# Hijacking Context in Large Multi-modal Models

Joonhyun Jeong
Korea Advanced Institute of Science
Graduate School of AI
<EMAIL_ADDRESS>

###### Abstract

Recently, Large Multi-modal Models (LMMs) have demonstrated their ability to
understand the visual contents of images given the instructions regarding the
images. Built upon the Large Language Models (LLMs), LMMs also inherit their
abilities and characteristics such as in-context learning where a coherent
sequence of images and texts are given as the input prompt. However, we
identify a new limitation of off-the-shelf LMMs where a small fraction of
incoherent images or text descriptions mislead LMMs to only generate biased
output about the hijacked context, not the originally intended context. To
address this, we propose a pre-filtering method that removes irrelevant
contexts via GPT-4V, based on its robustness towards distribution shift within
the contexts. We further investigate whether replacing the hijacked visual and
textual contexts with the correlated ones via GPT-4V and text-to-image models
can help yield coherent responses.

##  1 Introduction

Large Language Models (LLMs) pre-trained with huge amount of text corpus have
demonstrated remarkable generalization on various language tasks such as
human-like dialogues [1, 2, 3], reasoning-tasks [4, 5, 6], and few-shot in-
context learning [7, 8] where a few demonstration examples are given as a
context for answering the test question. By bootstrapping LLMs with several
add-on layers, Large Multi-modal Models (LMMs) [3, 9, 10, 11] were enabled to
understand the visual contents and answer the corresponding textual
instruction. LMMs also inherited the in-context learning capability of LLM,
where a coherent sequence of images and textual information are given as a
context for the input prompt [9, 3].

In this paper, we identify a new limitation of these off-the-shelf LMMs,
hijacking contexts, where a small fraction of incoherent images or text
descriptions mislead them to only generate biased output about the incoherent
contexts. In Figure 1, we provide the qualitative example of context hijacking
on visual story telling (VIST) dataset [12] with FROMAGe [9] as LMM. Given all
the former sequence of images and their corresponding captions, the LMM is
asked to provide a caption or description of the final query image. Normally,
when all the visual and textual information is coherent and consistently
aligned, LMM successfully provides a coherent response under consideration in
the given contexts. However, when only a single image-text pair that contains
an irrelevant subject is appended to the context, LMM falls into the hijacked
context and loses coherency concerning the bunch of formerly given original
contexts. In real-world applications and scenarios of LMMs, there is no
guarantee for the inexistence of such noises and irrelevant contexts. For
reliable usage of LMM, it should be robust to such distribution shift and
stick to the majority of context without confusion on the minority of
irrelevant contexts.

Figure 1: Hijacking context confuses LMMs to only generate responses with
incoherent contents. (top): Given a sequence of visual and textual story plot
1$\sim$4, LMM reasonably outputs a coherent caption for the final image.
(bottom): When a single pair of incoherent image and caption (highlighted with
red) is appended to the context, LMM only tells about the hijacked context
($i.e.,$ football game), disregarding all the aforementioned context of visual
story plots ($i.e.,$ emergency situation).

To address this, we propose a pre-filtering method that removes irrelevant and
hijacked contexts via prompting into GPT-4V [13], based on its robustness
towards distribution shift within the contexts. Notably, we observed that
GPT-4 and GPT-4V have the capability to filter irrelevant textual or visual
information via a simple few-shot prompting, respectively. Our goal is to
leverage this remarkable capability to enhance off-the-shelf LMMs that have a
smaller amount of parameters and are pre-trained on a much smaller scale of
dataset compared to GPT-4 and GPT-4V.

One step further, we investigate whether modifying the hijacked image and text
contexts with the correlated ones can help yield better coherent responses.
Firstly, we instructed GPT-4 to identify any irrelevant sentences and replace
them with appropriate alternative sentences that convey coherency given in the
contexts. Subsequently, we prompted text-to-image diffusion models 111We
employed DALLE-3 [14] for its detailed sensitiveness to the input prompts to
generate a coherent image corresponding to the newly replaced sentence under
consideration for the original sequence of sentences and images. As a result,
a new alternative image and caption that are coherent with the original story
plot are interleaved into inputs for LMMs. Notably, we observed that LMMs
still exhibit hijacked responses, due to two possible reasons: 1) the
generated captions are still confusing which seem to be the combination and
mixture between the original and hijacked context. 2) the generated images are
not realistic and do not have the consistent texture or styles with the
original image sequences.

We showcase our problem statement and aforementioned methods and
investigations with qualitative examples one by one222We tested with FROMAGe
[9] as the baseline of LMM.. In summary, our key contributions are outlined as
follows:

  * •

Identifying a new problem of off-the-shelf LMMs, context hijacking, that can
deteriorate reliable usage of LMMs.

  * •

Propose a straightforward remedy for suppressing hijacked contexts via the
enormous foundation models.

  * •

Investigate an open question of whether replacing hijacked contexts can help
to produce coherent responses, promoting a new future research frontier.

##  2 Related Works

### Large language models.

Recently, large language models (LLMs) have gained considerable versatility
based on enlarging the scale of pre-training datasets [15, 16, 3]. Along with
the development of human-aligned feedbacks [17], LLMs have further been well-
suited for conforming to the instruction rather than predicting simple
meaningless next tokens and words. These LLMs have also demonstrated
versatility on various language domain tasks including in-context learning
[16, 18] and long-term dialogues [19, 20, 21].

### Large multi-modal models.

Despite the compelling performance of LLMs on language tasks, LLMs could not
understand the visual contexts in an image. To bootstrap LLMs with vision
modality, several works [10, 11, 9] employed an additional pre-trained visual
encoder and leveraged a few linear layers to adapt the visual tokens to be
understandable by the frozen LLMs. FROMAGe [9] further demonstrated the in-
context learning capability of LMMs where consecutive sequences of image-text
pairs are given as a context to respond to the query image. In this paper, we
focus on the vulnerability of LMMs to incoherent and irrelevant context
information.

### Hijacking context in language and vision tasks.

In the language domain, several works empirically observed that LLMs are
easily hijacked by irrelevant context information [22], small perturbations of
characters ($e.g.,$ from "film" to "fi1m") [23], and adversarial addition of
suffix words [24]. In the vision domain, there have been several approaches
that investigated the adversarial robustness of LMMs [25, 26] where a small
adversarial tweak of pixel values significantly misled LMMs to output violent
content. To the best of our knowledge, our work is the first to investigate
hijacking contexts in an in-context learning scenario where the consecutive
sequence of images and texts along with irrelevant ones are given as the
context to provide answers about the query image.

##  3 Preliminaries

### Large Language Models.

An autoregressive LLM $p_{\theta}$ is pre-trained on a large corpus of text
tokens with the next-token prediction objective. Formally, given a text of
image caption $y$ and its sequence of tokens ($\bm{s}=s_{1},...,s_{t-1}$)
separated by a BPE tokenizer [27], the model is trained to maximize log-
likelihood for the next token $s_{t}$:

$\displaystyle\log p_{\theta}(y)=\sum_{t=1}^{t}\log p_{\theta}(s_{t}|\bm{s})$

### Large Multi-modal Models.

To bootstrap LLMs with vision modality, visual embeddings $V_{\phi}(x)$ from
input image $x$ are firstly extracted by a visual encoder model $V_{\phi}$
pre-trained on vision tasks. These visual embeddings are mapped into the same
dimension of text embedding space by applying several linear layers $W$. These
projected visual embeddings, $V_{\phi}(x)^{T}W$, are prepended to the text
tokens $s$ and trained to be understandable by the frozen LLMs, letting LLM
interpret the image and answer the corresponding captions or instructions.
Formally, the image-captioning objective given an image $x$ and a caption $y$
is defined as follows:

