# Adapting LLMs for Efficient Context Processing through Soft Prompt
Compression

Cangqing Wang1, Yutian Yang1
Ruisi Li2, Dan Sun2, Ruicong Cai2, Yuzhu Zhang2, Chengqian Fu3 and Lillian
Floyd∗ 1Cangqing Wang be with Boston University, MA 02215, USA
<EMAIL_ADDRESS>Yang be with University of California, Davis, CA
95616, USA<EMAIL_ADDRESS>Li is an independent researcher.
Correspondence to Ruisi Li via email<EMAIL_ADDRESS>Sun be with
Washington University in St. Louis, MO 63130, USA<EMAIL_ADDRESS>Cai is an independent researcher. Correspondence to Ruicong Cai via email:
<EMAIL_ADDRESS>Zhang be with Carnegie Mellon University,
Pittsburgh, PA 15213<EMAIL_ADDRESS>Fu is an independent
researcher. Correspondence to Chengqian Fu via email:
<EMAIL_ADDRESS>Floyd be with Georgia Institue of Technology, GA
30332, USA<EMAIL_ADDRESS>
###### Abstract

The rapid advancement of Large Language Models (LLMs) has inaugurated a
transformative epoch in natural language processing, fostering unprecedented
proficiency in text generation, comprehension, and contextual scrutiny.
Nevertheless, effectively handling extensive contexts, crucial for myriad
applications, poses a formidable obstacle owing to the intrinsic constraints
of the models’ context window sizes and the computational burdens entailed by
their operations. This investigation presents an innovative framework that
strategically tailors LLMs for streamlined context processing by harnessing
the synergies among natural language summarization, soft prompt compression,
and augmented utility preservation mechanisms. Our methodology, dubbed
SoftPromptComp, amalgamates natural language prompts extracted from
summarization methodologies with dynamically generated soft prompts to forge a
concise yet semantically robust depiction of protracted contexts. This
depiction undergoes further refinement via a weighting mechanism optimizing
information retention and utility for subsequent tasks. We substantiate that
our framework markedly diminishes computational overhead and enhances LLMs’
efficacy across various benchmarks, while upholding or even augmenting the
caliber of the produced content. By amalgamating soft prompt compression with
sophisticated summarization, SoftPromptComp confronts the dual challenges of
managing lengthy contexts and ensuring model scalability. Our findings point
towards a propitious trajectory for augmenting LLMs’ applicability and
efficiency, rendering them more versatile and pragmatic for real-world
applications. This research enriches the ongoing discourse on optimizing
language models, providing insights into the potency of soft prompts and
summarization techniques as pivotal instruments for the forthcoming generation
of NLP solutions.

###### Index Terms:

Knowledge Graph Reasoning, Reinforcement Learning, Reward Shaping, Transfer
Learning

##  I Introduction

The finite context window size inherent in most LLMs constrains their ability
to fully grasp and utilize extensive textual information, thereby limiting
their efficacy on tasks demanding profound comprehension of lengthy documents.
Additionally, the substantial computational resources requisite for LLM
processing present another obstacle, particularly for applications
necessitating swift responsiveness and high throughput. These challenges
underscore the imperative for innovative methodologies aimed at enhancing LLM
efficiency and context management capabilities without compromising
performance.

This paper introduces a pioneering framework, titled Soft Prompt Compression
for LLMs (SPC-LLM), which aims to overcome these constraints by amalgamating
the principles of soft prompt compression with natural language summarization
techniques. the integration of soft prompt compression with LLMs could benefit
from references to pioneering studies on prompt engineering or soft prompts in
NLP[1]. Our approach endeavors to tailor LLMs for streamlined context
processing, enabling them to navigate extensive textual data more adeptly
while alleviating computational burdens.

Our strategy comprises two primary facets: firstly, leveraging natural
language summarization to distill protracted texts into succinct, content-rich
summaries, and secondly, integrating these summaries into the model’s input
via trainable soft prompts. This dual-pronged approach extends the effective
context window of LLMs and fosters a nuanced comprehension and generation of
text predicated on diverse information reservoirs. By condensing the context
into a compact, information-dense format, SPC-LLM substantially diminishes
computational overheads, rendering the deployment of LLMs more viable across a
broad array of applications.

We delineate a comprehensive methodology for implementing soft prompt
compression alongside natural language summarization within LLMs, elucidating
how this amalgamation augments model performance on tasks necessitating
comprehension of extended contexts. Moreover, we furnish empirical
substantiation from a series of experiments evincing the efficacy of SPC-LLM
in enhancing the efficiency and precision of LLMs across various NLP tasks.

The structure of this paper is as follows: we commence with a review of
pertinent literature concerning LLM efficiency and context processing.
Subsequently, we expound upon the proposed SPC-LLM framework, delineating its
constituents and the integration process. We subsequently expound upon the
experimental setup and present our findings, highlighting the advantages of
our approach. Finally, we deliberate on the implications of our work and posit
avenues for future research endeavors in this domain.

##  II Prior Work

The endeavor to augment the efficiency and contextual processing capabilities
of Large Language Models (LLMs) stands as a focal point in contemporary
natural language processing (NLP) inquiry. This section provides an overview
of notable advancements in three pertinent domains: traditional methodologies
for managing lengthy contexts within LLMs, the genesis and utilization of soft
prompts, and innovations in text summarization techniques aimed at contextual
information compression.

Figure 1: An example of successful prompt compression with SPC. The compressed
prompt (green) in order to obtain a shorter length and maintain
transferability and utility simultaneously than the original long prompt
(red).

Figure 2: The illustration of SPC shows the compressed conversational answer
expect with question.

###  II-A Addressing Lengthy Contexts in LLMs

Initial efforts to expand the contextual scope of LLMs primarily concentrated
on architectural innovations. Transformer-based models such as Longformer [2]
and BigBird [3] introduced alterations to the conventional attention
mechanism, facilitating the efficient analysis of extended textual sequences.
These models employ sparse attention patterns and memory-efficient algorithms
to mitigate computational overheads associated with handling extensive
contexts. Nevertheless, notwithstanding their augmented capacity, these
architectural refinements frequently necessitate significant alterations to
pre-existing models and entail substantial retraining endeavors.

###  II-B Soft Prompts in NLP

The notion of soft prompts, first introduced by Li and Liang [4] and further
elucidated by Lester [5], signifies a departure towards more adaptable and
parameter-efficient approaches for tailoring LLMs to specific tasks. Unlike
rigidly defined textual prompts, soft prompts comprise trainable embeddings
optimized alongside the model to enhance performance on targeted tasks. This
adaptability facilitates effective fine-tuning of LLMs sans exhaustive
retraining, thereby enabling their adaptation across diverse applications.
Nonetheless, the amalgamation of soft prompts with the objective of enhancing
context compression and efficiency within LLMs remains a relatively
underexplored domain.

###  II-C Text Summarization for Contextual Compression

Text summarization methodologies, encompassing both extractive and abstractive
approaches, have been harnessed to condense protracted documents into succinct
representations while retaining salient information. Advancements in
summarization techniques, exemplified by models like BART [6], have exhibited
considerable promise in generating coherent and concise summaries. Although
these models adeptly distill lengthy texts, their direct utilization as a
preprocessing step for LLMs in tasks necessitating nuanced comprehension of
extended contexts has not been extensively investigated.

The fusion of soft prompts with summarization techniques to augment the
processing of lengthy contexts in LLMs epitomizes a novel convergence of these
research trajectories. Our study builds upon these foundational principles,
proposing a unified framework that capitalizes on the strengths of soft
prompts and advanced summarization methodologies to alleviate the constraints
of existing LLMs in efficiently handling extensive textual information.

##  III Methodology

Integrating summary vectors with natural language formatted prompts, the
fusion of utility preservation with information compression, and the
conceptualization of soft prompts are explored within a comprehensive
mathematical model. In this section, our endeavor lies in the construction of
a holistic mathematical framework, embodying the amalgamation of summary
vectors with prompts formatted in natural language, the convergence of utility
retention with information condensation, and the notion of soft prompts. Let
us consider the following components:

  * •

An original document $D$

  * •

A function $f_{NL}$ for generating prompts in natural language, yielding
$P_{NL}$

  * •

A function $f_{S}$ for the generation of summary vectors, yielding $V_{S}$

  * •

A parameter matrix $S$ for soft prompt

  * •

A predictive function of the language model, denoted as $f_{LM}$

The expression of the integrated model we aspire to formulate is as follows:

$C=f_{LM}(S\oplus f_{S}(f_{NL}(D)))$ (1)

Where:

  * •

$f_{NL}(D)$ denotes the conversion of the original document $D$ into the
prompt formatted in natural language, denoted as $P_{NL}$.

