# Task-Based Assessment for Neural Networks: Evaluating Undersampled MRI
Reconstructions based on Human Observer Signal Detection

Joshua D. Herman, Rachel E. Roca, Alexandra G. O’Neill, Marcus L. Wong
Mathematics Department
Manhattan College
NY Sajan G. Lingala
Roy J. Carver Department of Biomedical Engineering
University of Iowa
Iowa City Angel R. Pineda
Mathematics Department
Manhattan College
NY

###### Abstract

Recent research has explored using neural networks to reconstruct undersampled
magnetic resonance imaging (MRI) data. Because of the complexity of the
artifacts in the reconstructed images, there is a need to develop task-based
approaches of image quality. Common metrics for evaluating image quality like
the normalized root mean squared error (NRMSE) and structural similarity
(SSIM) are global metrics which average out impact of subtle features in the
images. Using measures of image quality which incorporate a subtle signal for
a specific task allow for image quality assessment which locally evaluates the
effect of undersampling on a signal. We used a U-Net to reconstruct under-
sampled images with 2x, 3x, 4x and 5x fold 1-D undersampling rates. Cross
validation was performed for a 500 and a 4000 image training set with both
structural similarity (SSIM) and mean squared error (MSE) losses. A two
alternative forced choice (2-AFC) observer study was carried out for detecting
a subtle signal (small blurred disk) from images with the 4000 image training
set. We found that for both loss functions and training set sizes, the human
observer performance on the 2-AFC studies led to a choice of a 2x
undersampling but the SSIM and NRMSE led to a choice of a 3x undersampling.
For this task, SSIM and NRMSE led to an overestimate of the achievable
undersampling using a U-Net before a steep loss of image quality when compared
to the performance of human observers in the detection of a subtle lesion.

_K_ eywords Task-based assessment $\cdot$ Undersampling $\cdot$ MRI $\cdot$
Neural network $\cdot$ 2-AFCF

##  1 Introduction

A major limiting factor in the use of MRI is the long acquisition time. One of
the approaches to decrease imaging time is undersampling the acquired k-space
data coupled with constrained reconstruction using total variation or wavelet
sparsity priors [1, 2, 3]. Another approach in reconstruction of undersampled
MRI data uses data driven priors [4, 5, 6, 7, 8, 9, 10] to estimate some of
the Fourier information not collected in the data acquisition process. Some of
these approaches incorporate a data agreement term but all introduce prior
information in the reconstruction process to compensate for the undersampling.
Since the priors are data driven, the artifacts depend both on the training
data and new data being reconstructed since the new data does not exactly
match the prior assumptions. Because of this mismatch, generalization of the
results of neural networks being used on new data is an active area of
research [11, 12]. The artifacts are non-trivial and are often challenging to
characterize and predict.

In a task-based approach to image quality, one needs to specify the task (e.g.
detection or estimation), the observer (e.g. human or machine) and the
ensemble of images which are being used to quantify the image content [13, 14,
15]. The image quality depends on these choices. This task-based approach is
particularly useful in evaluating neural network reconstructions because they
have artifacts that are non-trivial to characterize. For other medical imaging
modalities, there are recent results that neural networks outside may not lead
to improved performance and perhaps lead to decreased performance from a task-
based perspective [16, 17].

Evaluation of image quality regarding neural networks in MRI reconstruction
typically involves conventional measures of image quality like the normalized
root mean squared error (NRMSE) and structural similarity (SSIM) which are
simple to compute [18]. Another approach uses radiologist’s ratings for more
realistic tasks [19, 20, 21, 22] but which require a substantial amount of
radiologist’s time. This work explores the space in between using idealized
clinical tasks which capture the need of a local and signal dependent version
of image quality using non-radiologist observers. By utilizing these idealized
tasks, the objective is to increase the proportion of research which results
in clinical impact while minimizing the radiologist’s time required.

Preliminary results have been presented at conferences [23, 24]. Here, we
assess a U-Net based reconstruction scheme [25] from a task-based perspective
of detecting a subtle lesion. We determine the under-sampling rate that would
be chosen based on the proposed human observer studies, and compare it to the
undersampling rate that would be chosen based on existing NRMSE and SSIM
metrics.

##  2 Methods

###  2.1 Training and Validation Datasets

In this study, we consider retrospectively undersampling fluid-attenuated
inversion recovery (FLAIR) brain datasets from the fastMRI open source dataset
[25]. Coil sensitivity maps were estimated from central 16 k-space lines using
the sum of squares approach. The fully sampled slices were obtained via R=1
SENSE reconstruction [26] using the BART toolbox [27]. We ran five fold cross
validation [28] using two different sets of images, a 500 image dataset and a
4000 image dataset. The full data sets (500 and 4000 respectively) were used
in the training of the neural networks used for testing the human task
performance but the images did not contain the subtle signals used in the
human detection experiments. Those were only included only in the testing set.

We performed retrospective one-dimensional undersampling on fully sampled
images. Four 1-D kx undersampling factors: 2x, 3x, 4x, and 5x were
implemented. The undersampling pattern used samples the lowest 16 frequencies
(5% of the data), and sampling every k of the remaining spacial frequencies.
Figure 1(b) shows a 3x mask.

While the undersampling factors 2x, 3x, 4x and 5x refer to sampling every 2nd,
3rd, 4th, and 5th of the higher frequency bands respectively, the actual
undersampling as the percentage of total frequencies are shown in Figure 1(a).
Reconstructed images were normalized in image space to have a minimum pixel
value of 0 and maximum of 1.

kx undersampling | effective undersampling
---|---
2x | 1.90
3x | 2.71
4x | 3.48
5x | 4.16
(a) Effective undersampling factors for each kx undersampling

(b) 3x sampling mask

Figure 1: Undersampling factors and example undersampling mask. To simulate
under-sampled Fourier data, we performed the Fourier transform of the R=1
SENSE reconstruction and multiplied with the undersampling mask.

