diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzooje" "b/data_all_eng_slimpj/shuffled/split2/finalzzooje" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzooje" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nThe ability estimate position is key to most, if not all, robotics application involving mobility. In this paper we focus on the problem of visual odometry (VO), i.e. relative motion estimation based on visual information. This is the corner stone in vision-based SLAM systems, which will be the setting we will demonstrate our work in. As in our previous work, \\cite{GCN}, we estimate the motion using only an RGB-D sensor, and our target platform is a drone operating in an indoor environment. The RGB-D sensor makes scale directly observable without the need for visual-inertial integration or the computational cost of inferring depth using a neural network as in \\cite{Left-right,Single-stereo,SFMLearner}. This increases robustness, which is a key property when used on a drone and in particular for indoor environments where the margin for error is small and the there is typically less obvious textures than in outdoor environments. Our method is designed to be applicable to any system simply by adding an RGB-D sensor, without the need for complicated calibration and synchronisation routines with additional sensors, such as cameras or IMUs. Fusion can instead take place at a lower rate and with less need for precise timing, which, for example, makes integration with the flight control system of a drone simpler.\n\n\t\\begin{figure}\n\t \\centering\n\t \\resizebox{\\hsize}{!}{\n\t\t\\includegraphics[width=0.5\\textwidth]{uav.pdf}\n\t\t}\n\t\t\\caption{The top figure shows our drone preforming position hold using GCN-SLAM. The figures below show the intermediate output for comparison of binary features, ORB and GCNv2, in ORB-SLAM2 and GCN-SLAM respectively. GCNv2 (left) tends to predict more repeatable and evenly distributed features compared with ORB (right.)}\n\t\t\\label{fig:digest}\n\t\\end{figure}\n\nLike many other areas of research, there is a trend in SLAM to investigate deep learning-based methods. In \\cite{SuperPoint} a keypoint detector and descriptor called SuperPoint is presented. Experimental results show that this CNN-based method has more distinctive descriptors than classical descriptors such as SIFT, and a detector on par with them. However, evaluation on homography estimation in \\cite{SuperPoint} shows that it is only working on par with other keypoint extractors, classical or learning-based. In our previous work~\\cite{GCN}, presented before SuperPoint, we introduced the Geometric Correspondence Network, GCN, specifically tailored for producing keypoints for camera motion estimation, achieving better accuracy than classical methods. However, due to the computation and structure limitation of GCN, it is difficult to achieve real-time performance in a fully-operational system, e.g. on board a drone. Both keypoint extraction and matching are computationally too expensive. Furthermore, due to the multi-frame matching setup, integrating GCN into existing SLAM systems becomes non-trivial.\n\nIn this paper we introduce GCNv2, based on the conclusions from~\\cite{GCN}, to improve computational efficiency while still maintaining the high precision of GCN. Our contributions are:\n\\begin{itemize}\n\\item GCNv2 maintains the accuracy of GCN, providing significant improvements in motion estimation in comparison to related deep learning-based feature extraction methods as well as classical methods. \n\\item The inference of GCNv2 can run on embedded low-power hardware, such as a Jetson TX2, compared to GCN which requires a desktop GPU for real-time inference.\n\\item We design GCNv2 to have the same descriptor format as the ORB feature so that it can be drop-in substituted as the keypoint extractor in SLAM systems like ORB-SLAM2~\\cite{ORBSLAM2-TRO17} or SVO2~\\cite{SVO-TRO17}.\n\\item We demonstrate the effectiveness and robustness of our work by using GCN-SLAM\\footnote{Built on ORB-SLAM2 with ORB substituted for GCNv2} on a real drone for control, and show that it handles situations where ORB-SLAM2 fails.\n\\end{itemize}\n\n\\section{Related Work}\n\\label{sec:related}\nIn this section we cover related work in two areas. First VO and SLAM methods are covered and then we focus specifically on work on deep learning-based methods for image correspondence.\n\n\\subsection{VO and SLAM}\nIn direct methods for VO and SLAM, motion is estimated by aligning frames based directly on the pixel intensities, with~\\cite{Comport07} being an early example. DVO (Direct Visual Odometry), presented in~\\cite{DVO-Kerl13}, adds a pose graph to reduce the error. DSO~\\cite{DSO} is a direct and sparse method that adds joint optimisation of all model parameters. An alternative to the frame-to-frame matching is to match each new frame to a volumetric representation as in KinectFusion~\\cite{KinectFusion}, Kintinous~\\cite{Kintinous-IJRR14} and ElasticFusion~\\cite{ElasticFusion}. \n\nIn indirect methods, the first step in a typical pipeline is to extract keypoints, which are then matched to previous frames to estimate the motion. The matching is based on the keypoint descriptors and geometric constraints. The state-of-the-art in this category is still defined by ORB-SLAM2~\\cite{ORBSLAM-TRO15, ORBSLAM2-TRO17}. The ORB descriptor is a binary vector allowing high-performance matching. \n\nSomewhere between direct and indirect methods we find the semi-direct approaches. SVO2~\\cite{SVO-TRO17} is a sparse method in this category, and can run at hundreds of Hertz. There are also semi-dense methods, in which category LSD-SLAM~\\cite{LSD-SLAM} was one of the first. RGBDTAM~\\cite{RGBDTAM} combines both semi-dense photometric and dense geometric errors for pose estimation.\n\nThere are a number of recent deep learning-based mapping systems like \\cite{CNN-SLAM, DVSO}. The focus in these methods is deep learning-based single view depth estimation to reduce the scale drift inherent in monocular systems. CNN-SLAM~\\cite{CNN-SLAM} feeds the depth into LSD-SLAM. In DVSO~\\cite{DeepVO}, depth is predicted in a similar way to \\cite{Left-right}, using a virtual stereo view. CodeSLAM \\cite{code-slam} learns an optimizable representation from conditioned auto-encoding for 3D reconstruction. In S2D~\\cite{S2D}, we build on DSO~\\cite{DSO} and exploit both depth and normals predicted by a jointly optimised CNN. Some work on end-to-end training for motion estimation also exist. Image reconstruction loss is used for unsupervised learning in~\\cite{SFMLearner, Undeepvo}. However, geometry-based optimization methods still outperform end-to-end systems as shown in~\\cite{DeepVO}.\n\n\\subsection{Deep Correspondence Matching}\nThere is a an abundance of recent works that deploy variants of metric learning for training deep features for finding image correspondences~\\cite{feat_ref1, feat_ref2, feat_ref3, feat_ref4, feat_ref5, feat_ref6, LIFT, UCN, SuperPoint}.\nWorks in~\\cite{cd_ref1, cd_ref2} focus on improving learning-based detection with better invariances. \nAimed at a different aspect, \\cite{cd_ref3, cd_ref4, cd_ref5} use synthetic samples generated in a self-supervised manner to improve general feature matching.\n\nAmong the aforementioned methods, LIFT~\\cite{LIFT} in particular uses a patch-based method to perform both keypoint detection and descriptor extraction. SuperPoint~\\cite{SuperPoint} predicts the keypoints and descriptors using a single network together with the self-supervised strategy in~\\cite{cd_ref5}. Notably, \\cite{SuperPoint} shows that \\cite{SuperPoint, LIFT, UCN} work on par with classical methods like SIFT for motion estimation.\n\nIn GCN~\\cite{GCN}, we show that by learning keypoints and descriptors specifically targeting motion estimation, performance is improved -- contrary to what is reported for other more general deep learning-based keypoint extractor systems~\\cite{SuperPoint,UCN}. In this paper we introduce an extension to GCN, GCNv2. We demonstrate the applicability of these keypoints for SLAM and build on ORB-SLAM2 as it offers a comprehensive multi-threaded state-of-the-art indirect SLAM system with support for monocular as well as RGB-D cameras. ORB-SLAM2 complements the tracking front-end with a back-end that does both pose graph optimisation using g2o~\\cite{g2o} and loop closure detection using a binary bag of words representation~\\cite{Galves-BoW-TRO12}. To simplify this integration, we design the GCNv2 descriptor to have the same format as that of ORB.\n\n\\section{Geometric Correspondence Network}\n\nIn this section, we present the design of GCNv2, aimed at making it suitable for real-time SLAM applications running on embedded hardware rather than a powerful desktop computer. We first introduce the modifications to the network structure. Then, we present the training scheme for the binarized feature descriptor and keypoint detector.\n\n\\subsection{Network Structure}\nThe original GCN structure, proposed in~\\cite{GCN}, consists of two major parts: an FCN~\\cite{FCN} structure with a ResNet-50 backbone and a bidirectional convolutional network. The FCN is adopted for dense feature extraction and the bidirectional network is used for temporal keypoint prediction. Although impressive tracking performance has been achieved using GCN compared with existing methods, GCN has practical limitations when it comes to the use in a real-time SLAM system with limited hardware. \nThe network architecture requires relatively powerful computational hardware which renders it unable to run in real-time on board e.g. the Jetson TX2 used on our drone.\nFurthermore, the bidirectional structure requires matching between two or more frames at the same time. This significantly increases computational complexity for a window-based SLAM method, since keyframes then are updated dynamically based on the current camera position.\n\nInspired by SuperPoint~\\cite{SuperPoint}, which uses a simple structure to perform detection using a single frame,\nwe deploy a network with even fewer parameters and working on a lower scale than SuperPoint. Intuitively, the network performs an individual prediction for each grid cell of size $16\\times16$ pixels in the original image. \nIn GCNv2 all the pooling layers are replaced by convolutions with kernel size $4\\times4$, stride ${2}$ and padding $1$. As in SuperPoint, the network takes $320\\times240$ images as input. This is also the image size we used later for SLAM. Further details on the GCNv2 network specifics can be found in our publicly available source code\\footnote{\\url{https:\/\/github.com\/jiexiong2016\/GCNv2_SLAM}}. \n\nGCN-SLAM with GCNv2 runs at 20 Hz on Jetson TX2 and runs at around 80 Hz on a laptop with Intel i7-7700HQ and mobile version NVIDIA 1070. To achieve even higher frame rates we introduce a smaller version of GCNv2, called GCNv2-tiny, where we reduce the number of feature maps by half from the second layer and onward. GCN-SLAM with GCNv2-tiny runs at 40 Hz on a TX2 and is therefore well-suited for deployment on a drone.\n\n\n\\subsection{Feature Extractor}\n\n\\textbf{Binarized Descriptor}\nWe trained the features of GCNv2 to be binary for accelerating the matching procedure and to match those of ORB. To binarize the features, we add a binary activation layer on top the final output. It is essentially a hard sign function and is therefore not differentiable. The challenge is how to back-propagate the loss properly through this layer of the network. We used the method proposed in~\\cite{Binarization}. The binary activation layer can be written as follows:\n\\begin{equation} \\label{eq:bin_activation}\n\\begin{split}\n\\text{Forward: }\\,\\bm{b}(\\bm{x}) &= \\operatorname{sign}(\\bm{f}(\\bm{x})) = \n\\left\\{\n\\begin{array}{c l}\t\n +1 & \\bm{f}(\\bm{x}) \\ge 0\\\\\n -1 & \\text{otherwise}\n\\end{array}\\right. \\\\\n\\text{Backward: }\\,\\frac{\\partial \\bm{b}}{\\partial \\bm{f}} &= \\frac{\\partial L_{metric}}{\\partial \\bm{b}} \\bm{1}_{|\\bm{f}| \\le 1}\n\\end{split}\n\\end{equation}\nwhere $\\bm{x} = (u, v)$ is the 2D coordinates in the image and $\\bm{f}(\\cdot)$ is the feature vector at a given location. $L_{metric}$ is the loss for metric learning. $\\bm{1}$ is the indicator function.\nWe found that it is more efficient to train the network with the above method rather than forcing the network to directly predict a binary output by minimizing quantification loss as in \\cite{Lin-DeepBit-CVPR16}. One possible reason is that forcing the value to be clustered around $\n\\{+1,-1\\}$ conflicts with the metric learning that follows, which uses distance as margin making the training unstable.\n\nThe number of feature maps $\\bm{f}$ is set to $256$ to make the descriptor have the same bit width as ORB features so that the descriptor can be directly incorporated into existing ORB-based visual tracking systems.\n\n\\textbf{Nested Metric Learning} The pixel-wise metric learning is used for training the descriptor in a nearest-neighbour manner. The triplet loss for binarized features is as follows:\n\\begin{equation} \\label{eq:triplet}\n\\begin{split}\nL_{metric} &= \\sum_{i} \\operatorname{max} (0, d_{\\bm{x}_i, \\bm{x}_{i,+}} - d_{\\bm{x}_i, \\bm{x}_{i,-}} + m) \\\\\nd_{\\bm{x}_1, \\bm{x}_2} &= (\\operatorname{sign}(\\bm{f}_1 ( {\\bm{x}_1} ) ) - \\operatorname{sign}(\\bm{f}_2 ( {\\bm{x}_2} ) ) )^2\n\\end{split}\n\\end{equation}\nwhere $m$ is the distance margin for the truncation. $d$ is equivalent to the squared Hamming distance for the 32-byte descriptor. We use the squared distance since we found it leads to faster and better convergence for training. $(\\bm{x}_i, \\bm{x}_{i,+})$ is a matching pair obtained using the ground truth camera poses from the training data as follows:\n\\begin{equation} \\label{eq:cor_gt}\n\\bm{x}_{i,+} = \\pi^{-1}(\\mathbf{R}_{gt} \\cdot \\pi(\\bm{x}_i, d_i) + \\bm{t}_{gt})\n\\end{equation}\nwhere $\\mathbf{R}\\in \\mathbb{R}^{3 \\times 3}$ is the rotation matrix and $\\bm{t} \\in \\mathbb{R}^3$ is the translation vector. $(\\bm{x}_i, \\bm{x}_{i,-})$ is a non-matching pair retrieved by negative sample mining. The mining procedure is described in \\cref{alg:ENSM}. The exhaustive search will further penalize the already matched features with the relaxed criteria described in~\\cite{GCN}. The relaxed criteria is used to increase the tolerance to potentially noisy data.\n\n\\addtolength{\\topmargin}{2mm}\n\\begin{algorithm}[!t]\n\\caption{Exhaustive Negative Sample Mining}\n\\label{alg:ENSM}\n\\SetKwInOut{Input}{Input}\n\\SetKwInOut{Output}{Output}\n\\Input{Feature in 1st frame: $ \\bm{f}_1 ( \\bm{x}_{i} )$, \\\\ \n Features in 2nd frame: $ \\{\\bm{f}_2 ( \\bm{x}_{j} ) \\, | \\, j\\in[0,N] \\} $, \\\\ \n Correspondence Ground Truth: $\\bm{x}_{i, +}$, \\\\\n Relaxed Criteria: $\\bm{c} = (u_c, v_c)$}\n\\Output{Negative sample $\\bm{f}_2 ( \\bm{x}_{i,-} ) $ \\textbf{or} None}\n\\textbf{k-NN Matching}: Find top $k$ matched $\\bm{x}_j$ using Hamming distance\\\\\n\\For{$n=1;n<=k;n\\text{\\scshape++}$}{\n Query the $n$th nearest distance $\\bm{x}_{j}^{n}$ \\\\\n \\If{$\\operatorname{abs}( \\bm{x}_{j}^{n}- \\bm{x}_{i}) > c $}{\n \\Return $\\bm{f}_2 ( \\bm{x}_{j}^{n} )$\n }\n}\n\\Return None\n\\end{algorithm}\n \n\\subsection{Distributed Keypoint Detector}\nThe training loss for keypoint detection is the same as in the original GCN. It is computed using two consecutive frames as follows:\t\n\\begin{equation} \\label{eq:NLL}\n\\begin{gathered}\n\t\\resizebox{0.55\\hsize}{!}{$L_{mask} = L_{ce}(\\bm{o}_1, \\bm{x}) + L_{ce}(\\bm{o}_2, \\bm{x}_{+})$} \\\\\n\t\\resizebox{1.0\\hsize}{!}{\n $L_{ce}(\\bm{o}, \\bm{x}) = - \\sum_{i} ( \\alpha_1 c_{\\bm{x}_i} \\mathrm{log}(\\bm{o}({\\bm{x}_i})) +\n\t\\alpha_2 (1-c_{\\bm{x}_i}) \\mathrm{log}(1-\\bm{o}({\\bm{x}_i})) $}\n\t\\end{gathered}\n\t\\end{equation}\n\nA notable difference from SuperPoint is that GCNv2 specifically targets motion estimation. SuperPoint tries to accomplish SIFT-like corner detection and its performance gain can chiefly be attributed to its superior descriptor. However, a CNN is capable of generating more representative features with a larger receptive field than classical methods. We therefore generate the ground truth by detecting Shi-Tomasi corners in a $16x16$ grid and warp them to the next frame using \\cref{eq:cor_gt}. This leads to better distribution of keypoints and the objective function directly reflects the ability to track the keypoints based on texture.\n\n\\subsection{Training Details}\n\nThe triplet loss in \\cref{eq:triplet} and cross entropy in \\cref{eq:NLL} are weighted by ${100, 1}$ during training to provide a coarse normalization of the two terms. The learning rate for the adaptive gradient descent method, ADAM~\\cite{ADAM}, used for training is started from $10^{-4}$ and halved every 40 epoch for a total of 100 training epochs. The weights of GCNv2 are randomly initialized. \nWe mapped the squared Hamming distance to the $L_2$ unit sphere to perform the fast nearest neighbour matching as in \\cite{SuperPoint}. The margin for the triplet loss is set to $1$.\nThe weights $[\\alpha_1, \\alpha_2]$ of the weighted cross entropy is set to $[0.1, 1.0]$.\n\n\\section{GCN-SLAM}\n\n\\begin{figure}\n \\centering\n \n \\includegraphics{orbvsgcnpipe.pdf}\n }\n \\caption{Illustration of the original ORB-SLAM2 keypoint extraction process on the left side, and our method on the right. The keypoint extraction in GCN-SLAM is comparatively simple, in large part because it relies on 2D convolutions and matrix multiplication which is off-loaded to the GPU.}\n \\label{fig:piplines}\n\\end{figure}\n\nOne of the most important design choices for a keypoint-based SLAM system is the choice of keypoint extractor. The keypoints are often re-used at multiple stages in such systems. ORB features \\cite{ORB}, the namesake of ORB-SLAM2, are a particularly well-suited candidate as ORBs are invariant to rotation and scale, cheap to compute compared to other keypoint detectors with equivalent properties such SIFT or SURF, and have a binary feature vector to cater for fast matching. \n\nAs previously shown~\\cite{GCN}, GCN used in a naive motion estimation pipeline performs better than or on par with ORB-SLAM2~\\cite{ORBSLAM2-TRO17}. Notably, this is without higher-order SLAM functionality such as pose graph optimization, global bundle adjustment, or loop detection. Incorporating GCN into a system with such functionality would therefore be likely to yield better results. However, as mentioned, GCN is prohibitively expensive for real-time use on embedded hardware which we target in this work. In what follows we show how we modify ORB-SLAM2 to incorporate GCNv2, in a system we call GCN-SLAM.\n\n\n\nORB-SLAM2's motion estimation is based on frame-to-frame keypoint tracking and feature-based bundle adjustment. We will briefly describe the detection and description of these features. ORB-SLAM2 employs a \\textit{scale pyramid} where the input image is iteratively scaled down to enable multi-scale feature detection by running single-scale algorithms on the multiple rescaled images. For each scale level, the FAST corner detector is applied in a $30\\times{}30$ grid. If no detections are found in a cell, FAST is run again with a decreased threshold. After all detections have been gathered from all cells at a given level in the scale pyramid, a space partitioning algorithm is used to cull the keypoints first by their image coordinates, then by detection score. Finally, once typically 1000 keypoints have been selected in total, the viewing angle of each keypoint is computed, then each pyramid scale level is filtered with Gaussian blur, and the 256-bit ORB descriptor for each keypoint at each level is computed from the blurred image.\n\nOur method computes both keypoint locations and descriptors simultaneously in a single forward-pass of the network, and as stated before, its end result is designed to be a drop-in replacement for the ORB feature extractor outlined above. In GCNv2, we input a single grey image frame to the network which outputs two matrices: a $1\\times{}320\\times{}240$ keypoint mask, and a $256\\times{}320\\times{}240$ feature descriptor matrix. The keypoint mask is thresholded to obtain a set of keypoint locations, their confidences, and their corresponding 256-bit feature descriptors. As in \\cite{SuperPoint}, we apply non-maximum suppression with a grid size of $8\\times{}8$. As it is not possible to know the orientation of the detected features, we set the angle to zero. The two keypoint methods are illustrated in \\cref{fig:piplines}.\n\nOnce keypoints and their respective descriptors are found, ORB-SLAM2 relies primarily on two methods for frame-to-frame tracking: first, by assuming constant velocity and projecting the previous frame's keypoints into the current frame, and if that fails, by matching the keypoints of the current frame to the last-created keyframe using bag-of-words similarity. We have disabled the former so as to use only use the latter keypoint-based reference frame tracking. We have also replaced the matching algorithm with a standard nearest-neighbor search in our experiments. These modifications are made to examine the performance of our keypoint extraction method, rather than that of ORB-SLAM2's other tracking heuristics. \n\nFinally, we have left ORB-SLAM2's loop closure and pose graph optimization intact, apart from having regenerated the bag-of-words vocabulary to suit GCNv2 feature descriptors by computing them on the training dataset presented in \\cref{sec:trainingdata}.\n\n\n\n\\section{Experimental Results}\n\n\\addtolength{\\topmargin}{2mm}\n\\begin{table*}[!ht]\n\\caption{ATE USING FRAME TO FRAME TRACKING}\\label{tab:ate_open}\n\\centering\n\\begin{tabular}{ l | c | c | c | c || c | c | c | c}\n\\hline\nDataset (200 Frames) & GCN & ORB & SIFT & SURF & SuperPoint & GCNv2 & GCNv2-tiny & GCNv2-large\\\\ \n\\hline \\hline\nfr1\\textunderscore floor & \\textbf{0.015m} & 0.080m & 0.073m & 0.074m & - & - & -& -\\\\ \\hline\nfr1\\textunderscore desk & \\textbf{0.037m} & 0.151m & 0.144m & 0.148m & 0.166m & 0.049m & 0.084m & 0.038m \\\\ \\hline\nfr1\\textunderscore 360 & \\textbf{0.059m} & 0.278m & 0.305m & 0.279m & - & - & - & 0.097m \\\\ \\hline\nfr3\\textunderscore long\\textunderscore office& 0.061m & 0.090m & 0.076m & 0.070m & 0.105m & \\textbf{0.046m} & 0.085m & 0.067m\\\\ \\hline\nfr3\\textunderscore large\\textunderscore cabinet & 0.073m & 0.097m & 0.091m & 0.143m & 0.195m & 0.064m & 0.067m & \\textbf{0.056m} \\\\ \\hline\nfr3\\textunderscore nst & 0.020m & 0.061m & 0.036m & 0.030m & 0.055m & \\textbf{0.018m} & 0.024m & 0.021m\\\\ \\hline\nfr3\\textunderscore nnf & \\textbf{0.221m} & - & - & - & - & - & - & -\\\\ \\hline\n\\end{tabular}\n\\end{table*}\n\nIn this section, we present experimental results to justify our conclusions regarding the performance of our keypoint extraction method, and its embodiment in the GCN-SLAM system. We first introduce our training dataset, then four datasets on which we examined our method's performance and compare to some related methods, and finally we outline the quantitative and qualitative conclusions of these results. \n\nNote that our aim in this section is not to show that GCN-SLAM is better that ORB-SLAM2 but to show that GCNv2 is: i) better suited for accurate motion estimation, ii) computationally efficient, and iii) providing robustness for a SLAM system.\n\nIn the results below, evaluations on datasets were performed on a laptop with an Intel i7-7700HQ and a mobile version of NVIDIA 1070. For real-world experiments we used an NVIDIA Jetson TX2 embedded computer for processing and an Intel RealSense D435 RGB-D camera sensor on our drone (see \\cref{fig:digest}.)\n\n\n\\subsection{Training Data}\n\\label{sec:trainingdata}\nThe original GCN was trained using the TUM dataset from sensor fr2. It provides accurate pose through a motion capturing system. In GCNv2, we trained the network using a subset of the SUN-3D~\\cite{SUN-3D} dataset we created in our recent work~\\cite{S2D}. SUN-3D contains millions of real-world recorded RGB-D images in various typical indoor environments. A total $44,624$ frames were extracted by roughly one frame per second from the videos. It is very large and can potentially produce a more generalized network. However, the ground truth poses provided by SUN-3D are estimated by visual tracking with loop closure and so are relatively accurate globally, but have misalignments at frame level. To account for this local error, we extract SIFT features and use the provided poses as initial guesses for bundle adjustment to update the relative pose of each frame pair. In this sense, the training of GCNv2 is using self-annotated data with only vision information. \n\n\\subsection{Quantitative Results}\nFor comparison with the original GCN, we select the same sequences of the TUM datasets as in~\\cite{GCN} and evaluating tracking performance with an open and a closed loop system. We use the Absolute trajectory error~(ATE) as the metric. \n\nSince we trained GCNv2 on a different dataset than the original GCN~\\cite{GCN}, we also show results using the original recurrent structure for comparison. We have therefore also created GCNv2-large, with ResNet-18 as the backbone and deconvolutional up-sampling for the feature maps. The bidirectional feature detector is moved to the lowest scale as the other two versions of GCNv2.\n\n\\begin{table}[ht]\n\\centering\n\\caption{ATE USING CLOSED LOOP SYSTEM}\\label{tab:ate_closed}\n\\resizebox{\\hsize}{!}{\\begin{tabular}{ l | c | c | c | c || c } \n\\hline\n\\multirow{2}{*}{Dataset} & \\multirow{2}{*}{GCN} & ORB & Elastic & RGBD & GCN \\\\\n & & SLAM2 & Fusion & TAM & SLAM\\\\\n\\hline \\hline\nfr1\\textunderscore floor & 0.038m & 0.036m & - & - & \\textbf{0.021m} \\\\ \\hline\nfr1\\textunderscore desk & 0.029m & \\textbf{0.016m} & 0.020m & 0.027m & 0.031m \\\\ \\hline\nfr1\\textunderscore 360 & \\textbf{0.069m} & 0.213m & 0.108m & 0.101m & 0.155m \\\\ \\hline\nfr3\\textunderscore long\\textunderscore office & 0.040m & \\textbf{0.010m} & 0.017m & 0.027m & 0.021m \\\\ \\hline\nfr3\\textunderscore large\\textunderscore cabinet & 0.097m & - & 0.099m & \\textbf{0.070m} & \\textbf{0.070m} \\\\ \\hline\nfr3\\textunderscore nst & 0.020m & 0.019m & 0.016m & \\textbf{0.010m} & 0.014m\\\\ \\hline\nfr3\\textunderscore nnf & \\textbf{0.064m} & - & - & - & 0.086m \\\\ \\hline\n\\end{tabular}}\n\\end{table}\n\nFrame-to-frame tracking results are shown in \\cref{tab:ate_open}. The first columns, before SuperPoint, are from~\\cite{GCN} where 640x480 images were used and GCN was trained on TUM fr2 data only. All versions of GCNv2 use the same image resolution as SuperPoint, i.e. 320x240. The results are consistent with the results reported in \\cite{SuperPoint}, SuperPoint performs on par with classical method like SIFT. GCNv2 has a performance close to GCN, and like GCN, significantly better than both SuperPoint and classical keypoints. GCNv2 performance is on par with GCN, and even slightly better in two cases -- likely due to using a much larger dataset for training. The exceptions are fr1\\textunderscore floor and fr1\\textunderscore 360. These sequences require fine details, and the detection and descriptor extraction in GCNv2 is performed with a lower scale feature map for efficiency, though GCNv2-large handles one of these sequences. The smaller version of GCNv2, GCNv2-tiny, is only slightly less accurate than GCNv2. \n\nIn \\cref{tab:ate_closed}, we compare the closed loop performance of GCN-SLAM with our previous work, as well as ORB-SLAM2, Elastic Fusion, and RGBDTAM. GCN-SLAM successfully tracks the position in all sequences with an error similar to that of GCN, whereas ORB-SLAM2 fails on two sequences. GCNv2 has significantly reduced drift error compared to ORB-SLAM2 in the fast rotations of fr1\\textunderscore360. It is also noteworthy that for this particular sequence, the original GCN does significantly better than both ORB-SLAM2 and GCN-SLAM. ORB-SLAM2 is tracking well in all other sequences, and the errors of both GCN-SLAM and ORB-SLAM2 are small.\n\n\\subsection{Qualitative Results}\n\n\\begin{figure*}[hp]\n\\centering\n\\begin{subfigure}{0.45\\textwidth}\n\\centering\n\\includegraphics[width=\\textwidth]{cor_traj}\n\\caption{\\textit{Corridor}: indoor, handheld.}\n\\label{fig:trajs_cor}\n\\end{subfigure\n\\hfill\n\\begin{subfigure}{0.45\\textwidth}\n\\centering\n\\includegraphics[width=\\textwidth]{park_traj}\n\\caption{\\textit{Parking lot}: outdoor, handheld.}\n\\label{fig:trajs_park}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}{0.45\\textwidth}\n\\centering\n\\includegraphics[width=\\textwidth]{atrium_traj.pdf}\n\\caption{\\textit{Alcove}: indoor, flying with primitive optical flow sensor.}\n\\label{fig:trajs_atrium}\n\\end{subfigure\n\\hfill\n\\begin{subfigure}{0.45\\textwidth}\n\\centering\n\\includegraphics[width=\\textwidth]{kitchen_traj.pdf}\n\\caption{\\textit{Kitchen}: indoor, flying with GCN-SLAM for positioning.}\n\\label{fig:trajs_kitchen}\n\\end{subfigure\n\\caption{Estimated trajectory on each of the four datasets using the GCN-SLAM pipeline and illustrating the difference between GCNv2 and ORB features. Note that these trajectories do not have any ground truth but we mark track lost with a cross and we see that this happened in all four cases when using ORB features.