  * •

$f_{S}(P_{NL})$ signifies the further compression of $P_{NL}$ into the summary
vector $V_{S}$.

  * •

$S$ represents the parameter matrix of trained soft prompts, meticulously
optimized to augment the model’s versatility across distinct tasks.

  * •

$\oplus$ denotes the operation of concatenation, amalgamating the soft prompt
vector with the summary vector $V_{S}$.

  * •

$f_{LM}$ represents the predictive capability of the language model, which
takes the concatenated vector as input and generates the ultimate compressed
representation $C$.

In this expression:

  * •

Utility preservation is attained through the strategic design of functions
$f_{S}$ and $f_{NL}$ to uphold the maximum preservation of information.

  * •

The compression of information is executed via function$f_{S}$, which
compresses the NL formatted prompt $P_{NL}$ into the summary vector $V_{S}$.

  * •

The soft prompt is optimized during the training process to learn how to most
effectively utilize the compressed information $V_{S}$ to improve the model’s
performance on specific tasks.

Utilizing this methodology, we harness the robust information compression
potential inherent in NL formatted prompts and summary vectors. Additionally,
we bolster the adaptability and efficacy of the model for particular
downstream tasks by integrating soft prompts, thereby amplifying its capacity
for generalization [7].

##  IV Experiments

The principal aim of these experiments is to assess the efficacy of our
proposed methodology in enhancing the efficiency and performance of Large
Language Models (LLMs) during the processing of expansive textual contexts.
Our specific focus lies in evaluating: The efficacy of condensed contextual
representations produced via natural language summarization and the influence
of integrating soft prompt compression on the efficacy of LLMs across diverse
NLP tasks.

To achieve this aim, we meticulously designed our experiments to measure two
critical aspects of our methodology’s impact on LLMs. First, we evaluated the
quality and utility of the condensed contextual representations generated by
our advanced natural language summarization techniques. This assessment
focused on determining how effectively our approach could distill essential
information from extensive textual data, thereby facilitating a more efficient
processing by LLMs. The evaluation criteria included the coherence,
completeness, and relevance of the summaries in retaining the core message and
essential details from the original texts. Second, we investigated the role of
soft prompt compression in enhancing the overall performance of LLMs across a
variety of NLP tasks, including but not limited to text summarization,
sentiment analysis, text classification, and question answering. This part of
our experiment aimed at quantifying the improvements in task-specific metrics
such as accuracy, and processing time, attributable to the integration of soft
prompts[8]. Special attention was given to the adaptability of soft prompts in
capturing and utilizing the nuances of condensed contexts, thereby augmenting
the LLMs’ ability to derive accurate and relevant outcomes from the processed
data. Through a comprehensive set of experiments, we meticulously recorded and
analyzed the performance metrics, processing times, and resource utilization
patterns associated with each task, both with and without the application of
our methodology[9]. This data collection was instrumental in illustrating the
tangible benefits of our approach, particularly in terms of reducing
computational overhead and enhancing the adaptability and efficiency of LLMs
in handling extensive textual contexts.

By systematically addressing these focus areas, our experiments were designed
not only to validate the effectiveness of our proposed methodology but also to
explore its potential limitations and areas for further enhancement. The
findings from these experiments are anticipated to contribute significantly to
the ongoing efforts to optimize LLMs for a wider range of applications,
thereby pushing the boundaries of what is achievable in the field of natural
language processing.

###  IV-A Dataset Description

Summarization Task CNN/Daily Mail dataset, which pro- vides news articles and
associated highlights for evaluating summarization quality.

Sentiment Analysis The Stanford Sentiment Treebank (SST-2), offering movie
reviews with sentiment labels.

Text Classification The AG News dataset, containing news articles categorized
into four topics.

Question Answering The SQuAD v2.0 dataset, featuring questions posed on a set
of Wikipedia articles, where some questions are unanswerable.

###  IV-B Main Result

We refine the performance of a Transformer-based model renowned for its
summarization prowess through fine-tuning on the CNN/Daily Mail dataset. The
dataset is partitioned into distinct training, validation, and test splits
tailored to each specific task. Our summarization module effectively condenses
contextual information within the training and validation datasets. Soft
prompts are initialized as embeddings within the input LLMs, concurrently
trained with the model’s parameters to optimize task-specific performance.
Comparative evaluation of the LLMs’ performance, both with and without our
methodological intervention, is conducted using task-specific metrics.The
visualizations provided underscore the marked improvements in accuracy across
various NLP tasks such as text summarization, sentiment analysis, text
classification, and question answering, showcasing the efficacy of our
approach (refer to the generated images).

Additionally, our methodology significantly enhances computational efficiency,
as evidenced by the dramatic reduction in processing times. For instance, our
method reduces the processing time by up to 80.1 persent in the SQuAD2.0
dataset, and similarly substantial reductions are observed across other
datasets including CNN/Daily Mail, SST-2, and AG News, as detailed in the
accompanying table. These improvements in processing speed do not compromise
the quality of outcomes, illustrating the synergy between soft prompts and
summarization techniques in optimizing LLMs’ performance. Potential areas for
further inquiry may involve the refinement of soft prompt parameters and
summarization algorithms [7]to bolster performance across a wider array of NLP
tasks. Moreover, delving into the applicability of this methodology in
multilingual settings [10] and diverse domains could yield valuable insights
into its adaptability and global relevance. In summary, our research
represents a substantial advancement in the realm of NLP, presenting a
scalable and effective framework for augmenting the capabilities of LLMs. This
advancement establishes a foundation for future innovations aimed at
unleashing the full potential of LLMs in comprehending and processing
intricate and extensive textual data. In summary, our research represents a
substantial advancement in the realm of NLP, presenting a scalable and
effective framework for augmenting the capabilities of LLMs. This advancement
establishes a foundation for future innovations aimed at unleashing the full
potential of LLMs in comprehending and processing intricate and extensive
textual data. The efficiency gains, coupled with improved accuracy, position
our method as a transformative force in the field, paving the way for more
adaptable, faster, and precise language models.

| Claude2
---|---
Cost | Original | Capsule Prompt | Save
Mail dataset | 12.33 | 3.37 | -77.9%
SST-2 | 4.22 | 1.86 | -63.9%
AG News | 42.41 | 15.51 | -78.5%
SQuAD2.0 | 2.14 | 0.42 | -80.1%

Figure 3: comparison chart of processing times.

Figure 4: Performance difference compared to baseline model.

## Conclusion

There is a citing studies that have demonstrated the effectiveness of
summarization in enhancing model performance[11]. However, our comprehensive
investigation introduces a novel methodology that synergistically amalgamates
the functionalities of soft prompts, prompts formatted in natural language,
and advanced summarization techniques to augment the efficacy and efficiency
of Large Language Models (LLMs) in handling extensive textual contexts.
Through empirical validation across a varied spectrum of NLP tasks, we have
substantiated the substantial enhancements our approach provides in both
context compression and model adaptability.For instance, processing times were
reduced by up to 80.1 persent for tasks involving the SQuAD2.0 dataset, with
similar efficiencies noted across other datasets such as CNN/Daily Mail,
SST-2, and AG News. This not only underscores the efficiency of our approach
but also highlights its potential in making advanced NLP technologies more
accessible and feasible in resource-constrained scenarios.

The amalgamation of soft prompts with summary vectors, derived from prompts
formatted in natural language, not only optimizes information compression but
also conserves the utility of the original context. This dual emphasis ensures
the retention of essential information while diminishing the computational
overhead conventionally associated with processing lengthy texts. Furthermore,
the adaptability of our methodology is underscored by its performance
improvements across diverse NLP tasks, encompassing text summarization,
sentiment analysis, text classification, and question answering. In light of
these findings, our work not only contributes a significant leap forward in
the field of NLP by enhancing the performance and efficiency of LLMs but also
sets a new benchmark for future research in this area. The potential for our
methodology to be extended and applied in multilingual contexts and across
different domains offers exciting avenues for further exploration. As we
continue to push the boundaries of what is possible with LLMs, our research
lays the groundwork for a new era of NLP solutions that are more adaptable,
efficient, and accessible than ever before.

