We consider the largest observed sample including all intermediate-frequency peaked (IBL) and high-frequency peaked (HBL) flaring blazars above 100 GeV up to redshift $z = 0.6$. We show that the best-fit regression line of the emitted spectral indices $Γ_{\rm em} (z)$ is a concave parabola decreasing as $z$ increases, thereby implying a statistical correlation between the $\{Γ_{\rm em} (z) \}$ distribution and $z$. This result contradicts our expectation that such a distribution should be $z$-independent. We argue that the above correlation does not arise from any selection bias. We show that our expectation naturally emerges provided that axion-like particles (ALPs) are put into the game. Moreover, ALPs can also explain why flat spectrum radio quasars emit up to 400 GeV, in sharp contradiction with conventional physics. So, the combination of the two very different but consistent results -- taken at face value -- leads to a hint at an ALP with mass $m = {\cal O} (10^{-10} \, {\rm eV})$ and two-photon coupling in the range $2.94 \times 10^{- 12} \, {\rm GeV}^{- 1} < g_{a γγ} < 0.66 \times 10^{- 10} \, {\rm GeV}^{- 1}$. As a bonus, the Universe would become considerably more transparent above energies $E \gtrsim 1 \, {\rm TeV}$ than dictated by conventional physics. Our prediction can be checked not only by the new generation of observatories like CTA, HAWC, GAMMA-400, LHAASO, TAIGA-HiSCORE and HERD, but also thanks to the planned laboratory experiments ALPS II (upgraded), STAX, IAXO and with other techniques now being developed by Avignone and collaborators.
Blazar Constraints on Axions through New Spectral Modulation Searches in 1ES 1959+650 & B2 1811+31
Università di Bologna（博洛尼亚大学）；INFN, Sezione di Bologna（意大利国家核物理研究所博洛尼部分所）；University of California, Berkeley（加州大学伯克利分校）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
We propose that the interaction between the axionlike particles (ALPs) and photons can be a possible origin of optical circular polarization (CP) in blazars. Given that there is no definite detection of optical CP at $\sim0.1\%$ level, a rough limit on ALP-photon coupling can be obtained, specifically $g_{aγ}\cdot B_\mathrm{T0}\lesssim7.9\times10^{-12}~\mathrm{G\cdot GeV}^{-1}$ for $m_{a}\lesssim 10^{-13}~\mathrm{eV}$, depending on the magnetic field configuration of the blazar jet. Obviously, for the blazar models with a larger magnetic field strength, such as hadronic radiation models, this …
The oscillation of photons and axion-like particles (ALPs) in the astrophysical magnetic fields could modify the measured very high energy (VHE; $\mathcal{E}\gtrsim 100\, \rm GeV$) $γ$-ray spectra of the blazar sources. In this paper, we use the VHE $γ$-ray observations of the blazar Markarian 421 (Mrk 421) measured by MAGIC and Fermi-LAT in 2017 with four phases to constrain the ALP. We give the spectral energy distributions (SEDs) of these phases under the null and ALP hypotheses. We also test the effects of the $γ$-ray blazar intrinsic spectra models on the ALP constraints. No significant r…
Axion-like particles (ALPs) are a broad class of pseudo-scalar bosons that generically arise from broken symmetries in extensions of the standard model. In many scenarios, ALPs can mix with photons in regions with high magnetic fields. Photons from distant sources can mix with ALPs, which then travel unattenuated through the Universe, before they mix back to photons in the Milky Way galactic magnetic field. Thus, photons can traverse regions where their signals would normally be blocked or attenuated. In this paper, we study TeV $γ$-ray observations from distant blazars, utilizing the signific…
Very high energy (VHE) gamma-rays undergo pair production with low energy photons of background radiation fields. This leads to an attenuation of the primary gamma-ray flux of extragalactic sources in the interaction with the extragalactic background light (EBL) which stretches from ultraviolet to far-infrared wavelengths. In the presence of magnetic fields, gamma-rays could oscillate into hypothetical axion-like particles (ALPs). This might lead to a reduced opacity of the Universe for VHE gamma-rays, as ALPs circumvent pair production. Here, the impact of photon-ALP conversions on the spectr…
Decaying axion-like particles (ALP) with masses in the eV range which might occupy dark matter halos of the Milky Way and other galaxies produce a characteristic "bump" feature in the spectrum of extragalactic background light (EBL). This feature leaves an imprint on the gamma-ray spectra of distant extragalactic sources. We derive constraints on the ALP coupling to photons based on analysis of spectra of very-high-energy gamma-ray loud blazars. We combine gamma-ray spectral measurements by Fermi/LAT and Cherenkov telescopes and fit a model in which the intrinsic source spectrum is modified by…
