M.-N Avettand-Fènoël
email: marie-noelle.avettand-fenoel@univ-lille.fr

F De Geuser

A Deschamps

Effect of the ageing on precipitation spatial distribution in stationary shoulder friction stir welded AA2050 alloys

Keywords: Stationary shoulder friction stir welding, AA2050 alloy, ageing, microstructure, SAXS analyses, microhardness

The spatial distribution of the strengthening precipitates in stationary shoulder friction stir welds (SSFSW) of the AA2050 Al-Cu-Li alloy has been characterized by small-angle X-ray scattering (SAXS), combined with micro-hardness mapping and transmission electron microscopy (TEM). It is shown that welding the alloy before ageing (T3WT8) leads to a more homogeneous distribution of precipitates along the weld zones than welding after ageing (T8W). Further in-situ SAXS ageing (24 h at 155°C) has been performed on the nuggets of both welds, to better evaluate the subsequent precipitate evolution. The nugget zone is shown to be very inhomogeneous, which can be explained by the 2 passes sequence of SSFSW. The mid-depth zone shows no fine precipitation, because of the formation of coarser phase consuming the solutes. Irrespective of the welding sequence, further ageing does not yield homogeneous precipitation in the welds.

Introduction

Friction stir welding (FSW) which proceeds in the solid state, has been developed by the Welding Institute in 1991 [START_REF] Thomas | Improvements relating to friction welding[END_REF] and is particularly suitable to join precipitation strengthened aluminum alloys [START_REF] Threadgill | Friction stir welding of aluminum alloys[END_REF], among which AA2050 grade. This grade is an Al-Cu-Li alloy developed for its low density, high strength, high toughness and good corrosion resistance [START_REF] Lequeu | Aluminum-copper-lithium alloy 2050 developed for medium to thick plate[END_REF]. It can contain various precipitates such as θ' (Al2Cu), δ' (Al3Li), T2 (Al5Li3Cu), TB (Al7Cu4Li) and T1 (Al2CuLi) phase [START_REF] Lequeu | Aluminum-copper-lithium alloy 2050 developed for medium to thick plate[END_REF] which is known as the most effective strengthening precipitate in Al-Cu-Li alloys [4-7]. Few works dealing with thick AA2050 FSW joints are reported in literature and are focused on the investigation of material flow during welding [START_REF] Avettand-Fènoël | Experimental investigation of three-dimensional (3-D) material flow pattern in thick dissimilar 2050 friciton stir welds[END_REF], on the joints corrosion resistance [9-11] and on their microstructure and mechanical properties [3, [START_REF] Geuser | Mapping the mcirostructure of a friction stir welded (FSW) Al-Li-Cu alloy[END_REF][START_REF] Dhondt | Mechnaical behavior of periodical microstructure induced by friction stir welding on Al-Cu-Li 2050 alloy[END_REF][START_REF] Geuser | Microstructure mapping of a friction stir welded AA2050 Al-Li-Cu in the T8 state[END_REF][START_REF] Malard | Microstructure distribution in an AA2050 T34 friction stir welds and its evolution during post-welding heat treatment[END_REF][START_REF] Sidhar | Impact of thermal management on post weld heat treatment efficacy in friction stir welded 2050-T3 alloy[END_REF][START_REF] Avettand-Fènoël | Heterogeneity of the nugget microstructure in a thick 2050 Al friction stir weld[END_REF]. However, the use of a rotational shoulder in conventional FSW was shown to result in a thermal gradient along the thick AA2050 joint depth entailing various precipitation states and then different mechanical properties as illustrated by the spatial distribution of hardness [START_REF] Avettand-Fènoël | Heterogeneity of the nugget microstructure in a thick 2050 Al friction stir weld[END_REF].

A derivative of conventional FSW, called stationary shoulder friction stir welding (SSFSW), has been developed and consists in using a stationary shoulder, which slides over the surface of the workpiece, instead of a rotational shoulder [START_REF] Widener | High rotational speed friction stir welding with a fixed shoulder, 6 th international friction stir welding symposium[END_REF][START_REF] Russell | Recent developments in the stationary shoulder FSW of titatnium alloys[END_REF]. With SSFSW process, almost no shoulder heat is generated hence resulting in a reduction of the temperature gradient between the top and bottom surfaces of the thick joint; consequently, nearly the whole energy brought to the system results from the plastic work provided by the pin; the through-thickness weld profile is thus more parallel and uniform, the heat affected zone (HAZ) and the thermomechanical affected zone (TMAZ) are narrower compared to those obtained with conventional FSW [START_REF] Wu | Stationary shoulder FSW for joining high strength aluminum alloys[END_REF][START_REF] Sun | Comparison of residual stress distributions in conventional and stationary shoulder high-strength aluminum alloy friction stir welds[END_REF][START_REF] Sun | The effect of shoulder coupling on the residual stress and hardness distribution in AA7050 friction stir butt welds[END_REF] and the joint surface is smoother [START_REF] Wu | Stationary shoulder FSW for joining high strength aluminum alloys[END_REF][START_REF] Goebel | Semi-stationary shoulder bobbin tool friction stir welding of AA2198-T851[END_REF]. In addition, in terms of residual stresses, the tensile region is significantly narrower and the tensile residual stress magnitude is lower compared to conventional FSW [START_REF] Sun | Comparison of residual stress distributions in conventional and stationary shoulder high-strength aluminum alloy friction stir welds[END_REF][START_REF] Sun | The effect of shoulder coupling on the residual stress and hardness distribution in AA7050 friction stir butt welds[END_REF]. A higher rotational speed [START_REF] Sun | Comparison of residual stress distributions in conventional and stationary shoulder high-strength aluminum alloy friction stir welds[END_REF] or a lower advancing speed [START_REF] Li | Investigation of stationary shoulder friction stir welding of aluminum alloy 7075-T651[END_REF] than in conventional FSW is however required in SSFSW in order to bring a suitable heat input to join the materials. SSFSW has already been used to weld aluminum alloys such as A6N01 7075-T651 [START_REF] Li | Investigation of stationary shoulder friction stir welding of aluminum alloy 7075-T651[END_REF] or even dissimilar Al alloys AA2024 / AA7050 [START_REF] Barbini | Influence of non-rotating shoulder on heat generation, microstructure and mechanical properties of dissimilar AA2024/AA7050 FSW joints[END_REF]. A previous investigation was also devoted to SSFSW of AA2050 alloy and to the effect of a pretreatment or post-welding treatment on joint microstructure and mechanical properties [START_REF] Avettand-Fènoël | Effect of a pre or postweld heat treatment on microstructure and mechanical properties of an AA2050 weld obtained by SSFSW[END_REF]. It evidenced the presence of B and B shear texture components in the probe affected zone of the nugget irrespective of the process sequence and, in a pre-treated sample, a slightly greater grain size across the nugget depth, a larger shoulder affected zone, the presence of Guinier Preston zones, a smaller microhardness undermatching and a lower uniform deformation [START_REF] Avettand-Fènoël | Effect of a pre or postweld heat treatment on microstructure and mechanical properties of an AA2050 weld obtained by SSFSW[END_REF]. The present paper aims to report a complementary investigation dealing with the characterization by Small Angle X-ray Scattering (SAXS) of the distribution of phases in terms of size and volume fraction, relying on the methodology developed and applied previously to conventional AA2050 friction-stir welds [START_REF] Geuser | Mapping the mcirostructure of a friction stir welded (FSW) Al-Li-Cu alloy[END_REF][START_REF] Geuser | Microstructure mapping of a friction stir welded AA2050 Al-Li-Cu in the T8 state[END_REF][START_REF] Malard | Microstructure distribution in an AA2050 T34 friction stir welds and its evolution during post-welding heat treatment[END_REF]. Microstructure maps will be realized along these thick AA2050 stationary shoulder friction-stir welds, and will be linked to microhardness maps. A further aim of this study is to analyze the effect of pre or post-weld ageing on these spatial distributions of precipitates and on the kinetics of precipitation during subsequent artificial ageing of the welds at 155°C. Finally, the pros and the cons of SSFSW in connection with the phases distributions will be discussed.