Our discoveries indicate that the fusion of soft prompts with advanced
summarization techniques presents a promising avenue for future exploration
aimed at enhancing the efficiency and adaptability of LLMs. This approach not
only addresses the challenges associated with processing lengthy texts but
also unveils new prospects for tailoring LLMs for specific applications sans
the necessity for extensive retraining.This convergence of efficiency,
adaptability, and reduced computational demand represents a paradigm shift in
how LLMs can be optimized for a wide range of applications. Our findings
advocate for a continued exploration of soft prompt tuning and advanced
summarization techniques, suggesting that the future of NLP lies in the
strategic integration of these methodologies to overcome the inherent
limitations of current models. As we stand on the brink of this new frontier,
our research illuminates the path forward, offering a blueprint for the next
generation of language models that are not only more powerful but also more
practical for real-world applications.

## References

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# Absence of bursts between 4 – 8 GHz from FRB 20200120E located in an M 81
Globular Cluster

Vishal Gajjar Department of Astronomy, University of California Berkeley,
Berkeley, CA  Daniele Michilli Department of Physics, McGill University, 3600
rue University, Montréal, QC H3A 2T8, Canada  McGill Space Institute, McGill
University, 3550 rue University, Montréal, QC H3A 2A7, Canada  Jakob T. Faber
Department of Physics and Astronomy, Oberlin College, Oberlin, OH  Department
of Astronomy, University of California Berkeley, Berkeley, CA  Sabrina Berger
Department of Physics, McGill University, 3600 rue University, Montréal, QC
H3A 2T8, Canada  McGill Space Institute, McGill University, 3550 rue
University, Montréal, QC H3A 2A7, Canada  Steve Croft Department of Astronomy,
University of California Berkeley, Berkeley, CA  SETI Institute, Mountain
View, California  Aaron B. Pearlman Department of Physics, McGill University,
3600 rue University, Montréal, QC H3A 2T8, Canada  McGill Space Institute,
McGill University, 3550 rue University, Montréal, QC H3A 2A7, Canada  Division
of Physics, Mathematics, and Astronomy, California Institute of Technology,
Pasadena, CA 91125, USA  McGill Space Institute Fellow  FRQNT Postdoctoral
Fellow  Ketan R. Sand Department of Physics, McGill University, 3600 rue
University, Montréal, QC H3A 2T8, Canada  McGill Space Institute, McGill
University, 3550 rue University, Montréal, QC H3A 2A7, Canada  Paul Scholz
Dunlap Institute for Astronomy & Astrophysics, University of Toronto, 50 St.
George Street, Toronto, ON M5S 3H4, Canada  Andrew P. V. Siemion Department of
Astronomy, University of California Berkeley, Berkeley, CA  SETI Institute,
Mountain View, California  Jodrell Bank Centre for Astrophysics, Department of
Physics & Astronomy, Alan Turing Building, The University of Manchester, M13
9PL, United Kingdom  University of Malta, Institute of Space Sciences and
Astronomy

###### Abstract

We report the non-detection of dispersed bursts between $4-8$ GHz from 2.5
hours of observations of FRB 20200120E at 6 GHz using the Robert C. Byrd Green
Bank Telescope. Our fluence limits are several times lower than the average
burst fluences reported at 600 and 1400 MHz. We conclude that these non-
detections are either due to high-frequency bursts being weaker and/or
scintillation-induced modulated. It is also likely that our observations were
non-concurrent with any activity window of FRB 20200120E.

Radio Bursts – Radio Transient Sources

##  1 Introduction

Fast radio bursts (FRBs) are millisecond-duration pulses of extragalactic
origin. FRB 121102A has been detected across a wide frequency range, from 400
MHz (Josephy et al., 2019) to 8 GHz (Gajjar et al., 2018), giving insights
into the burst emission properties (Michilli et al., 2018); no other FRB has
been detected across such a large range of frequencies. The newly-discovered
FRB 20200120E was found to have a dispersion measure (DM) of 87.8 pc cm-3,
which is the lowest reported among all FRBs (Bhardwaj et al., 2021). Recently,
Kirsten et al. (2021) announced a precise localization of three bursts from
FRB 20200120E coinciding with a globular cluster in the M81 system. Thus, FRB
20200120E provides an excellent opportunity to conduct deep observations at
higher frequencies ($>4$ GHz). A detection at higher frequencies would offer
the opportunity to (a) compare frequency-dependent burst rates, (b) measure
scatter-independent rotation measure and polarization position angle, and (c)
investigate frequency-dependent burst widths, sub-burst components, and
downward-drifting behavior. Here, we report observations of FRB 20200120E from
the Green Bank Telescope (GBT), using the C-band receiver at $4-8$ GHz.

##  2 Observations and analysis

We have an ongoing observing campaign with the GBT to trigger C-band
observations for any repeating FRB reported by the CHIME/FRB
project111https://www.chime-frb.ca/repeaters with localization uncertainty
smaller than the GBT beam ($2\farcm 5$). The observations of FRB 20200120E
were taken for $2.5$ h. We utilized the Breakthrough Listen (BL) backend,
which is a state-of-the-art 64-node GPU cluster (MacMahon et al., 2018),
deployed primarily to conduct the most comprehensive search for evidence of
intelligent life in the Universe (Worden et al., 2017). Recorded baseband
voltages were converted to SIGPROC-formatted filterbank files (Lebofsky et
al., 2019). As FRBs are known to show limited spectral coverage (Gajjar et
al., 2018; Zhang et al., 2018), we divided our 4 GHz band into eight 500 MHz-
wide sub-bands, searched independently to improve the chance of detecting
spectrally-limited bursts (Faber et al., 2021). We used SPANDAK, a similar
search pipeline to that described by Gajjar et al. (2018), to blindly search
for single pulses, with search parameters listed in Table 1.

Observations and Search Parameters
---
Source Name | FRB 20200120E |  DMrange |  $0-1000$ pc cm-3
RA(J2000) | 09h57m56.7s |  Widths |  $0.3-76$ ms
Dec(J2000) | $+68\arcdeg 49\arcmin 31\farcs 8$ |  S/N | $>10$
Frequency |  $4-8$ GHz |  F${}_{\rm UL}^{\rm fullband}$ | 35 mJy ms
Tsampling |  $\sim 350\,\mu$s |  F${}_{\rm UL}^{\rm sub-band}$ ($\Delta\nu\sim 500$ MHz) | 100 mJy ms
Obs. MJD (59259+) | $0.969687500-1.072951388$ |  burst-rateUL |  $<0.4$ hr-1
Table 1: Summary of observations and single pulse searches of FRB 20200120E
across $4-8$ GHz. The F${}_{\rm UL}^{\rm fullband}$ and F${}_{\rm UL}^{\rm
sub-band}$ stand for minimum detectable fluence assuming a 1 ms-wide top-hat
burst for the full band and any one of the sub-bands, respectively.

##  3 Results and conclusion

We did not detect any significant dispersed burst across the full band ($4-8$
GHz) nor in any of the eight sub-bands. Upper limits on the fluences are
listed in Table 1. Bhardwaj et al. (2021) reported an average burst fluence
$\sim 2210$ mJy ms at 600 MHz. Kirsten et al. (2021) reported multiple bursts
at 1.4 GHz with an average burst fluence $\sim 500$ mJy ms and burst rate
$\sim 0.68$ hr-1. These fluences and burst rates are higher than our current
fluence limits and our inferred burst rate. However, it is plausible that
bursts at 6 GHz are intrinsically weaker (or absent) than those detected at
lower frequencies. Nimmo et al. (2021) and Majid et al. (2021) reported that
bursts from FRB 20200120E exhibit sub-burst structure at $<100$ ns scales,
similar to ‘nanoshots’ seen in giant pulses (GPs) from the Crab pulsar.
Hankins (2000) compared simultaneous detections of several hundred GPs from
the Crab pulsar at 1.4 and 5 GHz and found the average instantaneous spectral
index to be steep ($\alpha\sim-2$). If the bursts from FRB 20200120E are
indeed similar to GPs from the Crab pulsar, such a steep spectrum would render
burst fluences at 6 GHz below our detection threshold. Furthermore, the
expected diffractive scintillation bandwidth varies from 200 MHz to 3.5 GHz
across our observed band towards the source (see Majid et al. 2021). If the
bursts are spectrally narrower than this scintillation bandwidth then they are
likely to be 100% modulated, which could yield non-detections. It is also
possible that FRB 20200120E exhibits enhanced burst activity windows (seen
from other repeating FRBs; Zhang et al. 2018; Price et al. 2018) and our
observations were non-concurrent with such active phases. However, in the
absence of any other reported non-detection, we are unable to verify if FRB
20200120E indeed exhibits such prolonged inactive phases. Future simultaneous
multi-frequency multi-epoch observations may provide insights into the burst
properties around 6 GHz.