}\n\\label{fig:trajs}\n\\end{figure*}\n\n\\begin{figure*}[hp]\n\\centering\n\\begin{subfigure}{0.45\\textwidth}\n\\centering\n\\includegraphics[width=\\textwidth]{cor}\n\\end{subfigure\n\\begin{subfigure}{0.45\\textwidth}\n\\centering\n\\includegraphics[width=\\textwidth]{park}\n\\end{subfigure}\n\\caption{Keypoint extractor performance in terms of tracking shown for the \\textit{Corridor} (left) and \\textit{Kitchen} datasets (right.) Lines indicate the total number of keypoints detected per frame. The filled-in area under the lines shows the fraction of keypoints that were successfully used for local map tracking, i.e. contribute to tracking and is plotted against the right-hand axis.}\n\\label{fig:inliers}\n\\end{figure*}\n\nTo further verify the robustness of using GCNv2 in a real-world SLAM system, we show results on datasets collected in our environment under different conditions: a) going up a corridor, turning 180 degrees and walking back with a handheld camera, b) walking in a circle on an outdoor parking lot with a handheld sensor in daylight, c) flying in an alcove with windows and turning 180 degrees, and d) flying in a kitchen and turning 360 degrees while using GCN-SLAM for positioning.\n\n\\begin{figure*}[ht]\n\\centering\n\\begin{subfigure}[c]{0.49\\textwidth}\n\\centering\n\\includegraphics[width=\\textwidth]{room_crop.png}\n\\end{subfigure\n\\centering\n\\begin{subfigure}[c]{0.49\\textwidth}\n\\centering\n\\includegraphics[width=\\textwidth]{rpl_crop.png}\n\\end{subfigure}\n\\begin{subfigure}[c]{0.49\\textwidth}\n\\centering\n\\caption{\\textit{Lab room}: handheld.}\n\\end{subfigure\n\\centering\n\\begin{subfigure}[c]{0.49\\textwidth}\n\\centering\n\\caption{\\textit{Lab corridor}: handheld.}\n\\end{subfigure}\n\\caption{Mesh reconstruction results using GCN-SLAM as input to TSDF volume integration from Open3D. Loop closure detection in GCN-SLAM was disabled to demonstrate the accuracy in the tracking alone.}\n\\label{fig:mesh}\n\\end{figure*}\n\nSince they are without ground truth, the results are only qualitative.\nThese datasets were chosen to show that our method handles difficult scenarios, is robust, and can be used for positioning of a real drone. \\Cref{fig:trajs} shows the estimated trajectory of GCN-SLAM using ORB versus GCNv2 as keypoints. Note that both features are evaluated in exactly the same tracking pipeline for fair comparison, i.e. GCNv2 or ORB features is the \\textit{only} differences. Refer to the source-code for further details.\nusing GCN-SLAM as a basis for drone control improves performance as can be seen by comparing \\cref{fig:trajs_atrium,fig:trajs_kitchen}. In \\cref{fig:trajs_atrium}, the position is estimated using only an optical flow sensor whereas \\cref{fig:trajs_kitchen} uses GCN-SLAM as a source of position, and it is clear that the drone is able to hold its position better as there is significantly less jitter in this trajectory. In all four datasets, tracking is maintained with GCNv2, but lost with ORB. We used a remote control to send setpoints to the flight control unit on the drone for control, using the built-in position holding mode.\n\nIn \\cref{fig:inliers} we compare the performance of our keypoint extractor to the original ORB keypoint extractor. We plot the number of inliers during tracking of the local map for our adapted SLAM system, first with ORB keypoints, and then with GCNv2 keypoints. As the figure illustrates, while there are many more ORB features, our method has a higher percentage of inliers. In addition, as shown in \\cref{fig:digest}, GCNv2 results in better distributed features compared with ORB.\n\n\\Cref{fig:mesh} shows the mesh reconstruction using GCN-SLAM output poses from two additional sequences. The left sequence was from an office and the right was acquired walking between two floors using stairs. TSDF volume integration from Open3D\\footnote{\\url{http:\/\/www.open3d.org}} was used to create the mesh. To show the accuracy of our method, the loop closure detection of GCN-SLAM is disabled. \n\n\\section{Conclusions}\n \nIn our previous work~\\cite{GCN}, we showed that our method, GCN, achieves better performance in visual tracking compared with existing deep learning and classical methods. However, it cannot be directly deployed into a real-time SLAM system in an efficient way due to its deep recurrent structure. In this paper, we addressed these issues by proposing a smaller, more efficient version of GCN, called GCNv2, that is readily adaptable to existing SLAM systems. We showed that GCNv2 can be effectively used in a modern feature-based SLAM system to achieve state-of-the-art tracking performance. The robustness and performance of the method was verified by incorporating GCNv2 into GCN-SLAM and using it on-board for positioning on our drone.\n\n\\textbf{Limitations}\nGCNv2 is trained mainly for projective geometry and not generic feature matching.\nAs always with learning-based methods generalization is an important factor. GCNv2 works relatively well for outdoor scenes, as demonstrated in the experiments (Cf. \\cref{fig:trajs_park}). However, since no outdoor data was used for training, further improvements can likely be made. Our target here is an indoor setting and we did not investigate this further.\n\n\\textbf{Future work}\nIn this paper, we mainly improved the efficiency of GCN and achieve stable tracking perform on our platform, a drone with NVIDIA Jetson TX2. However, since the original recurrent structure is removed, there is trade-off between the accuracy and running speed. In the future, we would like to further investigate to use the self-supervised learning to improve our system.\n\n\n\n\n\n \\bibliographystyle{IEEEtran}\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nFor any given vector $\\bf{c} = [c_0, c_1, \\dots, c_{n-1}]^T$, the circulant matrix $C$ with the first row $\\bf{c}$ is defined as follows (\\cite{golub2013matrix,gray2006toeplitz,karner2003spectral}):\n\\begin{equation}\n\\label{Eq:C}\nC = \\left(\\begin{matrix}\nc_0 & c_{n-1} & \\cdots & c_2 & c_1 \\\\\nc_1 & c_0 & c_{n-1} & & c_2 \\\\\n\\vdots & c_1 & c_0 & \\ddots & \\vdots \\\\\nc_{n-2} & & \\ddots & \\ddots & c_{n-1} \\\\\nc_{n-1} & c_{n-2} & \\cdots & c_1 & c_0 \\\\\n\\end{matrix}\\right)\n\\end{equation}\nThe eigenspace of the circulant matrices are formed by Fourier matrices; therefore, the eigienvectors and the eigenvalues of $C$ are defined analytically in the following form of pairs:\nA $j$th ($j = 0, \\ldots, n-1$) eigenvector is given by\n\\begin{equation}\nv_j=\\frac{1}{\\sqrt{n}} \\left(1, \\omega^j, \\omega^{2j}, \\ldots, \\omega^{(n-1)j}\\right),\n\\end{equation}\nwhere $\\omega=\\exp \\left(\\tfrac{2\\pi i}{n}\\right)$.\nAnd its corresponding eigenvalue is\n\\begin{equation}\n\\label{Eq:EigenC}\n\\lambda_j = c_0+c_{n-1} \\omega^j + c_{n-2} \\omega^{2j} + \\dots + c_{1} \\omega^{(n-1)j}.\n\\end{equation}\nAs a result, the eigendecomposition of any circulant matrix can be written as $C = F\\Lambda F^{-1}$ where $F$ represents the normalized matrix for the discrete Fourier transform and $\\Lambda = diag({F \\bf c})$, which is a diagonal matrix representing the eigenvalues. \n\n\\subsection{Direct Implementation as a quantum circuit}\n\nOne can implement $C$ as a quantum circuit by using its eigendecomposition $C = F\\Lambda F^{-1}$: It is well known that $F$ can be implemented in $O(\\log(n)^2)$ time as the quantum Fourier transform \\cite{nielsen2002quantum}. \nIn addition, since any vector of dimension $n$ can be implemented in $O(n)$ time, we can also implement $\\Lambda$ in linear time. Therefore, $C$ can be implemented as a quantum circuit by using $O(n)$ two and single qubit quantum gates, which is less than the classical matrix size $O(n^2)$. \n\n\\subsection{Circuit implementation through permutations}\n$C$ can be represented as a function of the following permutation matrix, which is also circulant:\n\\begin{equation}\n\\label{EqMatrixP}\nP = \\left( \\begin{matrix}\n 0&0&\\cdots&0&1\\\\\n 1&0&\\cdots&0&0\\\\\n 0&\\ddots&&\\vdots&\\vdots\\\\\n \\vdots&\\ddots&\\ddots&0&0\\\\\n 0&\\cdots&0&1&0\n\\end{matrix} \\right)\n\\end{equation}\nThen, we can redefine $C$ as a polynomial function of $P$ with the coefficients given by $\\bf{c}$: \n\\begin{equation}\nC = c_0 I + c_1 P + c_2 P^2 + \\dots + c_{n-1} P^{n-1} = \\sum_{j=0}^{n-1}c_jP^j.\n\\end{equation}\nSince $P$ is an orthogonal matrix, we can construct its exact quantum circuit decomposition. \nMoreover, it is known that sum of unitary matrices can be implemented as a subpart of a larger quantum system with the help of ancilla registers (e.g., \\cite{childs2012hamiltonian,daskin2012universal,low2019hamiltonian}). \nHere, we shall show that $C$ can be constructed as a vector by using $V_P\\times \\ket{\\psi}$ where \\ket{\\psi} represents a combinations of the vector $\\bf{c}$ and $V_P$ includes the powers of $P$ used in the above polynomial. \nLater in the paper, we shall show that this way of implementation eases the way to solve some string problems. \n\n\\subsubsection{Circuit for $P$}\nSince $P$ is a circulant matrix, we can define its any $j$th eigenvalue by using Eq.\\eqref{Eq:EigenC}:\n\\begin{equation}\n\\lambda_j = w^j.\n\\end{equation}\nTherefore, its eigendecomposition can be written as:\n\\begin{equation}\n\\label{EqLambdaP}\nF\\times \n\\left(\\begin{matrix}\n1&&&&\\\\\n&w^1&&\\\\\n&&w^2&\\\\\n&&&\\ddots&\\\\\n&&&&w^{n-1}\n\\end{matrix}\\right)\\times F^{-1} =F \\Lambda_P F^{-1} .\n\\end{equation}\nAs a quantum circuit, we can implement $\\Lambda_P$ as follows:\n First, observe that the power $j$ in $w^j$ is equal to the row index which can be found from the binary expansion of $j = (b_{n-1}\\dots b_0)_2$. Therefore, if a qubit is in \\ket{1}, we apply a control phase gate to the first qubit. The phase of these gates are determined from the decimal index of the qubit in the binary expansion.\nWe define the phase gate as follows:\n\\begin{equation}\n\\label{Eq:Rw}\nR_w(k) = \\left(\\begin{matrix}\n 1 & 0\\\\\n 0 & w^{2^k}\n\\end{matrix}\\right),\n\\end{equation}\nwhere $k$ is the order of the control qubit in the binary expansion $(b_{n-1}\\dots b_0)_2$.\nThe resulting circuit is drawn in Fig.\\ref{FigCircuitLambdaP} for four qubits. The number of quantum gates for this implementation is equivalent to the number of qubits which is $(\\log n)$. \n\\begin{figure}[t]\n\t\\begin{center}\n\\colorbox{green!10}{\n\t\t\\Qcircuit @C=0.7em @R=1em {\n\\lstick{} &\\qw &\\qw & \\qw &\\ctrl{3} & \\qw & \\\\\n\\lstick{} &\\qw &\\qw& \\ctrl{2} & \\qw & \\qw & \\\\\n\\lstick{} & \\qw &\\ctrl{1}&\\qw & \\qw & \\qw & \\\\\n\\lstick{}&\\gate{R_w(0)} &\\gate{R_w(1)} &\\gate{R_w(2)} & \\gate{R_w(3)}&\\qw \\\\\n\t\t\t}\n}\n\t\\end{center}\n\t\\caption{\\label{FigCircuitLambdaP} Quantum circuit for $\\Lambda_P$}\n\\end{figure}\n\n\n\\subsubsection{Circuit for $V_P$}\nConsider the following matrix:\n\\begin{equation}\nV_P = \n\\left(\\begin{matrix}\nI&&&&\\\\\n&P^1&&\\\\\n&&P^2&\\\\\n&&&\\ddots&\\\\\n&&&&P^{n-1}\n\\end{matrix}\\right),\n\\end{equation}\nwhich highly resembles $\\Lambda_P$ given in Eq.\\eqref{EqLambdaP}. Therefore, we can implement the same way as we implemented $\\Lambda_P$: In this case, we only need to somehow replace $R_w({k})$ with the power of $P$: \n\\begin{equation}\nP^{2^k}=F\\Lambda_P^{2^k}F^{-1}.\n\\end{equation} \nSince all the powers of $P$s have the same eigenvectors, we apply $F$ and $F^{-1}$ only once at the beginning and at the end of the circuit. We use the following quantum operation in the circuit:\n\\begin{equation}\nU_P(k) = \\left(\\begin{matrix}\n I & 0\\\\\n 0 & \\Lambda_P^{2^k}\n\\end{matrix}\\right).\n\\end{equation}\nHere, when we look at the circuit for $\\Lambda_P$ in Fig.\\ref{FigCircuitLambdaP}, we can see that the power can be distributed to the quantum gates inside the circuit since the order of the gates is not important. \nThat means, we can obtain any power of $\\Lambda_P$ by simply adjusting the angle values of the quantum gates. \nTherefore, for power of $\\lambda_P^{2^k}$, we adjust the parameter of the rotation gates in Fig.\\ref{FigCircuitLambdaP} as: $R_w(0\\times k), R_w(1\\times k), \\dots$, and so on. The final circuit for $V_P$ is just composed of the controlled versions of these quantum gates which is shown in Fig.\\ref{FigCircuitV} and \\ref{FigUpk}.\n\n\\begin{figure}[t]\n\t\\begin{center}\n\\colorbox{green!10}{\n\t\t\\Qcircuit @C=0.7em @R=1em {\n\\lstick{} &\\qw&\\qw &\\qw & \\qw &\\ctrl{3} & \\qw & \\qw\\\\\n\\lstick{} &\\qw&\\qw &\\qw& \\ctrl{2} & \\qw & \\qw & \\qw \\\\\n\\lstick{} &\\qw& \\qw &\\ctrl{1}&\\qw & \\qw & \\qw & \\qw \\\\\n\\lstick{}&\\gate{F^{-1}}&\\gate{U_P(0)} &\\gate{U_P(1)} &\\gate{U_P(2)} & \\gate{U_P(3)}&\\gate{F}&\\qw \\\\\n}}\n\t\\end{center}\n\t\\caption{\\label{FigCircuitV} The final quantum circuit of $V_P$ illustrated for $n=4$. The explicit circuit for $U_P(k)$ is given in Fig.\\ref{FigUpk} below.}\n\\end{figure}\n\n\\begin{figure}[t]\n\\colorbox{green!10}{\n\t\t\\Qcircuit @C=0.7em @R=1em {\n\\lstick{} &\\ctrl{4}&\\ctrl{4} &\\ctrl{4} & \\ctrl{4} & \\qw & \\\\\n\\lstick{} &\\qw &\\qw & \\qw &\\ctrl{3} & \\qw & \\\\\n\\lstick{} &\\qw &\\qw& \\ctrl{2} & \\qw & \\qw & \\\\\n\\lstick{} & \\qw &\\ctrl{1}&\\qw & \\qw & \\qw & \\\\\n\\lstick{}&\\gate{R_w(0\\times k)} &\\gate{R_w(1\\times k)} &\\gate{R_w(2\\times k)} & \\gate{R_w(3\\times k)}&\\qw \\\\\n}}\n\t\\caption{\\label{FigUpk} The explicit circuit implementation of $U_P(k)$ which implements the $2^k$th power of $\\Lambda_P$ given in Eq.\\eqref{EqLambdaP}.}\n\\end{figure}\n\n\\subsection{The overall complexity}\nAs shown in Fig.\\ref{FigCircuitV} and \\ref{FigUpk} the number of quantum gates are limited to the number of qubits used in the main system. The size of the permutation matrix $P$ is $n$ by $n$, therefore the main system is described by $\\log n$ qubits. Then, an additional $\\log n$ qubit is used to construct $V_P$. Therefore, in total the circuit requires $(2\\times \\log n)$ qubits.\nThe quantum gates are at most controlled by two qubits, therefore the total number of required single and two qubit gates are bounded by $O(poly(\\log n))$ in general. More specifically, if we assume the general implementation of quantum Fourier transform requires $O((\\log n)^2)$ gates (with an optimization, it may require $O((\\log n)\\log\\log n)$ ) and the decomposition of each Toffoli gate requires less than $\\log n$ gates \\cite{nielsen2002quantum}, then the total complexity becomes bounded by $O((\\log n)^2)$.\nThis shows that we can form $V_P$ on quantum computers very efficiently since it requires a number of quantum gates which is a poly-logarithm of the dimension $n$.\n\n\n\\section{Applications}\n\\subsection{Suffix trees and arrays}\nMany string problems can be solved easier if they are stored by using well-known data structures such as suffix tries, trees, and arrays \\cite{manber1993suffix}. Below, we will follow the lecture notes in Ref.\\cite{langmeadburrows} to give a brief introduction of these data structures and explain how to implement them on quantum computers with the help of the circuits introduced in the previous section.\n\nFrom a given string, a suffix can be chosen by determining a starting position. For instance, we can choose the following suffixes from string \\textbf{``banana\"}:\n\\begin{equation}\n\\begin{bmatrix}\nb&a&n&a&n&a&\\$\\\\\na&n&a&n&a&\\$\\\\\nn&a&n&a&\\$\\\\\na&n&a&\\$\\\\\nn&a&\\$\\\\\na&\\$\\\\\n\\$\n \\end{bmatrix}\n\\end{equation}\nHere, ``\\$\" is used to indicate the end of the string and considered less than any letter of the alphabet in lexicographical order.\n\nIn suffix arrays, these strings are put in the buckets based on their orders and sorted. \nIn tries, the prefixes are put in the rooted tree in which every node has a child for each letter of the alphabet used in the string. \nIn the suffix tries, the suffixes are put in a rooted tree in which every node has a child for each letter of the alphabet used in the string. \nTherefore, by following the paths from the root of the tree, it is possible to determine if a letter follows another and if the prefix exists or not.\nIn the suffix trees, the tries are built for the suffixes and the number of nodes in the tries are optimized by compressing the nodes in the straight paths ( i.e. the paths that do not go more than one way in any nodes on the path.). \nEach path in the suffix tree ends with the letter ``\\$\".\n\nWhile constructing suffix arrays and trees can be done in $O(n\\log n)$ time \\cite{ukkonen1995line}, they can be used to solve many string problems more efficiently such as string searching and matching and sequence alignment \\cite{mielczarek2016review}. \n\n\\subsection{Burrows Wheeler Transform\\cite{burrows1994block}}\n\nThe main idea in Burrows Wheeler transform (BWT) is to first generate a circulant matrix by using the given string and then sort this matrix column by column. In the sorted matrix, the last column is used as the equivalent transformed string. As an illustration, consider the following example string \"banana\", which is used in most textbooks: \n\\begin{equation}\n \\begin{bmatrix}\nb&a&n&a&n&a&\\$\\\\\na&n&a&n&a&\\$&b\\\\\nn&a&n&a&\\$&b&a\\\\\na&n&a&\\$&b&a&n\\\\\nn&a&\\$&b&a&n&a\\\\\na&\\$&b&a&n&a&n\\\\\n\\$&b&a&n&a&n&a\n \\end{bmatrix} \\xrightarrow{sort}\n \\begin{bmatrix}\n\\$&b&a&n&a&n&\\bf a\\\\\na&\\$&b&a&n&a&\\bf n\\\\\na&n&a&\\$&b&a&\\bf n\\\\\na&n&a&n&a&\\$&\\bf b\\\\\nb&a&n&a&n&a&\\bf \\$\\\\\nn&a&n&a&\\$&b&\\bf a\\\\\nn&a&\\$&b&a&n&\\bf a\\\\\n \\end{bmatrix} \n\\end{equation}\nFrom the above, BWT(``banana\") = ``annb\\$aa\". In the above matrix, if one considers ``\\$\" as 1s and the rest of the letters as 0s, then BWT of any string can be represented as a permutation matrix. \n\\subsection{Quantum implementation of suffixes}\nIn the previous section we show how to construct $V_P$. \nHere, consider that we are given the text \\ket{c} including also the ``\\$\" sign.\nWe apply \\ket{c} in the main register and $H^{\\otimes \\log n}\\ket{0}$ in the ancilla register. This generates the following vector:\n\\begin{equation}\n(H^{\\otimes \\log n}\\ket{0}) \\ket{c} = \n\\frac{1}{\\sqrt{n}}\\left(\\begin{matrix}\n\\begin{bmatrix}\n c_0\\\\c_1\\\\ \\vdots \\\\c_{n-1}\\\\\n\\end{bmatrix}_{0\\ \\ \\ }\\\\\n\\vdots\\ \\ \\ \\ \\\\\n\\begin{bmatrix}\n c_0\\\\c_1\\\\ \\vdots \\\\c_{n-1}\\\\\n\\end{bmatrix}_{n-1}\\\\\n\\end{matrix}\\right).\n\\end{equation}\nIf we multiply this vector by $V_P$, we get $C$ in Eq.\\eqref{Eq:C} in vector form:\n\\begin{equation}\nV_P (H^{\\otimes \\log n}\\ket{0}) \\ket{c} = \n\\frac{1}{\\sqrt{n}}\\left(\\begin{matrix}\n\\begin{bmatrix}\n c_0\\\\c_1\\\\ \\vdots \\\\c_{n-1}\\\\\n\\end{bmatrix}_{0\\ \\ \\ }\\\\\n\\vdots\\ \\ \\ \\ \\\\\n\\begin{bmatrix}\n c_{n-1}\\\\c_{n-2}\\\\ \\vdots \\\\c_{0}\\\\\n\\end{bmatrix}_{n-1}\\\\\n\\end{matrix}\\right) \n= \n\\frac{1}{\\sqrt{n}}\\left(\\begin{matrix}\nS_{0}\\\\\n\\vdots \\\\\nS_{n-1}\\\\\n\\end{matrix}\\right),\n\\end{equation}\nwhere each $S_j$ represents a circularly permuted string.\nThe above vector provides the unsorted suffixes whose end indicated by some end of file character \"\\$\". \nSince the order of this character provides information about the suffixes, the amplitude of this char may be adjusted higher so that from the measurements, the order can be obtained in an efficient way.\n\nBy using the indices of the end of file characters, we can collapse the quantum state onto any desired direction. \nIn addition, we can draw some conclusion from the measurements on the collapsed state: e.g., . Therefore, the measurement statistics can be used to draw conclusions about the most common prefixes in the suffixes in the collapsed state. \n\nSince in most string algorithms sorted structures allow us to develop more efficient algorithms, the same sorting can be also done on this vector.\n\n\\section{Sorting}\nSorting problem simply can be described as finding an ordered list of items given as an unordered list.\nIt is known that comparison based sorting algorithms such as merge sort and quicksort have running time $\\Omega(n log n)$ for a list of $n$ items. \nThis lower bound can be broken by using bucket sorting (counting sort) where each item is considered as a direct pointer to the bucket; therefore, the sorting is done in $O(n)$ time. This simple approach is further improved to give $O(nd)$ time, where $d$ is the number of bits used to represent each item and $O(nlog log n)$ time. \nBecause of the memory constraints and requirements, in practice comparison sorts are used more often in practice than these kinds of sorting algorithms. \\cite{hagerup1998sorting}\n\nOn quantum computers, the sorting problem is considered based on registers $\\ket{x_1} \\dots \\ket{x_n}$. Here, each register represents an item in the unsorted list. \nTherefore, the algorithms try to prepare the registers in the output to encode the natural ordered list of the given items: i.e., $\\ket{q_1} \\geq \\dots \\geq \\ket{q_n}$.\nBased on this representation, it is shown that quantum algorithms based on comparison models have similar complexity bounds for the sorting problem: i.e. $\\Omega (\\sqrt{n}logn)$ \\cite{hoyer2002quantum,ambainis2002quantum}.\n\nAs done in the classical sorting algorithms, bucket sort with an order preserving hash function can be used to sort the items stored in memory as:\n\\begin{equation}\n\\sum_{i=1}^n\\ket{\\bf 0}\\ket{i}\\ket{x_i},\n\\end{equation}\nwhere $i$ is the index of the item $x_i$ in the given unordered list.\nThe sorting problem becomes constructing the following quantum state:\n\\begin{equation}\n\\sum_{i=1}^n\\ket{o_i}\\ket{i}\\ket{x_i}\n\\end{equation}\nwhere \\ket{o_i} represents the index of the item in the sorted list.\nWe can rewrite this in terms of a hash function $h(x_i)$ that maps an item $x_i$ to the index $o_i$:\n\\begin{equation}\n\\sum_{i=1}^n\\ket{h(x_i)}\\ket{i}\\ket{x_i}.\n\\end{equation}\n$h$ can be as simple as a direct map or more general hash function. \nIn particular consider\n$h$ as a partial order preserving hash function: i.e., if $x_i$ and $x_j$, their real ordered locations at some distance $d$ from each other so that $o_i + d < o_j$, then $h(x_i) < h(x_j)$. \nThen, since we can apply an operator $h$ simultaneously to all items, we can generate their orders in $O(1)$ time.\nSometimes knowing the elements' rough order in the array may be considered enough. \nIn those cases, the hash function need not be perfect; therefore,\none can use similar ideas to classical sorting algorithms such as bucket sort or the shell-sort \\footnote{Shell sort is a generalization of insertion sort algorithm, where items at certain distances are compared and if necessary swapped to have a k-sorted array: i.e. an array where the numbers are grouped into regions based on their orders. } to generate some k-sorted array in which buckets are sorted, however, the items in the same buckets are not sorted. \n\nAs mentioned above, if we use comparison based sorting algorithms, then the sorting is almost the same as classical sorting algorithm: \nLet us consider the following vector whose construction is given in the previous section:\n\\begin{equation}\n \\frac{1}{\\sqrt{n}}\\left(\\begin{matrix}\nS_{0}\\\\\n\\vdots \\\\\nS_{n-1}\\\\\n\\end{matrix}\\right),\n\\end{equation}\nIn Burrows Wheeler transform, the items are sorted by columns.\nWe can do the same sorting on this vector: A particular direct sorting may be as follows: \n\n\\begin{itemize}\n \\item \nFirst, we compare each element with its left neighbor, then if it is necessary we swap them. \\begin{itemize}\n \\item This step can be done in parallel, if the swap and comparison can be implemented in $O(poly(\\log n))$ number of quantum gates, then it takes $O(poly(\\log n))$ time \\footnote{Here, since the comparison and swap operations depend on the number of qubits, it may require controlled gates whose decomposition requires number of gates that are polynomial in the system size $O(\\log n)$. If this step can be done in $O(1)$, then sorting can be done more efficiently.}.\n \\end{itemize} \n\\item In the second step we do the same thing for the right neighbors (we group two elements together and compare them.).\n \\begin{itemize}\n \\item This step also requires $O(poly(\\log n))$ number of quantum gates.\n\\end{itemize} \n\\item If we repeat the above steps for O(n) time, we basically get a simple $O(n\\log n)$ time sorting algorithm.\n\\end{itemize}\n\n\n\\section{Conclusion}\nIn this paper, we describe how to generate suffix structures efficiently as a vector by using quantum circuits for circulant matrices.\nWe discuss how the generated vector can be used in the string algorithms and sorted if necessary.\nAs a future direction, we will apply this circuit to the sequence alignment and pattern matching problems.\nSince circulant matrices are used in convolutions, it can be also applied to problems in different areas such as convolution neural network, time series analysis \\cite{pollock2002circulant, daskin2022walk}.\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nColloidal metal chalcogenide NPLs offer well defined advantages over their quantum dot \nand rod counterparts as semiconductor building blocks for optical applications.\\cite{LhuillierACR,MinNR,DirollJMCc,SharmaIEEE}\nSome of the most distinctive features are order-of-magnitude shorter radiative lifetimes,\nwhich result from the strong exciton binding energies in quasi-2D systems\n(Giant Oscillator Strength effect),\\cite{FeldmannPRL,PlanellesACSph}\nand precisely controlled thickness of the nanostructure,\\cite{IthurriaNM,RiedingerNM,ChristodoulouNL,BhandariCM}\nwhich suppresses the emission broadening due to size dispersion usually observed in dots.\nThese properties give rise to bright and narrow emission lines, which\nis of interest for displays, lighting and lasers.\\cite{DirollJMCc,SharmaIEEE}\n\nUnfortunately, ligand passivation of surface dangling bonds is usually incomplete because of\n labile binding and steric hindrance between ligands. This can translate into significant \nnon-radiative losses.\\cite{TessierACS} To overcome this problem, core-only NPLs are sometimes\nreplaced by sandwich-like core\/shell heterostructures, where top and bottom facets of the \ncore material are coated with a higher band gap inorganic material. \nTypical core\/shell combinations are CdSe\/CdS,\\cite{TessierNL,AchtsteinACS} \nCdSe\/ZnS\\cite{PolovitsynCM,SaidzhonovJL} and their alloys.\\cite{RossinelliCM,KelestemurACS}\nThese heterostructures succeed in isolating the photogenerated carriers, \nwhich remain in and around the core, from top and bottom surfaces,\nthus translating into enhanced fluorescence quantum efficiency and \nphotostability.\\cite{LhuillierACR,SharmaIEEE,YadavJPCc}\nThe shell growth has however a negative side effect, namely the systematic broadening of\nthe emission line width, e.g. from $\\sim 35-40$ meV in CdSe NPLs to $\\sim 60-80$ meV in CdSe\/CdS NPLs.\\cite{TessierNL,RabouwNL}\n Linewidth broadening in core\/shell NPLs was initially ascribed to the presence of traps\ninduced upon shell coating.\\cite{TessierNL} Graded interface composition was then shown to \nnarrow the line width down to $\\sim 55$ meV,\\cite{RossinelliCM,KelestemurACS} \nbut this figure is still larger than in core-only NPLs, which suggests that interface \ndefects are not the only source of broadening.\n\nTo shed light into this problem, Antolinez and co-workers recently investigated the origin \nof the fluorescence line width broadening in CdSe\/CdS NPLs by means of single-particle spectroscopy.\\cite{AntolinezNL} \nThey observed that individual NPLs present a series of 2 to 4 narrow peaks split from each other by $\\sim 10$ meV.\nAltogether, the peaks fit well the asymmetric lineshape of ensemble NPLs at cryogenic temperatures.\\cite{TessierNL}\nA similar feature was soon after reported in core-only CdSe NPLs.\\cite{AntolinezNL2}\nThe nature of these peaks was tentatively ascribed to SU processes of negative trions (X$^-$).\nThese are partly radiative Auger processes, whereby an electron-hole pair recombines radiatively but \ntransfers part of its energy to the remaining electron by exciting it into a higher single-electron level\n(in-plane excitation).\n They have been previously reported in epitaxial quantum wells\\cite{NashPRL,FinkelsteinPRB,BryjaPRB,DzyubenkoPRB} \nand self-assembled quantum dots\\cite{PaskovPE} under the magnetic fields, corresponding \nto inter-Landau level excitations of the excess carrier.\\cite{HawrylakPRB}\n Clarifying the role of SU processes in the emission of colloidal NPLs is then a desirable step\nto fully understand and control the emission line width, which would be advantageous for optical \napplications.\n\nIn this work, we analyze the possible occurence of SU processes in colloidal CdSe-based NPLs\nfrom a theoretical perspective. The goal is to determine which physical conditions enable \nthese processes.\nTo this end we use effective mass models and full CI simulations,\nwhich provide an intuitive description of the underlying physics.\n We shall confirm that at least one intense SU replica can be expected for $X^-$\nupon electron-hole recombination, in both core-only and core\/shell NPLs, \ncorresponding to the excitation of the remaining electron into a higher orbital \nwith the same symmetry as the ground state.\nFor this to take place, the trion must be weakly bound to an off-centered acceptor impurity.