P Kervella
email: pkervell@das.uchile.cl.

F Mignard

A Mérand

F Thévenin

Close stellar conjunctions of α Centauri A and B until 2050 An m K = 7.8 star may enter the Einstein ring of α Cen A in 2028

Keywords: Astrometry, Proper motions, Stars: individual: α Cen, Stars: solar-type, Stars: binaries: visual, Ephemerides

The rapid proper motion of the α Cen pair (≈ 3.7 arcsec yr -1 ) and its location close to the galactic plane on a rich stellar background combine constructively to make them excellent candidates for close stellar conjunctions with more distant stars. Adding new differential astrometry to archival data, we have refined the orbital parameters, barycentric proper motion and parallax of α Cen and compute its apparent trajectory on sky over the coming decades. Based on NTT/SUSI2, NTT/SOFI and VLT/NACO maps of the field stars around the trajectories of α Cen A and B, we present a catalog of the expected close conjunctions until 2050. An exceptional event will take place in early May 2028, when α Cen A will come within ρ min = 0.015 ± 0.135 arcseconds of the m K = 7.8 star 2MASS 14392160-6049528 (hereafter S5). In terms of impact parameter and contrast, this is the most favorable stellar conjunction of α Cen within at least the next three decades. With an angular diameter of θ LD = 0.47 ± 0.05 mas, it is likely that S5 is a red giant or supergiant located at several kiloparsecs. The approached stars will act as moving light probes in transmission through the environment of α Cen. The observation of these close conjunctions holds great promises to search for planets and other low mass objects in the α Cen system using photometry and astrometry. The relativistic displacement of the approached star images will be observable, with significant deflection angles in the milliarcsecond range. The small impact parameter of the conjunction with S5 means that this star has a probability of 45% of entering the Einstein ring of α Cen A. The gravitational amplification of the flux of S5 could reach a factor five for the combination of the two lensed images. The proper motion, orbital parameters and parallax of α Cen will be measurable with an extreme accuracy from differential astrometry with the S stars. This will be valuable, for example to prepare the recently announced Breakthrough Starshot initiative to send interstellar nanocrafts to α Centauri.

Introduction

The triple star α Centauri (WDS J14396-6050AB, GJ559AB) is our closest stellar neighbor. The main stars α Cen A and B are located at a distance of only d = 1.338 ± 0.001 pc (π = 747.17 ± 0.61 mas; Sect. 3.1). Their orbital period is close to 80 years, with a projected semi-major axis of 17.6 . The main components are dwarf stars of G2V (component A, HD 128620) and K1V (component B, HD 128621) spectral types, while the third member is the red dwarf Proxima (M5.5V, GJ551). Proxima is actually slightly closer to Earth by approximately 7 500 AU, at d = 1.302 ± 0.002 pc (π = 768.13 ± 1.04 mas; Lurie et al. 2014). The proximity of α Cen A and B and their similarity to the Sun in terms of mass and spectral type make these stars extremely attractive from the standpoint of both stellar physics (Bazot et al. 2016) and extrasolar planet research (Demory et al. 2015).

As predicted by Paczynski (1996b) and discussed in details by Di Stefano (2008a,b) the close conjunction of a foreground Based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere under ESO programs 272.C-5010, 077.C-0587, 080.C-0775(A), 081.C-0940(A), 082.C-0615(A) and 296.D-5032. mass as α Cen with a distant background star will create a gravitational mesolensing event. Lists of stellar conjunctions have been predicted for example, by Salim & Gould (2000), Proft et al. (2011), and for specific objects by [START_REF] Lépine | [END_REF] (VB 10) and [START_REF] Sahu | [END_REF] (Proxima). But to our knowledge, none of the predicted lensing events has been observed successfully due to the very small observational signature of the expected gravitational deflection. Thanks to the large angular size of their Einstein rings and their fast displacement on a rich stellar background, α Cen A and B are prime candidates for such events.

In this paper, we use new and archival observations of α Cen to determine the conjunctions with background stars that will occur in the coming decades. These events are relatively rare and usually difficult to predict as star catalogs are poorly populated in the immediate vicinity of such very bright stars. Even in the deepest catalogs, only a handful of stars are typically present within 1 of α Cen, and many are artefacts. Measuring accurate astrometric and photometric parameters is also made difficult by the residual speckles and the spatially variable background. More than a decade ago, two of us obtained deep observations of the field surrounding α Cen using the (now decommissioned)

A&A proofs: manuscript no. Alpha-Cen-conjunctions_v3r1 NTT/SUSI2 instrument (Kervella & Thévenin 2007a). This allowed us to assemble a catalog of the field stars present within a few arcminutes of α Cen. While this catalog was far from complete within ≈ 30 of the two stars at the time, the very fast proper motion of α Cen has now brought the pair inside a much better known star field. As α Cen is located almost exactly in the plane of the Milky Way, background stars are extremely numerous, particularly at infrared wavelengths where the interstellar extinction is weaker. This has already been noted by Kervella et al. (2006) who observed hundreds of stars within ≈ 20 of α Cen using the VLT/NACO instrument.

We present in Sect. 2 the new VLT/NACO observations that we obtained in March 2016 as well as archival data from VLT/NACO, NTT/SOFI and ALMA. We take particular care to reference precisely the positions of α Cen and the celestial coordinate frame of our catalog. The trajectory of the α Cen pair is derived in Sect. 3, in which we have also refined the orbital parameters, parallax and masses of the two components. From our catalog of background stars, and taking into account the proper motion, the orbital motion and the parallactic motion of α Cen, we have derived in Sect. 4 the list of stellar conjunctions that will occur with α Cen A and B until 2050. We identify six conjunctions up to 2031 that we discuss individually, focusing in particular on the approach of star S5 in 2028. We estimate the angular diameters of the corresponding background sources, and obtain upper limits on theirs proper motions. We discuss in Sect. 5 the potential of the conjunctions in terms of gravitational deflection and amplification of the light from the approached background stars. We also address the scientific potential of such observations to search for low mass objects in the α Cen system.

Observations and data reduction

NTT/SUSI2 and NTT/SOFI imaging

The SUSI2 exposures of the field around α Cen in the VRIZ bands were presented by Kervella & Thévenin (2007a), where a catalog of the sources is also provided. This catalog is available through ViZieR with revised coordinates (see Sect. 2.4) and improved photometry. An example image in the I band is presented in Fig. 1. To complement these images, we retrieved from the ESO archive a series of exposures of α Cen that were obtained using the NTT/SOFI instrument (Moorwood et al. 1998). These exposures were taken in March 2008, August 2008and February 2009 in the infrared J s and K s bands for programs 080.C-0775(A), 081.C-0940(A) and 082.C-0615(A). The exposures in the K s band are generally saturated on S2 and S5, and we therefore selected the best exposure in the J s band for which S2 and S5 are unsaturated. This image was obtained on 25 February 2009 at UT 08:32:38. The exposure time is 20×1.2 s in the J s filter1 (λ 0 = 1.24 µm, ∆λ = 0.16 µm) using the large field of view (4.9 × 4.9 , 0.288 pix -1 ). The observatory seeing was 0.82 in the visible, and the observation took place at an airmass of 1.18. The raw image was classically corrected for flat field.

VLT/NACO adaptive optics

We observed a field located in front of α Cen in the direction of its proper motion on 27 March 2016 using the NACO instrument [START_REF] Girard | Proc. SPIE[END_REF]) installed at Unit Telescope 1 of the Very Large Telescope (VLT). This instrument comprises the Nasmyth Adaptive Optics System (NAOS; [START_REF] Rousset | Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series[END_REF] and the CONICA infrared camera [START_REF] Lenzen | Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series[END_REF]. We obtained a series of 60 × 3 s exposures using a K s broadband filter2 , at mean MJD=57474.4049. The wavefront sensor was tracking α Cen B. The airmass was 1.42 and the visible seeing 1.6 according to the observatory's seeing monitor (DIMM). We selected the S27 camera, which has a field of view of 27.7 . The astrometric calibration for the S27 camera is taken from Chauvin et al. (2006): the plate scale is 27.01 ± 0.05 mas/pix, and the position angle is +0.00 ± 0.20 degrees. These values are stable over several years within their stated errors, and they are in agreement with the calibrations of Masciadri et al. (2003) and Chauvin et al. (2015). We kept α Cen outside of the NACO field of view to reduce the bleeding on the detector and allow for relatively long exposures. Due to the complex interaction of the wings and speckles of the two point spread functions, α Cen is not a good candidate for classical coronagraphic techniques (e.g. as available in SPHERE or NACO).

In addition to this long exposure, we reduced the associated short exposure acquisition image to derive differential astrometry of components A and B (see Sect. 3). We also retrieved from the ESO archive one NACO acquisition image (Fig. 3, left panel) obtained on 10 April 2004(MJD = 53105.1976). It was taken at an airmass of 1.28 with a DIMM seeing of 0.57 using the S27 camera. The exposure time was 0.4 s, using a narrow band filter centered at 1.26 µm (NB_1.26) and a neutral density filter (ND_short). The cores of the two star images are saturated but the centroid position can be estimated to an accuracy of half a pixel (13 mas). These two acquisition images cannot be accurately tied to a global coordinate system, but they are nevertheless useful as they give a highly accurate relative position of the two stars.

ALMA millimeter interferometry

We took advantage of the observations of α Cen collected by Liseau et al. (2015) to improve the relatively limited recent astrometry of the α Cen system. The accuracy of the absolute referencing of the ALMA coordinate system of these images is limited by the position uncertainty and angular distance of the phase calibrator. Unfortunately, these observations were not designed to obtain the best possible absolute astrometry, and due to selfreferencing of the phases, variable offsets of 1 to 2 with respect to the ICRS are present in the images. However, the differential astrometry within the interferometric field of view is very accurate, and essentially limited by the signal-to-noise ratio of the images. This is particularly true for the shorter wavelengths in ALMA bands 7, 8 and 9, where the angular resolution is sufficiently high to obtain a very good relative position accuracy (Fig. 3, right panel). The systematics on the angular separation are linked to the uncertainty on the baseline length and on the effective wavelength of the observations, which are extremely well known. The position angle is defined by the geometry of the array, also very well known. As a result, both sources of systematic error are completely negligible compared to the statistical uncertainty of the measurement of the centroid of the star positions in the images. The relative positions of α Cen B with respect to component A are listed in Table A.1.