###  2.2 Observer Study Testing Dataset

The images used for the observer studies were based on a 50 image dataset
which was not included in either of the training sets. Four subimages were
extracted from each slice [29] to generate a 200 image dataset. The k-space of
a small disk (with radius = 0.25 pixels) blurred by a Gaussian kernel
($\sigma=1$ pixel) was added to the k-space of the background images to
generate images with a signal in the middle. The amplitude of the signal was
chosen in a pilot study so that the average human performance would be in a
range to show differences in undersampling. The multi-coil k-space data of the
signal was generated by appropriate coil sensitivity weighting and added to
the original multi-coil measurements in the test set. For each undersampling
factor, this led to a set of 200 background images and 200 images with a
signal that were used for the observer studies, but were not used in the
training or validation of the neural networks.

###  2.3 Neural Network Architecture

The neural network design parameters used in this study were chosen as part of
a study focusing on picking the preferred dropout and channel numbers for a
U-Net using NRMSE, SSIM and human 2-AFC for a fixed undersampling factor of 4x
[24]. In that work, using the SSIM loss function, it was found that when
considering SSIM, NRMSE, task-performance and artifacts, the model with 64
channels and 0.1 dropout performed best overall. Those are the parameters used
in this study.

The U-Net structure consists of two paths and their interconnections. A
contracting path is followed by an expanding path, as shown in Figure 2. The
network was based on the baseline network architecture in [25], except with
ReLU and sigmoid activation functions used respectively in the last two
layers. We trained our U-Nets with both SSIM and MSE loss functions.

Figure 2: U-Net with $x$ starting channels in the contracting path. The input
of the network is a 320 by 320 undersampled image and the output is the neural
network estimate of the 320 by 320 fully sampled image. The network
architecture was based on the baseline network in [25]. The network used in
this study had $x=64$ starting channels and a dropout of 0.1.

The model starts with a specified number of filters (initial channels), $x$,
in its first convolutional block and has 4 pooling layers in the contracting
path and 4 upsampling layers in the expanding path. It also contains
concatenation blocks that include the output features from each step in the
contracting path as input features for each corresponding step in the
expanding path.

###  2.4 Training Parameters, Loss Functions and Training Set Sizes

Each of the U-Net loss and undersampling schemes were trained using the
RMSProp algorithm [30] for 150 epochs and a batch size of 8. The different
versions of the U-Net were the SSIM and MSE training loss networks each
trained with either a small or large dataset (500 or 4000 training images
respectively). We let the SSIM between two images $a$ and $b$ be denoted by
$ssim({a,b})$ [18]. The ssim metric on which we based our training loss is:

$SSIM_{METRIC}({y},{\hat{y}})=\frac{1}{m}\sum_{i=1}^{m}{ssim({y_{i},\hat{y}_{i}})},$
(1)

where m stands for the number of images, $y$ stands for the array where each
entry is a different fully sampled image, while $\hat{y}$ stands for the array
where each entry is the corresponding reconstructed image. Since we want our
loss to correspond to best picture quality when minimized, we defined our loss
to be

$SSIM_{LOSS}=1-SSIM_{METRIC}.$ (2)

Our MSE loss was defined by the standard formula for MSE:

$MSE_{LOSS}=\frac{1}{m}{\sum_{i=1}^{m}||{y_{i}-\hat{y_{i}}}||_{2}^{2}}.$ (3)

The training of the networks was carried out in an ubuntu Linux workstation
with a CPU with 20 threads, 64 GB of DDR4 RAM, and 2 Quadro P5000 16 GB CUDA
GPUs.

###  2.5 Scoring Metrics

We used three scoring metrics to evaluate our reconstructions. Five fold cross
validation was used to calculate the mean and standard deviation of SSIM and
NRMSE for both the 4000 and 500 training and validation sets. For the NRMSE we
applied the following normalization:

$NRMSE_{METRIC}=\sqrt{\frac{\sum_{i=1}^{m}||{y_{i}-\hat{y_{i}}}||_{2}^{2}}{\sum_{i=1}^{m}||y_{i}||_{2}^{2}}}.$
(4)

The observer performance metric is the average percent correct of the human
observers in a two-alternative forced choice (2-AFC) study. In each 2-AFC
trial an observer is presented two images, one containing an artificial lesion
(signal image) and another containing no lesion (background image). The task
is then correctly selecting the image that contains the lesion. Since the
location of the lesion was always in the center but the backgrounds changed,
this is a signal-known-exactly with variable background study [14]. The human
observers carried out 200 2-AFC trials for each experimental condition. The
percentage of times the person chose correctly is the percent correct. Figure
3 is a sample 2-AFC trial.

Figure 3: Sample 2-AFC trial for human observer studies using images trained
with SSIM loss and large (4000 image) dataset using 3x undersampling. Image on
the right contains signal, with an arrow identifying the location. The image
in the middle is the signal. The image on the left is a different background
location with no signal. The trial consisted of determining which of the two
images contained the signal in the center of the image. The percent correct
reported is for 200 trials for each observer and condition.

All the observer studies were done using a Barco MDRC 2321 monitor in a dark
room. The observers were approximately 50 cm from the monitor but a restraint
was not used. The observers took breaks in between trials to avoid fatigue and
repeated the first condition until the performance stopped changing as
training.

