11institutetext: St. Francis Xavier University, Canada
11email<EMAIL_ADDRESS>22institutetext: Martinos Center for
Biomedical Imaging, Department of Radiology, Massachusetts General Hospital,
Charlestown, USA 33institutetext: Nova Scotia Health Authority
44institutetext: School of Biomedical Engineering & Imaging Sciences, King’s
College London, UK
44email: {walter.diaz_sanz, petru.tudosiu,
<EMAIL_ADDRESS>55institutetext: University of California, Santa
Cruz, USA
55email<EMAIL_ADDRESS>
# InverseSR: 3D Brain MRI Super-Resolution Using a Latent Diffusion Model

Jueqi Wang 11 Jacob Levman 112233 Walter Hugo Lopez Pinaya 44 Petru-Daniel
Tudosiu 44 M. Jorge Cardoso 44 Razvan Marinescu 55

###### Abstract

High-resolution (HR) MRI scans obtained from research-grade medical centers
provide precise information about imaged tissues. However, routine clinical
MRI scans are typically in low-resolution (LR) and vary greatly in contrast
and spatial resolution due to the adjustments of the scanning parameters to
the local needs of the medical center. End-to-end deep learning methods for
MRI super-resolution (SR) have been proposed, but they require re-training
each time there is a shift in the input distribution. To address this issue,
we propose a novel approach that leverages a state-of-the-art 3D brain
generative model, the latent diffusion model (LDM) from [23] trained on UK
BioBank, to increase the resolution of clinical MRI scans. The LDM acts as a
generative prior, which has the ability to capture the prior distribution of
3D T1-weighted brain MRI. Based on the architecture of the brain LDM, we find
that different methods are suitable for different settings of MRI SR, and thus
propose two novel strategies: 1) for SR with more sparsity, we invert through
both the decoder $\mathcal{D}$ of the LDM and also through a deterministic
Denoising Diffusion Implicit Models (DDIM), an approach we will call
InverseSR(LDM); 2) for SR with less sparsity, we invert only through the LDM
decoder $\mathcal{D}$, an approach we will call InverseSR(Decoder). These two
approaches search different latent spaces in the LDM model to find the optimal
latent code to map the given LR MRI into HR. The training process of the
generative model is independent of the MRI under-sampling process, ensuring
the generalization of our method to many MRI SR problems with different input
measurements. We validate our method on over 100 brain T1w MRIs from the IXI
dataset. Our method can demonstrate that powerful priors given by LDM can be
used for MRI reconstruction. Our source code is available online:
https://github.com/BioMedAI-UCSC/InverseSR.

###### Keywords:

MRI Super-Resolution Latent Diffusion Model Inverse Problem Optimization
Method.

##  1 Introduction

End-to-end convolutional neural networks (CNNs) have shown remarkable
performance compared to classical algorithms [15] on MRI SR. Deep CNNs have
been widely applied in a variety of MRI SR situations; for instance, slice
imputation on the brain, liver and prostate MRI [31] and brain MRI SR
reconstruction on scaling factors $\times 2$, $\times 3$, $\times 4$ [34].
Several techniques based on deep CNNs have been proposed to improve
performance, such as densely connected networks [7], adversarial networks [6],
and attention network [34]. However, their supervised training requires paired
images, which necessitates re-training every time there is a shift in the
input distribution [5, 17]. As a result, such methods are unsuitable for MRI
SR, as it is challenging to obtain paired training data that cover the
variability in acquisition protocols and resolution of clinical brain MRI
scans across institutions [15].
Building image priors through generative models has recently become a popular
approach in the field of image SR, for both computer vision [2, 1, 20, 8, 18]
as well as medical imaging [19, 27], as they do not require re-training in the
presence of several types of input distribution shifts. While these methods
have shown promise in MRI SR, they have so far been limited to 2D slices [19,
27], rendering them unsuitable for 3D brain MRIs slice imputation.
In this study, we propose solving the MRI SR problem by building powerful,
3D-native image priors through a recently proposed HR image generative model,
the latent diffusion model (LDM) [23, 24]. We solve the inverse problem by
finding the optimal latent code $z$ in the latent space of the pre-trained
generative model, which could restore a given LR MRI $I$, using a known
corruption function $f$. In this study, we focus on slice imputation, yet our
method could be applied to other medical image SR problems by implementing
different corruption functions $f$. We proposed two novel strategies for MRI
SR: Inverse(LDM), which additionally inverts the input image through the
deterministic DDIM model, and InverseSR(Decoder) which inverts the input image
through the corruption function $f$ and through the decoder $\mathcal{D}$ of
the LDM model. We found that for large sparsity, InverseSR(LDM) had a better
performance, while for low sparsity, InverseSR(Decoder) performed best. While
the LDM model was trained on UK BioBank, we demonstrate our methods on an
external dataset (IXI) which was inaccessible to the pre-trained generative
model. Both quantitative and qualitative results show that our method achieves
significantly better performance compared to two other baseline models.
Furthermore, our method can also be applied to tumour/lesion filling by
creating tumour/lesion shape masks.

###  1.1 Related work

####  1.1.1 MRI Super-resolution

End-to-end deep training [29, 31, 34] has been proposed recently for MRI SR,
which has achieved superior results compared to classical methods. However,
these methods require paired data to train, which is hard to acquire because
of the large variability present in clinical MRIs [25, 15]. To circumvent this
limitation, several unsupervised methods have been proposed without requiring
access to HR scans [9, 15, 3]. Dalca et al. [9] proposed a gaussian mixture
model for sparse image patches. Brudfors et al. [3] presented an algorithm
which could take advantage of multimodal MRI. Iglesias et al. [15] introduced
a method to train a CNN for MRI SR on any given combination of contrasts,
resolutions and orientations.

####  1.1.2 Solving inverse problems using generative models

A common way to solve the inverse problem using an LDM is to use the encoder
$\mathcal{E}$ to first encode the given image $x$ into the latent space
$z_{0}=\mathcal{E}(x)$ [11, 13, 22], followed by DDIM (Denoising Diffusion
Implicit Models) Inversion [10, 26] to encode $z_{0}$ into the noise latent
code $z_{T}$ [22]. However, this approach does not work for low-resolution
images, because the encoder $\mathcal{E}$ has only been trained on high-
resolution images.

Our work is also similar to the optimization-based generative adversarial
network (GAN) inversion approach [32], trying to find the optimal latent
representation $z^{*}$ in the latent space of GAN, which could be mapped to
represent the given image $x\approx G(z^{*})$. More recent works [8, 11, 14,
18, 27] have used diffusion models for inverse problems due to their superior
performance. However, all these methods require the diffusion model to operate
directly in the image space, which for large image resolutions can become GPU-
memory intensive.

##  2 Methods

####  2.0.1 3D Brain Latent Diffusion Models

We leverage a state-of-the-art LDM [23] to create high-quality priors for 3D
brain MRIs. There are two components in an LDM: an autoencoder and a diffusion
model [24]. An encoder $\mathcal{E}$ maps each high-resolution T1w brain MRI
$x\sim p_{data}(x)$ into a latent vector $z_{0}=\mathcal{E}(x)$ of size
$20\times 28\times 20$. The decoder $\mathcal{D}$ is trained to map the latent
vectors $z_{0}$ back into the MRI image domain $x$. The autoencoder was
trained on 31,740 T1w MRIs from the UK Biobank [28] using a combination of an
L1 loss, a perceptual loss [33], a patch-based adversarial loss [12] and a KL
regularization term in the latent space. The autoencoder was trained on pre-
processed MRIs using UniRes [3] into a common MNI space with a voxel size of
$1mm^{3}$ and was then kept unchanged during the LDM training. The latent
representations of the T1w brain MRIs were then used to train the LDM. A
conditional U-Net $\epsilon_{\theta}$ was then trained to predict the
artificial noise by the following objective:

$\theta^{*}=\operatorname*{arg\,min}_{\theta}\mathbb{E}_{z\sim\mathcal{E}(x),\epsilon\sim\mathcal{N}(0,1),t}||\epsilon-\epsilon_{\theta}(z_{t},t,\mathcal{C})||_{2}^{2}$
(1)

DDIM [26] has been used in brain LDM to replace the denoising diffusion
probabilistic models (DDPM) during inference to reduce the number of reverse
steps with minimal performance loss [23, 26]. This network
$\varepsilon_{\theta}$ is conditioned on four conditional variables
$\mathcal{C}$: age, gender, ventricular volume and brain volume, which are all
introduced by cross-attention layers [24]. Gender is a binary variable, while
the rest of the covariates are scaled to $[0,1]$. Finally, the pre-trained
decoder maps the latent vector into an HR MRI $\tilde{x}=\mathcal{D}(z_{0})$.
The architecture of the brain LDM can be found in Fig. 1.

Figure 1: (Left) The Brain LDM has two stages of training process. First, an
autoencoder is pre-trained to map T1w brain MRIs into a latent code
$z_{0}=\mathcal{E}(x)$. Then, a diffusion model is trained to generate $z_{0}$
latents from this learned latent space. During inference, DDIM has been
applied to reduce the sampling step with little performance drop. (Right) We
proposed two methods to handle different scenarios of MRI SR, based on the
architecture of brain LDM: 1) InverseSR(LDM): for SR with high sparsity, we
optimize the latent code $z_{T}^{*}$ and associated conditional variables
$\mathcal{C^{*}}$ using deterministic DDIM and decoder $\mathcal{D}$ to map
the latent code into brain MRI. 2) InverseSR(Decoder): for SR with low
sparsity, we optimize the $z_{0}^{*}$ which only use the decoder $\mathcal{D}$
to map the latent code into brain MRI.

####  2.0.2 Deterministic DDIM Sampling

In order to obtain a latent representation $z_{T}$ capable of reconstructing a
given noisy sample into a high-resolution image, we employ deterministic DDIM
sampling [26]:

$z_{t-1}=\sqrt{\alpha_{t-1}}\left(\frac{z_{t}-\sqrt{1-\alpha_{t}}\cdot\epsilon_{\theta}(z_{t},\mathcal{C},t)}{\sqrt{\alpha_{t}}}\right)+\sqrt{1-\alpha_{t-1}}\cdot\epsilon_{\theta}(z_{t},\mathcal{C},t)$
(2)

where $\alpha_{1:T}\in(0,1]^{T}$ is a time-dependent decreasing sequence,
$\frac{z_{t}-\sqrt{1-\alpha_{t}}\cdot\epsilon_{\theta}(z_{t},\mathcal{C},t)}{\sqrt{\alpha_{t}}}$
represents the “predicted $x_{0}$“, and
$\sqrt{1-\alpha_{t-1}}\cdot\epsilon_{\theta}(z_{t},\mathcal{C},t)$ can be
understood as the “direction pointing to $x_{t}$“ [26].

####  2.0.3 Corruption function $f$

We assume a corruption function $f$ known _a-priori_ that is applied on the HR
image $\tilde{x}$ obtained from the generative model, and compute the loss
function based on the corrupted image $f\circ\tilde{x}$ and the given LR input
image $I$. In clinical practice, a prevalent method for acquiring MR images is
prioritizing high in-plane resolution while sacrificing through-plane
resolution to expedite the acquisition process and reduce motion artifacts
[35]. To account for this procedure, we introduce a corruption function that
generates masks for non-acquired slices, enabling our method to in-paint the
missing slices. For instance, on $1\times 1\times 4\text{mm}^{3}$ undersampled
volumes, we create masks for three slices every four slices on the generated
HR $1\times 1\times 1\text{mm}^{3}$ volumes.

Algorithm 1 InverseSR(LDM)

1:Input: Low-resolution MR image $I$.

2:Output: Optimized noise latent code $z_{T}^{*}$ and conditional variables
$\mathcal{C}^{*}$

3:Initialize $z_{T}$ with gaussian noise from $N(0,\mathbb{I})$;

4:Initialize conditional variables $\mathcal{C}=0.5$;

5:for $j=0,\dots,N-1$ do

6: for $t=T,T-1,\dots,1$ do

7:
$z_{t-1}\leftarrow\sqrt{\alpha_{t-1}}\left(\frac{z_{t}-\sqrt{1-\alpha_{t}}\cdot\epsilon_{\theta}(z_{t},\mathcal{C},t)}{\sqrt{\alpha_{t}}}\right)+\sqrt{1-\alpha_{t-1}}\cdot\epsilon_{\theta}(z_{t},\mathcal{C},t)$;

8:
$L\leftarrow\lambda_{perc}L_{perc}(f\circ\mathcal{D}(z_{0}),I)+\lambda_{mae}\|f\circ\mathcal{D}(z_{0})-I\|$;

9: $\mathcal{C}\leftarrow\mathcal{C}-\alpha\nabla_{\mathcal{C}}L$;

10: $z_{T}\leftarrow z_{T}-\alpha\nabla_{z_{T}}L$;

11:Set $z_{T}^{*}\leftarrow z_{T}$; $\mathcal{C}^{*}\leftarrow\mathcal{C}$;

12:Return $z_{T}^{*},\mathcal{C}^{*}$;

####  2.0.4 InverseSR(LDM):

In the case of high sparsity MRI SR, we optimize the noise latent code
$z_{T}^{*}$ and its associated conditional variables $\mathcal{C^{*}}$ to
restore the HR image from the given LR input image $I$ using the optimization
method:

$\displaystyle\begin{split}z_{T}^{*},\mathcal{C}^{*}=\operatorname*{arg\,min}_{z_{T},\mathcal{C}}&\lambda_{perc}L_{perc}(f\circ\mathcal{D}(\text{DDIM}(z_{T},\mathcal{C},T)),I)+\\\
&\lambda_{mae}\|f\circ\mathcal{D}(\text{DDIM}(z_{T},\mathcal{C},T))-I\|\end{split}$
(3)

where $\text{DDIM}(z_{T},\mathcal{C},T)$ represents $T$ deterministic DDIM
sampling steps on the latent $z_{0}$ in Equation 2. We follow the brain LDM
model to use the perceptual loss $L_{perc}$ and the L1 pixelwise loss. The
loss function is computed on the corrupted image generated from the generative
model and the given LR input. A detailed pseudocode description of this method
can be found in algorithm 1.