Experimental procedure

The base material is a AA2050 aluminum alloy whose chemical composition is reported in Table 1.

Balance

Two plates (15 mm thick, 200 mm wide and 400 mm long) were abutted and welded perpendicularly to the rolling direction either in a T3 state or in a T8 state (precipitation heat treated). Friction stir welding was conducted with a 0.5° backward-tilted tool made of MP159 (35.7% Co, 25.5% Ni, 19% Cr, 9% Fe, 7% Mo, 3% Ti, 0.6% Nb, 0.2%Al (mass.%)) material. The tool is constituted of a 21.5 mm diameter stationary shoulder prolonged by a pin. The 10 mm long pin was a left hand threaded truncated flute tool with a diameter of 10 mm at its root. The backing plate was made of XC48 steel. Welding was performed under a mean axial effort of 20-22 kN with a clockwise rotation speed of 400 rpm and at a welding speed of 100 mm/min. Two passes with air cooling down to room temperature in-between were carried out, one on each surface of the plate. The welding direction of the first and second passes were opposite so that the advancing side after the first pass corresponds to the advancing side after the second pass. The T3 state was obtained according to the parameters defined in a patent [START_REF] Bordesoules | Method for assembling parts made of an aluminum alloy by welding, compressing sold deformation followed by the postwelding tempering of the entire welded area[END_REF]. The joint produced with the T3 material was then aged for a few tens of hours at a temperature close to 150°C and is referred to as the T3WT8 sample. The specimen aged in the aforementioned conditions before welding is referred to as the T8W sample.

The transverse sections of the joints were polished and etched for around 20 s with Keller's reagent (3 mL HNO3, 6 mL HCl, 6 mL HF, 150 mL H2O) in order to identify the various zones of the welds. For transmission electron microscopy (TEM) / X-ray energy dispersive spectroscopy (EDX) analyses under a 200 kV accelerating voltage, thin foils were sampled in the nuggets generated by the second pass at different depths from the stirred top surface (4 and 8 mm). They were first mechanically polished down to a thickness of 50 µm. They were subsequently electropolished in a 25 vol.% nitric acid -75 vol.% methanol solution at a temperature between -9 and -4°C with a voltage of 12.5 V and a current density in-between 18.1 and 22.6 A/cm². They were finally ion-milled with two 3 keV argon ion beams with a ±6° angle of incidence for 15 min. For SAXS analyses, a slice was sampled in both joints along their transverse cross-sections. The slices were mechanically polished down to a thickness of 150 µm. SAXS was performed on a custom built laboratory setup with a rotating anode source at the Kα energy of Cu (about 8 keV). The sample-to-detector distance was about 0.6 m and the SAXS patterns were recorded by a PILATUS 2 300k hybrid pixel detector. To map the microstructure along the weld, the samples were scanned along 2 directions with a step size of 0.5 mm, recording 1421 images, spanning a total surface of 24 mm x 14 mm. Since the beam size was roughly 1 mm, it should be noted that the SAXS maps are slightly oversampled and details smaller than 1 mm are likely to be smoothed out. For in situ experiments, a 4 mm vertical band along the nugget was cut from each sample. This band was then aged in situ for about 20 h at 155°C in a custom built furnace which allows scanning the sample while ageing. The temperature reached 155°C in less than a minute. We recorded a SAXS pattern every 1 mm at the centre of this band and repeated this scan for the duration of the in-situ experiments, resulting in a series of spatially resolved precipitation kinetics. The interpretation of the SAXS signal was performed according to [START_REF] Geuser | Microstructure mapping of a friction stir welded AA2050 Al-Li-Cu in the T8 state[END_REF]. The precipitates are considered as flat cylinders. Their long dimension being too large with respect to the characteristics of the SAXS set-up, it cannot be evaluated from the data. Thus the results of the data fitting consist in the relative volume fraction of platelets as well as their thickness. T1 and θ' can both be present and SAXS cannot distinguish between them. The results are thus to be interpreted as resulting from the combination of all platelets precipitates. The results are presented as 2D maps of size and volume fractions (in relative units). For the in-situ experiments, the maps axes are heat treatment time (X-axis) and position along the nugget height (Y-axis).

Vickers microhardness measurements were also carried out on the transverse cross sections of the welds under a 50 g load applied for 15 s. The microhardness measurements were performed every 200 µm along the X-axis and every 2 mm along the Y-axis. X-axis is along the weld height, Z-axis along the welding speed and Y-axis is defined such as the coordinate system (O, X, Y, Z) is direct (figure 1).

Results

Effect of the process sequence (T8W or T3WT8) on T1/θ' 2D distribution.

