diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzlafa" "b/data_all_eng_slimpj/shuffled/split2/finalzzlafa" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzlafa" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\vspace*{-5pt}\n\\begin{figure}[t]\n\t\\centering\n\t\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=2cm 1cm 2cm 1cm, width=0.45\\linewidth]{goodix.pdf}\n\t}\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=6.5cm 12cm 10cm 11.8cm, width=0.45\\linewidth]{minutiae.pdf}\n\t} \n\t\\hspace{0.2 cm}\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=4cm 8cm 4cm 8cm, width=0.55\\linewidth]{minu_distribution.pdf}\n\t}\n\t\\caption{Fingerprint texture for matching. (a) Only two minutiae in a fingerprint image (96$\\times $ 96 at 500 ppi) acquired by a capacitive sensor embedded in a smartphone (provided by Goodix), (b) five manually marked minutiae in a latent fingerprint image from NIST SD27, and (c) histogram of the number of minutiae in latent fingerprints in NIST SD27 database (consisting of 258 operational latents).}\n\t\\label{fig:introduction}\n\t\\vspace*{-5pt}\n\\end{figure}\n\nEver since minutiae were introduced for comparing fingerprints in 1888 by Sir Francis Galton \\cite{Galton}, minutiae are considered as the foundation for the science of fingerprint identification, which has expanded and transitioned to a wide variety of applications for person recognition over the past century \\cite{FingerprintHandbook}. The first Automated Fingerprint Identification System (AFIS) launched by FBI in early 1970s only stored type of fingerprint and its minutiae instead of digital images because of the compact and efficient representation offered by minutiae\\footnote{https:\/\/www.fbi.gov\/file-repository\/about-us-cjis-fingerprints\\_biometrics-biometric-center-of-excellences-fingerprint-recognition.pdf\/view}. With decades of research and development and advances in processor, memory and sensor design, fingerprint recognition systems have now been deployed in a broad set of applications such as border\ncontrol, employment background checks, secure facility access and national identity programs \\cite{FingerprintHandbook}.\n Aadhaar\\footnote{https:\/\/uidai.gov.in\/about-uidai\/about-uidai.html}, has the world's largest biometric ID system with an enrollment database that already exceeds 1.2 billion tenprints\n(along with corresponding irises and photos) of supposedly unique individuals. All of these systems are primarily based on minutiae based fingerprint matching algorithms. \n\n\n\n\nMinutiae based approaches, however, may not be effective in some cases, for example, in poor quality latent fingerprint matching and fingerprint images captured by small area sensors in mobile phones. Latent fingerprints (latents) are one of the\nmost important and widely used sources of evidence in law enforcement and forensic agencies worldwide \\cite{Hawthorne2002}. Due to the unintentional deposition of the print\nby a subject, latents are typically of poor quality in terms of ridge clarity, large background noise and small friction ridge area. Hence, the number of minutiae in a latent may be very small , e.g., $\\leq$10. \n Fig. \\ref{fig:introduction} (c) shows the distribution of the number of manually marked minutiae on NIST SD27 latent database and Fig. \\ref{fig:introduction} (b) shows a latent image with 5 manually marked minutiae overlaid on the image. Minutiae alone do not have enough information for latents.\n Another example where minutiae based matching does not work is for matching fingerprint images captured by the capacitive sensors embedded in smartphones. It is estimated that 67\\% of the smartphones in the world will have an embedded fingerprint sensor\\footnote{https:\/\/www.statista.com\/statistics\/522058\/global-smartphone-fingerprint-penetration\/} by 2018. The size of these embedded fingerprint sensors (only $88 \\times 88$ pixels for TocuhID\\footnote{https:\/\/assets.documentcloud.org\/documents\/1302613\/ios-security-guide-sept-2014.pdf}) are much smaller than the standalone sensors. Hence, the number of minutiae in these images is very few as shown in Fig. \\ref{fig:introduction} (a). For these reasons, accurate non-minutiae based (also called \\textit{texture based}) fingerprint matching algorithms are necessary. To our knowledge, all major latent AFIS vendors use texture templates and different quality latents are handled differently inside AFIS and that is why the processing time is different. \n \n\nA few non-minutiae based fingerprint matching algorithms have been proposed in literature. FingerCode by Jain et al. \\cite{Jain2000IP} uses a bank of Gabor filters to capture both the local and global details in a fingerprint. However, FingerCode relies on a reference point and, further, its accuracy is much lower than minutiae based approaches. Some approaches based on keypoints, e.g., SIFT \\cite{Park, Yamazaki} and AKAZE \\cite{AKAZE}, have been proposed to generate dense keypoints for fingerprint matching. But these keypoints as well as their descriptors are not sufficient to distinguish fingerprints from different fingers.\n\n\nDeep learning based approaches have also been proposed for fingerprint recognition. Zhang et al. \\cite{Zhang2017IJCB} proposed a deep learning based feature, called deep dense multi-level feature, for partial high resolution fingerprint recognition which achieved promising performance on their own database. However, their approach could not handle fingerprint rotation. Cao and Jain \\cite{Cao2018PAMI} proposed a virtual minutiae based approach for latent fingerprint recognition, where the virtual minutiae locations are determined by a raster scan with a stride of 16 pixels; the associated descriptors are obtained by three convolutional neural networks. Experimental results on two latent databases, NIST SD27 and WVU, showed that the recognition performance of virtual minutiae based ``texture template\" when fused with two different true minutiae templates boosts the rank-1 accuracy from 58.5\\% to 64.7\\% against 100K reference print gallery for NIST SD27 \\cite{Cao2018PAMI}. However, the virtual minutiae feature extractor and matcher are quite slow. \n\n\n\nThe objective of this paper is to improve both the accuracy and efficiency of virtual minutiae (texture template) based latent matching. The main contributions of this paper are as follows: \n \\begin{enumerate} \n\\itemsep0em \n\\item Reduce the average recognition time between a latent texture template and a rolled texture template from 11 ms (24 threads) to 7.7 ms (single thread);\n\\item Improve the rank-1 identification rate of the texture templates in \\cite{Cao2018PAMI} by 8.9\\% (from 59.3\\% to 68.2\\%) for 10K gallery;\n\\item Boost the rank-1 identification rate in \\cite{Cao2018PAMI} by 2.7\\% by fusion of the proposed three texture templates with three templates in \\cite{Cao2018PAMI} (from 75.6\\% to 78.3\\%) for 10K gallery. This means that out of the 258 latents in NIST SD27, improvements in the texture template will push 7 additional latents at rank 1.\n \\end{enumerate}\n \n \n\n\n\n\n\\section{Proposed Approach}\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[clip, trim=0cm 0cm 0cm 0cm, width=0.85\\linewidth]{overview4.pdf}\n\t\\caption{Overview of the proposed latent fingerprint recognition algorithm. }\n\t\\label{fig:overview}\n\t\\setlength{\\belowdisplayskip}{-6pt}\n\t\\setlength{\\belowdisplayshortskip}{-6pt}\n\t\\vspace*{-5pt}\n\\end{figure*}\n\nIn this section, we describe the proposed texture-based latent matching approach, including feature extraction and matching. Fig. \\ref{fig:overview} shows flowchart of the proposed approach. \n\n\n\\subsection{Virtual Minutiae Extraction}\nFor both latents and reference prints, the texture template is similar to that in \\cite{Cao2018PAMI} and consists of locations, orientations and descriptors of virtual minutiae. We first describe the virtual minutiae localization and then discuss the associated descriptors. \n\nFor reference fingerprints which are typically of good quality, the region of interest (ROI) is segmented by magnitude of the gradient and the orientation fields with a block size of $16\\times 16$ pixels as in \\cite{Chikkerur2007198}.\nThe locations of virtual minutiae are sampled by raster scan with a stride of $s$ and their orientations are the same as the orientations of their nearest blocks in the orientation field. The virtual minutiae close to the mask border are ignored. Fig. \\ref{fig:rolled_virtual} shows the virtual minutiae on two rolled prints.\n\nFor latents, the same manually marked ROIs and automatically\nextracted ridge flow with a block size of $16\\times 16$ pixels as in \\cite{Cao2015ICB} are used for\nvirtual minutiae extraction. Suppose that $(x, y)$ are the $x-$ and $y-$coordinates of a sampling point and $\\theta$ denotes the orientation of the block which is closest to $(x,y)$ in the ridge flow. Two virtual minutiae, i.e., $(x,y,\\theta)$ and $(x,y,\\theta+\\pi)$, are created to handle the ambiguity in ridge orientation. Fig. \\ref{fig:latent_virtual} show virtual minutiae\non two enhanced latents from NIST SD27 with $s = 32$.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=3cm 1cm 3cm 1cm, width=0.4\\linewidth]{Rolled_001.jpg}\n\t}\\hspace{1cm}\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=3cm 1cm 3cm 1cm, width=0.4\\linewidth]{Rolled_004.jpg}\n\t} \n\t\\caption{Virtual minutiae on two rolled prints with stride $s$=32. }\n\t\\label{fig:rolled_virtual}\n\t\\vspace*{-5pt}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=5cm 5.5cm 6cm 1.5cm, width=0.35\\linewidth]{Latent_001_32.jpg}\n\t}\n\\hspace{1cm}\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=5cm 3.5cm 6cm 3.5cm, width=0.35\\linewidth]{Latent_004_32.jpg}\n\t} \n\t\\caption{Virtual minutiae on two enhanced latent fingerprints with stride $s$=32. Note that each circle represents two virtual minutiae with opposite orientations to handle the ambiguity in ridge orientation.}\n\t\\label{fig:latent_virtual}\n\t\\vspace*{-5pt}\n\\end{figure}\n\n\n\\subsection{Descriptors for Virtual Minutiae}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\subfigure[][]{\n\t\t\\includegraphics[clip, trim=0cm 0cm 0cm 0cm, width=0.21\\linewidth]{3_3.jpeg}\n\t}\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=0cm 0cm 0cm 0cm, width=0.21\\linewidth]{3_8.jpeg}\n\t} \n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=0cm 0cm 0cm 0cm, width=0.21\\linewidth]{3_13.jpeg}\n\t} \n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=0cm 0cm 0cm 0cm, width=0.21\\linewidth]{3_18.jpeg}\n\t} \n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=0cm 0cm 0cm 0cm, width=0.21\\linewidth]{3_3_e.jpeg}\n\t}\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=0cm 0cm 0cm 0cm, width=0.21\\linewidth]{3_8_e.jpeg}\n\t} \n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=0cm 0cm 0cm 0cm, width=0.21\\linewidth]{3_13_e.jpeg}\n\t} \n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=0cm 0cm 0cm 0cm, width=0.21\\linewidth]{3_18_e.jpeg}\n\t} \n\t\\caption{Examples of training fingerprint patches.}\n\t\\label{fig:train_eg}\n\t\\vspace*{-5pt}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\centering\n\t\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=0cm 0cm 0cm 0cm, width=0.21\\linewidth]{scale_1.jpg}\n\t} \n\t\\subfigure[][]{\n\t\t\\includegraphics[clip, trim=0cm 0cm 0cm 0cm, width=0.21\\linewidth]{corner_3.jpg}\n\t}\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=0cm 0cm 0cm 0cm, width=0.21\\linewidth]{top.jpg}\n\t} \n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=0cm 0cm 0cm 0cm, width=0.21\\linewidth]{scale_2.jpg}\n\t} \n\t\\caption{Four different patch types used in \\cite{Cao2018PAMI} for descriptor extraction. Patch types in (a)-(c) were determined to be the best combination in terms of identification accuracy. The patches in (a), (b) and (d) are of sizes $96\\times 96$ pixels while the patch in (c) is of size $80\\times 80$ pixels. In this paper, we used patch types in (b), (c) and (d) for descriptor extraction because all of them are of the same size and hence no resizing is needed.}\n\t\\label{fig:patch_types}\n\t\\vspace*{-5pt}\n\\end{figure}\n\n\nA minutia descriptor contains attributes of the minutia\nbased on the image characteristics in its neighborhood.\nSalient descriptors are needed to establish \n minutiae correspondences and compute the similarity between a latent and\nreference prints. Instead of specifying the descriptor in an ad hoc manner, Cao and Jain \\cite{Cao2018PAMI} trained ConvNets to learn\nthe descriptor from local fingerprint patches around a minutia and showed its performance. In this paper, we improve both the distinctiveness of the descriptor and efficiency of descriptor extraction. \n\nCao and Jain \\cite{Cao2018PAMI} extracted around 800K $160 \\times 160$ fingerprint patches from around 50K minutiae from images in a large fingerprint longitudinal database \\cite{Yoon2015PNAS}, to train the ConvNets. On average, there are around 16 fingerprint patches around each minutia for training. Cao and Jain also reported that descriptors extracted from enhanced latent images give better performance. In order to augment the training dataset and improve the descriptor distinctiveness between enhanced latent and original rolled prints, we use patches from both the original and enhanced fingerprint patches for training ConvNets; this results in around 1.6 million fingerprint patches for training. Fig. \\ref{fig:train_eg} shows some example patches from the training dataset; Figs. \\ref{fig:train_eg} (a) to (d) are from the original image and (e) to (h) are from the corresponding enhanced images.\n\n\nLocation and size of the patches were also evaluated in \\cite{Cao2018PAMI}. The three patch types shown in Figs. \\ref{fig:patch_types} (a)-(c) were determined to be the best patch types via forward sequential selection. The two patches in Figs. \\ref{fig:patch_types} (b) and (c) are of the same size ($96\\times 96$ pixels) while the patch in Fig. \\ref{fig:patch_types} (a) is of size $80 \\times 80$ pixels. All the patches had to be resized to $160\\times 160$ pixels in \\cite{Cao2018PAMI}, by bilinear interpolation. In order to avoid resizing, we train a ConvNet on $96\\times 96$ images and use patch types in Figs. \\ref{fig:patch_types} (b)-(d) for descriptor extraction. Note that the patch in Fig. \\ref{fig:patch_types} (d) is minutiae centered whereas patches in Figs. \\ref{fig:patch_types} (b) and (c) are offset from the center minutiae.\n \nAmong the various ConvNet architectures \\cite{VGG}, \\cite{GoogLeNet}, \\cite{ResNet}, \\cite{Mobilenet}, MobileNet-v1 \\cite{Mobilenet} uses depth-wise separable convolutions, resulting in a \ndrastic reduction in model size and training\/evaluation times while providing good recognition performance. The number of original model parameters to be trained in MobileNet-v1 (4.24M), is significantly\nsmaller than the number of model parameters in Inception-v3 (23.2M) and VGG (138M), requiring significantly lower\nefforts in terms of regularization and data augmentation, to prevent overfitting. For these reasons, we utilize the MobileNet-v1 architecture due to its fast inference speed.\nIn order to feed $96 \\times 96$ fingerprint patches for training and to reduce the descriptor length, we make the following modifications to the MobileNet-v1 architecture: i) change the input image size to $96 \\times 96 \\times 1$, ii) remove the last two convolutional layers to accommodate the smaller input sizes, and iii) add a fully connected layer, also called the embedding layer, before the classifier layer to obtain a compact feature representation. The modified architecture is shown in Table \\ref{tab:MobileNet}.\n\nFor each of the three patch types shown in Figs. \\ref{fig:patch_types} (b)-(d), 1.6 million fingerprint patches are used to train a MobileNet. \nGiven a fingerprint patch around a virtual minutiae, the output ($l$-dimensional feature vector) of the last fully connected layer is considered as the virtual minutiae descriptor. In the experiments, three value of $l$, namely $l=32, 64$, and $128$, are investigated. The concatenation of the three outputs for the same virtual minutiae is regarded as the descriptor with length $l_d$ = $3\\times l$. \nThe set of virtual minutiae and the associated descriptors define a texture template. \n\n\n\\subsection{Texture Template Matching}\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[clip, trim=0cm 0cm 0cm 0cm, width=0.5\\linewidth]{H22.pdf}\n \\caption{Illustration of second-order graph matching \\cite{Cao2018PAMI}, where $m_{i_1}$ and $m_{j_1}$ are the two virtual minutiae, $d_{i_1,j_1}$ is the Euclidean distance between $m_{i_1}$ and $m_{j_1}$, and $\\theta_{i_1,j_1}$, $\\theta_{i_1}$ and $\\theta_{j_1}$ are the three angles formed by two virtual minutiae orientations and the line segment connecting $m_{i_1}$ and $m_{j_1}$. }\n \\label{fig:second-order}\n \\setlength{\\belowdisplayskip}{-6pt}\n\\setlength{\\belowdisplayshortskip}{-6pt}\n\\vspace*{-5pt}\n\\end{figure}\n\n\n\\begin{table}[tp]\n\\caption{MobileNet Architecture, where $l$ is the length of feature vector output by the MobileNet and $c$ is the number of classes used for training.}\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline\nType \/ Stride & Filter Shape & Input Size\\\\\n\\hline\nConv \/ s2 & 3$\\times$3$\\times$1$\\times$32 & 96$\\times$96$\\times$1 \\\\\n\\hline\nConv dw \/ s1 & 3$\\times$3$\\times$32 dw &48$\\times$48$\\times$32 \\\\\n\\hline\nConv \/ s1 &1$\\times$1$\\times$32$\\times$64 &48$\\times$48$\\times$32 \\\\\n\\hline\nConv dw \/ s2 & 3$\\times$3$\\times$64 dw &48$\\times$48$\\times$64 \\\\\n\\hline\nConv \/ s1 &1$\\times$1$\\times$64$\\times$128 &24$\\times$24$\\times$64 \\\\\n\\hline\nConv dw \/ s1 &3$\\times$3$\\times$128 dw &56$\\times$24$\\times$128 \\\\\n\\hline\nConv \/ s1 &1$\\times$1$\\times$128$\\times$128 &24$\\times$24$\\times$128 \\\\\n\\hline\nConv dw \/ s2 &3$\\times$3$\\times$128 dw &24$\\times$24$\\times$128 \\\\\n\\hline\nConv \/ s1 &1$\\times$1$\\times$128$\\times$256 &12$\\times$12$\\times$128 \\\\\n\\hline\nConv dw \/ s1 &3$\\times$3$\\times$256 dw &12$\\times$12$\\times$256 \\\\\n\\hline\nConv \/ s1 &1$\\times$1$\\times$256$\\times$256 &12$\\times$12$\\times$256 \\\\\n\\hline\nConv dw \/ s2 &3$\\times$3$\\times$256 dw &12$\\times$12$\\times$256 \\\\\n\\hline\nConv \/ s1 & 1$\\times$1$\\times$256$\\times$512 &6$\\times$6$\\times$256 \\\\\n\\hline\n5$\\times $Conv dw \/ s1& 3$\\times$3$\\times$512 dw &6$\\times$6$\\times$512 \\\\ \n\\hline\nConv \/ s1 &1$\\times$1$\\times$512$\\times$512 &6$\\times$6$\\times$512 \\\\\n\\hline\nConv dw \/ s2 &3$\\times$3$\\times$512 dw &6$\\times$6$\\times$512 \\\\\n\\hline\nAvg Pool \/ s1 & Pool 6$\\times$6 & 6$\\times$6$\\times$512\\\\\n\\hline\nFC \/ s1 & 512$\\times l$ & 1$\\times 1 \\times 512$ \\\\\n\\hline\nFC \/ s1 & $l \\times c $& 1$\\times 1 \\times l$ \\\\\n\\hline\nSoftmax \/ s1 & Classifier & 1 $\\times $ 1 $\\times c$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{tab:MobileNet}\n\\vspace*{-5pt}\n\\end{table}%\n\n\n\\label{sec:matching}\nThe algorithm for comparing two texture templates, one from latent and the other from a reference print, as proposed in \\cite{Cao2018PAMI} can be summarized as: i) compute pair-wise similarities between the latent and reference print virtual minutiae descriptors using cosine similarity; normalize the similarity matrix, ii) select the top $N$ ($N=200$) virtual minutiae correspondences based on the normalized similarity matrix, iii) remove false minutiae correspondences using second-order graph matching, iv) further remove false minutiae correspondences using third-order graph matching, and (v) finally compute the overall similarity between the two texture templates based on final minutiae correspondences. Although the second-order graph matching can remove most false correspondences, it is still time-consuming as it involves $N(N-1)\/2$ computations. Furthermore, since the locations of virtual minutiae do not have large variations due to the raster scan, the larger complexity third order graph matching does not help too much in virtual minutiae correspondences.\n\n\n\n\\begin{figure}[t]\n\t\\centering\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=2cm 1cm 1cm 1cm, width=0.85\\linewidth]{top_200.jpg}\n\t}\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=2cm 1cm 1cm 1cm, width=0.85\\linewidth]{top_200_dist.jpg}\n\t} \n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=2cm 1cm 1cm 1cm, width=0.85\\linewidth]{top_200_Fast2.jpg}\n\t} \n\t\\caption{Illustration of virtual minutiae correspondences by the proposed graph matching strategy. Figures in (a), (b) and (c) show the top $N=200$ minutiae correspondences, 88 virtual minutiae correspondences by using the modified second order graph matching and 73 virtual minutiae correspondences by applying the second order graph matching used in \\cite{Cao2018PAMI} to the 88 minutiae correspondences in (b).}\n\t\\label{fig:correspondences}\n\t\\vspace*{-5pt}\n\\end{figure}\n\nAs shown in Fig. \\ref{fig:second-order}, the computation of the compatibility between two virtual minutiae pairs in second-order graph matching involves one Euclidean distance computation and three angular distance computations. \nIn our preliminary experiments, we found that the Euclidean distance alone is good enough to remove most false correspondences as shown in Fig. \\ref{fig:correspondences}.\nThe approach we propose here is to use a modified second-order graph matching to remove most false virtual minutiae correspondences and use the second-order graph matching in \\cite{Cao2018PAMI} to get the final virtual minutiae correspondences\n\n\n An overview of the modified virtual minutiae matching algorithm is illustrated in $\\mathbf{Algorithm \\ \\ref{alg:search}}$. The details of the modified second-order graph matching are as following. Suppose $\\{i=(i_1,i_2)\\}_{i=1}^{N}$ is the set of $N$ selected minutiae correspondences between a latent $L$ and a rolled print $R$, where $i_1$ and $i_2$ denote the $i^{th}$ correspondence between the $i_1^{th}$ and $i_2^{th}$ virtual minutiae in the latent $F_l$ and the rolled print $F_r$, respectively. Given two minutiae correspondences $(i_1,i_2)$ and $(j_1,j_2)$, $H_{i,j}^2$ ($H^2 \\in R^{N\\times N}$) measures the compatibility between $(i_1,j_1)$ from the latent and $(i_2,j_2)$ from the rolled prints:\n\\begin{align}\nH^2_{i,j} &= Z(D_{i,j},\\mu,\\tau,t),\n\\end{align}\nwhere $D_{i,j} = |d_{i_1,j_1} - d_{i_2,j_2}|$ and $Z$ is a truncated sigmoid function:\n\n\\begin{equation}\nZ(v,\\mu,\\tau, t) =\t\\left.\n\\begin{cases}\n \\frac{1}{1+e^{-\\tau(v-\\mu)}}, & \\text{ if } v \\leq t, \\\\\n 0, & \\text{ otherwise. }\n\\end{cases}\\right.\n\\end{equation}\nHere $\\mu,\\tau$ and $t$ are the parameters of function $Z$.\n\n The goal of graph matching is to find an $N$-dimensional correspondence vector $Y$, where the $i^{th}$ element ($Y_i$) indicates whether $i_1$ is assigned to $i_2$ ($Y_i = 1$) or not ($Y_i = 0$). This can be represented in terms of maximizing the following objective function:\n \\begin{equation}\n\\label{eq:objective_simple_two}\nS_2(Y) = \\sum_{i,j} \\\\ H^2_{i,j} Y_i Y_j.\n\\end{equation}\nA strategy of power\niteration followed by discretization used in \\cite{Cao2018PAMI} is used to remove false minutiae correspondences.\n\nSuppose $\\{i=(i_1,i_2)\\}_{i=1}^{n}$ represent the final $n$ matched minutiae correspondences between $F_l$ and $F_r$. The similarity $S$ between $F_l$ and $F_r$ is defined as:\n\\begin{equation}\n\\label{eq:minutiae_similarity}\nS = \\sum_{i=1}^nDesSim(i_1,i_2),\n\\end{equation}\nwhere $DesSim(i_1,i_2)$ is the descriptor similarity between $i_1^{th}$ virtual minutiae in latent template $F_l$ and $i_2^{th}$ virtual minutiae in reference template $F_r$.\n\n\\begin{algorithm}\n\t\\caption{Modified virtual minutiae matching algorithm}\\label{alg:search}\n\t\\begin{algorithmic}[1]\n\t\n\t\t\\State \\textbf{Input:} Latent template $F_l$ and reference template $F_r$ \n\t\t\\State \\textbf{Output:} Similarity between $F_l$ and $F_r$ \n\t\t\\State \tCompute descriptor similarity matrix\\;\n\t\t\\State\tNormalize similarity matrix\\;\n\t\t\\State Select the top $N$ minutiae correspondences based on the normalized similarity matrix\\;\n\t\t\\State \tConstruct $H^2$ based on these $N$ minutiae correspondences\\;\n\t\t\\State\tRemove false correspondences using modified second-order graph matching\\;\n\t\t\\State Further remove false correspondences using original second-order graph matching\\;\n\t\t\\State Compute similarity between $F_l$ and $F_r$\\;\n\t\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\n\n \\begin{figure}[t] \n\t\\centering\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=6cm 8cm 7cm 8cm, width=0.35\\linewidth]{019_original.jpg}\n\t} \\hspace{1 cm}\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=6cm 8cm 7cm 8cm, width=0.35\\linewidth]{019_enhanced_1.jpg}\n\t} \\hspace{1 cm}\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=0cm 0cm 0cm 0cm, width=0.22\\linewidth]{019_enhanced_3.jpg}\n\t} \\hspace{1.5 cm}\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=6cm 8cm 7cm 8cm, width=0.35\\linewidth]{019_texture.jpg}\n\t} \n\t\\caption{Illustration of processing strategies applied to the input latent shown in (a) for virtual minutiae descriptor extraction. (b) and (c) are the two enhanced latent fingerprint images used for minutiae template 1 and 2 extraction in \\cite{Cao2018PAMI}, and (d) is the texture component after decomposition \\cite{Cao2014PAMI}. }\n\t\\label{fig:input_images}\n\t\\vspace*{-5pt}\n\\end{figure}\n\n\\subsection{Constructing texture templates}\n\\label{sec:templates}\nThe locations and orientations of virtual minutiae are determined by the ROI and ridge flow, while the descriptors depend on the images input to the trained ConvNets. Three different processed latent images for each latent are investigated for texture template construction. Two different enhancement algorithms were proposed in \\cite{Cao2018PAMI} and the resulting two minutiae sets were extracted, one per enhanced images. Both of the enhanced images for virtual minutiae descriptor extraction are also investigated here. However, the fingerprint enhancement performance critically depends on the estimates of ridge flow and ridge spacing. If ridge flow or ridge spacing are not estimated correctly, spurious ridge structures are created in the enhanced images. In addition to the two enhanced images, we also consider the texture image obtained by image decomposition \\cite{Cao2014PAMI}, which essentially removes large scale background noise and enhances ridge contrast. Figs. \\ref{fig:input_images} (b)-(d) illustrate the three processed images for the input latent image in Fig. \\ref{fig:input_images} (a). For each latent, three different texture templates, $T_{e_1}$, $T_{e_2}$ and $T_{t}$, can then be extracted. Given a latent to reference print pair, three texture template similarities are computed which are then fused to improve the overall latent recognition performance.\n\n\n\\section{Experimental Results}\nThe proposed texture template matching algorithm is evaluated on NIST SD27 latent database, which consists of 88 good quality, 85 bad quality and 85 ugly quality latent fingerprint images. For latent search experiments, 10,000 reference prints\\footnote{Results for the larger gallery of 100K reference prints are not yet available at the time of submitting this paper.}, including the 258 mates of NIST SD27 and others from NIST SD14, are used as the gallery.\n\n\n\\subsection{Descriptor length evaluation}\nThe performance of the new virtual minutiae descriptors with different feature lengths, namely, $l_d$ = 96 ($l=32$), 192 ($l=64$) and 384 ($l=128$), are evaluated by verification performance based on manually marked minutiae correspondences on NIST SD27\\footnote{NIST SD27 dataset is no longer available for download from the NIST site.} \\cite{NISTDB27}. A total of 5,460 minutiae correspondences between the latent images and the mated rolled fingerprint images were provided with the NIST SD27 database. The average numbers of manually marked minutiae correspondences on 88 good quality, 85 bad quality and 85 ugly quality latent images are 31, 18 and 14, respectively. For a fair comparison with the descriptors used in \\cite{Cao2018PAMI}, the descriptors for the latents are extracted on the same enhanced images as in \\cite{Cao2018PAMI} while the descriptors of the mated rolled prints are extracted on the original rolled prints. Fig. \\ref{fig:manual_minutiae} shows the manually marked minutiae on an enhanced latent image and its mated rolled prints. The genuine scores are computed using the similarities of descriptors from the manually marked minutiae correspondences while the impostor scores are computed using the similarities of descriptors from the different minutiae. Thus, a total of 5,460 genuine scores and around 30 million impostor scores ($5,460\\times 5,460$) are computed. Fig. \\ref{fig:verification} compares the Receiver Operating Characteristic (ROC) curves of different descriptor lengths as well as the descriptors from \\cite{Cao2018PAMI} which used a descriptor length of 384. Note that the ROC curves of descriptors with feature lengths 384 and 192 are very close to each other, slightly better than descriptors with feature length 96 and significantly better than the descriptors in \\cite{Cao2018PAMI}. This can be explained by the use of both enhanced and original fingerprint patches along with a more appropriate ConvNet architecture for training.\n \\begin{figure}[t] \n\t\\centering\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=5cm 2cm 4cm 2cm, width=0.45\\linewidth]{003_Latent.jpg}\n\t}\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=5cm 2cm 4cm 2cm, width=0.45\\linewidth]{Rolled_minutiae_003.jpg}\n\t} \n\t\\caption{Manually marked minutiae correspondences are used to evaluate the proposed virtual minutiae descriptors. (a) Manually marked latent minutiae shown on the enhanced latent image and (b) manually marked rolled minutiae shown on mated rolled print image.}\n\t\\label{fig:manual_minutiae}\n\t\\vspace*{-5pt}\n\\end{figure}\n\n\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[clip, trim=0cm 0cm 0cm 0cm, width=0.9\\linewidth]{ROC.pdf}\n \\caption{Receiver Operating Characteristic (ROC) curves under different descriptor lengths.}\n \\label{fig:verification}\n \n\\vspace*{-5pt}\n\\end{figure}\n\n\n\n\n \\begin{figure*}[t] \n\t\\centering\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=4cm 8.5cm 4cm 9cm, width=0.35\\linewidth]{All.pdf}\n\t} \\hspace{1cm}\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=4cm 8.5cm 4cm 9cm, width=0.35\\linewidth]{Good.pdf}\n\t} \\hspace{1cm}\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=4cm 8.5cm 4cm 9cm, width=0.35\\linewidth]{Bad.pdf}\n\t} \\hspace{ 1cm}\n\t\\subfigure[]{\n\t\t\\includegraphics[clip, trim=4cm 8.5cm 4cm 9cm, width=0.35\\linewidth]{Ugly.pdf}\n\t} \n\t\\caption{Cumulative Match Characteristic (CMC) curves of the three texture templates proposed here and their fusion with three templates used in \\cite{Cao2018PAMI} on (a) all 258 latents in NIST SD27, (b) subset of 88 good latents, (c) subset of 85 bad latents and (d) subset of 85 ugly latents. Note that the scales of the y-axis in these four plots are different to show the differences between the two curves.}\n\t\\label{fig:CMC}\n\t\\vspace*{-5pt}\n\\end{figure*}\n\n\n\n\\begin{table}[htp]\n\\caption{Latent search accuracies under different scenarios for NIST SD27. Reference database size is 10K. }\n\\vspace*{-10pt}\n\\begin{center}\n\\begin{tabular}{|p{1.9cm}|p{1.3cm}|p{1.cm}|p{1.cm}|p{1.1cm}|}\n\\hline\nInput templates & descriptor length &rank-1 (\\%)& rank-5 (\\%)& rank-10 (\\%) \\\\\n\\hline\nCao\\&Jain \\cite{Cao2018PAMI} & 384 & 59.30 & 70.16& 73.26 \\\\\n\\hline\n\\hline\n$T_{e_1}$ & 192 & 68.22 & 73.64 & 74.81 \\\\\n\\hline\n$T_{e_2}$& 192 & 66.67 & 72.48 & 74.42 \\\\\n\\hline\n$T_{t}$ & 192 & 60.47 & 67.83 & 70.93 \\\\\n\\hline\n$T_{e_1}$+$T_{e_2}$ & 192 & $\\mathbf{70.93}$ & 74.81 & 77.91 \\\\\n\\hline\n$T_{e_1}$+$T_{t}$ & 192 & $\\mathbf{70.93}$ & 76.36 & 79.07 \\\\ \n\\hline\n$T_{e_2}$+$T_{t}$ & 192 & 67.05 & 75.19 & 77.13\\\\\n\\hline\n$T_{e_1}$+$T_{e_2}$+$T_{t}$ & 192 & 70.16 & $\\mathbf{76.74}$ & $\\mathbf{81.40}$ \\\\\n\\hline\n$T_{e_1}$ & 384 &69.38 & 75.58 & 77.13 \\\\\n\\hline\n$T_{e_2}$ & 384 & 66.28 & 72.48 & 73.64 \\\\\n\\hline\n$T_{t}$ & 384 & 58.91 & 66.28 & 69.77\\\\\n\\hline\n$T_{e_1}$+$T_{e_2}$ & 384 & 70.16 & 75.58 & 78.29 \\\\\n\\hline\n$T_{e_1}$+$T_{t}$ & 384 & 69.38 & 76.74 & 77.91 \\\\ \n\\hline\n$T_{e_2}$+$T_{t}$ & 384 & 67.83 & 74.03 & 75.97 \\\\\n\\hline\n$T_{e_1}$+$T_{e_2}$+$T_{t}$ & 384 & $\\mathbf{70.93 }$ & 75.97 & 78.68 \\\\\n\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{tab:accuracy}\n\\vspace*{-10pt}\n\\end{table}%\n\n\\subsection{Texture template search}\nIn this section, we evaluate the search performance of the proposed three texture templates, i.e., $T_{e_1}$, $T_{e_2}$ and $T_{t}$ extracted in section \\ref{sec:templates}, as well as their fusion. The following seven scenarios are considered:\n\\begin{enumerate}\n\\itemsep0em \n\\item $T_{e_1}$: Texture template with enhanced image 1 (Fig. \\ref{fig:input_images} (b)) for descriptor extraction;\n\\item $T_{e_2}$: Texture template with enhanced image 2 (Fig. \\ref{fig:input_images} (c)) for descriptor extraction;\n\\item $T_{t}$: Texture template with texture image (Fig. \\ref{fig:input_images} (d)) for descriptor extraction;\n\\item $T_{e_1}$ + $T_{e_2}$: Score level fusion of $T_{e_1}$ and $T_{e_2}$;\n\\item $T_{e_1}$ + $T_{t}$: Score level fusion of $T_{e_1}$ and $T_{t}$;\n\\item $T_{e_2}$ + $T_{t}$: Score level fusion of $T_{e_2}$ and $T_{t}$;\n\\item $T_{e_1}$ + $T_{e_2}$ + $T_{t}$: Score level fusion of $T_{e_1}$, $T_{e_2}$ and $T_{t}$.\n\\end{enumerate}\nFor each one of the above seven scenarios, all three descriptor lengths, i.e., $l_d$ = 96, 192 and 384, are considered. The latent search accuracies of different scenarios at three different ranks for NIST SD27 against 10K reference fingerprints are shown in Table \\ref{tab:accuracy}. In addition, the performance of texture template used in \\cite{Cao2018PAMI} is also included for a comparison (row 1 of Table \\ref{tab:accuracy}). Note that the enhanced images used for $T_{e_1}$ extraction are the same as those in \\cite{Cao2018PAMI}. A descriptor length 96 performs much lower than the other two lengths (192 and 384) so its performance is not reported in Table \\ref{tab:accuracy}. The findings from Table \\ref{tab:accuracy} can be summarized as follows: i) among the three texture templates, $T_{e_1}$ performs the best for both descriptor lengths; ii) the performance of descriptor length of 192 is sufficiently close to that of descriptor length 384 in different scenarios; iii) rank-1 accuracy of $T_{e_1}$ with descriptor length 192 is 8.98\\% higher than that in \\cite{Cao2018PAMI}; and iv) fusion of any two out of the three proposed texture templates is higher than that of any single template; the fusion of all three templates boosts the performance slightly at rank-1 (for $l_d$=192) and rank-5 (for $l_d$=384). Fusion of all three proposed templates with $l_d=192$ is preferred because of its smaller feature length and higher accuracy at rank-10. As examiners typically evaluate top 10 candidate matches to identify the source of a latent \\cite{Busey2010}, this will ensure that the true mate is frequently available in the candidate list.\n\n\n\n\n\\subsection{Fusion with minutiae templates}\n\n \\begin{figure}[t] \n\t\\centering\n\t\\subfigure[]{\n\t\t\\includegraphics[width=0.4\\linewidth]{217.jpg}\n\t} \\hspace{0.5cm}\n\t\\subfigure[]{\n\t\t\\includegraphics[ width=0.4\\linewidth]{218.jpg}\n\t} \\hspace{1 cm}\n\t\\caption{Two ugly latents whose true mates were not retrieved at rank 1 by the algorithm in \\cite{Cao2018PAMI}. The use of proposed texture templates and their fusion with templates in \\cite{Cao2018PAMI} correctly retrieves their true mates at rank 1. }\n\t\\label{fig:failure}\n\t\\vspace*{-10pt}\n\\end{figure}\n\nIn order to determine if our new texture templates can boost the performance over the results in \\cite{Cao2018PAMI} we fuse the proposed three texture templates with the three templates used in \\cite{Cao2018PAMI}. Fig. \\ref{fig:CMC} compares the Cumulative Match Characteristic (CMC) curves of the fusion scheme on all 258 latents in NIST SD27 as well as subsets of latents of three different quality levels (good, bad and ugly). Plots in Fig. \\ref{fig:CMC} show the proposed three texture templates when fused with the three templates in \\cite{Cao2018PAMI} can boost the overall performance by 2.7\\% at rank-1 (from 75.6\\% to 78.3\\%). In particular, the fusion of six templates (three proposed + three from \\cite{Cao2018PAMI}) improves the rank-1 accuracy by 4.7\\% on the subset of ugly latents, some of the most challenging latents with an average of only 5 minutiae per latent. Fig. \\ref{fig:failure} shows two ugly latents whose true mates were not retrieved at rank-1 by the method in \\cite{Cao2018PAMI}, but are now correctly retrieved at rank-1 with the introduction of three new texture templates.\n\n\\subsection{Computation time}\nThe texture template matching algorithm was implemented in tensorflow and python and executed on a desktop with i7-6700K CPU@4.00GHz, GTX 1080 Ti (GPU), 32 GB RAM and Linux operating system. The average computation time for comparing a latent texture template to a rolled texture template is 7.7ms (single thread) compared to 11.0 ms (24 threads) in \\cite{Cao2018PAMI}. The average times for extracting one proposed latent texture template and one proposed rolled texture template are 0.7s and 1.5s (GPU), respectively; when fused with the three templates in \\cite{Cao2018PAMI} are 1.2s and 2.2s (24 threads).