorana masses $M$’s that couple $\nu_{R}$’s by
themselves, it leads to canonical seesaw mechanism producing appropriate
neutrino masses, suppressed by $m_{\nu}\simeq-m^{2}/M$ where Dirac neutrino
masses $m$’s linking $\nu_{L}$ with $\nu_{R}$ are proportional to the weak
scale, obeying $m\ll M$ [3, 4, 5, 6, 7]. However, the canonical seesaw
mechanism in itself does not manifestly explain the existence of dark matter,
similarly to the standard model [8, 9].

If right-handed neutrinos $\nu_{R}$’s are odd under an exact symmetry $Z_{2}$,
the Dirac masses $m$’s are suppressed, while the Majorana masses $M$’s are
allowed. Additionally, the left-handed neutrinos $\nu_{L}$’s only couple to
$\nu_{R}$’s via an inert Higgs doublet $\eta$ that is odd under $Z_{2}$ too.
In this way, neutrino masses are radiatively induced by dark fields
$\nu_{R}$’s and $\eta^{0}$’s that run in the loop, called scotogenic mechanism
[10]. This approach is well motivated since the lightest of dark fields
$\nu_{R}$’s and $\eta^{0}$’s is stabilized by $Z_{2}$ responsible for dark
matter. However, the existence of the $Z_{2}$ symmetry—which is ad hoc
input—is a mystery.

The $Z_{2}$ symmetry arises as a residual matter parity from gauged $B-L$
(see, e.g., [11, 12]); alternatively, it results from breaking of a dark gauge
charge (see, e.g., [13, 14]). To our best knowledge, most gauge extensions of
the scotogenic setup that properly stabilizes a dark matter candidate require
extra degrees of fermions of freedom beyond the conventional three right-
handed neutrinos—the counterparts of usual neutrinos.

This work looks for a novel dark (gauge) symmetry that governs just three
right-handed neutrinos as in the minimal scotogenic setup, by contrast.
Interestingly, dark symmetry breaking implies neutrino mass generation and
dark matter stability as interplay between canonical seesaw and scotogenic
scheme, so-called scotoseesaw.

Let us remind the reader that the simplest scotoseesaw set by two right-handed
neutrinos was first studied in [15] and late in [16] by ad hoc imposing an
anomalous $U(1)$ or a $Z_{2}$ symmetry, by contrast. And, this scheme has been
extensively studied in [17, 18, 19, 20]. Hence, our proposal indicates that a
gauge completion of scotoseesaw requires at least three right-handed neutrinos
in order for it properly working.

##  II Proposal

This work for the first time assumes that the three right-handed neutrinos
$\nu_{aR}(a=1,2,3)$ by themselves transform nontrivially under a dark gauge
symmetry $U(1)_{D}$, i.e. $\nu_{aR}$ each is assigned to a dark charge $D_{a}$
according to $a=1,2,3$, whereas all the standard model fields possess $D=0$.
The cancellation of $[\mathrm{Gravity}]^{2}U(1)_{D}$ and $[U(1)_{D}]^{3}$
anomalies constrains $D_{1}+D_{2}+D_{3}=0$ and
$D^{3}_{1}+D^{3}_{2}+D^{3}_{3}=0$, respectively. It leads to $D_{1}=0$,
$D_{2}=-1$, and $D_{3}=+1$ to be the unique solution, if indistinguishing
$\nu_{1,2,3R}$ and normalizing the nonzero charge to unity, without loss of
generality. Indeed, since $U(1)_{D}$ theory is invariant under a scale
symmetry $D\to cD$ for charge and $g_{D}\to g_{D}/c$ for coupling, one can
choose $c$ so that the charge normalization takes place.

Field | $SU(3)_{C}$ | $SU(2)_{L}$ | $U(1)_{Y}$ | $U(1)_{D}$ | $P_{D}$
---|---|---|---|---|---
$l_{aL}=\begin{pmatrix}\nu_{aL}\\\ e_{aL}\end{pmatrix}$ | 1 | 2 | $-1/2$ | $0$ | $+$
$\nu_{1R}$ | 1 | 1 | 0 | $0$ | $+$
$\nu_{2R}$ | 1 | 1 | 0 | $-1$ | $-$
$\nu_{3R}$ | 1 | 1 | 0 | $+1$ | $-$
$e_{aR}$ | 1 | 1 | $-1$ | $0$ | $+$
$q_{aL}=\begin{pmatrix}u_{aL}\\\ d_{aL}\end{pmatrix}$ | 3 | 2 | $1/6$ | $0$ | $+$
$u_{aR}$ | 3 | 1 | $2/3$ | $0$ | $+$
$d_{aR}$ | 3 | 1 | $-1/3$ | $0$ | $+$
$H=\begin{pmatrix}H^{+}\\\ H^{0}\end{pmatrix}$ | 1 | 2 | $1/2$ | 0 | $+$
$\eta=\begin{pmatrix}\eta^{0}\\\ \eta^{-}\end{pmatrix}$ | 1 | 2 | $-1/2$ | $+1$ | $-$
$\chi=\begin{pmatrix}\chi^{0}\\\ \chi^{-}\end{pmatrix}$ | 1 | 2 | $-1/2$ | $-1$ | $-$
$\phi$ | 1 | 1 | 0 | 2 | $+$
Table 1:  Field presentation content of the model.

Further, $\nu_{1R}$ couples to leptons via usual Higgs doublet $H$, while
$\nu_{2,3R}$ couple to leptons via new Higgs doublets $\eta,\chi$ with charge
$D=+1,-1$, respectively. $D$-charge is broken by a new Higgs singlet $\phi$
with charge $D=2$, which couples to $\nu_{2,3R}$ generating Majorana masses
for $\nu_{2,3R}$ too. Notice that $\nu_{1R}$ can pick up a Majorana mass by
itself, unsuppressed by any symmetry. It is noteworthy that $D$-charge is
broken by two units due to $\langle\phi\rangle$ down to a residual dark parity
$P_{D}=(-1)^{D}$. Particle representation content under the full gauge
symmetry and residual dark parity is collected to Tab. 1 for brevity. The dark
parity conservation implies that $\eta,\chi$ have vanishing VEVs, while
$H,\phi$ develop VEVs, $\langle H\rangle=(0,v/\sqrt{2})$ and
$\langle\phi\rangle=\Lambda/\sqrt{2}$, such that $\Lambda\gg v=246$ GeV for
consistency with the standard model.

Figure 1:  Scotoseesaw neutrino mass generation governed by dark gauge
symmetry, where interchanging vertices of $\nu_{aL}$ and $\nu_{bL}$ leads to
three new one-loop diagrams (unplotted).

It is further stressed that the neutrinos gain a suitable mass via a so-called
scotoseesaw mechanism as described by Feynman diagrams in Fig. 1. The tree-
level diagram is a part of the canonical seesaw governed by $\nu_{1R}$, while
the loop-level diagrams are a part of the scotogenic scheme set by
$\nu_{2,3R}$. For comparison, the latter that contains Majorana fermions in
the loop differs from [13] which interprets Dirac fermions by contrast.
Additionally, the present scotoseesaw recognized for three families naturally
arises as a result of dark charge breaking, significantly overhauling the
simplest models previously studied in [16, 15]. Let us remind the reader that
a recent approach [21] based upon gauged $B-L$ that assigns $B-L=-4,-4,+5$ to
$\nu_{1,2,3R}$ also hints at existence of the scotoseesaw; furthermore, the
scotogenic part may alternatively come from [22].