Figure 1 displays the features of the transverse cross sections of the T3WT8 and T8W joints. Keller etching makes it possible to distinguish the different zones on both joints, namely the nugget, the thermomechanical affected zone (TMAZ), the heat affected zone (HAZ) whose microstructure differs from that of the base material. Given the pin length and its penetration depth, the nugget zone at workpiece mid-depth has been stirred twice by the pin during the 2 passes. The 2D distribution of the volume fraction and thickness of T1/θ' in both joints are respectively depicted in Figures 2 and3. These maps clearly highlight the effect of welding on the precipitates distribution, particularly the drastic effect of the welding sequence. In the sample aged after welding (T3WT8, Figure 2), there are precipitates in each zone of the weld, except at the centre of the nugget where no precipitates are detected. The rest of the nugget does not reach the precipitate fraction of the base material, with a smaller fraction by a factor of ~2. The HAZ area shows a very similar distribution of precipitates to that of the base material, confirming that the moderate heating experienced during welding has had no influence on the subsequent precipitation. In the sample welded after precipitation treatment (T8W, figure 3), the welding process has dissolved nearly all precipitates in the nugget. A low fraction of precipitates is detected in the area of the first pass, which can be attributed to the heat provided by the second pass. These precipitates are thicker than in the base material. The nugget bottom created by the second pass is shown to have much thicker precipitates than elsewhere, but with a negligible volume fraction, showing a strong coarsening in this area having undergone the 2 passes. These large sizes with negligible fractions are also observed at the borders of the nugget, mostly at the top part (second pass), suggesting strong coarsening of the precipitates when reaching a temperature just below their solvus. In the HAZ area, the fraction of precipitates is slightly higher than in the base material, with a thickness relatively larger.

Figure 4 shows TEM observations within the weld. According to the observations of Figure 4 a and b, a mixture of T1 and θ' is present in the center of the nugget of the T3WT8 sample generated by second pass., Figures 4a,b and c also depict the presence of T1 phases on dislocations. Bbut no T1 and θ' could be evidenced at the bottom of theis nugget of T3WT8 (not shown here). Regarding the T8W sample, TEM analyses did not enable to detect T1 and θ' phases neither in the centre of the nugget generated by the second pass (not shown here) nor at its bottom (figure 4cd). Besides, many Cu rich precipitates containing Fe, Mn or Mg and with a size close to 0.5 µm were identified at an 8 mm depth from the top surface in both joints (see example for T8W joint in figures 4de to ij). Other smaller Cu bearing particles without Fe, Mn and Mg were also observed at this location (see white arrows in figure 4g).

Microhardness maps of joints T3WT8 and T8W.

Figure 5 shows microhardness maps of the section of the T3WT8 and T8W welds. It evidences that the softest zone in both joints is located in the nugget at the plate midthickness. This zone has experienced twice the effect of the pin and corresponds to the bottom of the nuggets generated by the two passes. Besides, in the remaining parts of the nugget, the microhardness of the nugget is greater in the T3WT8 sample than in the T8W specimen which is in agreement with the precipitates distribution obtained by SAXS.

Effect of isothermal artificial ageing at 155°C on T1 and θ' distribution in the nugget zone along the joints thickness.

From the in-situ SAXS experiments while heating the nugget zones, one can build time-dependent maps of the volume fraction of T1 and θ' along the depth of the median part of the nugget in the welds transverse cross section (figure 6). For the T3WT8 joint (left figure), the results around position 0 show that precipitation of T1 and θ' does not occur at nugget mid-depth even after an isothermal treatment at 155°C for 20 h. However, in the remaining part of the nugget of the same sample, the fraction of T1 and θ' increases with holding time at 155°C which suggests that the material was still underaged after the initial T8 treatment. In the T8W weld, there is very little T1 and θ' precipitation during isothermal ageing indicating a much slower kinetics. In both cases, the final state shows essentially no precipitates at mid-thickness.

Discussion

Effect of pre-or post-ageing on the joint microstructure

According to SAXS (figure 3) and TEM analyses, no T1 phase was observed in the central zone of the T8W joint formed during the second pass. It suggests that SSFSW leads to the dissolution of T1 in the nugget. This observation is consistent with previous studies reported in literature on FSW joints of 2050 Al alloy in T8 state [14] or T3 state [START_REF] Malard | Microstructure distribution in an AA2050 T34 friction stir welds and its evolution during post-welding heat treatment[END_REF]. The combination of the homologous temperature (defined as the ratio between the absolute welding temperature and the absolute melting point of the material) reached in the nugget during FSW, at least of 0.6-0.7 [START_REF] Mishra | Friction stir welding and processing[END_REF], which is usually in-between 400 and 550°C [2,21,35-36] in aluminum alloys, and of the strain, which is in the range in FSW [37-42], is very likely suitable to dissolve the T1 phase. Moreover, the temperature reached in the nugget during FSW is consistent with the temperature of dissolution of T1 which is around 450°C according to DSC analyses performed with a heating rate of 10°C/min [7,14,16]. The heating rate during SSFSW is however greater than 10°C/min, which should increase the dissolution temperature of T1 while the latter temperature may be decreased by the strain.

For the T3WT8, the post-welding heat treatment effectively triggers the precipitation of T1 in the base material as expected. In the nugget, however, the precipitation treatment results in a much lower fraction of precipitates than in the base material. T1 is known to preferentially nucleate at dislocations and subgrain boundaries [5,7,43-46]. Al-Cu-Li alloys are thus generally pre-stretched before ageing to accelerate the kinetics of T1 precipitation, whose rate is monotonically increasing with the level of predeformation [47-49]. Since the dislocation density is much lower in the nugget than in the base material due to the high temperature reached and associated dynamic recrystallization, the amount of nucleation sites is lower, and thus the fraction of precipitates is lower. Some T1 phases could indeed be observed on dislocations, whose density remains low (figure 4 a andc). It is also likely that the respective relative proportion of T1 and θ' has changed, since the absence of dislocations has been shown to favour θ' compared to T1 [START_REF] Gable | The role of plastic deformation on the competitive microstructural evolution and mechanical properties of a novel Al-Li-Cu-X alloy[END_REF]. The proportion of T1 and θ' phases has however not been estimated.

In both joints, SAXS analyses show the asymmetrical features of the nugget along the joint thickness (figures 2 and 3) which is generated by sequence of 2 passes.