\nThe role of the impurity is to lower the system symmetry, thus relaxing selection rules, \nand to stimulate electron-electron repulsion (quench electron-hole attraction) in the \nground orbital.\n By doing so, SU peaks can reach intensities exceeding 10\\% of the fundamental (band edge, fully radiative) transition. \nThis is one order of magnitude higher than in epitaxial quantum wells, which can be rationalized\nfrom the stronger Coulomb interactions, which result from the pronounced dielectric confinement,\nand the presence of lateral sidewalls, which are prone to surface traps.\n We discuss connections with experiments in the literature and propose potential \n strategies to suppress these processes.\n \n\n\\section{Results}\n\nWe analyze the emission spectra of trions in core-only and core\/shell NPLs. \nNegative trions are studied unless otherwise noted, as it is the\nmost frequently reported species in these structures, but the conclusions do not\ndepend on the sign of the charged exciton (see Fig.~S2 in the supporting information, SI).\nOnce the general behavior of SU processes in these systems is understood, \nwe discuss how our conclusions fit the interpretation of different experimental \nobservations and the practical implications of our findings.\n\n\n\\subsection{Core-only NPLs}\n\nWe start by studying core-only CdSe NPLs. The NPLs are chosen to have $4.5$ monolayer (ML) thickness \nand a lateral size of $20 \\times 20$ nm$^2$, for similarity with the core dimensions \nof Ref.~\\cite{AntolinezNL}. They have a pronounced dielectric mismatch with the organic \nenvironment, which we model with $\\epsilon_{in}=6$ and $\\epsilon_{out}=2$ as dielectric \nconstants inside and outside the NPL, unless otherwise stated.\\cite{Sadao_book,AchtsteinNL}\n The presence of few-meV spectral jumps in photoluminescence experiments\\cite{AntolinezNL} \nsuggests that the trion is subject to the influence of carriers temporarily trapped on \nthe surface.\\cite{RabouwNL,BeylerPRL} \nTo model this phenomenon, a fractional point charge is placed on the surface, \nwith charge $Q=e \\, Q_X$ ($|Q_X| \\leq 1$ and $e$ the full electron charge).\nThe fractional value of $Q_X$ accounts for the screening of trapped charged (e.g. hole)\nby the trap defect itself (e.g. surface dangling bond).\\cite{CalifanoNL}\nTwo scenarios are considered: a charge centered on the top facet ($Q_{top}$) \nand an off-centered charge, located along the edge of a lateral facet ($Q_{edge}$). \nThe latter setup is suggested by studies showing that edge and vertex atoms in CdSe \nstructures have weaker binding to oleate ligands.\\cite{DrijversCM}\nThe two systems are represented in Figure \\ref{fig1}a and \\ref{fig1}b.\nThe corresponding emission spectra are shown in Fig.~\\ref{fig1}c and \\ref{fig1}d. \nThe figure reveals a number of important observations.\n(i) In the absence of surface charge ($Q_X=0$, thick lines),\nonly the fundamental transition shows up, with no sizable SU replica. \n(ii) Charges on the top facet induce SU peaks (see arrow in Fig.~\\ref{fig1}{c}),\nbut their strength is two orders of magnitude smaller than that of the \nfundamental transition (main line). This is similar to the case of epitaxial quantum wells.\\cite{NashPRL,FinkelsteinPRB,BryjaPRB,DzyubenkoPRB}\n(iii) Stronger SU replica are however obtained for charges located on the \nlateral sidewall, provided the charge is attractive (acceptor impurity) \nand binding to the trion is moderately weak, see Fig.~\\ref{fig1}d.\nFor $Q_{edge}=0.4$ (marked with a star in the figure),\nthe SU peak reaches $\\sim 25$\\% of the main peak height.\nThis ratio is about 20 times higher than in epitaxial quantum wells,\nand it holds despite the Giant Oscillator Strength enhancing the band edge \nrecombination,\\cite{FeldmannPRL,PlanellesACSph,IthurriaNM,AchtsteinNL}\nwhich suggests that SU satellites also benefit from this phenomenon.\nFor $Q_{edge}>0.4$, however, the SU peak intensity is lowered again\nand the energy splitting (redshift) with respect to the main line increases.\nSecond and third SU lines are built for strong surface charges \n(see inset in Fig.~\\ref{fig1}d at $Q_{edge}=0.7$), \nbut their magnitude is negligible.\nWe have also explored different locations of the charge, obtaining intermediate\nresults between those shown in Fig.~\\ref{fig1} (see Fig.~S3 in SI). \nThese results point out the potentially significant role of lateral sidewalls, \nwhich are characteristic feature of colloidal quantum wells as compared to epitaxial ones, \nin obtaining high SU peaks.\n\n\n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=8cm]{fig1.eps}\n\\caption{\n(a,b) Schematics of core-only NPLs with different location of the surface charge. \n(c,d) Corresponding X$^-$ emission spectrum for charge strength $Q=Q_X\\,e$. \nThe arrows point at the SU satellites \n(dotted lines are guides to the eyes). The highest SU peak is observed \nfor off-centered acceptor charges weakly bound to the trion ($Q_{edge} = 0.4$, \nmarked with a star in (d)). \nThe spectra are normalized to the intensity of the fundamental transition\nat $Q_X=0$, and offset vertically for clarity.\nThe insets for $Q_{edge}=0.7$ in (d) show amplified SU peaks.\n}\\label{fig1}\n\\end{figure}\n\nTo gain understanding on the origin of strong SU peaks when trions bind\nto lateral surface acceptors, beyond the full numerical calculation\nof Fig.~\\ref{fig1}, in Fig.~\\ref{fig2}a and \\ref{fig2}b we \ncompare sketches of the SU processes,\nin the absence and presence of an attractive edge charge.\nWithin effective mass theory, the conduction band \nand valence band energy levels of (non-interacting) \nelectrons and holes can be described as particle-in-the-box states, \nwith quantum numbers $(n_x,n_y,n_z)$. \nIt is useful however to label the states by their symmetry \n(irreducible representation).\nWhen $Q_{edge}=0$, because the NPL has squared shape, the \npoint group is $D_{4h}$. When $Q_{edge}\\neq 0$, the electrostatic\npotential yields a symmetry descent to $C_s$.\nAs a consequence, degeneracies are lifted and additional\nstates with the same symmetry as the ground orbital ($A'$)\nare obtained. This is important because after electron-hole\nrecombination, the excess electron can only be excited to\nan orbital with the same symmetry as the initial one\n(vertical arrows in Fig.~\\ref{fig2}a and \\ref{fig2}b).\nTherefore, lowering the system symmetry opens new channels\nfor SU processes. Furthermore, these can involve low-energy \norbitals, which have fewer nodes and will then have larger\noverlap with the trion ground state, as we shall see below. \nBoth the number and the intensity of the SU processes \nare in principle enhanced. \nBy contrast, a centered charge on the top surface barely affects \nthe system symmetry, which remains high ($C_{4v}$), and\nSU processes are only slightly stronger than in the $Q_{edge}=0$ case.\n\n\n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=13cm]{fig2.eps}\n\\caption{\n(a,b) Sketch of SU processes in NPLs with (a) and without (b) an edge charge.\nLabels on the left are $(n_x, n_y, n_z)$ quantum numbers\nfor the (independent particle) energy levels.\nLabels on the right are the corresponding irreducible representation. \nThe surface charge lowers the point group symmetry, \nfrom $D_{4h}$ to $C_s$, lifting degeneracies and enabling\nnew channels for SU transitions (vertical arrows).\n(c,d) Two main configurations $|m_{X^-}\\rangle$ in the CI expansion of $|GS_{X^-}\\rangle$,\nwith and without edge charge. Thin (thick) arrowsheads denote electron (hole) spin.\nOnly when $Q_{edge}\\neq 0$ a SU process is expected.\n(e) Energy splitting between $|1_{X^-}\\rangle$ and $|2_{X^-}\\rangle$ at an \nindependent particle level.\n(f) average value of electron-electron repulsion and (g) electron-hole\nattraction in configurations $|1_{X^-}\\rangle$ and $|2_{X^-}\\rangle$.}\\label{fig2}\n\\end{figure}\nThe qualitative reasoning above can be substantiated with a \nCI formalism on the basis of independent particle (non-interacting) \nelectron and hole states, which has the additional advantage of giving \nintuitive insight on how Coulomb interactions affect the likelihood of SU processes.\nWe consider that the transition rate from the trion ground state \n$|GS_{X^-}\\rangle$ to an electron spin-orbital $|f_e\\rangle$, \nis proportional to:\\cite{Pawel_book}\n\\begin{equation}\nP_{GS \\rightarrow f} = \\left| \\langle f_e | \\,{\\hat P}\\, | GS_{X^-} \\rangle \\right|^2 .\n\\label{eq:trans}\n\\end{equation}\n\\noindent ${\\hat P}$ is the dipolar transition operator,\n${\\hat P} = \\sum_{i_e,i_h} \\, \\langle i_e | i_h \\rangle \\, e_{i_e}\\,h_{i_h}$,\nwhere $e_{i_e}$ and $h_{i_h}$ are annihilation operators for independent electron\nand hole spin-orbitals $|i_e\\rangle$ and $|i_h\\rangle$, respectively.\nWe describe the trion ground state with a CI expansion,\n\\begin{equation}\n|GS_{X^-} \\rangle = \\sum_m c_m \\, | m_{X^-} \\rangle, \n\\end{equation}\n\\noindent where $|m_{X^-}\\rangle$ is a trion configuration:\n$|m_{X^-} \\rangle = e_{r_e}^\\dagger e_{s_e}^\\dagger |0 \\rangle_e \\, h_{t_h}^\\dagger |0 \\rangle_h$,\n with $e_{r_e}^\\dagger$ and $h_{t_h}^\\dagger$ creator operators,\n $|0\\rangle_e$ and $|0\\rangle_h$ the vacuum occupation vectors of electron and hole, \nand $c_m$ the coefficient in the expansion.\n Inserting ${\\hat P}$ and $|GS_{X^-}\\rangle$ into Equation (\\ref{eq:trans}), \n one obtains:\n\\begin{equation}\nP_{GS \\rightarrow f} = \n\\left| \\sum_m c_m \\, \\left( \\langle r_e | t_h \\rangle\\, \\delta_{f_e\\, s_e} \n- \\langle s_e | t_h \\rangle \\delta_{f_e\\, r_e} \\right) \\right|^2.\n\\label{eq:trans2}\n\\end{equation}\n\\noindent In SU processes, $|f_e\\rangle$ is an excited spin-orbital. \nIt then follows from Equation (\\ref{eq:trans2}) that such a transition will \nonly take place if $|GS_{X^-}\\rangle$ contains at least one configuration $|m_{X^-}\\rangle$\nin the CI expansion where one electron is in the excited spin-orbital \nand the other electron has finite overlap with the hole ground state \n($|s_e\\rangle = |f_e\\rangle$ and $\\langle r_e | t_h \\rangle \\neq 0$\nor $|r_e\\rangle = |f_e\\rangle$ and $\\langle s_e | t_h \\rangle \\neq 0$). \nThe larger the weight of this configuration, $|c_m|^2$, the more likely the SU process.\nIt is worth noting that in the strong confinement limit, the trion ground state is well\ndescribed by a single configuration where all carriers are in the lowest-energy spin-orbitals\n(configuration $|1_{X^-}\\rangle$ in Fig.~\\ref{fig2}c and \\ref{fig2}d).\nThat is, $c_1 \\approx 1$ and $c_m \\approx 0$ for $m > 1$. \nSU transitions are then forbidden, which is why SU peaks are rarely reported \nin quantum dots.\nOn the contrary, in systems where Coulomb interaction energies exceed quantum\nconfinement energies, the CI expansion contains mono- and biexcitations of electrons.\nSU processes are then enabled.\nColloidal NPLs constitute an ideal system at this regard, because they combine weak\nconfinement in the lateral direction with strong Coulomb \ninteractions.\\cite{RajadellPRB,RichterPRM}\nHereafter, we refer to this condition ($c_m \\neq 0$ for $m > 1$) as Coulomb admixture.\n\nThe role of Coulomb correlation and symmetry breaking in activating SU processes\ncan be illustrated, in the simplest approximation, by considering the two lowest-energy \nconfigurations of the\ntrion ground state, \n\\begin{equation}\n|GS_{X^-}\\rangle \\approx c_1 |1_{X^-}\\rangle + c_2 |2_{X^-}\\rangle. \n\\end{equation}\nIn Fig.~\\ref{fig2}c and \\ref{fig2}d we depict such configurations\nin the absence and presence of an edge charge, respectively.\nThese can be expected to be the two most important configurations in the full CI expansion.\nNotice that the two configurations must have the same symmetry, \nfor Coulomb interaction to couple them.\nBecause the lowest-energy configuration, $|1_{X^-}\\rangle$, is always totally symmetric,\nso must be $|2_{X^-}\\rangle$.\nThus, when $Q_{edge}=0$ ($D_{4h}$ group), the electronic configuration of $|1_{X^-}\\rangle$ is\n$[A_{1g}^2]_e \\, [A_{1g}]_h$, and that of $|2_{X^-}\\rangle$ is $[E_{u}^2]_e \\, [A_{1g}]_h$. \nThe recombination of the $E_u$ electrons with the hole, which stays in a $A_{1g}$ orbital, is then \nsymmetry forbidden ($\\langle r_e | t_h \\rangle = \\langle s_e | t_h \\rangle = 0$ in Eq.~(\\ref{eq:trans2})).\nBy contrast, when $Q_{edge} \\neq 0$ ($C_s$ group), $|2_{X^-}\\rangle$ is formed by a monoexcitation where\none electron is placed in the $(n_x,n_y,n_z)=(2,1,1)$ orbital, which also has $A'$ symmetry, resulting\nin an electronic configuration $[A'\\,A']_e \\, [A']_h$ (see Fig.\\ref{fig2}d).\n\n The hole can then recombine with the ground orbital electron, as both have $A'$ symmetry \n ($\\langle r_e | t_h \\rangle \\neq 0$ or $\\langle s_e | t_h \\rangle \\neq 0$ in Eq.~(\\ref{eq:trans2}))\nand leave the excited electron as the final state. \nThis constitutes a SU process. \n Because both SU and fundamental transition rely on the recombination of the same electron-hole pair\n (same overlap integral, e.g. $\\langle r_e | t_h \\rangle$), the ratio between SU and fundamental\n radiative rates can be approximated as:\n %\n \\begin{equation}\n\t \\frac{ P_{GS \\rightarrow (2,1,1)_e} }{ P_{GS \\rightarrow (1,1,1)_e} } \\approx \n\t \\frac{|c_2|^2}{|c_1|^2}.\n \\end{equation}\n %\n \\noindent i.e. it is set exclusively by the degree of Coulomb admixture.\n\nOne can guess the requirements that maximize $|c_2|^2$ by looking which\nconditions favor energetically $|2_{X^-}\\rangle$ over $|1_{X^-}\\rangle$. \nThese include: (i) small energy splitting between the two configurations, \nat an independent particle level, $\\Delta_{sp}$ in Fig.~\\ref{fig2}d, \n(ii) weaker electron-electron repulsion ($V_{ee}$) and \n(iii) stronger electron-hole attraction ($V_{eh}$) in $|2_{X^-}\\rangle$ as compared to $|1_{X^-}\\rangle$.\nFigures ~\\ref{fig2}e-g show that these conditions are met for moderately attractive (positive) \ncharges ($Q_{edge} \\sim 0.3-0.4$).\nWhen the off-centered charge is switched on, $\\Delta_{sp}$ rapidly decreases (see Fig.~\\ref{fig2}e) because \nthe symmetry descent turns one of the $E_u$ ($p$-like) electron orbitals into a $A'$ ($s$-like) one.\nHowever, the surface charge brings about electrostatic confinement and hence $\\Delta_{sp}$ increases again soon after.\nAs for inter-electron repulsion, $\\langle 1_{X^-} | V_{ee} | 1_{X^-} \\rangle$ \nincreases more rapidly than $\\langle 2_{X^-} | V_{ee} | 2_{X^-} \\rangle$\n(see Fig.~\\ref{fig2}f) because the former involves placing the two electrons in \nidentical orbitals, while the latter does not.\n\nLast, $\\langle 1_{X^-} | V_{eh} | 1_{X^-} \\rangle$ is rapidly quenched (see Fig.~\\ref{fig2}g) \nbecause it involves the ground orbitals of electron and hole --$(1,1,1)_e$ and $(1,1,1)_h$--, \nwhich dissociate rapidly under an external charge.\n$\\langle 2_{X^-} | V_{eh} | 2_{X^-} \\rangle$ stays strong up to $Q_{edge} \\sim 0.3$ because it involves the $(2,1,1)_e$ orbital,\nwhich is spatially more extended and then keeps significant overlap with the $(1,1,1)_h$ hole. \nFigs.~\\ref{fig2}e-f further evidence that $Q_{edge} > 0.3-0.4$ is inconvenient for SU processes, \nbecause the electrostatic potential increases lateral \nquantum confinement ($\\Delta_{sp}$ increases) and because electrons and hole in configuration $|2_{X^-}\\rangle$ are\neventually dissociated as well ($\\langle 2_{X^-} | V_{eh} | 2_{X^-} \\rangle$ is quenched in Fig.~\\ref{fig2}g).\n\n\n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=8cm]{fig3.eps}\n\\caption{In-plane charge density of the two electrons and hole in the X$^-$ ground state (top rows), \nand wave functions of the two lowest electron orbitals with $A'$ symmetry (bottom rows), \nas a function of the edge charge magnitude. The edge charge is located on the top edge, in this view. \nThe strongest SU peak corresponds to $Q_{edge}\\approx 0.4$,\nwhen the X$^-$ electron charge density reveals a clear contribution from $(2,1,1)_e$, and the hole\nis not yet fully dissociated from electrons.\n\t}\\label{fig3}\n\\end{figure}\n\nMuch of the above observations can be visualized by analyzing the evolution\nof charge densities and wave functions under $Q_{edge}$. \nIn Figure \\ref{fig3} we show the two-electron (first row)\nand one-hole (second row) charge densities of $|GS_{X^-}\\rangle$, \nobtained from the CI calculations of Fig.~\\ref{fig1}.\nThe wave functions of the two lowest electron orbitals which can constitute \nconfiguration $|2_{X^-}\\rangle$, --$(n_x,n_y,n_z)_e=(1,1,1)_e$ and $(2,1,1)_e$-- \nare also plotted (bottom rows).\nAt $Q_{edge}\\approx 0$, the two orbitals are quasi-orthogonal. \nAs a result, Coulomb interaction cannot couple configurations \n$|1_{X^-}\\rangle$ and $|2_{X^-}\\rangle$.\n and $c_2 \\approx 0$. This is why the two-electron charge density \nclosely resembles the $(1,1,1)_e$ orbital. SU processes are not expected in this case.\n\nAt $Q_{edge} \\approx 0.4$, symmetry lowering and energetic considerations \nenable efficient Coulomb coupling. \nThe oval shape of the two-electron charge density reflects a significant contribution \nfrom $(2,1,1)_e$ to $|GS_{X^-}\\rangle$ (i.e. $|c_2| > 0$). \nAt the same time, the electron $(1,1,1)_e$ orbital and the hole ground state have \nsizable overlap. This is an optimal situation for the appearance for\nthe transition $P_{GS \\rightarrow (2,1,1)_e}$ to show up as a SU process,\naccording to Equation (\\ref{eq:trans2}).\nFurther increasing $Q_{edge}$ separates the $(2,1,1)_e$ electron orbital from\nthe hole. Coulomb attraction is then weaker, making $c_2$ and consequently \n$P_{GS \\rightarrow (2,1,1)_e}$ small again.\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=6cm]{fig4.eps}\n\\caption{Normalized X$^-$ emission as a function of the environment dielectric constant.\nWith increasing dielectric contrast, the SU peak increases and becomes more redshifted.\nFor every value of $\\epsilon_{out}$, the value of $Q_{edge}$ that maximizes SU transitions\nis shown. In all cases, $\\epsilon_{in}=6$.}\\label{fig4}\n\\end{figure}\n \nWe have argued above that strong Coulomb admixture of configurations facilitates the appearance of SU processes.\nA distinct feature of colloidal NPLs when compared to epitaxial quantum wells is the presence\nof a prounounced dielectric contrast with the organic ligands surrounding the NPL,\nwhich enhances Coulomb interactions by effectively reducing the system dielectric screening.\\cite{AchtsteinNL,RajadellPRB,BenchamekhPRB}\nTo study the influence of this phenomenon over SU transitions, in Figure \\ref{fig4}\nwe compare the trion emission spectrum for different values of the environment\ndielectric constant $\\epsilon_{out}$, while fixing that of the NPL to the high-frequency\nCdSe value, $\\epsilon_{in}=6$. For the sake of comparison, the emission spectrum is \nnormalized so that the band edge peak has the same intensity in all cases. \nAlso, we have selected the value of $Q_{edge}$ that maximizes the relative size of the SU peak in each case.\nBecause $\\epsilon_{out}$ screens the surface charge electrostatic field, \nlarger $Q_{edge}$ values are needed when $\\epsilon_{out}$ increases.\nThe figure evidences that lowering $\\epsilon_{out}$ increases the SU peak height \nand energetic redshift.\nFor typical ligands of CdSe NPLs (e.g. oleic acid), $\\epsilon_{out} \\sim 2$.\\cite{AchtsteinNL,EvenPCCP}\nWe then conclude that dielectric confinement makes SU processes in colloidal NPLs\nmore conspicuous.\n\n\\subsection{Core\/shell NPLs}\n\nWe next consider heterostructured core\/shell NPLs. The first case under study are CdSe\/CdS NPLs.\\cite{TessierNL,AchtsteinACS,PolovitsynCM,LlusarJPCc} \n The NPLs have the same CdSe core as in the previous section and 6 ML thick CdS shells\non top and bottom (see inset in Figure \\ref{fig5}a).\nIn general, the behavior of SU replicas is found to be analogous to that of core-only NPLs. \nAn off-centered acceptor impurity is needed to yield sizable SU replicas,\nwith an optimal value of $Q_{edge}$ maximizing the relative size of the SU peak. \n\nFigure \\ref{fig5}a shows the emission spectrum of X$^-$ for the optimal $Q_{edge}$ value,\nin CdSe\/CdS NPLs (green line) against CdSe core-only NPLs (black, dashed line).\nOne can see that the SU replica of the CdSe\/CdS structure is again significant\n (11\\% of the main transition), but less pronounced than in the core-only structure \n (26\\%). \n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=13cm]{fig5.eps}\n\\caption{(a) Normalized X$^-$ emission spectrum a CdSe\/CdS NPL with 6 ML-thick shell (solid line), \ncompared to that of a core-only CdSe NPL (dotted line). \nThe spectra are centered at the energy of band edge transition.\n$Q_{edge}=0.6$ ($0.4$) is used for the CdSe\/CdS NPL (core-only NPL), to maximize the relative height of SU lines.\nThe SU peak for the core\/shell system is smaller than for core-only NPLs. \n\n(b,c) Average Coulomb integrals of $|GS_{X^-}\\rangle$ configurations \n$|1_{X^-}\\rangle$ and $|2_{X^-}\\rangle$: (b) electron-electron repulsion, (c) electron-hole attraction.\nSolid (dotted) lines are used for core\/shell (core-only) NPLs. The interactions\nare weaker in the core\/shell structure.\n(d) Charge densities of\nelectrons (left) and hole (right) for the trion ground state in the CdSe\/CdS NPL at $Q_{edge}=0.6$. \nThe electron stays in the vicinity of the core, despite the shallow band offset.}\\label{fig5}\n\\end{figure}\nThe smaller SU replica in the core\/shell structures is a robust result, which holds\nfor different shell thickness and surface charge locations. It is a consequence of the\nweaker Coulomb interactions. The electron leakage into the CdS shell reduces \nelectron-electron repulsions and electron-hole attractions. \n The quenching of dielectric confinement by the CdS shell, which pushes organic ligands far from\nthe core, further contributes to the weakening.\nThis observation is reflected by Figs.~\\ref{fig5}b and \\ref{fig5}c, \nwhich show that Coulomb interactions (especially $V_{ee}$) are weakened in core\/shell NPLs \n(solid lines) as compared to core-only NPLs (dotted lines). \nConfiguration $|2_{X^-}\\rangle$ is then less stabilized with respect to $|1_{X^-}\\rangle$,\nwhich implies smaller $|c_2|$ coefficient in the CI expansion.\n\nFigure \\ref{fig5}d compares the charge density of the two electrons (left) \nand hole (right) in $|GS_{X^-}\\rangle$.\n The trion electrons are found to stay in the vicinity of the core, rather than delocalizing all over the structure, \n to benefit from interaction with the hole. \n This is consistent with the observed behavior of CdSe\/CdS NPLs being similar to that\n of core-only structures, albeit with weakened Coulomb interactions due to the lessened confinement.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=6cm]{fig6.eps}\n\\caption{(a) Normalized X$^+$ emission spectrum in a CdSe\/CdTe NPL\nwith 6 ML thick shells, as a function of the lateral charge strength.\nDotted line is a guide to the eye. SU peaks and fundamental transition\nhave comparable intensities.\n(b) Two main $|GS_{X^+}\\rangle$ configurations in the CI expansion\nin the presence of a charge. The weight of $|2_{X^+}\\rangle$ is comparable\nto that of $|1_{X^+}\\rangle$ in this system, which explains the high SU peaks in (a).\n(c) Wave function of $(1,1,1)_h$ and $(1,1,2)_h$ hole orbitals under $Q_{edge}=-0.5$.\nThe states have the same symmetry but localize on opposite sides of the core\nto stay orthogonal.}\\label{fig6}\n\\end{figure}\n\nUnderstanding the conditions which promote SU processes allows us to devise\nstructures where their impact would be maximal. In Fig.~\\ref{fig6} we consider \na core\/shell NPL with the same dimensions as before, but CdSe\/CdTe composition.\nThe NPL is chosen to be charged with a positive trion (X$^+$).\nBecause of the type-II band alignment, the electron stays in the CdSe\ncore and the holes in the CdTe region, as observed in related core\/crown structures.\\cite{AntanovichNS,KelestemurJPCc}\nIn the absence of external charges, the two first hole orbitals are $(1,1,1)_h$ and $(1,1,2)_h$, \ni.e. the symmetric ($A_{1g}$) and antisymmetric ($A_{1u}$) solutions of the double well potential, respectively,\nwhich are almost degenerate because tunneling across the core is negligible\n(i.e. $\\Delta_{sp} \\rightarrow 0$). Switching on a negative surface charge, \n$Q_{edge} < 0$, lifts the inversion symmetry so that both orbitals acquire $A'$ symmetry\nand can be Coulomb coupled. \nThe admixture between configurations $|1_{X^+}\\rangle$ and $|2_{X^+}\\rangle$, \ndepicted in Fig.~\\ref{fig6}b, is then very strong.\n In the presence of the charge, the two hole orbitals tend to localize \non opposite shell sides to remain orthogonal, as shown in Fig.~\\ref{fig6}c. \nThis implies that configuration $|1_{X^+}\\rangle$, which has two holes\nin the same orbital, has much stronger repulsion than configuration $|2_{X^+}\\rangle$, \nwhich distributes the two electrons on opposite sides of the core. This makes\n$\\langle 1_{X^+} | V_{hh} | 1_{X^+} \\rangle \\gg \\langle 2_{X^+} | V_{hh} | 2_{X^+}\\rangle$. \nAltogether, the small $\\Delta_{sp}$ value and the large difference in hole-hole repulsion\nexplain the strong admixture between configurations $|1_{X^+}\\rangle$ and $|2_{X^+}\\rangle$.\nAs shown in Fig.~\\ref{fig6}a, this gives rise to SU peaks whose magnitude is almost\nas large as that of the fundamental transition ($72\\%$ for $Q_{edge}=-0.5$).\n\n\n\n\\section{Discussion}\n\nOur simulations show that SU processes can be expected for trions in core-only \nand core\/shell NPLs, if off-centered impurities are present. %\nWe discuss here the potential relationship of this finding with experiments\nand practical implications.\n\n\\subsection{Relationship with experiments}\n\nIn core-only CdSe NPLs, the low temperature photoluminescence is thought to arise \nfrom subpopulations of excitons and negative trions.\\cite{ShornikovaNL,YuAMI,AntolinezNL2} \nVery recently, Antolinez and co-workers have reported that the X$^-$ \nemission shows a distinct peak or a shoulder (depending on the film thickness)\nredshifted from the trion band edge transition. The redshift is $\\sim 19$ meV \nand the relative height $15-25\\%$ that of the main peak.\\cite{AntolinezNL2} \nThey speculated that the origin could be a SU process of the kind we study.\nOur calculations support the feasibility of this interpretation.\nFigure \\ref{fig1}a shows excellent agreement with the experimental measurements,\nboth in energy and relative intensity of the SU peak, \nassuming a lateral charge with $Q_{edge}=0.3-0.4$,\nwhich gives a redshift of $19-25$ meV and a relative height of $15-23$ \\%.\n\nThe presence of acceptor impurities in CdSe NPLs likely originates when\nthe hole of a photoexcited electron-hole pair is trapped by a surface\ndefect. The next electron-hole pair generated in the NPL joins the\nresidual electron to form X$^-$, while the trapped hole exerts \na screened electrostatic potential.\\cite{RabouwNL,FengNL,CalifanoNL}\nThe coexistence of X$^-$ and trapped surface charges in CdSe NPLs \nis backed up by studies reporting correlation between surface-to-volume ratio, \nlaser irradiation time and trion emission intensity.\\cite{YuAMI}\nA plausible location for surface charges are the\nlateral sidewalls of the NPL (as in Fig.~\\ref{fig1}b).\nThis possibility is suggested by studies showing that Z-type \nligand desorption --and hence surface traps-- in CdSe NPLs is more \nfrequent on these facets,\\cite{LeemansJPCL}\nand by the fact that CdSe\/CdS core\/crown NPLs generally improve the\nphotoluminescence quantum yield as compared to core-only structures,\ndespite having larger surfaces on top and bottom.\\cite{TessierNL2}\nBecause off-centered charges are needed to originate SU peaks, \nlateral charges are candidates to trigger such processes.\\\\\n\n\n\nIn core\/shell CdSe\/CdS NPLs, SU processes have been also proposed \nas the origin of multi-peaked fluorescence emission \n--and hence broadened line width--.\\cite{AntolinezNL}\nOur simulations in Fig.~\\ref{fig5}a confirm one can indeed expect \na sizable SU peak in such structures.\nWe note that earlier experimental studies had so far interpreted the\nline width broadening as a result of either SU processes\\cite{AntolinezNL}\nor of surface defects.\\cite{TessierNL} By showing that the second effect\nis a prerequesite for the first one, our study helps to reconcile both\ninterpretations.\nNonetheless, two remarkable disagreements are observed between our simulations\nand Ref.~\\cite{AntolinezNL} measurements. \nFirst, the experiments show from 2 to 4 emission peaks, \nwhich are interpreted as the X$^-$ fundamental transition plus up to three redshifted, SU peaks. \nIn our calculations, however, we fail to see more than one significant SU replica.\nSecond, the highest-energy peak in the experiment is never the brightest one. \nThis is inconsistent with our results and with earlier studies on epitaxial\nquantum wells and dots, where the higher-energy peak corresponds to the fundamental\ntransition, which is the most likely recombination channel.\\cite{NashPRL,FinkelsteinPRB,BryjaPRB,DzyubenkoPRB,PaskovPE}\n\nTentatively, one may suspect that a large number of SU peaks in core\/shell CdSe\/CdS NPLs \ncould be connected with the thick CdS shell (12 ML in Ref.