capbtabboxtable[][]

# Testing a Neural Network for Anomaly Detection in the CMS Global Trigger
Test Crate during Run 3

Noah Zipper11footnotetext: Corresponding author.

###### Abstract

We present the deployment and testing of an autoencoder trained for unbiased
detection of new physics signatures in the CMS Level-1 Global Trigger (GT)
test crate during LHC Run 3. The GT test crate is a copy of the main GT
system, receiving the same input data, but whose output is not used to trigger
the readout of CMS, providing a platform for thorough testing of new trigger
algorithms on live data, but without interrupting data taking. We describe the
integration of the Neural Network into the GT test crate, and the monitoring,
testing, and validation of the algorithm during proton collisions.

##  1 Introduction

The CMS detector [1, 2] reads out far more data than can be processed,
reconstructed, and analyzed. In order to use any of the TB/s being generated,
a reduction of more than 99% is necessary. The job of the CMS Level-1 trigger
(L1T), which it does in real-time on a chain of field programmable gate arrays
(FPGAs) [3], is to perform this data reduction without missing interesting
physics events. Operating on the clock of the LHC, where collisions occur
every 25 nanoseconds, requires the entire system to adhere to microsecond
latency constraints. Furthermore, stability is crucial for this system. Any
error can lead to detector "dead time", where data is lost forever.

A potential problem of traditional trigger strategies is that they rely either
on a priori knowledge of signal or generic kinematic selections. This problem
is addressed by triggering on how anomalous an event is. A variational
autoencoder (VAE) trained on real unbiased CMS data to detect outliers offers
a solution that is both signal agnostic (applicable to signatures we have not
had the foresight to target specifically) and highly sensitive (effectively
boosts signal efficiency for multiple physics signatures) [4, 5].

##  2 Anomaly Detection Trigger Algorithm

Figure 1: (Left) A typical design of a VAE, utilizing both an encoding and
decoding network to reconstruct event information. (Right) The VAE model
visualization for AXOL1TL, showing the layers, inputs, outputs, and the
calculation of the anomaly score, our metric for triggering on interesting
physics.

The VAE design uses an information bottleneck created by a small-dimensional
latent space, which enforces an efficient data encoding, and leads the model
to learn what makes an event anomalous. For this Anomaly Detection
implementation in the CMS Level-1 Global Trigger, called Anomaly eXtraction
Online Level-1 Trigger aLgorithm (AXOL1TL), the inputs are taken from a set of
standard L1T objects ($p_{\mathrm{T}}^{\mathrm{miss}}$, 4 e/$\gamma$, 4 $\mu$,
and 10 jets) as $(p_{\mathrm{T}},\eta,\phi)$ vectors. The design of the VAE,
visualized in figure 1, was driven by the constraints of the L1T system.
Multiple steps were taken to minimize latency and resource utilization,
including the removal of the decoder network, and the simplification of the
latent space loss term, shown in (2.1). The reconstruction term is computed
from the difference between the input (x) and output ($\hat{x}$) of the VAE.
The second, full regularization term, is the Kullback–Leibler divergence (KL-
divergence) between the latent space distribution and a standard normal
distribution with mean $\mu$ and standard deviation $\sigma$. The parameter
$\beta$ can be tuned to balance the reconstruction performance with more
efficient latent space encoding. At inference time, the loss is approximated
by the mean-squared term $\Sigma_{i}\mu^{2}$ of the KL-divergence for latency
considerations. This approximation has no impact on performance.

$\displaystyle\text{Loss}=(1-\beta)||x-\hat{x}||^{2}+\beta\frac{1}{2}(\mu^{2}+\sigma^{2}-1-\log\sigma^{2})$
(2.1)

AXOL1TL is trained with unbiased data collected by the CMS Experiment during
2023 with proton collisions at a centre of mass energy $\sqrt{s}=13.6$ TeV.
From this dataset, 10.5 million events were used: 50% for training and 50% for
setting thresholds on the anomaly score. Quantization aware training, where
inference-time quantization is emulated during training using the package
QKeras [6], allowed for optimal performance in the final hardware
implementation. A set of thresholds to demonstrate the range of performance
possible with AXOL1TL was chosen, plotted in figure 2, estimating the L1T rate
for different thresholds.

[capbesideposition=right,top,capbesidewidth=.5]figure[]

Figure 2: Anomaly score distributions for 2023 Ephemeral ZeroBias events.
Individual event scores/losses for the QKeras model in Python (orange) and
standalone High-Level Synthesis (HLS) emulator (blue). Dotted lines represent
scores that correspond to trigger paths in the $\mu$GT test crate.

For the 2023-trained model, an example of the significant performance
improvement for Beyond the Standard Model (BSM) signals, measured in simulated
events, by adding AXOL1TL to the 2023 trigger menu is shown in table 1.

Table 1: Efficiency improvement of AXOL1TL trigger bits to 2023 L1 Menu with respect to the BSM signal of a Higgs decaying to two (pseudo)scalars of mass 15 GeV, where (pseudo)scalars decay to bottom quark pairs. The model used is trained on Run 3 ZeroBias events (Run 367883). Efficiency gains from AXOL1TL at various triggering rates are compared. AXOL1TL Rate | 1 kHz | 5 kHz | 10 kHz
---|---|---|---
$H\rightarrow aa\rightarrow 4b$ Signal Efficiency Gain | 46% | 100% | 133%

##  3 CMS Global Trigger Firmware

Figure 3: Schematic of the $\mu GT\ $board, showing the AXOL1TL score
calculation in yellow.

The firmware for the Anomaly Detection algorithms had to be integrated into
the existing CMS Global Trigger ($\mu GT\ $) board structure [7], shown in
in figure 3. To meet timing, the score is calculated in concert with other
global trigger quantities, fed into the trigger combination logic, and its
corresponding trigger decisions are read out via the standard links.

High-level synthesis (HLS) was used to generate the hardware code (VHDL) for
the $\mu GT\ $FPGA, utilizing the hls4ml package to efficiently synthesize the
neural network [8]. To incorporate the trigger information into the common CMS
software tools, an external CERN Gitlab repository for HLS dependencies was
made. This code generates the bitfiles that configure the trigger boards.

Vivado co-simulation tests, shown in table 2, demonstrate that the synthesized
VHDL code meets the L1T latency requirement of 50 ns and takes up only a small
fraction of the resources available on the Xilinx Virtex-7 chip [9].

Table 2: Vivado timing and resource utilization report for Anomaly Detection trigger on Xilinx Virtex-7 FPGA. Results show that firmware build meets L1 latency requirements, fitting within 2 clock cycles @ 40 MHz. The resource utilization of the module is also relatively small. Columns refer to different FPGA components: look-up tables (LUTs), flip-flops (FFs), digital signal processing (DSPs) slices, and block random access memory (BRAMs) | Latency | LUTs | FFs | DSPs | BRAMs
---|---|---|---|---|---
AXOL1TL | 2 ticks (50 ns) | 2.1% |  $\sim$ 0% | 0% | 0%

Further validation was performed to ensure trigger decisions were being
computed correctly for the given thresholds. A ModelSim emulator, the standard
CMS tool for L1T menu [10] validation was used. Test vector files with the
Level-1 objects, detector conditions, and an independently emulated reference
decision were read into the environment, and table 3 shows the perfect trigger
bit agreement observed. This successful ModelSim test confirmed a functional
firmware module that could be implemented in hardware, with final triggers
decisions that could be accurately emulated and verified.

Table 3: Test Crate firmware validation. The table shows trigger bits for the
L1 menu of 4 Anomaly Detection thresholds: scores >1250, >250, >25, and >5
from top to bottom (factor of 16 differentiates physical anomaly score and
hardware integer). Test vector counts are generated with standalone emulator
and hardware counts come from $\mu GT\ $ModelSim firmware validation workflow,
using the same events from Run 368566. Perfect bit agreement is observed.  L1
Menu
Index |  L1 Menu
Algorithm Name |  Test Vector
Count |  Hardware
Emulation Count |  Agreement
---|---|---|---|---
94 |  L1_ADT_20000 |  0 |  0 |  ✓
95 |  L1_ADT_4000 |  29 |  29 |  ✓
103 |  L1_ADT_400 |  2618 |  2618 |  ✓
108 |  L1_ADT_80 |  3331 |  3331 |  ✓

##  4 Test Crate Implementation

Once validated, the $\mu GT\ $firmware was implemented on the CMS Global
Trigger Test Crate (TC); The TC is a set of identical MP7 boards that are used
as backup for the production system, as well as for testing new trigger
strategies. In this configuration, the TC is set to read in the same inputs,
but without actually triggering on events and saving data. The TC is connected
to the CMS Prometheus monitoring server that can query trigger metrics in
real-time allowing us to monitor trigger rates for the anomaly trigger paths
while data is being taken, shown in figure 4.

Figure 4: Test crate rate monitoring time series. L1 trigger rates shown for 4
Anomaly Detection threshold triggers, overall L1 physics rate, and the
L1_SingleMu22 un-prescaled single muon reference trigger. Time-averaged rates
are read from $\mu GT\ $test crate monitoring software via Prometheus server
at a $\sim$20s buffer rate during good data-taking conditions in 2023. AXOL1TL
model is trained on 2018 ZeroBias data and thresholds are chosen to test
possible range of accessible trigger rate. Thresholds are not meant to model
realistic trigger rates. Consistent performance is shown over the course of
partial fill-cycle. The turn-on corresponds to the beginning of an LHC fill
and the sawtooth pattern corresponds to luminosity levelling.

For particular runs, events that were triggered and saved by the production
system contain TC information, showing which trigger bits were fired. This
allows for a final validation of the anomaly score performance with respect to
emulation. Figure 6 and table 6 show that for such events we see minimal
mismatches and reasonable agreement between hardware and emulation.

Figure 5: Anomaly Detection hardware and emulator trigger counts for 2023
Ephemeral ZeroBias data where hardware bits are recorded from configured $\mu
GT\ $test crate. Red segments represent mismatches between hardware and
emulation. Clustering near decision boundaries implies issue is due to
precision/rounding problem. Minimal mismatches in hardware vs. emulation (<1%)
are observed.

L1 Menu
Algorithm Name |  Test Crate
Count |  Standalone
Emulation Count |  Mismatches
---|---|---|---
L1_ADT_20000 |  1 |  1 |  0
L1_ADT_4000 |  742 |  741 |  19
L1_ADT_400 |  21236 |  21229 |  253
L1_ADT_80 |  25468 |  25481 |  93

Figure 6: Anomaly Detection hardware and emulator trigger counts for 2023
Ephemeral ZeroBias data where hardware bits are recorded from configured $\mu
GT\ $test crate. Test Crate Count shows events triggered in hardware and read
out into data and Standalone Emulator Count is evaluated via offline inference
with L1 objects.

##  5 Summary

We have shown a signal-agnostic trigger model sensitive to interesting
physics. A firmware implementation for this trigger algorithm was successfully
integrated into the CMS Level-1 trigger architecture. Using the CMS Global
Trigger test crate, we showed an active hardware trigger that performed
consistently during 2023 collisions. Finally, validation was performed at each
steps using HLS emulation. We plan to update this algorithm and prepare
downstream trigger logic to implement the Anomaly eXtraction Online Level-1
Trigger aLgorithm (AXOL1TL) in the Level-1 trigger for 2024 data-taking.