The workpiece mid-depth is a very particular zone, since it can be considered as the thermomechanically affected zone (TMAZ) of both the 1 st pass and the 2 nd pass. This zone is characterized in both T3WT8 and T8W welds by the absence of T1 and θ' according to SAXS analyses (figures 2 and 3) and TEM characterization (figure 4cd to ij). In this zone, coarse Cu rich precipitates sometimes containing Mn, Fe or Mg together with smaller Cu bearing ones which are Mn, Fe or Mg free (see white arrows in figure 4g) are detected have been formed (figure 4de to ij). The smaller Cu bearing precipitates and have very likely consumed the Cu solute required to form T1 and θ'. Concerning the coarser precipitates, the same kinds of Mn, Fe or Mg rich precipitates have already been detected in the case of FSW of 2050 alloy [START_REF] Malard | Microstructure distribution in an AA2050 T34 friction stir welds and its evolution during post-welding heat treatment[END_REF]. Given their size, Tthese large precipitates are very likely native particles of the base material, which were not dissolved during SSFSW., which were formed either during welding or during the cooling stage after welding, They are thermally stable at processing temperature and do not seem to be affected by strain. Below the pin, the material downflow is blocked by the cold die effect [8] of the mid-thickness of AA2050 thick workpiece which presents a low thermal conductivity of 76.6 W/m/K (at 20°C) in T3 state compared to that of many aluminum alloys [START_REF] Brandes | Smithells Metals Reference Book[END_REF]. It plays the role of a backing plate and induces some vortices and a subsequent material upflow leading to the formation of onion half cups in the nugget [START_REF] Avettand-Fènoël | Experimental investigation of three-dimensional (3-D) material flow pattern in thick dissimilar 2050 friciton stir welds[END_REF]. The thermal diffusivity of AA2050 alloy in T3 state close to 3.15 10 -5 m²/s, which is intermediate between that of Al of 9.7 10 -5 m²/s and that of steel of 1.2-1.5 10 -5 m²/s, suggests that at the nugget root, the temperature is colder than at the stirred surface. Another study about FSW of 25 mm thick AA 6061 alloy has already proved the importance of the nature of the material playing the role of the backing plate on the temperature evolution along the joint depth [START_REF] Upadhyay | Effect of backing plate thermal property on friciton stir welding of 25 mm thick AA6061[END_REF]. The assumption according to which the temperature is reduced along the SSFSW depth during welding is also consistent with thermal modeling results performed in the case of SSFSW of 6.5 mm thick AA7050-T7651 [START_REF] Wu | Stationary shoulder FSW for joining high strength aluminum alloys[END_REF]. Finally, this hypothesis of temperature reduction along the SSFSW joint depth seems to be realistic since the grain size near the stirred surface was shown to be greater than at the plate mid-depth [START_REF] Avettand-Fènoël | Effect of a pre or postweld heat treatment on microstructure and mechanical properties of an AA2050 weld obtained by SSFSW[END_REF]. The lower temperature at nugget root during a pass, together with a possible lower cooling rate could thus explain why phases other than T1 and θ' were formed are detected at joint mid-depth.

In the T8W joint, the T1 and θ' precipitates have coarsened and their fraction has increased in the HAZ, indicating that the material was still in an underaged condition, which is consistent with previous studies [START_REF] Dorin | Quantification and modelling of the microstructure/strength relationship by tailoring the morphological parameters of the T1 phase in an Al-Cu-Li alloy[END_REF][START_REF] Dorin | Quantitative description of the T1 formation kinetics in an Al-Cu-Li alloy using differential scanning calorimetry, small angle X-ray scattering and transmission electron microscopy[END_REF] which have shown that these types of alloys undergo additional precipitation, and especially an increase in thickness, when aged at a temperature above their T8 ageing temperature. These results are also consistent with other FSW studies reported in the literature [7,16]. In the nugget, while it seems clear that the heat and strain due to the SSFSW have dissolved most of the T1 and θ' precipitates (figure 3), the 2 nd pass seems to have affected the 1 st pass zone of the nugget, since some precipitates are present. This suggests that the 1 st pass area has been affected by the heat input of the second pass.

In T3WT8 specimen, the volume fraction of T1 and θ' is more symmetrical from either side of the joint mid-depth (figure 2). The volume fraction of these phases is however more important in the 1 st pass part of the nugget and closer to the stirred surface, because of the heat received during the 2 nd pass, which may have initiated the nucleation of T1 and θ', before their further development during T8 post-weld ageing. It is worth noting that, contrarily to T8W, no coarsening of T1 and θ' is observed in the HAZ of T3WT8 sample (figure 3). This can be explained by the sequence of treatments. Since the precipitates were formed after the welding process, they were not affected by the additional heat during the weld, whereas in the T8W process, the precipitates in the HAZ were already present when it was heat-affected.

To summarize, the post-weld ageing leads to a more homogeneous microstructure along the weld depth except at mid-depth.

According to Hall-Petch relationship, the hardening due to a reduction of the grain size is expected to be significant for aluminum alloys [START_REF]Strengthening Methods in Crystals[END_REF]. Nevertheless, according to a previous paper dealing with the current joints [33], the mean grain size in the nuggets generated by the second pass is rather similar for both joints; actually it evolves from 14.3 (11.9) µm just below the stirred surface to 4.8 (2.9) µm at the nugget bottom for the T8W and T3WT8 samples respectively; the widths of the grains distribution are also roughly similar for both samples [33]; thus the marked differences of hardening between T3W and T3WT8 joints cannot be explained by the very slight differences of grain size, but rather by the different states of precipitation. Actually, the T1 phase is known as the most effective strengthening phase in Al-Cu-Li alloy [4-7] among which AA2050 alloy [START_REF] Lequeu | Aluminum-copper-lithium alloy 2050 developed for medium to thick plate[END_REF]. And figures 2, 3 and 5 obviously show that Tthe distribution maps of the volume fraction of T1 and θ' along the weld depth (figures 2 and 3) are consistent with the hardness maps of the transverse section of the joints in figure 5; the absence of T1 at joint mid-depth in the nugget is linked to a marked softening. The stationary shoulder which is generally used to reduce the thermal gradient along the plate thickness [START_REF] Widener | High rotational speed friction stir welding with a fixed shoulder, 6 th international friction stir welding symposium[END_REF][START_REF] Russell | Recent developments in the stationary shoulder FSW of titatnium alloys[END_REF] does not enable to homogenize the microstructure and the hardness. Indeed, the present case is concerned by SSFSW of an age hardenable alloy whose precipitation state is sensitive to the different temperatures, local strains and strain rates experienced along the plate thickness during welding. This result has already been faced in terms of grain size and microstructure along the joint depth in the case of SSFSW of AA2050 [33], of AA7050 [20] or of AA7010 [START_REF] Sun | Comparison of residual stress distributions in conventional and stationary shoulder high-strength aluminum alloy friction stir welds[END_REF]. These observations can very likely be explained by the use of a slightly conical Triflat pin shape which entails more heat at the stirred surface and by the cold die effect of the mid-thick AA2050 workpiece (which plays the role of backing plate since, in the present case, welding is performed on mid-thickness), which leads to heat loss at the weld root. Another explanation could be also related to the decreasing dislocations densities along the joint depth from the weld top to the nugget root, even if they are low. These dislocations could have been introduced in the nugget by the internal stresses entailed during the cooling stage of welding, in greater amounts at the weld top since the higher temperature peak during welding entails a higher cooling rate at this location.