~\\cite{AntolinezNL}), which\nmakes surface defects more likely than in core-only structures. \nA significant presence of defects in these structures has been hinted \nby studies showing that the long radiative lifetime is not due to electron delocalization \nbut to the influence of impurities.\\cite{AchtsteinACS}\nHowever, Coulomb interactions are weaker than in core-only structures\n(Fig.~\\ref{fig5}c,d), where only one SU peak has been measured.\\cite{AntolinezNL2} \nIt is then not surprising that, despite investigating different charge locations\n(Figs. S3, S6 and S7 in SI), conduction band-offset values (Fig. S4) and\nshell thicknesses (Fig.S5), we see at most one significant SU satellite.\n \nRegarding the relative intensity of the peaks,\n as mentioned in the previous section, the highest-energy one (fundamental transition) \n is proportional to the weight of configuration $|1_{X^-}\\rangle$ in the CI expansion, $|c_1|^2$, \nwhile subsequent (SU) peaks would be proportional to $|c_2|^2$, $|c_3|^2$, \\ldots\nConfiguration $|1_{X^-}\\rangle$ (all carriers in the ground orbital, Fig.~\\ref{fig2}c) \nis nodeless and hence naturally expected to be the dominant one, \nso the highest-energy peak is also the brightest one.\nWe have not observed SU peaks exceeding the fundamental transition height\n despite considering different charge locations and shell thicknesses (see SI). \n Even in CdSe\/CdTe NPLs, which constitute a limit case, \n SU peaks never exceed the height of the main transition, see Fig.~\\ref{fig6}a.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=6cm]{fig7.eps}\n\\caption{(a) Normalized X$^-$ emission spectrum in a CdSe\/CdS NPL,\n as a function of the lateral charge strength.\nAn electron spin relaxation bottleneck is imposed, so that emission comes from \nthe lowest singlet ($S_e=0$, ground state) and triplet ($S_e=1$) states.\nDotted lines are guides to the eyes.\n(b-c) Sketches showing the relevant electron-hole recombination channels\nof singlet and triplet states. (b) The singlet can give rise to one fully radiative \n($s$-$R1$) plus one SU transition ($s$-$SU$). \n(c) The triplet can give rise to two fully radiative transitions, $t$-$R1$\nand $t$-$R2$.}\\label{fig7}\n\\end{figure}\n\nAs an alternative interpretation for the experiments, a multi-peaked emission spectrum could \nresult from stacking of colloidal NPLs,\\cite{DirollNL} which leads to electronic \ncoupling through dielectric confinement.\\cite{MovillaJPCL}\nHowever, the time-dependent spectral shifts observed by Antolinez \\emph{et al.} \nsuggest that all peaks arise from the same NPL, and significant stacking was not expected\nin the experiment samples.\\cite{AntolinezNL}\nWe thus propose a different interpretation. Namely, simultaneous emission from the X$^-$ \nground state, with singlet electron spin ($S_e=0$), and a metastable excited state with triplet\nelectron spin ($S_e=1$). The decay from the triplet to the singlet state is slowed down \nby spin selection rules, as phonons are spinless. This could allow simultaneous occupation\nof the two states even if the energy splitting exceeds thermal energy.\n \nTo illustrate this point, in Figure \\ref{fig7}a we show the calculated emission of X$^-$ \nassuming equipopulation of $S_e=0$ and $S_e=1$ trion states. One can see that the number \nof sizable peaks in the spectrum ranges from two to four, depending on the \nstrength of surface charge, $Q_{edge}$.\n The origin of these peaks is summarized in the sketches of Fig.~\\ref{fig7}b and \\ref{fig7}c.\nThe singlet (Fig.~\\ref{fig7}b) can give rise to a fully radiative transition ($s$-$R1$) \nand a SU transition ($s$-$SU$), as described in the previous sections. \nIn turn, the triplet (Fig.~\\ref{fig7}c) can give rise to two fully radiative transitions \n($t$-$R1$ and $t$-$R2$), depending on which electron recombines with the hole.\n$t$-$R2$ is readily visible at $Q_{edge}=0$, but $t$-$R1$ requires recombining the hole \nwith an excited electron, a process which is again activated when the surface charge \nlifts symmetry restrictions.\nHowever, unlike in SU processes, the two triplet transitions come from the main configuration \nof the trion CI expansion. Therefore, their intensity can be comparable to that of the band\nedge transition, $s$-$SU$, even if Coulomb admixture is weak.\nThe triplet transitions are built on both sides of $s$-$SU$, with inter-peak energy \nsplittings up to few tens of meV.\nThe relative sizes of the peaks will be further modulated in realistic situations\nby a finite triplet-singlet decay rate. This relaxation channel would possibly reduce the relative \npopulation of $S_e=1$, and hence the intensity of $t$-$R1$. \n\nAltogether, the number of peaks, the magnitude of the energy splitting between the peaks\nand the flexible intensities provide a framework to explain the \nmulti-peaked photoluminescence of Ref.~\\cite{AntolinezNL}.\nSeveral other aspects of this proposal are consistent with the experiments.\nFor example, because all peaks in Fig.~\\ref{fig7}a arise from the same NPL, \nthey will experience simultaneous spectral shifts when surface impurities migrate.\\cite{AntolinezNL} \nAlso, the hot trion emission is expected to vanish when the impurities are removed, \nas $t$-$R1$ becomes deactivated and $t$-$R1$ almost merges with the singlet emission, $s$-$R1$,\nsee Fig.~\\ref{fig7}a for $Q_{edge}=0$. This fits the transition from asymmetric to symmetric band shape\nas temperature increases.\\cite{TessierNL}\n\nThe fact that triplet emission is observed in CdSe\/CdS NPLs, but not in CdSe ones,\nmay be explained from the strong spin-spin interaction of resident carriers and \nsurface dangling bonds in the latter case,\\cite{ShornikovaNN}\nwhich should speed up spin relaxation through flip\/flop processes.\nThis mechanism is expected to be inhibited in core\/shell structures, because\nX$^-$ carriers stay far from the surface, as shown in Fig.~\\ref{fig5}d.\nOn the other hand, the triplet trion is expected to have fine structure\nthrough electron-hole exchange interaction\\cite{WarePRL}, \nwhich may not fit the mono-exponential photoluminescence decay reported in Ref.~\\cite{AntolinezNL}. \nFurther experiments are needed, e.g. on polarisation of the different peaks under external fields\\cite{ShornikovaNL,JovanovPRB},\nto confirm the different spin of the emissive states in CdSe\/CdS NPLs. \n\nThe observation of metastable triplet trion photoluminescence has been previously reported \nin epitaxial quantum wells\\cite{BryjaPRB,ShieldsPRB} and dots\\cite{JovanovPRB},\nand more recently in transition metal chalcogenide monolayers.\\cite{VaclavkovaNT\nTo our knowledge, however, its presence in colloidal nanostructures has not been confirmed.\n\n\n\\subsection{Control of SU processes}\n\nInasmuch as SU processes can be responsible for the line width broadening NPLs,\ntheir supression is desirable to improve color purity in optical applications.\nIt has been suggested that this job could be achieved by increasing quantum confinement,\nreducing either lateral dimensions or shell thickness --the latter would favor \nelectrostatic confinement.\\cite{AntolinezNL} \nBoth strategies have the drawback of introducing size dispersion in ensemble\nluminescence. \nFrom our theoretical analysis, we confirm that reducing Coulomb admixture \nwould minimize SU processes, but this can be achieved by weakening Coulomb interactions \ninstead of increasing quantum confinement. For example, reducing dielectric \nconfinement or using thinner cores to enhance the quasi-type-II character \nshould contribute to this goal. Obviously, this approach would have the \ndrawback of reducing the band edge recombination rate as well.\n\nAlternatively, since our study shows that impurities are ultimately responsible\nfor SU processes, experimental routes to suppress SU processes could be directed \nto control of traps. Appropriate choice of surface ligands\\cite{LeemansJPCL}, electrochemical \npotentials\\cite{GallandNAT} and interface alloying\\cite{RossinelliCM,KelestemurACS} \ncould contribute to this end. \n\nBecause we find surface charges on lateral sidewalls particularly suited to induce SU processes,\nthe growth of core\/crown heterostructures is expected to reduce their influence by \nkeeping the outer rim away from the photogenerated carriers. \nThis suggestion seems to agree with experimental observations by Kelestemur and co-workers, \nindicating that core\/crown\/shell CdSe\/CdS NPLs have more symmetric emission behavior than core\/shell \nones at cryogenic temperatures,\\cite{KelestemurAFM}\nThis can be understood as a consequence of the suppression of SU processes in \nthe low-energy tail of the emission band. \nIt is also consistent with recent single-particle studies showing that the \nline width in CdTe\/CdSe core\/crown NPLs is set by LO phonon replica, \nrather than SU ones.\\cite{SteinmetzJPCc}\n\n\nShould the role of metastable triplet states be confirmed in CdSe\/CdS NPLs, \nstrategies to control the line width should rather focus on enhancing the interaction of\nconfined carriers with surface spins\\cite{ShornikovaNN} or intrinsic spin-orbit interaction\\cite{TadjinePRB}, \nto shorten their lifetime.\nReplacing trion by neutral exciton emission through thermal dissociation,\\cite{AyariNS} \nis yet another possibility to avoid SU and high spin peaks.\n\nThus, our calculations propose a wealth of experiments targeted at material design to \ntune quantum and dielectric confinement, and exciton-surface\/interface interactions, \nand set suitable temperature ranges to control SU\/triplet emission.\n\n\\section{Conclusions}\n\nWe have shown that SU processes in colloidal NPLs are enabled by severe Coulomb admixture \n--which results from strong Coulomb interactions and weak lateral confinement-- \nand the presence of off-centered electrostatic traps, which suppress the\n protection against Auger processes provided by symmetry conservation. \nSurface charges on lateral sidewalls seem particularly efficient to this end. \n\nUnder typical experimental conditions, core-only and core\/shell NPLs are susceptible of showing a SU peak\nwith oscillator strength $0.1$-$0.3$ times that of the band edge transition.\nThis is at least one order magnitude larger than in epitaxial quantum wells.\nThe SU peak is redshifted from the band edge peak by up to few tens of meV,\nthus providing a source of line width broadening.\n \nThese results are in excellent agreement with recent experimental findings in CdSe NPLs\\cite{AntolinezNL2}\nin terms of number of emission peaks, energy splitting and relative intensity,\nbut only partially so with those of core\/shell CdSe\/CdS NPLs.\\cite{AntolinezNL}\nExperiments in the latter structure are however in line with an alternative interpretation \ninvolving simultaneous participation from trion singlet and metastable triplet states.\n\nStrategies to narrow the line width of NPLs through suppression of SU processes should\naim at controlling electrostatic impurities or Coulomb admixture.\n\n\n\n\\section{Methods}\n\nCalculations are carried within k$\\cdot$p-continuum elastic theory framework.\nIndependent electron and hole states are calculated with single-band Hamiltonians\nincluding strain and self-energy potential terms. \nModel details and material parameters are given in Ref.~\\cite{LlusarJPCc}.\nPoint charge electrostatic potentials and Coulomb integrals for \nCI matrix elements, including dielectric mismatch effects, \nare calculated solving Poisson Equation with Comsol $4.2$.\nThe CI basis set is formed by all possible combinations of the first\n22 single-electron and 22 single-hole spin-orbitals. For X$^-$,\nthese are combined to form configurations $|m_{X^-}\\rangle$ \nas the Hatree product of one hole spin-orbital with a two-electron \nSlater determinant.\n\n\n\n\\begin{acknowledgement}\nThe authors acknowledge support from MICINN project CTQ2017-83781-P.\nWe are grateful to I. Moreels, A. Achtstein and F. Rabouw for useful discussions.\n\\end{acknowledgement}\n\n{\\bf Supporting Information Available:} \n\nAdditional calculations on the influence of trion charge, surface charge\nlocation and conduction band offset over the formation of SU processes are\nprovided.\n\n\n\n\n\\section{Additional calculations}\n\nWe present here additional calculations for further understanding of SU processes. \n\n\\subsection{Convergence of CI calculations}\n\nConfiguration Interaction (CI) calculations on the basis of independent particle (or Hartree-Fock) orbitals provide an excellent description of repulsions in few- and many-fermion systems\\cite{JacakSpringer,RontaniJCP}. However, large basis sets are needed to describe strong attractions,\\cite{RontaniJPB,ShumwayPRB} which are certainly present in colloidal NPLs\\cite{RajadellPRB} and are involved in a correct description of SU processes.\n\n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=8cm]{convergence_su.eps}\n\t\t\\caption{X$^-$ emission spectrum for $Q_{edge}=0.4$ (see main text). Zero energy is set for the fundamental transition with $ne=nh=22$.\n$ne$ and $nh$ are the number of single-electron and single-hole spin-orbitals, respectively, used to build the CI basis sets.}\n\n\\label{figS}\n\\end{figure}\n \n In Fig.~\\ref{figS} we compare the X$^-$ emission spectrum calculated for CdSe NPLs --same dimensions as in main text-- \nusing different basis sizes. The basis is formed by all possible combinations of the first $ne$ ($nh$) independent particle \n spin-orbital states of electrons (holes). \n With increasing basis dimensions, the band edge transition peak redshiftes and gains intensity, which reveals an improved\n description of electron-hole correlation. The intensity of the SU peak height and its redshift with respect to the \n band edge transition are however less sensitive to the basis dimensions.\n %\n It follows from the figure that quantitative assessment on the ratio of fundamental vs SU peak heights requires \n large basis sets. In the main text we use $ne=nh=22$. By comparing with smaller values of $ne\/nh$ in the figure,\n it is clear that for this value --which involves very time-consuming computations-- the ratio is reaching saturation.\n This validates the order of ratios provided in the main text.\n For the calculations in this Supporting Information, however, we may resort to $ne=nh=12$, \n which overestimates the relative height of SU peaks, but suffices to provide qualitative assessment.\n\n\\subsection{Positive trion behaviour} \\label{positive_trion}\n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=8cm]{emission_X+.eps}\n\\caption{$X^{+}$ normalized spectrum emission for different charge intensities. The arrows are pointing to SU satellites (dotted lines are guides to the eyes). The highest SU peak ($Q_{edge}=-0.3$) is marked with a star. The origin of energies is set at the band edge recombination peak. The insets correspond to $Q_{edge}=0.5$ amplified SU peaks.}\n\\label{figSXXX3}\n\\end{figure}\n\n\nIn the main text, we have mostly considered the case of negative trions.\nWe show here that the same behavior holds for positive ones.\nTo illustrate this point, we choose the case of the core-only NPL with an edge charge,\nequivalent to Fig.1d of the main text.\nFigure \\ref{figSXXX3} shows that the presence of SU peaks in the emission \nspectrum is again strongly dependent on the value of the surface charge.\nFor $Q_{edge}=0$, no SU peak is observed. For repulsive ($Q_{edge}>0$) charges,\nSU are formed but very small in magnitude. The highest SU peaks are formed\nfor weakly bound donor charges ($Q_{edge}<0$), which attract the holes of X$^+$,\nmarked with a star in the figure. As in the X$^-$ case, if the attractive charge\nfurther increases it starts dissociating the trion. \nConsequently, SU peaks are quenched again.\nNotice however energy splittings for X$^+$ (Fig.~\\ref{figSXXX3}) \nare smaller than for X$^-$ (Fig.~1d in the main text).\nThis is expected from the heavier masses of holes.\n\n\n\\subsection{Effect of charge impurity location}\n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=16cm]{effect_charge_location.eps}\n\\caption{X$^-$ emission spectra for different locations of surface charges. \nThe spectra are normalized with respect to the energy and intensity of the $Q_{edge}$ fundamental transition.\n(a) Edge-located vs. corner-located impurity. Blue and red lines stand for edge and corner, respectively. \n(b) Edge-located vs. edge-top-located vs. corner-top-located. Blue, green and pink lines stand for edge, \ntop-corner and top-edge, respectively. $ne=nh=12$.}\n\\label{figSX}\n\\end{figure}\n\nIn the main text we present the representative cases of a surface charge centered on top of the NPL ($Q_{top}$), \nand centered and that of a charge on the edge of lateral sidewall ($Q_{edge}$). In Figure ~\\ref{figSX} we compare with\ndifferent locations. One can see that the effect of a charge located in the corner, red line in Fig.~\\ref{figSX}a, \nprovides similar SU peaks to that of the edge charge, blue line in the figure, both in energy and intensity.\nWe recall that these traps seem to be particularly likely according to recent studies on ligand desorption.\\cite{LeemansJPCL,DrijversCM}\nOff-centered charges on top and bottom surfaces are studied in Fig.~\\ref{figSX}b. They give rise to SU peaks\nof similar height to that of $Q_{edge}$, although they reach the optimal charge value sooner than $Q_{edge}$\n($Q_{top-edge} \\sim Q_{top-corner} \\approx 0.2$ versus $Q_{edge}=0.4$), \nbecause they lie closer to the center of the NPL, where photogenerated carriers tend to localize.\n\n\n\n\n\n\\subsection{Effect of conduction band offset in CdSe\/CdS NPLs} \\label{red_cbo}\n\n\nThe value of the CdSe\/CdS conduction band offset (CBO) has been a subject of debate in nanocrystal heterostructures.\\cite{AIPZunger,SteinerNL,LlusarJPCc}. We used, along our main text, an upper-bound unstrained value of $0.48$ eV\\cite{AIPZunger}, which is partly reduced by compressive strain in the core.\\cite{LlusarJPCc} Here we explore the scenario where we use a lower-bound\\cite{SteinerNL} value as well, to see the possible effect of enhancing electron delocalization over the CdS shell. Figure \\ref{figSXX} compares the two cases.\nLowering the CBO gives rise to slightly weaker electron-electron repulsion ($V_{ee}$) and electron-hole attraction ($V_{eh}$),\nhowever the differences are very small. One can then expect similar role of SU processes as in the main text.\n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=16cm]{CBO.eps}\n\t\\caption{(a,b) Average Coulomb integrals for $Q_{edge}=0.6$: (a) electron-electron repulsion and (b) electron-hole attraction for every CBO. $ne=nh=22$}\n\\label{figSXX}\n\\end{figure}\n\n\\newpage\n\n\\subsection{Effect of shell thickness in CdSe\/CdS NPLs} \\label{correspondence}\n\n\nAlong the main text, core\/shell NPLs under study had a shell thickness of 6ML on each side of the core. \nThe experiments of Antolinez et al.\\cite{AntolinezNL} however used thicker shells (12 ML). \nIn this section we compare qualitatively the response in the two cases using a moderate basis set ($ne=nh=12$),\nwhich permits addressing the experimental dimensions without the computational burden of the \nlarge basis set (for 12 ML thickness, the extended CI computation is beyond our current resources).\n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=16cm]{fig_correspondence.eps}\n\\caption{(a) Sketch of the NPLs we are comparing: 12ML shell (top) and 6ML (bottom). \nThe charge is located at same coordinates. \n(b,c) Coulomb interactions: (b) repulsions e-e and (c) attractions e-h for $Q=0.5$ and $Q=0.8$. \n(d,e) Normalized emission spectra of 6ML vs 12ML: (d) $Q=0.5$ and (e) $Q=0.8$; \n$Q=0$ is centred at band edge recombination energy for 6ML in both cases. $ne=nh=12$}\n\\label{figSXXX}\n\\end{figure}\n\nIf we focus on the charge location in both systems, Fig.~\\ref{figSXXX}a, one may expect similar behaviour. \nThe main difference, as can be seen in Fig.~\\ref{figSXXX}b (left panel) occurs for repulsive electron-electron interactions,\nwhich are slightly weaker for thick shells. This is a consequence of the larger electron delocalization,\nwhich translates into smaller $|c_2|$ coefficients in the CI expansion (see main text) and hence slightly smaller SU satellite,\nas observed in Fig.~\\ref{figSXXX}c.\n\n\n\\subsection{Effect of inserting multiple impurities in CdSe\/CdS} \\label{impurity_effect}\n\nWe consider here the possibility that two surface traps, instead of one, \nare acting as electrostatic impurities in CdSe\/CdS NPLs.\nSince there is a general preference of forming defects in the heterostructure interfaces \n-- because of lattice mismatch\\cite{LlusarJPCc,LiACS} -- and on lateral facets \n-- where ligand desorption is more likely to happen\\cite{LeemansJPCL}--, we choose the charges\nto be located as shown in Fig.~\\ref{figSXXX1}a.\nThe presence of two charges, combined with the weak in-plane confinement, easily dissociates the trion \nby driving one electron to each surface impurity. This can be seen in the charge densities of Fig.~\\ref{figSXXX1}b. The number of visible SU peaks, however, remains one (see Fig.~\\ref{figSXXX1}c).\nIn the case of strong surface charges ($Q=1.0$), the trion triplet (discussed in the main text) becomes so close in \nenergy to the singlet ground state that it shows up in the spectrum at 4 K, see right panel in Fig.~\\ref{figSXXX1}c.\n\n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=16.5cm]{fig_bicharged.eps}\n\\caption{(a) Schematic of a CdSe\/CdS NPL with 2 charges on edges intersecting interface and sidewall facet. \n$Q_{edge(1)}=Q_{edge(2)}=Q_{edge}$. The NPL shell is 12 ML thick. \n(b) In-plane electrons and hole charge densities for the X$^{-}$ singlet ($S=0$) ground state at $Q=0.5$ and $Q=1.0$; \n(c) Normalized emission spectra at $Q=0.5$ (left) and $Q=1.0$ (right). $ne=nh=12$.}\n\\label{figSXXX1}\n\\end{figure}\n\nIf we further increase the charge $Q$ (e.g. by assuming double point charges on each side of the NPL, see Fig.~\\ref{figSXXX2}a),\nadditional peaks start showing up in the emission spectrum, which is shown in Fig.~\\ref{figSXXX2}g.\nThe sketches in Fig.~\\ref{figSXXX2}e-f assign each peak to a corresponding recombination process.\nTwo transitions come from the X$^-$ singlet ground state, namely its band edge ($s$-$R1$) and first SU ($s$-$SU$) recombinations.\nThe other transitions are fully radiative recombinations arising from the triplet state, $t$-$R1$ and $t$-$R2$.\nThe picture is analogous to that proposed in the Discussion section of the main text to explain the multi-peaked emission of\nRef.~\\cite{AntolinezNL}, but in this case the triplet state is thermally occupied at 4 K, so there is no need to assume\nslow spin relaxation. The top panel in Fig~\\ref{figSXXX2}g qualitatively resembles the clusters of four peaks often\nobserved by Antolinez and co-workers in their photoluminescence measurements\\cite{AntolinezNL}, \nalthough the inter-peak energy splittings here are one order \nof magnitude smaller. As mentioned in the main text, assuming the triplet state is metastable even if it is beyond $kT$\nfrom the singlet ground state, and varying trapped charge location, it may be possible to retrieve the experimental spectra. \n\n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=16cm]{fig_tetracharged.eps}\n\\caption{(a) Schematic of a CdSe\/CdS NPL with 2 double charges on edges intersecting interface and sidewall facet. \n(b,c) Coulomb interactions: (b) electron-electron repulsion and (c) electron-hole attraction for configurations $\\ket{1}$ and $\\ket{2}$. \n(d-f) Recombination processes involved in each transition. \n(g) Normalized emission spectrum. The energy origin is set at position of the brightest peak, $t$-$R2$. \n$Q_{edge}$ is the net charge on each edge (times the fundamental electron charge). $ne=nh=12$. }\n\n\\label{figSXXX2}\n\\end{figure}\n\n\\newpage\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe neutrino mixing information is encapsulated in the unitary PMNS mixing matrix which, in the standard PDG parameterisation~\\cite{PDG}, is given by\n\\begin{equation}\nU_\\text{PMNS}=\\left(\\begin{matrix}\nc_{12} c_{13} &s_{12} c_{13} &s_{13}e^{-i\\delta} \\\\\n-s_{12} c_{23}-c_{12} s_{23} s_{13} e^{i\\delta} &c_{12}c_{23} -s_{12}s_{23} s_{13} e^{i\\delta} &s_{23} c_{13} \\\\\ns_{12} s_{23}-c_{12} c_{23} s_{13}e^{i\\delta} &-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\\delta} &c_{23}c_{13}\n\\end{matrix}\\right)\\left(\\begin{matrix}\n1 &0 &0 \\\\\n0 &e^{i\\frac{\\alpha_{21}}{2}} &0 \\\\\n0 &0 &e^{i\\frac{\\alpha_{31}}{2}}\n\\end{matrix}\\right)\n\\label{eq:pmns}\n\\end{equation}\nwhere $s_{ij}=\\sin \\theta_{ij}, c_{ij}=\\cos \\theta_{ij}$. The three mixing angles $\\theta_{12}$ (solar angle), $\\theta_{23}$ (atmospheric angle) and $\\theta_{13}$ (reactor angle) along with the $CP$-violating complex phases (the Dirac phase, $\\delta$, and the two Majorana phases, $\\alpha_{21}$ and $\\alpha_{31}$) parameterise $U_{PMNS}$. In comparison to the small mixing angles observed in the quark sector, the neutrino mixing angles are found to be relatively large~\\cite{NeutrinoGlobalFit}:\n\\begin{align}\n\\sin^2 \\theta_{12} &= 0.313_{-0.012}^{+0.013}\\,,\\label{eq:anglevalues1}\\\\\n\\sin^2 \\theta_{23} &= 0.444_{-0.031}^{+0.036}\\,\\, \\text{and} \\,\\, 0.600_{-0.026}^{+0.019}\\,,\\label{eq:anglevalues2}\\\\\n\\sin^2 \\theta_{13} &= 0.0244_{-0.0019}^{+0.0020}\\,. \\label{eq:anglevalues3}\n\\end{align}\nThe values of the complex phases are unknown at present. Besides measuring the mixing angles, the neutrino oscillation experiments also proved that neutrinos are massive particles. These experiments measure the mass-squared-differences of the neutrinos and currently their values are known at about $3\\%$ precision~\\cite{NeutrinoGlobalFit},\n\\begin{gather}\n\\Delta m_{21}^2=75.0_{-1.7}^{+1.9}~\\text{meV}^2,\\label{eq:massvalues1}\\\\\n|\\Delta m_{31}^2|=2429_{-54}^{+55}~\\text{meV}^2.\\label{eq:massvalues2}\n\\end{gather}\n\nSeveral mixing ansatze with a trimaximally mixed second column for $U_\\text{PMNS}$, i.e.~$|U_{e2}|=|U_{\\mu2}|=|U_{\\tau2}|=\\frac{1}{\\sqrt{3}}$, were proposed during the early 2000s~\\cite{TM, TBM, TXPM, Xing, Demo}. Here we briefly revisit two of those, the tri-chi-maximal mixing ($\\txm$) and the tri-phi-maximal mixing ($\\tpm$),\\footnote{$TM_i$ ($TM^i$) has been proposed~\\cite{TM2a, TM2b} as a nomenclature to denote the mixing matrices that preserve various rows (columns) of the tribimaximal mixing~\\cite{TBM}. Under this notation, both $\\txm$ and $\\tpm$ fall under the category of $TM_2$. To be more specific, $TM_2$ which breaks $CP$ maximally is $\\txm$ and $TM_2$ which conserves $CP$ is $\\tpm$.} which are relevant to our model. They can be conveniently parameterised~\\cite{TXPM} as follows \n\\begin{align}\nU_{\\txm}&=\\left(\\begin{matrix}\\sqrt{\\frac{2}{3}}\\cos \\chi & \\frac{1}{\\sqrt{3}} & \\sqrt{\\frac{2}{3}}\\sin \\chi\\\\\n-\\frac{\\cos \\chi}{\\sqrt{6}}-i\\frac{\\sin \\chi}{\\sqrt{2}} & \\frac{1}{\\sqrt{3}} & i\\frac{\\cos \\chi}{\\sqrt{2}}-\\frac{\\sin \\chi}{\\sqrt{6}}\\\\\n-\\frac{\\cos \\chi}{\\sqrt{6}}+i\\frac{\\sin \\chi}{\\sqrt{2}} & \\frac{1}{\\sqrt{3}} & -i\\frac{\\cos \\chi}{\\sqrt{2}}-\\frac{\\sin \\chi}{\\sqrt{6}}\n\\end{matrix}\\right),\\label{eq:txmform}\\\\\nU_{\\tpm}&=\\left(\\begin{matrix}\\sqrt{\\frac{2}{3}}\\cos \\phi & \\frac{1}{\\sqrt{3}} & \\sqrt{\\frac{2}{3}}\\sin \\phi\\\\\n-\\frac{\\cos \\phi}{\\sqrt{6}}-\\frac{\\sin \\phi}{\\sqrt{2}} & \\frac{1}{\\sqrt{3}} & \\frac{\\cos \\phi}{\\sqrt{2}}-\\frac{\\sin \\phi}{\\sqrt{6}}\\\\\n-\\frac{\\cos \\phi}{\\sqrt{6}}+\\frac{\\sin \\phi}{\\sqrt{2}} & \\frac{1}{\\sqrt{3}} & -\\frac{\\cos \\phi}{\\sqrt{2}}-\\frac{\\sin \\phi}{\\sqrt{6}}\n\\end{matrix}\\right)\\label{eq:tpmform}.\n\\end{align}\nBoth $\\txm$ and $\\tpm$ have one free parameter each ($\\chi$ and $\\phi$) which directly corresponds to the reactor mixing angle, $\\theta_{13}$, through the $U_{e3}$ elements of the mixing matrices. The three mixing angles and the Dirac $CP$ phase obtained by relating Eq.~(\\ref{eq:pmns}) with Eqs.~(\\ref{eq:txmform},~\\ref{eq:tpmform}) are shown in Table~\\ref{tab:anglesandphase}.\n{\\renewcommand{\\arraystretch}{1.6}\n\\begin{table}[H]\n\\begin{center}\n\\begin{tabular}{||c||c|c|c|c||}\n\\hline\n\\hline\n\t&$\\sin^2 \\theta_{13}$\t&$\\sin^2 \\theta_{12}$\t&$\\sin^2 \\theta_{23}$\t&$\\delta$\t\\\\\n\\hline\n\\hline\n$\\txm$\t&$\\frac{2}{3} \\sin^2 \\chi$\t&$\\frac{1}{\\left(3-2\\sin^2 \\chi\\right)}$\t&$\\frac{1}{2}$\t&$\\pm\\frac{\\pi}{2}$\t\\\\\n\\hline\n$\\tpm$\t&$\\frac{2}{3} \\sin^2 \\phi$\t&$\\frac{1}{\\left(3-2\\sin^2 \\phi\\right)}$\t&$\\frac{2 \\sin^2 \\left(\\frac{2\\pi}{3}+\\phi\\right)}{\\left(3-2\\sin^2 \\phi\\right)}$\t&$0,~\\pi$\t\\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{The standard PDG observables $\\theta_{13}$, $\\theta_{12}$, $\\theta_{23}$ and $\\delta$ in terms of the parameters $\\chi$ and $\\phi$. Note that the range of $\\chi$ as well as $\\phi$ is $-\\frac{\\pi}{2}$ to $+\\frac{\\pi}{2}$. In $\\txm$ ($\\tpm$), the parameter $\\chi$ ($\\phi$) being in the first and the fourth quadrant correspond to $\\delta$ equal to $+\\frac{\\pi}{2}$ ($0$) and $-\\frac{\\pi}{2}$ ($\\pi$) respectively.