## References

  * [1] CMS Collaboration, _The CMS experiment at the CERN LHC_ , _Journal of Instrumentation_ 3 (2008) S08004.
  * [2] CMS Collaboration, _Development of the CMS detector for the CERN LHC Run 3_ , Tech. Rep. CMS-PRF-21-001, CERN-EP-2023-136, CMS-PRF-21-001-003, CERN, Geneva (2023).
  * [3] CMS Collaboration, _The Phase-2 Upgrade of the CMS Level-1 Trigger_ , Tech. Rep. CERN-LHCC-2020-004, CMS-TDR-021, CERN, Geneva (2020).
  * [4] E. Govorkova et al., _Autoencoders on field-programmable gate arrays for real-time, unsupervised new physics detection at 40 MHz at the Large Hadron Collider_ , _Nature Mach. Intell._ 4 (2022) 154 [2108.03986].
  * [5] J. Duarte, S. Han, P. Harris, S. Jindariani, E. Kreinar, B. Kreis et al., _Fast inference of deep neural networks in FPGAs for particle physics_ , _Journal of Instrumentation_ 13 (2018) P07027 [1804.06913].
  * [6] C.N. Coelho, A. Kuusela, S. Li, H. Zhuang, T. Aarrestad, V. Loncar et al., _Automatic heterogeneous quantization of deep neural networks for low-latency inference on the edge for particle detectors_ , _Nature Mach. Intell._ 3 (2021) 675 [2006.10159].
  * [7] M. Jeitler et al., _The level-1 global trigger for the CMS experiment at LHC_ , _JINST_ 2 (2007) P01006.
  * [8] FastML Team, _hls4ml_ , May, 2023.  10.5281/zenodo.7933047.
  * [9] AMD Xilinx, “Virtex 7 FPGA.” https://www.xilinx.com/products/silicon-devices/fpga/virtex-7.html.
  * [10] B. Herbert, “l1menus.” https://github.com/herbberg/l1menus, 2020.
  * [11] CMS Collaboration, _Performance of missing transverse momentum reconstruction in proton-proton collisions at $\sqrt{s}=13$ TeV using the CMS detector_, _JINST_ 14 (2019) P07004 [1903.06078].
  * [12] CMS Collaboration, _Anomaly Detection in the CMS Global Trigger Test Crate for Run 3_ , Tech. Rep. CMS-DP-2023-079, CERN, Geneva (2023).


# Gauge embedding procedure: classical and quantum equivalence between dual
models

B. Alves Marques(a,b<EMAIL_ADDRESS>B. Z. Felippe(a,c)
<EMAIL_ADDRESS>A. P. Baeta Scarpelli(a<EMAIL_ADDRESS>L. C.
T. Brito(d<EMAIL_ADDRESS>A. Yu. Petrov(e<EMAIL_ADDRESS>(a)Centro
Federal de Educação Tecnológica - MG
Avenida Amazonas, 7675 - 30510-000 - Nova Gameleira - Belo Horizonte -MG -
Brazil  (b) Instituto Federal de Minas Gerais - Campus Sabará
Rodovia MGC 262, Sobradinho, Minas Gerais, Brazil  (c) Universidade Federal de
Itajubá - Campus Itabira
Rua Irmã Ivone Drumond, 200 - Distrito Industrial II, Itabira, Minas Gerais,
Brazil  (d) Universidade Federal de Lavras - Departamento de Física
Caixa Postal 3037, 37.200-000, Lavras, Minas Gerais, Brazil  (e) Departamento
de Física, Universidade Federal da Paraíba
Caixa Postal 5008, 58051-970, João Pessoa, Paraíba, Brazil

###### Abstract

In this paper the gauge embedding procedure of dualization is reassessed
through a deeper analysis of the mutual equivalence of vector field models of
more generic forms, explicitly, a general modified massive gauge-breaking
extension of electrodynamics and its dual gauge-invariant model we derive in
the paper. General relations between the vector field propagators and
interaction terms of these models are obtained. Further, these models are
shown to be equivalent at tree-level and one-loop physical calculations.
Finally, we discuss extension of this equivalence to all loop orders.

###### pacs:

11.15.-q, 11.10.Kk, 11.10.Ef

##  I Introduction

The concept of duality has been explored for several decades in different
contexts. It is basically a mapping of the structures of models that, at first
glance, seem to be distinct, but which, after a deeper analysis, turn out to
be equivalent. We can say that a duality corresponds to equivalent
descriptions of a model by using distinct Lagrangian formulations (for
reviews, see review-1  and review-2 ). One of the most interesting aspects of
duality, with important implications in the perturbative context, concerns the
exchange of the coupling regime from $g$ to $1/g$. In this case, the duality
relation allows the mapping of a weak coupling in a strong coupling and vice
versa, as in the case of the implementation of the relation between electric
and magnetic couplings. It is important to note that duality can relate a
theory without gauge invariance to a gauge invariant theory.

In view of these relevant features, different methods have been developed to
establish and investigate duality. We highlight especially the master action
approach Malacarne , whose starting point is an action depending on two
fields, from which, two dual models are obtained by eliminating one field with
the use of its field equations. The gauge embedding procedure ilha-1 , ilha-2
, also known as Noether Dualization Method (NDM), was developed to handle
difficulties when dealing with the non-Abelian version of the self-dual model
nonab1 , nonab2 , nonab3 . The well known equivalence in $2+1$ dimensions
between the self-dual Townsend  and the topological massive topological
models, first shown by Deser and Jackiw Deser , is easily obtained with the
use of the gauge embedding approach.

The procedure of NDM is carried out by transforming the self-dual model into a
gauge theory with the addition of on-shell vanishing terms. It is an iterative
embedding of Noether counterterms, which is based on the idea of local lifting
a global symmetry and which follows the concepts used in the papers by
Freedman and van Nieuwenhuizen Nieuw2  and subsequent works by Ferrara,
Freedman and van Nieuwenhuizen Ferrara1  and Ferrara and Scherk Ferrara2 ,
which played an important role for constructing the component-field
supergravity actions.

The gauge embedding method has been shown to be an efficient tool in the study
of different field theory models. In particular, it allows to find new
couplings for vector fields, as in the case of the self-dual theory minimally
coupled to the spinor matter, which has been shown to generate the magnetic
(nonminimal) and the Thirring-like current-current couplings ilha-2 . Further,
with the cited methods, dualities have been established between the nonlinear
generalizations of self-dual and Maxwell-Chern-Simons theories nlin , in
higher-rank tensor generalizations of these theories highrank  and in their
higher-derivative extensions HD . Besides, noncommutative extensions of the
duality have been discussed in NC . There were also several works elaborated
extending duality to Lorentz-violating (LV) models, like for the LV extension
of the 3D self-dual theory Fur  and in its promotion to four-dimensional
space-time Bota , Wo , with the role of the Chern-Simons mass term played by
the LV Carroll-Field-Jackiw (CFJ) term. The dual embedding of a four-
dimensional Proca-like theory with a CPT-even Lorentz-breaking mass term was
carried out in Scarp , with the resulting theory involving very interesting
higher-derivative terms.

At the same time, one aspect of duality discussed in scarp-epl  was lacking a
deeper analysis. In that paper, a relation between the propagators of the two
models interrelated by duality was derived, and it is potentially dangerous
for the theory constructed through the gauge embedding. Indeed, since the dual
propagator of the gauge field looks like the difference between two
propagators, one still faces the question whether the new model is
contaminated by ghost modes of propagation of the vector field or not.
Alternatives to avoid the emergency of ghosts in the process of dualization
were developed Dalm-1 , Dalm-2 , Dalm-3 , Dalm-4 . In some approaches, the
price to be paid is the lost of locality Dalm-1 . In this paper, we show that
the complete analysis needs to take in consideration the new interaction terms
which show up in the constructed model. It becomes evident that the
traditional study of the saturated propagator should be improved to consider
also the current-current interaction. When this additional contribution is
considered, there occurs a cancellation of the spurious nonphysical terms.
This conclusion is shown to be extended for quantum calculations.

The structure of the paper looks like follows. In the section II, we perform
the gauge embedding of our gauge-breaking model for a vector field, obtaining
a new gauge invariant theory. In the section III, we compare propagators of
these theories. In the section IV, we prove tree-level equivalence of the
models, and, in the section V, we prove their perturbative equivalence at the
one-loop order and discuss the situation at higher loops. Our results are
discussed in section VI.

##  II The gauge embedding procedure

Let us consider the following Lagrangian density, which, in general, can
represent itself as a Lorentz-violating extension of quantum electrodynamics
(QED),

${\cal L}={\cal
L}_{A}+\frac{1}{2}\mu^{2}A^{\mu}M_{\mu\nu}A^{\nu}-J_{\mu}A^{\mu}+{\cal
L}_{F},$ (1)

where ${\cal L}_{A}$ is the gauge invariant part of the photon sector, which
can encompass CPT-odd or even Lorentz-breaking contributions, ${\cal L}_{F}$
is the free fermion Lagrangian density, $M_{\mu\nu}$ is a dimensionless tensor
which depends on constant background tensors and $\mu$ is a parameter with the
mass dimension 1. It is possible to construct, on the base of (1), a gauge-
invariant model which describes the same physical solutions by means of a
dualization method. Here, we use a gauge embedding technique also known as
Noether dualization method (NDM), initially formulated in AnaWo  and further
applied in some other papers, f.e. Fur .

The first step is the calculation of the first variation of the Lagrangian
density,

$\delta{\cal L}[A_{\mu}]=K_{\nu}\,\delta A^{\nu},$ (2)

with $K^{\mu}$ the Noether current, so that we construct the first iterated
Lagrangian by introducing an auxiliary field $B^{\mu}$,

${\cal L}^{(1)}={\cal L}-K^{\mu}B_{\mu}.$ (3)

For $B^{\mu}$ transforming as $\delta B_{\mu}=\delta
A_{\mu}=\partial_{\mu}\eta$, we have the following variation of the first-
iterated Lagrangian:

$\delta\,{\cal L}^{(1)}\,=\,-(\delta\,K_{\mu})\,B^{\mu}.$ (4)

Since the breaking of gauge symmetry occurs exclusively due to the mass term,
i.e. $\delta{\cal L}_{A}=0$, we have

$\delta K^{\mu}=\mu^{2}M^{\mu\nu}\delta A_{\nu},$ (5)

and

$\delta{\cal L}^{(1)}=-\mu^{2}B_{\mu}M^{\mu\nu}\delta A_{\nu}.$ (6)

If we define the second iterated Lagrangian as

${\cal L}^{(2)}\,=\,{\cal
L}^{(1)}\,+\,\frac{\mu^{2}}{2}\,B^{\mu}M_{\mu\nu}B^{\nu}$ (7)

and use the variation of $B_{\mu}$ and (6), we find that its total variation
vanishes, $\delta{\cal L}^{(2)}\,=\,0$. Let us write down the explicit form of
this action,

${\cal L}^{(2)}={\cal
L}-K_{\mu}B^{\mu}+\frac{\mu^{2}}{2}\,B^{\mu}M_{\mu\nu}B^{\nu}.$ (8)

After carrying out the variation of this action with relation to $B_{\mu}$, we
get the following equation of motion:

$K_{\mu}-\mu^{2}M_{\mu\nu}B^{\nu}=0.$ (9)

Plugging this back into (8), we obtain the dual gauge invariant theory:

${\cal L}_{D}={\cal L}-\frac{1}{2\mu^{2}}K^{\mu}L_{\mu\nu}K^{\nu},$ (10)

in which $L_{\mu\nu}=(M^{-1})_{\mu\nu}$. This is a general relation between
the Lagrangians of the gauge-breaking model and its dual gauge-invariant
theory, for the case when the mass term is the unique one responsible for the
violation of this symmetry.