Kinetics of precipitation in the nugget during artificial ageing at 155°C

During artificial ageing, no precipitation was observed by SAXS in the nugget central zone (at plate mid-thickness) in both samples (figure 6). As previously discussed, a large part of the solute must have been trapped during SSFSW to form coarse Cu rich precipitates sometimes containing Mn, Fe or Mg (figure 4cd to ij) and was not available any more to lead to the precipitation of T1 and θ' during artificial ageing.

In the remaining parts of the nugget, T1 and θ' precipitation kinetics is slower in T8W compared to T3WT8 during artificial ageing (figure 6), which is consistent with DSC results related to T1 precipitation and obtained with a heating rate of 10°C/min [START_REF] Avettand-Fènoël | Effect of a pre or postweld heat treatment on microstructure and mechanical properties of an AA2050 weld obtained by SSFSW[END_REF]. Indeed, in the remaining parts of the nugget in T8W, neither natural ageing nor artificial aging up to 18 h at 155°C leads to significant T1 and θ' precipitation. These results can be explained by the longer duration of ageing in the case of T3WT8 due to the addition of the T8 treatment and further in-situ aging at 155°C. The T8 treatment thus permitted a preliminary precipitation to occur in the T3WT8 nugget, which further develops during artificial ageing at 155°C. This observation highlights the fact that T1 precipitation, which is known to occur at temperatures between ~100 to ~350°C according to a time-temperature-precipitation diagram of an alloy similar to AA2050 [5354], occurs with a very slow kinetics at 155°C in the absence, or very low, density of dislocations [15, 5455], which are the preferential nucleation sites for this phase [5]. In the weld nugget, the low dislocations density can be explained by dynamic recrystallization occurring during welding.

Moreover, the precipitate distribution of T1 and θ' is heterogeneous along the nugget depth (figure 6). The fraction of precipitates reached at the end of the investigated heat treatment increases with the distance to the nugget centre. This observation confirms the DSC results obtained with a heating rate of 10°C/min on the T3WT8 joint sampled at different depths in the nugget get by the second pass [START_REF] Avettand-Fènoël | Effect of a pre or postweld heat treatment on microstructure and mechanical properties of an AA2050 weld obtained by SSFSW[END_REF]. It suggests that there are more nucleation sites of T1 at the top surface than at joint mid-depth before artificial ageing. It seems to confirm once again and as previously explained, that, before artificial ageing, the temperature during SSFSW decreased from the stirred surfaces towards the joint mid-depth. The temperature reduction along the joint depth is all the more important that the plate is thick and that welding was performed on mid-thickness. As aforesaid, the big grains near the stirred surface very likely contain slightly more dislocations entailed by internal stresses during welding cooling stage than those at plate mid-depth, because of higher cooling rate. Near the stirred surface, the greater heat input during welding and the higher density of dislocations introduced during cooling stage of welding have very likely contributed to the formation of more T1 nucleation sites and could thus explain the faster kinetics of T1 precipitation. Previous studies on similar alloys such as AA2198 or AF/C 458 alloy have already shown that increasing stretch levels result in increasing T1 precipitate density [5,47-49] and smaller T1 size [START_REF] Gable | The role of plastic deformation on the competitive microstructural evolution and mechanical properties of a novel Al-Li-Cu-X alloy[END_REF][START_REF] Dorin | Quantification and modelling of the microstructure/strength relationship by tailoring the morphological parameters of the T1 phase in an Al-Cu-Li alloy[END_REF]. The in-situ SAXS experiment for a 20 h holding time thus enables to highlight the heterogeneity of the microstructure in the nugget of T3WT8.

For both joints, the additional ageing treatment at 155°C for 20 h is insufficient to reach the base material microstructure because of the various precipitation states developed in the nuggets following the different thermomechanical histories they experienced during FSW.

It is finally worth noting that the T1 and θ' volume fraction map of the T8W nugget central area after 20 h of artificial ageing at 155°C corresponds to that of T3WT8 nugget central part. This observation suggests that T8 treatment corresponds to an artificial ageing of about 20 h at 155°C. The material in the T8 state does not seem to be more difficult to weld than in the T3 state, although the AA2050 in the T8 state is harder than in the T3 state at room temperature. Indeed, to be welded, the material must reach a homologous temperature of 0.6-0.7, as mentioned above. At this temperature, the dissolution of the T1 strengthening phase occurs in both T3 and T8 state and the material to be welded is soft. The constitutive laws of AA2050 in T3 and T8 state are very likely similar at the welding temperature.

Conclusions

SAXS analyses have evidenced the asymmetry of precipitation along the 15 mm depth of the nuggets generated by the 2 SSFSW passes. This is true for both sequences of processing (T3WT8 and T8W). Because of the 2 passes, the central zone of the nugget experienced different thermomechanical histories along its depth. In fact, the tool movement during the second pass generated some additional strain under the pin as well as extra heat input in the nugget resulting from the 1 st pass leading to a change of its precipitation state. In addition, even after artificial ageing at 155°C for 20 h and whatever the joint, no precipitation of T1 and θ' phases occurred at the soft mid-depth of the nugget zone which experienced the FSW tool movement twice, very likely because of the formation of other Cu-rich precipitates in this zone which trapped the solute required for the precipitation of T1 and θ'.

Although friction stir welding was performed with a stationary shoulder to join a 15 mm thick plate, it did not give rise to a homogenization of the microstructure along the thickness of joint. This microstructural heterogeneity indeed directly results from the precipitation states linked to the thermomechanical history generated during welding.

Aknowledgements

The authors are very thankful to Constellium C-TEC Voreppe Research Centre, France for providing the joints.

  [25], AA2198-T851 [23], AA2219-T6 [26-27], AA6061-T6 [28], 6005A-T6 [29], AA6082 [30], AA7010 [21], AA7050-T7651 [20,22,31],

Figure 1 :

 1 Figure 1: Transverse cross-sections (X-Y plane) of T3WT8 and T8W joints etched with Keller solutions (AS and RS are respective abbreviations for advancing and retreating side). It is worthy to note that the S-line in the nugget is less obvious in micrograph of T8W very likely because of differences of etching duration.

Figure 2 :

 2 Figure 2: SAXS maps showing the 2D distribution of T1/θ' in the X-Y plane for the T3WT8 sample. Compared to figure 1, the images in figure 2 are slightly shifted towards RS.