##  III Scalar potential

We decomposes the scalar potential into
$V=V(H,\phi)+V(\eta,\chi,\mathrm{mix})$, where the first part includes the
normal fields that induce gauge symmetry breaking,

$V(H,\phi)=\mu^{2}_{1}H^{\dagger}H+\mu^{2}_{2}\phi^{*}\phi+\lambda_{1}(H^{\dagger}H)^{2}+\lambda_{2}(\phi^{*}\phi)^{2}+\lambda_{3}(H^{\dagger}H)(\phi^{*}\phi),$
(1)

while the second part contains the dark fields and their couplings with normal
fields,

$\displaystyle V(\eta,\chi,\mathrm{mix})$ $\displaystyle=$
$\displaystyle\mu^{2}_{3}\eta^{\dagger}\eta+\mu^{2}_{4}\chi^{\dagger}\chi+\lambda_{4}(\eta^{\dagger}\eta)^{2}+\lambda_{5}(\chi^{\dagger}\chi)^{2}+\lambda_{6}(\eta^{\dagger}\eta)(\chi^{\dagger}\chi)+\lambda_{7}(\eta^{\dagger}\chi)(\chi^{\dagger}\eta)$
(2)
$\displaystyle+(\lambda_{8}\eta^{\dagger}\eta+\lambda_{9}\chi^{\dagger}\chi)(H^{\dagger}H)+\lambda_{10}(\eta^{\dagger}H)(H^{\dagger}\eta)+\lambda_{11}(\chi^{\dagger}H)(H^{\dagger}\chi)$
$\displaystyle+(\lambda_{12}\eta^{\dagger}\eta+\lambda_{13}\chi^{\dagger}\chi)(\phi^{*}\phi)+\left[\lambda_{14}(H\eta)(H\chi)+\mu_{5}(\eta^{\dagger}\chi)\phi+H.c.\right].$

Hereafter, $\lambda_{14}$ and $\mu_{5}$ are assumed to be real, since
otherwise their phases can be absorbed by redefining appropriate scalar
fields. The expected vacuum structure acquires $\mu^{2}_{1,2}<0$ and
$\mu^{2}_{3,4}>0$. Additionally, necessary condition for the potential bounded
from below is $\lambda_{1,2,4,5}>0$, which is obtained by requiring $V>0$ for
$H,\phi,\eta,\chi$ separately tending to infinity, respectively. Supplemental
conditions would exist by requiring $V>0$ for a number of scalar fields
simultaneously tending to infinity, which is skipped for brevity.

Expanding $H^{0}=(v+S+iA)/\sqrt{2}$,
$\phi=(\Lambda+S^{\prime}+iA^{\prime})/\sqrt{2}$, $\eta^{0}=(R+iI)/\sqrt{2}$,
and $\chi^{0}=(R^{\prime}+iI^{\prime})/\sqrt{2}$, we find the condition of
potential minimum,

$v^{2}=\frac{2(\lambda_{3}\mu^{2}_{2}-2\lambda_{2}\mu^{2}_{1})}{4\lambda_{1}\lambda_{2}-\lambda^{2}_{3}},\hskip
14.22636pt\Lambda^{2}=\frac{2(\lambda_{3}\mu^{2}_{1}-2\lambda_{1}\mu^{2}_{2})}{4\lambda_{1}\lambda_{2}-\lambda^{2}_{3}}.$
(3)

Physical scalar fields relevant to the normal sector are identified as

$H=\begin{pmatrix}G^{+}_{W}\\\
\frac{1}{\sqrt{2}}(v+c_{\varphi}H_{1}+s_{\varphi}H_{2}+iG_{Z})\end{pmatrix},\hskip
14.22636pt\phi=\frac{1}{\sqrt{2}}(\Lambda-
s_{\varphi}H_{1}+c_{\varphi}H_{2}+iG_{Z^{\prime}}),$ (4)

where $G^{+}_{W}\equiv H^{+}$, $G_{Z}\equiv A$, and $G_{Z^{\prime}}\equiv
A^{\prime}$ are massless Goldstone bosons coupled to gauge bosons $W^{+}$,
$Z$, and $Z^{\prime}$, respectively. The standard model like Higgs boson
$H_{1}\equiv c_{\varphi}S-s_{\varphi}S^{\prime}$ and the new Higgs boson
$H_{2}\equiv s_{\varphi}S+c_{\varphi}S^{\prime}$ are determined via a
$S$-$S^{\prime}$ mixing angle,

$t_{2\varphi}=\frac{\lambda_{3}v\Lambda}{\lambda_{2}\Lambda^{2}-\lambda_{1}v^{2}}\simeq\frac{\lambda_{3}}{\lambda_{2}}\frac{v}{\Lambda}\ll
1,$ (5)

as well as obtaining respective masses,

$\displaystyle
m^{2}_{H_{1}}=\lambda_{1}v^{2}+\lambda_{2}\Lambda^{2}-\sqrt{(\lambda_{1}v^{2}-\lambda_{2}\Lambda^{2})^{2}+\lambda^{2}_{3}v^{2}\Lambda^{2}}\simeq\left(2\lambda_{1}-\frac{\lambda^{2}_{3}}{2\lambda_{2}}\right)v^{2},$
(6) $\displaystyle
m^{2}_{H_{2}}=\lambda_{1}v^{2}+\lambda_{2}\Lambda^{2}+\sqrt{(\lambda_{1}v^{2}-\lambda_{2}\Lambda^{2})^{2}+\lambda^{2}_{3}v^{2}\Lambda^{2}}\simeq
2\lambda_{2}\Lambda^{2}.$ (7)

Relevant to the dark scalar sector, we conveniently define
$M^{2}_{\eta}\equiv\mu^{2}_{3}+\frac{\lambda_{8}}{2}v^{2}+\frac{\lambda_{12}}{2}\Lambda^{2}$
and
$M^{2}_{\chi}\equiv\mu^{2}_{4}+\frac{\lambda_{9}}{2}v^{2}+\frac{\lambda_{13}}{2}\Lambda^{2}$.
The charged dark scalars $\eta^{-}$ and $\chi^{-}$ are physical fields by
themselves with respective masses, obtained by

$m^{2}_{\eta^{-}}=M^{2}_{\eta}+\frac{\lambda_{10}}{2}v^{2},\hskip
14.22636ptm^{2}_{\chi^{-}}=M^{2}_{\chi}+\frac{\lambda_{11}}{2}v^{2}.$ (8)

By contrast, the neutral dark scalars $R,R^{\prime}$ and $I,I^{\prime}$ mix in
each pair, such as

$\displaystyle V$ $\displaystyle\supset$
$\displaystyle\frac{1}{2}\begin{pmatrix}R&R^{\prime}\end{pmatrix}\begin{pmatrix}M^{2}_{\eta}&\frac{\mu_{5}\Lambda}{\sqrt{2}}+\frac{\lambda_{14}v^{2}}{2}\\\
\frac{\mu_{5}\Lambda}{\sqrt{2}}+\frac{\lambda_{14}v^{2}}{2}&M^{2}_{\chi}\end{pmatrix}\begin{pmatrix}R\\\
R^{\prime}\end{pmatrix}$ (9)
$\displaystyle+\frac{1}{2}\begin{pmatrix}I&I^{\prime}\end{pmatrix}\begin{pmatrix}M^{2}_{\eta}&\frac{\mu_{5}\Lambda}{\sqrt{2}}-\frac{\lambda_{14}v^{2}}{2}\\\
\frac{\mu_{5}\Lambda}{\sqrt{2}}-\frac{\lambda_{14}v^{2}}{2}&M^{2}_{\chi}\end{pmatrix}\begin{pmatrix}I\\\
I^{\prime}\end{pmatrix}.$