##  3 Results & Discussion

Tables 1 and 2 show results for the 500 image (Small) and 4000 image (Large)
5-fold cross validation SSIM and NRMSE metric scores for the 4 undersampling
rates. The human 2-AFC scores are reported for the 4000 image data-set. Scores
are also included for the full reconstruction, which uses fully sampled images
and no U-Net reconstruction. The format is mean/standard deviation, and the
mean and standard deviation were computed using the five test-train splits for
SSIM and NRMSE. Mean and standard deviation for human 2-AFC were computed
using the four human observer results. This approach to estimate the standard
deviation of observer studies represents an over-estimate of the variability
[31]. The scores in bold correspond to the preferred undersampling for each
metric and the 2-AFC study.

These tables display a small change in mean SSIM and NRMSE from 2x to 3x
undersampling, while a larger drop from 3x to 4x. This is true for both
training set sizes. Since 3x is the greatest extent you can undersample
without a large drop in mean SSIM or NRMSE, 3x is thus the preferred
undersampling rate. For human 2-AFC average percent correct, 2x is the
preferred amount, since a large drop occurs for mean human 2-AFC from 2x to 3x
undersampling. These results suggest that a human needs a more conservative
undersampling when compared to NRMSE and SSIM.

Unet 64 channels 0.1 dropout | SSIM | SSIM | NRMSE | NRMSE | Human 2-AFC
---|---|---|---|---|---
Undersampling | 500 image | 4000 image | 500 image | 4000 image | 4000 image
1x | 1 | 1 | 0 | 0 | 94.500/1.225
2x | 0.885/0.004 | 0.901/0.003 | 0.135/0.019 | 0.114/0.005 | 92.750/1.299
3x | 0.892/0.006 | 0.910/0.003 | 0.144/0.020 | 0.108/0.008 | 79.875/4.174
4x | 0.815/0.007 | 0.825/0.006 | 0.202/0.021 | 0.153/0.007 | 77.625/3.110
5x | 0.792/0.012 | 0.805/0.007 | 0.204/0.027 | 0.163/0.006 | 67.625/2.880
Table 1: MSE loss U-Net image quality metric results across four undersampling factors and full sampling. Unet 64 channels 0.1 dropout | SSIM | SSIM | NRMSE | NRMSE | Human 2-AFC
---|---|---|---|---|---
Undersampling | 500 image | 4000 image | 500 image | 4000 image | 4000 image
1x | 1 | 1 | 0 | 0 | 94.500/1.225
2x | 0.914/0.004 | 0.917/0.002 | 0.118/0.012 | 0.106/0.003 | 91.625/2.012
3x | 0.912/0.005 | 0.922/0.001 | 0.136/0.023 | 0.109/0.015 | 77.625/1.746
4x | 0.836/0.008 x | 0.857/0.002 | 0.161/0.009 | 0.148/0.004 | 76.000/3.354
5x | 0.814/0.015 | 0.844/0.003 | 0.178/0.006 | 0.155/0.017 | 64.125/4.762
Table 2: SSIM loss U-Net image quality metric results across four
undersampling factors and full sampling.

Subjective evaluation of reconstructed images shown in Figures 4 and 5 for the
MSE and SSIM loss functions with the large training set shows that higher
undersampling rates affect the visibility of the signal and the artifacts in
the images. However, judging the detectability of a signal cannot be done just
using a single image because of the significant variability across images. The
need for an evaluation using an ensemble of images is one of the
justifications of our approach.

Figure 4: Comparison of undersampled MSE loss images with and without the
signal for the large training set. The white arrow points to the signal in the
original image and the black arrow to an artifact in the 5x undersampled
image. The signal becomes more faint as the undersampling factor increases.
Figure 5: Comparison of undersampled SSIM loss images with and without the
signal for the large training set. The white arrow points to the signal in the
original image. This particular image does not have an artifact which
resembled anatomy like in the MSE image but similar artifacts were also found
for large undersampling factors in the SSIM images.

Subjective evaluation of the sample images shown in Figures 4 and 5 suggests
that an undersampling rate of 3x might be acceptable for those images even in
a detection task. Our task performance study is not consistent with that
subjective assessment. This is in part because of the variability of the
detection of the signal on the varying backgrounds. The sample image included
an artifact for the 5x undersampling rate for the MSE loss function and not
for the SSIM loss function, but those types of artifacts were also present in
the SSIM loss images for other subimages. The artifact looks like a structure
and this is sometimes referred to as a hallucination [10, 32]. The variability
in the use of sample images for assessment and the lack of connection of the
NRMSE and SSIM metrics to a task are two of the motivations for the
development of task-based approaches.

While current neural network research in image reconstruction mostly focuses
on neural networks which enforce data agreement, image domain U-Nets are of
interest for analyzing the behavior of neural networks [33] and are being used
for specific applications [34]. In future work we plan to apply a multicoil
image implementation of this study which incorporates data agreement using
MODL [8] instead of a U-Net. We also plan on applying model observers for a
2-AFC task [14, 29] to reduce the number of human observer studies needed for
evaluation in this study and incorporate a task where an observer searches for
the signal [35, 36]. This is part of an overall program to develop task-based
assessment in MRI.

##  4 Conclusion

This study introduced human detection performance to evaluate neural networks
for undersampled image reconstruction across different undersampling schemes.
Scores for two standard image quality metrics, SSIM and NRMSE, and the average
human observer performance were calculated for four different undersampling
schemes and the fully sampled scheme. This was done using a 4000 image dataset
for training and cross validation, with two different loss functions, SSIM and
MSE. For NRMSE and SSIM, the 3x undersampling was preferred, while the human
observer preferred a more conservative undersampling of 2x. This work suggests
that commonly used metrics for evaluating image quality may over-estimate the
amount of undersampling that can be done compared with human detection
performance.