Figure 3 :

 3 Figure 3: SAXS maps showing the 2D distribution of T1/θ' in the X-Y plane for the T8W sample. Compared to figure 1, the images in figure 3 are slightly shifted towards RS.

Figure 4 :

 4 Figure 4: Examples of bright field transmission electron micrographs (close to <110>Al) in the nugget central zone (X-Z plane) generated by the second pass at 4 mm from the stirred surface of T3WT8 joint (a, b, c), and at 8 mm from the stirred surface of T8W joint (cd). Details of figure bd (de) and linked X-ray maps of Al Kα (ef), Cu Kα (fg), Fe Kα (gh), Mn Kα (hi) and Mg Kα (ij).

Figure 5 :

 5 Figure 5: HV0.05 microhardness maps of the transverse cross-sections (X-Y plane) of T3WT8 and T8W samples. The location of the faying interface corresponds to Y = 0 mm; the values on the X-axis are related to the distance from the stirred top surface during the first pass. The borders of the nugget and the HAZ are delineated in black and refer to their limits deduced from chemical etching in figure 1.

Figure 6 :

 6 Figure 6: SAXS maps showing the evolution of the distribution of the volume fraction T1 and θ' in the central zone of the nugget for both joints in the X-Y plane during isothermal treatment at 155°C. Position at 0 mm corresponds to the central nugget area mid-depth.

Table 1 :

 1 Chemical composition range of the AA2050 alloy.

	Element	Si	Fe	Cu	Mn	Mg	Zr	Li	Ag	Al
	Mass.% <0.08	<0.10	3.2 to	0.2 to	0.2 to	0.06 to	0.7 to	0.2 to
				3.9	0.5	0.6	0.14	1.3	0.7

Data availability

All the data in this manuscript are available upon request to the corresponding author.


Yu Feng

Jibo He

Jean-Philippe Lansberg

Hua-Sheng Shao

Andrii Usachov

Hong-Fei Zhang

Phenomenological NLO analysis of η c production at the LHC in the collider and fixed-target modes

In view of the good agreement between the LHCb prompt-η c data at √ s = 7 and 8 TeV and the NLO colour-singlet model predictions -i.e. the leading v 2 NRQCD contribution-, we provide predictions in the LHCb acceptance for the forthcoming 13 TeV analysis bearing on data taken during the LHC Run2. We also provide predictions for √ s = 115 GeV for proton-hydrogen collisions in the fixed-target mode which could be studied during the LHC Run3. Our predictions are complemented by a full theoretical uncertainty analysis. In addition to cross section predictions, we elaborate on the uncertainties on the p p branching ratio -necessary for data-theory comparisonand discuss other usable branching fractions for future studies.

Introduction

In 2014, LHCb released the first experimental study of prompt-η c hadroproduction at the LHC [START_REF] Aaij | Measurement of the η c (1S ) production cross-section in proton-proton collisions via the decay η c (1S ) → p p[END_REF] at √ s = 7 and 8 TeV. It was found that the cross section measured by LHCb was compatible with a negligible contribution of Colour-Octet (CO) transitions. More quantitatively, this observation combined with Heavy-Quark-Spin Symmetry (HQSS) yielded severe constraints on the corresponding CO transitions at work on J/ψ production [START_REF] Han | η c production at LHC and indications on the understanding of J/ψ production[END_REF][START_REF] Zhang | Impact of η c hadroproduction data on charmonium production and polarization within NRQCD framework[END_REF][START_REF] Butenschoen | η c production at the LHC challenges nonrelativistic-QCD factorization[END_REF][START_REF] Likhoded | Production of η Q meson at LHC[END_REF]. These are so stringent that only one fit [START_REF] Han | η c production at LHC and indications on the understanding of J/ψ production[END_REF] currently survives these constraints at the expense of a slight tension with the CDF polarisation data [START_REF] Abulencia | Polarization of J/ψ and ψ 2S mesons produced in p p collisions at √ s = 1.96-TeV[END_REF] 1 . For reviews on quarkonium production, the reader is referred to Refs. [START_REF] Lansberg | New Observables in Inclusive Production of Quarkonia[END_REF][START_REF] Andronic | Heavy-flavour and quarkonium production in the LHC era: from protonproton to heavy-ion collisions[END_REF][START_REF] Brambilla | Heavy quarkonium: progress, puzzles, and opportunities[END_REF][START_REF] Rapp | Charmonium and bottomonium production in heavy-ion collisions[END_REF][START_REF] Lansberg | On the mechanisms of heavy-quarkonium hadroproduction[END_REF][START_REF] Lansberg | J/ψ, ψ ' and Υ production at hadron colliders: A Review[END_REF][START_REF] Kraemer | Quarkonium production at high-energy colliders[END_REF] In this paper, we provide predictions for prompt-η c hadroproduction at √ s = 13 TeV to further test the compatibility between the Colour-Singlet (CS) contributions and the data and then in turn to refine the constraints on the Long-Distance Matrix Elements (LDMEs) associated with the dominant CO contributions. See [START_REF] Lansberg | η Hadroproduction at Next-to-Leading Order and its Relevance to ψ Production[END_REF] for a recent similar study for the η c case for which forthcoming data will also be invaluable. Since such constraints need to be extracted by taking a proper account of both theoretical and experimental uncertainties, we also elaborate on our knowledge of the branching fractions of the decay channels which can be used by LHCb as well as on the scale, the charm-quark mass and the Parton-Distribution Function (PDF) uncertainties of the CS cross-section predictions.

The structure of the article is as follows. Section 2 is devoted to the discussion on the decay channels. Section 3 explains the theory framework we have used to provide CS NLO predictions and gathers our predictions both for the collider kinematics and for the fixed-target kinematics. Section 4 gathers our conclusion.