Defining the mixing angles of the dark real and imaginary sectors,

$t_{2\theta_{R}}=\frac{\sqrt{2}\mu_{5}\Lambda+\lambda_{14}v^{2}}{M^{2}_{\chi}-M^{2}_{\eta}},\hskip
14.22636ptt_{2\theta_{I}}=\frac{\sqrt{2}\mu_{5}\Lambda-\lambda_{14}v^{2}}{M^{2}_{\chi}-M^{2}_{\eta}},$
(10)

we obtain physical fields,

$\displaystyle R_{1}=c_{\theta_{R}}R-s_{\theta_{R}}R^{\prime},\hskip
14.22636ptR_{2}=s_{\theta_{R}}R+c_{\theta_{R}}R^{\prime},$ (11) $\displaystyle
I_{1}=c_{\theta_{I}}I-s_{\theta_{I}}I^{\prime},\hskip
14.22636ptI_{2}=s_{\theta_{I}}I+c_{\theta_{I}}I^{\prime},$ (12)

with respective masses,

$\displaystyle m^{2}_{R_{1}}\simeq
M^{2}_{\eta}+\frac{\frac{1}{4}(\sqrt{2}\mu_{5}\Lambda+\lambda_{14}v^{2})^{2}}{M^{2}_{\eta}-M^{2}_{\chi}},\hskip
14.22636ptm^{2}_{R_{2}}\simeq
M^{2}_{\chi}-\frac{\frac{1}{4}(\sqrt{2}\mu_{5}\Lambda+\lambda_{14}v^{2})^{2}}{M^{2}_{\eta}-M^{2}_{\chi}},$
(13) $\displaystyle m^{2}_{I_{1}}\simeq
M^{2}_{\eta}+\frac{\frac{1}{4}(\sqrt{2}\mu_{5}\Lambda-\lambda_{14}v^{2})^{2}}{M^{2}_{\eta}-M^{2}_{\chi}},\hskip
14.22636ptm^{2}_{I_{2}}\simeq
M^{2}_{\chi}-\frac{\frac{1}{4}(\sqrt{2}\mu_{5}\Lambda-\lambda_{14}v^{2})^{2}}{M^{2}_{\eta}-M^{2}_{\chi}},$
(14)

where the approximations assume $\mu_{5}\Lambda\sim\lambda_{14}v^{2}\ll
M^{2}_{\eta,\chi}\sim\Lambda^{2}$, thus $\mu_{5}\ll\Lambda$. It is clear that
$R_{1},I_{1}$, as well as $R_{2},I_{2}$, are now separated in mass, namely
$(m^{2}_{R_{i}}-m^{2}_{I_{i}})/(m^{2}_{R_{i}}+m^{2}_{I_{i}})\sim\lambda^{2}_{14}(v/\Lambda)^{4}\ll
1$, for $i=1,2$. It is stressed that this mass splitting is in the same
magnitude with the mixing angles-squared,
$\theta^{2}_{R,I}\sim\lambda^{2}_{14}(v/\Lambda)^{4}\ll 1$.

##  IV Neutrino mass

Every charged lepton and quark gain an appropriate mass through interacting
with the Higgs field $H$ identical to the standard model. Concerning left-
handed and right-handed neutrinos, their Yukawa Lagrangian is given by

$\displaystyle\mathcal{L}$ $\displaystyle\supset$ $\displaystyle
h_{a1}\bar{l}_{aL}\tilde{H}\nu_{1R}+h_{a2}\bar{l}_{aL}\eta\nu_{2R}+h_{a3}\bar{l}_{aL}\chi\nu_{3R}$
(15)
$\displaystyle-\frac{1}{2}M_{1}\bar{\nu}^{c}_{1R}\nu_{1R}+\frac{1}{2}f_{2}\bar{\nu}^{c}_{2R}\nu_{2R}\phi+\frac{1}{2}f_{3}\bar{\nu}^{c}_{3R}\nu_{3R}\phi^{*}-M_{23}\bar{\nu}^{c}_{2R}\nu_{3R}+H.c.$

When $H,\phi$ develop VEVs, we obtain a tree-level mass Lagrangian,

$\mathcal{L}\supset-\frac{1}{2}\begin{pmatrix}\bar{\nu}_{aL}&\bar{\nu}^{c}_{1R}\end{pmatrix}\begin{pmatrix}0&m_{a1}\\\
m_{b1}&M_{1}\end{pmatrix}\begin{pmatrix}\nu^{c}_{bL}\\\
\nu_{1R}\end{pmatrix}-\frac{1}{2}\begin{pmatrix}\bar{\nu}^{c}_{2R}&\bar{\nu}^{c}_{3R}\end{pmatrix}\begin{pmatrix}M_{2}&M_{23}\\\
M_{23}&M_{3}\end{pmatrix}\begin{pmatrix}\nu_{2R}\\\
\nu_{3R}\end{pmatrix}+H.c.,$ (16)

where $b=1,2,3$ is a family index as $a$ is. Additionally,
$m_{a1}=-h_{a1}v/\sqrt{2}$ is a Dirac mass, while
$M_{2}=-f_{2}\Lambda/\sqrt{2}$ and $M_{3}=-f_{3}\Lambda/\sqrt{2}$ are Majorana
masses as $M_{1}$ and $M_{23}$ are.

Without loss of generality, we assume $M_{23}=0$, i.e. neither mixing between
$\nu_{2R}$ and $\nu_{3R}$ nor the corresponding loop diagram (right-upper) in
Fig. 1 exist. Diagonalizing the mass matrix of $(\nu_{L},\nu_{1R})$ in (16),
which has a seesaw form due to $v\ll M_{1}$, we get

$\mathcal{L}\supset-\frac{1}{2}(m_{\nu})^{\mathrm{tree}}_{ab}\bar{\nu}_{aL}\nu^{c}_{bL}-\frac{1}{2}M_{1}\bar{\nu}^{c}_{1R}\nu_{1R}-\frac{1}{2}M_{2}\bar{\nu}^{c}_{2R}\nu_{2R}-\frac{1}{2}M_{3}\bar{\nu}^{c}_{3R}\nu_{3R}+H.c.,$
(17)

where the first term,

$(m_{\nu})^{\mathrm{tree}}_{ab}\simeq-\frac{m_{a1}m_{b1}}{M_{1}},$ (18)

is the seesaw-induced neutrino mass identically to the contribution of the
tree-level diagram (left-upper) in Fig. 1, while $\nu_{1,2,3R}$ are apparently
decoupled, behaving as physical Majorana fields with masses $M_{1,2,3}$,
respectively.