###  4.1 Code Availability

The code used for creating, training and running the cross-validation of the
UNET as well as for generating the 2-AFC images can be found at:
https://github.com/MoMI-Manhattan-College/UNET-2AFC

## Acknowledgements

We acknowledge support from the National Institute of Biomedical Imaging and
Bioengineering of the National Institute of Health under award number
R15-EB029172. We also acknowledge support from the Kakos Center for Scientific
Computing and thank Michael Rozycki for his technical support. The authors
thank Krishna Nayak, and Dr. Craig Abbey for their helpful insights.

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# Propagation of field disturbances in the Yang-Mills theory

Vitorio A. De Lorenci<EMAIL_ADDRESS>Institute of Science, Federal
University of Itajubá, 37500-903 Itajubá, M. G., Brazil,  PH Department, TH
Unit, CERN, 1211 Geneva 23, Switzerland  Shi-Yuan Li<EMAIL_ADDRESS>School
of Physics, Shandong University, Jinan, 250100, P. R. China,  PH Department,
TH Unit, CERN, 1211 Geneva 23, Switzerland

###### Abstract

The propagation of field disturbances is examined in the context of the
effective Yang-Mills Lagrangian, which is intended to be applied to QCD
systems. It is shown that birefringence phenomena can occur in such systems
provided some restrictive conditions, as causality, are fulfilled. Possible
applications to phenomenology are addressed.

###### pacs:

42.25.Lc, 12.38.Mh, 24.70.+s

##  I Introduction

Small disturbances on nonlinear fields propagate with velocity depending on
the polarization states. In general there will be two polarization modes,
leading to the existence of two waves propagating with different velocities.
This phenomenon is known in the literature as birefringence. In the Maxwell
theory (i.e., Abelian gauge field) it can be found when light propagates
inside certain material media landau . It can also appear in the context of
nonlinear spin-one fields birula70 ; iacopini1979 ; delorenci00 , as it occurs
in the quantum electrodynamics (QED). The effective Lagrangian for QED was
derived long ago heisenberg36  for slowly varying but arbitrary strong
electromagnetic fields. Its nonlinearities lead to effects like birefringence
and photon splitting birula70 . Some other investigations on this issue can be
found in rikken2000 ; adler2007 ; biswas2007 ; kruglov2007 .

For the case of non-Abelian gauge theories, the quantum fluctuations lead to a
vaccum state which does not coincides with the vacuum coming from the
perturbation theories. The structure of the vacuum state was discussed for
several models in savvidy1977 ; pagels78 ; nielsen1979 ; shuryak1980 ;
ambjorn1980 ; ambjorn1980b . The one-loop effective action for Yang-Mills
theory was presented and discussed in, e.g., savvidy1977 ; pagels78 . For the
asymptotically free theory in the regime of large mean fields the effective
action is controlled by perturbation theory. In this context the issue of
event horizon formation in the physical vacuum associated with color
confinement was considered in castorina07 .

The mathematical formalism (see Section II for details) to deal with the
propagation of small disturbances in nonlinear spin-one fields birula70 ;
delorenci00  depends on the Lagrangian as a general function of the field
invariant. Thus, it can be used to examine the wave propagation in systems
governed by an Yang-Mills effective Lagrangian. Particularly, it is worthwhile
to analyze if effects like birefringence can occur in this context. Quantum
Chromodynamics (QCD) could be taken as a specific application, since it
presents strong nonlinear properties.

In this manuscript the one-parameter effective Lagrangian presented in
pagels78  is used as a ‘working model’ when discussing Yang-Mills fields. Due
to the possibility of two polarization modes presenting different velocities,
as derived from our theoretical framework, it is shown that the birefringence
phenomena can occur provided that causal conditions are fulfilled. In QCD
case, we discuss how to observe the birefringence phenomena associated with
the propagation of small disturbances of the gluon field. Though the gluon is
not directly observable due to confinement, a bulk of deconfined hot (and/or
dense) quark-gluon matter is expected to exist in the ultra-relativistic heavy
ion interactions, as well as in the early phase of the universe collins1975 .
In those cases when the gluon propagates in the quark gluon matter, it is
argued that the birefringence of gluon field leads to local polarization of
gluons. The polarization correlation is suggested to be measured in Gold-Gold
collision on Relativistic Heavy Ion Collider (RHIC) at Brookhaven National
Laboratory and Lead-Lead collision on Large Hadron Collider (LHC) at CERN.

The paper is organized as follows. In the Section II the light cone conditions
associated with the propagation of small disturbances in one-parameter spin-
one theories are reviewed. In the Section III the effective Lagrangian for
Yang-Mills field pagels78  is presented and the procedure of taking volumetric
spatial average is defined. Then it is discussed the non-trivial behavior of
the phase velocity. The conditions on the causal propagation are stated and
some limiting cases from the effective Lagrangian are examined. Sections IV
and V are dedicated to the possible applications to phenomenology and the
effective geometry issue, respectively. Finally, some final remarks are
presented in the conclusion section.

The present investigation is considered under the regime of the eikonal
approximation, as addressed in delorenci00 . Latin indices run in the range
$(1,2,3)$ and Greek indices run in the range $(0,1,2,3)$. The Minkowski
spacetime is used, employing a Cartesian coordinate system. The background
metric is denoted by $\eta_{\mu\nu}$, which is defined by diag(+1,-1,-1,-1).
Heaviside non-rationalized units are used and $c=1=\hbar$. The completely
anti-symmetric tensor $\epsilon^{\alpha\beta\mu\nu}$ is defined such that
$\epsilon^{0123}=1$.