$\displaystyle l(x,y)=\sum_{t=1}^{t}\log
p_{\theta}(s_{t}|V_{\phi}(x)^{T}W,\bm{s})$

For in-context learning scenarios where multiple images and corresponding text
descriptions are given, FROMAGe [9] is trained with a random concatenation
strategy where multiple image-text pairs are piled for attending and
responding more explicitly to each image. Hence, FROMAGe exhibited superior
capability for answering the query image $x_{n}$ given former visual contexts
($x_{1},\ldots,x_{n-1}$) and textual contexts
$(\bm{s}_{1},\ldots,\bm{s}_{n-1})$:

$\displaystyle\sum_{n=1}^{n}\log
p_{\theta}(s_{n}|V_{\phi}(x_{1})^{T}W,\bm{s}_{1},\ldots,V_{\phi}(x_{n-1})^{T}W,\bm{s}_{n-1})$

However, when the contexts in the former images ($x_{1},\ldots,x_{n-1}$) or
texts ($\bm{s}_{1},\ldots,\bm{s}_{n-1}$) are composing an incoherent visual
story due to the existence of some irrelevant contexts $(x^{*},\bm{s}^{*})$,
the output response tends to be biased towards the irrelevant context, as
shown in Figure 1.

##  4 Method

### Filtering hijacked contexts.

To address the context hijacking issue, we first propose a simple yet
effective method that can preemptively remove hijacked visual or textual
contexts. We observed that GPT-4 and GPT-4V have the capability to filter
irrelevant textual or visual contexts via a simple few-shot prompting, based
on the powerful zero-shot performance and its robustness towards distribution
shift within the contexts. As in Figure 2, we labeled the sequence number for
each image for GPT-4 to understand both the image and text pair
correspondingly. Then, we prompted GPT-4 to remove any irrelevant images or
textual contexts based on the story plot. GPT-4 appropriately identifies the
hijacked contexts (sequence number 5) and hence these filtered outputs will
prevent LMMs (such as FROMAGe) from yielding hijacked responses.

Figure 2: Filtering hijacked contexts via GPT-4(V). Given multiple image-text
pairs of a visual story plot from VIST dataset [12], GPT-4 is instructed to
filter any irrelevant images or text descriptions based on the context.

### Reforming hijacked contexts

We further investigate whether reforming the identified irrelevant visual and
textual contexts into coherent ones can help LMMs yield better coherent
responses. As in Figure 3, to reform the hijacked text description, we first
instructed GPT-4 to provide appropriate sentences that convey coherency given
in the contexts. Subsequently, we instructed DALLE-3 to generate a synthetic
image that corresponds to the newly reformed text, with the consideration of
underlying visual and textual context information. Consequently, the arranged
sequence of visual and textual contexts with the newly reformed one might help
to provide a more sufficient and coherent context for LMMs not to be confused.

Figure 3: Reforming hijacked text and images via GPT-4 and DALLE-3. Given
multiple image-text pairs of a visual story plot from VIST dataset [12], GPT-4
is instructed to replace any irrelevant images or text descriptions with
coherent ones. Subsequently, DALLE-3 is instructed to generate an image
corresponding to the newly reformed text, under consideration of the
underlying context.

##  5 Discussions

### Location of hijacked context.

We investigate the effect of the sequential location of hijacked context on
the coherency of the response from LMMs. In Figure 4, when the hijacking
context is interleaved in the former part, the final response still tends to
adhere to the original context, "family". In contrast, as the hijacking
context is interleaved closer to the query image, the response tends to be
incoherent to the former contexts and biased towards the hijacked context,
"football".

Figure 4: Effect of location for the hijacking context. We ablated to insert
the hijacking context in between the sequences 1$\sim$5 and visualize the
LMM’s response with regard to the query image.

### Robustness of GPT-4V.

We further tested GPT-4V whether it is robust to such context-hijacking
issues. In Figure 4, we inserted hijacked context between sequences 4 and 5
and instructed GPT-4V to answer the caption corresponding to the query image.
Notably, GPT-4V’s response was not biased towards the hijacked context, rather
sticking to the original context:

Playtime continues with Auntie Marie, as she supervises the little adventurers
on their indoor swing set, making memories to cherish.

This result is indicative of the robustness of GPT-4V toward the distribution
shift, shedding light on a future direction of explicitly distilling the
robust knowledge from GPT-4V.

### Limited impact of reforming hijacked contexts.

In Figure 5, we observed that our reforming strategy introduced in Section 4
still does not ensure LMMs to give coherent responses. We conjecture that the
reformed captions still contain the hijacked context ("football") while also
adhering to the original context ("family"). The generated image also
correspondingly contains both visual contexts of "family" and "football",
confusing LMMs to produce biased responses. Also, we conjecture that the style
and texture of the reformed image differ to the original images, which might
also induce LMMs to be confused.

Figure 5: Effect of reformed context on the LMM response. We replaced the
hijacked context with the one reformed by GPT-4 and DALLE-3.

##  6 Limitations

The limitations of this work are that we provide only qualitative example
results on the VIST dataset since quantitative evaluation is too labor-
demanding; the generated texts from LMMs require a human to check if they are
aligned with the image and coherent with the story plot. Therefore, we leave
thorough and rigorous quantitative evaluation using various LMMs as future
work. Also, our proposed scheme is a straightforward solution yet fragile when
the irrelevant contexts become the majority within the entire context where
even GPT-4 would also be confused. Therefore, developing a more secure way of
filtering or replacing a bunch of the hijacked contexts is a promising future
research direction.

##  7 Conclusion

In conclusion, we newly identified the context-hijacking problem of existing
LMM where the irrelevant visual and textual contexts induce the LMM to
generate a biased and incoherent response, disregarding the original context.
To mitigate this issue, we introduced a simple filtering technique to
preemptively remove such hijacked contexts based on the versatile foundation
models. We further investigated reforming the hijacked contexts into coherent
ones, which promotes future research direction to generate more coherent
visual and textual information.

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# New Approximate Multiplier for Low Power
Digital Signal Processing

Farzad Farshchi, Muhammad Saeed Abrishami, and Sied Mehdi Fakhraie  School of
Electrical and Computer Engineering
University of Tehran
Tehran, Iran
{f.farshchi, msabrishami<EMAIL_ADDRESS>
###### Abstract

In this paper a low power multiplier is proposed. The proposed multiplier
utilizes Broken-Array Multiplier approximation method on the conventional
modified Booth multiplier. This method reduces the total power consumption of
multiplier up to 58% at the cost of a small decrease in output accuracy. The
proposed multiplier is compared with other approximate multipliers in terms of
power consumption and accuracy. Furthermore, to have a better evaluation of
the proposed multiplier efficiency, it has been used in designing a 30-tap
low-pass FIR filter and the power consumption and accuracy are compared with
that of a filter with conventional booth multipliers. The simulation results
show a 17.1% power reduction at the cost of only 0.4dB decrease in the output
SNR.

###### Index Terms:

Approximate computimg, low power, DSP systems, FIR filter, inaccurate hardware
units

##  I Introduction

Power consumption is one of the most important characteristics of any
electronic device especially for battery powered hand-held devices. A lot of
efforts have been put into reducing power consumption of systems at different
design levels.

One of the favorite techniques for power reduction is trading accuracy for
power consumption. Different designs have been proposed in this regard. One of
these approaches is using approximate computing in applications showing
inherent error resilience. Some DSP, multimedia, fuzzy logic, neural networks,
wireless communications, recognition, and data mining algorithms are examples
of such applications [1], [2].