The tree-level mass matrix (18) gives only a nonzero (mass) eigenvalue
according to mass eigenstate $\sim m_{a1}\nu_{aL}$ inappropriate to experiment
[23]. Interestingly, the tree-level vanished masses, or exactly the mass
matrix of ($\nu_{L},\nu_{1R}$) in (16), receive radiative corrections by dark
fields $(\eta,\chi,\nu_{2,3R})$ as in the loop diagrams (lower-left and
-right) in Fig. 1. In mass basis, the loop diagrams are commonly contributed
by one of $\nu_{2,3R}$ fermions and one of $R_{1,2},I_{1,2}$ scalars,
determined by the Lagrangian,

$\displaystyle\mathcal{L}$ $\displaystyle\supset$
$\displaystyle\frac{h_{a2}}{\sqrt{2}}\bar{\nu}_{aL}\nu_{2R}(c_{R}R_{1}+s_{R}R_{2}+ic_{I}I_{1}+is_{I}I_{2})$
(19)
$\displaystyle+\frac{h_{a3}}{\sqrt{2}}\bar{\nu}_{aL}\nu_{3R}(-s_{R}R_{1}+c_{R}R_{2}-is_{I}I_{1}+ic_{I}I_{2})+H.c.$

plus kinetic and mass terms of the relevant physical fields, where we have
defined $c_{R}\equiv c_{\theta_{R}}$ and $s_{R}\equiv s_{\theta_{R}}$ for
short. Therefore, the loop-induced neutrino masses are computed as

$\displaystyle(m_{\nu})^{\mathrm{rad}}_{ab}$ $\displaystyle=$
$\displaystyle\frac{h_{a2}h_{b2}M_{2}}{32\pi^{2}}\left(\frac{c^{2}_{R}m^{2}_{R_{1}}\ln\frac{M^{2}_{2}}{m^{2}_{R_{1}}}}{M^{2}_{2}-m^{2}_{R_{1}}}-\frac{c^{2}_{I}m^{2}_{I_{1}}\ln\frac{M^{2}_{2}}{m^{2}_{I_{1}}}}{M^{2}_{2}-m^{2}_{I_{1}}}+\frac{s^{2}_{R}m^{2}_{R_{2}}\ln\frac{M^{2}_{2}}{m^{2}_{R_{2}}}}{M^{2}_{2}-m^{2}_{R_{2}}}-\frac{s^{2}_{I}m^{2}_{I_{2}}\ln\frac{M^{2}_{2}}{m^{2}_{I_{2}}}}{M^{2}_{2}-m^{2}_{I_{2}}}\right)$
(20) $\displaystyle+\nu_{3R}\mathrm{-contribution},$

where the $\nu_{3R}$–contribution is translated from that of $\nu_{2R}$
(previous term) by replacing $h_{a(b)2}\to h_{a(b)3}$, $M_{2}\to M_{3}$,
$c_{R,I}\to-s_{R,I}$, and $s_{R,I}\to c_{R,I}$. Notice that the divergences
are manifestly cancelled out due to
$c^{2}_{R}-c^{2}_{I}+s^{2}_{R}-s^{2}_{I}=0$ for each $\nu_{2,3R}$
contribution. If the mixing between $\nu_{2R}$ and $\nu_{3R}$ is included, the
above result is easily generalized by changing $h_{a(b)k}\to h_{a(b)k}U_{kl}$
for $k,l=2,3$, where $U$ relates $\nu_{2,3R}$ to mass eigenstates with mass
eigenvalues $M_{2,3}$ as retained, without confusion.

With the aid of dark scalar mass splittings and mixing angles, i.e.
$(m^{2}_{R_{i}}-m^{2}_{I_{i}})/(m^{2}_{R_{i}}+m^{2}_{I_{i}})\sim\theta^{2}_{R,I}\sim\lambda^{2}_{14}(v/\Lambda)^{4}$
for $i=1,2$, we estimate the magnitude of radiative masses,
$(m_{\nu})^{\mathrm{rad}}\sim
0.1(M_{2,3}/\mathrm{TeV})(h\lambda_{14}/10^{-3})^{2}\ \mathrm{eV}$, assuming
$v/\Lambda\sim 10^{-1}$, where $h\sim h_{a(b)k}$. Comparing with measured
value $m_{\nu}\sim 0.1$ eV implies $M_{2,3}\sim$ TeV and $h\lambda_{14}\sim
10^{-3}$, thus $h,\lambda_{14}$ are not too small. The last conclusion is due
to a suppression in dark scalar mass splittings and mixing angles as
$(v/\Lambda)^{4}$, which is more stronger than that of the conventional
scotogenic model [10]. On the other hand, the tree-level neutrino mass in (18)
obeys $(m_{\nu})^{\mathrm{tree}}\sim 0.1$ eV, indicating that either $M_{1}$
is close to the GUT scale for $h_{a1}\sim h\sim 0.1$, or alternatively
$h_{a1}$ is close to the Yukawa coupling of electron for $M_{1}\sim
M_{2,3}\sim$ TeV. The first case is possible since $M_{1}$ is not suppressed
by dark symmetry, while the second case is available too but it requires a
hierarchy $h_{a1}\ll h_{ak}$ like those of the Yukawa couplings of charged
leptons. In summary, the scotoseesaw neutrino mass described in Fig. 1 is
given by

$(m_{\nu})_{ab}=(m_{\nu})^{\mathrm{tree}}_{ab}+(m_{\nu})^{\mathrm{rad}}_{ab}.$
(21)

##  V Dark matter

The model contains two kinds of dark matter, dark Majorana fermion
($\nu_{2,3R}$) and dark doublet scalar ($R_{1,2},I_{1,2}$), communicated with
normal matter via $U(1)_{D}$ gauge boson $Z^{\prime}$ and usual/new Higgs
portals $H_{1,2}$. Kinetic mixing effect [24] between $U(1)_{D}$ and
$U(1)_{Y}$ is too small and is not signified in this work. The dark matter
candidate—the lightest of dark fields by appropriate parameter choice—is
studied in order.

###  V.1 Majorana dark matter

Figure 2:  Annihilation of Majorana dark matter to normal matter that sets the
relic abundance. Figure 3:  Scattering of Majorana dark matter with nuclei in
direct detection.

Assuming $\nu_{3R}$ to be the lightest of dark fields, it is absolutely
stabilized by the residual dark parity $P_{D}$ responsible for dark matter. In
the early universe, $\nu_{3R}$’s potentially annihilate to pair of
$Z^{\prime}$—given that $Z^{\prime}$ is lighter than $\nu_{3R}$—via
$t,u$-channel diagrams as well as to usual fermions ($ff^{c}$) via $s$-channel
$H_{1,2}$ portals due to a mixing between the standard model Higgs field ($H$)
and new Higgs field ($\phi$). The usual Higgs $H_{1}$ portal negligibly
contributes to the relic density since $\nu_{3R}$ is heavy at TeV, necessarily
set by $H_{2}$ resonance. Also, the annihilation to $Z^{\prime}$ by
$t,u$-channels is suppressed similarly to the $H_{1}$ portal, not
significantly compared to $H_{2}$ portal (cf. [25]). That said, the $\nu_{3R}$
annihilation that sets the relic density is governed by $H_{2}$ as described
by the Feynman diagram in Fig. 2. Although the $H_{1}$ portal does not
contributes to the relic density, it would set the spin-independent (SI)
cross-section presenting scattering of $\nu_{3R}$ with nuclei in direct
detection experiment as determined by the Feynman diagram in Fig. 3. This is
because the $H_{2}$ contribution to the effective interaction
$(\bar{\nu}^{c}_{3R}\nu_{3R})(\bar{q}q)$ is suppressed by
$m^{2}_{H_{1}}/m^{2}_{H_{2}}\ll 1$ as compared to that of $H_{1}$.