##  II Light cone conditions

###  II.1 Field equations for one-parameter spin-one theories

The strength tensor field $F_{\mu\nu}^{(a)}$ and the gauge field
$A_{\mu}^{(a)}$ are related by

$F_{\mu\nu}^{(a)}=\partial_{\mu}A_{\nu}^{(a)}-\partial_{\nu}A_{\mu}^{(a)}+C^{abc}A_{\mu}^{(b)}A_{\nu}^{(c)},$
(1)

where $C^{abc}$ represent the structure constant for a compact Lie group $G$.
This tensor field can be conveniently defined in terms of the (non-Abelian)
electric $E_{\mu}^{(a)}$ and magnetic $H_{\mu}^{(a)}$ fields as

$F_{\mu\nu}^{(a)}=V_{\mu}E_{\nu}^{(a)}-V_{\nu}E_{\mu}^{(a)}-\epsilon_{\mu\nu}{}^{\alpha\beta}V_{\alpha}H_{\beta}^{(a)},$
(2)

where $V_{\mu}$ represents the four-velocity of an observer at rest with
respect to the laboratory. In Cartesian coordinates it is given by
$V_{\mu}=\delta_{\mu}^{0}$. In order to alleviate the notation, the ‘color’
indices in the upper brackets will be omitted in what follows.

Let us assume the gauge invariant density of Lagrangian as a general function
of the Lorentz invariant $F\doteq F^{\mu\nu}F_{\mu\nu}$ as $L=L(F)$. From the
minimal action principle we get the equation of motion

$\left(L_{F}F^{\mu\nu}\right){}_{,\nu}=0.$ (3)

where a comma denotes partial derivatives with respect to the Cartesian
coordinates. $L_{F}$ represents the derivative of $L$ with respect to the
invariant $F$. $L_{FF}$ is the second derivative. Using the relation
$F_{,\nu}=2F^{\alpha\beta}F_{\alpha\beta,\nu}$ in equation (3) we obtain:

$\displaystyle
2L_{FF}F^{\mu\nu}F^{\alpha\beta}F_{\alpha\beta,\nu}+L_{F}F^{\mu\nu}{}_{,\nu}=0.$
(4)

The field strength $F_{\mu\nu}$ must satisfy the identity

$F_{\alpha\beta,\lambda}+F_{\beta\lambda,\alpha}+F_{\lambda\alpha,\beta}=0.$
(5)

Let us now derive the expression for the light cone conditions for this class
of theories.

###  II.2 The propagation of the field disturbances

In this section we analyze the propagation of waves associated with the
discontinuities of the field Hadamard . Let us consider a surface of
discontinuity $\Sigma$ defined by ${\cal Z}(x^{\mu})=0$. Whenever $\Sigma$ is
a global surface, it divides the spacetime in two distinct regions $U^{-}$ and
$U^{+}$ (${\cal Z}<0$ and ${\cal Z}>0$, respectively). Given an arbitrary
function of the coordinates, $f(x^{\mu})$, we define its discontinuity on
$\Sigma$ as

$\left[f(x^{\alpha})\right]_{\Sigma}\doteq\lim_{\\{P^{\pm}\\}\rightarrow
P}\left[f(P^{+})-f(P^{-})\right]$ (6)

where $P^{+},\,P^{-}$ and $P$ belong to $U^{+},\,U^{-}$ and $\Sigma$
respectively. Applying the conditions Hadamard  for the tensor field
$F_{\mu\nu}$ and its derivative we set

$\displaystyle\left[F_{\alpha\beta}\right]_{\Sigma}$ $\displaystyle=$
$\displaystyle 0$ (7a)
$\displaystyle\left[F_{\alpha\beta,\lambda}\right]_{\Sigma}$ $\displaystyle=$
$\displaystyle f_{\alpha\beta}k_{\lambda}$ (7b)

where $f_{\alpha\beta}$ represents the discontinuities of field on the surface
$\Sigma$ and $k_{\mu}\doteq(\omega,\vec{k})$ represents the components of the
wave 4-vector. The discontinuity of equations (4) and (5) yields,
respectively,

$\displaystyle
f_{\beta\lambda}k^{\lambda}=-\frac{2}{L_{F}}L_{FF}F_{\beta}{}^{\mu}F^{\nu\rho}f_{\nu\rho}k_{\mu},$
(8) $\displaystyle
f_{\alpha\beta}k_{\lambda}+f_{\beta\lambda}k_{\alpha}+f_{\lambda\alpha}k_{\beta}=0.$
(9)

For the case where $f_{\alpha\beta}$ is the wave propagation tensor given by
equation (7b), for which equation (9) applies, it follows that

$f_{\alpha\beta}=\sigma(\epsilon_{\alpha}k_{\beta}-\epsilon_{\beta}k_{\alpha}),$
(10)

where $\sigma$ is the strength of the wavelet and $\epsilon_{\beta}$
represents the polarization vector. Working with Eqs. (8)-(10) we obtain the
eigenvalue equation

$Z^{\mu}{}_{\nu}\epsilon^{\nu}=0,$ (11)

where we defined

$Z^{\mu}{}_{\nu}\doteq\delta^{\mu}{}_{\nu}+\frac{4}{L_{F}k^{2}}L_{FF}F^{\mu\alpha}F^{\nu\beta}k_{\alpha}k_{\beta},$
(12)

with $k^{2}\doteq k^{\mu}k_{\mu}$. The eigenvectors of $Z^{\mu}{}_{\nu}$
represent the dynamically allowed polarization modes $(e_{+},e_{-})$. The
general solution for the eigenvalue equation is formally given by
$det|Z^{\mu}{}_{\beta}|=0$, and results in the following light cone conditions
boillat70 ; birula70 ; delorenci00 :

$\displaystyle k_{+}^{2}$ $\displaystyle=$ $\displaystyle\gamma
F^{\lambda\mu}F^{\nu}{}_{\lambda}k^{+}_{\mu}k^{+}_{\nu},\qquad\gamma\doteq\frac{4L_{FF}}{L_{F}}$
(13) $\displaystyle k_{-}^{2}$ $\displaystyle=$ $\displaystyle 0,$ (14)