##  III Relation between propagators and new interaction terms

Now let us obtain simple relations between the wave operators of the models
involved in the duality relation (see, for example, scarp-epl ). The
Lagrangian density for the gauge-breaking model can be written as

${\cal L}=\frac{1}{2}A^{\mu}{\cal O}_{\mu\nu}A^{\nu}-J^{\mu}A_{\mu}+{\cal
L}_{F},$ (11)

being ${\cal O}_{\mu\nu}$ its wave operator. The variation of the above
expression is given by

$\delta{\cal L}=\left({\cal O}A-J\right)\delta A,$ (12)

where, for compactness, we suppressed the Lorentz indices. The equation above
was obtained after partial integrations in the action. For this, it is
important to note that this differential operator ${\cal O}$ involves only
terms of first and second orders in derivatives. We require the first-order
term to be antisymmetric in the Lorentz indices, as it occurs in the Chern-
Simons case.

Comparing equations (2) and (12), we get

$K={\cal O}A-J,$ (13)

such that we have

$\frac{1}{2\mu^{2}}KLK=\frac{1}{2\mu^{2}}\left({\cal O}A-J\right)L\left({\cal
O}A-J\right).$ (14)

The same argument used before is valid now for new integrations by parts in
the action, which will provide us with the result

$\frac{1}{2\mu^{2}}KLK=\frac{1}{2}A\left(\frac{1}{\mu^{2}}{\cal
O}L\cal{O}\right)A-\frac{1}{\mu^{2}}J\,L({\cal
O}A)+\frac{1}{2\mu^{2}}J\,L\,J.$ (15)

Now, let us consider a model with the quadratic part $\frac{1}{2}A{\cal
O}_{0}A$, such that ${\cal O}_{0}={\cal O}-\mu^{2}M$, which is simply the wave
operator of our model without the mass term. With this definition, we have

${\cal O}L{\cal O}={\cal O}L\left({\cal O}_{0}+\mu^{2}M\right)={\cal O}L{\cal
O}_{0}+\mu^{2}{\cal O}.$ (16)

When we plug this result back into equation (10), we get

${\cal L}_{D}=\frac{1}{2}A\left(-\frac{1}{\mu^{2}}{\cal O}L{\cal
O}_{0}\right)A+{\cal L}^{\prime}_{I}+{\cal L}^{\prime}_{F},$ (17)

with

${\cal L}^{\prime}_{I}=-J\,A+\frac{1}{\mu^{2}}J\,L\,({\cal O}A)$ (18)

and

${\cal L}^{\prime}_{F}={\cal L}_{F}-\frac{1}{2\mu^{2}}J\,L\,J.$ (19)

From (17), we identify the wave operator for the dualized model as

${\cal O}_{D}=-\frac{1}{\mu^{2}}{\cal O}L{\cal O}_{0},$ (20)

which is an interesting relation between the wave operators of the two models.
As we will see, this has strong implications when the field dynamics is
studied. It is also worth to emphasize that the higher order derivative terms
will bring new modes of field propagation which may affect the physical
equivalence between the dual models. Besides, it is important to note that new
interaction terms were generated: a nonminimal one, responsible for a
magnetic-like interaction; and a current-current Thirring-like coupling.

Let us now take care of the quadratic part of the dualized gauge action.
Equation (20) allows a simple procedure for the calculation of the gauge
propagator of the dualized model. First, a gauge-fixing term is necessary,
since ${\cal O}_{0}$ has no inverse. But here we have two possibilities: the
gauge-fixing can be added to ${\cal O}_{0}$ or to ${\cal O}_{D}$. In the first
situation, we use

$\tilde{{\cal O}}_{0}={\cal O}_{0}+\frac{\Box}{\tau}\omega$ (21)

in the place of ${\cal O}_{0}$, where $\tau$ is the gauge-fixing parameter and
$\omega_{\mu\nu}=\partial_{\mu}\partial_{\nu}/\Box$ is the longitudinal spin
projector. The gauge-fixed wave operator of the dualized model is then given
by

$\displaystyle{\cal O}_{D}$ $\displaystyle=$
$\displaystyle-\frac{1}{\mu^{2}}{\cal O}L\tilde{{\cal O}}_{0}$ (22)
$\displaystyle=$ $\displaystyle-\frac{1}{\mu^{2}}{\cal O}L{\cal
O}_{0}+\frac{\Box}{\tau}\left(-\frac{1}{\mu^{2}}{\cal O}\,L\,\omega\right).$

Effectively, we have a new gauge-fixing term in which the projector
$\omega_{\mu\nu}$ is replaced by another longitudinal operator
$-\frac{1}{\mu^{2}}\left({\cal O}\,L\,\omega\right)_{\mu\nu}$. This way of
dealing with the problem is very interesting, as it allows a very simple
calculation of the propagator for the vector field, which is given by

$\left<A_{\mu}A_{\nu}\right>_{D}=i\left({\cal O}_{D}^{-1}\right)_{\mu\nu}.$
(23)

The inversion of the dual wave operator is straightforward, so that we have

${\cal O}_{D}^{-1}=-\mu^{2}\tilde{\cal O}_{0}^{-1}\,M\,{\cal O}^{-1}.$ (24)

It is possible to obtain a simpler relation. First, we have

${\cal O}=\tilde{\cal O}_{0}+\mu^{2}M-\frac{\Box}{\tau}\omega,$ (25)

from which, after making the product of $\tilde{\cal O}_{0}^{-1}$, on the
right, and of ${\cal O}^{-1}$, on the left, we obtain

$\tilde{\cal O}_{0}^{-1}={\cal O}^{-1}+\mu^{2}\tilde{\cal
O}_{0}^{-1}\,M\,{\cal O}^{-1}-\frac{\Box}{\tau}\tilde{\cal
O}_{0}^{-1}\,\omega\,{\cal O}^{-1}.$ (26)

We then identify the second term on the right hand side as $-{\cal
O}_{D}^{-1}$ and, in the third, we write

$\tilde{\cal O}_{0}^{-1}=\tilde{\cal O}_{0\,Tr}^{-1}+\frac{\tau}{\Box}\omega,$
(27)

where $\tilde{\cal O}_{0\,Tr}^{-1}$ is the transversal part of $\tilde{\cal
O}_{0}^{-1}$. This decomposition is explained by the fact that the model which
originates the wave operator ${\cal O}_{0}$ is gauge invariant. So, we have

$\displaystyle-\frac{\Box}{\tau}\tilde{\cal O}_{0}^{-1}\,\omega\,{\cal
O}^{-1}$ $\displaystyle=$ $\displaystyle-\frac{\Box}{\tau}\left(\tilde{\cal
O}_{0\,Tr}^{-1}+\frac{\tau}{\Box}\omega\right)\,\omega\,{\cal O}^{-1}$ (28)
$\displaystyle=$ $\displaystyle-\omega{\cal O}^{-1}=\theta{\cal O}^{-1}-{\cal
O}^{-1},$

in which $\theta_{\mu\nu}=\eta_{\mu\nu}-\omega_{\mu\nu}$ is the transversal
spin operator projector. If we use the above relation in equation (26), we get

${\cal O}_{D}^{-1}=\theta{\cal O}^{-1}-\tilde{\cal O}_{0}^{-1}.$ (29)

The relation between propagators shown in Eq. (29) is remarkably simple and
more general than the one obtained in scarp-epl . It is worth to note that it
does not depend on the explicit form of $M_{\mu\nu}$. In addition, the
expression makes clear the presence of the same modes of propagation as the
original theory, which are naturally assumed to describe the same particles.
Earlier, the arising of modes characterized by the same dispersion relations
has been observed in Fur , sdual-even , for three-dimensional LV self-dual and
Maxwell-Chern-Simons (MCS) models. However, attention should be paid to the
presence of new poles in the propagator, coming from the massless theory. As
they appear with the sign exchanged in the expression, this may indicate the
presence of ghost modes in the dual model. This possibility should be verified
in detail.

##  IV Tree level equivalence

In the previous section, we derived a relation between the propagators of the
original model and its dual gauge invariant partner. From this relation, which
is a difference of two propagators, there emerges the dangerous possibility of
nonphysical modes, ghosts, which could spoil the energy spectrum of the model.
In this section, we show that the traditional analysis of the saturated
propagator, similar to that one performed, e.g. in Scarp , needs, in this
case, to be extended by the inclusion of the current-current interaction,
which will cause the exact cancellation of the non-physical contributions.

Let us write the dual Lagrangian density in the following form

${\cal L}_{D}=\frac{1}{2}A^{\mu}{\cal
O}_{D\mu\nu}A^{\nu}-\left[J_{\mu}-\frac{1}{\mu^{2}}J_{\alpha}L^{\alpha\beta}{\cal
O}_{\beta\mu}\right]A^{\mu}-\frac{1}{2\mu^{2}}J_{\mu}L^{\mu\nu}J_{\nu}+{\cal
L}_{F},$ (30)

from which we define the new current as

$\tilde{J}_{\mu}=J_{\nu}\left(\delta_{\mu}^{\nu}-\frac{1}{\mu^{2}}L^{\nu\beta}{\cal
O}_{\beta\mu}\right),$ (31)

which is easily shown to be conserved. In the momentum space, we have

$p^{\mu}\tilde{J}_{\mu}=p^{\mu}\left(J_{\mu}-\frac{1}{\mu^{2}}{\cal
O}_{\mu\beta}L^{\beta\nu}J_{\nu}\right),$ (32)

and, since ${\cal O}$ includes a transverse part and the $M$ dependent one, we
write

$-\frac{1}{\mu^{2}}p^{\mu}{\cal
O}_{\mu\beta}L^{\beta\nu}J_{\nu}=-\frac{1}{\mu^{2}}p^{\mu}({\cal
O}_{0\mu\beta}+\mu^{2}M_{\mu\beta})L^{\beta\nu}J_{\nu}=-p^{\nu}J_{\nu},$ (33)

where we have used the fact that $L=M^{-1}$ and that ${\cal O}_{0}$ is
transverse (the original massless model is gauge-invariant). We then have
$p^{\mu}\tilde{J}_{\mu}=0$, a relation which will be useful in our
demonstration.

We will apply the electron-electron scattering in order to show the tree-level
equivalence between the two models. For the original gauge-violating model,
the amplitude is represented by the graph depicted in Fig. 1 added to its
crossed diagram. The on-shell amplitude corresponding to Fig. 1 is given by

$F=-iJ{\cal O}^{-1}J,$ (34)

in which, again, we omitted the Lorentz indices.