The annihilation cross-section of $\nu_{3R}$ is evaluated as

$\displaystyle\langle\sigma v_{\mathrm{rel}}\rangle_{\nu_{3R}}$
$\displaystyle=$
$\displaystyle\sum_{f}\frac{s^{2}_{\varphi}c^{2}_{\varphi}m^{2}_{f}M^{4}_{3}}{8\pi
v^{2}\Lambda^{2}(4M^{2}_{3}-m^{2}_{H_{2}})^{2}}\left(1-\frac{m^{2}_{f}}{M^{2}_{3}}\right)^{\frac{3}{2}}$
(22) $\displaystyle\simeq$
$\displaystyle\frac{s^{2}_{\varphi}m^{2}_{t}M^{4}_{3}}{8\pi
v^{2}\Lambda^{2}(4M^{2}_{3}-m^{2}_{H_{2}})^{2}},$

where $v_{\mathrm{rel}}$ is dark matter relative velocity, and the biggest
contribution comes from the annihilation to top quark as enhanced by its
Yukawa coupling. The usual-new Higgs mixing leads to a shift in the standard
model Higgs couplings, which are well-measured at present; and, the projected
HL-LHC would make stringent bounds on this shift [23]. To satisfy all the
limits, we take the mixing angle to be $s_{\varphi}=10^{-2}$. Additionally,
let $m_{t}=173$ GeV, $v=246$ GeV, and $v/\Lambda=0.1$. We plot the relic
density $\Omega_{\nu_{3R}}h^{2}\simeq 0.1\ \mathrm{pb}/\langle\sigma
v_{\mathrm{rel}}\rangle_{\nu_{3R}}$ as function of the dark matter mass in
Fig. 4 for $m_{H_{2}}=2$ TeV for example. It is clear from the figure that the
$H_{2}$ resonance $m_{\nu_{3R}}=\frac{1}{2}m_{H_{2}}$ is crucial to set the
dark matter relic density.

Figure 4:  Relic density of Majorana dark matter plotted as a function of its
mass, where the dashed line is the measured value
$\Omega_{\mathrm{DM}}h^{2}\simeq 0.12$ [23].

The effective Lagrangian describing scattering of $\nu_{3R}$ with quark is
induced directly from Fig. 3 to be
$\mathcal{L}_{\mathrm{eff}}\supset\alpha^{S}_{q}(\bar{\nu}^{c}_{3R}\nu_{3R})(\bar{q}q)$,
where

$\alpha^{S}_{q}=-\frac{s_{\varphi}c_{\varphi}m_{q}M_{3}}{v\Lambda
m^{2}_{H_{1}}}.$ (23)

Hence, the scattering cross-section of dark matter with a nucleon $N=p,n$ is
given by [26]

$\sigma^{\mathrm{SI}}_{\nu_{3R}-N}=\frac{4m^{2}_{N}}{\pi}(f^{N})^{2},$ (24)

where the nucleon form factor obeys

$\frac{f^{N}}{m_{N}}=\sum_{q=u,d,s}\frac{\alpha^{S}_{q}}{m_{q}}f^{N}_{Tq}+\frac{2}{27}f^{N}_{TG}\sum_{q=c,b,t}\frac{\alpha^{S}_{q}}{m_{q}}\simeq-\frac{0.35s_{\varphi}c_{\varphi}M_{3}}{v\Lambda
m^{2}_{H_{1}}},$ (25)

where we have used $f^{N}_{TG}=1-\sum_{q=u,d,s}f^{N}_{Tq}$, $f^{p}_{Tu}=0.02$,
$f^{p}_{Td}=0.026$, $f^{p}_{Ts}=0.118$, $f^{n}_{Tu}=f^{p}_{Td}$,
$f^{n}_{Td}=f^{p}_{Tu}$, and $f^{n}_{Ts}=f^{p}_{Ts}$ [27]. It follows that

$\sigma^{\mathrm{SI}}_{\nu_{3R}-N}\simeq\frac{0.49m^{4}_{N}s^{2}_{\varphi}M^{2}_{3}}{\pi
v^{2}\Lambda^{2}m^{4}_{H_{1}}}\simeq\left(\frac{s_{\varphi}}{10^{-2}}\right)^{2}\left(\frac{M_{3}}{\mathrm{TeV}}\right)^{2}\times
0.68\times 10^{-46}\ \mathrm{cm}^{2},$ (26)

where we have taken $m_{N}=1$ GeV, $m_{H_{1}}=125$ GeV, $v=246$ GeV, and
$v/\Lambda=0.1$. The SI cross-section predicted is in agreement with the
latest data $\sigma^{\mathrm{SI}}_{\nu_{3R}-N}\sim 10^{-46}\ \mathrm{cm}^{2}$
[28] for $\nu_{3R}$ dark matter mass $M_{3}\sim$ TeV and usual-new Higgs
mixing $s_{\varphi}\sim 10^{-2}$.

###  V.2 Scalar dark matter

Since the dark scalars $R_{1,2}$ and $I_{1,2}$ are heavy at TeV, their dark
matter phenomenology is quite similar to the Majorana dark matter interpreted
above. Indeed, the lightest of dark fields, $R_{1}$ for instance, is now
stabilized by $P_{D}$. In the early universe, it may annihilate to a pair of
$Z^{\prime}$, as well as of weak bosons, through $t,u$-channel diagrams or to
usual fermions via $H_{1,2}$ portals. However, only the $H_{2}$ portal is
enhanced due to the mass resonance that sets $R_{1}$ relic density. Concerning
direct detection, $R_{1}$ scatters with nucleon via the standard model Higgs
portal $H_{1}$. The new Higgs portal gives a negligible contribution, while
the $Z$ portal is inaccessible due to the dark scalar mass splitting.

##  VI Conclusion and outlook

We have shown that if the three right-handed neutrinos—the right-handed
parters of the usual left-handed neutrinos—possess a dark gauge charge while
the standard model particles do not, their charges would be $0,-1,+1$ in order
for the theory to be consistent at quantum level. This new theory implies
naturally the scotoseesaw mechanism which induces appropriate neutrino masses
via both contributions of DM (scotogenic) and non-DM (seesaw) right-handed
neutrinos. The dark matter candidate is either one of the dark right-handed
neutrinos or one of the neutral dark scalars, obtaining a mass at TeV,
possessing a correct abundance enhanced by the new Higgs mass resonance
$m_{\mathrm{DM}}=\frac{1}{2}m_{H_{2}}$, and potentially scattering with
nucleon via the standard model Higgs exchange.

Since the present work has not signified a kinetic mixing, or this mixing is
small, $Z^{\prime}$ does not significantly couple to usual fermions. Hence,
the current colliders [29, 30, 31] do not make a constraint on $Z^{\prime}$.
Similarly, the other new particles, such as DM ($\nu_{3R}$ or $R_{1}$) and
non-DM ($\nu_{1R}$ and $H_{2}$), would exhibit negligible signals at the
present colliders because they interact rather weak with the normal matter.
However, the projected HL-LHC and ILC may be worth exploring for them when the
energy and sensitivity are enhanced.

