# Color Me Intrigued: Quantifying Usage of Colors in Fiction

Siyan Li

###### Abstract

We present preliminary results in quantitative analyses of color usage in
selected authors’ works from LitBank. Using Glasgow Norms, human ratings on
5000+ words, we measure attributes of nouns dependent on color terms. Early
results demonstrate a significant increase in noun concreteness over time. We
also propose future research directions for computational literary color
analytics. 111All code and data used are available at
https://github.com/siyan-sylvia-li/ColorLit.

##  1 Introduction

All great writers are great colourists, Virginia Woolf once stated (Woolf
1934). Analyzing colors in literary work across time and authors has
fascinated the field of literature, philosophy, and psychology (Skard 1946).

Most literary analyses of colors focus on only one author, one work, or one
historical era. There has been very few large-scale analyses of color usage
shifts. Recently, natural language processing (NLP) has progressed in fields
potentially relevant to literary color analyses, such as dependency parsing
(Qi et al. 2020) and named entity recognition (Li et al. 2020). Leveraging
these tools expedites localization of spans of interest, increasing efficiency
and ease of larger-scale literary analyses.

What makes literary color analyses interesting for natural language
processing? Authors utilize colors in numerous ways, and NLP tools should
capture this variety. While Goethe uses colors as a backdrop of his
narratives, only using them to emphasize the plastic shapes of objects (Bruns
1928), Dante’s coloring in his Divine Comedy displays more symbolic
undertones. The colors on the three faces of Dante’s Lucifer can be related to
the three horses of the Apocalypse (Skard 1946). The sudden absence of green
in Heavenly Paradise may stem from green’s association with hope, and Dante’s
Paradise eliminates the need for hope since it fulfills all wishes (Austin
1933). In contrast, James Joyce’s green can be interpreted to symbolize
absinthe (Earle 2003) and the author’s frustration with the Irish Catholic
Church (Xie 2015). For more contemporary writers, Virginia Woolf’s blue in To
the Lighthouse accompanies Mrs. Ramsay for her Madonna role and her mixture of
radiance and somberness (Stewart 1985). The same blue manifests cholera and
illness in Edgar Allan Poe’s The Masque of the Red Death (Yoon 2021). Despite
some subjectivity in these interpretations, the existence of differences in
color usages are absolute. We want to examine whether current NLP tools
“understand” these differences.

We propose a novel line of research using word embeddings and pre-trained
language models to quantify color usages in literature. Specifically, we
measure the attributes of nouns dependent on color adjectives according to the
Glasgow Norms (Scott et al. 2019). Preliminary results demonstrate
statistically significant trends over time for certain colors’ Glasgow Norm
attributes. We present future research directions and plausible experiments.

Our proposed framework can supplement literary color analyses research and
provide additional insight for color usages comparisons. Looking at literature
and creativity through the lens of colors is informative because of the
prevalence of color terms in literature. Color terms can serve as anchoring
points of comparison between authors, and potentially between humans and
language models.

##  2 Related Work

The most similar work to ours would be Rabinovich and Carmeli (2022), a study
of color term usage by both non-color-blind and color-blind individuals on
Reddit. The authors discover significant differences in certain color terms.
They then concentrate on the nouns that are modified by color words, using
dependency parsing to obtain NOUN words in an AMOD dependency pair with an ADJ
color term. The authors identify significant discrepancies between the two
populations in imageability (Scott et al. 2019) values of color-modified
nouns. Our preliminary work is methodologically similar, but we study
literature instead. Additionally, our work more extensively leverages labels
from the Glasgow Norms by using three dimensions instead of one.

Word embeddings play a crucial role in computational social science. Garg et
al. (2018) leverages Word2vec (Mikolov et al. 2013) to reflect changes in
relationships between the embedding representing women and different
adjectives as potentially a result of the feminist movement. A similar work,
Bailey, Williams, and Cimpian (2022), showcases that people = man by comparing
distances between word embeddings of trait words of people, men, and women
respectively. We are interested in similar techniques in a literary color
analysis context.

##  3 Dataset

We use LitBank (Bamman, Popat, and Shen 2019), a Euro-centric collection of
100 English fictions from 75 authors. We conduct a scrape of Project Gutenberg
using LitBank’s Gutenberg ID’s to obtain the full text of each work. The
genres consist primarily of realistic novels, with few exceptions of science
fiction (H.G. Wells, Mary Shelley), fantasy (Bram Stoker, Oscar Wilde), and
horror (Edgar Allan Poe).

##  4 Methodology

###  4.1 Extracting Modified Nouns

#### Colors and Synonyms

We select common colors and curate a list of their synonyms. The colors
include “red”, “green”, “black”, “white”, “blue”, “brown”, “gray”, “yellow”,
“pink”, and “purple”. All color terms and their synonyms are in Appendix A.
Each set of sentences from a Project Gutenberg E-book are split into
lemmatized words. We choose sentences containing either our specified color
adjectives or their synonyms for dependency parsing.

#### Dependency Parsing

Although Rabinovich and Carmeli (2022) strictly studies nouns modified by
color terms through the AMOD dependency, this would be limiting in lyrical
writing. For instance, “she has eyes of sapphire” describes blue eyes and
should be included in our analysis, but dependency parsing would categorize
“eyes” and “saphire” to linked by NMOD instead of AMOD. Therefore, we expand
upon our pool of nouns by including all nouns with a dependency link to our
color terms. We employ Stanza’s Dependency Parser (Qi et al. 2020). Upon
obtaining dependencies on a sentence, we perform a filtering process to retain
the relevant head-dependent pairs. The specific filtering process is as
follows. For each head-dependent pair: (1) Lemmatize both the head and the
dependent. (2) Iterate through all color words and their synonyms; if none of
them is present in either the head or the dependent, prune out this pair. (3)
If the other word in the dependency pair is not a noun or a proper noun, prune
out this pair.

Color | # of Occurrences | Color | # of Occurrences
---|---|---|---
red | 2888 | green | 1839
black | 3325 | white | 4990
blue | 1622 | brown | 1206
gray | 1575 | yellow | 848
pink | 648 | purple | 545
Table 1: The total numbers of occurrences of our selected color terms in the
100 LitBank novels. These are instances where the color terms act as either a
dependent or a dependency head.

###  4.2 Glasgow Norm Models

The Glasgow Norms are a list of 5,553 words with corresponding normative human
ratings on different psycholinguistic dimensions. Our ongoing work
concentrates on: (1) Imageability (IMAG), the ease of summoning a mental image
from a word; (2) Concreteness (CNC), the extent to which words can be
experienced to our senses; and (3) Valence (VAL), how positive or negative a
word’s value is. We hypothesize that different authors differ on the
imageability/concreteness/valence values of color-dependent nouns.

Although the Glasgow Norms vocabulary is extensive, we still hope to handle
unseen words. FastText embeddings (Joulin et al. 2016) reduced to 100
dimensions are used to train separate 1-layer Multi-Layer Perceptron (MLP)
models to predict these values. We choose FastText for its adaptability for
unseen words. Prior to training, all scores are normalized to the 0 to 1 range
for better interpretability, consistent with Rabinovich and Carmeli (2022).
Three neural networks with sigmoid activations are trained on these data, and
evaluated on a held-out test set with an 8:1:1 split. We use Pearson’s
correlation between predictions and ground truths as our metric. Rabinovich
and Carmeli (2022) reports a Pearson’s correlation of 0.76 for their IMAG
model on a random held-out set, while ours achieves 0.79 on the test set. We
understand that the held-out test set may be different, but this indicates
that our IMAG model should as potent as the prior model. Our CNC model and VAL
model achieve correlation scores of 0.83 and 0.76, respectively.

To prevent repeated occurrences of a word affecting the average Glasgow Norm
values, the dependent nouns are deduplicated when computing the averages.

##  5 Preliminary Results

Despite our analyses on LitBank yield statistically significant results when
analyzed across time, this could stem from an imbalance in the distribution of
publication time in LitBank. This paper aims to establish a preliminary
framework for studying color usages in literature, and current results would
need corroboration from additional texts from different eras and genres.

###  5.1 Color-dependent Nouns

We conduct both quantitative and qualitative analyses on color-dependent nouns
in LitBank through both computing average Glasgow Norm values and through
inspection of most frequently associated nouns. Additional analyses of color
term frequencies are in Appendix B.

Figure 1: Imageability plots for the color terms with significant results. We
omit the concreteness plots here because IMAG and CNC are highly correlated,
but we provide the full set of plots in Appendix C.

Out of all unique nouns, 1299 are within the Glasgow Norm vocabulary, and 1924
are OOV. We use our trained MLPs to infer Glasgow Norm scales of the out-of-
vocabulary nouns. After recognizing an upward trend in IMAG and CNC, we
compute Pearson’s correlations between publication year and average IMAG and
CNC values for all color terms in novels where the color is present (Table 2,
Figure 1). The IMG and CNC values increase significantly over time for black,
white, yellow, and pink. This indicates that the nouns associated with these
color terms become increasingly concrete easier to conjure mental images of.
This is consistent with some conclusions from Skard (1946) that early color
usages are obscure and abstract.

IMAG Results

Color | Pearson’s r | Color | Pearson’s r
---|---|---|---
red | -0.095* | green | 0.059
black | 0.257** | white | 0.303***
pink | 0.534*** | yellow | 0.340**

CNC Results

Color | Pearson’s r | Color | Pearson’s r
---|---|---|---
black | 0.237** | white | 0.239**
pink | 0.517*** | yellow | 0.318***

VAL Results

Color | Pearson’s r | Color | Pearson’s r
---|---|---|---
green | 0.273*** | purple | 0.210*

Table 2: Pearson’s correlations between published years and Glasgow Norm
values of the 100 fictions for color terms with sig. results. *** $p<0.001$,
** $p<0.05$, and * $p<0.1$. Full results are in Appendix C.

#### Nouns in Individual Works

We observe interesting data points in our plots (the full sets of figures are
available in Appendix C). For instance, we notice a significantly lower
valence for red in Edgar Allan Poe’s The Masque of the Red Death, because the
only nouns associated with red are “death”, “stain”, and “horror”. Similarly,
an abnormally low green valence arises in Henry Fielding’s History of Tom
Jones, a Foundling, because green is attached to “slut”, “witch”, and
“monster”.

#### Nouns over Publication Time

We divide our fictions into pre-1800’s, 1800 - 1900, and post-1900 based on
patterns in our data. We then deduplicate dependent nouns in each work so that
we can measure most frequent nouns in each era without thematic motifs biasing
our analyses. A list of selected color terms and their most dependent nouns in
each of the eras are in Table 3.

Color | Era | Frequent Nouns
---|---|---
pink | Pre-1800 |  | shame, guilt, folly,
---
ribbon, indignation
| 1800 - 1900 | cheek, face, ribbon, rose, lip
| Post-1900 | cheek, face, rose, paper, bud
black | Pre-1800 | color, eye, grain, hair, wave
| 1800 - 1900 | hair, eye, shadow, dress, face
| Post-1900 | hair, eye, dress, figure, man
white | Pre-1800 |  | face, cheek, countenance,
---
cliff, cave
| 1800 - 1900 | face, cheek, hand, hair, man
| Post-1900 | face, hand, eye, hair, man
yellow | Pre-1800 |  | appearance, complexion,
---
blossom, lustre, mist
| 1800 - 1900 | hair, light, face, glove, skin
| Post-1900 | hair, light, eye, flower, house
Table 3: Most frequently color-modified nouns from each era for selected color
terms. Example sentences are in Appendix D

From the table we observe a significant shift in frequently used nouns across
time. Pre-1800 dependent nouns are more abstract and complex compared to
post-1800 nouns dependent on the same colors, while there is no significant
difference between 1800 - 1900 and post-1900. This is a possible explanation
for the increase in imageability and concreteness over time among LitBank
works.

###  5.2 Inter-Author Differences

We plot the Word2vec embeddings of nouns dependent on the same colors from
different authors to decipher how the color terms are used. Out-of-vocabulary
nouns are discarded. This serves as a crude visualization of different topics
associated with these color terms. For instance, when comparing nouns modified
by yellow in works of Fitzgerald and Joyce in LitBank, the topic of facial
hair (hair, beard, pompadour) only manifests in Fitzgerald’s, while food items
(soup, cheese) appear in Joyce’s. Additional examples are in Appendix D.

##  6 Proposal of Future Work

###  6.1 Further Analyses

#### Fine-grained Timeline Analyses

Similar to Garg et al. (2018), we can train separate Word2vec models on each
decade of literature in our collection for fine-grained analyses. Preliminary
results indicate that certain colors become increasingly associated with
concrete descriptions (pink associated with cheeks and face). We can compute
cosine similarities between Word2vec embeddings of certain colors with words
such as “face” and “lips”; this should increase over time as we observe
increasing presence of colors in character descriptions. A similar metric can
further quantize inter-author differences as well.