}\n\\label{tab:anglesandphase}\n\\end{table}}\n\\noindent In $\\txm$, since $\\delta = \\pm \\frac{\\pi}{2}$, $CP$ violation is maximal for a given set of mixing angles. The Jarlskog $CP$ violating invariant~\\cite{JCP1, JCP2, JCP3, JCP4, JCP5} in the context of $\\txm$~\\cite{TXPM} is given by \n\\begin{equation}\\label{eq:jcp}\nJ=\\frac{\\sin 2\\chi}{6\\sqrt{3}}.\n\\end{equation} \nOn the other hand, $\\tpm$ is $CP$ conserving, i.e.~$\\delta = 0,~\\pi$, and thus $J=0$. Since the reactor angle was discovered to be non-zero in 2012~\\cite{DayaBay}, there has been a resurgence of interest~\\cite{TM21, TM22, TM23, TM24, TM25, S4Paper, LIS, SteveFour, Delta16, Thomas} in $\\txm$ and $\\tpm$ and their equivalent forms\\footnote{Any $CP$-conserving ($\\delta = 0,~\\pi$) mixing matrix with non-zero $\\theta_{13}$ and trimaximally mixed $\\nu_2$ column is equivalent to $\\tpm$. Observationally they differ only with respect to the Majorana phases. Similarly any mixing matrix with $\\delta = \\pm \\frac{\\pi}{2}$, $\\theta_{13}\\neq0$ and trimaximal $\\nu_2$ column is equivalent to $\\txm$.}. \n\nRecently~\\cite{LIS} it was shown that $\\txm_{(\\chi=\\pm \\frac{\\pi}{16})}$ as well as $\\tpm_{(\\phi=\\pm \\frac{\\pi}{16})}$ results in a reactor mixing angle, \n\\begin{equation}\n\\begin{split}\n\\sin^2 \\theta_{13} &= \\frac{2}{3} \\sin^2 \\frac{\\pi}{16} \\\\\n&= 0.025,\n\\end{split}\n\\end{equation} \nconsistent with the experimental data. The model was constructed in the Type-1 see-saw framework. Four cases of Majorana mass matrices were discussed:\n\\begin{align}\nM_\\text{Maj} & \\propto \\left(\\begin{matrix}2-\\sqrt{2} & 0 & \\frac{1}{\\sqrt{2}}\\\\\n0 & 1 & 0\\\\\n\\frac{1}{\\sqrt{2}} & 0 & 0\n\\end{matrix}\\right), & M_\\text{Maj} & \\propto \\left(\\begin{matrix}0 & 0 & \\frac{1}{\\sqrt{2}}\\\\\n0 & 1 & 0\\\\\n\\frac{1}{\\sqrt{2}} & 0 & 2-\\sqrt{2}\n\\end{matrix}\\right),\\label{eq:txmmat}\\\\\nM_\\text{Maj} & \\propto \\left(\\begin{matrix}i+\\frac{1-i}{\\sqrt{2}} & 0 & 1-\\frac{1}{\\sqrt{2}}\\\\\n0 & 1 & 0\\\\\n1-\\frac{1}{\\sqrt{2}} & 0 & -i+\\frac{1+i}{\\sqrt{2}}\n\\end{matrix}\\right), & M_\\text{Maj} & \\propto \\left(\\begin{matrix}-i+\\frac{1+i}{\\sqrt{2}} & 0 & 1-\\frac{1}{\\sqrt{2}}\\\\\n0 & 1 & 0\\\\\n1-\\frac{1}{\\sqrt{2}} & 0 & i+\\frac{1-i}{\\sqrt{2}}\n\\end{matrix}\\right)\\label{eq:tpmmat}\n\\end{align}\nwhere $M_\\text{Maj}$ is the coupling among the right-handed neutrino fields, i.e.~$\\overline{(\\nu_R)^c}M_\\text{Maj} \\nu_R$. In Ref.~\\cite{LIS}, the mixing matrix was modelled in the form\n\\begin{equation}\nU_\\text{PMNS} = \\mathcal{T} U_\\nu\n\\end{equation}\nwhere the $3\\times3$ trimaximal contribution,\n\\begin{equation}\n\\mathcal{T}=\\frac{1}{\\sqrt{3}}\\left(\\begin{matrix}1 & 1 & 1\\\\\n1 & \\omega & \\bar{\\omega}\\\\\n1 & \\bar{\\omega} & \\omega\n\\end{matrix}\\right) \\quad \\text{with} \\quad \\omega=e^{i\\frac{2\\pi}{3}}, \\,\\,\\, \\bar{\\omega}=e^{\\text{-}i\\frac{2\\pi}{3}},\n\\end{equation}\ncame from the charged-lepton sector. $U_\\nu$, on the other hand, was the contribution from the neutrino sector. The four $U_\\nu$s vis-a-vis the four Majorana neutrino mass matrices given in Eqs.~(\\ref{eq:txmmat}) and Eqs.~(\\ref{eq:tpmmat}), gave rise to $\\txm_{(\\chi=\\pm \\frac{\\pi}{16})}$ and $\\tpm_{(\\phi=\\pm \\frac{\\pi}{16})}$ respectively. All the four mass matrices, Eqs.~(\\ref{eq:txmmat},~\\ref{eq:tpmmat}), have the eigenvalues $\\frac{1+\\sqrt{2(2+\\sqrt{2})}}{\\left(2+\\sqrt{2}\\right)}$, $1$ and $\\frac{-1+\\sqrt{2(2+\\sqrt{2})}}{\\left(2+\\sqrt{2}\\right)}$. Due to the see-saw mechanism, the neutrino masses become inversely proportional to the eigenvalues of the Majorana mass matrices. As a result we obtained the mass relation\n\\begin{equation}\\label{eq:numass}\nm_1:m_2:m_3=\\frac{\\left(2+\\sqrt{2}\\right)}{1+\\sqrt{2(2+\\sqrt{2})}}:1:\\frac{\\left(2+\\sqrt{2}\\right)}{-1+\\sqrt{2(2+\\sqrt{2})}}\\,\\,.\n\\end{equation}\nUsing this mass relation and given the experimentally measured mass-squared differences, we also predicted the light neutrino mass to be around $25~\\text{meV}$.\n\nIn this paper we use the discrete group $\\Sigma(72\\times3)$ to construct a flavon model that essentially reproduces the above results. Unlike the original paper~\\cite{LIS} where the neutrino mass matrix was decomposed into a symmetric bi-product, here a single representation of the flavour group is used to build the symmetric mass matrix. A brief discussion of the group $\\Sigma(72\\times3)$ and its representations is provided in Section~2. Appendix~A contains further details such as the tensor product expansions of its various irreducible representations (irreps) and the corresponding Clebsch-Gordan (C-G) coefficients. In Section~3, we describe the model with its fermion and flavon field content in relation to these irreps. The flavons are assigned specific Vacuum Expectation Values (VEVs) to obtain the required mass matrices. How we may construct suitable flavon potentials to generate the given set of VEVs is demonstrated in Appendix~B. In Section~4, we obtain the phenomenological predictions and compare them with the current experimental data along with the possibility of further validation from future experiments. Finally the results are summarised in Section~5.\n\n\\section{The Group $\\Sigma(72\\times3)$ and its Representations}\n\nDiscrete groups have been used extensively in the description of flavour symmetries. Historically, the study of discrete groups can be traced back to the study of symmetries of geometrical objects. Tetrahedran, cube, octahedran, dodecahedran and icosahedran, which are the famous Platonic solids, were known to the ancient Greeks. These objects are the only regular polyhedra with congruent regular polygonal faces. Interestingly, the symmetry groups of the platonic solids are the most studied in the context of flavour symmetries too - $A_4$ (tetrahedron), $S_4$ (cube and its dual octahedron) and $A_5$ (dodecahedron and its dual icosahedron). These polyhedra live in the three-dimensional Euclidean space. In the context of flavour physics, it might be rewarding to study similar polyhedra that live in three-dimensional complex Hilbert space. In fact, five such complex polyhedra that correspond to the five Platonic solids exist as shown by Coxeter~\\cite{Coxpoly}. They are $3\\{3\\}3\\{3\\}3$, $2\\{3\\}2\\{4\\}p$, $p\\{4\\}2\\{3\\}2$, $2\\{4\\}3\\{3\\}3$, $3\\{3\\}3\\{4\\}2$ where we have used the generalised schlafli symbols~\\cite{Coxpoly} to represent the polyhedra. The polyhedron $3\\{3\\}3\\{3\\}3$ known as the Hessian polydehron can be thought of as the tetrahedron in the complex space. Its full symmetry group has 648 elements and is called $\\Sigma(216\\times3)$. Like the other discrete groups relevant in flavour symmetry, $\\Sigma(216\\times3)$ is also a subgroup of the continuous group $SU(3)$. \n\nThe principal series of $\\Sigma(216\\times3)$~\\cite{Sigma1} is given by\n\\begin{equation}\n\\{e\\} \\triangleleft Z_3 \\triangleleft \\Delta(27) \\triangleleft \\Delta(54) \\triangleleft \\Sigma(72\\times3) \\triangleleft \\Sigma(216\\times3).\n\\end{equation}\nOur flavour symmetry group, $\\Sigma(72\\times3)$, is the maximal normal subgroup of $\\Sigma(216\\times3)$. So we get $\\Sigma(216\\times3)\/\\Sigma(72\\times3)=Z_3$. Various details about the properties of the group $\\Sigma(72\\times3)$ and its representations can be found in Refs.~\\cite{Sigma1, Sigma2, Smallgroup, SigmaHagedorn, Merle}. Note that $\\Sigma(72\\times3)$ is quite distinct from $\\Sigma(216)$ which is defined using the relation $\\Sigma(216\\times3)\/Z_3=\\Sigma(216)$. In other words, $\\Sigma(216\\times3)$ forms the triple cover of $\\Sigma(216)$. $\\Sigma(216\\times3)$ as well as $\\Sigma(216)$ is sometimes referred to as the Hessian group. In terms of the GAP~\\cite{GAP4} nomenclature, we have $\\Sigma(216\\times3)\\equiv \\text{SmallGroup(648,532)},\\,$ $\\Sigma(72\\times3)\\equiv \\text{SmallGroup(216,88)}\\,$ and $\\,\\Sigma(216)\\equiv \\text{SmallGroup(216,153)}$.\n\nWe find that, in the context of flavour physics and model building, $\\Sigma(72\\times3)$ has an appealing feature: it is the smallest group containing a complex three-dimensional representation whose tensor product with itself results in a complex six-dimensional representation, i.e.\n\\begin{equation}\\label{eq:tensor0}\n\\boldsymbol{3}\\otimes\\boldsymbol{3}=\\boldsymbol{6}\\oplus\\boldsymbol{\\tb}.\n\\end{equation}\nWith a suitably chosen basis for $\\boldsymbol{6}$ we get\n\\begin{equation}\\label{eq:tensor1exp}\n\\boldsymbol{6}\\equiv\\left(\\begin{matrix}\\frac{1}{\\sqrt{3}}\\left(a_1 b_1 + a_2 b_2 + a_3 b_3\\right)\\\\\n\t\\frac{1}{\\sqrt{6}} a_1 b_1 - \\sqrt{\\frac{2}{3}} a_2 b_2 + \\frac{1}{\\sqrt{6}} a_3 b_3\\\\\n\t\\frac{1}{\\sqrt{2}}\\left(a_1 b_1 - a_3 b_3\\right)\\\\\n\t\\frac{1}{\\sqrt{2}}\\left(a_2 b_3 + a_3 b_2\\right)\\\\\n\t\\frac{1}{\\sqrt{2}}\\left(a_3 b_1 + a_1 b_3\\right)\\\\\n\\frac{1}{\\sqrt{2}}\\left(a_1 b_2 + a_2 b_1\\right)\n\\end{matrix}\\right), \\quad \\quad \\boldsymbol{\\bar{3}} \\equiv\n\\left(\\begin{matrix}\\frac{1}{\\sqrt{2}}\\left(a_2 b_3 - a_3 b_2\\right)\\\\\n\t\\frac{1}{\\sqrt{2}}\\left(a_3 b_1 - a_1 b_3\\right)\\\\\n\\frac{1}{\\sqrt{2}}\\left(a_1 b_2 - a_2 b_1\\right)\n\\end{matrix}\\right)\n\\end{equation}\nwhere $(a_1, a_2, a_3)^T$ and $(b_1, b_2, b_3)^T$ represent the first triplet and the second triplet respectively appearing in the LHS of Eq.~(\\ref{eq:tensor0}). All the symmetric components of the tensor product together form the representation $\\boldsymbol{6}$ and the antisymmetric components form $\\boldsymbol{\\tb}$. For the $SU(3)$ group it is well known that the tensor product of two $\\boldsymbol{3}$s gives rise to a symmetric $\\boldsymbol{6}$ and an antisymmetric $\\boldsymbol{\\tb}$. $\\Sigma(72\\times3)$ being a subgroup of $SU(3)$, of course, has its $\\boldsymbol{6}$ and $\\boldsymbol{\\tb}$ embedded in the $\\boldsymbol{6}$ and $\\boldsymbol{\\tb}$ of $SU(3)$. \n\nConsider the complex conjugation of Eq.~(\\ref{eq:tensor0}), i.e.~$\\boldsymbol{\\tb}\\otimes\\boldsymbol{\\tb}=\\boldsymbol{\\xb}\\oplus\\boldsymbol{3}$. Let the right-handed neutrinos form a triplet, $\\nu_R=(\\nu_{R1},\\nu_{R2},\\nu_{R3})^T$, which transforms as a $\\boldsymbol{\\tb}$. A symmetric (and also Lorentz invariant) combination of two such triplets leads to a sextet, $X_\\nu$, which transforms as a $\\boldsymbol{\\xb}$,\n\\begin{equation}\\label{eq:Xnu}\nX_\\nu = \\left(\\begin{matrix}\\frac{1}{\\sqrt{3}}\\left(\\nu_{R1}.\\nu_{R1} + \\nu_{R2}.\\nu_{R2} + \\nu_{R3}.\\nu_{R3}\\right)\\\\\n\t\\frac{1}{\\sqrt{6}} \\nu_{R1}.\\nu_{R1} - \\sqrt{\\frac{2}{3}} \\nu_{R2}.\\nu_{R2} + \\frac{1}{\\sqrt{6}} \\nu_{R3}.\\nu_{R3}\\\\\n\t\\frac{1}{\\sqrt{2}}\\left(\\nu_{R1}.\\nu_{R1} - \\nu_{R3}.\\nu_{R3}\\right)\\\\\n\t\\frac{1}{\\sqrt{2}}\\left(\\nu_{R2}.\\nu_{R3} + \\nu_{R3}.\\nu_{R2}\\right)\\\\\n\t\\frac{1}{\\sqrt{2}}\\left(\\nu_{R3}.\\nu_{R1} + \\nu_{R1}.\\nu_{R3}\\right)\\\\\n\\frac{1}{\\sqrt{2}}\\left(\\nu_{R1}.\\nu_{R2} + \\nu_{R2}.\\nu_{R1}\\right)\n\\end{matrix}\\right)\\equiv \\boldsymbol{\\xb}\n\\end{equation}\nwhere $\\nu_i.\\nu_j$ is the Lorentz invariant product of the right-handed neutrino Weyl spinors. We may couple $X_\\nu$ to a flavon field $\\phi=(\\phi_1,\\phi_2,\\phi_3,\\phi_4,\\phi_5,\\phi_6)^T$ which transforms as a $\\boldsymbol{6}$ to construct the invariant term\n\\begin{equation}\\label{eq:Tnu}\nX_\\nu^T \\phi = \\left(\\begin{matrix}\\nu_{R1}\\\\\n\t\\nu_{R2}\\\\\n\\nu_{R3}\n\\end{matrix}\\right)^T \\left(\\begin{matrix}\\frac{\\phi_1}{\\sqrt{3}}+\\frac{\\phi_2}{\\sqrt{6}}+\\frac{\\phi_3}{\\sqrt{2}} & \\frac{\\phi_6}{\\sqrt{2}} & \\frac{\\phi_5}{\\sqrt{2}}\\\\\n \\frac{\\phi_6}{\\sqrt{2}} & \\frac{\\phi_1}{\\sqrt{3}}-\\frac{\\sqrt{2}\\phi_2}{\\sqrt{3}} & \\frac{\\phi_4}{\\sqrt{2}}\\\\\n \\frac{\\phi_5}{\\sqrt{2}} & \\frac{\\phi_4}{\\sqrt{2}} & \\frac{\\phi_1}{\\sqrt{3}}+\\frac{\\phi_2}{\\sqrt{6}}-\\frac{\\phi_3}{\\sqrt{2}}\n\\end{matrix}\\right)\\left(\\begin{matrix}\\nu_{R1}\\\\\n\t\\nu_{R2}\\\\\n\\nu_{R3}\n\\end{matrix}\\right).\n\\end{equation}\nIn general, the $3\\times3$ Majorana mass matrix is symmetric and has six complex degrees of freedom. Therefore, using Eq.~(\\ref{eq:Tnu}), any required mass matrix can be obtained through a suitably chosen Vacuum Expectation Value (VEV) for the flavon field. Constructing the symmetric Majorana neutrino mass matrix with the help of flavon sextets has been attempted before, eg. scalar fields transforming as the antisextets of $SU(3)_L$ are used in Refs.~\\cite{Long1, Long2}.\n\nTo describe the representation theory of $\\Sigma(72\\times3)$ we largely follow Ref.~\\cite{Sigma1}. $\\Sigma(72\\times3)$ can be constructed using four generators, namely $C$, $E$, $V$ and $X$~\\cite{Sigma1}. For the three-dimensional representation, we have\n\\begin{equation}\\label{eq:gen3}\nC \\equiv\n\\left(\\begin{matrix}1 & 0 & 0\\\\\n 0 & \\om & 0\\\\\n 0 & 0 & \\ob\n\\end{matrix}\\right), \\quad E \\equiv\n\\left(\\begin{matrix}0 & 1 & 0\\\\\n 0 & 0 & 1\\\\\n 1 & 0 & 0\n\\end{matrix}\\right), \\quad V\\equiv\n-\\frac{i}{\\sqrt{3}}\\left(\\begin{matrix}1 & 1 & 1\\\\\n 1 & \\om & \\ob\\\\\n 1 & \\ob & \\om\n\\end{matrix}\\right), \\quad X\\equiv\n-\\frac{i}{\\sqrt{3}}\\left(\\begin{matrix}1 & 1 & \\ob\\\\\n 1 & \\om & \\om\\\\ \n \\om & 1 & \\om\n \\end{matrix}\\right).\n\\end{equation}\nThe characters of the representations of $\\Sigma(72\\times3)$ are given in Table~\\ref{tab:charactertable}. From the character table it is easy to infer that the one-dimensional representations $\\boldsymbol{1^p}$, $\\boldsymbol{1^q}$ and $\\boldsymbol{1^r}$ involve a multiplication with $\\pm1$ only. For these representations, the generators $C$, $E$, $V$ and $X$ are given by\n\\begin{align}\n\\boldsymbol{1^p}:\\quad \\quad &C \\equiv 1, \\quad E \\equiv 1, \\quad V \\equiv -1, \\quad X \\equiv 1 \\label{eq:gen1p},\\\\ \n\\boldsymbol{1^q}:\\quad \\quad &C \\equiv 1, \\quad E \\equiv 1, \\quad V \\equiv 1, \\quad X \\equiv -1 \\label{eq:gen1q},\\\\ \n\\boldsymbol{1^r}:\\quad \\quad &C \\equiv 1, \\quad E \\equiv 1, \\quad V \\equiv -1, \\quad X \\equiv -1 \\label{eq:gen1r}.\n\\end{align}\nThe representations $\\boldsymbol{1^p}$, $\\boldsymbol{1^q}$ and $\\boldsymbol{1^r}$ along with the representation $\\boldsymbol{3}$ can be used to construct $\\boldsymbol{3^p}$, $\\boldsymbol{3^q}$ and $\\boldsymbol{3^r}$:\n\\begin{equation}\\label{eq:3times1}\n\\boldsymbol{3^p} = \\boldsymbol{1^p} \\otimes \\boldsymbol{3}, \\quad \\boldsymbol{3^q} = \\boldsymbol{1^q} \\otimes \\boldsymbol{3}, \\quad \\boldsymbol{3^r} = \\boldsymbol{1^r} \\otimes \\boldsymbol{3}.\n\\end{equation}\nFor $\\boldsymbol{3^p}$, $\\boldsymbol{3^q}$ and $\\boldsymbol{3^r}$, we use the basis defined using the generator matrices given in Eqs.~(\\ref{eq:gen3}) multiplied with $\\pm1$ in accordance with Eqs.~(\\ref{eq:gen1p},~\\ref{eq:gen1q},~\\ref{eq:gen1r},~\\ref{eq:3times1}). Tensor product expansions of various representations relevant to our model along with the $SU(3)$ embeddings (branching rules) are given in the Appendix~A. We have also provided the C-G coefficients and the generator matrices in the bases corresponding to those coefficients.\n\n\n\\begin{table}[H]\n\\begin{center}\n\\scriptsize\n\\begin{tabular}{||c||c c c|c|c c c|c c c|c c c|c c c||}\n\\hline\n\\hline\n$\\Sigma(72\\times3)$&$C_1$&$C_2$&$C_3$&$C_4$&$C_5$&$C_6$&$C_7$&$C_8$&$C_9$&$C_{10}$&$C_{11}$&$C_{12}$&$C_{13}$&$C_{14}$&$C_{15}$&$C_{16}$\\\\\n$\\#C_k$&$1$&$1$&$1$&$24$&$9$&$9$&$9$&$18$&$18$&$18$&$18$&$18$&$18$&$18$&$18$&$18$\\\\\n$ord(C_k)$&$1$&$3$&$3$&$3$&$2$&$6$&$6$&$4$&$12$&$12$&$4$&$12$&$12$&$4$&$12$&$12$\\\\\n\\hline\n\\hline\n$\\boldsymbol{1}$&$1$&$1$&$1$\t\t&$1$&$1$&$1$&$1$\t\t&$1$&$1$&$1$\t\t&$1$&$1$&$1$\t\t&$1$&$1$&$1$\\\\\n\n\\hline\n$\\boldsymbol{1^p}$&$1$&$1$&$1$\t\t&$1$&$1$&$1$&$1$\t\t&$-1$&$-1$&$-1$\t\t&$1$&$1$&$1$\t\t&$-1$&$-1$&$-1$\\\\\n$\\boldsymbol{1^q}$&$1$&$1$&$1$\t\t&$1$&$1$&$1$&$1$\t\t&$1$&$1$&$1$\t\t&$-1$&$-1$&$-1$\t\t&$-1$&$-1$&$-1$\\\\\n$\\boldsymbol{1^r}$&$1$&$1$&$1$\t\t&$1$&$1$&$1$&$1$\t\t&$-1$&$-1$&$-1$\t\t&$-1$&$-1$&$-1$\t\t&$1$&$1$&$1$\\\\\n\\hline\n$\\boldsymbol{2}$&$2$&$2$&$2$\t\t&$2$&$-2$&$-2$&$-2$\t\t&$0$&$0$&$0$\t\t&$0$&$0$&$0$\t\t&$0$&$0$&$0$\\\\\n\\hline\n$\\boldsymbol{3}$&$3$&$3\\om$&$3\\ob$\t&$0$&$-1$&$-\\om$&$-\\ob$\t\t&$1$&$\\om$&$\\ob$\t&$1$&$\\om$&$\\ob$\t&$1$&$\\om$&$\\ob$\\\\\n\\hline\n$\\boldsymbol{3^p}$&$3$&$3\\om$&$3\\ob$\t&$0$&$-1$&$-\\om$&$-\\ob$\t\t&$-1$&$-\\om$&$-\\ob$\t&$1$&$\\om$&$\\ob$\t&$-1$&$-\\om$&$-\\ob$\\\\\n$\\boldsymbol{3^q}$&$3$&$3\\om$&$3\\ob$\t&$0$&$-1$&$-\\om$&$-\\ob$\t\t&$1$&$\\om$&$\\ob$\t&$-1$&$-\\om$&$-\\ob$\t&$-1$&$-\\om$&$-\\ob$\\\\\n$\\boldsymbol{3^r}$&$3$&$3\\om$&$3\\ob$\t&$0$&$-1$&$-\\om$&$-\\ob$\t\t&$-1$&$-\\om$&$-\\ob$\t&$-1$&$-\\om$&$-\\ob$\t&$1$&$\\om$&$\\ob$\\\\\n\\hline\n$\\boldsymbol{\\tb}$&$3$&$3\\ob$&$3\\om$\t&$0$&$-1$&$-\\ob$&$-\\om$\t\t&$1$&$\\ob$&$\\om$\t&$1$&$\\ob$&$\\om$\t&$1$&$\\ob$&$\\om$\\\\\n\\hline\n$\\boldsymbol{\\ab}$&$3$&$3\\ob$&$3\\om$&$0$&$-1$&$-\\ob$&$-\\om$\t&$-1$&$-\\ob$&$-\\om$\t&$1$&$\\ob$&$\\om$\t&$-1$&$-\\ob$&$-\\om$\\\\\n$\\boldsymbol{\\bb}$&$3$&$3\\ob$&$3\\om$&$0$&$-1$&$-\\ob$&$-\\om$\t&$1$&$\\ob$&$\\om$\t&$-1$&$-\\ob$&$-\\om$\t&$-1$&$-\\ob$&$-\\om$\\\\\n$\\boldsymbol{\\cb}$&$3$&$3\\ob$&$3\\om$&$0$&$-1$&$-\\ob$&$-\\om$\t&$-1$&$-\\ob$&$-\\om$\t&$-1$&$-\\ob$&$-\\om$\t&$1$&$\\ob$&$\\om$\\\\\n\\hline\n$\\boldsymbol{6}$&$6$&$6\\ob$&$6\\om$&$0$&$2$&$2\\ob$&$2\\om$\t\t&$0$&$0$&$0$\t\t&$0$&$0$&$0$\t\t&$0$&$0$&$0$\\\\\n$\\boldsymbol{\\xb}$&$6$&$6\\om$&$6\\ob$\t&$0$&$2$&$2\\om$&$2\\ob$\t\t&$0$&$0$&$0$\t\t&$0$&$0$&$0$\t\t&$0$&$0$&$0$\\\\\n\\hline\n$\\boldsymbol{8}$&$8$&$8$&$8$\t\t&$-1$&$0$&$0$&$0$\t\t&$0$&$0$&$0$\t\t&$0$&$0$&$0$\t\t&$0$&$0$&$0$\\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Character table of $\\Sigma(72\\times3)$.}\n\\label{tab:charactertable}\n\\end{table} \n\n\\section{The Model}\n\\addtocontents{toc}{\\protect\\setcounter{tocdepth}{1}}\n\nIn this paper we construct our model in the Standard Model framework with the addition of heavy right-handed neutrinos. Through the type~I see-saw mechanism, light Majorana neutrinos are produced. The fermion and flavon content of the model with the representations to which they belong is given in Table~\\ref{tab:flavourcontent}. The Standard Model Higgs field is assigned to the trivial (singlet) representation of $\\Sigma(72\\times3)$.\n\n\\begin{table}[H]\n\\begin{center}\n\\begin{tabular}{|c|c c c c c c c c c|}\n\\hline\n\t&$e_R$\t&$\\mu_R$&$\\tau_R$&$L$\t&$\\nu_R$&$\\phi_e$&$\\phi_\\mu$&$\\phi_\\tau$&$\\phi$\\\\\n\\hline\n\n$\\Sigma(72\\times3)$\t&$\\boldsymbol{1^p}$&$\\boldsymbol{1^q}$&$\\boldsymbol{1^r}$&$\\boldsymbol{\\tb}$&$\\boldsymbol{\\tb}$&$\\boldsymbol{\\ab}$&$\\boldsymbol{\\bb}$&$\\boldsymbol{\\cb}$&$\\boldsymbol{6}$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{The flavour structure of the model. The three families of the left-handed-weak-isospin lepton doublets form the triplet $L$ and the three right-handed heavy neutrinos form the triplet $\\nu_R$. The flavons $\\phi_e$, $\\phi_\\mu$, $\\phi_\\tau$ and $\\phi$, are scalar fields and are gauge invariants. On the other hand, they transform non-trivially under the flavour group.}\n\\label{tab:flavourcontent}\n\\end{table}\n\nFor the charged leptons, we obtain the mass term\n\\begin{equation}\\label{eq:mclept}\n\\left(y_e L^\\dagger e_R \\frac{\\phi_e}{\\Lambda}+y_\\mu L^\\dagger \\mu_R \\frac{\\phi_\\mu}{\\Lambda}+y_\\tau L^\\dagger \\tau_R \\frac{\\phi_\\tau}{\\Lambda}\\right)H+H.C.\n\\end{equation}\nwhere $H$ is the Standard Model Higgs, $\\Lambda$ is the cut-off scale and $y_i$ are the coupling constants. The VEV of the Higgs, $(0, h_o)$, breaks the weak gauge symmetry. For the flavons $\\phi_e$, $\\phi_\\mu$ and $\\phi_\\tau$, we assign the vacuum alignments\\footnote{Refer to Appendix~B for the details of the flavon potential that leads to these VEVs.} \n\\begin{equation}\\label{eq:leptvev}\n\\langle\\phi_e\\rangle= \\frac{i}{\\sqrt{3}}(1,1,1),\\quad\\langle\\phi_\\mu\\rangle= \\frac{i}{\\sqrt{3}}(1,\\ob,\\om),\\quad \\langle\\phi_\\tau\\rangle= \\frac{i}{\\sqrt{3}}(1,\\om,\\ob).\n\\end{equation}\nAs a result of these vacuum alignments we get the following charged-lepton mass term\n\\begin{equation}\\label{eq:leptcontrib}\n\\left(\\begin{matrix} e_L\\\\\n\\mu_L\\\\\n\\tau_L\n\\end{matrix}\\right)^\\dagger V^\\dagger \\left(\\begin{matrix} m_e & 0 & 0\\\\\n0 & m_\\mu & 0\\\\\n0 & 0 & m_\\tau\n\\end{matrix}\\right)\\left(\\begin{matrix} e_R\\\\\n\\mu_R\\\\\n\\tau_R\n\\end{matrix}\\right)\n\\end{equation}\nwhere $m_e=\\frac{y_e h_o}{\\Lambda}$ etc. $V$ is the $3\\times3$ trimaximal matrix and is one of the generators of $\\Sigma(72\\times3)$ as given in Eqs.~(\\ref{eq:gen3}).\n\nNow, we write the Dirac mass term for the neutrinos:\n\\begin{equation}\\label{eq:mdl2dirac}\n2 y_w L^\\dagger \\nu_R \\tilde{H}+H.C.\n\\end{equation}\nwhere $\\tilde{H}$ is the conjugate Higgs and $y_w$ is the coupling constant. With the help of Eq.~(\\ref{eq:Tnu}), we also write the Majorana mass term for the neutrinos:\n\\begin{equation}\\label{eq:mnurnur}\ny_G X_\\nu^T\\frac{\\phi}{\\Lambda}\n\\end{equation}\nwhere $y_G$ is the coupling which gives rise to the heavy right-handed Majorana masses. Let $\\langle\\phi\\rangle$ be the VEV acquired by the sextet flavon $\\phi$, and let $\\boldsymbol{ \\langle\\phi\\rangle}$ be the corresponding $3\\times 3$ symmetric matrix of the form given in Eq.~(\\ref{eq:Tnu}). Combining the mass terms, Eq.~(\\ref{eq:mdl2dirac}) and Eq.~(\\ref{eq:mnurnur}), and using the VEVs of the Higgs and the flavon, we obtain the Dirac-Majorana mass matrix:\n\\begin{equation}\nM=\\left(\\begin{matrix}0 & y_w h_o I\\\\\n y_w h_o I & \\,\\,y_G \\frac{1}{\\Lambda}\\boldsymbol{ \\langle\\phi\\rangle}\n\\end{matrix}\\right).\n\\end{equation}\nThe $6\\times6$ mass matrix $M$, forms the coupling\n\\begin{equation}\nM_{ij} \\,\\nu_i.\\nu_j \\quad \\text{with} \\quad \\nu=\\left(\\begin{matrix}\\nu_{L}^*\\\\\n\t\\nu_{R}\n\\end{matrix}\\right)\n\\end{equation}\nwhere $\\nu_L=(\\nu_e,\\nu_\\mu,\\nu_\\tau)^T$ are the left-handed neutrino flavour eigenstates.\n\nSince $y_w h_o$ is at the weak scale and $y_G$ is at the GUT scale, small neutrino masses are generated through the see-saw mechanism. It can be shown~\\cite{Majorana} that the resulting effective see-saw mass matrix is of the form\n\\begin{equation}\\label{eq:seesaw}\nM_\\text{ss}=-\\left(y_w h_o\\right)^2\\left(y_G \\frac{{\\boldsymbol{ \\langle\\phi\\rangle}}}{\\Lambda}\\right)^{-1}.\n\\end{equation}\nFrom Eq.~(\\ref{eq:seesaw}) it is clear that the see-saw mechanism makes the light neutrino masses inversely proportional to the eigenvalues of the matrix $\\boldsymbol{ \\langle\\phi\\rangle}$. As foreseen in the Introduction, we now construct the four cases of the mass matrices, Eqs.~(\\ref{eq:txmmat},~\\ref{eq:tpmmat}), all of which resulting in the neutrino mass relation, Eq.~(\\ref{eq:numass}). To achieve this we choose suitable vacuum alignments\\footnote{Refer to Appendix~B for the details of the flavon potentials that lead to these VEVs.} for the sextet flavon $\\phi$. \n\n\\subsection{$\\txm_{(\\chi=+\\frac{\\pi}{16})}$} \n\nHere we assign the vacuum alignment\n\\begin{equation}\\label{eq:vevtxmp}\n\\langle\\phi\\rangle = \\left(\\frac{3-\\sqrt{2}}{\\sqrt{3}},-\\frac{1}{\\sqrt{3}},-1+\\sqrt{2},0,1,0\\right).\n\\end{equation}\nUsing the symmetric matrix form of the sextet given in Eq.~(\\ref{eq:Tnu}), we obtain\n\\begin{equation}\\label{eq:vevsym1}\n\\boldsymbol{ \\langle\\phi\\rangle} =\\left(\\begin{matrix}2-\\sqrt{2} & 0 & \\frac{1}{\\sqrt{2}}\\\\\n0 & 1 & 0\\\\\n\\frac{1}{\\sqrt{2}} & 0 & 0\n\\end{matrix}\\right).\n\\end{equation}\nDiagonalising the corresponing effective see-saw mass matrix $M_{ss}$, Eq.~(\\ref{eq:seesaw}), we get\n\\begin{equation}\\label{eq:nucontrib}\nU_\\nu^\\dagger M_{ss} U_\\nu^* = \\left(y_w h_o\\right)^2\\frac{\\Lambda}{y_G} \\text{Diag}\\left({\\textstyle\\frac{\\left(2+\\sqrt{2}\\right)}{1+\\sqrt{2(2+\\sqrt{2})}},1,\\frac{\\left(2+\\sqrt{2}\\right)}{-1+\\sqrt{2(2+\\sqrt{2})}}}\\right)\n\\end{equation}\nleading to the neutrino mass relation, Eq.~(\\ref{eq:numass}). The unitary matrix $U_\\nu$ is given by\n\\begin{equation}\\label{eq:unu1}\nU_\\nu = i \\left(\\begin{matrix}\\cos\\left(\\frac{3\\pi}{16}\\right) & 0 & -i\\sin\\left(\\frac{3\\pi}{16}\\right)\\\\\n0 & 1 & 0\\\\\n\\sin\\left(\\frac{3\\pi}{16}\\right) & 0 & i\\cos\\left(\\frac{3\\pi}{16}\\right)\n\\end{matrix}\\right).\n\\end{equation}\nThe product of the contribution from the charged-lepton sector i.e.~$V$ from Eqs.~(\\ref{eq:leptcontrib},~\\ref{eq:gen3}) and the contribution from the neutrino sector i.e.~$U_\\nu$ from Eqs.~(\\ref{eq:nucontrib},~\\ref{eq:unu1}) results in the $\\txm_{(\\chi=+\\frac{\\pi}{16})}$ mixing:\n\\begin{equation}\\label{eq:mix1}\nU_\\text{PMNS}=V.U_\\nu=\\left(\\begin{matrix}1 & 0 & 0\\\\\n0 & \\om & 0\\\\\n0 & 0 & \\ob\n\\end{matrix}\\right).\\left(\\begin{matrix}\\sqrt{\\frac{2}{3}}\\cos \\chi & \\frac{1}{\\sqrt{3}} & \\sqrt{\\frac{2}{3}}\\sin \\chi\\\\\n-\\frac{\\cos \\chi}{\\sqrt{6}}-i\\frac{\\sin \\chi}{\\sqrt{2}} & \\frac{1}{\\sqrt{3}} & i\\frac{\\cos \\chi}{\\sqrt{2}}-\\frac{\\sin \\chi}{\\sqrt{6}}\\\\\n-\\frac{\\cos \\chi}{\\sqrt{6}}+i\\frac{\\sin \\chi}{\\sqrt{2}} & \\frac{1}{\\sqrt{3}} & -i\\frac{\\cos \\chi}{\\sqrt{2}}-\\frac{\\sin \\chi}{\\sqrt{6}}\n\\end{matrix}\\right).\\left(\\begin{matrix}1 & 0 & 0\\\\\n0 & 1 & 0\\\\\n0 & 0 & i\n\\end{matrix}\\right)\n\\end{equation}\nwith $\\chi=+\\frac{\\pi}{16}$.\n\n\\subsection{$\\txm_{(\\chi=-\\frac{\\pi}{16})}$} \nHere we assign the vacuum alignment\n\\begin{equation}\\label{eq:vevtxmm}\n\\langle\\phi\\rangle = \\left(\\frac{3-\\sqrt{2}}{\\sqrt{3}},-\\frac{1}{\\sqrt{3}},1-\\sqrt{2},0,1,0\\right)\n\\end{equation}\nresulting in the symmetric matrix\n\\begin{equation}\\label{eq:vevsym2}\n\\boldsymbol{ \\langle\\phi\\rangle}=\\left(\\begin{matrix}0 & 0 & \\frac{1}{\\sqrt{2}}\\\\\n0 & 1 & 0\\\\\n\\frac{1}{\\sqrt{2}} & 0 & 2-\\sqrt{2}\n\\end{matrix}\\right).\n\\end{equation}\nIn this case, the diagonalising matrix is\n\\begin{equation}\nU_\\nu =i \\left(\\begin{matrix}\\cos\\left(\\frac{5\\pi}{16}\\right) & 0 & i\\sin\\left(\\frac{5\\pi}{16}\\right)\\\\\n0 & 1 & 0\\\\\n\\sin\\left(\\frac{5\\pi}{16}\\right) & 0 & -i\\cos\\left(\\frac{5\\pi}{16}\\right)\n\\end{matrix}\\right)\n\\end{equation}\nand the corresponding mixing matrix is\n\\begin{equation}\nU_\\text{PMNS}=V.U_\\nu=\\left(\\begin{matrix}1 & 0 & 0\\\\\n0 & \\om & 0\\\\\n0 & 0 & \\ob\n\\end{matrix}\\right).\\left(\\begin{matrix}\\sqrt{\\frac{2}{3}}\\cos \\chi & \\frac{1}{\\sqrt{3}} & \\sqrt{\\frac{2}{3}}\\sin \\chi\\\\\n-\\frac{\\cos \\chi}{\\sqrt{6}}-i\\frac{\\sin \\chi}{\\sqrt{2}} & \\frac{1}{\\sqrt{3}} & i\\frac{\\cos \\chi}{\\sqrt{2}}-\\frac{\\sin \\chi}{\\sqrt{6}}\\\\\n-\\frac{\\cos \\chi}{\\sqrt{6}}+i\\frac{\\sin \\chi}{\\sqrt{2}} & \\frac{1}{\\sqrt{3}} & -i\\frac{\\cos \\chi}{\\sqrt{2}}-\\frac{\\sin \\chi}{\\sqrt{6}}\n\\end{matrix}\\right).\\left(\\begin{matrix}1 & 0 & 0\\\\\n0 & 1 & 0\\\\\n0 & 0 & -i\n\\end{matrix}\\right)\\label{eq:mix2}\n\\end{equation}\nwith $\\chi=-\\frac{\\pi}{16}$.\n\n\\subsection{$\\tpm_{(\\phi=+\\frac{\\pi}{16})}$} \nHere we assign the vacuum alignment\n\\begin{equation}\\label{eq:vevtpmp}\n\\langle\\phi\\rangle = \\left(\\frac{1+\\sqrt{2}}{\\sqrt{3}},\\frac{1-\\sqrt{2}}{\\sqrt{3}}, -i \\left(1-\\sqrt{2}\\right),0, -1+\\sqrt{2}, 0\\right)\n\\end{equation}\nresulting in the symmetric matrix\n\\begin{equation}\\label{eq:vevsym3}\n\\boldsymbol{ \\langle\\phi\\rangle}=\\left(\\begin{matrix}i+\\frac{1-i}{\\sqrt{2}} & 0 & 1-\\frac{1}{\\sqrt{2}}\\\\\n0 & 1 & 0\\\\\n1-\\frac{1}{\\sqrt{2}} & 0 & -i+\\frac{1+i}{\\sqrt{2}}\n\\end{matrix}\\right).\n\\end{equation}\nIn this case, the diagonalising matrix is\n\\begin{equation}\nU_\\nu =i \\left(\\begin{matrix}\\frac{1}{\\sqrt{2}} e^{-i\\frac{\\pi}{16}} & 0 & -\\frac{1}{\\sqrt{2}} e^{-i\\frac{\\pi}{16}}\\\\\n0 & 1 & 0\\\\\n\\frac{1}{\\sqrt{2}} e^{i\\frac{\\pi}{16}} & 0 & \\frac{1}{\\sqrt{2}} e^{i\\frac{\\pi}{16}}\n\\end{matrix}\\right)\n\\end{equation}\nand the corresponding mixing matrix is\n\\begin{equation}\\label{eq:mix3}\nU_\\text{PMNS}=V.U_\\nu=\\left(\\begin{matrix}1 & 0 & 0\\\\\n0 & \\om & 0\\\\\n0 & 0 & \\ob\n\\end{matrix}\\right).\\left(\\begin{matrix}\\sqrt{\\frac{2}{3}}\\cos \\phi & \\frac{1}{\\sqrt{3}} & \\sqrt{\\frac{2}{3}}\\sin \\phi\\\\\n-\\frac{\\cos \\phi}{\\sqrt{6}}-\\frac{\\sin \\phi}{\\sqrt{2}} & \\frac{1}{\\sqrt{3}} & \\frac{\\cos \\phi}{\\sqrt{2}}-\\frac{\\sin \\phi}{\\sqrt{6}}\\\\\n-\\frac{\\cos \\phi}{\\sqrt{6}}+\\frac{\\sin \\phi}{\\sqrt{2}} & \\frac{1}{\\sqrt{3}} & -\\frac{\\cos \\phi}{\\sqrt{2}}-\\frac{\\sin \\phi}{\\sqrt{6}}\n\\end{matrix}\\right).\\left(\\begin{matrix}1 & 0 & 0\\\\\n0 & 1 & 0\\\\\n0 & 0 & i\n\\end{matrix}\\right)\n\\end{equation}\nwith $\\phi=+\\frac{\\pi}{16}$.\n\n\\subsection{$\\tpm_{(\\phi=-\\frac{\\pi}{16})}$} \nHere we assign the vacuum alignment\n\\begin{equation}\\label{eq:vevtpmm}\n\\langle\\phi\\rangle = \\left(\\frac{1+\\sqrt{2}}{\\sqrt{3}},\\frac{1-\\sqrt{2}}{\\sqrt{3}}, i (1-\\sqrt{2}),0, -1+\\sqrt{2}, 0\\right)\n\\end{equation}\nresulting in the symmetric matrix\n\\begin{equation}\\label{eq:vevsym4}\n\\boldsymbol{ \\langle\\phi\\rangle}=\\left(\\begin{matrix}-i+\\frac{1+i}{\\sqrt{2}} & 0 & 1-\\frac{1}{\\sqrt{2}}\\\\\n0 & 1 & 0\\\\\n1-\\frac{1}{\\sqrt{2}} & 0 & i+\\frac{1-i}{\\sqrt{2}}\n\\end{matrix}\\right).