(150,70)2 (150,50)2 (150,70)(150,50)33 (150,70)(130,90) (170,90)(150,70)
(150,50)(130,30) (170,30)(150,50)

FIG. 1. Electron-electron scattering for the original massive model.

The same calculation, when performed for the dual model, must also include the
contribution of the tree graph coming from the four-fermion vertex. This
contribution is presented graphically in Fig. 2, which does not show the
crossed diagrams.

(80,70)2 (80,50)2 (80,70)(80,50)33 (80,70)(60,90)(100,90)(80,70)
(80,50)(60,30) (100,30)(80,50) (80,20)[t]$(a)$(220,60)(200,90)
(240,90)(220,60) (220,60)2 (220,60)(200,30) (240,30)(220,60) (220,20)[t]$(b)$

FIG. 2. Contributions to the fermion-fermion scattering for the dual model.

We then have, for the dual model,

$\tilde{F}=-i\tilde{J}{\cal O}_{D}^{-1}\tilde{J}-iJ\frac{L}{\mu^{2}}J.$ (35)

Let us calculate the first term. Since the current $\tilde{J}$ is conserved,
we can discard terms of ${\cal O}_{D}^{-1}$ which produces contractions of the
photon momentum with the modified current yielding zero result. We can then
use the identity

$\displaystyle{\cal O}_{D}^{-1}$ $\displaystyle=$ $\displaystyle\theta{\cal
O}^{-1}-\tilde{\cal O}_{o}^{-1}$ (36) $\displaystyle=$ $\displaystyle{\cal
O}^{-1}-\tilde{\cal O}_{o}^{-1}-\omega{\cal O}^{-1}\to{\cal
O}^{-1}-\tilde{\cal O}_{o}^{-1}$

in our calculation, in which we took into account that $\theta+\omega=\eta$.
So, we write

$\displaystyle-i\tilde{J}{\cal O}_{D}^{-1}\tilde{J}$ $\displaystyle=$
$\displaystyle-iJ\left(\eta-\frac{1}{\mu^{2}}L{\cal O}\right)\left({\cal
O}^{-1}-\tilde{\cal O}_{o}^{-1}\right)\left(\eta-\frac{1}{\mu^{2}}{\cal
O}L\right)J$ (37) $\displaystyle=$ $\displaystyle-iJ{\cal
O}^{-1}J+\frac{i}{\mu^{2}}JLJ+iJ\tilde{\cal
O}_{0}^{-1}J-\frac{i}{\mu^{2}}J\tilde{\cal O}_{0}^{-1}{\cal O}LJ+$
$\displaystyle+$ $\displaystyle\frac{i}{\mu^{2}}JLJ-\frac{i}{\mu^{4}}JL{\cal
O}LJ-\frac{i}{\mu^{2}}JL{\cal O}\tilde{\cal
O}_{0}^{-1}J+\frac{i}{\mu^{4}}JL{\cal O}\tilde{\cal O}_{0}^{-1}{\cal O}LJ.$

We now substitute the relation ${\cal O}=\tilde{\cal
O}_{0}+\mu^{2}M+\frac{p^{2}}{\tau}\omega$ into the last two terms of Eq. (37)
to obtain

$\displaystyle-\frac{i}{\mu^{2}}JL{\cal O}\tilde{\cal O}_{0}^{-1}J$
$\displaystyle=$ $\displaystyle-\frac{i}{\mu^{2}}J\left[L+\tilde{\cal
O}_{0}^{-1}+\frac{p^{2}}{\tau}L\omega\tilde{\cal O}_{0}^{-1}\right]J;$ (38)
$\displaystyle\frac{i}{\mu^{4}}JL{\cal O}\tilde{\cal O}_{0}^{-1}{\cal O}LJ$
$\displaystyle=$ $\displaystyle\frac{i}{\mu^{4}}J\left[L{\cal
O}L+\mu^{2}\tilde{\cal O}_{0}^{-1}{\cal O}L+\frac{p^{2}}{\tau}L\omega
L+\frac{p^{2}}{\tau}\mu^{2}L\omega\tilde{\cal
O}_{0}^{-1}+\frac{p^{4}}{\tau^{2}}L\omega\tilde{\cal O}_{0}^{-1}\omega
L\right]J.$ (39)

We then use (27) in the momentum space, i.e. $\tilde{\cal
O}_{0}^{-1}=\tilde{\cal O}_{0\,Tr}^{-1}-\frac{\tau}{p^{2}}\omega$, in the last
term of (39), and the relations $\omega\tilde{\cal O}_{0\,Tr}^{-1}=0$ and
$\omega^{2}=\omega$ to write

$\frac{p^{4}}{\tau^{2}}L\omega\tilde{\cal O}_{0}^{-1}\omega
L=-\frac{p^{2}}{\tau}L\omega L,$ (40)

so that

$\frac{i}{\mu^{4}}JL{\cal O}\tilde{\cal O}_{0}^{-1}{\cal
O}LJ=\frac{i}{\mu^{4}}J\left[L{\cal O}L+\mu^{2}\tilde{\cal O}_{0}^{-1}{\cal
O}L+\frac{p^{2}}{\tau}\mu^{2}L\omega\tilde{\cal O}_{0}^{-1}\right]J.$ (41)

After substituting (38) and (41) in (37), we finally get

$-i\tilde{J}{\cal O}_{D}^{-1}\tilde{J}=-iJ{\cal
O}^{-1}J+\frac{i}{\mu^{2}}JLJ,$ (42)

such that $F=\tilde{F}$. This is a on-shell result, which is maintained for
the crossed diagrams. In the next section we will show with an one-loop
example that this equivalence remains. Besides, we will show this is also true
for the off-shell calculation.

##  V Equivalence at higher loop orders

The equivalence we showed in the previous section is an on-shell result. To
generalize it, in this section, we first demonstrate the on-shell equivalence
for a simple one-loop amplitude and, then, we show the equivalence is valid
off-shell and argue that this implies that the result can be extended to all
loop orders. Let us begin by considering the electron self-energy of some
extended massive electrodynamics and of its corresponding gauge-invariant dual
model. The amplitude for the massive model, which is graphically represented
in Fig. 3, is given by

$i\Sigma(p)=-q^{2}\int^{\Lambda}\frac{d^{4}k}{(2\pi)^{4}}\frac{\gamma^{\mu}(\hbox{$p\\!\\!\\!{/}$}-\hbox{$k\\!\\!\\!{/}$}+m)\gamma^{\nu}}{[(p-k)^{2}-m^{2}]}({\cal
O}^{-1})_{\mu\nu}(k).$ (43)

On the other hand, the fermion self-energy for the dual model receives the
contributions from the graphs (a) and (b) of Fig. 4,

$i\tilde{\Sigma}(p)=i\tilde{\Sigma}_{a}(p)+i\tilde{\Sigma}_{b}(p),$ (44)

for which we have

$i\tilde{\Sigma}_{a}(p)=\textasciiacute-q^{2}\int^{\Lambda}\frac{d^{4}k}{(2\pi)^{4}}\frac{\gamma^{\mu}\left(\eta-\frac{1}{\mu^{2}}L{\cal
O}\right)_{\mu\alpha}(\hbox{$p\\!\\!\\!{/}$}-\hbox{$k\\!\\!\\!{/}$}+m)\left(\eta-\frac{1}{\mu^{2}}{\cal
O}L\right)_{\beta\nu}\gamma^{\nu}}{[(p-k)^{2}-m^{2}]}({\cal
O}^{-1})_{D}^{\alpha\beta}(k)$ (45)

and

$i\tilde{\Sigma}_{b}(p)=\textasciiacute\frac{q^{2}}{\mu^{2}}\int^{\Lambda}\frac{d^{4}k}{(2\pi)^{4}}\frac{\gamma^{\mu}L_{\mu\nu}\gamma^{\nu}(\hbox{$k\\!\\!\\!{/}$}+m)}{[k^{2}-m^{2}]}=\textasciiacute\frac{mq^{2}}{\mu^{2}}\int^{\Lambda}\frac{d^{4}k}{(2\pi)^{4}}\frac{\gamma^{\mu}L_{\mu\nu}\gamma^{\nu}}{[k^{2}-m^{2}]}.$
(46) (180,50)2 (120,50)2 (90,50)(210,50) (140,55)(160,55) (100,55)(110,55)
(190,55)(200,55) (150,50)(20,120,60) (150,50)(30,0,180)48.5
(150,40)[t]$i\Sigma(p)$

FIG. 3. Self-energy diagram for the fermion in the original massive model.

(110,50)2 (50,50)2 (20,50)(140,50) (70,55)(90,55) (30,55)(40,55)
(120,55)(130,55) (80,50)(20,120,60) (80,50)(30,0,180)48.5
(80,40)[t]$(a)\,\,\,\,i\tilde{\Sigma}_{a}(p)$(220,70)(20,0,360)
(220,40)[t]$(b)\,\,\,\,i\tilde{\Sigma}_{b}(p)$(220,70)(15,130,50)
(160,50)(280,50) (220,50)2 (185,55)(195,55) (245,55)(255,55)

FIG. 4. Contributions to the one-loop self-energy of the fermion in the dual
model.

Note that the operator ${\cal O}$ acts on the vector field and, therefore,
depends only on the momentum $k^{\mu}$. Let us consider first the amplitude
$i\tilde{\Sigma}_{a}(p)$. We have to deal with the product of operators

$\left(\eta-\frac{1}{\mu^{2}}L{\cal O}\right)_{\mu\alpha}({\cal
O}^{-1})_{D}^{\alpha\beta}(k)\left(\eta-\frac{1}{\mu^{2}}{\cal
O}L\right)_{\beta\nu}.$ (47)

In the tree level calculation of the last section, we disregarded the term in
$\omega^{\alpha\beta}$ in $({\cal O}^{-1})_{D}^{\alpha\beta}$ due to the
conservation of the current $\tilde{J}$, $k^{\mu}\tilde{J}_{\mu}=0$. Let us
consider this term here and show how the on-shell calculation fixes this
contribution to be zero. The possible non-transverse parts of the vertices are
proportional to $\omega$. Then, since $\omega^{2}=\omega$ because $\omega$ is
a projector, the contribution due to the disregarded term, in the numerator,
involves the factor

$\gamma^{\mu}\omega_{\mu\nu}(\hbox{$p\\!\\!\\!{/}$}-\hbox{$k\\!\\!\\!{/}$}+m)\gamma^{\nu}=\frac{1}{k^{2}}\hbox{$k\\!\\!\\!{/}$}(\hbox{$p\\!\\!\\!{/}$}-\hbox{$k\\!\\!\\!{/}$}+m)\hbox{$k\\!\\!\\!{/}$},$
(48)

and is proportional to

$I=\int^{\Lambda}\frac{d^{4}k}{(2\pi)^{4}}\frac{\hbox{$k\\!\\!\\!{/}$}(\hbox{$p\\!\\!\\!{/}$}-\hbox{$k\\!\\!\\!{/}$}+m)\hbox{$k\\!\\!\\!{/}$}}{k^{2}[(p-k)^{2}-m^{2}]}.$
(49)

In order to calculate $I$, we use Implicit Regularization (IReg) methodology
described in IR-Bruno . Assuming the integral is regularized, we perform
Feynman parametrization to obtain