#### Additional Clustering

We demonstrate the prowess of Word2vec for visualizing color-related topics in
this proposal, but word embeddings fail to account for contexts. Further
clustering analyses can include embeddings from context-dependent pre-trained
language models such as SentenceBERT (Reimers and Gurevych 2019) and BERT
(Devlin et al. 2018).

Figure 2: Word2vec embeddings of nouns modified by yellow in novels by F.
Scott Fitzgerald (red points) and James Joyce (blue points).

###  6.2 RQs: From a Literature Perspective

#### How do colors differ across genres?

We chose LitBank for its ease of access and thorough documentations, but one
emerging issue is that LitBank skews heavily towards realistic fictions. Due
to this imbalance, we cannot compare color usages meaningfully across genres.
We will include more books in future analyses. If we observe significant
differences in frequencies of different color words or in the concepts and
objects associated with the colors, we can conclude that there exists cross-
genre differences in color usage.

We already observe such differences. Bram Stoker’s Dracula (Stoker 1897), a
pioneering work in vampire literature, features numerous descriptions of pale
maidens with their red lips, as well as of scarlet blood and crimson eyes.
Therefore, we notice much more frequent usage of the color red in this work,
compared to H. G. Wells’s The War of the Worlds (Wells 1897), where red most
commonly modifies weeds. We want to inspect whether the same general pattern
would persist on larger-scale analyses.

#### How do colors differ across literary forms?

Literary color analyses often separate novelists and poets. Comparisons are
often drawn only between two poets or two novelists, but rarely both. Our
hypothesis is that colors usage in poetry differs significantly from color
usage in prose and novels. Such differences can manifest as a discrepancy in
concreteness (e.g. in poetry, colors can more often associate with a concept
instead of a concrete object). Expanding our research to include poets,
preferably those contemporary to our novelists, should enable us to address
this systematically.

###  6.3 RQs: From a Social Science Perspective

#### How do colors differ across cultures and classes?

Different cultures can have different associations with the same color; for
instance, white is often a symbol of purity and a staple at Western weddings,
whereas the same color is used more traditionally in Chinese funerals. These
associations may reflect socio-economic classes as well (e.g. white-collar and
blue-collar jobs) through colors frequently co-occurring with characters from
different cultures and societal classes. To analyze this, we will utilize
named entity recognition to link colors to characters, operating with more
context. Pursuing this research direction would involve translated work,
instead of using only Euro-centric collections of literature. The social class
of a character can either be looked up online or inferred by a model to
automate the process. We can then cluster character colors by cultures and
classes.

#### How do colors affect biases and stereotypes?

Current-day color associations, such as pink with girls and blue with boys,
can fuel biases. For instance, boys who enjoy wearing pink may be regarded as
“girly” and overly feminine. Certain colors also contain associations with
LGBTQ+ communities. We are interested in identifying how these color-based
biases (going beyond race) manifest in literature and online communities. We
can study this by finding colors associated with characters of a demographic
group of interest. Tracing through past literature may also shed light upon
the evolution of color associations.

##  7 Discussion

Our work serves as a step towards more systematic analyses of color usages in
literature using natural language processing tools. Following prior work, we
propose using The Glasgow Norms and word embeddings as tools for quantifying
color usage differences. We demonstrate significant increasing trends in
imageability and concreteness in color-dependent nouns over time.

One limitation is the range of language we are capable of handling is
constrained by the language models we employ. Our current collection does not
have many pre-1800 pieces. While it is possible to increase the representation
of pre-1800s literature, the domain shift in English style and word
conventions may require different word embedding and pre-trained models to
embed narratives (e.g. Chaucer often uses “red” in place of “read”, standard
of his time, but causes ambiguity when processing texts on a large scale).
Given such shifts in vocabulary and sentence structures, we may fail to
provide meaningful insights into earlier literature, since the word embeddings
may have disjoint vocabularies, and models such as SentenceBERT are trained on
more modern texts.

## References

  * Austin (1933) Austin, H. D. 1933.  Heavenly Gold; A Study of the Use of Color in Dante.  _Philological Quarterly_ , 12: 44.
  * Bailey, Williams, and Cimpian (2022) Bailey, A. H.; Williams, A.; and Cimpian, A. 2022.  Based on billions of words on the internet, people= men.  _Science Advances_ , 8(13): eabm2463.
  * Bamman, Popat, and Shen (2019) Bamman, D.; Popat, S.; and Shen, S. 2019.  An annotated dataset of literary entities.  In _Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers)_ , 2138–2144.
  * Bruns (1928) Bruns, F. 1928.  Auge und Ohr in Goethes Lyrik.  _The Journal of English and Germanic Philology_ , 27(3): 325–360.
  * Devlin et al. (2018) Devlin, J.; Chang, M.-W.; Lee, K.; and Toutanova, K. 2018.  Bert: Pre-training of deep bidirectional transformers for language understanding.  _arXiv preprint arXiv:1810.04805_.
  * Earle (2003) Earle, D. M. 2003.  ”Green Eyes, I See You. Fang, I Feel”: The Symbol of Absinthe in ”Ulysses”.  _James Joyce Quarterly_ , 40(4): 691–709.
  * Garg et al. (2018) Garg, N.; Schiebinger, L.; Jurafsky, D.; and Zou, J. 2018.  Word embeddings quantify 100 years of gender and ethnic stereotypes.  _Proceedings of the National Academy of Sciences_ , 115(16): E3635–E3644.
  * Joulin et al. (2016) Joulin, A.; Grave, E.; Bojanowski, P.; Douze, M.; Jégou, H.; and Mikolov, T. 2016.  FastText.zip: Compressing text classification models.  _arXiv preprint arXiv:1612.03651_.
  * Li et al. (2020) Li, J.; Sun, A.; Han, J.; and Li, C. 2020.  A survey on deep learning for named entity recognition.  _IEEE Transactions on Knowledge and Data Engineering_ , 34(1): 50–70.
  * Mikolov et al. (2013) Mikolov, T.; Chen, K.; Corrado, G.; and Dean, J. 2013.  Efficient estimation of word representations in vector space.  _arXiv preprint arXiv:1301.3781_.
  * Qi et al. (2020) Qi, P.; Zhang, Y.; Zhang, Y.; Bolton, J.; and Manning, C. D. 2020.  Stanza: A Python natural language processing toolkit for many human languages.  _arXiv preprint arXiv:2003.07082_.
  * Rabinovich and Carmeli (2022) Rabinovich, E.; and Carmeli, B. 2022.  Exploration of the Usage of Color Terms by Color-blind Participants in Online Discussion Platforms.  _arXiv preprint arXiv:2210.11905_.
  * Reimers and Gurevych (2019) Reimers, N.; and Gurevych, I. 2019.  Sentence-bert: Sentence embeddings using siamese bert-networks.  _arXiv preprint arXiv:1908.10084_.
  * Scott et al. (2019) Scott, G. G.; Keitel, A.; Becirspahic, M.; Yao, B.; and Sereno, S. C. 2019.  The Glasgow Norms: Ratings of 5,500 words on nine scales.  _Behavior research methods_ , 51(3): 1258–1270.
  * Skard (1946) Skard, S. 1946.  The Use of Color in Literature: A Survey of Research.  _Proceedings of the American Philosophical Society_ , 90(3): 163–249.
  * Stewart (1985) Stewart, J. F. 1985.  Color in To the Lighthouse.  _Twentieth Century Literature_ , 31(4): 438–458.  Publisher: [Duke University Press, Hofstra University].
  * Stoker (1897) Stoker, B. 1897.  Dracula: 1897.
  * Wells (1897) Wells, H. G. 1897.  _The war of the worlds_.  Broadview Press.
  * Woolf (1934) Woolf, V. 1934.  _Walter Sickert: A Conversation_.  L. and Virginia Woolf at the Hogarth Press.
  * Xie (2015) Xie, Y. 2015.  Color as Metaphor-A Study of Joyce’s Use of” Black” and” Green” in Dubliners and A Portrait of the Artist as a Young Man.  _English Language and Literature Studies_ , 5(4): 61.
  * Yoon (2021) Yoon, S. 2021.  Color Symbolisms of Diseases: Edgar Allan Poe’s “The Masque of the Red Death”.  _The Explicator_ , 79(1-2): 21–24.

##  Appendix A Color terms and their synonyms used

We list the color terms used in this work, along with their corresponding
synonyms obtained through manual inspection on Thesaurus.com entries, in Table
4.

Color Word | Synonyms
---|---
red |  | cardinal, coral, crimson, maroon,
---
burgundy, flaming, scarlet, fuchsia
green |  | emerald, olive, aquamarine, beryl,
---
jade, lime
black |  | ebony, jet, obsidian, onyx, inky
---
white |  | alabaster, ashen, blanched, bleached,
---
cadaverous, doughy, pale, pallid,
pasty, ivory, pearly, beige
blue |  | azure, indigo, sapphire, cerulean,
---
cobalt, turquoise, teal
brown |  | amber, khaki, tan, umber, hazel
---
gray | grey
yellow | N/A
pink |  | rosy, blush, magenta
---
purple |  | lavender, lilac, mauve, periwinkle,
---
plum, violet, amethyst
Table 4: List of color words used and their synonyms

##  Appendix B Normalized Frequencies Over Time

After filtering through sentences in LitBank for relevant dependence pairs,
the absolute frequencies of each color term are listed in Table 1. We also
calculate the normalized frequencies (absolute frequency divided by the total
number of words) for the 100 fictions. We notice an increasing trend in the
normalized frequencies over time for LitBank, and compute Pearson’s
correlations between normalized frequencies of color terms and time to verify
this trend (Table 5).

Color | Pearson’s r | Color | Pearson’s r
---|---|---|---
red | 0.173* | green | 0.134
black | 0.139 | white | 0.168*
blue | 0.148 | brown | 0.202**
gray | 0.162 | yellow | 0.184*
pink | 0.173* | purple | 0.150
Table 5: Pearson’s correlations between published years and normalized
frequencies of the 100 fictions for each color term. ** indicates $p<0.05$,
and * $p<0.1$.

We present the scatter plots of normalized frequencies in Figure 3.

##  Appendix C Glasgow Norm Plots of Modified Nouns Over Time

The plots corresponding to how color-modified nouns change over time with
respect to their Glasgow Norm IMAG, CNC, and VAL values are presented here in
Figures 4, 5, and 6. We also include the full list of Pearson correlation
results for all color terms in Table 6.

Figure 3: Log-scale scatter plots of normalized color term frequencies for
each publication in LitBank.

Figure 4: IMAG scatter plots of nouns modified by color terms for each
publication in LitBank.

Figure 5: CNC scatter plots of nouns modified by color terms for each
publication in LitBank.

Figure 6: VAL scatter plots of nouns modified by color terms for each
publication in LitBank.

IMAG Results

Color | Pearson’s r | Color | Pearson’s r
---|---|---|---
red | -0.095* | green | 0.059
black | 0.257** | white | 0.303***
blue | -0.163 | brown | 0.129
gray | -0.081 | yellow | 0.340**
pink | 0.534*** | purple | 0.162

CNC Results

Color | Pearson’s r | Color | Pearson’s r
---|---|---|---
red | -0.054 | green | 0.160
black | 0.237** | white | 0.239**
blue | -0.133 | brown | 0.175
gray | -0.044 | yellow | 0.318***
pink | 0.517*** | purple | 0.055

VAL Results

Color | Pearson’s r | Color | Pearson’s r
---|---|---|---
red | -0.025 | green | 0.273***
black | 0.059 | white | 0.161
blue | -0.104 | brown | -0.148
gray | 0.072 | yellow | -0.01
pink | 0.081 | purple | 0.210*

Table 6: Full list of Pearson’s correlations between published years and
Glasgow Norm values of the 100 fictions for each color term. *** $p<0.001$, **
$p<0.05$, and * $p<0.1$.

##  Appendix D Example of Color Term Usages

In the following usage examples, the color terms are underlined, bolded and
colored, while the dependent noun is bolded.

###  D.1 Pink

HENRY FIELDING, 1749

Adorned with all the charms in which nature can array her; bedecked with
beauty, youth, sprightliness, innocence, modesty, and tenderness, breathing
sweetness from her  rosy lips, and darting brightness from her sparkling eyes,
the lovely Sophia comes!

FANNY BURNEY, 1778

We were sitting in this manner, he conversing with all gaiety, I looking down
with all foolishness, when that fop who had first asked me to dance, […] I
interrupted him I  blush for my folly, with laughing; yet I could not help it;
for, added to the man’s stately foppishness, […] that I could not for my life
preserve my gravity.