\n\\end{equation}\nIn this case, the diagonalising matrix is\n\\begin{equation}\nU_\\nu =i \\left(\\begin{matrix}\\frac{1}{\\sqrt{2}} e^{i\\frac{\\pi}{16}} & 0 & \\frac{1}{\\sqrt{2}} e^{i\\frac{\\pi}{16}}\\\\\n0 & 1 & 0\\\\\n\\frac{1}{\\sqrt{2}} e^{-i\\frac{\\pi}{16}} & 0 & -\\frac{1}{\\sqrt{2}} e^{-i\\frac{\\pi}{16}}\n\\end{matrix}\\right)\n\\end{equation}\nand the corresponding mixing matrix is\n\\begin{equation}\nU_\\text{PMNS}=V.U_\\nu=\\left(\\begin{matrix}1 & 0 & 0\\\\\n0 & \\om & 0\\\\\n0 & 0 & \\ob\n\\end{matrix}\\right).\\left(\\begin{matrix}\\sqrt{\\frac{2}{3}}\\cos \\phi & \\frac{1}{\\sqrt{3}} & \\sqrt{\\frac{2}{3}}\\sin \\phi\\\\\n-\\frac{\\cos \\phi}{\\sqrt{6}}-\\frac{\\sin \\phi}{\\sqrt{2}} & \\frac{1}{\\sqrt{3}} & \\frac{\\cos \\phi}{\\sqrt{2}}-\\frac{\\sin \\phi}{\\sqrt{6}}\\\\\n-\\frac{\\cos \\phi}{\\sqrt{6}}+\\frac{\\sin \\phi}{\\sqrt{2}} & \\frac{1}{\\sqrt{3}} & -\\frac{\\cos \\phi}{\\sqrt{2}}-\\frac{\\sin \\phi}{\\sqrt{6}}\n\\end{matrix}\\right).\\left(\\begin{matrix}1 & 0 & 0\\\\\n0 & 1 & 0\\\\\n0 & 0 & -i\n\\end{matrix}\\right)\\label{eq:mix4}\n\\end{equation}\nwith $\\phi=-\\frac{\\pi}{16}$.\n\nAs stated earlier, the four cases, Eqs.~(\\ref{eq:vevsym1},~\\ref{eq:vevsym2},~\\ref{eq:vevsym3},~\\ref{eq:vevsym4}), result in the same neutrino mass relation Eq.~(\\ref{eq:numass}).\n\n\\section{Predicted Observables}\n\nFor comparing our model with the neutrino oscillation experimental data, we use the global analysis done by the NuFIT group and their latest results reproduced in Eqs.~(\\ref{eq:anglevalues1}-\\ref{eq:massvalues2}). The prediction $\\sin^2 \\theta_{13} = \\frac{2}{3} \\sin^2 \\frac{\\pi}{16} = 0.025$,\\footnote{Besides in Ref.~\\cite{LIS}, this value was predicted in the context of $\\Delta(6n^2)$ symmetry group in Ref.~\\cite{Delta16} and later obtained in Ref.~\\cite{Thomas}} is within $1\\sigma$ errors. For the solar angle, using the formula given in Table~\\ref{tab:anglesandphase}, we get \n\\begin{equation}\n\\begin{split}\n\\sin^2 \\theta_{12} &= \\frac{1}{3-2\\sin^2\\left(\\frac{\\pi}{16}\\right)}\\\\ \n&= 0.342\\,.\n\\end{split}\n\\end{equation}\nThis has a small tension with the experimental value, but it is still within $3\\sigma$ errors. For the atmospheric angle, $\\txm$ predicts maximal mixing:\n\\begin{equation}\n\\sin^2 \\theta_{23} = \\frac{1}{2}\\,.\n\\end{equation}\nThe NuFIT data as well as other global fits~\\cite{Global1,Global1b,Global2} are showing a preference for non-maximal atmospheric mixing. As a result there has been a recent interest in the problem of octant degeneracy of $\\theta_{23}$~\\cite{Octant1,Octant2,Octant3}. $\\tpm$ predicts this non-maximal scenario of atmospheric mixing. $\\tpm_{(\\phi=\\frac{\\pi}{16})}$ and $\\tpm_{(\\phi=-\\frac{\\pi}{16})}$ correspond to the first and the second octant solutions respectively. Using the formula for $\\theta_{23}$ given in Table~{tab:anglesandphase}, we get \n\\begin{gather}\n\\tpm_{(\\phi=+\\frac{\\pi}{16})}:\\,\n\\begin{split}\n\\quad \\sin^2 \\theta_{23} &= \\frac{2\\sin^2\\left(\\frac{2\\pi}{3}+\\frac{\\pi}{16}\\right)}{3-2\\sin^2\\left(\\frac{\\pi}{16}\\right)} \\\\\n&= 0.387\\,,\\\\\n\\end{split}\\\\\n\\tpm_{(\\phi=-\\frac{\\pi}{16})}:\\,\n\\begin{split}\n\\quad \\sin^2 \\theta_{23} &= \\frac{2\\sin^2\\left(\\frac{2\\pi}{3}-\\frac{\\pi}{16}\\right)}{3-2\\sin^2\\left(\\frac{\\pi}{16}\\right)}\\\\\n&= 0.613\\,.\n\\end{split}\n\\end{gather}\nThe dirac $CP$ phase, $\\delta$, has not been measured yet. The discovery that the reactor mixing angle is not very small has raised the possibility of a relatively earlier measurement of $\\delta$~\\cite{MHCP1,MHCP2}. $\\txm$ having $\\delta=\\pm\\frac{\\pi}{2}$ should lead to large observable $CP$ violating effects. Substituting $\\chi=\\pm\\frac{\\pi}{16}$ in Eq.~(\\ref{eq:jcp}), our model gives\n\\begin{equation}\n\\begin{split}\nJ&=\\pm\\frac{\\sin \\frac{\\pi}{8}}{6\\sqrt{3}}\\\\\n&= \\pm0.0368.\n\\end{split}\n\\end{equation} \nwhich is about $40\\%$ of the maximum value of the theoretical range, $-\\frac{1}{6\\sqrt{3}}\\leq J\\leq +\\frac{1}{6\\sqrt{3}}$.\nOn the other hand, $\\tpm$, with $\\delta=0,\\,\\pi$ and $J=0$, is $CP$ conserving. \n\nFrom Figure~\\ref{fig:neutrinopredict2}, it is clear that the neutrino mass relation~Eq.~(\\ref{eq:numass}), is consistent with the measured mass-squared differences, Eqs.~(\\ref{eq:massvalues1},~\\ref{eq:massvalues2}). Here we have assumed the normal mass hierarchy. Using Eq.~(\\ref{eq:numass}) and Eqs.~(\\ref{eq:massvalues1},~\\ref{eq:massvalues2}), we predict the neutrino masses:\\footnote{The best fit values correspond to $\\chi^2_\\text{min}=1.1$ and the error ranges correspond to $\\Delta\\chi^2=1$ where $\\chi^2=\\displaystyle\\sum_{{\\displaystyle x}=\\Delta m^2_{21}, \\Delta m^2_{31}} \\left(\\frac{x_\\text{model}-x_\\text{expt}}{\\sigma_{x\\,\\text{expt}}}\\right)^2$ and $\\Delta\\chi^2=\\chi^2-\\chi^2_\\text{min}$.}\n\\begin{gather}\nm_1=24.77^{+0.20}_{-0.19}~\\text{meV},\\notag\\\\\nm_2=26.22^{+0.21}_{-0.20}~\\text{meV},\\label{eq:m123}\\\\\nm_3=55.49^{+0.45}_{-0.42}~\\text{meV}.\\notag\n\\end{gather}\nNote that the mass relation Eq.~(\\ref{eq:numass}), is incompatible with the inverted mass hierarchy. Considerable experimental studies are being conducted to determine the mass hierarchy~\\cite{MHCP2,MH1,MH2,MH3} and we may expect a solution in the not-too-distant future. Observation of inverted hierarchy will obviously rule out the model.\n\n\\begin{figure}[]\n\\begin{center}\n\\includegraphics[scale=1.0]{massplot.pdf}\n\\caption[$\\Delta m_{31}^2$ vs $\\Delta m_{21}^2$ plane]{$\\Delta m_{31}^2$ vs $\\Delta m_{21}^2$ plane. The straight line shows the neutrino mass relation Eq.~(\\ref{eq:numass}). As a parametric plot, the line can be represented as $\\Delta m_{21}^2=(r_{21}^2-1)m_1^2$ and $\\Delta m_{31}^2=(r_{31}^2-1)m_1^2$ where $r_{21}=\\frac{m_2}{m_1}=\\frac{1+\\sqrt{2(2+\\sqrt{2})}}{\\left(2+\\sqrt{2}\\right)}$ and $r_{31}=\\frac{m_3}{m_1}=\\frac{1+\\sqrt{2(2+\\sqrt{2})}}{-1+\\sqrt{2(2+\\sqrt{2})}}$ are the mass ratios obtained from Eq.~(\\ref{eq:numass}). The parametric values of the light neutrino mass, $m_1$, (denoted by the black dots on the line) are in terms of meV. The red marking indicates the experimental best fit for $\\Delta m_{21}^2$ and $\\Delta m_{31}^2$ along with $1\\sigma$ and $3\\sigma$ errors.}\n\\label{fig:neutrinopredict2}\n\\end{center}\n\\end{figure}\n\nCosmological observations can give us limits on the sum of the neutrino masses. The strongest such limit has been set recently by the data collected using the Planck satellite~\\cite{Planck}:\n\\begin{equation}\n\\sum_i m_i < 230~\\text{meV}.\n\\end{equation}\nOur predictions Eqs.~(\\ref{eq:m123}), give a sum \n\\begin{equation}\n\\sum_i m_i = 106.48^{+0.86}_{-0.81}~\\text{meV}\n\\end{equation}\nwhich is not far below the current cosmological limit. By including the data from future projects such as Polarbear and SKA, we may be able to lower the cosmological limit to around $100~\\text{meV}$~\\cite{FutureCosmo}. Such a result may support or rule out our model.\n\nNeutrinoless double beta decay experiments seek to determine the Majorana nature of the neutrinos. These experiments have so far set limits on the effective electron neutrino mass~\\cite{DBD} $|m_{\\beta\\beta}|$, where\n\\begin{align}\n\\begin{split}\nm_{\\beta\\beta}&=m_1 U_{e1}^2+m_2 U_{e2}^2+m_3 U_{e3}^2\\\\\n&=m_1 |U_{e1}|^2+m_2 |U_{e2}|^2 e^{i\\alpha_{21}}+m_3 |U_{e3}|^2 e^{i\\alpha_{31}}\n\\end{split}\n\\end{align}\nwith $U$ representing $U_\\text{PMNS}$. In all the four mixing scenarios predicted by the model, Eqs.~(\\ref{eq:mix1},~\\ref{eq:mix2},~\\ref{eq:mix3},~\\ref{eq:mix4}), we have $|U_{e1}|=\\sqrt{\\frac{2}{3}}\\cos \\frac{\\pi}{16}$, $|U_{e2}|=\\frac{1}{\\sqrt{3}}$ and $|U_{e3}|=\\sqrt{\\frac{2}{3}}\\sin \\frac{\\pi}{16}$. Also, all of them result in the Majorana phases:\\footnote{Note that both $+i$ and $-i$ appearing in the diagonal phase matrices in Eqs.~(\\ref{eq:mix1},~\\ref{eq:mix2},~\\ref{eq:mix3},~\\ref{eq:mix4}) correspond to $\\alpha_{31}=\\pi$}\n\\begin{equation}\n\\alpha_{21}=0, \\quad \\alpha_{31}=\\pi. \n\\end{equation}\nTherefore the model predicts\n\\begin{equation}\\label{eq:modelbb}\nm_{\\beta\\beta}=\\frac{2}{3}m_1\\cos^2 \\frac{\\pi}{16}+\\frac{1}{3}m_2-\\frac{2}{3}m_3\\sin^2 \\frac{\\pi}{16}. \n\\end{equation}\nSubstituting the neutrino masses from Eqs.~(\\ref{eq:m123}) in Eq.~(\\ref{eq:modelbb}) we get \n\\begin{equation}\nm_{\\beta\\beta}=23.22^{+0.19}_{-0.18}~\\text{meV}. \n\\end{equation}\n\nThe most stringent upper bounds on the value of $|m_{\\beta\\beta}|$ have been set by Heidelberg-Moscow, Cuoricino, NEMO3 and GERDA experiments. These bounds are of the order of a few hundreds of meV. In future, experiments such as GERDA-2, CUORE and EXO can improve the bounds on $|m_{\\beta\\beta}|$ to a few tens~\\cite{DBD} of meV and thus may support or rule out our model.\n\n\\section{Summary}\n\nIn this paper we utilise the group $\\Sigma(72\\times3)$ to construct fully-constrained Majorana mass matrices for the neutrinos. These mass matrices reproduce the results obtained in Ref.~\\cite{LIS} i.e.~$\\txm_{(\\chi=\\pm\\frac{\\pi}{16})}$ and $\\tpm_{(\\phi=\\pm\\frac{\\pi}{16})}$ mixings along with the neutrino mass relation Eq.~(\\ref{eq:numass}). The mixing observables as well as the neutrino mass relation are shown to be consistent with the experimental data. $\\txm_{(\\chi=\\pm\\frac{\\pi}{16})}$ and $\\tpm_{(\\phi=\\pm\\frac{\\pi}{16})}$ predict the Dirac $CP$ violating effect to be maximal (at fixed $\\theta_{13}$) and null respectively. Using our neutrino mass relation in conjunction with the experimentally-observed neutrino mass-squared differences, we calculate the individual neutrino masses. We note that the neutrino mass relation Eq.~(\\ref{eq:numass}) is incompatible with the inverted mass hierarchy. We also predict the effective electron neutrino mass for the neutrinoless double beta decay, $|m_{\\beta\\beta}|$. We briefly discuss the current status and future prospects of experimentally determining the neutrino observables leading to the confirmation or the falsification of our model. In the context of model building, we carry out an in-depth analysis of the representations of $\\Sigma(72\\times3)$ and develop the necessary groundwork to construct the flavon potentials satisfying the $\\Sigma(72\\times3)$ flavour symmetry. In the charged-lepton sector, we use three triplet flavons with a suitably chosen set of VEVs which provide a $3\\times 3$ trimaximal contribution towards the PMNS mixing matrix. In the neutrino sector, we discuss four cases of Majorana mass matrices. The $\\Sigma(72\\times3)$ sextet acts as the most general placeholder for a fully constrained Majorana mass matrix. The intended mass matrices are obtained by assigning appropriate VEVs to the sextet flavon. It should be noted that we need additional symmetries to `explain' any specific texture in the mass matrix. Further research in this direction is a work in progress.\n\nI would like to thank Paul Harrison and Bill Scott for helpful discussions. This work was supported by the UK Science and Technology Facilities Council (STFC). I acknowledge the hospitality of the Centre for Fundamental Physics (CfFP) at the Rutherford Appleton Laboratory and support from the University of Warwick. \n\n\\section{Appendix A: Tensor Product Expansions of Irreps of $\\Sigma(72\\times3)$}\n\n\\begin{flalign}\\label{eq:tensor288}\n&\\text{{\\textit {\\textbf {i}}})\\,\\,\\,}\\boldsymbol{3}\\otimes\\boldsymbol{3}=\\boldsymbol{6}\\oplus\\boldsymbol{\\tb}&\n\\end{flalign}\nThe generator matrices for the triplet representation were provided in Eq.~(\\ref{eq:gen3}). We define the basis for the sextet representation using Eqs.~(\\ref{eq:tensor1exp}). The resulting generator matrices are \n\\begin{align}\\label{eq:gen6}\n\\begin{split}\n&C \\equiv\n\\left(\\begin{matrix}0 & -\\frac{\\ob}{\\sqrt{2}} & \\frac{i\\ob}{\\sqrt{2}} & 0 & 0 & 0\\\\\n\t-\\frac{\\ob}{\\sqrt{2}} & \\frac{\\ob}{2} & \\frac{i\\ob}{2} & 0 & 0 & 0\\\\\n\t\\frac{i\\ob}{\\sqrt{2}} & \\frac{i\\ob}{2} & -\\frac{\\ob}{2} & 0 & 0 & 0\\\\\n\t0 & 0 & 0 & 1 & 0 & 0\\\\\n\t0 & 0 & 0 & 0 & \\ob & 0\\\\\n\t0 & 0 & 0 & 0 & 0 & \\om\n\\end{matrix}\\right), \\quad E \\equiv\n\\left(\\begin{matrix}1 & 0 & 0 & 0 & 0 & 0\\\\\n\t0 & -\\frac{1}{2} & \\frac{\\sqrt{3}}{2} & 0 & 0 & 0\\\\\n\t0 & -\\frac{\\sqrt{3}}{2} & -\\frac{1}{2} & 0 & 0 & 0\\\\\n\t0 & 0 & 0 & 0 & 1 & 0\\\\\n\t0 & 0 & 0 & 0 & 0 & 1\\\\\n\t0 & 0 & 0 & 1 & 0 & 0\n\\end{matrix}\\right),\\\\\n&V\\equiv\n\\left(\\begin{matrix}-\\frac{1}{3} & -\\frac{1}{3\\sqrt{2}} & -\\frac{1}{\\sqrt{6}} & -\\sqrt{\\frac{2}{3}} & 0 & 0\\\\\n\t-\\frac{1}{3\\sqrt{2}} & \\frac{1}{3}+\\frac{\\om}{2} & -\\frac{\\om}{2\\sqrt{3}} & 0 & \\frac{\\ob}{\\sqrt{3}} & \\frac{\\om}{\\sqrt{3}}\\\\\n\t-\\frac{1}{\\sqrt{6}} & -\\frac{\\om}{2\\sqrt{3}} & -\\frac{i\\om}{2\\sqrt{3}} & 0 & -\\frac{i\\ob}{\\sqrt{3}} & \\frac{i\\om}{\\sqrt{3}}\\\\\n\t-\\sqrt{\\frac{2}{3}} & 0 & 0 & \\frac{1}{3} & \\frac{1}{3} & \\frac{1}{3}\\\\\n\t0 & \\frac{\\ob}{\\sqrt{3}} & -\\frac{i\\ob}{\\sqrt{3}} & \\frac{1}{3} & \\frac{\\ob}{3} & \\frac{\\om}{3}\\\\\n\t0 & \\frac{\\om}{\\sqrt{3}} & \\frac{i\\om}{\\sqrt{3}} & \\frac{1}{3} & \\frac{\\om}{3} & \\frac{\\ob}{3}\n\\end{matrix}\\right),\\\\\n&X\\equiv\n\\left(\\begin{matrix}\\frac{\\om}{3} & \\frac{1}{3\\sqrt{2}} & -\\frac{1}{\\sqrt{6}} & -\\frac{i\\om\\sqrt{2}}{3} & -\\frac{i\\om\\sqrt{2}}{3} & -\\frac{i\\sqrt{2}}{3}\\\\\n\t\\frac{\\ob}{3\\sqrt{2}} & \\frac{1}{2}+\\frac{\\om}{6} & -\\frac{\\ob}{2\\sqrt{3}} & -\\frac{i}{3} & \\frac{i2}{3} & -\\frac{i\\ob}{3}\\\\\n\t\\frac{\\ob}{\\sqrt{6}} & \\frac{\\ob}{2\\sqrt{3}} & -\\frac{i\\ob}{2\\sqrt{3}} & \\frac{i}{\\sqrt{3}} & 0 & -\\frac{i\\ob}{\\sqrt{3}}\\\\\n\t\\frac{i\\ob\\sqrt{2}}{3} & \\frac{i}{3} & -\\frac{i}{\\sqrt{3}} & \\frac{1}{3} & \\frac{1}{3} & \\frac{\\om}{3}\\\\\n\t\\frac{i\\om\\sqrt{2}}{3} & \\frac{i\\ob}{3} & \\frac{i\\ob}{\\sqrt{3}} & \\frac{1}{3} & \\frac{\\ob}{3} & \\frac{\\ob}{3}\\\\\n\t\\frac{i\\om\\sqrt{2}}{3} & -\\frac{i\\ob2}{3} & 0 & \\frac{\\ob}{3} & \\frac{1}{3} & \\frac{\\ob}{3}\n\\end{matrix}\\right).\n\\end{split}\n\\end{align}\n\n\\newpage\n\\begin{flalign}\\label{eq:tensor3pqr}\n&\\text{{\\textit {\\textbf {ii}}})\\,\\,\\,}\\boldsymbol{3^p}\\otimes\\boldsymbol{3^p}=\\boldsymbol{6}\\oplus\\boldsymbol{\\tb}, \\quad \\boldsymbol{3^q}\\otimes\\boldsymbol{3^q}=\\boldsymbol{6}\\oplus\\boldsymbol{\\tb}, \\quad \\boldsymbol{3^r}\\otimes\\boldsymbol{3^r}=\\boldsymbol{6}\\oplus\\boldsymbol{\\tb}&\n\\end{flalign}\nWe have already defined the bases of $\\boldsymbol{3}$, $\\boldsymbol{3^p}$, $\\boldsymbol{3^q}$, $\\boldsymbol{3^r}$ and $\\boldsymbol{6}$. Since the representation matrices corresponding to $\\boldsymbol{3}$, $\\boldsymbol{3^p}$, $\\boldsymbol{3^q}$ and $\\boldsymbol{3^r}$ differ only with respect to the multiplication with $\\pm1$, the C-G coefficients for the tensor product expansions in Eqs.~(\\ref{eq:tensor3pqr}) are exactly those given in Eqs.~(\\ref{eq:tensor1exp}). However that is not the case for tensor products involving different types of triplets, \n\\begin{flalign}\\label{eq:tensor3pqr2}\n&\\quad \\,\\,\\,\\boldsymbol{3^p}\\otimes\\boldsymbol{3^q}=\\boldsymbol{6}\\oplus\\boldsymbol{\\cb}, \\quad \\boldsymbol{3^q}\\otimes\\boldsymbol{3^r}=\\boldsymbol{6}\\oplus\\boldsymbol{\\ab}, \\quad \\boldsymbol{3^r}\\otimes\\boldsymbol{3^p}=\\boldsymbol{6}\\oplus\\boldsymbol{\\bb}.&\n\\end{flalign}\nThe C-G coefficients for the sextets appearing in the first, the second and the third tensor product expansions in Eqs.~(\\ref{eq:tensor3pqr2}) are given by\n\\begin{equation}\\label{eq:tensorpqrexp}\n\\boldsymbol{6}\\equiv\\left(\\begin{matrix}\\frac{1}{3}\\left(a_1 b_1 + a_2 b_2 + a_3 b_3+\\om\\left(a_2 b_3+a_3 b_2+a_3 b_1 + a_1 b_3 +a_1 b_2 + a_2 b_1\\right)\\right)\\\\\n\t\\frac{1}{3\\sqrt{2}} a_1 b_1 - \\frac{\\sqrt{2}}{3} a_2 b_2 + \\frac{1}{3\\sqrt{2}} a_3 b_3+\\frac{\\om}{3\\sqrt{2}} (a_2 b_3+a_3 b_2) - \\frac{\\om\\sqrt{2}}{3} (a_3 b_1 + a_1 b_3) + \\frac{\\om}{3\\sqrt{2}} (a_1 b_2 +a_2 b_1)\\\\\n\t\\frac{1}{\\sqrt{6}}\\left(a_1 b_1 - a_3 b_3\\right)+\\frac{\\om}{\\sqrt{6}}\\left(a_2 b_3 + a_3 b_2 - a_1 b_2 - a_2 b_1\\right)\\\\\n\t\\frac{\\ob \\sqrt{2}}{\\sqrt{3}}a_1 b_1 - \\frac{1}{\\sqrt{6}}\\left(a_2 b_3+a_3 b_2\\right)\\\\\n\t\\frac{\\ob \\sqrt{2}}{\\sqrt{3}}a_2 b_2 - \\frac{1}{\\sqrt{6}}\\left(a_3 b_1+a_1 b_3\\right)\\\\\n\\frac{\\ob \\sqrt{2}}{\\sqrt{3}}a_3 b_3 - \\frac{1}{\\sqrt{6}}\\left(a_1 b_2+a_2 b_1\\right),\n\\end{matrix}\\right),\n\\end{equation}\n\\begin{equation}\\label{eq:tensorqrexp}\n\\boldsymbol{6}\\equiv\\left(\\begin{matrix}\\frac{1}{3}\\left(a_1 b_1 + a_2 b_2 + a_3 b_3+\\ob\\left(a_2 b_3+a_3 b_2+a_3 b_1 + a_1 b_3 +a_1 b_2 + a_2 b_1\\right)\\right)\\\\\n\t\\frac{1}{3\\sqrt{2}} a_1 b_1 - \\frac{\\sqrt{2}}{3} a_2 b_2 + \\frac{1}{3\\sqrt{2}} a_3 b_3+\\frac{\\ob}{3\\sqrt{2}} (a_2 b_3+a_3 b_2) - \\frac{\\ob\\sqrt{2}}{3} (a_3 b_1 + a_1 b_3) + \\frac{\\ob}{3\\sqrt{2}} (a_1 b_2 +a_2 b_1)\\\\\n\t\\frac{1}{\\sqrt{6}}\\left(a_1 b_1 - a_3 b_3\\right)+\\frac{\\ob}{\\sqrt{6}}\\left(a_2 b_3 + a_3 b_2 - a_1 b_2 - a_2 b_1\\right)\\\\\n\t\\frac{\\om \\sqrt{2}}{\\sqrt{3}}a_1 b_1 - \\frac{1}{\\sqrt{6}}\\left(a_2 b_3+a_3 b_2\\right)\\\\\n\t\\frac{\\om \\sqrt{2}}{\\sqrt{3}}a_2 b_2 - \\frac{1}{\\sqrt{6}}\\left(a_3 b_1+a_1 b_3\\right)\\\\\n\\frac{\\om \\sqrt{2}}{\\sqrt{3}}a_3 b_3 - \\frac{1}{\\sqrt{6}}\\left(a_1 b_2+a_2 b_1\\right)\n\\end{matrix}\\right)\n\\end{equation}\nand\n\\begin{equation}\\label{eq:tensorrpexp}\n\\boldsymbol{6}\\equiv\\left(\\begin{matrix}\\frac{1}{3}\\left(a_1 b_1 + a_2 b_2 + a_3 b_3+a_2 b_3+a_3 b_2+a_3 b_1 + a_1 b_3 +a_1 b_2 + a_2 b_1\\right)\\\\\n\t\\frac{1}{3\\sqrt{2}} a_1 b_1 - \\frac{\\sqrt{2}}{3} a_2 b_2 + \\frac{1}{3\\sqrt{2}} a_3 b_3+\\frac{1}{3\\sqrt{2}} (a_2 b_3+a_3 b_2) - \\frac{\\sqrt{2}}{3} (a_3 b_1 + a_1 b_3) + \\frac{1}{3\\sqrt{2}} (a_1 b_2 +a_2 b_1)\\\\\n\t\\frac{1}{\\sqrt{6}}\\left(a_1 b_1 - a_3 b_3\\right)+\\frac{1}{\\sqrt{6}}\\left(a_2 b_3 + a_3 b_2 - a_1 b_2 - a_2 b_1\\right)\\\\\n\t\\frac{\\sqrt{2}}{\\sqrt{3}}a_1 b_1 - \\frac{1}{\\sqrt{6}}\\left(a_2 b_3+a_3 b_2\\right)\\\\\n\t\\frac{\\sqrt{2}}{\\sqrt{3}}a_2 b_2 - \\frac{1}{\\sqrt{6}}\\left(a_3 b_1+a_1 b_3\\right)\\\\\n\\frac{\\sqrt{2}}{\\sqrt{3}}a_3 b_3 - \\frac{1}{\\sqrt{6}}\\left(a_1 b_2+a_2 b_1\\right)\n\\end{matrix}\\right)\n\\end{equation}\nrespectively. On the other hand, the C-G coefficients for $\\boldsymbol{\\cb}$, $\\boldsymbol{\\ab}$ and $\\boldsymbol{\\bb}$ (the triplets in the RHS of Eqs.~(\\ref{eq:tensor3pqr2})) are given by the same expression as the one for $\\boldsymbol{\\tb}$ in Eq.~(\\ref{eq:tensor1exp}), i.e.~\n\\begin{equation}\n\\boldsymbol{\\cb},\\, \\boldsymbol{\\ab},\\, \\boldsymbol{\\bb} \\equiv\n\\left(\\begin{matrix}\\frac{1}{\\sqrt{2}}\\left(a_2 b_3 - a_3 b_2\\right)\\\\\n\t\\frac{1}{\\sqrt{2}}\\left(a_3 b_1 - a_1 b_3\\right)\\\\\n\\frac{1}{\\sqrt{2}}\\left(a_1 b_2 - a_2 b_1\\right)\n\\end{matrix}\\right).\n\\end{equation}\n\n\\newpage\n\\begin{flalign}\\label{eq:tensor18}\n&\\text{{\\textit {\\textbf {iii}}})\\,\\,\\,}\\boldsymbol{3}\\otimes\\boldsymbol{\\bar{3}}=\\boldsymbol{1}\\oplus\\boldsymbol{8}&\n\\end{flalign}\n\\begin{equation}\\label{eq:tensor18exp}\n\\boldsymbol{1}\\equiv \\frac{1}{\\sqrt{3}}\\left(a_1 b_1 + a_2 b_2 + a_3 b_3\\right), \\quad \\quad \\boldsymbol{8}\\equiv\\left(\\begin{matrix}\\frac{1}{\\sqrt{6}} a_1 b_1 - \\sqrt{\\frac{2}{3}} a_2 b_2 + \\frac{1}{\\sqrt{6}} a_3 b_3\\\\\n\t\\frac{1}{\\sqrt{2}}\\left(a_1 b_1 - a_3 b_3\\right)\\\\\n\t\\frac{1}{\\sqrt{2}}\\left(a_2 b_3 + a_3 b_2\\right)\\\\\n\t\\frac{1}{\\sqrt{2}}\\left(a_3 b_1 + a_1 b_3\\right)\\\\\n\t\\frac{1}{\\sqrt{2}}\\left(a_1 b_2 + a_2 b_1\\right)\\\\\n\t\\frac{1}{\\sqrt{2}}\\left(a_2 b_3 - a_3 b_2\\right)\\\\\n\t\\frac{1}{\\sqrt{2}}\\left(a_3 b_1 - a_1 b_3\\right)\\\\\n\\frac{1}{\\sqrt{2}}\\left(a_1 b_2 - a_2 b_1\\right)\n\\end{matrix}\\right).\n\\end{equation}\nWe define the basis for the octet representation using Eqs.~(\\ref{eq:tensor18exp}). The resulting generator matrices are \n\\begin{align}\n&C \\equiv\n\\left(\\begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n\t0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n\t0 & 0 & -\\frac{1}{2} & 0 & 0 & -\\frac{i \\sqrt{3}}{2} & 0 & 0\\\\\n\t0 & 0 & 0 & -\\frac{1}{2} & 0 & 0 & -\\frac{i \\sqrt{3}}{2} & 0\\\\\n\t0 & 0 & 0 & 0 & -\\frac{1}{2} & 0 & 0 & -\\frac{i \\sqrt{3}}{2}\\\\\n\t0 & 0 & -\\frac{i \\sqrt{3}}{2} & 0 & 0 & -\\frac{1}{2} & 0 & 0\\\\\n\t0 & 0 & 0 & -\\frac{i \\sqrt{3}}{2} & 0 & 0 & -\\frac{1}{2} & 0\\\\\n\t0 & 0 & 0 & 0 & -\\frac{i \\sqrt{3}}{2} & 0 & 0 & -\\frac{1}{2}\n\\end{matrix}\\right), \\quad E \\equiv\n\\left(\\begin{matrix}-\\frac{1}{2} & \\frac{\\sqrt{3}}{2} & 0 & 0 & 0 & 0 & 0 & 0\\\\\n\t-\\frac{\\sqrt{3}}{2} & -\\frac{1}{2} & 0 & 0 & 0 & 0 & 0 & 0\\\\\n\t0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\\\\n\t0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\\\\n\t0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\\\\n\t0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\\\\n\t0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\\\\n\t0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\n\\end{matrix}\\right),\\notag \\\\\n&V \\equiv\n\\left(\\begin{matrix}0 & 0 & \\frac{1}{2\\sqrt{3}} & \\frac{1}{2\\sqrt{3}} & \\frac{1}{2\\sqrt{3}} & \\frac{i}{2} & \\frac{i}{2} & \\frac{i}{2}\\\\\n\t0 & 0 & \\frac{1}{2} & \\frac{1}{2} & \\frac{1}{2} & -\\frac{i}{2\\sqrt{3}} & -\\frac{i}{2\\sqrt{3}} & -\\frac{i}{2\\sqrt{3}}\\\\\n\t\\frac{1}{2\\sqrt{3}} & \\frac{1}{2} & \\frac{2}{3} & -\\frac{1}{3} & -\\frac{1}{3} & 0 & 0 & 0\\\\\n\t\\frac{1}{2\\sqrt{3}} & \\frac{1}{2} & -\\frac{1}{3} & \\frac{1}{6} & \\frac{1}{6} & -\\frac{i}{\\sqrt{3}} & \\frac{i}{2\\sqrt{3}} & \\frac{i}{2\\sqrt{3}}\\\\\n\t\\frac{1}{2\\sqrt{3}} & \\frac{1}{2} & -\\frac{1}{3} & \\frac{1}{6} & \\frac{1}{6} & \\frac{i}{\\sqrt{3}} & -\\frac{i}{2\\sqrt{3}} & -\\frac{i}{2\\sqrt{3}}\\\\\n\t\\frac{i}{2} & -\\frac{i}{2\\sqrt{3}} & 0 & -\\frac{i}{\\sqrt{3}} & \\frac{i}{\\sqrt{3}} & 0 & 0 & 0\\\\\n\t\\frac{i}{2} & -\\frac{i}{2\\sqrt{3}} & 0 & \\frac{i}{2\\sqrt{3}} & -\\frac{i}{2\\sqrt{3}} & 0 & -\\frac{1}{2} & \\frac{1}{2}\\\\\n\t\\frac{i}{2} & -\\frac{i}{2\\sqrt{3}} & 0 & \\frac{i}{2\\sqrt{3}} & -\\frac{i}{2\\sqrt{3}} & 0 & \\frac{1}{2} & -\\frac{1}{2}\n\\end{matrix}\\right),\\label{eq:gen8}\\\\\n&X \\equiv\n\\left(\\begin{matrix}0 & 0 & -\\frac{1}{\\sqrt{3}} & \\frac{1}{2\\sqrt{3}} & \\frac{1}{2\\sqrt{3}} & 0 & -\\frac{i}{2} & \\frac{i}{2}\\\\\n\t0 & 0 & 0 & -\\frac{1}{2} & \\frac{1}{2} & \\frac{i}{\\sqrt{3}} & -\\frac{i}{2\\sqrt{3}} & -\\frac{i}{2\\sqrt{3}}\\\\\n\t\\frac{1}{2\\sqrt{3}} & -\\frac{1}{2} & \\frac{1}{6} & \\frac{1}{6} & \\frac{2}{3} & -\\frac{i}{2\\sqrt{3}} & \\frac{i}{2\\sqrt{3}} & 0\\\\\n\t-\\frac{1}{\\sqrt{3}} & 0 & -\\frac{1}{3} & -\\frac{1}{3} & \\frac{1}{6} & 0 & \\frac{i}{\\sqrt{3}} & \\frac{i}{2\\sqrt{3}}\\\\\n\t\\frac{1}{2\\sqrt{3}} & \\frac{1}{2} & -\\frac{1}{3} & -\\frac{1}{3} & \\frac{1}{6} & -\\frac{i}{\\sqrt{3}} & 0 & -\\frac{i}{2\\sqrt{3}}\\\\\n\t-\\frac{i}{2} & -\\frac{i}{2\\sqrt{3}} & \\frac{i}{2\\sqrt{3}} & -\\frac{i}{2\\sqrt{3}} & 0 & \\frac{1}{2} & \\frac{1}{2} & 0\\\\\n\t0 & \\frac{i}{\\sqrt{3}} & \\frac{i}{\\sqrt{3}} & 0 & \\frac{i}{2\\sqrt{3}} & 0 & 0 & -\\frac{1}{2}\\\\\n\t\\frac{i}{2} & -\\frac{i}{2\\sqrt{3}} & 0 & -\\frac{i}{\\sqrt{3}} & -\\frac{i}{2\\sqrt{3}} & 0 & 0 & -\\frac{1}{2}\n\\end{matrix}\\right).\\notag\\\\ \\notag\n\\end{align}\nNote that for the $SU(3)$ group, the tensor product of a $\\boldsymbol{3}$ and a $\\boldsymbol{\\tb}$ gives a $\\boldsymbol{1}$ and an $\\boldsymbol{8}$, i.e.~the tensor product expansion Eqs.~(\\ref{eq:tensor18},~\\ref{eq:tensor18exp}) is applicable to the $SU(3)$ group as well.\n\\begin{flalign}\\label{eq:tensor288}\n&\\text{{\\textit {\\textbf {iv}}})\\,\\,\\,}\\boldsymbol{6}\\otimes\\boldsymbol{3}=\\boldsymbol{2}\\oplus\\boldsymbol{8}\\oplus\\boldsymbol{8}&\n\\end{flalign}\n\\begin{align}\\label{eq:tensor288exp}\n\\begin{split}\n&\\boldsymbol{2}\\equiv\\left(\\begin{matrix}\\frac{a_4 b_1}{\\sqrt{3}}+\\frac{a_5 b_2}{\\sqrt{3}}+\\frac{a_6 b_3}{\\sqrt{3}}\\\\\n\t\\frac{a_1 b_1}{3}+\\frac{a_1 b_2}{3} +\\frac{a_1 b_3}{3}+\\frac{a_2 b_1}{3 \\sqrt{2}}-\\frac{\\sqrt{2}a_2 b_2}{3}+\\frac{a_2 b_3}{3 \\sqrt{2}}+\\frac{a_3 b_1}{\\sqrt{6}} -\\frac{a_3 b_3}{\\sqrt{6}}\n\\end{matrix}\\right),\\\\\n&\\boldsymbol{8}\\equiv\\left(\\begin{matrix}\\frac{a_1 b_1}{\\sqrt{6}}-\\frac{a_1 b_3}{\\sqrt{6}}+\\frac{a_2 b_1}{2 \\sqrt{3}}-\\frac{a_2 b_3}{2 \\sqrt{3}}+\\frac{a_3 b_1}{2}+\\frac{a_3 b_3}{2}\\\\\n\t-\\frac{a_1 b_1}{3 \\sqrt{2}}+\\frac{\\sqrt{2} a_1 b_2}{3}-\\frac{a_1 b_3}{3\\sqrt{2}}-\\frac{a_2 b_1}{6}-\\frac{2 a_2 b_2}{3}-\\frac{a_2 b_3}{6}-\\frac{a_3 b_1}{2 \\sqrt{3}}+\\frac{a_3 b_3}{2 \\sqrt{3}}\\\\\n\t\\frac{a_1 b_2}{3 \\sqrt{2}}-\\frac{a_1 b_3}{3 \\sqrt{2}}+\\frac{a_2 b_2}{6}+\\frac{a_2 b_3}{3}-\\frac{a_3 b_2}{2 \\sqrt{3}}-\\frac{a_4 b_2}{\\sqrt{3}}+\\frac{a_4 b_3}{\\sqrt{3}}\\\\\n\t-\\frac{a_1 b_1}{3 \\sqrt{2}}+\\frac{a_1 b_3}{3 \\sqrt{2}}-\\frac{a_2 b_1}{6}+\\frac{a_2 b_3}{6}+\\frac{a_3 b_1}{2 \\sqrt{3}}+\\frac{a_3 b_3}{2 \\sqrt{3}}+\\frac{a_5 b_1}{\\sqrt{3}}-\\frac{a_5 b_3}{\\sqrt{3}}\\\\\n\t\\frac{a_1 b_1}{3 \\sqrt{2}}-\\frac{a_1 b_2}{3 \\sqrt{2}}-\\frac{a_2 b_1}{3}-\\frac{a_2 b_2}{6}-\\frac{a_3 b_2}{2\\sqrt{3}}-\\frac{a_6 b_1}{\\sqrt{3}}+\\frac{a_6 b_2}{\\sqrt{3}}\\\\\n\t\\frac{a_1 b_2}{3 \\sqrt{2}}+\\frac{a_1 b_3}{3 \\sqrt{2}}+\\frac{a_2 b_2}{6}-\\frac{a_2 b_3}{3}-\\frac{a_3 b_2}{2 \\sqrt{3}}+\\frac{a_4 b_2}{\\sqrt{3}}+\\frac{a_4 b_3}{\\sqrt{3}}\\\\\n\t\\frac{a_1 b_1}{3 \\sqrt{2}}+\\frac{a_1 b_3}{3 \\sqrt{2}}+\\frac{a_2 b_1}{6}+\\frac{a_2 b_3}{6}-\\frac{a_3 b_1}{2 \\sqrt{3}}+\\frac{a_3 b_3}{2 \\sqrt{3}}+\\frac{a_5 b_1}{\\sqrt{3}}+\\frac{a_5 b_3}{\\sqrt{3}}\\\\\n\\frac{a_1 b_1}{3 \\sqrt{2}}+\\frac{a_1 b_2}{3 \\sqrt{2}}-\\frac{a_2 b_1}{3}+\\frac{a_2 b_2}{6}+\\frac{a_3 b_2}{2\\sqrt{3}}+\\frac{a_6 b_1}{\\sqrt{3}}+\\frac{a_6 b_2}{\\sqrt{3}}\n\\end{matrix}\\right), \\\\ &\\boldsymbol{8}\\equiv\\left(\\begin{matrix}\\frac{a_4 b_1}{\\sqrt{2}}-\\frac{a_6 b_3}{\\sqrt{2}}\\\\\n\t-\\frac{a_4 b_1}{\\sqrt{6}}+\\frac{\\sqrt{2}a_5 b_2}{\\sqrt{3}} -\\frac{a_6 b_3}{\\sqrt{6}}\\\\\n\t-\\frac{a_2 b_1}{\\sqrt{2}}+\\frac{a_3 b_1}{\\sqrt{6}}+\\frac{a_5 b_3}{\\sqrt{6}}-\\frac{a_6 b_2}{\\sqrt{6}}\\\\\n\t-\\frac{\\sqrt{2} a_3 b_2}{\\sqrt{3}}-\\frac{a_4 b_3}{\\sqrt{6}}+\\frac{a_6 b_1}{\\sqrt{6}}\\\\\n\t\\frac{a_2 b_3}{\\sqrt{2}}+\\frac{a_3 b_3}{\\sqrt{6}}+\\frac{a_4 b_2}{\\sqrt{6}}-\\frac{a_5 b_1}{\\sqrt{6}}\\\\\n\t\\frac{2 a_1 b_1}{3}-\\frac{a_2 b_1}{3 \\sqrt{2}}-\\frac{a_3 b_1}{\\sqrt{6}}-\\frac{a_5 b_3}{\\sqrt{6}}-\\frac{a_6 b_2}{\\sqrt{6}}\\\\\n\t\\frac{2 a_1 b_2}{3}+\\frac{\\sqrt{2} a_2 b_2}{3} -\\frac{a_4 b_3}{\\sqrt{6}}-\\frac{a_6 b_1}{\\sqrt{6}}\\\\\n\t\\frac{2 a_1 b_3}{3}-\\frac{a_2 b_3}{3 \\sqrt{2}}+\\frac{a_3 b_3}{\\sqrt{6}}-\\frac{a_4 b_2}{\\sqrt{6}}-\\frac{a_5 b_1}{\\sqrt{6}}\n\\end{matrix}\\right).\n\\end{split}\n\\end{align}\nWe define the basis for the doublet representation using Eqs.~(\\ref{eq:tensor288exp}). The resulting generator matrices are \n\\begin{equation}\\label{eq:gen2}\nC \\equiv\n\\left(\\begin{matrix}1 & 0\\\\\n0 & 1\n\\end{matrix}\\right), \\quad E \\equiv\n\\left(\\begin{matrix}1 & 0\\\\\n0 & 1\n\\end{matrix}\\right), \\quad V \\equiv\ni\\left(\\begin{matrix}-\\frac{1}{\\sqrt{3}} & \\sqrt{\\frac{2}{3}}\\\\\n\\sqrt{\\frac{2}{3}} & \\frac{1}{\\sqrt{3}}\n\\end{matrix}\\right), \\quad X \\equiv\ni\\left(\\begin{matrix}-\\frac{1}{\\sqrt{3}} & \\sqrt{\\frac{2}{3}}\\om\\\\\n\\sqrt{\\frac{2}{3}}\\ob & \\frac{1}{\\sqrt{3}}\n\\end{matrix}\\right).\n\\end{equation}\nIn relation to the tensor product expansion Eq.~(\\ref{eq:tensor288}), we may embed the irreps of $\\Sigma(72\\times3)$ in the irreps of $SU(3)$:\n\\begin{equation}\n \\label{eq:orig}\n \\begin{tikzpicture}[>=stealth,baseline=(current bounding box.center),anchor=base,inner sep=0pt]\n \\matrix (foil) [matrix of math nodes,nodes={minimum height=1em}] {\n SU(3)&\\,:\\,&\\boldsymbol{6}&\\otimes&\\boldsymbol{3}&=&\\boldsymbol{10}&\\oplus&\\boldsymbol{8}\\\\\n \\,\\\\\n \\Sigma(72\\times3)&\\,:\\,&\\boldsymbol{6}&\\otimes&\\boldsymbol{3}&=&\\boldsymbol{2}&\\oplus&\\boldsymbol{8}&\\oplus&\\boldsymbol{8}.