\n\n\\section{Summary and future work}\n\nTexture template is critical to improve the search accuracy of latent fingerprints, especially for latents with small friction ridge area and large background noise. We have proposed a set of three texture templates, defined as a set of virtual minutiae along with their descriptors. Different virtual minutiae descriptors lead to different texture templates. The contributions of this paper are as follows. i) Use patches from original fingerprints and enhanced fingerprints to improve the distinctiveness of virtual minutiae descriptors, ii) three different texture templates, and iii) a modified second-order graph matching. \nIdentification results on NIST SD27 latent database demonstrate that the proposed texture templates when used alone can improve the rank-1 accuracy by 8.9\\% (from 59.3\\% in \\cite{Cao2018PAMI} to 68.2\\%). Our ongoing research includes i) improving ridge flow estimation, ii) using latent-rolled pairs for learning minutiae descriptors and similarities, and iii) using multicore processors to improve the search speed. \n\\section*{Acknowledgement}\nThis research is based upon work supported in part by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via IARPA R\\&D Contract No. 2018-18012900001. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright annotation therein.\n\n{\\footnotesize\n\\bibliographystyle{ieee}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\\subsection{Small sumsets in LCA groups} Given subsets $A$ and $B$ of an abelian group $G$, we consider their \\textit{sumset} $A+B:=\\{a+b: a\\in A, b\\in B\\}$. A general theme in additive combinatorics is that $A$ and $B$ must be highly structured when $A+B$ is not very large in comparison to $A$ and $B$; see \\cite{GrynkiewiczKemperman}, \\cite{Nathansoninverse}, or \\cite{TaoVu} for many instances of this theme. We consider the case where $G$ is a locally compact abelian (LCA) group, extending the investigations of \\cite{Kneser56}. Theorem 1 of \\cite{Kneser56} describes the pairs $(A,B)$ of Haar measurable subsets of an LCA group $G$ with Haar measure $m$ satisfying $m_*(A+B)0$ and $m(B)>0$. We call such a pair $(A,B)$ a \\textit{sur-critical} pair. Here $\\mathbb T$ is the group $\\mathbb R\/\\mathbb Z$; see \\S \\ref{intervalsdef} for our usage of the term ``interval.\"\n\n\n\n\\begin{proposition}[\\cite{Kneser56}, Theorem 3]\\label{Kconnected} Let $G$ be a compact connected abelian group with Haar measure $m$, and $A$, $B\\subseteq G$ measurable sets satisfying $m(A)>0$, $m(B)>0$, and $m_*(A+B)=m(A)+m(B)<1$. Then there is a continuous surjective homomorphism $\\chi:G\\to \\mathbb T$, and there are intervals $I$, $J\\subseteq \\mathbb T$ such that $A\\subseteq \\chi^{-1}(I)$, $B\\subseteq \\chi^{-1}(J)$, $m(A)=m(\\chi^{-1}(I))$, and $m(B)=m(\\chi^{-1}(J))$.\n\\end{proposition}\n\n\nKemperman \\cite{KempermanTopological} generalized part of Proposition \\ref{Ktopinequality} to the case where $G$ is not abelian, showing in particular that $m_*(A\\cdot B)\\geq m(A)+m(B)$ for subsets of a general locally compact connected group (here $A\\cdot B$ denotes $\\{ab:a\\in A, b\\in B\\}$). Bilu \\cite{Bilu} investigates the inequality $m(A+B)0$, $m(B)>0$, and $m_*(A+B)=m(A)+m(B)$, generalizing Proposition \\ref{Kconnected} to the case where $G$ may be disconnected. Our main result is Theorem \\ref{main}, where $G$ is assumed to be compact. See \\S \\ref{terminology} for notation and terminology. In \\S \\ref{lc} we discuss the case where $G$ is not compact.\n\n\\begin{theorem}\\label{main} Let $G$ be a compact abelian group with Haar measure $m$, and let $A$, $B\\subseteq G$ be measurable sets such that $m(A)>0$, $m(B)>0$, and $m_*(A+B)=m(A)+m(B)$. Then at least one of the following is true\\textup:\n\\begin{enumerate}\n\\item[(P)] There is a compact open subgroup $K\\leqslant G$ with $A+K\\sim A$ and $B+K\\sim B$.\n\n\\smallskip\n\n\\item[(E)] There are measurable sets $A'\\supseteq A$ and $B'\\supseteq B$ such that\n\\begin{align*}\nm(A')+m(B')>m(A)+m(B),\n\\end{align*} and $m_*(A'+B')=m_*(A+B)$.\n\n\\smallskip\n\n\\item[(K)] There is a compact open subgroup $K\\leqslant G$, a continuous surjective homomorphism $\\chi:K\\to \\mathbb T$, intervals $I$, $J\\subseteq \\mathbb T$, and elements $a$, $b\\in G$ such that $A\\subseteq a+\\chi^{-1}(I)$, $B\\subseteq b+\\chi^{-1}(J)$, $m(A)=m(\\chi^{-1}(I))$, and $m(B)=m(\\chi^{-1}(J))$.\n\n\\smallskip\n\n\\item[(QP)] There is a compact open subgroup $K\\leqslant G$ and partitions $A=A_1\\cup A_0$, $B=B_1\\cup B_0$ such that $A_0\\neq \\varnothing$, $B_0\\neq \\varnothing$, at least one of $A_1\\neq \\varnothing$, $B_1\\neq \\varnothing$, and\n\n\\begin{enumerate}\n\\item[(QP.1)]\n$A_1+K\\sim A_1$, $B_1+K\\sim B_1$, while $A_0$ and $B_0$ are each contained in a coset of $K$ and $(A_1+K)\\cap A_0=(B_1+`K)\\cap B_0=\\varnothing$\\textup{;}\n\n\\smallskip\n\n\\item[(QP.2)] $A_0+B_0+K$ is a unique expression element of $A+B+K$ in $G\/K$\\textup{;}\n\n\\smallskip\n\n\\item[(QP.3)] $m_*(A_0+B_0)=m(A_0)+m(B_0)$.\n\\end{enumerate}\n\\end{enumerate}\n\\end{theorem}\n\\begin{remarks} (i) Corollary \\ref{QPfour} provides more detail in conclusion (QP); conclusion (E) is examined in \\S \\ref{Eremark}. Corollary \\ref{roman} is a convenient rephrasing of Theorem \\ref{main}.\n\n\\smallskip\n\n\\noindent (ii) We do not provide explicit constructions of all pairs satisfying conclusion (QP), so the statement of Theorem \\ref{main} is not a complete characterization of the sur-critical pairs for a compact abelian group.\n\n\\smallskip\n\n\\noindent (iii) The labels (P), (E), (K), and (QP) stand for ``periodic,\" ``extendible,\" ``Kneser,\" and ``quasi-periodic,\" respectively. See \\S \\ref{setsandpairs} for elaboration.\n\n\\smallskip\n\n\\noindent (iv) Theorem \\ref{main} was developed partly to help answer Question 4.1 of \\cite{Jin10}. \\hfill $\\blacksquare$ \\end{remarks}\n\n\n\n\nTheorem \\ref{main} is proved in \\S \\ref{proof}. A sequence of lemmas in \\S \\ref{lemmas} reduces the proof to a special case. We then apply the $e$-transform (\\S \\ref{etsection}) as in the proof of Proposition \\ref{Kconnected} from \\cite{Kneser56}. Lemma \\ref{transformcase1} handles difficulties related to applying the $e$-transform in a disconnected group; its statement and proof are inspired by arguments from \\cite{Grynkiewicz}.\n\n\\smallskip\n\nSome easy consequences of Theorem \\ref{main} are discussed in \\S \\ref{corollarysection}. Examples are given in \\S \\ref{examples} showing that each conclusion in Theorem \\ref{main} may occur, and none can be omitted. \\S \\ref{corollarysection} and \\S \\ref{examples} are technically irrelevant to the proof of Theorem \\ref{main}.\n\n\n\n\\subsection{When one of $m(A)=0$ or $m(B)=0$.} When $m(B)=0$, $00$. Consequently, if $\\chi:G\\to \\mathbb T$ is a continuous surjective homomorphism and $t\\notin \\chi^{-1}(I)$, then $m_G((t+\\tilde{J})\\setminus (\\tilde{I}+\\tilde{J}))>0$, where $\\tilde{I}=\\chi^{-1}(I)$ and $\\tilde{J}=\\chi^{-1}(J)$.\n\n\\subsubsection{Subgroups and quotients} If $K\\leqslant G$ is a closed subgroup, the quotient $G\/K$ with the quotient topology is a locally compact abelian group. We may identify subsets of $G$ of the form $A+K$ with subsets of $G\/K$, and conversely subsets of $G\/K$ may be identified with subsets of $G$.\n\n\\smallskip\n\nA \\textit{$K$-coset decomposition} of a set $A\\subseteq G$ is the collection of sets $A_i=A\\cap K_i$, where $K_i$ ranges over the cosets of $K$ having nonempty intersection with $A$.\n\n\\smallskip\n\nWe use without comment the following well-known facts:\n\n\n\\begin{itemize}\n\\item If $K\\leqslant G$ is a measurable subgroup of a compact group, then $m(K)>0$ if and only if $K$ is open, if and only if $K$ has finite index in $G$. The index of $K$ in $G$ is $1\/m(K)$.\n\n\n\\item If $K\\leqslant G$ is an open subgroup, the group $G\/K$ is discrete.\n\\end{itemize}\n\n\\subsubsection{Disintegration of Haar measure} This subsection regards technicalities occurring only in \\S \\ref{trivializing}. For simplicity we assume $G$ is compact. Given a closed subgroup $K\\leqslant G$, we consider the Haar measure $m_K$ as a measure on $G$. If $A\\subseteq G$, and $x\\in G$, we consider $m_K(A-x)$, which may be regarded as the $m_K$-measure of $A$ in the coset $x+K$ of $K$. Although the function $g(x):= m_K(A-x)$ depends only on the coset $K+x$, and may therefore be regarded as a function whose domain is (a subset of) $G\/K$, we prefer to regard $g$ as a function whose domain is (a subset of) $G$. The measure $m_G(A)$ can be recovered in a natural way from the measures $m_K(A-x)$; in other words, Haar measure can be disintegrated over the cosets of a closed subgroup. This is the content of the following proposition; cf.~\\S 2 of \\cite{Kneser56}. It may be obtained by specializing Theorem 3.4.6 of \\cite{ReiterStegeman} to the case where $G$ is compact and abelian.\n\n\\begin{proposition}\\label{Weil} Let $K\\leqslant G$ be a closed subgroup of $G$ with Haar measure $m_K$, and let $f\\in L^1(m)$ be a real-valued function. Then the function $\\tilde f: G\\to \\mathbb R$ given by $\\tilde{f}(x)= \\int f(x+t)\\, dm_K(t)$ is defined for $m$-almost every $x$, $\\tilde{f}$ is $m_G$-measurable, and $\\int \\tilde f\\, dm=\\int f\\, dm$. In particular, if $A\\subseteq G$ is measurable, then the function $x\\mapsto m_K(A-x)$ is measurable, and $\\int m_K(A-x)\\, dm(x) = m(A)$. \\hfill $\\blacksquare$\n\\end{proposition}\n\n\n\\subsubsection{Unique expression elements}\\label{uees} We say $c\\in A+B$ is a \\textit{unique expression element of} $A+B$ if $c=a_0+b_0$ for some $a_0\\in A$, $b_0\\in B$, and $a+b=c$ implies $a=a_0$ and $ b=b_0$ when $a\\in A$ and $b\\in B$. Unique expression elements play an important role in \\cite{Kemperman60}, the classification of pairs $(A,B)$ satisfying $|A+B|=|A|+|B|-1$; Corollary \\ref{QPfour} connects that classification to conclusion (QP) of Theorem \\ref{main}. If $K\\leqslant G$ is a subgroup, the phrase ``$C+K$ is a unique expression element of $A+B+K$ in $G\/K$\" means that $C+K= a_0+b_0+K$ for some $a_0\\in A$, $b_0\\in B$, and whenever $a\\in A$, $b\\in B$, and $a+b+K=C+K$, then $a+K=a_0+K$ and $b+K=b_0+K$.\n\n\n\n\n\\subsubsection{Critical and sur-critical pairs}\\label{critdef} If $A$, $B\\subseteq G$ are measurable sets satisfying $m_*(A+B)=m(A)+m(B)$, we call $(A,B)$ a \\textit{sur-critical pair} for $G$. We call a pair $(A,B)$ satisfying $m_*(A+B)0$. We will exploit the following relation between the essential stabilizer and periodicity. This observation is a consequence of Lemma \\ref{H(S)}, although it may be obtained by more elementary means.\n\n\\begin{observation} If $m(H(C))>0$, then for all cosets $H_i$ of $H(C)$, either $m(C\\cap H_i)=0$ or $m(C\\cap H_i)=m(H_i)$. Consequently, if $m(H(C))>0$, there is a measurable subset $C'\\subseteq C$ such that $C\\sim C'\\sim C'+H(C)$. \\hfill $\\blacksquare$\n\\end{observation}\n\n\\begin{remark} When $G$ is discrete, every subset of $G$ is periodic according to our definition. This differs from the terminology of \\cite{Grynkiewicz}, where a periodic set $S$ must satisfy $S+t \\sim S$ for some $t\\neq 0$. \\hfill $\\blacksquare$\n\\end{remark}\n\n\n\\subsubsection{Extendibility and nonextendibility} Let $A$ and $B$ be measurable subsets of a compact abelian group $G$. We say that $A$ is \\textit{extendible with respect to} $B$ if there is a measurable set $A'\\supseteq A$ with $m(A')>m(A)$ and $m_*(A'+B)=m_*(A+B)$. We say that the pair $(A,B)$ is \\textit{extendible} if $A$ is extendible with respect to $B$ or $B$ is extendible with respect to $A$. Otherwise, we say that $(A,B)$ is \\textit{nonextendible}. The nonextendibility of a pair $(A,B)$ may be expressed as follows: if $A'\\supseteq A$ and $B'\\supseteq B$ are measurable and $m_*(A'+B')=m_*(A+B)$, then $A'\\sim A$ and $B'\\sim B$.\n\n\\smallskip\n\n When $m_*(A+B)=m(A)+m(B)$ and $(A,B)$ is extendible, Proposition \\ref{Ktopinequality} implies $H(A+B)$ is compact and open, and $A+B\\sim A+B+H(A+B)$. Consequently, $A+B$ is measurable when $(A,B)$ is extendible. The following example lemma illustrates how nonextendibility will be exploited in subsequent proofs.\n\n \\begin{lemma} If $K\\leqslant G$ is a compact open subgroup, $A+B\\sim A+B+K$, and $(A,B)$ is nonextendible, then $A+K\\sim A$.\n \\end{lemma}\n\n\\begin{proof} The similarity $A+B\\sim A+B+K$ can be rewritten as $(A+K)+B\\sim A+B$. Since $A+K\\supseteq A$ and $(A,B)$ is nonextendible, we have $A+K\\sim A$. \\end{proof}\n\nIn subsequent proofs, such as those of Lemmas \\ref{reduciblelemma1}, \\ref{reduciblelemma2}, \\ref{QP3}, and \\ref{transformcase1}, we will omit the above reasoning.\n\n\\subsubsection{Complementary pairs}\\label{complementarydef} If $G$ is compact and $m(A+B)=m(A)+m(B)=1$, call $(A,B)$ a \\textit{complementary} pair. When $G$ is infinite, it is easy to construct such pairs $(A,B)$ with $A+B\\neq G$: let $A\\subseteq G$ be any measurable set meeting every coset of every finite index subgroup of $G$ with $00$ and $m(B)>0$ satisfying $A+B\\neq G$ can be described as follows. If $(A,B)$ is a complementary pair, and $A+B\\neq G$, then $A\\cap (t-B)=\\varnothing$ for some $t\\in G$, so $t-B\\sim A^c$. If $s\\notin H(t-B)$ ($=H(A^c)=H(A)$), then $m(A\\cap (t+s-B))>0$, so $t+s\\in A+B$. It follows that $A+B$ contains a translate of $G\\setminus H(A)$.\n\n\\smallskip\n\nIf $K\\leqslant G$ is a compact open subgroup, $A$ and $B$ are each contained in a coset of $K$, and $m(A+B)=m(A)+m(B)=m(K)$, we say that $(A,B)$ is \\textit{complementary with respect to} $K$. Such pairs form a subclass of the extendible pairs.\n\n\n\\subsubsection{Reducibility}\\label{reducibledef} If there are measurable subsets $A'\\subseteq A$ and $B'\\subseteq B$ such that $m(A')=m(A)$, $m(B')=m(B)$, and $m_*(A'+B')0$, $m(B)>0$, and $m(A+B)=m(A)+m(B)$. Corollary \\ref{measurableremark} below shows that restricting the measures of $A$ and $B$ in Theorem \\ref{main} can guarantee that $(A,B)$ is essentially regular. For this we need the following lemma.\n\n\\begin{lemma}\\label{interior} If for all $\\varepsilon>0$, $(A,B)$ has a quasi-periodic decomposition \\textup{(\\S \\ref{qpdef})} with respect to a compact open subgroup $K$ having $00$. For a given quasi-period $K$ of $(A,B)$ having $00$ and $m(B_0)>0$, (QP.3) guarantees that Theorem \\ref{main} applies to the pair $(A_0,B_0)$. This observation yields the following corollary.\n\n\\begin{corollary}\\label{roman} With the hypotheses and notation of Theorem \\textup{\\ref{main}}, at least one of the following is true\\textup{:}\n\\begin{enumerate}\n\\item[\\textup{(I)}] One of \\textup{(P)}, \\textup{(E)}, or \\textup{(K)} holds.\n\\item[\\textup{(II)}] Conclusion \\textup{(QP)} holds, and $(A_0,B_0)$ satisfies \\textup{(K)}.\n\\item[\\textup{(III)}] For all $\\varepsilon>0$, there is a compact open subgroup $H\\leqslant G$ having $m(H)<\\varepsilon$ and $(A,B)$ satisfies \\textup{(QP)} with $K=H$.\n\\item[\\textup{(IV)}] $(A,B)$ satisfies \\textup{(QP)} with one at least one of $m(A_0)=0$ or $m(B_0)=0$.\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{proof} If $(A,B)$ satisfies (P), (E), or (K), we have (I). If not, inductively form a sequence of pairs $(A^{(n)}, B^{(n)})$ and subgroups $K^{(n)}\\leqslant G$ as follows: let $(A^{(0)}, B^{(0)})=(A,B)$, and let $K^{(0)}=G$. Suppose $(A^{(j)},B^{(j)})$ and $K^{(j)}$ are defined for $j=0,\\dots,n$ and for each $j=1,\\dots, n$,\n\\begin{enumerate}\n\\item[(1)]\n$A=A'_1\\cup A^{(j)}$, $B=B'_1\\cup B^{(j)}$ is a quasi-periodic decomposition of $(A,B)$ with respect to $K^{(j)}$ satisfying (QP) of Theorem \\ref{main}, so that\n\\item[(2)] $A^{(j)}$ and $B^{(j)}$ are each contained in cosets of $K^{(j)}$,\n\\item[(3)] $m_*(A^{(j)}+B^{(j)})=m(A^{(j)})+m(B^{(j)})$, and\n\\item[(4)] $K^{(j)}$ is a proper subgroup of $K^{(j-1)}$.\n\\end{enumerate}\nWe will construct $(A^{(n+1)},B^{(n+1)})$ and $K^{(n+1)}\\leqslant K^{(n)}$ satisfying (1)-(4), or show that one of (I), (II), or (IV) holds.\n\n\\smallskip\n\nIf one or both of $m(A^{(n)})=0$ or $m(B^{(n)})=0$, we have conclusion (IV). Otherwise, apply Theorem \\ref{main} to $(A^{(n)},B^{(n)})$. If $(A^{(n)},B^{(n)})$ satisfies (P) or (E), then so does $(A,B)$, and we have (I). If ($A^{(n)}, B^{(n)})$ satisfies (K), we have (II). Otherwise, $(A^{(n)},B^{(n)})$ satisfies (QP), so take $K^{(n+1)}$ to be corresponding subgroup $K$, and let $A^{(n)}= A^{(n)}_1\\cup A^{(n)}_0$, $B^{(n)}=B^{(n)}_1\\cup B^{(n)}_0$ be the corresponding decompositions. Observe that $K^{(n+1)}$ must be a proper subgroup of $K^{(n)}$, so that $m(K^{(n+1)})\\leq \\frac{1}{2}m(K^{(n)})$. Now take $A^{(n+1)}=A^{(n)}_0$ and $B^{(n+1)}=B^{(n)}_0$, so that $(A^{(n+1)},B^{(n+1)})$ satisfies (1)-(4) with $j=n+1$ (cf.~\\S \\ref{repeateddef}).\n\n\\smallskip\n\nIf the above construction terminates, we conclude (I), (II), or (IV). Otherwise, we have for each $n$ quasi-periodic decompositions with respect to $K^{(n)}$, and we conclude (III).\n\\end{proof}\n\n\n\\begin{corollary}\\label{measurableremark} Suppose $(A,B)$ satisfies the hypotheses of Theorem \\textup{\\ref{main}}.\n\\begin{enumerate}\n\\item[\\textup{(a)}] If $A+B$ is not measurable, then $(A,B)$ satisfies \\textup{(QP)} with at least one of $m(A_0)=0$ or $m(B_0)=0$.\n\n\\smallskip\n\n\\item[\\textup{(b)}] If $(A,B)$ is not essentially regular, then $(A,B)$ satisfies conclusion \\textup{(E)} of Theorem \\textup{\\ref{main}}, or $(A,B)$ satisfies conclusion \\textup{(QP)} of Theorem \\textup{\\ref{main}} with one or both of $m(A_0)=0$ or $m(B_0)=0$.\n\n\\smallskip\n\n\\item[\\textup{(c)}] If $m(A)$, $m(B)$, and $m(A+B)$ are all irrational numbers, then $(A,B)$ is essentially regular.\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{proof} (a) In conclusions (I) and (II) of Corollary \\ref{roman}, it is routine to check that $A+B$ is measurable. If (III) holds in Corollary \\ref{roman}, Lemma \\ref{interior} implies $A+B$ is measurable.\n\n\\smallskip\n\n\\noindent (b) If (P) or (K) holds, then $(A,B)$ is essentially regular. If (II) or (III) of Corollary \\ref{roman} holds, then $(A,B)$ is essentially regular. The only remaining alternatives are (E) of Theorem \\ref{main} or (IV) of Corollary \\ref{roman}.\n\n\\smallskip\n\n\\noindent (c) If $m(A+B)$ is irrational, then neither conclusion (P) nor (E) can hold in Theorem \\ref{main}. If $m(A)$ and $m(B)$ are irrational, then (QP) cannot hold with one of $m(A_0)=0$ or $m(B_0)=0$. Now $(A,B)$ is essentially regular, by Part (b) of the present corollary. \\end{proof}\n\n\n\\subsection{Further description of extendible pairs}\\label{Eremark} If $(A,B)$ is a sur-critical pair satisfying conclusion (E) of Theorem \\ref{main}, Proposition \\ref{Ktopinequality} implies the group $H:=H(A+B)$ is compact and open and $A+B\\sim A+B+H$. Furthermore, exactly one of the following holds:\n\\begin{enumerate}\n\\item[(E.1)] $A+B=A+B+H$.\n\n\\smallskip\n\n\\item[(E.2)] $(A,B)$ is complementary with respect to $H$ and $A+B\\neq A+B+H$, or $(A,B)$ satisfies conclusion (QP) of Theorem \\ref{main}, with $K=H$, $m(A_0)+m(B_0)=m(H)$, and $A_0+B_0\\neq A_0+B_0+H.$\n\\end{enumerate}\nWhen $m(A_0)>0$ and $m(B_0)>0$ in (E.2), the pair $(A_0,B_0)$ is complementary with respect to the subgroup $H$; see \\S \\ref{complementarydef} for further description.\n\nTo obtain this classification, fix $A'\\supseteq A$ and $B'\\supseteq B$ such that $m(A')+m(B')>m(A)+m(B)$ and $m(A'+B')=m(A+B)$, and write $H$ for $H(A'+B')$. By Proposition \\ref{Ktopinequality},\n\\begin{align*}\nm(A'+B')=m(A'+H)+m(B'+H)-m(H),\n\\end{align*}\nso $m(A)+m(B)=m(A+B)=m(A'+H)+m(B'+H)-m(H)$. Rearranging, we get\n\\begin{align}\\label{holes}\nm(A'+H)-m(A)+m(B'+H)-m(B)=m(H).\n\\end{align}\nLet $A=\\bigcup_{i=1}^n A_i$ and $B=\\bigcup_{j=1}^m B_j$ be $H$-coset decompositions of $A$ and $B$. Since $A\\subseteq A'+H$ and $B\\subseteq B'+H$, (\\ref{holes}) implies\n\\begin{align}\\label{surholes}\nm(A_i)+m(B_j)\\geq m(H)\n\\end{align}\n for each $i$ and $j$ (otherwise the left-hand side of (\\ref{holes}) would be larger than $m(H)$). If the inequality (\\ref{surholes}) is strict for each pair $i$ and $j$, we have (E.1). Otherwise, take $i$ and $j$ so that equality holds in (\\ref{holes}) and set $A_0=A_i$, $B_0=B_j$. If $A=A_0$ and $B=B_0$, then $(A,B)$ is complementary with respect to $H$. If not, we set $A_1=A\\setminus A_0$ and $B_1=B\\setminus B_0$, we verify (QP.1)-(QP.3) in Theorem \\ref{main}. Equation (\\ref{holes}) implies $A_1\\sim A_1+H$ and $B_1\\sim B_1+H$, so (QP.1) holds. If $A+B\\neq A+B+H$, we conclude that $A_0+B_0+H$ is a unique expression element of $A+B+H$ in $G\/H$ and we have (QP.2). (QP.3) now follows from $A+B\\sim A+B+H$.\n\n\\section{Examples}\\label{examples} We list some examples of sur-critical pairs to show that each alternative in Theorem \\ref{main} can occur, and that none of the alternatives can be omitted. We do not attempt to exhaustively construct all possible sur-critical pairs.\n\n\n\\subsection{Periodic pairs}\\label{periodicexamples} If $F$ is a finite group, every sur-critical pair for $F$ satisfies (P) and not (K), and every nonextendible sur-critical pair satisfies (P) but not (E). A specific example with $F=\\mathbb Z\/11\\mathbb Z$ is $A=\\{0,1\\}$, $B=\\{0,3\\}$, so that $A+B=\\{0,1,3,4\\}$, and $(A,B)$ satisfies (P) but not (E) or (K). Every periodic sur-critical pair $(A,B)$ for a compact group $G$ has the form $A=\\phi^{-1}(C)\\setminus N$, $B=\\phi^{-1}(D)\\setminus N$, where $(C,D)$ is a sur-critical pair for a finite group $F$, $\\phi:G\\to F$ is a continuous surjective homomorphism, and $m(N)=0$, so the periodic sur-critical pairs for an arbitrary compact group $G$ are classified in \\cite{Grynkiewicz}.\n\n\\subsection{Extendible pairs} The following example satisfies (E), but not (P), (K), or (QP). Let $G=(\\mathbb Z\/ 17\\mathbb Z)\\times \\mathbb T$ and let $A=\\{1,3,5,7\\}\\times [0,0.8]$, $B=\\{0,2\\}\\times [0,0.9]$, so that $A+B=\\{1,3,5,7,9\\}\\times \\mathbb T$. Then $m(A)=3.2\/17$, $m(B)=1.8\/17$, and $m(A+B)=5\/17$. Also $A+B=A'+B'$, where $A'=\\{1,3,5,7\\}\\times \\mathbb T$ and $B'=\\{0,2\\} \\times \\mathbb T$.\n\n\n\\smallskip\n\nTo find examples satisfying (E.1) (\\S \\ref{Eremark}), fix a pair of sets $A'$, $B'\\subseteq G$ satisfying $m(A'+B')m(H)$ for all $a\\in A'$ and $b\\in B'$. Then $A+B=A'+B'$, so $m(A+B)=m(A)+m(B)$.\n\n\\smallskip\n\nTo find examples satisfying (E.2) but not (E.1), recall the construction in \\S \\ref{remark2}. Let $(A,B)$ satisfy conclusion (QP) of Theorem \\ref{main} with $m(A_0)+m(B_0)=m(K)$, but $A_0+B_0\\neq A_0+B_0+K$. To form a specific example of this construction with $G=(\\mathbb Z\/ 15\\mathbb Z)\\times \\mathbb T$, let $A=(\\{1,3,5\\}\\times \\mathbb T)\\cup (\\{7\\}\\times S)$, where $S\\subseteq \\mathbb T$ is any measurable set, and let $B=(\\{0,2\\}\\times \\mathbb T)\\cup (\\{4\\}\\times (-S^c))$. This example satisfies (E) and (QP) but not (P) or (K).\n\n\\subsection{Pairs arising from $\\mathbb T$} Let $G$ be a compact group which is not totally disconnected. Let $(A,B)$ have the form $A=\\chi^{-1}(I)$, $B=\\chi^{-1}(J)$, where $H\\leqslant G$ is a compact open subgroup, $\\chi:H\\to \\mathbb T$ is a continuous surjective homomorphism, $I,J\\subseteq \\mathbb T$ are intervals, and $m_{\\mathbb T}(I)+m_{\\mathbb T}(J)<1$. One can verify that $(A,B)$ is a sur-critical pair satisfying (K) but not (P), (E), or (QP).\n\n\\subsection{A quasi-periodic pair} This example will satisfy (QP) but not (P), (E), or (K).\n\n\\smallskip\n\nLet $G=\\mathbb Z_7$, the $7$-adic integers with the usual topology, and consider $\\mathbb Z$ as a subset of $\\mathbb Z_7$ in the usual way. Define the set $C\\subseteq \\mathbb Z$ by\n\\begin{align*}\nC:=(\\{0,1\\}+7\\mathbb Z)\\cup\\Bigl(2+7((\\{0,1\\}+7\\mathbb Z)\\cup (2+7(\\cdots)))\\Bigr),\n\\end{align*}\nso that $C=(\\{0,1\\}+7\\mathbb Z)\\cup (2+7C)$. Let $A$ be the closure of $C$ in $\\mathbb Z_7$, and let $B=A$. Then $\nm(A)=m(B)=\\sum_{n=1}^\\infty \\frac{2}{7^n}=1\/3$. Note that $A$ has a quasi-periodic decomposition $A_1\\cup A_0$ where $A_1=\\overline{\\{0,1\\}+7\\mathbb Z}$, $A_0=A\\setminus A_1$. The sumset $A+B$ is the closure of $C+C$ in $\\mathbb Z_7$. Note that\n \\begin{align}\\label{C+C}\nC+C=(\\{0,1,2,3\\}+7\\mathbb Z)\\cup\\Bigl(4+7((\\{0,1,2,3\\}+7\\mathbb Z)\\cup (4+7(\\cdots)))\\Bigr),\n \\end{align}\nso $m(A+B)=2\/3=m(A)+m(B)$. From (\\ref{C+C}) one can check that $H(A+B)=\\{0\\}$, so (P) and (E) fail. Since $G$ is totally disconnected, (K) cannot hold.\n\n\n\\subsection{More quasi-periodic pairs.} Let $G'$ be an infinite compact abelian group with Haar measure $m'$, and let $G=(\\mathbb Z\/4\\mathbb Z)\\times G'$. Fix sets $A_0'$, $B_0'\\subseteq G'$ satisfying $m_{*}'(A_0'+B_0')=m'(A_0')+m_0(B_0')$, let $A_1=\\{0\\}\\times G'$, $B_1=\\{0\\}\\times G'$, and let $A_0=\\{1\\}\\times A_0'$, $B_0=\\{1\\}\\times B_0'$. Let $A=A_1\\cup A_0$, $B=B_1\\cup B_0$. Then $m_*(A+B)=m(A)+m(B)$, but depending on our choice of $A_0$ and $B_0$, $A+B$ may not be measurable. If $A_0'$ is a singleton, then $B_0'$ may be an arbitrary measurable subset of $G'$.\n\n\n\n\n\\section{Lemmas}\\label{lemmas}\n\n\n\nWe fix a compact abelian group $G$ with Haar measure $m$. Unless stated otherwise, sets $A$ and $B$ are assumed to be subsets of $G$.\n\n\n\n\\subsection{Consequences of Proposition \\ref{Ktopinequality}}\n\nWe need Lemma \\ref{overspill} and Corollary \\ref{consequence} in many subsequent proofs. See \\S \\ref{Hdef} for notation.\n\n\\begin{lemma}\\label{overspill}\nIf $m_*(A+B)m(H).\n\\end{align}\n\\end{lemma}\n\n\\begin{proof} If inequality (\\ref{organize}) fails, then $m(A+H)+m(B+H)\\geq m(A)+m(B)+m(H)$, so inequality (\\ref{critical}) in Proposition \\ref{Ktopinequality} implies $m_*(A+B)\\geq m(A)+m(B)$, contradicting the hypothesis.\n\\end{proof}\n\nIn the next corollary we use the elementary fact that if $G$ is compact, and $C$, $D\\subseteq G$ have $m(C)+m(D)>1$, then $C+D=G$. This fact follows from the observation that $t\\in C+D$ if and only if $(t-C)\\cap D\\neq \\varnothing$, while $t-C$ and $D$ cannot be disjoint if $m(C)+m(D)>m(G)$. Consequently, if $H\\leqslant G$ is a compact open subgroup and $C,$ $D\\subset G$ are each contained in a coset of $H$ while $m(C)+m(D)>m(H)$, then $C+D=C+D+H$.\n\n\\begin{corollary}\\label{differencetosum} If $m_*(A+B)m(H)$ for each $i$ and $j$, so $A_i-B_j= A_i-B_j+H$ for each $i$ and $j$, implying $A-B=A-B+H$.\n\\end{proof}\n\n\\subsection{Reducible pairs} We now dispense with the case of Theorem \\ref{main} where $(A,B)$ is reducible (\\S \\ref{reducibledef}). For the next two lemmas and the following corollary, we assume that $m(A)>0$, $m(B)>0$, and $(A,B)$ satisfies $m_*(A+B)=m(A)+m(B)$.\n\n\\begin{lemma}\\label{reduciblelemma1} If $A'\\subseteq A$ is such that $m(A')=m(A)$ and $m_*(A'+B)0,\n\\end{align}\nwhere $B_0:=(b_0+H)\\cap B$. Fix $a_0\\in A$, $b_0\\in B$, and $B_0$ satisfying (\\ref{Bover2}), and let $B_1=B\\setminus B_0$. Lemma \\ref{overspill} and the hypothesis $m(A')=m(A)$ imply\n\\begin{align}\\label{Bover1}\nm(B_0)+m(A'+B)\\geq m(A')+m(B) =m_*(A+B).\n\\end{align}\n\n\\begin{claim1} $A'+H \\subset_m A$ and $B_1+H\\subset_m B$.\n\\end{claim1}\n\n\\noindent\\textit{Proof of Claim \\textup{1}.} By the definition of $B_0$ and the fact that $A'+B=A'+B+H$, $A'+B$ is disjoint from $a_0+B_0$. Using (\\ref{Bover1}) we write\n\\begin{align}\\label{innermeasure}\nm_*(A+B)=m(A'+B)+m(B_0).\n\\end{align}\nThen\n\\begin{align}\\label{arranging}\\begin{split}m(A')+m(B)&=m_*(A+B)\\\\\n&=m(A'+B)+m(B_0)\\\\\n&=m(A'+H)+m(B+H)-m(H)+m(B_0)\\\\\n&=m(A'+H)+m(B_1+H) +m(B_0),\\end{split}\n\\end{align}\nwhere the second line is (\\ref{innermeasure}), the third line is from Proposition \\ref{Ktopinequality}, and the fourth line is due to $B=B_1\\cup B_0$ being a disjoint union and $B_0$ being contained in a coset of $H$. Subtracting $m(A')+m(B)$ from the first and last lines of (\\ref{arranging}), we find\n\\begin{align}\\label{arranged}\n[m(A'+H)-m(A')] + [m(B_1+H)+m(B_0)-m(B)]=0\n\\end{align}\nEach bracketed summand in (\\ref{arranged}) is nonnegative, so\n\\begin{align}\\label{A'}\nm(A')=m(A'+H)\n\\end{align}\nand\n\\begin{align}\\label{B'}\nm(B)=m(B_1+H)+m(B_0).\n\\end{align}\nSince $(B_1+H) \\cap B_0=\\varnothing$ and $B=B_1\\cup B_0$, (\\ref{A'}) and (\\ref{B'}) prove the claim. \\hfill $\\square$\n\n\\smallskip\n\nWe proceed based on whether $m(B_0)m(A'+B)+m(B_0)$, contradicting (\\ref{innermeasure}). Furthermore, $A'+H$ is the set $\\{z\\in G: z+B\\subset_m A'+B\\}$, by Corollary \\ref{consequence}. Thus, $A_0$ is the set of $a\\in A$ such that $a+B_0\\not\\subseteq A'+B$. Now if $a$, $a'\\in A_0$, then $a+B_0\\sim a'+B_0$, by (\\ref{innermeasure}). Since $B_0$ is contained in a coset of $H$ and $m(B_0)>0$, the similarity $a+B_0\\sim a'+B_0$ implies $a-a'\\in H$. We conclude that $A_0$ is contained in a coset of $H$. Now the containment $A+B_1\\subseteq A'+B$ implies $A_0+B_0+H$ is a unique expression element of $A+B+H$ in $G\/H$, as claimed. \\hfill $\\square$\n\n\\smallskip\n\nWriting $A_1=A\\cap(A'+H)$, and maintaining $A_0$, $B_1$, and $B_0$ as defined above, the partition $A=A_1\\cup A_0$ is a quasi-periodic decomposition with quasi-period $H$, while $B_1\\sim B_1+H$, and $B_0$ is contained in a coset of $H$ (possibly $B_1=\\varnothing$), so we have verified that (QP.1) and (QP.2) hold. To verify (QP.3), consider the partition $A+B= (A_1+B)\\cup (A_0+B_0)$, so that (\\ref{innermeasure}) implies $m_*(A_0+B_0)=m(B_0)$. Since $m(A_0)=0$, we then have $m_*(A_0+B_0)=m(A_0)+m(B_0)$. This concludes the analysis of Case 2.\n\n\\smallskip\n\n Now suppose $(A,B)$ is nonextendible. If Case 2 holds, we have conclusion (i) as desired. If Case 1 holds we derive a contradiction: from $B+H\\sim B$ and the nonextendibility of $(A,B)$ we find $A\\sim A+H$. Since $H$ is compact and open, the similarity $A\\sim A+H$ implies $m(A''+B)=m_*(A+B)$ whenever $A''\\subseteq A$ has $m(A'')=m(A)$, contradicting the hypothesis of the lemma. \\end{proof}\n\n\n\\begin{lemma}\\label{reduciblelemma2}\n If $(A,B)$ is reducible and nonextendible, then $(A,B)$ has a quasi-periodic decomposition $A=A_1\\cup A_0$, $B=B_1\\cup B_0$ satisfying conclusion \\textup{(QP)} of Theorem \\textup{\\ref{main}}, and at least one of $m(A_0)=0$ or $m(B_0)=0$.\n\\end{lemma}\n\n\\begin{proof} Let $A'\\subseteq A$ and $B'\\subseteq B$ be such that $m(A')+m(B')=m(A)+m(B)$ and $m_*(A'+B')0$, as the hypothesis of Case 2 implies $A+H\\neq A'+H$.\n\n\\smallskip\n\nConsider the pair $(A+H,B')$. By (\\ref{simsim}), $A+H+B'\\sim A+B'\\sim A+B$. Then\n\\begin{align}\\label{computation}\n\\begin{aligned}m((A+H)+B') &=m_*(A+B)\\\\\n&=m(A)+m(B) && \\text{(by hypothesis)}\\\\\n&=m(A'+H)+m(B') && \\text{(by (\\ref{simsim}))}\\\\\n&=m(A+H)-t\\cdot m(H)+m(B') && \\text{(by (\\ref{tee}))}\\\\\n&0$.\n\n\\smallskip\n\nWe have shown that $A_0$ is contained in a coset of $H$, so $A_1:=A\\setminus A_0$ gives the desired quasi-periodic decomposition of $A$. Reversing the roles of $A$ and $B$, we find the corresponding quasi-periodic decomposition of $B$: $B_0=B\\setminus (B'+H)$, $B_1=B\\setminus B_0$. We now show that $A_1\\cup A_0$ and $B_1\\cup B_0$ satisfy (QP.2) and (QP.3).\n\n\\smallskip\n\nObserve that $m(A_0)=m(B_0)=0$, by our choice of $A_0$ and $B_0$. Nonextendibility of $(A,B)$ implies $A_0+B_0+H$ is a unique expression element of $A+B+H$ in $G\/H$. To see that $m_*(A_0+B_0)=0$, write $A+B$ as $(A_1+B)\\cup (A_0+B_0)$, so that $m_*(A+B)=m(A_1+B)+m_*(A_0+B_0)$. By the hypothesis of Case 2, we have $m_*(A_1+B)=m_*(A+B)$, so $m_*(A_0+B_0)=0=m(A_0)+m(B_0)$. \\end{proof}\n\n\\begin{corollary}\\label{reduciblecor} If $(A,B)$ is reducible and $m(H(A+B))=0$, then $(A,B)$ satisfies conclusion \\textup{(QP)} of Theorem \\textup{\\ref{main}} with at least one of $m(A_0)=0$ or $m(B_0)=0$.\n\\end{corollary}\n\n\n\\begin{proof} If $m(H(A+B))=0$, then $(A,B)$ is nonextendible, by Proposition \\ref{Ktopinequality}. The conclusion now follows from Lemma \\ref{reduciblelemma2}. \\end{proof}\n\n\n\\subsection{Auxiliary lemmas} The next two lemmas handle technicalities arising repeatedly in the next subsection. We continue to assume $(A,B)$ satisfies the hypotheses of Theorem \\ref{main}.\n\n\n\\begin{lemma}\\label{easy} Suppose $(A,B)$ is irreducible and nonextendible. Assume further that $H:=H(\\tilde{A}+\\tilde{B})$ has positive Haar measure for some $\\tilde{A}\\subseteq A$, $\\tilde{B}\\subseteq B$ such that $\\tilde{A}+\\tilde{B}$ is measurable, $m(\\tilde{A})=m(A)$ and $m(\\tilde{B})=m(B)$. Then $A$ and $B$ are periodic with period $H$, or $(A,B)$ satisfies \\textup{(QP)} of Theorem \\textup{\\ref{main}} with $K=H$.\n\\end{lemma}\n\n\\begin{proof} Let $\\delta=m(H)$. Since $m(H)>0$, we have $m((\\tilde{A}+\\tilde{B})\\cap H_i)\\in \\{0,\\delta\\}$ for every coset $H_i$ of $H$ (cf.~\\S \\ref{periodicdef}). Irreducibility of $(A,B)$ then implies\n\\begin{align}\\label{0delta}\nm_*((A+B)\\cap H_i)\\in\\{0,\\delta\\} \\text{ for every coset } H_i \\text{ of } H.\n\\end{align} Combining (\\ref{0delta}) with the nonextendibility of $(A,B)$, we have $m(A\\cap H_i)\\in \\{0,\\delta\\}$ and $m(B\\cap H_i)\\in \\{0,\\delta\\}$ for every coset $H_i$ of $H$. Now by the irreducibility of $(A,B)$, there are sets $A'\\subseteq A$, $B'\\subseteq B$ such that\n\\begin{align}\\label{AsimsBsims} A\\sim A'+H\\sim A' \\text{ and } B\\sim B'+H\\sim B',\n \\end{align}\n $A'+B'$ is measurable, and $m_*(A+B)=m(A'+B')$. Setting $A_0:=A\\setminus (A'+H)$, we claim $A_0$ is contained in a coset of $H$. To prove this claim, assume $A_0$ is nonempty, let $a\\in A_0$, and consider the pair $(A'\\cup (a+H), B')$. Write $C$ for $A'\\cup (a+H)$, so $m(C)=m(A')+m(H)$. By (\\ref{AsimsBsims}) we have $a+B'+H\\subset_m A+B$, so $C+B'\\subset_m A+B$. The irreducibility of $(A,B)$ then implies\n\\begin{align}\\label{C+B'}\nC+B' \\sim \\tilde{A}+\\tilde{B},\n\\end{align} so that $H(C+B')=H(\\tilde{A}+\\tilde{B})$. We also have\n$$\nm(C+B')=m_*(A+B)=m(A)+m(B)0$ and $m(B_0')>0$. If $(A,B)$ is another irreducible nonextendible sur-critical pair such that $A\\sim A'$ and $B\\sim B'$, then $(A,B)$ also satisfies \\textup{(QP)}.\n\\end{lemma}\n\n\\begin{proof} We may assume $A'\\subseteq A$ and $B'\\subseteq B$, since we can replace $A'$ and $B'$ with $A'\\cap A$ and $B'\\cap B$ while maintaining the hypotheses of the lemma -- this follows from the irreducibility of $(A,B)$ and $(A',B')$. Let $K$ be the quasi-period of $(A',B')$ from conclusion (QP). Assume, without loss of generality, that $A_1'\\neq \\varnothing$. Define $A_0:=A\\cap (A_0'+K)$ and $A_1:= A\\setminus A_0$. We will show that $A_1\\sim A_1+K$.\n\n\\smallskip\n\nLet $a\\in A_1$. We aim to show that $a+K\\subseteq A_1'+K$. Irreducibility of $(A,B)$ implies $a+B'\\subset_m A'+B'$. Since $a\\notin A_0'+K$, we have $a+B'\\subset_m (A_1'+B')\\cup (A+B_1')$, so that $a+B'+K\\subset_m A'+B'$. Then $a+K\\subset_m A'$ by the nonextendibility of $(A',B')$. Consequently, $a+K\\subseteq A_1'+K$, and we find $A_1+K=A_1'+K$. The similarity $A\\sim A'$ then implies $A_1\\sim A_1'$, and $A=A_1\\cup A_0$ is a quasi-periodic decomposition. Reversing the roles of $A$ and $B$, we define $B_0=B\\cap (B_0'+K)$ and $B_1=B\\setminus B_0$. Now it is routine to verify that $(A,B)$ also satisfies (QP) of Theorem \\ref{main}. \\end{proof}\n\n\\subsection{Measurability and trivializing $H(A+B)$.}\\label{trivializing}\n\n\\begin{lemma}\\label{measurability}\nIf Theorem \\textup{\\ref{main}} holds under the additional assumption that $A+B$ is measurable, then Theorem \\textup{\\ref{main}} holds in general.\n\\end{lemma}\n\n\\begin{proof} We first list cases where we obtain the conclusion of Theorem \\ref{main} without assuming $A+B$ is measurable. Excluding those cases, we then show how to deduce the conclusion of Theorem \\ref{main} from the special case where $A+B$ is assumed to be measurable.\n\n\\smallskip\n\nIf $(A,B)$ is extendible, then $(A,B)$ satisfies conclusion (E) of Theorem \\ref{main}. If $(A,B)$ is nonextendible and reducible, then $(A,B)$ satisfies conclusion (QP) of Theorem \\ref{main}, by Lemma \\ref{reduciblelemma2}. We may therefore assume that $(A,B)$ is nonextendible and irreducible. Let $A'\\subseteq A$ and $B'\\subseteq B$ be countable unions of compact sets with $m(A')=m(A)$ and $m(B')=m(B)$, so that $A'+B'$ is measurable.\n\n\\smallskip\n\nIrreducibility of $(A,B)$ implies $m(A'+B')=m_*(A+B)$, so $(A',B')$ satisfies the hypotheses of Theorem \\ref{main}. If $(A',B')$ satisfies conclusion (P) or conclusion (E) of Theorem \\ref{main}, then $m(H(A'+B'))>0$ by Proposition \\ref{Ktopinequality}. We apply Lemma \\ref{easy} and conclude that $(A,B)$ satisfies one of conclusion (P) or (QP). If $(A',B')$ satisfies (K), so that $A'\\sim a+\\chi^{-1}(I)$ and $B'\\sim b+\\chi^{-1}(J)$, where $I$, $J\\subseteq \\mathbb T$ are intervals and $\\chi:K\\to \\mathbb T$ is a continuous surjective homomorphism from a compact open subgroup $K\\leqslant G$, then a routine argument (\\S \\ref{intervalsdef}) shows that $A$ and $B$ must be contained in $a+\\chi^{-1}(I)$, and $b+\\chi^{-1}(J)$, respectively. We conclude that $(A,B)$ satisfies (K).\n\n\\smallskip\n\nThe only remaining possibility is that $(A',B')$ satisfies (QP), but not (P) or (E). Then $(A',B')$ is nonextendible, while irreducibility of $(A,B)$ implies that of $(A',B')$. Writing $A'=A_1'\\cup A_0'$ and $B'=B_1'\\cup B_0'$ for the decomposition of $(A',B')$ in (QP) and $K$ for the corresponding subgroup, we consider different cases based on the measures of $A_0'$ and $B_0'$.\n\n \\smallskip\n\n If $m(A_0')>0$ and $m(B_0')>0$, Lemma \\ref{QPtoQP} implies $(A,B)$ satisfies (QP).\n\n \\smallskip\n\n If $m(A_0')=0$ and $m(B_0')>0$, or vice versa, we derive a contradiction. Under these conditions, $(A',B')$ is reducible, since $A_0'+B_0'+K$ is a unique expression element of $A'+B'+K$ in $G\/K$. Reducibility of $(A',B')$ contradicts our present assumptions.