Discussion on the decay channels

The decays of non-1 --charmonium states to the experimentally clean di-muon channel are strongly suppressed and hence these states can only be reconstructed using decays to hadrons or their radiative transitions to underlying charmonium states. In this section we discuss possible decay channels to study η c , h c and η c , which cannot be accessed using their decays to µ + µ -or J/ψγ. The known branching fractions [START_REF] Patrignani | Review of particle physics[END_REF] of the decays discussed below are summarised in Tab. 1. Many of these branching fractions can be measured more precisely at Belle, Belle II, BES III, or the super tau-charm experiments. The p p decays of charmonia have been investigated as a possible channel to measure charmonium production at the LHC [START_REF] Barsuk | Investigating charmonium production at LHC with the p pbar final state[END_REF]. The measurement of the η c production at the LHCb experiment has been performed using the η c → p p decay [START_REF] Aaij | Measurement of the η c (1S ) production cross-section in proton-proton collisions via the decay η c (1S ) → p p[END_REF], which demonstrated that the p p final state is powerful to reconstruct the η c state and measure the η c production rate relative to that of the J/ψ, even though the η c hadroproduction rate is measured only for η c with transverse momenta (P T ) larger than 6.5 GeV due to the available trigger bandwidth. Also, this decay is used to study exotic candidates decaying to η c π - [START_REF] Aaij | Evidence for an η c (1S )πresonance in B 0 → η c (1S )K + πdecays[END_REF]. The branching fraction of the η c → p p is known to about 10% precision [START_REF] Patrignani | Review of particle physics[END_REF]. The studies of the η c would benefit from a more precise measurement of B(η c → p p) or B(η c → p p)/B(J/ψ → p p). Branching fractions of χ cJ → p p decays and ψ → p p have been measured to about 3-5% precision. Recently, LHCb has observed the decay η c → p p using a data sample of exclusive B + → p pK + decays [START_REF] Aaij | Observation of η c (2S ) → p p and search for X(3872) → p p decays[END_REF]. Together with the measurement of B(B + → η c K + ) by Belle [START_REF] Kato | Measurements of the absolute branching fractions of B + → X cc K + and B + → D( * )0 π + at Belle[END_REF], the branching fraction of η c → p p is indirectly determined to be about 0.7 × 10 -4 . Therefore, the decay η c → p p is promising for the η c hadroproduction measurement.

B × 10 3 p p φφ φK + K - φπ + π - ΛΛ Ξ + Ξ - Λ(1520)Λ(1520) η c γ p pπ + π - η c 1.
The other promising final state to study prompt production of charmonium is φφ. The 1 - charmonium states are forbidden to decay to φφ. LHCb measured the χ c0,1,2 and η c production in inclusive b-hadron decays using the φφ final state with the first evidence of the η c → φφ decay [START_REF] Aaij | Study of charmonium production in b-hadron decays and first evidence for the decay B 0 s → φφφ[END_REF]. In the latter analysis, a possible problem was highlighted, namely the PDG fit value of B(η c → φφ) differs from the PDG average value [START_REF] Patrignani | Review of particle physics[END_REF] by a factor close to 2. In addition, the ratio of the branching fractions B(η c → φφ)/B(η c → p p) was measured. More measurements are needed to establish a robust value of the B(η c → φφ). Also, due to the evidence of the η c → φφ, this channel is promising to measure the hadroproduction of the η c . Similarly, the φK + K -and the φπ + π -final states could be used.

The branching fractions of charmonium decays to long-lived baryons such as Λ Λ and Ξ + Ξ -are measured for most charmonium states. The reconstruction of these decay channels is challenging for LHCb due to the large lifetimes of these baryons such that they escape the Vertex Locator (VELO), which cause a small reconstruction and trigger efficiency. Decays involving short-lived baryons are reconstructed by LHCb with better efficiency.

The decays χ c0,2 → Λ(1520) Λ(1520) have been observed by the BES III collaboration [START_REF] Besiii Collaboration | Observation of χ cJ decaying into the p pK + Kfinal state[END_REF] while the J/ψ → Λ(1520) Λ(1520) decay is not observed so far. This channel becomes another candidate to measure hadroproduction of charmonium states [23].

The least studied charmonium state is the h c meson, and not many of h c decays have been observed so far. The h c meson is expected to decay to p p, however, the upper limit on the B(h c → p p) reported by the BES III collaboration [START_REF] Ablikim | Search for η c (2S )/h c → pp decays and measurements of the χ cJ → pp branching fractions[END_REF] is more than one order of magnitude smaller than the theoretical prediction [START_REF] Barsuk | Investigating charmonium production at LHC with the p pbar final state[END_REF]. Also, the h c can be measured using its radiative transition h c → η c γ with branching fraction about 50%, which requires reconstruction of the η c state. Recently, LHCb observed the very clean decays χ c1,2 → J/ψµ + µ -, and precisely measured the χ c2 mass and its natural width [START_REF] Aaij | χ c1 and χ c2 Resonance Parameters with the Decays χ c1,c2 → J/ψµ + µ[END_REF]. The h c → η c µ + µ -decay can be searched similarly. Also, the BES III has observed the h c → p pπ + π -decay and measured its branching fraction [START_REF] Ablikim | First observations of h c → hadrons[END_REF] to be (2.89 ± 0.32 ± 0.55) × 10 -3 , which is promising for searches by LHCb.

Framework and results

Framework

The present NLO analysis was performed thanks to the FDC framework [START_REF] Wang | Progress in FDC project[END_REF][START_REF] Gong | Automatic calculation in quarkonium physics[END_REF] 3 which generates the Born, real-emission and virtual contributions, ensures the finiteness of their sum, performs the partonic-phase-space integration and that over the PDFs. As announced, we performed a full study of the scale uncertainty by varying both µ R and µ F about the default value µ 0 = m 2 η c + P 2 T as (µ R , µ F ) = µ 0 × (1, 1; 0.5, 0.5; 2, 2; 0.5, 1; 1, 0.5; 1, 2; 2, 1). We have also studied the mass uncertainty by varying the charm quark mass m c by 0.1 GeV about a default value of 1.5 GeV.

As for the CS LDME, we have taken O η c ( 1 S [1] 0 ) = 0.39 GeV 3 which corresponds to |R(0)| 2 = 0.81 GeV 3 for the radial wave function at the origin. In order to study the impact of the PDF uncertainties at NLO, we have used the CT14 set [START_REF] Dulat | New parton distribution functions from a global analysis of quantum chromodynamics[END_REF] which is included in LHAPDF5 [START_REF] Whalley | The Les Houches accord PDFs (LHAPDF) and LHAGLUE[END_REF]. The corresponding uncertainties follow from the 57 eigensets of CT14.

Results for the collider mode at

√ s = 13 TeV Our predictions at √ s = 13 TeV follow from the expected kinematical range of the forthcoming LHCb study performed on data taken during the Run2 in 2015-2016. They correspond to 2 fb -1 of data at √ s = 13 TeV. We have therefore considered the same rapidity acceptance as that used for the first LHCb analysis [START_REF] Aaij | Measurement of the η c (1S ) production cross-section in proton-proton collisions via the decay η c (1S ) → p p[END_REF], namely 2 < y cms < 4.5 without any additional fiducial cuts on the decay product of the η c . Fig. 1a displays our predictions for the P T -differential prompt-η c cross section at NLO accuracy along with their associated scale, mass and PDF uncertainties. It is clear that the latter are negligible in this energy range as compared to those from the scales. Fig. 1b shows the ratio of the NLO/LO cross sections with the scale and mass uncertainties only and points at a K factor slightly increasing with P T . This is the expected behaviour with leading P T channels opening up at α 4 s . It also shows that the scale uncertainty is as large as 50%, whereas the mass uncertainty is rather on the order of 25% in the P T region which we considered.