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# COMBINING LANGUAGE MODELS FOR SPECIALIZED DOMAINS: A COLORFUL APPROACH

###### Abstract

General purpose language models (LMs) encounter difficulties when processing
domain-specific jargon and terminology, which are frequently utilized in
specialized fields such as medicine or industrial settings. Moreover, they
often find it challenging to interpret mixed speech that blends general
language with specialized jargon. This poses a challenge for automatic speech
recognition systems operating within these specific domains. In this work, we
introduce a novel approach that integrates domain-specific or secondary LM
into general-purpose LM. This strategy involves labeling, or “coloring”, each
word to indicate its association with either the general or the domain-
specific LM. We develop an optimized algorithm that enhances the beam search
algorithm to effectively handle inferences involving colored words. Our
evaluations indicate that this approach is highly effective in integrating
jargon into language tasks. Notably, our method substantially lowers the error
rate for domain-specific words without compromising performance in the general
domain.

Index Terms—  language modeling, decoding, speech recognition, jargon language
model

##  1 Introduction

Specialized jargon is crucial in numerous tasks and real-world scenarios,
facilitating precise and efficient communication among experts or
professionals. Jargon language is prevalent in sectors such as law, industry,
business, and healthcare. The sentence “The patient suffered from
_formication_ due to _neuropathy_ caused by _diabetes mellitus_ ” is an
example from healthcare jargon language where the jargon term are highlighted
in italics. In the subsequent sections, we will use the term _mixed speech_ to
refer to spoken expressions that combine general language with specialized
jargon.

Jargon language poses significant challenges for automatic speech recognition
(ASR) and natural language processing (NLP) systems, which are predominantly
trained on more general language datasets. While some jargon words and terms
are publicly accessible (e.g., flight control or some medical terms) and hence
can be used in training or fine-tuning LMs, many jargon words are used
exclusively within industrial companies. As a result, they are not available
for training in the widely used general-purpose language models.

Fig. 1: Illustration of the probability calculation in language models for
mixed speech using different approaches: (A) static interpolation, and (B)
dynamic interpolation — both methods facilitate interpolation between LMs.
Conversely, in approach (C), which is our proposed method, each LM is
responsible for handling either the general or the jargon segment exclusively.

One common approach to address this challenge involves interpolating between
multiple LMs, each specializing in a different domain and estimated using
distinct corpus [1, 2, 3, 4]. However, this approach suffers from three
limitations. First, it often relies on fixed interpolation coefficients, which
may be sub-optimal [5, 6, 7]. Second, it does not address mixed speech within
a sentence, resulting in low probability scores for such sentences from all
LMs. Third, utilizing the average outputs of several LMs can lead to an over-
smoothing effect [2].

The problem of incorporating several LMs also relates to topic-model LMs [8,
9, 10]. Topic-model LM is a mixture model where each mixture component is
represented by a topic-dependent word probability, and each mixture weight
corresponds to a topic proportion probability. These methods assume an
underlying parametric distribution and are often implemented using a unigram
word probabilities [10].

In this study, we introduce a novel method for integrating multiple LMs by
dynamically switching between general and jargon-specific models, mitigating
the limitations of previous interpolation methods. We demonstrate the
effectiveness of our approach in the realm of ASR, which has seen remarkable
advancements in recent years. This development can be attributed to the
emergence of self-supervised representation techniques [11, 12]. However,
precise language modeling is crucial for achieving high accuracy rates in ASR
systems.

We describe an efficient procedure for implementing our method, based on the
beam search algorithm. As our approach requires only minor changes to the
original algorithm, it is easy to integrate into existing systems. Our
approach provides a practical solution for handling domain-specific jargon in
speech recognition, with important implications for a wide range of ASR
applications that rely on the accurate transcription of specialized
terminology. Importantly, our approach can easily extend and be utilized in
other NLP tasks, such as MT [13, 14] and text generation [15].

This paper makes the following contributions: (i) it introduces a novel
approach for incorporating general and domain-specific LMs; (ii) it develops
an efficient and scalable algorithm, grounded in beam search techniques, for
approximating the proposed method; and (iii) it undertakes a set of
experiments to empirically validate the efficacy of our approach.

##  2 Method

A language model is defined as the probability of a sequence of words. Denote
the sequence of words by $\mathbf{y}=(y_{1},\ldots,y_{T})$. Each word $y_{t}$
is associated to one or more of the $C$ vocabularies
$\mathcal{V}^{1},\ldots,\mathcal{V}^{C}$. For simplicity of presentation we
will discus our solution for combining two LMs namely a general and jargon
models. We denote the input sequence by $\mathbf{x}=(x_{1},\ldots,x_{L})$.
This sequence can be a speech utterance for ASR [11, 12], a sequence of word
tokens for (conditional) language generation [15], or a sequence of words from
a different language in the case of MT [13]. Let $\mathcal{Y}(\mathbf{x})$
denote the set of all valid output sequences for input $\mathbf{x}$. Given an
input $\mathbf{x}$, our goal is to obta in the maximum a posteriori (MAP),

$\displaystyle\mathbf{\hat{y}}=\arg\max_{\mathbf{y}\in\mathcal{Y}(\mathbf{x})}\mathrm{P}(\mathbf{y}\mid\mathbf{x})$
(1)

In this work, we focus on additive scoring rules as commonly used in MAP
decoding of neural sequence-to-sequence models. Concretely, at each position
$t$ we wish to obtain,

$\hat{y}_{t}=\arg\max_{y_{t}}\mathrm{P}(y_{t}\mid\mathbf{y}_{<t},\mathbf{x})$
(2)

where $\mathbf{y}_{<t}=(y_{1},\ldots,y_{t-1})$.

We introduce an additional sequence $\mathbf{c}=(c_{1},...,c_{T})$ where
$c_{t}\in[C]$, that controls the source from which $y_{t}$ is derived, i.e.,
$c_{t}=j$ if $y_{t}\in\mathcal{V}^{j}$. We refer to $\mathbf{c}$ as the
coloring variable. Our approach modifies Eq. (1) to account for the additional
coloring variable and jointly optimize $\mathbf{y},\mathbf{c}$. Assuming
$c_{t}$ is independent of the colored history
$(\mathbf{y},\mathbf{c})_{<t}=((y_{1},c_{1}),\ldots,(y_{t-1},c_{t-1}))$ we
write,

$\displaystyle(\hat{\mathbf{y}},\hat{\mathbf{c}})$
$\displaystyle=\arg\max_{\mathbf{y},\mathbf{c}}\prod_{t}\mathrm{P}(y_{t},c_{t}\mid(\mathbf{y},\mathbf{c})_{<t},\mathbf{x})$
(3)
$\displaystyle=\arg\max_{\mathbf{y},\mathbf{c}}\prod_{t}\mathrm{P}(c_{t}\mid(\mathbf{y},\mathbf{c})_{<t})\mathrm{P}(y_{t}\mid\mathbf{c}_{\leq
t},\mathbf{y}_{<t},\mathbf{x})$
$\displaystyle=\arg\max_{\mathbf{y},\mathbf{c}}\prod_{t}\mathrm{P}(c_{t})\mathrm{P}(y_{t}\mid\mathbf{c}_{\leq
t},\mathbf{y}_{<t},\mathbf{x})~{},$

where the last equality follows from our independence assumption. For each
time $t$, our method optimizes for the upcoming word and its source lexicon,
conditioned on the “colored” history. In the experiments, we fixed coloring
probabilities $\mathrm{P}(c_{t}=j)=\mathrm{P}(c=j)=1/C$ for each $j\in[C]$. We
choose to use a uniform distribution for the colors as a reasonable
assumption, given the scarcity of mixed speech texts.