Approximation can be performed using different techniques such as allowing
some timing violations (e.g., voltage over-scaling or over-clocking) and
function approximation techniques (e.g., modifying the Boolean function of a
circuit) or a mixture [2]. Reference [1] proposed an approximate adder and an
approximate multiplier based on a technique named Broken-Array Multiplier
(BAM) and demonstrated their benefits in terms of delay and area when
exploited to implement a face recognition neural network and defuzzification
block of a fuzzy processor. In [3], another approximate multiplier was
proposed. It consisted of some $2\times 2$ inaccurate building blocks and
could save power between 31.8% and 45.4% over an accurate multiplier. The
proposed multiplier was used to filter an image. The approximate filter saved
power by 41.5% over an accurate one and achieved a Signal to Noise Ratio (SNR)
of 20.4dB. Reference [4] designed an approximate signed 32-bit multiplier for
speculation purposes in pipelined processors. The multiplier is 20% faster,
with a probability of error around 14%. In [5], Error Tolerant Multiplier
(ETM) was introduced. It computed the approximate result by dividing
multiplication into one accurate and one approximate part. Accuracy for
various bit-width multipliers was reported. Power saving of more than 50% was
reported for a 12-bit multiplier. The authors did not demonstrate any
application for their design. The authors in [6] proposed the Error Tolerant
Adder (ETA) by dividing the operation into precise and approximate parts and
proposed a new circuit for the approximate part. They improved Power-Delay
Product (PDP) more than 65% comparing to conventional adders. An FFT processor
was implemented with ETA to compare the quality reduction of the output and
results showed the output quality reduction. A quantitative criterion on the
quality loss and power saving of the whole system was not reported. Three
forms of approximate Full Adders (FAs) were introduced in [7]. These FAs were
used to build adders of a DCT-IDCT processor in image compression
applications. The proposed approximate blocks improved the power consumption
of the system by about 50% while at the same time leaded to about 6dB Peak
Signal to Noise Ratio (PSNR) reduction. In [8], a Java extension for a
compiler was proposed to map some parts of a code to approximate hardware, so
that less power is consumed. An approximate hardware architecture was also
introduced. The authors in [9] introduced meta-functions that behave
gracefully under voltage over scaling. These meta-functions construct the main
parts of some multimedia, recognition, and data mining algorithms. Reference
[2] proposed a methodology for modeling and analysis of circuits for
approximate computing. This method can be used to analyze how an approximate
circuit behaves with reference to an accurate implementation.

Most of the previous work utilized inefficient arithmetic units for applying
their approximation techniques which obviously leads to inefficient
approximate arithmetic units. In addition, multipliers have the greatest share
of arithmetic unit power consumption in most DSP systems, but many of the
previous works focused on the less power consuming units like adders
neglecting the total system power reduction. In this paper, an approximate
modified Booth multiplier is proposed. The approximation technique is based on
BAM [1]. Due to better efficiency of modified Booth multiplier comparing to
other multipliers, it is expected that the approximate version is also more
efficient comparing to other approximate multipliers. To examine the proposed
multiplier, it is utilized in design of a low-pass 30-tap FIR filter. The
results of implementation are compared with filters built out of accurate
multipliers with different Word Lengths (WLs).

Rest of this paper is organized as follows. In Section II, the proposed
Broken-Booth Multiplier is introduced and the output error is analyzed.
Section III shows the synthesis and simulation results and compares the
results with other multipliers. Section IV concludes this paper.

##  II Broken-Booth Multiplier

In this section we introduce the approximation algorithm and discuss the
statistical parameters of the output error. The evaluation method of the
proposed multiplier is also discussed in this section.

###  II-A Approximation Algorithm

Fig. 1 shows the dot diagram notation of the proposed approximate modified
Booth signed multiplier [10]. Every row demonstrates one of the Partial
Products (PPs) of the multiplication. In this approximation method, all the
dot products positioned at the right hand side of the Vertical Breaking Level
(VBL) are replaced by zero. In modified Booth algorithm, 2’s complements of
some of the PPs are required. This means, complementing the PP then adding one
to it. In this figure, ’S’ will be one if the 2’s complement is required and
otherwise it will be zero. According to this, two breaking algorithms are
possible. Fig. 1 (a) shows the first possible method. In this method, which we
will call it Broken-Booth Multiplier Type0 through the rest of this paper, the
required PPs are complemented and added with one then breaking procedure is
applied. Fig. 1 (b) shows the second possible method. We will call this method
Broken-Booth Multiplier Type1 through the rest of this paper. In this method
the required PPs are complemented but are not added with one at this stage.
The breaking procedure is applied after this stage and then the result is
added with one if that one is not replaced by zero during the breakage. In
both methods after this stage the PPs are added according to their positions.
Complementing a row needs an increment operation, therefore nullifying some
sign bits Type1 results in less increment operations, thus more power saving.
The weakness of this method is higher inaccuracy penalty in comparison with
Type0.

Figure 1: The Broken-Booth Multiplier Type0 (a) and Type1 (b) for
$\mathit{WL}=12$ and $\mathit{VBL}=7$. Adopted from [10].

###  II-B Statistical Parameters of the Output Error

The statistical parameters of the output error of the Broken-Booth Multiplier
Type0 with WL of 12, are reflected in Table I for different VBLs. Throughout
this paper we use the term “error” instead of “output error”. In this table,
the error is calculated according to Eq. (1) and Mean Squared Error (MSE) to
Eq. (2).

$error=approximate\>output-accurate\>output.$ (1)

TABLE I: MSE, Error Mean and Probability and Minimum Error of the Broken-Booth Multiplier Type0 with WL = 12. | Error Mean | MSE | Error Prob. | Min-Error
---|---|---|---|---
VBL = 3 | -3.50 | 2.22$\times 10^{1}$ | 0.6875 | -1.10$\times 10^{1}$
VBL = 6 | -6.15$\times 10^{1}$ | 5.05$\times 10^{3}$ | 0.9375 | -1.71$\times 10^{2}$
VBL = 9 | -7.89$\times 10^{2}$ | 7.52$\times 10^{5}$ | 0.9893 | -2.22$\times 10^{3}$
VBL = 12 | -8.53$\times 10^{3}$ | 8.33$\times 10^{7}$ | 0.9983 | -2.32$\times 10^{4}$

To obtain these parameters, the arithmetic behavior of the multiplier is
modeled and in a simulation environment, all the possible input vectors are
exhaustively applied to it. For example, error percentage distribution of the
Broken-Booth multiplier with WL = 10 and VBL = 9 is shown in Fig. 2. It should
be noticed that in this figure the error is normalized to $2^{19}$ which is
the maximum possible output of a $10\times 10$ signed multiplier.

Figure 2: The percentage of error distribution of Broken-Booth Multiplier
Type0 for $\mathit{WL}=10$ and $\mathit{VBL}=9$.

In [11], in order to analyze the quantization error induced on the output of a
DSP system, an analytic method is described. In this method the quantization
error is assumed to be a white noise and as a result a power level is defined
for it. We have evaluated the output error of our proposed multiplier and
compare it to the previous work, based on this suggestion. Therefore, the most
important parameter reported in Table I is MSE which is calculated using Eq
(2).

$\mathit{MSE}=\frac{1}{N}\sum_{i=0}^{N-1}error^{2}(i).$ (2)

In this equation, i is the representative of input vector number and $N$ is
the number of applied input vectors which is equal to $2^{24}$ for a $12\times
12$ multiplier.

As seen in Table I, all the error parameters increase proportional to VBL. A
similar trend exists for other word lengths.