\n\n \\smallskip\n\nIf $m(A_0')=m(B_0')=0$, the irreducibility of $(A,B)$ implies $A+B\\sim A_1'+B_1'$. Since $H(A_1'+B_1')$ has positive Haar measure, we apply Lemma \\ref{easy} with $\\tilde{A}=A_1'$, $\\tilde{B}=B_1'$ and conclude that $(A,B)$ satisfies (P) or (QP) of Theorem \\ref{main}. \\end{proof}\n\n\n\nThe next lemma is a rephrasing of Lemma 7 of \\cite{Kneser56}; we include a proof for completeness. Under the standing assumption that $G$ is compact, the group $H(S)$ below is compact, so the convention that Haar measure is normalized applies: $m_{H(S)}(H(S))=1$.\n\n\n\\begin{lemma}\\label{H(S)}\nIf $S\\subseteq G$ is measurable, set $H:=H(S)$ and let\n$$\nE:=\\{x\\in G: 00\\},\\ \\ S_2:=\\{x\\in G: m_H(B-x)>0\\}.\n\\end{align*}\nBy Proposition \\ref{Weil}, $S_1$ and $S_2$ are measurable. Let $A''=A\\cap S_1$, $B''=B\\cap S_2$; Proposition \\ref{Weil} implies $m(A'')=m(A)$ and $m(B'')=m(B)$. Let $A'\\subseteq A''$ and $B'\\subseteq B''$ be countable unions of compact sets having $m(A')=m(A)$ and $m(B')=m(B)$. The irreducibility of $(A,B)$ implies $A'+B'\\sim A''+B''\\sim A+B$, so that $H(A'+B')=H$. By Lemma \\ref{H(S)}, the set\n\\begin{align*}\nE=\\{x\\in G: 0m(A')$, so $m((A'+H)+B')=m(A'+B')0$. Since $A'+H+B'\\sim A+B$, we then have $m(H(A+B))>0$ contradicting the hypothesis. Similarly $B'+H\\sim B'$. \\end{proof}\n\n\n\\begin{lemma}\\label{specialtogeneral}\nIf Theorem \\textup{\\ref{main}} holds under the additional assumption that $A+B$ is measurable and $H(A+B)=\\{0\\}$ then Theorem \\textup{\\ref{main}} holds in general.\n\\end{lemma}\n\n\n\n\n\\begin{proof} By Lemma \\ref{measurability} we may assume $A+B$ is measurable. As in the proof of Lemma \\ref{measurability}, we may assume that $(A,B)$ is irreducible and nonextendible. Write $K:=H(A+B)$. We first dispense with the case $m(K)>0$. In that case, Lemma \\ref{easy} implies $(A,B)$ satisfies conclusion (P) or (QP) of Theorem \\ref{main}. Now assume $m(K)=0$. Let $A'\\subseteq A$ and $B'\\subseteq B$ be as in the conclusion of Lemma \\ref{H(A+B)}, so that $A'+K\\sim A'$, $B'+K\\sim B'$, and $A'+B'+K\\sim A'+B'\\sim A+B$. Observe that $G\/K$ is infinite, since $m(K)=0$. For the remainder of the proof, write $G'$ for $G\/K$, and write $m'$ for $m_{G'}$.\n\n\\begin{claim}\nViewing $A'+B'+K$ as a subset of $G'$, the pair $(A'+K,B'+K)$ is an irreducible, nonextendible, sur-critical pair for $G'$, and $H(A'+B'+K)=\\{0_{G'}\\}$. In particular $m'(H(A'+B'+K))=0$, since $G'$ is infinite.\n\\end{claim}\n\n\\noindent\\textit{Proof of Claim.} We first show $H(A'+B'+K)=\\{0_{G'}\\}$ by contradiction. Supposing otherwise, there exists $z\\neq 0_{G'}\\in G'$ such that \\begin{align}\\label{triangle1}\nm'((A'+B'+K)\\triangle(A'+B'+K+z))=0.\n \\end{align} If $t\\in G$ is such that $t+K=z$, then $t\\notin K$, and by Proposition \\ref{Weil} and the choice of $A'$ and $B'$,\n \\begin{align*}\nm_G((A+B)\\triangle (A+B+t)) &= m_G((A'+B'+K)\\triangle (A'+B'+K+t))\\\\\n&=m'((A'+B'+K)\\triangle(A'+B'+K+z))\\\\\n&=0 ,\n\\end{align*} contradicting the definition of $K$.\n\n\\smallskip\n\nIf $(A'+K,B'+K)$ is extendible, then $m'(H(A'+B'+K))>0$ by Proposition \\ref{Ktopinequality}, contradicting the previous paragraph.\n\n\\smallskip\n\nIf $(A'+K,B'+K)$ is reducible, there exist $A''\\subseteq A'+K$ and $B''\\subseteq B'+K$ such that $m'(A''+K)=m'(A'+K)$, $m'(B''+K)=m'(B'+K)$, and\n\\begin{align*}\nm'(A''+K+B''+K)0$, so that $m(H(A+B))>0$, contradicting our present assumptions. Thus we may assume $m(A_0'+K)>0$ and $m(B_0'+K)>0$, so that Lemma \\ref{QPtoQP} applies to $(A',B')$ and $(A,B)$. We conclude $(A,B)$ satisfies (QP). \\end{proof}\n\n\n\n\\subsection{Quasi-periodicity and complements} Throughout this subsection we make the standing assumption that $A+B$ is measurable in addition our previous assumption that $G$ is compact.\n\n\\begin{lemma}\\label{complements} If $(A,B)$ is an irreducible nonextendible sur-critical pair satisfying $m(A)>0$, $m(B)>0$, and $m(H(A+B))=0$, then\n\\begin{align}\\label{comp1}\n-B+ (A+ B)^c \\sim A^c,\n\\end{align}\n\\begin{align}\\label{comp2}\n-A+ (A+ B)^c \\sim B^c,\n\\end{align}\n and both $(-B,(A+B)^c)$ and $(-A,(A+B)^c)$ are irreducible nonextendible sur-critical pairs with $m(H(-B+(A+B)^c))=0$ and $m(H(-A+(A+B)^c))=0$.\n\n\\end{lemma}\n\nNote that (\\ref{comp1}) and (\\ref{comp2}) imply $-B+(A+B)^c$ and $-A+(A+B)^c$ are measurable; the proof is complicated somewhat by the hypothetical possibility that these sets are not measurable.\n\n\\smallskip\n\nLemma \\ref{complements} is an analogue of Lemma 2.4 of \\cite{Grynkiewicz}. As shown in \\cite{Grynkiewicz}, the containment $-B+(A+B)^c\\subseteq A^c$ holds unconditionally.\n\n\n\n\n\n\\begin{proof} We first establish (\\ref{comp1}). Write $D$ for $-B+(A+B)^c$. From the unconditional containment $D\\subseteq A^c$, we have $m_*(D)\\leq m(A^c)$. If $m_*(D)=m(A^c)$, then (\\ref{comp1}) holds, so we assume\n\\begin{align}\\label{compcontradiction}\nm_*(D)< m(A^c)\n\\end{align} and derive a contradiction. Now\n\\begin{align}\\label{oneminus}\\begin{split}m(-B)+m((A+B)^c)&= m(B)+1-m(A+B)\\\\\n&=m(B)+1-(m(A)+m(B))\\\\\n&=m(A^c),\\end{split}\n\\end{align}\nso (\\ref{compcontradiction}) means $m_*(-B+(A+B)^c)0$ by (\\ref{comp2}), and consequently $m(H(B))>0$. Now irreducibility of $(A,B)$ implies $m(H(A+B))>0$, contradicting our assumptions on $(A,B)$.\n\n\\smallskip\n\n\n\\noindent\\textit{\\textup{2}. $(-A,(A+B)^c)$ is a sur-critical pair.} By a computation similar to (\\ref{oneminus}), we get $m(-A)+m((A+B)^c)=m(B^c)$,\nso $(-A,(A+B)^c)$ is a sur-critical pair by (\\ref{comp2}).\n\n\\smallskip\n\n\n\\noindent\\textit{\\textup{3}. $(-A,(A+B)^c)$ is irreducible.} Assume otherwise. By Corollary \\ref{reduciblecor}, reducibility of $(-A,(A+B)^c)$ yields decompositions $A=A_1\\cup A_0$ and $(A+B)^c=C_1\\cup C_0$ satisfying (QP), and one of $m(A_0)=0$ or $m(C_0)=0$. If $m(A_0)=0$, the irreducibility of $(A,B)$ implies $m(H(A+B))>0$, contradicting the hypothesis. If $m(C_0)=0$ then $m(H((A+B)^c))>0$, and therefore $m(H(A+B))>0$, contradicting the hypothesis.\n\n\\smallskip\n\n\n\\noindent\\textit{\\textup{4}. $(-A,(A+B)^c)$ is nonextendible.} Assume otherwise. Then Proposition \\ref{Ktopinequality} implies $m(H(-A+(A+B)^c))>0$, contradicting the previous claim to the contrary.\n\n\\smallskip\n\nWe have shown that $(-A,(A+B)^c)$ is an irreducible nonextendible sur-critical pair satisfying $m(H(-A+(A+B)^c))=0$. Reversing the roles of $A$ and $B$ yields the corresponding description of $(-B,(A+B)^c)$. \\end{proof}\n\n\n\n\\begin{lemma}[cf.~\\cite{Grynkiewicz}, Lemma 5.1]\\label{QP1} Let $(A,B)$ be a sur-critical pair with $m(A)>0$ and $m(B)>0$, and let $K$ be the subgroup of $G$ generated by $A$ \\textup{(}so $K$ is compact and open\\textup{)}. If $A+B$ is aperiodic then $B$ is quasi-periodic with respect to $K$, or $B$ is contained in a coset of $K$.\n\\end{lemma}\n\n\\begin{proof} Let $B= B_1\\cup \\dots \\cup B_{l}$ be a $K$-coset decomposition of $B$. Since $A\\subseteq K$, $A+B$ is the disjoint union $\\bigcup_{i=1}^l A+B_i$. Then\n\\begin{align}\\label{sumpartition}\nm(A+B)=\\sum_{i=1}^l m(A+B_i),\n\\end{align} while\n\\begin{align}\\label{originalpartition}\nm(A+B)=m(A)+m(B)=m(A)+\\sum_{i=1}^l m(B_i).\n\\end{align}\nEquating the right-hand sides of (\\ref{sumpartition}) and (\\ref{originalpartition}) we get\n \\begin{align}\\label{collectsplitsum}\n \\sum_{i=1}^l m(A+B_{i})-m(B_{i})=m(A).\n \\end{align}\n If $m(A+B_{i})< m(A)+m(B_{i})$ for all $i$, then the group $W:=K\\cap \\bigcap_{i=1}^l H(A+B_i)$ is open by Proposition \\ref{Ktopinequality}, and $A+B$ is periodic with period $W$, contradicting the hypothesis. Thus there is an $i$ such that $m(A+B_{i})\\geq m(A)+m(B_{i})$, and (\\ref{collectsplitsum}) implies $m(A+B_{j})=m(B_{j})$ for $j\\neq i$. Consequently, $A\\subseteq H(B_{j})$ and $m(B_j)>0$ for all $j\\neq i$. This implies $K\\leqslant H(B_{j})$ and $B_j\\sim B_j+K$ for all $j\\neq i$. We then have that $B$ is quasi-periodic with respect to $K$ if $l>1$, and $B$ is contained in a coset of $K$ if $l=1$. \\end{proof}\n\n\\begin{corollary}\\label{QP1cor} With the hypotheses of Theorem \\textup{\\ref{main}}, if $m(H(A+B))=0$ and $A$ is contained in a coset of a compact open subgroup $K_0\\leqslant G$, then $B$ has a quasi-periodic decomposition with respect to a compact open subgroup $K\\leqslant K_0$, or $B$ is contained in a coset of $K_0$.\n\\end{corollary}\n\n\\begin{proof} By translation we may assume $A\\subseteq K_0$. The hypothesis $m(H(A+B))=0$ implies $A+B$ is aperiodic. Now apply Lemma \\ref{QP1} and let $K$ be the subgroup generated by $A$. \\end{proof}\n\nIn the next lemma, we assume that $A=A_1\\cup A_0$ has a quasi-periodic decomposition with respect to the maximal period $K$ of $A_1$. This is no less general than assuming that $A$ has a quasi-periodic decomposition $A=A_1'\\cup A_0'$ with respect to some compact open subgroup $L$, since the maximal period $K$ of $A_1'$ must contain $L$, and then $A_0'$ is contained in a coset of $K$, since $A_0'$ is contained in a coset of $L$.\n\n\\begin{lemma}[cf.~\\cite{Grynkiewicz}, Lemma 5.3]\\label{QP3} Let $(A,B)$ be an irreducible nonextendible sur-critical pair with $m(A)>0$, $m(B)>0$, and $m(H(A+B))=0$. Let $A=A_1\\cup A_0$ be a quasi-periodic decomposition with $A_1$ periodic with maximal period $K$. Then $B=B_1\\cup B_0$, where $B_1\\sim B_1+K$, $B_0$ is contained in a coset of $K$, and\n\\begin{enumerate}\n\\item[(i)] $A_0+B_0+K$ is a unique expression element of $A+B+K$ in $G\/K$,\n\n \\smallskip\n\n\n\\item[(ii)] $m(A_0+B_0)=m(A_0)+m(B_0)$,\n\\end{enumerate}\nso $(A,B)$ satisfies conclusion \\textup{(QP)} of Theorem \\textup{\\ref{main}}.\n\n\\end{lemma}\n\n\n\n\\begin{proof} Observe that\n\\begin{align}\\label{A1smaller}\nm(A_1+B)0$, while $m(H(A+B))=0$. Inequality (\\ref{A1smaller}) implies\n\\begin{align}\\label{1step}\n\\begin{split}\nm(A_1+B)&\\leq m(A+B)-m(A_0)\\\\\n&=m(A)+m(B)-m(A_0)\\\\\n&=m(A_1)+m(B),\n\\end{split}\n\\end{align}\nwhere the first line results from $A_1+B\\sim A_1+B+K$, while $A_0$ is contained in a coset of $K$, the second line is due to $m(A+B)=m(A)+m(B)$, and the third results from $A=A_1\\cup A_0$ being a partition of $A$. We also have\n\\begin{align}\\label{inbetween}\n00$, (\\ref{A0plusb0}) implies $K\\cap H(A_0)\\subseteq H(A+B)$, so $m(H(A+B))>0$, again contradicting our hypothesis.\n\n\\smallskip\n\n\nStrict inequality in (\\ref{1step}) implies $m(A_1+B)0$, $m(B)>0$, and $m(H(A+B))=0$. If $A+B\\sim D$, where $D$ is quasi-periodic with respect to a compact open subgroup $K\\leqslant G$, then $(A,B)$ satisfies conclusion \\textup{(QP)} of Theorem \\textup{\\ref{main}}. \\end{lemma}\n\n\n\\begin{proof} Write $C$ for $(A+B)^c$. The assumption that $A+B\\sim D$, where $D$ is quasi-periodic with quasi-period $K$, implies $C\\sim C'$ where $C'\\subseteq C$ is quasi-periodic with respect to $K$, or $C'$ is contained in a coset of $K$. Lemma \\ref{complements} implies $(-B,C)$ is an irreducible nonextendible sur-critical pair with $m(H(-B+C))=0$, so $(-B,C')$ is such a pair with $m(H(-B+C'))=0$. If $C'$ is contained in a coset of $K$, then Corollary \\ref{QP1cor} implies $-B$, and therefore $B$, is quasi-periodic with respect to $K$, or is contained in a coset of $K$. If $B$ is quasi-periodic with respect to $K$, then Lemma \\ref{QP3} implies $(A,B)$ satisfies (QP). If $B$ is contained in a coset of $K$, then Corollary \\ref{QP1cor} implies $A$ is contained in a coset of $K$ or $A$ is quasi-periodic with respect to $K$. Since $A+B\\sim D$, which is quasi-periodic with respect to $K$, the sets $A$ and $B$ cannot both be contained in a coset of $K$, so $A$ is quasi-periodic with respect to $K$. Again we apply Lemma \\ref{QP3} and find $(A,B)$ satisfies (QP).\n\n\\smallskip\n\nIf $C'$ is not contained in a coset of $K$, then $C'$ is quasi-periodic with respect to $K$, and Lemma \\ref{QP3} implies $(-B,C')$ satisfies conclusion (QP) of Theorem \\ref{main}. Then $-B$ is contained in a coset of $K$ or $-B$ is quasi-periodic with respect to $K$, and as above we conclude that $(A,B)$ satisfies (QP). \\end{proof}\n\n\n\\subsection{The $e$-transform}\\label{etsection} For $A$, $B\\subseteq G$ and $e\\in G$, form the pair $(A_e,B_e)$, where\n\\begin{align*}\nA_e:= A\\cup (B+e),\\ B_e:=(A-e)\\cap B.\n\\end{align*}\nThis is the Dyson $e$-transform, whose properties are well-documented. In particular,\n\\begin{align}\\label{et1}\nA_e+B_e&\\subseteq A+B, \\\\\n\\label{et2}m(A_e)+m(B_e)&=m(A)+m(B),\n\\end{align}\nwhenever $B_e$ is nonempty. See \\cite{Kneser56}, \\cite{Nathansoninverse}, or \\cite{TaoVu} for further exposition.\n\n\\smallskip\n\nThe proof of Proposition \\ref{Kconnected} in \\cite{Kneser56} relies on a sequence of pairs $(A^{(n)}, B^{(n)})$ successively derived by $e$-transform, where $\\lim_{n\\to \\infty} m(B^{(n)})=0$ and $m(B^{(n)})>0$ for all $n$. The next lemma facilitates the same construction for some pairs $(A,B)$ where the ambient group is disconnected.\n\n\\begin{lemma}\\label{transformestimate}\nIf $A$, $B\\subseteq G$ have positive Haar measure, $A+B$ is measurable, $m(A)+m(B)\\leq 1$, and $A+B$ is aperiodic, then there is an $e\\in G$ such that $B_e:=(A-e)\\cap B\\neq \\varnothing$ and $m(B_e)\\leq (1-m(B))m(B)$.\n\\end{lemma}\n\nIn particular, the conclusion holds when $m(H(A+B))=0$. The hypothesis ``$A+B$ is aperiodic\" cannot be omitted; the conclusion fails when $B$ is a coset of a compact open subgroup $K\\leqslant G$ and $A$ is a union of cosets of $K$.\n\n\\begin{proof} Write $f(z)=m((A-z)\\cap B)$, so that $f:G\\to [0,1]$ is a continuous function and $\\int f\\, dm=m(A)m(B)$ (\\cite{Kneser56}, Lemma 1). Note that $(A-z)\\cap B$ is nonempty exactly when $z\\in A-B$. Consider $S:=\\{z:f(z)>0\\}$, which is contained in $A-B$. If $S\\neq A-B$, then there is an $e\\in A-B$ with $f(e)=0$, meaning $B_e\\neq \\varnothing$ and $m(B_e)=0$, so we are done. If $S=A-B$, consider the average\n\\begin{align}\\label{Saverage}\n\\frac{1}{m(S)}\\int_S f dm =\\frac{1}{m(A-B)} m(A)m(B).\n\\end{align}\n We estimate $m(A-B)$. If $m(A-B)0$, $m(B)>0$, $m(H(A+B))=0$, and there exists $e\\in G$ such that $m_*(A_e+B_e) m(A_e+B_e)+m(A_{0,i})+m(B_{0,i})\\\\\n&\\geq m(A)+m(B),\n\\end{aligned}\n\\end{align}\nwhere the last line follows from (\\ref{e-coset}). The strict inequality in (\\ref{ijij}) is the desired contradiction. This concludes the analysis of Case 3 and the proof of the lemma. \\end{proof}\n\n\n\\subsection{Intersections and measure}\n\n\n\n\\begin{lemma}\\label{intersections} If $A$, $B\\subseteq G$ have $m(A)>0$ and $m(B)>0$, there are sets $A'\\subseteq A$, $B'\\subseteq B$ with $m(A')=m(A)$ and $m(B')=m(B)$, such that $m((A'-t)\\cap B')>0$ whenever $(A'-t)\\cap B'$ is nonempty.\n\\end{lemma}\n\n\n\\begin{proof} By Theorem A of \\cite{Mueller65}, there is a sequence of neighborhoods $U_n$ of the identity $0\\in G$ such that for $m$-almost all $a\\in A$ and $b\\in B$,\n\\begin{align}\\label{PoD}\n\\lim_{n\\to \\infty} \\frac{m(A\\cap (U_n+a) )}{m(U_n)}=1 \\text{ and } \\lim_{n\\to \\infty} \\frac{m(B\\cap (U_n+b) )}{m(U_n)} =1.\n\\end{align}\nLet $A'\\subseteq A$ and $B'\\subseteq B$ be the sets of points in $A$ and $B$, respectively, satisfying (\\ref{PoD}), so that $m(A')=m(A)$ and $m(B')=m(B)$. If $t\\in G$ and $(A'-t)\\cap B'$ is nonempty, assume without loss of generality that $0\\in (A'-t)\\cap B'$. Now (\\ref{PoD}) implies that for $n$ sufficiently large, \\begin{align*}\nm((A'-t)\\cap U_n)>m(U_n)\/2 \\text{ and } m(B'\\cap U_n)>m(U_n)\/2,\n\\end{align*}\nso $m((A'-t)\\cap B'\\cap U_n)>0$. Thus $m((A'-t)\\cap B')>0$. \\end{proof}\n\n\\begin{remark} Theorem A of \\cite{Mueller65} is proved in \\cite{Mueller62}. We outline a way to find neighborhoods $U_n$ satisfying (\\ref{PoD}): by Proposition 2.42 of \\cite{FollandACAHA} and the ensuing remarks, there is a sequence of neighborhoods $V_n$ of the identity $0\\in G$ such that the functions\n\\begin{align*}\nf_n(z):= m(A\\cap (V_n+z))\/m(V_n),\\ g_n(z):=m(B\\cap (V_n+z))\/m(V_n)\n\\end{align*}\nconverge in $L^1(m)$ to the functions $1_A$ and $1_B$, respectively. One may then choose a subsequence $\\{U_n\\}_{n\\in \\mathbb N}$ of $\\{V_n\\}_{n\\in \\mathbb N}$ so that the corresponding subsequences of $f_n$ and $g_n$ converge $m$-almost everywhere. These $U_n$ will satisfy (\\ref{PoD}). \\hfill $\\blacksquare$ \\end{remark}\n\n\\section{The main argument}\\label{main argument}\n\n\\subsection{The sequence of $e$-transforms}\\label{transforms} As in \\cite{Kneser56}, we construct a sequence of pairs by successively applying the $e$-transform (\\S \\ref{etsection}).\n\n\\begin{lemma}\\label{esequence}\nLet $(A,B)$ be an irreducible, nonextendible pair satisfying the hypotheses of Theorem \\textup{\\ref{main}} such that $m(H(A+B))=0$. Then at least one of the following holds.\n\\begin{enumerate}\n\\item[(i)] There is a sequence of pairs $(A^{(n)}, B^{(n)})$, $n=0,1,2,\\dots$ such that $A^{(0)}=A$, $B^{(0)}=B$, and for all $n\\geq 1$,\n\n\\smallskip\n\n \\begin{enumerate}\n \\item[(i.1)] the pair $(A^{(n)},B^{(n)})$ is derived from $(A^{(n-1)},B^{(n-1)})$ by $e$-transform,\n\n \\item[(i.2)] $A^{(n)}+B^{(n)}\\sim A+B$, and\n\n \\item[(i.3)] $00$. By Lemma \\ref{transformestimate}, there is an $e\\in G$ such that\n\\begin{align}\\label{Destimate}\n0m(K)$. Then $B^{(n)}$ is contained in a coset of $K$, while $A^{(n)}$ is not. Then Corollary \\ref{QP1cor} implies $A^{(n)}$ is quasi-periodic, and Lemma \\ref{QP3} implies $A^{(n)}+B^{(n)}$ is quasi-periodic. Since $A+B\\sim A^{(n)}+B^{(n)}$, Lemma \\ref{QPsumtosummand} implies $(A,B)$ satisfies (QP) of Theorem \\ref{main}.\n\n\\subsubsection*{Case \\textup{2}. $G$ has an open connected subgroup.}\n\nLet $G_0$ be an open connected subgroup of $G$. By the Claim, we can choose $n$ sufficiently large that $B^{(n)}$ is contained in a coset of $G_0$. If $A^{(n)}$ is contained in a coset of $G_0$, then by the construction of $A^{(n)}$, $A$ and $B$ are each contained in a coset of $G_0$. Then Proposition \\ref{Kconnected} implies the existence of a continuous surjective homomorphism $\\chi:G_0\\to \\mathbb T$ and intervals $I,J\\subseteq \\mathbb T$ such that $A\\subseteq a+\\chi^{-1}(I)$, $B\\subseteq b+\\chi^{-1}(J)$, for some $a$, $b\\in G$, such that $m(A)=m(\\chi^{-1}(I))$ and $m(B)=m(\\chi^{-1}(J))$. From this we conclude (K) in Theorem \\ref{main}.\nIf $A^{(n)}$ is not contained in a coset of $G_0$, we argue as in Case 1 and find that $(A,B)$ satisfies (QP). \\end{proof}\n\n\\subsection{Proof of Theorem \\textup{\\ref{main}}.}\\label{proof} If $(A,B)$ is extendible, we have conclusion (E). If $(A,B)$ is nonextendible and reducible, Lemma \\ref{reduciblelemma2} implies $(A,B)$ satisfies conclusion (QP). If $(A,B)$ is irreducible and nonextendible, Lemma \\ref{specialtogeneral} allows us to assume that $A+B$ is measurable and $H(A+B)=\\{0\\}$. In that case, Proposition \\ref{specialprop} implies the conclusion of Theorem \\ref{main}. \\hfill $\\square$\n\n\\section{When $G$ is not compact}\\label{lc} A structure theorem for locally compact abelian groups (\\cite{HewittandRoss1}, Theorem 24.30) says that such a group $G$ is isomorphic (as a topological group) to $\\mathbb R^n\\times L$, where $L$ is a locally compact group containing a compact open subgroup, and $n$ is a nonnegative integer. The study of the equation \\begin{align}\\label{equation}\nm_*(A+B)=m(A)+m(B)\n\\end{align}\ndepends heavily on $n$. Theorem 5 of \\cite{Kneser56} says that when $n\\geq 1$,\n\\begin{align}\\label{BM}\nm_*(A+B)^{1\/n} \\geq m(A)^{1\/n}+m(B)^{1\/n}\n \\end{align}\n for measurable subsets $A$ and $B$ of $G$, and Theorem 6 of \\cite{Kneser56} classifies the pairs for which equality holds in (\\ref{BM}), generalizing Theorem 2 of \\cite{HenstockMacbeath} to the case where $L$ is nontrivial. When $n>1$ and $A$ and $B$ both have positive measure, (\\ref{BM}) implies $m_*(A+B)>m(A)+m(B)$, so there are no such pairs satisfying (\\ref{equation}) when $n>1$. When $n=1$, Theorem 6 of \\cite{Kneser56} says that if $A$ and $B$ have positive measure then (\\ref{equation}) holds if and only if there are closed intervals $I$, $J\\subseteq \\mathbb R$, a compact open subgroup $K\\leqslant L$, and $a$, $b\\in G$ such that $A\\subseteq a+(I\\times K)$, $B\\subseteq b+(J\\times K)$, and $m(A)=m(I\\times K)$, $m(B)=m(J\\times K)$.\n\n\\smallskip\n\nOur study of equation (\\ref{equation}) is thus reduced to groups $G$ having a compact open subgroup. We assume $A$ and $B$ have finite measure, since otherwise (\\ref{equation}) is satisfied. One can easily check that $A$ and $B$ have compact closures under these assumptions. Replacing $G$ by the group generated by $A\\cup B$, we can assume that $G$ is compactly generated. A compactly generated group with a compact open subgroup is isomorphic to $\\mathbb Z^d\\times K$ for some nonnegative integer $d$ and some compact group $K$ (\\cite{HewittandRoss1}, Theorem 9.8). For such groups $G$, it is then routine to verify that Theorem \\ref{main} holds with no modification of the conclusion. This is the content of the following corollary.\n\n\\begin{corollary}\\label{generalization} The conclusion of Theorem \\textup{\\ref{main}} holds under the weaker assumption that $G$ has a compact open subgroup and $m_*(A+B)=m(A)+m(B)<\\infty$. \\hfill $\\square$\n\\end{corollary}\n\n\n\n\n\\section{Acknowledgements} This work was conducted mostly while the author was a postdoctoral fellow in the Department of Mathematics at the University of British Columbia. The author thanks Izabella {\\L}aba and Malabika Pramanik for financial support and helpful discussions. Discussions with Michael Bj\\\"orklund and Alexander Fish provided motivation for this work.\n\n\n\n\\bibliographystyle{amsplain}\n\\frenchspacing\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\n \nThe spectacular discovery of a boson with a mass around $\\sim 125 \\,\\, \\mathrm{GeV}$ by\nthe ATLAS and CMS experiments~\\cite{Aad:2012tfa,Chatrchyan:2012xdj} at\nCERN constitutes a milestone in the quest for understanding the physics\nof electroweak symmetry breaking (EWSB). While within the present experimental\nuncertainties the properties of the observed Higgs boson are compatible with\nthe predictions of the Standard Model (SM)~\\cite{Aad:2015zhl},\nmany other interpretations are possible as well, in particular as a\nHiggs boson of an extended Higgs sector. Consequently, any model describing \nelectroweak physics needs to provide a state that can be identified\nwith the observed signal. \n\nOne of the prime candidates for physics beyond the SM is supersymmetry\n(SUSY), which doubles the particle degrees of freedom by predicting\ntwo scalar partners for all SM fermions, as well as fermionic partners\nto all bosons. The simplest SUSY extension is the Minimal Supersymmetric\nStandard Model (MSSM)~\\cite{Nilles:1983ge,Haber:1984rc}.\nIn contrast to the single Higgs doublet of the SM, the Higgs sector of\nthe MSSM contains two Higgs doublets, which in the ${\\CP}$ conserving\ncase leads to a physical spectrum consisting of two ${\\CP}$-even, one\n${\\CP}$-odd and two charged Higgs bosons. The light (or the heavy)\n${\\CP}$-even MSSM Higgs boson can be interpreted as the signal discovered\nat $\\sim 125 \\,\\, \\mathrm{GeV}$~\\cite{Heinemeyer:2011aa}. \n \nGoing beyond the MSSM, a well-motivated extension is given by\nthe Next-to-Minimal Supersymmetric Standard Model (NMSSM), see\ne.g.~\\cite{Ellwanger:2009dp,Maniatis:2009re} for reviews. In particular\nthe NMSSM provides a solution for the so-called ``$\\mu$ problem'' by\nnaturally associating an adequate scale to the $\\mu$ parameter appearing\nin the MSSM superpotential~\\cite{Ellis:1988er,Miller:2003ay}.\nIn the NMSSM a new singlet superfield is introduced, which only couples to the\nHiggs- and sfermion-sectors, giving rise to an effective $\\mu$-term,\nproportional to the vacuum expectation value (vev) of the scalar singlet.\nAssuming ${\\CP}$ conservation, as we do throughout the paper, the states in the\nNMSSM Higgs sector can be classified as three ${\\CP}$-even Higgs bosons,\n$h_i$ ($i = 1,2,3$), two ${\\CP}$-odd Higgs bosons, $a_j$ ($j = 1,2$),\nand the charged Higgs boson pair $H^\\pm$. In addition, the SUSY\npartner of the singlet Higgs (called the singlino) extends the\nneutralino sector to a total of five neutralinos. In the NMSSM the\nlightest but also the second lightest ${\\CP}$-even neutral Higgs boson\ncan be interpreted as the signal observed at about $125~\\,\\, \\mathrm{GeV}$, see,\ne.g., \\cite{King:2012is,Domingo:2015eea}.\n\nA natural extension of the NMSSM is the \\ensuremath{\\mu\\nu\\mathrm{SSM}}, in which the \nsinglet superfield is interpreted as a right-handed neutrino \nsuperfield~\\cite{LopezFogliani:2005yw,Escudero:2008jg} \n(see \\citeres{Munoz:2009an,Munoz:2016vaa,Ghosh:2017yeh} for reviews). \nThe \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ is the simplest extension of the MSSM that can provide \nmassive neutrinos through a see-saw mechanism at the electroweak scale.\nIn this paper we will focus on the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ with one family of \nright-handed neutrino superfields, and the case of three families will\nbe studied in a future publication.%\n\\footnote{The \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ with three families of right-handed neutrinos\nextends the ${\\CP}$-even and ${\\CP}$-odd scalar sector and the\nneutral fermion sector by two additional\nparticles each, in particular\nallowing a more viable reproduction of neutrino\ndata~\\cite{LopezFogliani:2005yw,Escudero:2008jg,Ghosh:2008yh,Bartl:2009an,Fidalgo:2009dm,Ghosh:2010zi}.}\n~The $\\mu$ problem is solved analogously to the NMSSM by the coupling of the right-handed neutrino \nsuperfield to the Higgs sector, and a trilinear coupling of the \nright-handed neutrino generates an effective Majorana mass at the \nelectroweak scale. The unique feature of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ is the introduction \nof a Yukawa coupling for the right-handed neutrino of the order of the\nelectron Yukawa coupling that induces the explicit\nbreaking of $R$-parity. One of the consequences is that there is no lightest stable \nSUSY particle anymore. Nevertheless, the model can still provide a dark matter\ncandidate with a gravitino that has a life time longer \nthan the age of the observable universe~\\cite{Choi:2009ng,GomezVargas:2011ph,Albert:2014hwa,Gomez-Vargas:2016ocf}. \nSince the lightest particle beyond the SM is not stable, it can \ncarry electrical charge or even \nbe coloured.\nThe explicit violation of lepton number and lepton flavor \ncan modify the \nspectrum of the neutral and charged fermions in comparison to the NMSSM.\nThe three families of charged leptons will mix with the chargino and \nthe Higgsino and form five massive charged fermions. However, the mixing will naturally be tiny since the breaking of $R$-parity is governed by the \nsmall neutrino Yukawa couplings. In the neutral fermion sector the three \nleft-handed neutrinos mix with the right-handed neutrino and the four \nMSSM-like neutralinos. When just one family of right-handed neutrino is \nconsidered (as we do in this paper), the mass matrix of the\nneutral fermions is of rank six, so \njust one light neutrino mass is generated at tree-level, while the other \ntwo light-neutrino masses will be generated by quantum corrections.\nFor the Higgs sector the breaking of $R$-parity has dramatic \nconsequences. The three left-handed and the right-handed sneutrinos \nwill mix with the doublet Higgses and form six massive ${\\CP}$-even \nand five massive ${\\CP}$-odd states, assuming that there is no \n${\\CP}$-violation. Additionally, since the vacuum\nof the model is not protected anymore by lepton number, the sneutrinos \nwill acquire a vev after spontaneous EWSB.\nWhile the vev of the right-handed sneutrino can easily take values up to \nthe TeV-scale, the stability of the vacuum together with the smallness \nof the neutrino Yukawa couplings force the vevs of the left-handed \nsneutrinos to be several orders of magnitude \nsmaller~\\cite{Escudero:2008jg,LopezFogliani:2005yw}.\nAs in the NMSSM, the couplings of the \ndoublet-like Higgses to the gauge-singlet right-handed sneutrino provide \nadditional contributions to the tree-level mass of the SM-like Higgs \nboson, relaxing the prediction of the MSSM, that it is bounded from above by the $Z$ boson mass. Still it was shown in the\nNMSSM~\\cite{Drechsel:2016ukp}\nthat a consistent \ntreatment of the quantum corrections is necessary for accurate\nHiggs mass predictions (see also\n\\citeres{Goodsell:2014bna,Goodsell:2014pla,Staub:2015aea}).\nIn this paper we will investigate if this is also\nthe case in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ and if its unique couplings generate significant \ncorrections to the SM-like Higgs mass, that go beyond the corrections \narising in the NMSSM.\n\n\nThe experimental accuracy of the measured mass of the observed Higgs\nboson has already reached the level of a precision observable, with an\nuncertainty of less than $300~\\,\\, \\mathrm{MeV}$~\\cite{Aad:2015zhl}.\nIn the MSSM the masses of the ${\\CP}$-even Higgs bosons can be\npredicted at lowest order in terms of two SUSY parameters\ncharacterising the MSSM Higgs sector, e.g.\\ $\\tb$, the ratio of\nthe vevs of the two doublets, and the mass of the\n${\\CP}$-odd Higgs boson, $\\MA$, or the charged Higgs boson, $\\MHp$. This\nresults in particular in an upper bound on the mass of the light\n${\\CP}$-even Higgs boson given by the $Z$-boson mass.\nHowever, these relations receive large higher-order corrections. \nBeyond the one-loop level, the dominant two-loop corrections of\n$\\order{\\alpha_t\\alpha_s}$~\\cite{Heinemeyer:1998jw,Heinemeyer:1998kz,Heinemeyer:1998np,Zhang:1998bm,Espinosa:1999zm,Degrassi:2001yf}\nand \\order{\\alpha_t^2}~\\cite{Espinosa:2000df,Brignole:2001jy} as well as\nthe corresponding corrections of \n$\\order{\\alpha_b\\alpha_s}$~\\cite{Brignole:2002bz,Heinemeyer:2004xw} and\n\\order{\\alpha_t\\alpha_b}~\\cite{Brignole:2002bz} are known since more than a\ndecade. (Here we use $\\al_f = (Y^f)^2\/(4\\pi)$, with $Y^f$ denoting the\nfermion Yukawa coupling.) These corrections, together with a\nresummation of leading and subleading logarithms from the top\/scalar\ntop sector~\\cite{Hahn:2013ria} (see also~\\cite{Draper:2013oza,Lee:2015uza} \nfor more details on this type of approach),\na resummation of leading contributions from the bottom\/scalar bottom\nsector~\\cite{Brignole:2002bz,Heinemeyer:2004xw,Hempfling:1993kv,Hall:1993gn,Carena:1994bv,Carena:1999py}\n(see also~\\cite{Noth:2008tw, Noth:2010jy}) and momentum-dependent two-loop\ncontributions~\\cite{Borowka:2014wla, Borowka:2015ura} (see\nalso~\\cite{Degrassi:2014pfa}) are included in the public\ncode~\\fh~\\cite{Heinemeyer:1998yj,Hahn:2009zz,Heinemeyer:1998np,Degrassi:2002fi,Frank:2006yh,Hahn:2013ria,Bahl:2016brp,Bahl:2017aev,feynhiggs-www}. \nA (nearly) full two-loop EP calculation, including even the leading\nthree-loop corrections, has also been\npublished~\\cite{Martin:2005eg,Martin:2007pg}, which is, however, not\npublicly available as a computer code. Furthermore, another leading\nthree-loop calculation of \\order{\\alpha_t\\alpha_s^2}, depending on the various\nSUSY mass hierarchies, has been\nperformed~\\cite{Harlander:2008ju,Kant:2010tf}, resulting in the code \n{\\tt H3m} and is now available as a stand-alone\n code~\\cite{Harlander:2017kuc}. \nThe theoretical uncertainty on the lightest\n${\\CP}$-even Higgs-boson mass within the MSSM from unknown higher-order\ncontributions is still at the level of about $2-3~\\,\\, \\mathrm{GeV}$ for scalar top\nmasses at the TeV-scale, where the actual uncertainty\ndepends on the considered parameter region~\\cite{Degrassi:2002fi,Heinemeyer:2004gx,Hahn:2013ria,Buchmueller:2013psa}.\n\nIn the NMSSM the status of the higher-order corrections to the Higgs-boson\nmasses (and mixings) is the following. \nFull one-loop calculations including the momentum dependence have been\nperformed in the \\DRbar\\ renormalization scheme in \n\\citere{Degrassi:2009yq,Staub:2010ty}, or in a mixed on-shell (OS)-\\DRbar\\ \nscheme in \\citere{Ender:2011qh,Graf:2012hh,Drechsel:2016jdg}.\nTwo-loop corrections of \\order{\\alpha_t\\alpha_s,\\alpha_t^2} have been included in the\nNMSSM in the leading logarithmic approximation (LLA) in\n\\citeres{Yeghian:1999kr,Ellwanger:1999ji}. \nIn the EP approach at the two-loop level, the dominant \n\\order{\\alpha_t\\alpha_s, \\alpha_b\\alpha_s} in the \\DRbar\\ scheme became available in \n\\citere{Degrassi:2009yq}. The two-loop corrections involving only\nsuperpotential couplings such as Yukawa and singlet interactions were given in\n\\cite{Goodsell:2014pla}. A two-loop calculation of the \\order{\\alpha_t\\alpha_s}\ncorrections with the top\/stop sector renormalized in the OS scheme or in the\n\\DRbar\\ scheme were provided in \\citere{Muhlleitner:2014vsa}.\nA consistent combination of a full one-loop calculation with all corrections\nbeyond one-loop in the MSSM approximation was given in\n\\citere{Drechsel:2016jdg}, which is included in the (private) version of\n\\fh\\ for the NMSSM. \nA detailed comparison of the various higher-order corrections up to the\ntwo-loop level involving a \\DRbar\\ renormalization was performed in\n\\citere{Staub:2015aea}, and involving an OS renormalization of the top\/stop\nsector for the \\order{\\alpha_t\\alpha_s} corrections in \\citere{Drechsel:2016htw}.\nAccordingly, at present the theoretical uncertainties from\nunknown higher-order corrections in the NMSSM are expected to be still\nlarger than for the MSSM.\n\nIn this paper we go one step beyond and investigate the scalar sector of\nthe \\ensuremath{\\mu\\nu\\mathrm{SSM}}, containing (mixtures of) Higgs bosons and scalar neutrinos.\nAs a first step we present the renormalization at the one-loop level of the\nneutral scalar sector in detail. Here a crucial point is that the NMSSM\npart of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ is treated exactly in the same way as in\n\\citere{Drechsel:2016jdg}. Consequently, differences (at the one-loop\nlevel) appearing for, e.g., mass relations or couplings can be directly\nattributed to the richer structure of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}. As for the NMSSM in\n\\citere{Drechsel:2016jdg}, the full one-loop calculation is supplemented\nwith higher-order corrections in the MSSM limit (as provided by \\fh~\\cite{Heinemeyer:1998yj,Hahn:2009zz,Heinemeyer:1998np,Degrassi:2002fi,Frank:2006yh,Hahn:2013ria,Bahl:2016brp,Bahl:2017aev,feynhiggs-www}).\\footnote{A\ncorresponding calculation using a pure \\DRbar\\ renormalization could\nin principle be performed using \\texttt{SARAH} and\n\\texttt{SPheno}~\\cite{Goodsell:2014bna}.}\nIn our numerical analysis we evaluate several ``representative''\nscenarios using the full one-loop results together with the MSSM-type\nhigher-order contributions. Differences found w.r.t.\\ the NMSSM can be\ninterpreted in a two-fold way. On the one hand, if non-negligible\ndifferences are found, they might serve as a probe to distinguish the\ntwo models experimentally. On the other hand, they indicate the level of\ntheoretical uncertainties of the Higgs-boson\/scalar neutrino mass\ncalculation in the \n\\ensuremath{\\mu\\nu\\mathrm{SSM}}, which should be brought to the same level of accuracy as in the\n(N)MSSM. \n\nThe paper is organized as follows. In \\refse{sec:mnSSM} we describe\nthe \\ensuremath{\\mu\\nu\\mathrm{SSM}}, including the details for all sectors relevant in this paper.\nThe full one-loop renormalization of the neutral scalar potential \nis presented in \\refse{sec:renopot}. \nWe will establish a convenient set of free parameters and fix their \ncounterterms in a mixed OS-\\DRbar\\ scheme. The counterterms are \ncalculated and applied in the renormalized ${\\CP}$-even and\n${\\CP}$-odd one-loop scalar\nself-energies in \\refse{sec:getmasses}. \nIn this work we focus on the application to the renormalized\n${\\CP}$-even\nself-energies, but the calculation of the renormalized ${\\CP}$-odd\nones constitutes a good additional test for the counterterms. \nWe also describe the \nincorporation of higher-order contributions taken over from the MSSM.\nOur numerical analysis, including an analysis of differences w.r.t.\\ \nNMSSM, is presented in \\refse{sec:numanal}. \nWe conclude in section \\refse{sec:concl}.\n\n\n\n\n\n\n\n\n\n\\section{\\protect\\boldmath The model: \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ with one generation of right handed neutrinos.}\n\\label{sec:mnSSM}\n\nIn the three-family notation of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ with one generation of\nright-handed neutrinos the superpotential is written as\n\\begin{align}\\label{superpotential}\nW = & \\;\n\\epsilon_{ab} \\left(\nY^e_{ij} \\, \\hat H_d^a\\, \\hat L^b_i \\, \\hat e_j^c +\nY^d_{ij} \\, \\hat H_d^a\\, \\hat Q^{b}_{i} \\, \\hat d_{j}^{c} \n+\nY^u_{ij} \\, \\hat H_u^b\\, \\hat Q^{a}_{i} \\, \\hat u_{j}^{c}\n\\right)\n\\nonumber\\\\\n&+ \n\\epsilon_{ab} \\left(\nY^{\\nu}_{i} \\, \\hat H_u^b\\, \\hat L^a_i \\, \\hat \\nu^c \n-\n\\lambda \\, \\hat \\nu^c\\, \\hat H_u^b \\hat H_d^a\n\\right)\n+\n\\frac{1}{3}\n\\kappa \n\\hat \\nu^c\\hat \\nu^c\\hat \\nu^c\n\\ .\n\\end{align}\nwhere $\\hat H_d^T=(\\hat H_d^0, \\hat H_d^-)$ and \n$\\hat H_u^T=(\\hat H_u^+, \\hat H_u^0)$ are the MSSM-like doublet Higgs\nsuperfields, $\\hat Q_i^T=(\\hat u_i, \\hat d_i)$ and \n$\\hat L_i^T=(\\hat \\nu_i, \\hat e_i)$ are the left-chiral\nquark and lepton superfield doublets,\nand $\\hat u_{j}^{c}$, $\\hat d_{j}^{c}$, $\\hat e_j^c$ and $\\hat{\\nu}^c$ are\n the right-chiral quark and lepton superfields.\n$i$ and $j$ are family indices running from one to three and \n$a,b=1,2$ are indices of the fundamental representation of SU(2) \nwith $\\epsilon_{ab}$ the totally antisymmetric tensor and \n$\\varepsilon_{12}=1$. The colour indices are undisplayed. \n$Y^u$, $Y^d$ and $Y^e$ are the usual Yukawa couplings\nalso present in the MSSM. \nThe right-handed neutrino is a gauge singlet, which permits us to \nwrite \nthe gauge-invariant \ntrilinear self coupling $\\kappa$ and the trilinear \ncoupling with the Higgs doublets $\\lambda$ in the second row, \nwhich are analogues to the couplings of the singlet in the \nsuperpotential of the trilinear NMSSM.