Astrometric referencing

Due to the extreme brightness of α Cen, the region around the two stars is poorly covered by astrometric catalogs, and the referencing of the images from the different instruments is a delicate task. We adopted the approach to consistently reference all our images to the 2MASS coordinate system, that is tied to the ICRS and whose astrometric accuracy is ±0.08 for individual stars brighter than K s = 14 (Skrutskie et al. 2006, Roeser et al. 2010). We first matched the World Coordinate System (WCS) of the archival NTT/SOFI J s band image to the 2MASS Atlas K s band image with reference ki1260044.fits retrieved from the All-Sky Release Survey at NASA/IPAC Infrared Science Archive3 . The binding of the SOFI image to the 2MASS frame relies on a sample of 14 bright unsaturated stars spread over the SOFI field. These stars were selected to be unblended in the higher resolution SOFI images, and they were identified unambiguously in the 2MASS image. The standard deviation of the residuals of the star positions with respect to the adjusted SOFI WCS is 0.07 . We conservatively adopted an uncertainty of half of this value (σ SOFI = 0.035 ) as our relative astrometric uncertainty in binding the SOFI coordinates to the 2MASS frame. Following a similar procedure with a set of nine reference stars, we referenced the coordinates of the SUSI2 images from Kervella & Thévenin (2007a) to the 2MASS frame with a residual standard deviation of 0.11 . We defined the uncertainty in the coordinate referencing as σ SUSI2 = 0.06 . Finally, using 11 stars detected both on the NACO and SUSI2 images, we set the WCS of our NACO image to the SUSI2 coordinates with a standard deviation of 0.04 , giving an uncertainty of σ NACO-SUSI2 = 0.02 . We preferred to reference the infrared NACO images in the K s band to the SUSI2 image rather than the SOFI image (J s band) because the SOFI frame is not as deep as the SUSI2 I band image and therefore shows fewer stars in common. We checked on the brightest stars that the agreement between the coordinate systems of NACO and SOFI is well within ±0.05 . We adopted the same uncertainty on the relative coordinate shift between SOFI and NACO as between SUSI2 and NACO: σ NACO-SOFI = 0.02 .

A&A proofs: manuscript no. Alpha-Cen-conjunctions_v3r1

In summary, we obtained 1σ coordinate accuracies of ±0.035 for the SOFI image with respect to the 2MASS frame and ±0.06 for the NACO image. As we referenced our positions of α Cen and the S stars to the SOFI frame, we consider only the difference uncertainty that is σ NACO-SOFI = 0.02 .

Astrometry of α Centauri A and B

The apparent trajectories of the two components of α Cen on the sky are defined by the combination of their orbital motion, their parallactic wobble and the proper motion of their center of mass. We explain in this section how we computed these trajectories.

Revised orbital elements and parallax

The quality of the orbital elements of α Cen is essential to predict accurately the positions of the two stars several decades from now. We therefore compared the predictions in angular separation ρ and position angle θ obtained using the orbital elements recently published by Pourbaix & Boffin (2016) with our new astrometry complemented by archival positions from the literature. These relative astrometric measurements are listed in Table A.1. The agreement was found to be good at a level of 50 mas on the separation, but we find a systematically increasing deviation of the position angle θ from the expected orbit over the last five years, reaching in 2016 approximately +1 • (more than 3σ significance). This shift is shown with dashed lines in the lower panels of Fig. 4. This deviation is also visible in the recent CCD camera astrometry by Anton (2015). Our improved astrometric model is the one shown with the blue lines in Fig. 4.

Taking our latest astrometric measurement into account, we thus recomputed the orbital parameters of α Cen using the same high precision radial velocity data set as Pourbaix & Boffin (2016). We downloaded the radial velocities from the Ninth Catalog of Spectroscopic Binary Orbits4 (SB9, Pourbaix et al. 2004). The archival astrometric measurements were retrieved from the Washington Double Star catalog (Mason et al. 2016;Mason et al. 2001), and the associated uncertainties were taken from the original articles (when available). We rejected the measurements with large uncertainties as their weight in the fit was negligible. We chose to give a dominant weight to the ALMA band 9 astrometry obtained at epoch 2014.5426, to make it our fiducial astrometric point. This is to our knowledge the most accurate differential measurement of the relative position of α Cen A and B, with an unbiased precision of 10 mas on the separation and 0.15 • on the position angle.

The calculated, projected orbital trajectory of α Cen B relative to α Cen A is shown in Fig. 5, together with the corresponding radial velocity residuals for the two stars. Only the recent CES and HARPS radial velocity data are shown in the plot of the residuals, but all the available measurements were included in the fit. Following Pourbaix & Boffin (2016), we added constant velocity terms ∆V A and ∆V B for α Cen A and B to account for their gravitational redshift and convective shift. Pourbaix & Boffin (2016) considered only a shift for component B, but we preferred to consider both components separately as otherwise we observe a systematic residual of the HARPS velocities of A with respect to the best fit orbit. We obtained a very comparable number as Pourbaix & Boffin (2016) for α Cen B (∆V B = 322±5 m s -1 ), and a small shift of ∆V A = 8 ± 5 m s -1 for component A. We also corrected the zero point of the CES spectrograph with respect to HARPS by -70 m s -1 for α Cen A and -29 m s -1 for α Cen A. This different velocity zero point is not unexpected as the two stars have significantly different spectral types.

Although the uncertainties of the individual HARPS measurements are extremely small (a few 10 cm s -1 ), a much larger dispersion is observed from one measurement to the other. This dispersion can be either of instrumental of astrophysical origin. As the two components of α Cen are close on sky and very bright, contamination from one star on the other can affect the measurements depending on the seeing (Bergmann et al. 2015). Shifts could also be induced for example by the asteroseismic oscillations of the two stars, or (on a longer timescale) by the presence of star spots. The rotation periods of α Cen A and B are of 22 and 41 days (Morel et al. 2000), and although this corresponds to a relatively slow rotation for a field star, the evolving star spots on the two stars will affect the radial velocities through the Rossiter-McLaughlin effect. For this reason, we added quadratically a uniform uncertainty of σ = 3 m s -1 to all radial velocity measurement errors, including the HARPS measurements. This corresponds to the observed dispersion of the HARPS points around the best fit orbit. This operation does not change significantly the derived parameters, but it results in a more balanced weight to the different observables (astrometry and radial velocities) and produces a more stable fit.

The resulting orbital parameters, parallax and masses of α Cen are listed in Table 1. The overall quality of the fit of the astrometric positions can be assessed from Fig. 4, which compares the model to the observations. The best-fit reduced χ 2 is 0.95, confirming that the adjusted orbit is a faithful representation of the observations. Most of the orbital elements are statistically compatible with the values found by Pourbaix & Boffin (2016), but we find a smaller semi-major axis a (this parameter is well constrained by our new astrometry). The smaller uncertainty on a combined with the high precision radial velocities results in a better defined parallax. We find π = 747.17 ± 0.61 mas, that is slightly larger than the parallax quoted in Pourbaix & Boffin (2016) (743 ± 1.3 mas) but in line with the one from Söderhjelm (1999) (π = 747.1±1.2 mas). Our parallax is also consistent with the indirect value π = 745.3±2.5 mas predicted by Kervella et al. (2003) and is almost in between the original (ESA 1997) and revised (van Leeuwen 2007) Hipparcos values. We obtain significantly lower masses for α Cen A and B than Pourbaix & Boffin (2016), in particular for component B (-3.6%). Our values are however in excellent agreement with the previous determinations by Pourbaix et al. (2002), who adopted the parallax value of Söderhjelm (1999). We plan to obtain sub-milliarcsecond differential astrometry of α Cen A and B in the coming years (using e.g. VLTI/GRAVITY or VLT/NACO) to further improve the orbital elements and masses.

The excellent accuracy of the derived orbital parameters (Table 1) allow us to confidently predict the relative position of α Cen B with respect to A to an accuracy better than 0.020 at least over the next 100 years. We therefore conclude that for the conjunctions that will occur until 2050, the uncertainty on the orbital displacement of α Cen A and B has a negligible contribution to the astrometric error budget.

Apparent trajectories of α Cen A & B

To determine the barycentric proper motion vector of α Cen, we used the very accurate positions of α Cen A and B measured by Hipparcos at epoch 1991.25 coupled to the positions of α Cen A and B we measured in the SOFI image at epoch 2009.1517 (Fig. 6). We set the masses of the two stars to the values we determined in Sect 3.1 to compute the position of the barycenter of the The predictions from our best-fit orbit (Table 1) are represented with solid blue curves. The residuals compared to our best-fit orbit are shown in the bottom panels, together with the differences between the orbit determined by Pourbaix & Boffin (2016) and our best-fit orbit (dashed gray lines, for comparison). The time scale runs from 1st January 2000 to 31 December 2031, and the magenta vertical markers indicate the epochs of the conjunctions of S1 to S6.

Table 1. Parallax, proper motion, orbital elements and masses of α Cen.

Parameter

P16 a K16 b π (mas) 743 ± 1.3 747.17 ± 0.61 µ α (mas yr -1 ) c -3619.9 ± 3.9 µ δ (mas yr -1 ) c +693.8 ± 3.9 v α (km s -1 ) c -22.9666 ± 0.0193 v δ (km s -1 ) c +4.4019 ± 0.0037 v r (km s system. The SOFI image provides a robust link between α Cen and the S star coordinates, as the positions of both α Cen and the S stars can be measured in the same image. We therefore took the SOFI image coordinates as our reference point for the barycenter position (Table 2). In spite of the saturation of the images of α Cen A and B, their individual positions can be measured in the image to ±0.07 (1/4 of the 0.288 pixel) thanks to the long and thin diffraction spikes that are detectable almost to the edge of the image. We determined the position of their barycenter to an absolute accuracy of ±0.07 . This slightly conservative error estimate also accounts for the absolute referencing uncertainty of the SOFI coordinates to the 2MASS frame (σ SOFI = 0.035 ).