####  2.0.5 InverseSR(Decoder):

For low sparsity MRI SR, we directly find the optimal latent code $z_{T}^{*}$
using the decoder $\mathcal{D}$:

$z_{0}^{*}=\operatorname*{arg\,min}_{z_{0}}\lambda_{perc}L_{perc}(f\circ\mathcal{D}(z_{0}),I)+\lambda_{mae}\|f\circ\mathcal{D}(z_{0})-I\|$
(4)

##  3 Experimental Design

####  3.0.1 Dataset For Validation:

We use 100 HR T1 MRIs from the IXI dataset (http://brain-development.org/ixi-
dataset/) to validate our method, after filtering out those scans where
registration failed. We note that subjects in the IXI dataset are around 10
years younger on average than those in UK Biobank. The MRI scans from UK
Biobank also had the faces masked out, while the scans from IXI did not. This
caused the faces of our reconstructions to appear blurred.

Figure 2: Qualitative results of our approach (InverseSR) and the cubic and
UniRes baselines on scans with 4 $mm$ and 8 $mm$ thickness.

####  3.0.2 Implementation:

Conditional variables are all initialized to 0.5. Voxels in all input volumes
are normalized to [0,1]. When sampling the pre-trained brain LDM with the DDIM
sampler, we run $T=46$ timesteps due to computational limitations on our
hardware. For InverseSR(LDM), $z_{T}$ is initialized with random gaussian
noise. For InverseSR(Decoder), we compute the mean latent code $\bar{z_{0}}$
as $\bar{z_{0}}=\sum_{i=1}^{S}\frac{1}{S}\text{DDIM}(z_{T}^{i},\mathcal{C},T)$
by first sampling $S=10,000$ $z_{T}^{i}$ samples from $N(0,\mathbb{I})$, then
passing them through the DDIM model. $N=600$ gradient descent steps are used
for InverseSR(LDM) to guarantee converging (Algorithm 1, line 5). 600
optimization steps are also utilized in InverseSR(Decoder). We use the Adam
optimizer with $\alpha=0.07$, $\beta_{1}=0.9$ and $\beta_{2}=0.999$.

##  4 Results

Fig. 2 shows the qualitative results on the coronal slices of SR from 4 and 8
$mm$ axial scans. The advantage of our approach is clear compared to baseline
methods because it is capable of restoring HR MRIs with smoothness even when
the slice thickness is large (i.e., 8 $mm$). This is the case because the pre-
trained LDM we use is able to build a powerful prior over the HR T1w MRI
domain. Therefore, the generated images of our method are HR MRIs with
smoothness in 3 directions: axial, sagittal and coronal, no matter how sparse
the input images $I$ are. Qualitative results of applying our method on tumour
and lesion filling are available in the supplementary material.

Table 1 shows quantitative results on 100 HR T1 scans from the IXI dataset,
which the brain LDM did not have access to during training. We investigated
mean peak signal-to-noise ratio (PSNR), and structural similarity index
measure (SSIM) [30] values and their corresponding standard deviation. We
compare our method to cubic interpolation, as well as a similar unsupervised
approach, UniRes [3]. We show our approach and the two compared methods on two
different settings of slice imputation: 4 $mm$ and 8 $mm$ thick-sliced axial
scans representing low sparsity and high sparsity LR MRIs, respectively. All
the metrics are computed on a 3D volume around the brain of size $160\times
224\times 160$. For SR at 4 $mm$, InverseSR(Decoder) achieves the highest mean
SSIM and PSNR scores among all compared methods, which are slightly higher
than the scores for InverseSR(LDM). For SR at 8 $mm$, Inverse(LDM) achieves
the highest mean SSIM and PSNR and lowest standard error than the two baseline
methods, which could be attributed to the stronger prior learned by the DDIM
model.

Table 1: Quantitative evaluation results (mean $\pm$ standard error) of our approach (InverseSR) and two baselines on 1 mm scans and corresponding SR counterpart - from 4 and 8 $mm$ axial scans. Slice Thickness | Methods | SSIM $\uparrow$ | PSNR $\uparrow$
---|---|---|---
4 $mm$ | InverseSR(LDM) | 0.797 $\pm$ 0.037 | 28.59 $\pm$ 1.61
InverseSR(Decoder) | 0.803 $\pm$ 0.030 | 29.64 $\pm$ 1.64
Cubic | 0.760 $\pm$ 0.052 | 23.84 $\pm$ 2.36
UniRes [3] | 0.688 $\pm$ 0.079 | 21.49 $\pm$ 2.61
8 $mm$ | InverseSR(LDM) | 0.754 $\pm$ 0.038 | 27.92 $\pm$ 1.60
Cubic | 0.632 $\pm$ 0.067 | 21.80 $\pm$ 2.36
UniRes [3] | 0.633 $\pm$ 0.053 | 20.91 $\pm$ 2.29

##  5 Limitations

One key limitation of our method is the need for large computational resources
to perform the image reconstruction, in particular the long Markov chain of
sampling steps required by the diffusion model to generate samples. An entire
pass through the diffusion model (lines 6-8 in Algorithm 1) is required for
every step in the gradient descent method. Another limitation of our method is
that it is limited by the capacity and output heterogeneity of the LDM
generator.

##  6 Conclusions

In this study, we have developed an unsupervised technique for MRI super-
resolution. We leverage a recent pre-trained Brain LDM [23] for building
powerful image priors over T1w brain MRIs. Unlike end-to-end supervised
approaches, which require retraining each time there is a distribution shift
over the input, our method is capable of being adapted to different settings
of MRI SR problems at test time. This feature is suitable for MRI SR since the
acquisition protocols and resolution of clinical brain MRI exams vary across
or even within institutions. We proposed two novel strategies for different
settings of MRI SR: InverseSR(LDM) for low sparsity MRI and InverseSR(Decoder)
for high sparsity MRI. We validated our method on 100 brain T1w MRIs from the
IXI dataset through slice imputation using input scans of 4 and 8 $mm$ slice
thickness, and compared our method with cubic interpolation and UniRes [3].

Experimental results have shown that our approach achieves superior
performance compared to the unsupervised baselines, and could create smooth HR
images with fine detail even on an external dataset (IXI). Experiments in this
paper focus on slice imputation, but our method could be adapted to other MRI
under-sampling problems by implementing different corruption functions $f$.
For instance, for reconstructing k-space under-sampled MR images, a new
corruption function could be designed by first converting the HR image into
k-space, then masking a chosen set of k-space measurements, and then
converting back to image space. Instead of estimating a single image, future
work could also estimate a distribution of reconstructed images through either
variational inference (like the BRGM model [19]) or through sampling methods
such as Markov Chain Monte Carlo (MCMC) or Langevin dynamics [16].

####  6.0.1 Acknowledgements

This work was fund by an NSERC Discovery Grant to JL. Funding was also
provided by a Nova Scotia Graduate Scholarship and a StFX Graduate Scholarship
to JW. Computational resources were provided by Compute Canada.

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Supplementary Material

Figure 3: Qualitative results of our approach (InverseSR) on lesion and tumour
filling. Lesion and tumour masks are separately from the ISBI2015 multiple
sclerosis dataset [4] and BraTS2020 [21].


# Fermion and meson mass generation in non-Hermitian Nambu–Jona-Lasinio models

Alexander Felski<EMAIL_ADDRESS>S. P. Klevansky
<EMAIL_ADDRESS>Institut für Theoretische Physik, Universität
Heidelberg, Philosophenweg 12, 69120 Heidelberg, Germany

###### Abstract

We investigate the effects of non-Hermiticity on interacting fermionic
systems. We do this by including non-Hermitian bilinear terms into the 3+1
dimensional Nambu–Jona-Lasinio (NJL) model. Two possible bilinear
modifications give rise to $\mathcal{PT}$ symmetric theories; this happens
when the standard NJL model is extended either by a pseudovector background
field $ig\bar{\psi}\gamma_{5}B_{\mu}\gamma^{\mu}\psi$ or by an antisymmetric-
tensor background field $g\bar{\psi}F_{\mu\nu}\gamma^{\mu}\gamma^{\nu}\psi$.
The three remaining bilinears are anti-$\mathcal{PT}$-symmetric in nature,
$ig\bar{\psi}B_{\mu}\gamma^{\mu}\psi,ig\bar{\psi}\gamma_{5}\psi$ and
$ig\bar{\psi}\mathbbm{1}\psi$, so that the Hamiltonian then has no overall
symmetry. The pseudovector $ig\bar{\psi}\gamma_{5}B_{\mu}\gamma^{\mu}\psi$ and
the vector $ig\bar{\psi}B_{\mu}\gamma^{\mu}\psi$ combinations, are, in
addition, chirally symmetric. Thus, within this framework we are able to
examine the effects that the various combinations of non-Hermiticity,
$\mathcal{PT}$ symmetry, chiral symmetry and the two-body interactions of the
NJL model have on the existence and dynamical generation of a real effective
fermion mass (a feature which is absent in the corresponding modified massless
free Dirac models) as well as on the masses of the composite particles, the
pseudoscalar and scalar mesonic modes ($\pi$ and $\sigma$ mesons). Our
findings demonstrate that $\mathcal{PT}$ symmetry is not necessary for real
fermion mass solutions to exist, rather the two-body interactions of the NJL
model supersede the non-Hermitian bilinear effects. The effects of chiral
symmetry are evident most clearly in the meson modes, the pseudoscalar of
which will always be Goldstone in nature if the system is chirally symmetric.
Second solutions of the mesonic equations are also discussed.

##  I Introduction

The occurrence of real eigenvalue spectra in non-Hermitian systems that are
symmetric under combined parity reflection $\mathcal{P}$ and time reversal
$\mathcal{T}$ has, since its first demonstration by Bender and Boettcher in
1998 bb , inspired a wide variety of theoretical and experimental studies in
$\mathcal{PT}$-symmetric physics. While initially focusing on classical and
semi-classical systems these studies were soon extended to include bosonic
field theories as well bhkss . For fermionic systems, Jones-Smith et al. jsm
brought attention to the property of odd time-reversal symmetry
($\mathcal{T}^{2}=-\mathbbm{1}$) in $3+1$ dimensions and its importance in the
context of $\mathcal{PT}$ symmetry, in contrast to the even time-reversal
symmetry of bosonic systems.

In further studies involving fermions, unexpected behavior has emerged: In bkb
the free Dirac Lagrangian was modified by the addition of non-Hermitian, but
$\mathcal{PT}$-symmetric, bilinears of the fermionic field $\psi$ and its
conjugate $\bar{\psi}$. It was found that an unbroken $\mathcal{PT}$ symmetry
phase, that is, a regime with a real spectrum, does not simply exist: without
a finite bare mass these modified Dirac fermions generally have complex
physical masses. However, including higher-order interactions can override
this breakdown of an unbroken $\mathcal{PT}$ symmetry regime. We demonstrated
in fbk  that a region of real mass solutions exists, when the bilinear term
that includes the $\mathcal{PT}$-symmetric pseudovector background field
$ig\bar{\psi}\gamma_{5}B_{\mu}\gamma^{\mu}\psi$, is incorporated into the
$3+1$ dimensional Nambu–Jona-Lasinio (NJL) model, which contains two-body
interactions as well. The gap equation of this modified system with
Hamiltonian density

$\mathcal{H}=\bar{\psi}(-i\gamma^{k}\partial_{k}+m_{0}+ig\gamma_{5}B_{\mu}\gamma^{\mu})\psi-G[(\bar{\psi}\psi)^{2}+(\bar{\psi}i\gamma_{5}\vec{\tau}\psi)^{2}]$
(1)

has real physical mass solutions even in the chiral limit of vanishing bare
mass $m_{0}$, provided the coupling constant $g$ of the non-Hermitian bilinear
does not exceed a critical value $g_{crit}$. Moreover, when compared to the
standard NJL model ($g=0$), increased fermion masses could be generated
dynamically, mimicking the effect of a small bare mass $m_{0}$ without
breaking chiral symmetry.

Similar results have been demonstrated in ams ; ms ; ccr  for models with
Yukawa-type interactions that couple different fields, and can at least in
principle be obtained from four-point interactions through partial
bosonization.

The existence of an unbroken $\mathcal{PT}$ symmetry regime in the modified
NJL model (1) without a bare mass $m_{0}$ is remarkable, given that in the
underlying modified Dirac theory ($G=0$) this symmetry is realized in the
broken regime exclusively. It raises the question as to whether the addition
of a non-Hermitian bilinear has to uphold $\mathcal{PT}$ symmetry at all. We
thus ask what specific role the combination of chiral symmetry, $\mathcal{PT}$
symmetry and higher-order interactions play in the generation of real masses
within fermionic models. The first aim of this study is to expand the analysis
of the modified NJL model (1) to cover all possible non-Hermitian bilinear
extensions, including but not limited to the only other
$\mathcal{PT}$-symmetric case of an antisymmetric-tensor background field
$g\bar{\psi}F_{\mu\nu}\gamma^{\mu}\gamma^{\nu}\psi$ bkb . The analysis of
these modifications to the NJL model furthermore covers cases in which chiral
symmetry is preserved as well as cases in which it is broken explicitly.