ANN WARD RADCLIFFE, 1794

As they descended, they saw at a distance, on the right, one of the grand
passes of the Pyrenees into Spain, gleaming with its battlements and towers to
the splendour of the setting rays, yellow tops of woods colouring the steeps
below, while far above aspired the snowy points of the mountains, still
reflecting a  rosy hue.

Conscious innocence could not prevent a blush from stealing over Emily’s
cheek; she trembled, and looked confusedly under the bold eye of Madame
Cheron, who blushed also; but hers was the  blush of triumph, such as
sometimes stains the countenance of a person, congratulating himself on the
penetration which had taught him to suspect another, and who loses both pity
for the supposed criminal, and indignation of his guilt, in the gratification
of his own vanity

CHARLOTTE BRONTE, 1847

Her beauty, her  pink cheeks and golden curls, seemed to give delight to all
who looked at her, and to purchase indemnity for every fault.

Its garden, too, glowed with flowers: hollyhocks had sprung up tall as trees,
lilies had opened, tulips and roses were in bloom; the borders of the little
beds were gay with  pink thrift and crimson double daisies; the sweetbriars
gave out, morning and evening, their scent of spice and apples; and these
fragrant treasures were all useless for most of the inmates of Lowood, except
to furnish now and then a handful of herbs and blossoms to put in a coffin.

She pulled out of her box, about ten minutes ago, a little  pink silk frock;
rapture lit her face as she unfolded it; coquetry runs in her blood, blends
with her brains, and seasons the marrow of her bones.

ARTHUR CONAN DOYLE, 1892

That trick of staining the fishes’ scales of a delicate  pink is quite
peculiar to China.

Her violet eyes shining, her lips parted, a  pink flush upon her cheeks, all
thought of her natural reserve lost in her overpowering excitement and
concern.

As we approached, the door flew open, and a little blonde woman stood in the
opening, clad in some sort of light mousseline de soie, with a touch of fluffy
pink chiffon at her neck and wrists.

H.G. WELLS, 1897

It was the fact that all his forehead above his blue glasses was covered by a
white bandage, and that another covered his ears, leaving not a scrap of his
face exposed excepting only his  pink, peaked nose.

P. G. WODEHOUSE, 1919

Jimmy turned into that drug store at the top of the Haymarket at which so many
Londoners have found healing and comfort on the morning after, and bought the
pink drink for which his system had been craving since he rose from bed.

The clerk had finished writing the ticket, and was pressing labels and a  pink
paper on him.

F. SCOTT FITZGERALD, 1920:

Myra sprang up, her cheeks  pink with bruised vanity, the great bow on the
back of her head trembling sympathetically.

###  D.2 White

FANNY BURNEY, 1778

She told her niece, that she had been indulging in fanciful sorrows, and
begged she would have more regard for decorum, than to let the world see that
she could not renounce an improper attachment; at which Emily’s  pale cheek
became flushed with crimson, but it was the blush of pride, and she made no
answer.

MARY WOLLSTONECRAFT, 1788

Henry started at the sight of her altered appearance; the day before her
complexion had been of the most  pallid hue; but now her cheeks were flushed,
and her eyes enlivened with a false vivacity, an unusual fire.

JANE AUSTEN, 1813

Her  pale face and impetuous manner made him start, and before he could
recover himself to speak, she, in whose mind every idea was superseded by
Lydia’s situation, hastily exclaimed, “I beg your pardon, but I must leave
you.

From his garden, Mr. Collins would have led them round his two meadows; but
the ladies, not having shoes to encounter the remains of a  white frost,
turned back

ELIZABETH CLEGHORN GASKELL, 1855

She lay curled up on the sofa in the back drawing room in Harley Street
looking very lovely in her  white muslin and blue ribbons.

She found out that the water in the urn was cold, and ordered up the great
kitchen tea kettle; the only consequence of which was that when she met it at
the door, and tried to carry it in, it was too heavy for her, and she came in
pouting, with a black mark on her muslin gown, and a little round  white hand
indented by the handle, which she took to show to Captain Lennox, just like a
hurt child, and, of course, the remedy was the same in both cases.

But he had the same large, soft eyes as his daughter, eyes which moved slowly
and almost grandly round in their orbits, and were well veiled by their
transparent  white eyelids.

CHARLES DICKENS, 1861

I lighted my fire, which burnt with a raw  pale flare at that time of the
morning, and fell into a doze before it.

She’s all in white,’ he says, ‘wi’  white flowers in her hair, and she’s awful
mad, and she’s got a shroud hanging over her arm, and she says she’ll put it
on me at five in the morning.’

When we was put in the dock, I noticed first of all what a gentleman Compeyson
looked, wi’ his curly hair and his black clothes and his  white pocket
handkercher, and what a common sort of a wretch I looked.

ELIZABETH VON ARNIM, 1898

There are so many bird cherries round me, great trees with branches sweeping
the grass, and they are so wreathed just now with  white blossoms and
tenderest green that the garden looks like a wedding.

When that time came, and when, before it was over, the acacias all blossomed
too, and four great clumps of  pale, silvery pink peonies flowered under the
south windows, I felt so absolutely happy, and blest, and thankful, and
grateful, that I really cannot describe it.

###  D.3 Black

JONATHAN SWIFT, 1726

In his right waistcoat pocket we found a prodigious bundle of white thin
substances, folded one over another, about the bigness of three men, tied with
a strong cable, and marked with black figures; which we humbly conceive to be
writings, every letter almost half as large as the palm of our hands.

In the left pocket were two black pillars irregularly shaped: we could not,
without difficulty, reach the top of them, as we stood at the bottom of his
pocket.

In one of these cells were several globes, or balls, of a most ponderous
metal, about the bigness of our heads, and requiring a strong hand to lift
them: the other cell contained a heap of certain black grains, but of no great
bulk or weight, for we could hold above fifty of them in the palms of our
hands.

About two or three days before I was set at liberty, as I was entertaining the
court with this kind of feat, there arrived an express to inform his majesty,
that some of his subjects, riding near the place where I was first taken up,
had seen a great black substance lying on the ground, […] they would undertake
to bring it with only five horses.

FANNY BURNEY, 1778

You can’t think how oddly my head feels; full of powder and black pins, and a
great cushion on the top of it.

if you’ll say that, you’ll say anything: however, if you swear till you’re
black in the face, I sha’n’t believe you; for nobody sha’n’t persuade me out
of my senses, that I’m resolved.

Ridiculous, I told him, was a term which he would find no one else do him the
favour to make use of, in speaking of the horrible actions belonging to the
old story he made so light of; ’actions’ continued I, ’which would dye still
deeper the black annals of Nero or Caligula.

JANE AUSTEN, 1813

The ladies were somewhat more fortunate, for they had the advantage of
ascertaining from an upper window, that he wore a blue coat and rode a black
horse.

CHARLOTTE BRONTE, 1847

And then she had such a fine head of hair; raven black and so becomingly
arranged: a crown of thick plaits behind, and in front the longest, the
glossiest curls I ever saw.

Afternoon arrived: Mrs. Fairfax assumed her best black satin gown, her gloves,
and her gold watch; for it was her part to receive the company,- to conduct
the ladies to their rooms, Adele, too, would be dressed: though I thought she
had little chance of being introduced to the party that day at least.

Fluttering veils and waving plumes filled the vehicles; two of the cavaliers
were young, dashing looking gentlemen; the third was Mr. Rochester, on his
black horse, Mesrour, Pilot bounding before him; at his side rode a lady, and
he and she were the first of the party.

The noble bust, the sloping shoulders, the graceful neck, the dark eyes and
black ringlets were all there; - but her face?

Mrs. Fairfax was summoned to give information respecting the resources of the
house in shawls, dresses, draperies of any kind; and certain wardrobes of the
third storey were ransacked, and their contents, in the shape of brocaded and
hooped petticoats, satin sacques, black modes, lace lappets, etc, were brought
down in armfuls by the abigails; then a selection was made, and such things as
were chosen were carried to the boudoir within the drawing room.

NATHANIEL HAWTHORNE, 1850

Doubtless, however, either of these stern and black browed Puritans would have
thought it quite a sufficient retribution for his sins that, after so long a
lapse of years, the old trunk of the family tree, with so much venerable moss
upon it, should have borne, as its topmost bough, an idler like myself.

The Custom House marker imprinted it, with a stencil and black paint, on
pepper bags, and baskets of anatto, and cigar boxes, and bales of all kinds of
dutiable merchandise, in testimony that these commodities had paid the impost,
and gone regularly through the office.

Before this ugly edifice, and between it and the wheel track of the street,
was a grass plot, much overgrown with burdock, pig weed, apple pern, and such
unsightly vegetation, which evidently found something congenial in the soil
that had so early borne the black flower of civilised society, a prison.

CHARLES DICKENS, 1861

I still held her forcibly down with all my strength, like a prisoner who might
escape; and I doubt if I even knew who she was, or why we had struggled, or
that she had been in flames, or that the flames were out, until I saw the
patches of tinder that had been her garments no longer alight but falling in a
black shower around us.

The sudden exclusion of the night, and the substitution of black darkness in
its place, warned me that the man had closed a shutter.

He had a boat cloak with him, and a black canvas bag; and he looked as like a
river pilot as my heart could have wished.

The marshes were just a long black horizontal line then, as I stopped to look
after him; and the river was just another horizontal line, not nearly so broad
nor yet so black; and the sky was just a row of long angry red lines and dense
black lines intermixed.

G.K. CHESTERTON, 1908

The tall hat and long frock coat were black; the face, in an abrupt shadow,
was almost as dark.

He wore an old fashioned black chimney pot hat; he was wrapped in a yet more
old fashioned cloak, black and ragged; and the combination gave him the look
of the early villains in Dickens and Bulwer Lytton.

A long, lean, black cigar, bought in Soho for twopence, stood out from between
his tightened teeth, and altogether he looked a very satisfactory specimen of
the anarchists upon whom he had vowed a holy war.

He had a black French beard cut square and a black English frock coat cut even
squarer.

SOMERSET W. MAUGHAM, 1915

She wore a black dress, and her only ornament was a gold chain, from which
hung a cross.

It was a large black stove that stood in the hall and was only lighted if the
weather was very bad and the Vicar had a cold.

And the poor lady, so small in her black satin, shrivelled up and sallow, with
her funny corkscrew curls, took the little boy on her lap and put her arms
around him and wept as though her heart would break.

###  D.4 Yellow

DANIEL DEFOE, 1719

The colour of his skin was not quite black, but very tawny; and yet not an
ugly,  yellow, nauseous tawny, as the Brazilians and Virginians, and other
natives of America are, but of a bright kind of a dun olive colour, that had
in it something very agreeable, though not very easy to describe.

JONATHAN SWIFT, 1726

The projector of this cell was the most ancient student of the academy; his
face and beard were of a pale  yellow; his hands and clothes daubed over with
filth.

The hair of both sexes was of several colours, brown, red, black, and  yellow.

I forgot another circumstance (and perhaps I might have the reader’s pardon if
it were wholly omitted), that while I held the odious vermin in my hands, it
voided its filthy excrements of a  yellow liquid substance all over my
clothes; but by good fortune there was a small brook hard by, where I washed
myself as clean as I could; although I durst not come into my master’s
presence until I were sufficiently aired.

WILLIAM MAKEPEACE THACKERAY, 1848

Joseph still continued a huge clattering at the poker and tongs, puffing and
blowing the while, and turning as red as his  yellow face would allow him.

Why, he had the  yellow fever three times; twice at Nassau, and once at St.

At one end of the hall is the great staircase all in black oak, as dismal as
may be, and on either side are tall doors with stags’ heads over them, leading
to the billiard room and the library, and the great  yellow saloon and the
morning rooms.

He’s never content unless he gets my  yellow sealed wine, which costs me ten
shillings a bottle, hang him!

JAMES JOYCE, 1914

I liked the last best because its leaves were  yellow.

Breakfast was over in the boarding house and the table of the breakfast room
was covered with plates on which lay  yellow streaks of eggs with morsels of
bacon fat and bacon rind.

An immense scarf of peacock blue muslin was wound round her hat and knotted in
a great bow under her chin; and she wore bright  yellow gloves, reaching to
the elbow.

His face, shining with raindrops, had the appearance of damp  yellow cheese
save where two rosy spots indicated the cheekbones.

While the point was being debated a tall agile gentleman of fair complexion,
wearing a long  yellow ulster, came from the far end of the bar.

I saw that he had great gaps in his mouth between his  yellow teeth.

On the closed square piano a pudding in a huge  yellow dish lay in waiting and
behind it were three squads of bottles of stout and ale and minerals, drawn up
according to the colours of their uniforms, the first two black, with brown
and red labels, the third and smallest squad white, with transverse green
sashes.

A dull  yellow light brooded over the houses and the river; and the sky seemed
to be descending.