$I=\int_{0}^{1}\,dx\int^{\Lambda}\frac{d^{4}k}{(2\pi)^{4}}\frac{(\hbox{$k\\!\\!\\!{/}$}+x\hbox{$p\\!\\!\\!{/}$})[(1-x)\hbox{$p\\!\\!\\!{/}$}-\hbox{$k\\!\\!\\!{/}$}+m](\hbox{$k\\!\\!\\!{/}$}+x\hbox{$p\\!\\!\\!{/}$})}{(k^{2}+H^{2})^{2}},$
(50)

with $H^{2}=p^{2}x(1-x)-m^{2}x$. Selecting the even terms in the numerator and
using the on-shell conditions $\bar{u}(p)\hbox{$p\\!\\!\\!{/}$}=\bar{u}(p)m$,
$\hbox{$p\\!\\!\\!{/}$}u(p)=mu(p)$ and $p^{2}=m^{2}$, we get

$\displaystyle I$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\,dx\left\\{-mx\int^{\Lambda}\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{(k^{2}+H^{2})}+2(1-x)p^{\alpha}\gamma^{\beta}\int^{\Lambda}\frac{d^{4}k}{(2\pi)^{4}}\frac{k_{\alpha}k_{\beta}}{(k^{2}+H^{2})^{2}}+\right.$
(51) $\displaystyle+$
$\displaystyle\left.\left[mxH^{2}+m^{3}x^{2}(2-x)\right]\int^{\Lambda}\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{(k^{2}+H^{2})^{2}}\right\\}.$

Now we use the relation

$\int^{\Lambda}\frac{d^{4}k}{(2\pi)^{4}}\frac{k_{\alpha}k_{\beta}}{(k^{2}+H^{2})^{2}}=\frac{\eta_{\alpha\beta}}{2}I_{quad}(-H^{2})$
(52)

with the aim of eliminating the surface terms, which is a condition for gauge
invariance, and the on-shell result $H^{2}=-m^{2}x^{2}$, to obtain

$I=\int_{0}^{1}\,dx\left\\{(1-2x)mI_{quad}(-H^{2})+2m^{3}x^{2}(1-x)I_{log}(-H^{2})\right\\},$
(53)

where we use the definitions of the basic divergences of IReg,

$I_{log}(m^{2})=\int^{\Lambda}\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{(k^{2}-m^{2})^{2}}$
(54)

and

$I_{quad}(m^{2})=\int^{\Lambda}\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{(k^{2}-m^{2})}.$
(55)

Next, the following scale relations are used in order to have the basic
divergences free from the external momentum,

$I_{log}(-H^{2})=I_{log}(m^{2})-\frac{i}{16\pi^{2}}\ln{\left(-\frac{H^{2}}{m^{2}}\right)}$
(56)

and

$I_{quad}(-H^{2})=I_{quad}(m^{2})-(m^{2}+H^{2})I_{log}(m^{2})-\frac{i}{16\pi^{2}}\left[m^{2}+H^{2}-H^{2}\ln{\left(-\frac{H^{2}}{m^{2}}\right)}\right].$
(57)

We get

$\displaystyle I$ $\displaystyle=$ $\displaystyle
I_{quad}(m^{2})\int_{0}^{1}\,dx\,m(1-2x)+I_{log}(m^{2})\int_{0}^{1}\,dx\,m^{3}(1-x)(4x^{2}+x-1)+$
(58) $\displaystyle-$
$\displaystyle\frac{i}{16\pi^{2}}\int_{0}^{1}\,dx\,m^{3}\left[(1-2x)(1-x^{2})+x^{2}(3-4x)\ln{x^{2}}\right]=0.$

This shows that even in the presence of a non-transverse vertex in the dual
model this spurious term does not contribute to physical calculations.
However, as it was already indirectly shown in the previous section, we have

$\left(\eta-\frac{1}{\mu^{2}}L{\cal
O}\right)^{\mu\nu}k_{\nu}=k^{\mu}-\frac{1}{\mu^{2}}\left[L({\cal
O}_{0}+\mu^{2}M)\right]^{\mu\nu}k_{\nu}=0,$ (59)

since ${\cal O}_{0}$ is transverse and $M=L^{-1}$. We then turn our attention
to the Eq. (46). Given that the vertices are transverse, we follow exactly the
same steps of the calculations from Eq. (37) to Eq. (42), to get

$\left(\eta-\frac{1}{\mu^{2}}L{\cal O}\right)_{\mu\alpha}({\cal
O}^{-1})_{D}^{\alpha\beta}(k)\left(\eta-\frac{1}{\mu^{2}}{\cal
O}L\right)_{\beta\nu}=\left({\cal O}^{-1}+\frac{1}{\mu^{2}}L\right)_{\mu\nu}.$
(60)

As a consequence, we can write

$i\tilde{\Sigma}_{a}=-q^{2}\int^{\Lambda}\frac{d^{4}k}{(2\pi)^{4}}\frac{\gamma^{\mu}(\hbox{$p\\!\\!\\!{/}$}-\hbox{$k\\!\\!\\!{/}$}+m)\gamma^{\nu}}{[(p-k)^{2}-m^{2}]}({\cal
O}^{-1})_{\mu\nu}-\frac{q^{2}}{\mu^{2}}\int^{\Lambda}\frac{d^{4}k}{(2\pi)^{4}}\frac{\gamma^{\mu}(\hbox{$p\\!\\!\\!{/}$}-\hbox{$k\\!\\!\\!{/}$}+m)\gamma^{\nu}L_{\mu\nu}}{[(p-k)^{2}-m^{2}]}.$
(61)

The first term is $i\Sigma(p)$ and, in the second, we make the shift $k\to
k+p$ and discard the surface term, as prescribed by IReg. The amplitude reads

$\displaystyle i\tilde{\Sigma}_{a}$ $\displaystyle=$ $\displaystyle
i\Sigma-\frac{q^{2}}{\mu^{2}}\int^{\Lambda}\frac{d^{4}k}{(2\pi)^{4}}\frac{\gamma^{\mu}(-\hbox{$k\\!\\!\\!{/}$}+m)\gamma^{\nu}L_{\mu\nu}}{(k^{2}-m^{2})}$
(62) $\displaystyle=$ $\displaystyle
i\Sigma-\frac{mq^{2}}{\mu^{2}}\int^{\Lambda}\frac{d^{4}k}{(2\pi)^{4}}\frac{\gamma^{\mu}\gamma^{\nu}L_{\mu\nu}}{(k^{2}-m^{2})}=i\Sigma-i\tilde{\Sigma}_{b},$

and we have

$i\tilde{\Sigma}=i\tilde{\Sigma}_{a}+i\tilde{\Sigma}_{b}=i\Sigma.$ (63)

We emphasize that this result takes place off-shell. It is natural to expect
that the result of equation (60) assures that the equivalence between the two
models interrelated through the dualization process can be extended to all
loop orders. This occurs since every tree diagram of the type shown in Figure
1, referring to the original massive model, which composes an amplitude, will
correspond a pair of diagrams of the types shown in Figure 2, referring to the
dual gauge invariant model. As a result of this, we hope that it will be
possible to construct a diagrammatic proof of the equivalence to all loop
orders between the models interconnected by dualization. For example, for a
pair of graphs of $(n-1)$ loop order that are correlate, and that have $l$
external legs of fermions, it is possible, by joining two of them, to
construct a new pair of diagrams of $n$ loop order and $l-2$ external fermion
legs. We expect that an all-loop order equivalence proof can be carried out
using a procedure similar to the subtraction of divergences and subdivergences
with the use of the forests formula mathematically established by the
Bogoliubov– Parasiuk–Hepp–Zimmermann (BPHZ) theorem BPHZ-1 , BPHZ-2 . The
quartic vertices of the gauge-invariant dual model will play the role of
collapsed subgraphs in “counterterms” to cancel out the spurious terms.

##  VI Summary

We studied the gauge embedding methodology for a completely generic gauge-
breaking model of the vector field, which can be defined in a spacetime of an
arbitrary dimension, can include higher derivatives or not, is Lorentz-
invariant or Lorentz-breaking, etc. This methodology, as usual, implies in the
arising of a new gauge invariant model for the vector field. We obtained the
relationship between propagators of these models, which, in particular,
implies that they display common modes, describing therefore the same
particles (earlier a similar study has been performed for a certain LV
extension of the three-dimensional self-dual model and the dual theory
representing itself as some extension of the MCS theory in Fur ). Then, we
demonstrated that the duality also holds for tree-level scattering amplitudes
and one-loop contributions, and, moreover, can be naturally expected to occur
for loop corrections of any order. Effectively, in this paper we not only
presented a prescription for obtaining the dual gauge invariant theory for an
arbitrary gauge-breaking one, but also argued that the duality keeps both at
classical and quantum levels.

Now, let us briefly discuss possible extensions of our studies. First of all,
we hope that it will be possible to construct a diagrammatic proof of the
equivalence to all-loop orders between the models interconnected by
dualization, by using a procedure similar to that prescribed by the forests
formula of the BPHZ theorem. It is also interesting to investigate the
relation between this kind of duality and the equivalence theorems described
in Salam . Besides, we intend, in forthcoming papers, to perform a better
study of duality in noncommutative field theory models, generalizing results
of NC  and investigate the extension of the concept of duality to higher spin
theories, especially, to gravitational ones.

Acknowledgments. Authors are grateful to J. R. Nascimento for important
discussions. A. P. B. S. acknowledges CNPq for financial support. The work by
A. Yu. P. has been partially supported by the CNPq project No. 301562/2019-9.

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# Impact of the Hole Gas on Optically Detected Magnetic Resonance in (Cd,Mn)Te
Based Quantum Wells

A. Łopion<EMAIL_ADDRESS>Institute of Experimental Physics, Faculty
of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland
Institute of Physics, University of Münster, 48149 Münster, Germany  A.
Bogucki  Institute of Experimental Physics, Faculty of Physics, University of
Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland  M. Raczyński  Institute of
Experimental Physics, Faculty of Physics, University of Warsaw, ul. Pasteura
5, 02-093 Warsaw, Poland  Z. Śnioch  Institute of Experimental Physics,
Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw,
Poland  K. E. Połczyńska  Institute of Experimental Physics, Faculty of
Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland  W.
Pacuski  Institute of Experimental Physics, Faculty of Physics, University of
Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland  T. Kazimierczuk  Institute of
Experimental Physics, Faculty of Physics, University of Warsaw, ul. Pasteura
5, 02-093 Warsaw, Poland  A. Golnik  Institute of Experimental Physics,
Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw,
Poland  P. Kossacki  Institute of Experimental Physics, Faculty of Physics,
University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland

###### Abstract

Optically detected magnetic resonance (ODMR) is a useful technique for
studying interactions between local spins (magnetic ions) and carrier gas. We
present the ODMR study of single (Cd,Mn)Te/(Cd,Mg)Te quantum wells (QWs) with
the hole gas. We observe different characteristics of the ODMR signals
obtained simultaneously using the optical signals of neutral and positively
charged exciton. From that, we infer an existence of local fluctuations of
carrier gas density resulting in separate populations of Mn2+ ions. At the
same time, the shape of the ODMR signal contains information about the
temperature of the magnetic ions involved in the absorption of the MW.
Studying it in detail provides even more information about the interactions
with charge carriers. In the QW, two separate ensembles of ions are
thermalized differently in the presence of carriers.