In order to understand the advantages of our approach, let’s examine the
challenge of integrating jargon LMs into a more general LM. Assuming the
general LM is optimized for standard language, and the jargon model is
optimized for domain-specific language, any interpolation method will fall
short when it comes to mixed speech. However, our approach dynamically
switches between the two LMs, allowing for an accurate estimation of the
likelihood of mixed sentences. Additionally, our method assigns the source
lexicon for each word in the decoded text. This supplementary information can
be advantageous for downstream NLP tasks.

Algorithm 1 CTC Beam Search with Coloring

Input: AM output scores matrix $S$, beam width $w$.

Initialize: $B\leftarrow\\{\varnothing\\}$, $\mathrm{P}^{-}(\varnothing,0)=1$

for $t=1,...,L$ do

$\hat{B}\leftarrow GetBestBeams(B,w)$

$B\leftarrow\\{\\}$

for $\tilde{\mathbf{y}}=(\mathbf{y},\mathbf{c})\in\hat{B}$ do

if $\mathbf{y}\neq\varnothing$ then

$\mathrm{P}^{+}(\mathbf{y},t)\leftarrow\mathrm{P}^{+}(\mathbf{y},t-1)\cdot
S[y_{-1},t]$

end if

$\mathrm{P}^{-}(\mathbf{y},t)\leftarrow\mathrm{P}^{\text{tot}}(\mathbf{y},t-1)\cdot
S[-,t]$

$B\leftarrow B\cup\\{\tilde{\mathbf{y}}\\}$

$NextChars\leftarrow GetNextChars(\tilde{\mathbf{y}})$

for $(\ell,c)\in NextChars$ do

$\mathbf{y}^{\prime}\leftarrow\mathbf{y}\circ\ell$

$\mathbf{c}^{\prime}\leftarrow\mathbf{c}\circ c$

$\tilde{\mathbf{y}}^{\prime}\leftarrow(\mathbf{y}^{\prime},\mathbf{c}^{\prime})$

$\mathrm{P}^{\text{text}}(\tilde{\mathbf{y}}^{\prime})\leftarrow
ScoreBeam(\tilde{\mathbf{y}},\ell,c)$

if $y_{t}\mathrel{=}\mathrel{=}\ell$ then

$\mathrm{P}^{+}(\mathbf{y}^{\prime},t)\leftarrow\mathrm{P}^{-}(\mathbf{y},t-1)\cdot
S[i_{\ell},t]$

else

$\mathrm{P}^{+}(\mathbf{y}^{\prime},t)\leftarrow\mathrm{P}^{\text{tot}}(\mathbf{y},t-1)\cdot
S[i_{\ell},t]$

end if

$B\leftarrow B\cup\\{\tilde{\mathbf{y}}^{\prime}\\}$

end for

end for

end for

Return: $GetBestBeams(B,1)$

##  3 Implementation Details

In this section, we describe an approach to utilize the beam search algorithm
and efficiently scaling with the number of lexicons, denoted by $C$.

It may not be possible to solve the exact exhaustive search of the decoding in
Eq. (3). Therefore, beam-search decoding approximation is often utilized [16,
17].

In our settings, the input for the beam search process is the acoustic model’s
output, i.e., the probability scores (logits) for each token (characters in
our case) at each time step. To accommodate the different lexicon sources, we
identify each token score with the corresponding tokens in all lexicons.
During the beam search, each word’s first letter can come from either of the
lexicons, once the first letter has been set, only letters from the same
lexicon can be appended, thus alleviating the computational cost.

We now provide a detailed description of the implementation details and
modifications to the beam search algorithm. For simplicity and ease of
exposition, we consider having two lexicons, a general and a jargon-specific,
denoted $\mathcal{V}^{G},\mathcal{V}^{J}$, respectively. Our method, however,
trivially extends to multiple lexicons. We denote the characters set by
$\mathcal{A}$ of size $|\mathcal{A}|=K$. To achieve word coloring and identify
jargon terms, we modify the character set of the jargon lexicon. We denote the
jargon characters set $\mathcal{A}^{\prime}$, and identify each
$\ell^{\prime}\in\mathcal{A}^{\prime}$ with $\ell\in\mathcal{A}$ in a one-to-
one manner by color each character in $\mathcal{A}$. We further identify each
character $\ell\in\mathcal{A}$ with an integer $i_{\ell}\in\\{1,2,\ldots,K\\}$
such that $i_{\ell}=i_{\ell^{\prime}}$. The importance of having $\mathcal{A}$
and $\mathcal{A}^{\prime}$ as disjoint sets lies in the fact that the n-grams
of the models are also necessarily disjoint in this scenario. This makes it
straightforward to merge the original n-gram models simply by appending the
n-grams. We let $\varnothing$ denote the empty sequence, and
$S=S(\mathbf{x})\in\mathbb{R}^{L\times(K+1)}$ the output from the acoustic
model ($K$ characters and one blank symbol denoted $-$). $S$ is arranged such
that the $i_{\ell}$th column corresponds to $\ell\in\mathcal{A}$. Furthermore,
let $\mathrm{P}^{-}$, $\mathrm{P}^{+}$ and $\mathrm{P}^{\text{tot}}$ denote
the _blank_ , _non-blank_ and _total_ probabilities, with
$\mathrm{P}^{\text{tot}}=\mathrm{P}^{-}+\mathrm{P}^{+}$. We denote
$\tilde{\mathbf{y}}$ the joint sequence $(\mathbf{y},\mathbf{c})$, and let
$\circ$ denote the concatenation operation. Last, we denote by $y_{-1}$ the
last character of the sequence $\mathbf{y}$.

We define three functions $GetNextChars$, $ScoreBeam$ and $GetBestBeams$. The
function $GetNextChars(\tilde{\mathbf{y}})$ return a set of possible
characters given $\tilde{\mathbf{y}}=(\mathbf{y},\mathbf{c})$. At each time
$t$, it returns the characters corresponding to lexicon $c_{t-1}$ if the
current beam ends with a sub-word. Otherwise, it returns all possible
characters. The function $ScoreBeam(\tilde{\mathbf{y}},\ell,c)$ returns the
probability of seeing $(\ell,c)$ as an extension of the beam
$\tilde{\mathbf{y}}$. Using $ScoreBeam$, we define
$\mathrm{P}^{\text{text}}(\tilde{\mathbf{y}}^{\prime})=ScoreBeam(\tilde{\mathbf{y}},\ell,c)$
for $\tilde{\mathbf{y}}^{\prime}=(\mathbf{y}\circ\ell,\mathbf{c}\circ c)$.
Finally, the function $GetBestBeams(B,M)$ returns the top $M$ beams from $B$
according to $\mathrm{P}^{\text{tot}}\cdot\mathrm{P}^{\text{text}}$.

Alg. 1 describes our modification to the standard CTC beam search algorithm.
The modifications are marked _green_. Our method involves making slight
adjustments to the original CTC beam search algorithm. Furthermore, the number
of beams in our approach should remain the same as the original CTC beam
search, as $|GetNextChars(\tilde{y})|=K$ for each sub-word. As a result, our
approach can be easily scaled with the number of lexicons $C$.