###  II-C Evaluation Method

To evaluate the proposed multiplier and also make an analogy between the
previous designs and the proposed one, the hardware related parameters such as
delay, area, and power consumption should be extracted. To do that, a
parametric Verilog description of the design is developed. In this model,
setting the VBL to 0 will result in an accurate version of the multiplier.
Moreover, PPs are generated based on modified Booth algorithm and the
summation of them is described at high level description and the details of
implementation are left to synthesis tool. The design is synthesized in
standard cells of 90nm CMOS technology using Synopsys Design Compiler. To
calculate the power consumption of the synthesized circuit, the post-synthesis
simulation is employed and a Value-Change-Dump (VCD) file is extracted. The
extracted VCD file is fed into PrimeTime PX and the average total power - the
sum of dynamic and leakage power - is reported.

##  III Simulation Results

In this section, first, a comparison between the hardware characteristics of
the proposed multiplier and an accurate version is made. Next, the proposed
multiplier is compared to the previous designs in the literature; finally, as
an application an FIR filter is implemented once using an accurate multiplier
and another time with the proposed approximate multiplier. To come up with an
analogy, the output SNR and total power consumption of the filter for
different cases are compared.

###  III-A Comparison with the Accurate Booth Multiplier

At the first step, an accurate $16\times 16$ Booth multiplier is obtained by
setting the VBL to 0 in the developed Verilog model of Broken-Booth
Multiplier. Next, the model is synthesized and the minimum possible delay
(Tmin) is obtained. After that, both the approximate (Type0) and accurate
models are synthesized with timing constraints of Tmin and four different
timing constraints more relaxed than Tmin. It should be noticed that in the
approximate model, the VBL parameter is set to 15, i.e. from 32 columns, 15
are nullified. Furthermore to compare the performance of the proposed
multiplier and the accurate one, the proposed multiplier is synthesized again
for minimum possible delay. The power consumption is calculated for both
multipliers and reflected in Fig. 3 for each delay setting. The simulation is
done on synthesized models of the multipliers and in this process the circuits
are tested with 5$\times 10^{5}$ random input vectors. Since the input vectors
are generated randomly, the rate of switching activity for internal nodes is
relatively higher than applying a runtime workload. However, as the condition
is the same for all models, the comparison between power consumptions remains
valid.

Figure 3: Total power vs. delay for accurate and approximate multipliers with
input $\mathit{WL}=16$.

It can be inferred from Fig. 3 that the power consumption of the Broken-Booth
Multiplier is about half of the power consumption of the accurate one. The
power consumption of both multipliers grows suddenly as the delay reaches its
minimum value. The minimum possible delays for the accurate and Broken-Booth
multipliers are 1.21ns and 1.13ns, respectively. Therefore, the Broken-Booth
multiplier is 6.6% faster than the accurate one.

The Broken-Booth Multiplier is also compared to the accurate one in this way
for different WLs and Tables II and III demonstrate the percent of power and
area reduction of the Broken-Booth Multiplier respectively, comparing to the
accurate multiplier. As shown in Table II, the power consumption of the
Broken-Booth Multiplier is reduced by 28.4% to 58.6% and the area is reduced
by 19.7% to 41.8%. As the multipliers hardware almost halved on average, it is
expected that the area and power consumption reduce by this rate. For example,
in the case WL = 12 and VBL = 11, 36 bits out of 77 are nullified which
results in removing some parts of PP generators and PP adders, therefore we
expect that area and power consumption almost reduce by 47% ($36\div 77$).
Moreover, as the power reduction is more than the area reduction and total
capacitance is proportional to area, it is concluded that in Broken-Booth
Multiplier the switching activities of the internal nodes are also reduced.

TABLE II: Percentage of Power Reduction for Various WLs and Delay Constraints. | $1\\!\times\\!T_{min}$ | $1.25\\!\times\\!T_{min}$ | $1.5\\!\times\\!T_{min}$ | $1.75\\!\times\\!T_{min}$ | $2\\!\times\\!T_{min}$ | Mean
---|---|---|---|---|---|---
| (%) | (%) | (%) | (%) | (%) | (%)
WL=4, | 18.2 | 35.8 | 33.9 | 27.6 | 24.7 | 28.0
VBL=3
WL=8, | 44.8 | 47.7 | 58.0 | 64.2 | 66.9 | 56.3
VBL=7
WL=12, | 52.7 | 52.2 | 60.0 | 57.3 | 70.9 | 58.6
VBL=11
WL=16, | 58.1 | 52.8 | 53.2 | 59.0 | 64.0 | 57.4
VBL=15
TABLE III: Percentage of Area Reduction for Various WLs and Delay Constraints. | $1\\!\times\\!T_{min}$ | $1.25\\!\times\\!T_{min}$ | $1.5\\!\times\\!T_{min}$ | $1.75\\!\times\\!T_{min}$ | $2\\!\times\\!T_{min}$ | Mean
---|---|---|---|---|---|---
| (%) | (%) | (%) | (%) | (%) | (%)
WL=4, | 14.0 | 25.0 | 19.0 | 21.6 | 18.9 | 19.7
VBL=3
WL=8, | 45.3 | 31.6 | 28.9 | 29.9 | 31.2 | 33.4
VBL=7
WL=12, | 54.0 | 41.8 | 39.9 | 33.3 | 40.0 | 41.8
VBL=11
WL=16, | 53.8 | 42.8 | 38.3 | 37.1 | 36.0 | 41.6
VBL=15

###  III-B Comparison to Previous Designs

In order to compare the proposed multiplier with other approximate
multipliers, two approximate multipliers presented in [1] and [3], are
modeled, synthesized, and simulated in the same technology and compared to the
proposed one in terms of Power-Delay Product (PDP) and MSE. The multiplier
with lower PDP and error power is preferred. In [1], one of the proposed
methods which is named Broken-Array Multiplier (BAM) is an unsigned
approximate multiplier. In BAM, in addition to VBL, there is another parameter
called Horizontal Breaking Level (HBL) for adjusting precision and hardware
saving. In this comparison we set the HBL to 0 and only manipulate the VBL. It
should be noticed that there is no difference between BAM and its signed
counterpart, in terms of MSE. The multiplier presented in [3] is another
unsigned approximate multiplier made up from basic blocks of $2\times 2$
approximate multipliers. In [3], there is no defined parameter to adjust the
precision of multiplier. Hence, in our implementation, we modified the design
and defined the K parameter as illustrated in Fig. 4. In this method, an
imaginary vertical line is introduced between the PPs, and the blocks
positioned entirely on the right hand side of this line are replaced by
approximate blocks and K controls the position of this line. In fact, the K
parameter acts so alike to VBL in our proposed method and enhances the
versatility of the design in [3]. To compare the multipliers, the Verilog
descriptions of both models are developed in a parametric manner.

Figure 4: The PP diagram of multiplier [3] with our added parameter K for WL =
6. Blocks in gray are approximate.

To calculate the PDP over MSE, the following procedure is taken:

  1. 1.

Using the method introduced in Section II.A, the MSE of the multipliers is
calculated over five different precision settings.

  2. 2.

All multipliers with each precision setting are synthesized for minimum delay;
the power consumption and PDP of each synthesis result is calculated
afterwards.

  3. 3.

The synthesis procedure is repeated once again with timing constraint of
1.75ns. In this step the PDP would be the product of calculated power
consumption and 1.75ns.

  4. 4.

The average PDP is calculated from the results of steps 2 and 3.

Fig. 5 shows the different PDPs as calculated in steps 2 through 4 over the
MSE and adjusting parameter, corresponding to each of them. It shows that the
variation of PDP over MSE is different for the steps 2 and 3.

Figure 5: PDP vs. MSE logarithm for studied multipliers, Broken-Booth
Multiplier Type0 (a), Type1 (b), BAM (c), and The multiplier in [3] (d).