ODMR, QW, spin-carrier interaction

##  I INTRODUCTION

Effective coupling between magnetic ions and charge carriers is crucial for
the realization of electrical control of magnetism, which is the foundation of
the spintronics Dietl and Ohno (2014). Cadmium manganese telluride is a
perfect model system in which carrier density doping is independent of the
concentration of incorporated magnetic ions Gaj and Kossut (2010). In this
work, we use the optically detected magnetic resonance (ODMR) technique as a
very selective tool to study interactions between magnetic ions and charge
carriers in the semimagnetic quantum wells (QWs) with hole gas.

In the QWs being the subject of the research presented here, it is possible to
control the concentration of carriers by using the additional illumination
Kossacki  _et al._ (1999); Maślana  _et al._ (2003); Kossacki (2003); Kossacki
_et al._ (2004); Aku-Leh  _et al._ (2007). Using the illumination above the
energy of the barrier results in increase of holes concentration. While the
sample is illuminated only below the barrier energy – the quantum well is more
neutral. This effect can be observed in both photoluminescence and reflection
measurements. This additional above-bandgap illumination is usually realized
with the excitation laser for the photoluminescence study. During reflectivity
measurements, apart from incident light in the range corresponding to exciton
structures, the sample should be illuminated additionally above the barrier
energy gap – to increase carrier density in the QW. Moreover, carrier
concentration can be estimated based on the ratio of the amplitudes of the
spectral lines corresponding to the neutral X and the charged X+ exciton
present in the optical spectrum measured in a zero magnetic field. That allows
one to simultaneously change and probe the carrier concentration in the
nanostructure on an optical basis.

The ODMR technique is a selective tool to study various properties of the
magnetic ions placed in the crystal lattice. Using ODMR studies, we can
investigate the local strain or spin-lattice relaxation time Baranov  _et al._
(2008); Gurin  _et al._ (2015); Łopion  _et al._ (2022). In this work, we show
that the shape of the ODMR signal can also provide information about the
carrier density interacting with the ions.

##  II Samples and characterization

The studied structures were single 10 nm (Cd, Mn)Te QWs grown by molecular
beam epitaxy (MBE) between symmetrical 50-nm (Cd,Mg)Te barrier layers with Mg
content of about 20 %. They were prepared on GaAs substrate with a thick (5500
nm to 6000 nm) CdTe buffer layer. The Mg content in the barrier layer was
determined by the reflectance measurements from the bandgap energy of the
barrier Waag  _et al._ (1993). The Mn2+ content in the QW material close to
0.3 % was chosen to ensure sufficiently large Zeeman splitting while
preserving narrow excitonic lines. The exact value was confirmed by fitting
the Brillouin function Gaj  _et al._ (1994) to the exciton Zeeman shift
obtained from reflectance measurements in a magnetic field, see Fig.1(b).

Generally, (Cd,Mn)Te/(Cd,Mg)Te QWs are naturally p-type, even without
intentional doping. In such a case, hole gas originates from background doping
of the barrier and/or from the surface states, which can be especially
influential for the QW placed near the surface Maślana  _et al._ (2003).
Indeed, the presented samples contain hole gas in the QW. The sign of the
carriers can be confirmed by using measurements of the photoluminescence
signal in the magnetic field and observation of the spin singlet-triplet
transition Kossacki  _et al._ (2004); Łopion  _et al._ (2020). The same group
of samples was presented in Łopion  _et al._ (2020), where their optical
properties (along with the field of the spin singlet-triplet transition) were
studied as a function of external illumination.

Here, we also used external illumination to increase carrier density in the
samples. In this case, a blue LED (wavelength approx. 405 nm) was used for
providing such additional illumination. The LED was placed in front of the
optical cryostat where the samples were located. With the presence of an
additional illumination above the gap energy, apart from the line of the
neutral exciton X, we can observe an additional line in the reflection
spectrum associated with the positively charged exciton X+, Fig. 1(a). Based
on the relative intensities of the lines in the zero magnetic field X and X+,
the density of holes is estimated to be below 1$\cdot 10^{10}$ cm-2 (without
additional illumination) and 3 $\cdot 10^{10}$ cm-2 (with additional
illumination). In this paper, the two levels of carrier density are denoted as
low and increased, accordingly.

In the magnetic field, both exciton lines show a giant Zeeman shift. One can
consider how introducing carriers into the system affects the thermalization
of magnetic ions excited by microwave radiation in the ODMR experiment. Figure
1(b) shows the measurement of giant Zeeman splitting as a function of the
magnetic field with low carrier density (without additional illumination). The
modified Brillouin function was fitted to the splitting of exciton lines X in
a magnetic field. From that fit, the temperature of the system of polarizing
ions is estimated at 1.73 K. An analogous measurement was carried out for the
increased carrier density – the effective temperature of the ions increased
slightly in this situation, up to 1.81 K.

Figure 1(c) shows reflection spectra measured with increased carrier density
for different magnetic fields. In one circular polarization, the X+ line
disappears. The positively charged exciton consists of two holes with opposite
spins. The optical transition X+ associated with the formation of a hole with
a specific spin depends on the prior existence of a hole with an opposite spin
Huard  _et al._ (2000). Hence, the oscillator strength of the charged exciton
X+ is proportional to the concentration of holes in the opposite spin subband.
By tracking the changes in the relative intensity of the reflection lines X
and X+, it can be seen that in this case, the hole gas is completely polarized
already in the field of about 0.1 T, Fig. 1(d). The following function,
derived from Boltzmann distribution of holes between two spin levels, was
fitted to the intensity changes in the field presented in the graph:

$A_{\pm}(\Delta_{v,Z})=\frac{A_{\mathrm{max}}}{1+\exp\left(\frac{\pm\Delta_{v,Z}}{k_{\mathrm{B}}\mathrm{T}_{\mathrm{h}}}\right)},$
(1)

where $A_{\mathrm{max}}$ is the maximum intensity of a given line,
$\Delta_{v,Z}$ giant Zeeman splitting of the valence band, $T_{\mathrm{h}}$ is
the carriers’ temperature. This results in a temperature of the carriers
slightly higher than the temperature of the ion system without microwave
radiation. The magnetic fields used for ODMR measurements are larger for all
the cases presented in this paper. Thus, it can be assumed that the ions
interact with the polarized hole gas in our experiments.

Figure 1: (a) Difference in the reflection spectrum in zero magnetic fields,
in the absence and the presence of additional illumination with a blue LED.
(b) Zeeman splitting of the neutral exciton line for the unilluminated
situation (black points) with the fitted modified Brillouin function (gray
curve) and the sample subjected to additional illumination (increased carrier
density) - yellow points and yellow curve, respectively. The temperature of
the ion system was a fitting parameter and was set to 1.73 K in the non-
illuminated situation and 1.81 K in the illuminated situation. The black curve
represents the ODMR signal obtained without the blue LED backlight for the
frequency of microwave radiation $f$ = 12.9 GHz, the red curve symbolizes the
modified Brillouin function for the remaining parameters exactly as in the
case of the gray curve but the temperature set to 2.68 K. (c) The dependence
of the reflection spectrum for different magnetic fields with increased
carrier density; (d) Relative intensity of the lines of the neutral X and
positively charged exciton X+ depending on the magnetic field with fitted
relationships – according to the formula (1).

##  III ODMR - setup and method of the measurements

The transitions between Mn2+ energy levels due to absorption of the applied
microwave (MW) radiation lead to a decrease in magnetization. This can be
evidenced by the decrease of the giant Zeeman shift. The effect can also be
expressed in terms of increased effective spin temperature, which will be
discussed later in this work. All the ODMR measurements presented in this work
were conducted in the pulsed mode of MW radiation and optical excitation to
minimize non-resonant thermal drifts. Here, we define the ODMR signal as the
difference in the energy position of the excitonic line in the reflectance
spectrum for microwave and light pulses overlapping (ON) and shifted (OFF).
The duration of the MW pulse is about a few milliseconds. A representative
ODMR signal as a function of magnetic field (ODMR spectrum) for fixed
microwave frequency is present in Fig. 1(b). Careful study of ODMR spectra can
provide information about strain, what was discussed in detail by Bogucki  _et
al._ (2022). Moreover, the ODMR spectrum can provide information about the
impact of interactions with carriers, which is the core of the work presented
in this paper.

The amplitude of the ODMR signal (shift of the excitonic line) corresponds
directly to the temperature of the magnetic ions, which create net
magnetization. After the sample is excited with pulsed microwave radiation of
frequency of 12.9 GHz, a decrease in Zeeman splitting for a resonant magnetic
field of 0.46 T can be observed, see Fig.1(b). This reduction in Zeeman
splitting corresponds to an increase in the temperature of the ion system to
2.7 K as shown by the red curve in the figure.

An important detail is that the measurement conditions are set to avoid non-
resonant signals. It can be evidenced by the fact that outside the resonance
pattern, the position of the excitonic line with and without microwaves is the
same – see inset of Fig. 1(b). In some cases, the offset can be seen. It is
connected to the non-resonant heating of the sample, Fig. 2(a). The overall
temperature increase results from the resonant and non-resonant heating of the
magnetic ions system, Fig. 2(b).

##  IV Shape and position of the ODMR signal

The ODMR spectrum has a specific shape determined directly by the structure of
Mn2+ energy levels. They can be described by the spin Hamiltonian:

Figure 2: (a) ODMR signal power dependence, both amplitude of the signal and
the offset is increasing with the applied MW power (b) Temperature derived
from Zeeman splitting change vs. MW excitation power dependence. Full Giant
Zeeman splitting for this sample, bath temperature, and the magnetic field are
about 3 meV, marked with a red line. The temperature of magnetic ions derived
from Zeeman splitting measured without MW radiation is marked. The MW
excitation power in the ODMR measurements was set to -10 dBm to avoid non-
resonant signals, which are represented by increase of the offset. Figure 3:
(a) Reflection spectrum of this sample measured at temperature $\sim$1.6 K in
magnetic field $B=0.463$ T with microwave radiation resonant for this field
(12.9 GHz) at in the "ON" and "OFF" states, the change of position is visible
on both visible exciton complexes; (b) The ODMR signal as the difference in
the positions of the energy lines observed in the reflection spectrum in the
presence and absence of microwave radiation at the frequency $f$ = 12.9 GHz
depending on the value of the applied external magnetic field; the signal
observed on the line of the charged exciton CX is shifted towards lower
magnetic fields compared to the signal obtained on the neutral exciton X. o
$\Delta B_{\mathrm{res}}$ = 14 mT

$H_{\mathrm{Mn^{2+}}}=g_{Mn}\mu_{\mathrm{B}}\boldsymbol{B}\cdot\widehat{\boldsymbol{S}}+A\widehat{\boldsymbol{I}}\cdot\widehat{\boldsymbol{S}}\\\
+\frac{a}{6}\left(\widehat{S}_{x}^{4}+\widehat{S}_{y}^{4}+\widehat{S}_{z}^{4}-\frac{S(S+1)(3S^{2}+3S-1)}{5}\right)\\\
+D\left(\widehat{S}_{z}^{2}-\frac{S(S+1)}{3}\right),$ (2)

The first term of the Hamiltonian describes the Zeeman splitting of energy
levels in a magnetic field due to the electrons d-shell spin ($S$).
$g_{\text{Mn}}$ is the g-factor of the manganese ion. The second term
describes hyperfine splitting, where $\widehat{\boldsymbol{I}}$ denotes the
manganese ion nuclear spin operators. Mn2+ ion has electronic spin S = 5/2 and
nuclear spin I = 5/2. The third term for the level-splitting part of the
Hamiltonian due to the placement of the ion in the crystal field with the
symmetry Td.