F. SCOTT FITZGERALD, 1920

The Gothic halls and cloisters were infinitely more mysterious as they loomed
suddenly out of the darkness, outlined each by myriad faint squares of  yellow
light.

His face was cast in the same  yellow wax as in the cafe, neither the dull,
pasty color of a dead man—rather a sort of virile pallor—nor unhealthy, you’d
have called it; but like a strong man who’d worked in a mine or done night
shifts in a damp climate.

The two hours’ ride were like days, and he nearly cried aloud with joy when
the towers of Princeton loomed up beside him and the  yellow squares of light
filtered through the blue rain.

Browsing in her library, Amory found a tattered gray book out of which fell a
yellow sheet that he impudently opened.

There was that shade of glorious  yellow hair, the desire to imitate which
supports the dye industry.

F. SCOTT FITZGERALD, 1922

That this feeble, unintelligent old man was possessed of such power that,
yellow journals to the contrary, the men in the republic whose souls he could
not have bought directly or indirectly would scarcely have populated White
Plains, seemed as impossible to believe as that he had once been a pink and
white baby.

He bulges in other places his paunch bulges, prophetically, his words have an
air of bulging from his mouth, even his dinner coat pockets bulge, as though
from contamination, with a dog eared collection of time tables, programmes,
and miscellaneous scraps on these he takes his notes with great screwings up
of his unmatched  yellow eyes and motions of silence with his disengaged left
hand.

It was a crackling dusk when they turned in under the white façade of the
Plaza and tasted slowly the foam and  yellow thickness of an egg nog.

He had fixed his aunt with the bright  yellow eye, giving her that acute and
exaggerated attention that young males are accustomed to render to all females
who are of no further value.

On the appointed Wednesday in February Anthony had gone to the imposing
offices of Wilson, Hiemer and Hardy and listened to many vague instructions
delivered by an energetic young man of about his own age, named Kahler, who
wore a defiant  yellow pompadour, and in announcing himself as an assistant
secretary gave the impression that it was a tribute to exceptional ability.

Joe Hull had a  yellow beard continually fighting through his skin and a low
voice which varied between basso profundo and a husky whisper.


# The mass of charged pions in neutron star matter

Bryce Fore  Institute for Nuclear Theory and Department of Physics, University
of Washington, Seattle, WA 98195.  Norbert Kaiser  Physik-Department T39,
Technische Universität München, D-85747 Garching, Germany  Sanjay Reddy
Institute for Nuclear Theory and Department of Physics, University of
Washington, Seattle, WA 98195.  Neill C. Warrington  Institute for Nuclear
Theory and Department of Physics, University of Washington, Seattle, WA 98195.

###### Abstract

We examine the behavior of charged pions in neutron-rich matter using heavy-
baryon chiral perturbation theory. This study is motivated by the prospect
that pions, or pion-like, excitations, may be relevant in neutron-rich matter
encountered in core-collapse supernovae and neutron star mergers. We find, as
previously expected, that the $\pi^{-}$ mass increases with density and
precludes s-wave condensation, and the mass of the $\pi^{+}$ mode decreases
with increasing density. The energy difference between neutrons and protons in
neutron-rich matter related to the nuclear symmetry energy alters the power
counting. It enhances higher-order contributions to the pion self-energy.
Previously unimportant but attractive diagrams are now enhanced. The net
effect of this is the appearance of a new collective mode with the quantum
numbers of the $\pi^{+}$, and a pronounced reduction of the $\pi^{+}$ mass.

††preprint: INT-PUB-21-035

##  I Introduction

The study of pions in dense nuclear matter has a long and storied history. In
a pioneering work in the 1970s, Sawyer and Scalapino proposed that pion
condensation might lower the energy of nuclear matter at high density Sawyer
(1972); Scalapino (1972); Sawyer and Scalapino (1973). In subsequent years,
several authors studied its manifestation and implications for nuclei and
neutron stars in some detail using simple models for the pion-nucleon
interaction Migdal (1973); Baym (1973); Baym and Flowers (1974); Chi-Kwan Au
and Baym (1974); Dashen and Manassah (1974); Barshay and Brown (1973); Weise
and Brown (1975); Backman and Weise (1975). Pion condensation occurs at low
temperatures when the energy to produce a pion is less than its associated
chemical potential. Earlier work has addressed the possibility of condensation
of negatively charged pions in neutron stars because the chemical potential
for negative charge, which we denote throughout as $\hat{\mu}$, increases with
density and is typically in the range $100-300$ MeV in the neutron star core.
In the absence of interactions, a pion condensate with zero-momentum will
occur when $\hat{\mu}\geq m_{\pi}$, where $m_{\pi}\simeq 140$ MeV is the mass
of the pion. However, repulsive s-wave interactions between $\pi^{-}$ and
neutrons increased the pion energy in neutron stars and were believed to
disfavor condensation of zero-momentum pions.

In contrast, a strongly attractive p-wave interaction between pions and
nucleons was shown to favor condensation of pions with momentum $k_{\pi}\simeq
m_{\pi}$ and lead to a non-uniform ground state Chi-Kwan Au and Baym (1974).
In the 1980s, more sophisticated but model-dependent analyses, including many-
body corrections and correlations between nucleons at short distances, found
that even p-wave condensation may not be robust at the densities encountered
in neutron stars (for a comprehensive review see Ericson and Weise (1988)).
Consequently, the interest in pion condensation has waned over the years.

In this paper, we revisit the calculation of the energy of zero-momentum
charged pions in dense neutron-rich matter using heavy-baryon chiral
perturbation theory (HB$\chi$PT) at two-loop order. We do so for the following
reasons. First, earlier calculations of the energy shift due to the nuclear
medium were computed in the mean-field approximation and based on simple
models for the pion-nucleon interaction, which were poorly constrained by
pion-nucleon scattering data. Second, earlier calculations also neglected the
effect of the nuclear symmetry energy, which induces a large energy difference
between neutrons and protons in neutron-rich matter. Finally, even in the
absence of pion condensation, the mass of pionic excitations in the medium is
relevant to the description of the ground state and response properties of
dense nuclear matter at finite temperatures realized in extreme astrophysical
phenomena such as neutron star mergers and core-collapse supernovae Fore and
Reddy (2020).

We present results for all two-loop diagrams that contribute to the pion self-
energy in dense neutron-rich matter using heavy-baryon chiral perturbation
theory (HB$\chi$PT) and augment these diagrams with two pieces of modeling.
First, we incorporate a model of the nucleon self-energy into all nucleon
lines. This nucleon self-energy incorporates energy shifts of the neutrons
$\&$ protons in the dense nuclear medium. We additionally replace bare $\pi
N\rightarrow\pi N$ vertices with their full T-matrix counterparts to account
for the energy dependence of the pion self-energy. Such T-matrix modifications
have proven crucial in understanding the binding energies of pionic atoms
Kolomeitsev  _et al._ (2003); Kaiser and Weise (2001). We find that
previously-neglected two-loop diagrams (involving p-wave interactions in the
intermediate state) and the inclusion of in-medium nucleon self-energies lower
the pion self-energy relative to previous estimates. In particular, small
energy denominators in perturbation theory are produced when the energy
difference between neutrons and protons becomes of the order of the pion mass.
These small energy denominators promote the importance of certain attractive
Feynman diagrams, which lower the $\pi^{-}$ energy; this modified power
counting is one of the central findings of this work. Modified power counting
is further supported by a recent analysis of experimental data suggesting that
the nuclear symmetry energy - the in-medium energy difference between protons
and neutrons - can be large at densities of interest to neutron stars.
Finally, we find a collective mode with the quantum numbers of the $\pi^{+}$
in the medium.

The calculation presented here improves upon the systematic calculations of
the pion self-energy in asymmetric matter using $\chi$PT presented in Refs.
Kaiser and Weise (2001); Kolomeitsev  _et al._ (2003). We include all diagrams
considered by previous authors while including several others. Interestingly,
we find that the self-energy for a pion at rest is insensitive to the short-
distance, repulsive, spin-isospin correlations between nucleons that are known
to play a key role in inhibiting p-wave condensation Barshay and Brown (1973).
This feature, again, lowers the $\pi^{-}$ self-energy relative to previous
estimates.

The material in this paper is organized as follows. In Section II, we describe
our two-loop HB$\chi$PT calculation of the pion self-energy in isospin-
asymmetric dense matter. In Section III, we present a parametric model to
account for strong interactions between nucleons and study its effect on the
pion self-energy. Finally, in Section V, we summarize our findings and discuss
their limitations.

##  II Pion self-energy in neutron-rich matter from HB$\chi$PT

The dispersion relation of the pion and pion-like collective excitations in
dense neutron-rich matter is obtained from the poles of the in-medium pion
propagator. Employing $\phi(x)$ as the field operator that destroys a
$\pi^{-}$ particle, we adopt the standard definition for the propagator:

$iD(\omega,k)=\int d^{4}x~{}\langle
T\phi(x)\phi^{\dagger}(0)\rangle~{}e^{i(\omega
x_{0}-\vec{k}\cdot\vec{x})}=\frac{i}{\omega^{2}-k^{2}-m_{\pi}^{2}-\Pi(\omega,k)}~{},$
(1)

with $k=|\vec{k}|$. The quantity $\Pi(\omega,k)$ is the pion self-energy in
nuclear matter and depends implicitly on the baryon and isospin densities. The
dispersion relations for both the $\pi^{-}$ and $\pi^{+}$, as well as that of
pion-like collective modes, can be obtained as solutions of the equation

$\omega^{2}-k^{2}-m_{\pi}^{2}-\Pi(\omega,k)=0\,,$ (2)

where $\omega$ and $k$ are the pion energy and momentum. The sign of the
residue at the pole determines the particle’s charge corresponding to a given
pole Ericson and Weise (1988).

In this study, we will consider the dispersion relation of a pion at rest in
the medium, denoting $\Pi(\omega,0)=\Pi(\omega)$. This restriction comes from
our initial motivation of exploring s-wave pion condensation in neutron stars;
however, as we will see later on, we find no evidence for this. Nevertheless,
$\Pi(\omega)$ gives access to long-wavelength properties of pion-like
excitations in dense matter. To calculate $\Pi(\omega)$, we will need a
description of pion-nucleon, pion-pion, and nucleon-nucleon interactions,
choosing HB$\chi$PT as a systematic framework to accomplish this. This
effective field theory organizes all interactions consistent with chiral
symmetry in a derivative expansion and is expected to be reliable when the
pion and nucleon momenta are much smaller than $\Lambda\simeq 4\pi
f_{\pi}\simeq 1$ GeV, where $f_{\pi}=92.2$ MeV is the pion decay constant.

The energy and momentum scales relevant to our calculation are set by the pion
energy $\omega=\hat{\mu}\simeq m_{\pi}$, and the nucleon Fermi momentum
$k_{F}=(3\pi^{2}n_{B}/2)^{1/3}\simeq
263\,$MeV$\,(n_{B}/n_{\mathrm{sat}})^{1/3}$, with $n_{\mathrm{sat}}\simeq
0.16\,$fm-3 the saturation density. In what follows, we calculate
$\Pi(\omega)$ by retaining terms up to order $k_{F}^{5}$ and
$\omega_{\pi^{-}}^{2}$. For this purpose, it is adequate to work with the
following heavy-baryon Lagrangian Fettes  _et al._ (1998); Epelbaum  _et al._
(2009)

$\begin{split}\mathcal{L}^{(2)}_{\pi\pi}+\mathcal{L}^{(1)}_{\pi
N}+\mathcal{L}^{(2)}_{\pi N}+\mathcal{L}^{(1)}_{\pi NN}\,,\end{split}$ (3)

where the superscript denotes the chiral dimension, i.e., the number of
derivatives and/or insertions of the pion mass. Interactions between pions at
leading order are given by

$\begin{split}\mathcal{L}^{(2)}_{\pi\pi}=&\frac{f_{\pi}^{2}}{4}~{}{\rm
tr}\big{(}\partial_{\mu}U\partial^{\mu}U^{\dagger}+\chi_{+}\big{)}\,,\end{split}$
(4)

with the pion fields $(\pi^{+},\pi^{0},\pi^{-})$ contained in the special
unitary $2\times 2$ matrix

$\displaystyle U$
$\displaystyle=\exp{\left(\frac{i}{f_{\pi}}\begin{bmatrix}\pi^{0}&\sqrt{2}\pi^{+}\\\
\sqrt{2}\pi^{-}&-\pi^{0}\end{bmatrix}\right)}\,,$

and the term $\chi_{+}=m_{\pi}^{2}(U+U^{\dagger})$ introduces explicit chiral
symmetry breaking. Interaction terms between pions and nucleons in leading
order are given by

$\mathcal{L}_{\pi N}^{(1)}=\bar{N}\big{(}iv\cdot D+g_{A}S\cdot u\big{)}N\,,$
(5)

where $v_{\mu}=(1,\vec{0})$ is the four-velocity and
$S^{\mu}=(0,\vec{\sigma}/2)$ the spin-vector of the nucleon. Furthermore,
$g_{A}=1.27$ denotes the axial-vector coupling constant. The chiral covariant
derivative and axial-vector quantity are defined as
$D_{\mu}=\partial_{\mu}+\frac{1}{2}[\xi^{\dagger},\partial_{\mu}\xi]$ and
$u_{\mu}=i(\xi^{\dagger}\partial_{\mu}\xi-\xi\partial_{\mu}\xi^{\dagger})$,
with $\xi=\sqrt{U}$.