From the combination of the Hipparcos and SOFI barycentric coordinates corrected for parallax, which are separated by almost 19 years, we obtain a measurement of the equatorial coordinate components of the barycentric proper motion vector: µ α = -3619.9 ± 3.9 mas yr -1 and µ δ = +693.8 ± 3.9 mas yr -1 .

Combining the barycentric proper motion and the parallax, we derived the tangential components of the barycentric velocity: v α = -22.9666 ± 0.0193 km s -1 and v δ = +4.4019 ± 0.0037 km s -1 . The radial component is v r = -22.3930 ± 0.0043 km s -1 . As a consistency check, we observed that the ephemeris positions of components A and B match the observed positions with SOFI (Table 2) to 0.025 and 0.029 , respectively. These offsets are smaller than the ±0.07 uncertainty on the measurement of the SOFI positions of A and B. The relative position of components A and B is therefore in agreement with the predicted orbital position, which gives confidence in the accuracy of our position measurements in the SOFI images. The barycenter position is propagated in time using the 3-D velocity vector components to allow for the full 3-D rectilinear motion. The orbital elements determined in Sect. 3.1 provide the α Cen A-B separation vector as a function of time. We eventually obtained the celestial coordinates of A and B from the barycenter coordinates and the masses with a local inverse gnomonic projection centered at the barycenter direction. The resulting trajectories of the two stars are represented in Figs. 1, 2 and 6, as well as for the individual S1, S2 and S5 star conjunctions (see Sect 4.2).

Near stellar conjunctions with α Cen A & B

List of close approaches until 2050

The identified close approaches of α Cen A and B with background sources are listed in Table 3. We denote in this table the approached stars as "S stars" in order of time of closest approach. The contrast between the approached star and the corresponding α Cen component was computed using the following magnitudes for α Cen A and B: m Due to the high density of stars in the plane of the Galaxy, many close conjunctions are detected with very faint stars. We limited our selection to impact parameters smaller than 5 (6.5 AU at the distance of α Cen), with stars brighter than magnitude 16 in the visible (VRIZ) and 15 in the infrared (JHK bands). This angular distance corresponds to the accessible field for narrow-angle astrometry with the VLTI/GRAVITY instrument, and approxi-mately one quarter of the maximum apparent separation between α Cen A and B. Considering the capabilities of the current and coming generation of high contrast imaging instrumentation, the selected magnitude limits correspond to a contrast level with the two components of α Cen that is potentially accessible down to sub-arcsecond angular separations. We define the minimum approach distance ρ min as the geometrical approach in absence of gravitational lensing. In most cases it is very close to the lensed minimum distance, but not for the closest angular separations (within ≈ 0.1 ) below which the relativistic deflection of the background source light becomes significant (see Sect. 5.1 for details). We do not have proper motion measurements for the S stars that will be approached after 2034 (starting with S7, see Table 3). To estimate the corresponding uncertainty on ρ min , we considered an arbitrary uncertainty on their proper motion of 10 mas yr -1 that was scaled considering the date at which their position was last measured and the predicted date of closest approach. Such a proper motion corresponds to a sky velocity of 20 km s -1 at a distance of 400 pc, and is therefore a reasonable statistical upper limit for the faint and probably distant S stars. The resulting uncertainty was added quadratically to the error bar of ρ min and generally dominates the total error budget on this parameter. This uncertainty will be reduced in the future, as Gaia will provide accurate positions and proper motions for the S stars.

V (A) = 0.01, m V (B) = 1.33,
As the proper motion of α Cen in sky coordinates is approximately 3.685 arcsec yr -1 (Table 1), the uncertainty on the dates of the closest approaches is ±10 days for a typical position uncertainty of the S stars of 0.1 .

Approaches of S1 to S6 until 2031

We discuss here the cases of stars S1 to S6 that will be approached by the components of α Cen prior to 2031. Table B.1 lists their existing photometric measurements and Table B.2 gives their positions measured at different epochs, which we used to estimate their proper motions. None of the proper motions that we obtained is statistically significant. For this reason, we considered a null proper motion for the computation of the impact parameter ρ min . We however scaled the proper motion uncertainty for each star S1 to S6, taking into account the time of the foreseen closest approach and the date at which their position was last measured. The resulting uncertainty was added quadratically to the impact parameter error bar. We adjusted to the photometry the spectral energy distribution (SED) models from the library by Castelli & Kurucz (2004) 5 parametrized by the effective temperature T eff , the limb darkened angular diam- Kervella & Thévenin (2007b). e

The proper motion and associated uncertainty is listed in mas yr -1

.

A&A proofs: manuscript no. Alpha-Cen-conjunctions_v3r1 eter θ LD and the color excess E(B -V). There is a partial degeneracy between the reddening (computed assuming a classical Galactic extinction law with R V = A V /E(B -V) = 3.1 and the intrinsic color of the source (i.e., its effective temperature). As a consequence, broadband photometry alone is insufficient to determine precisely the nature of the S stars, but we could obtain reasonably good fits of their SEDs. Taking advantage of the current relatively large angular separation of the S stars with α Cen, we plan to better characterize the S stars in the future (in particular S5) with spectroscopic observations. The positions and proper motions of all the approached stars will also be refined by Gaia.

Approach of α Cen B to Star S1 in May 2021

Star S1 will be approached by α Cen B, with a minimum separation of ρ min (S1) = 0.205 ± 0.177 on 2021-05-03 ±18 days.

Based on astrometry between 2000 and 2009, the derived proper motion of S1 is µ α (S1) = -5 ± 12 mas yr -1 and µ δ (S1) = -12 ± 12 mas yr -1 . The SED of S1 is poorly constrained as the star is faint at visible wavelengths and infrared photometry beyond the K band is not available. As shown in Fig. 7, we obtained a correct fit of the SED for an angular diameter of θ LD (S1) = 0.012 ± 0.002 mas, an effective temperature of T eff (S2) = 4500±1500 K and a color excess E(B-V) = 1.6±0.5.

Approach of α Cen A to Star S2 in April 2023

Star S2 will be approached by α Cen A at a minimum separation of ρ min (S2) = 1.643 ± 0.112 on 2023-04-27 ±11 days (Fig. 8, left panel). Based on astrometry spread between 2000 and 2016 (Table B.2), the proper motion of S2 is found to be very slow at µ α (S2) = +2 ± 5 mas yr -1 and µ δ (S2) = -2 ± 5 mas yr -1 . From the fit of its photometry (Fig. 8, right panel) S2 is probably a hot (T eff (S2) = 8000 ± 1000 K) and relatively nearby star as its reddening is E(B -V) = 1.0 ± 0.1. The Spitzer/IRAC measurements of S2 at 5.8 and 8.0 µm appear to be affected by the close presence of α Cen, and although they are represented in Fig. 8, they were not used to constrain the fit. Its angular diameter of θ LD (S2) = 0.025 ± 0.003 mas is compatible with a main sequence A-type star (R ≈ 1.5 R ) located at a distance of 550 pc.

Approach of α Cen B to Star S3 in December 2023

The minimum distance between S3 and α Cen B will be reached on 2023-12-12 ±19 days at ρ min = 1.354 ± 0.186 . The impact parameter is relatively large and the contrast of 13.5 magnitudes in the K band with α Cen B is high. α Cen will be difficult to observe as it will be relatively close to the Sun at the time of the closest approach.

The SED fit gives an angular diameter θ LD (S3) = 0.018 ± 0.002 mas and an effective temperature of T eff (S3) = 4500 ± 1000 K for E(B -V) = 1.4 ± 1.0 (Fig. 9). It is interesting to remark that S3 was possibly detected by XMM-Newton with an X-ray flux of 1.2 ± 0.2 10 -13 W m -2 between 0.2 and 12 keV (Xmm-Newton Survey Science Centre 2013).

Approach of α Cen B to Star S4 in October 2024

The closest angular distance of S4 and α Cen B will be reached on 2024-10-26 ±12 days at ρ min = 2.433 ± 0.119 . The impact parameter is large and S4 will unfortunately not be easily observable at the time of the closest approach. The SED fit is made difficult and uncertain by the absence of infrared photometry. We obtained a probable angular diameter θ LD (S4) = 0.008 ± 0.003 mas and an effective temperature of T eff (S4) = 4500 ± 1500 K for E(B-V) = 0.3±0.3. The low (but poorly constrained) reddening and (marginally) significant proper motion would be compatible with a relatively nearby dwarf star.

Approach of α Cen A to Star S5 in May 2028

We adopted for the position of S5 (2MASS 14392160-6049528, WISE J143921.46-604952.9) at epoch 2000.0 the coordinates from the 2MASS Point Source Catalog and not the positions we measure on the 2MASS Atlas image (Table B.2). The catalog coordinates are more accurate, since they are derived from the raw images (Skrutskie et al. 2006). We note however that the difference between the position we measure in the image and the catalog value is very small, within 0.10 . The positional uncertainty listed in the 2MASS Point source catalog is ±0.08 . We derived a proper motion for S5 of µ α (S5) = -5 ± 5 mas yr -1 and µ δ (S5) = -5 ± 5 mas yr -1 .

We obtained a minimum approach distance of ρ min (S5) = 0.015 ± 0.135 from α Cen A on 2028-05-06 ±14 days (Fig. 10, left panel). Fortunately, the date of the closest approach of S5 is well within the observability period of α Cen (January-August), and the two objects will therefore be in a favorable position in the sky. In addition, the separation on sky of α Cen A and B at the time of the event will be larger than 10 , and the light contamination from α Cen B in the field of S5 will therefore be limited. The magnitude difference in the K band between α Cen A (m K = -1.5) and S5 (m K = 7.8) is ∆m K = 9.3 corresponding to a contrast of f (S5)/ f (A) = 2 10 -4 . This is the most favorable contrast for the stellar conjunctions with α Cen within at least the next 30 years.