\nThe $\\mu$-term is generated dynamically after the spontaneous EWSB,\nwhen the right-handed sneutrino obtains a vev.\nThe $\\kappa$-term forbids a global U(1) symmetry and we\navoid the existence of a Goldstone boson in the ${\\CP}$-even sector. \nThe remarkable difference to the\nNMSSM is the additional Yukawa coupling $Y^\\nu_i$, which induces\nexplicit breaking of $R$-parity through the $\\lambda$- and\n$\\kappa$-term,\nand which justifies the interpretation of the singlet\nsuperfield as a right-handed neutrino superfield.\nIt should be pointed out\nthat in this case lepton number is not conserved anymore, and also the\nflavor symmetry in the leptonic sector is broken. \nA more complete motivation of this superpotential can be found in\n\\citere{LopezFogliani:2005yw,Escudero:2008jg,Ghosh:2017yeh}. \n\nWorking in the framework of low-energy SUSY the\ncorresponding soft SUSY-breaking Lagrangian can be written as\n\\begin{eqnarray}\n-\\mathcal{L}_{\\text{soft}} =&&\n\\epsilon_{ab} \\left(\nT^e_{ij} \\, H_d^a \\, \\widetilde L^b_{iL} \\, \\widetilde e_{jR}^* +\nT^d_{ij} \\, H_d^a\\, \\widetilde Q^b_{iL} \\, \\widetilde d_{jR}^{*} \n+\nT^u_{ij} \\, H_u^b \\widetilde Q^a_{iL} \\widetilde u_{jR}^* \n+ \\text{h.c.}\n\\right)\n\\nonumber \\\\\n&+&\n\\epsilon_{ab} \\left(\nT^{\\nu}_{i} \\, H_u^b \\, \\widetilde L^a_{iL} \\widetilde \\nu_{R}^* \n- \nT^{\\lambda} \\, \\widetilde \\nu_{R}^*\n\\, H_d^a H_u^b\n+ \\frac{1}{3} T^{\\kappa} \\, \\widetilde \\nu_{R}^*\n\\widetilde \\nu_{R}^*\n\\widetilde \\nu_{R}^*\n\\\n+ \\text{h.c.}\\right)\n\\nonumber \\\\\n&+& \n\\left(m_{\\widetilde{Q}_L}^2\\right)_{ij} \n\\widetilde{Q}_{iL}^{a*}\n\\widetilde{Q}^a_{jL}\n+\\left(m_{\\widetilde{u}_R}^{2}\\right)_{ij} \\widetilde{u}_{iR}^* \n\\widetilde u_{jR}\n+ \\left(m_{\\widetilde{d}_R}^2\\right)_{ij} \\widetilde{d}_{iR}^* \n\\widetilde d_{jR}\n+\n\\left(m_{\\widetilde{L}_L}^2\\right)_{ij} \n\\widetilde{L}_{iL}^{a*} \n\\widetilde{L}^a_{jL}\n\\nonumber\\\\\n&+&\n\\left( m_{H_d\\widetilde{L}_L}^2 \\right)_{i}H_d^{a*} \\widetilde{L}_{iL}^a\n+\nm_{\\widetilde{\\nu}_R}^2 \\widetilde{\\nu}_{R}^*\n\\widetilde\\nu_{R} \n+\n\\left(m_{\\widetilde{e}_R}^2\\right)_{ij} \\widetilde{e}_{iR}^* \n\\widetilde e_{jR}\n+ \nm_{H_d}^2 {H^a_d}^*\nH^a_d + m_{H_u}^2 {H^a_u}^*\nH^a_u\n\\nonumber \\\\\n&+& \\frac{1}{2}\\, \\left(M_3\\, {\\widetilde g}\\, {\\widetilde g}\n+\nM_2\\, {\\widetilde{W}}\\, {\\widetilde{W}}\n+M_1\\, {\\widetilde B}^0 \\, {\\widetilde B}^0 + \\text{h.c.} \\right)\\ ,\n\\label{2:Vsoft}\n\\end{eqnarray}\nIn the first four lines\nthe fields denote the scalar component of the\ncorresponding superfields.\nIn the last line the fields denote the fermionic\nsuperpartners of the gauge bosons.\nThe scalar trilinear parameters \n$T^{e,\\nu,d,u,\\lambda,\\kappa}$ correspond to the trilinear couplings \nin the superpotential. The soft mass parameters \n$m_{\\widetilde{Q}_L,\\widetilde{u}_R,\\widetilde{d}_R,\n\\widetilde{L}_L,\\widetilde{e}_R}^2$ are hermitian $3\\times 3$ matrices in family space. \n$m_{H_d,H_u,\\widetilde{\\nu}_R}^2$ are the soft masses\nof the doublet Higgs fields and the right-handed sneutrino, and \n$ m_{H_d\\widetilde{L}_L}^2$ is a 3-dimensional\nvector in family space allowed\nby gauge symmetries since the left-handed lepton fields and the down-type \nHiggs field share the same quantum numbers. In the last row \nthe parameters $M_{3,2,1}$ define Majorana masses for the gluino, wino \nand bino, where the summation\nover the gauge-group indices in the \nadjoint representation\nis undisplayed. While all the soft parameters except\n$m_{H_d}^2$, $m_{H_u}^2$ and $m_{\\widetilde{\\nu}_R}^2$ can \nin general be complex,\nthey are assumed to be real in the following to avoid \n${\\CP}$-violation.\nAdditionally, we will neglect flavor mixing at tree-level in the squark\nand the quark sector, so the soft masses will be diagonal and we write\n$m_{\\widetilde{Q}_{iL} }^2$, $m_{\\widetilde{u}_{iR}}^2$ and\n$m_{\\widetilde{d}_{iR}}^2$, as well as for the soft trilinears\n$T^u_i=A^u_i\nY^u_i$, $T^d_i=A^d_i Y^d_i$, where the summation convention on repeated\nindices is not implied, and the quark Yukawas $Y^u_{ii}=Y^u_i$ and\n$Y^d_{ii}=Y^d_i$ are diagonal. For the sleptons we define \n$T^e_{ij}=A^e_{ij}Y^e_{ij}$ and $T^\\nu_{i}=A^\\nu_i Y^\\nu_i$, again without summation over repeated indices.\n\nSome care has to be taken with the parameters\n$(m_{\\widetilde{L}_L}^2)_{ij}$\ncontributing to the tree-level neutral scalar potential, because\nthese parameters cannot be set flavor-diagonal a priori. The reason is that\nduring the renormalization procedure (see \\refse{sec:condis})\nthe non-diagonal elements receive a\ncounterterm. Of course, the tree-level value of the non-diagonal\nelements can and should be set to zero to avoid too large flavor mixing. This assures that the contributions generated by virtual corrections\nwill always be small.\n\nSimilarly to the off-diagonal elements of the squared\nsfermion mass matrices,\nthe parameters $( m_{H_d\\widetilde{L}_L}^2 )_{i}$ are usually\nnot included in the tree-level Lagrangian of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}.\nIn the latter case because\nthey contribute to the minimization equations of the left-handed\nsneutrinos and spoil the electroweak seesaw mechanism that\ngenerates neutrino masses of the correct order of\nmagnitude. Theoretically, the absence of these\nparameters mixing different fields\nat tree level, $( m_{H_d\\widetilde{L}_L}^2 )_{i}$,\n$(m_{\\widetilde{L}_L}^2)_{ij}$,\n$(m_{\\widetilde{Q}_L}^2)_{ij}$, etc., \ncan be justified by the diagonal structure of the K\\\"ahler metric in\ncertain supergravity models, or when the dilaton field\nis the source of SUSY breaking in string\nconstructions~\\cite{Ghosh:2017yeh}.\nNotice also that when the\ndown-type Higgs doublet superfield is interpreted as a fourth family\nof leptons, the parameters\n$m_{H_d\\widetilde{L}_L}^2$ can be seen as\nnon-diagonal elements\nof $m_{\\widetilde{L}_L}^2$~\\cite{Lopez-Fogliani:2017qzj}.\nNevertheless, we include them in the soft\nSUSY-breaking Lagrangian in this paper,\nbecause these terms are generated at (one-)loop level, and\nin our renormalization approach we need the functional dependence\nof the scalar potential on $m_{H_d\\widetilde{L}_L}^2$.\n\nAfter the electroweak symmetry breaking the neutral scalar fields will acquire\na vev. This includes the left- and right-handed\nsneutrinos, because they are not protected by lepton number conservation as in\nthe MSSM and the NMSSM. We define the decomposition \n\\begin{align}\n\\label{eq:vevdecompH}\nH_d^0 =& \\frac{1}{\\sqrt 2} \\left(H_{d}^{\\mathcal{R}} + v_d + \\text{i}\\ H_{d}^{\\mathcal{I}}\\right)\\ , \\\\\nH^0_u =& \\frac{1}{\\sqrt 2} \\left(H_{u}^{\\mathcal{R}} + v_u +\\text{i}\\ H_{u}^{\\mathcal{I}}\\right)\\ , \\\\\n\\widetilde{\\nu}_{R} =&\n \\frac{1}{\\sqrt 2} \\left(\\widetilde{\\nu}^{\\mathcal{R}}_{R}+ v_{R} + \\text{i}\\ \\widetilde{\\nu}^{\\mathcal{I}}_{R}\\right), \\\\\n\\label{eq:vevdecompL}\n \\widetilde{\\nu}_{iL} =& \\frac{1}{\\sqrt 2} \\left(\\widetilde{\\nu}_{iL}^{\\mathcal{R}} \n + v_{iL} +\\text{i}\\ \\widetilde{\\nu}_{iL}^{\\mathcal{I}}\\right)\\ ,\n\\end{align}\nwhich is valid assuming ${\\CP}$-conservation, as we will do\nthroughout this paper.\n\n\n\n\n\\subsection{The \\boldmath{\\ensuremath{\\mu\\nu\\mathrm{SSM}}} Higgs potential}\n\nThe neutral scalar potential $V_H$ of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ with one generation of right-handed neutrinos is given at tree-level with all parameters chosen to be real by the soft terms and the $F$- and $D$-term contributions of the superpotential. We find\n\\begin{equation}\nV^{(0)} = V_{\\text{soft}} + V_F + V_D\\ , \n\\label{finalpotential}\n\\end{equation}\nwith\n\\begin{eqnarray}\nV_{\\text{soft}} =&&\n\\left(\nT^{\\nu}_{i} \\, H_u^0\\, \\widetilde \\nu_{iL} \\, \\widetilde \\nu_{R}^* \n- T^{\\lambda} \\, \\widetilde \\nu_{R}^*\\, H_d^0 H_u^0\n+ \\frac{1}{3} T^{\\kappa} \\, \\widetilde \\nu_{R}^* \\widetilde \\nu_{R}^* \n\\widetilde \\nu_{R}^*\\\n+\n\\text{h.c.} \\right)\n\\\\\n&+&\n\\left(m_{\\widetilde{L}_L}^2\\right)_{ij} \\widetilde{\\nu}_{iL}^* \\widetilde\\nu_{jL}\n+\n\\left( m_{H_d\\widetilde{L}_L}^2 \\right)_{i}H_d^{0*}\\widetilde{\\nu}_{iL}\n+\nm_{\\widetilde{\\nu}_R}^2 \\widetilde{\\nu}_{R}^* \\widetilde\\nu_{R} +\nm_{H_d}^2 {H^0_d}^* H^0_d + m_{H_u}^2 {H^0_u}^* H^0_u \\nonumber\n\\ ,\n\\label{akappa}\n\\\\\n\\nonumber\n\\\\\nV_{F} =&&\n \\lambda^2 H^0_{d}H_d^0{^{^*}}H^0_{u}H_u^0{^{^*}}\n +\n\\lambda^2\\tilde{\\nu}^{*}_{R}\\tilde{\\nu}_{R}H^0_{d}H_d^0{^*}\n +\n\\lambda^2\n\\tilde{\\nu}^{*}_{R}\\tilde{\\nu}_{R} H^0_{u}H_u^0{^*} \n\\nonumber\\\\ \n&+&\n\\kappa^2\\left(\\tilde{\\nu}^*_{R}\\right)^2\n\t\\left(\\tilde{\\nu}_{R}\\right)^2\n- \\left(\\kappa\\lambda\\left(\\tilde{\\nu}^{*}_{R}\\right)^2 H_d^{0*}H_u^{0*} \n -Y^{\\nu}_{i}\\kappa\\tilde{\\nu}_{iL}\\left(\\tilde{\\nu}_{R}\\right)^2H^0_{u}\n \\right.\n \\nonumber\\\\\n &+&\n \\left.\n Y^{\\nu}_{i}\\lambda\\tilde{\\nu}_{iL} H_d^{0*}H_{u}^{0*}H^0_{u}\n+{Y^{\\nu}_{i}}\\lambda \\tilde{\\nu}_{iL}^{*}\\tilde{\\nu}_{R}\\tilde{\\nu}_{R}^* H^0_{d}\n + \\text{h.c.}\\right) \n\\nonumber \\\\\n &+& \nY^{\\nu}_{i}{Y^{\\nu}_{i}} \\tilde{\\nu}^{*}_{R}\n\\tilde{\\nu}_{R}H^0_{u}H_u^0{^*} \n +\nY^{\\nu}_{i}{Y^{\\nu}_{j}}\\tilde{\\nu}_{iL}\\tilde{\\nu}_{jL}^{*}\\tilde{\\nu}_{R}^{*}\n \\tilde{\\nu}_{R} \n +\nY^{\\nu}_{i}{Y^{\\nu}_{j}}\\tilde{\\nu}_{i}\\tilde{\\nu}_{j}^* H^0_{u}H_u^{0*}\\, ,\n\\\\\n\\nonumber\n\\\\\nV_D =&&\n\\frac{1}{8}\\left(g_1^{2}+g_2^{2}\\right)\\left(\\widetilde\\nu_{iL}\\widetilde{\\nu}_{iL}^* \n+H^0_d {H^0_d}^* - H^0_u {H^0_u}^* \\right)^{2}\\, .\n\\label{dterms}\n\\end{eqnarray}\nUsing the decomposition from\n\\refeqs{eq:vevdecompH} - (\\ref{eq:vevdecompL}) \nthe linear and bilinear terms\nin the fields define the tadpoles $T_{\\varphi}$ and the scalar ${\\CP}$-even\nand ${\\CP}$-odd neutral mass matrices $m_{\\varphi}^2$ and\n$m_{\\sigma}^2$ after electroweak symmetry breaking,\n\\begin{align}\nV_H=\\cdots - T_{\\varphi_i}\\varphi_i + \\frac{1}{2} \\varphi^T m_{\\varphi}^2\n \\varphi + \\frac{1}{2} \\sigma^T m_{\\sigma}^2 \\sigma + \\cdots \\; .\n\\end{align}\nwhere we collectively denote with \n$\\varphi^T=(H_d^{\\mathcal{R}},H_u^{\\mathcal{R}},\n\\widetilde{\\nu}_{R}^{\\mathcal{R}},\\widetilde{\\nu}_{iL}^{\\mathcal{R}})$ \nand \n$\\sigma^T=(H_d^{\\mathcal{I}},H_u^{\\mathcal{I}},\n\\widetilde{\\nu}_{R}^{\\mathcal{I}},\\widetilde{\\nu}_{iL}^{\\mathcal{I}})$ \nthe ${\\CP}$-even and ${\\CP}$-odd scalar fields.\nThe linear terms are only allowed for ${\\CP}$-even fields and given by:\n\\begin{align}\nT_{H_d^{\\mathcal{R}}}=&-m_{H_d}^2v_d -\n\\left( m_{H_d\\widetilde{L}_L}^2 \\right)_{i} v_{iL} -\n\\frac{1}{8}\\left(g_1^2+g_2^2\\right)v_d \\left( v_d^2 + v_{iL}v_{iL} - v_u^2\\right)\n\\notag \\\\\n&-\\frac{1}{2}\\lambda\\left(v_R^2+v_u^2\\right)\\left(\\lambda v_d-v_{iL}Y^\\nu_i\\right)+\n\\frac{1}{\\sqrt{2}}T^\\lambda v_R v_u+\n\\frac{1}{2}\\kappa\\lambda v_R^2 v_u \\; , \\label{eq:tp1}\\\\[4pt]\nT_{H_u^{\\mathcal{R}}}=&-m_{H_u}^2v_u+\n\\frac{1}{8}\\left( g_1^2+g_2^2\\right)v_u\\left( v_d^2+v_{iL}v_{iL}-v_u^2\\right)\n\\notag \\\\\n&-\\frac{1}{2}\\lambda^2\\left( v_d^2+v_R^2\\right)+\n\\frac{1}{\\sqrt{2}}T^\\lambda v_d v_R+\n\\lambda v_d v_u v_{iL}Y^\\nu_i+\n\\frac{1}{2}\\kappa\\lambda v_d v_R^2-\n\\frac{1}{2}\\kappa v_R^2 v_{iL}Y^\\nu_i\n\\notag \\\\\n&-\\frac{1}{2}v_u\\left( v_{iL}Y^\\nu_i\\right)^2-\n\\frac{1}{\\sqrt{2}}v_R v_{iL}T^\\nu_i-\n\\frac{1}{2}v_R^2v_u Y^\\nu_i Y^\\nu_i \\; , \\label{eq:tp2}\\\\[4pt]\nT_{\\widetilde{\\nu}_R^{\\mathcal{R}}}=&-m_{\\widetilde{\\nu}_R}^2 v_R-\n\\frac{1}{\\sqrt{2}}T^\\kappa v_R^2-\\kappa^2 v_R^3+\n\\frac{1}{\\sqrt{2}}T^\\lambda v_d v_u-\n\\frac{1}{2}\\lambda^2 v_R\\left( v_d^2+v_u^2\\right)\n\\notag \\\\\n&+\\lambda v_d v_R v_{iL}Y^\\nu_i +\n\\kappa \\lambda v_d v_R v_u-\n\\kappa v_R v_u v_{iL} Y^\\nu_i-\n\\frac{1}{2}v_R\\left( v_{iL}Y^\\nu_i\\right)^2 \n\\notag \\\\\n&-\\frac{1}{\\sqrt{2}}v_u v_{iL}T^\\nu_i-\n\\frac{1}{2}v_Rv_u^2 Y^\\nu_i Y^\\nu_i \\; , \\label{eq:tp3}\\\\[4pt]\nT_{\\widetilde{\\nu}_{iL}^{\\mathcal{R}}}=&-\n\\left(m_{\\widetilde{L}_L}^2\\right)_{ij}v_{jL}-\n\\left( m_{H_d\\widetilde{L}_L}^2 \\right)_{i}v_d-\n\\frac{1}{8}\\left( g_1^2+g_2^2\\right)v_{iL}\\left( v_d^2+v_{jL}v_{jL}-v_u^2\\right)\n\\notag \\\\\n&+\\frac{1}{2}\\lambda v_d v_R^2 Y^\\nu_i-\n\\frac{1}{\\sqrt{2}}v_R v_u T^\\nu_i-\n\\frac{1}{2}\\kappa v_R^2v_uY^\\nu_i+\n\\frac{1}{2}\\lambda v_d v_u^2 Y^\\nu_i\n\\notag \\\\\n&-\\frac{1}{2}v_R^2Y^\\nu_i v_{jL}Y^\\nu_j-\n\\frac{1}{2}v_u^2Y^\\nu_i v_{jL}Y^\\nu_j \\label{eq:tp4} \\; .\n\\end{align}\nThe tadpoles vanish in the true vacuum of the model. During the\nrenormalization procedure they will be treated as OS parameters,\ni.e., finite corrections will be canceled by their corresponding\ncounterterms. This guarantees that the vacuum is\nstable w.r.t.\\ quantum corrections.\n\nThe bilinear terms \n\\begin{align}\nm_{\\varphi}^2= \n \\left( \\begin{array}{cccc} \n m_{H_{d}^{\\mathcal{R}}H_{d}^{\\mathcal{R}}}^{2}& \nm_{H_{d}^{\\mathcal{R}}H_{u}^{\\mathcal{R}}}^{2} & \nm_{H_{d}^{\\mathcal{R}}\\widetilde{\\nu}_{R}^{\\mathcal{R}}}^{2} & \nm_{H_{d}^{\\mathcal{R}}\\widetilde{\\nu}_{jL}^{\\mathcal{R}}}^{2}\n\\\\\nm_{H_{u}^{\\mathcal{R}}H_{d}^{\\mathcal{R}}}^{2} & \nm_{H_{u}^{\\mathcal{R}}H_{u}^{\\mathcal{R}}}^{2}& \nm_{H_{u}^{\\mathcal{R}}\\widetilde{\\nu}_{R}^{\\mathcal{R}}}^{2} & \nm_{H_{u}^{\\mathcal{R}}\\widetilde{\\nu}_{jL}^{\\mathcal{R}}}^{2}\n\\\\\n m_{\\widetilde{{\\nu}}^{\\mathcal{R}}_{R} H_{d}^{\\mathcal{R}}}& \nm_{\\widetilde{{\\nu}}^{\\mathcal{R}}_{R} H_{u}^{\\mathcal{R}}}& \nm_{\\widetilde{{\\nu}}^{\\mathcal{R}}_{R} \\widetilde{{\\nu}}^{\\mathcal{R}}_{R}}^{2} & \nm_{\\widetilde{{\\nu}}^{\\mathcal{R}}_{R} \\widetilde{{\\nu}}^{\\mathcal{R}}_{jL}}^{2} \n\\\\\nm_{\\widetilde{\\nu}_{iL}^{\\mathcal{R}} H_{d}^{\\mathcal{R}}}^{2} & \n m_{\\widetilde{\\nu}_{iL}^{\\mathcal{R}} H_{u}^{\\mathcal{R}}}^{2} & \nm_{\\widetilde{\\nu}_{iL}^{\\mathcal{R}} \\widetilde{\\nu}^{\\mathcal{R}}_{R}}^{2} & \nm_{\\widetilde{\\nu}_{iL}^{\\mathcal{R}} \\widetilde{\\nu}_{jL}^{\\mathcal{R}}}^{2} \n \\end{array} \\right)\\ , \n\\label{matrixscalar1}\n\\end{align}\nand\n\\begin{align}\nm^2_{\\sigma}= \n \\left( \\begin{array}{cccc} \n m_{H_{d}^{\\mathcal{I}}H_{d}^{\\mathcal{I}}}^{2} & \nm_{H_{d}^{\\mathcal{I}}H_{u}^{\\mathcal{I}}}^{2} & \nm_{H_{d}^{\\mathcal{I}}\\widetilde{\\nu}_{R}^{\\mathcal{I}}}^{2} & \nm_{H_{d}^{\\mathcal{I}}\\widetilde{\\nu}_{jL}^{\\mathcal{I}}}^{2}\n\\\\\nm_{H_{u}^{\\mathcal{I}}H_{d}^{\\mathcal{I}}}^{2} & \nm_{H_{u}^{\\mathcal{I}}H_{u}^{\\mathcal{I}}}^{2}& \nm_{H_{u}^{\\mathcal{I}}\\widetilde{\\nu}_{R}^{\\mathcal{I}}}^{2} & \nm_{H_{u}^{\\mathcal{I}}\\widetilde{\\nu}_{jL}^{\\mathcal{I}}}^{2}\n\\\\\n m_{\\widetilde{{\\nu}}^{\\mathcal{I}}_{R} H_{d}^{\\mathcal{I}}}^2& \nm_{\\widetilde{{\\nu}}^{\\mathcal{I}}_{R} H_{u}^{\\mathcal{I}}}^2& \nm_{\\widetilde{{\\nu}}^{\\mathcal{I}}_{R} \\widetilde{{\\nu}}^{\\mathcal{I}}_{R}}^{2} & \nm_{\\widetilde{{\\nu}}^{\\mathcal{I}}_{R} \\widetilde{{\\nu}}^{\\mathcal{I}}_{jL}}^{2} \n\\\\\nm_{\\widetilde{\\nu}_{iL}^{\\mathcal{I}} H_{d}^{\\mathcal{I}}}^{2} & \n m_{\\widetilde{\\nu}_{iL}^{\\mathcal{I}} H_{u}^{\\mathcal{I}}}^{2} & \nm_{\\widetilde{\\nu}_{iL}^{\\mathcal{I}} \\widetilde{\\nu}^{\\mathcal{I}}_{R}}^{2} & \nm_{\\widetilde{\\nu}_{iL}^{\\mathcal{I}} \\widetilde{\\nu}_{jL}^{\\mathcal{I}}}^{2}\n \\end{array} \\right)\\ , \n\\label{matrixscalar2} \n\\end{align}\nare $6\\times 6$ matrices in family space whose rather lengthy entries\nare given in the appendix in \\refse{app:cpeven} and \\refse{app:cpodd}.\nWe transform to the mass eigenstate basis\nof the ${\\CP}$-even scalars through a\nunitary transformation defined by the matrix $U^H$,\nthat diagonalizes the mass matrix $m_{\\varphi}^2$,\n\\begin{eqnarray}\nU^H\nm^2_{\\varphi}\\ {U^H}^{^T}=\nm^2_{h}\n\\ ,\n\\label{eq:scalarhiggs}\n\\end{eqnarray}\nwith \n\\begin{equation}\n\\varphi = {U^H}^{^T} h\\, ,\n\\label{physscalarhiggses}\n\\end{equation}\nwhere the $h_i$ are the ${\\CP}$-even scalar fields in the mass eigenstate basis. Without ${\\CP}$-violation in the scalar sector the matrix $U^H$ is real. Similarly, for the ${\\CP}$-odd scalar we define the rotation matrix $U^A$, that diagonalizes the mass matrix $m_{\\sigma}^2$,\n\\begin{eqnarray}\nU^A\nm^2_{\\sigma}\\ {U^A}^{^T}=\nm^2_{A}\n\\ , \\qquad \\text{with } \\sigma={U^A}^T A \\; .\n\\label{eq:scalaroddhiggs}\n\\end{eqnarray}\nBecause of the smallness of the neutrino Yukawa couplings $Y^\\nu_i$, which also implies that the left-handed sneutrino vevs $v_{iL}$ have to be small, so that the tadpole coefficients vanish at tree-level \\cite{Escudero:2008jg}, the mixing of the left-handed sneutrinos with the doublet fields and the singlet will be small. \n\nIt is a well known fact that the quantum corrections to the Higgs\npotential are highly significant in supersymmetric models, see\ne.g.\\ \\citeres{Heinemeyer:2004ms,Heinemeyer:2004gx,Djouadi:2005gj} \nfor reviews. As in the NMSSM~\\cite{Ellwanger:2009dp}, the upper bound on \nthe lowest Higgs mass squared at tree-level is relaxed through additional\ncontributions from the \nsinglet~\\cite{Escudero:2008jg};\n\\begin{equation}\nM_Z^2 \\left( \\cos^2 2\\beta +\n\\frac{2\\lambda^2}{g_1^2+g_2^2}\\sin^2 2\\beta \\right) \\; .\n\\end{equation}\nNevertheless, quantum corrections were \nstill shown to contribute significantly especially in the prediction of \nthe SM-like Higgs boson\nmass~\\cite{Draper:2016pys,Ellwanger:1993hn,Elliott:1993uc,Elliott:1993bs,Ender:2011qh,Drechsel:2016htw,Drechsel:2016ukp,Drechsel:2016jdg}.\nIn this paper we will investigate how\nimportant the unique loop corrections of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ beyond the NMSSM \nare in realistic scenarios. Before that we briefly describe the other\nrelevant sectors of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}.\n\n\n\n\\subsection{Squark sector}\n\nThe numerically most important one-loop corrections to the scalar\npotential are expected from the stop\/top-sector, analogous to the\n(N)MSSM~\\cite{ELLIS199183,Ellwanger:1993hn,Elliott:1993ex,Elliott:1993uc,Elliott:1993bs,Pandita:1993tg} due to the huge Yukawa\ncoupling of the (scalar) top. The tree-level mass matrices of the squarks differ\nslightly from the ones in the MSSM. Neglecting flavor mixing in the\nsquark sector, one finds for the up-type squark mass matrix\n$M^{\\widetilde{u}_i}$ of flavor $i$, \n\\begin{align}\n M^{\\widetilde{u}_i}_{11}&=m_{\\widetilde{Q}_{iL}}^2 + \\frac{1}{24}(3g_2^{2}-g_1^2)(v_d^2+v_{jL}v_{jL}-v_u^2)+\\frac{1}{2}v_u^2{Y^{u}_i}^2\n \\label{eq:usquarks11} \\\\\n M^{\\widetilde{u}_i}_{12}&=\\frac{1}{2}(\\sqrt{2}A^u_i v_u + v_RY^u v_{jL}Y^\\nu_j -\\lambda v_d v_R) \\label{eq:usquarks12} \\\\\n M^{\\widetilde{u}_i}_{22}&=m_{\\widetilde{u}_{iR}}^2+\\frac{1}{6}g_1^2(v_d^2+v_{jL}v_{jL}-v_u^2)+\\frac{1}{2}v_u^2{Y^u_i}^2 \\; .\n \\label{eq:usquarks22}\n\\end{align}\nIt should be noted that in the non-diagonal element explicitly\nappear the neutrino \nYukawa couplings. This term arises in the F-term contributions of the\nsquark potential through the quartic coupling of up-type quarks and one \nleft-handed and the right-handed sneutrino after EWSB.\nThe mass eigenstates $\\widetilde{u}_{i1}$ and $\\widetilde{u}_{i2}$ are\nobtained by the unitary transformation\n\\begin{equation}\n\\begin{pmatrix}\n \\widetilde{u}_{i1} \\\\\n \\widetilde{u}_{i2}\n\\end{pmatrix}\n=U^{\\widetilde{u}}_i\n\\begin{pmatrix}\n \\widetilde{u}_{iL} \\\\\n \\widetilde{u}_{iR}\n\\end{pmatrix}\\; ,\n\\qquad U^{\\widetilde{u}}_i{U^{\\widetilde{u}}_i}^\\dagger=\\mathbbm{1} \\; .\n\\end{equation}\nSimilarly, for the down-type squarks it is\n\\begin{align}\n M^{\\widetilde{d}_i}_{11}&=m_{\\widetilde{Q}_{iL}}^2 - \\frac{1}{24}(3g_2^{2}+g_1^2)(v_d^2+v_{jL}v_{jL}-v_u^2)-\\frac{1}{2}v_d^2{Y^{d}_i}^2 \n \\label{eq:dsquarks11} \\\\\n M^{\\widetilde{d}_i}_{12}&=\\frac{1}{2}(\\sqrt{2}A^d_i v_d -\\lambda v_d v_R) \\\\\n M^{\\widetilde{d}_i}_{22}&=m_{\\widetilde{d}_{iR}}^2-\\frac{1}{12}g_1^2(v_d^2+v_{jL}v_{jL}-v_u^2)+\\frac{1}{2}v_d^2{Y^d_i}^2 \\; .\n \\label{eq:dsquarks22}\n\\end{align}\nThe mass eigenstates $\\widetilde{d}_{i1}$ and $\\widetilde{d}_{i2}$ are obtained by the unitary transformation\n\\begin{equation}\n\\begin{pmatrix}\n \\widetilde{d}_{i1} \\\\\n \\widetilde{d}_{i2}\n\\end{pmatrix}\n=U^{\\widetilde{d}}_i\n\\begin{pmatrix}\n \\widetilde{d}_{iL} \\\\\n \\widetilde{d}_{iR}\n\\end{pmatrix}\\; ,\n\\qquad U^{\\widetilde{d}}_i{U^{\\widetilde{d}}_i}^\\dagger=\\mathbbm{1} \\; .\n\\end{equation}\n\n\n\n\n\\subsection{Charged scalar sector}\n\nSince $R$-parity, lepton number and lepton-flavor are broken, the six charged left- and right-handed sleptons mix with each other and with the two charged scalars from the Higgs doublets. In the basis $C^T=\n({H^-_d}^*,{H^+_u},\\widetilde{e}_{iL}^*,\\widetilde{e}_{jR}^*)$ we find\nthe following mass terms in the Lagrangian:\n\\begin{equation}\n\\cL_{C} \\; = \\; \n-{C^*}^T {m}^2_{H^+} C\\, ,\n\\label{matrix122}\n\\end{equation}\nwhere ${m}^2_{H^+}$ assuming ${\\CP}$ conservation is a symmetric\nmatrix of dimension 8,\n\\begin{align}\nm_{H^+}^2= \n \\left( \\begin{array}{cccc} \n m_{H_{d}^-{H^{-}_d}^{*}}^{2} & m_{H_{d}^- H_{u}^+}^{2} & \nm_{{H_d^-} \\widetilde{e}^*_{jL}}^2 & m_{{H_d^-} \\widetilde{e}^*_{jR}}^2 \\\\\n m_{{H_{u}^+}^* {H_d^-}^*}^{2} & m_{{H^{+}_u}^{*} H_{u}^{+}}^{2} & \nm_{{H_u^+}^*\\widetilde{e}^*_{jL}}^{2} & m_{{H_u^+}^*\\widetilde{e}^*_{jR}}^{2} \\\\\n m_{\\widetilde{e}_{iL} {{H^{-}_d}}^{*}}^2 & m_{\\widetilde{e}_{iL} H_u^+}^2 & \nm_{\\widetilde{e}_{iL} \\widetilde{e}_{jL}^{*}}^{2} & \nm_{\\widetilde{e}_{iL} \\widetilde{e}_{jR}^{*}}^{2}\\\\\n m_{\\widetilde{e}_{iR} {{H^{-}_d}}^{*}}^2 & m_{\\widetilde{e}_{iR} H_u^+}^2 & \n m_{\\widetilde{e}_{iR} \\widetilde{e}_{jL}^{*}}^{2} & \nm_{\\widetilde{e}_{iR} \\widetilde{e}_{jR}^{*}}^{2}\n \\end{array} \\right)\\ . \n \\label{matrixcharged2}\n\\end{align}\nThe entries are given\nin appendix \\ref{app:charged}.\nThe mass matrix is diagonalized by an orthogonal matrix $U^+$:\n\\begin{eqnarray}\nU^+\nm^2_{H^+}\\ {U^+}^{^T}\n=\n\\left(m^2_{H^+}\\right)^{\\text{diag}}\n\\ ,\n\\label{scalarhiggs22}\n\\end{eqnarray}\nwhere the diagonal elements of $\\left(m^2_{H^+}\\right)^{\\text{diag}}$ are the squared masses of the mass eigenstates\n\\begin{equation}\nH^+=U^+\\; C \\; ,\n\\end{equation}\nwhich include the charged Goldstone boson $H^+_1=G^\\pm_0$.\n\n\n\n\n\\subsection{Charged fermion sector}\n\nThe charged leptons mix with the charged gauginos and the charged higgsinos. Following the notation of \\citere{Ghosh:2017yeh} we write the relevant part of the Lagrangian in terms of two-component spinors \n$(\\chi^-)^T = \\left({({e_{iL})}^{c}}^{^*}, \\widetilde{W}^-, \\widetilde{H}^-_d\\right)$\nand \n$(\\chi^+)^T = \\left(({e_{jR}})^c, \\widetilde{W}^+, \\widetilde{H}^+_u\\right)$:\n\\begin{equation}\n\\cL_{\\chi^\\pm} \\; = \\; -({\\chi^{-}})^T \n{m}_{e}\n\\chi^+ + \\mathrm{h.c.}\\, .\n\\label{matrixcharginos0}\n\\end{equation}\nThe $5\\times5$ mixing matrix $m_e$ is defined by\n\\begin{equation}\nm_e=\n\\begin{pmatrix}\n \\frac{v_d Y^e_{11}}{\\sqrt{2}} & \\frac{v_d Y^e_{12}}{\\sqrt{2}} & \\frac{v_d Y^e_{13}}{\\sqrt{2}} & \\frac{g_2 v_{1L}}{\\sqrt{2}} &\n \t-\\frac{v_R Y^\\nu_1}{\\sqrt{2}} \\\\\n \\frac{v_d Y^e_{21}}{\\sqrt{2}} & \\frac{v_d Y^e_{22}}{\\sqrt{2}} & \\frac{v_d Y^e_{23}}{\\sqrt{2}} & \\frac{g_2 v_{2L}}{\\sqrt{2}} & \n \t-\\frac{v_R Y^\\nu_2}{\\sqrt{2}} \\\\\n \\frac{v_d Y^e_{31}}{\\sqrt{2}} & \\frac{v_d Y^e_{32}}{\\sqrt{2}} & \\frac{v_d Y^e_{33}}{\\sqrt{2}} & \\frac{g_2 v_{3L}}{\\sqrt{2}} &\n \t-\\frac{v_R Y^\\nu_3}{\\sqrt{2}} \\\\\n 0 & 0 & 0 & M_2 & \\frac{g_2 v_{u}}{\\sqrt{2}} \\\\\n -\\frac{v_{iL}Y^e_{1i}}{\\sqrt{2}} & -\\frac{v_{iL}Y^e_{2i}}{\\sqrt{2}} & -\\frac{v_{iL}Y^e_{3i}}{\\sqrt{2}} & \\frac{g_2 v_{d}}{\\sqrt{2}} & \\frac{\\lambda v_R}{\\sqrt{2}}\n\\end{pmatrix} \\; .\n\\end{equation}\nIt is diagonalized by two unitary matrices $U^e_L$ and $U^e_R$:\n\\begin{equation}\n{U_R^e}^{^*} m_{e} {U_L^e}^{^\\dagger} = m_{e}^{\\text{diag}}\n\\, ,\n\\label{diagmatrixneutralinosn}\n\\end{equation}\nwhere $m_{e}^{\\text{diag}}$ contains the masses of the charged fermions in the mass eigenstate base\n\\begin{eqnarray}\n\\chi^{+} = {U_L^e}^{^\\dagger} \\lambda^+\\, ,\n\\\\\n\\chi^{-} = {U_R^e}^{^\\dagger} \\lambda^-\\, .\n\\label{physcharginos}\n\\end{eqnarray}\nThe smallness of the left-handed sneutrino vevs in comparison to the doublet ones assures the decoupling of the three leptons from the Higgsino and the wino.\n\n\n\n\n\\subsection{Neutral fermion sector}\n\nThe three left-handed neutrinos and the right-handed neutrino mix with\nthe neutral Higgsinos and gauginos. Again, following\n\\citere{Ghosh:2017yeh} we write the relevant part of the Lagrangian in\nterms of two-component spinors\n$({\\chi^{0}})^T=\\left({(\\nu_{iL})^{c}}^{^*},\\widetilde B^0, \n\\widetilde W^{0},\\widetilde H_{d}^0,\\widetilde H_{u}^0,\\nu_{R}^*\\right)$\nas\n\\begin{equation}\n\\cL_{\\chi^0} \\; = \\;-\\frac{1}{2} ({\\chi^{0}})^T \n{m}_{\\nu} \n\\chi^0 + \\mathrm{h.c.}\\, ,\n\\label{matrixneutralinos}\n\\end{equation}\nwhere ${m}_{\\nu}$ is the $8\\times 8$ symmetric mass matrix. The neutral fermion mass matrix is determined by\n\\begin{equation}\n{m}_{\\nu}=\n\\begin{pmatrix}\n 0 & 0 & 0 & -\\frac{g_1 v_{1L}}{2} & \\frac{g_2 v_{1L}}{2} &\n 0 & \\frac{v_{R} Y^\\nu_1}{\\sqrt{2}} & \\frac{v_{u}Y^\\nu_1}{\\sqrt{2}} \\\\\n 0 & 0 & 0 & -\\frac{g_1 v_{2L}}{2} & \\frac{g_2 v_{2L}}{2} &\n 0 & \\frac{v_{R} Y^\\nu_2}{\\sqrt{2}} & \\frac{v_{u}Y^\\nu_2}{\\sqrt{2}} \\\\\n 0 & 0 & 0 & -\\frac{g_1 v_{3L}}{2} & \\frac{g_2 v_{3L}}{2} &\n 0 & \\frac{v_{R} Y^\\nu_3}{\\sqrt{2}} & \\frac{v_{u}Y^\\nu_3}{\\sqrt{2}} \\\\\n -\\frac{g_1 v_{1L}}{2} & -\\frac{g_1 v_{2L}}{2} & -\\frac{g_1 v_{3L}}{2} &\n M_1 & 0 & -\\frac{g_1 v_{d}}{2} & \\frac{g_1 v_{u}}{2} & 0 \\\\\n \\frac{g_2 v_{1L}}{2} & \\frac{g_2 v_{2L}}{2} & \\frac{g_2 v_{3L}}{2} &\n 0 & M_2 & \\frac{g_2 v_{d}}{2} & -\\frac{g_2 v_{u}}{2} & 0 \\\\\n 0 & 0 & 0 & -\\frac{g_1 v_{d}}{2} & \\frac{g_2 v_{d}}{2} & 0 &\n -\\frac{\\lambda v_{R}}{\\sqrt{2}} & -\\frac{\\lambda v_{u}}{\\sqrt{2}} \\\\\n \\frac{v_R Y^\\nu_1}{\\sqrt{2}} & \\frac{v_R Y^\\nu_2}{\\sqrt{2}} &\n \\frac{v_R Y^\\nu_3}{\\sqrt{2}} & \\frac{g_1 v_u}{2} & -\\frac{g_2 v_u}{2} &\n -\\frac{\\lambda v_R}{\\sqrt{2}} & 0 &\n \\frac{-\\lambda v_d + v_{kL}Y^\\nu_k}{\\sqrt{2}} \\\\\n \\frac{v_u Y^\\nu_1}{\\sqrt{2}} & \\frac{v_u Y^\\nu_2}{\\sqrt{2}} &\n \\frac{v_u Y^\\nu_3}{\\sqrt{2}} & 0 & 0 & -\\frac{\\lambda v_u}{\\sqrt{2}} &\n \\frac{-\\lambda v_d +v_{iL}Y^\\nu_i}{\\sqrt{2}} &\n \\sqrt{2}\\kappa v_R\n\\end{pmatrix}~.\n\\end{equation}\nBecause of the Majorana nature of the neutral fermions we can \ndiagonalize ${m}_{\\nu}$ with the help of just a single - but complex - \nunitary matrix $U^V$,\n\\begin{equation}\n{U^V}^{^*} m_{\\nu}\\ {U^V}^{^\\dagger} = m_{\\nu}^{\\text{diag}}\\, ,\n\\label{diagmatrixneutralinos}\n\\end{equation}\nwith\n\\begin{equation}\n\\chi^{0} = {U^V}^{^\\dagger} \\lambda^0\\, ,\n\\label{physneutralinos}\n\\end{equation}\nwhere $\\lambda^0$ are the two-component spinors in the mass basis. The eigenvalues of the diagonalized mass matrix $m_{\\nu}^{\\text{diag}}$ are the masses of the neutral fermions in the mass eigenstate basis. It turns out that the matrix $m_\\nu$ is of rank six, so it can only generate a single neutrino mass at tree-level.\\footnote{Including three generations of right-handed neutrinos, three light tree-level neutrino masses are generated.} The remaining two light neutrino masses can be generated by loop-effects.\n\n\n\n\n\n\\section{Renormalization of the Higgs potential at One-Loop}\n\\label{sec:renopot}\n\n\nThe first step in renormalizing the neutral scalar potential is to choose\nthe set of free parameters. These free parameters will receive a counter\nterm fixed by consistent renormalization conditions to cancel all\nultraviolet divergences that are produced by higher-order corrections. \n\nAt tree-level the relevant part of the Higgs potential $V_H$ is given by\nthe tadpole coefficients \\refeqs{eq:tp1}-(\\ref{eq:tp4}) and the ${\\CP}$-even and\n${\\CP}$-odd mass matrix elements in \\refeqs{matrixscalar1}\nand (\\ref{matrixscalar2}). The following\nparameters appear in the Higgs potential:\n\\begin{itemize}\n\t\\setlength\\itemsep{0.25em}\n\t\\renewcommand\\labelitemi{--}\n\t\\item Scalar soft masses: $m_{H_d}^2$, $m_{H_u}^2$, $m_{\\widetilde{\\nu}_R}^2$,\n\t\t\t$\\left( m_{\\widetilde{L}_L}^2 \\right)_{ij}$,\n\t\t\t$\\left( m_{H_d\\widetilde{L}_L}^2 \\right)_{i}$\n\t\t\t$\\quad$ (12 parameters)\n\t\\item Vacuum expectation values: $v_d$, $v_u$, $v_R$, $v_{iL}$\n\t\t\t$\\quad$ (6 parameters)\n\t\\item Gauge couplings: $g_1$, $g_2$ $\\quad$ (2 parameters)\n\t\\item Superpotential parameters: $\\lambda$, $\\kappa$, $Y^\\nu_i$\n\t\t\t$\\quad$ (5 parameters)\n\t\\item Soft trilinear couplings: $T^\\lambda$, $T^\\kappa$, $T^\\nu_i$\n\t\t\t$\\quad$ (5 parameters)\n\\end{itemize}\nThe complexity of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ Higgs scalar\nsector becomes evident when we\ncompare the numbers of free parameters (30) with the one in the real\nMSSM (7) \\cite{Frank:2006yh} and the NMSSM (12)\n\\cite{Drechsel:2016ukp}. While the number of free parameters is fixed,\nwe are free to replace some of the parameters by physical parameters. We\nchose to make the following replacements: \n\nThe soft masses $m_{H_d}^2$, $m_{H_u}^2$, $m_{\\widetilde{\\nu}_R}^2$, and\nthe diagonal elements of the matrix $m_{\\widetilde{L}_L}^2$ will\nbe replaced \nby the tadpole coefficients. The substitution is defined by\nthe tadpole~\\refeqs{eq:tp1}-(\\ref{eq:tp4})\nsolved for the soft mass parameters just\nmentioned. This will give us the possiblity\nto define the\nrenormalization scheme in a way that the true vacuum is not spoiled by\nthe higher-order corrections. The Higgs doublet vevs $v_d$ and $v_u$ will be replaced by the MSSM-like parameters\n$\\tan\\beta$ and $v$ according to \n\\begin{equation}\\label{eq:tanbetadef}\n\\tan\\beta=\\frac{v_u}{v_d} \\qquad \\text{and} \\qquad\nv^2=v_d^2+v_u^2+v_{iL}v_{iL} \\; .\n\\end{equation}\nNote that the definition of $v^2$ differs from the one in the MSSM by\nthe term $v_{iL}v_{iL}$. This allows to maintain the relations\nbetween $v^2$ and the gauge boson masses as they are in the\nMSSM. Numerically, the difference in the definition of $v^2$ is\nnegligible, since the $v_{iL}$ are of the order of $10^{-4}\\,\\, \\mathrm{GeV}$ in\nrealistic scenarios. Analytically, however, maintaining the\nfunctional form of $\\tan\\beta$ as it is in the (N)MSSM is convenient\nto facilitate the comparison of the quantum corrections in the\n\\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ and the NMSSM. In particular, we can still express the one-loop\ncounterterm of $\\tan\\beta$ without having to include the counterterms\nfor the left-handed sneutrino vevs.\nFor the vev of the\nright-handed sneutrino we chose to make the same substitution as was\ndone in previous calculations in the NMSSM \\cite{Drechsel:2016ukp} \n\\begin{equation}\\label{eq:mudef}\n\\mu=\\frac{v_R \\lambda}{\\sqrt{2}} \\; ,\n\\end{equation}\nwhere we make use of the fact that when the sneutrino obtains the vev,\nthe $\\mu$-term of the MSSM is dynamically generated.\nThe gauge couplings $g_1$ and $g_2$ will be replaced by the\ngauge boson masses $\\MW$ and $\\MZ$ via the definitions\n\\begin{equation}\\label{eq:gaguebosonmasses}\n\\MW^2=\\frac{1}{4}g_2^2v^2 \\qquad \\text{and} \\qquad\n\\MZ^2=\\frac{1}{4}\\left( g_1^2+g_2^2\\right) v^2 \\; .\n\\end{equation}\nThis is reasonable because the gauge boson masses are well measured\nphysical observables, so we can define them as OS parameters.\nInterestingly, the mass counterterm for $\\MW^2$ drops out at one-loop,\nbut it will contribute in the definition of the counterterm for $v^2$,\nso it is not a redundant parameter.\nFor the soft trilinear couplings we chose to adopt the redefinitions\n\\begin{equation}\nT^\\lambda = A^\\lambda \\lambda \\; , \\qquad T^\\kappa = A^\\kappa \\kappa\n\\; , \\qquad T^\\nu_i=A^\\nu_i Y^\\nu_i \\; .\n\\end{equation} \nThe reparametrization from the initial to the physical set of independent parameters is summarized in \\refta{tab:setpara}.\n\\begin{table}\n\\centering\n\\begin{tabular}{c c c c c}\nSoft masses & VEVs & Gauge cpl. & Superpot. & Soft trilinears \\\\\n\\hline\n$m_{H_d}^2$, $m_{H_u}^2$, $m_{\\widetilde{\\nu}_R}^2$,\n${m_{\\widetilde{L}_L}^2}_{ij}$,\n\t\t\t${m_{H_d\\widetilde{L}_L}^2}_i$ \n & $v_d$, $v_u$, $v_R$, $v_{iL}$ &\n$g_1$, $g_2$ &\n$\\lambda$, $\\kappa$, $Y^\\nu_i$ &\n$T^\\lambda$, $T^\\kappa$, $T^\\nu_i$ \\\\\n$\\downarrow$ & $\\downarrow$ & $\\downarrow$ & $\\downarrow$ & $\\downarrow$ \\\\\n $T_{H_d^{\\mathcal{R}}}$, $T_{H_u^{\\mathcal{R}}}$,\n $T_{\\widetilde{\\nu}_{R}^{\\mathcal{R}}}$,\n $T_{\\widetilde{\\nu}_{iL}^{\\mathcal{R}}}$,&\n $\\tan\\beta$, $v$, $\\mu$, $v_{iL}$ &\n $\\MW$, $\\MZ$ &\n $\\lambda$, $\\kappa$, $Y^\\nu_i$ &\n $A^\\lambda$, $A^\\kappa$, $A^\\nu_i$ \\\\\n ${m_{\\widetilde{L}_L}^2}_{i\\neq j}$,\n ${m_{H_d\\widetilde{L}_L}^2}_i$ & & & & \n\\end{tabular}\n\\caption{Set of independent parameters initially entering the tree-level Higgs potential of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ in the first row, and final choice of free parameters after the substitutions mentioned in the text.}\n\\label{tab:setpara}\n\\end{table}\n\nIn the following we will regard the entries of the neutral scalar mass matrix as functions of the final set of parameters,\n\\begin{align}\nm_{\\varphi}^2&=m_{\\varphi}^2\\left( \\MZ^2, v^2, \\tan\\beta, \\lambda, \\dots \\right) \\; , \\\\\nm_{\\sigma}^2&=m_{\\sigma}^2\\left( \\MZ^2, v^2, \\tan\\beta, \\lambda, \\dots \\right) \\; ,\n\\end{align}\nand we define their renormalization as\n\\begin{align}\nm_{\\varphi}^2&\\to m_{\\varphi}^2+\\delta m_{\\varphi}^2 \\; , \\\\\nm_{\\sigma}^2&\\to m_{\\sigma}^2+\\delta m_{\\sigma}^2 \\; .\n\\end{align}\nThe mass counterterms $\\delta m_{\\varphi}^2$ and $\\delta m_{\\sigma}^2$\nenter the renormalized one-loop scalar self-energies. They have to be\nexpressed as a linear combination of the counterterms of the independent\nparameters. We define their one-loop renormalization as\n\\begin{equation}\n \\begin{split}\nT_{H_d^{\\mathcal{R}}} &\\to\nT_{H_d^{\\mathcal{R}}}+\\delta T_{H_d^{\\mathcal{R}}} \\; , \\\\\nT_{H_u^{\\mathcal{R}}} &\\to\nT_{H_u^{\\mathcal{R}}}+\\delta T_{H_u^{\\mathcal{R}}} \\; , \\\\\nT_{\\widetilde{\\nu}_{R}^{\\mathcal{R}}} &\\to\nT_{\\widetilde{\\nu}_{R}^{\\mathcal{R}}}+\\delta T_{\\widetilde{\\nu}_{R}^{\\mathcal{R}}}\n\\; , \\\\\nT_{\\widetilde{\\nu}_{iL}^{\\mathcal{R}}} &\\to\nT_{\\widetilde{\\nu}_{iL}^{\\mathcal{R}}}+ \\delta T_{\\widetilde{\\nu}_{iL}^{\\mathcal{R}}}\n\\; , \\\\\n{m_{\\widetilde{L}_L}^2}_{i\\neq j} &\\to\n{m_{\\widetilde{L}_L}^2}_{i\\neq j}+ \\delta {m_{\\widetilde{L}_L}^2}_{i\\neq j}\n\\; , \\\\\n{m_{H_d\\widetilde{L}_L}^2}_i &\\to\n{m_{H_d\\widetilde{L}_L}^2}_i+\\delta {m_{H_d\\widetilde{L}_L}^2}_i \\; ,\n \\end{split}\n\\qquad\n \\begin{split}\n\\tan\\beta &\\to \\tan\\beta+\\delta \\tan\\beta \\; , \\\\\nv^2 &\\to v^2+\\delta v^2 \\; , \\\\\n\\mu &\\to \\mu +\\delta \\mu \\; , \\\\\nv_{iL}^2 &\\to v_{iL}^2+\\delta v_{iL}^2 \\; , \\\\\n\\MW^2 &\\to \\MW^2 +\\delta \\MW^2 \\; , \\\\\n\\MZ^2 &\\to \\MZ^2 +\\delta \\MZ^2 \\; , \\\\\n \\end{split}\n \\qquad\n \\begin{split}\n\\lambda &\\to \\lambda + \\delta \\lambda \\; , \\\\\n\\kappa &\\to \\kappa + \\delta \\kappa \\; , \\\\\nY^\\nu_i &\\to Y^\\nu_i + \\delta Y^\\nu_i \\; , \\\\\nA^\\lambda &\\to A^\\lambda+\\delta A^\\lambda \\; , \\\\\nA^\\kappa &\\to A^\\kappa + \\delta A^\\kappa \\; , \\\\\nA^\\nu_i &\\to A^\\nu_i + \\delta A^\\nu_i \\; .\n \\end{split}\n\\end{equation}\nSince the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ is a renormalizable theory, the divergent parts of the\ncounterterms are fixed to cancel the UV divergences.\nThe finite pieces, and thus the meaning of the parameters have to be\nfixed by renormalization conditions. \nWe will adopt a mixed renormalization scheme, where tadpoles\nand gauge boson masses are fixed OS, and the other parameters are\nfixed in the $\\overline{\\text{DR}}$ scheme. The exact renormalization\nconditions will be given in \\refse{sec:condis}. The dependence of the\nmass counterterms $\\delta m_{\\varphi}^2$ and $\\delta\nm_{\\sigma}^2$ on the counterterms of the free parameters is given\nat one-loop by \n\\begin{equation}\n\\delta m_{\\varphi}^2=\\sum_{X\\in\\text{Free param.}} \\left(\n\\frac{\\partial}{\\partial X} m_{\\varphi}^2 \\right)\\delta X \\; , \n\\qquad\n\\delta m_{\\sigma}^2=\\sum_{X\\in\\text{Free param.}} \\left(\n\\frac{\\partial}{\\partial X} m_{\\sigma}^2 \\right)\\delta X \\; . \n\\label{eq:masscounterderive}\n\\end{equation}\nIn our calculation the mixing matrices are defined in a way to diagonalize the renormalized mass matrices, so they do not have to be renormalized, because they are defined exclusively by renormalized quantities. The expressions for the counterterms of the scalar mass matrices in the mass eigenstate basis are then simply\n\\begin{equation}\\label{eq:masscounterrot}\n\\delta m_{h}^2=U^H \\delta m_{\\varphi}^2 {U^H}^T \\; , \n\\qquad\n\\delta m_{A}^2=U^A \\delta m_{\\sigma}^2{U^A}^T \\; .\n\\end{equation}\nIt should be noted at this point that the counterterm matrices in the mass eigenstate basis $\\delta m_{h}^2$ and $\\delta m_{A}^2$ are not diagonal, as they would be in a purely OS renormalization procedure, which is often used in theories with flavor mixing \\cite{Grimus:2016hmw}.\n\nIn the following chapter we will discuss the field renormalization, which is necessary to obtain finite scalar self-energies at arbitrary momentum.