Seven terms contribute to the Lagrangian at next-to-leading order (NLO) Fettes
_et al._ (1998); however, we display here only those operators that contribute
to the self-energy of a zero-momentum pion. This part of the chiral $\pi N$
Lagrangian reads

$\mathcal{L}^{(2)}_{\pi N}=\bar{N}\Big{(}-{ig_{A}\over 2M}\\{S\cdot D,v\cdot
u\\}+c_{1}{\rm tr}\chi_{+}+\Big{(}c_{2}-{g_{A}^{2}\over 8M}\Big{)}(v\cdot
u)^{2}+c_{3}\,u\cdot u\Big{)}N\,,$ (6)

where the first term is a relativistic $1/M$-correction to the pion-nucleon
vertex, and $c_{1},c_{2},c_{3}$ are low-energy constants (LEC’s). Pions also
couple to a two-nucleon axial current at NLO and this interaction is described
by

$\mathcal{L}_{\pi NN}^{(1)}=\frac{c_{D}}{2f_{\pi}^{2}\Lambda_{\chi}}\\\
\left(\bar{N}N\right)\left(\bar{N}S\cdot uN\right)\,.$ (7)

with the scale $\Lambda_{\chi}\simeq 700$ MeV and $c_{D}$ a dimensionless
parameter. The interaction in Eq.(7) generates a one-pion exchange three-
nucleon force, and therefore $c_{D}$ has been determined from the triton
binding energy and the neutron-deuteron doublet scattering length Epelbaum
_et al._ (2002). Its value is sensitive to the regularization scheme and
choice of cutoff in the chiral two-nucleon potential. For momentum-space
cutoffs between $500$ MeV and $600$ MeV, $c_{D}$ falls into the range
$-4<c_{D}<1$ Epelbaum  _et al._ (2002).

Figure 1: Feynman diagrams contributing to the pion self-energy. Solid and
dahed lines represent nucleons and pions, respectively. There are several
types of vertices: the Weinberg-Tomozawa $2\pi$-vertex is undecorated, the dot
represents the axial pion-nucleon vertex, the hatched-box represents T-matrix
insertions, the open-box is the pion coupling to four nucleons, and the cross-
dot indicates a short-distance nucleon-nucleon interaction.

The Feynman diagrams contributing to the pion self-energy at two-loop order
are shown in Fig. 1. Some time ago the contributions due to the diagrams
labelled $(ld)$, $(\tilde{ds})$, $(ph)$, and $(cr)$ have been calculated in
Refs.Kaiser and Weise (2001); Kolomeitsev  _et al._ (2003). The diagram
labeled ($ld$) includes contributions linear in the proton and neutron density
and is given by

$\Pi_{ld}(\omega)=T^{-}(\omega)~{}\frac{k_{n}^{3}-k_{p}^{3}}{3\pi^{2}}-T^{+}(\omega)~{}\frac{k_{n}^{3}+k_{p}^{3}}{3\pi^{2}}\,,$
(8)

where $k_{n}$ and $k_{p}$ denote the neutron and proton Fermi momenta, and
$T^{\mp}(\omega)$ are the isospin-odd and isospin-even s-wave off-shell $\pi
N$ scattering amplitudes. The isospin-odd amplitude (restricted to its real
part) is given up to order $\omega^{3}$ by:

$T^{-}(\omega)=\frac{\omega}{2f_{\pi}^{2}}+\frac{\gamma\omega^{3}}{8\pi^{2}f_{\pi}^{4}}-\frac{\omega^{2}Q}{8\pi^{2}f_{\pi}^{4}}\ln{\frac{\omega+Q}{m_{\pi}}}\,,$
(9)

where $Q=\sqrt{\omega^{2}-m_{\pi}^{2}}$ and $\gamma$ is chosen such that the
empirical value $T^{-}(m_{\pi})=1.85$ fm is obtained. The leading term of
$T^{-}$, first predicted by Weinberg and Tomozawa Weinberg (1966); Tomozawa
(1966), is remarkably close to the empirical value, and chiral perturbation
theory converges well for this quantity. In contrast, the convergence of
$T^{+}(\omega)$ is known to be poor. Up to order $\omega^{3}$, its real part
is given by

$T^{+}(\omega)=\frac{\sigma_{N}-\beta\omega^{2}}{f_{\pi}^{2}}+\frac{3g_{A}^{2}m_{\pi}^{3}}{16\pi
f_{\pi}^{4}}+\frac{3g_{A}^{2}m_{\pi}Q^{2}\zeta}{64\pi f_{\pi}^{4}}\,,$ (10)

where $\sigma_{N}=-4c_{1}m_{\pi}^{2}-9g_{A}^{2}m_{\pi}^{3}/64\pi f_{\pi}^{2}$
is the pion-nucleon sigma term Sainio (2002) and
$\beta=-2(c_{2}+c_{3})+g_{A}^{2}/4M$. In Ref. Kolomeitsev  _et al._ (2003) the
low-energy constant $c_{1}$ is chosen to reproduce $\sigma_{N}\simeq 45\pm 15$
MeV, the range probed by phenomenological and lattice calculations Hoferichter
_et al._ (2015); Borsanyi  _et al._ (2020), and the combination $c_{2}+c_{3}$
is determined by the empirical value $T^{+}(m_{\pi})\simeq 0$. The off-shell
parameter $\zeta$ encodes the freedom of choosing the interpolating pion
field. Its value is scheme-dependent and needs to be estimated by fits to
data. In Ref. Kolomeitsev  _et al._ (2003), the authors set $\zeta=0$ to
ensure that the residue at the pole of the pion propagator in the vacuum is
one. However, this choice introduces systematic uncertainties because the
convergence of the $\chi$PT expansion for $T^{+}(\omega)$ is not well-
understood. To incorporate this uncertainty, and assuming that $Q^{2}\ll
m_{\pi}^{2}$, we rewrite the scattering amplitude as

$\tilde{T}^{+}(\omega)=-\frac{\tilde{\sigma}}{f_{\pi}^{2}}~{}\frac{Q^{2}}{m_{\pi}^{2}}\,,$
(11)

where $\tilde{\sigma}\simeq\sigma_{N}$ is treated as model parameter with
values in the range $40\dots 80$ MeV. Later, we will use constraints on the
$\pi^{-}$ optical potential in 208Pb to justify this choice.

The contribution to $\Pi(\omega)$ from the two-loop diagram labeled
$\tilde{ds}$ accounts for double-scattering due to the Weinberg-Tomozawa
interaction and is given by

$\Pi_{\tilde{ds}}=\frac{\omega^{2}}{3(4\pi
f_{\pi})^{4}}\Big{\\{}L(\omega;k_{n},k_{n})+L(\omega;k_{p},k_{p})+2L(\omega;k_{n},k_{p})\Big{\\}}\,,$
(12)

with the logarithmic function

$\begin{split}L(\omega;k_{n},k_{p})=&\,\,4k_{n}k_{p}(3k_{n}^{2}+3k_{p}^{2}+Q^{2})+8Q(k_{n}^{3}-k_{p}^{3})\ln{\frac{Q+k_{n}-k_{p}}{|Q-k_{n}+k_{p}|}}-8Q(k_{n}^{3}+k_{p}^{3})\ln{\frac{Q+k_{n}+k_{p}}{|Q-k_{n}-k_{p}|}}\\\
&+\Big{[}3(k_{n}^{2}-k_{p}^{2})^{2}+6Q^{2}(k_{n}^{2}+k_{p}^{2})-Q^{4}\Big{]}\ln{\frac{|(k_{n}-k_{p})^{2}-Q^{2}|}{|(k_{n}+k_{p})^{2}-Q^{2}|}}\,,\end{split}$
(13)

with $Q=\sqrt{\omega^{2}-m_{\pi}^{2}}$ for $\omega>m_{\pi}$. The diagram
labeled ($ds$) includes higher-order contributions to double-scattering. This
contains contributions up to order $\omega^{3}$ that were included in the off-
shell scattering amplitudes $T^{\pm}(\omega)$ defined earlier. Including these
contributions, we find

$\displaystyle\Pi_{ds}=\frac{1}{192\pi^{4}}\Big{\\{}T_{nn}(\omega)^{2}L(\omega;k_{n},k_{n})+T_{pp}(\omega)^{2}L(\omega;k_{p},k_{p})+T_{np}(\omega)^{2}L(\omega;k_{n},k_{p})\Big{\\}}\,,$
(14)

where $T_{nn}(\omega)=T^{+}(\omega)-T^{-}(\omega)$,
$T_{pp}(\omega)=T^{+}(\omega)+T^{-}(\omega)$, and
$T_{np}(\omega)=-\sqrt{2}T^{-}(\omega)$.

The particle-hole diagram (labeled (ph) in Fig. 1) contains the contribution
due to the pseudovector interaction between nucleons and pions. Since the pion
momentum is zero, this contribution appears first as relativistic correction
suppressed by the factor $(k_{p,n}/M)^{2}$, and is given by

$\Pi_{ph}(\omega)=\frac{g_{A}^{2}\omega}{f_{\pi}^{2}}\bigg{(}\frac{k_{p}^{5}-k_{n}^{5}}{10\pi^{2}M^{2}}\bigg{)}\,.$
(15)

We note that the particle-hole contribution is also suppressed in systems with
small isospin asymmetries. For these reasons, the attractive contribution from
$\Pi_{ph}$ is relatively unimportant in the analysis of pionic atoms
Kolomeitsev  _et al._ (2003). However, we shall find that this diagram
contributes to the pion self-energy in neutron stars.

The two-loop diagram labeled $(pw)$ includes the attractive p-wave pion-
nucleon interaction in the intermediate state. In Fig. 1 we only show one of
several possible intermediate states. Others are obtained by connecting the
p-wave vertex to the lower Fermion line and by switching the location of the
Weinberg-Tomazawa vertex. The intermediate pion can be either a neutral pion
or charged pion. We have calculated the contribution from all of these
diagrams, and their sum is given by

$\displaystyle\Pi_{\tilde{pw}}(\omega)=\frac{g_{A}^{2}\omega}{4Mf_{\pi}^{4}}\Big{\\{}2I(\omega;k_{p},k_{p})-2I(\omega;k_{n},k_{n})-2K(\omega;k_{n},k_{p})+I(0;k_{p},k_{p})-I(0;k_{n},k_{n})-K(0;k_{n},k_{p})\Big{\\}}\,,$
(16)

where the sum originates from all possible arrangements of the Weinberg-
Tomozawa (WT) and dotted vertices. The functions $I(\omega;k_{n},k_{p})$ and
$K(\omega;k_{n},k_{p})$ arise from the following principal-value integrals of
a pion propagator over two Fermi spheres:

$\displaystyle I(\omega;k_{n},k_{p})$
$\displaystyle=\int\frac{d^{3}l_{1}d^{3}l_{2}}{(2\pi)^{6}}\theta\big{(}k_{n}-|\vec{l}_{1}|\big{)}\theta\big{(}k_{p}-|\vec{l}_{2}|\big{)}\frac{(\vec{l}_{1}-\vec{l}_{2})^{2}}{(\vec{l}_{1}-\vec{l}_{2})^{2}+m_{\pi}^{2}-\omega^{2}}\,,$
$\displaystyle K(\omega;k_{n},k_{p})$
$\displaystyle=\int\frac{d^{3}l_{1}d^{3}l_{2}}{(2\pi)^{6}}\theta\big{(}k_{n}-|\vec{l}_{1}|\big{)}\theta\big{(}k_{p}-|\vec{l}_{2}|\big{)}\frac{2({\vec{l}_{1}}^{\,\,2}-{\vec{l}_{2}}^{\,\,2})}{(\vec{l}_{1}-\vec{l}_{2})^{2}+m_{\pi}^{2}-\omega^{2}}\,.$
(17)

We provide explicit analytical expressions for these functions in the
appendix.