We obtained a satisfactory fit of the photometry of S5 considering an intrinsically red star with T eff (S5) = 3500 ± 500 K, and a color excess of E(B -V) = 6.5 ± 1.0 (Fig. 10, right panel). We excluded the WISE measurements in the W1 (3.35 µm) as it is clear from the original WISE image that this measurement (reported by Gutermuth & Heyer 2015) is affected by contamination from α Cen. The WISE W4 (22.1 µm) and MIPS 24 µm points show an excess compared to the adjusted SED, but it is uncertain if this excess is also due to contamination from α Cen, so we did not take these points into account in the fit. The very high value we obtained for E(B -V) is consistent with the strong reddening expected for a distance of several kpc and the position in the galactic plane. As a consequence of this particularly red color (V -K = 13.7), the predicted angular diameter is large with θ LD (S5) = 0.47 ± 0.05 mas. These properties may indicate that S5 is a very distant red supergiant, for instance a twin of Betelgeuse (θ LD ≈ 45 mas, d ≈ 200 pc) located at a distance of the order of 20 kpc. It could in principle also be an intrinsically smaller red giant star located at a few kiloparsec. A photomet-A&A proofs: manuscript no. Alpha-Cen-conjunctions_v3r1 ric variability of this source is not excluded, and we foresee a thorough characterization in the coming years.

Approach of α Cen B to Star S6 in May 2031

The minimum distance of α Cen B from S6 is ρ min (S6) = 0.269 ± 0.277 on 2031-05-25±28 days. As for S4, the SED fit gives only limited information on the object due to the absence of infrared photometry. We obtained an angular diameter θ LD (S6) = 0.010 ± 0.003 mas and an effective temperature of T eff (S3) = 4500 ± 2000 K for E(B -V) = 0.5 ± 0.5.

Potential of future stellar conjunctions

Relativistic deflection and amplification

Following [START_REF] Sahu | [END_REF] (see also Di Stefano 2008a, Paczynski 1996a), the angular radius θ E of the Einstein ring of a point source is given by the expression:

θ E = 4 GM c 2 d π with 1/d π = 1/D L -1/D S . (1)
M is the mass of the lens (α Cen A or B in our case), d π the parallax distance of the lens, D L and D S the distances of the lens and of the source, respectively. We can compute the Einstein ring radius of α Cen A accurately as we know the mass of the star M A = 1.1055 ± 0.0022 M and its parallax π L = 747.17 ± 0.61 mas from Sect 3.1. The parallaxes of the S stars cannot be estimated directly, but considering their high reddening, they are very likely negligible compared to that of α Cen (S4 being possibly an exception). Assuming that the approximation of a point source is valid for α Cen A (θ LD = 8.511 ± 0.020 mas, from Kervella et al. 2003), we obtained an Einstein radius of θ E (A) = 81.97 ± 0.09 mas, corresponding to approximately 20 times the angular radius of the photosphere of α Cen A (Fig. 10).

The predicted impact parameter of the conjunction with S5 is ρ min = 0.015 ± 0.135 giving a probability of 45% that S5 will enter the Einstein ring of α Cen A. For α Cen B, θ E (B) is estimated to 75.48 ± 0.08 mas.

The gravitational lensing of the light from the background source results in the creation of two images, whose apparent shifts are given by [START_REF] Sahu | [END_REF]:

θ ± = 0.5 u ± √ u 2 + 4 θ E , (2)
with θ E the Einstein ring angular radius and u = ρ min /θ E . Following the definition adopted in the rest of this article, ρ min is the undeflected impact parameter (i.e., the minimum distance between the source and the lens in absence of gravitational lensing). The photometric amplification factor A is given by the expression:

A ± = 0.5 u 2 + 2 u √ 4 + u 2 ± 1 . (3)
When the source is located at a large separation compared to θ E (u 1), the secondary image contribution is negligible. We thus consider only A + for S2, S3 and S4, for which the apparent shift δθ + is between 2 and 5 mas.

The deflection angles δθ ± and amplification factors A ± for the conjunctions of α Cen with stars S1 to S6 are listed in Table 4. For these approximate computations, we consider only the gravitational lensing by the approached component of α Cen and we neglect the influence of the other. Due to the uncertainty on the coordinates and proper motions of S5 and S6, an almost perfect alignment with the lens cannot be excluded, and significantly higher amplification factors are possible. For S5, the amplification could reach a significant factor for the sum of the two images (Table 4). As we will precise the impact parameter in the future, we could in principle observe a near-perfect alignment and a very large gravitational amplification by positioning a spacecraft precisely where the impact parameter will be zero. Shifting the apparent position of α Cen A on the sky by 0.05 requires to send an instrument at a distance of 0.067 AU ≈ 10 million kilometers from the Earth (26 times the Earth-Moon distance). The future availability of more accurate coordinates for S5 and the surrounding field (thanks e.g., to the Gaia mission) will help in refining the value of the impact parameter at the Earth and the zero impact parameter projected trajectory in the solar system.

The close environment of α Cen A and B

The question of the stability of planetary orbits in the α Cen system has been studied by several authors. The conclusion reached by Quarles & Lissauer (2016) is that long-term stable orbits exist up to 3 (4 AU) of each component, including the habitable zone (Andrade-Ines & Michtchenko 2014). Stable circumbinary orbits are also found at distances larger than about 50 (70 AU) from the center of mass, in agreement with Wiegert & Holman (1997). So exoplanets may exist around α Cen, but their probability of presence is higher either close to each component or at large distances from the pair. Through the photometric and astrometric monitoring of the S stars, we plan to probe the environment of α Cen A and B for the presence of high and low mass planets, asteroids, comets or dust. The K band is particularly interesting for this type of observations as adaptive optics systems are very efficient in this wavelength range, providing excellent correction of the atmospheric turbulence. Using a combination of coronagraphy and advanced processing techniques, the VLT/SPHERE instrument has demonstrated that sources with a contrast of 10 -4 or smaller can be detected down to an angular separation of ≈ 3λ/D = 0.2 with a high signal-to-noise ratio (Paul et al. 2014). It has also been proposed recently that Notes. a L is the lens (one of the α Cen components). b S is the lensed source (S stars).

optimized correction techniques may significantly improve the accessible contrast (N'Diaye et al. 2016), particularly in view of the E-ELT instrumentation. In addition to the photometric and astrometric observations, it will be very valuable to monitor spectroscopically the S stars and the α Cen pair continuously before, during and after the conjunctions.

Transits of dark low mass objects

By analogy with stellar occultations by asteroids (e.g. Braga-Ribas et al. 2014), the transit of very low mass objects (e.g., planets, comets or asteroids) across gravitationally lensed images of the S sources will cause significant variations of their apparent flux. The photometric stability of the approached stars will have to be checked before the event, but transits are expected to be achromatic and brief, facilitating their identification. The determined angular diameters of the S1 to S6 sources range between 0.008 mas (S4) and 0.47 mas (S5), corresponding to projected linear diameters of 1600 to 95 000 km at α Cen (Table 4). Due to their small projected diameters, the probability to observe the transit of a planet in front of one of the S stars is low. However, the conjunctions will allow us to efficiently search for the pres-ence of the potentially more numerous asteroids and comets in the system through continuous monitoring of the photometry of the S stars. Their small angular diameters decreases the probability of transit, but will result in a large relative change of the apparent brightness of the S star during the transit, even for a very small body. For instance, the transit of a Ceres-sized asteroid (φ ≈ 950 km) would induce a relative photometric change between 4% on S2 and 30% on S4 (only 0.01% on S5). Due to the extreme brightness of the α Cen stars, the photometric monitoring of the conjunctions will be difficult for the fainter S stars at their closest approach. Depending on the available instrumentation at the time of the transits, it may only be possible to follow their photometry down to a few arcseconds from the main stars. They will in any case be useful to probe the circumbinary orbits domain, but thanks to the large parallax of α Cen, even the inner regions within a few AU will be accessible.

The case of S5 is remarkable as its large linear diameter of 95 000 km projected at α Cen A results in a much more extended covered area in the α Cen system during the approach. The transit of an Earth-like planet (φ ≈ 12 700 km) would cause a drop of 1.8% of the flux of S5. As S5 is bright at infrared wavelengths, it will be possible to obtain highly accurate photometry down to very small angular separations from α Cen A. In addition, the small impact parameter of the conjunction will result in the appearance of two amplified images while S5 evolves inside the Einstein ring of α Cen A. These two images will move rapidly in the inner 0.1 AU around the star and therefore probe a relatively large area within this short range. They will be located symmetrically from α Cen A at angular separations around 80 mas, within reach of the ALMA, E-ELT and VLTI/GRAVITY instruments. When more accurate astrometry will be available for S5, we plan to conduct a detailed simulation of the conjunction taking into account the gravitational lensing from both α Cen A and B.

Secondary gravitational lensing by a massive planet

As discussed by Di Stefano et al. (2013) and Nucita et al. (2016), and more specifically in the case of Proxima by [START_REF] Sahu | [END_REF], secondary gravitational lensing could occur due to the presence of a massive planet in the α Cen system. Such events are routinely observed in gravitational microlensing surveys (see e.g. Skowron et al. 2015;Han et al. 2016;Beaulieu et al. 2006). As the mass of a planet is much lower than the mass of a star, the radius of its Einstein ring is also smaller. But its dependence with the square root of the mass results in a non-negligible extension: for a Jupiter-mass planet orbiting α Cen, the angular radius of the Einstein ring is θ E = 8 mas. The VLTI/GRAVITY instrument will be able to measure the relative positions of α Cen and the S stars with an accuracy of 10 µas (Eisenhauer et al. 2011;Lacour et al. 2014). For a jovian mass planet, a deflection angle of 30 µas corresponds to an impact parameter of more than 2 . This means that the astrometric monitoring of the conjunctions with GRAVITY, independently of the impact parameter with the main α Cen components, gives us a 3σ sensitivity to the presence of a Jupiter-like planet over a ±2 band around the S star trajectory relatively to α Cen. Alternatively, the presence of a Neptune mass planet (0.05 M J ) will be detectable within ±0.1 from the relative trajectory of the approached S star. One potential difficulty of this observations is that the duration of the approach between the S star and the planet will be relatively short. The proper motion of α Cen is 3.7 yr -1 , so the crossing of the planet's Einstein ring will last less than two days. But the deflection will be observable over a much longer period, including if the S star does not enter in the Einstein ring of the planet. Considering a jovian mass planet, the duration during which the lensing event will be detectable will typically be of half a year.