The last term refers to zero field splitting related to lowering the symmetry
of the crystal and corresponds to the uniaxial stress associated with
deformation in the $z$ direction. The stress-induced parameter of the axial
component of the crystal field $D$ is a parameter characteristic for a given
sample, resulting from the parameters of growth, composition and thickness of
individual layers. $D$ can be determined from the mismatch of the material
lattice constants of the quantum well $a_{\mathrm{QW}}$ and the buffer
$a_{\mathrm{buf}}$ by determining the strain
$\varepsilon=\frac{a_{\mathrm{QW}}-a_{\mathrm{buf}}}{a_{\mathrm{QW}}}$.
Finally, the $D$ parameter Qazzaz  _et al._ (1995) can be determined:
$D=-\frac{3}{2}G_{11}\left(1+\frac{2C_{12}}{C_{11}}\right)\varepsilon$, where
$G_{11}$ = $72{.}2$ neV and $\frac{2C_{12}}{C_{11}}=0.7$Bogucki  _et al._
(2022); Greenough and Palmer (1973).

The electronic and nuclear spin coupling for S and I equal to 5/2 results in
36 energetic states of Mn2+ ion. In a zero magnetic field, the splitting of
those states corresponds mainly to the strain ($D$ parameter). For low
magnetic fields ($B<0.05$ T), the evolution of the states is complicated, and
the eigenstates of the Hamiltonian are still not eigenstates of the $S_{z}$
operator. The Zeeman term dominates Hamiltonian for higher magnetic fields,
and the energy levels begin to have a well-defined spin projection to magnetic
field quantization $z$ direction. Considering the selection rules for magnetic
dipole transitions ($\Delta S_{z}=\pm 1$, $\Delta I_{z}=0$), the ODMR spectrum
contains 6 sets with 5 transitions in each.

The shape of the ODMR spectrum depends on the separation and the population of
energy levels participating in microwave absorption. The intensity of a single
transition corresponds to the occupation of the states between which the
transition occurs—thus, it is strictly connected to the temperature of the
magnetic ions. Qualitatively, the features observed in the higher magnetic
field correspond to the transitions between lower energy levels than features
observed at the lower magnetic field.

For situations without additional illumination the carrier density is low and
the neutral exciton transition dominates the reflectance spectrum. When
additional illumination is set on the sample the carrier density is increased.
That is evidenced in the reflectance spectrum, which shows another excitonic
transition in the energy below X: charged exciton (X+). Both excitonic lines
observed in the reflectance are sensitive to microwave radiation, Fig. 3(a).
We can see that both X and CX lines move towards lower energies when the light
and microwave pulses coincide in time ("ON" state) relative to the position of
the lines when the light and microwave pulses are significantly temporarily
separated ("OFF" state).

Figure 3(b) shows in detail the ODMR signal measured as a shift of both X and
CX lines. In the ODMR signal measured at the neutral exciton line, many narrow
features can be seen related to the transitions between different states of
the manganese ion split by the magnetic field, hyperfine interaction and
strain. At the same time, we can notice that such a structure is not visible
in the signal monitored on the charged exciton line and the shape of the
signal seems to be smoother. Moreover, the maximum of the ODMR signal for CX
line is shifted relative to the maximum of the signal measured on the neutral
exciton towards lower magnetic fields. These differences suggest that the two
lines probe different populations of Mn2+ ions, exhibiting different
paramagnetic resonance broadenings and shifts. The shift of the resonance
observed on the charged state line results from interactions between magnetic
ions and charge carriers. The $\Delta B_{\mathrm{res}}$ is connected to the
density of carriers. Here, and in a similar situation in the following parts
of the paper, the center of the resonance for the excitonic state is
calculated as the center of mass of the ODMR spectrum (ODMR signal vs.
magnetic field). That approach gives clear information about the position of
the resonance when it is more complicated in shape, however it is important
for the calculation to subtract the offset of the ODMR spectrum properly. A
very similar result can be obtained by simply fitting a peak function (e.g.,
Gaussian) to the data.

It should be noted here that both signals are observed when measuring the
optical spectrum simultaneously. The ODMR signal obtained by observing the X
line appears to be unaffected by the interaction with the charge carriers,
while that obtained by observing the CX line is apparently modified. This
result suggests that the absorption associated with the neutral exciton occurs
in different, spatially separated regions than the absorption associated with
the charged exciton. At the same time the spot of light used for reflectance
measurements covers and probes both types of areas of different local carrier
density. In the scale above the laser spot size ($\sim$100 $\mu m$), the
sample is rather homogenous in terms of the optical response. That suggests
the discussed inhomogeneities of the carrier density distribution are
relatively small, below the spatial resolution of this optical measurement.

##  V Estimation of effective temperature of magnetic ions Mn2+ and carriers

Figure 4: (a) Low-temperature (helium bath temperature $T$ = 1.6 K) reflection
spectrum measured in the presence of additional backlight and microwave
radiation (a microwave pulse overlapped with a light pulse) with frequency $f$
= 12.9 GHz in a resonant magnetic field $B$ = 0.462 T measured in two circular
polarizations; (b) Differential ODMR signal for measurement (a), (a,b) were
corrected for giant Zeeman shift; (c) Curves representing the normalized
signal of a charged exciton X+ in a magnetic field, according to the formula
(1) – carrier polarizability, for two different carrier gas temperatures 2 K
and 6 K; (d) Dependence of the exciton signal X+ on the temperature of the
hole gas, according to the formula (1)

In order to elucidate the effect of carrier concentration on the ODMR signal
obtained on both excitonic features, we exploited possibility to control
carrier gas density with additional illumination. The question can be asked
whether applying microwave radiation to the sample affects the temperature of
the carrier gas, and whether it does not depolarize the carriers by heating
them in the typical ODMR measurement. Figure 4(a) shows the spectra of
reflectivity in the presence of microwaves ($f$ = 12.9 GHz) in a resonant
magnetic field for two different circular polarizations, corrected for the
giant Zeeman shift, so as to emphasize the difference in the intensity of the
exciton structures. It can be seen that in the $\sigma^{-}$ polarization there
are two excitonic features – while in the opposite polarization we do not
observe the lines of the charged exciton X+. The lack of reflection of the
exciton charged X+ in this polarization also confirms the lack of a
differential signal for the corresponding wavelength 4(b). These results prove
that even in the presence of microwave radiation, the carrier gas is not
heated sufficiently to depolarize its spin in the $B$ = 0.462 T field. This
gives us an upper estimate for the temperature of the carrier gas
$T_{\mathrm{h}}$. Assuming that the uncertainty of the reflection measurement
determines the minimum observable intensity X+ (gray line in the Figure 4(c),
we can use the formula (1) to find that the temperature of holes under the
influence of microwave radiation $T_{\mathrm{h}}<6$ K, Fig. 4(d). Whether we
are dealing with a gas that is fully or only partially polarized obviously
also depends on the value of the magnetic field. However, in most of the
measurements presented in this paper the smallest magnetic fields of the ODMR
measurement had a value of about 0.4 - 0.5 T. Hence, it seems reasonable to
assume that in all cases, we are dealing with a polarized hole gas, even when
the carrier gas temperature is increased as a result of the sample being under
microwave radiation.

As discussed in the Introduction, the most straightforward way to determine
the temperature of the Mn subsystem is fitting the Brillouin function to the
observed Zeeman shift. However, we could also employ an alternative approach
based on the results of the ODMR measurements. This method relies on the
observation that the shape of the ODMR spectrum obtained by observing the
neutral exciton with lower and higher carrier density is slightly different,
but the position of the sharp transition lines is the same. As such they can
be therefore used to extract the temperature of the system. Quantitatively,
the value of the temperature was extracted from numerical calculations of the
whole ODMR spectrum. The figure 5(a) shows the comparison between measured and
calculated ODMR spectra for the magnetic field perpendicular to the plane of
the QW and two different illumination conditions, which corresponds to
different temperatures – it can be seen that they quite well reflect the
differences observed in the spectra measured with low and increased carrier
gas concentration.

As it was mentioned before, the strict indicator of the temperature is the
shape of the ODMR spectrum. As the spectrum itself can be complex, we propose
using the measure given by the data central moments. The first central moment
(M1) provides information about the center of mass of the spectrum. The third
one (M3) is connected to the asymmetry of the spectrum (regarding M1). Both of
them are changing with the temperature as the occupation of the manganese
levels and then transitions are changing the intensities, Fig. 5(b). The exact
values of M1 and M3 are changing with the properties of the sample (strain,
composition), however the quantitative behavior with temperature is the same.
For each sample and the given set of data it is much easier to calculate the
moments and estimate the temperature of the magnetic ions on their basis.

Figure 5: (a) ODMR signal as a shift of neutral exciton, with MW frequency $f$
= 12.9 GHz measured without (X) and with increased carrier density (X’),
helium bath temperature 1.6 K; numerical calculation of ODMR spectra for $D$ =
780 neV, $C$ =0 and Mn ion temperature $T$ = 2.68 K i $T$ = 7.5 K; (b) black
connected points - change of first (M1) and third (M3) central moment in the
function of Mn ion temperature calculated from numerical calculated ODMR
spectra, blue and red lines represent M1 and M3 calculated for data measured
for low (blue) and increased (red) carrier density and corresponding
temperatures of Mn ions. (c) Center of the resonance for neutral exciton
measured without (black spheres) and with increased carrier density (blue
spheres) for different angle of external magnetic field, colored curves mark
the numerically calculated evolution corresponding to different temperatures
of Mn ions.

##  VI Dependence on magnetic field angle

To further test our interpretation, we also performed measurements with
varying angles of the external magnetic field. The shape of the ODMR spectrum
obtained on the neutral exciton line was examined depending on the angle of
the external magnetic field $\theta$ with respect to growth axis. The zero
angle corresponds to the measurement in the Faraday configuration (magnetic
field perpendicular to the QW plane). The results obtained for the situation
with low (X) and increased carrier density (X’) are presented in the figure
5(c). Due to the increased carrier density, a significant change in the
thermal distribution of manganese ion states leads to changes in the shape of
the ODMR signal and an effective shift of their center. The dependence of the
resonance field for two different temperatures, 2.68 K and 7.5 K, determined
on the basis of numerically calculated ODMR spectra, was plotted on the graph
5(b) obtaining a very good agreement with the experimental points. Such
changes in the resonance field depending on the $\theta$ angle confirm that
the temperatures of the ion system in the low and increased carrier density
conditions are significantly different. Measurement of the ODMR spectrum and
its center as a function of a magnetic field angle confirms the accuracy of
the used model and the high temperature of the magnetic ions under MW
radiation and with increased carrier density.