Table 1: Incorporating jargon LMs using four datasets: Industrial English, two medical datasets [18, 19], and Industrial Thai. | Industrial English |  Medical [18] |  Medical [19] | Industrial Thai
---|---|---|---|---
|  CER $\downarrow$ |  WER $\downarrow$ |  CER $\downarrow$ |  WER $\downarrow$ |  CER $\downarrow$ |  WER $\downarrow$ |  CER $\downarrow$ |  WER $\downarrow$
General LM | $12.6$ | $36.8$ | $6.1$ | $13.0$ | $12.1$ | $23.6$ | $6.0$ | $10.6$
Jargon LM | $9.6$ | $21.9$ | $24.9$ | $62.7$ | $21.5$ | $56.4$ | $22.1$ | $51.8$
Linear interpolation | $6.9$ | $15.9$ | $6.2$ | $12.8$ | $6.2$ | $12.6$ | $5.7$ | $\mathbf{9.9}$
Log-linear interpolation | $10.2$ | $23.9$ | $6.1$ | $13.7$ | $7.2$ | $17.3$ | $6.1$ | $11.3$
Bin estimation [2] | $9.2$ | $22.6$ | $8.2$ | $21.5$ | $10.9$ | $39.8$ | $6.2$ | $12.6$
Bayes [20] | $9.0$ | $23.3$ | $5.6$ | $12.8$ | $7.4$ | $18.3$ | $6.0$ | $11.5$
Ours | $\mathbf{5.7}$ | $\mathbf{11.9}$ | $\mathbf{6.1}$ | $\mathbf{12.3}$ | $\mathbf{5.8}$ | $\mathbf{11.4}$ | $\mathbf{5.5}$ | $\mathbf{9.9}$

##  4 Experiments

In this section, we compare our method with natural baselines for combining
language models. In all experiments, we used two lexicons, a general and a
domain-specific or secondary LM. We denote the probability density functions
under the two LMs as $\mathrm{P}_{G}$ and $\mathrm{P}_{J}$.

Datasets. We assess our approach using four datasets: (i) _Industrial English_
: $\sim\\!\\!\\!2.5$ hours, $267$ audio files, featuring Australian English
utterances on mechanical equipment conditions (jargon) and inspector
discussions (general) with machinery background noise. (ii) _Industrial Thai_
: $\sim\\!\\!\\!1$ hour, $90$ Thai audio files, similar to the Industrial
English dataset. (iii) _Medical_ [18]: $\sim\\!\\!55$ hours of simulated
patient-physician interviews. We segmented recordings into 30-second audio
files, using $\sim\\!\\!1.5$ hours for our experiment (0.5 hours for
validation, 1 hour for testing). (iv) _Medical_ [19]: $1,700$ brief doctor-
patient conversations. We extracted 400 sentences for medical terms, created a
3-gram LM, and augmented the dataset with 401 audio files (totaling $80$
minutes). For evaluation, we split it into a validation set (149 files,
$\sim\\!\\!25$ minutes) and a test set (252 files, $\sim\\!\\!55$ minutes).

Comparison Methods. We evaluate several baseline approaches for combining
language models: (i) _Linear Interpolation_ : Combines language models using
$\lambda\mathrm{P}_{J}+(1-\lambda)\mathrm{P}_{G}$ for $\lambda\in[0,1]$. (ii)
_Log-Linear Interpolation_ [1]: Interpolates log probabilities:
$\mathrm{P}_{J}^{\lambda}\mathrm{P}_{G}^{(1-\lambda)}$. (iii) _Bin Estimation
Method_ [2]: Maps word probabilities from different LMs into a single
probability by binning the space and calibrating the output. (iv) _Bayes
Approach_ [20]: A dynamic interpolation method based on Bayes’ theorem,
assigning interpolation weights for each word. (v) _General LM_ : Uses only
the general LM. (vi) _Jargon LM_ : Uses only the jargon LM. (vii) _Ours_ : Our
proposed approach as described in Section 2.

Table 2: Qualitative examples from the medical dataset [19]. General LM marked in _blue_ , while domain-specific LM in _red_. | Ground Truth | Linear Interpolation | Ours
---|---|---|---
Success |  She had no fever or chills cough congestion nausea vomiting chest pain chest pressure |  She had no fever or chills of congestion nausea vomiting chest pain chest pressure |  She had no fever or chills cough congestion nausea vomiting chest pain chest pressure
Success |  He has had years in which he did better on clozaril and also his other medications |  He has had year in which he did better on care and other medications |  He has had years in which he did better on clozaril and also his other medications
Failure |  He has not been treated for diabetes since his last weight since he stopped taking zyprexa |  He has not been treated for diabetes since his weight since he stopped taking the pre |  He has not been treated for diabetes since his last weight since he stopped taking the prex

Hyperparameter Search (HP). We performed grid searches with validation splits
to optimize hyperparameters. For all methods, we explored $\alpha$ and $\beta$
in $\\{0.5,0.75,1.0,1.25,1.5\\}$, representing language model contribution and
text length scaling. As well as unknown word penalties in $\\{-10,-50\\}$ for
each LM. For linear and log-linear interpolation, we tuned
$\lambda\in\\{0.25,0.5,0.75\\}$. For the bin estimation method [2], we
considered the number of bins in $\\{53,100\\}$ (where $53$ was optimal [2]).
In our method, we explored an unknown sub-word penalty in
$\\{-7,-5,-3,-1,0\\}$.

###  4.1 Incorporation Domain-Specific Jargon

Here we evaluate our method on the task of incorporating domain-specific
jargon into general LMs. We used the (i) Industrial EN, the (ii) medical
dataset [18] and (iii) medical dataset [19]. We employed a pre-trained XLSR-
Wav2Vec 2.0 [21] based model fine-tuned using the Common Voice dataset [22]111
https://huggingface.co/jonatasgrosman/wav2vec2-large-xlsr- 53-english.. We use
a $3-$gram English LM222 https://catalog.ngc.nvidia.com/orgs/nvidia/teams/tlt-
jarvis/models /speechtotext_english_lm. as the general LM. Furthermore, we
constructed domain-specific $3-$gram LMs using held-out data. (iv) Industrial
TH dataset containing Thai sentences with specialized terms. We used a pre-
trained Thai XLSR-Wav2Vec 2.0 based model [23] fine-tuned using the Thai
Common Voice corpus V8 [22]. We trained two $3$-gram LMs; The first was
trained on a standard Thai, while the second contained factory terms.

The results are presented in Table 1. Our method achieves a significant
reduction in the WER and CER compared with the natural baselines for combining
language models. Notably, the significant error reduction in the Industrial
dataset which was recorded in a challenging acoustic environment.

###  4.2 Qualitative Examples

To gain insights into the successes and failures of our method, this section
presents a collection of qualitative examples that showcase the efficacy of
our proposed approach. The examples, presented in Table 2, are selected from
the medical dataset proposed in [19]. The first two examples demonstrate how
our method accurately produces the ground truth (GT) transcription. On the
other hand, the linear interpolation model fails to produce some of the
domain-specific terms; We highlight these terms in the GT column for clarity.
The last example presents a failure case: both our approach and the linear
interpolation method fail to produce the domain-specific word _zyprexa_.

##  5 Conclusions

In this work, we introduce a novel method for combining language models,
demonstrating its effectiveness in ASR with domain-specific jargon. We present
an efficient algorithm based on the common beam search, requiring only minor
modifications to integrate into ASR systems. Our approach offers a practical
solution for accurate transcription of specialized terminology in diverse ASR
applications.

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