Breakthrough Listen is managed by the Breakthrough Initiatives, sponsored by
the Breakthrough Prize Foundation. The Green Bank Observatory is a facility of
the National Science Foundation, operated under cooperative agreement by
Associated Universities, Inc. J.T.F. and S.C. were supported by the National
Science Foundation under Grant No. 1950897. D.M. is a Banting Fellow. A.B.P is
a McGill Space Institute (MSI) Fellow and a Fonds de Recherche du Quebec -
Nature et Technologies (FRQNT) postdoctoral fellow. P.S. is a Dunlap Fellow
and an NSERC Postdoctoral Fellow. The Dunlap Institute is funded through an
endowment established by the David Dunlap family and the University of
Toronto.

## References

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  * Faber et al. (2021) Faber, J. T., Gajjar, V., Siemion, A. P. V., et al. 2021, Research Notes of the American Astronomical Society, 5, 17, doi: 10.3847/2515-5172/abde48
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[a]Michał Praszałowicz

# Scaling properties of elastic pp cross-section

Cristian Baldenegro
Christophe Royon  Anna M. Staśto

###### Abstract

We show that the elastic differential $pp$ cross-section has a unique
universal property that the ratio of bump-to-dip position is constant from the
energies of the ISR to the LHC. We explore this property to compare Geometric
Scaling present at the ISR with the recently proposed scaling law at the LHC.
We argue that at the LHC, within present experimental uncertainties, there is
fact a family of scaling laws.

##  1 Introduction

In this report we discuss scaling properties of the differential elastic $pp$
cross-section at low and high energies following Refs. [1, 2]. Differential
$pp$ cross-sections have very characteristic structure. One observes a rapid
decrease for small $|t|$, then a minimum at $t_{\rm d}$, called a dip,
followed by a broad maximum at $t_{\rm b}$, dubbed as a bump. It turns out
that the bump-to-dip cross-section ratio

$\mathcal{R}_{\mathrm{bd}}(s)=\frac{d\sigma_{\mathrm{el}}/d|t|_{\rm{b}}}{d\sigma_{\mathrm{el}}/d|t|_{\mathrm{d}}}\,,$
(1)

which seems to saturate at LHC energies at a value of approximately 1.8, is
rather strongly energy dependent at the ISR (see e.g. Fig. 2 in Ref. [3] ).

In Ref. [2] we explored an interesting regularity of
$d\sigma_{\mathrm{el}}/d|t|$, which, to the best of our knowledge, has not
been used in phenomenological studies of the $pp$ elastic scattering. It turns
out that the ratio of bump-to-dip positions in $|t|$

$\mathcal{T}_{\rm bd}(s)=|t_{\rm b}|/|t_{\rm d}|\,,$ (2)

is constant, within the experimental uncertainties, at all energies from the
ISR to the LHC and equal approximately to 1.355 [2]. It is striking and
unexpected that this ratio is constant over such large range of energies.

In the following, w discuss first Geometric Scaling (GS) at the ISR, and then
new scaling laws at the LHC. For details see Refs. [1, 2].

##  2 Geometric Scaling at the ISR

Fifty years ago, in 1973-74, Jorge Dias de Deus proposed and developed the
idea of GS in elastic $pp$ scattering [4, 5]. It was based on a
phenomenological observation that elastic, inelastic and total cross-sections
have (almost) the same energy dependence over the ISR energy span, see Tab. 1.

| elastic | inelastic | total | $\frac{\mathrm{elastic}}{\mathrm{inelastic}}$ | $\rho$
---|---|---|---|---|---
ISR | $W^{0.1142\pm 0.0034}$ | $W^{0.1099\pm 0.0012}$ | $W^{0.1098\pm 0.0012}$ | $W^{0.0043\pm 0.0036}$ | $0.02-0.095$
LHC | $W^{0.2279\pm 0.0228}$ | $W^{0.1465\pm 0.0133}$ | $W^{0.1729\pm 0.0163}$ | $W^{0.0814\pm 0.0264}$ | $0.15-0.10$
Table 1: Energy dependence of the integrated cross-sections for the energies
$W=\sqrt{s}$ at the ISR [6] and at the LHC [7] and the $\rho$ parameter [6,
8]. Fits from [2].

Elastic cross-sections can be parametrized in terms of the opacity
$\Omega(s,b)$ and phase $\chi(b,s)$, which are both real functions of the
impact parameter and scattering energy [9, 10]

$\sigma_{\mathrm{el}}={\displaystyle\int}d^{2}\boldsymbol{b}\,\left|1-e^{-\Omega(s,b)+i\chi(s,b)}\right|^{2}\,,~{}~{}~{}~{}~{}~{}\sigma_{\mathrm{inel}}={\displaystyle\int}d^{2}\boldsymbol{b}\,\left[1-\left|e^{-\Omega(s,b)}\right|^{2}\right]$
(3)

and

$\sigma_{\mathrm{tot}}=2{\displaystyle\int}d^{2}\boldsymbol{b}\,\operatorname{Re}\left[1-e^{-\Omega(s,b)+i\chi(s,b)}\right]\,.$
(4)

Since the real part of the scattering amplitude (related to the $\rho$
parameter) is small, one can safely neglect $\chi$. GS is a hypothesis that
$\Omega(s,b)=\Omega\left(b/R(s)\right)$ where $R(s)$ is the interaction radius
[4] increasing with energy. Changing the integration variable from
$\boldsymbol{b}\rightarrow\boldsymbol{B}=\boldsymbol{b}/R(s)$, one obtains
that

$\sigma_{\mathrm{inel}}=R^{2}(s){\displaystyle\int}d^{2}\boldsymbol{B}\,\left[1-\left|e^{-\Omega(B)}\right|^{2}\right]\,,$
(5)

where the integral in (5) is an energy independent constant. If we neglect the
phase $\chi(s,b)$, both elastic and total cross-sections should scale the same
way, which means that their ratios should be energy independent. As can be
seen from Table 1, this is indeed the case.

GS of total cross-sections has important consequences for the differential
cross-section:

$\frac{d\sigma_{\mathrm{el}}}{d|t|}\sim\left|{\displaystyle\int\limits_{0}^{\infty}}db^{2}A_{\mathrm{el}}(b^{2},s)\,J_{0}\left(b\sqrt{\left|t\right|}\right)\right|^{2}=\sigma_{\mathrm{inel}}^{2}(s)\left|{\displaystyle\int\limits_{0}^{\infty}}dB^{2}A_{\mathrm{el}}(B^{2})\,J_{0}\left(B\sqrt{\tau}\right)\right|^{2}$
(6)

where $A_{\mathrm{el}}(b^{2},s)$ is the elastic scattering amplitude and
$J_{0}$ denotes the Bessel function originating from the Fourier transform.
Equation (6) implies that the scaled cross-section

$\frac{1}{\sigma_{\mathrm{inel}}^{2}(s)}\frac{d\sigma_{\mathrm{el}}}{d|t|}(s,t)=\Phi(\tau)$
(7)

should be a universal, energy independent function of the scaling variable
$\tau$:

$\tau=R^{2}(s)|t|\times\mathrm{const.}\,=\sigma_{\mathrm{inel}}(s)\,|t|\,.$
(8)

GS at the ISR was confirmed in Ref. [5], except for the dip region [11], see
also [2].