Our second aim in this paper goes beyond the discussion of fermionic mass
generation to study the effects on the mesonic composite states that can be
generated within the context of these models. For those systems in which we
find dynamically generated real fermion masses, we investigate the effect that
the non-Hermitian bilinear terms have on the mass of the scalar and
pseudoscalar bound states, i.e. these are the $\sigma$ and $\pi$ mesons in the
context of quantum chromodynamics (QCD).

This paper is structured as follows: In Sec. II the modified NJL model is
introduced and all possible bilinear additions leading to a non-Hermitian
model are identified. Their behavior under $\mathcal{PT}$ symmetry and chiral
symmetry is presented. In Sec. III we first recapitulate the algebraic form of
the gap equation and its solution for the system previously studied in fbk .
We then proceed to analyze the structure of the gap equations and the
existence of real fermion mass solutions for the other non-Hermitian
extensions of the NJL model. In Sec. IV we investigate the corresponding mass
of the scalar and pseudoscalar bound states for models in which a dynamical
fermion mass generation could be identified. We conclude in Sec. V.

##  II The modified NJL model

In its two-flavor version ($N_{f}=2$), the standard NJL model njl  is given by
the Hamiltonian density

$\mathcal{H}_{\textrm{NJL}}=\bar{\psi}(-i\gamma^{k}\partial_{k}+m_{0})\psi-G[(\bar{\psi}\psi)^{2}+(\bar{\psi}i\gamma_{5}\vec{\tau}\psi)^{2}],$

in terms of the Dirac matrices $\gamma$ with $k=1..3$, the isospin SU(2)
matrices $\vec{\tau}$, a coupling strength $G$, and the bare mass $m_{0}$.
While the mass term $m_{0}$ breaks chiral symmetry explicitly, the specific
combination of four-point interaction terms preserves it. In the limit of
vanishing bare mass the model can thus be used to study the spontaneous
breaking of chiral symmetry occurring through a mechanism that parallels pair
condensation in the Bardeen-Cooper-Schrieffer (BCS) theory of
superconductivity bcs . As such it has been developed into an effective field
theory of QCD in the chiral limit. In this context, including a small bare
mass $m_{0}$ leads to a model with an _approximate_ spontaneously broken
chiral symmetry that has a small current quark mass and a light pseudo-
Goldstone boson, identified as the pion. The numerical analysis throughout
this paper will be based on established quantities within the QCD
interpretation, using a four-momentum Euclidean cutoff scale of $\Lambda=1015$
MeV for the purpose of regularization and a coupling strength satisfying
$G\Lambda^{2}=3.93$, cf. spk .

In continuation of our previous study fbk , we introduce the bilinear
extension of the NJL model

$\mathcal{H}=\bar{\psi}(-i\gamma^{k}\partial_{k}+m_{0}+g\Gamma)\psi-G[(\bar{\psi}\psi)^{2}+(\bar{\psi}i\gamma_{5}\vec{\tau}\psi)^{2}],$
(2)

where $\Gamma$ generally denotes a complex $4\times 4$ matrix and $g$ is the
associated real coupling constant. In the following, we consider the complete
set of $4\times 4$ matrices generated by the Dirac matrices and identify all
those that are non-Hermitian, irrespective of whether they are
$\mathcal{PT}$-symmetric or not. The matrix $\Gamma$ in (2) will then take on
the corresponding role in the subsequent analysis. In addition, we consider
the behavior of the non-Hermitian bilinears under chiral symmetry
transformations, and thus infer the symmetries present in the modified NJL
models given by (2).

Any bilinear of the fermionic field $\psi$ and its conjugate $\bar{\psi}$ is
of the form $\bar{\psi}\Gamma\psi$. As such, they can be written as real
superpositions based on the following $32$ matrices:

$\displaystyle\mathbbm{1},$ $\displaystyle\quad\gamma_{5},$
$\displaystyle\quad B_{\mu}\gamma^{\mu},$
$\displaystyle\quad\gamma_{5}B_{\mu}\gamma^{\mu},$ $\displaystyle\quad
F_{\mu\nu}\gamma^{\mu}\gamma^{\nu},$ (3) $\displaystyle i\mathbbm{1},$
$\displaystyle\quad i\gamma_{5},$ $\displaystyle\quad iB_{\mu}\gamma^{\mu},$
$\displaystyle\quad i\gamma_{5}B_{\mu}\gamma^{\mu},$ $\displaystyle\quad
iF_{\mu\nu}\gamma^{\mu}\gamma^{\nu},$

where $\mu\leq\nu$ denote spin indices, $B_{\mu}$ are real vector elements,
and $F_{\mu\nu}$ are real elements of an antisymmetric matrix. The
corresponding bilinears $\bar{\psi}\Gamma\psi$ formed by these behave as
scalars, pseudoscalars, vectors, pseudovectors and antisymmetric $2$nd-rank
tensors respectively under Lorentz transformations (from left to right).
Including an imaginary unit into any bilinear will change it from being
symmetric to being antisymmetric under Hermitian conjugation, and vice versa.
Thus we identify half of the bilinear terms associated with the matrices in
(3) to be anti-Hermitian, namely those for which $\Gamma$ takes one of the
forms

$i\mathbbm{1},\quad\gamma_{5},\quad iB_{\mu}\gamma^{\mu},\quad
i\gamma_{5}B_{\mu}\gamma^{\mu},\quad F_{\mu\nu}\gamma^{\mu}\gamma^{\nu}.$ (4)

These terms result in the non-Hermitian extensions of the NJL model considered
throughout this paper.

To determine the behavior of $\bar{\psi}\Gamma\psi$, with $\Gamma$ being one
of the terms in (4), under combined parity reflection $\mathcal{P}$ and time
reversal $\mathcal{T}$, recall the usual definition of these transformations
in $3+1$ dimensions bd :

$\displaystyle\mathcal{P}:$
$\displaystyle\,\psi(t,\mathbf{x})\to\mathcal{P}\psi(t,\mathbf{x})\mathcal{P}^{-1}$
$\displaystyle=\gamma^{0}\psi(t,-\mathbf{x}),$ (5) $\displaystyle\mathcal{T}:$
$\displaystyle\,\psi(t,\mathbf{x})\to\mathcal{T}\psi(t,\mathbf{x})\mathcal{T}^{-1}$
$\displaystyle=i\gamma^{1}\gamma^{3}\psi^{*}(-t,\mathbf{x}).$ (6)

We find that only two types of non-Hermitian bilinears are
$\mathcal{PT}$-symmetric; specifically, $[\mathcal{PT},\Gamma]=0$ for

$\displaystyle\Gamma_{PT_{1}}$ $\displaystyle=i\gamma_{5}B_{\mu}\gamma^{\mu},$
(7) $\displaystyle\Gamma_{PT_{2}}$
$\displaystyle=F_{\mu\nu}\gamma^{\mu}\gamma^{\nu}.$ (8)

The remaining non-Hermitian terms are anti-$\mathcal{PT}$-symmetric,
$\\{\mathcal{PT},\Gamma\\}=0$, and will in the following be referred to as

$\displaystyle\Gamma_{aPT_{1}}$ $\displaystyle=iB_{\mu}\gamma^{\mu},$ (9)
$\displaystyle\Gamma_{aPT_{2}}$ $\displaystyle=\gamma_{5},$ (10)
$\displaystyle\Gamma_{aPT_{3}}$ $\displaystyle=i\mathbbm{1}.$ (11)

Notice that in these anti-$\mathcal{PT}$-symmetric cases the Hamiltonian
density (2) has no such overall symmetry. The system as a whole is thus non-
Hermitian and non-$\mathcal{PT}$-symmetric.

Moreover, only two of the terms in (4) anticommute with $\gamma_{5}$, leading
to an axial flavor symmetry and thus an overall chiral symmetry of the
Hamiltonian density (2) in the limit of vanishing bare mass: $\Gamma_{PT_{1}}$
and $\Gamma_{aPT_{1}}$. In contrast, even without a bare mass $m_{0}$, chiral
symmetry is broken explicitly in the theories based on including the terms
$\Gamma_{PT_{2}}$, $\Gamma_{aPT_{2}}$, and $\Gamma_{aPT_{3}}$.

Thus, we have identified five possible non-Hermitian bilinear extensions of
the NJL model: One that is $\mathcal{PT}$-symmetric and chirally symmetric
($\Gamma_{PT_{1}}$), one that is $\mathcal{PT}$-symmetric but breaks chiral
symmetry explicitly ($\Gamma_{PT_{2}}$), one that is
anti-$\mathcal{PT}$-symmetric but chirally symmetric ($\Gamma_{aPT_{1}}$), and
two which break both $\mathcal{PT}$ and chiral symmetry ($\Gamma_{aPT_{2}}$
and $\Gamma_{aPT_{3}}$).

##  III Gap equation and fermion masses

A self-consistent approximation to the effective fermion mass of the standard
NJL model is expressed by the gap equation, which can be obtained following
Feynman-Dyson perturbation theory, cf. spk ; fw . We established in fbk  that
the gap equation of an extended NJL model such as (2), which includes
additional bilinear terms, has the same general structure as that of the
standard NJL model:

$m=m_{0}+\frac{2iGN_{c}N_{f}}{(2\pi)^{4}}\,I,$ (12)

where $m$ denotes the effective fermion mass, $N_{c}=3$ the number of colors,
$N_{f}=2$ is the number of flavors, and $I$ is the momentum integral over the
spinor trace of the fermion propagator:

$I=\int\\!\\!d^{4}p\,\,{\rm tr}[S(p)].$ (13)

Here, the full fermion propagator $S(p)$ has the same form as the propagator
$S^{0}(p)$ of the corresponding modified Dirac theory (obtained at $G=0$) and
can be found by replacing the bare mass $m_{0}$ with the effective mass $m$:

$S(p)=(\not{p}-m-g\Gamma)^{-1}.$ (14)

Analyzing the effective fermion mass of the non-Hermitian NJL models
introduced in the previous section thus relies on the evaluation of the spinor
trace and the momentum integration in (13) for these models.

In this section we derive the algebraic gap equation for the five non-
Hermitian NJL models and study the existence and behavior of real fermion mass
solutions. Section III.1 summarizes the results for the
$\mathcal{PT}$-symmetric model based on $\Gamma_{PT_{1}}$, which were
previously obtained in fbk . In Sec. III.2 the second $\mathcal{PT}$-symmetric
case (based on $\Gamma_{PT_{2}}$) is analyzed, and Secs. III.3-E discuss the
three remaining non-$\mathcal{PT}$-symmetric models.

###  III.1 $\Gamma_{PT_{1}}=i\gamma_{5}B_{\mu}\gamma^{\mu}$

In the case of the $\mathcal{PT}$-symmetric model based on including the
pseudovector background field
$\Gamma_{PT_{1}}=i\gamma_{5}B_{\mu}\gamma^{\mu}$, as studied in fbk , the full
fermion propagator (14) was rationalized,

$S(p)=\frac{(\not{p}+m+i\gamma_{5}B_{\mu}\gamma^{\mu})\bigl{[}(p^{2}-m^{2}-g^{2}B^{2})+2igm\gamma_{5}B_{\nu}\gamma^{\nu}+2igB_{\nu}p^{\nu}\gamma_{5}\bigr{]}}{(p^{2}-m^{2}-g^{2}B^{2})^{2}-4g^{2}m^{2}B^{2}+4g^{2}(B_{\mu}p^{\mu})^{2}},$
(15)

in order to evaluate the spinor trace

$\mathrm{tr}[S(p)]=\frac{4m(p^{2}-m^{2}+g^{2}B^{2})}{(p^{2}-m^{2}-g^{2}B^{2})^{2}-4g^{2}m^{2}B^{2}+4g^{2}(B_{\mu}p^{\mu})^{2}}.$

The introduction of a Euclidean four-momentum cutoff $\Lambda$ for the purpose
of regularization then allows the momentum integration in (13). To that end we
change to Euclidean coordinates with $p_{0}=ip_{4}$ and $B_{0}=iB_{4}$, such
that $p^{2}=-p_{E}^{2}$, $B^{2}=-B_{E}^{2}$, and $B_{\mu}p^{\mu}=-B_{E}\cdot
p_{E}$. In the chiral limit of vanishing bare mass $m_{0}$, the gap equation
(12) then takes on the algebraic form

$\displaystyle\begin{split}\frac{2\pi^{2}}{\tilde{G}N_{c}N_{f}}=\frac{1}{4\tilde{g}^{2}}\bigg{\\{}&\sqrt{(1+\tilde{m}^{2}-\tilde{g}^{2})^{2}+4\tilde{g}^{2}\tilde{m}^{2}}(1+\tilde{m}^{2}+7\tilde{g}^{2})-(\tilde{m}^{2}+\tilde{g}^{2})(2+\tilde{m}^{2}+7\tilde{g}^{2})-1\\\
&+4\tilde{g}^{2}(2\tilde{g}^{2}-\tilde{m}^{2})\ln\Big{[}\frac{1}{2\tilde{m}^{2}}\Big{(}\sqrt{(1+\tilde{m}^{2}-\tilde{g}^{2})^{2}+4\tilde{g}^{2}\tilde{m}^{2}}+1+\tilde{m}^{2}-\tilde{g}^{2}\Big{)}\Big{]}\bigg{\\}}\end{split}$
(16)

in terms of the rescaled parameters $\tilde{G}=G\Lambda^{2}$,
$\tilde{m}=m\Lambda^{-1}$, and $\tilde{g}=g\lvert B_{E}\rvert\Lambda^{-1}$,
which is proportional to the amplitude of the pseudovector background field.