In Fig. 6 the calculated average PDP of each multiplier is depicted over MSE
in a single diagram. The multiplier in [3] has the best PDP at lower MSE but
as the error power increases, it does not show any PDP improvement. The
Broken-Booth Multipliers Type0 and Type1 have better PDP for high MSE values
comparing to [3] and the PDP of them decreases almost steadily as the MSE
grows. The reduction of PDP for Type0 is more graceful than Type1. The reason
could be ability of synthesis tool to optimize the implied circuits in step 3.

Figure 6: Average PDP vs. MSE logarithm for studied multipliers.

###  III-C FIR Filter

The application we have used is a low-pass FIR filter which is introduced in
[12]. Fig. 7 shows the block diagram of the testbed, frequency response of the
filter $H(\omega)$, and test signals $d_{i}(\omega)$. The testbed is designed
based on using the filter in a real situation. The bandwidth and guard
bandwidth of signals $d_{i}[n]$ are 0.25$\pi$ and 0.1$\pi$, respectively. The
FIR filter input is the sum of signals $d_{i}[n]$ in the presence of white
Gaussian noise source with -30dB power spectral density, $\eta[n]$. The
$d_{2}[n]$ and $d_{3}[n]$ signals are located on transition and stop bands
respectively. The desired signal is $d_{1}[n]$ which is located on pass band
region. The SNR at the output and input of filter is defined as,
SNR${}_{out}=10\log_{10}\frac{\sigma^{2}_{d_{1}}}{\sigma^{2}_{d_{1}-y}}$ and
SNR${}_{in}=10\log_{10}\frac{\sigma^{2}_{d_{1}}}{\sigma^{2}_{d_{1}-x}}$,
respectively, where $\sigma^{2}_{d_{1}-y}=E[|d_{1}-y|^{2}]$ and
$\sigma^{2}_{d_{1}-x}=E[|d_{1}-x|^{2}]$.

Figure 7: Testbed for simulating the FIR filter (a) frequency response of the
filter and input signals (b) [12].

A 30-tap order Parks-McClellan low-pass filter is modeled with double
precision arithmetic. Simulation results show SNRout = 25.7dB and SNRin =
-3.47dB. It means this filter increases SNRout up to 29.1dB in comparison with
SNRin. Next, a fixed-point filter is modeled with variable WL. Fig. 8 (a)
shows SNRout for different WLs. Since Booth multipliers are optimum for even
WLs, simulations are done based on even WLs. For the sake of efficient
hardware implantation, the least possible WL should be chosen. Therefore, we
choose WL = 16 and SNRout = 25.4dB, as is implied from Fig. 8 (a) that lower
WLs lead to significant SNRout reduction.

Figure 8: SNRout vs. WL (a) SNRout vs. VBL (b).

Next, the Broken-Booth Multiplier Type0 is used as filter’s multipliers. Fig.
8 (b) illustrates SNRout for different VBLs. It is seen that increasing VBL,
leads to steady SNRout reduction. The desired operating point is defined by
VBL = 13 and SNRout = 25dB for realization of the filter using Broken-Booth
Multiplier, as higher VBL values leads to significant SNRout reduction. In
order to demonstrate hardware reduction, the filter is modeled in Verilog with
parametric WL and VBL. The model is synthesized for 3 cases:

  1. 1.

WL = 16, VBL = 0,

  2. 2.

WL = 16, VBL = 13,

  3. 3.

WL = 14, VBL = 0.

The case 3 is considered to evaluate further WL reduction effects on filter
hardware and comparing with using Broken-Booth Multiplier at higher WL.
Synthesis results, power consumption, and SNRout are reported in Table IV. The
value of QUAP in this table is introduced in [7] and defined as:

$\displaystyle\mathit{QUAP}=$ $\displaystyle QUality\times
Area\>savings\>(\%)\times Power\>savings\>(\%)$ (3)

TABLE IV: Synthesis Results, QUAP of FIR Filter for 3 Different Implemeted Cases. Power Reduction Is Measured with Respect to Case 1. Case | SNRout | Clock | Area | Power | Power | QUAP
---|---|---|---|---|---|---
(dB) | Period (ns) | ($\mu m^{2}$) | (mW) | Reduction (%) | $\div 10^{4}$
WL = 16, | 25.35 | 4.78 | 1.22$\times 10^{5}$ | 3.63 | N.A. | N.A.
VBL = 0
WL = 16, | 25.0 | 4.78 | 1.07$\times 10^{5}$ | 3.01 | 17.1 | 13.1
VBL = 13
WL = 14, | 23.1 | 4.78 | 1.13$\times 10^{5}$ | 2.91 | 19.8 | 7.73
VBL = 0

The quality is assumed to be (SNRout)2, as did the reference [7]. Since in Eq.
3, power saving and area saving are both variables that related to hardware
characteristics, SNRout should increase by the power of 2, to give an equal
weight to quality. As seen in Table IV, the implementation which utilizes
Broken-Booth Multiplier reduces power consumption by 17.1% at the cost of
0.4dB SNR reduction and comparing to case number 3, it improves QUAP by 70%.

##  IV Conclusion

In this paper an approximate signed Booth multiplier is proposed which saves
power consumption form 28% to 58.6% and area from 19.7% to 41.8% for different
word lengths in comparison to a regular Booth multiplier. The Mean Squared
Error (MSE) introduced by this approximate multiplier varies with word length
and approximation level, from 0.25 to 8.33$\times 10^{7}$. To compare the
proposed approximate multiplier with two previous works in the literature,
Verilog models were prepared and synthesized and then compared using the
average power-delay product and MSE criteria. The proposed multiplier shows a
reasonable and acceptable performance. To demonstrate an application for the
proposed multiplier, an FIR filter was implemented once utilizing accurate
multiplier and again using the proposed multiplier. A previously introduced
figure of merit (FOM) was used to compare the performance of these versions of
the filter implementation. The filter which is using our multiplier has the
best FOM.

## References

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# Neutron oscillation and Baryogenesis from six dimensions

Mathew Thomas Arun1, Debajyoti Choudhury2 1School of Physics, IISER
Thiruvananthapuram, Vithura, Kerala, 695551, India
2Department of Physics and Astrophysics, University of Delhi, Delhi 110005,
India

(January 2022)

###### Abstract

Considering a six-dimensional geometry orbifolded on $S^{1}/Z_{2}\times
S^{1}/Z_{2}$ with quarks and leptons localised on orthogonal branes, we show
that the construction admits observable $n-\bar{n}$ oscillation while
naturally suppressing the proton decay rates. Consistent with other low-energy
observables, the model also accommodates baryogenesis at $\mathcal{O}$(10 TeV)
scale.

Baryon number ($B$) violation, a key ingredient for generating the observed
baryon to photon ratio in the universe ($n_{B}/n_{\gamma}=(6.19\pm 0.14)\times
10^{-10}$), is allowed in the Standard Model (SM) only through non-
perturbative processes ’t Hooft (1976a); *tHooft:1976snw; *Kuzmin:1985mm;
*Dolgov:1991fr. Presumably important for baryogenesis at temperatures near the
electro-weak scale Shaposhnikov (1987); *Babu:2006xc; *Babu:2006wz;
*Babu:2008rq; *Morrissey:2012db; *Dev:2015uca, such processes are highly
suppressed at low temperatures, and $B$ is seemingly a good global symmetry.
With $\cancel{B}$ and CP violation within the SM not being large enough to
generate the $n_{B}/n_{\gamma}$, other sources must be explored.