##  3 Scaling at the LHC

At the LHC elastic, inelastic and total $pp$ cross-sections have different
energy dependence, see Tab. 1, and therefore no GS described in Sect. 2 is
expected. Nevertheless, the fact that both (2) and (1) saturate at the LHC
energies, suggests that the transformation

$t\rightarrow\tau=f(s)t$ (9)

should align dips and bumps of different energies, and rescaling

$\frac{d\sigma_{\mathrm{el}}}{dt}(t)\rightarrow
g(s)\frac{d\sigma_{\mathrm{el}}}{dt}(\tau)$ (10)

should superimpose the cross-section values, at least in the dip and bump
regions. At the ISR $g\sim 1/f^{2}$, but at the LHC both functions $f$ and $g$
seem to be independent [1, 2].

In Ref. [2] we have analyzed TOTEM data [8, 12, 13, 14, 15] summarized in [7]
with the following result:

$f(s=W^{2})=W^{\beta},~{}~{}\beta=0.1686~{}~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}~{}g(s=W^{2})=W^{-\alpha},~{}~{}\alpha\simeq 0.66.$ (11)

The result is plotted in Fig. 1 where the cross-section scaling is clearly
seen within the experimental uncertainties.

Figure 1: Left: elastic $pp$ cross-section $d\sigma_{\mathrm{el}}/dt$
[mb/GeV2] at the LHC energies in terms of $|t|$ [GeV2] in the dip and bump
region. Right: scaled cross-section in tems scaling variable
$\tau=W^{\beta}\,|t|$. One can see that the cross-sections at different
energies are aligned after scaling.

In Ref. [1] a more general scaling variable was proposed

$\tilde{\tau}=s^{a}t^{b}$ (12)

with the result and $a\simeq 0.065$, $b\simeq 0.72$. Aligning dips and bumps
alone (rather than the entire cross-sections) requires the value of $\beta$
given in (11). This imposes a constraint

$a-b\,\beta/2=0,$ (13)

which is roughly satisfied by $a$ and $b$ of Ref. [1]. The assumption of
energy independence of ${\cal T}_{\rm bd}$ (2) leads to a family of scaling
laws (12) constrained by (13) where the determination of $a$ (or equivalently
of $b$) must follow from the alignment of the points outside of the immediate
vicinity of the dip and bump regions. The quality of the present LHC data does
not allow for a precise determination of $a$ and $b$.

## Acknowledgements

CB is supported by the European Research Council consolidator grant no.
101002207. AMS is supported by the U.S. Department of Energy grant No. DE-
SC-0002145 and within the framework of the Saturated Glue (SURGE) Topical
Theory Collaboration, as well as in part by National Science Centre in Poland,
grant 2019/33/B/ST2/02588.

## References

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# Dark Lepton Superfluid in Proto-Neutron Stars

Sanjay Reddy<EMAIL_ADDRESS>Department of Physics, University of Washington,
Seattle, WA 98195  Institute for Nuclear Theory, University of Washington,
Seattle, WA 98195  Dake Zhou<EMAIL_ADDRESS>Department of Physics, University of
Washington, Seattle, WA 98195  Institute for Nuclear Theory, University of
Washington, Seattle, WA 98195

###### Abstract

We find that sub-GeV neutrino portal bosons that carry lepton number can
condense inside a proto-neutron star (newly born neutron star). These bosons
are produced copiously and form a Bose-Einstein condensate for a range of as
yet unconstrained coupling strengths to neutrinos. The condensate is a lepton
number superfluid with transport properties that differ dramatically from
those encountered in ordinary dense baryonic matter. We discuss how this phase
could alter the evolution of proto-neutron stars and comment on the
implications for neutrino signals and nucleosynthesis.

††preprint: INT-PUB-21-016

##  I Introduction

Hidden sectors (HS) that couple to the standard model (SM) often contain a
rich spectrum of particles that can constitute dark matter (DM) [1]. Their
self-interactions and their interactions with the SM lead to interesting
phenomenology and strategies for detection. In particular, scenarios where HS
particles interact strongly with neutrinos has sparked recent interest [2, 3,
4, 5, 6, 7, 8, 9]. These interactions can generate stronger self-interactions
between neutrinos mediated by new degrees of freedom [7, 8] and have been
invoked to explain hints of anomalies in neutrino oscillation experiments [10,
11] and puzzling cosmological observations [12, 13, 14]. In this article we
consider HS bosons that carry lepton number $L$ but is otherwise uncharged
with respect to the SM gauge group and demonstrate its novel implications in
extreme astrophysical environments.

A newly born neutron star, also called a proto-neutron star (PNS), provides a
unique laboratory to study DM coupled to neutrinos. A PNS born in the
aftermath of core-collapse supernovae explosion is hot ($T\simeq
30~{}\text{MeV}$) and contains large densities ($\rho\simeq 5\times
10^{14}~{}\text{g}/\text{cm}^{3}$) of baryons and leptons. Detailed
simulations have shown that neutrinos are trapped inside the PNS, and that
their diffusion takes $\tau_{\text{diff}}\sim~{}$tens of seconds [15, 16].
During this time, the hot and dense PNS matter supports a large chemical
potential for lepton number ($\mu_{L}=\mu_{\nu_{e}}\simeq
100-200~{}\text{MeV}$) and a corresponding large excess of electron neutrinos.
These conditions, which are not realized in any other environment in the
universe, presents an opportunity to study DM candidates that carry lepton
number.

In this article we show that lepton number scalars, denoted by $\phi$, will
have important implications for PNSs. For a wide range of parameter space
compatible with existing constraints, $\phi$ particles are rapidly produced in
the PNS and thermalize to form a Bose-Einstein condensate. The condensate is a
superfluid that transports lepton number from the core to the surface of PNS
on timescales that are negligible compared to neutrino diffusion. This alters
the radial distribution of lepton number (see fig. 1) and modifies the
composition and neutrino transport properties of dense matter inside the PNS.

Figure 1: Superfluid transport of lepton number due to $\phi$ condensation
rapidly quenches the spatial gradients in the lepton number chemical
potential. It transforms the initial state with approximately constant lepton
fraction $Y_{L}\simeq 0.3$ (depicted on the left) to the state on the right.

We introduce the lepton number scalar $\phi$ through a minimal model defined
by the low-energy effective Lagrangian [3]

$\mathcal{L}_{\text{eff,
int}}\supset-\frac{g_{\alpha\beta}}{2}\nu_{\alpha}\nu_{\beta}\phi^{*}+\text{h.c.}-m_{\phi}^{2}\phi^{*}\phi-\frac{\lambda}{4}(\phi^{*}\phi)^{2}.$
(1)

Here, $g_{\alpha\beta}$ is the coupling between left-handed Weyl neutrinos
$\nu_{\alpha,\beta}$ and the scalar $\phi$ that carries lepton number 2.
$\alpha,\beta$ are flavor indices and the quartic coupling $\lambda$
characterizes the strength of self-interaction among $\phi$’s. In general,
$g_{\alpha\beta}$ may involve all 3 flavors $e,\mu,\tau$. However, in what
follows we focus on the flavor conserving coupling to electron neutrinos
$g_{ee}=g$, and comment on the possible role of couplings to $\mu$ and $\tau$
neutrinos in our concluding remarks.

This coupling $g$ may arise after electroweak symmetry breaking from the
effective operator $(l\tilde{H})(l\tilde{H})\phi^{*}/\Lambda^{2}$ [17, 3],
where $l=(\nu_{e},e)^{T}$ is the SM lepton doublet,
$\tilde{H}=i\sigma_{2}H^{*}$ is the flipped SM Higgs doublet, and $\Lambda$ is
the high-energy scale associated with new physics. Since the Weinberg operator
$(l\tilde{H})(l\tilde{H})$[18] carries lepton number 2, the Lagrangian eq. 1
conserves $L$. This effective operator is one of few dimension six operators
that generates neutrino self-interactions. In contrast to conventional majoron
models where the coupling is highly suppressed by neutrino Majorana masses
[19, 20], in this scenario, the natural strength for the coupling
$g\simeq(v/\Lambda)^{2}$ ($v=246~{}\text{GeV}$ is the electroweak scale) is
expected to be large from a model building perspective [17, 3].