In the context of QCD, this equation has real fermion mass solutions $m$ as
long as $\tilde{g}<\tilde{g}_{crit}\approx 1.261$. The behavior of the
solution in this regime is shown in Fig. 1. The largest mass of $m\approx
460.870$ MeV is found at a coupling value of $\tilde{g}\approx 0.702$. Notice
that masses larger than the $m_{NJL}\approx 238.487$ MeV of the standard NJL
model are dynamically generated for all coupling values below
$\tilde{g}_{dyn}\approx 1.183$. A current quark mass in the range of the bare
up quark mass, $m_{u}=(1.7-3.3)$ MeV, is already generated at small coupling
values $\tilde{g}\approx(0.025-0.034)$ and in the range of the bare down quark
mass, $m_{d}=(4.1-5.8)$ MeV, for $\tilde{g}\approx(0.038-0.046)$.

Figure 1:  Behavior of the effective fermion mass $m$ in the modified NJL
model based on $\Gamma_{PT_{1}}$ as a function of the scaled coupling constant
$\tilde{g}=g\lvert B_{E}\rvert\Lambda^{-1}$.

Altogether, the $\mathcal{PT}$ and chirally symmetric modified NJL model based
on $\Gamma_{PT_{1}}$ admits a finite region of unbroken $\mathcal{PT}$
symmetry, even for vanishing bare mass $m_{0}$. Real fermion mass solutions
are obtained for coupling constant values up to a critical value
$\tilde{g}_{crit}$. Fermion mass is dynamically generated for
$\tilde{g}<\tilde{g}_{dyn}<\tilde{g}_{crit}$; larger coupling values lead to
an effective mass loss.

###  III.2 $\Gamma_{PT_{2}}=F_{\mu\nu}\gamma^{\mu}\gamma^{\nu}$

For the $\mathcal{PT}$-symmetric model based on
$\Gamma_{PT_{2}}=F_{\mu\nu}\gamma^{\mu}\gamma^{\nu}$, where
$F_{\mu\nu}=-F_{\nu\mu}$ is real, the first goal is to rationalize the full
fermion propagator

$S(p)=(\not{p}-m-gF_{\mu\nu}\gamma^{\mu}\gamma^{\nu})^{-1}$

in order to evaluate the spinor trace in (13). We expand $S(p)$ with
$(\not{p}-m+gF_{\alpha\beta}\gamma^{\alpha}\gamma^{\beta})$. The denominator
then takes the form

$\displaystyle(\not{p}-m-gF_{\mu\nu}\gamma^{\mu}\gamma^{\nu})\,(\not{p}-m+gF_{\alpha\beta}\gamma^{\alpha}\gamma^{\beta})=$
$\displaystyle(p^{2}+m^{2}-2g^{2}f_{1})-2m\not{p}-4gF_{\mu\nu}p^{\nu}\gamma^{\mu}+8ig^{2}f_{2}\gamma_{5},$

where we have used the relations
$F_{\mu\nu}(\gamma^{\mu}\gamma^{\nu}\not{p}-\not{p}\gamma^{\mu}\gamma^{\nu})=4F_{\mu\nu}p^{\nu}\gamma^{\mu}$
and
$F_{\mu\nu}F_{\alpha\beta}\gamma^{\mu}\gamma^{\nu}\gamma^{\alpha}\gamma^{\beta}=2f_{1}\mathbbm{1}-8if_{2}\gamma_{5}$,
with

$\displaystyle f_{1}$
$\displaystyle=2(F_{01}^{2}+F_{02}^{2}+F_{03}^{2}-F_{12}^{2}-F_{13}^{2}-F_{23}^{2})=\mathrm{tr}[F^{2}],$
(17) $\displaystyle f_{2}$
$\displaystyle=(F_{01}F_{23}-F_{02}F_{13}+F_{03}F_{12})=\mathrm{Pf}(F).$ (18)

The denominator still contains non-scalar contributions, thus we expand $S(p)$
again with a factor that has the opposite sign in those terms: the denominator
of the resulting expression becomes

$\displaystyle\bigl{[}(p^{2}+m^{2}-2g^{2}f_{1})-2m\not{p}-4gF_{\mu\nu}p^{\nu}\gamma^{\mu}+8ig^{2}f_{2}\gamma_{5}\bigr{]}$
$\displaystyle\times$
$\displaystyle\bigl{[}(p^{2}+m^{2}-2g^{2}f_{1})+2m\not{p}+4gF_{\alpha\beta}p^{\beta}\gamma^{\alpha}-8ig^{2}f_{2}\gamma_{5}\bigr{]}$
$\displaystyle=(p^{2}+m^{2}-2g^{2}f_{1})^{2}-4m^{2}p^{2}+64g^{4}f_{2}^{2}$
$\displaystyle\qquad\qquad+16g^{2}p^{\mu}F_{\mu\nu}F^{\nu}_{\,\,\,\alpha}p^{\alpha},$

where we utilized $\\{\gamma^{\mu},\gamma^{\nu}\\}=2\eta^{\mu\nu}$,
$\\{\gamma^{\mu},\gamma_{5}\\}=0$, and $F_{\mu\nu}=-F_{\nu\mu}$. It is now a
scalar and the full propagator can then be written in the rationalized form

$\displaystyle
S(p)=\frac{(\not{p}-m+gF_{\mu\nu}\gamma^{\mu}\gamma^{\nu})\bigl{[}(p^{2}+m^{2}-2g^{2}f_{1})+2m\not{p}+4gF_{\alpha\beta}p^{\beta}\gamma^{\alpha}-8ig^{2}f_{2}\gamma_{5}\bigr{]}}{(p^{2}+m^{2}-2g^{2}f_{1})^{2}-4m^{2}p^{2}+64g^{4}f_{2}^{2}+16g^{2}p^{\mu}F_{\mu\nu}F^{\nu}_{\,\,\,\alpha}p^{\alpha}}.$
(19)

In multiplying out the numerator, we notice that almost all resulting terms
have a vanishing spinor trace. Using that $\mathrm{tr}[\mathbbm{1}]=4$,
$\mathrm{tr}[\gamma^{\mu}\gamma^{\nu}]=4\eta^{\mu\nu}$, and
$F_{\mu}^{\,\,\mu}=0$ we find the trace of the full propagator to be

$\displaystyle\mathrm{tr}[S(p)]=\frac{4m(p^{2}-m^{2}+2g^{2}f_{1})}{(p^{2}+m^{2}-2g^{2}f_{1})^{2}-4m^{2}p^{2}+64g^{4}f_{2}^{2}+16g^{2}p^{\mu}F_{\mu\nu}F^{\nu}_{\,\,\,\alpha}p^{\alpha}}.$
(20)

For the momentum integration in (13) we now regularize the integral in the
Euclidean four-momentum cutoff method. That is, we first change to a Euclidean
system by denoting $p_{0}=ip_{4}$ and $F_{0k}=iF_{4k},\forall k\in[1,3]$, such
that $d^{4}p=id^{4}p_{E}$, $p^{2}=-p_{E}^{2}$ and
$p^{\mu}F_{\mu\nu}F^{\nu}_{\,\,\,\alpha}p^{\alpha}=-p_{E}\cdot F_{E}\cdot
F_{E}\cdot p_{E}$, where the (complex) matrix $F_{E}$ is

$F_{E}=\begin{bmatrix}0&F_{41}&F_{42}&F_{43}\\\ -F_{41}&0&F_{12}&F_{13}\\\
-F_{42}&-F_{12}&0&F_{23}\\\ -F_{43}&-F_{13}&-F_{23}&0\end{bmatrix}.$ (21)

In this form the dependence on $F_{E}$ is somewhat unwieldy. When $F_{E}$ is
diagonalizable, it is orthogonally so hj , such that a transformation $Q$
exists, with $Q^{T}Q=\mathbbm{1}$ and
$Q^{T}F_{E}\,Q=\mathrm{diag}(\lambda_{1},-\lambda_{1},\lambda_{2},-\lambda_{2})$,
where

$\displaystyle\lambda_{1,2}=$
$\displaystyle\tfrac{1}{2}\sqrt{f_{1}\mp\sqrt{f_{1}^{2}+16f_{2}^{2}}}.$ (22)

Thus transforming the Euclidean four-momentum according to
$p_{E}\rightarrow\leavevmode\nobreak\ Q\cdot\leavevmode\nobreak\ p_{E}$ leaves
the integral measure invariant and results in a much more convenient
dependence of (20) on the momentum components. This of course relies on the
diagonalizability of $F_{E}$, which from the eigenvalues (22) we see to be so,
provided that $f_{2}$ does not vanish. Since $f_{2}$ as given in (18) implies
that $f_{2}^{2}=\mathrm{Det}(F)$, this is a minor restriction and will be
considered to be the case in what follows. With the introduction of the four-
momentum cutoff $\Lambda$, the regularized integral (13) over the trace (20)
then has the form

$I_{PT_{2}}=-4im\int^{\Lambda}\\!\\!d^{4}p_{E}\,\frac{(p_{E}^{2}+m^{2}-2g^{2}f_{1})}{(p_{E}^{2}-m^{2}+2g^{2}f_{1})^{2}+4m^{2}p_{E}^{2}+64g^{4}f_{2}^{2}-16g^{2}\lambda_{1}^{2}(p_{4}^{2}+p_{1}^{2})-16g^{2}\lambda_{2}^{2}(p_{2}^{2}+p_{3}^{2})}.$
(23)

To evaluate this, we first express the momentum components in two sets of
polar coordinates, where $p_{4}=R_{1}\cos\phi_{1}$, $p_{1}=R_{1}\sin\phi_{1}$
and $p_{2}=R_{2}\cos\phi_{2}$, $p_{3}=R_{2}\sin\phi_{2}$. Equation (23)
becomes

$I_{PT_{2}}=-16i\pi^{2}m\int^{\Lambda}\\!\\!dR_{1}dR_{2}\frac{R_{1}R_{2}(R_{1}^{2}+R_{2}^{2}+m^{2}-2g^{2}f_{1})}{(R_{1}^{2}+R_{2}^{2}-m^{2}+2g^{2}f_{1})^{2}+4m^{2}(R_{1}^{2}+R_{2}^{2})+64g^{4}f_{2}^{2}-16g^{2}(\lambda_{1}^{2}R_{1}^{2}+\lambda_{2}^{2}R_{2}^{2})}.$
(24)

Now, combining $R_{1}$ and $R_{2}$ in polar coordinates $R_{1}=r\cos\theta$
and $R_{2}=r\sin\theta$ with $\theta\in[0,\pi/2]$ and $r\in[0,\Lambda]$ yields

$\displaystyle\begin{split}I_{PT_{2}}=&-16i\pi^{2}m\int_{0}^{\Lambda}\\!\\!drr^{3}(r^{2}+m^{2}-2g^{2}f_{1})\\\
&\times\int_{0}^{\pi/2}\\!\\!d\theta\frac{\cos\theta\sin\theta}{A(r)-B(r)\cos^{2}\theta},\end{split}$
(25)

where

$\displaystyle\begin{split}A(r)=&(r^{2}+m^{2}-2g^{2}f_{1}^{2})+64g^{4}f_{2}^{2}\\\
&+4g^{2}r^{2}(f_{1}-\sqrt{f_{1}^{2}+16f_{2}^{2}}),\end{split}$ (26)
$\displaystyle B(r)=$ $\displaystyle-8g^{2}r^{2}\sqrt{f_{1}^{2}+16f_{2}^{2}}.$
(27)

The angular part is a standard integral gr , and $I_{PT_{2}}$ becomes

$\displaystyle\begin{split}I_{PT_{2}}=&-i\pi^{2}m\int_{0}^{\Lambda}\\!\\!dr\frac{r(r^{2}+m^{2}-2g^{2}f_{1})}{g^{2}\sqrt{f_{1}^{2}+16f_{2}^{2}}}\ln\Big{[}1-\frac{B(r)}{A(r)}\Big{]}\end{split}$
(28)

and the remaining radial integration can then be rewritten as

$\displaystyle\begin{split}&I_{PT_{2}}=\frac{i\pi^{2}m}{2g^{2}\sqrt{f_{1}^{2}+16f_{2}^{2}}}\int_{0}^{\Lambda^{2}}\\!\\!dz\,(z+m^{2}-2g^{2}f_{1})\\\
&\times\bigl{\\{}\ln(z+a_{1})+\ln(z+a_{2})-\ln(z+a_{3})-\ln(z+a_{4})\bigr{\\}},\end{split}$
(29)

with

$\displaystyle a_{1,2}=$ $\displaystyle
m^{2}-2g^{2}\sqrt{f_{1}^{2}+16f_{2}^{2}}\pm\sqrt{4m^{2}g^{2}(f_{1}-\sqrt{f_{1}^{2}+16f_{2}^{2}})},$
(30) $\displaystyle a_{3,4}=$ $\displaystyle
m^{2}+2g^{2}\sqrt{f_{1}^{2}+16f_{2}^{2}}\pm\sqrt{4m^{2}g^{2}(f_{1}+\sqrt{f_{1}^{2}+16f_{2}^{2}})}.$
(31)