The simplest gauge invariant and $\cancel{B}$ effective operator also violates
lepton number ($L$) by one unit, and non-observation of proton decay pushes
the corresponding scale to well above $10^{15}$ GeV. On the other hand, if the
leading operator were a $\Delta B=2$ one, the scale could be much lower and be
probed by looking for either $n-\bar{n}$ oscillation in nuclei, or
annihilations brought about in collisions of cold neutrons against a target.
The characteristic time scales are related through $T_{\rm nucl}=\tau^{2}_{\rm
free}R$, where $R$ characterises the strong interaction $\bar{n}$ annihilation
time, and is, typically, ${\cal O}(100\,{\rm MeV})$ Friedman and Gal (2008).
Using different nuclei, experiments at SOUDAN-II (Fe56) Chung  _et al._
(2002), Super-Kamiokande (O16) Abe  _et al._ (2015) and SNO (deuteron)
Bellerive  _et al._ (2016) have constrained $\tau_{\rm free}$ to be larger
than $1.3\times 10^{8}$s, $2.7\times 10^{8}$s and $1.23\times 10^{8}$s
respectively, each at 90% C.L. Relating to the underlying theory is best
doneMohapatra and Marshak (1980); *Ozer:1982qh; *Kuo:1980ew by considering the
matrix element of the effective six-quark (dimension-9) operator viz. $\Delta
m\equiv\tau_{\rm free}^{-1}=\langle\bar{n}|\mathcal{O}_{9}|n\rangle$. The
transition probability for pure state $|n\,;t=0\rangle$ to evolve to
$|\bar{n}\,;t\rangle$ is given by $P(t)=(t/\tau_{\rm free})^{2}e^{-\lambda t}$
where $\lambda^{-1}=880\,s$ is the mean life of a free neutron. The bound on
$\tau_{\rm free}$ implies $\Delta m\lesssim 6\times 10^{-33}$ GeV, or, for an
${\cal O}(1)$ Wilson coefficient, a new physics scale ($\gtrsim 500$ TeV
Arnold  _et al._ (2013)) much lower than the proton decay scale. A roadblock
to a UV-complete model for $n-\bar{n}$ oscillation is that it, generically,
needs two new fields with gauge symmetry allowing one of these to couple to a
$\Delta L=1$ current as well, thereby requiring an unnatural suppression for
the said coupling.

In our quest for a well-motivated scenario that naturally circumvents all such
constraints, we propose a six-dimensional space-time orbifolded on
$S^{1}/Z_{2}\times S^{1}/Z_{2}$ Choudhury and SenGupta (2007); Arun and
Choudhury (2017); Arun  _et al._ (2017) and a highly-warped $x_{5}-$direction.
With quarks and leptons localised on orthogonal branes, the geometry supports
substantial $\Delta\mathcal{B}=2$ while evading proton-decay constraints
without any hierarchy/un-naturalness in the parameters.

With successive warpings along the two compactified dimensions,
($x_{4}\in[0,\pi R_{y}]$ and $x_{5}\in[0,\pi r_{z}]$) that are individually
$Z_{2}$-orbifolded with 4-branes sitting at each of the edges, the geometry is
described by the line element Choudhury and SenGupta (2007)

$ds^{2}_{6}=b^{2}(x_{5})[a^{2}(x_{4})\eta_{\mu\nu}dx^{\mu}dx^{\nu}+dx_{4}^{2}]+dx_{5}^{2}\
,$

where $\eta_{\mu\nu}$ is the flat metric. Denoting the fundamental scale in
six dimensions by $M_{6}$ and the negative bulk cosmological constant by
$\Lambda_{6}$, the total bulk-brane Lagrangian is, thus,

$\begin{array}[]{rcl}{\cal
L}&=&\displaystyle\sqrt{-g_{6}}\,(M_{6}^{4}R_{6}-\Lambda_{6})\\\\[4.30554pt]
&+&\displaystyle\sqrt{-g_{5}}\,[V_{1}(x_{5})\,\delta(x_{4})+V_{2}(x_{5})\,\delta(x_{4}-\pi
R_{y})]\\\\[4.30554pt]
&+&\displaystyle\sqrt{-\tilde{g}_{5}}\,[V_{3}(x_{4})\,\delta(x_{5})+V_{4}(x_{4})\,\delta(x_{5}-\pi
r_{z})]\,.\end{array}$

The five-dimensional metrics ($g_{5},\tilde{g}_{5}$) are those induced on the
appropriate 4-branes, and the brane potentials $V_{i}$ encode the Israel
junction conditions.

Generalizing from the restrictive case studied in Ref.Choudhury and SenGupta
(2007), we admit a five-dimensional induced non-zero cosmological constant
$\widetilde{\Omega}<0$ on the 4-branes, allowing these to be bent. The four-
dimensional cosmological constant, though, is held to be zero (the generic
case Arun and Choudhury (2017) only leads to algebraic complications). With
this, the Einstein equations lead to

$b(x_{5})=b_{1}\cosh(k|x_{5}|+b_{2})\ ,\,b_{1}={\rm sech}(k\pi r_{z}+b_{2})\
,\qquad$ (1)

with $k=\sqrt{-\Lambda_{6}/10\,M_{6}^{4}}$. Similarly,

$a(x_{4})=\exp(-c|x_{4}|)\,,\text{where}\ \ c\equiv kb_{1}R_{y}/r_{z}\,.$ (2)

Unlike ref.Choudhury and SenGupta (2007), we consider a very large $b_{2}$
Arun and Choudhury (2017); Arun  _et al._ (2017) leading to
$b(x_{5})\propto\exp(k|x_{5}|)$, and one is forced to $c\ll k$, unless a
large, and unpleasant, hierarchy between $R_{y}$ and $r_{z}$ is to be
admitted. This also implies that the 4-brane tensions are pairwise almost
equal and opposite ( $V_{1}=-V_{2}\approx\displaystyle 0$ and
$V_{3}=-V_{4}\approx\displaystyle-8M^{4}k$) thereby ensuring the near
vanishing of the induced cosmological constant on the 4-branes at the ends of
the world. In this limit, the metric is nearly conformally flat, and, along
with the $AdS_{6}$ bulk, resembles a generalization of the Randall-Sundrum
geometry to one dimension higher.

In this space-time, the Lagrangian for a free massless fermion is given by

$\begin{array}[]{rcl}{\cal L}_{\rm
fermion}&=&\displaystyle\frac{1}{2}b^{4}(x_{5})\,\Big{(}\bar{Q}\,i\Gamma^{a}\,E^{M}_{a}\,D_{M}\,Q\
\Big{)}\overline{\Delta}(x_{4})\\\\[4.30554pt] &+&\displaystyle
b^{5}(x_{5})\,\Big{(}\bar{L}\,i\Gamma^{a}\,E^{M}_{a}\,D_{M}\,L\
\Big{)}\delta(x_{5})\\\\[4.30554pt]
\displaystyle\overline{\Delta}(x_{4})&\equiv&\displaystyle\delta(x_{4})+\delta(x_{4}-\pi
R_{y})\end{array}$

where $Q\,(L)$ denote the quark (lepton) fields and $E^{M}_{a}$ are the
appropriate fünfbeins. The low-energy phenomenology is, of course, dictated by
only the zero-modes of the fermions. Since the $x_{4}-$direction is nearly
flat, so would be the wave profile of the leptonic zero mode. On the other
hand, the quark zero mode is given, to a very good approximation, by an
exponential function.