We explore the axion-like particle (ALP)-photon oscillation effect in the $γ$-ray spectra of the blazars Markarian 421 (Mrk 421) and PG 1553+113, which are measured by the Major Atmospheric Gamma Imaging Cherenkov Telescopes (MAGIC) and Fermi Large Area Telescope (Fermi-LAT) with high precision. We investigate the constraints on the ALP parameter space using the Mrk 421 and PG 1553+113 observations of 15 and 5 phases, respectively. We find that the combined analysis with all the 15 phases improves the limits from the Mrk 421 observations. The combined limit set by the Mrk 421 observations has …
Observations with the Imaging Atmospheric Cherenkov Telescopes H.E.S.S., MAGIC, CANGAROO III and VERITAS have shown that the Universe is more transparent than expected to gamma rays above 100GeV. As a natural explanation, the DARMA scenario has previously been proposed, wherein photons can oscillate into a new very light axionlike particle and vice-versa in the presence of cosmic magnetic fields. Here we demonstrate that the most recent observations further support the DARMA scenario, thereby making the existence of a very light axion-like particle more likely.
We review and update the current status of the observed anomalies in stellar cooling and their interpretation in terms of axions and axion like particles.
The high-energy Universe is potentially a great laboratory for searching new light bosons such as axion-like particles (ALPs). Cosmic sources are indeed the scene of violent phenomena that involve strong magnetic field and/or very long baselines, where the effects of the mixing of photons with ALPs could lead to observable effects. Two examples are archetypal of this fact, that are the Universe opacity to gamma-rays and the imprints of astrophysical magnetic turbulence in the energy spectra of high-energy sources. In the first case, hints for the existence of ALPs can be proposed whereas the s…
Axionlike particles (ALPs) could mix with photons in the presence of astrophysical magnetic fields. Searching for this effect in gamma-ray observations of blazars has provided some of the strongest constraints on ALP parameter space so far. Previously, photon-photon dispersion of gamma-rays off of the CMB has been shown to be important for these calculations, and is universally included in ALP-photon mixing models. Here, we assess the effects of dispersion off of other photon fields within the blazar (produced by the accretion disk, the broad line region, the dust torus, starlight, and the syn…
Probing Extragalactic Ultralight Axion Dark Matter with Decade-long Blazar Optical Polarimetry of 1ES 1959+650
Purple Mountain Observatory, Chinese Academy of Sciences（中国科学院紫金山天文台）；School of Astronomy and Space Sciences, University of Science and Technology of China（中国科学技术大学天文与空间科学学院）
We examine a model for the observed temporal variability of powerful blazars in the $γ$-ray band in which the dynamics is described in terms of a stochastic differential equation, including the contribution of a deterministic drift and a stochastic term. The form of the equation is motivated by the current astrophysical framework, accepting that jets are powered through the extraction of the rotational energy of the central supermassive black hole mediated by magnetic fields supported by a so-called \emph{magnetically arrested} accretion disk. We apply the model to the $γ$-ray light curves of …
It has been recently claimed by two different groups that the spectral modulation observed in gamma rays from Galactic pulsars and supernova remnants can be due to conversion of photons into ultra-light axion-like-particles (ALPs) in large-scale Galactic magnetic fields. While we show the required best-fit photon-ALP coupling, $g_{aγ} \sim 2 \times 10^{-10}$ GeV${}^{-1}$, to be consistent with constraints from observations of photon-ALPs mixing in vacuum, this is in conflict with other bounds, specifically from the CAST solar axion limit, from the helium-burning lifetime in globular clusters, …
Many theories beyond the Standard Model of particle physics predict the existence of axionlike particles (ALPs) that mix with photons in the presence of a magnetic field. One prominent indirect method of searching for ALPs is to look for irregularities in blazar gamma-ray spectra caused by ALP-photon mixing in astrophysical magnetic fields. This requires the modelling of magnetic fields between Earth and the blazar. So far, only very simple models for the magnetic field in the blazar jet have been used. Here we investigate the effects of more complicated jet magnetic field configurations on th…
The spectral energy distribution and variability of several blazars (Mkn 501, 1 ES 2344+51.4, PKS 2155-30.4, 1 ES 1218+30.4, 3C 454.3) along the blazar sequence is investigated using a selfconsistent and timedependent lepto-hadronic hybrid emission model. The BL Lac objects in the list are successfully modelled with purely leptonic radiation processes (essentially Synchrotron Self-Compton emission), while the Flat Spectrum Radio Quasar requires highly relativistic hadrons to be present within the jet. Variability is exploited as well with our model to distinguish between Self-Compton and hadro…