It is straightforward to include higher-order terms in $\omega$ contributing
to the $(pw)$ diagram. The net effect of inserting further pion-nucleon
vertices is to replace the WT vertex with the off-shell amplitudes, as was
obtained in Eq. 14 for the double-scattering contribution. This also allows
for the incorporation of the isospin-even contributions, and we find

$\displaystyle\Pi_{pw}(\omega)=$
$\displaystyle\,\,T^{-}(\omega)\frac{g_{A}^{2}}{Mf_{\pi}^{2}}\Big{\\{}I(\omega;k_{p},k_{p})-I(\omega;k_{n},k_{n})-K(\omega;k_{n},k_{p})+\frac{1}{2}\Big{[}I(0;k_{p},k_{p})-I(0;k_{n},k_{n})-K(0;k_{n},k_{p})\Big{]}\Big{\\}}$
$\displaystyle+T^{+}(\omega)\frac{g_{A}^{2}}{Mf_{\pi}^{2}}\Big{\\{}I(\omega;k_{n},k_{n})+I(\omega;k_{p},k_{p})-I(\omega;k_{n},k_{p})\Big{\\}}\,.$
(18)

where again, the sum indicates all possible arrangements of vertices.

The contributions due to diagrams labeled $(cr)$ have also been calculated and
are found to be negligibly small (the analytic expressions can be found in
Ref. Kolomeitsev  _et al._ (2003)). We have calculated the contribution due to
the diagram $(ce)$ and find that its contribution is also negligible. We,
therefore, neglect both in our analysis.

The contribution to the pion self-energy due to the pion-two-nucleon current
operator in ${\cal L}^{(1)}_{\pi NN}$ reads

$\Pi_{cd}(\omega)=\frac{g_{A}c_{D}\omega}{f_{\pi}^{4}\Lambda_{\chi}}\bigg{(}\frac{k_{n}^{3}+k_{p}^{3}}{3\pi^{2}}\bigg{)}\bigg{(}\frac{k_{p}^{5}-k_{n}^{5}}{10\pi^{2}M^{2}}\bigg{)}\,,$
(19)

and it appears at order $k_{F}^{8}\omega$. At this order, other diagrams
contribute that we are not including. We retain this diagram only to study the
sensitivity to higher-order corrections and the low-energy constant $c_{D}$.

The only remaining diagrams of Fig. 1 to discuss are the (gp) diagrams, which
include one insertion of leading-order nucleon-nucleon (NN) contact
interactions between the pion absorption and emission. Due to the isospin flip
at the $\pi N$ vertex, the set of contributing interactions is restricted to
the following:

$\delta\mathcal{L}=g_{1}^{\prime}\,(\bar{n}n)(\bar{p}p)+g_{2}^{\prime}\,(\bar{n}\vec{\sigma}n)\cdot(\bar{p}\vec{\sigma}p)+g_{3}^{\prime}\,(\bar{n}\vec{\sigma}p)\cdot(\bar{n}\vec{\sigma}p)\,.$
(20)

In the past, these short-range interactions, especially in the spin-isospin
channel, were important for understanding p-wave condensation, and their
repulsive nature ultimately led to the conclusion that p-wave condensation is
unlikely. The last low-energy constant $g_{3}^{\prime}$ in Eq.(20) is
conventionally written as $g_{3}^{\prime}=g^{\prime}f^{2}/m_{\pi}^{2}$ with
$f\simeq 1$ and $g^{\prime}\simeq 0.6$ is the Landau-Migdal parameter Ericson
and Weise (1988). However, when the pion’s momentum is zero, all (gp) diagrams
vanish in the non-relativistic limit. This contrasts the p-wave case, where
(gp) diagrams arise from the pion coupling to nucleon spin-isospin
fluctuations at finite momentum.

The pion self-energy at two-loop order is obtained by summing over the
individual contributions

$\Pi_{\rm
tot}(\omega)=\Pi_{ld}(\omega)+\Pi_{ds}(\omega)+\Pi_{ph}(\omega)+\Pi_{pw}(\omega)+\Pi_{cd}(\omega)\,.$
(21)

At the typical densities and isospin asymmetries encountered inside nuclei,
the most relevant contributions are due to the terms linear in density and the
double-scattering diagram. For example, for $k_{n}=280.5$ MeV and $k_{p}=243$
MeV, which approximately correspond to the ambient conditions in the central
regions of 208Pb, the individual contributions from the diagrams are listed in
Table 1.

$\omega$ |  $\Pi_{ld}/2m_{\pi}$ |  $\Pi_{ds}/2m_{\pi}$ |  $\Pi_{ph}/2m_{\pi}$ |  $\Pi_{pw}/2m_{\pi}$ |  $\Pi_{cd}/2m_{\pi}$
---|---|---|---|---|---
$m_{\pi}$ |  $8.8$ MeV |  $7.5$ MeV |  $-0.96$ MeV |  $-0.76$ MeV |  $-0.16$ MeV
165 MeV |  $23.0$ MeV |  $9.5$ MeV |  $-1.14$ MeV |  $-1.3$ MeV |  $-0.18$ MeV
Table 1: Contributions from the individual diagrams to the pion self-energy at
$k_{n}=280.5$ MeV and $k_{p}=243$ MeV using $\tilde{\sigma}=60$ MeV and
$c_{D}=1$.

The numerical values in the table are obtained by setting $\tilde{\sigma}=60$
MeV and $c_{D}=1$. Since $\Pi_{ld}(\omega)$ and $\Pi_{ds}(\omega)$ make the
dominant contributions, below we shall use pionic atom data to constrain the
range for the parameter $\tilde{\sigma}$ defined in Eq. (11).

Experimentally measured energy levels of pionic atoms provide useful
constraints on the $\pi^{-}$ optical potential in neutron-rich heavy-nuclei
such as Pb. In the local density approximation, the optical potential $V_{s}$
at the center of the pionic atom is related to the local pion-self energy by
$V_{s}=\Pi_{\rm tot}(\omega-V_{C},k_{n},k_{p})/2\omega$ with
$\omega=m_{\pi}-B$, where $B$ is the binding energy and $V_{C}$ the Coulomb
potential of the nucleus, and $k_{n},k_{p}$ are (local) neutron and proton
Fermi momenta, respectively Kolomeitsev  _et al._ (2003).

Phenomenological fits to data suggest that the s-wave component of the
$\pi^{-}$ optical potential at the center of 208Pb, which we denote as
$V_{s}(^{208}\rm Pb)$, is in the range $28\pm 3$ MeV Friedman and Gal (1998);
Friedman (2002). In Table 2 we present our results for the optical potential
for several values of $\tilde{\sigma}$ obtained using our two-loop calculation
of the $\pi^{-}$ self-energy.

$\tilde{\sigma}$ |  $40$ MeV |  $50$ MeV |  $60$ MeV |  $70$ MeV |  $80$ MeV
---|---|---|---|---|---
$V_{s}(^{208}\rm Pb)$ |  $24.1$ MeV |  $26.0$ MeV |  $27.9$ MeV |  $29.7$ MeV |  $31.6$ MeV
Table 2: S-wave optical potential at the center of 208Pb. Pionic atoms data
indicate that $V_{s}\simeq 28\pm 3$ MeV Friedman (2002).

These values are obtained by assuming that at the center of 208Pb, the baryon
density $n_{B}=0.16$ fm-3, and the proton density $n_{p}=0.063$ fm-3. We also
assumed that $B\simeq 7$ MeV for the 1S state, and $V_{C}\simeq-29$ MeV. From
Table 2 we can conclude that experimental constraints on the $\pi^{-}$ optical
potential require $\tilde{\sigma}$ to be in the range $45-75$ MeV. We note
that if one assumes that $\tilde{\sigma}\approx\sigma_{N}$, a recent analysis
of pion-nucleon scattering data that favor $\sigma_{N}\simeq 60$ MeV
Hoferichter  _et al._ (2016) is contained in our range for $\tilde{\sigma}$.

##  III Modeling NN Interactions

The calculation of the pion self-energy presented in the preceding section
neglected interactions between nucleons and assumed that the energies of the
nucleons were unaffected by the medium. However, the energy difference between
neutrons and protons can be quite large in dense matter. It is important to
include this energy difference in calculating the pion self-energy, in
particular, because it alters energy denominators associated with diagrams
that include intermediate neutron-proton particle-hole states. A consistent
approach to include this effect would be based on Chiral Effective Field
Theory ($\chi$EFT )Epelbaum  _et al._ (2009), as it systematically accounts
for short-range interactions and interactions mediated by pions. However, the
predictive power $\chi$EFT for $n_{B}\gtrsim 2n_{\mathrm{sat}}$ is not well
understood because truncation errors can grow rapidly at these densities (see
Ref. Drischler  _et al._ (2021) for a recent review). Instead, we adopt a
phenomenological model to perform a parametric study of the pion self-energy
in dense matter.

The model we adopt is based on a mean-field description of nuclear
interactions. It is characterized by just two key parameters: the nuclear
symmetry energy and the effective mass of the nucleons. In general, the
strength of phenomenological, short-range interactions in mean field models is
chosen to reproduce nuclear masses and bulk properties of matter such as its
energy density, pressure, and susceptibilities. A common feature of mean-field
models is the modification of the single-particle nucleon energies due to
their coupling to the mean field generated by other nucleons in the medium. In
a large class of these models, the neutron and proton energies are given by

$\displaystyle E_{n}(p)={p^{2}\over 2M^{*}_{n}}+\Sigma_{n},\,\qquad
E_{p}(p)={p^{2}\over 2M^{*}_{p}}+\Sigma_{p}\,.$ (22)

Here, $\Sigma_{n,p}$ are the mean-field energy shifts, and $M^{*}_{n,p}$ are
the effective masses of neutrons and protons in the medium. The mean field
energy shift and the effective masses depend on the baryon density and the
isospin asymmetry. For simplicity, we neglect the difference between neutron
and proton effective masses and assume that $M^{*}_{n}=M^{*}_{p}=M^{*}$.

Using nucleon propagators that include mean field energy shifts and a common
nucleon effective mass $M^{*}$, we have calculated the pion self-energy
discussed in the previous section. We find these modifications only affect
those diagrams that contain both neutron and proton intermediate states. In
particular, the diagrams in Fig. 1 labeled $(ph)$, $(pw)$, and $(cd)$ are the
only ones changed, and the modification is to multiply the corresponding
expressions in Eqs. (15), (18), and (19) by the factor

$\xi(\omega)=\frac{\omega}{\omega-\big{(}\Sigma_{n}-\Sigma_{p}\big{)}}\,,$
(23)

and to make the substitution $M\rightarrow M^{*}$. In our parametric study,
the density dependence of the nucleon effective mass is given by

$M^{*}=M\left(1-\alpha\frac{n_{B}}{n_{\mathrm{sat}}}\right)\,,$ (24)

and we vary the parameter $\alpha$ in the range $0\dots 0.1$ to explore the
range predicted by microscopic calculations Shang  _et al._ (2020); Li  _et
al._ (2018). The factor $\xi(\omega)$ can resonantly enhance the contribution
from these diagrams when $\omega\simeq\Sigma_{n}-\Sigma_{p}$. Interestingly,
in beta-equilibrated neutron star matter, the enhancement factor $\xi(\omega)$
can be simplified by reexpressing the energy difference
$\Sigma_{n}-\Sigma_{p}$ in terms of the nuclear symmetry energy and the
nucleon effective mass. This is accomplished by first noting that the isospin
chemical potential $\hat{\mu}$, and the proton, neutron, and electron chemical
potentials are related through

$\hat{\mu}=\mu_{e}=\mu_{n}-\mu_{p}~{},$ (25)

and that furthermore $\mu_{e}=4S(n_{B})(1-2x)$, where $S(n_{B})$ is the
nuclear symmetry energy and $x$ the proton fraction. Substituting these
relations into $\xi(\omega)$ yields

$\xi(\omega)=\frac{\omega}{\omega-4S(n_{B})(1-2x)+(k_{n}^{2}-k_{p}^{2})/2M^{*}}~{},$
(26)

where $k_{n}=(3\pi^{2}n_{B}(1-x))^{1/3}$ and $k_{p}=(3\pi^{2}n_{B}x)^{1/3}$
are the neutron and proton Fermi momenta. As stated previously, $\xi(\omega)$,
when large, can change the importance of Feynman diagrams. To estimate this,
note that if a pion is produced in the medium, then its frequency is forced to
$\omega=\mu_{e}$. Making this substitution into $\xi(\omega)$ one finds that
the diagram labelled $(ph)$, which originally contributed at ${\cal
O}(k_{F}^{5}\mu_{e})$ now contributes at ${\cal O}(k_{F}^{3}\mu_{e}^{2})$; the
diagram labelled $(pw)$, which originally contributed at ${\cal
O}(k_{F}^{6}\mu_{e})$ now contributes at ${\cal O}(k_{F}^{4}\mu_{e}^{2})$; and
the diagram labelled $(cd)$, which originally contributed at ${\cal
O}(k_{F}^{8}\mu_{e})$ now contributes at ${\cal
O}(k_{\text{F}}^{6}\mu_{e}^{2})$. Crucially, in neutron-rich matter these
enhanced diagrams are attractive for positive frequencies.