To induce $\cancel{\cal B}$, we begin by introducing two colored scalar fields
$\phi(3,1,-1/3)$ and $\omega(6,1,2/3)$, localised, of course, on the two
4-branes at $x_{4}=0$ and $x_{4}=\pi R_{y}$. With leptons being confined to
the 4-brane at $x_{5}=0$, the scalar interactions are given by

$\displaystyle\mathcal{L}_{\rm scal}$ $\displaystyle=$
$\displaystyle\displaystyle\sqrt{2r_{z}}\Big{[}y_{ud}\phi\overline{u^{c}}d+\sqrt{4r_{z}R_{y}}y_{ue}\phi^{*}\overline{u^{c}}e\
\delta(x_{5})$ (3) $\displaystyle+$ $\displaystyle\displaystyle
z_{dd}\omega\overline{d^{c}}d+\lambda
M\phi^{2}\omega\Big{]}\overline{\Delta}(x_{4})+h.c.\,.$

All fermions here are $SU(2)$ singlets and we have included just the relevant
term of the potential. Introduced for convenience, the scale $M$ is equated to
the mass of the heaviest scalar field. Since, post compactification, we would
be interested only in the lightest KK-modes, the five dimensional fields
(${\cal F}=u,d,\phi,\omega$) could be decomposed as ${\cal
F}(x_{\mu},x_{5})=(2r_{z})^{-1/2}{\cal F}(x_{\mu})\,\chi_{\cal F}(x_{5})$,
where the zero-mode wavefunctions $\chi_{\cal F}(x_{5})$ satisfy the
normalization condition $\int dx_{5}b^{s}\chi^{2}_{\cal F}=1$, with $(s=2,3)$
for scalars and fermions respectively, yielding canonically normalized four-
dimensional fields. In contrast, thanks to the $x_{4}$-direction being nearly
flat, the wavefunction for the lepton zero-mode is a trivial one. To obtain
the effective four-dimensional theory, one needs to integrate over both
$x_{5}$ and $x_{4}$ (given the smallness of the warping $c$, the second
integration is essentially trivial), resulting in

$\begin{array}[]{rcl}\mathcal{L}_{int}&=&\displaystyle\eta_{ud}\phi\overline{u^{c}}d+\eta_{ue}\phi^{*}\overline{u^{c}}e+\zeta_{dd}\omega^{*}\overline{d^{c}}d\\\
&+&\displaystyle\rho M\phi^{2}\omega+h.c.\ ,\end{array}$ (4)

where

$\begin{array}[]{rcl}\zeta_{dd}&=&\displaystyle(2r_{z})^{-1}z_{dd}\int
dx_{5}b^{2}\chi_{\phi}\chi_{d}^{2}\\\
\eta_{ud}&=&\displaystyle(2r_{z})^{-1}y_{ud}\int
dx_{5}b^{2}\chi_{\phi}\chi_{u}\chi_{d}\\\
\rho&=&\displaystyle(2r_{z})^{-1}\lambda\int
dx_{5}b^{2}\chi^{2}_{\phi}\chi_{\omega}\\\ \eta_{ue}&=&\displaystyle
y_{ue}b^{2}(0)\chi_{\phi}(0)\chi_{u}(0)\end{array}$ (5)

The relations above may include any possible flavour structure, a point that
we return to later. Now, for fermions with vanishing bulk masses (an excellent
approximation for the first and second generation fields), the wave profiles
would be expected to be nearly identical, and the ratio
$\zeta_{dd}/\eta_{ud}\approx z_{dd}/y_{ud}$.

The Wilson coefficients for proton-decay and neutron oscillation are now
straightforward, viz.

$C^{p}=\frac{\eta_{ud}\eta_{ue}}{m_{\phi}^{2}}\ ,\quad C^{nn}=\frac{\rho
M}{m_{\phi}^{2}m_{\omega}^{4}}\eta_{ud}^{2}\,\zeta_{dd}\,.$ (6)

Qualitative features of this result are best appreciated by assuming that all
the masses are of the same order. Despite the fact that $C^{nn}$ is fourth
order in the couplings while $C^{p}$ is only bilinear, $C^{p}\ll C^{nn}$ on
account of the factor $b^{2}(0)$ in $\eta_{ue}$. Note that this relative
suppression does not need any hierarchy, either of couplings or between
masses. Rather, it is engendered dynamically on account of the warping.

The neutron oscillation rate is determined by $|\Delta
m|=|\langle\bar{n}|H_{eff}|n\rangle|=8\xi^{2}C^{nn}/3$ where the matrix
element is computed using vacuum insertion approximation Arnold  _et al._
(2013) with $\xi$ parametrizing the reduced matrix element for the three quark
operator. Lattice computations Tsutsui  _et al._ (2004) give $\xi\approx
0.0096{}^{+6}_{-20}\,{\rm GeV}^{3}$, with this error being the main
uncertainty in the calculation.

The observed limits on $|\Delta m|$ can now be translated to constraints on
the parameter space. In view of the large dimensionality of the latter, it is
instructive to make the simplifying assumption that
$\eta_{ud}=\zeta_{dd}=\rho\equiv\kappa$ and $m_{\phi}=m_{\omega}=M$. The
saturation of the limit can then be expressed as

$\frac{|\Delta m|}{|\Delta m|_{\rm
max}}\approx\left[\frac{\kappa}{0.01}\right]^{4}\,\left[\frac{M}{15\,{\rm
TeV}}\right]^{-5}\,.$ (7)

More precise values are listed in Table 1 for a few benchmark points. It is
worth noting here that, for $k\,r_{z}\sim 8.5$ (a value that that leads to
$C^{p}\raisebox{-1.72218pt}{\hbox
to0.0pt{$\sim$\hss}}\raisebox{1.72218pt}{$<$}10^{-30}$, thereby suppressing
proton decay to rates well below the current limit) also implies that for
anarchic bulk-couplings $z_{dd},y_{ud},\lambda\sim\mathcal{O}(1)$, the
corresponding 4-dimensional ones ($\zeta_{dd},\eta_{ud},\rho$), obtained on
integrating over $x_{5}$, are all $\sim{\cal O}(10^{-2})$. Further suppression
of the effective couplings could arise in two ways: $(i)$ starting with
smaller bulk couplings or $(ii)$ localising the light quark states away from
the scalars. In contrast, the scenario with $\kappa\sim 1$ is achievable only
for a flat extra dimension, at the cost of very large scalar masses and self-
consistency of the treatment would imply that the scales of the KK-excitations
be at least as large, thereby taking the entire scenario beyond detection.

$\kappa$ | 1 | 0.1 | 0.01 | 0.001
---|---|---|---|---
$m_{\phi}({\rm TeV})=m_{\omega}=M$ | 670 | 106 | 17 | 2.6
$m_{\phi}({\rm TeV})=m_{\omega}/3=M/3$ | 345 | 55 | 9 | 1.4
Table 1: Lower limits on the mass parameter $m_{\phi}$ as a function of the
common coupling.

We now examine other possible phenomenological consequences. For light (a few
TeVs) $\phi$ or $\omega$, resonance production at the LHC is possible, leading
to a peak in the dijet invariant mass (while a lepton-jet decay is notionally
available to the $\phi$, it is highly suppressed). With the QCD background
being very large, the only hope is to concentrate on high-$p_{T}$, high
invariant-mass events. For $m_{\phi}=3TeV$ and $\eta_{ud}=10^{-2}$, the
$\phi$-production cross-section is $\lesssim 0.4$fb (and somewhat lower for
the $\omega$) even without accounting for efficiencies. In other words, even
in the most optimistic scenario, direct detection would have to wait. The only
caveat to this would be to consider a hierarchy in the scalar masses whereby
one of them could be made significantly lighter and brought into the current
reach of the LHC.