\n\n\n\n\n\\subsection{Field renormalization}\n\nWe write the renormalization of the neutral scalar-component fields as\n\\begin{equation}\\label{eq:fieldrenodef}\n\\begin{pmatrix}\nH_d \\\\ H_u \\\\ \\widetilde{\\nu}_R \\\\ \\widetilde{\\nu}_{iL}\n\\end{pmatrix}\n\\to \\sqrt{Z} \n\\begin{pmatrix}\nH_d \\\\ H_u \\\\ \\widetilde{\\nu}_R \\\\ \\widetilde{\\nu}_{iL}\n\\end{pmatrix}=\n\\left( \\mathbbm{1} + \\frac{1}{2} \\delta Z \\right)\n\\begin{pmatrix}\nH_d \\\\ H_u \\\\ \\widetilde{\\nu}_R \\\\ \\widetilde{\\nu}_{iL}\n\\end{pmatrix} \\; ,\n\\end{equation}\nwhere $\\sqrt{Z}$ and $\\delta Z$ are $6\\times6$ dimensional matrices and the equal sign is valid at one-loop. It should be emphasized that in contrast to the MSSM and the NMSSM these matrices cannot be made diagonal even in the interaction basis. The reason is that the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ explicitly breaks lepton number and lepton flavor, so the fields $H_d$ and $\\widetilde{\\nu}_{iL}$ share exactly the same quantum numbers and kinetic mixing terms are already generated at one-loop order.\n\nFor the ${\\CP}$-even and ${\\CP}$-odd neutral scalar fields the \ndefinition in \\refeq{eq:fieldrenodef} implies the following field\nrenormalization in the mass eigenstate basis: \n\\begin{equation}\nh \\to \\left( \\mathbbm{1} + \\frac{1}{2} \\delta Z^H \\right) h \\; , \n\\qquad\nA \\to \\left( \\mathbbm{1} + \\frac{1}{2} \\delta Z^A \\right) A \\; ,\n\\end{equation}\nwith\n\\begin{equation}\\label{eq:fieldrenorotate}\n\\delta Z^H = U^H \\left(\\delta Z \\right) {U^H}^T \\; \\qquad \\text{and} \\qquad\n\\delta Z^A = U^A \\left(\\delta Z \\right) {U^A}^T \\; .\n\\end{equation}\nAs renormalization conditions for the field renormalization counterterms we chose to adopt the $\\overline{\\text{DR}}$ scheme. We calculate the UV-divergent part of the derivative of the scalar ${\\CP}$-even self-energies in the interaction basis and define\n\\begin{equation}\\label{eq:drfieldcounterdef}\n\\delta Z_{ij}=-\\left.\\frac{d}{dp^2}\\Sigma_{\\varphi_i \\varphi_j} \\right|^{\\rm div} \\; .\n\\end{equation}\nHere $\\mbox{}^{\\rm div}$ denotes taking the divergent part only,\nproporional to $\\Delta$, \n\\begin{equation}\n\\Delta = \\frac{1}{\\varepsilon}-\\gamma_E+\\ln{4\\pi} \\; ,\n\\end{equation}\nwhere loop integral are solved in $4-2\\varepsilon$ dimensions and\n$\\gamma_E=0.5772\\dots$ is the Euler-Mascharoni constant.\nSince the field renormalization constants contribute only via divergent\nparts, they do not contribute to the finite result after canceling \ndivergences in the self-energies.\nAs regularization\nscheme we chose dimensional reduction~\\cite{SIEGEL1979193,CAPPER1980479}, \nwhich was shown to be SUSY\nconserving at one-loop~\\cite{Stockinger:2005gx}. In contrast to the\nOS renormalization scheme our field renormalization matrices are\nhermitian.\nThis holds also\ntrue for the field renormalization in the mass eigenstate basis, because\nas already mentioned the rotations in \\refeq{eq:scalarhiggs} and\n\\refeq{eq:scalaroddhiggs} diagonalize the renormalized tree-level scalar\nmass matrices, so \\refeqs{eq:fieldrenorotate} do not introduce\nnon-hermitian parts into the field renormalization, that would have to\nbe canceled by a renormalization of the mixing matrices $U^H$ and $U^A$\nthemselves. \n\nIn appendix \\ref{app:fieldcounters} we list our field renormalization\ncounterterms $\\delta Z_{ij}$ in terms of the divergent quantity $\\Delta$.\nNote that the field counterterms mixing the down-type Higgs and the left-handed sleptons are proportional to the neutrino Yukawa couplings $Y^\\nu_i$, while the counterterms mixing different flavors of left-handed sneutrinos contain terms proportional to non-diagonal lepton Yukawa couplings $Y^e$ and terms proportional to $Y^\\nu_i Y^\\nu_j$. This is why their numerical impact is negligible, but they are needed for a consistent renormalization of the scalar self-energies.\n\n\n\n\n\\subsection{Renormalization conditions for free parameters}\n\\label{sec:condis}\n\nIn this section we describe our choice for the renormalization\nconditions, where we stick to the one-loop level everywhere. \nWe start\nwith the OS conditions for the gauge boson mass parameters and the\ntadpole coefficients followed by our definitions for the\n\\DRbar\\ renormalized parameters.\n\nThe SM gauge boson masses are renormalized OS requiring\n\\begin{equation}\n\\text{Re}\\left[{\\hat{\\Sigma}_{ZZ}}^T\\left( \\MZ^2 \\right) \\right]=0 \\qquad\n\\text{and} \\qquad\n\\text{Re}\\left[{\\hat{\\Sigma}_{WW}}^T\\left( \\MW^2 \\right) \\right]=0 \\; ,\n\\end{equation}\nwhere $\\hat{\\Sigma}^T$ stands for the transverse part of the renormalized gauge boson self-energy. For their mass counterterms these conditions yield\n\\begin{equation}\n\\delta \\MZ^2 = \\text{Re}\\left[ {\\Sigma}_{ZZ}^T\\left( \\MZ^2 \\right) \\right] \\qquad\n\\text{and} \\qquad\n\\delta \\MW^2 = \\text{Re}\\left[ {\\Sigma}_{WW}^T\\left( \\MW^2 \\right) \\right] \\; .\n\\end{equation}\nHere the ${\\Sigma}^T$ (without the hat) denote the transverse part of the unrenormalized gauge boson self-energies.\n\nFor the tadpole coefficients $T_{\\varphi_i}$ the OS conditions read\n\\begin{equation}\nT_{\\varphi_i}^{(1)}+\\delta T_{\\varphi_i}=0 \\; ,\n\\end{equation}\nwhere $T_{\\varphi_i}^{(1)}$ are the one-loop contributions to the linear terms of the scalar potential, stemming from tadpole diagrams shown in \\reffi{fig:tadpoles}. The tadpole diagrams are calculated in the mass eigenstate basis $h$. The one-loop tadpole contributions in the interaction basis $\\varphi$ are then obtained by the rotation\n\\begin{equation}\nT_\\varphi^{(1)} = {U^H}^T T_h^{(1)} \\; .\n\\end{equation}\nAccordingly we find for the one-loop tadpole counterterms\n\\begin{equation}\n\\delta T_{\\varphi_i}=-T_{\\varphi_i}^{(1)} \\; .\n\\end{equation}\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.6\\textwidth]{tadpoles.pdf}\n \\caption{Generic Feynman diagrams for the tadpoles $T_{h_i}$.}\n \\label{fig:tadpoles}\n\\end{figure}\n\nFor practical purposes we\ndecided to renormalize all remaining parameters in the $\\DRbar$\nscheme (reflecting the fact that there are no physical observables that\ncould be directly related to them). \nThe counterterms of each parameter were obtained by\ncalculating the divergent parts of one-loop corrections to different\nscalar and fermionic two- and three-point functions. \nWe state the determination of the counterterms in the (possible) order \nin which they can be successively derived.\nWe start with the counterterms that\nwere obtained by renormalizing certain neutral fermion self-energies. \n\n\\paragraph{\\protect\\boldmath Renormalization of $\\mu$:}\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.8\\textwidth]{chiSE.pdf}\n \\caption{Diagrams contributing to the neutral\n fermion self-energies in the interaction basis.}\n\\label{fig:neufermdiags}\n\\end{figure}\nThe $\\mu$ parameter appears isolated in the Majorana-type mass matrix of\nthe neutral fermions \n\\begin{equation}\n\\left(m_\\nu\\right)_{67}=-\\frac{\\lambda v_R}{\\sqrt{2}}=-\\mu \\; ,\n\\end{equation}\nwhich is the element mixing the down-type and the up-type Higgsinos\n$\\widetilde{H}_d$ and $\\widetilde{H}_u$.\nThe entries\n$\\left(m_\\nu\\right)_{ij}$ get one-loop corrections via the neutral\nfermion self-energies ${\\sum}_{\\neu{i} \\neu{j}}$, that for Majorana\nfermions can be decomposed as\\footnote{Left-handed components and\n right-handed components are the same for Majorana fields.} \n\\begin{equation}\n\\Sigma_{\\neu{i} \\neu{j}}\\left( p^2 \\right) = \\slashed{p}\\Sigma_{\\neu{i} \\neu{j}}^F\\left( p^2 \\right)+\\Sigma_{\\neu{i} \\neu{j}}^S\\left( p^2 \\right) \\; .\n\\end{equation}\nThe part $\\Sigma_{\\neu{i} \\neu{j}}^F$ is renormalized through field\nrenormalization and the part $\\Sigma_{\\neu{i} \\neu{j}}^S$ is\nrenormalized by both the field renormalization and a mass counter\nterm. Since we are interested in the mass renormalization we focus on\n$\\Sigma_{\\neu{i} \\neu{j}}^S$ and write for the renormalized self-energy\nat zero momentum \n\\begin{equation}\\label{eq:neufermreno}\n\\hat{\\Sigma}_{\\neu{i} \\neu{j}}^S\\left(0\\right)=\\Sigma_{\\neu{i} \\neu{j}}^S\\left(0\\right)-\\frac{1}{2}\n\t\\left( \\delta Z_{ki}^\\chi \\left( m_\\nu \\right)_{kj} +\n\t\t\t\\left( m_\\nu \\right)_{ik} \\delta Z_{kj}^\\chi \\right)\n\t\t-\\delta\\left(m_\\nu\\right)_{ij} \\; .\n\\end{equation}\nThe field renormalization constants can be obtained by calculating the divergent part of $\\Sigma_{\\neu{i} \\neu{j}}^F$:\n\\begin{equation}\n\\delta Z_{ij}^\\chi=-\\left.\\Sigma_{\\neu{i}\\neu{j}}^F\\right|^{\\rm div} \\; ,\n\\end{equation}\nwhere we make use of the fact that there are no divergences proportional\nto $p^2$ in our case. The divergent parts of the self-energies of the\nneutral fermions are calculated diagrammatically in the interaction\nbasis, where diagrams with mass insertions have to be included. In\n\\reffi{fig:neufermdiags} we show the generic diagrams potentially\ncontributing to the divergent part of the self-energies.\nDiagrams with a scalar mass insertion or more than\none fermionic mass insertion are power-counting finite, so we do\nnot depict them.\nThe diagram shown in \\reffi{fig:neufermdiags} with a mass\ninsertion on the chargino propagator\ncan be divergent depending on the expressions for the couplings of the\ncharginos.\n\nWe checked that our results for the field renormalization counterterms\nfor the neutral fermions are consistent with the one-loop anomalous\ndimensions $\\gamma_{ij}^{(1)}$ of the corresponding superfields, i.e., \n\\begin{equation}\n\\delta Z_{ij}^\\chi=\\frac{\\gamma_{ij}^{(1)}\\Delta}{16 \\pi^2} \\; .\n\\end{equation}\nTo extract $\\delta \\mu$ we now just have to identify \n\\begin{equation}\n\\delta\\left(m_\\nu\\right)_{67}=-\\delta\\mu \\; ,\n\\end{equation}\nand calculate the divergent part of $\\Sigma_{\\widetilde{H}_d \\widetilde{H}_u}^S$, which again is not momentum dependent. $\\delta\\mu$ is then given by\n\\begin{equation}\n\\delta\\mu=\\frac{1}{2}\\mu\\left(\n\t-\\left(\n\t\t\\delta Z_{66}^\\chi+\n\t\t\\delta Z_{77}^\\chi\n\t\\right)+\n\t\\frac{1}{\\lambda}\\left(\n\t\\delta Z_{16}^\\chi Y^\\nu_1+\n\t\\delta Z_{26}^\\chi Y^\\nu_2+\n\t\\delta Z_{36}^\\chi Y^\\nu_3 \\right)\\right) -\n\t\\left.\\Sigma_{\\widetilde{H}_d \\widetilde{H}_u}^S\\right|^{\\rm div} \\; ,\n\\end{equation}\nwhere we made us of the fact that the matrix $\\delta Z_{ij}^\\chi$ is real and symmetric and that components mixing left-handed neutrinos and the down-type Higgsino are the only non-diagonal elements contributing here.\n\nExplicit formulas for the counterterms of the parameters renormalized in the $\\DRbar$ scheme are listed in the appendix \\ref{app:paracounters}. \nFor the $\\DRbar$ counterterms we checked that in the limit $Y^\\nu_i\\rightarrow 0$ our results coincide with the one in the \nNMSSM~\\cite{Ellwanger:2009dp}.\n\n\\paragraph{\\protect\\boldmath Renormalization of $\\kappa$:}\nThe parameters $\\kappa$ appears isolated at tree-level in the three-point vertex that couples the right-handed neutrino to the right-handed sneutrino,\n\\begin{equation}\n\\Gamma_{\\nu_R \\nu_R \\widetilde{\\nu}_R}^{(0)}=-\\sqrt{2}\\kappa \\; .\n\\end{equation}\nThe divergences induced to this coupling at one-loop have to be absorbed by the field renormalization of the right-handed neutrino and sneutrino and the counterterm for $\\kappa$, which is the only parameter in the tree-level expression. We find\n\\begin{equation}\n\\delta \\kappa = \n\\frac{1}{\\sqrt{2}}\n\\left.\\Gamma{\\nu_R \\nu_R \\widetilde{\\nu}_R}^{(1)}\\right|^{\\rm div} -\n\\frac{1}{2}\\kappa\\left( \\delta Z_{33}+2 \\delta Z^\\chi_{88}\\right) \\; ,\n\\end{equation}\nwhere $\\Gamma{\\nu_R \\nu_R \\widetilde{\\nu}_R}^{(1)}\\rvert^{\\rm div}$ is the divergent part of the corresponding one-loop three-point function, and the terms containing the field renormalization is trivial, because there is only one singlet-like superfield so that no non-diagonal field renormalization constants appear. The divergent one-loop contributions to the vertex are calculated diagrammatically in the interaction basis. The only contributing generic diagrams are shown in \\reffi{fig:kappadiags}.\n\nAll other topologies, including diagrams with one or more\nmass insertion, are finite, and there are no diagrams with gauge bosons\ninstead of scalars in the loop, because there are three gauge-singlet\nfields on the outer legs. It turns out that the sum over the diagrams\nshown in \\reffi{fig:kappadiags} is also finite, so that \n$\\Gamma_{\\nu_R \\nu_R\\widetilde{\\nu}_R}^{(1)}\\rvert^{\\rm div}$ vanishes. \n\\begin{figure}[tb]\n \\centering\n \\includegraphics[width=0.6\\textwidth]{kappadiags.pdf}\n \\caption{Potentially divergent one-particle irreducible diagrams\n contributing to the three-point vertex between two right-handed\n neutrinos and one right-handed sneutrino.}\n \\label{fig:kappadiags}\n\\end{figure}\n\n\\paragraph{\\protect\\boldmath Renormalization of $\\lambda$:}\nHaving calculated $\\delta \\mu$ and $\\delta \\kappa$ we can extract the counterterm for $\\lambda$ in the neutral fermion sector. $\\lambda$ appears in the mass matrix element\n\\begin{equation}\n\\left( m_\\nu \\right)_{88}=\\frac{2\\kappa\\mu}{\\lambda} \\; .\n\\end{equation}\nMaking use of \\refeq{eq:neufermreno} we find\n\\begin{equation}\n\\delta\\lambda=\n\\lambda\\left( \\delta Z^\\chi_{88}+\\frac{\\delta\\kappa}{\\kappa}\n+\\frac{\\delta\\mu}{\\mu} \\right)-\\frac{\\lambda^2}{2\\mu\\kappa}\n\\left.\\Sigma_{\\nu_R \\nu_R}^S\\right|^{\\rm div} \\; ,\n\\end{equation}\nwhere we calculated the divergent part of the right-handed neutrino self-energie $\\Sigma_{\\nu_R \\nu_R}^S\\rvert^{\\rm div}$ diagrammatically in the interaction basis using the diagrams already shown in \\reffi{fig:neufermdiags}.\n\n\\paragraph{\\protect\\boldmath Renormalization of $A_\\kappa$:}\nThe counterterm for the parameter $A_\\kappa$ can be extracted from the one-loop corrections to the scalar three-point vertex of right-handed sneutrinos when $\\delta\\kappa$ is known and using the one-loop relation\n\\begin{equation}\n\\left[ \\frac{\\delta\\mu}{\\mu} - \\frac{\\delta\\lambda}{\\lambda} \\right]^{\\rm div}\n=\\left.\\frac{1}{2}\\delta Z_{33}\\right|^{\\rm div} \\; ,\n\\end{equation}\nwhich was found in the NMSSM~\\cite{Sperling:2013eva} and confirmed\nfor this work also in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}.\nFor the trilinear singlet vertex we have at tree-level \n\\begin{equation}\n\\Gamma_{\\widetilde{\\nu}_R\\widetilde{\\nu}_R\\widetilde{\\nu}_R}^{(0)}=\n-\\sqrt{2}\\kappa\\left(A_\\kappa+\\frac{6\\kappa\\mu}{\\lambda}\\right) \\; .\n\\end{equation}\nThe tree-level vertex does not depend on the momentum, so the one-loop counterterm for $A_\\kappa$ can be calculated through\n\\begin{equation}\n\\delta A_\\kappa = \\frac{1}{\\sqrt{2}\\kappa}\\left( \n\\left.\\Gamma_{\\widetilde{\\nu}_R\\widetilde{\\nu}_R\\widetilde{\\nu}_R}^{(1)}\\right|^{\\rm div} +\n\\frac{3}{2} \\delta Z_{33}\n\\Gamma_{\\widetilde{\\nu}_R\\widetilde{\\nu}_R\\widetilde{\\nu}_R}^{(0)} \\right)\n-A_\\kappa \\frac{\\delta \\kappa}{\\kappa}-\n\\frac{6\\kappa\\mu}{\\lambda} \\left( 2\\frac{\\delta\\kappa}{\\kappa}+\n\\frac{1}{2}\\delta Z_{33} \\right) \\; .\n\\end{equation}\nHere $\\Gamma_{\\widetilde{\\nu}_R\\widetilde{\\nu}_R\\widetilde{\\nu}_R}^{(1)}\\rvert^{\\rm div}$ is the divergent part of the one-loop corrections to the three-point vertex, which was calculated diagrammatically in the interaction basis. The number of contributing diagrams is rather high, so for simplicity we just show the topologies of the diagrams contributing, that potentially lead to divergences, in \\reffi{fig:triplescalardiags}.\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.7\\textwidth]{triplescalardiags.pdf}\n\\caption{Potentially divergent one-particle irreducible topologies\n contributing to a scalar three-point vertex at one-loop in the\n interaction basis. Diagrams with scalar mass insertions or more than\n one fermionic mass insertions are finite.} \n\\label{fig:triplescalardiags}\n\\end{figure}\nIn the case of the vertex\n$\\Gamma_{\\widetilde{\\nu}_R\\widetilde{\\nu}_R\\widetilde{\\nu}_R}$ we can\nneglect the diagrams with gauge bosons, because the right-handed\nsneutrinos are gauge singlets. \n\n\\paragraph{\\protect\\boldmath Renormalization of $A_\\lambda$:}\nThe counterterm for the parameter $A_\\lambda$ is like in the previous case extracted from the one-loop corrections to a scalar three-point function. Here we consider $\\Gamma_{H_d H_u \\widetilde{\\nu}_R}$, the coupling between the two doublet-type Higgses and the right-handed sneutrino. At tree-level it is\n\\begin{equation}\n\\Gamma_{H_d H_u \\widetilde{\\nu}_R}^{(0)}=\n\\frac{A_\\lambda \\lambda}{\\sqrt{2}}+\n\\sqrt{2}\\kappa\\mu \\; ,\n\\end{equation}\nso we will make use of the fact that we already know the counterterms for $\\lambda$, $\\kappa$ and $\\mu$.\n\nThe final expression defining $\\delta A_\\lambda$ will also contain the\ntree-level expressions for the couplings where the down-type Higgs is\nreplaced by one of the left-handed sneutrinos. They are induced by the\nnon-diagonal field renormalization of $H_d$ and\n$\\widetilde{\\nu}_{iL}$ and enter the\nrenormalization of $\\Gamma_{H_d H_u \\widetilde{\\nu}_R}$ at one-loop. We\nfind\n\\begin{align}\n\\delta A_\\lambda&=\n-\\frac{\\sqrt{2}}{\\lambda}\n\\left.\\Gamma_{H_d H_u \\widetilde{\\nu}_R}^{(1)}\\right|^{\\rm div}-\n\\frac{1}{\\sqrt{2}\\lambda}\\left(\n\\delta Z_{11}\\Gamma_{H_d H_u \\widetilde{\\nu}_R}^{(0)}+\n\\delta Z_{14}\\Gamma_{\\widetilde{\\nu}_{1L}H_u\\widetilde{\\nu}_R}^{(0)}+\n\\delta Z_{15}\\Gamma_{\\widetilde{\\nu}_{2L}H_u\\widetilde{\\nu}_R}^{(0)}\n \\right. \\notag \\\\ +& \\left.\n\\delta Z_{16}\\Gamma_{\\widetilde{\\nu}_{3L}H_u\\widetilde{\\nu}_R}^{(0)}+\n\\delta Z_{22}\\Gamma_{H_d H_u \\widetilde{\\nu}_R}^{(0)}+\n\\delta Z_{33}\\Gamma_{H_d H_u \\widetilde{\\nu}_R}^{(0)} \\right)-\n\\frac{A_\\lambda}{\\lambda}\\delta\\lambda-\n\\frac{2\\kappa}{\\lambda}\\delta\\mu-\n\\frac{2\\mu}{\\lambda}\\delta\\kappa \\; ,\n\\end{align}\nwith\n\\begin{equation}\n\\Gamma_{\\widetilde{\\nu}_{iL}H_u\\widetilde{\\nu}_R}^{(0)}=\n\\frac{-Y^\\nu_i\\left( A^\\nu_i+\\frac{2\\kappa\\mu}{\\lambda} \\right)}{\\sqrt{2}}\n\\; .\n\\end{equation}\n\n\\paragraph{\\protect\\boldmath Renormalization of $v^2$:}\nThe SM-like vev is renormalized via the\nrenormalization of the electromagnetic coupling in the Thompson limit,\nwhich can be done when the counterterms for the gauge boson masses are\nfixed. We follow here the approach of \\citere{Drechsel:2016ukp} used in\nthe NMSSM to be able to compare the results in both models as best as\npossible.\n\nThe renormalization of the electromagnetic coupling is defined by\n\\begin{equation}\ne \\to e \\left( 1+ \\delta Z_e \\right) \\; ,\n\\end{equation}\nand the counterterm $\\delta Z_e$ can be calculated via\n\\begin{equation}\n\\left.\\delta Z_e\\right|^{\\rm div}=\n\\left[\n\\frac{1}{2} \\left( \\frac{\\partial \\Sigma_{\\gamma\\gamma}^T}{\\partial p^2} \\left( 0 \\right) \\right) +\n\\frac{\\SW}{\\CW \\MZ^2} \\Sigma_{\\gamma Z}^T \\left( 0 \\right)\n\\right]^{\\rm div} \\; ,\n\\end{equation}\nwhere $\\Sigma_{\\gamma\\gamma}^T(0)$ is the transverse part of the photon\nself-energy and $\\Sigma_{\\gamma Z}^T$ is the transverse part of the\nmixed photon-Z boson self-energy. $\\SW$ and $\\CW$ are defined as \n$\\SW = \\sqrt{1 - \\CW^2}$ with $\\CW = \\MW\/\\MZ$. $v^2$ and $e$ are related by\n\\begin{equation}\nv^2=\\frac{2\\SW^2 \\MW^2}{e^2} \\; ,\n\\end{equation}\nso the counterterm $\\delta v^2$ can be obtained through\n\\begin{equation}\\label{eq:dv2def}\n\\delta v^2 = \\frac{4 \\SW^2 \\MW^2}{e^2}\\left.\\left(\n\\frac{\\delta \\SW^2}{\\SW^2}+\\frac{\\delta \\MW^2}{\\MW^2}-\n2\\delta Z_e \\right)\\right|^{\\rm div} \\; ,\n\\end{equation}\nwhere\n\\begin{equation}\\label{eq:dsw2def}\n\\SW^2 \\rightarrow \\SW^2 + \\delta \\SW^2 \\; , \\quad\n\\text{with } \\quad\n\\delta \\SW^2=-\\CW^2 \\left( \\frac{\\delta \\MW^2}{\\MW^2}\n-\\frac{\\delta \\MZ^2}{\\MZ^2} \\right) .\n\\end{equation}\nHere we take only the divergent parts of the counterterms \n$\\delta \\MZ^2$,\n$\\delta \\MW^2$ and $\\delta Z_e$, so that $\\delta v^2$ is\nrenormalized in the $\\DRbar$ scheme. \nThis implies that the counterterm $\\delta Z_e$ is not a\nfree parameter, even if we calculated it as if it\nwould be to determine $\\delta v^2$. Instead $\\delta Z_e$ is a\ndependent parameter defined by $\\delta v^2$ in the $\\DRbar$ scheme\nand $\\delta M_Z^2$ and $\\delta M_W^2$ in the OS scheme through\n\\refeq{eq:dv2def} and \\refeq{eq:dsw2def},\n\\begin{equation}\n\\delta Z_e = \\frac{1}{2 \\SW^2}\n\\left( \\CW^2 \\frac{\\delta \\MZ^2}{\\MZ^2}\n +\\left( \\SW^2-\\CW^2 \\right)\\frac{\\delta \\MW^2}{\\MW^2}\n -\\frac{e^2}{4\\MW^2}\\delta v^2 \\right) \\; .\n\\end{equation}\n\n\\paragraph{\\protect\\boldmath Renormalization of $v_{iL}^2$:}\nThe counterterms for the three vevs of the left-handed sneutrinos\n$v_{iL}$ can be extracted from the divergent part of the one-loop\nself-energies $\\Sigma_{\\widetilde{B}\\nu_{iL}}$ between the bino and the\ncorresponding left-handed neutrino. The tree-level mass matrix entries\nwe renormalize are defined by\n\\begin{equation}\n\\left( m_\\nu \\right)_{4i}=-\\frac{g_1 v_{iL}}{2} \\; ,\n\\end{equation}\nso it is necessary to have the counterterm of the gauge \ncoupling $g_1$, whose renormalization we define as \n$g_1\\rightarrow g_1+\\delta g_1$. We then can obtaine \n$\\delta g_1$ from $\\delta \\MW^2$, $\\delta \\MZ^2$\nand $\\delta v^2$ through the definitions of the gauge boson masses in\n\\refeq{eq:gaguebosonmasses},\n\\begin{equation}\n\\delta g_1 = \\frac{2}{g_1 v^2}\\left( \\delta M_Z^2-\\delta M_W^2 \\right)\n\t-\\frac{g_1}{2}\\frac{\\delta v^2}{v^2} \\; .\n\\end{equation}\nRenormalizing the self-energies\n$\\Sigma_{\\widetilde{B}\\nu_{iL}}$ using \\refeq{eq:neufermreno} we\nfind the following expression for the $\\delta v_{iL}^2$:\n\\begin{equation}\n\\delta v_{iL}^2=\\frac{4 v_{iL}}{g_1}\n\\left.\\Sigma_{\\widetilde{B}\\nu_{iL}}^S\\right|^{\\rm div}-\nv_{iL}\\left( \\delta Z_{44}^\\chi v_{iL} +\n\\delta Z_{ij}^\\chi v_{jL} \n+ \\delta Z_{i6}^\\chi v_d \\right) -\n2v_{iL}^2\\left.\\frac{\\delta g_1}{g_1}\\right|^{\\rm div} \\; ,\n\\end{equation}\nwhere again the divergent contributions of $\\Sigma_{\\widetilde{B}\\nu_{iL}}^S$ are calculated diagrammatically in the interaction basis.\n\n\\paragraph{\\protect\\boldmath Renormalization of $Y^\\nu_i$:}\nThe counterterm for the neutrino Yukawas $Y^\\nu_i$ can be extracted in the neutral fermion sector as well. We decide to use the renormalization of the tree-level masses\n\\begin{equation}\n\\left( m_\\nu \\right)_{i7}=\\frac{\\mu Y^\\nu_i}{\\lambda} \\; ,\n\\end{equation}\nthat mix the left-handed neutrinos and the up-type Higgsino. Since we already found $\\delta\\lambda$ and $\\delta\\mu$ we can get $\\delta Y^\\nu_i$ from the divergent part of the one-loop self-energies $\\Sigma_{\\nu_{iL}\\widetilde{H}_u}^S$,\n\\begin{equation}\n\\delta Y^\\nu_i = \\frac{1}{2}\\left( \\delta Z_{16}^\\chi \\lambda -\n\\delta Z_{77}^\\chi Y^\\nu_i - \\delta Z_{ij}^\\chi Y^\\nu_j \\right)\n- \\left( \\frac{\\delta\\mu}{\\mu}-\\frac{\\delta\\lambda}{\\lambda} \\right)\n+\\frac{\\lambda}{\\mu} \\left.\\Sigma_{\\nu_{iL}\\widetilde{H}_u}^S\\right|^{\\rm div} \\; .\n\\end{equation}\n\n\\paragraph{\\protect\\boldmath Renormalization of $\\tb$:}\nWe adopted the usual definition for $\\tb$ as in the MSSM (see\n\\refeq{eq:tanbetadef}). \nIf we define the renormalization for the vevs of the doublet fields as\n\\begin{equation}\\label{eq:vudreno}\n v_d^2 \\rightarrow v_d^2 + \\delta v_d^2 \\; , \\quad\n v_u^2 \\rightarrow v_u^2 + \\delta v_u^2 \\; ,\n\\end{equation}\nthe counterterm for $\\tb$ can be written at one-loop as a linear\ncombination of the counterterms for the vevs of\nthe doublet Higgses, \n\\begin{equation}\\label{eq:dtanbetadef}\n\\delta \\tb = \\frac{1}{2}\\tb\\left( \\frac{\\delta v_u^2}{v_u^2} -\n\\frac{\\delta v_d^2}{v_d^2} \\right) \\; .\n\\end{equation}\nNote that our renormalization of $v_u^2$ and $v_d^2$ in \\refeq{eq:vudreno} \nincludes the contributions from the field renormalization constants inside \nthe counterterms $\\delta v_u^2$ and $\\delta v_d^2$. This approach is \nequivalent as defining \n\\begin{equation}\n v_d \\rightarrow \\sqrt{Z_{11}}\n \t\\left(v_d + \\delta \\hat{v}_d\\right) \\; , \\quad\n v_u \\rightarrow \\sqrt{Z_{22}}\n \t\\left(v_u + \\delta \\hat{v}_u\\right) \\; ,\n\\end{equation}\nand writing the counterterm of $\\tan\\beta$ as\n\\begin{equation}\\label{eq:tanfieldreno}\n \\delta \\tan\\beta = \\frac{1}{2}\\tan\\beta\n \t\\left( \\delta Z_{22} - \\delta Z_{11} \\right) +\n \t\\tan\\beta \\left( \\frac{\\delta\\hat{v}_u}{v_u} -\n \t\t\\frac{\\delta\\hat{v}_d}{v_d} \\right) \\; .\n\\end{equation}\nThis notation was convenient in the MSSM and the NMSSM, because the \nsecond bracket in \\refeq{eq:tanfieldreno} is finite at \none-loop~\\cite{Chankowski:1992er,Dabelstein:1994hb,Ender:2011qh,Drechsel:2016ukp}\nand can be set to zero in the $\\DRbar$ scheme, \nso that $\\delta \\tan\\beta$ can be expressed exclusively by the\nfield renormalization constants. In contrast, \nin the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ we find\n\\begin{equation}\n\\left.\\left( \\frac{\\delta\\hat{v}_u}{v_u} -\n \t\t\\frac{\\delta\\hat{v}_d}{v_d} \\right)\\right|^{\\rm div}\n=-\\frac{\\Delta \\lambda v_{iL} Y^\\nu_i }{32 \\pi^2 v_d} \\; .\n\\end{equation}\nThere are several possibilities to extract the counterterms $\\delta\nv_d^2$ and $\\delta v_u^2$. \nA convenient choice is to extract $\\delta v_d^2$\nfrom the renormalization of the entry of the neutral fermion\nmass matrix\nmixing the up-type Higgsino and\nthe right-handed neutrino, \n\\begin{equation}\n\\left( m_\\nu \\right)_{78}=\\frac{-\\lambda v_d + v_{iL}Y^\\nu_i}{\\sqrt{2}} \\; ,\n\\end{equation}\nbecause in this case no non-diagonal field renormalization counterterms are needed. Calculating the divergent part of $\\Sigma_{\\widetilde{H}_u v_R}^S$ and using the counterterms previously calculated we can extract $\\delta v_d^2$ via the expression\n\\begin{align}\n\\delta v_d^2 =& -\\frac{2\\sqrt{2}v_d}{\\lambda}\n\\left.\\Sigma_{\\widetilde{H}_u v_R}^S\\right|^{\\rm div}+\n\\frac{v_d}{\\lambda}\\left( \\delta Z_{77}^\\chi+ \\delta Z_{88}^\\chi \\right)\n\\left( -v_d\\lambda + v_{iL}Y^\\nu_i \\right) -\n2v_d^2\\frac{\\delta\\lambda}{\\lambda} \\notag \\\\\n&+\\frac{v_d}{\\lambda} Y^\\nu_i \\left( \\frac{\\delta v_L^2}{v_L} \\right)_i\n+\\frac{2v_d}{\\lambda}v_{iL}\\delta Y^\\nu_i \\; . \\label{eq:dvd2def}\n\\end{align}\nSince all counterterms appearing in \\refeq{eq:dvd2def}\nare renormalized in the $\\DRbar$ scheme also\n$\\delta v_d^2$ has no finite part.\nThere are now two ways to determine $\\delta v_u^2$. Firstly, we could similarly to $\\delta v_d^2$ extract the counterterm $\\delta v_u^2$ by renormalizing the up-type Higgsino self-energy $\\Sigma_{\\widetilde{H}_u \\widetilde{H}_u}^S$. Alternatively, we can deduce $\\delta v_u^2$ from the definition of $v^2$ in \\refeq{eq:tanbetadef} and simply write\n\\begin{equation}\\label{eq:dvu2def}\n\\delta v_u^2 = \\delta v^2 -\\delta v_d^2 - \\delta v_{1L}^2 - \\delta v_{2L}^2 - \\delta v_{3L}^2 \\; .\n\\end{equation}\nWe verified that both options yield the same result, which constitutes a\nconsistency test for the counterterms $\\delta v_{iL}^2$, which are\nunique for the \\ensuremath{\\mu\\nu\\mathrm{SSM}}. Inserting $\\delta v_d^2$ from \\refeq{eq:dvd2def}\nand $\\delta v_u^2$ from \\refeq{eq:dvu2def} into \\refeq{eq:dtanbetadef}\nfinally gives the counterterm for $\\tb$.\nWe checked that the final expression for $\\tb$ in \\refeq{eq:dtbdrbar}\nagrees with the NMSSM result in the limit\n$Y^\\nu_i\\rightarrow 0$.\n\nThe renormalization of $\\tb$ in the $\\DRbar$ scheme is manifestly process-independent and has shown to give stable numerical results in the MSSM \\cite{Frank:2002qf,Freitas:2002um} and the NMSSM \\cite{Drechsel:2016ukp,Ender:2011qh}.\n\n\\paragraph{\\protect\\boldmath Renormalization of $A^\\nu_i$:}\nThe soft trilinears $A^\\nu_i$ can be renormalized through the calculation of the radiative corrections to the corresponding scalar vertex in the interaction basis. The tree-level expression for the interaction between the up-type Higgs, one left-handed sneutrinos and the right-handed sneutrino is given by\n\\begin{equation}\n\\Gamma_{H_u \\widetilde{\\nu}_R \\widetilde{\\nu}_{iL}}^{(0)}=\n-\\left( \\frac{A^\\nu_i}{\\sqrt{2}}+\\frac{\\sqrt{2}\\kappa\n\\mu}{\\lambda} \\right) Y^\\nu_i \\; .\n\\end{equation}\nThe renormalized one-loop corrected vertex will define the counterterm for $A^\\nu_i$ since the counterterms for $\\kappa$, $\\mu$ and $\\lambda$ were already determined. We showed in \\reffi{fig:triplescalardiags} the topologies of the diagrams that have to be calculated in the interaction basis to get the divergent part of one-loop corrections\n$\\Gamma_{H_u \\widetilde{\\nu}_R \\widetilde{\\nu}_{iL}}^{(1)}$.\nAs in the case of the renormalization of $A^\\lambda$ the renormalization of the scalar vertex will contain the tree-level expressions of all the vertices with the same quantum numbers of the external fields, because of the non-diagonal field renormalization. Solved for $\\delta A^\\nu_i$ the renormalization of the vertex leads to\n\\begin{align}\n\\delta A^\\nu_i=&\n\\frac{\\sqrt{2}}{Y^\\nu_i}\\left.\\Gamma_{H_u \\widetilde{\\nu}_R\n\\widetilde{\\nu}_{iL}}^{(1)}\\right|^{\\rm div}+\n\\frac{1}{\\sqrt{2}Y^\\nu_i}\\left( \\left( \\delta Z_{22}+ \\delta Z_{33}\n\\right) \\Gamma_{H_u \\widetilde{\\nu}_R\n\\widetilde{\\nu}_{iL}}^{(0)}+\n\\delta Z_{1,3+i} \\Gamma_{H_u \\widetilde{\\nu}_R H_d}^{(0)} \\right. \\notag \\\\ \n&\\left.+\\delta Z_{3+j,3+i}\\Gamma_{H_u \\widetilde{\\nu}_R\n\\widetilde{\\nu}_{jL}}^{(0)}\\right)-\n\\frac{A^\\nu_i}{Y^\\nu_i}\\delta Y^\\nu_i -\n\\frac{2\\mu}{\\lambda}\\delta \\kappa -\n\\frac{2\\kappa}{\\lambda}\\delta \\mu -\n\\frac{2\\kappa\\mu}{\\lambda Y^\\nu_i}\\delta Y^\\nu_i +\n\\frac{2\\kappa\\mu}{\\lambda^2}\\delta\\lambda \\; ,\n\\end{align}\nwith\n\\begin{equation}\n\\Gamma_{H_u \\widetilde{\\nu}_R H_d}^{(0)} =\n\\frac{\\lambda A^\\lambda}{\\sqrt{2}}+\\sqrt{2}\\kappa\\mu \\; .\n\\end{equation}\n\n\\paragraph{\\protect\\boldmath Renormalization of ${m_{H_d\\widetilde{L}_L}^2}_i$:}\nThe soft scalar masses appear in the bilinear terms of the Higgs\npotential. They can be renormalized by calculating radiative corrections\nto scalar self-energies. It proved to be convenient \nto calculate the ${\\CP}$-odd scalar self-energies in the mass basis,\nand then to rotate the self-energies back to the interaction basis. \n\nWe find ${m_{H_d\\widetilde{L}_L}^2}_i$ at tree-level in\n\\begin{equation}\nm_{\\widetilde{\\nu}_{iL}^{\\mathcal{I}} H_{d}^{\\mathcal{I}}}^{2}=\n\\left(m_{H_d\\widetilde{L}_L}^2\\right)_i-\n\\frac{1}{2}v_R^2\\lambda Y^\\nu_i-\n\\frac{1}{2}v_u^2\\lambda Y^\\nu_i \\; .\n\\end{equation}\nThe general form of the renormalized scalar self-energies at one-loop is\n\\begin{align}\n\\hat{\\Sigma}_{X_i X_j}\\left( p^2 \\right) =&\n\\Sigma_{X_i X_j}\\left( p^2 \\right)+\n\\frac{1}{2} p^2 \\left( \\delta Z_{ji} + \\delta Z_{ij} \\right) \\notag \\\\\n-&\\frac{1}{2} \\left( \\delta Z_{ki} \\left( m_{X}^2 \\right)_{kj} +\n\\left( m_{X}^2 \\right)_{ik} \\delta Z_{kj} \\right) -\n\\delta \\left( m_{X}^2 \\right)_{ij} \\; ,\n\\end{align}\nwhere $X=(\\varphi,\\sigma)$ represents either the ${\\CP}$-even or the ${\\CP}$-odd scalar fields and we made use of the fact that the field renormalization constants $\\delta Z$ and the mass matrix $m_{X}^2$\nare real.\nDemanding that the renormalized self-energies $\\hat{\\Sigma}_{A_i A_j}$ are finite in the mass eigenstate basis we can define the divergent parts of the mass counterterms via\n\\begin{equation}\n\\left.\\delta \\left( m_{A}^2 \\right)_{ij}\\right|^{\\rm div} =\n\\left.\\Sigma_{A_i A_j}\\left( 0 \\right) \\right|^{\\rm div} -\n\\frac{1}{2}\\left( \\left(\\delta Z^A\\right)_{ji} m_{A_j}^2 +\nm_{A_i}^2 \\left(\\delta Z^A\\right)_{ij} \\right)\\; \n\\label{eq:oddcounters} ,\n\\end{equation}\nwhere the field counterterms in the mass eigenstate basis were defined in \\refeq{eq:fieldrenorotate} and the masses $m_{A_i}^2$ are the eigenvalues of the diagonal ${\\CP}$-odd scalar mass matrix $m_{A}^2$. In \\reffi{fig:scalarselfs} we show the diagrams that have to be calculated to get the quantum corrections to scalar self-energies at one-loop in the mass eigenstate basis.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.76\\textwidth]{scalarselfscpodd.pdf}\n \\caption{Generic diagrams for the ${\\CP}$-even (h) and \n ${\\CP}$-odd (A) scalar self-energies in the mass\n eigenstate basis.}\n\\label{fig:scalarselfs}\n\\end{figure}\n\n\nWe calculated all diagrams in the 't Hooft-Feynman gauge, in which the Goldstone bosons $A_1$ and $H^\\pm_1$ and the ghost fields $u^\\pm$ and $u^Z$ have the same masses as the corresponding gauge bosons.\nCalculating the ${\\CP}$-odd self-energies $\\Sigma_{A_i A_i}$ diagrammatically, we get the mass counterterms in mass eigenstate basis through the \\refeq{eq:oddcounters}. Now inverting the rotation in \n\\refeq{eq:masscounterrot} we can get the mass counterterms for the ${\\CP}$-odd self-energies in the interaction basis via \n\\begin{equation}\\label{eq:oddcountersrot}\n\\left.\\delta m_{\\sigma}^2\\right|^{\\rm div}=\n{U^A}^T \\left.\\delta m_{A}^2\\right|^{\\rm div} U^A \\; ,\n\\end{equation}\nRecognizing that\n\\begin{equation}\n\\left(\\delta m_{\\sigma}^2\\right)_{3+i,1}=\n\\delta m_{\\widetilde{\\nu}_{iL}^{\\mathcal{I}} H_{d}^{\\mathcal{I}}}^{2} \\; ,\n\\end{equation}\nand that $m_{\\widetilde{\\nu}_{iL}^{\\mathcal{I}} H_{d}^{\\mathcal{I}}}^{2}$\n depends on $(m_{H_d\\widetilde{L}_L}^2)_i$, we can extract\n$\\delta (m_{H_d\\widetilde{L}_L}^2)_i$ through\n\\begin{align}\n\\delta \\left(m_{H_d\\widetilde{L}_L}^2\\right)_i=&\n\\left.\\left(\\delta m_{\\sigma}^2\\right)_{3+i,1}\\right|^{\\rm div}+\n\\frac{2\\mu Y^\\nu_i}{\\lambda}\\delta\\mu +\n\\lambda \\left( v_d^2+v_u^2\\right) Y^\\nu_i \\cos^3\\beta\\sin\\beta\n\\; \\delta\\tb \\notag \\\\\n&+\\frac{1}{2}\\lambda Y^\\nu_i \\sin^2\\beta \\; \\delta v^2 -\n\\frac{1}{2}\\lambda\\sin^2\\beta Y^\\nu_i\\left( \\delta v_{1L}^2 +\n\\delta v_{2L}^2 + \\delta v_{3L}^2 \\right) \\notag \\\\\n+& \\left( \\frac{\\mu^2}{\\lambda}+\\frac{1}{2}\\lambda\\left(\nv_d^2+v_u^2 \\right) \\sin^2\\beta \\right) \\delta Y^\\nu_i \\notag \\\\\n-&\\left( \\frac{\\mu^2 Y^\\nu_i}{\\lambda^2}-\\frac{1}{2}\\left( v_d^2+v_u^2 \\right)Y^\\nu_i\\sin^2\\beta\\right)\\delta\\lambda \\; .\n\\end{align}\n\n\\paragraph{\\protect\\boldmath Renormalization of ${m_{\\widetilde{L}_L}^2}_{ij}$:}\nSince we neglect ${\\CP}$-violation the counterterms for the \nnon-diagonal elements of the hermitian matrix\n${m_{\\widetilde{L}_L}^2}_{ij}$ \nare symmetric under the exchange of the indices $i$ and $j$.\nThen we can extract the counterterms for the non-diagonal elements in the same way as the ones for ${m_{H_d\\widetilde{L}_L}^2}_i$ in the \n${\\CP}$-odd scalar sector. They appear in the tree-level mass matrix in \n\\begin{equation}\nm_{\\widetilde{\\nu}_{iL}^{\\mathcal{I}} \\widetilde{\\nu}_{jL}^{\\mathcal{I}}}^{2}=\n\\left(m_{\\widetilde{L}}^2\\right)_{ij}+\n\\frac{1}{2}\\left(v_R^2+v_u^2\\right)Y^\\nu_iY^\\nu_j \\quad \\text{for }\ni\\neq j \\; .\n\\end{equation}\nHence, the counterterms\n$\\delta\\left({m_{\\widetilde{L}_L}^2}\\right)_{ij}$ for $i\\neq j$ \nare given by\n\\begin{align}\n\\delta\\left({m_{\\widetilde{L}_L}^2}\\right)_{ij}=&\n\\left.\\left(\\delta m_{\\sigma}^2\\right)_{3+i,3+j}\\right|^{\\rm div} -\n\\frac{1}{2}\\left( v_R^2+v_u^2 \\right)\\left(Y^\\nu_i\\delta Y^\\nu_j -\nY^\\nu_j\\delta Y^\\nu_i \\right) \\\\\n&-\\frac{2\\mu Y^\\nu_i Y^\\nu_j}{\\lambda^2}\\left( \\frac{\\delta\\mu}{\\mu}-\n\\frac{\\delta\\lambda}{\\lambda} \\right)-\n\\frac{1}{2}Y^\\nu_i Y^\\nu_j \\sin^2\\beta\\; \\delta v^2 \\notag \\\\\n&-\\left( v_d^2+v_u^2 \\right)Y^\\nu_i Y^\\nu_j \\cos^3\\beta\\sin\\beta\n\\;\\delta\\tb+\n\\frac{1}{2}Y^\\nu_i Y^\\nu_j\\sin^2\\beta\n\\left( \\delta v_{1L}^2 + \\delta v_{3L}^2 + \\delta v_{3L}^2 \\right)\n\\; . \\notag\n\\end{align} \n\\paragraph{\\protect\\boldmath \\FA\\ modelfile:}\nThe diagrams and their amplitudes\nthat had to be calculated to obtain the counterterms, as described\nin this section,\nwere generated using the Mathematica package\n\\FA~\\cite{Hahn:2000kx} and further evaluated with the package\n\\FC~\\cite{Hahn:1998yk}. The \\FA\\ model file for the\n\\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ was created with the Mathematica program \\texttt{SARAH}\n\\cite{Staub:2009bi}. We modified the model file to neglect ${\\CP}$-violation \nby choosing all relevant parameters to be real. We also neglected \nflavor-mixing in the squark- and the quark-sector \nin this work. The \\FA\\ model file can be provided by the authors upon\nrequest.\nThe calculation of renormalized two- and three-point\nfunctions of the neutral scalars of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ at one-loop accuracy is\nthereby fully automated. \n(as it is in the MSSM~\\cite{Fritzsche:2013fta}).\n\nIn \\refse{sec:numanal} we will present our predictions for the Higgs\nmasses in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ compared to the ones of the NMSSM.\nTo be able to make this comparison, we had to calculate the\nNMSSM-predictions in the same renormalization scheme and using\nthe same conventions as were used in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}.\nThis is why we calculated the one-loop self-energies in the NMSSM \nwith our own NMSSM-modelfile for \\FA\/\\FC\\ created with\n\\texttt{SARAH} using the same procedure as for the \\ensuremath{\\mu\\nu\\mathrm{SSM}}.\nWe verified that the results\ncalculated in the NMSSM with our modelfile are equal to the results\ncalculated with the modelfile presented in \\citere{Paehr:276451}, which\nwas a good check that the generation of the modelfiles for the NMSSM\nand the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ was correct.