Proper motion of α Cen, orbital parameters, parallax

Due to its extreme apparent brightness, α Cen's distance will not be measured accurately by the Gaia mission. But from high precision astrometry of the S star conjunctions, for instance with the GRAVITY instrument, it will be possible to derive its parallax with an accuracy of the order of a few 10 µas. As star S5 is likely an evolved massive star, there are good prospects that it is observable in the radio domain with e.g. ALMA or VLBI.

The GRAVITY instrument can measure the differential position of two stars separated by up to 5 with the 1.8 m VLTI Auxiliary Telescopes (ATs), and up to 2 with the 8 m Unit Telescopes (UTs). Thanks to its brightness, S5 will be observable with the ATs. It will enter the 5 accessible range of GRAV-ITY in dual field approximately 18 months before the closest approach, and remain observable for another 18 months afterward (i.e., from late 2026 to late 2029). It will be possible to follow the differential position of α Cen A and S5 with an accuracy of 10 µas per measurement epoch. Over these three years, S5 will sweep through the area where stable planetary orbits were predicted by Quarles & Lissauer (2016). Except for the approach of S4, the other conjunctions will be observable with the UTs in dual field mode. Relativistic effects will have to be taken into account to correct the measured astrometric positions, but we already have accurate masses of α Cen A and B with uncertainties of 0.4% (Sect. 3.1) so we can model the combined gravitational lens of the two stars with a high accuracy. We also plan to obtain differential astrometry of α Cen A and B in the coming years using GRAVITY to improve our estimate of the masses.

Following the conjunctions of α Cen and the S stars will enable us to derive extremely accurate parallaxes for the components of α Cen, potentially at the 10 µas level (10 -5 relative accuracy), that is, estimate their distances to ±3 AU. Combining the microarcsecond differential astrometry with ultra-precise radial velocities at a few cm s -1 level obtained, for instance, using the frequency comb technology (Yi et al. 2016) will enable us to ascertain the orbital parameters and masses of α Cen A and B with an exquisite accuracy.

Conclusions

We report that the two components of α Cen will approach several distant stars within small angular separations in the coming decades. An extraordinary conjunction of α Cen A and the m K = 7.8 star S5 will occur in early May 2028 at a minimum predicted separation of 0.015 ± 0.135 . These events provide remarkable opportunities to study the close environment of the two stars in "transmission" using the background approached star as a light probe. During the approaches, astrometric measurements will reveal the relativistic deflection of the background star light with a high signal to noise ratio. For the conjunction with S5, we may be able to directly observe the gravitational splitting of the distant source image using the E-ELT, VLTI/GRAVITY or ALMA. The astrometric monitoring of the relative positions of α Cen and the S stars may reveal the presence of planets through secondary gravitational lensing. In addition, the light from the background star will possibly be subject to photometric variations induced by transits of low mass objects present in the α Cen system. Finally, following the S star conjunctions will give access to extremely accurate proper motion, orbital parameters and parallax values for α Cen. This accuracy will be valuable for high-precision modeling of the two stars of α Cen, and for the preparation of the recently announced Breakthrough Starshot6 initiative to send ultra-fast light-driven nanocrafts to α Centauri.

Fig. 1 .

 1 Fig. 1. SUSI2 field in the I band on 2 April 2004 and trajectories of α Cen A (orange curve) and B (red curve) until 2050. The positions of the approached S stars are circled in green (Sect. 4.1).

Fig. 2 .

 2 Fig. 2. NACO exposure obtained on 27 March 2016 of the field that will be crossed by α Cen between 2023 and 2032.

Fig. 3 .

 3 Fig. 3. Left panel: NACO acquisition image of α Cen A and B obtained on 10 April 2004 (the color scale is in ADU). Right panel: ALMA image of α Cen A and B in band 7 (345 GHz) obtained on 7 July 2014 (the color scale is in Jy/beam). The separation of the two stars is 11.12 in the NACO image and 4.24 in the ALMA image. The scale and orientation of the two images is the same (North is up and East to the left).

PFig. 4 .

 4 Fig. 4. Angular separation ρ (left) and position angle θ (right) of α Cen B relative to α Cen A. Our new astrometric measurements are shown as red dots, and the literature values from the WDS are shown as gray dots.The predictions from our best-fit orbit (Table1) are represented with solid blue curves. The residuals compared to our best-fit orbit are shown in the bottom panels, together with the differences between the orbit determined byPourbaix & Boffin (2016) and our best-fit orbit (dashed gray lines, for comparison). The time scale runs from 1st January 2000 to 31 December 2031, and the magenta vertical markers indicate the epochs of the conjunctions of S1 to S6.

  Fig. 5. Top: Adjusted astrometric orbit of α Cen B relative to α Cen A. The top right panel shows an enlargement of measurements since 2002. The predicted orbital positions are shown with cyan disks, and the new measurements with red points. Bottom: Residual of the radial velocities of α Cen A and B with respect to the adjusted orbit.

  m I (A) = -0.68, m I (B) = 0.45, m K (A) = -1.49, m K (B) = -0.60 (the V -I colors of α Cen were taken from van Leeuwen 2007).

=

  5 of α Cen A or B between 1st January 2016 and 25 December 2049 for background stars brighter than magnitude m = 16 at visible wavelengths (VRIZ) and m = 15 in the near-infrared (JHK). The listed S star coordinates (ICRS) correspond to the indicated measurement epoch (no proper motion correction is included). The contrast between the corresponding α Cen star is listed in the ∆V, ∆I and ∆K columns. The error bars on the photometry are given in subscript of the corresponding values.

Fig. 6 .

 6 Fig. 6. Apparent trajectories of α Cen A (orange curve) and B (red curve) superimposed on the NTT/SOFI image in the J s band obtained on 25 February 2009. The positions of the approached S stars are circled in green (Sect. 4.1).

Fig. 7 .Fig. 8 .

 78 Fig. 7. Left panel: Trajectory of α Cen B relative to S1 (red curve) on the SOFI image obtained on 25 February 2009. The uncertainty on the impact parameter is represented with blue segments crossing at S1's position. Right panel: Photometry of S1 (red dots) compared to a Kurucz SED model of a θ LD = 0.012 mas star with T eff = 4500 K (blue curve) and E(B -V) = 1.6 (green curve).

Fig. 9 .

 9 Fig. 9. Photometry of S3 (red dots) compared to a Kurucz SED with θ LD = 0.018 mas, T eff = 4500 K (blue curve) reddened with E(B -V) = 1.4 (green curve).

Fig. 10 .

 10 Fig. 10. Left panel: Trajectory of α Cen A relative to S5 (orange curve) on the NACO image obtained on 27 March 2016. The position of the Einstein ring of α Cen A is represented with a light blue circle at the time of the closest approach, and the uncertainty on the impact parameter is represented with blue segments crossing at S5's position. Right panel: Photometry of S5 (red dots) compared to a Kurucz SED with θ LD = 0.47 mas, T eff = 3500 K (blue curve) reddened with E(B -V) = 6.5 (green curve).

Table 4 .

 4 Maximum apparent displacement δθ ± and amplification A ± of the two images of S stars formed by the gravitational lens during the conjunctions with α Cen A and B. The displacements are relative to the non-deflected (pre-lensing) positions. We also list the angular sizes θ LD of the S stars, and their projected linear diameters φ at α Centauri. The error bars are given in super/subscripts position next to each value.

Table 2 .

 2 Positions of α Cen A and B in the ICRS coordinate frame. MJD is the average modified julian date of the measurement. b σ is the estimated position uncertainty in arcseconds. c Positions taken from the original Hipparcos catalog (ESA 1997). d The listed 2MASS coordinates are measured from the Atlas image in the K s band.

	Date	Band	MJD a	Star	RA	Dec	σ[ ] b Instrument
	1991-04-02	H p	48348.250 α Cen A 14:39:40.8985 -60:50:06.530 0.001 Hipparcos c
	2000-03-12	K s	51615.383 α Cen A 14:39:36.394	-60:50:03.31	0.20	2MASS d
	2009-02-25	J s	54887.356 α Cen A 14:39:31.813	-60:50:00.01	0.07	SOFI
	1991-04-02	H p	48348.250 α Cen B 14:39:39.3908 -60:50:22.097 0.046 Hipparcos c
	2000-03-12	K s	51615.383 α Cen B 14:39:35.103	-60:50:13.56	0.20	2MASS d
	2009-02-25	J s	54887.356 α Cen B 14:39:30.919	-60:50:03.48	0.07	SOFI

Notes. a

Table 3 .

 3 Close approaches within ρ

Table B .

 B 1. Photometry of stars S1 to S6. The flux is expressed in units of 10 -16 W m -2 µm -1 . Kervella et al.: Close stellar conjunctions with α Centauri Table B.2. Astrometry of the stars approached by α Cen A and B until 2031. The listed coordinates are referenced to the 2MASS frame (ICRS).Notes. a MJD is the average modified julian date of the measurement. b σ rel is the position uncertainty with respect to the 2MASS frame in arcseconds, including statistical uncertainty and systematic offset between instruments (see Sect. 2.4). c The coordinates are measured on the 2MASS Atlas image in the K s band. d The coordinates of S5 are taken from the 2MASS Point Source catalog.