Much more important are the flavour sector observables. While $n$-$\bar{n}$
oscillation needs only the $d-d-\omega$ coupling, it is conceivable that the
$s-s-\omega$ coupling is unsupressed as well. This would result in an
effective flavour changing Hamiltonian of the form ${\cal H}^{\Delta
S=2}=m_{\phi}^{-2}(\bar{s^{c}}P_{R}s)(\bar{d}P_{L}d^{c})$, thereby
contributing to $K^{0}$–$\bar{K^{0}}$ oscillation (which, within, the SM,
proceeds through the charm-dominated box diagram). To bring this to the usual
form, one has to effect Fierz rotations Chakraverty and Choudhury (2001), both
in the Dirac space and the color space, using $(6_{c}\otimes
6_{c})_{1}=(2/3)(1_{c}\otimes 1_{c})+(1/2)(8_{c}\otimes 8_{c})_{1}$,
encapsulating both color-unsuppressed and suppressed contributions. Comparing
with the SM contribution (which saturates the observed mass difference), we
have

$\frac{Re(\zeta_{dd}\zeta_{ss}^{*})}{8m_{\omega}^{2}}\raisebox{-1.72218pt}{\hbox
to0.0pt{$\sim$\hss}}\raisebox{1.72218pt}{$<$}\frac{G_{F}^{2}M_{W}^{2}}{16\pi^{2}}\mathcal{F}^{0}\sim
10^{-7}{\rm TeV}^{-2}\,,$ (8)

where $\mathcal{F}^{0}$ includes the box-diagram computation Aoki  _et al._
(2020) (note that the hadronic matrix element calculation is common to both).
Assuming that the couplings are of similar size, for $m_{\omega}=3\,$TeV, this
translates to $|\zeta_{dd}|<0.0026$. In other words, the coupling values
required for observable neutron-oscillation rates more than easily satisfy the
bounds from kaon oscillations. This exercise, though, tells us that any
introduction of hierarchy in the scalar masses needs to be approached
carefully.

As is well-known, the creation of a baryon asymmetry needs not only
$\cancel{B}$ and $\cancel{CP}$ (much larger than that we have in the SM) but
also an accompanying epoch with out-of-equilibrium condition. To create an
environment amenable for this, let us augment the model with the inclusion of
a further copy of $\phi$ (henceforth called $\phi_{1}$) and a singlet $S$
(which, for simplicity, we consider to be localized on the same branes as the
colored scalars). Similarly, we also assume, that the $\phi$ mass matrix is
diagonal and that there is no substantial mass hierarchy between these. As for
the potential in 4, it now includes additional terms such as ($i=1,2$)

$\begin{array}[]{rcl}-V&\ni&\displaystyle
M\Big{[}\rho_{ij}\phi_{i}\phi_{j}\omega+\widetilde{\rho}_{ij}\phi_{i}^{*}\phi_{j}S\Big{]}+h.c.\,,\end{array}$
(9)

where, for the sake of simplicity, we retain only the zero-modes. While the
terms in eq.(9) could have extra factors, of $S/M$, these do not add anything
to our discussion, and we omit them.

At the tree level, we forbid any Yukawa couplings for the new field $\phi_{2}$
(this could be arranged simply by introducing a softly broken discrete
symmetry). Thus, the dominant decay available to it would be one through
$\rho_{12}$ leading to $\phi_{2}\to uddd$, or a final state of $B=4/3$. On the
other hand, $\phi_{1}\to\bar{u}\bar{d}$ ($B=-2/3$).

For $2m_{\phi}<m_{S}\raisebox{-1.72218pt}{\hbox
to0.0pt{$\sim$\hss}}\raisebox{1.72218pt}{$<$}2m_{\omega}$, the dominant decay
of the $S$ would be $S\to\phi_{i}^{*}\phi_{j}$. While the tree-level diagram
is obvious, one-loop corrections are brought about by the diagram in Fig.1.

Figure 1: One-loop diagram contributing to $B$-asymmetry generated from a
decaying $S$.

The latter evidently has a non-zero absorptive part, receiving contributions
from possibly three ways of cutting a pair of internal lines. For example, for
$m_{\phi_{1}}\sim m_{\phi_{2}}\raisebox{-1.72218pt}{\hbox
to0.0pt{$\sim$\hss}}\raisebox{1.72218pt}{$>$}m_{\omega}/2$, only the vertical
cut in Fig. 1 contributes. Since a net baryon number $\epsilon_{B}$ (per
$S$-decay) can arise only from the difference
$\Gamma(S\to\phi_{1}\phi_{2}^{*})-\Gamma(S\to\phi_{1}^{*}\phi_{2})$, we have
(denoting $x_{a}\equiv m_{a}^{2}/m_{S}^{2}$)

$\displaystyle\epsilon_{B}$ $\displaystyle\approx$
$\displaystyle\displaystyle\frac{M^{2}}{4\pi^{2}m_{S}^{2}\beta}\,\log\Big{(}\frac{x_{\omega}-x_{\phi}+1+\beta}{x_{\omega}-x_{\phi}+1-\beta}\Big{)}\;{\cal
A}_{\rho}$ $\displaystyle{\cal A}_{\rho}$ $\displaystyle\equiv$
$\displaystyle\displaystyle
Im\left[\widetilde{\rho}_{12}^{*}(\rho^{\dagger}\widetilde{\rho}\rho)_{12}\right]/\sum_{i,j}|\widetilde{\rho}_{ij}|^{2}$
(10)

where $\beta=(1-4x_{\phi})^{1/2}$ and we have neglected terms proportional to
the mass difference $\delta m_{\phi}^{2}$.

Thus, the number density $n_{S}$ of the decaying field is related to those for
the (anti-)baryons through, $n_{b}-n_{\bar{b}}=n_{S}\epsilon_{B}B_{q}$, where
$B_{q}=2$ is the baryon number created per decay. Since the photons in the
universe far outnumber the baryons, they dominate the entropy density $s$ of
the universe and we may write Cline (2006)

$\frac{n_{b}-n_{\bar{b}}}{s}=\frac{n_{S}B_{q}\epsilon_{B}}{4\pi^{2}/45g_{*}T^{3}}=\frac{180\zeta(3)}{\pi^{4}}\epsilon_{B}\
,$ (11)

where $g_{*}\approx 107$ is the weighted number of degrees of freedom
operative at that temperature.

Using Eq.11, the constraint on baryon asymmetry in the universe as,
$\epsilon_{B}=3.8\times 10^{-11}$. This could be arranged quite easily in the
model. In the parameter space where the nuetron-anti-neutron oscillation is
observable, $m_{\phi}\sim m_{\omega}=3TeV$, and $M\sim m_{S}=10TeV$, the
asymmetry required for baryogenesis is obtained for ${\cal A}_{\rho}\lesssim
2.2\times 10^{-9}$. Note that the very structure of ${\cal A}_{\rho}$ implies
cancellations between the nominal phases in the couplings. Consequently, the
residual phase is naturally smaller. Indeed, such a phase is also stipulated
by the $\epsilon^{\prime}$ parameter in kaon oscillations. To summarise, we
have investigated baryon-number violation in a six-dimensional world
compactified on $(S^{1}/Z_{2})\otimes(S^{1}/Z_{2})$ wherein one direction is
highly warped with the other being almost flat (minuscule warping). Boasting
stabilised moduli Arun and Choudhury (2017), such models are more natural than
their flat extra dimension counterparts. As we have demonstrated here,
augmenting the minimal SM extension by just two extra colored scalars allows
for low-energy $\Delta B=2$ processes, thereby offering the tantalizing
prospect of an observable $n-\bar{n}$ oscillation signal while proton decay is
dynamically suppressed (without the need for any hierarchy). While, for
natural sizes of Yukawa couplings, the masses of the scalars are just beyond
the reach of the LHC, these are likely to be visible in the next generation of
hadronic colliders or their presence inferred from more sensitive flavour
probes. Most interestingly, a minor addition to the spectrum opens the
possibility for multi-TeV scale baryogenesis. The parameter space amenable to
reproducing the observed baryon asymmetry is large and no hierarchy or
unnatural assumptions need be invoked. Furthermore, these scalars hold the
prospect of generating strong phase transitions, signals of which are likely
to be observable in the next generation of gravitational wave detectors.

M.T.A. acknowledges financial support of the DST through INSPIRE Faculty grant
[DST/INSPIRE/04/2019/002507].

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