The $\pm$ signs are related with the two possible polarization modes
associated with the wave propagation birula70 . The existence of these two
solutions shows that birefringence effects may appear, provided that
$L_{FF}/L_{F}\neq 0$. In the formalism of geometrical optics it is usually
said that generally there will be two rays inside the medium, the ordinary ray
($o$-ray) and the extraordinary ray ($e$-ray). The former does not depend on
the direction of wave propagation and its velocity is equal to the light
velocity in the classical vacuum of electrodynamics. The latter presents an
explicit dependence on the direction of propagation. The light cone conditions
for two-parameters Lagrangians can be obtained in the same lines. For further
details see Refs. birula70 ; delorenci00 .

##  III Wave propagation in the Yang-Mils field

###  III.1 The effective Lagrangian

The effective Lagrangian for QCD in terms of the parameter background field
$F$ can be presented pagels78  in the form

$\displaystyle{\cal L}_{eff}=\frac{1}{4}\frac{F}{\bar{g}(t)^{2}},\qquad
t\doteq\log\frac{F}{\mu^{4}}$ (15)

where the effective coupling $\bar{g}(t)$ is implicitly given by

$t=\int_{g}^{\bar{g}(t)}\\!\\!\\!dg\;\frac{1}{\beta(g)},$ (16)

with $\beta(g)$ the Callan-Symanzik $\beta$-function and $g$ the gauge field
coupling constant appearing in the basic QCD Lagrangian.

In fact, there are many invariants of the Yang-Mills field with the number
dependent on the specific gauge group roskies1977 . The ansatz used to derive
this effective Lagrangian takes in consideration only the algebraic invariant
$F$ and imposes consistency with the trace anomaly for the energy-momentum
tensor collins1977 .

For the present proposes the system described by Eq. (15) is assumed to
satisfy the following requirements:

  1. 1.

the volumetric spatial average of the color field strength is independent of
direction;

  2. 2.

it is equally probable that the products ${E^{i}E^{j}}$ and ${H^{i}H^{j}}$
(with $i\neq j$), at any time, take positive or negative values;

  3. 3.

there is no net flow of energy as measured by a observer at rest with respect
to the system.

The above mentioned volumetric spatial average of an arbitrary quantity $X$
for a given instant of time $t$ is defined as

$\overline{X}\stackrel{{\scriptstyle.}}{{=}}\lim_{V\rightarrow
V_{o}}\frac{1}{V}\int X\,\sqrt{-g}\,d^{3}\\!x^{i},$ (17)

with $V=\int\sqrt{-g}\,d^{3}\\!x^{i}$, and $V_{o}$ stands for the time
dependent volume of the whole space. Similar average procedure has been
already considered in the context of general relativity solutions tolman1930 ;
delorenci2002 ; kunze2008 .

In terms of the color fields, these requirements imply that:

$\displaystyle\overline{\rule{0.0pt}{8.61108pt}E_{i}}=0,\qquad\overline{\rule{0.0pt}{8.61108pt}H_{i}}$
$\displaystyle=$ $\displaystyle
0,\qquad\overline{\rule{0.0pt}{8.61108pt}E_{i}\,H_{j}-H_{i}\,E_{j}}=0,$ (18)
$\displaystyle\overline{\rule{0.0pt}{8.61108pt}E_{i}\,E_{j}}$ $\displaystyle=$
$\displaystyle-\,\frac{1}{3}E^{2}\,\eta_{ij},$ (19)
$\displaystyle\overline{\rule{0.0pt}{8.61108pt}H_{i}\,H_{j}}$ $\displaystyle=$
$\displaystyle-\,\frac{1}{3}H^{2}\,\eta_{ij},$ (20)

where we have defined $E^{2}\doteq-E^{i}E_{i}$ and $H^{2}\doteq-H^{i}H_{i}$.
The above average procedure consists in an idealization to deal with systems
like a quark-gluon plasma. Nevertheless, it has adequate elements for our
discussions. From the point of view of a statistical ensemble, we can assume
that the field average over the whole bulk is vanishing comparing to the
fluctuation, so it is isotropic as whole, while anisotropic for each local
area.

###  III.2 Application to the effective Yang-Mills Lagrangian

For the special Lagrangian presented in Eq. (15), the factor $\gamma$ in Eq.
(13) is given by

$\gamma=\frac{-4}{E^{2}(Z^{2}-1)}G(\bar{g}),$ (21)

where we have defined the quantities:

$\displaystyle Z^{2}$ $\displaystyle\doteq$ $\displaystyle\frac{H^{2}}{E^{2}}$
(22) $\displaystyle G(\bar{g})$ $\displaystyle\doteq$
$\displaystyle\frac{\bar{g}\dot{\bar{g}}-3\dot{\bar{g}}^{2}+\bar{g}\ddot{\bar{g}}}{\bar{g}^{2}-2\bar{g}\dot{\bar{g}}},$
(23)

with $\dot{\bar{g}}\doteq\partial\bar{g}/\partial t$.