\n\n\n\n\\section{Loop corrected Higgs boson masses}\n\\label{sec:getmasses}\n\nIn the previous section we have derived an\nOS\/\\DRbar\\ renormalization scheme for the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ Higgs sector. This\ncan be applied (via the future \\FA\\ model file, once the\ncounterterms are implemented) to any higher-order\ncorrection in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}. As a first application, we evaluate the full\none-loop corrections to the ${\\CP}$-even scalar sector in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}.\nDue to the still missing implementation of counterterms\nin the \\FA\\ model file,\nthe calculation of the renormalized scalar self-energies is done in\ntwo steps. Firstly, the unrenormalized self-energies are calculated\nusing \\FA\\ and \\FC, and subsequently the self-energies are renormalized\nsubtracting (by hand) the field renormalization and\nmass counterterms, as will\nbe described in the next section.\n\n\n\\subsection{Evaluation at one-loop} \n\nHere we describe the final form of the renormalized ${\\CP}$-even scalar\nself-energies \n$\\hat{\\Sigma}_{hh}$ and how the loop corrected physical masses of the\nHiggs boson masses are evaluated.\n\nThe one-loop renormalized self-energies in the mass eigenstate basis are\ngiven by\n\\begin{equation}\\label{eq:renomself}\n\\hat{\\Sigma}_{h_i h_j}^{(1)}\\left( p^2 \\right) = \\Sigma_{h_i h_j}^{(1)}\\left( p^2 \\right) + \\delta Z^H_{ij}\n\\left( p^2 - \\frac{1}{2}\\left( m_{h_i}^2 + m_{h_j}^2 \\right) \\right)\n- \\left(\\delta m_{h}^2\\right)_{ij} \\; ,\n\\end{equation}\nwith the field renormalization constants $\\delta Z^H$ \nand the mass counter terms $\\delta m_{h}^2$\nin the mass eigenstate basis\ndefined by the rotations in \\refeq{eq:fieldrenorotate}\nand \\refeq{eq:masscounterrot}.\n$\\Sigma_{h_i h_j}$ is the unrenormalized self-energy obtained by\ncalculating the diagrams shown in \\reffi{fig:scalarselfs} \nwith the ${\\CP}$-even states $h$ on the external legs.\nThe self-energies were calculated in the Feynman\ngauge, so that gauge-fixing terms do not yield counterterm contributions\nin the Higgs sector at one-loop. The loop integrals were regularized\nusing dimensional reduction \\cite{SIEGEL1979193,CAPPER1980479} and\nnumerically evaluated for arbitrary real momentum using\n\\LT~\\cite{Hahn:1998yk}. The contributions from complex values of $p^2$\nwere approximated using a Taylor expansion with respect to the imaginary part\nof $p^2$ up to first order. \n\nIn \\refeq{eq:renomself} we already made use of the fact that $\\delta\nZ^H$ is real and symmetric in our renormalization scheme. The mass\ncounterterms are defined as functions of the counterterms of the free\nparameters following \\refeq{eq:masscounterderive} and\n\\refeq{eq:masscounterrot}. They contain finite contributions from the\ntadpole counterterms and from the counterterm for the gauge boson mass\n$\\MZ^2$. The matrix $\\delta m_{h}^2$ is real and symmetric.\n\nThe renormalized self-energies enter the inverse propagator matrix\n\\begin{equation}\\label{eq:renoselfdef}\n\\hat{\\Gamma}_{h}=\\text{i} \\left[ p^2\\; \\mathbbm{1} -\n\\left( m_{h}^2 - \\hat{\\Sigma}_{h}\\left(p^2\\right) \\right)\n\\right] \\; , \\qquad \\text{with }\n\\left(\\hat{\\Sigma}_{h}\\right)_{ij} = \n\\hat{\\Sigma}_{h_i h_j} \\; .\n\\end{equation}\nThe loop-corrected scalar masses squared are the zeroes of the\ndeterminant of the inverse propagator matrix. The determination of\ncorrected masses has to be done numerically when we want to account for\nthe momentum-dependence of the renormalized self-energies. This is done\nby an iterative method that has to be carried out for each of the six\nsquared loop-corrected masses \\cite{Fuchs:2015jwa}. \n\n\n\n\n\\subsection{Inclusion of higher orders}\\label{sec:higherorders}\n\nIn \\refeq{eq:renoselfdef} we did not include the superscript $^{(1)}$ in\nthe self-energies. Restricting the numerical evaluation to a pure\none-loop calculation would lead to very large theoretical uncertainties.\nThese can be avoided by the inclusion of corrections beyond the one-loop\nlevel. Here we follow the approach of \\citere{Drechsel:2016jdg} and\nsupplement the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ one-loop results by higher-order corrections in\nthe MSSM limit as provided by \\fh\\ (version 2.13.0)~\\cite{Heinemeyer:1998yj,Hahn:2009zz,Heinemeyer:1998np,Degrassi:2002fi,Frank:2006yh,Hahn:2013ria,Bahl:2016brp,feynhiggs-www}.\nIn this way the leading and subleading two-loop corrections are\nincluded, as well as a resummation of large logarithmic terms, see the\ndiscussion in \\refse{sec:intro}, \n\\begin{equation}\n\\hat{\\Sigma}_{h}\\left( p^2 \\right) = \n\\hat{\\Sigma}_{h}^{(1)}\\left( p^2 \\right) +\n\\hat{\\Sigma}_{h}^{(2')} +\n\\hat{\\Sigma}_{h}^{\\rm resum} \\; .\n\\end{equation}\nIn the partial two-loop contributions $\\hat{\\Sigma}_{h}^{(2')}$ we take\nover the corrections of\n\\order{\\alpha_s\\alpha_t,\\alpha_s\\alpha_b,\\alpha_t^2,\\alpha_t\\alpha_b}, \nassuming that the MSSM-like corrections are also valid in the\n\\ensuremath{\\mu\\nu\\mathrm{SSM}}. This assumption \nis reasonable since the only difference between the squark sector of the\n\\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ in comparison to the MSSM are the terms proportional to \n$Y^\\nu_i v_{iL}$ in the non-diagonal element of the up-type squark mass \nmatrices (see \\refeq{eq:usquarks12})\nand the terms proportional to $v_{iL}v_{iL}$ in the diagonal\nelements of the up- and down-type squark mass matrices\n(see \\refeq{eq:usquarks11}, \\refeq{eq:usquarks22},\n\\refeq{eq:dsquarks11} and \\refeq{eq:dsquarks22}),\nwhich numerically will always be negligible in\nrealistic scenarios since $v_{iL}\\ll v_d,v_u,v_R$.\nFurthermore,. in \\citere{Drechsel:2016ukp} the quality of the MSSM\napproximation was tested in the NMSSM, showing that the genuine NMSSM\ncontributions are in most cases sub-leading. The same is expected for\nthe contributions stemming from the resummation of large logarithmic\nterms given by $\\hat{\\Sigma}_{h}^{\\rm resum}$. \n\n\n\n\n\\section{Numerical analysis}\n\\label{sec:numanal}\n\nIn the following we present for the first time the full one-loop\ncorrections to the scalar masses in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}, \nwith one generation of right-handed neutrinos obtained in the\nFeynman-diagrammtic approach, taking into account all parameters of the\nmodel and the complete dependence on the external momentum, which\nincludes a consistent treatment of the imaginary parts of the scalar\nself-energies. \nOur results extend the known ones in the literature of\nthe MSSM and the NMSSM to a model, which has a\nrich and unique phenomenology through explicit $R$-parity breaking. The\none-loop results are supplemented by known higher-loop results from the\nMSSM (see the previous section) to reproduce the Higgs mass value of \n$\\sim 125 \\,\\, \\mathrm{GeV}$~\\cite{Aad:2015zhl}. \nHere the theory uncertainty must be kept in mind. In the\nMSSM it is estimated to be at the level of \n$2-3 \\,\\, \\mathrm{GeV}$~\\cite{Degrassi:2002fi,Bahl:2017aev}, and in extended models\nit is naturally slightly larger. \n\nWe will present results in several different scenarios, in\nall of which one scalar with the correct SM-like Higgs mass\nis reproduced.\nTo get an estimation of the significance of quantum\ncorrections to the Higgs masses that are unique for the \\ensuremath{\\mu\\nu\\mathrm{SSM}}, we\ncompare the results to the corresponding ones in the NMSSM.\nThe results in the NMSSM are obtained by a calculation based on\n\\citere{Drechsel:2016ukp}, but with slightly changed renormalization\nconditions to be as close as possible to the calculation in the\n\\ensuremath{\\mu\\nu\\mathrm{SSM}}. While Ref.~\\cite{Drechsel:2016ukp} uses the mass squared of\nthe charged Higgs mass as input parameter and renormalizes it as OS\nparameter we instead use $\\DRbar$ conditions for $A^\\lambda$.\n\nThe benchmark points used in the following were not tested in detail\nagainst experimental bounds including the $R$-parity violating effects of\nthe \\ensuremath{\\mu\\nu\\mathrm{SSM}}. They have been chosen to exemplify the potential magnitude\nof unique \\ensuremath{\\mu\\nu\\mathrm{SSM}}-like corrections. Nevertheless, the values\nwe picked for the free parameters should be close to realistic and\nexperimentally allowed scenarios: the parameters in the scalar sector\nare taken over from calculations in the NMSSM~\\cite{Drechsel:2016ukp}, and\nunique \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ parameters are chosen in a range to reproduce\nneutrino masses of the correct order of magnitude. That means that the\nneutrino Yukawas $Y^\\nu_i$ should be of the order $10^{-6}$ to generate\nneutrino masses of the order less than $1\\,\\, \\mathrm{eV}$. \nFor the left-handed sneutrino\nvevs this directly implies $v_{iL}\\ll v_d, v_u$ so\nthat the tadpole coefficients vanish at tree-level\n\\cite{Escudero:2008jg}. We will leave a more detailed discussion of\nnumerical results for a future publication, in which we will also include\nthree generations of right-handed neutrinos.\n\n\n\n\\subsection{NMSSM-like crossing point scenario}\n\\label{sec:nmssmpoint}\n\n\nThe first scenario we want to analyze is one studied in the NMSSM with a\nsinglet becoming the LSP in the region of $\\lambda > \\kappa$\ntaken from Ref.~\\cite{Drechsel:2016ukp}. \nThis scenario was tested therein against the experimental limits \nimplemented in \\texttt{HiggsBounds 4.1.3}~\\cite{Bechtle:2008jh,Bechtle:2011sb,Bechtle:2013gu,Bechtle:2013wla,Bechtle:2015pma}.\nIt has the nice feature that there is a crossing point when\n$\\lambda\\approx\\kappa$ in the neutral scalar sector, in which the masses\nof the singlet and the SM-like Higgs become degenerate and NMSSM-like\nloop corrections become significant \\cite{Drechsel:2016jdg}. \n\nIn \\refta{tab:crossingpoint} we list the values chosen for the\nparameters. The SM-like parameters from the electroweak sector and\nthe lepton and quark masses are given in appendix \\ref{app:smvalues} \nin \\refta{tab:smsum}. The\nparameters present in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ and the NMSSM are of course chosen\nequally in both models. \nThe region $\\lambda < 0.026$ is excluded\nbecause the left-handed sneutrinos become tachyonic at tree-level. The\nflavor-changing non-diagonal elements in the slepton sector are\nzero. The value for $A^\\lambda$ is chosen to correspond to a mass of\n$m_{H^\\pm}= 1000 \\,\\, \\mathrm{GeV}$ for the charged Higgs mass in the NMSSM with\n$m_{H^\\pm}$ renormalized OS and $A^\\lambda$ not being a free\nparameter. $A^\\kappa$ should be chosen to be negative in our convention\n(when $\\kappa$ is positive) to avoid false vacua~\\cite{Escudero:2008jg}\nor tachyons in the pseudo-scalar sector~\\cite{Ghosh:2014ida}. It should\nbe kept in mind \nthat the diagonal soft scalar masses in the neutral sector are extracted\nfrom the values for\n$v_{iL}$, $\\tb$ and\n$\\mu$ via the tadpole equations, and their non-diagonal,\nflavor-violating elements are always set to zero at tree-level.\nThis is of crucial importance for the comparison of the\nscalar masses\nin the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ and the NMSSM, since in the NMSSM the soft slepton masses\n$m_{\\widetilde{L}}^2$ are independent parameters, while in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ the\ndiagonal elements are dependent parameters fixed by the tadpole\n\\refeqs{eq:tp4}, \nwhen the vevs are used as input.\nThe latter strategy is particularly convenient since the order\nof magnitude of the vevs is roughly fixed through the electroweak seesaw mechanism\nby demanding neutrino masses below the eV scale,\nwhile the soft scalar masses are not directly related to any physical observable.\nConsequentially, for each parameter point\ncalculated in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}, the corresponding values that have to be chosen for\n$m_{\\widetilde{L}}^2$ in the NMSSM have to be adjusted accordingly,\ndefined as a function of all the free parameters appearing in the\nHiggs potential.\n\n\n\\begin{table}\n\\renewcommand{\\arraystretch}{1.7}\n\\centering\n\\begin{tabular}{c c c c c c c c c c c}\n $v_{iL}\/\\sqrt{2}$ & $Y^\\nu_i$ & $A^\\nu_i$ & $\\tb$ & $\\mu$ & $\\lambda$ &\n \t$A^\\lambda$ & $\\kappa$ & $A^\\kappa$ & $M_1$ & \\\\\n \\hline\n $10^{-4}$ & $10^{-6}$ & $-1000$ & $8$ & $125$ & $[0.026;0.3]$ &\n \t$897.61$ & $0.2$ & $-300$ & $143$ & \\\\\n \\hline\n \\hline\n $M_2$ & $M_3$ & $m_{\\widetilde{Q}_{iL}}^2$ &\n \t$m_{\\widetilde{u}_{iR}}^2$ & $m_{\\widetilde{d}_{iR}}^2$ &\n \t$A^u_3$ & $A^u_{1,2}$ & $A^{d}_{1,2,3}$ & $(m_{\\widetilde{e}}^2)_{ii}$ &\n \t$A^e_{33}$ & $A^e_{11,22}$ \\\\\n \\hline\n $300$ & $1500$ & $1500^2$& $1500^2$ & $1500^2$ & $-2000$ & $-1500$ &\n \t$-1500$ & $200^2$ & $-1500$ & $-100$ \n\\end{tabular}\n\\caption{Input parameters for the NMSSM-like crossing point scenario;\n all masses and values for trilinear parameters are in GeV.}\n\\label{tab:crossingpoint}\n\\renewcommand{\\arraystretch}{1.0}\n\\end{table}\n\nIn \\reffi{fig:crossingspec} we show the resulting spectrum of the\n${\\CP}$-even scalars at tree-level and including the full one-loop and\ntwo-loop contributions.%\n\\footnote{Here and in the following we denote with ``two-loop'' result the\none-loop plus partial two-loop plus resummation corrected masses.}%\n~The standard model Higgs mass value\nis reproduced accurately when the quantum corrections are included. The\nheavy MSSM-like Higgs $H$ and the left-handed sneutrinos are at the\nTeV-scale and rather decoupled from the SM-like Higgs boson. The three\nleft-handed sneutrinos are degenerate because the \\ensuremath{\\mu\\nu\\mathrm{SSM}}-like parameters\nare set equal for all flavors. \n\\begin{figure}\n \\centering\n \\includegraphics[width=0.7\\textwidth]{nmssmspec.pdf}\n \\vspace{-0.4cm}\n \\caption{Spectrum of ${\\CP}$-even scalar masses in NMSSM-like\n crossing point scenario. The three left-handed sneutrinos\n $\\widetilde{\\nu}_{iL}$ are degenerate.\\protect\\footnotemark} \n \\label{fig:crossingspec}\n\\end{figure}\n\\footnotetext{All plots\n have been produced using \\texttt{ggplot2}~\\cite{ggplot2}\n and \\texttt{tikzDevice}~\\cite{tikzDevice} in \\texttt{R}~\\cite{Rlanguage}.}\nThe singlet-like scalar mass heavily depends on $\\lambda$,\nbecause when $\\mu$\nis fixed, increasing $\\lambda$ leads to\na smaller value for $v_R$ (see \\refeq{eq:mudef}).\nAs was observed in \\citere{Drechsel:2016ukp}, the loop-corrected\nmass of the singlet becomes smaller than the SM-like Higgs boson mass\nat about $\\lambda\\approx\\kappa$.\nWe observe non-negligible loop-corrections to the singlet in the\nregion of $\\lambda$ where the singlet is the lightest neutral scalar.\n\nDue to the similarity of the Higgs sectors of the\nNMSSM and the \\ensuremath{\\mu\\nu\\mathrm{SSM}}, the masses of the doublet-like Higgs bosons and\nthe right-handed sneutrino will be of comparable size as the masses\npredicted for the doublet-like Higgses and the singlet in the\nNMSSM.\nIn \\reffi{fig:nmssmdif} we show the tree-level and the one- and\ntwo-loop corrected mass of the SM-like Higgs boson in the\ncrossing-point scenario. One can see that, as expected, \nthe two-loop corrections are crucial to predict a SM-like Higgs mass of $125\\,\\, \\mathrm{GeV}$.\nIndeed, our analysis confirmed that differences in the prediction of the SM-like\nHiggs boson mass are negligible compared to the current experimental\nuncertainty~\\cite{Aad:2015zhl}\nand the anticipated\nexperimental accuracy of the ILC of about\n$\\lsim 50 \\,\\, \\mathrm{MeV}$~\\cite{Moortgat-Picka:2015yla}, even when there is a\nsubstantial mixing between left-handed sneutrinos and the SM-like\nHiggs at tree-level or one-loop.\nApart from that, they\nare clearly exceeded by the (future) parametric uncertainties in the Higgs-boson\nmass calculations. Consequently, the\nHiggs sector alone will not be sufficient to distinguish the\n\\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ from the NMSSM. On the other hand, we can regard\nthe theoretical uncertainties\nin the NMSSM and the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ to be at the same level of accuracy.\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.6\\textwidth]{nmssmcompare.pdf}\n \\vspace{-0.5cm}\n \\caption{Tree-level, one-loop and two-loop corrected\n masses of the SM-like Higgs boson in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ in the NMSSM-like\n crossing point scenario.} \n \\label{fig:nmssmdif}\n\\end{figure}\n\n\n\n\n\\subsection{\\protect\\boldmath Light $\\tau$-sneutrino scenario}\n\\label{sec:lighttaulsp}\n\nIn the previous scenario we observed that, in\na scenario where the left-handed sneutrinos where practically \ndecoupled from the SM-like\nHiggs boson, the unique \\ensuremath{\\mu\\nu\\mathrm{SSM}}-like\ncorrections do not account for a substantial deviation of the SM-like Higgs\nmass prediction compared to the NMSSM.\nIn this section we will investigate a scenario in which one of\nthe left-handed sneutrinos has a small mass close to SM Higgs boson\nmass. The phenomenology of such a spectrum was recently studied in\ndetail,\nincluding a comparison of its predictions with the\nLHC searches~\\cite{Ghosh:2017yeh,Lara:2018rwv}. It was found that a light\nleft-handed sneutrino as the LSP can give rise to distinct\nsignals for the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ (for instance, final states with\ndiphoton plus missing energy, diphoton plus leptons and multileptons).\n\\begin{table}\n\\renewcommand{\\arraystretch}{1.7}\n\\centering\n\\begin{tabular}{c c c c c c c c c c c}\n $v_{1,2L}\/\\sqrt{2}$ & $v_{3L}\/\\sqrt{2}$ & $Y^\\nu_i$ & $A^\\nu_i$ & $\\tb$ & $\\mu$ & $\\lambda$ &\n \t$A^\\lambda$ & $\\kappa$ & $A^\\kappa$ \\\\\n \\hline\n $10^{-5}$ & $4\\cdot 10^{-4}$ & $5\\cdot 10^{-7}$ & $-400$ & $10$ & $270$ & $[0.19;0.3]$ &\n \t$1000$ & $0.3$ & $-1000$\n\\end{tabular}\n\\caption{Input parameters for the light $\\tau$-sneutrino scenario;\n\t\t all masses and values for trilinear parameters are in GeV.}\n\\label{tab:lsptaupoint}\n\\renewcommand{\\arraystretch}{1.0}\n\\end{table}\n\nIn \\refta{tab:lsptaupoint} we list the relevant parameters that were\nchosen to\nobtain a light left-handed $\\tau$-sneutrino. The parameters not shown\nhere are chosen to be the same as in the previous case, shown in\n\\refta{tab:crossingpoint}. One can see that the vev\n$v_{3L}$ (corresponding to $\\widetilde{\\nu}_{3L}$) was increased w.r.t.\\\nthe NMSSM-like scenario. The reason for this becomes clear when one\nextracts the leading terms of the diagonal tree-level mass matrix\nelement of the left-handed sneutrinos,\n\\begin{equation}\\label{eq:approxnumass}\nm_{\\widetilde{\\nu}_{iL}^{\\mathcal{R}} \\widetilde{\\nu}_{iL}^{\\mathcal{R}}}^{2}\n\\approx \n\\frac{Y^\\nu_i v_R v_u}{2v_{iL}}\n\\left( - \\sqrt{2}A^\\nu_i-\\kappa v_R \n\t\t\t\t+\\frac{\\sqrt{2}\\mu}{\\tan\\beta} \\right) \t\n\t\t\t\t\\; .\n\\end{equation}\nThe tree-level masses of the left-handed sneutrinos are roughly\nproportional to the inverse of their vev. We also\ndecreased $A^\\nu_3$ in comparison to the previous scenario, keeping it\nnegative, so that it is of order $\\kappa v_R$ and the sum in the brackets of \\refeq{eq:approxnumass} becomes small.\n\n\\begin{figure}[!]\n \\centering\n \\hspace{1cm}\\includegraphics[width=0.8\\textwidth]{sntauLSPspec.pdf}\n \\vspace{-0.5cm}\n \\caption{${\\CP}$-even scalar mass spectrum of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ in the light\n $\\tau$-sneutrino scenario, see \\refta{tab:lsptaupoint}.\n On the right side we state the dominant composition\n of the mass eigenstates.}\n \\label{fig:sntaulspspec}\n\\end{figure}\n\nIn \\reffi{fig:sntaulspspec} we show the tree-level and loop-corrected\nspectrum of the scalars in the region of $\\lambda$ where there are no\ntachyons at tree-level.\nFor too small $\\lambda$ the tree-level mass of\n$\\widetilde{\\nu}_{3L}$ becomes tachyonic,\nbecause when $\\mu=(v_R \\lambda)\/ \\sqrt{2}$ is\nfixed $v_R$ has to grow and the second term in the bracket\nof \\refeq{eq:approxnumass} will grow larger than the sum of the first and the\nthird term.\nFor too large $\\lambda$,\nthe tree-level mass of the SM-like Higgs boson\nbecomes tachyonic.\nIn particular, it starts to mix with the tree-level singlet mass, which\nbecomes tachyonic because $v_R$ decreases when $\\lambda$\nincreases.\nThe central value of the\nSM Higgs boson mass is reproduced in this scenario\nup to values of $\\lambda\\leq\\num{0.22}$. However, \nconsidering the theoretical uncertainty even higher values of $\\lambda$ \ncan be viable. For $\\lambda=0.236$ the prediction for the SM-like \nHiggs mass decreases below $m_{h_1}\\approx 122\\,\\, \\mathrm{GeV}$. \nAs discussed in the introduction we assume a theory uncertainty of\n$\\sim 3 \\,\\, \\mathrm{GeV}$ on the mass evaluation, \nso we consider in this scenario the region $\\lambda\\leq0.236$ to be valid \nregarding the SM Higgs boson mass.\nAn interesting observation is\nthat the masses of light left-handed sneutrinos are\nmainly induced via quantum corrections, while the tree-level mass\napproaches 0 for small values of $\\lambda$.\nThis indicates that a consistent treatment of\nquantum corrections to light sneutrino masses is of\ncrucial importance.\n\nThe large upward shift of the left-handed sneutrino masses through\nthe one-loop \ncorrections is due to the fact that in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ the sneutrino fields\nare part of the Higgs potential, each with an associated\ntadpole coefficient $T_{\\widetilde{\\nu}_{iL}}$. To ensure the stability of the\nvacuum \\wrt quantum corrections, the tadpoles are renormalized OS, absorbing\nall finite corrections into the counterterms $\\delta T_{\\widetilde{\\nu}_{iL}}$\n(see \\refse{sec:condis}). In the mass counterterms for the left-handed\nsneutrinos the finite parts $\\delta T_{\\widetilde{\\nu}_{iL}}^{\\rm fin}$\nintroduce the main finite contribution in the form\n\\begin{equation}\n\\delta m_{\\widetilde{\\nu}_{iL}^{\\mathcal{R}}\n \\widetilde{\\nu}_{iL}^{\\mathcal{R}}}^{2\\;{\\rm fin}}\n = -\\frac{\\delta T_{\\widetilde{\\nu}_{iL}}^{\\rm fin}}{v_{iL}} + \\cdots \\; ,\n\\end{equation}\nwhich is enhanced by the inverse of the vev of $\\widetilde{\\nu}_{iL}$.\nIt is these terms inside the counterterms of the renormalized self-energies\n$\\hat{\\sum}_{\\widetilde{\\nu}_{iL}^{\\mathcal{R}}\n \\widetilde{\\nu}_{iL}^{\\mathcal{R}}}^{(1)}$ that shift the poles of the\npropagator matrix and increase the masses of the left-handed sneutrinos,\nespecially in cases where the tree-level masses are small.\n\n\\begin{figure}[!]\n \\centering\n \\includegraphics[width=0.9\\textwidth]{sntauLSPcompare.pdf}\n \\vspace{-1.0cm}\n \\caption{Light $\\tau$-sneutrino scenario, see \\refta{tab:lsptaupoint}.\n In the shaded region the prediction for the SM-like Higgs mass is \n below $122\\,\\, \\mathrm{GeV}$. \\textit{Left:} Masses of the SM-like Higgs, the left-handed\n $\\tau$-sneutrino and the right-handed sneutrino in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ at\n tree-level and one-loop. \\textit{Right:} Masses of the SM-like\n Higgs and the singlet in the NMSSM at tree-level and\n one-loop.}\n \\label{fig:sntaulspcompare}\n\\end{figure}\n\nThis behavior is a peculiarity of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}, meaning that the leptonic\nsector and the Higgs sector are mixed through the breaking of $R$-parity.\nThe relations between the vevs $v_{iL}$ and the soft masses\n$m_{\\widetilde{L}}^2$ via the tadpole equations automatically lead to\ndependences between the sneutrino masses and, for instance, the neutrino\nor the Higgs sector.\nIn the NMSSM, on the other hand, the sneutrinos are not part of the\nHiggs potential, since the fields are protected by lepton-number conservation.\nThere, the soft masses $m_{\\widetilde{L}}^2$ are, without further assumptions,\nfree parameters that can be chosen without taking into account\nany leptonic observable (such as neutrino masses and mixings).\nIn principle, the additional dependences of the\n\\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ scalar (neutrino) masses on the neutrino sector could be\nused (e.g.\\ when all neutrino masses and mixing angles will be\nknown with sufficient experimental accuracy)\nto restrict the possible range of $m_{\\widetilde{L}}^2$, and thus the\npossible values for the left-handed sneutrino masses. However, with\nour current experimental knowledge on the neutrino masses,\nthe possible values for the vevs $v_{iL}$, and hence the\npossible range of left-handed sneutrino masses, are\neffectively not yet constrained.\n\nIt should be noted as well, that the soft masses\n$m_{\\widetilde{L}}^2$ also appear \nin the mass matrix of the charged scalars (see \\refeq{eq:mlinmch})\nand the pseudoscalars (see \\refeq{eq:mlinmps}).\nIn many cases they are the dominant\nterm in the tree-level masses of the left-handed sleptons and\nsneutrinos, so the values of the masses of charged sleptons and\nsneutrino of the same family will be close. A precise treatment of\nquantum corrections of the size observed in \\reffi{fig:sntaulspspec} is\nextremely important in those cases, since \nthey might easily change the relative sign of their\nmass differences. This can result in a complete change of the\nphenomenology of the corresponding benchmark point, for instance when\neither the neutral (pseudo)scalar or the charged scalar is the\nLSP~\\cite{Ghosh:2017yeh,Lara:2018rwv}.\n\nWe compare the relevant spectrum of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ to the\ncorresponding one in the NMSSM in \\reffi{fig:sntaulspcompare}.\nWe show the tree-level\nand one-loop corrected masses of the light scalars in the \\ensuremath{\\mu\\nu\\mathrm{SSM}},\nand the masses of the SM-like Higgs boson and the singlet in the\nNMSSM on the right, with parameters set accordingly.\nWe shade in grey the region of $\\lambda$ where the the prediction for\nthe SM-like Higgs boson mass is below $122\\,\\, \\mathrm{GeV}$ if two-loop corrections\nare included. As expected, the SM-like Higgs-boson mass and\nthe mass of \nthe singlet turn out to be equal in both models. Even in regions\nwhere there is a substantial mixing\nof the SM-like Higgs boson with the left-handed sneutrinos, something that\ncannot occur in the NMSSM, the differences in the SM-like Higgs\nmass prediction are not larger than a few keV.\n\n\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=1.0\\textwidth]{mixingsntau.pdf}\n \\vspace*{-1.0cm}\n \\caption{Light $\\tau$-sneutrino scenario, see \\refta{tab:lsptaupoint}.\n We show the absolute values of the mixing matrix elements at\n tree-level $|U_{1i}^{H(0)}|$ (left) and\n $|U_{2i}^{H(0)}|$ (right), whose squared value\n define the admixture of the two-lightest\n ${\\CP}$-even scalar mass eigenstate $h_{1,2}$ with the fields\n $\\varphi_i=(H_d,H_u,\\widetilde{\\nu}_R,\\widetilde{\\nu}_{1L},\n \\widetilde{\\nu}_{2L},\\widetilde{\\nu}_{3L})$ in the\n interaction basis. A substantial mixing of the $\\tau$-sneutrino\n $\\widetilde{\\nu}_{3L}$ with the SM-like Higgs boson $h^{125}$ and\n with the singlet $\\widetilde{\\nu}_R$ is present in the narrow region\n where the corresponding tree-level masses are degenerate\n (for example in the \\textit{ right} plot at\n $\\lambda\\sim 0.20237$ and $\\lambda\\sim0.29692$).}\n \\label{fig:sntaulsmixing}\n\\end{figure}\n\nIt is rather surprising that the SM-like Higgs masses coincide this precisely\nin both models, considering the fact\nthat a substantial mixing with the sneutrino is\npossible at tree-level, as we show in \\reffi{fig:sntaulsmixing}.\nWe individually plot the mixing matrix elements of\nthe two lightest ${\\CP}$-even scalars, whose\nsquared values define the composition of each mass eigenstate\nat tree-level. In the cross-over point of the $\\tau$-sneutrino and the\nSM-like Higgs boson the lightest scalar results to be a mixture\nof $\\widetilde{\\nu}_\\tau$ and the doublet-components $H_u$ and $H_d$, as\none can see in the upper left plot of \\reffi{fig:sntaulsmixing}.\nFor example, if we fine-tune $\\lambda = 0.20237$ we find that the lightest Higgs boson\nis composed of approximately\n\\begin{align}\nH_d \\quad \\rightarrow &\\quad |U_{11}^{H(0)}|^2\\sim 1\\% \\; , \\\\\nH_u \\quad \\rightarrow &\\quad |U_{12}^{H(0)}|^2\\sim 80\\% \\; , \\\\\n\\widetilde{\\nu}_{3L} \\quad \\rightarrow &\\quad |U_{16}^{H(0)}|^2\\sim 19\\% \\; .\n\\end{align}\nNevertheless, due to the upward shift, as explained before,\nthe one-loop corrections break the degeneracy and no trace\non the SM-like Higgs mass remains, which would deviate it from the NMSSM\nprediction.\n\n\n\n\n\n\n\n\\subsection{\\protect\\boldmath The \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ and the CMS $\\gamma\\ga$ excess at\n $96 \\,\\, \\mathrm{GeV}$}\n\\label{sec:gaga95}\nIn this section we will investigate a scenario in which the SM-like Higgs\nboson is not the lightest ${\\CP}$-even scalar. This is inspired by\nthe reported excesses of LEP~\\cite{Barate:2003sz}\nand CMS~\\cite{CMS:2015ocq,CMS:2017yta}\nin the mass range around\n$\\sim 96\\,\\, \\mathrm{GeV}$, that (as we will show) can be explained\nsimultaneously by the presence\nof a light scalar in this mass window.\nWhile in the NMSSM the light scalar can be interpreted as the ${\\CP}$-even\nscalar singlet and can accommodate both excesses at $1\\sigma$ level\nwithout violating any known experimental\nconstraints~\\cite{Cao:2016uwt,Domingo:2017},\\footnote{Other possible\nexplanations of the CMS excess were analyzed in\n\\citere{Cacciapaglia:2016tlr,Mariotti:2017vtv,Crivellin:2017upt}. On the other hand, in the MSSM the CMS excess cannot be\nrealized~\\cite{Bechtle:2016kui}.}\nwe will interpret the\nlight scalar as the ${\\CP}$-even right-handed sneutrino of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}.\nSince the singlet of the NMSSM and the right-handed sneutrino of\nthe \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ are both gauge-singlets, they share very similar properties.\nHowever, the explanation of the excesses in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ avoids\nbounds from direct detection experiments, because $R$-parity is broken\nin the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ and the dark matter candidate is not a neutralino as in\nthe NMSSM but a gravitino with a lifetime longer than the age of\nthe universe~\\cite{Munoz:2016vaa}. This is important because the direct\ndetection measurements were shown to be very constraining in the NMSSM\nwhile trying to explain the dark matter abundance on top of the\nexcesses from LEP and CMS~\\cite{Cao:2016uwt}.\n\n\\begin{table}\n\\renewcommand{\\arraystretch}{1.7}\n\\centering\n\\begin{tabular}{c c c c c c c c c c}\n $v_{iL}\/\\sqrt{2}$ & $Y^\\nu_i$ & $A^\\nu_i$ & $\\tb$ & $\\mu$ & $\\lambda$ &\n \t$A^\\lambda$ & $\\kappa$ & $A^\\kappa$ & $M_1$ \\\\\n \\hline\n $10^{-5}$ & $10^{-7}$ & $-1000$ & $2$ & $[413;418]$ & $0.6$ &\n \t$956$ & $0.035$ & $[-300;-318]$ & $100$ \\\\\n \\hline\n \\hline\n $M_2$ & $M_3$ & $m_{\\widetilde{Q}_{iL}}^2$ &\n \t$m_{\\widetilde{u}_{iR}}^2$ & $m_{\\widetilde{d}_{iR}}^2$ &\n \t$A^u_i$ & $A^{d}_i$ & $(m_{\\widetilde{e}}^2)_{ii}$ &\n \t$A^e_{33}$ & $A^e_{11,22}$ \\\\\n \\hline\n $200$ & $1500$ & $800^2$& $800^2$ & $800^2$ & $0$ &\n \t$0$ & $800^2$ & $0$ & $0$ \n\\end{tabular}\n\\caption{Input parameters for the scenario featuring the right-handed\n\t\t sneutrino in the mass range of the LEP and CMS excesses and\n\t\t a SM-like Higgs boson as next-to-lightest ${\\CP}$-even scalar;\n \t\t all masses and values for trilinear parameters are in GeV.}\n\\label{tab:lepcms}\n\\renewcommand{\\arraystretch}{1.0}\n\\end{table\n\nIn \\refta{tab:lepcms} we list the values of the parameters we used\nto account for the lightest ${\\CP}$-even scalar as the right-handed\nsneutrino and the second lightest one the SM-like Higgs boson.\n$\\lambda$ is chosen to be large to account for a sizable mixing of\nthe right-handed sneutrino and the doublet Higgses.\nIn the regime where the SM-like Higgs boson is not the lightest scalar,\none does not need large quantum corrections to the Higgs boson mass,\nbecause the tree-level mass is already well above $100\\,\\, \\mathrm{GeV}$. This is\nwhy $\\tan\\beta$ can be low and the soft trilinears $A^{u,d,e}$\nare set to zero. The values of $A^{\\lambda}$ and $\\left|A^\\nu\\right|$\nare chosen to be around $1\\,\\, \\mathrm{TeV}$ to get masses for the\nheavy MSSM-like Higgs and the left-handed sneutrinos of this order, so\nthey do not play an important role in the following discussion.\nOn the other hand, $\\kappa$ is small to bring the mass\nof the right-handed sneutrino below the SM-like Higgs boson mass.\nFinally, the two parameters that are varied are $\\mu$ and $A^\\kappa$.\nBy increasing $\\mu$ the mixing of the right-handed sneutrino with\nthe SM-like Higgs boson is increased, which is needed to couple\nthe gauge-singlet to quarks and gauge-bosons. At the same time we\nused the value of $A^\\kappa$ to keep the mass of the\nright-handed sneutrino in the correct range. Accordingly, the\nresults in this chapter will all be displayed in the scanned\n$A^\\kappa$-$\\mu$ plane.\n\nThe process measured at LEP was the production of a Higgs boson\nvia Higgstrahlung associated with the Higgs decaying to\nbottom-quarks:\n\\begin{equation}\n\\mu_{\\rm LEP}=\\frac{\\sigma\\left( e^+e^- \\to Z h_1 \\to Zb\\bar{b} \\right)}\n\t\t\t {\\sigma^{SM}\\left( e^+e^- \\to Z h \n\t\t\t \t\t\\to Zb\\bar{b} \\right)}\n\t\t\t = 0.117 \\pm 0.057 \\; ,\n\\end{equation}\nwhere $\\mu_{\\rm LEP}$ is called the signal strength, which is the\nmeasured cross section normalized to the standard model expectation,\nwith the SM Higgs boson mass at $\\sim 96\\,\\, \\mathrm{GeV}$.\nThe value for $\\mu_{\\rm LEP}$ was extracted in \\citere{Cao:2016uwt}\nusing methods described in \\citere{Azatov:2012bz}.\nWe can find an\napproximate expression for $\\mu_{\\rm LEP}$ factorizing the production\nand the decay of the scalar and expressing it in terms of couplings\nto the massive gauge bosons $C_{h_1VV}$ and the up- and down-type\nquarks $C_{h_1u\\bar{u}}$ and $C_{h_1d\\bar{d}}$, respsectively, normalized to\nthe SM predictions for the corresponding couplings (where with\n\\mbox{}$^{\\mu\\nu}$ we denote the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ prediction, and $\\Gamma$ is the\nHiggs-boson decay width):\n\n\\begin{align}\n\\mu_{\\rm LEP}^{\\mu\\nu}&=\\frac{\\sigma^{\\mu\\nu}\\left(Z^*\\to Zh_1 \\right)}\n {\\sigma^{\\rm SM}\\left(Z^*\\to Zh \\right)}\\times\n \\frac{\\text{BR}^{\\mu\\nu}\\left( h_1\\to b \\bar b\\right)}\n {\\text{BR}^{\\rm SM}\\left( h\\to b \\bar b\\right)} \n \\notag \\\\\n &\\approx\\left| C_{h_1VV} \\right|^2\\times\n \\frac{\\Gamma^{\\mu\\nu}_{b\\bar{b}}}{\\Gamma^{\\rm SM}_{b\\bar{b}}}\n \\times\n \\frac{\\Gamma^{\\rm SM}_{\\rm tot}}{\\Gamma^{\\mu\\nu}_{\\rm tot}} \n \\notag \\\\\n &\\approx \\frac{\n \\left| C_{h_1VV} \\right|^2\\times\\left| C_{h_1d\\bar{d}} \\right|^2\n }{\n \\left| C_{h_1d\\bar{d}} \\right|^2\n (\\text{BR}^{\\rm SM}_{b\\bar{b}}+\n \\text{BR}^{\\rm SM}_{\\tau\\bar{\\tau}}) +\n \\left| C_{h_1u\\bar{u}} \\right|^2\n (\\text{BR}^{\\rm SM}_{gg}+\n \\text{BR}^{\\rm SM}_{c\\bar{c}})} \\; .\n\\label{eq:mulep}\n\\end{align}\n\nThe SM branching ratios dependent on the Higgs boson mass can be\nobtained from \\citere{Heinemeyer:2013tqa}.\nThe denominator is the ratio of the total decay width of $h_1$\nin the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ and $h$ in the SM when all SM\nbranching ratios larger than $1\\%$ are considered.\nThe off-shell decay to $W$ and $Z$ bosons is in\nprinciple also possible, but the BRs are very small for a SM Higgs\nboson with a mass around $95\\,\\, \\mathrm{GeV}$ ($\\text{BR}^{\\rm SM}_{WW}\\sim 0.5\\%$ \nand $\\text{BR}^{\\rm SM}_{ZZ}\\sim 0.06\\%$)~\\cite{Denner:2011mq,Heinemeyer:2013tqa}.\nIt is worth noticing that although the right-handed neutrino mass\nis small, $m_{\\nu_R}\\sim62-63\\,\\, \\mathrm{GeV}$,\nin the investigated parameter region, it is nevertheless\nlarger than half of\nthe SM-like Higgs boson mass\nin all benchmark points, so the decay of the Higgs\nto the right-handed neutrino is kinematically forbidden and\ncannot spoil the properties of the SM-like Higgs.\nNeglecting the vevs $v_{iL}$ the normalized couplings of the\nscalars are given at\nleading order by the admixture of the mass eigenstate $h_i$ with\nthe doublet like Higgs $H_d$ and $H_u$ via\n\\begin{equation}\nC_{h_i d \\bar{d}}=\\frac{U^{H,(2')}_{i1}}{\\cos\\beta} \\; , \\quad\nC_{h_i u \\bar{u}}=\\frac{U^{H,(2')}_{i2}}{\\sin\\beta} \\; , \\quad\nC_{h_i V V}=U^{H,(2')}_{i1}\\cos\\beta+U^{H,(2')}_{i2}\\sin\\beta \\; ,\n\\end{equation}\nwhere the partial two-loop plus resummation corrected mixing matrix elements $U^{H,(2')}_{ij}$\nwere calculated\nin the approximation of vanishing momentum,\nsee the discussion in \\refse{sec:higherorders}.\nWe show in \\reffi{fig:lep1} the masses (top row) and the normalized\ncouplings ($|C_{h_1 d \\bar d}|$ second row, $|C_{h_1 u \\bar b}$| third\nrow, $|C_{h_1 VV}|$ lowest row) \nof the lightest and the next-to-lightest ${\\CP}$-even scalar. The lower\nright corner (marked in gray) results in the right-handed sneutrino\nbecoming tachyonic (at tree-level).\nThe largest mixing of the right-handed sneutrino and the SM-like\nHiggs boson is achieved where $\\mu$ is largest\nand $|A^\\kappa|$ is smallest. The mass of $h_2$ is in the allowed\nregion for a SM-like Higgs boson at $\\sim 125\\,\\, \\mathrm{GeV}$ if we assume a\ntheory uncertainty of up to $3\\,\\, \\mathrm{GeV}$ (see the previous subsections).