	Band Bessel V Bessel R Bessel I Bessel Z 2MASS H 2MASS K Bessel V Bessel R Bessel I Bessel Z 2MASS J 2MASS H 2MASS K IRAC 5.8 IRAC 8.0 Bessel V Bessel R Bessel I Bessel Z 2MASS J 2MASS H 2MASS K Bessel V Bessel R Bessel I Bessel Z Bessel V Bessel R Bessel I Bessel Z 2MASS J 2MASS H 2MASS K s 2.16 λ 0 [µm] 0.54 0.64 0.79 0.84 1.62 2.16 0.54 0.64 0.79 0.84 1.25 1.62 2.16 5.8 8.0 0.54 0.64 0.79 0.84 1.25 1.62 2.16 0.54 0.64 0.79 0.84 0.54 0.64 0.79 0.84 1.25 1.62 IRAC 3.6 3.6 IRAC 4.5 4.5 WISE W2 4.6 IRAC 5.8 5.8 IRAC 8.0 8.0 MSX6C A 8.28 WISE W3 11.6 WISE W4 22.1 MIPS 24 24 Bessel V 0.54 Bessel R 0.64 Bessel I 0.79 Bessel Z 0.84 Dec σ rel [ ] b Instrument Mag. Star S1 20.04 ± 0.46 19.35 ± 0.34 17.61 ± 0.22 16.80 ± 0.20 14.13 ± 0.31 13.58 ± 0.17 Star S2 15.66 ± 0.20 14.26 ± 0.20 1020 ± 207 K07 Flux a Ref. b 3.5 ± 1.6 K07 3.9 ± 1.3 K07 10.2 ± 2.2 K07 21.0 ± 4.2 K07 25.2 ± 7.8 K07 15.8 ± 2.7 K07 782 ± 158 K07 13.26 ± 0.20 973 ± 197 K07 13.29 ± 0.20 965 ± 195 K07 11.97 ± 0.04 510 ± 19 K07 11.38 ± 0.05 320 ± 15 K07 11.14 ± 0.03 150 ± 4.2 K07 10.59 ± 0.20 6.0 ± 1.2 S09 10.30 ± 0.20 2.3 ± 0.4 S09 Star S3 19.72 ± 0.35 4.7 ± 1.8 K07 17.57 ± 0.21 20.3 ± 4.3 K07 16.04 ± 0.20 43.5 ± 8.8 K07 15.74 ± 0.20 55.6 ± 11 K07 13.90 ± 0.19 86.2 ± 17 K07 12.85 ± 0.08 82.1 ± 6.3 K07 12.89 ± 0.11 29.9 ± 3.2 K07 Star S4 17.51 ± 0.20 35.9 ± 7.3 K07 16.59 ± 0.20 50.1 ± 10 K07 15.95 ± 0.20 47.2 ± 10 K07 16.22 ± 0.20 35.8 ± 7.2 K07 Star S5 21.53 ± 0.95 0.9 ± 0.9 K07 20.00 ± 0.41 2.2 ± 1.0 K07 18.66 ± 0.27 3.9 ± 1.1 K07 18.00 ± 0.22 6.9 ± 1.6 K07 11.60 ± 0.03 717 ± 20 K07 9.125 ± 0.027 2540 ± 64 G15 7.756 ± 0.024 3380 ± 76 G15 7.161 ± 0.058 888 ± 49 G15 6.912 ± 0.052 458 ± 22 G15 6.089 ± 0.025 886 ± 21 G15 6.500 ± 0.027 258 ± 6.4 G15 6.257 ± 0.027 94.5 ± 2.5 G15 6.05 ± 0.06 89.4 ± 4.7 E03 5.725 ± 0.017 33.4 ± 0.5 G15 3.924 ± 0.029 13.7 ± 0.4 G15 4.22 ± 0.02 7.8 ± 0.2 G15 Star S6 16.93 ± 0.20 41.2 ± 12 K07 15.98 ± 0.20 87.8 ± 18 K07 15.38 ± 0.20 79.8 ± 16 K07 15.65 ± 0.20 60.4 ± 12 K07 Star S1 RA 51615.383 14:39:25.277 -60:49:49.57 MJD a 0.30 2MASS c 53064.271 14:39:25.228 -60:49:49.60 0.05 SUSI2 53097.121 14:39:25.224 -60:49:49.60 0.05 SUSI2 53097.243 14:39:25.223 -60:49:49.64 0.05 SUSI2 54887.356 14:39:25.224 -60:49:49.67 0.05 SOFI Star S2 51615.383 14:39:24.291 -60:49:53.66 0.20 2MASS c 53061.325 14:39:24.284 -60:49:53.79 0.10 SUSI2 53064.271 14:39:24.285 -60:49:53.80 0.05 SUSI2 53064.329 14:39:24.284 -60:49:53.77 0.10 SUSI2 53097.121 14:39:24.284 -60:49:53.83 0.05 SUSI2 53097.181 14:39:24.287 -60:49:53.77 0.10 SUSI2 53097.243 14:39:24.284 -60:49:53.75 0.05 SUSI2 54887.356 14:39:24.288 -60:49:53.80 0.05 SOFI 57474.405 14:39:24.287 -60:49:53.80 0.06 NACO Star S3 51615.383 14:39:24.279 -60:49:46.03 0.20 2MASS c 53060.277 14:39:24.271 -60:49:46.02 0.05 SUSI2 53064.271 14:39:24.266 -60:49:46.05 0.05 SUSI2 53097.121 14:39:24.264 -60:49:46.07 0.05 SUSI2 53097.243 14:39:24.269 -60:49:46.07 0.05 SUSI2 54887.356 14:39:24.269 -60:49:46.07 0.05 SOFI Star S4 53060.277 14:39:23.472 -60:49:42.97 0.05 SUSI2 53064.271 14:39:23.482 -60:49:42.97 0.05 SUSI2 53064.329 14:39:23.477 -60:49:42.97 0.10 SUSI2 53097.121 14:39:23.467 -60:49:43.04 0.05 SUSI2 53097.181 14:39:23.482 -60:49:42.97 0.10 SUSI2 53097.243 14:39:23.477 -60:49:42.97 0.05 SUSI2 54887.356 14:39:23.481 -60:49:42.89 0.05 SOFI 57474.405 14:39:23.485 -60:49:42.91 0.06 NACO Star S5 51615.383 14:39:21.600 -60:49:52.84 0.08 2MASS d 53097.121 14:39:21.589 -60:49:52.82 0.05 SUSI2 54887.356 14:39:21.585 -60:49:52.93 0.05 SOFI 57474.405 14:39:21.586 -60:49:52.89 0.06 NACO Star S6 53060.277 14:39:20.523 -60:49:41.41 0.05 SUSI2 53064.271 14:39:20.528 -60:49:41.37 0.05 SUSI2 53064.329 14:39:20.528 -60:49:41.37 0.10 SUSI2 53097.121 14:39:20.513 -60:49:41.48 0.05 SUSI2 53097.181 14:39:20.523 -60:49:41.41 0.10 SUSI2 53097.243 14:39:20.523 -60:49:41.37 0.05 SUSI2 Notes. a Date Band 2000-03-12 K s 2004-02-29 I 2004-04-02 Z 2004-04-02 I 2009-02-25 J s 2000-03-12 K s 2004-02-26 V 2004-02-29 I 2004-02-29 V 2004-04-02 Z 2004-04-02 V 2004-04-02 I 2009-02-25 J s 2016-03-27 K s 2000-03-12 K s 2004-02-25 R 2004-02-29 I 2004-04-02 Z 2004-04-02 I 2009-02-25 J s 2004-02-25 R 2004-02-29 I 2004-02-29 V 2004-04-02 Z 2004-04-02 V 2004-04-02 I 2009-02-25 J s 2016-03-27 K s 2000-03-12 K s 2004-04-02 Z 2009-02-25 J s 2016-03-27 K s 2004-02-25 R 2004-02-29 I 2004-02-29 V 2004-04-02 Z 2004-04-02 V 2004-04-02 I 2009-02-25 J s 54887.356 14:39:20.518 -60:49:41.41 0.05 SOFI

b E03: Egan et al. (2003); G15: Gutermuth & Heyer (2015); K07: Kervella & Thévenin (2007a); S09: Spitzer Science (2009).

Article number, page 14 of 15 P.

https://www.eso.org/sci/facilities/lasilla/ instruments/sofi/inst/Imaging.html

http://www.eso.org/sci/facilities/paranal/ instruments/naco/

http://irsa.ipac.caltech.edu/

http://sb9.astro.ulb.ac.be

http://www.stsci.edu/hst/observatory/crds/castelli_ kurucz_atlas.html

https://breakthroughinitiatives.org/Initiative/3

http://www.astropy.org/ Article number, page 12 of 15 P. Kervella et al.: Close stellar conjunctions with α Centauri

Acknowledgements. We thank ESO's Director General, Dr Tim de Zeeuw, for the allocation of Director's Discretionary Time with NACO to our program 296.D-5032, and the Paranal staff for the successful execution of the observations. This research received the support of PHASE, the partnership between ONERA, Observatoire de Paris, CNRS and University Denis Diderot Paris 7. This research made use of Astropy 7 (Astropy Collaboration et al. 2013). This research made use of the SIMBAD (Wenger et al. 2000) and VIZIER databases (CDS, Strasbourg, France) and NASA's Astrophysics Data System. This research has made use of the NASA/ IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. This paper makes use of the following ALMA data: ADS/JAO.ALMA#2013.1.00170.S. ALMA is a partnership of ESO (representing its member states), NSF (USA), and NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO, and NAOJ. This research has made use of the Washington Double Star Catalog maintained at the U.S. Naval Observatory.

A&A proofs: manuscript no. Alpha-Cen-conjunctions_v3r1

Appendix A: Relative astrometry of α Cen A and B

The new and archival positions of α Cen B relative to A are listed in Table A.1 since 1980. The complete list of measurements is available from the Washington Double Star catalog (Mason et al. 2016;Mason et al. 2001). We adopted for each measurement the error bars indicated in the original paper, when they were available. For the visual micrometer measurements, we adopted constant cartesian position uncertainties σ = 0.10 between 1960 and 1976. We took σ = 0.15 between 1930 and 1959, 0.20 between 1900 and 1929, 0.40 between 1850 and 1899, and σ = 0.80 before 1850. As a remark, the observation recorded in 1752 by Nicolas-Louis de Lacaille from Cape Town (South Africa) is well in agreement with the modern orbit, with deviations of only dα = 1.5 ± 0.8 and dδ = 1.0 ± 0.8 with respect to the expected position (see also Lequeux 2014).