Laboratory constraints on the coupling $g_{\alpha\beta}$ are flavor dependent.
For $g\equiv g_{ee}$, limits from exotic meson decays demand $g\lesssim
10^{-2}$ when $m_{\phi}\lesssim$ GeV, and double beta-decay experiments
require $g\lesssim 10^{-4}$ when $m_{\phi}\lesssim\text{MeV}$ [21, 3, 6].
Astrophysical and cosmological bounds on the parameter space are also quite
stringent. For $m_{\phi}\lesssim 10$ MeV and $g\gtrsim 10^{-10}$, a thermal
population of $\phi^{\prime}s$ alters the expansion history of the early
universe during and after weak decoupling. This spoils the concordance between
the predictions of Big Bang nucleosynthesis (BBN) and observations of light
element abundances [22, 8], and is also in tension with the observed cosmic
microwave background (CMB) spectrum [7]. The supernovae cooling bound based on
SN1987a – the core-collapse supernova occurred in the Large Magellanic Cloud
in 1987 – excludes $g^{-11}\lesssim g\lesssim 10^{-6}$ when $m_{\phi}\lesssim
50$ MeV [23, 4]. This constraint arises because the duration of the observed
neutrino signal is shortened when free streaming weakly interacting particles
rapidly cool the PNS core subsequent to their thermal production [24].

In this article, we will focus on the, as yet, unconstrained region for
$\phi$’s with mass $m_{\phi}\gtrsim 10$ MeV and a coupling to electron
neutrinos in the range $10^{-6}\lesssim g\lesssim 10^{-2}$ to show that novel
phenomena arise in the PNS that can probe new physics at energy scale
$\Lambda\sim 1-10^{4}$ TeV.

##  II $\phi$ condensation in the PNS

As noted earlier, the PNS is characterized by high baryon number density
$n_{B}\simeq 1-3~{}n_{0}$, where $n_{0}=0.16~{}\text{fm}^{-3}$ is the nuclear
saturation density, high temperature $T\sim 20-60$ MeV, and a lepton fraction
$Y_{L}=n_{L}/n_{B}\simeq 0.3$. In this hot and dense environment lepton number
scalars are produced rapidly through $\nu\nu\rightarrow\phi$. The rate of
production is given by

$\frac{\mathrm{d}n_{\nu\nu\rightarrow\phi}}{\mathrm{d}t}=\int\frac{\mathrm{d}^{3}p_{1}}{2E_{p_{1}}(2\pi)^{3}}\frac{\mathrm{d}^{3}p_{2}}{2E_{p_{2}}(2\pi)^{3}}\frac{\mathrm{d}^{3}k}{2E_{k}(2\pi)^{3}}\\\
\times\frac{\left|\mathcal{M}_{\nu\nu\rightarrow\phi}\right|^{2}}{2}f_{1}f_{2}(2\pi)^{4}\delta^{4}(p_{1}+p_{2}-k)\,,$
(2)

where $\left|\mathcal{M}_{\nu\nu\rightarrow\phi}\right|^{2}=4g^{2}(p_{1}\cdot
p_{2})$ is the square of the tree level matrix element, and
$f_{{1,2}}=1/\left(1+\exp\left[(E_{p_{1,2}}-\mu_{L})/T\right]\right)$ are the
Fermi-Dirac distribution functions for neutrinos. For $g>10^{-6}$ and $T\simeq
10$ MeV, the timescale for transferring lepton number to the $\phi$ particles
$\tau_{\text{prod}}\simeq{n_{L}}/{\dot{n}_{\nu\nu\rightarrow\phi}}\ll\tau_{\text{diff}}$.
Further, most of the $\phi$’s produced have short mean free paths due to the
inverse decay process hence are trapped and subsequently thermalize with the
star. Qualitatively, the relaxation timescale can be calculated as

$\displaystyle\frac{1}{\tau}=\frac{1}{2E_{k}g_{k}}\int\frac{\mathrm{d}^{3}p_{1}}{2E_{p_{1}}(2\pi)^{3}}\frac{\mathrm{d}^{3}p_{2}}{2E_{p_{2}}(2\pi)^{3}}$
(3)
$\displaystyle\times\frac{\left|\mathcal{M}_{\nu\nu\leftrightarrow\phi}\right|^{2}}{2}f_{1}f_{2}(2\pi)^{4}\delta^{4}(p_{1}+p_{2}-k),$

where $g_{k}=1/(\exp((E_{k}-\mu_{\phi})/T)-1)$ denotes the Bose-Einstein
distribution function of $\phi$’s. For a simple estimate we set all the
distribution functions to $1$ and find that the mean free path of $\phi$’s

$\displaystyle\lambda_{\phi}=v\tau\sim
10^{-8}~{}\left(\frac{10^{-3}}{g}\right)^{2}\left(\frac{50~{}\text{MeV}}{m_{\phi}}\right)^{2}\left(\frac{T}{30~{}\text{MeV}}\right)~{}\text{km}\,.$
(4)

Consequently, for the range of couplings considered in this study, $\phi$’s
produced in PNS are in chemical and thermal equilibrium with neutrinos.

Since $\phi$ carries two units of lepton number, its chemical potential
$\mu_{\phi}=2\mu_{L}$. When $\mu_{\phi}>m_{\phi}$, there will be a macroscopic
occupations of $\phi$’s in the zero-momentum state, and the ground state is a
Bose-Einstein condensate. In the PNS where $\mu_{L}\simeq 200$ MeV at early
times, we can expect condensation for $m_{\phi}\lesssim 2\mu_{L,max}\simeq
400$ MeV.

In the absence of repulsive forces between $\phi$’s, the density of the
condensate grows rapidly in the core of the PNS. When the total mass of bosons
in the core exceeds $M_{crit}\simeq
m^{2}_{pl}/m_{\phi}\approx(100~{}\text{MeV}/m_{\phi})~{}10^{-18}~{}M_{\odot}$
the bosons collapse to form a black hole. The black hole once formed is
expected to subsequently consume the entire star [25, 26]. Thus, the very
existence of of neutron stars rules out such scenarios and viable models for
$\phi$ must include repulsive self-interactions. Even a small quartic coupling
$\lambda$ in Eq. 1 alters the effective potential for $\phi$’s [27, 28, 29,
30]

$V_{\text{eff}}(\phi)=(m_{\phi}^{2}-\mu_{\phi}^{2})\phi^{*}\phi+\frac{\lambda}{4}(\phi^{*}\phi)^{2}\,,$
(5)

and provides the necessary stabilization. When condensation occurs the finite
vacuum expectation value (vev) of the scalar field that minimizes the
effective potential above is given by

$\langle\phi\rangle=\sqrt{2(\mu_{\phi}^{2}-m_{\phi}^{2})/\lambda}\,.$ (6)

This vev breaks the global $U(1)_{L}$ symmetry and leads to superfluidity.

In the symmetry broken phase, the excitation spectrum is modified. The two
degrees of freedom associated with the complex $\phi$ field manifest as
fluctuations around the vev. Denoting the massless Goldstone mode as $J$, and
the massive mode as $\sigma$, in the condensed phase we can write
$\phi=(f+\sigma+iJ)/\sqrt{2}$, where $f=\sqrt{2}\langle\phi\rangle$. The
dispersion relations for the goldstone mode ($\omega_{-}^{\phi}$) and the
massive mode ($\omega_{+}^{\phi}$) are [27, 28, 29, 30]

$\omega_{\pm}^{\phi}=\sqrt{E_{p}^{2}+\mu_{\phi}^{2}+\frac{\lambda
f^{2}}{2}\pm\sqrt{\frac{\lambda^{2}f^{4}}{16}+\mu_{\phi}^{2}(4E_{p}^{2}+2\lambda
f^{2})}}\,,$ (7)

where $E_{p}=\sqrt{p^{2}+m_{\phi}^{2}}$.

The propagation of neutrinos is also modified in the superfluid phase because
$\nu$’s directly couple to the condensate. When $\phi$ acquires a vev, the
Yukawa term in eq. 1 becomes

$\mathcal{L}_{\text{Yukawa}}\supset\frac{g}{2\sqrt{2}}\left(f+\sigma+iJ\right)\nu\nu+\text{h.c.}.$
(8)

The term proportional to $f$, which would have led to neutrino Majorana mass
in the vacuum ($\mu_{L}=0$), leads to pairing of electron neutrinos near their
Fermi surface. This BCS-like pairing introduces a gap
$\Delta=g\langle\phi\rangle$ in the neutrino excitation spectrum

$\omega_{\pm}^{\nu}=\sqrt{\Delta^{2}+(p\pm\mu_{L})^{2}}\,.$ (9)

For the viable range of coupling $g\lesssim 10^{-2}$, this gap is negligible
compared to the temperature in PNS.