After integration and some further simplification this then becomes

$\displaystyle\begin{split}I_{PT_{2}}=\frac{i\Lambda^{3}\pi^{2}\tilde{m}}{2\tilde{g}^{2}\sqrt{1+f^{2}}}\Bigl{[}&(\tilde{a}_{1}+1)\ln(\tfrac{1+\tilde{a}_{1}}{\tilde{a}_{1}})(\tilde{m}^{2}-2\tilde{g}^{2}+\tfrac{1-\tilde{a}_{1}}{2})\\\
+&(\tilde{a}_{2}+1)\ln(\tfrac{1+\tilde{a}_{2}}{\tilde{a}_{2}})(\tilde{m}^{2}-2\tilde{g}^{2}+\tfrac{1-\tilde{a}_{2}}{2})\\\
-&(\tilde{a}_{3}+1)\ln(\tfrac{1+\tilde{a}_{3}}{\tilde{a}_{3}})(\tilde{m}^{2}-2\tilde{g}^{2}+\tfrac{1-\tilde{a}_{3}}{2})\\\
-&(\tilde{a}_{4}+1)\ln(\tfrac{1+\tilde{a}_{4}}{\tilde{a}_{4}})(\tilde{m}^{2}-2\tilde{g}^{2}+\tfrac{1-\tilde{a}_{4}}{2})\\\
-&4\tilde{g}^{2}\sqrt{1+f^{2}}\Bigr{]}\end{split}$ (32)

in terms of the rescaled quantities $\tilde{m}=m\Lambda^{-1}$,
$\tilde{a}=a\Lambda^{-2}$, $\tilde{g}^{2}=g^{2}f_{1}\Lambda^{-2}$, and
$f=4f_{2}/f_{1}$. Note that $f$ is a real valued parameter. When we consider
the limit of vanishing bare mass $m_{0}$, denote $\tilde{G}=G\Lambda^{2}$, and
use the result for the momentum integral $I_{PT_{2}}$, the general gap
equation (12) takes the algebraic form

((a))

((b))

((c))

Figure 2:  (a) Behavior of the right hand side of the gap equation (III.2)
based on $\Gamma_{PT_{2}}$ as a function of the mass $\tilde{m}$ and for
different values of the coupling $\tilde{g}$. The value for $f=0.5$ is fixed,
but this case is representative of the general behavior. The horizontal line
represents the left hand side of the gap equation. (b) Behavior of the
critical coupling value $\tilde{g}_{crit}$ up to which the gap equation
(III.2) has real mass solutions as a function of $f$. (c) Behavior of the
effective fermion mass $m$ in MeV as a function of the coupling constant up to
the corresponding critical values for different values of $f$.

$\displaystyle\frac{2\pi^{2}}{\tilde{G}N_{c}N_{f}}=\frac{1}{8\tilde{g}^{2}\sqrt{1+f^{2}}}\Big{\\{}$
$\displaystyle
4\tilde{g}^{2}\sqrt{1+f^{2}}-(\tilde{a}_{1}+1)\ln(\tfrac{1+\tilde{a}_{1}}{\tilde{a}_{1}})(\tilde{m}^{2}-2\tilde{g}^{2}+\tfrac{1-\tilde{a}_{1}}{2})-(\tilde{a}_{2}+1)\ln(\tfrac{1+\tilde{a}_{2}}{\tilde{a}_{2}})(\tilde{m}^{2}-2\tilde{g}^{2}+\tfrac{1-\tilde{a}_{2}}{2})$
$\displaystyle+(\tilde{a}_{3}+1)\ln(\tfrac{1+\tilde{a}_{3}}{\tilde{a}_{3}})(\tilde{m}^{2}-2\tilde{g}^{2}+\tfrac{1-\tilde{a}_{3}}{2})+(\tilde{a}_{4}+1)\ln(\tfrac{1+\tilde{a}_{4}}{\tilde{a}_{4}})(\tilde{m}^{2}-2\tilde{g}^{2}+\tfrac{1-\tilde{a}_{4}}{2})\Big{\\}}$
(33)

in the limit of vanishing bare mass $m_{0}$. We note that in the limit
$\tilde{g}\to 0$ one recovers the known gap equation of the conventional
Hermitian NJL model in this regularization scheme spk :

$\frac{2\pi^{2}}{\tilde{G}N_{c}N_{f}}=1-\tilde{m}^{2}\ln\Big{(}\frac{1+\tilde{m}^{2}}{\tilde{m}^{2}}\Big{)}.$
(34)

The behavior of the right hand side of (III.2) is shown in Fig. 2(a) as a
function of $\tilde{m}$ and for different values of $\tilde{g}$. The parameter
$f=0.5$ is fixed, but is representative of the general behavior. The
horizontal line denotes the constant real positive left hand side of (III.2)
for $\tilde{G}=3.93$, $N_{c}=3$ and $N_{f}=2$. For sufficiently small coupling
values of $\tilde{g}$ an intersection, and therefore a real fermion mass
solution $\tilde{m}$ to the gap equation, can always be found. However, beyond
a critical value $\tilde{g}_{crit}$ the maximum of the right hand side no
longer exceeds the constant given by the left hand side and real solutions no
longer exist. The value of this critical coupling depends on the parameter $f$
and its behavior is shown in Fig. 2(b). It approaches zero asymptotically as
$f$ increases to infinity. Figure 2(c) displays the behavior of the fermion
mass solution $m$ (in MeV) as a function of $\tilde{g}$ up to the
corresponding critical couplings for different values of $f$. We note that,
independent of $f$, the effective mass decreases with increasing coupling
$\tilde{g}$. Thus a current quark mass can not be generated dynamically in
this model.

Altogether, we find that - similar to the model discussed in Sec. III.1 \-
this $\mathcal{PT}$-symmetric modified (non-Hermitian) NJL model admits a
finite region of unbroken $\mathcal{PT}$ symmetry, even for vanishing bare
mass $m_{0}$. However, while real fermion mass solutions do exist for coupling
values up to the critical value $\tilde{g}_{crit}$, these solutions always
describe an effective mass loss; an additional dynamical mass generation over
and above that obtained through the four-fermion interaction is not possible.

###  III.3 $\Gamma_{aPT_{1}}=iB_{\mu}\gamma^{\mu}$

The case of the non-$\mathcal{PT}$-symmetric model based on including the
vector background field $\Gamma_{aPT_{1}}=iB_{\mu}\gamma^{\mu}$ is
structurally very similar to the $\mathcal{PT}$-symmetric one discussed in
Sec. III.1. It is also the only other non-Hermitian, bilinear extension of the
NJL model that preserves chiral symmetry, as discussed in Sec. II.

To find the effective fermion mass of the model, we first rationalize the full
propagator

$S(p)=(\not{p}-m-igB_{\mu}\gamma^{\mu})^{-1},$

finding that it can be written as

$S(p)=\frac{\not{p}+m-iB_{\mu}\gamma^{\mu}}{p^{2}-m^{2}-g^{2}B^{2}-2igB_{\mu}p^{\mu}},$
(35)

from which the spinor trace is readily obtained:

((a))

((b))

Figure 3:  In (a) the behavior of the right hand side of the gap equation (39)
based on $\Gamma_{aPT_{1}}$ is shown as a function of the mass $\tilde{m}$ for
different values of the coupling $\tilde{g}$. The horizontal line represents
the left hand side of the gap equation. Figure (b) displays the behavior of
the real mass solutions $m$ in MeV as function of the coupling constant.

$\displaystyle\mathrm{tr}[S(p)]=\frac{4m}{p^{2}-m^{2}-g^{2}B^{2}-2igB_{\mu}p^{\mu}}.$
(36)

Integrating out the four-momentum dependence of this expression results in an
algebraic form of the gap equation of the model, cf. Eq. (13). As described
previously in Sec. III.1, we evaluate this integral in the Euclidean four-
momentum cutoff regularization scheme: The cutoff $\Lambda$ is introduced
after transforming to coordinates $p_{0}=ip_{4}$ and $B_{0}=iB_{4}$, such that
$p^{2}=-p_{E}^{2}$, $B^{2}=-B_{E}^{2}$, and $B_{\mu}p^{\mu}=-B_{E}\cdot
p_{E}$. Thus

$I_{aPT_{1}}=\int^{\Lambda}\\!\\!d^{4}p_{E}\,\frac{-4im}{p_{E}^{2}+m^{2}-g^{2}B_{E}^{2}-2ig(B_{E}\cdot
p_{E})}.$

In spherical coordinates with the zenith direction along $B_{E}$, the
Euclidean scalar product $B_{E}\cdot p_{E}=\lvert B_{E}\rvert r\cos\theta$
contains only the radius $r=\lvert p_{E}\rvert$ and the zenith angle $\theta$,
so that we can write

$I_{aPT_{1}}=\frac{8\pi m}{g\lvert
B_{E}\rvert}\int_{0}^{\Lambda}\\!\\!dr\,r^{2}\int_{0}^{\pi}\\!d\theta\,\frac{\sin^{2}\theta}{A(r)+\cos\theta},$
(37)

where $A(r)=(r^{2}+m^{2}-g^{2}\lvert B_{E}\rvert^{2})/(-2igr\lvert
B_{E}\rvert)$. Both the angular and the resulting radial integrals are
standard gr . In terms of the rescaled quantities $\tilde{m}=m\Lambda^{-1}$
and $\tilde{g}=g\lvert B_{E}\rvert\Lambda^{-1}$, the latter of which is
proportional to the amplitude of the vector background field, (37) becomes:

$\displaystyle\begin{split}I_{aPT_{1}}=\frac{i\Lambda^{3}\pi^{2}\tilde{m}}{\tilde{g}^{2}}\Bigl{\\{}&1+(\tilde{m}^{2}-\tilde{g}^{2})(2+\tilde{m}^{2}+\tilde{g}^{2})-(1+\tilde{m}^{2}+\tilde{g}^{2})\sqrt{(1+\tilde{m}^{2}+\tilde{g}^{2})^{2}-4\tilde{g}^{2}\tilde{m}^{2}}\\\
&+4\tilde{m}^{2}\tilde{g}^{2}\ln\Big{[}\frac{1}{2\tilde{m}^{2}}\Bigl{(}\sqrt{(1+\tilde{m}^{2}+\tilde{g}^{2})^{2}-4\tilde{g}^{2}\tilde{m}^{2}}+1+\tilde{m}^{2}+\tilde{g}^{2}\Bigr{)}\Big{]}\Bigr{\\}}.\end{split}$
(38)

Denoting $\tilde{G}=G\Lambda^{2}$, the general gap equation (13) in the limit
of vanishing bare mass $m_{0}$ thus takes the algebraic form

$\displaystyle\begin{split}\frac{2\pi^{2}}{\tilde{G}N_{c}N_{f}}=\frac{1}{4\tilde{g}^{2}}\Bigl{\\{}&(1+\tilde{m}^{2}+\tilde{g}^{2})\sqrt{(1+\tilde{m}^{2}+\tilde{g}^{2})^{2}-4\tilde{g}^{2}\tilde{m}^{2}}-1-(\tilde{m}^{2}-\tilde{g}^{2})(2+\tilde{m}^{2}+\tilde{g}^{2})\\\
&-4\tilde{m}^{2}\tilde{g}^{2}\ln\Big{[}\frac{1}{2\tilde{m}^{2}}\Bigl{(}\sqrt{(1+\tilde{m}^{2}+\tilde{g}^{2})^{2}-4\tilde{g}^{2}\tilde{m}^{2}}+1+\tilde{m}^{2}+\tilde{g}^{2}\Bigr{)}\Big{]}\Bigr{\\}}.\end{split}$
(39)

The gap equation (34) of the standard NJL model is recovered in the limit
$\tilde{g}\to 0$.

The right hand side of (39) is a real function of the scaled mass $\tilde{m}$
and coupling constant $\tilde{g}$. Its behavior is shown in Fig. 3(a).
Intersections with the real constant left hand side, visualized by a
horizontal line, can always be found when evaluated within the context of QCD.
The corresponding fermion mass solution $m$ (in MeV) is shown in Fig. 3(b) as
a function of the coupling constant $\tilde{g}$. Note that, contrary to the
$\mathcal{PT}$-symmetric models discussed in III.1 and III.2, real mass
solutions are not limited to a finite coupling constant regime and increase
monotonously with the value of the coupling constant. A current quark mass is
generated dynamically in this model: the equivalent of a bare up quark mass,
$m_{u}=(1.7-3.3)$ MeV, requires coupling values in the range
$\tilde{g}\approx(0.059-0.083)$ and for the bare down quark mass,
$m_{d}=(4.1-5.8)$ MeV, coupling values $\tilde{g}\approx(0.092-0.110)$ are
necessary.

Altogether, we find that real fermion mass solutions are obtained for all
values of the coupling constant $\tilde{g}$ in this chirally symmetric non-
Hermitian extension of the NJL model, even though $\mathcal{PT}$ symmetry is
broken explicitly. Moreover, the generation of an additional dynamical mass
mimicking a bare fermion mass occurs.

###  III.4 $\Gamma_{aPT_{2}}=\gamma_{5}$

Figure 4:  Behavior of the fermion mass solution $m$ of the modified NJL model
based on $\Gamma_{aPT_{2}}$ as a function of the scaled coupling constant
$\tilde{g}=g\Lambda^{-1}$.