Axion-like particles (ALPs), predicted in theories beyond the Standard Model, can have observational effects on the transparency of the Universe to $γ$ rays in the presence of magnetic fields. In this work, we search for effects compatible with the existence of ALPs with 80 months of data from the ${\it Fermi}$ Large Area Telescope, by comparing the distributions of observed highest energy photons from sources beyond redshifts of z $\geq$ 0.1 with theoretical predictions in the presence of ALPs. We find no evidence for an increased $γ$-ray transparency due to ALPs and therefore we set limits o…
We compile 23 Gamma-ray Bursts (GRBs) and 21 blazars with estimated Doppler factors, and the Doppler factors of GRBs are estimated from their Lorentz factors by assuming their jet viewing angles θ->0 degree. Using the conventional assumption that the prompt emission of GRBs is dominated by the synchrotron radiation, we calculate the synchrotron luminosity of GRBs from their total isotropic energy and burst duration. Intriguingly, we discover a uniform correlation between the synchrotron luminosity and Doppler factor, L_syn \propto D^3.1, for GRBs and blazars, which suggests that they may share…
We report detection of a line-like feature in the $γ$-ray spectrum of the blazar B0516$-$621, for which the data obtained with the Large Area Telescope onboard {\it Fermi Gamma-Ray Space Telescope (Fermi)} are analyzed. The feature is at $\sim$7\,GeV and different analyses are conducted to check its real presence. We determine that it has a significance of 2.5--3.0$σ$, and cautiously note the presence of possible systematics in the data which could reduce the significance. This putative feature is too narrow to be explained with radiation processes generally considered for jet emission of blaz…
Flat spectrum radio quasars (FSRQs) are a particular class of blazars rich of optical/ultraviolet photons inside the broad line region (BLR), necessarily implying a huge optical depth for $γ$ rays above 20 GeV. As a consequence, photons with energy above such a threshold should not be emitted. However, photons in the energy range 70 - 400 GeV have been observed by MAGIC from the FSRQ PKS 1222+216. Several astrophysical explanations exist in the literature, but they are all ad hoc, namely devised only for that specific purpose. We show that such a surprising discovery can be explained within st…
Axion-Like Particles (ALPs), if exist in nature, are expected to mix with photons in the presence of an external magnetic field. The energy range of photons which undergo strong mixing with ALPs depends on the ALP mass, on its coupling with photons as well as on the external magnetic field and particle density configurations. Recent observations of blazars by the Fermi Gamma-Ray Space Telescope in the 0.1-300 GeV energy range show a break in their spectra in the 1-10 GeV range. We have modeled this spectral feature for the flat-spectrum radio quasar 3C454.3 during its November 2010 outburst, a…
Context. Most of the studies on extragalactic γ-ray propagation performed up to now only accounted for primary gamma-ray absorption and adiabatic losses ("absorption-only model"). However, there is growing evidence that this model is oversimplified and must be modified in some way. In particular, it was found that the intensity extrapolated from the optically-thin energy range of some blazar spectra is insufficient to explain the optically-thick part of these spectra. This effect was interpreted as an indication for γ-axion-like particle (ALP) oscillation. On the other hand, there are many hin…
Axion-like particles (ALPs) are attracting increasing interest since, among other things, they are a prediction of many extensions of the standard model of elementary particles physics and in particular of superstrings and superbranes. Remarkably, depending on the set of their parameter space, they strongly increase the photon transparency in the very-high energy band. The recent discovery of photon dispersion on the CMB requires a substantial modification of the previous picture: this is indeed the goal of the present paper. We compute the photon survival probability from a blazar to us exact…
Blazars are often characterized by a spectral break at soft X-rays, whose origin is still debated. While most sources show a flattening, some exhibit a blackbody-like soft excess with temperatures of the order of $\sim$0.1 keV, similar to low-luminosity, non-jetted Seyferts. Here we present the analysis of the simultaneous XMM-Newton and NuSTAR observation of the luminous FSRQ 4C+25.05 ($z=2.368$). The observed 0.3-30 keV spectrum is best described by the sum of a hard X-ray power law ($Γ= 1.38_{-0.03}^{+0.05}$) and a soft component, approximated by a blackbody with $kT_{\rm BB} = 0.66_{-0.04}…