\\\\\n };\n\\path[->] ($(foil-1-1.south)$) edge[] ($(foil-3-1.north)$);\n\\path[->] ($(foil-1-3.south)$) edge[] ($(foil-3-3.north)$);\n\\path[->] ($(foil-1-5.south)$) edge[] ($(foil-3-5.north)$);\n\\path[->] ($(foil-1-7.south)$) edge[] ($(foil-3-7.north)$);\n\\path[->] ($(foil-1-7.south)$) edge[] ($(foil-3-9.north)$);\n\\path[->] ($(foil-1-9.south)$) edge[] ($(foil-3-11.north)$);\n \\end{tikzpicture}\n \\end{equation}\n\\begin{flalign}\\label{eq:tensor315}\n&\\text{{\\textit {\\textbf {v}}})\\,\\,\\,}\\boldsymbol{6}\\otimes\\boldsymbol{\\tb}=\\boldsymbol{3}\\oplus\\boldsymbol{\\xb}\\oplus\\boldsymbol{3^p}\\oplus\\boldsymbol{3^q}\\oplus\\boldsymbol{3^r}&\n\\end{flalign}\n\\begin{equation*}\n\\boldsymbol{3}\\equiv\\left(\\begin{matrix}\\frac{a_1 b_1}{\\sqrt{6}}+\\frac{a_2 b_1}{2 \\sqrt{3}}+\\frac{a_3 b_1}{2}+\\frac{a_5 b_3}{2}+\\frac{a_6 b_2}{2}\\\\\n\t\\frac{a_1 b_2}{\\sqrt{6}}-\\frac{a_2 b_2}{\\sqrt{3}}+\\frac{a_4 b_3}{2}+\\frac{a_6 b_1}{2}\\\\\n\t\\frac{a_1 b_3}{\\sqrt{6}}+\\frac{a_2 b_3}{2 \\sqrt{3}}-\\frac{a_3 b_3}{2}+\\frac{a_4 b_2}{2}+\\frac{a_5 b_1}{2}\n\\end{matrix}\\right),\n\\end{equation*}\n\\newpage\n\\begin{align}\n\\begin{split}\n&\\boldsymbol{\\xb}\\equiv\\left(\\begin{matrix}-\\frac{a_4 b_2}{\\sqrt{6}}+\\frac{a_4 b_3}{\\sqrt{6}}+\\frac{a_5 b_1}{\\sqrt{6}}-\\frac{a_5 b_3}{\\sqrt{6}}-\\frac{a_6 b_1}{\\sqrt{6}}+\\frac{a_6 b_2}{\\sqrt{6}}\\\\\n\t-\\frac{a_4 b_2}{2 \\sqrt{3}}-\\frac{a_4 b_3}{\\sqrt{3}}+\\frac{a_5 b_1}{2 \\sqrt{3}}-\\frac{a_5 b_3}{2 \\sqrt{3}}+\\frac{a_6 b_1}{\\sqrt{3}}+\\frac{a_6 b_2}{2 \\sqrt{3}}\\\\\n\t\\frac{a_4 b_2}{2}-\\frac{a_5 b_1}{2}-\\frac{a_5 b_3}{2}+\\frac{a_6 b_2}{2}\\\\\n\t-\\frac{a_1 b_2}{\\sqrt{6}}+\\frac{a_1 b_3}{\\sqrt{6}}-\\frac{a_2 b_2}{2 \\sqrt{3}}-\\frac{a_2 b_3}{\\sqrt{3}}+\\frac{a_3 b_2}{2}\\\\\n\t\\frac{a_1 b_1}{\\sqrt{6}}-\\frac{a_1 b_3}{\\sqrt{6}}+\\frac{a_2 b_1}{2 \\sqrt{3}}-\\frac{a_2 b_3}{2 \\sqrt{3}}-\\frac{a_3 b_1}{2}-\\frac{a_3 b_3}{2}\\\\\n-\\frac{a_1 b_1}{\\sqrt{6}}+\\frac{a_1 b_2}{\\sqrt{6}}+\\frac{a_2 b_1}{\\sqrt{3}}+\\frac{a_2 b_2}{2 \\sqrt{3}}+\\frac{a_3 b_2}{2}\n\\end{matrix}\\right),\\\\\n&\\boldsymbol{3^p}\\equiv\\left(\\begin{matrix} -\\frac{a_1 b_1}{3 \\sqrt{2}}-\\frac{\\om a_1 b_2 }{3 \\sqrt{2}}-\\frac{\\om a_1 b_3}{3 \\sqrt{2}}-\\frac{a_2 b_1}{6}-\\frac{\\om a_2 b_2}{6}+\\frac{\\om a_2 b_3}{3} -\\frac{a_3 b_1}{2 \\sqrt{3}}+\\frac{\\om a_3 b_2}{2 \\sqrt{3}} -\\frac{\\ob a_4 b_1}{\\sqrt{3}} +\\frac{a_5 b_3}{2 \\sqrt{3}}+\\frac{a_6 b_2}{2\\sqrt{3}} \\\\\n\t-\\frac{\\om a_1 b_1}{3 \\sqrt{2}}-\\frac{a_1 b_2}{3 \\sqrt{2}}-\\frac{\\om a_1 b_3}{3 \\sqrt{2}}-\\frac{\\om a_2 b_1}{6}+\\frac{a_2 b_2}{3}-\\frac{\\om a_2 b_3}{6} +\\frac{\\om a_3 b_1}{2 \\sqrt{3}} -\\frac{\\om a_3 b_3}{2 \\sqrt{3}}+\\frac{a_4 b_3}{2 \\sqrt{3}}-\\frac{\\ob a_5 b_2}{\\sqrt{3}}+\\frac{a_6 b_1}{2 \\sqrt{3}}\\\\\n\t-\\frac{\\om a_1 b_1}{3 \\sqrt{2}}-\\frac{\\om a_1 b_2}{3 \\sqrt{2}}-\\frac{a_1 b_3}{3 \\sqrt{2}}+\\frac{\\om a_2 b_1}{3} -\\frac{\\om a_2 b_2}{6}-\\frac{a_2 b_3}{6} -\\frac{\\om a_3 b_2}{2\\sqrt{3}}+\\frac{a_3 b_3}{2 \\sqrt{3}}+\\frac{a_4 b_2}{2 \\sqrt{3}}+\\frac{a_5 b_1}{2 \\sqrt{3}}-\\frac{\\ob a_6 b_3}{\\sqrt{3}} \n\\end{matrix}\\right),\\\\\n&\\boldsymbol{3^q}\\equiv\\left(\\begin{matrix}-\\frac{a_1 b_1}{3 \\sqrt{2}}-\\frac{a_1 b_2}{3 \\sqrt{2}}-\\frac{a_1 b_3}{3 \\sqrt{2}}-\\frac{a_2 b_1}{6}-\\frac{a_2 b_2}{6}+\\frac{a_2 b_3}{3}-\\frac{a_3 b_1}{2 \\sqrt{3}}+\\frac{a_3 b_2}{2 \\sqrt{3}}-\\frac{a_4 b_1}{\\sqrt{3}}+\\frac{a_5 b_3}{2 \\sqrt{3}}+\\frac{a_6 b_2}{2 \\sqrt{3}}\\\\\n\t-\\frac{a_1 b_1}{3 \\sqrt{2}}-\\frac{a_1 b_2}{3 \\sqrt{2}}-\\frac{a_1 b_3}{3 \\sqrt{2}}-\\frac{a_2 b_1}{6}+\\frac{a_2 b_2}{3}-\\frac{a_2 b_3}{6}+\\frac{a_3 b_1}{2 \\sqrt{3}}-\\frac{a_3 b_3}{2 \\sqrt{3}}+\\frac{a_4 b_3}{2 \\sqrt{3}}-\\frac{a_5 b_2}{\\sqrt{3}}+\\frac{a_6 b_1}{2 \\sqrt{3}}\\\\\n-\\frac{a_1 b_1}{3 \\sqrt{2}}-\\frac{a_1 b_2}{3 \\sqrt{2}}-\\frac{a_1 b_3}{3 \\sqrt{2}}+\\frac{a_2 b_1}{3}-\\frac{a_2 b_2}{6}-\\frac{a_2 b_3}{6}-\\frac{a_3 b_2}{2\\sqrt{3}}+\\frac{a_3 b_3}{2 \\sqrt{3}}+\\frac{a_4 b_2}{2 \\sqrt{3}}+\\frac{a_5 b_1}{2 \\sqrt{3}}-\\frac{a_6 b_3}{\\sqrt{3}}\n\\end{matrix}\\right),\\\\\n&\\boldsymbol{3^r}\\equiv\\left(\\begin{matrix}\\frac{\\om a_1 b_1}{3 \\sqrt{2}}+\\frac{a_1 b_2}{3 \\sqrt{2}}+\\frac{a_1 b_3}{3\\sqrt{2}}+\\frac{\\om a_2 b_1}{6}+\\frac{a_2 b_2}{6}-\\frac{a_2 b_3}{3} +\\frac{\\om a_3 b_1}{2 \\sqrt{3}} -\\frac{a_3 b_2}{2 \\sqrt{3}}+\\frac{\\ob a_4 b_1}{\\sqrt{3}}-\\frac{\\om a_5 b_3}{2\\sqrt{3}}-\\frac{\\om a_6 b_2}{2 \\sqrt{3}} \\\\\n\t\\frac{a_1 b_1}{3 \\sqrt{2}}+\\frac{\\om a_1 b_2}{3 \\sqrt{2}}+\\frac{a_1 b_3}{3 \\sqrt{2}}+\\frac{a_2 b_1}{6}-\\frac{\\om a_2 b_2}{3}+\\frac{a_2 b_3}{6} -\\frac{a_3 b_1}{2 \\sqrt{3}} +\\frac{a_3 b_3}{2 \\sqrt{3}}-\\frac{\\om a_4 b_3}{2 \\sqrt{3}}+\\frac{\\ob a_5 b_2}{\\sqrt{3}}-\\frac{\\om a_6 b_1}{2 \\sqrt{3}}\\\\\n\\frac{a_1 b_1}{3 \\sqrt{2}}+\\frac{a_1 b_2}{3 \\sqrt{2}}+\\frac{\\om a_1 b_3}{3 \\sqrt{2}}-\\frac{a_2 b_1}{3} +\\frac{a_2 b_2}{6}+\\frac{\\om a_2 b_3}{6}+\\frac{a_3 b_2}{2 \\sqrt{3}}-\\frac{\\om a_3 b_3}{2 \\sqrt{3}} -\\frac{\\om a_4 b_2}{2 \\sqrt{3}}-\\frac{\\om a_5 b_1}{2 \\sqrt{3}}+\\frac{\\ob a_6 b_3}{\\sqrt{3}}\n\\end{matrix}\\right).\n\\end{split}\n\\end{align}\nThe $SU(3)$ embedding corresponding to Eq.~(\\ref{eq:tensor315}) is \n\\begin{equation}\n \\label{eq:orig}\n \\begin{tikzpicture}[>=stealth,baseline=(current bounding box.center),anchor=base,inner sep=0pt]\n \\matrix (foil) [matrix of math nodes,nodes={minimum height=1em}] {\n SU(3)&\\,:\\,&\\boldsymbol{6}&\\otimes&\\boldsymbol{\\tb}&=&\\boldsymbol{3}&\\oplus&\\boldsymbol{15}\\\\\n \\,\\\\\n \\Sigma(72\\times3)&\\,:\\,&\\boldsymbol{6}&\\otimes&\\boldsymbol{\\tb}&=&\\boldsymbol{3}&\\oplus&\\boldsymbol{\\xb}&\\oplus&\\boldsymbol{3^p}&\\oplus&\\boldsymbol{3^q}&\\oplus&\\boldsymbol{3^r}.\\\\\n };\n\\path[->] ($(foil-1-1.south)$) edge[] ($(foil-3-1.north)$);\n\\path[->] ($(foil-1-3.south)$) edge[] ($(foil-3-3.north)$);\n\\path[->] ($(foil-1-5.south)$) edge[] ($(foil-3-5.north)$);\n\\path[->] ($(foil-1-7.south)$) edge[] ($(foil-3-7.north)$);\n\\path[->] ($(foil-1-9.south)$) edge[] ($(foil-3-9.north)$);\n\\path[->] ($(foil-1-9.south)$) edge[] ($(foil-3-11.north)$);\n\\path[->] ($(foil-1-9.south)$) edge[] ($(foil-3-13.north)$);\n\\path[->] ($(foil-1-9.south)$) edge[] ($(foil-3-15.north)$);\n \\end{tikzpicture}\n \\end{equation}\n\\begin{flalign}\\label{eq:tensor66}\n&\\text{{\\textit {\\textbf {vi}}})\\,\\,\\,}\\boldsymbol{6}\\otimes\\boldsymbol{6}=\\underbrace{\\boldsymbol{\\xb}\\oplus\\boldsymbol{\\xb}\\oplus\\boldsymbol{\\xb}\\oplus\\boldsymbol{3}}_\\text{symm}\\oplus\\underbrace{\\boldsymbol{\\xb}\\oplus\\boldsymbol{3^p}\\oplus\\boldsymbol{3^q}\\oplus\\boldsymbol{3^r}}_\\text{antisymm}&\n\\end{flalign}\nHere the sextet, $\\boldsymbol{\\xb}$, appears more than once in the symmetric part. So there is no unique way of decomposing the product space into the sum of the irreducible sextets, i.e.~the C-G coefficients are not uniquely defined. To solve this problem, we utilise the group $\\Sigma(216\\times3)$ which has $\\Sigma(72\\times3)$ as one of its subgroups. $\\Sigma(216\\times3)$ has three distinct types of sextets~\\cite{Sigma1}, $\\boldsymbol{6^x}$, $\\boldsymbol{6^y}$, $\\boldsymbol{6^z}$. The sextet of $\\Sigma(72\\times3)$ can be embedded in any of these three sextets of $\\Sigma(216\\times3)$. A tensor product expansion for $\\Sigma(216\\times3)$, equivalent to Eq.~(\\ref{eq:tensor66}), is given by \n\\begin{equation}\\label{eq:tensor66hes}\n\\boldsymbol{6^x}\\otimes\\boldsymbol{6^x}=\\underbrace{\\boldsymbol{\\xb^x}\\oplus\\boldsymbol{\\xb^y}\\oplus\\boldsymbol{\\xb^z}\\oplus\\boldsymbol{3^x}}_\\text{symm}\\oplus\\underbrace{\\boldsymbol{\\xb^x}\\oplus\\boldsymbol{9}}_\\text{antisymm}.\n\\end{equation}\nIn Eq.~(\\ref{eq:tensor66hes}), the decomposition of the symmetric part into the irreducible sextets is unique. Hence we embed the irreps of $\\Sigma(72\\times3)$ in the irreps of $\\Sigma(216\\times3)$,\n\\begin{equation}\\label{eq:orig}\n\\begin{tikzpicture}[>=stealth,baseline=(current bounding box.center),anchor=base,inner sep=0pt]\n\\matrix (foil) [matrix of math nodes,nodes={minimum height=1em}] {\n\\Sigma(216\\times3)&\\,:\\,& \\boldsymbol{6^x}&\\otimes&\\boldsymbol{6^x}&=&\\boldsymbol{\\bar{6^x}}&\\oplus&\\boldsymbol{\\bar{6^y}}&\\oplus&\\boldsymbol{\\bar{6^z}}&\\oplus&\\boldsymbol{3^x}&\\oplus&\\boldsymbol{\\bar{6^x}}&\\oplus&\\boldsymbol{9}\\\\\n\\,\\\\\n\\Sigma(72\\times3)&\\,:\\,& \\boldsymbol{6}&\\otimes&\\boldsymbol{6}&=&\\boldsymbol{\\xb}&\\oplus&\\boldsymbol{\\xb}&\\oplus&\\boldsymbol{\\xb}&\\oplus&\\boldsymbol{3}&\\oplus&\\boldsymbol{\\xb}&\\oplus&\\boldsymbol{3^p}&\\oplus&\\boldsymbol{3^q}&\\oplus&\\boldsymbol{3^r}.\\\\};\n\\path[->] ($(foil-1-1.south)$) edge[] ($(foil-3-1.north)$);\n\\path[->] ($(foil-1-3.south)$) edge[] ($(foil-3-3.north)$);\n\\path[->] ($(foil-1-5.south)$) edge[] ($(foil-3-5.north)$);\n\\path[->] ($(foil-1-7.south)$) edge[] ($(foil-3-7.north)$);\n\\path[->] ($(foil-1-9.south)$) edge[] ($(foil-3-9.north)$);\n\\path[->] ($(foil-1-11.south)$) edge[] ($(foil-3-11.north)$);\n\\path[->] ($(foil-1-13.south)$) edge[] ($(foil-3-13.north)$);\n\\path[->] ($(foil-1-15.south)$) edge[] ($(foil-3-15.north)$);\n\\path[->] ($(foil-1-17.south)$) edge[] ($(foil-3-17.north)$);\n\\path[->] ($(foil-1-17.south)$) edge[] ($(foil-3-19.north)$);\n\\path[->] ($(foil-1-17.south)$) edge[] ($(foil-3-21.north)$);\n\\end{tikzpicture}\n\\end{equation}\nto obtain a unique decomposition for the case of $\\Sigma(72\\times3)$ as well.\n\\newpage\n\n{\\small\n\\begin{align}\\label{eq:tensor66exp}\n\\begin{split}\n&\\boldsymbol{\\xb}\\equiv\\left(\\begin{matrix}\\frac{2}{3}a_1 b_1-\\frac{1}{3}a_2 b_2-\\frac{1}{3}a_3 b_3-\\frac{1}{3}a_4 b_4-\\frac{1}{3}a_5 b_5-\\frac{1}{3}a_6 b_6\\\\\n\t-\\frac{\\sqrt{2}}{3}a_2 b_2 +\\frac{\\sqrt{2}}{3}a_3 b_3 -\\frac{1}{3 \\sqrt{2}}a_4 b_4+\\frac{\\sqrt{2}}{3}a_5 b_5 -\\frac{1}{3 \\sqrt{2}}a_6 b_6-\\frac{1}{3}a_{\\{1} b_{2\\}}\\\\\n\t-\\frac{1}{\\sqrt{6}}a_4 b_4+\\frac{1}{\\sqrt{6}}a_6 b_6 -\\frac{1}{3}a_{\\{1} b_{3\\}}+\\frac{\\sqrt{2}}{3}a_{\\{2} b_{3\\}}\\\\\n\t-\\frac{1}{3}a_{\\{1} b_{4\\}}-\\frac{1}{3 \\sqrt{2}}a_{\\{2} b_{4\\}}-\\frac{1}{\\sqrt{6}}a_{\\{3} b_{4\\}}+\\frac{1}{\\sqrt{6}}a_{\\{5} b_{6\\}}\\\\\n\t-\\frac{1}{3}a_{\\{1} b_{5\\}}+\\frac{\\sqrt{2}}{3}a_{\\{2} b_{5\\}}+\\frac{1}{\\sqrt{6}}a_{\\{4} b_{6\\}}\\\\\n\t-\\frac{1}{3}a_{\\{1} b_{6\\}}-\\frac{1}{3 \\sqrt{2}}a_{\\{2} b_{6\\}}+\\frac{1}{\\sqrt{6}}a_{\\{3} b_{6\\}}+\\frac{1}{\\sqrt{6}}a_{\\{4} b_{5\\}}\n\\end{matrix}\\right),\\\\\n&\\boldsymbol{\\xb}\\equiv\\left(\\begin{matrix}\n\\frac{1}{3}(a_1 b_1+a_2 b_2+a_3 b_3)-\\frac{\\sqrt{2}}{3\\sqrt{3}}(a_{\\{1}b_{4\\}}+a_{\\{1}b_{5\\}}+a_{\\{1}b_{6\\}}) +\\frac{1}{6\\sqrt{3}}(a_{\\{2}b_{4\\}}+a_{\\{2}b_{6\\}})-\\frac{1}{3 \\sqrt{3}}a_{\\{2}b_{5\\}}+\\\\\n\t\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\frac{1}{6}(a_{\\{3}b_{4\\}}-a_{\\{3}b_{6\\}})\\\\\n\t\\frac{1}{3 \\sqrt{2}}(-a_2 b_2+a_3 b_3)+\\frac{1}{3} a_{\\{1} b_{2\\}}+\\frac{1}{6 \\sqrt{3}}(a_{\\{1} b_{4\\}}+a_{\\{1} b_{6\\}})-\\frac{1}{3 \\sqrt{3}}a_{\\{1} b_{5\\}}+\\frac{\\sqrt{2}}{3\\sqrt{3}} (a_{\\{2} b_{4\\}}+a_{\\{2} b_{6\\}})+\\\\\n\t\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad-\\frac{1}{3 \\sqrt{6}}a_{\\{2} b_{5\\}}+\\frac{1}{3 \\sqrt{2}}(-a_{\\{3} b_{4\\}}+a_{\\{3} b_{6\\}})\\\\\n\t\\frac{1}{3} a_{\\{1} b_{3\\}}+\\frac{1}{6} (a_{\\{1} b_{4\\}}-a_{\\{1} b_{6\\}})+\\frac{1}{3 \\sqrt{2}}(a_{\\{2} b_{3\\}}-a_{\\{2} b_{4\\}}+a_{\\{2} b_{6\\}})+\\frac{1}{\\sqrt{6}}a_{\\{3} b_{5\\}}\\\\\n\t\\frac{\\sqrt{2}}{3\\sqrt{3}} \\left(-a_1 b_1+a_2 b_2\\right)-\\frac{\\sqrt{2}}{\\sqrt{3}} a_4 b_4+\\frac{1}{6 \\sqrt{3}}a_{\\{1} b_{2\\}}+\\frac{1}{6} a_{\\{1} b_{3\\}}-\\frac{1}{3 \\sqrt{2}} a_{\\{2} b_{3\\}}\\\\\n\t-\\frac{\\sqrt{2}}{3\\sqrt{3}}a_1 b_1 -\\frac{1}{3\\sqrt{6}} a_2 b_2+\\frac{1}{\\sqrt{6}} a_3 b_3-\\frac{\\sqrt{2}}{\\sqrt{3}} a_5 b_5-\\frac{1}{3\\sqrt{3}}a_{\\{1} b_{2\\}}\\\\\n\\frac{\\sqrt{2}}{3\\sqrt{3}}(-a_1 b_1+a_2 b_2)-\\frac{\\sqrt{2}}{\\sqrt{3}} a_6 b_6+\\frac{1}{6\\sqrt{3}}a_{\\{1} b_{2\\}}-\\frac{1}{6} a_{\\{1} b_{3\\}}+\\frac{1}{3\\sqrt{2}}a_{\\{2} b_{3\\}}\n\\end{matrix}\\right),\\\\ \n&\\boldsymbol{\\xb}\\equiv\\left(\\begin{matrix}\\frac{1}{3 \\sqrt{6}}(a_{\\{1} b_{4}\\}+a_{\\{1} b_{5\\}}+a_{\\{1} b_{6\\}})+\\frac{1}{6 \\sqrt{3}}(a_{\\{2} b_{4\\}}+a_{\\{2} b_{6\\}})-\\frac{1}{3 \\sqrt{3}}a_{\\{2} b_{5\\}}+\\frac{1}{6} \\left(a_{\\{3} b_{4\\}}-a_{\\{3} b_{6\\}}\\right)+\\\\\n\t\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\frac{1}{3} \\left(a_{\\{4} b_{5\\}}+a_{\\{4} b_{6\\}}+a_{\\{5} b_{6\\}}\\right)\\\\\n\t\\frac{1}{6 \\sqrt{3}}(a_{\\{1} b_{4\\}}+a_{\\{1} b_{6\\}})-\\frac{1}{3\\sqrt{3}}a_{\\{1} b_{5\\}}+\\frac{1}{6 \\sqrt{6}}(a_{\\{2} b_{4\\}}+a_{\\{2} b_{6\\}})+\\frac{\\sqrt{2}}{3\\sqrt{3}}a_{\\{2} b_{5\\}}+\\frac{1}{6\\sqrt{2}}(a_{\\{3} b_{4\\}}-a_{\\{3} b_{6\\}})+\\\\\n\t\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\frac{1}{3\\sqrt{2}}(a_{\\{4} b_{5\\}}+a_{\\{5} b_{6\\}})-\\frac{\\sqrt{2}}{3}a_{\\{4} b_{6\\}}\\\\\n\t\\frac{1}{6}(a_{\\{1} b_{4\\}}-a_{\\{1} b_{6\\}})+\\frac{1}{6\\sqrt{2}}(a_{\\{2} b_{4\\}}-a_{\\{2} b_{6\\}})+\\frac{1}{2\\sqrt{6}}(a_{\\{3} b_{4\\}}+a_{\\{3} b_{6\\}})+\\frac{1}{\\sqrt{6}}(-a_{\\{4} b_{5\\}}+a_{\\{5} b_{6\\}})\\\\\n\t-\\frac{\\sqrt{2}}{3\\sqrt{3}}a_1 b_1-\\frac{1}{3\\sqrt{6}}a_2 b_2-\\frac{1}{\\sqrt{6}}a_3 b_3-\\frac{1}{3\\sqrt{3}}a_{\\{1} b_{2\\}}-\\frac{1}{6}(a_{\\{1} b_{5\\}}+a_{\\{1} b_{6\\}})-\\frac{1}{3}a_{\\{1} b_{3\\}}+\\frac{1}{3\\sqrt{2}}(-a_{\\{2} b_{3\\}}+a_{\\{2} b_{6\\}})+\\\\\n\t\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\quad\\,\\,+\\frac{1}{2\\sqrt{6}}a_{\\{3} b_{5\\}}-\\frac{1}{6\\sqrt{2}}a_{\\{2} b_{5\\}}\\\\\n\t-\\frac{\\sqrt{2}}{3\\sqrt{3}}a_1 b_1-\\frac{2\\sqrt{2}}{3\\sqrt{3}}a_2 b_2+\\frac{2}{3\\sqrt{3}}a_{\\{1}b_{2\\}}-\\frac{1}{6}(a_{\\{1}b_{4\\}}+a_{\\{1}b_{6\\}})-\\frac{1}{6\\sqrt{2}}(a_{\\{2}b_{4\\}}+a_{\\{2}b_{6\\}})+\\frac{1}{2\\sqrt{6}}(a_{\\{3}b_{4\\}}-a_{\\{3}b_{6\\}})\\\\\n\t-\\frac{\\sqrt{2}}{3\\sqrt{3}}a_1 b_1-\\frac{1}{3\\sqrt{6}}a_2 b_2-\\frac{1}{\\sqrt{6}}a_3 b_3-\\frac{1}{3\\sqrt{3}}a_{\\{1} b_{2\\}}+\\frac{1}{3}a_{\\{1} b_{3\\}}-\\frac{1}{6}(a_{\\{1} b_{4\\}}+a_{\\{1} b_{5\\}})+\\frac{1}{3\\sqrt{2}}(a_{\\{2} b_{3\\}}+a_{\\{2} b_{4\\}})+\\\\\n\t\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\quad-\\frac{1}{6\\sqrt{2}}a_{\\{2} b_{5\\}}-\\frac{1}{2\\sqrt{6}}a_{\\{3} b_{5\\}}\n\\end{matrix}\\right),\\\\\n&\\boldsymbol{3}\\equiv\\left(\\begin{matrix}\\frac{1}{2\\sqrt{3}}(-a_{\\{1} b_{5\\}}+a_{\\{1} b_{6\\}})-\\frac{1}{2\\sqrt{6}}a_{\\{2} b_{5\\}}-\\frac{1}{\\sqrt{6}}a_{\\{2} b_{6\\}}+\\frac{1}{2\\sqrt{2}}a_{\\{3} b_{5\\}}\\\\\n\t\\frac{1}{2\\sqrt{3}}(a_{\\{1} b_{4\\}}-a_{\\{1} b_{6\\}})+\\frac{1}{2\\sqrt{6}}(a_{\\{2} b_{4\\}}-a_{\\{2} b_{6\\}})-\\frac{1}{2\\sqrt{2}}(a_{\\{3} b_{4\\}}+a_{\\{3} b_{6\\}})\\\\\n\t\\frac{1}{2\\sqrt{3}}(-a_{\\{1} b_{4\\}}+a_{\\{1} b_{5\\}})+\\frac{1}{\\sqrt{6}}a_{\\{2} b_{4\\}}+\\frac{1}{2\\sqrt{6}}a_{\\{2} b_{5\\}}+\\frac{1}{2\\sqrt{2}}a_{\\{3} b_{5\\}}\n\\end{matrix}\\right),\\\\\n&\\boldsymbol{\\xb}\\equiv\\left(\\begin{matrix}\\frac{1}{2\\sqrt{3}}(a_{[6} b_{3]}+a_{[3} b_{4]})+\\frac{1}{3\\sqrt{2}}(a_{[1} b_{4]}+a_{[1} b_{5]}+a_{[1} b_{6]})+\\frac{1}{6}(a_{[2} b_{4]}+a_{[2} b_{6]})+\\frac{1}{3}a_{[5} b_{2]}\\\\\n\t\\frac{1}{2\\sqrt{6}}(a_{[3} b_{4]}+a_{[6} b_{3]})+\\frac{1}{6\\sqrt{2}}(a_{[2} b_{4]}+a_{[2} b_{6]})+\\frac{1}{3}a_{[5} b_{1]}+\\frac{\\sqrt{2}}{3}a_{[2} b_{5]}+\\frac{1}{6}(a_{[1} b_{4]}+a_{[1} b_{6]})\\\\\n\t\\frac{1}{2\\sqrt{3}}(a_{[1} b_{4]}+a_{[6} b_{1]})+\\frac{1}{2\\sqrt{6}}(a_{[2} b_{4]}+a_{[6} b_{2]})+\\frac{1}{2\\sqrt{2}}(a_{[3} b_{4]}+a_{[3} b_{6]})\\\\\n\t\\frac{1}{2\\sqrt{3}}(a_{[1} b_{5]}+a_{[1} b_{6]})+\\frac{1}{2\\sqrt{6}}a_{[2} b_{5]}+\\frac{1}{\\sqrt{6}}a_{[6} b_{2]}+\\frac{1}{2\\sqrt{2}}a_{[5} b_{3]}\\\\\n\t\\frac{1}{2\\sqrt{3}}(a_{[1} b_{6]}+a_{[1} b_{4]})+\\frac{1}{2\\sqrt{6}}(a_{[2} b_{4]}+a_{[2} b_{6]})+\\frac{1}{2\\sqrt{2}}(a_{[3} b_{6]}+a_{[4} b_{3]})\\\\\n\t\\frac{1}{2\\sqrt{3}}(a_{[1} b_{4]}+a_{[1} b_{5]})+\\frac{1}{2\\sqrt{6}}a_{[2} b_{5]}+\\frac{1}{\\sqrt{6}}a_{[4} b_{2]}+\\frac{1}{2\\sqrt{2}}a_{[3} b_{5]}\n\\end{matrix}\\right),\\\\\n&\\boldsymbol{3^p}\\equiv\\left(\\begin{matrix}\\frac{\\om}{2\\sqrt{3}}a_{[2} b_{1]}+\\frac{1}{3\\sqrt{2}}(a_{[6} b_{2]}+\\om a_{[3} b_{2]})+\\frac{1}{6}(a_{[1} b_{6]}+a_{[5} b_{1]}+\\om a_{[1} b_{3]})+\\frac{1}{6\\sqrt{2}}a_{[5} b_{2]}+\\frac{\\ob}{\\sqrt{6}}a_{[5} b_{6]}+\\frac{1}{2\\sqrt{6}}a_{[3} b_{5]}\\\\\n\t\\frac{\\om}{3\\sqrt{2}}a_{[3} b_{2]}+\\frac{1}{6}(a_{[1} b_{4]}+a_{[6} b_{1]})+\\frac{\\om}{3}a_{[3} b_{1]}+\\frac{1}{6\\sqrt{2}}(a_{[2} b_{4]}+a_{[6} b_{2]})+\\frac{\\ob}{\\sqrt{6}}a_{[6} b_{4]}+\\frac{1}{2\\sqrt{6}}(a_{[4} b_{3]}+a_{[6} b_{3]})\\\\\n\t\\frac{\\om}{2\\sqrt{3}}a_{[1} b_{2]}+\\frac{1}{3\\sqrt{2}}(a_{[2} b_{4]}+\\om a_{[3} b_{2]})+\\frac{1}{6}(a_{[1} b_{5]}+a_{[4} b_{1]}+\\om a_{[1} b_{3]})+\\frac{1}{6\\sqrt{2}}a_{[2} b_{5]}+\\frac{\\ob}{\\sqrt{6}}a_{[4} b_{5]}+\\frac{1}{2\\sqrt{6}}a_{[3} b_{5]}\n\\end{matrix}\\right),\\\\\n&\\boldsymbol{3^q}\\equiv\\left(\\begin{matrix}\\frac{1}{2\\sqrt{3}}a_{[2} b_{1]}+\\frac{1}{3\\sqrt{2}}(a_{[3} b_{2]}+a_{[6} b_{2]})+\\frac{1}{6}(a_{[1} b_{3]}+a_{[1} b_{6]}+a_{[5} b_{1]})+\\frac{1}{6\\sqrt{2}}a_{[5} b_{2]}+\\frac{1}{\\sqrt{6}}a_{[5} b_{6]}+\\frac{1}{2\\sqrt{6}}a_{[3} b_{5]}\\\\\n\t\\frac{1}{3\\sqrt{2}}a_{[3} b_{2]}+\\frac{1}{6}(a_{[1} b_{4]}+a_{[6} b_{1]})+\\frac{1}{3}a_{[3} b_{1]}+\\frac{1}{6\\sqrt{2}}(a_{[2} b_{4]}+a_{[6} b_{2]})+\\frac{1}{\\sqrt{6}}a_{[6} b_{4]}+\\frac{1}{2\\sqrt{6}}(a_{[4} b_{3]}+a_{[6} b_{3]})\\\\\n\t\\frac{1}{2\\sqrt{3}}a_{[1} b_{2]}+\\frac{1}{3\\sqrt{2}}(a_{[2} b_{4]}+a_{[3} b_{2]})+\\frac{1}{6}(a_{[1} b_{3]}+a_{[1} b_{5]}+a_{[4} b_{1]})+\\frac{1}{6\\sqrt{2}}a_{[2} b_{5]}+\\frac{1}{\\sqrt{6}}a_{[4} b_{5]}+\\frac{1}{2\\sqrt{6}}a_{[3} b_{5]}\n\\end{matrix}\\right),\\\\\n&\\boldsymbol{3^r}\\equiv\\left(\\begin{matrix}\\frac{\\ob}{2\\sqrt{3}}a_{[2} b_{1]}+\\frac{1}{3\\sqrt{2}}(a_{[6} b_{2]}+\\ob a_{[3} b_{2]})+\\frac{1}{6}(a_{[5} b_{1]}+a_{[1} b_{6]}+\\ob a_{[1} b_{3]})+\\frac{1}{6\\sqrt{2}}a_{[5} b_{2]}+\\frac{\\om}{\\sqrt{6}}a_{[5} b_{6]}+\\frac{1}{2\\sqrt{6}}a_{[3} b_{5]}\\\\\n\t\\frac{\\ob}{3\\sqrt{2}}a_{[3} b_{2]}+\\frac{1}{6}(a_{[1} b_{4]}+a_{[6} b_{1]})+\\frac{\\ob}{3}a_{[3} b_{1]}+\\frac{1}{6\\sqrt{2}}(a_{[2} b_{4]}+a_{[6} b_{2]})+\\frac{\\om}{\\sqrt{6}}a_{[6} b_{4]}+\\frac{1}{2\\sqrt{6}}(a_{[4} b_{3]}+a_{[6} b_{3]})\\\\\n\t\\frac{\\ob}{2\\sqrt{3}}a_{[1} b_{2]}+\\frac{1}{3\\sqrt{2}}(a_{[2} b_{4]}+\\ob a_{[3} b_{2]})+\\frac{1}{6}(a_{[1} b_{5]}+a_{[4} b_{1]}+\\ob a_{[1} b_{3]})+\\frac{1}{6\\sqrt{2}}a_{[2} b_{5]}+\\frac{\\om}{\\sqrt{6}}a_{[4} b_{5]}+\\frac{1}{2\\sqrt{6}}a_{[3} b_{5]}\n\\end{matrix}\\right).\\\\\n\\end{split}\n\\end{align}\n}\n\nIn Eqs.~(\\ref{eq:tensor66exp}) we have used the curly bracket and the square bracket to denote the symmetric sum and the antisymmetric sum respectively, i.e.~$a_{\\{i} b_{j\\}}=a_i b_j + a_j b_i$ and $a_{[i} b_{j]}=a_i b_j - a_j b_i$ . \n\n\\section{Appendix B: Flavon Potentials}\n\nHere we discuss the flavon potentials that lead to the vacuum alignments assumed in our model. The potentials we construct contain only up to the sixth-order flavon terms. It should be noted that even though our construction results in the required VEVs, we are not doing an exhaustive analysis of the most general flavon potentials involving all the possible invariant terms. However, the content we include is sufficient to realise our VEVs.\n\n\\subsection{The triplet flavons: $\\phi_e$, $\\phi_\\mu$, $\\phi_\\tau$}\\label{sec:triplets}\n\nFirst we consider the triplet flavons $\\phi_e$, $\\phi_\\mu$ and $\\phi_\\tau$. Our target is to obtain the VEVs $\\langle\\phi_e\\rangle=\\frac{i}{\\sqrt{3}}(1,1,1)^T$, $\\langle\\phi_\\mu\\rangle=\\frac{i}{\\sqrt{3}}(1,\\ob,\\om)^T$ and $\\langle\\phi_\\tau\\rangle=\\frac{i}{\\sqrt{3}}(1,\\om,\\ob)^T$, Eqs.~(\\ref{eq:leptvev}). The flavons $\\phi_e$, $\\phi_\\mu$ and $\\phi_\\tau$ transform as $\\boldsymbol{\\ab}$, $\\boldsymbol{\\bb}$ and $\\boldsymbol{\\cb}$ respectively. The $3\\times3$ maximal matrix $V$, Eqs.~(\\ref{eq:gen3}), is one of the generators of $\\boldsymbol{3}$. The corresponding generators of $\\boldsymbol{\\ab}$, $\\boldsymbol{\\bb}$ and $\\boldsymbol{\\cb}$ are $-V^*$, $V^*$ and $-V^*$ respectively, Eqs.~(\\ref{eq:gen1p}-\\ref{eq:3times1}). If the potentials of $\\phi_e$, $\\phi_\\mu$ and $\\phi_\\tau$ have minima at $(-1,0,0)^T$, $(0,1,0)^T$ and $(0,0,-1)^T$, then they have minima also at $-V^*(-1,0,0)^T=\\frac{i}{\\sqrt{3}}(1,1,1)^T$, $V^*(0,1,0)^T=\\frac{i}{\\sqrt{3}}(1,\\ob,\\om)^T$ and $-V^*(0,0,-1)^T=\\frac{i}{\\sqrt{3}}(1,\\om,\\ob)^T$ as required. The $3\\times3$ cyclic matrix $E$, Eqs.~(\\ref{eq:gen3}), is another generator of $\\boldsymbol{3}$ and thus of $\\boldsymbol{\\ab}$, $\\boldsymbol{\\bb}$ and $\\boldsymbol{\\cb}$ as well. Therefore, if the potential has minima at $(\\pm1,0,0)^T$, then it has minima also at $(0,\\pm1,0)^T$ and $(0,0,\\pm1)^T$. In a nutshell, for obtaining the required VEVs, Eqs.~(\\ref{eq:leptvev}), all we need to do is to construct potentials with minima at $(\\pm1,0,0)^T$.\n\nWe start with the term $\\left(\\phi_e^\\dagger\\phi_e-1\\right)^2$ which is $SU(3)$-invariant and which leads to a continuous set of minima that corresponds to unit magnitude for $\\phi_e$, i.e.~$\\phi_e^\\dagger\\phi_e=1$. Now we add terms which are invariant under $\\Sigma(72\\times3)$, but which break $SU(3)$ and result in a discrete set of points of minima including $(\\pm1,0,0)^T$.\n\nWith two $\\phi_e$ triplets which transform as $\\boldsymbol{\\ab}$, we may construct a conjugate sextet $\\boldsymbol{\\xb}$, i.e.~$\\boldsymbol{\\bar{3^p}}\\otimes\\boldsymbol{\\bar{3^p}}=\\boldsymbol{\\bar{6}}\\oplus\\boldsymbol{3}$, Eqs.~(\\ref{eq:tensor3pqr}). Note that the antisymmetric part, $\\boldsymbol{3}$, vanishes. With the help of Eqs.~(\\ref{eq:tensor288},~\\ref{eq:tensor288exp}), we combine the symmetric part, $\\boldsymbol{\\xb}$, with another $\\phi_e$ ($\\boldsymbol{\\bar{3^p}}$) to obtain\\footnote{Here we have omitted the distinction among $\\boldsymbol{\\tb}$, $\\boldsymbol{\\ab}$, $\\boldsymbol{\\ab}$ and $\\boldsymbol{\\ab}$. These triplets differ only with respect to a multiplication with $\\pm1$. In the potential terms constructed subsequently in Sec.~(\\ref{sec:triplets}), each type of triplet appears an even number of times and hence a sign flip does not have any impact.}\n\\begin{equation}\\label{eq:tensor288conj}\n\\boldsymbol{\\xb}\\otimes\\boldsymbol{\\tb}=\\boldsymbol{2}\\oplus\\boldsymbol{8}\\oplus\\boldsymbol{8}.\n\\end{equation} \nIn terms of $\\phi_e=(a_1,a_2,a_3)^T$, we provide the explicit expressions for the doublet,\n\\begin{equation}\\label{eq:doubletexplicit}\n\\boldsymbol{2} \\equiv \\frac{1}{\\sqrt{3}}\\left(3\\sqrt{2}a_1 a_2 a_3, a_1^3+a_2^3+a_3^3\\right)^T,\n\\end{equation}\nand the first octet,\n\\begin{equation}\\label{eq:octetexplicit}\n\\begin{split}\n\\boldsymbol{8}&\\equiv\\frac{\\sqrt{3}}{\\sqrt{2}}\\left(\\frac{\\left(a_1^3-a_3^3\\right)}{\\sqrt{3}}, \\frac{\\left(-a_1^3+2a_2^3-a_3^3\\right)}{3},a_3a_2(a_3-a_2),a_1a_3(a_1-a_3),a_2a_1(a_2-a_1),\\right.\\\\\n&\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\left. \\vphantom{\\frac{1}{\\sqrt{3}}}a_3a_2(a_3+a_2),a_1a_3(a_1+a_3),a_2a_1(a_2+a_1)\\right)^T.\\\\\n\\end{split}\n\\end{equation}\nIt can be shown that the second octet in Eq.~(\\ref{eq:tensor288conj}) is totally antisymmetric with respect to the permutation of the indices of the $\\phi_e$ triplets and therefore it vanishes. Using the doublets\\footnote{$\\boldsymbol{2}$ as well as $\\boldsymbol{8}$ are quaternionic representations. Their characters are real, but the use complex numbers can not be avoided in their representation matrices, eg.~Eqs.~(\\ref{eq:gen8},~\\ref{eq:gen2}). Real, complex and quaternionic representations can be identified by calculating the Frobenius-Schur indicator which gets the values $+1$, $0$ and $-1$ respectively.} from Eq.~(\\ref{eq:doubletexplicit}) we obtain the invariant term $T_{e\\boldsymbol{2}}\\equiv\\boldsymbol{2}^\\dagger \\boldsymbol{2}$. Similarly using the octets from Eq.~(\\ref{eq:octetexplicit}) we obtain the invariant term $T_{e\\boldsymbol{8}} \\equiv \\boldsymbol{8}^\\dagger \\boldsymbol{8}$.\n\nIn the previous paragraph, we obtained a conjugate sextet $\\boldsymbol{\\xb}$ from two $\\phi_e$ triplets. With the help of Eqs.~(\\ref{eq:tensor66},~\\ref{eq:tensor66exp}), we take the tensor product of two of these $\\boldsymbol{\\xb}$s to construct three $\\boldsymbol{6}$s and a $\\boldsymbol{\\tb}$:\n\\begin{equation}\\label{eq:tensor66conj}\n\\boldsymbol{\\xb}\\otimes\\boldsymbol{\\xb}=\\underbrace{\\boldsymbol{6}\\oplus\\boldsymbol{6}\\oplus\\boldsymbol{6}\\oplus\\boldsymbol{\\tb}}_{sym}\\oplus\\underbrace{\\boldsymbol{6}\\oplus\\boldsymbol{\\ab}\\oplus\\boldsymbol{\\bb}\\oplus\\boldsymbol{\\cb}}_{antisym}.\n\\end{equation}\nOf course, the antisymmetric part vanishes. It can also be shown that the first sextet in the symmetric part is antisymmetric with respect to the permutation of the indices of the original triplets ($\\phi_e$) and therefore it vanishes too. We combine the second sextet in the symmetric part with the original $\\boldsymbol{\\xb}$ in the LHS of Eq.~(\\ref{eq:tensor66conj}) to obtain an invariant term $T_{e\\boldsymbol{6}}\\equiv\\boldsymbol{6}^T\\boldsymbol{\\xb}$. $T_{e\\boldsymbol{6}}$ in terms of the components of $\\phi_e$ is given by\\footnote{The invariant of lowest degree that breaks $SU(3)\\rightarrow\\Sigma(72\\times3)$ as calculated in \\cite{Merle} agrees with our expression for $T_{e\\boldsymbol{6}}$.}:\n\\begin{equation}\nT_{e\\boldsymbol{6}}=\\frac{1}{\\sqrt{3}}\\left(a_1^6+a_2^6+a_3^6-10a_1^3a_2^3-10a_2^3a_3^3-10a_3^3a_1^3 \\right).\n\\end{equation} \nNote that $T_{e\\boldsymbol{6}}$ is complex. A similar invariant term can be constructed using the third sextet in the RHS and the $\\boldsymbol{\\xb}$ in LHS. However this term, when viewed as the tensor product of the three $\\boldsymbol{\\xb}$s, is totally antisymmetric with respect to the permutation of indices of the $\\boldsymbol{\\xb}$s and thus vanishes. Note that $T_{e\\boldsymbol{2}}$, $T_{e\\boldsymbol{8}}$ and $T_{e\\boldsymbol{8}}$ are sixth-order flavon terms. \n\nCombining all the non-vanishing invariant terms we write the potential\n\\begin{equation}\\label{eq:leptpotinit}\n\\left(\\phi_e^\\dagger\\phi_e-1\\right)^2+k_{e1} T_{e\\boldsymbol{2}} +k_{e2} T_{e\\boldsymbol{8}} +k_{e3} \\text{Re}\\left(T_{e\\boldsymbol{6}}\\right)+k_{e4} \\text{Im}\\left(T_{e\\boldsymbol{6}}\\right). \n\\end{equation}\nBy the suitable choice of coefficients $k_{e1}$, $k_{e2}$, $k_{e3}$ and $k_{e4}$, the above potential can be made to have minima at the required points, eg.~at $(\\pm1,0,0)^T$. The first partial derivates at the points of extrema must be zero. To make sure that these points are minima, we do the second derivative test using the Hessian partial derivative matrix. Such a procedure has been followed in previous works eg. in~\\cite{Derivative}. The first partial derivatives vanishing at $(\\pm1,0,0)^T$ leads to the conditions\n\\begin{equation}\\label{eq:leptpotfirstder}\nk_{e1}+2 k_{e2} + \\sqrt{3} k_{e3} = 0, \\quad k_{e4}=0.\n\\end{equation}\nSubstituting Eqs.~(\\ref{eq:leptpotfirstder}) in Eq.~(\\ref{eq:leptpotinit}), we obtain the potential\n\\begin{equation}\\label{eq:leptpot}\n\\left(\\phi_e^\\dagger\\phi_e-1\\right)^2-\\left(2k_{e2}+\\sqrt{3} k_{e3}\\right) T_{e\\boldsymbol{2}} +k_{e2} T_{e\\boldsymbol{8}} +k_{e3} \\text{Re}\\left(T_{e\\boldsymbol{6}}\\right).\n\\end{equation}\nNow we apply the second partial derivative test which gives the constraints\n\\begin{equation}\\label{eq:leptpotcond}\nk_{e2}>0, \\quad k_{e3}<0.\n\\end{equation}\nTo sum up our discussion; the potential, Eq.~(\\ref{eq:leptpot}), with the constraints, Eq.~(\\ref{eq:leptpotcond}), has minima at $(\\pm1,0,0)^T$, $(0,\\pm1,0)^T$, $(0,0,\\pm1)^T$ and also at $\\pm\\frac{i}{\\sqrt{3}}(1,1,1)^T$, $\\pm\\frac{i}{\\sqrt{3}}(1,\\ob,\\om)^T$, $\\pm\\frac{i}{\\sqrt{3}}(1,\\om,\\ob)^T$.\n\nPotentials similar to Eq.~(\\ref{eq:leptpot}) can be written for the flavons $\\phi_\\mu$ and $\\phi_\\tau$ also, i.e.\n\\begin{gather}\n\\left(\\phi_\\mu^\\dagger\\phi_\\mu-1\\right)^2-\\left(2k_{\\mu2}+\\sqrt{3} k_{\\mu3}\\right) T_{\\mu\\boldsymbol{2}} +k_{\\mu2} T_{\\mu\\boldsymbol{8}} +k_{\\mu3} \\text{Re}\\left(T_{\\mu\\boldsymbol{6}}\\right)\\label{eq:potleptmu}\\\\\n\\left(\\phi_\\tau^\\dagger\\phi_\\tau-1\\right)^2-\\left(2k_{\\tau2}+\\sqrt{3} k_{\\tau3}\\right) T_{\\tau\\boldsymbol{2}} +k_{\\tau2} T_{\\tau\\boldsymbol{8}} +k_{\\tau3} \\text{Re}\\left(T_{\\tau\\boldsymbol{6}}\\right).\\label{eq:potlepttau}\n\\end{gather}\nWe need to ensure that the vacuum alignments of $\\phi_e$, $\\phi_\\mu$ and $\\phi_\\tau$ are orthogonal to each other, Eqs.~(\\ref{eq:leptvev}). For that purpose, we construct the cross term\n\\begin{equation}\\label{eq:leptcross}\nk_{e \\mu}|\\phi_e^\\dagger \\phi_\\mu|^2 + k_{\\mu \\tau} |\\phi_\\mu^\\dagger \\phi_\\tau|^2 + k_{\\tau e} |\\phi_\\tau^\\dagger \\phi_e|^2,\n\\end{equation}\nwhere $k_{e \\mu}$, $k_{\\mu \\tau}$ and $k_{\\tau e}$ are positive constants. Therefore, the complete potential for the triplet flavons is the sum of Eq.~(\\ref{eq:leptpot}), Eq.~(\\ref{eq:potleptmu}), Eq.~(\\ref{eq:potlepttau}) and Eq.~(\\ref{eq:leptcross}). Such a potential has minima at $\\phi_e=(\\pm1,0,0)^T$, $\\phi_\\mu=(0,\\pm1,0)^T$ and $\\phi_\\tau=(0,0,\\pm1)^T$, and also at $\\phi_e=\\pm\\frac{i}{\\sqrt{3}}(1,1,1)^T$, $\\phi_\\mu=\\pm\\frac{i}{\\sqrt{3}}(1,\\ob,\\om)^T$ and $\\phi_\\tau=\\pm\\frac{i}{\\sqrt{3}}(1,\\om,\\ob)^T$ as originally proposed.\n\n\\subsection{The sextet flavon: $\\phi$}\n\nNow we turn our attention towards constructing the potentials for the sextet flavon $\\phi$. We discussed four different cases of VEVs and here the construction of potentials corresponding to all these cases are done in a rather similar framework. Let us list all the four VEVs:\\footnote{Note that compared to Eqs.~(\\ref{eq:vevtxmp},~\\ref{eq:vevtxmm}), there is an extra negative sign in Eqs.~(\\ref{eq:vevlisttxmp}, \\ref{eq:vevlisttxmm}). The reason for this becomes apparent later in our discussion, but it is clear that such a choice does not have any observable consequence.}.\n\\begin{align}\n\\txm_{(\\chi=+\\frac{\\pi}{16})}:\\quad \\langle\\phi\\rangle &= \\left(\\frac{-3+\\sqrt{2}}{\\sqrt{3}},\\frac{1}{\\sqrt{3}},1-\\sqrt{2},0,-1,0\\right),\\label{eq:vevlisttxmp}\\\\\n\\txm_{(\\chi=-\\frac{\\pi}{16})}:\\quad \\langle\\phi\\rangle &= \\left(\\frac{-3+\\sqrt{2}}{\\sqrt{3}},\\frac{1}{\\sqrt{3}},-1+\\sqrt{2},0,-1,0\\right),\\label{eq:vevlisttxmm}\\\\\n\\txm_{(\\phi=+\\frac{\\pi}{16})}:\\quad \\langle\\phi\\rangle &= \\left(\\frac{1+\\sqrt{2}}{\\sqrt{3}},\\frac{1-\\sqrt{2}}{\\sqrt{3}}, -i \\left(1-\\sqrt{2}\\right),0, -1+\\sqrt{2}, 0\\right),\\label{eq:vevlisttpmp}\\\\\n\\txm_{(\\phi=-\\frac{\\pi}{16})}:\\quad \\langle\\phi\\rangle &= \\left(\\frac{1+\\sqrt{2}}{\\sqrt{3}},\\frac{1-\\sqrt{2}}{\\sqrt{3}}, i (1-\\sqrt{2}),0, -1+\\sqrt{2}, 0\\right).\\label{eq:vevlisttpmm}\n\\end{align}\nThese VEVs have the same magnitude, i.e.