Appendix B: Positions of stars S1 to S6

We present in Tables B.1 and B.2 the photometric and astrometric measurements of the S stars S1 to S6, that we used to determine their angular diameters and place upper limits to their proper motions.


Damien Gredat

Hugues Chaté

Ivan Dornic

Bertrand Delamotte

Finite scale singularity in the renormalization group flow of a reaction-diffusion system

We study the nonequilibrium critical behavior of the pair contact process with diffusion (PCPD) by means of nonperturbative functional renormalization group techniques. We show that usual perturbation theory fails because the effective potential develops a nonanalyticity at a finite length scale: Perturbatively forbidden terms are dynamically generated and the flow can be continued once they are taken into account. Our results suggest that the critical behavior of PCPD can be either in the directed percolation or in a new (conjugated) universality class.

Reaction-diffusion systems involving one particle species (also known as branching and annihilating random walks (BARW)) are stochastic out of equilibrium systems important from both a phenomenological and a theoretical viewpoint. They consist of identical particles A, diffusing on a d-dimensional lattice, that can branch (nA → (n + p)A) or annihilate (nA → (n -q)A). The competition between these two types of reaction is generically responsible for the existence of transitions between an active phase where the density of particles is finite, and an "absorbing" phase where all particles, and thus all fluctuations, have disappeared. Such models provide the building blocks of a large variety of applications and models in physics and beyond, and are therefore of fundamental importance [START_REF] Henkel | Non-Equilibrium Phase Transitions: Volume 1: Absorbing Phase Transitions[END_REF]. They also have the advantage of providing a relatively simple theoretical framework for the study of the different universality classes of absorbing phase transitions.

Our understanding of out-of-equilibrium critical phenomena in general and absorbing phase transitions in particular has benefited from perturbative approaches [5,6], but important advances were brought recently by the application of nonperturbative renormalization group (NPRG) methods [2][3][4]7]. For the two prominent cases of BARW, A → 2A, 2A → ∅ which represents the directed percolation (DP) class, and A → 3A, 2A → ∅ which belongs to the parity-conserving, or generalized voter class, the success of the NPRG owed to the presence of nonperturbative features.

The case of the "pair contact process with diffusion" (PCPD) has largely resisted analysis so far [8] but was not studied with NPRG methods. The simplest BARW model in this class consists of reactions 2A → 3A and 2A → ∅ with rates σ and λ. (A limiting reaction such as 3A → ∅, with rate λ , is actually needed to ensure a finite density active phase [START_REF] Howard | When this reaction is absent the model is soluble but this solution does not help understanding the phase transition we study here[END_REF].) The distinctive feature of the PCPD is that two particles must meet to trigger branching. On general grounds, this is not expected to be a relevant ingredient defining universality classes, hence the interest raised by results obtained so far on the critical behavior of PCPD: It has been intensively stud-ied numerically in d = 1 but remains unclear because of the presence of slow dynamics and/or strong corrections to scaling [13]. The debate, ongoing still recently, is to know whether PCPD belongs to the DP universality class [13][14][15] or not [16][17][18][19].

Even the status of d c , the upper critical dimension of PCPD, is unclear: numerically, d = 3 seems beyond it [20], but in d = 2, the presence of large corrections to scaling is difficult to disentangle from logarithmic terms preventing clear conclusions in spite of indications of mean-field behavior [21,22]. In perturbation theory the RG flow of PCPD goes to the Gaussian fixed point for d > 2 and sufficiently small coupling constants, whereas it blows up at a finite scale for larger couplings or for d < 2 [9]. This suggests d c = 2. Note that the explosive flow forbids the exploration of the long-distance physics of the model. This is also known to occur in quantum chromodynamics (at the confinement scale), in the O(N ) nonlinear sigma model (at the scale of the correlation length) [START_REF] Polyakov | Gauge Fields and Strings[END_REF], and in pinned elastic manifolds (at the Larkin length) [START_REF] Le Doussal | [END_REF]. In this last case, NPRG methods at the functional level allowed to treat the problem [12].

In this Letter, we examine the PCPD field theory in the light of the NPRG, explain why perturbation theory fails, and how to avoid its problems. We show that the potential in the running effective action develops a singularity at a finite scale, signalling that couplings that are perturbatively forbidden are dynamically generated. Once taken into account the RG flow can be continued and a fixed point can be found. Our results suggest that the critical behavior of the model is either in the DP class or possibly in a new class characterized by a "conjugated" symmetry, and that d c = 2 only at small coupling. Our study indicates that NPRG is a powerful tool for dealing with similar situations beyond reaction-diffusion systems.

The field theory associated with PCPD. By using the usual Doi-Peliti formalism [23] it is possible to derive the action associated with PCPD from first principles:

S = x φ ∂ t φ -D∇ 2 φ + φ 2 g 1 φ + g 2 φ2 + g 3 φ3 +φ 3 3λ φ + O( φ2 ) (1)
arXiv:1211.1274v1 [cond-mat.stat-mech] 6 Nov 2012

where φ and φ are complex conjugates, x = (t, x), x = d d x dt, and g 1 = 2λ -σ, g 2 = λ -2σ, g 3 = -σ. Within perturbation theory, one finds that initializing the RG flow with the (bare) couplings of the action S, Eq.( 1), the running coupling g 2 (k) (k being a momentum scale) diverges at a finite scale k c for d < 2 which bars from exploring scales below k c . Moreover, the φφ response function does not receive any loop correction and thus the dynamical exponent z remains equal to 2 which is clearly invalidated by numerical results. For d > 2, all couplings are irrelevant around the Gaussian fixed point which is thus (locally) attractive. (Hence the conclusion that d c = 2.) We now present a calculation where the flow no longer diverges but develops a singularity that can naturally be taken into account at the price of working functionally.

The nonperturbative renormalization group. NPRG follows Wilson's idea [24] of partial integration over fluctuations. It builds a one-parameter family of models indexed by a momentum scale k such that fluctuations are smoothly included as k is lowered from the inverse lattice spacing Λ down to k = 0 where they have all been summed over. To this aim, we add to the original action a momentum-dependent mass-like term [25,26]

∆S k = 1 2 q φ i (-q) [R k (q)] ij φ j (q) (2)
where q = (ω, q), q = | q|, i = 1, 2, φ 1 = φ, φ 2 = φ and repeated indices are summed over. With, e.g.,

[R k ] 12 = [R k ] 21 = (k 2 -q 2 )θ(k 2 -q 2 ) and [R k ] 11 = [R k ] 22 = 0,
the fluctuation modes φ i (q > k) are unaffected by ∆S k while the others with q < k are essentially frozen. The k-dependent generating functional of correlation and response functions thus reads

Z k [J 1 , J 2 ] = DφD φ e -S-∆S k + x Jiφi . (3)
The effective action Γ k [ψ i ], where

ψ 1 = ψ = φ 1 , ψ 2 = ψ = φ 2 , is given by the Legendre transform of W k = log Z k (up to the term proportional to R k ): Γ k [ψ, ψ] + W k = x J i ψ i - q R k (q)ψ(q) ψ(-q). ( 4
)
The two-point functions can be computed from Γ k by differentiating:

[ Γ (2) k ] i1i2 [x 1 , x 2 , ψ, ψ] = δ 2 Γ k δψ i1 (x 1 )δψ i2 (x 2 ) (5)
and the exact flow equation for Γ k [ψ, ψ] reads [25]:

∂ k Γ k = 1 2 Tr q ∂ k R k . G k with G k = [Γ (2) k + R k ] -1 . (6)
When k decreases from Λ to 0, Γ k varies between the (bare) action: Γ k=Λ = S, and the full effective action:

Γ k=0 = Γ. Solving the flow equation ( 6) is thus equivalent to solving the model. This is however impossible to do exactly and approximations must be made. We perform here the local potential approximation (LPA) which is known to work well for the determination of the critical behavior of models either at equilibrium or outof-equilibrium [2,3,25,27]:

Γ k → Γ LPA k = x ψ(∂ t -D∇ 2 )ψ + U k (ψ, ψ) . (7)
Substituting Eq.( 7) into Eq.( 6), choosing the R k function described above and integrating over q, we find :

∂ k U k = C d             k 2 + U (1,1) k k 2 + U (1,1) k 2 -U (0,2) k U (2,0) k -1             (8)
where

C d = 4k 2+d [2 d+1 π d/2 dΓ(d/2)] -1
and the upper indices code for derivatives in ψ and ψ. It is easy to verify that performing a field expansion of U k (ψ, ψ) around (0, 0) leads, for the couplings in front of the monomials ψ m ψn , to the same difficulties as encountered perturbatively: the flow blows up at a finite scale for d < 2. We thus need to work functionally. However, we can further simplify our ansatz Eq.( 7) by performing an expansion of U k (ψ, ψ) in ψ only while remaining functional in the ψ-direction. We thus replace U k (ψ, ψ) by

U k (ψ, ψ) → N n=1 ψn V n,k (ψ) (9)
and we typically truncate the sum at order N = 3 or 4. Our approximation scheme is therefore based on the assumption that the LPA is sufficient, and that the nontrivial features of the model lie in the ψ direction (which is not spoiled by the expansion in ψ). Inserting Eq.( 9) into Eq.( 8) we find the flow equations for the functions V n,k at order N = 3 :

V1 = BV 2 V 1 (10) V2 = B 1 2 V 1 (AV 2 (3AV 2 V 1 -8V 2 )+6V 3 ) + V 2 V 2 (11) V3 = B 1 2 (A (8V 2 (3V 1 (AV 2 V 2 -V 3 ) -V 2 V 2 ) +V 2 V 1 (A (V 1 (AV 2 (5AV 2 V 1 -24V 2 ) + 18V 3 ) + 6V 2 V 2 ) -12V 3 )) + 6V 3 V 2 ) + V 2 V 3 (12)
where, for simplicity, we have omitted the index k in the functions V n,k , V n and V n are the derivatives of these functions with respect to ψ,

Vn = k∂ k V n , A = (k 2 +V 1 ) -1 and B = (k d+2 4v d /d)A 3 |A -1 |.
The initial conditions of these flow equations are provided by the action S with 9). Right: Same when U k is expanded in both ψ and ψ and infinitesimal linear couplings are considered (σ1(k = Λ) = 0, σ2(k = Λ) = 10 -8 , all other rates are of order 1). Only terms at most quartic in the fields have been retained. The RG flow of all couplings remains finite but shows an abrupt change around kc 0.6. φ i → ψ i . They impose that ∀n: V n,k=Λ (ψ = 0) = 0: the bare potential has no linear term in φ. Perturbatively, it is easy to show that these terms cannot be generated in U k since all Feynman diagrams involve at least two incoming particles and thus two fields ψ.