The analysis of the non-$\mathcal{PT}$-symmetric model based on the
pseudoscalar term $\Gamma_{aPT_{2}}=\gamma_{5}$ is straight forward: The full
fermion propagator

$S(p)=(\not{p}-m-g\gamma_{5})^{-1}$

can be rationalized and simplified to the form

$S(p)=\frac{\not{p}+m-g\gamma_{5}}{p^{2}-m^{2}+g^{2}},$ (40)

which has the spinor trace

$\displaystyle\mathrm{tr}[S(p)]=\frac{4m}{(p^{2}-m^{2}+g^{2})}.$ (41)

Since this function only depends on the square of the four-momentum, the
momentum integral (13) can be evaluated directly: in spherical Euclidean
coordinates, with $p^{2}=-p_{E}^{2}$, we introduce the Euclidean four-momentum
cutoff $\Lambda$ as the bound of the radial integration and subsequently find
that

$I_{aPT_{2}}=4i\Lambda^{3}\pi^{2}\tilde{m}\Big{[}(\tilde{m}^{2}-\tilde{g}^{2})\ln\big{(}\frac{1+\tilde{m}^{2}-\tilde{g}^{2}}{\tilde{m}^{2}-\tilde{g}^{2}}\big{)}-1\Big{]},$
(42)

in terms of the rescaled parameters $\tilde{m}=m\Lambda^{-1}$ and
$\tilde{g}=g\Lambda^{-1}$. The gap equation (12) of this model thus takes the
algebraic form

$\frac{2\pi^{2}}{\tilde{G}N_{c}N_{f}}=1-(\tilde{m}^{2}-\tilde{g}^{2})\ln\big{(}\frac{1+\tilde{m}^{2}-\tilde{g}^{2}}{\tilde{m}^{2}-\tilde{g}^{2}}\big{)}$
(43)

in the limit $m_{0}\rightarrow 0$, which reduces to the gap equation (34) of
the standard NJL model in the limit of vanishing coupling constant values
$\tilde{g}$.

Equation (43) can be solved algebraically in terms of the Lambert $W$ function
cghjk . We find the real fermion mass solution

$\tilde{m}=\sqrt{\tilde{g}^{2}+[\tfrac{1}{c}W_{-1}(ce^{c})-1]^{-1}}=\sqrt{\tilde{g}^{2}+\tilde{m}_{NJL}^{2}},$
(44)

where, in the context of QCD, $c=2\pi^{2}/\tilde{G}N_{c}N_{f}-1$ is a negative
real constant and $\tilde{m}_{NJL}$ is the (scaled) fermion mass of the
standard NJL model. The behavior of this solution is shown in Fig. 4. Similar
to the case discussed in the previous Sec. III.3, we find that the existence
of real fermion mass solutions is, in contrast to the $\mathcal{PT}$-symmetric
models, not limited to a finite coupling constant regime and the mass
increases again monotonously with the coupling constant value $\tilde{g}$. A
current quark mass equivalent to that of a bare up quark mass,
$m_{u}=(1.7-3.3)$ MeV, is dynamically generated for coupling values in the
range $\tilde{g}\approx(0.028-0.039)$ and for the equivalent of a the bare
down quark mass, $m_{d}=(4.1-5.8)$ MeV, coupling values
$\tilde{g}\approx(0.044-0.052)$ are required.

We remark that the analytical mass solution (44) coincides with that found for
the renormalized, modified Gross-Neveu (GN) model in fbk , which can be seen
as a $1+1$ dimensional version of the NJL model. We emphasize that while the
pseudoscalar term $\Gamma_{aPT_{2}}=\gamma_{5}$ does not give rise to a
$\mathcal{PT}$-symmetric bilinear in $3+1$ dimensions, its $1+1$ dimensional
equivalent is, in fact, the only possible $\mathcal{PT}$-symmetric, non-
Hermitian, bilinear extension of the GN model bkb ; fbk .

Altogether, we find that the fermion mass solution of this model resembles
that found in the previous Sec. III.3: even though $\mathcal{PT}$ symmetry is
explicitly broken, real mass solutions are found for all coupling constant
values and an additional dynamical mass generation, over and above that
generated by the standard NJL model, occurs. In contrast to the model based on
$\Gamma_{aPT_{1}}$, this model does, however, not preserve chiral symmetry.

###  III.5 $\Gamma_{aPT_{3}}=i\mathbbm{1}$

Modifying the NJL model by adding a bilinear term based on the scalar
$\Gamma_{aPT_{3}}=i\mathbbm{1}$ amounts to including an imaginary mass
contribution proportional to the coupling constant $g$ into the propagator:

$S(p)=(\not{p}-m-ig)^{-1}.$

Correspondingly, the gap equation (12) is obtained as that of the standard NJL
model with an additional contribution to the mass terms arising in the
momentum integral on the right hand side. In the four-momentum cutoff
regularization scheme it reads

$\frac{2\pi^{2}}{\tilde{G}N_{c}N_{f}}=\frac{\tilde{m}+i\tilde{g}}{\tilde{m}}\Bigl{\\{}1-(\tilde{m}+i\tilde{g})^{2}\ln\Big{[}\frac{1+(\tilde{m}+i\tilde{g})^{2}}{(\tilde{m}+i\tilde{g})^{2}}\Big{]}\Bigr{\\}}$
(45)

in terms of the rescaled parameters $\tilde{m}=m\Lambda^{-1}$ and
$\tilde{g}=g\Lambda^{-1}$. For a vanishing coupling constant $\tilde{g}$ this
simplifies to the gap equation (34) of the standard NJL model in this
regularization scheme. Otherwise, the right hand side of (45) takes on
inherently complex values and an intersection with the purely real left hand
side, and thus a real fermion mass solution can not be found.

The modified NJL model based on $\Gamma_{aPT_{3}}$ is thus the only one of the
non-Hermitian systems considered here that does not admit real fermionic mass
solutions at all.

_In summary_ : For $\mathcal{PT}$-symmetric theories based on
$\Gamma_{PT_{1}}$ and $\Gamma_{PT_{2}}$ that symmetry was realized exclusively
as a broken symmetry phase in the underlying modified free Dirac models,
generally giving rise to complex fermion masses bkb . However, the addition of
two-body interactions via the associated modified NJL models restores this
symmetry, leading to a finite region where $\mathcal{PT}$ is unbroken. This is
manifested through the real fermion mass solutions found in both cases (cf.
III.1 and III.2). Nevertheless, an additional dynamical mass generation is
only possible in the model extended by a pseudovector background field
$\Gamma_{PT_{1}}$ (III.1), and in this case only for specific values of the
coupling strength $\tilde{g}$.

In contrast, it is suprising that one can find real fermion masses in the non-
Hermitian non-$\mathcal{PT}$-symmetric models that include $\Gamma_{aPT_{1}}$
and $\Gamma_{aPT_{2}}$ at all, cf. III.3 and III.4, as these can not be
attributed to an unbroken symmetry phase of any kind. Furthermore, in both
models these real mass solutions are not restricted to finite coupling
regions, but are obtained for all possible values of the coupling constant
$\tilde{g}$. In addition, these solutions always allow for increasing
dynamically generated masses, over and above those obtained from the standard
NJL model. In the model containing $\Gamma_{aPT_{3}}$ in III.5 no real fermion
masses were found.

Thus, the effect of a small bare mass $m_{0}$ in the standard NJL model can be
mimicked by including any of the three non-Hermitian bilinear terms
$\Gamma_{PT_{1}}$, $\Gamma_{aPT_{1}}$ or $\Gamma_{aPT_{2}}$ into the NJL
model.

##  IV Meson masses

We have identified four non-Hermitian bilinear extensions of the NJL model
that give rise to purely real fermion mass solutions, three of which allow for
an additional dynamical generation of fermion mass as a function of the
coupling strength $\tilde{g}$. In this section we investigate the masses of
the corresponding scalar and pseudoscalar bound states for these three models.
In the context of QCD these correspond to the $\sigma$ and $\pi$ mesons
respectively. First we recall the established self-consistent mass equations
for these bound states and how they can be obtained from the effective meson
interaction, cf. njl ; spk ; fw . In Sec. IV.1 we call to mind the resulting
meson mass solutions for the standard NJL model, followed by a discussion of
the results found for the modified NJL model based on including the
pseudoscalar term $\Gamma_{aPT_{2}}=\gamma_{5}$, in which both chiral and
$\mathcal{PT}$ symmetry are broken explicitly, c.f. Sec. IV.2. In Secs. IV.3
and IV.4, the scalar and pseudoscalar meson masses are analyzed for the
modified NJL models based on the vector background field
$\Gamma_{aPT_{1}}=iB_{\mu}\gamma^{\mu}$ and the pseudovector background field
$\Gamma_{PT_{1}}=i\gamma_{5}B_{\mu}\gamma^{\mu}$, both of which are chirally
symmetric.

Following the discussions in spk  and fw  for the standard NJL model, one must
contruct the effective meson interaction
$U_{\alpha\alpha^{\prime},\beta\beta^{\prime}}$, where all relevant degrees of
freedom are subsumed in the indices
$\alpha,\alpha^{\prime},\beta,\beta^{\prime}$. The effective interaction is
expressed in terms of the (proper) polarization insertion
$\Pi_{\lambda\lambda^{\prime},\mu\mu^{\prime}}$:

$\displaystyle\begin{split}iU_{\alpha\alpha^{\prime},\beta\beta^{\prime}}(k)&=iU^{0}_{\alpha\alpha^{\prime},\beta\beta^{\prime}}(k)\\\
&+iU^{0}_{\alpha\alpha^{\prime},\lambda\lambda^{\prime}}(k)[-i\Pi_{\lambda\lambda^{\prime},\mu\mu^{\prime}}(k)]iU_{\mu\mu^{\prime},\beta\beta^{\prime}}(k).\end{split}$
(46)

Here $U^{0}_{\alpha\alpha^{\prime},\beta\beta^{\prime}}$ denotes the bare
four-point interaction of the model, which (in position space) has the form

$\displaystyle\begin{split}U^{0}_{\alpha\alpha^{\prime},\beta\beta^{\prime}}(x,y)&=2G\delta^{(4)}(x-y)\\\
&\times[\delta_{\alpha\alpha^{\prime}}\delta_{\beta\beta^{\prime}}+(i\gamma_{5}\vec{\tau})_{\alpha\alpha^{\prime}}(i\gamma_{5}\vec{\tau})_{\beta\beta^{\prime}}].\end{split}$
(47)

Choosing
$U^{0}_{\alpha\alpha^{\prime},\beta\beta^{\prime}}(k)=U^{0}(k)\delta_{\alpha\alpha^{\prime}}\delta_{\beta\beta^{\prime}}$
or
$U^{0}_{\alpha\alpha^{\prime},\beta\beta^{\prime}}(k)=U^{0}(k)(i\gamma_{5}\vec{\tau})_{\alpha\alpha^{\prime}}(i\gamma_{5}\vec{\tau})_{\beta\beta^{\prime}}$
(46) simplifies to

$iU(k)=iU^{0}(k)+iU^{0}(k)[-i\Pi^{s/ps}(k)]iU(k),$ (48)

where
$\Pi^{s}(k)=\delta_{\lambda\lambda^{\prime}}\Pi^{s}_{\lambda\lambda^{\prime},\mu\mu^{\prime}}(k)\delta_{\mu\mu^{\prime}}$
and
$\Pi^{ps}(k)=(i\gamma_{5}\vec{\tau})_{\lambda\lambda^{\prime}}\Pi^{ps}_{\lambda\lambda^{\prime},\mu\mu^{\prime}}(k)(i\gamma_{5}\vec{\tau})_{\mu\mu^{\prime}}$
for the scalar and pseudoscalar case respectively. Equation (48) is a
geometric progression that can be summed to the form

$iU(k)=\frac{iU^{0}(k)}{1-2G\Pi^{s/ps}(k)}.$

The pole of this expression corresponds to that of the general scalar or
pseudoscalar mode propagator, that is it occurs when $k^{2}=m_{s/ps}^{2}$.
Thus, in order to determine the mass $m_{s/ps}$, we have to solve the equation

$1-2G\Pi^{s/ps}(k)=0.$ (49)

The lowest order polarization insertion is a closed fermion loop. One can
improve this approximation by implementing self-consistency, i.e. we replace
the free fermion propagator $S^{0}$ by the full propagator $S$. Evaluating
this contribution with the corresponding vertex functions gives

$\displaystyle-i\Pi^{s}(k)=$ $\displaystyle
N_{c}N_{f}\int\\!\\!\frac{d^{4}p}{(2\pi)^{4}}\mathrm{tr}[S(p+k)S(p)]$ (50)

and similarly

$-i\Pi^{ps}(k)=-N_{c}N_{f}\int\\!\\!\frac{d^{4}p}{(2\pi)^{4}}\mathrm{tr}[\gamma_{5}S(p+k)\gamma_{5}S(p)]$
(51)

for each pseudoscalar channel.

Equation (49) at $k^{2}=m_{s/ps}^{2}$, together with (50) and (51), thus
determines the mass of the scalar and pseudoscalar bound states. To simplify
notation, we use the general form of the gap equation (12), as discussed in
Sec. III, in the limit of vanishing bare mass $m_{0}$, and rewrite Eq. (49)
as:

$R^{s/ps}(p,k)-I/m=0\quad\text{at}\,\,\,k^{2}=m_{s/ps}^{2},$ (52)

where

$\displaystyle R^{s}(k)$
$\displaystyle=\int\\!\\!d^{4}p\,\mathrm{tr}[S(p+k)S(p)],$ (53) $\displaystyle
R^{ps}(k)$
$\displaystyle=-\int\\!\\!d^{4}p\,\mathrm{tr}[\gamma_{5}S(p+k)\gamma_{5}S(p)],$
(54)

and $I$ defined in (13) is evaluated through

$\frac{I}{m}=\,\frac{1}{2m}\int\\!\\!d^{4}p\,\bigl{(}{\rm tr}[S(p)]+{\rm
tr}[S(p+k)]\bigr{)}.$ (55)

As was the case for the gap equation, the equation for the self-consistent
meson masses (52) of the standard NJL model retains the same general structure
for the modified NJL models constructed with the Hamiltonian (2), since the
influence of the additional bilinear terms is accounted for in the full
fermion propagator $S(p)$, cf. fbk . Similar to the discussion of the gap
equation in Sec. III, our analysis of the meson masses thus relies on the
evaluation of spinor traces and momentum integrations in (53) and (54)
involving the appropriate fermion propagators of the modified NJL models.