The phase velocity for the wave perturbation can be obtained from the
dispersion relation as $v_{e}^{2}=\omega/|\vec{k}|$, where the index $e$
stands for the $e$-ray. The $o$-ray propagates with the light velocity, as
determined by Eq. (14). Now using the previous results, we obtain

$v_{e}^{2}=1-\frac{8}{3}\frac{(Z^{2}+1)G(\bar{g})}{(Z^{2}-1)+4G(\bar{g})}.$
(24)

In order to guarantee causality, the physical solutions must satisfy the
requirement $0\leq v_{e}\leq 1$, which implies that:

$0\leq\frac{8}{3}\frac{(Z^{2}+1)G(\bar{g})}{(Z^{2}-1)+4G(\bar{g})}\leq 1.$
(25)

From the analysis of the energy density for the effective action associated
with this problem, it can be inferred that the case $E^{2}>H^{2}$ leads to a
metastability of the vaccum. The interpretation for this behavior is that if a
region in the system develops a large $E$ field, it will decay fastly to a
configuration where $H^{2}>E^{2}$ pagels78 . Therefore it is adopted here that
$H^{2}>E^{2}$ (which means $Z^{2}>1$ and $F>0$) and the condition stated by
Eq. (25) yields in

$0\leq G(\bar{g})\leq\frac{3(Z^{2}-1)}{4(2Z^{2}-1)}.$ (26)

Now we are going to examine two cases in which the explicit form of the
effective coupling was presented in the literature. The first one is obtained
when the regime of small coupling is taken in consideration, and the second
one was previously proposed in gross1973  as a suggestion for the case of
large coupling constant.

For the case of small coupling the beta function can be expanded as gross1973
; poli1973 ; jone1974 ; casw1974

$\beta(g)=-\frac{1}{2}b_{0}g^{3}+b_{1}g^{5}+\cdots$ (27)

where $b_{0}$ and $b_{1}$ are constants. Now, taking the limit of large mean
fields ($F\rightarrow\infty$) we obtain from Eq. (16) that

$\frac{1}{\bar{g}(t)}=b_{0}t-2\frac{b_{1}}{b_{0}}\log t+\cdots$ (28)

Introducing these results in the effective Lagrangian we obtain pagels78 ;
nielsen1978

$\displaystyle{\cal L}_{eff}=\frac{1}{4}b_{0}F\log\frac{F}{\mu^{4}}.$ (29)

For this case the function $G(\bar{g})$ results to be

$G(\bar{g})=-\frac{1}{2}\frac{1}{t+1}.$ (30)

Since $t>1$ at the large mean field regime we conclude that $G(\bar{g})<0$,
and there will be no propagation associated with the $e$-ray. The $o$-ray
travels with the unperturbed velocity $v_{-}=1$ and does not depend on the
direction of propagation.

The non-perturbative expression for the beta function is still unknown.
Nevertheless, a suggestion about the strong coupling form of the beta function
was presented long ago gross1973 , and is given by

$\beta(g)=-\frac{a}{2}g,$ (31)

with $a$ a positive constant. If we assume this form for the beta function, we
obtain from Eqs. (16) and (31):

$\frac{1}{\bar{g}(t)^{2}}=\frac{1}{g^{2}}\left(\frac{F}{\mu^{4}}\right)^{a}.$
(32)

In terms of the parameter $t$ it follows that

$\bar{g}(t)^{2}=g^{2}e^{-at}.$ (33)

Introducing these results in the effective Lagrangian we obtain

$\displaystyle{\cal
L}_{eff}=\frac{1}{4g^{2}}F\left(\frac{F}{\mu^{4}}\right)^{a}.$ (34)

For this case the function $G(\bar{g})$ results in

$G(\bar{g})=-\frac{1}{2}a.$ (35)

Thus since $a>0$ we conclude that $G(\bar{g})<0$, and again there will be no
propagation associated with the $e$-ray. In the both cases, once the
superluminal propagation is suppressed, the medium allows just one
polarization mode to propagate. So it seems to behave like a polarizer.

##  IV Observable for birefringence

As shown in the previous section, birefringence effects can occur in the Yang-
Mills fields. Nevertheless, due to confinement phenomenon, a direct
measurement of these effects on gluons propagating in an external color field
seems to be improbable. However, deconfined quark-gluon matter, also known as
quark-gluon plasma (QGP), is expected to be produced in the Gold-Gold
interaction at RHIC or Lead-Lead interaction at LHC. This provides an
opportunity to investigate the issue of gluon propagation in QGP systems.

The different velocities in which the field disturbances can propagate are
associated with different polarization modes. It can be simply understood as a
quantum measurement on the gluon, by which the gluon fall onto the eigenstate
of the polarization mode. In the special cases where the $e$-ray gluon is
forbidden by causality, the external field works like a polarizer, and only
the $o$-ray gluon is allowed there. The polarization is assigned by the
external field. When one gluon propagates in a QGP, it is ‘measured’ time and
time by the local field. Hence, the last polarization direction before its
hadronization into hadrons is completely out of control. This will also
destroy any global polarization information 111One of the examples is that
originated from the non-central interaction, as suggested by the Shandong
Group liangw . Corresponding to those cases, the phenomenon suggested here
appears as a ‘local polarization’. Hence this local polarization destroys the
information of the global one and itself is not measurable if we simply take
the average in the experiment. The STAR Collaboration of RHIC has recently
measured the global polarization effect s1 ; s2 . From the above discussion,
it seems very natural to adopt the negative results from STAR, still with
confidence that both of the polarization effects can exist.. However, due to
the complete polarization at any local area, the polarization correlation will
be very strong. A crucial point is how to identify two particles at the same
local area with the same polarization. In order to explore this point let us
consider, for simplicity, a ‘gluon plasma’. When the hard parton propagates in
the medium, it works as a source of small disturbances on the external field,
and could lead to the emission of Cherenkov radiation. The $o$-gluon can exist
within the Cherenkov cone, but the $e$-gluon must be outside. If the gluon is
not confined it can be measured that the polarizations for two gluons inside
the cone are parallel. More specifically
$\hat{\epsilon}_{1}\cdot\hat{\epsilon}_{2}=1$. The experimental results depend
not only on the polarization transfer from the gluon to a certain kind of
hadron in hadronization process, but also on the identification of the
Cherenkov cone. So, one suggestion is to measure the polarization correlation
of two particles of the same kind (2 vector mesons, 2 hyperons, etc.) with
almost parallel momenta and within the same jet cone (when the jet can be
identified, as expected in LHC). By studying the correlation dependent on the
jet cone angle, one may have a way to measure the Cherenkov angle.