The grand potential per unit volume for neutrinos is

$\Omega_{\nu}=\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\left[\frac{1}{2}(\omega_{+}^{\nu}+\omega_{-}^{\nu})\right.\\\
\left.+T\log\left(1+e^{-\omega_{+}^{\nu}/T}\right)+T\log\left(1+e^{-\omega_{-}^{\nu}/T}\right)\right],$
(10)

and for $\phi$’s is

$\Omega_{\phi}=\frac{f^{2}}{4}(m_{\phi}^{2}-\mu_{\phi}^{2})-\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\left[\frac{1}{2}(\omega_{+}^{\phi}+\omega_{-}^{\phi})\right.\\\
\left.+T\log\left(1-e^{-\omega_{+}^{\phi}/T}\right)+T\log\left(1-e^{-\omega_{-}^{\phi}/T}\right)\right].$
(11)

The first term on the RHS of Eq. 11 accounts for the condensate, and the
second term contains the contribution from thermal excitations. UV divergences
in eqs. 10 and 11 are regulated by imposing a momentum cutoff while demanding
the Goldstone mode remains gapless with the vev given in eq. 6.

Figure 2: Critical temperature of the superfluid in a gas comprised solely of
$\phi$’s.

In fig. 2 we show the critical temperature as a function of the total lepton
number density $n_{L}=\partial(\Omega_{\nu}+\Omega_{\phi})/\partial\mu_{L}$.
For $T>T_{c}$, the lepton number resides in the thermal population and the
condensate amplitude vanishes $f=0$. Note that the lepton number density in
Fig. 2 is in units of $n_{0}$, and in the PNS where the lepton fraction
$Y_{L}\simeq 0.3$ we expect $n_{L}\simeq 0.1-0.3~{}n_{B}$, and $n_{B}\simeq
1-3~{}n_{0}$. Under these conditions the large $T_{c}$ indicates that
condensation will persist even at the highest temperatures encountered in the
PNS when the lepton fraction is large.

By accommodating a large fraction of the lepton number, the condensate alters
the composition of hot and dense matter in the PNS. At given $n_{B}$, $T$, and
$Y_{L}$, the baryonic and leptonic components are related by the conditions of
beta-equilibrium and charge neutrality:

$\displaystyle\mu_{n}-\mu_{p}$ $\displaystyle=\mu_{e}-\mu_{L}\,,$ (12)
$\displaystyle n_{p}$ $\displaystyle=n_{e}\,.$ (13)

Above, subscripts $n$, $p$, $e$ denotes neutrons, protons, and electrons.
Number densities $n_{i}$ include anti-particle contributions, e.g.,
$n_{e}=n_{e^{-}}-n_{e^{+}}$. In this work we treat electrons as non-
interacting relativistic Fermi gas, and neglect muons for simplicity.

To obtain the composition of hot and dense matter in equilibrium we need to
specify an baryonic equation of state $\mathcal{E}(n_{n},n_{p},T)$, the
nuclear energy density as a function of the neutron and proton number
densities, and the temperature. For a given $\mathcal{E}(n_{n},n_{p},T)$, the
nucleon chemical potentials are given by
$\mu_{n}=(\partial\mathcal{E}(n_{n},n_{p})/\partial n_{n})_{n_{p},T}$, and
$\mu_{p}=(\partial\mathcal{E}(n_{n},n_{p})/\partial n_{p})_{n_{n},T}$. In what
follows we employ the Skyrme energy functional $\mathcal{E}(n_{n},n_{p},T)$
discussed in Ref. [31].

In Fig. 3 we show the constituent particle fractions in the presence of
$\phi$’s at $n_{B}=n_{0}$, $T=20$ MeV, and $Y_{L}=0.3$. The individual number
densities are determined by chemical potentials $\mu_{n}$, $\mu_{p}$,
$\mu_{e}$, and $\mu_{L}$ which are obtained by solving eqs. 12 and 13, subject
to the lepton number budget

$n_{L}=n_{B}Y_{L}=n_{e}+n_{L}^{\nu}+n_{L}^{\phi}$

where $n_{L}^{\nu}={\partial\Omega_{\nu}}/{\partial\mu_{L}}$ and
$n_{L}^{\phi}={\partial\Omega_{\phi}}/{\partial\mu_{L}}$ denote lepton number
densities carried by $\nu$’s and $\phi$’s, respectively.

Effects of $\phi$’s on dense matter composition shown in fig. 3 can be
understood by noting that the condensate amplitude decreases with increasing
$m_{\phi}$. For a given value of $m_{\phi}$, the condensate amplitude is
calculated using eq. 6 and we have set $\lambda=1$. For small $m_{\phi}$, the
condensate amplitude is large and it accommodates a large mount of lepton
number as shown by the green curve. Consequently, the populations of $e$’s
(blue curve) and $\nu$’s (orange curve) are suppressed. Further, owing to
charge neutrality eq. 13, the condensate lowers the proton fraction and
amplifies the isospin asymmetry of PNS matter at early times.

Figure 3: PNS composition as a function of $m_{\phi}$ at $n_{B}=n_{0}$
assuming $T=20$ MeV and $Y_{L}=0.3$. The green and red curve represent lepton
number carried by the condensate and thermal $\phi$’s respectively. We took
$\lambda=1$. For $m_{\phi}\gtrsim 230$ MeV the $L$ budget can no longer
support the superfluid in this scenario.

At larger $m_{\phi}$, the lepton number in the condensate is smaller and
vanishes when $m_{\phi}\geq 2\mu_{L}$. For the chosen parameters, this occurs
at $m_{\phi}\simeq 230$ MeV. At finite temperature, a thermal bath of
excitations also contributes to the lepton number density, and this
contribution is shown by the red curve. It is interesting to note that for
$m_{\phi}\lesssim 100$ MeV, $\phi$ condensation reduces the proton fraction by
about a factor 2.

##  III Implications for neutrino transport

In the standard scenario, the evolution of the PNS and the associated neutrino
signal is largely determined by the diffusion of neutrinos [32, 33, 34].
Neutrino trapping during collapse leads to $Y_{L}\simeq 0.3$ in the bulk of
the PNS, and $\mu_{L}\simeq 200$ MeV in the PNS core. The baryon density,
temperature and lepton fraction profiles that roughly correspond to the state
of the PNS at $t\simeq 1$ s (see Fig. 1 of Ref. [34]) are shown in the top
panel of fig. 4. The spatial gradient of $\mu_{L}$ drives deleptonization by
neutrino diffusion and simulations suggest that it takes about $20$ seconds
for $\mu_{L}$ to decrease substantially [34]. During this time, the lepton
number current carried by neutrinos leads to heating in the core akin to Joule
heating in ordinary conductors [32]. As we discuss below, $\phi$ condensation
fundamentally alters lepton number transport in PNS.

A defining characteristic of a superfluid is that it supports dissipationless
supercurrents that eliminates gradients in the chemical potential. Superflow
redistributes the lepton number in the PNS on a short timescale
$\tau\ll\tau_{\rm diff}$ without the associated Joule heating. Consequently,
the PNS is characterized by a constant lepton number chemical potential in the
star’s gravitational field

$\mu_{L}(r)\sqrt{g_{tt}(r)}=\mu_{L}^{0}=\text{const},$ (14)

where $\mu_{L}(r)$ is the local chemical potential, and $g_{tt}(r)$ is the
time-time component of the metric tensor. The value of $\mu_{L}^{0}$ is set by
the total lepton number ($N_{L,tot}$) in the star which at early times is
$N_{L,tot}\approx 3-5\times 10^{56}$.

Figure 4: Superfluid transports $L$ towards the surface of the star. Solid
curves in the lower panel are obtained by distributing $N_{L,tot}\approx
4\times 10^{56}$ according to eq. 14 in a $1.4M_{\odot}$ star. The underlying
temperature, baryon number, and lepton chemical potential profiles are shown
in the top panel. We chose $m_{\phi}=50$ MeV, $g=10^{-3}$, and $\lambda=1$.