###  IV.1 Standard NJL model

In the standard NJL model the evaluation of the self-consistent meson mass
equation (52) is based on the full Dirac fermion propagator

$S(p)=(\not{p}-m)^{-1}=\frac{\not{p}+m}{p^{2}-m^{2}},$

with $m=m_{NJL}$ being the effective fermion mass determined by the gap
equation (12) in the chiral limit. From this, the spinor traces in Eqs.
(53)-(55) are readily calculated, leading to the expressions

$R^{s}(k)=\int\\!\\!d^{4}p\frac{4(p^{2}+p_{\mu}k^{\mu}+m^{2})}{((p+k)^{2}-m^{2})(p^{2}-m^{2})}$

and

$R^{ps}(k)=\int\\!\\!d^{4}p\frac{4(p^{2}+p_{\mu}k^{\mu}-m^{2})}{((p+k)^{2}-m^{2})(p^{2}-m^{2})},$

as well as

$\frac{I}{m}=\int\\!\\!d^{4}p\,\frac{4(p^{2}+p_{\mu}k^{\mu}-m^{2})+2k^{2}}{((p+k)^{2}-m^{2})(p^{2}-m^{2})}.$

Equation (52) then takes the form of the well-known conditions

$\displaystyle 0=$
$\displaystyle\,(k^{2}-4m^{2})\int\\!\\!d^{4}p\frac{1}{((p+k)^{2}-m^{2})(p^{2}-m^{2})},$
(56) $\displaystyle 0=$
$\displaystyle\,k^{2}\int\\!\\!d^{4}p\frac{1}{((p+k)^{2}-m^{2})(p^{2}-m^{2})}$
(57)

at $k^{2}=m_{s/ps}^{2}$ for the scalar and pseudoscalar meson masses
respectively, cf. spk .

These conditions give rise to the apparent solutions that
$m_{s}^{2}=4m^{2}=4m_{NJL}^{2}$ and $m_{ps}^{2}=0$, the latter of which is the
Nambu-Goldstone mode in this model, in which chiral symmetry is spontaneously
broken. In addition, a simultaneous solution of $m_{s}$ and $m_{ps}$ arises
when the integral in (56) and (57) vanishes. In the Euclidean four-momentum
cutoff regularization scheme, this happens at $\lvert k\rvert=m_{s/ps}\approx
0.712\Lambda$. The mass spectrum of the scalar and pseudoscalar modes in the
NJL model thus not only contains the Nambu-Goldstone mode and its chiral
partner, but also an additional finite mass solution, which describes a
scalar/pseudoscalar mode degeneracy.

###  IV.2 $\Gamma_{aPT_{2}}=\gamma_{5}$

When the NJL model is modified by the pseudoscalar term
$\Gamma_{aPT_{2}}=\gamma_{5}$ the resulting extended model is neither
Hermitian, nor $\mathcal{PT}$-symmetric, nor chirally symmetric. Nevertheless,
the model gives rise to real fermion masses as was shown in Sec. III.4.
Looking now at the self-consistent mass equation (52) for the mesonic bound
states, we find that evaluating the spinor traces in (53) to (55) for the
fermion propagator associated with this model and given in (40), yields

$\displaystyle R^{s}_{aPT_{2}}(k)$
$\displaystyle=\int\\!\\!d^{4}p\frac{4(p^{2}+p_{\mu}k^{\mu}+m^{2}+g^{2})}{((p+k)^{2}-m^{2}+g^{2})(p^{2}-m^{2}+g^{2})},$
(58) $\displaystyle R^{ps}_{aPT_{2}}(k)$
$\displaystyle=\int\\!\\!d^{4}p\frac{4(p^{2}+p_{\mu}k^{\mu}-m^{2}-g^{2})}{((p+k)^{2}-m^{2}+g^{2})(p^{2}-m^{2}+g^{2})},$
(59) $\displaystyle\frac{I_{aPT_{2}}}{m}$
$\displaystyle=\int\\!\\!d^{4}p\,\frac{4(p^{2}+p_{\mu}k^{\mu}-m^{2}+g^{2})+2k^{2}}{((p+k)^{2}-m^{2}+g^{2})(p^{2}-m^{2}+g^{2})}.$
(60)

The condition (52) for the scalar and pseudoscalar meson modes thus takes the
form

$\displaystyle 0=$
$\displaystyle\,(k^{2}-4m^{2})\int\\!\\!d^{4}p\frac{1}{((p+k)^{2}-m^{2}+g^{2})(p^{2}-m^{2}+g^{2})},$
(61) $\displaystyle 0=$
$\displaystyle\,(k^{2}+4g^{2})\int\\!\\!d^{4}p\frac{1}{((p+k)^{2}-m^{2}+g^{2})(p^{2}-m^{2}+g^{2})},$
(62)

respectively at $k^{2}=m_{s/ps}^{2}$. This closely resembles the conditions
(56) and (57) of the standard NJL model.

We note, that (unsurprisingly) the appearance of a Nambu-Goldstone mode in the
standard NJL model breaks down with the inclusion of the chiral symmetry
breaking term $\Gamma_{aPT_{2}}$. In its place we find the apparent solution
to (62) that $m_{ps}^{2}=-4g^{2}$, i.e. the pseudoscalar solution is a
_tachyonic_ state with mass $m_{ps}=\pm 2ig$. The apparent solution to (61)
for the scalar meson, however, remains structurally identical to that of the
standard NJL model, cf. (56); but the effective fermion mass $m$ is now a
function of the coupling constant $g$ in this modified model, see (44). Thus
$m_{s}^{2}=4m^{2}=4(m_{NJL}^{2}+g^{2})$. The dynamical generation of fermion
mass thus also affects the scalar meson. In addition, the dependence of the
fermion mass on the coupling constant $g$ also affects the appearance of a
simultaneous solution of $m_{s}$ and $m_{ps}$ when the momentum integral in
(61) and (62) vanishes. In fact, it counteracts the explicit $g$ dependence of
the momentum integrals, leaving the same expression that is found in the
standard NJL model, cf. (56) and (57). We thus find a scalar/pseudoscalar mode
degeneracy with the same mass, $m_{s/ps}\approx\pm 0.712\Lambda$ in the
Euclidean four-momentum cutoff regularization, which is unaffected by the
explicit breaking of chiral symmetry in this model.

Altogether, the bilinear based on $\Gamma_{aPT_{2}}$, while causing a
dynamical mass gain of the fermion and the scalar meson, seems to act as a
tachyonic instability for the pseudoscalar meson. The scalar/pseudoscalar mode
degeneracy of the standard NJL model remains unchanged by this extension. We
point out that, while the Nambu-Goldstone mode of the standard NJL model
becomes tachyonic in this extended model, the combination of the non-
degenerate scalar and pseudoscalar meson masses
$m_{s}^{2}+m_{ps}^{2}=4m_{NJL}^{2}$ remains valid.

###  IV.3 $\Gamma_{aPT_{1}}=iB_{\mu}\gamma^{\mu}$

Having analyzed the modified NJL model, which breaks chiral symmetry
explicitly, we now turn to the remaining two models in which chiral symmetry
is conserved: the non-Hermitian models based on the vector background field
$\Gamma_{aPT_{1}}=iB_{\mu}\gamma^{\mu}$, which we discuss in this section, and
the pseudovector background field
$\Gamma_{PT_{1}}=i\gamma_{5}B_{\mu}\gamma^{\mu}$, which is discussed in Sec.
IV.4. As per the results of Sec. III.3, real fermion masses are dynamically
generated for all values of the coupling constant in the former,
non-$\mathcal{PT}$-symmetric case. In the latter, $\mathcal{PT}$-symmetric
case this happens in the finite region of coupling values smaller than
$\tilde{g}_{dyn}\approx 1.183$, cf. Sec. III.1.

For the NJL model that is extended by an anti-$\mathcal{PT}$-symmetric scalar
background field term $\Gamma_{aPT_{1}}=iB_{\mu}\gamma^{\mu}$ calculating the
spinor traces in the terms (53) and (54), with the fermion propagator as given
in (35), results in the expressions

((a))

((b))

Figure 5:  In (a) the mass of the scalar/pseudoscalar mode degeneracy in the
modified NJL model based on $\Gamma_{aPT_{1}}$ is shown as function of the
angle $\alpha$ between the Euclidean background field and the Euclidean meson
momentum for fixed values of the coupling constant $\tilde{g}$. The region of
accessible mass over this angle interval is visualized as function of
$\tilde{g}$ in (b).

$R^{s}_{aPT_{1}}(k)=\int\\!\\!d^{4}p\frac{4(p^{2}+p_{\mu}k^{\mu}+m^{2}-g^{2}B^{2}-2igB_{\mu}p^{\mu}-igB_{\mu}k^{\mu})}{((p+k)^{2}-m^{2}-g^{2}B^{2}-2igB_{\mu}(p+k)^{\mu})(p^{2}-m^{2}-g^{2}B^{2}-2igB_{\mu}p^{\mu})}$

and

$R^{ps}_{aPT_{1}}(k)=\int\\!\\!d^{4}p\frac{4(p^{2}+p_{\mu}k^{\mu}-m^{2}-g^{2}B^{2}-2igB_{\mu}p^{\mu}-igB_{\mu}k^{\mu})}{((p+k)^{2}-m^{2}-g^{2}B^{2}-2igB_{\mu}(p+k)^{\mu})(p^{2}-m^{2}-g^{2}B^{2}-2igB_{\mu}p^{\mu})}.$

With the integral $I_{aPT_{1}}$ in the form (55) being

$\frac{I_{aPT_{1}}}{m}=\int\\!\\!d^{4}p\frac{4(p^{2}+p_{\mu}k^{\mu}-m^{2}-g^{2}B^{2}-2igB_{\mu}p^{\mu}-igB_{\mu}k^{\mu})+2k^{2}}{((p+k)^{2}-m^{2}-g^{2}B^{2}-2igB_{\mu}(p+k)^{\mu})(p^{2}-m^{2}-g^{2}B^{2}-2igB_{\mu}p^{\mu})},$

we obtain the following form for the self-consistent mass equation (52) for
the scalar and pseudoscalar modes:

$\displaystyle 0=$
$\displaystyle\,(k^{2}-4m^{2})\int\\!\\!d^{4}p\frac{1}{((p+k)^{2}-m^{2}-g^{2}B^{2}-2igB_{\mu}(p+k)^{\mu})(p^{2}-m^{2}-g^{2}B^{2}-2igB_{\mu}p^{\mu})},$
(63) $\displaystyle 0=$
$\displaystyle\,k^{2}\int\\!\\!d^{4}p\frac{1}{((p+k)^{2}-m^{2}-g^{2}B^{2}-2igB_{\mu}(p+k)^{\mu})(p^{2}-m^{2}-g^{2}B^{2}-2igB_{\mu}p^{\mu})}.$
(64)

Both of the apparent solutions that $m_{ps}^{2}=0$ and $m_{s}^{2}=4m^{2}$ are
structurally identical to those of the standard NJL model, cf. Eqs. (56) and
(57). The existence of a massless Nambu-Goldstone mode remains unaffected
throughout the extension of the model by the non-Hermitian
anti-$\mathcal{PT}$-symmetric term $\Gamma_{aPT_{1}}$. One can understand
this, as chiral symmetry continues to be broken spontaneously.

The mass of the scalar meson, $m_{s}=2m$, remains structurally identical as
well, although once again the fermion mass $m$ now depends on the coupling
constant $g$, cf. Sec. III.3. The dynamical mass generation of the fermion is
thus reflected in the mass of the scalar meson.

Furthermore, the appearance of the identical four-momentum integrals in both
conditions (63) and (64) suggests that an additional simultaneous solution of
$m_{ps}$ and $m_{s}$ can be found here as well. However, the appearance of a
term $B_{\mu}k^{\mu}$ in the denominator of this integral indicates that this
solution depends on the angle between the background field $B_{\mu}$ and the
meson momentum $k_{\mu}$. The roots of the momentum integral can be determined
numerically in the Euclidean four-momentum cutoff regularization scheme for
various values of the angle $\alpha$ between the Euclidean background field
$B_{E}$ (where $B_{0}=iB_{4}$) and the Euclidean meson momentum $k_{E}$ (where
$k_{0}=ik_{4}$). Figure 5(a) shows the behavior of the degenerate meson mass
$\tilde{m}_{s/ps}$ as a function of the angle $\alpha$ for small values of the
scaled coupling constant $\tilde{g}=g\lvert B_{E}\rvert\Lambda^{-1}<1$, which
describes the amplitude of the vector background field. In Fig. 5(b) the
extremal values of the meson mass in the angular interval are visualized as a
function of the coupling $\tilde{g}$ in the same region. Since the meson mass
is a continuous function in $\alpha$, all mass values between these extrema
can be found for some value of $\alpha$. This is shown in Fig. 5(b) as shaded
regions. We note that for coupling values larger than $\tilde{g}\approx 0.425$
arbitrarily small (or vanishing) meson masses $m_{\/ps}$ can be realized.