One experimental observable for the polarization correlation can be suggested.
This can be extracted from the ideal case of two $\Lambda$ particles with the
same polarization $\vec{P}$, with $P=|\vec{P}|$ representing the polarization
rate. The conventional way to measure $\vec{P}$ of a single $\Lambda$ is by
measuring the direction vector (denoted as $\hat{p}$) of the momenta of the
daughter particles, e.g., proton or pion from the $\Lambda$ decay, at the rest
frame of $\Lambda$. Then the angular distribution,

$\frac{dN}{dcos\theta}\sim 1+\alpha\hat{p}\cdot\vec{P}=1+\alpha Pcos\theta,$
(36)

can give the information on the polarization. Here $\alpha$ is the hyperon
decay parameter. From this equation we see that if the direction of $\vec{P}$
is random, the average of all $\Lambda^{\prime}s$ gives zero, then $P$ is not
able to be measured. However, for the two $\Lambda^{\prime}s$ with the same
polarization, in the rest frame of each $\Lambda$, respectively, the direction
vectors $\hat{p}_{1}$ and $\hat{p_{2}}$ can be measured. Then we calculate the
expectation value $<\hat{p}_{1}\cdot\hat{p}_{2}>$, which results in

$P=3\sqrt{<\hat{p}_{1}\cdot\hat{p}_{2}>}/\alpha.$ (37)

Because $<\hat{p}_{1}\cdot\hat{p}_{2}>$ is a SO(3) scalar, it does not depend
on the direction of polarization. Thus, it can be averaged for all the jets of
all the (QGP) events in order to get the scalar value of averaged polarization
from Eq (37). In an experiment, the $\Lambda$ pair of the same jet 222If the
jet is not defined it can be considered the near-parallel pairs of the same
kind of hyperon. are expected to have the same polarization with larger
probability, as discussed above.

##  V The effective geometry issue

The effective theory approach has long been considered as a possible way to
understand confinement phenomena. One of the possible ways to investigate
these phenomena consists in the construction of analogue models in which
confinement would be related with an event horizon formation, as it occurs in
the black hole physics. Such a interpretation was considered in castorina07 ;
elbaz  (see also the references therein).

The results discussed in this paper can be read in the context of the optical
geometry. In this context some comments are in order. Eqs. (13)–(14) can be
presented in the appealing form

$g_{\pm}^{\mu\nu}k_{\mu}k_{\nu}=0,$ (38)

where we defined the two symmetric contravariant tensors

$\displaystyle g^{\mu\nu}_{+}$ $\displaystyle=$
$\displaystyle\eta^{\mu\nu}-\gamma F^{\lambda\mu}F^{\nu}{}_{\lambda},$ (39)
$\displaystyle g^{\mu\nu}_{-}$ $\displaystyle=$ $\displaystyle\eta^{\mu\nu}.$
(40)

The inverse symmetric tensor $g_{\mu\nu}$ is defined in such way that
$g^{\mu\alpha}g_{\alpha\nu}=\delta^{\mu}_{\nu}$. Therefore, for each
propagation vector $k_{\nu}$, the corresponding tensor $g^{\mu\nu}$ plays the
role of an effective metric tensor. Indeed, $k_{\nu}$ is a light-like (or
null) vector with respect to the associated metric tensor. It can be shown
delorenci02  that it also satisfies a geodesic equation in terms of the
Christoffel symbols $\Gamma^{\alpha}{}_{\\!\mu\nu}$ associated with this
metric. In this way we can refer to $k_{\nu}$ as a geodesic null vector with
respect to the effective metric tensor $g^{\mu\nu}$. This geometric
interpretation could be used in order to produce an analogue model for
confinement based on the possible formation of an event horizon. In this case,
the requirement of causal propagation would assume a nontrivial role. Inside
the hadron, where quark and gluons can interact and propagate, the velocity of
the field disturbances would be expected to be smaller than 1. On the other
hand, if the vacuum outside the hadron is described by the effective
Lagrangian, the larger than 1 velocity could be interpreted as an indication
of confinement, since no physical observable could propagate there.

##  VI Conclusions

In this paper the propagation of field disturbances was investigated in the
context of the effective Yang-Mills Lagrangian. The general dispersion
relations for one-parameter Lagrangians, Eqs. (13) and (14), were derived
employing the method presented in delorenci00 ; birula70 . It was shown that
birefringence phenomena can occur.

Let us remark some points. First, it should be stressed that the method
depends on the effective Lagrangian as $L(F)$, so the conclusions are quite
general. Second, the assumption of causal propagation of the signals sets non-
trivial constraint when exploring specific solutions. Finally, for the case of
a deconfined quark gluon system, which naturally provides an effective
external field, the birefringence phenomena with gluons and its local
polarization effects are expected to be observed at RHIC and LHC by measuring
the strong local spin correlations of various hadrons from QGP. The
measurements of the spin correlations suggested here, at the same time, can be
useful in assigning the details of the effective fields in QGP. These
informations provide opportunities to develop the effective Lagrangian
framework and hence the better understanding of QCD.

###### Acknowledgements.

This work is partially supported by the Brazilian research agencies CNPq and
FAPEMIG (V.A.D.L.); and National Natural Science Foundation of China (NSFC)
with grant No.10775090, and China Scholarship Council (CSC) (S.Y.L.).

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