~$\\langle\\phi\\rangle^\\dagger\\langle\\phi\\rangle=8-4\\sqrt{2}$ for every given $\\langle\\phi\\rangle$ in Eqs.~(\\ref{eq:vevlisttxmp}-\\ref{eq:vevlisttpmm}). Therefore we may write an $SU(3)$-invariant term $|\\phi^\\dagger\\phi-(8-4\\sqrt{2})|^2$. Just as we did in the case of the triplet flavons, here also we add terms that break $SU(3)$ but respect the $\\Sigma(72\\times3)$ symmetry to obtain a discrete set of minima. \n\nWith the help of Eqs.~(\\ref{eq:tensor66},~\\ref{eq:tensor66exp}), We analyse the tensor product of three $\\phi$s ($\\boldsymbol{6}$s):\n\\begin{align}\n\\boldsymbol{6}\\otimes\\boldsymbol{6}\\otimes\\boldsymbol{6}&=\\left(\\underbrace{\\boldsymbol{\\xb}\\oplus\\boldsymbol{\\xb}\\oplus\\boldsymbol{\\xb}\\oplus\\boldsymbol{3}}_{sym}\\oplus\\underbrace{\\xcancel{\\boldsymbol{\\xb}\\oplus\\boldsymbol{3^p}\\oplus\\boldsymbol{3^q}\\oplus\\boldsymbol{3^r}}}_{antisym}\\right)\\otimes\\boldsymbol{6}\\label{eq:tensor666a}\\\\\n&=(\\boldsymbol{\\xb}\\otimes\\boldsymbol{6})\\oplus(\\boldsymbol{\\xb}\\otimes\\boldsymbol{6})\\oplus(\\boldsymbol{\\xb}\\otimes\\boldsymbol{6})\\oplus(\\boldsymbol{3}\\otimes\\boldsymbol{6})\\label{eq:tensor666b}.\n\\end{align}\nEach tensor product $(\\boldsymbol{\\xb}\\otimes\\boldsymbol{6})$ in Eq.~(\\ref{eq:tensor666b}) contains the invariant term $\\boldsymbol{\\xb}^T\\boldsymbol{6}$. Such invariants constructed from the first $(\\boldsymbol{\\xb}\\otimes\\boldsymbol{6})$ and the second $(\\boldsymbol{\\xb}\\otimes\\boldsymbol{6})$ in Eq.~(\\ref{eq:tensor666b}) are named $T_{\\boldsymbol{6}}$ and $T'_{\\boldsymbol{6}}$ respectively. In terms of the components of the sextet flavon, $\\phi=(a_1,a_2,a_3,a_4,a_5,a_6)^T$, they are given by\n\n\\begin{align}\\label{eq:t6invar}\n\\begin{split}\nT_{\\boldsymbol{6}} &=\\frac{2}{3}a_1^3-\\frac{\\sqrt{2}}{3}a_2^3-a_1\\left(a_2^2+a_3^2+a_4^2+a_5^2+a_6^2\\right)\\\\\n&\\qquad\\qquad+\\sqrt{2}a_2\\left(a_3^2+a_5^2\\right)-\\frac{1}{\\sqrt{2}}a_2\\left(a_4^2+a_6^2\\right)-\\frac{\\sqrt{3}}{\\sqrt{2}}a_3\\left(a_4^2+a_6^2\\right)+\\sqrt{6} a_4 a_5 a_6,\n\\end{split}\\\\\n\\begin{split}\nT'_{\\boldsymbol{6}}&=\\frac{1}{3}a_1^3 -\\frac{1}{3\\sqrt{2}}a_2^3-\\frac{\\sqrt{2}}{\\sqrt{3}}\\left(a_4^3+a_5^3+a_6^3\\right)+a_1\\left(a_2^2+a_3^2\\right)+\\frac{1}{\\sqrt{2}}a_2 a_3^2\\\\\n&\\qquad\\qquad-\\frac{\\sqrt{2}}{\\sqrt{3}}a_1^2\\left(a_4-a_5+a_6\\right)+\\frac{\\sqrt{2}}{\\sqrt{3}}a_2^2\\left(a_4+a_6\\right)-\\frac{1}{\\sqrt{6}}a_2^2a_5+\\frac{\\sqrt{3}}{\\sqrt{2}}a_3^2 a_5\\\\\n&\\qquad\\qquad+a_1 a_3 \\left(a_4-a_6\\right) + \\sqrt{2}a_2 a_3\\left(a_6-a_4\\right) + \\frac{1}{\\sqrt{3}}a_1 a_2\\left(a_6+a_4\\right) +\\frac{2}{\\sqrt{3}} a_1 a_2 a_5.\n\\end{split}\n\\end{align}\n\nThe invariant term constructed from the third $(\\boldsymbol{\\xb}\\otimes\\boldsymbol{6})$ in Eq.~(\\ref{eq:tensor666b}) is totally antisymmetric under the permutation of the indices of the three $\\boldsymbol{6}$s in the LHS of Eq.~(\\ref{eq:tensor666a}) and therefore it vanishes. \n\nConsider the tensor product space $\\phi\\otimes\\phi\\otimes\\phi$. Any specific alignment of $\\phi$, eg.~the VEV $\\langle\\phi\\rangle$, has a corresponding alignment in the tensor product space, eg.~$\\langle\\phi\\rangle\\otimes\\langle\\phi\\rangle\\otimes\\langle\\phi\\rangle$. We have shown that the tensor product space $\\phi\\otimes\\phi\\otimes\\phi$ contains two non-vanishing invariant directions, the ones that correspond to $T_{\\boldsymbol{6}}$ and $T'_{\\boldsymbol{6}}$. We take the projection of the alignment $\\langle\\phi\\rangle\\otimes\\langle\\phi\\rangle\\otimes\\langle\\phi\\rangle$ along these directions and call them $\\langle\\phi\\rangle\\otimes\\langle\\phi\\rangle\\otimes\\langle\\phi\\rangle_{T_{\\boldsymbol{6}}}$ and $\\langle\\phi\\rangle\\otimes\\langle\\phi\\rangle\\otimes\\langle\\phi\\rangle_{T'_{\\boldsymbol{6}}}$ respectively. It can be shown that, for every VEV given in Eqs.~(\\ref{eq:vevlisttxmp}-\\ref{eq:vevlisttpmm})\\footnote{The reason for changing the signs of Eqs.~(\\ref{eq:vevlisttxmp},~\\ref{eq:vevlisttxmm}) compared to Eqs.~(\\ref{eq:vevtxmp},~\\ref{eq:vevtxmm}) was to ensure that for all the VEVs, Eqs.~(\\ref{eq:vevlisttxmp}-\\ref{eq:vevlisttpmm}), the values of $\\langle\\phi\\rangle\\otimes\\langle\\phi\\rangle\\otimes\\langle\\phi\\rangle_{T_{\\boldsymbol{6}}}$ (and $\\langle\\phi\\otimes\\langle\\phi\\rangle\\otimes\\langle\\phi\\rangle_{T'_{\\boldsymbol{6}}}$) have the same sign.}, we get $\\langle\\phi\\rangle\\otimes\\langle\\phi\\rangle\\otimes\\langle\\phi\\rangle_{T_{\\boldsymbol{6}}}=\\sqrt{3}$ and $\\langle\\phi\\rangle\\otimes\\langle\\phi\\rangle\\otimes\\langle\\phi\\rangle_{T'_{\\boldsymbol{6}}}=\\sqrt{3}\\left(5\\sqrt{2}-7\\right)$. In other words $|T_{\\boldsymbol{6}}-\\sqrt{3}|^2$ and $|T'_{\\boldsymbol{6}}-\\sqrt{3}\\left(5\\sqrt{2}-7\\right)|^2$ have the minimum value zero when the flavon field acquires any of the four VEVs. Thus we construct the flavon potential\n\\begin{equation}\\label{eq:commonpot}\n\\left|\\phi^\\dagger\\phi-(8-4\\sqrt{2})\\right|^2 + k_{\\nu1} \\left|T_{\\boldsymbol{6}}-\\sqrt{3}\\right|^2 + k_{\\nu2} \\left|T'_{\\boldsymbol{6}}-\\sqrt{3}\\left(5\\sqrt{2}-7\\right)\\right|^2\n\\end{equation}\nwhere $k_{\\nu 1}$ and $k_{\\nu 2}$ are positive constants. This potential, Eq.(\\ref{eq:commonpot}), has minima at the required VEVs, Eqs.~(\\ref{eq:vevlisttxmp}-\\ref{eq:vevlisttpmm}), as required. Further analysis shows that the points of minima are not discrete, but rather they form a continuous set. In order to remove this ambiguity and ensure a discrete set of minima we add more invariant terms to the potential. We construct such terms by coupling the triplet flavons $\\phi_e$, $\\phi_\\mu$ and $\\phi_\\tau$ with sextet flavon $\\phi$.\n\nFirst we construct sextets ($\\boldsymbol{\\xb}$s) by combining two triplets ($\\boldsymbol{\\tb}$s) using the conjugate forms of Eqs.~(\\ref{eq:tensor3pqr},~\\ref{eq:tensor3pqr2}). The various possiblilities are $\\phi_e\\otimes \\phi_e$, $\\phi_\\mu\\otimes \\phi_\\mu$, $\\phi_\\tau\\otimes \\phi_\\tau$, $\\phi_\\mu\\otimes \\phi_\\tau$, $\\phi_\\tau\\otimes \\phi_e$ and $\\phi_e\\otimes \\phi_\\mu$. The sextets so constructed are combined with the sextet flavon $\\phi$ to obtain invariants, namely $T_{ee}$, $T_{\\mu\\mu}$, $T_{\\tau\\tau}$, $T_{\\mu\\tau}$, $T_{\\tau e}$ and $T_{e\\mu}$. In the tensor product space $\\phi_\\alpha\\otimes \\phi_\\beta \\otimes \\phi$ where $\\alpha, \\beta = e, \\mu, \\tau$, we consider the specific alignment $\\langle\\phi_\\alpha\\rangle\\otimes \\langle\\phi_\\beta\\rangle \\otimes \\langle\\phi\\rangle$ which corresponds to the required VEVs of the flavons given in Eqs.~(\\ref{eq:leptvev},~\\ref{eq:vevlisttxmp}-\\ref{eq:vevlisttpmm}). As was done previously, we take the projection of this alignment along the direction of the invariant $T_{\\alpha\\beta}$, i.e.~$\\langle\\phi_\\alpha\\rangle \\otimes \\langle\\phi_\\beta\\rangle \\otimes \\langle\\phi\\rangle_{T_{\\alpha\\beta}}$. Finally we construct the potential term\n\\begin{equation}\\label{eq:uniquepot} \n\\displaystyle \\sum_{\\alpha,\\beta} k'_{\\alpha\\beta} \\left|T_{\\alpha\\beta}-\\langle\\phi_\\alpha\\rangle \\otimes \\langle\\phi_\\beta\\rangle \\otimes \\langle\\phi\\rangle_{T_{\\alpha\\beta\\nu}}\\right|^2\n\\end{equation}\nwhere $k'_{\\alpha\\beta}$ are positive constants and the summation is over $(\\alpha,\\beta)=$ $(e,e)$, $(\\mu,\\mu)$, $(\\tau,\\tau)$, $(\\mu,\\tau)$, $(\\tau,e)$ and $(e,\\mu)$. The values of $\\langle\\phi_\\alpha\\rangle \\otimes \\langle\\phi_\\beta\\rangle \\otimes \\langle\\phi\\rangle_{T_{\\alpha\\beta}}$ corresponding to the VEVs, Eqs.~(\\ref{eq:vevlisttxmp}-\\ref{eq:vevlisttpmm}), are given in Table~\\ref{tab:vev}. \n\n\\begin{table}[]\n\\begin{center}\n\\begin{tabular}{||c||c|c|c|c||}\n \\hline \\hline\n \\multirow{2}{*}{} & \\multicolumn{4}{c||}{$\\langle\\phi_\\alpha\\rangle \\otimes \\langle\\phi_\\beta\\rangle \\otimes \\langle\\phi\\rangle_{T_{\\alpha\\beta\\nu}}$} \\\\\n \t\t\t\t \\cline{2-5}\n $(\\alpha,\\beta)$&$\\txm_{(\\chi=+\\frac{\\pi}{16})}$& $\\txm_{(\\chi=-\\frac{\\pi}{16})}$& $\\tpm_{(\\phi=+\\frac{\\pi}{16})}$& $\\tpm_{(\\phi=-\\frac{\\pi}{16})}$\\\\\n &({\\small Eq.~(\\ref{eq:vevlisttxmp})}) & ({\\small Eq.~(\\ref{eq:vevlisttxmm})}) & ({\\small Eq.~(\\ref{eq:vevlisttpmp})}) & ({\\small Eq.~(\\ref{eq:vevlisttpmm})}) \\\\\n \\hline \\hline\n $(e,e)$\t\t&$1$ \t & $1$ & $-1$ & $-1$ \\\\\n \\hline \n $(\\mu,\\mu)$\t\t&$\\frac{1-\\sqrt{2}}{2}+i\\frac{1+\\sqrt{2}}{2\\sqrt{3}}$ \t& $-\\frac{1}{2}+i\\frac{-1+2\\sqrt{2}}{2\\sqrt{3}}$ & $\\frac{(-1+\\sqrt{2})(1+\\sqrt{3})(1-i\\sqrt{3})}{2\\sqrt{6}}$ & $\\frac{(-1+\\sqrt{2})(-1+\\sqrt{3})(1-i\\sqrt{3})}{2\\sqrt{6}}$ \\\\\n \\hline \n $(\\tau,\\tau)$\t& $\\frac{1-\\sqrt{2}}{2}-i\\frac{1+\\sqrt{2}}{2\\sqrt{3}}$\t& $-\\frac{1}{2}-i\\frac{-1+2\\sqrt{2}}{2\\sqrt{3}}$ & $\\frac{(-1+\\sqrt{2})(-1+\\sqrt{3})(1+i\\sqrt{3})}{2\\sqrt{6}}$ & $\\frac{(-1+\\sqrt{2})(1+\\sqrt{3})(1+i\\sqrt{3})}{2\\sqrt{6}}$ \\\\\n \\hline \n $(\\mu,\\tau)$\t&$\\frac{3-\\sqrt{2}}{2\\sqrt{3}}-i\\frac{1}{2}$ \t \t& $\\frac{3-\\sqrt{2}}{2\\sqrt{3}}-i\\frac{1}{2}$ & $-\\frac{1+\\sqrt{2}}{2\\sqrt{3}}+i\\frac{1}{2}$ & $-\\frac{1+\\sqrt{2}}{2\\sqrt{3}}+i\\frac{1}{2}$ \\\\\n \\hline\n $(\\tau,e)$\t\t&$-\\frac{1}{2\\sqrt{6}}+i\\frac{1}{2\\sqrt{2}}$ \t \t& $-\\frac{1}{2\\sqrt{6}}+i\\frac{1}{2\\sqrt{2}}$ & $\\frac{-1+\\sqrt{2}}{2\\sqrt{6}}-i\\frac{-1+\\sqrt{2}}{2\\sqrt{2}}$ & $\\frac{-1+\\sqrt{2}}{2\\sqrt{6}}-i\\frac{-1+\\sqrt{2}}{2\\sqrt{2}}$ \\\\\n \\hline \n $(e,\\mu)$\t\t&$\\frac{-1+2\\sqrt{2}}{2\\sqrt{6}}-i\\frac{1}{2\\sqrt{2}}$ \t& $\\frac{3-2\\sqrt{2}}{2\\sqrt{6}}-i\\frac{1}{2\\sqrt{2}}$ & $\\frac{1-\\sqrt{2}}{2\\sqrt{6}}+i\\frac{(-1+\\sqrt{2})(3-2\\sqrt{3})}{6\\sqrt{2}}$ & $\\frac{1-\\sqrt{2}}{2\\sqrt{6}}+i\\frac{(-1+\\sqrt{2})(3+2\\sqrt{3})}{6\\sqrt{2}}$ \\\\\n \\hline \\hline\n \\end{tabular}\n\\end{center}\n\\caption{The projection of the tensor products of VEVs along the corresponding invariant directions.}\n\\label{tab:vev}\n\\end{table}\n\nThe terms given in Eq.~(\\ref{eq:commonpot}) and Eq.~(\\ref{eq:uniquepot}) form a potential with a discrete set of minima which includes the requied VEV, Eqs.~(\\ref{eq:vevlisttxmp}-\\ref{eq:vevlisttpmm}). Note that, we use highly specific constants in our potential, eg. $(8-4\\sqrt{2})$, $\\sqrt{3}$ and $\\sqrt{3}\\left(5\\sqrt{2}-7\\right)$ in Eq.~(\\ref{eq:commonpot}). Put another way, we may be able to construct any given mass matrix by suitably tweaking such constants. This situation can be made a lot less arbitrary by imposing additional symmetries on top of $\\Sigma(72\\times3)$. In this context, we can not help noticing the appearance of $\\frac{\\pi}{16}$ and the factor $-1+\\sqrt{2}$ ($=\\tan \\frac{\\pi}{8}$) throughout this paper. They give hints towards the presence of additional symmetries like $Z_{16}$. Such topics are beyond the scope of this paper, but will be discussed in a future publication.\n\n\\providecommand{\\href}[2]{#2}\\begingroup\\raggedright","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nWhile the standard model proves an excellent framework for fundamental interactions at energy scales up to at least the sub TeV range, it nevertheless leaves a number of fundamental problems. Theoretically, one of the most outstanding puzzles centers on the origin and mechanism of electroweak symmetry breaking, and the quantum mechanical stability of the hierarchy generated between the electroweak scale and the Planck scale. In addition, recent astrophysical observations \\cite{DarkMatter} concord with $0.094 < \\Omega_{CDM}h^2 < 0.129$, indicating the presence of cold non-baryonic dark matter as the principle form of matter in the Universe, of which the standard model provides no explanation. The most popular candidate for dark matter assumes a non-standard model, stable, electrically neutral, and weakly interacting particle -- the WIMP hypothesis. Clearly, from both a theoretical and phenomenological perspective, the standard model requires extension. In this letter we wish to explore the consequences of the universal extra dimension (UED) \\cite{Appelquist:2000nn} extension of the standard model.\n\nIn the UED scenario, all standard model particles can freely propagate in the bulk of one or more extra dimensions and thus each standard model particle is associated with a Kaluza-Klein (KK) tower of states. Each state in the KK tower has the same spin as its standard model counterpart. An important consequence of UED models concerns the existence of a conserved discrete symmetry, KK-parity, which guarantees the stability of the lightest KK particle (LKP) and thus provides a dark matter candidate. Suitable thermal relic dark matter candidates that have been studied extensively \\cite{Servant:2002aq} include the first KK-excitations of the hypercharge boson, the photon, and the neutrino {\\it i.e. } $B^{(1)}$, $\\gamma^{(1)}$, or $\\nu^{(1)}$.\n\nThe tree-level mass spectrum of the KK-excitations of UED models reveals a nearly degenerate spectrum. As an example, a UED model with one extra-dimension compactified on an $S_1\/Z_2$ orbifold of radius $R$, leads to the tree level mass relation \n\\begin{equation}\\label{mass} m^{(n)} = \\sqrt{(n\/R)^2 + (m^{(0)})^2} \n\\end{equation} \nfor the n-th KK mode, where $m^{(0)}$ constitutes the zero-mode mass ({\\it i.e. } the standard model particle value). Quantum corrections typically dominate over zero mode level contributions and therefore the resulting mass spectrum depends crucially on radiative effects. In general, a moderately split UED mass spectrum \\cite{Cheng:2002ej} develops. In the minimal UED model (MUED) \\cite{Cheng:2002ej}, one-loop calculations suggest that the LKP is well approximated by the KK hypercharge boson, $B^{(1)}$, and numerous studies have examined the thermal production and prospects of direct and indirect detection of $B^{(1)}$ and $\\gamma^{(1)}$ LKP dark matter \\cite{Servant:2002aq,KKDMcollection}. \n\nHowever, the non-renormalizability of UED models imply the existence of an ultraviolet cut-off, typically of the order of a few tens of TeV, at which point the model requires UV completion. As such, UED models must be regarded as an effective theory. The presence of incalculable boundary terms arising from the UV complete theory can potentially change the mass spectrum, resulting in different LKP candidates. Recent studies of the relic density of $B^{(1)}$ LKP dark matter with the full MUED spectrum \\cite{coannmat,coannkribs} reveal substantial observational tension with constraints from electroweak precision data \\cite{Flacke:2005hb}. Thus, non-minimal models with brane-localized terms appear as a likely alternative if UED models are to provide a successful phenomenology. Furthermore, model independent studies \\cite{Servant:2002aq} show that $B^{(1)}$, $\\gamma^{(1)}$, and $\\nu^{(1)}$ can all be thermally produced with abundances sufficient to provide the dark matter. Constraints on minimal UED models from limits on weak neutral current nucleon-$\\nu^{(1)}$ elastic scattering in direct searches, together with thermal dark matter production mechanisms, disfavour $\\nu^{(1)}$ dark matter \\cite{Servant:2002hb}. However, given the need for possible new non-minimal interactions, we consider further consequences of KK-neutrino dark matter model-independently.\n\nWhile there exists compelling evidence for dark matter in the form of WIMPS, there also exists strong constraints on possible electromagnetic interactions of dark matter, even in the limit of complete charge neutrality. A neutral Dirac fermion can posses both a permanent magnetic dipole moment, $\\mu$, and a permanent electric dipole moment, $d$, arising from the dimension-five operator, \n\\begin{equation} \n\\mathcal{L}_{D} = \\frac{i}{2} \\bar f\n\\sigma_{\\mu\\nu}\\left(\\mu +\\gamma_5 d \\right)f F^{\\mu\\nu}. \n\\end{equation}\nWhile the presence of the magnetic dipole moment does not violate any discrete symmetries, the electric dipole moment requires the violation of parity and CP. Severe constraints exist on the dipole moments of $\\sim 1$ TeV dark matter WIMPS \\cite{Sigurdson:2004zp}. (We are aware that the authors of \\cite{Sigurdson:2004zp} are currently revising their estimates and, as a result, the strength of the constraints in \\cite{Sigurdson:2004zp} may change substantially \\cite{private}.) Thus, if a model predicts Dirac fermionic dark matter, it is important to determine the strength of the induced dipole moment.\n\nSince the KK-neutrino of UED models is a Dirac fermion from the 4-dimensional perspective, UED models that assume KK-neutrino dark matter are constrained by the strength of the induced dipole operator.\n\nIn this letter we derive model independent bounds on KK-neutrino dark matter by examining the induced dipole moment and comparing the predictions with the current observational bound. Section II provides a brief review on the properties of KK-fermions in UED models along with a discussion on dipole moments relevant to the calculation presented in section III. Finally, in section IV we present our conclusions.\n\n\\section{KK-neutrino LKP and induced dipole moments}\n\nIn UED models, standard model fields become identified with the zero modes of KK towers of states once the extra dimensions are integrated out of the theory. For concreteness, we will restrict our discussion to five dimensional UED models compactified on an $S_1\/Z_2$ orbifold. Five dimensional theories do not posses a chirality condition since $\\gamma_5$ becomes part of the five dimensional Clifford Algebra. Thus, in order to arrive at a chiral theory at the zero mode level, we require one five dimensional \\emph{Dirac} spinor for every \\emph{Weyl} spinor of the standard model. By use of the orbifold boundary conditions half the number of states project out of the spectrum leaving a zero mode level chiral theory. In the case of the lepton doublet, the decomposition of the corresponding five dimensional Dirac spinor reads ({\\it c.f. } {\\it e.g. }\n\\cite{Appelquist:2000nn}),\n\\begin{equation} \\label{fermiondecomp}\n\\begin{split}\n\\hat{\\mathcal{L}}(x^{\\mu},y) = \\frac{1}{\\sqrt{\\pi R}}\nP_L\\mathcal{L}(x^\\mu) + \\sqrt{\\frac{2}{\\pi\nR}}\\sum_{n=1}\\left[P_L\\mathcal{L}(x^\\mu)\\cos(\\frac{ny}{R})\\right.&\\\\\n \\left.+P_R\\mathcal{L}(x^\\mu)\\sin(\\frac{ny}{R})\\right]&\\\\\n\\end{split}\n\\end{equation}\nwhere $\\hat{\\mathcal{L}}$ denotes the five dimensional Dirac spinor, $\\mathcal{L}$ denotes a four dimensional \\emph{4\ncomponent} spinor with $P_{L,R} = (1\\pm\\gamma_5)\/2$, and $\\hat{\\mathcal{L}}$ is associated with the (4D left-handed) lepton doublet via the identification $P_L\\mathcal{L}(x^\\mu)\\leftrightarrow (\\nu_L,e_L)$. We see from eq(\\ref{fermiondecomp}) that $\\hat{\\mathcal{L}}$ contains a purely left-handed zero mode while all higher KK modes contain both chiralities. Therefore, the non-zero mode KK-level fermions appear as Dirac spinors from the 4-dimensional perspective. Specifically, the KK-neutrino charged under the weak $SU(2)$ appears with both chiralities.\n\nTranslational invariance in the fifth direction implies (once the extra dimension becomes integrated out) that a UED model compactified solely on $S_1$ conserves KK-excitation-number in every vertex. The orbifold reduces the conserved KK-number to a discrete $Z_2$ symmetry, called KK-parity. The conservation of KK-parity in every vertex implies the stability of the lightest KK-particle.\n\nSince the fermions receive mass through Yukawa couplings after electroweak symmetry breaking and through the KK-level expansion itself, the theory, in general, requires a unitary transformation to connect mass and gauge eigenstates. For example, in the lepton sector at the $j$th KK-level we have,\n\\begin{equation} \n\\left(\\begin{array}{c} \\mathcal{E}^j \\\\ \\mathcal{L}^j \\end{array} \\right) =\n\\left(\\begin{array}{cc}-\\gamma_5\\cos\\alpha_j & \\sin\\alpha_j\\\\\n\\gamma_5\\sin\\alpha_j & \\cos\\alpha_j\n\\end{array}\\right)\\left(\\begin{array}{c} \\mathcal{E}^{\\prime j} \\\\\n\\mathcal{L}^{\\prime j} \\end{array} \\right) \n\\end{equation} \nwhere $\\mathcal{E}^j$ and $\\mathcal{L}^j$ denote the $j^{\\mbox{th}}$ KK mode of the lepton $SU(2)$ singlet and doublet respectively, and where \\begin{equation} \\tan (2\\alpha_j) = \\frac{m_l^{(0)}}{j\/R}. \\end{equation} As the lepton masses are small compared to $j\/R$, we ignore the effects of the mixing matrix for the remainder of this letter. Furthermore, we ignore the effects of lepton flavour violation and neutrino mixing.\n\nA Dirac fermion can posses a dipole moment, derived from the transition amplitude ({\\it c.f. } {\\it e.g. } \\cite{neutrinobook}), \n\\begin{equation} \nT =\n-i\\epsilon^\\mu q^\\nu \\bar f (p^\\prime) \\sigma_{\\mu\\nu} \\left(F_2 + G_2\n\\gamma_5\\right) f(p) \n\\end{equation} \nwhere $q=p^\\prime -p$. The magnetic moment is defined by $\\mu = F_2(0)$ while the electric dipole is defined by $ d = G_2(0)$. At low energies compared to the mass of the particle, the photon does not distinguish between $\\mu$ or $d$ provided that one ignores other time-reversal violating observables. We will focus on the limits established on $\\mu$ throughout. On dimensional grounds, we naively expect the induced magnetic dipole moment to scale as, \n\\begin{equation} \n\\mu \\lesssim e \\frac{M_{\\nu^{(1)}}}{R^{-2}} \\simeq \\frac{e}{M_{\\nu^{(1)}}} = 1.022\\times\n10^{-6}\\mu_B\\left(\\frac{\\mathrm{TeV}}{M_{\\nu^{(1)}}}\\right) \n\\end{equation}\nwhere $M_{\\nu^{(1)}}$ denotes the mass of KK-neutrino, $\\nu^{(1)}$, and $R$ indicates the radius of compactification.\n\n\\section{Calculation}\n\nThe KK-neutrino develops a magnetic dipole moment through the diagrams tabulated in figure \\ref{Figgraphs}.\n\\begin{figure}[h]\n \\begin{center}\n \t\\vspace{5pt}\n \t\\hspace{-100pt}{\\includegraphics[height=120pt,width=120pt]{testgraph.ps}}\n \t\\vspace{45pt}\n \t\\caption{Magnetic dipole moment inducing one loop Feynman graphs for\n the first KK-excitation of the neutrino contained in the $\\mathrm{SU(2)}_L$ doublet in UED models.\n KK-number assignments are denoted in graph a), to be understood analogously in graphs b) - f).\n Graphs with dashed lines denote the Goldstone modes along their\n KK-excitations, and the KK-excitation of the Higgs scalar. The Feynman rules are taken from \\cite{Buras:2002ej}.}\n\t\\label{Figgraphs} \n \\end{center}\n\\end{figure}\nIn general, the entire Kaluza-Klein tower of states participate, however we estimate the leading order effect by considering only the first level KK excitations. Furthermore, we restrict our calculation to KK-number conserving graphs since KK-number violation proceeds with volume suppression. For simplicity, we ignore flavour violation in the lepton sector and we assume that the lightest KK-neutrino is $\\nu_e^{(1)}$. Relaxing these assumptions will not significantly alter our conclusions.\n\nAs we make no assumptions on the exact UED spectrum, we consider the mass of the KK-$W$ ($W^{(1)}$) and the KK-electron ($e^{(1)}$) as free parameters. In our numerical calculations we do not consider a KK-electron\/KK-neutrino mass difference in excess of 5\\% as any substantial splitting will lead to unacceptably large contributions to the $T$ parameter.\n\nThe relevant UED Feynman rules are listed in \\cite{Buras:2002ej} and we calculate in the Feynman-'t Hooft gauge. In the limit of exact $M_{\\nu^{(1)}}$-$M_{e^{(1)}}$ degeneracy and where the effects of Yukawa couplings are ignored, we arrive at the semi-analytic result,\n\\begin{equation} \n\\label{dipole_result} \n\\begin{split}\n\\mu = & \\frac{eg^2}{(4\\pi)^2}\\frac{1}{M_{\\nu^{(1)}}} \\times\\\\\n\t&\\left\\{\\frac{3}{2}\\ln (\\epsilon)+r+\\frac{7}{2}+\\frac{1}{2r}-\\frac{5}{2}(r-1)\\ln \\left(\\frac{r}{r-1}\\right)\\right.\\\\\n\t&\\left. -(r-1)^2\\ln \\left(\\frac{r}{r-1}\\right) + \\mathcal{O}(\\sqrt{\\epsilon})\\right\\}\t\n\\end{split}\n\\end{equation} \nwith the approximation $\\epsilon\\equiv M_{W^{(0)}}^2\/M_{\\nu^{(1)}}^2\\ll 1$ and $r\\equiv M_{W^{(1)}}^2\/M_{\\nu^{(1)}}^2\\simeq 1$.\nNumerically, we find agreement with our semi-analytical result as seen in figure \\ref{plot_1}.\n\\begin{figure}[ht!]\n \\newlength{\\picwidtha}\n \\setlength{\\picwidtha}{3.5in}\n \\begin{center}\n \\resizebox{\\picwidtha}{!}{\\includegraphics{plot_v2.eps}}\n \\end{center}\n \\caption{Magnetic moment of the KK-neutrino $\\nu^{(1)}$ vs.\n KK dark matter mass $M_{\\nu^{(1)}}$ for varying $M_{e^{(1)}}$, and $M_{W^{(1)}}$.\n Upper line (solid): $M_{e^{(1)}}$-$M_{\\nu^{(1)}}$ degeneracy with tree-level $M_{W^{(1)}}$ ({\\it c.f. } analytical result eq(\\ref{dipole_result})).\n Middle line (dashed-dotted): 5\\% splitting of $M_{e^{(1)}} \/ M_{\\nu^{(1)}}$ with with tree-level $M_{W^{(1)}}$.\n Lower line (dashed): $M_{e^{(1)}}$-$M_{\\nu^{(1)}}$ degeneracy with 5\\% splitting of $M_{W^{(1)}} \/ M_{\\nu^{(1)}}$}\n\t \\label{plot_1}\n\\end{figure}\n\n \nIn figure \\ref{plot_1} we display the result of the dipole moment as a function of the KK-neutrino mass. The upper curve illustrates the magnetic dipole moment with exact degeneracy $M_{e^{(1)}}$-$M_{\\nu^{(1)}}$ while holding the KK-$W$ mass at its tree level value. The middle curve plots the effect of a KK-electron 5\\% heavier than the KK-neutrino. This has an $\\mathcal{O}(1)$ effect on the dipole moment. The predicted value of the dipole moment exceeds the current upper bound for a KK-neutrino mass $M_{\\nu^{(1)}}\\sim 1 \\hspace{1mm}\\mathrm{TeV}$ by more than five orders of magnitude \\cite{Sigurdson:2004zp}. The lower curve displays the effect of maintaining $M_{\\nu^{(1)}}$-$M_{e^{(1)}}$ degeneracy while varying the KK-$W$ mass. Allowing the mass difference, $M_{W^{(1)}}$-$M_{\\nu^{(1)}}$, to vary by up to 5\\% has at most an $\\mathcal{O}(10)$ effect.\n\nWe should note that the calculation presented above determines only the radiatively induced part of the KK-neutrino magnetic dipole moment. The presence of boundary terms or effects arising from the UV complete theory may also contribute a non-renormalizable dimension-five dipole operator which, a priori, may be of the same order as the radiative part itself.\n\n\\section{Conclusion}\n\nUED models have attracted attention as a possible extension to the standard model. A particular appealing feature of the model class centers on the existence of plausible dark matter candidates as the result of KK-parity conservation. While the minimal UED model suggests $B^{(1)}$ dark matter \\cite{Cheng:2002ej}, detailed studies of the relic abundance in the minimal UED model \\cite{coannmat,coannkribs, Matsumoto:2005uh} in combination with electroweak precision constraints \\cite{Flacke:2005hb} show strong observational tension, and thus provide motivation for new possible UED model building avenues. The need for non-minimality has been reported \\cite{Hewett:2004py}. Extensions of the MUED scenario by incalculable boundary terms arising from the UV completion of the model can lead to a different LKP and therefore different possible dark matter candidates. We have taken a model independent approach, following \\cite{Servant:2002aq}, and examined the consequences of UED KK-neutrino dark matter.\n\nAs the KK-neutrino is a Dirac fermion, UED models predict an induced KK-neutrino dipole moment. We find that the induced dipole moment, typically $\\mu \\lesssim 10^{-7} \\mu_B$, strongly conflicts -- by over five orders of magnitude -- with the current observational bounds stated in \\cite{Sigurdson:2004zp} for TeV scale dipole dark matter. We reiterate that the constraints provided by \\cite{Sigurdson:2004zp} are currently under revision, and the strength of the stated bounds are expected to change \\cite{private}. The constraints on magnetic dipole moments given in \\cite{Sigurdson:2004zp} would provide the strongest limits on KK-neutrino dark matter. Even in the absence of the strong limits provided by \\cite{Sigurdson:2004zp}, the bounds on dipole moments remain an important constraint on future model building. Not only will new, non-minimal models that predict KK-neutrino LKP need to circumvent the constaints provided by \\cite{Servant:2002hb}, but will also have to evade the constraints provided by radiatively induced magnetic dipole moments, which are generically at least as large as the current experimental bounds. \n\nWhile we have restricted our discussion to five-dimensional models compactified on $S_1\/Z_2$, we expect that the qualitative features carry over to UED models with multiple extra dimensions. We have also assumed the absence of any fine-tuning between the radiatively induced magnetic dipole moment and possible non-renormalizable dimension-five dipole operators arising from the UV complete theory or boundary terms.\n\nOur results indicate that observational limits on dipole dark matter can place significant constraints on UED scenarios where the KK-neutrino is the LKP dark matter candidate.\n\n\n\n\\section{Acknowledgments} \n\nWe would like to thank J.~March-Russell, G.~Starkman, B.~A.~Campbell, and K.~Sigurdson for useful discussions. DM wishes to acknowledge the support of the Natural Science and Engineering Research Council of Canada and the Canada-United Kingdom Millennium Research Fellowship. The work of TF is supported by ``Evangelisches Studienwerk Villigst e.V.\" and PPARC Grant No. PPA\/S\/S\/2002\/03540A. This work was also supported by the ``Quest for Unification\" network, MRTN 2004-503369.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}