Many extensions of the Standard Model predict the existence of ALPs, which are very light spin-zero bosons with a two-photon coupling. Photon-ALP oscillations occur in the presence of an external magnetic field, and ALPs can lead to observable effects on the measured photon spectrum of astrophysical sources. An intriguing situation arises when blazars are observed with the Cherenkov Telescopes H.E.S.S., MAGIC, CANGAROO III and VERITAS. The extragalactic background light (EBL) produced by galaxies during cosmic evolution gives rise to a source dimming which becomes important in the VHE band. Th…
Axion production from astrophysical bodies is a topic in continuous development, because of theoretical progress in the estimate of stellar emission rates and, especially, because of improved stellar observations. We carry out a comprehensive analysis of the most informative astrophysics data, revisiting the bounds on axion couplings to photons, nucleons and electrons, and reassessing the significance of various hints of anomalous stellar energy losses. We confront the performance of various theoretical constructions in accounting for these hints, while complying with the observational limits …
Photons may convert into axion like particles and back in the magnetic field of various astrophysical objects, including active galaxies, clusters of galaxies, intergalactic space and the Milky Way. This is a potential explanation for the candidate neutral ultra-high-energy (E>10^18 eV) particles from distant BL Lac type objects which have been observed by the High Resolution Fly's Eye experiment. Axions of the same mass and coupling may explain also TeV photons detected from distant blazars.
Axion-like particles (ALPs), which are very light neutral spin zero elusive particles primarily interacting with two photons and predicted by superstring and superbrane theories, have come to help by solving two distinct problems about blazars (a type of active galactic nuclei), thus providing two hints at the existence of ALPs themselves. In the presence of an external magnetic field, ALPs produce: (i) photon-ALP oscillations, (ii) the change of the polarization state of photons. The former effect has many consequences in the astrophysical contest, such as the modification of the transparency…
The first measurements from the CIBER experiment of extragalactic background light (EBL) in near-infrared (NIR) band exhibit a higher intensity than those inferred through $γ$-ray observations. Recent theoretical-EBL intensities are typically consistent with the very high energy (VHE) $γ$-ray observations. Yet, it is possible that the excess NIR radiation is a new component of EBL and not in tension with the TeV spectra of distant blazars, since the hypothetical axion-like particle (ALP) may lead to a reduced opacity of the Universe for VHE $γ$-rays. In order to probe whether the excess compon…
The idea that photons can convert into axion-like particles (ALPs) $γ\to a$ in or around an AGN and reconvert back into photons $a \to γ$ in the Milky Way magnetic field has been put forward in 2008 and has recently attracted growing interest. Yet, so far nobody has estimated the conversion probability $γ\to a$ as carefully as allowed by present-day knowledge. Our aim is to fill this gap. We first remark that AGN that can be detected above 100 GeV are blazars, namely AGN with jets, with one of them pointing towards us. Moreover, blazars fall into two well defined classes: BL Lac objects (BL La…
One of the most important problem of the blazar astrophysics is to understand the physical origin of the blazar sequence. In this study, we focus on the GeV gamma-ray variability of blazars and evolution perspective we search the relation between the redshift and the variability amplitude of blazars for each blazar subclass. We analyzed the Fermi-LAT data of the TeV blazars and the bright AGNs (flux $\geq$ 4$\times10^{-9}$ cm$^{-2}$s$^{-1}$) selected from the 2LAC (the 2nd LAT AGN catalog) data base. As a result, we found a hint of the correlation between the redshift and the variability ampli…
D. Wouters and P. Brun in Phys. Rev. D 86, 043005 (2012) claim that an observable effect in the spectra of distant very-high-energy blazars arises as a consequence of oscillations of photons into axion-like particles (ALPs) in the presence of turbulent extra-galactic magnetic fields. The main objective of this comment is to demonstrate that such a result is physically incorrect. We also show that a physically correct treatment of the same issue leads to a much less relevant conclusion, which makes the effect pointed out by WB likely unobservable with the present capabilities.