Figure 6: (a) Dependence of the ODMR signal for microwave frequency 12.9 GHz
observed by examining the position of the neutral (X) and charged (CX) exciton
lines on the magnetic field for different field angles relative to the UW1467
sample plane – and simulated spectra ODMR for parameters: $T$ = 7.5 K, $C$ = 0
(X) and $C$ = -3240 neV (CX) , $D$ = 780 neV. The constant $C$ is the carrier-
ion interaction constant. (b) Positions of the ODMR signal maxima for the
charged and neutral exciton depending on the angle of the magnetic field
relative to the sample plane (0∘ means Faraday configuration, 90∘ – Voigt);
(c) The distance of the resonance magnetic fields measured by observing the
lines X and CX (Knight shift) depending on the angle of the magnetic field;
(d) ODMR signal width measured by observing X and CX, black curve represents
simulated values.

Figure 6(a) shows the ODMR signals depending on the magnetic field and for
different angles of the field relative to the plane of the sample for charged
exciton. Both the ODMR signal obtained by observing the X and CX lines change
the location of the center of gravity depending on the direction of the
magnetic field, fig. 6(b).

For the interactions with the carriers (hole gas), a simple Heisenberg model
was used, adding to the Hamiltonian mean field exchange term:

$H_{p-d}=-\beta
n|\varphi(z)|^{2}\langle\hat{s}\rangle\cdot\hat{S}_{Mn}=C\langle\hat{s}\rangle\cdot\hat{S}_{Mn}$
(3)

where $\beta$ is exchange integral between hole and Mn ion, $n$ – carrier
density, $|\varphi(z)|^{2}$ \- distribution of carriers along growth axis $z$.
Here, we make a justified assumption that the magnetic ions in the QW are
localized and interacting with the local density of delocalized carriers –
$n|\varphi(z)|^{2}$. It is noteworthy that in this model we assume that Mn2+
ion is coupled with the fully polarized hole gas of a certain density, not to
the single, localized carrier. Besides the aforementioned polarizability
measurements, this assumption is also justified because we do not see any ODMR
signal shifted towards higher magnetic fields. Such signal could be a result
of the interaction with the hole (i.e. localized one) anti-parallel to the
ground spin state. It is not our case, therefore, we use mean-field
approximation, and $\langle\hat{s}\rangle$ denotes the average over the whole
gas of holes. The shift towards lower magnetic fields is connected only to the
effective magnetic field derived from polarized holes.

In the case of a magnetic field perpendicular to the plane of the well
($\theta=0^{\circ}$), the shift of the resonances measured between both
exciton complexes is about 23 mT – which is reproduced for the constant $C$ =
-3240 neV. This value, according to the equation (3), corresponds to a hole
gas concentration of 4.05 $\cdot$1010 cm-2. From the fitting of excitonic
lines of neutral and charged state in zero-field reflectance spectrum we
estimated based on Kossacki  _et al._ , 1999 and Kossacki  _et al._ , 2004 the
carrier concentration to be about 3 $\cdot 10^{10}$ cm-2 (with additional
illumination). This value obtained from the Knight shift is slightly higher
than estimated from reflection spectra. Assuming that the carrier density
obtained from reflectance spectrum is a mean over the laser spot area and at
the same time, the Knight shift is a local result for areas with the excess of
carriers we can estimate what is the percentage of those areas – it is about
75%.

##  VII Discussion

Even for such a simple toy model, the agreement with the experimental data is
satisfactory. One of the possible reasons for the remaining deviations is the
strong assumption of the equal temperature of the magnetic ions probed by X
and CX, or the parameters remain constant with changing magnetic field angle.
The exchange integral in this system is isotropic. However, the $C$ parameter
contains the local density of the carrier wavefunction, which, in contrast,
can depend on the $\theta$ angle. We consider a uniform distribution of
charges in the plane of the well. At the same time, the literature data Golnik
_et al._ (2004); Maślana  _et al._ (2006); Łopion  _et al._ (2020) and the
fact of simultaneous observation of different ODMR signals on the X and CX
lines show that this is not true. Most likely, in the plane of the well, there
are areas with an excess of carriers, from which the optical signal comes from
the recombination of charged excitons, and areas with a lower carrier density.
In this case, the size of the areas is submicron and in the measurement of
photoluminescence or reflection in the macro scale it is impossible to
separate them.

The observation about the spatial separation of X and CX goes along with the
long-running discussion on that topic. Significant works, such as Moody  _et
al._ , 2014; Brunhes  _et al._ , 1999, have confirmed the correlations between
charged and neutral excitons, which is serious evidence for the lack of (or
significantly small) separation between the two. However, the level of the
separation or its absence can be connected to the (in-) homogeneity of the
Coulomb potential in the QW layer. This Coulomb disorder may be a result of
the inhomogeneity of the surface of the sample for the systems with low
separation between QW and the surface Łopion  _et al._ (2020); Maślana  _et
al._ (2006) or inhomogeneity of the QW layer itself. In our case we believe it
is a first case scenario – as we observe the possibility of change of the
Coulomb disorder with changing the additional illumination and screening the
surface potential.

In the next part we discuss the temperatures of the subsystems having a part
in the spin relaxation processes observed with the ODMR signal.

When the density of hole gas is low, the temperature of the magnetic ions
derived from the giant Zeeman curve is similar to the temperature that can be
derived from the ODMR measurement – both from the intensity of the single MW
transitions and the center of mass of the ODMR spectrum vs. magnetic field
angle.

For increased carrier density, the situation becomes more complex. In such a
case the two different transitions visible the reflectance spectrum – neutral
(X) and positively charged exciton (CX) give different characteristics of ODMR
signal. In this case, we assume that the optical signal from charged and
neutral exciton comes from spatially separated (in plane of QW) regions
differing in the carrier density. In the regions with the higher density of
carriers, where the optical signal of charged exciton comes from, the ions
experience an effective exchange field of polarized hole gas, inducing ODMR
signals shifting towards lower external magnetic field. However, even the ions
probing the "neutral" regions experience interactions with the increased
density of carriers. In this case, the change of the ODMR spectrum is fine but
significant. It becomes even more clear when we observe the behavior of the
center of mass of the ODMR spectrum with magnetic field angle. The temperature
of the magnetic ions estimated based on the ODMR measurements seems to be
higher than the temperature derived from giant Zeeman splitting in the same
conditions. This means that these two measurements (ODMR and Zeeman splitting)
examine different groups of ions. Part of the ions absorbing radiation
microwave has a greater contribution to the ODMR spectrum and, simultaneously,
a higher temperature than other ions, contributing mainly to building
magnetization in the sample. In general, all ions in the QW should absorb
microwave radiation the same way. However, there may be differences in rates
of energy and spin transfer. A less effectively relaxing population would be
favored in creating the ODMR signal. The others, due to increased interactions
between them, would be less active in resonance studies, as they relax much
faster and their lines are broadened.

| Brillouin | Brillouin | ODMR | hole polarizability ($T_{\mathrm{h}})$
---|---|---|---|---
| ($T_{\mathrm{Mn}}^{\mathrm{B}}$) | ODMR ($\Delta$$T_{\mathrm{Mn}}^{\mathrm{B}}$) | shape | (without and with MW)
Low hole density | 1.73 $\pm$ 0.02 K | 2.68 $\pm$ 0.05 K | 2.68 K | -
Increased hole density | 1.81 $\pm$ 0.01 K |  $\sim$ 2.7 K | 7.5 K | 1.82/<6 K
Table 1: Table summarizing the temperatures of different subsystems in the QW
measured or estimated with different methods. The temperature of the helium
bath in all cases was about 1.6 K and in the case of pulsed microwave
excitation it slightly increased by about 5-10%. In order to be able to
compare possibly similar measurement conditions in the table the microwave
excitation means excitation with $f$ = 12.9 GHz, $P_{\mathrm{MW}}$ = $-10$
dBm. As the power emitted by the antenna and absorbed by the sample may differ
for different frequencies, it is difficult to compare carrier or ion
temperatures in such cases.

Previous works have shown that different subsystems present in the sample –
manganese ions, charge carriers, and crystal lattice –can vary in effective
temperature Scherbakov  _et al._ (2001); König  _et al._ (2000); Scherbakov
_et al._ (1999). Here we deal with even more complex situation in which
manganese ions do not form homogenous system. A summary of the temperature
results obtained in this paper is provided in the table 1.

If the system is not exposed to microwave radiation, the series of
temperatures is as follows: the lowest temperature is the temperature of the
helium bath $\sim$1.6 K, then the temperature of the ions
$T^{B}_{\mathrm{Mn}}$ obtained from Brillouin function fitted to magnetization
in the magnetic field (without microwaves), the temperature of the holes
$T_{\mathrm{h}}$ from the measurement of polarizability is only slightly
higher Huard  _et al._ (2000).

With increased carrier density, the temperatures concluded from the ODMR
measurements – also from the measurements with different magnetic field angles
– are higher than the temperatures derived from the giant Zeeman splitting
with the same conditions. As mentioned before, presumably, a group of ions in
the sample is somehow separated and needs a longer time to relax (reorient
itself in the magnetic field) after being excited by a microwave pulse, but
also to transfer energy in the form of temperature. This population would thus
be more active in magnetic resonance signals. The rest would be less active in
resonance studies, as they return to equilibrium much faster and their lines
are significantly broadened. Nevertheless, this difference is observed only in
the presence of carriers. For increased carrier density the ions which are
contributing in ODMR signal thermalize in the ambient of hot carriers, which
are excited by MW radiation non-resonantly.

The influence of interaction with carriers on the ODMR resonance field
depending on the angle of the magnetic field has already been analyzed in the
works Dietl  _et al._ (2000); Gurin  _et al._ (2015). In those works, the
resonance field increases with increasing magnetic angle, which was explained
by the interaction of a single, localized carrier with more than one Mn2+ ion
– which may be related to the fact that the concentration of manganese ions in
these samples was higher $\sim$1 -2% and the system was probably already in
the magnetic polaron regime (Chapter 7 in Ref.Gaj and Kossut, 2010,Godejohann
_et al._ (2022); Kavokin  _et al._ (1999)). In the case of the results
presented here, the concentration of manganese ions is lower $\sim$0.3%, and
the model of delocalized carrier gas interacting with a multitude of Mn2+ ions
is apparently more appropriate here Haury  _et al._ (1997); Dietl  _et al._
(2000); Kossacki  _et al._ (2002); Boukari  _et al._ (2002).

##  VIII Summary

In this work, we showed that the ODMR spectra can provide information about
the temperature of the magnetic ions and interacting carrier density.

We simultaneously observe the ODMR signal as a change of the neutral and
charged excitonic lines. Those two have different characteristics, proving the
inhomogeneity of the hole gas distribution in the QW plane.

Moreover, we found out that in the presence of holes, the temperature of the
magnetic ions contributing in the ODMR signal is slightly different than the
temperature of the ions contributing in building net magnetization. That
suggests existence of two separated ensembles of magnetic ions, which are
thermalized differently in the presence of carriers, which are excited by
microwave radiation.

##  IX Acknowledgements

This work was supported by the Polish National Science Centre under decisions
DEC-2016/23/B/ST3/03437, DEC-2021/41/B/ST3/04183, DEC-2020/38/E/ST3/00364,
DEC-2020/39/B/ST3/03251.

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