Figure 2: Experimental and theoretical constraints on the density dependence
of the nuclear symmetry energy. Predictions of the parametrization in Eq.31
are shown for reference.

Through its appearance in the enhancement factor $\xi(\omega)$, the nuclear
symmetry energy $S(n_{B})$ plays a central role in our calculation. The
nuclear symmetry energy is defined through the difference

$S(n_{B})=E(n_{B},x=0)-E(n_{B},x=1/2)\,,$ (27)

where $E(n_{B},x)$ is the energy per baryon at baryon density $n_{B}$ and
proton fraction $x$. Microscopic calculations and fits to phenomenological
models indicate that the energy per particle at arbitrary proton fraction is
well approximated by

$E(n_{B},x)\approx E(n_{B},x=1/2)+S(n_{B})(1-2x)\,,$ (28)

since higher order terms in the expansion have been found to be small, even
for $x\ll 1/2$ Li  _et al._ (2008). In this case, the electron chemical
potential in neutron star matter is

$\mu_{e}(n_{B})=4S(n_{B})(1-2x)\,.$ (29)

Recently, there has been much interest in determining the density dependence
of the symmetry energy. However, despite progress in both theory and
experiment, this dependence remains poorly known at densities reached in
neutron stars. In the vicinity of nuclear saturation density, $S(n_{B})$
impacts nuclear structure. Nuclear masses, measurements of the neutron-skin
thickness, and the electric-dipole polarizability of neutron-rich nuclei such
as 208Pb provide useful constraints on $S_{0}=S(n_{\mathrm{sat}})$. Its
density dependence is characterized by the slope parameter
$L=3n_{B}\left(dS(n_{B})/dn_{B}\right)$ at $n_{B}=n_{\mathrm{sat}}$ Steiner
_et al._ (2005). Until recently, experiments, when combined with theoretical
models, suggested the empirical range $S_{0}=32\pm 2$ MeV and $L=50\pm 15$
MeV. Theoretical calculations using nucleon-nucleon interactions determined by
$\chi$EFT predict $S_{0}$ and $L$ compatible with this empirical range. For
example, a recent calculation that combines many-body perturbation theory
(MBPT) and Bayesian estimates for the truncation errors predicts
$S_{0}=31.7\pm 1.1$ MeV and $L=59.8\pm 4.1$ MeV Drischler  _et al._ (2020).
However, the recent measurement of the neutron-skin thickness of 208Pb using
parity-violating electron scattering imply larger values: $S_{0}=38.1\pm 4.7$
MeV, and $L=106\pm 37$ MeV Adhikari  _et al._ (2021); Reed  _et al._ (2021).

The symmetry energy at higher density can be accessed in heavy-ion experiments
but is not presently well-determined Horowitz  _et al._ (2014). At
$n_{B}\simeq 1.5~{}n_{\mathrm{sat}}$, a recent analysis by Estee et al. of
charged pion yields from intermediate-energy heavy-ion collisions suggests
that $S(n_{B}\simeq 1.5n_{\mathrm{sat}})=52\pm 13$ MeV Estee  _et al._ (2021).
Earlier studies by Russoto et al. of heavy-ion collisions at GSI indicate that
$S(n_{B}\approx 2n_{\mathrm{sat}})=50\pm 7$ MeV Russotto  _et al._ (2016).

Theoretical calculations of the equation of state using potentials derived
from $\chi$EFT also provide useful constraints on the symmetry energy in the
region $n_{\mathrm{sat}}<n_{B}<2n_{\mathrm{sat}}$. Quantum Monte Carlo
calculations by Lonardoni et al., using local $\chi$EFT potentials predict
$S(1.5n_{\mathrm{sat}})\approx 37\pm 5$ MeV and $S(2n_{\mathrm{sat}})\approx
46\pm 11$ MeV Lonardoni  _et al._ (2020). In Ref. Lim and Holt (2018), Lim and
Holt use many-body perturbation theory (MBPT) to predict
$S(2n_{\mathrm{sat}})\approx 49\pm 12$ MeV, and Drischler et al., combine MBPT
and Bayesian estimates of the $\chi$EFT truncation errors (but neglect errors
associated with low energy constants) to predict $S(2n_{\mathrm{sat}})\approx
45\pm 3$ MeV Drischler  _et al._ (2020). These constraints are shown in Fig.
2.

The density dependence of the symmetry energy is typically written as an
expansion about $n_{B}=n_{\mathrm{sat}}$:

$S(n_{B})=S_{0}+L\left(\frac{u-1}{3}\right)+\frac{K_{\rm
sym}}{2!}\left(\frac{u-1}{3}\right)^{2}+\frac{Q_{\rm
sym}}{3!}\left(\frac{u-1}{3}\right)^{3}\,,$ (30)

where $u=n_{B}/n_{\mathrm{sat}}$, and the parameters $S_{0}$, $L$, $K_{\rm
sym}$, and $Q_{\rm sym}$ are chosen to ensure compatibility with experimental
constraints. However, because the parameters $K_{\rm sym}$ and $Q_{\rm sym}$
are poorly constrained, the predictions of Eq. (30) at $n_{B}\gtrsim
2n_{\mathrm{sat}}$ vary over a wide range. In this study, we adopt a simple
ansatz for the density dependence of the symmetry energy

$\tilde{S}(n_{B})=\frac{2S_{2}u}{2+u}\,,$ (31)

which depends on a single parameter $S_{2}=S(2n_{\mathrm{sat}})$, the symmetry
energy at twice saturation density. From Fig. 2 it is clear that $S_{2}$ in
the range $45-60$ MeV satisfies all existing constraints. The high value
$S_{2}=60$ MeV is compatible with the large $S_{0}$ and $L$ predicted by PREX
and is implied by the heavy-ion data, and the intermediate value $S_{2}\simeq
50$ MeV is compatible with the empirical range. The low value $S_{2}=45$ MeV
is in reasonable agreement with the $\chi$EFT predictions made in Ref.
Drischler  _et al._ (2020).

##  IV Results $\&$ Discussion

In this section, we gather our results. We begin by examining the behavior of
the inverse propagator as a function of frequency. A typical plot of the
inverse propagator at baryon density $n_{B}=2n_{\mathrm{sat}}$ is shown in
Fig. 3. Here we have taken the following “typical” values of the scattering
parameters: $\tilde{\sigma}=60$ MeV, $\gamma=2.6$, $\Lambda_{\chi}=700$ MeV,
$c_{D}=2.6$. We have also taken $S_{2}=50$ MeV, and $\alpha=0.07$ in our mean-
field description of nucleon-nucleon interactions, which lie roughly in the
middle of the acceptable range. The plot shows that interactions significantly
alter the self-energy compared to the free theory (shown by the dashed curve),
which produces a parabolic dispersion with zeros at $\pm m_{\pi}$.

The poles of the propagator, or the zeros of the inverse propagator,
correspond to real excitations in the medium. The energy and charge associated
with these excitations are determined by examining the residue at the pole.
When the mean field splitting of the neutron and proton energies is neglected
(blue curve), the propagator has two poles, as expected in the vacuum. The
pole at positive frequency has a positive residue, as can be inferred from
examining the slope of the inverse propagator, and is denoted by $\omega_{+}$.
Since we are examining the two-point function of a _negatively_ charged field,
this implies that the charge associated with the pole at $\omega_{+}$ is
negative and its effective mass $m^{*}_{\pi^{-}}=\omega_{+}$. The pole at
negative frequency denoted by $\omega_{-}$ has a negative residue. It thereby
corresponds to the $\pi^{+}$ state with effective mass
$m^{*}_{\pi^{+}}=-\omega_{-}$. At this density, $m^{*}_{\pi^{-}}\simeq 400$
MeV, and this significant increase is largely due to the energy-dependent
Weinberg-Tomozawa term that results in a strong and repulsive s-wave
interaction at leading order. In contrast, the reduction of the $\pi^{+}$
effective mass is modest. This is also largely due to the energy-dependent
Weinberg-Tomozawa term. The contributions of the individual Feynman diagrams
depicted in Fig. 1 to the pion self-energy are shown in the right panel. In
the vicinity of the poles corresponding to the $\pi^{\pm}$ modes, beyond-
leading-order contributions to the self-energy are modest, and there are large
cancellations between various two-loop contributions. When mean-field effects
are included, the poles of the propagator are shifted. Both $\pi^{+}$ and
$\pi^{-}$ effective masses are reduced. However, the reduction is modest
because, as mentioned earlier, they are both dominated by the leading order
contribution, which is insensitive to the mean-field energies. Another
noteworthy feature of the result shown in the left panel of Fig. 3 is that the
inverse propagator is small (compared with $m_{\pi}^{2}$) over a fairly wide
interval in $\omega$ around $\omega=0$.

Figure 3: Left panel: Inverse propagator at two loops with and without mean
field corrections as a function of frequency. Here, $n_{B}=2n_{\mathrm{sat}}$
and the interaction parameters $\tilde{\sigma}=60$ MeV, $\gamma=2.6$,
$\Lambda_{\chi}=700$ MeV, $c_{D}=2.6$, $S_{2}=50$ MeV, $\alpha=0.07$. Right
panel: Individual contribution from the diagrams depicted in Fig. 1 to the
pion self-energy.

A striking feature of mean field effects is that it predicts a third pole
corresponding to a collective excitation with a positive charge but negative
energy. This collective state has the quantum numbers of the positively
charged pion and arises as a coherent proton-particle–neutron-hole excitation.
We denoted this mode by the symbol $\pi_{s}^{+}$ to be consistent with the
notation used in earlier work to describe a similar mode at finite momentum
Chi-Kwan Au and Baym (1974). We note that the negative energy of the
$\pi_{s}^{+}$ does not imply instability. The large electron chemical
potential ensures that energy associated with producing $\pi_{s}^{+}$ through
the reactions $p\rightarrow n\pi_{s}^{+}$, given by
$\delta\omega_{s}=\mu_{n}-\mu_{p}+\omega_{\pi_{s}^{+}}=\omega_{\pi_{s}^{+}}+\mu_{e}$
is positive. The ground state is generally unstable when
$\delta\omega_{i}=m^{*}_{i}-\mu_{i}<0$, where $\mu_{i}$ is the chemical
potential associated with the conserved charge labeled $i$, and $m^{*}_{i}$ is
the effective mass of the particle in the ground state. In charge neutral
neutron-rich matter in beta-equilibrium, the electric charge chemical
potential $\mu_{Q}=-\mu_{e}$, where $\mu_{e}=\mu_{n}-\mu_{p}>0$. From the
right panel of Fig. 3 we can infer that in the vicinity of the pole associated
with the $\pi_{s}^{+}$, the two-loop contribution to the pion self-energy is
not much suppressed compared to the leading-order term.

The density dependence of the in-medium masses of the charged pions are shown
in Fig. 4. The leading order s-wave WT contribution to the self-energy
dominates the $\pi^{-}$ mode for the full range of densities considered. Its
rapid increase with increasing density precludes s-wave condensation of
negatively charged pions. In contrast, the mass of the $\pi^{+}$ mode
decreases with density. In this case, the s-wave WT interaction dominates at
low density. Higher-order contributions become important for
$n_{B}>n_{\mathrm{sat}}$ resulting in a rapid decrease of the $\pi^{+}$ mass
at higher density. As discussed earlier, the mass of the collective mode
labeled $\pi_{s}^{+}$ is negative and decreases with density. Its value is
sensitive to the mean-field energy shifts of the nucleons and the two-loop
contributions to the pion self-energy. For the density range considered here,
$\delta\omega_{\pm}>0$ and $\delta\omega_{s}>0$.Thus the normal ground state
without pion condensation is stable, and the energy to create a
$\pi^{-}-\pi_{s}^{+}$ pair is also positive.

Figure 4: Masses of the charged pions and the pion-like collective excitation
in neutron-rich matter (model parameters used are described in the text). The
leading-order (in the density expansion) results are shown by dashed-curves,
and the solid curves show the two-loop calculation.

##  V Conclusions

We have constructed a model for the pion self-energy in neutron star matter
which accounts for important aspects of the pion-nucleon interaction
discounted in earlier investigations. The model consists of a two-loop
calculation of the pion self-energy within heavy-baryon chiral perturbation
theory, which is augmented by in-medium modifications to the nucleon
propagators. These modifications include energy shifts and effective masses of
the nucleons due to interactions with the nuclear medium. Qualitative changes
arise from these modifications. In particular, small energy denominators are
produced, which modify the power counting of Feynman diagrams. Certain
attractive two-loop diagrams (in this paper denoted $\Pi_{ph}(\omega)$,
$\Pi_{pw}(\omega)$, $\Pi_{cd}(\omega)$) are enhanced, especially at energies
comparable to the energy splitting between neutrons and protons.