The synergy between the Fermi-LAT and ground-based Cherenkov telescope arrays gives us the opportunity for the first time to characterize the high energy emission (100 MeV - 10 TeV) from more than 30 blazars. In this study we performed a Fermi-LAT spectral analysis for all TeV-detected blazars and combined it with archival TeV spectra. Our results for low synchrotron-peaked BL Lacs (LBL) show hints of absorption features in the GeV band that could be interpreted as internal opacity at the source. We note that simple or broken power laws cannot describe all the observed GeV-TeV spectra and more…
We offer a pedagogical introduction to axion-like particles (ALPs) as far as their relevance for high-energy astrophysics is concerned, from a few MeV to 1000 TeV. This review is self-contained, in such a way to be understandable even to non-specialists. Among other things, we discuss two strong hints at a specific ALP that emerge from two very different astrophysical situations. More technical matters are contained in three Appendices.
Axion Like Particles (ALPs) are predicted to couple with photons in the presence of magnetic fields. This effect may lead to a significant change in the observed spectra of gamma-ray sources such as AGNs. Here we carry out a detailed study that for the first time simultaneously considers in the same framework both the photon/axion mixing that takes place in the gamma-ray source and that one expected to occur in the intergalactic magnetic fields. An efficient photon/axion mixing in the source always means an attenuation in the photon flux, whereas the mixing in the intergalactic medium may resu…
Several extensions of the Standard Model predict the existence of Axion-Like Particles (ALPs), very light spin-zero bosons with a two-photon coupling. ALPs can give rise to observable effects in very-high-energy astrophysics. Above roughly 100 GeV the horizon of the observable Universe progressively shrinks as the energy increases, due to scattering of beam photons off background photons in the optical and infrared bands, which produces e+e- pairs. In the presence of large-scale magnetic fields photons emitted by a blazar can oscillate into ALPs on the way to us and back into photons before re…
The Cherenkov Telescope Array (CTA) is about to enter construction phase and one of its main key science projects is to perform an unbiased survey in search of extragalactic sources. We make use of both the latest blazar gamma--ray luminosity function and spectral energy distribution to derive the expected number of detectable sources for both the planned Northern and Southern arrays of the CTA observatory. We find that a shallow, wide survey of about 0.5 hour per field of view would lead to the highest number of blazar detections. Furthermore, we investigate the effect of axion-like particles…
We compiled the radio, optical, and X-ray data of blazars from the Sloan Digital Sky Survey (SDSS) database, and presented the distribution of luminosities and broad band spectral indices. The distribution of luminosities shows that the averaged luminosity of flat-spectral radio quasars (FSRQs) is larger than that of BL Lacs objects. On the other hand, the broad band spectral energy distribution reveals that FSRQs and low energy peaked BL Lac objects (LBLs) objects have similar spectral properties, but high energy peaked BL Lac objects (HBLs) have a distinct spectral property. This may be due …
We re-examine possible dependencies on redshift of the spectral parameters of blazars observed at very-high energies (VHEs) with Imaging Atmospheric Cherenkov telescopes (IACTs). This is relevant to assess potential effects with the source distance of the photon to axion-like particle (ALP) mixing, that would deeply affect the propagation of VHE photons across the Universe. We focus our spectral analysis on 38 BL Lac objects (32 high-peaked and 6 intermediate-peaked) up to redshift $z\simeq 0.5$, and a small sample of 5 Flat Spectrum Radio Quasars up to $z=1$ treated independently to increase …
Very-high-energy (VHE) BL Lac spectra extending above $10 \, \rm TeV$ provide a unique opportunity for testing physics beyond the standard model of elementary particle and alternative blazar emission models. We consider the hadron beam, the photon to axion-like particle (ALP) conversion, and the Lorentz invariance violation (LIV) scenarios by analyzing their consequences and induced modifications to BL Lac spectra. In particular, we consider how different processes can provide similar spectral features (e.g. hard tails) and we discuss the ways they can be disentangled. We use HEGRA data of a h…