We have numerically integrated the coupled flow equations of the functions V n,k (ψ) with N = 4 (see Eqs. [START_REF] Polyakov | Gauge Fields and Strings[END_REF] for the N = 3 case) together with the initial conditions provided by S, Eq.( 1). In the early stage of this flow, the linear term of each of these functions remains identically zero as naively expected. However, at a finite scale k c which is typically the scale where the perturbative flow blows up, a linear term is generated in all these functions (Fig. 1): the potential U k develops a corner at ψ = 0 and its analytic structure is changed below k c .

A detailed study of the emergence of the linear terms reveals that for k k c , a boundary layer appears in the V n,k (ψ) functions such that in the inner part of the layer, that is, at small ψ, these functions are expandable around ψ = 0 (for ψ ≥ 0) and start quadratically in ψ. In the outer region, a linear part appears in the V n,k (ψ) functions. As k approaches k c from above, the width of the layer decreases and vanishes at k c , leaving the linear term as the dominant term around ψ = 0 (Fig. 2). Below k c , the linear terms remain present in the vicinity of ψ = 0 and the flow can be continued all the way to k = 0. We have checked that this emerging scenario holds at order N = 3 and 4 of the ψ-expansion. Converged, higherorder results are unfortunately difficult to obtain [START_REF]reliably the emergence of nonanalytic behavior in ψ at a finite scale using grids both in the RG[END_REF].

We have confirmed our scenario by considering the usual set of reactions 2A → 3A, 2A → ∅ and 3A → ∅ complemented by A → 2A and A → ∅ with infinitesimal rates. Expanding the potential U k in both ψ and ψ we now have (at least) two new couplings that are linear in ψ: σ 1 (k)ψ ψ and σ 2 (k)ψ ψ2 with σ 1,2 (k = Λ) infinitesimal. If the limit σ 1,2 (k = Λ) → 0 were regular then σ 1,2 (k) would start playing a significant role in the RG flows of the other couplings only below a very small scale k that would go to 0 as σ 1,2 (k = Λ) → 0. We find on the contrary that while for k just below Λ the flows of all the couplings are indeed almost insensitive to σ 1,2 (when they are initially extremely small), this is no longer the case for k k c since the dramatic increase of g 2 (k) makes σ 1,2 (k) grow abruptly around k c , independently of their initial smallness, see Fig. 1. For k close to k c , the back reactions of σ 1,2 (k) on the flows of all the other couplings start to be significant and eventually modify them completely since σ 1,2 are the most relevant couplings. The RG flow is no longer singular (g 2 (k c ) remains finite) but lives below k c in a larger functional space involving the couplings linear in ψ. This result is fully consistent with what is found in the functional viewpoint.

Criticality in d = 1. Our two different approaches both conclude that terms linear in ψ are generated below a nonuniversal scale k c in d = 1. We can therefore consider the field theory obtained just below k c as a new field theory that can be studied per se, the difficulty being that its action is non polynomial since the functions V n,k (ψ) are not. In a perturbative analysis only the terms of lowest degrees in ψ and ψ would be retained in the bare action, i.e. ψ ψ, ψ 2 ψ and ψ ψ2 . Depending on the relative sign of the two cubic terms, this action, truncated at order three, exhibits one of the following symmetries (after a trivial rescaling of the fields): ψ(t) ±ψ(-t). The minus sign corresponds to the cubic terms having opposite signs. This is the "rapidity" symmetry defining the DP class. The other sign defines a new, "conjugated", symmetry. We call the corresponding class DP'. It is easy to show in our framework, Eq.( 8), that when only the above terms are kept in U k , only two nontrivial fixed points exist and that they show either one or the other of the above symmetries (we call them DP and DP'). No such result exists beyond this simple truncation. To the best of our knowledge, the DP' symmetry has never appeared nor been studied before.

Two difficulties appear when studying the existence and the nature of the fixed point. First, when using a complete field expansion of U k , it is difficult to initialize the flow below k c with what has been found functionally using Eq.( 9) just after the singularity because this amounts to projecting functions of ψ onto polynomials, setting infinitely many terms of high degree to 0. Second, if we work directly with our semi-functional approximation, Eq.( 9), the roles of ψ and ψ are dissymmetric which hinders the search of a fixed point exhibiting a symmetry that exchanges ψ and ψ as in DP or DP'. We have nevertheless studied the existence of a fixed point using both approaches. Within richer and richer polynomial approximations of U k (where ψ and ψ play symmetric roles) initialized with couplings found with the semi-functional approximation just after the singularity, we have found a range of initial reaction rates leading to the DP fixed point. In the other, semi-functional approximation truncated at order N = 4, we have recorded the

flow of r k = (U (2,1) k /U (1,2) k ) U (1,3) k /U
(3,1) k (calculated at ψ = ψ = 0). This quantity is an indicator of the nature of the fixed point because if r k = -1 the expansion of U k up to order four in the fields exhibits the rapidity symmetry (up to a rescaling of the fields). If r k = +1 the same holds true for the DP' symmetry. Notice that just below k c , r k is neither 1 nor -1. We show in Fig. 3 the evolution of r k computed from Eq.( 9) with N = 4. Although the instabilities of our numerical code prevent us from finding a true fixed point value for r k there is little doubt that it indeed reaches a plateau at the value -1 so that the expected fixed point should be that of DP in agreement with what is found in the polynomial approximation. Notice however that if we work with a field truncation of U k , starting at k = Λ with a very small coupling σ 2 (k = Λ) as explained above, we find, at least in the simplest truncation of U k , either the DP or DP' fixed points depending on the initial rates (we have not been able to confirm this result with truncations involving higher powers of the fields because of the extreme sensitivity of the flow at small k on the choice of the initial rates). Interestingly, we find that the vicinity of both fixed points is reached after a very long RG "time" (log k/Λ ∼ -9) which means that the scaling regime only appears at very large lengthscales. This could explain why it is so difficult to observe the asymptotic regime in numerical simulations. Our general conclusion is therefore that the critical behavior of PCPD in d = 1 should be either in the DP or DP' universality class and our best results are in favor of DP. The upper critical dimension. The determination of d c is the same in perturbation theory and in our scheme for small initial couplings: for d > 2, the conditions g 1,Λ = 0 and g 2,Λ small make the system critical and the flow is driven towards the Gaussian fixed point. However, for large enough g 2,Λ the flow always becomes singular, even when g 1,Λ = 0. We show in Fig. 3 the basin of attraction of the Gaussian fixed point in this case as a function of g 2,Λ obtained within perturbation theory. The result is similar to what is found in the Kardar-Parisi-Zhang equation [28][34]. Note that within perturbation theory, criticality is reached when g 1,Λ = 0. It is no longer clear in our approach that this condition is necessary for large initial g 2,Λ and d > 2 because the finite scale singularity emerges for generic g 1,Λ . Our results suggest that there could exist a nontrivial critical behavior at large initial g 2,Λ even above one dimension that could be in the DP (or possibly DP') universality class.

To conclude, we believe our results represent a breakthrough for the understanding of the critical behavior of PCPD. Even though we do not have yet a complete solution of the problem, we have unlocked an heretofore blocked situation and offered new lines of further research. The recourse to functional nonperturbative renormalization was essential because it allows us to address questions such as the generation of linear terms. Going beyond our semi-functional approximation remains the main challenge that should lead, once controlled numerically, to satisfactory results. From a physical point of view, a final answer to the question of the phase diagram and its accessible fixed points in all dimensions remains of course the main goal. But understanding the meaning of the scale k c is also challenging. It could be related to the existence of relevant "elementary excitations" made of pairs of particles that would involve an intrinsic scale. Disentangling the roles of the particles and of the pairs has already been studied by effectively taking them into account through the introduction of another species B and the reactions: 2A → B, B → A, B → 2B, 2B → B, B → ∅. Such two-species PCPD models, which are believed to exhibit the same critical behavior as the one species case studied here [31], could be the starting point of more complicated NPRG approaches (involving four fields, but possibly deprived of singularities in the flow). These endeavors are left for future work.

2 FIG. 1 :

 21 FIG.1:(color online) Left: Flows in d = 1 of the linear couplings σn(k) = V n,k (0) in the semi-functional approach, Eq.(9). Right: Same when U k is expanded in both ψ and ψ and infinitesimal linear couplings are considered (σ1(k = Λ) = 0, σ2(k = Λ) = 10 -8 , all other rates are of order 1). Only terms at most quartic in the fields have been retained. The RG flow of all couplings remains finite but shows an abrupt change around kc 0.6.

FIG. 2 :

 2 FIG.2:(color online) Boundary layer for V 3,k (ψ), whose behavior around the origin is displayed for k kc and several values of k above kc. For all k > kc, V 3,k starts quadratically at ψ = 0 whereas a linear term appears at k = kc. Inset: width of the boundary layer as a function of log k/Λ.

FIG. 3 :

 3 FIG. 3: (color online) Left: Flow of the symmetry-testing ratio r k computed in the semi-functional approach with N = 4. For k < kc, r k flows toward the DP value r (DP ) k = -1. Right: line separating the basin of attraction of the Gaussian fixed point and the domain where the singularity is present in the (d, g2(Λ)) plane when g1(Λ) = 0.

We thank Matthieu Tissier and Gilles Tarjus for illuminating discussions.