\nThe LHC measurements of the SM-like Higgs boson\ncouplings to fermions and massive gauge bosons are still not\nvery precise~\\cite{Khachatryan:2016vau}, with uncertainties\nbetween 10 and $20\\%$ at the $1\\sigma$ confidence level\n(obtained with\nthe assumption that no beyond-the-SM decays modify the total\nwidth of the SM-like Higgs boson). Therefore, it would be\nchallenging to exclude parts of the parameter space by considering\nthe deviations of the normalized couplings of $h_2$.\nHowever, possible future lepton colliders\nlike the ILC could measure these couplings to a\n$\\%$-level~\\cite{Dawson:2013bba,Moortgat-Picka:2015yla}, which\ncould exclude (or confirm) most of the parameter space presented here.\nSeen from a more optimistic perspective, the precise measurement of\nthe SM-like Higgs boson couplings at future colliders could\nbe used to make predictions for the properties of the\nlighter right-handed sneutrino in this scenario.\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.6\\textwidth]{lep1.pdf}\n \\vspace{-0.5cm}\n \\caption{Properties of the lightest (left) and next-to-lightest (right)\n ${\\CP}$-even scalar in the $\\mu$--$A^\\kappa$ plane.\n The couplings are normalized to the SM-prediction\n of a Higgs particle of the same mass. The gray area is\n excluded because the right-handed sneutrino becomes\n tachyonic at tree-level.\n \\textit{First row:} two-loop masses,\n \\textit{second row:} coupling to down-type quarks,\n \\textit{third row:} coupling to up-type quarks,\n \\textit{fourth row:} coupling to massive gauge bosons.}\n \\label{fig:lep1}\n\\end{figure}\n\nThe CMS excess was observed in the diphoton channel\nwith a signal strength of~\\cite{Shotkin:2017}\n\\begin{equation}\n\\mu_{\\rm CMS}=\\frac{\\sigma\\left( gg\\to h_1 \\to \\gamma\\gamma \\right)}\n {\\sigma^{\\rm SM}\\left( gg\\to h \\to \\gamma\\gamma \\right)}\n = 0.6 \\pm 0.2 \\; .\n\\end{equation}\nWe calculate the signal strength using the approximation that the\nHiggs production via gluonfusion is described at leading order\nexclusively by the loop-diagram with a\ntop quark running in the loop, and that\nthe diphoton decay is described by the diagrams with $W$~bosons or a\ntop quark in the loop, which is sufficient in the investigated mass\nrange of $h_1$. One can then write\n\\begin{align}\n\\mu_{\\rm CMS}^{\\mu\\nu}&=\\frac{\\sigma^{\\mu\\nu}\\left( gg\\to h_1 \\right)}\n {\\sigma^{\\rm SM}\\left( gg\\to h \\right)} \\times\n \\frac{\\text{BR}^{\\mu\\nu}_{\\gamma\\gamma}}\n {\\text{BR}^{\\rm SM}_{\\gamma\\gamma}} \\notag \\\\\n &\\approx \\left|C_{h_1 u \\bar{u}}\\right|^2 \\times\n \\frac{\\Gamma^{\\mu\\nu}_{\\gamma\\gamma}}\n {\\Gamma^{\\rm SM}_{\\gamma\\gamma}} \\times\n \\frac{\\Gamma^{\\rm SM}_{\\rm tot}}{\\Gamma^{\\mu\\nu}_{\\rm tot}} \\notag \\\\\n &\\approx\\frac{\\left|C_{h_1 u \\bar{u}}\\right|^2 \\times\n \\left|C^{\\rm eff}_{h_1 \\gamma\\gamma}\\right|^2}\n {\\left| C_{h_1d\\bar{d}} \\right|^2\n (\\text{BR}^{\\rm SM}_{b\\bar{b}}+\n \\text{BR}^{\\rm SM}_{\\tau\\bar{\\tau}}) +\n \\left| C_{h_1u\\bar{u}} \\right|^2\n (\\text{BR}^{\\rm SM}_{gg}+\n \\text{BR}^{\\rm SM}_{c\\bar{c}})} \\; .\n\\label{eq:mucms}\n\\end{align}\nThe effective coupling of the neutral scalars to photons\n$C^{\\rm eff}_{h_i\\gamma\\gamma}$ has to be\ncalculated in terms of the couplings to the $W$ boson and the up-type\nquarks. In the SM the dominant contributions to the decay to photons\ncan be written as~\\cite{Djouadi:2005gi}\n\\begin{equation}\n\\Gamma^{\\rm SM}_{\\gamma\\gamma} =\n\\frac{G_\\mu\\,\\alpha^2 \\, m_h^3}{128\\, \\sqrt{2}\\, \\pi^3}\n\\left| \\frac{4}{3} A_{1\/2}\\left( \\tau_t \\right) +\n A_{1}\\left( \\tau_W \\right) \\right|^2 \\; ,\n\\end{equation}\nwhere $G_\\mu$ is the Fermi-constant and the form factors $A_{1\/2}$\nand $A_{1}$ are defined as\n\\begin{align}\nA_{1\/2}\\left(\\tau\\right)&=2\\left( \\tau+\\left(\\tau -1\\right)\n \\arcsin^2\\sqrt{\\tau} \\right)\\tau^{-2} \\; , \\\\\nA_{1}\\left(\\tau\\right)&=-\\left( 2\\tau^2+3\\tau\n +3\\left( 2\\tau-1 \\right)\n \\arcsin^2\\sqrt{\\tau} \\right)\\tau^{-2} \\; ,\n\\end{align}\nfor $\\tau \\leq 1$, and the arguments of these functions are\n$\\tau_t = m_h^2\/(4 m_t^2)$ and $\\tau_W = m_h^2\/(4 M_W^2)$.\nIn our approximation the only difference between the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ and the\nSM will be\nthat the couplings of $h_i$ to the top quark and the $W$ boson is\nmodified by the factors $C_{h_i t\\bar{t}}$ and $C_{h_i VV}$, so the\neffective coupling of the Higgses to photons in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\\nnormalized to the SM predictions\ncan be written as\n\\begin{equation}\n\\left|C^{\\rm eff}_{h_i \\gamma\\gamma}\\right|^2=\n\\frac{\\left| \\frac{4}{3} C_{h_i t\\bar{t}} A_{1\/2}\\left( \\tau_t \\right) +\n C_{h_i VV} A_{1}\\left( \\tau_W \\right) \\right|^2}\n {\\left| \\frac{4}{3} A_{1\/2}\\left( \\tau_t \\right) +\n A_{1}\\left( \\tau_W \\right) \\right|^2} \\; .\n\\end{equation}\n\nUsing \\refeq{eq:mulep} and \\refeq{eq:mucms} we can\ncalculate the two signal strengths. \nThe result are shown in \\reffi{fig:lep2}, the LEP (left) and the CMS\nexcesses (right) in the $\\mu$--$A^{\\kappa}$ plane.\nWhile the LEP excess is easily reproduced\nin the observed parameter space, we cannot achieve the central\nvalue for $\\mu_{\\rm CMS}$, but only slightly smaller values.\nAs already observed in \\citere{Cao:2016uwt},\nthe reason for this is that for explaining the LEP excess\na sizable coupling to the bottom quark is needed.\nOn the contrary, the CMS\nexcess demands a small value for $C_{h_1 d\\bar{d}}$ so that the\ndenominator in \\refeq{eq:mucms} becomes small and $\\mu_{\\rm CMS}$\nis enhanced. Nevertheless, considering the large\nexperimental uncertainties in $\\mu_{\\rm CMS}$ and $\\mu_{\\rm LEP}$,\nthe scenario presented in this section\naccommodates both excesses comfortably well\n(at approximately $1\\sigma$),\nand it is a good motivation to keep on\nsearching for light Higgses in the allowed mass window\nbelow the SM-like Higgs mass. Apart from that, this scenario\nillustrates the importance of an accurate calculation of the\nloop-corrected scalar masses and mixings, since already small changes\nin the parameters can have a big impact on the production and the\ndecay modes of the ${\\CP}$-even Higgs bosons.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{lep2.pdf}\n \\vspace{-0.5cm}\n \\caption{Signal strengths for the lightest $\\widetilde{\\nu}_R$-like\n neutral scalar at CMS ($pp\\to h_1 \\to \\gamma\\gamma$)\n (\\textit{left}) and\n LEP (${e^+e^-\\to h_1 Z \\to b\\bar{b} Z}$)\n (\\textit{right}) in the\n $\\mu$-$A^\\kappa$ plane. The gray area is\n excluded because the right-handed sneutrino becomes\n tachyonic at tree-level.}\n \\label{fig:lep2}\n\\end{figure}\n\n\n\\section{Conclusion and Outlook}\n\\label{sec:concl}\nThe \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ is a simple SUSY extension of the SM that is capable of predicting neutrino physics in agreement with experimental data. As in other SUSY models, higher-order corrections are crucial to reach a\ntheoretical uncertainty at the same level of (anticipated) experimental\naccuracy. So far, higher-order corrections in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ had been restricted\nto \\DRbar\\ calculations, which suffer from the disadvantage that they cannot\nbe directly connected to (possibly future observed) new BSM particles.\n\nIn this paper we have performed the complete one-loop renormalization of the\nneutral scalar sector of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ with one generation of right-handed\nneutrinos in a mixed on-shell\/\\DRbar\\ scheme. The renormalization procedure\nwas discussed in detail for each of the free parameters appearing in the\n\\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ Higgs sector. We have emphasized the conceptual differences to the MSSM and the NMSSM regarding the field\nrenormalization and the treatment of non-flavor-diagonal soft mass parameters, which have their\norigin in the breaking of $R$-parity in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}. However, we have ensured that the renormalization of the relevant (N)MSSM\nparts in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ are in agreement with previous calculations in those\nmodels. Consequently, numerical differences found can directly be attributed\nto the extended structure of the \\ensuremath{\\mu\\nu\\mathrm{SSM}}.\nThe derived renormalization can be applied\nto any higher-order correction in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}.\nThe one-loop counterterms derived in this paper are implemented into\nthe \\FA\\ model file, so the computation of these corrections\ncan be done fully automatically.\n\nWe have applied the newly derived renormalization to the calculation of the full one-loop corrections to the neutral scalar masses\nof the \\ensuremath{\\mu\\nu\\mathrm{SSM}}, where we found that all UV-divergences cancel.\nIn our numerical analysis the newly derived full one-loop contributions are\nsupplemented by available MSSM higher-order corrections as provided by the\ncode \\fh\\ (leading and subleading fixed-order corrections as well as resummed large logarithmic\ncontributions obtained in an EFT approach.)\nWe investigated various representative scenarios, in which we obtained\nnumerical results for a SM-like Higgs boson mass consistent with experimental bounds. We compared our results to predictions of the various\nneutral scalars in the NMSSM to\ninvestigate the relevance of genuine \\ensuremath{\\mu\\nu\\mathrm{SSM}}-like contributions.\nWe find negligible\ncorrections w.r.t.\\ the NMSSM,\nindicating that the Higgs boson mass calculations in the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ are at \nthe same level of\naccuracy as in the NMSSM.\n\nFinally we showed that the \\ensuremath{\\mu\\nu\\mathrm{SSM}}\\ can accommodate a right-handed\n(${\\CP}$-even) scalar neutrino \nin a mass regime of $\\sim 96 \\,\\, \\mathrm{GeV}$, where the full Higgs sector is in\nagreement with the Higgs-boson measurements obtained at the LHC, as well as\nwith the Higgs exclusion bounds obtained at LEP, the Tevatron and the LHC.\nThis includes in particular a SM-like Higgs boson at $\\sim 125 \\,\\, \\mathrm{GeV}$.\nWe have demonstrated that the light right-handed sneutrino can explain\nan excess of $\\gamma\\ga$ events at $\\sim 96 \\,\\, \\mathrm{GeV}$ as reported \nrecently by CMS in their Run~I and Run~II date. It can\nsimultaneously describe the $2\\,\\sigma$ excess of $b \\bar b$ events\nobserved at LEP at a similar mass scale. We are eagerly awaiting the \ncorresponding ATLAS Higgs-boson search results. \n\n\n\n\\subsection*{Acknowledgements}\nWe thank F.~Domingo for helpful discussions.\nThis work was supported in part by the Spanish Agencia Estatal de\nInvestigaci\\'on through the grants\nFPA2016-78022-P MINECO\/FEDER-UE (TB and SH) and\nFPA2015-65929-P MINECO\/FEDER-UE (CM), and IFT Centro de\nExcelencia Severo Ochoa SEV-2016-0597. The work of TB was\nfunded by Fundaci\\'on La Caixa under `La Caixa-Severo Ochoa' international\npredoctoral grant. We also acknowledge the support of the MINECO's\nConsolider-Ingenio 2010 Programme under grant MultiDark CSD2009-00064.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\n \n \n \n \n \n \n \n \n \n\nGrammatical error correction (GEC) is the task of automatically identifying and correcting the grammatical errors in the written text. \nRecent work treats GEC as a translation task that use sequence-to-sequence models \\citep{SutskeverVL14seq2seq,BahdanauCB14} to rewrite sentences with grammatical errors to grammatically correct sentences. As with machine translation models, GEC models benefit largely from the amount of parallel training data. Since it is expensive and time-consuming to create annotated parallel corpus for training, there is research into generating sentences with artificial errors from grammatically correct sentences with the goal of simulating human-annotated data in a cost-effective way \\citep{YuanB16,XieAAJN16,chollampatt2018mlconv}. \n\n\nRecent work in artificial error generation (AEG) is inspired by the back-translation approach of machine translation systems \\citep{DBLP:conf\/acl\/SennrichHB16,DBLP:journals\/corr\/abs-1804-06189}. In this framework, an intermediate model is trained to translate correct sentences into errorful sentences. A new parallel corpus is created using the largely available grammatically correct sentences and the corresponding synthetic data generated by this intermediate model. The newly created corpus with the artificial errors is then used to train a GEC model \\citep{DBLP:conf\/bea\/ReiFYB17,DBLP:conf\/naacl\/XieGXNJ18,Ge2018HumanGEC}.\n\nTo date, there is no work that compares how different base model architectures perform in the AEG task. In this paper, we investigate how effective are \ndifferent model architectures in generating artificial, parallel data to improve a GEC model. Specifically, we train four recent neural models (and one rule-based model \\cite{Bryant2018LanguageMB}), including two new syntax-based models, for generating as well as correcting errors. We analyze which models are effective in the AEG and correction conditions as well as by data size. Essentially, we seek to understand how effective are recent sequence-to-sequence (seq2seq) neural model as AEG mechanisms ``out of the box.\" \n\n\n\n\n\\section{Related Work}\n\n\n\nBefore the adoption of neural models, early approaches to AEG involved identifying error statistics and patterns in the corpus and applying them to grammatically correct sentences \\citep{BrockettDG06,RozovskayaR10}. Inspired by the back-translation approach, recent AEG approaches inject errors into grammatically correct input sentences by adopting methods from neural machine translation \\citep{FeliceY14,KasewaS018}. \\citet{DBLP:conf\/naacl\/XieGXNJ18} propose an approach that adds noise to the beam-search phase of an back-translation based AEG model to generate more diverse errors. They use the synthesized parallel data generated by this method to train a multi-layer convolutional GEC model and achieve a 5 point $F_{0.5}$ improvement on the CoNLL-2014 test data \\cite{ng-EtAl:2014:W14-17}. \\citet{Ge2018HumanGEC} propose a fluency-boosting learning method that generates less fluent sentences from correct sentences and pairs them with correct sentences to create new error-correct sentence pairs during training. Their GEC model trained with artificial errors approaches human-level performance on multiple test sets. \n\n\n\n\\section{Approach}\n\n\\subsection{Correction and Generation Tasks}\nWe train our models on the two tasks---error correction and error generation. In \\textit{error correction}, the encoder of the sequence-to-sequence model takes an errorful sentence as input and the decoder outputs the grammatically correct sentence. The process is reversed in the \\textit{error generation} task, where the model takes a correct sentence as input and produces an errorful sentence as the output of the decoder. \n\nWe investigate four recent neural sequence-to-sequence models---(i) multi-layer convolutional model \\citep[MLCONV;][]{chollampatt2018mlconv}, (ii) Transformer \\citep{DBLP:conf\/nips\/VaswaniSPUJGKP17}, (iii) Parsing-Reading-Predict Networks \\citep[PRPN;][]{shen2018neural}, (iv) Ordered Neurons \\citep[ON-LSTM;][]{shen2018ordered}---as error correction models as well as error generation models. The PRPN and ON-LSTM models are originally designed as recurrent language models that jointly learn to induce latent constituency parse trees. We use the adaption of PRPN and ON-LSTM models as decoders of machine translation systems \\citep{anon2018syntaxtranslation}: In this setting, a 2-layer LSTM is used as the encoder of the syntactic seq-to-seq models, and the PRPN and ON-LSTM are implemented as the decoders with attention \\citep{BahdanauCB14}. We hypothesize that syntax is important in GEC and explore whether models that incorporate syntactic bias would help with GEC task. We provide a brief description of each model in \\S\\ref{sec:models} and refer readers to the original work for more details. \n\n\\subsection{Models}\n\\label{sec:models}\n\\paragraph{Multi-layer Convolutional Model}\nWe use the multi-layer convolutional encoder-decoder base model (MLCONV) of \\citet{chollampatt2018mlconv} using the publicly available code from the authors.\\footnote{\\url{https:\/\/github.com\/nusnlp\/mlconvgec2018}} \nAs our aim is to only compare the performance of different architectures and not to achieve state-of-the-art performance, we make few changes to their code.\nThe model of \\citet{chollampatt2018mlconv} produces 12 possible correct sentences for each input sentences with error. They also train an N-gram language model as a re-ranker to score the generated sentences and pick the corrected sentence with the best score as final output. We did not use this re-ranking step in our model, nor did we perform ensembling or use the pre-trained embeddings as in the original work. We do not observe improvement in models like transformer and PRPN using re-ranking with an N-gram language model. Additionally, there's only a slight improvement in MLCONV using re-ranking. The reason might be because the N-gram language model is not very powerful.\n\n\n\\paragraph{Transformer Model} We use the publicly available Fairseq framework which is built using Pytorch for training the Transformer model. We apply the same hyper-parameters used for training the IWSLT'14 German-English translation model in the experiments of \\citet{DBLP:conf\/nips\/VaswaniSPUJGKP17}. \n\n\n\\paragraph{PRPN Model} is a language model that jointly learns to parse and perform language modeling \\citep{shen2018neural}. It uses a recurrent module with a self-attention gating mechanism and the gate values are used to construct the constituency tree. \nWe use the BiLSTM model as the encoder and PRPN as the decoder of the sequence-to-sequence model.\n\n\\paragraph{ON-LSTM Model}\n is follow-up work of PRPN, which incorporates syntax-based inductive bias to the LSTM unit by imposing hierarchical update order on the hidden state neurons \\citep{shen2018ordered}. ON-LSTM assumes that different nodes of a constituency trees are represented by the different chunks of adjacent neurons in the hidden state, and introduces a master forget gate and a master input gate to dynamically allocate the chunks of hidden state neurons to different nodes. \n\n We use a BiLSTM model as encoder and ON-LSTM model as decoder.\n\n\n\n\n\n\\section{Experiments}\n\n\\subsection{Data}\nWe use the NUS Corpus of Learner English \\citep[NUCLE;][]{DBLP:conf\/bea\/DahlmeierNW13} and the Cambridge Learner Corpus \\citep[CLC;][]{Nicholls2003} as base data for training both the correction and generation models. We remove sentence pairs that do not contain errors during preprocessing resulting in 51,693 sentence pairs from NUCLE and 1.09 million sentence pairs from the CLC .\nWe append the CLC data to the NUCLE training set (henceforth NUCLE-CLC) to use as training data for both AEG and correction. We use the standard NUCLE development data as our validation set and we early-stop the training based on the cross-entropy loss of the seq-to-seq models for all models. For the generation of synthetic errorful data, we use the 2017 subsection of the LDC New York Times corpus also employed in the error generation experiments of \\citet{DBLP:conf\/naacl\/XieGXNJ18} which contains around 1 million sentences.\\footnote{\\url{https:\/\/catalog.ldc.upenn.edu\/LDC2008T19} }\n\n\n\\begin{table*}[]\n\\small\n\\centering\n\\begin{center}\n\\begin{tabular}{llccccccc} \n\\toprule\n \\bf GEC Model & \\bf AEG model & \\bf NUCLE-CLC & \\bf 10K & \\bf 50K & \\bf 100K & \\bf 500K & \\bf 1M & \\bf 2M \\\\\n \\midrule\n\n\n MLCONV & MLCONV & 35.2 & 35.1 & 34.7 & 34.6 & 38.9 & 39.4 & 34.0 \\\\\n Transformer & MLCONV & 36.3 & 43.9 & \\bf 44.1 & \\bf 45.4 & \\bf 44.4 & \\bf 45.5 & \\bf 42.0 \\\\\n PRPN & MLCONV & \\bf 43.6 & \\bf 45.4 & 42.8 & 43.2 & 39.6 & 38.6 & 31.7 \\\\\n ON-LSTM & MLCONV & 36.6 & 39.8 & 35.6 & 38.4 & 36.9 & 24.2 & 20.1 \\\\\n \\midrule\n \n\n MLCONV & Transformer & 35.2 & 36.1 & 35.2 & 39.4 & 36.6 & 36.6 & 36.1 \\\\\n Transformer & Transformer & 36.3 & 20.1 & \\bf 43.9 & \\bf 42.9 & \\bf 43.7 & \\bf 44.0 & \\bf 41.0 \\\\\n PRPN & Transformer & \\bf 43.6 & \\bf 43.1 & 40.9 & 40.6 & 41.4 & 29.4 & 31.7 \\\\\n ON-LSTM & Transformer & 36.6 & 39.8 & 38.2 & 39.6 & 24.0 & 21.3 & 20.1 \\\\\n \\midrule\n MLCONV & Rule-based & 35.2 & 6.0 & 7.8 & 10.5 & 13.7 & 13.9 & -- \\\\\n Transformer & Rule-based & 36.3 & \\bf 13.5 & \\bf 14.4 & \\bf 21.8 & \\bf 14.5 & \\bf 21.6 & -- \\\\\n PRPN & Rule-based & \\bf 43.6 & 2.8 & 4.9 & 2.6 & 3.9 & 8.9 & -- \\\\\n ON-LSTM & Rule-based & 36.6 & 4.7 & 3.9 & 5.5 & 4.2 & 5.3 & -- \\\\\n \\bottomrule \n\\end{tabular}\n\\end{center}\n\\caption{\\label{tab:rb-table} (Exp2) Evaluating the impact of MLCONV, Transformer and the rule-based AEG systems. \nNUCLE-CLC column represents the F0.5 score of GEC models trained on the base NUCLE-CLC data. \\textit{10K, 50K, 100K, 500K, 1M, and 2M} represents the amount of artificial data added to the NUCLE-CLC during training. \n}\n\\end{table*}\n\n\n\\subsection{Setup}\nWe conduct four experiments in this paper. In \\textbf{Exp1}, we train all the AEG models and intermediate GEC models on NUCLE-CLC. We use the NYT dataset as input to the AEG models to generate sentences with artificial errors. We then create new parallel training sets for correction by combining the sentences from CLC and NUCLE with the errorful sentences generated by each model. We then train the GEC models using these parallel datasets. \n\nThe three other experiments are variants of the first. In \\textbf{Exp2} we train all correction models on artificial errors generated by the top neural AEG systems and a rule-based system for comparison. In \\textbf{Exp3}, we train the GEC models on NUCLE to analyze models built on real data. Finally, in \\textbf{Exp4}, we train all GEC models on artificial data to determine how well correction models can perform without any real data.\n\n\nAll our experiments are tested on the CoNLL-2014 test set and we use the sentence-level $F0.5$ score from the MaxMatch ($M^2$) scorer \\citep{DahlmeierN12} for evaluation. All models are implemented using the Fairseq framework.\\footnote{\\url{https:\/\/github.com\/pytorch\/fairseq}}\n\n\n\n\n\n\n\n\n\n\\subsection{Results}\n\n\\noindent\\textbf{Exp1}: Figure~\\ref{fig:table1a} shows the performance of GEC models trained on the base NUCLE-CLC set and then retraining with various amounts of artificial data. We first observe that PRPN performs substantially higher than the rest of the models when trained only with the base CLC-NUCLE data. However, its performance drops when artificial data generated by the corresponding PRPN AEG model is added. As for ON-LSTM, the performance improves slightly when the amount of added data is less than 100k but the performance drops drastically otherwise. Conversely, the performance of MLCONV and Transformer improves with the added artificial data but the improvement is not linear with the amount of added data. \n\n\\begin{figure}[htbp]\n \\centering\n \\includegraphics[height=1.8in] {Nucle-CLC-table1a.png}\n \\caption{(Exp1) Models trained on the artificial data generated by the corresponding AEG model. The X-axis represents the amount of artificial data added to NUCLE-CLC during training. \n }\n \\label{fig:table1a}\n\\end{figure}\n\n\\noindent\\textbf{Exp2}: Since the performance of MLCONV and Transformer GEC models improve with the addition of artificial data generated by corresponding AEG models, we hypothesize that the artificial error generated by these models are useful. To test this hypothesis, we train all the GEC models with various amount of artificial error generated by MLCONV and Transformer AEG models. We also compare these AEG models to a rule-based one inspired by the confusion set generation method in \\citet{Bryant2018LanguageMB}. We subsequently score each sentence with a language model (GPT-2 \\citep{noauthororeditor}) in order not to select the most probable sentence. This method generates a confusion set for prepositions (set of prepositions plus an empty element), determiners, and morphological alternatives (cat $\\rightarrow$ cats).\n\n\n\nThe results of these experiments are found in Table~\\ref{tab:rb-table}. Nearly all correction models improve when using MLCONV or Transformer AEG data with the biggest performances yielded using the Transformer model. Interestingly, when using 1M or 2M samples, performance starts to decline. We believe that over 1M samples, the noisiness of the artificial data overwhelms the contributions of the real data (roughly over 1M samples). The performance of all models drops when trained with the errors generated by the rule-based model. It is interesting to observe that the performance drops significantly just by adding 10K artificial sentences to the base data. \n\n\n\n\n\n\n\\noindent\\textbf{Exp3}: Table \\ref{tab:nucle-table} shows the performance of the models trained on NUCLE dataset with additional artificial data generated by corresponding AEG models trained on NUCLE-CLC. We can see that the performance of all models, except ON-LSTM, improves significantly when 1 million artificial sentence pairs are added to the NUCLE training data, even though the errors in these sentences do not resemble natural errors. This contrasts with the result in Figure~\\ref{fig:table1a} where the performance of the GEC models trained with the combination of artificial error and CLC-NUCLE base data drops. This suggests that artificial data is helpful when the base data size is relatively small.\n\n\\begin{table}[htbp]\n\\small\n\\centering\n\\setlength{\\tabcolsep}{9 pt} \n\\begin{center}\n\\begin{tabular}{lcccc} \n\\toprule\n \\bf Model & \\bf NUCLE & \\bf +10K & \\bf +50K & \\bf +1M \\\\\n \\midrule\n\n\n MLCONV & 10.1 & 12.3 & 12.9 & 16.1 \\\\\n Transformer & 11.2 & \\bf 28.1 & \\bf 16.9 & 22.8 \\\\\n PRPN & 8.3 & 6.9 & 12.5 & \\bf 26.2 \\\\\n ON-LSTM & 9.4 & 11.3 & 11.8 & 6.0 \\\\\n \\bottomrule \n\\end{tabular}\n\\end{center}\n\\caption{\\label{tab:nucle-table} (Exp3) Using only NUCLE as base training for correction. The AEG models are trained using NUCLE-CLC data as in other experiments. \n}\n\\end{table}\n\n\\noindent\\textbf{Exp4}: The GEC models trained only on artificial data perform very poorly. The best setups, Transformer and MLCONV, achieve F0.5 scores of 12.8 and 12.4 respectively when trained with 2 million sentences generated by the corresponding AEG model.\nThis outcomes suggests that AEG data should be paired with some sample of real data to be effective.\n\n\\subsection{Manual Evaluation}\nWe performed a manual analysis of the generated error sentences and found that many of the errors did not always resemble those produced by humans. For example, \\textit{The situation with other types is not much (better $\\rightarrow$ downward)}. This shows that despite the noisiness of the error-generated data, some models (namely MLCONV and Transformer) were robust enough to improve. This also suggests that we may achieve better improvement by controlling artificial errors to resemble the errors produced by humans. The performance of syntax-based models goes down significantly with the addition of artificial errors, which indicates that these models may be sensitive to poor artificial data.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\nWe investigated the potential of recent neural architectures, as well as rule-based one, to generate parallel data to improve neural GEC. We found that the Multi-Layer Convolutional and Transformer models tended to produce data that could improve several models, though too much of it would begin to dampen performance. The most substantial improvements could be found when the size of the real data for training was quite small. We also found that the syntax-based models, PRPN and ONLSTM, are very sensitive to the quality of artificial errors and their performance drops substantially with the addition of artificial error data. Our experiments suggest that, unlike in machine translation, it is not very straightforward to use a simple back-translation approach for GEC as unrealistic errors produced by back-translation can hurt the correction performance substantially.\n\nWe believe this work shows the promise of using recent neural methods in an out-of-the-box framework, though with care. Future work will focus on ways of improving the quality of the synthetic data. Ideas include leveraging recent developments in powerful language models or better controlling for diversity and frequency of specific error types.\n\n\\section*{Acknowledgements}\nWe would like to thank the Grammarly Research Team, especially Maria Nadejde, Courtney Napoles, Dimitris Alikaniotis, Andrey Gryschuck, Maksym Bezva and Oleksiy Syvokon. We would also like to thank Sam Bowman, Kyunghyun Cho, and the three anonymous reviewers for their helpful discussion and feedback.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction and Summary}\n\\label{introduction}\n\nThree dimensional gauge theories, in contrast to four dimensional ones, are always asymptotic free and strongly coupled in the infra-red (IR) regime. \nIt is a central problem to investigate properties of their IR fixed points, and for this analysis instanton or monopole operators often play a key role. \n\nIndeed, (BPS) monopole operators played a crucial role in classifying a class of $\\cN=4$ supersymmetric linear quiver gauge theories from their long distance behavior \\cite{Gaiotto:2008ak}.\nIn this class it was already studied how to compute the R-charge of an arbitrary monopole operator \\cite{Borokhov:2002ib,Borokhov:2002cg}, which depends on the number of hypermultiplets charged under the associated gauge group. \nIf the number of hypermultiplets is great enough so that the R-charge of any monopole operator is always greater than half, then the system flows to a standard critical point.\nIn this case the theory was called ``good''.\nOn the other hand, if the number of hypermultiplets decreases to saturate some bound so that there exists a monopole operator whose R-charge is equal to half, then it decouples at the IR fixed point. \nThis was called an ``ugly'' theory.\nFinally if the number of hypermultiplets is so small that there exists a BPS monopole operator whose R-charge is less than half, then the IR fixed point of the system is not a standard one. This case was called a ``bad'' theory. \nWe refer to such monopole operators in the ugly\/bad theory as ugly\/bad ones.\nIn a bad theory the R-symmetry in the IR has to be different from that in the UV in order for the system to preserve the unitarity. \n\nThe infra-red physics of a bad quiver as well as the fate of such a bad monopole operator in the IR were further studied in \\cite{Kapustin:2009kz,Bashkirov:2013dda,Yaakov:2013fza,Hwang:2015wna} (See \\cite{Assel:2017jgo,Dey:2017fqs} for recent study).\nThe simplest example to study a bad theory is ${\\cal N}=4$ SQCD with $U(N_\\text{c})$ gauge group and $N_\\text{f}$ fundamental hypermultiplets.\\footnote{ \nIn this case the theory is good if $2N_c\\le N_f$, ugly if $N_f=2N_c-1$ and bad if $N_c < N_f\\le 2N_c -2$ \\cite{Gaiotto:2008ak}.} \nIt was pointed out in \\cite{Kapustin:2009kz} that this classification by using a monopole operator is related to the convergence property of the three-sphere partition function, which was shown to reduce to a finite dimensional integration by the supersymmetric localization, so that the partition function of a good\/bad theory is absolutely convergent\/divergent. \nThe three-sphere partition function in a bad case was further studied by converting into a good or ugly theory by adding extra hypermultiplets with real mass so as to flow to the original bad theory in the IR \\cite{Yaakov:2013fza}. \nThis modified partition function was used to propose a Seiberg-like duality between a good theory and a bad one in $\\cN=4$ SQCD, where the bad monopole operators decouple in the IR. \nThis duality was further supported by a direct comparison of the factorized form of the superconformal index for both sides \\cite{Hwang:2015wna}.\\footnote{\nRecent argument by \\cite{Assel:2017jgo} based on the analysis of the moduli space suggests that the duality holds only around the special singular locus called ``symmetric vacuum'' in the moduli space of the bad theory.\nThis is, however, still consistent with the observations in \\cite{Yaakov:2013fza,Hwang:2015wna}.\n}\n\nIt is natural to ask whether this classification of $\\cN=4$ supersymmetric linear quiver gauge theories works also for the theories including Chern-Simons interaction. \nThis question may not be trivial in the following sense. \nFirst the three-sphere partition function for a supersymmetric Chern-Simons theory has the Fresnel factor, which plays a role of a damping factor by performing the analytic continuation for the Chern-Simons coupling, so that there may be no issue on the convergence or at least completely different convergence property from the case without Chern-Simons terms. \nThis suggests that the classification by using the convergence property of the three-sphere partition function does not work or at least needs speculation. \nOn the other hand, the classification by employing monopole operators seems to work, though the modification needs to be done once Chern-Simons interaction is introduced. \nThis is because the gauge charge of bare monopole operators changes accordingly, and so does the total R-charge of the gauge invariant monopole operator due to the dress by compensating matter operators. \n\nThe above question is strategically good to be addressed first by using the simplest model in the $\\cN=4$ linear quiver Chern-Simons theories, that is ${\\cal N}=4$ $U(N_1)_k\\times U(N_2)_{-k}$ Chern-Simons theory coupled with single bi-fundamental hypermultiplet.\nIn our previous publication we studied the three-sphere partition function in this minimal $\\cN=4$ Chern-Simons theory, where we performed the remaining integrations in the localization formula explicitly \\cite{Nosaka:2017ohr}.\nAs a result we found that the resulting partition function diverges for $k -N_1 -N_2 \\le -2$. This divergence is rather the IR one than the UV one since it can be cured by introducing FI terms or equivalently mass terms. \nWe have further found that the resulting partition functions of the theory connected by the Hanany-Witten transition coincide up to an overall factor, which is given by a trigonometric function to the power of $k-N_1-N_2$.\nThese results are reminiscent of those for $\\cN=4$ SQCD in \\cite{Yaakov:2013fza} and suggests that the minimal $\\cN=4$ Chern-Simons theory is bad for $k -N_1 -N_2 \\le -2$ in the terminology of \\cite{Gaiotto:2008ak}.\nIn this paper we study this behavior of minimal $\\cN=4$ Chern-Simons theories by computing their superconformal indices \\cite{Kim:2009wb,Imamura:2011su} (see \\cite{Yokoyama:2011qu} for a review and \\cite{Aharony:2015pla} for related works). \n\nThe rest of this paper is organized as follows.\nIn section \\ref{N4theory} we review the results of \\cite{Nosaka:2017ohr} and compare them with the observations in \\cite{Kapustin:2009kz,Yaakov:2013fza} without Chern-Simons terms.\nWe provide additional evidence for this duality by counting the dimension of the moduli spaces.\nIn section \\ref{generalexpression} we introduce the superconformal index, and in section \\ref{perturbative} we estimate the contributions from dressed monopole operator.\nAs a result we find that for $k-N_1-N_2\\le -2$ there is a family of infinitely many monopole operator which contribute with non-positive power of $x$, the fugacity of $D+J_3$.\nThis implies that $U(N_1)_k\\times U(N_2)_{-k}$ theory with $k-N_1-N_2\\le -2$ is indeed a bad theory.\nIn section \\ref{perturbative_results} we further compute the superconformal index of non-bad theories with $k-N_1-N_2=\\pm 1$ explicitly in small $x$ expansion.\nBy comparing those for pairs of theories related through the Hanany-Witten transition, we find that the ratio of the superconformal indices have a completely same expression as the superconformal index of a hypermultiplet.\nThis suggests that the theory with $k-N_1-N_2=-1$ is ugly and dual to the paired good theory with $k-N_1-N_2=1$ plus a hypermultiplet.\n\n\n\n\n\n\n\n\n\\section{Minimal $\\cN=4$ Chern-Simons matter theories }\n\\label{N4theory}\n\n\nIn this section we briefly review the minimal $\\cN=4$ Chern-Simons matter theory or $\\text{U}(N_1)_k\\times \\text{U}(N_2)_{-k}$ Chern-Simons theory interacting with one bifundamental hypermultiplet \\cite{Gaiotto:2008sd}. (See also \\cite{Hosomichi:2008jd,Nosaka:2017ohr}.)\nThis theory is the simplest example in a class of ${\\cal N}=4$ linear quiver Chern-Simons matter theories, which are realized by taking the low energy limit of a UV field theory on D3-branes in a type IIB brane configuration. \nThe type IIB brane configuration of the simplest linear quiver theory is given by Table.\\ref{GWN4BraneSetup}.\n\\begin{table}[htbp]\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\n & $012$ & $3$ & $456789$ \\\\ \n\\hline \nNS5-brane & $\\bigcirc $ & $\\times$ & [456] \\\\ \n\\hline\n$N_1$ D3-branes & $\\bigcirc $ & $\\bigcirc $ & $\\times$ \\\\ \n\\hline\n$(1,k)$5-brane & $\\bigcirc $ & $\\times$ & $[(47)_\\theta(58)_\\theta(69)_\\theta]$ \\\\ \n\\hline\n$N_2$ D3-branes & $\\bigcirc $ & $\\bigcirc $ & $\\times$ \\\\ \n\\hline\nNS5'-brane & $\\bigcirc $ & $\\times$ & [456] \\\\ \n\\hline\n\\end{tabular} \n\\caption{Type IIB Brane configuration describing the minimal linear quiver theory.