Background radiation fields pervade the Universe, and above a certain energy any $γ$-ray flux emitted by an extragalactic source should be attenuated due to $e^+e^-$ pair production. The opacity could be alleviated if photons oscillated into hypothetical axion-like particles (ALPs) in ambient magnetic fields, leading to a $γ$-ray excess especially at high optical depths that could be detected with imaging air Cherenkov telescopes (IACTs). Here, we introduce a method to search for such a signal in $γ$-ray data and to estimate sensitivities for future observations. Different magnetic fields clos…
Photons may convert to axion-like particles (ALPs) in external magnetic fields. Under certain conditions, this effect should result in irregular features in observed spectra of astrophysical sources. Lack of such irregularities in particular spectra was used to constrain ALP parameters, with two most popular sources being the radio galaxy NGC 1275 and the blazar PKS 2155-304. The effect and, consequently, the constraints, depend on the magnetic fields through which the light from the source is propagated. Here, we revisit ALP constraints from gamma-ray observations of NGC 1275 taking into acco…
We explore the correlation of $γ$-ray emitting blazars with IceCube neutrinos by using three very recently completed, and independently built, catalogues and the latest neutrino lists. We introduce a new observable, namely the number of neutrino events with at least one $γ$-ray counterpart, $N_ν$. In all three catalogues we consistently observe a positive fluctuation of $N_ν$ with respect to the mean random expectation at a significance level of $0.4 - 1.3$ per cent. This applies only to extreme blazars, namely strong, very high energy $γ$-ray sources of the high energy peaked type, and implie…
Prompted by the increasing interest of axion-like particles (ALPs) for very-high-energy (VHE) astrophysics, we have considered a full scenario for the propagation of a VHE photon/ALP beam emitted by a BL Lac and reaching us in the light of the most up-to-date astrophysical information and for energies up to above $100 \, \rm TeV$. During its trip, the beam -- generated in a small region of a BL Lac jet -- crosses a variety of magnetic structures in very different astronomical environments: the BL Lac jet, the host elliptical galaxy, the extragalactic space and the Milky Way. We have taken an e…
In this work, we investigate the axion-like particle (ALP)-photon oscillation effect in the spectra of the blazar Markarian 421 (Mrk 421) using 4.5 years of the Astrophysical Radiation with Ground-based Observatory at YangBaJing (ARGO-YBJ) and Fermi Large Area Telescope (Fermi-LAT) data. These data are collected during the common operation time, which cover ten activity phases of Mrk 421. No significant ALP-photon oscillation effect is confirmed. On the other hand, not all the observations of the ten phases can be individually used to set the 95% confidence level ($\rm C.L.$) constraint on the…
The surprising discovery by MAGIC of an intense, rapidly varying emission in the energy range 70 - 400 GeV from the flat spectrum radio quasar PKS 1222+216 represents a challenge for all interpretative scenarios. Indeed, in order to avoid absorption of γrays in the dense ultraviolet radiation field of the broad line region (BLR), one is forced to invoke some unconventional astrophysical picture, like for instance the existence of a very compact (r\sim 10^{14} cm) emitting blob at a large distance (R \sim10^{18} cm) from the jet base. We offer the investigation of a scenario based on the standa…
A Tentative Line-like GeV Excess in Fermi Blazar 4FGL J1754.2+3212: Implications for Jet Physics and Beyond
School of Physics and Electrical Engineering, Liupanshui Normal University（六盘水师范学院物理与电气工程学院）；Department of Physics, Yunnan Normal University（云南师范大学物理系）；Department of Astronomy, School of Physics, Huazhong University of Science and Technology（华中科技大学物理学院天文学系）
After NANOGrav, the IPTA collaboration also reports a strong evidence of a stochastic gravitation wave background. This hint has very important implications for fundamental physics. With the recent IPTA data release two, we attempt to search signals of light new physics. and give new constraints on the audible axion, domain walls and cosmic strings models. We find that the best fit point corresponding to a decay constant $F\approx5\times10^{17}$ GeV and an axion mass $m_a\approx2\times10^{-13}$ eV from NANOGrav data is ruled out by IPTA at beyond $2σ$ confidence level. Fixing the coupling stre…