Our parametric study of the self-energy has revealed general trends which we
describe in the following. The pion self-energy at zero momentum is found to
be strongly energy dependent and the dressed pion propagator has a complex
structure with three poles. Two of these poles coincide with the charged pion
modes in vacuum in the limit of vanishing density. The third pole corresponds
to a new collective excitation that contains an admixture of nucleon particle-
hole states with quantum numbers of a positively charged pion. At the poles
corresponding to the charged pions modes, the one-loop leading-order (in the
density expansion) s-wave diagram remains the dominant contribution. This is
especially true for the $\pi^{-}$ where the repulsive s-wave interaction
greatly increases the in-medium mass and precludes s-wave pion condensation
even for the case of a large symmetry energy. The in-medium mass of the
$\pi^{+}$ is reduced due to the attractive s-wave interaction. At high
density, the two-loop contribution becomes relevant and accelerates its
decrease with increasing density. Our prediction of a positively charged
collective mode in the long-wavelength limit is new. The energy of this mode
is negative, and its value is sensitive to the energy splitting of neutrons
and protons in the medium, the one-loop particle-hole diagram, and the two-
loop diagram that involves a p-wave interaction between pions and nucleons.

The phenomenological implications of the low-energy collective mode and the
large splitting between the masses of the charged pion excitations warrant
further study. We speculate that the collective mode can play a role in
neutrino interactions, and the strongly modified and energy-dependent pion
propagator can alter nuclear interactions at large distances.

## Acknowledgements

The work of S. R. and N. C. W. was supported by the U.S. DOE under Grant No.
DE-FG02- 00ER41132. B. F. acknowledges support from the SciDAC Grant No.
A18-0354-S002 (de-sc0018232). We thank Paulo Bedaque, Gordon Baym, Evgeny
Epelbaum, David Kaplan, Thomas Schafer, Achim Schwenk, and Corbinian
Wellenhofer for helpful conversations. We also thank Avraham Gal and Eli
Friedman for useful correspondence relating to pionic atoms. The work of S. R.
was performed in part while attending the ”Exploring Extreme Matter in the Era
of Multimessenger Astronomy: from the Cosmos to Quarks” workshop at the Aspen
Center for Physics.

## References

  * Sawyer (1972) R. F. Sawyer, Phys. Rev. Lett. 29, 382 (1972).
  * Scalapino (1972) D. J. Scalapino, Phys. Rev. Lett. 29, 386 (1972).
  * Sawyer and Scalapino (1973) R. F. Sawyer and D. J. Scalapino, Phys. Rev. D 7, 953 (1973).
  * Migdal (1973) A. Migdal, Physics Letters B 45, 448 (1973).
  * Baym (1973) G. Baym, Phys. Rev. Lett. 30, 1340 (1973).
  * Baym and Flowers (1974) G. Baym and E. Flowers, Nucl. Phys. A 222, 29 (1974).
  * Chi-Kwan Au and Baym (1974) N. Chi-Kwan Au and G. Baym, Nucl. Phys. A 236, 500 (1974).
  * Dashen and Manassah (1974) R. F. Dashen and J. Manassah, Phys. Lett. B 50, 460 (1974).
  * Barshay and Brown (1973) S. Barshay and G. Brown, Phys. Lett. B 47, 107 (1973).
  * Weise and Brown (1975) W. Weise and G. Brown, Physics Letters B 58, 300 (1975).
  * Backman and Weise (1975) S.-O. Backman and W. Weise, Physics Letters B 55, 1 (1975).
  * Ericson and Weise (1988) T. E. O. Ericson and W. Weise,  _Pions and Nuclei_ (Clarendon Press, Oxford, UK, 1988).
  * Fore and Reddy (2020) B. Fore and S. Reddy, Phys. Rev. C 101, 035809 (2020).
  * Kolomeitsev  _et al._ (2003) E. E. Kolomeitsev, N. Kaiser, and W. Weise, Phys. Rev. Lett. 90, 092501 (2003), arXiv:nucl-th/0207090 .
  * Kaiser and Weise (2001) N. Kaiser and W. Weise, Phys. Lett. B 512, 283 (2001), arXiv:nucl-th/0102062 .
  * Fettes  _et al._ (1998) N. Fettes, U.-G. Meissner, and S. Steininger, Nucl. Phys. A 640, 199 (1998), arXiv:hep-ph/9803266 .
  * Epelbaum  _et al._ (2009) E. Epelbaum, H.-W. Hammer, and U.-G. Meissner, Rev. Mod. Phys. 81, 1773 (2009), arXiv:0811.1338 [nucl-th] .
  * Epelbaum  _et al._ (2002) E. Epelbaum, A. Nogga, W. Gloeckle, H. Kamada, U. G. Meissner, and H. Witala, Phys. Rev. C 66, 064001 (2002), arXiv:nucl-th/0208023 .
  * Weinberg (1966) S. Weinberg, Phys. Rev. Lett. 17, 616 (1966).
  * Tomozawa (1966) Y. Tomozawa, Nuovo Cim. A 46, 707 (1966).
  * Sainio (2002) M. E. Sainio, PiN Newslett. 16, 138 (2002), arXiv:hep-ph/0110413 .
  * Hoferichter  _et al._ (2015) M. Hoferichter, J. Ruiz de Elvira, B. Kubis, and U.-G. Meißner, Phys. Rev. Lett. 115, 092301 (2015), arXiv:1506.04142 [hep-ph] .
  * Borsanyi  _et al._ (2020) S. Borsanyi, Z. Fodor, C. Hoelbling, L. Lellouch, K. K. Szabo, C. Torrero, and L. Varnhorst, (2020), arXiv:2007.03319 [hep-lat] .
  * Friedman and Gal (1998) E. Friedman and A. Gal, Phys. Lett. B 432, 235 (1998), arXiv:nucl-th/9805004 .
  * Friedman (2002) E. Friedman, Phys. Lett. B 524, 87 (2002), arXiv:nucl-th/0107050 .
  * Hoferichter  _et al._ (2016) M. Hoferichter, J. Ruiz de Elvira, B. Kubis, and U.-G. Meißner, Phys. Lett. B 760, 74 (2016), arXiv:1602.07688 [hep-lat] .
  * Drischler  _et al._ (2021) C. Drischler, J. W. Holt, and C. Wellenhofer,  (2021), 10.1146/annurev-nucl-102419-041903, arXiv:2101.01709 [nucl-th] .
  * Shang  _et al._ (2020) X. L. Shang, A. Li, Z. Q. Miao, G. F. Burgio, and H. J. Schulze, Phys. Rev. C 101, 065801 (2020), arXiv:2001.03859 [nucl-th] .
  * Li  _et al._ (2018) B.-A. Li, B.-J. Cai, L.-W. Chen, and J. Xu, Prog. Part. Nucl. Phys. 99, 29 (2018), arXiv:1801.01213 [nucl-th] .
  * Li  _et al._ (2008) B.-A. Li, L.-W. Chen, and C. M. Ko, Phys. Rept. 464, 113 (2008), arXiv:0804.3580 [nucl-th] .
  * Steiner  _et al._ (2005) A. W. Steiner, M. Prakash, J. M. Lattimer, and P. J. Ellis, Phys. Rept. 411, 325 (2005), arXiv:nucl-th/0410066 [nucl-th] .
  * Drischler  _et al._ (2020) C. Drischler, R. J. Furnstahl, J. A. Melendez, and D. R. Phillips, Phys. Rev. Lett. 125, 202702 (2020), arXiv:2004.07232 [nucl-th] .
  * Adhikari  _et al._ (2021) D. Adhikari _et al._ (PREX), Phys. Rev. Lett. 126, 172502 (2021), arXiv:2102.10767 [nucl-ex] .
  * Reed  _et al._ (2021) B. T. Reed, F. J. Fattoyev, C. J. Horowitz, and J. Piekarewicz, Phys. Rev. Lett. 126, 172503 (2021), arXiv:2101.03193 [nucl-th] .
  * Horowitz  _et al._ (2014) C. J. Horowitz, E. F. Brown, Y. Kim, W. G. Lynch, R. Michaels, A. Ono, J. Piekarewicz, M. B. Tsang, and H. H. Wolter, J. Phys. G 41, 093001 (2014), arXiv:1401.5839 [nucl-th] .
  * Estee  _et al._ (2021) J. Estee _et al._ (S RIT), Phys. Rev. Lett. 126, 162701 (2021), arXiv:2103.06861 [nucl-ex] .
  * Russotto  _et al._ (2016) P. Russotto _et al._ , Phys. Rev. C 94, 034608 (2016), arXiv:1608.04332 [nucl-ex] .
  * Lonardoni  _et al._ (2020) D. Lonardoni, I. Tews, S. Gandolfi, and J. Carlson, Phys. Rev. Res. 2, 022033 (2020), arXiv:1912.09411 [nucl-th] .
  * Lim and Holt (2018) Y. Lim and J. W. Holt, Phys. Rev. Lett. 121, 062701 (2018), arXiv:1803.02803 [nucl-th] .

##  Appendix A Loop functions

In this appendix, we give analytical expressions for the pertinent functions
$I(\omega;k_{n},k_{p}),I(0,k_{n},k_{n}),K(\omega;k_{n},k_{p})$ and
$K(0;k_{n},k_{p})$ that appear in the contribution $\Pi_{pw}(\omega)$ to the
pion selfenergy. In order to evaluate the Fermi sphere integrals in Eq.(17),
three nontrivial integrations over the cosine of an inclined angle and two
radii have to be performed. The angular integral leads to logarithms and by
treating these as logarithms of absolute values the principal-value
prescription is effectively implemented. One finds:

$\displaystyle 384\pi^{4}I(\omega;k_{n},k_{p})={32\over
3}k_{n}^{3}k_{p}^{3}+Q^{2}L(\omega;k_{n},k_{p})\,,$ (32)

with $Q=\sqrt{\omega^{2}-m_{\pi}^{2}}$ and the function
$L(\omega;k_{n},k_{p})$ is written in Eq.(13). The other loop functions read:

$\displaystyle 384\pi^{4}I(0;k_{n},k_{n})={32\over
3}k_{n}^{6}-24k_{n}^{4}m_{\pi}^{2}+4k_{n}^{2}m_{\pi}^{4}+32k_{n}^{3}m_{\pi}^{3}\arctan{2k_{n}\over
m_{\pi}}-m_{\pi}^{4}(12k_{n}^{2}+m_{\pi}^{2})\ln\Big{(}1+{4k_{n}^{2}\over
m_{\pi}^{2}}\Big{)}\,,$ (33) $\displaystyle 192\pi^{4}K(\omega;k_{n},k_{p})=$
$\displaystyle\,\,4k_{n}k_{p}(k_{n}^{2}-k_{p}^{2})(k_{n}^{2}+k_{p}^{2}-5Q^{2})+(k_{n}^{2}-k_{p}^{2})\Big{[}(k_{n}^{2}-k_{p}^{2})^{2}-6Q^{2}(k_{n}^{2}+k_{p}^{2})-3Q^{4}\Big{]}$
$\displaystyle\times\ln{|(k_{n}-k_{p})^{2}-Q^{2}|\over|(k_{n}+k_{p})^{2}-Q^{2}|}+8Q^{3}\bigg{[}(k_{n}^{3}+k_{p}^{3})\ln{|k_{n}-k_{p}-Q|\over|k_{n}-k_{p}+Q|}+(k_{n}^{3}-k_{p}^{3})\ln{k_{n}+k_{p}+Q\over|k_{n}+k_{p}-Q|}\bigg{]}\,,$
(34) $\displaystyle 192\pi^{4}K(0;k_{n},k_{p})=$
$\displaystyle\,\,4k_{n}k_{p}(k_{n}^{2}-k_{p}^{2})(k_{n}^{2}+k_{p}^{2}+5m_{\pi}^{2})+(k_{n}^{2}-k_{p}^{2})\Big{[}3m_{\pi}^{4}-6m_{\pi}^{2}(k_{n}^{2}+k_{p}^{2})-(k_{n}^{2}-k_{p}^{2})^{2}\Big{]}$
$\displaystyle\times\ln{(k_{n}+k_{p})^{2}+m_{\pi}^{2}\over(k_{n}-k_{p})^{2}+m_{\pi}^{2}}+16m_{\pi}^{3}\bigg{[}(k_{n}^{3}+k_{p}^{3})\arctan{k_{n}-k_{p}\over
m_{\pi}}+(k_{p}^{3}-k_{n}^{3})\arctan{k_{n}+k_{p}\over m_{\pi}}\bigg{]}\,.$
(35)