The bottom panel of fig. 4 shows how the superfluid transport of lepton number
alters the composition of the PNS. In the presence of the condensate, it
ensures that the chemical potential $\mu_{L}(r)$ satisfies eq. 14. To obtain
metric functions we solved the Tolman–Oppenheimer–Volkoff equation assuming
the Skyrme equation of state mentioned earlier. At fixed $N_{L,tot}$, curves
shown in the top panel reveals that $\mu_{L}(r)$ in the core is greatly
reduced relative to the standard scenario, and is enhanced in the surface
regions. Thus, in stark contrast to the standard scenario, the electron and
the neutrino fractions are reduced in the core, and increase towards the
surface of the PNS. A comparison between the solid and dashed curves in the
lower panel reveals that the electron (and proton) fraction is reduced by
about a factor of 2 in the core, and is enhanced in the low-density surface
regions. The rapid increase in the neutrino fraction at intermediate radius is
due to thermal effects and is sensitive to chosen temperature profile. We
experimented with a few different initial profiles of $T$ and $\mu_{L}$, and
found that qualitative features of lepton number transport by the superfluid
seen in fig. 4 are robust.

The results shown in fig. 4 are obtained by setting $m_{\phi}=50$ MeV and
$\lambda=1$. It is interesting that the $\phi$ superfluid can extend to large
radii at early times and this is a generic feature for $m_{\phi}\lesssim 50$
MeV. For larger $m_{\phi}$, the condensate recedes to smaller radius and the
region of the PNS influenced by the lepton superfluid is diminished. We can
also expect the spatial extent of the superfluid to decrease with time as
neutrino diffusion in the outer (non-superfluid) regions of the PNS
continuously saps lepton number, causing $\mu_{L}^{0}$ in the condensate to
decrease.

Condensation accelerates deleptonization of the PNS, especially during the
early stages since the timescale for neutrino diffusion is now set by the
neutrino mean free paths in the low density outer regions. However, a
quantitative description of this time evolution will require self-consistent
PNS simulations and is beyond the scope of this study. During this phase,
temperature gradients in the PNS drives neutrino diffusion and all flavors of
neutrinos contribute to the energy flux. We can expect the transport of energy
in the condensate to differ from the standard scenario because the neutrino
mean free paths are altered. In particular, we find that the mean free path
due to the Cherenkov process $\nu\rightarrow\nu J$

$\frac{1}{\lambda(E_{p})}=\frac{1}{E_{p}}\int\frac{\mathrm{d}^{3}p^{\prime}}{2E_{p^{\prime}}(2\pi)^{3}}\frac{\mathrm{d}^{3}k^{\prime}}{2E_{k^{\prime}}(2\pi)^{3}}\\\
\times\left|\mathcal{M}_{\nu\rightarrow\nu
J}\right|^{2}(1-f_{p^{\prime}})(2\pi)^{4}\delta^{4}(p-p^{\prime}-k^{\prime})\,,$
(15)

is significantly shorter than other processes previously considered. In eq.
15, $p=(E_{p},\mathbf{p})$ is the four momentum of the initial neutrino, and
$p^{\prime}$, $k^{\prime}$ are the four-momenta of the neutrino in the final
state and the Goldstone boson, respectively. This process is kinematically
feasible because the Goldstone mode (c.f. eq. 7) exhibits a linear dispersion
with velocity
$v_{J}=\sqrt{(\mu_{\phi}^{2}-m_{\phi}^{2})/(3\mu_{\phi}^{2}-m_{\phi}^{2})}<1$
at small momenta. However, with increasing momenta, when
$|\mathbf{k^{\prime}}|\gtrsim\langle\phi\rangle$ the dispersion relation
resembles the vacuum mode of $\phi$’s and the process is kinematically
forbidden. We therefore require that
$|\mathbf{k^{\prime}}|\lesssim\langle\phi\rangle$ in eq. 15.

Figure 5: Neutrino mean free paths at $n_{B}=n_{0}$ in the condensate depicted
in fig. 4, where $T\approx 27$ MeV and $\mu_{L}\approx 30$ MeV. As $E_{\nu}$
increases, $\nu\rightarrow\nu+J$ becomes forbidden at sufficiently large
Goldstone momenta (see main text), resulting in the rising tail for the
Cherenkov mean free path.

Figure 5 shows the Cherenkov mean free paths for neutrinos and anti-neutrinos
as functions of their energy. For reference, the mean free path due to neutral
and charged current reactions on nucleons are also shown. These mean free
paths are calculated assuming non-relativistic nucleons under the influence of
mean field potentials [35] obtained from the Skyrme model discussed earlier.
Since $\mu$ and $\tau$ neutrinos do not couple to the Goldstone modes in our
simple model we do not expect a similar reduction in their mean free paths.

The drastic shortening, by orders of magnitude, of the electron and anti-
electron mean free paths suggests that energy transport in the PNS are
affected by the condensate. It appears likely that while deleptonization is
accelerated by condensation, the cooling of the PNS during the early times may
be delayed.

The condensate could alter the neutrino energy spectrum at early time when the
superfluid extends to low baryon density (such as the case shown in fig. 4).
We find that for $m_{\phi}\lesssim 50$ MeV, condensation of $\phi$’s is likely
to impact the location and temperature of the $\nu_{e}$ and $\bar{\nu}_{e}$
decoupling surfaces. Short mean free paths in condensed regions imply that
neutrino decoupling must occur at the outer edge of the condensate. In this
region, we can expect a large thermal population of $\nu_{e}$’s and $\phi$’s
due to the finite lepton chemical potential. Further, due to the strong
coupling between $\nu_{e}$’s and $\phi$’s, we speculate that the decoupling of
$\nu_{e}$ could occur in a thin layer and that the $\nu_{e}$ spectrum might
contain a non-thermal component with energy close to $m_{\phi}/2\sim 10$ MeV
arising from decaying $\phi$’s. Note that because the population of
$\phi^{*}$’s is suppressed by $-2\mu_{L}$, the anti-electron neutrino signal
does not receive similar non-thermal contributions. This enhanced difference
between fluxes of $\nu_{e}$ and $\bar{\nu}_{e}$ would alter the neutron-to-
proton ratio in the neutrino driven wind and therefore may have implications
for nucleosynthesis [36]. However, as we have noted earlier these speculations
neglect feedback effects and self-consistent PNS simulations [33, 16] will be
needed to draw definitive conclusions about how the aforementioned effects
will impact the temporal and spectral features of the neutrino signal.

##  IV Conclusion

Our main conclusion is that if lepton number scalars that couple to neutrinos
exist in nature they will condense to form a superfluid in the PNS. For the
conditions realized inside the PNS, condensation is favored for an interesting
range of masses and couplings, and is likely to persist for several seconds
during which lepton number remains trapped. We find that condensation
dramatically alters the composition and transport properties of hot and dense
PNS matter.

This first study of lepton number superfluidity in PNSs suggests that the
thermodynamic and transport properties of matter are dramatically altered. The
large spatial extent of the condensate, and shortened neutrino mean free paths
in the condensed phase suggest that the energy spectrum and the luminosity of
neutrinos will be affected, especially at early times. However, identifying
unique observable signatures in the neutrino signal will require self-
consistent PNS simulations that include lepton number scalars. Such studies
could address how neutrino data from SN1987a, and future galactic supernovae
can help either discover or constrain a dark lepton sector.

Finally, we note that the model employed here neglected the diagonal and off-
diagonal couplings of the $\phi$ to $\mu$ and $\tau$ neutrinos. Their
inclusions could be interesting as it provides additional degrees to
accommodate lepton number. It could have an impact during the infall phase of
core-collapse [37] and the PNS evolution studied here. These issues are being
investigated and will be reported in future work.

## Acknowledgment

We thank Andre de Gouvea and George Fuller for useful comments on the
manuscript. This work is supported by Grant No. DE-FG02-00ER41132 from the
Department of Energy , and the Grant No. PHY-1430152 (JINA Center for the
Evolution of the Elements), and PHY-1630782 (Network for Neutrinos, Nuclear
Astrophysics, and Symmetries (N3AS)) from the National Science Foundation.

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