\nHere $\\bigcirc$ indicates that the corresponding brane extends in all directions specified on its above column and $\\times$ means that it does not in any. The numbers inside the bracket represent the directions of the extension, where $(ij)_\\theta$ stands for a single direction in the $x^ix^j$-plane with the angle $\\theta=\\arctan k$ from the $x^{i}$ axis. }\n\\label{GWN4BraneSetup}\n\\end{center}\n\\end{table} \nThe field contents of such a UV field theory of the simplest model are the two $\\cN=4$ vector multiplets associated with U$(N_1)\\times \\text{U}(N_2)$ and one hypermultiplet in the bi-fundamental representation in U$(N_1)\\times\\text{U}(N_2)$.\nIn the terminology of 3d ${\\cal N}=2$ representations, here a 3d ${\\cal N}=4$ vector multiplet consists of a vector multiplet plus a adjoint chiral multiplet, while a hypermultiplet consists of a pair of chiral multiplets.\n\nThese massless supersymmetric multiplets arise from open strings ending on the $N_1$ D3-branes, those on the $N_2$ D3-branes and the ones connecting the two stacks of D3-branes, respectively.\nThe $(1,k)$5-brane induces the `twisted' mass for the $\\cN=4$ vector multiplets \\cite{Kitao:1998mf,Bergman:1999na}: for the U$(N_1)$ vectormultiplet the ${\\cal N}=2$ Chern-Simons term with level $k$ plus the cocmplex mass term proportional to $k$ for the adjoint chiral multiplet; for the U$(N_2)$ vectormultiplet the same terms with $k$ replaced with $-k$.\nThe twisted mass breaks SO$(4)_{\\rm UV}$ R-symmetry to SO$(3)_{\\rm UV}$. \n\nIn the low energy limit this system gets strongly coupled and flow to the conformal fixed point, which is nothing but the minimal $\\cN=4$ Chern-Simons theory mentioned above. \nIn the IR limit, the $\\text{SO}(3)_{\\rm UV}$ R-symmetry is enhanced to SO$(4)_{\\rm R}$, which can be explicitly seen by integrating out the massive vector multiplet and adjoint chiral multiplets except the Chern-Simons gauge field. \nIn addition to the other global symmetry this system enjoys the parity invariance which exchanges the two gauge fields when $N_1=N_2$.\nWhen the two ranks are different, the system is invariant under the exchange of the ranks as well as the levels.\n\n\n\n\\subsection{Moduli space} \n\\label{modulispace}\n\nIn this subsection we briefly comment on the moduli space of the minimal $\\text{U}(N_1)_k\\times \\text{U}(N_2)_{-k}$ $\\cN=4$ Chern-Simons theory.\nWe will use the result of this analysis for a quick check of Seiberg-like duality discussed in the next subsection.\nFor convenience we write down the Lagrangian in the Euclidean space following the convention used in \\cite{Yokoyama:2013pxa} ($\\kappa=k\/(4\\pi)$)\n\\beal{\n\\cL\n=& \\tr \\bigg[\\kappa \\Bigl(-\\varepsilon^{\\mu\\nu\\rho}\\bigl(A_\\mu\\partial_\\nu A_\\rho +{2\\over3}A_\\mu A_\\nu A_\\rho\\bigr)+ \\varepsilon^{\\mu\\nu\\rho}\\bigl(\\wt A_\\mu\\partial_\\nu \\wt A_\\rho +{2\\over3}\\wt A_\\mu \\wt A_\\nu \\wt A_\\rho\\bigr) \\Bigr) +D_\\mu Q_A ^\\dagger D^\\mu Q^A - {\\psi^\\dagger}^A \\sla D \\psi_A \\nn\n&+ {1\\over 2\\kappa}(- {\\psi^\\dagger}^A \\psi_A Q^\\dagger_C Q^C + \\psi_A {\\psi^\\dagger}^A Q^B Q^\\dagger _B + \\varepsilon_{CD} \\varepsilon_{AB} {\\psi^\\dagger}^A Q^C {\\psi^\\dagger}^B Q^D +\\varepsilon^{AB} \\varepsilon^{CD} Q^\\dagger_C \\psi_A Q^\\dagger_D \\psi_B )\\bigg] +V_q.\n\\label{N4GWaction}\n}\nHere $A,B=1,2$ and the covariant derivative for $Q^A$ and $\\psi_B$ is defined as $D_\\mu X = \\partial_\\mu X +A_\\mu X -X \\wt A_\\mu $.\n$V_q$ is the scalar potential given b\n\\footnote{ \nThis form of the scalar potential is obtained after rewriting the one given in \\cite{Yokoyama:2013pxa} using an identity for SU(2) indices. \n}\n\\beal{ \nV_{q} =\\tr [ T_C (T_C)^\\dagger ]\n}\nwhere $T_C = {1\\over 2\\kappa} \\varepsilon_{AB} Q^A Q^\\dagger_C Q^B$. \n\nThe vacuum moduli space is the solutions $(Q^1,Q^2)$ of $T_1=T_2=0$, modded by the U$(N_1)\\times \\text{U}(N_2)$ gauge transformations.\nThe generic point on the moduli spcae can be characterized as follows.\nDue to the (generalized) parity invariance we can assume $N_1\\le N_2$ without loss of generality.\nBy using the gauge degrees of freedom the complex scalar $Q^1{}^a_{i}\\; (1\\leq a \\leq N_1, 1\\leq i \\leq N_2 )$ can be diagonalized so that\n\\beal{ \nQ^1{}^a_{ i} = \\left\\{\n\\begin{array}{cl}\n r_a \\delta^a_{ i} & (1 \\leq i \\leq N_1) \\\\\n 0 & (N_1 < i \\leq N_2) \\\\\n\\end{array}\n\\right.\n}\nwhere $r_a$ are real positive numbers.\nThe residual gauge symmetry is the diagonal $U(1)^{N_1}$. \nNow, in the generic situation where all $r_a$ are different with each other, the vacuum equation $T_C=0$ requires also $Q^2$ to be diagonal\n\\beal{ \nQ^2{}^a_{ i} = \\left\\{\n\\begin{array}{cl}\nq_a \\delta^a_{ i} & (1 \\leq i \\leq N_1) \\\\\n0 & (N_1 < i \\leq N_2) \\\\\n\\end{array}\n\\right.\n}\nwith $q_a$ some complex numbers.\n\nThe diagonal U(1)$^{N_1}$ gauge symmetry fixes all the degrees of freedom of the gauge fields except its zero modes in the Cartan part, which span the extra directions in the moduli space since the abelian gauge field does not couple with the bi-fundamental matter fields.\nThe gauge flux quantization gives the $2\\pi$ periodicity for the range of each zero mode, while the level $k$ Chern-Simons interaction breaks the U(1)$^{N_1}$ to $\\mathbb{Z}_k^{N_1}$ (see \\cite{Imamura:2008nn,Martelli:2008si} for example).\nThese zero modes become coordinates of the moduli space as $(S^1\/\\mathbb{Z}_k)^{N_1}$.\n\nAs a result, the classical moduli space is generically given by {$(\\mathbb{C}^2\/\\mathbb{Z}_k)^{\\text{min}(N_1,N_2)}\/{\\cal S}_{\\text{min}(N_1,N_2)}$. We suspect that the moduli space does not receive any quantum correction and is classically exact at least for non-bad theories as in the cases with the Higgs phase in a linear quiver non-Chern-Simons theory and with $\\cN=6$ ABJM theory. \nThis is indeed supported from the superconformal index computed in section \\ref{SCindex}.\nWe leave the proof thereof to a future work. In what follows, we confirm that this moduli space is consistent with the analysis below that there exists a decoupled sector of real dimension $4$ in the duality between pairs of non-bad theories\n\\footnote{ \nPrecisely speaking the decoupled sector turns out to couple to the other sector through the topological current. \n}\n\n\\subsection{Comments on level-rank duality} \n\\label{levelrankduality}\n\nThis minimal $\\cN=4$ Chern-Simons theory is expected to enjoy the level-rank (or Seiberg-like) duality \\cite{Giveon:2008zn}, that is, the duality between the $\\text{U}(N_1)_k\\times \\text{U}(N_2)_{-k}$ theory and the $\\text{U}(k-N_2)_k\\times \\text{U}(k-N_1)_{-k}$ theory.\nThe two theories are indeed related under the exchange of the two NS5-branes in the tpe IIB brane setup given in table \\ref{GWN4BraneSetup} by taking into account the Hanany-Witten brane creation\/annihilation \\cite{Hanany:1996ie}.\\footnote{\nSee \\cite{Yokoyama:2013pxa} for the first scratch of evidence of this self-duality from the large $N$ thermal free energy.\n}\n\n{\nOn the other hand, there is a dicrepancy in the $S^3$ partition functions between the two theories \\cite{Nosaka:2017ohr}; in some cases the partition function even diverges though it is finite in the other theory in the supposed dual.\nBy introducing the FI parameters which regularizes the divergences, we instead found that in any Hanany-Witten pair the ratio of the partition functions takes the same expression which depends on $k$, $N_1$, $N_2$ only through $k-N_1-N_2$.\n}\nThis mismatch of the overall factor may be a signal of the existence of some decoupled sector. Indeed, it was pointed out that such decoupled sector appears if there exists a monopole operator whose dimension computed in the UV theory saturates or violates the unitarity bound in the case without Chern-Simons interaction \\cite{Yaakov:2013fza,Hwang:2015wna}.\nIn the terminology used in \\cite{Gaiotto:2008ak} such theories with (naively) unitarity violating monopole operators are called {\\it bad} theories, and the divergences discovered in \\cite{Nosaka:2017ohr} suggests the following classification of the minimal ${\\cal N}=4$ Chern-Simons theories:\n\\begin{itemize}\n\\item [] The minimal ${\\cal N}=4$ $\\text{U}(N_1)_k\\times \\text{U}(N_2)_{-k}$ Chern-Simons theory is respectively\n\\begin{align}\n\\text{good\/ugly\/bad if }\nk-N_1-N_2 \\text{ greater than\/equal to\/smaller than } -1.\n\\label{goodbaduglyfromZS3}\n\\end{align}\n\\end{itemize}\nThe main motivation of this paper is to investigate whether this classification is valid and what such decoupled sector in the minimal $\\cN=4$ Chern-Simons theories is by using a superconformal index. \n \nThe existence of the decoupled sector can be understood from the dimension of the moduli space.\nFor example, let us consider the minimal $\\cN=4$ theory with $\\text{U}(1)_3\\times \\text{U}(1)_{-3}$, whose moduli space is of real dimension $4$.\nThis theory is suggested to be dual to the one with $\\text{U}(2)_3\\times \\text{U}(2)_{-3}$.\nThe dimension of the moduli space in the latter theory, however, is $8$ and does not match with the former theory.\nWe suspect that the discrepancy is explained as the presence of some decoupled sector in the latter theory, as in the cases without Chern-Simons terms.\n\nIn what follows, we study such decoupled sectors from a superconformal index, which encodes the BPS spectrum of the theory and is useful to analyze the system more precisely.\n\n\\section{Superconformal index}\n\\label{SCindex} \n\nIn this section we examine the IR aspects of the ${\\cal N}=4$ minimal $\\text{U}(N_1)_k\\times \\text{U}(N_2)_{-k}$ superconformal Chern-Simons theory proposed in section \\ref{levelrankduality} by studing the superconformal index.\nAfter introducing the definition of the superconformal index in section \\ref{generalexpression}, in section \\ref{analytic} we first provide an analytic computation of the superconformal index.\nThough this analytic computation works well only for the abelian case $N_1=N_2=1$, it and also tells us a technical obstacle associated with the bad theory.\nIn section \\ref{perturbative} we adopt the small $x$ expansion.\nThrough an argument on its convergence we provide the good\/bad classifiation for the theory, which coincides with the one suggested from the convergence of the $S^3$ partition function \\eqref{goodbaduglyfromZS3}.\nFinally we compare the superconformal indices of the two theories related under the Hanany-Witten transition in several examples.\nAs we have expected, we observe that their ratio takes the identical form as the superconformal index of a hypermulitplet.\n\n\\subsection{General expression} \n\\label{generalexpression}\n\nIn this section we study the superconformal index of minimal $\\cN=4$ $\\text{U}(N_1)_k\\times \\text{U}(N_2)_{-k}$ Chern-Simons matter theories.\nFirst let us define a superconformal index by introducing fugacities of the global symmetry which commutes with the chosen $\\cN=2$ supersymmetry.\nThis requires a special care as we shall explain below. \n\nWe compute the superconformal index by the supersymmetric localization so that \nwe deform a minimal $\\cN=4$ Chern-Simons theory to be an $\\cN=3$ UV field theory described in section \\ref{N4theory} in the free theory limit. \nWe emphasize that the SO(4)$_\\text{R}$ symmetry is broken to SO(3)$_{\\text{UV}}$ under the deformation. \nHence we shall turn on the fugacity only for this SO(3)$_{\\rm UV}$ symmetry to define a superconformal index of the system, which we denote by $\\sigma$. \nFurthermore, there is a conserved topological current for each gauge group associated with any node in the quiver diagram. \nWe also turn on a fugacity for each topological current, which we denote by $y,z$ respectively for $\\text{U}(N_1)$ and $\\text{U}(N_2)$.\nTaking these into account the superconformal index for this system is defined by the following trace over the spectrum of the theory:\n\\begin{align}\n{\\cal I}_{k,N_1,N_2}(x,\\sigma,y,z)=\\Tr[ (-1)^F e^{-\\beta'\\{Q,Q^\\dagger\\}} x^{D+J_3} \\sigma^A y^{m_{\\rm tot}} z^{n_{\\rm tot}} ]\n\\label{IndexDefinition}\n\\end{align}\nwhere $Q$ is a nilpotent supercharge and $Q^\\dagger$ is its BPZ conjugate, while $D$, $J_3$, $A$, $(m_\\text{tot},n_\\text{tot})$ are respectively the dilatation, the angular momentum, the Cartan generator for SO$(3)_\\text{UV}$ and the total magnetic charge of U$(N_1)\\times \\text{U}(N_2)$.\nOne can show that \\eqref{IndexDefinition} is actually independent of $\\beta'$ so that only BPS states, which satisfy $\\{Q,Q^\\dagger\\}=D-J_3-R = 0$, contribute to the index.\n\nTo perform the supersymmetric localization, \nwe first rewrite \\eqref{IndexDefinition} as the path integral form over $\\mbf S^2 \\times \\mbf S^1$. \nThen we take the weak coupling limit so that the evaluation of the path integral by the WKB approximation becomes exact. As a result each supersymmetric multiplet contributes to the index independently under the saddle points parametrized by magnetic charges and holonomies for U$(N_1) \\times {\\rm U}(N_2)$, which \nwe denote by $(m_a,n_i)$ and $(u_a,v_i)$ with $a=1,\\cdots, N_1, i=1,\\cdots, N_2$, respectively. \nThen the 1-loop contribution of $\\cN=4$ U$(N_1) \\times {\\rm U}(N_2)$ vector multiplet is given by\n\\beal{\n{\\cal I}_\\text{vec}&=\n\\prod_{a\\neq b}^{N_1}(1-x^{|m_a-m_b|}u_au_b^{-1})\n\\prod_{i\\neq j}^{N_2}(1-x^{|n_i-n_j|}v_iv_j^{-1}), \n\\label{Ivec}\n}\nand that of the bi-fundamental hypermultiplets is \n\\beal{\n{\\cal I}_\\text{hyp}&=\n\\prod_{a=1}^{N_1}\n\\prod_{j=1}^{N_2}\n\\frac{\n(x^{|m_a-n_j|+\\frac{3}{2}}\\sigma^{-1}u_a^{-1}v_j;x^2)_\\infty\n}{\n(x^{|m_a-n_j|+\\frac{1}{2}}\\sigma u_a v_j^{-1};x^2)_\\infty\n}\n\\frac{\n(x^{|m_a-n_j|+\\frac{3}{2}}\\sigma u_a v_j^{-1};x^2)_\\infty\n}{\n(x^{|m_a-n_j|+\\frac{1}{2}}\\sigma^{-1}u_a^{-1}v_j;x^2)_\\infty\n},\n\\label{Ihyp}\n}\nwhere we have used the Pochhammer symbol defined by \n\\beal{\n(z;x)_m :=& \\prod_{n=0}^{m-1} (1-zx^n). \n}\nNote that the contribution of the adjoint chiral multiplet in the $\\cN=4$ vector multiplet completely cancels.\nIncluding the vacuum and classical contribution the index is finally given by \n\\begin{align}\n{\\cal I}_{k,N_1,N_2}(x,\\sigma,y,z)\n&=\\frac{1}{N_1!N_2!}\\sum_{\\overrightarrow{m}\\in \\mathbb{Z}^{N_1} \\atop \\overrightarrow{n}\\in\\mathbb{Z}^{N_2}}\n\\int \\prod_{a=1}^{N_1}\\frac{du_a}{2\\pi iu_a} \n\\prod_{i=1}^{N_2}\\frac{dv_i}{2\\pi iv_i} y^{m_\\text{tot}}z^{n_\\text{tot}}\nx^{\\epsilon_0} u_a^{km_a} v_i^{-kn_i}\n{\\cal I}_\\text{vec}\n{\\cal I}_\\text{mat}\n\\end{align}\nwhere $m_\\text{tot}=\\sum_{a=1}^{N_1}m_a$, $n_\\text{tot}=\\sum_{i=1}^{N_2}n_i$ and\n\\begin{align}\n\\epsilon_0&=\\frac{1}{2}\\sum_{a=1}^{N_1}\\sum_{j=1}^{N_2}|m_a-n_j|\n-\\frac{1}{2}\\sum_{a,b=1}^{N_1}|m_a-m_b|\n-\\frac{1}{2}\\sum_{i,j=1}^{N_2}|n_i-n_j|. \n\\end{align}\nThis expression may be obtained from the superconformal index of ABJM theory determined in \\cite{Kim:2009wb} by excluding the contribution of one bifundamental hypermultiplet. \n\nThe above expression of the index can be rewritten as a more compact form \nby using its invariance under the action of the Weyl group for ${\\rm U}(N_1) \\times {\\rm U}(N_2)$:\n\\begin{align}\n{\\cal I}_{k,N_1,N_2}(x,\\sigma,y,z)\n&=\\sum_{\\overrightarrow{m}\\in \\mathbb{Z}^{N_1}\/{\\cal S}_{N_1} \\atop \\overrightarrow{n}\\in\\mathbb{Z}^{N_2}\/{\\cal S}_{N_2}}\\frac{y^{m_\\text{tot}}z^{n_\\text{tot}}}{|W_{(\\overrightarrow{m},\\overrightarrow{n})}|}{\\cal I}^{(\\overrightarrow{m},\\overrightarrow{n})}(x,\\sigma)\n\\end{align}\nwhere $\\cS_N$ is the permutation group for $N$ elements and $|W_{(\\overrightarrow{m},\\overrightarrow{n})}|$ is the number of the permutations which fix the monopole configuration specified by $(\\overrightarrow{m},\\overrightarrow{n})$, and \n\\begin{align}\n{\\cal I}^{(\\overrightarrow{m},\\overrightarrow{n})}(x,\\sigma)=\nx^{\\epsilon_0}\n\\int\n\\prod_{a=1}^{N_1}\\frac{du_a}{2\\pi iu_a} u_a^{km_a}\n\\prod_{i=1}^{N_2}\\frac{dv_i}{2\\pi iv_i} v_i^{-kn_i}\n{\\cal I}_\\text{vec}\n{\\cal I}_\\text{mat}.\n\\end{align}\n\n\\subsection{Analytic computation in the abelian case}\n\\label{analytic}\n\n\nBefore considering the case with general ranks, let us start with the abelian case with $N_1=N_2=1$ with a general Chern-Simons coupling constant.\nIn this case we can compute the superconformal index analytically without difficulty.\n\nIn the minimal Gaiotto-Witten theory, the diagonal U(1) gauge field does not couple to the bi-fundamental matter, so that the corresponding holonomy integration can be trivially performed. \nIn the abelian case, the only one non-trivial integration remains. \nWe perform the remaining residue integration by deforming the integration contour to either the origin or infinity so as to avoid the poles arising due to the Chern-Simons interaction, as shown below. \n\nLet us first perform the integration of the diagonal U(1). This can be done by changing integration variables such that \n\\be \nw = \\sigma^{-1} v_1\/u_1. \n\\ee\nWe fix the region of variables by $\\sigma x^\\half <1, x<1,\\sigma x^{-\\half} >1 $.\nThen the integral form of the index becomes \n\\beal{\n{\\cal I}_{k,1,1}^{(m_{\\rm tot},n_{\\rm tot})}(x,\\sigma)\n=& \\oint {d{u} \\over (2\\pi i{u}) }u^{-k(m_{\\rm tot} -n_{\\rm tot})} \\oint {dw \\over (2\\pi iw) } \\sigma^{km_{\\rm tot}} w^{km_{\\rm tot}} {(wx^{3\/2} ; x^2)_\\infty (w^{-1}x^{3\/2}; x^2)_\\infty \\over (w^{-1} x^{\\half} ;x^2)_\\infty (wx^{\\half} ;x^2)_\\infty }, \n\\notag\n}\nin which the integration of $u$ variable decouples from the other part. \nPerforming this integration over $u$ gives \n\\beal{\n{\\cal I}_{k,1,1}^{(m_{\\rm tot},n_{\\rm tot})}(x,\\sigma)\n=&\\delta_{m_{\\rm tot},n_{\\rm tot}} \\sigma^{km_{\\rm tot}} \\oint {dw \\over (2\\pi iw) } w^{km_{\\rm tot}} {(wx^{3\/2} ; x^2)_\\infty (w^{-1}x^{3\/2}; x^2)_\\infty \\over (w^{-1} x^{\\half} ;x^2)_\\infty (wx^{\\half} ;x^2)_\\infty }. \n}\nIt turns out that the contribution associated with the different monopole charges for two U(1) gauge groups vanishes\n\\footnote{ \nThis statement may hold in the non-abelian case as well by considering the total monopole charges for U($N_1$) and U($N_2$). \n}\nWe perform the remaining integral by deforming the integration contour to either the origin or infinity so as to avoid the poles generated by the classical contribution, $w^{km_{\\rm tot}}$ at the origin or infinity, which depends on the value of $m_{\\rm tot}$.\n\nWhen $m_{\\rm tot} \\geq 0$, the term $w^{km_{\\rm tot}}$ is a pole around $w\\sim\\infty$. \nTo avoid this pole, we deform the integration contour to the origin so that we pick up the poles inside the unit circle. \n\\be \n{\\cal I}_{k,1,1}^{(m_{\\rm tot},n_{\\rm tot})}(x,\\sigma)\n= \\delta_{m_{\\rm tot},n_{\\rm tot}} \\sigma^{km_{\\rm tot}} \\oint_{|w|<1} {dw \\over (2\\pi iw) } w^{km_{\\rm tot}} {(wx^{3\/2} ; x^2)_\\infty (w^{-1}x^{3\/2}; x^2)_\\infty \\over (w^{-1} x^{\\half} ;x^2)_\\infty (wx^{\\half} ;x^2)_\\infty }\n\\ee\nWe pick up poles inside the unit circle at $w= x^{2n+\\half}$ with $n\\geq0$, which come from the term $(w^{-1} x^{\\half} ;x^2)_\\infty$ in the denominator.\\footnote{ \nAs is the case without Chern-Simons interaction, we do not pick up the pole at the origin in this evaluation, which arises when $m_{\\rm tot} = 0$ (see \\cite{Krattenthaler:2011da} for example). This can be justified in the following way. Since the contribution coming from the pole at the origin is clearly ill-defined, one needs to regularize the contribution with $m_{\\rm tot}=0$ for its evaluation. We regularize it by introducing a (discrete) background magnetic flux studied in \\cite{Kapustin:2011jm} so that the pole at the origin disappears, which is turned off after the evaluation. Then the evaluation can be done as described in the main text.\n}\nPerforming the residue integral we obtain \n\\beal{\n{\\cal I}_{k,1,1}^{(m_{\\rm tot},n_{\\rm tot})}(x,\\sigma)\n=&\\delta_{m_{\\rm tot},n_{\\rm tot}}\\sigma^{km_{\\rm tot}} \\sum_{n\\geq0} (x^{2n+\\half})^{km_{\\rm tot}} {((x^{2n+\\half}) x^{3\/2} ; x^2)_\\infty ((x^{2n+\\half})^{-1}x^{3\/2}; x^2)_\\infty \\over (x^{-2n};x^2)_n(x^2;x^2)_\\infty ((x^{2n+\\half})x^{\\half} ;x^2)_\\infty } \\nn\n=& \\delta_{m_{\\rm tot},n_{\\rm tot}}\\sigma^{km_{\\rm tot}} f^{m_{\\rm tot}}(x)\n}\nwhere we set \n\\be \nf^{m_{\\rm tot}}(x)\n= {1\\over (x^2;x^2)_\\infty} \\sum_{n\\geq0} (x^{2n+\\half})^{km_{\\rm tot}} {(x^{2n+2}; x^2)_\\infty (x^{-2n+1} ; x^2)_\\infty \\over (x^{-2n};x^2)_n (x^{2n+1} ;x^2)_\\infty }. \n\\ee\n\nWhen $m_{\\rm tot} < 0$, on the other hand, the term $w^{km_{\\rm tot}}$ is a pole around $w\\sim0$. To avoid this pole we deform the integration contour to the infinity so that we pick up the poles outside the unit circle. \n\\be \n{\\cal I}_{k,1,1}^{(m_{\\rm tot},n_{\\rm tot})}(x,\\sigma)\n= \\delta_{m_{\\rm tot},n_{\\rm tot}} \\sigma^{km_{\\rm tot}} \\oint_{|w|>1} {dw \\over (2\\pi iw) } w^{km_{\\rm tot}} {(wx^{3\/2} ; x^2)_\\infty (w^{-1}x^{3\/2}; x^2)_\\infty \\over (w^{-1} x^{\\half} ;x^2)_\\infty (wx^{\\half} ;x^2)_\\infty }.\n\\ee\nExchanging the integration variable such that $w \\to \\bar w = 1\/w$, we find \n\\beal{\n{\\cal I}_{k,1,1}^{(m_{\\rm tot},n_{\\rm tot})}(x,\\sigma)=&\\delta_{m_{\\rm tot},n_{\\rm tot}}\\sigma^{km_{\\rm tot}} \\oint_{|\\bar w|<1} {d\\bar w \\over (2\\pi i\\bar w) }\\bar w^{-km_{\\rm tot}} {(\\bar w^{-1}x^{3\/2} ; x^2)_\\infty (\\bar w x^{3\/2}; x^2)_\\infty \\over (\\bar w x^{\\half} ;x^2)_\\infty (\\bar w^{-1} x^{\\half} ;x^2)_\\infty } \n}\nwhich is the same as $\\delta_{m_{\\rm tot},n_{\\rm tot}}\\sigma^{km_{\\rm tot}} f^{-m_{\\rm tot}}(x)$.\n\nAs a result, we obtain \n\\beal{\n{\\cal I}_{k,1,1}(x,\\sigma, y,z)\n=&\\sum_{m_{\\rm tot} \\geq 0 } (yz)^{m_{\\rm tot}} \\sigma^{km_{\\rm tot}} f^{m_{\\rm tot}}(x) + \\sum_{m_{\\rm tot} < 0 } (yz)^{m_{\\rm tot}} \\sigma^{km_{\\rm tot}} f^{-m_{\\rm tot}}(x) \\nn\n=&\\sum_{m_{\\rm tot} > 0 } ( (yz\\sigma^k)^{m_{\\rm tot}} + (yz\\sigma^k)^{-m_{\\rm tot}} ) f^{m_{\\rm tot}}(x) + f^{0}(x). \n} \nWe have verified this expression by Taylor-expansion in terms of small $x$\n\\footnote{ \nTo expand the superconformal index up to $x^{\\nu_\\text{th}}$ we truncate the summation over $m_{\\rm tot}$, $n$ and the infinite product in the Pochhammer symbols as follows.\n\\begin{align}\n\\sum_{m_{\\rm tot}=1}^\\infty&\\longrightarrow \\sum_{m_{\\rm tot}=1}^{m_\\text{max}},\\quad\\quad\\quad\\quad \\Bigl(m_\\text{max}=\\Bigl[\\frac{2\\nu_\\text{th}}{k}\\Bigr]\\Bigr),\\nonumber \\\\\n\\sum_{n=0}^\\infty&\\longrightarrow \\sum_{n=0}^{n_\\text{max}},\\quad\\quad\\quad\\quad \\Bigl(n_\\text{max}=\\Bigl[\\frac{1}{2km_{\\rm tot}+1}\\Bigl(\\nu_\\text{th}-\\frac{km_{\\rm tot}}{2}\\Bigr)\\Bigr]\\Bigr),\\nonumber \\\\\n\\frac{(x;x^2)_\\infty}{(x^{1+2n};x^2)_\\infty}\n&\\longrightarrow \\frac{\\prod_{j=0}^{\\text{Floor}[\\frac{x_\\text{th}-n-1}{2}]}(1- x^{1+2j})\n}{\n\\prod_{j=0}^{\\text{Floor}[\\frac{x_\\text{th}-3n-1}{2}]}(1- x^{1+2n+2j})\n}.\n\\end{align}\n}\n\n\n\n\\subsubsection{Obstacle in bad theory for higher ranks}\nIn the case without Chern-Simons terms \\cite{Hwang:2015wna} the benefit of such computation was that it works also for bad theories and enable us the direct comparison of the superconformal indices between the Hanany-Witten pairs.\nIn the same motivation below we show an attempt to generalize the above computation for the theories with higher ranks, though we end up with a difficulty for bad cases.\n\nFor simplicity we consider the case with $N_1=N_2=N$, which satisfy the s-rule bound $k\\le N$.\nFirst we consider the $u_a$-integrations, estimating the poles at the origin and at the infinity as (${\\widetilde u}_a=u_a^{-1}$)\n\\begin{align}\n{\\cal I}\\sim\n\\begin{cases}\n\\int du_a u_a^{km_a-(N-1)-1}=\\int du_a u_a^{km_a-N}&\\quad(u_a\\sim 0)\\\\\n\\int du_a u_a^{km_a+(N-1)-1}=\\int d{\\widetilde u}_a e^{-km_a-N} &\\quad(u_a\\sim\\infty)\n\\end{cases}\n\\end{align}\nwhere in the middle $u_a^{\\mp (N-1)}$ comes from ${\\cal I}_\\text{vec}$, which increases the singularity at $u_a=0,\\infty$.\n\nTo see the difficulty we focus on the case $m_a>0$ and the contribution for the following choice of the poles: $u_a=(const)\\cdot v_{i=a}$ which amounts to\n\\begin{align}\n{\\cal I}^{(\\overrightarrow{m};\\overrightarrow{n})}\\sim \\int \\frac{dv_a}{2\\pi iv_a} v_a^{-k(n_a-m_a)}{\\cal I}_\\text{vec}{\\cal I}_\\text{mat}.\n\\end{align}\nSince the integrand is still a complicated function of $\\{v_i\\}$, we may want to perform the $v_i$-integration one by one by classifying $n_i$ in the same way as we did in $u_a$ integration.\nHere we encounter a problem.\nDue to the substitution $u_a=(const)\\cdot v_a$, the $\\text{U}(N)_k$ part of ${\\cal I}_\\text{vec}$ contributes in the same way as $\\text{U}(N)_{-k}$ part.\nHence the estimation of poles at $v_i=0,\\infty$ is modified as\n\\begin{align}\n{\\cal I}\\sim \n\\begin{cases}\n\\int dv_i v_i^{-k(n_i-m_i)-2(N-1)-1}=\\int dv_i v_i^{-k(n_i-m_i)-2N+1} &\\quad(v_i\\sim 0)\\\\\n\\int dv_i v_i^{-k(n_i-m_i)+2(N-1)-1}=\\int dv_i {\\widetilde v}_i^{k(n_i-m_i)-2N+1} &\\quad (v_i\\sim \\infty)\n\\end{cases},\n\\end{align}\nwhich indicates:\nfor $n_i-m_i<-(2N-2)\/k$ the integration has no pole at $v_i=0$ hence can be computed from the residues at the poles in $|v_i|<1$;\nfor $n_i-m_i>(2N-2)\/k$ the integration has no pole at $v_i=\\infty$ hence can be computed from the residues at the poles in $|v_i|>1$;\nfor $-(2N-2)\/k\\le n_i-m_i\\le (2N-2)\/k$ both poles are present with of order $N$ hence such computation does not work.\nIf $2N-2-2$, the expansion starts from a positive power of $x$. \n\\end{enumerate}\nNotice that the singularities for $k-N_1-N_2\\le -2$ cannot be cured by introducing chemical potential for $m_\\text{tot}=n_\\text{tot}$.\nTo see this, let us work out similar power estimation for the following monopole charge\n\\begin{align}\n(\\overrightarrow{m};\\overrightarrow{n})=(m+m_\\text{tot},-m,0,\\cdots,0;m+m_\\text{tot},-m,0,\\cdots,0),\n\\end{align}\nwith $m$ being an arbitrary integer, which contributes to the sector of $m_\\text{tot}$.\nFor this monopole charge the contribution from bare monopole is $x^{\\epsilon_0}=x^{-|2m+m_\\text{tot}|-\\frac{N_1+N_2-4}{2}(|m+m_\\text{tot}|+|m|)}$.\nThe bare monopole is charged as $u_1^{k(m+m_\\text{tot})} u_2^{-km} v_1^{-k(m+m_\\text{tot})} v_2^{km}$ due to the Chern-Simons term.\nAssuming $m>0$ and $|m|$ being large enought compared with $m_\\text{tot}$ we find that the most economical way to cancel this charge and form a gauge singlet is to bring $(x^{\\frac{1}{2}}u_1^{-1}v_1)^{k|m+m_\\text{tot}|}\\cdot (x^{\\frac{1}{2}}u_2v_2^{-1})^{k|m|}$ from ${\\cal I}_\\text{hyp}$.\nIn total, the leading power of the superconformal index is\n\\begin{align}\n{\\cal I}_{k,N_1,N_2}^{(m+m_\\text{tot},-m,0,0,\\cdots,0;m+m_\\text{tot},-m,0,0,\\cdots,0)}(x,\\sigma)\\sim \nx^{\\epsilon_0+\\frac{k|m+m_\\text{tot}|}{2}+\\frac{k|m|}{2}}=x^{(k-N_1-N_2+2)(m+\\frac{m_\\text{tot}}{2})}.\n\\label{badforeachmtot}\n\\end{align}\nHere we have used $m>0$ and $m\\gg m_\\text{tot}$ to reduce $|m+m_\\text{tot}|=m+m_\\text{tot}$, $|m|=m$, $|2m+m_\\text{tot}|=2m+m_\\text{tot}$.\nThe result \\eqref{badforeachmtot} implies that for $k-N_1-N_2\\le -2$ the non-analyticity of the superconformal index at $x=0$ appears already at each sector of $m_\\text{tot}$.\n\nThis preliminary analysis implies that the classification \\eqref{goodbaduglyfromZS3} done by using the convergence property of the three-sphere partition function works also for a superconformal index. \nIn what follows we display the result of the perturbative computation of the superconformal index for several good or ugly theories.\n\n\\subsubsection{Results}\n\\label{perturbative_results}\n\nWe display the result of the perturbative computation of the superconformal index for various $k$, $N_1$, $N_2$.\nWe are especially interested in the pairs of $\\text{U}(N_1)_k\\times \\text{U}(N_2)_{-k}$ theory and $\\text{U}(k-N_2)_k\\times \\text{U}(k-N_1)_{-k}$ theory, which are suggested to be dual with each other from the Hanany-Witten transition.\nSince $k-N_1-N_2=k-N_1-N_2$ transforms $k-N_1-N_2\\rightarrow -k-N_1-N_2$ under the Hanany-Witten transition, such comparison is possible only for the pairs of $k-N_1-N_2=k-N_1-N_2=\\pm 1$ because of the singularity for $k-N_1-N_2<-2$ argued in section \\ref{levelrankduality}.\\footnote{\nFor $k-N_1-N_2=0$ the pairs are different only in the sign of $k$.\nIn this case the superconformal indices trivially coincide in the dual pair.\n}\nFor $|k-N_1-N_2|=1$ case, the explicit computation shows that the superconformal indices do not coincide in the proposed dual pair.\nNevertheless, their ratio simplifies and allows a physical interpretation as the contribution from an extra hypermultiplet.\n\nWe have computed the superconformal index for the proposed dual pairs with $N_1,N_2\\le 2$, namely, $(k,N_1,N_2)=(1,0,0),(1,1,1),(2,0,1),(2,1,2),(3,0,2),(3,1,1),(3,1,3),(3,2,2)$ up to $x^5$.\nHere for the cases with $N_1=0$ or $N_2=0$ the theory is the pure Chern-Simons theory and the superconformal index is trivially ${\\cal I}(x,\\sigma)=1$.\nWe have taken into account all the monopole charges in $|m_a|,|n_i|\\le 20$, and found only small number of those in $|m_a|,|n_i|\\le 10$ displayed in table \\ref{contributingmn} contributes to the superconformal index, which supports this truncation is indeed exact up to $x^5$.\n\\begin{table}\n\\begin{tabular}{|c|c|c|l|}\n\\hline\n$k$&$N_1$&$N_2$&$(\\overrightarrow{m};\\overrightarrow{n})$ (up to permutations and $(\\overrightarrow{m},\\overrightarrow{n})\\rightarrow (-\\overrightarrow{m},-\\overrightarrow{n})$)\\\\ \\hline\\hline\n$1$&$1$&$1$&$(0,0),(1,1),(2,2),(3,3),(4,4),(5,5),(6,6),(7,7),(8,8),(9,9),(10,10)$\\\\ \\hline\n$2$&$1$&$2$&$\n(0;-1,1),\n(0;0,0),\n(1;0,1),\n(2;0,2),\n(3;0,3),\n(4;0,4),\n(5;0,5),\n(6;0,6),\n(7;0,7),\n$\\\\\n&&&$\n(8;0,8),\n(9;0,9),\n(10;0,10)\n$\\\\ \\hline\n$3$&$1$&$1$&$(0,0),(1,1),(2,2),(3,3)$\\\\ \\hline\n$3$&$2$&$2$&$(-5,5;-5,5),\n(-4,4;-4,4),\n(-3,3;-3,3),\n(-2,2;-2,2),\n(-1,1;-1,1),\n$\\\\\n&&&$\n(0,0 ;0,0),\n(-4,5;-4,5),\n(-3,4;-3,4),\n(-2,3;-2,3),\n(-1,2;-1,2),\n(0,1 ;0,1),\n$\\\\\n&&&$\n(-4,6;-4,6),\n(-3,5;-3,5),\n(-2,4;-2,4),\n(-1,3;-1,3),\n(0,2 ;0,2),\n(1,1 ;1,1),\n$\\\\\n&&&$\n(-3,6;-3,6),\n(-2,5;-2,5),\n(-1,4;-1,4),\n(0,3 ;0,3),\n(1,2 ;1,2),\n(-3,7;-3,7),\n$\\\\\n&&&$\n(-2,6;-2,6),\n(-1,5;-1,5),\n(0,4 ;0,4),\n(1,3 ;1,3),\n(-2,7;-2,7),\n(-1,6;-1,6),\n$\\\\\n&&&$\n(0,5 ;0,5),\n(1,4 ;1,4),\n(-2,8;-2,8),\n(-1,7;-1,7),\n(0,6 ;0,6),\n(1,5 ;1,5),\n$\\\\\n&&&$\n(-1,8;-1,8),\n(0,7 ;0,7),\n(-1,9;-1,9),\n(0,8 ;0,8),\n(0,9 ;0,9),\n(0,10;0,10)\n$\\\\ \\hline\n\\end{tabular}\n\\caption{\n{\nWe list the monopole charges with non-vanishing contribution to the superconformal index up to ${\\cal O}(x^5)$.\nHere we have denoted only one element among each family generated by permutations and $(m_a;n_i)\\rightarrow (-m_a;-n_i)$ whose contributions are identical (up to $(y,z,\\sigma)\\rightarrow (y^{-1},z^{-1},\\sigma^{-1})$).\n}\n}\n\\label{contributingmn}\n\\end{table}\n\nFor our purpose of comparison between HW pairs it is convenient to express the superconformal index ${\\cal I}$ in the letter index ${\\widetilde {\\cal I}}$ defined by the plethystic exponential\n\\begin{align}\n{\\cal I}(x,\\sigma,y,z)=\\PE[x,\\sigma,y,z;{\\widetilde {\\cal I}}(x,\\sigma,y,z)]=\\exp[\\sum_{n=1}^\\infty \\frac{1}{n}{\\widetilde {\\cal I}}(x^n,\\sigma^n,y^n,z^n)],\n\\end{align}\nwith which the ratio is mapped to the difference\n\\begin{align}\n\\frac{{\\cal I}}{{\\cal I}'}=\\PE[{\\widetilde {\\cal I}}-{\\widetilde{\\cal I}'}].\n\\end{align}\nWe have obtained the following letter indices\n\\begin{align}\n{\\widetilde {\\cal I}}_{1,0,0}&=0,\\nonumber \\\\\n{\\widetilde {\\cal I}}_{1,1,1}&=(yz\\sigma +y^{-1}z^{-1}\\sigma^{-1})(x^{\\frac{1}{2}}-x^{\\frac{3}{2}}+x^{\\frac{5}{2}}-x^{\\frac{7}{2}}+x^{\\frac{9}{2}})+{\\cal O}(x^{\\frac{11}{2}}),\\nonumber \\\\\n{\\widetilde {\\cal I}}_{2,0,1}&=0,\\nonumber \\\\\n{\\widetilde {\\cal I}}_{2,1,2}&=(yz\\sigma^2+y^{-1}z^{-1}\\sigma^{-2})(x^{\\frac{1}{2}}-x^{\\frac{3}{2}}+x^{\\frac{5}{2}}-x^{\\frac{7}{2}}+x^{\\frac{9}{2}})+{\\cal O}(x^{\\frac{11}{2}}),\\nonumber \\\\\n{\\widetilde {\\cal I}}_{3,0,2}&=0,\\nonumber \\\\\n{\\widetilde {\\cal I}}_{3,1,1}&=x+(yz\\sigma^3+y^{-1}z^{-1}\\sigma^{-3})x^{\\frac{3}{2}}-2x^2-(yz\\sigma^3+y^{-1}z^{-1}\\sigma^{-3})x^{\\frac{5}{2}}+2x^3\\nonumber \\\\\n&\\quad +2(yz\\sigma^3+y^{-1}z^{-1}\\sigma^{-3})x^{\\frac{7}{2}}-3x^4-5(yz\\sigma^3+y^{-1}z^{-1}\\sigma^{-3})x^{\\frac{9}{2}}+4x^5+{\\cal O}(x^{\\frac{11}{2}}),\\nonumber \\\\\n{\\widetilde {\\cal I}}_{3,1,3}&=(yz\\sigma^3+y^{-1}z^{-1}\\sigma^{-3})(x^{\\frac{1}{2}}-x^{\\frac{3}{2}}+x^{\\frac{5}{2}}-x^{\\frac{7}{2}}+x^{\\frac{9}{2}})+{\\cal O}(x^{\\frac{11}{2}}),\\nonumber \\\\\n{\\widetilde {\\cal I}}_{3,2,2}&=(yz\\sigma^3+y^{-1}z^{-1}\\sigma^{-3})x^{\\frac{1}{2}}\n+x-2x^2\n+2x^3\n+(yz\\sigma^3+y^{-1}z^{-1}\\sigma^{-3})x^{\\frac{7}{2}}\n-3x^4\\nonumber \\\\\n&\\quad -4(yz\\sigma^3+y^{-1}z^{-1}\\sigma^{-3})x^{\\frac{9}{2}}\n+4x^5\n+{\\cal O}(x^{\\frac{11}{2}}).\n\\label{pertresults}\n\\end{align}\n\nInterestingly, we find that the ratio of the superconformal index between Hanany-Witten pair is the same for all choices of $(k,N_1,N_2)$.\nThat is,\n\\begin{align}\n\\frac{{\\cal I}_{k,N_1,N_2}}{{\\cal I}_{k,k-N_2,k-N_1}}=\\PE[x,\\sigma,y,z;{\\widetilde {\\cal I}}_{(k,N_1,N_2)\/(k,k-N_2,k-N_1)}] \n\\end{align}\nfor $N_1>k-N_1$, with\n\\begin{align}\n{\\widetilde {\\cal I}}_{(1,1,1)\/(1,0,0)}=\n{\\widetilde {\\cal I}}_{(2,1,2)\/(2,0,1)}=\n{\\widetilde {\\cal I}}_{(3,1,3)\/(3,0,2)}=\n{\\widetilde {\\cal I}}_{(3,2,2)\/(3,1,1)}=(yz\\sigma^k+y^{-1}z^{-1}\\sigma^{-k})\\frac{x^{\\frac{1}{2}}-x^{\\frac{3}{2}}}{1-x^2},\n\\label{ratioisfreehyper}\n\\end{align}\nup to ${\\cal O}(x^{\\frac{11}{2}})$. \nAs a result the indices of Hanany-Witten dual pairs turned out to coincide up to the contribution of one hypermultiplet up to the order. \nThis result indicates that the Hanany-Witten duality holds up to a hypermultiplet. \nThis implies that an ugly monopole operator decouples in the IR to form a hypermultiplet\n\\footnote{ \nStrictly speaking this ugly monopole operator does not totally decouple but couples to the other sector through the topological current. \n}\n\n\\section{Discussion}\n\nIn this paper we have considered the 3d ${\\cal N}=4$ $\\text{U}(N_1)_k\\times \\text{U}(N_2)_{-k}$ superconformal Chern-Simons theory coupled with a bifundamental hypermultiplet.\nIt was observed that the three-sphere partition function of this theory diverges if $k-N_1-N_2\\le -2$ \\cite{Nosaka:2017ohr}.\nThis suggests, following the argument \\cite{Kapustin:2010mh,Yaakov:2013fza} in the case without Chern-Simons interactions, that the theory is {\\it bad} when $k-N_1-N_2\\le -2$ according to the good\/ugly\/bad classification in \\cite{Gaiotto:2008ak}.\nTo check this classification we have studied the superconformal index of the theory and we have indeed found that there exists monopole operators with unitarity violating R-chage if $k-N_1-N_2\\le -2$.\n\nWe have further computed the superconformal indices in small $x$ expansion for the pairs of theories with $k-N_1-N_2=\\pm 1$ related by the Hanany-Witten transition.\nAs a result we have found that the superconformal indices of the two theories in pair coincide with each other up to an overall factor which is the same as the contribution of the hypermultiplet up to a certain order.\nNotably, this is consistent with the dimension of moduli space $4\\text{min}(N_1,N_2)$ of the theory: the difference between the dimensions the moduli space of the theories in a Hanany-Witten pair is $4|k-N_1-N_2|$, which coincides with the number of degrees of freedom of a hypermultiplet for $k-N_1-N_2=\\pm 1$.\nThese results are again natural generalizations of what occurs in the case without Chern-Simons term.\n\nIt would be interesting to ask what happens in a pair with $|k-N_1-N_2|\\ge 2$, that is, a pair of a good theory and a bad one.\nFrom the dimension of the moduli space it is natural to expect that the number of decoupled hypermultiplets is $|k-N_1-N_2|$.\nUnfortunately so far we do not have a method to compute the superconformal index of the bad theory.\nThe perturbative approach with the truncation of the summation over the monopole charges does not work for a bad theory.\nIn the case without Chern-Simons interactions the superconformal index is obtained in factorized form which is valid also for bad theories \\cite{Hwang:2015wna}.\nThe computation, however, requires the theory to satisfy $|k|<|N_\\text{f}-N_\\text{a}|\/2$ where $N_\\text{f}$ and $N_\\text{a}$ are the number of fundamental and anti-fundamental chiral multiplet in the UV field content (called as ``maximally chiral'') \\cite{Benini:2013yva}.\nThis condition is not satisfied in our setup and the computation does not work due to the non-trivial poles at the origin and infinity.\nIt is desirable to establish an alternative technique to compute the superconformal index for our theory with $k-N_1-N_2\\le -2$.\n\nOne possible approach is to introduce the fugacities for all the components of monopole charge $(m_1,m_2,\\cdots,m_{N_1};n_1,n_2,\\cdots,n_{N_2})$ not only for $m_\\text{tot}=n_\\text{tot}$.\nThis might remedy the singularity in the perturbative computation argued in section \\ref{sec_singularity} and make the monopole summation convergent.\nOnce we obtain a resummed expression the original superconformal index will be obtained by sending all the fugacities to unity except for the one for $m_\\text{tot}=n_\\text{tot}$.\n\nIt would also be interesting to study the moduli space in more detail along the line of \\cite{Nakajima:2015txa,Bullimore:2015lsa,Assel:2017jgo}.\nSeveral generalizations of our setup could be studied in a similar manner.\nWe can also add an arbitrary number of fundamental hypermultiplets coupling with each gauge node.\nAs the extra hypermultiplets lift the R-charge of the monopole operators up, such generalizations are not only interesting by themselves but also can be easier than the original theory and would be helpful to understand the original theory.\n\n\\section*{Acknowledgement}\n\nWe would like to thank Chiung Hwang for helpful discussions and valuable comments.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}