Overall, we find a Nambu-Goldstone mode and a scalar meson with mass
$m_{s}=2m$ reflecting the dynamical mass generation of the fermion, plus the
existence of an additional scalar/pseudoscalar degeneracy in the modified NJL
model based on $\Gamma_{aPT_{1}}=iB_{\mu}\gamma^{\mu}$. The simultaneous
solution shows an intricate dependence on the amplitude of the scalar
background field and its angle relative to the meson momentum. For
sufficiently large values of the coupling this mass solution may be
arbitrarily small (or vanish entirely), but such values of the coupling also
generate a significant fermion mass in the model.

###  IV.4 $\Gamma_{PT_{1}}=i\gamma_{5}B_{\mu}\gamma^{\mu}$

((a))

((b))

Figure 6:  In (a) the mass of the pseudoscalar meson in the modified NJL model
based on $\Gamma_{PT_{1}}$ is shown as function of the angle $\alpha$ between
the Euclidean background field and the Euclidean meson momentum for fixed
values of the coupling constant $\tilde{g}$. The region of accessible mass
over this angle interval is visualized as function of $\tilde{g}$ in (b).

When the standard NJL model is extended by a $\mathcal{PT}$-symmetric
pseudovector background field
$\Gamma_{PT_{1}}=i\gamma_{5}B_{\mu}\gamma^{\mu}$, its rationalized fermion
propagator is given by Eq. (15), which does not simplify as much as in the
models discussed previously. The evaluation of the spinor traces in the terms
(53), (54) and (55) thus yields somewhat cumbersome results. However, when
inserted into the self-consistent meson mass equation (52) some contributions
simplify and the conditions for the scalar and pseudoscalar meson masses
become

$\displaystyle
0=\int\\!\\!d^{4}p\,\frac{(k^{2}-4m^{2})A(p,k)+8g^{2}B(p,k)}{\bigl{[}((p+k)^{2}-m^{2}-g^{2}B^{2})^{2}-4g^{2}m^{2}B^{2}+4g^{2}(B_{\mu}(p+k)^{\mu})^{2}\bigr{]}\bigl{[}(p^{2}-m^{2}-g^{2}B^{2})^{2}-4g^{2}m^{2}B^{2}+4g^{2}(B_{\mu}p^{\mu})^{2}\bigr{]}}$
(65)

and

$\displaystyle
0=\int\\!\\!d^{4}p\,\frac{k^{2}A(p,k)+8g^{2}B(p,k)}{\bigl{[}((p+k)^{2}-m^{2}-g^{2}B^{2})^{2}-4g^{2}m^{2}B^{2}+4g^{2}(B_{\mu}(p+k)^{\mu})^{2}\bigr{]}\bigl{[}(p^{2}-m^{2}-g^{2}B^{2})^{2}-4g^{2}m^{2}B^{2}+4g^{2}(B_{\mu}p^{\mu})^{2}\bigr{]}}$
(66)

respectively at $k^{2}=m_{s/ps}^{2}$, where

$\displaystyle A(p,k)=$
$\displaystyle((p+k)^{2}-m^{2}+g^{2}B^{2})(p^{2}-m^{2}+g^{2}B^{2})+4g^{2}[B^{2}p_{\mu}(p+k)^{\mu}-(B_{\mu}p^{\mu})(B_{\nu}(p+k)^{\nu})],$
(67) $\displaystyle B(p,k)=$ $\displaystyle
2(p_{\mu}k^{\mu})(B_{\nu}k^{\nu})(B_{\alpha}p^{\alpha})-B^{2}(p_{\mu}k^{\mu})^{2}+p^{2}k^{2}B^{2}-k^{2}(B_{\mu}p^{\mu})^{2}-p^{2}(B_{\mu}k^{\mu})^{2}.$
(68)

When considering the vector products in $B(p,k)$, we notice an overall
proportionality to $\lvert k\rvert^{2}$. Thus the condition (66) for the
pseudoscalar meson mass has an apparent solution that $m_{ps}^{2}=0$. Once
again, a Nambu-Goldstone mode appears as a manifestation of chiral symmetry in
this modified NJL model. A second solution for the pseudoscalar meson mass
$m_{ps}$ can be found by evaluating the momentum integral in (66) numerically
in the Euclidean four-momentum cutoff regularization scheme. Similar to the
case of the model based on the vector background field $\Gamma_{aPT_{1}}$
discussed in Sec. IV.3, this solution depends on the amplitude of the
(pseudovector) background field $B_{\mu}$ and its angle with the pion momentum
$k_{\mu}$. Figure 6(a) shows the behavior of this pseudoscalar meson mass as a
function of the angle $\alpha$ between the Euclidean background field $B_{E}$
(where $B_{0}=iB_{4}$) and the Euclidean meson momentum $k_{E}$ (where
$k_{0}=ik_{4}$) at fixed values of the coupling constant $\tilde{g}=g\lvert
B_{E}\rvert\Lambda^{-1}$ for which a fermion mass is dynamically generated,
i.e. $\tilde{g}<\tilde{g}_{dyn}\approx 1.183$. In Fig. 6(b) the extremal
values of the pseudoscalar meson mass in the angular interval are visualized
as a function of the coupling constant $\tilde{g}$ in the same region. Since
again the mass is a continuous function of $\alpha$, all values between these
extrema can be obtained for some value of $\alpha$, which is indicated in Fig.
6(b) as shaded regions. Notice that independent of the values of $\alpha$ and
$\tilde{g}$ the second pseudoscalar mass solution $m_{ps}$ has finite real
values.

((a))

((b))

Figure 7:  In (a) the masses of the scalar meson in the modified NJL model
based on $\Gamma_{PT_{1}}$ are shown as function of the angle $\alpha$ between
the Euclidean background field and the Euclidean meson momentum for fixed
values of the coupling constant $\tilde{g}$. The region of accessible mass
over this angle interval is visualized as function of $\tilde{g}$ in (b).

For the scalar meson mass as determined by condition (65), we do not generally
find an apparent solution that factors out of the momentum integral. In
particular, we do not find a scalar meson mass that is only proportional to
the fermion mass $m$ of the model, in contrast to the solutions of the other
non-Hermitian models discussed so far. We also do not generally find a
simultaneous solution of the scalar and pseudoscalar meson modes. The two
solutions $m_{s}$ that one can find by evaluating (65) numerically in the
Euclidean four-momentum cutoff regularization scheme _both_ depend on the
angle $\alpha$ between $B_{E}$ and $k_{E}$, and on the value of the scaled
coupling constant $\tilde{g}=g\lvert B_{E}\rvert\Lambda^{-1}$, which is
proportional to the amplitude of the background field. Their behavior is shown
in Fig. 7(a) as a function of the angle $\alpha$ and for fixed values of
$\tilde{g}$. We point out that only for small coupling values below
$\tilde{g}\approx 0.107$ do real mass solutions $m_{s}$ exist for all angles
$\alpha$. For larger coupling values the solutions around $\alpha=\pi/2$ begin
to break down. However, the region of real masses $m_{s}$ never breaks down
entirely and real-valued solutions can always be found for some value of
$\alpha$.

The mass of the scalar mode is a continuous function of $\alpha$ and any real
mass value between the extremal values over the angle interval is realized at
some value of $\alpha$. The behavior of the extremal values is visualized in
Fig. 7(b) as a function of coupling constant values for which a fermion mass
is dynamically generated ($\tilde{g}<\tilde{g}_{dyn}\approx 1.183$). The
region of accessible mass values in between minima and maxima is indicated by
the shaded area. As remarked above, the two scalar meson mass solutions remain
real and separate solutions at all angles $\alpha$ for coupling values below
$\tilde{g}\approx 0.107$. For higher coupling values the solutions combine as
shown in Fig. 7(a).

_Special case $\alpha=0$_. We remark that when the pseudovector background
field is parallel to the meson momenta, i.e. $\alpha=0$, the contribution
$B(p,k)$ in the conditions (65) and (66) vanishes. In this special case we
once again find the apparent solutions $m_{ps}^{2}=0$ and $m_{s}^{2}=4m^{2}$
as in the case of the standard NJL model and the extended model based on the
vector field $\Gamma_{aPT_{1}}$.

We further note that $\alpha=0$ is a special case of the underlying modified
Dirac theory as well: The dispersion relation of the model with Hamiltonian
density

$\mathcal{H}=\bar{\psi}(-i\gamma^{k}\partial_{k}+ig\gamma_{5}B_{\mu}\gamma^{\mu})\psi$

has the form

$0=(p^{2}-g^{2}B^{2})^{2}+4g^{2}(B_{\mu}p^{\mu})^{2}$

at $p^{2}=m^{2}$. Thus the effective mass of the modified Dirac fermion has
the form $m^{2}=g^{2}B^{2}\exp[\mp 2i\alpha]$. This generally describes
complex mass solutions, as noted in bkb , but in the special case of
$\alpha=0$ we find a purely real fermion mass of $m^{2}=g^{2}B^{2}$.

Overall, we find a Nambu-Goldstone mode for the pseudoscalar meson of this
chirally symmetric theory as well as an additional finite pseudoscalar meson
mass solution that depends in particular on the angle $\alpha$ between its
momentum and the pseudovector background field. Independently, we have found
two scalar meson mass solutions which display an intricate dependence on the
angle $\alpha$ as well and which, in particular, may break down depending on
the specific value of $\alpha$ and the coupling constant $g$.

##  V Concluding remarks

It is surprising that the free Dirac equation in 3+1 dimensions, augmented by
any $\mathcal{PT}$-symmetric bilinears does not lead to a real energy spectrum
for the fermions unless a bare quark mass is present bkb . (This is
counterintuitive, since for bosonic systems, a region in which
$\mathcal{PT}$-symmetry is unbroken leading to a real mass spectrum can
usually be found b .) However, it was shown in fbk  that in a fermionic system
such as the NJL model, which contains additional two-body interactions a phase
of unbroken $\mathcal{PT}$ symmetry having real energies can occur within a
specific parameter range. In that case, the $\mathcal{PT}$-symmetric bilinear
included into the Hamiltonian density $\mathcal{H}$ given in (2) was
$\Gamma_{PT_{1}}$. Having determined that such a Hamiltonian can lead to an
additional dynamical mass generation, mimicking the effect of having a bare
fermion mass $m_{0}$, we have examined in detail in this paper the effects of
including all other possible non-Hermitian bilinear interactions into the
Hamiltonian, not only in view of their ability to generate a real fermion
spectrum, but also in view of the implications for constructing composite
particles. In particular, our interest has been to understand the roles of
both $\mathcal{PT}$ and chiral symmetry, which are both relevant in this
context. We have thus considered non-Hermitian scalar ($\Gamma_{aPT_{3}}$),
pseudoscalar ($\Gamma_{aPT_{2}}$), vector ($\Gamma_{aPT_{1}}$), pseudovector
($\Gamma_{PT_{1}}$) and tensor ($\Gamma_{PT_{2}}$) additions to the NJL
Hamiltonian.

We find the following results:

(a) $\mathcal{PT}$ symmetry is not necessary for real fermion mass solutions
to exist, rather the two-body interactions supersede the non-Hermitian
bilinear effects. To be specific, real solutions for the fermion mass occur
for all interactions, with the exception of $\Gamma_{aPT_{3}}$. While the
presence of $\Gamma_{PT_{1}},\Gamma_{aPT_{1}}$ and $\Gamma_{aPT_{2}}$
(pseudovector, vector and pseudoscalar contributions) can lead to an increase
in the dynamically generated mass, mimicking the role of a current quark mass,
this is not true when $\Gamma_{PT_{2}}$ is included: then a loss in mass
results.

(b) The role of chiral symmetry becomes most evident in the mesonic sector.
Both chirally symmetric bilinears, $\Gamma_{aPT_{1}}$ (vector) and
$\Gamma_{PT_{1}}$ (pseudovector), as mentioned already in (a), allow for the
dynamical generation of fermion masses. In addition, the Goldstone mode
remains unaffected, lying again at zero, manifesting the chiral symmetry of
the model. For the vector interaction, the scalar (or $\sigma$) meson gains
mass dynamically as does the fermion, while a second degenerate solution for
the composite scalar and pseudoscalar modes depends on the relation of the
background field and the meson momentum. It is possible to generate small
masses at values of the coupling $\tilde{g}>0.425$, but this again would imply
that there is a significant fermion mass.

For the pseudovector interaction no general scalar/pseudoscalar degeneracy is
found. Furthermore, the additional pseudoscalar and scalar meson masses depend
on the relation of the background field and the meson momentum and remain
"large" (of the same size as the degenerate state of the standard NJL model)

The inclusion of a pseudoscalar bilinear leads to a tachionic pseudoscalar
mode (which is unphysical) while the scalar mode gains mass dynamically in the
same fashion as the fermion, the degenerate solution remaining unaffected.

We thus conclude that the mechanism of mass generation due to four-fermion
interactions can occur in many different ways. There seems to be no argument
for discarding non-Hermitian bilinears that also give rise to the generation
of real fermion masses. With the exception of $\Gamma_{aPT_{3}}$, all other
forms do this when bare fermion masses are absent. However, in order to give
the Goldstone mode a finite mass, another mechanism is required. An additional
scalar/pseudoscalar degenerate mode can often be found that is tunable with
the coupling strength.

We point out that this analysis is applicable generally to Hamiltonians that
include four-fermion interactions, as used for example in discussions of the
generation of neutrino masses ams ; ms .

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