Assuming a recorded luminosity of 2 fb -1 , B(η c → p p) = 1.52 × 10 -3 and an efficiency on the order of 2 %, the 100-count limit per GeV correspond to 2 pb and is located around P T 20 GeV. Without any surprise, the increase in the energy should allow LHCb to push their measurements at 13 TeV to slightly larger P T compared to 7 and 8 TeV. Limitations may come from the range where the J/ψ → p p yield is measured as well as from systematical uncertainties. In view of the other branching fractions on Table [START_REF] Aaij | Measurement of the η c (1S ) production cross-section in proton-proton collisions via the decay η c (1S ) → p p[END_REF], let us add that other decays are also within the reach of LHCb measurements.

Results for the fixed-target mode at √ s = 115 GeV

The use of the proton LHC beam in the fixed-target mode has lately be the object of intense investigation both in terms of feasibility and in terms of physics reach, see e.g. . In particular, a wide variety of measurements in different possible implementations were discussed in [START_REF] Hadjidakis | A Fixed-Target Programme at the LHC: Physics Case and Projected Performances for Heavy-Ion, Hadron, Spin and Astroparticle Studies[END_REF]. We will limit ourselves here to a few statements on the kinematics. First, 7 TeV protons impinging fixed targets release a center-of-mass system (cms) energy close to 115 GeV ( √ s = 2E p m N ). Second, the boost between the cms and the laboratory is γ lab cms = √ s/(2m p ) 60 yielding a rapidity shift as large as tanh -1 β lab cms 4.8. As such, the nominal acceptance of the LHCb detector in the cms approximates to -2.8 < y cms < -0.3. Physics wise, the LHCb detector probes backward physics in the fixed-target mode.

Nowadays, the first analysed fixed-target data based on the SMOG LHCb system -initially designed to improve the luminosity determination in LHCb, now used as a low-density-unpolarisedgas target-are coming in [START_REF] Aaij | First Measurement of Charm Production in its Fixed-Target Configuration at the LHC[END_REF][START_REF] Aaij | Measurement of Antiproton Production in pHe Collisions at √ s NN = 110 GeV[END_REF] and confirm that the particle multiplicity in the LHCb detector is such that its performance in the fixed-target mode remains intact. One can thus consider that similar decay channels of the η c as those discussed for the collider mode could be studied if sufficient luminosities can be achieved.

Until now, the LHCb-SMOG statistical samples for J/ψ taken with different noble gases (He, Ar, Ne) remain too small -on the order of hundreds-to expect any η c counts. The situation could significantly get better in the future with proposed SMOG2 system [START_REF] Redaelli | Studies for Future Fixed-Target Experiments at the LHC in the Framework of the CERN Physics Beyond Colliders Study[END_REF][START_REF] Dainese | Physics Beyond Colliders: QCD Working Group Report[END_REF] with achievable yearly luminosities on the order of 10 pb -1 during the LHC Run3. It is however crucial to further constrain NRQCD LDMEs -as we propose here-to have a H target available as opposed as to nuclear -noble gas-targets. For the LHC Run4, yearly luminosities as high as few fb -1 will be within experimental reach as discussed in [START_REF] Hadjidakis | A Fixed-Target Programme at the LHC: Physics Case and Projected Performances for Heavy-Ion, Hadron, Spin and Astroparticle Studies[END_REF].

Fig. 2a displays the P T -differential η c cross section at √ s = 115 GeV in the expected acceptance of LHCb in the fixed-target mode. As above, we separated out the uncertainties from the scale variations (µ F and µ R ), from the charm-quark mass and from the PDFs which are a little larger here since one probes slightly larger x values. As a matter of fact, dedicated rapiditydifferential measurements at very negative y cms could provide specific constraints on the gluon PDFs [START_REF] Hadjidakis | A Fixed-Target Programme at the LHC: Physics Case and Projected Performances for Heavy-Ion, Hadron, Spin and Astroparticle Studies[END_REF][START_REF] Lansberg | Quarkonium Physics at a Fixed-Target Experiment using the LHC Beams[END_REF][START_REF] Brodsky | Physics Opportunities of a Fixed-Target Experiment using the LHC Beams[END_REF]. Fig. 2b shows the ratio of the NLO/LO cross sections with the mass and scale uncertainties only and points at a K factor slightly increasing with P T . This is the expected behaviour with leading P T channels opening up at α 4 s . It also shows that the scale uncertainty is as large as 5 at low energies, whereas that from the mass remains on the order of 25% like at LHC-collider energies. Assuming an integrated luminosity of 10 pb -1 , B(η c → p p) = 1.52 × 10 -3 and an efficiency on 50 %, the one-count limit per 2.5 GeV for dσ/dP T is on the order 0.08 pb, which corresponds according to our results to a P T upper limit of 8.5 ÷ 10 GeV. It precisely happens to be the range accessed at 7 and 8 TeV. With 10 fb -1 , the reach would simply be equivalent to that of the collider mode. We further note that, thanks to the reduced multiplicities in the fixed-target mode, lower P T 's should be accessible. This would allow one to measure the gluon Transverse-Momentum-Dependent functions (TMDs) along the lines of [START_REF] Boer | Polarized gluon studies with charmonium and bottomonium at LHCb and AFTER[END_REF][START_REF] Ma | Transverse momentum dependent factorization for quarkonium production at low transverse momentum[END_REF][START_REF] Signori | Gluon TMDs in quarkonium production[END_REF].

Conclusions and outlook

We have computed the prompt η c -production cross section at one loop accuracy in QCD and in the CSM (LO in v 2 of NRQCD) for the LHCb kinematics in the collider mode at √ s = 13 TeV and in the fixed-target mode at √ s = 115 GeV. In addition, we have provided an up-to-date discussion of the possible decay channels to be used for such studies and performed an original analysis of the theoretical analysis including that from the factorisation and renormalisation scales, from the charm quark mass and from the PDFs.

In addition, let us stress that the understanding and the measurements of η c production go well beyond the determination of NRQCD LDMEs. Its production in proton-nucleus collisions (see [START_REF] Lansberg | Towards an automated tool to evaluate the impact of the nuclear modification of the gluon density on quarkonium, D and B meson production in protonnucleus collisions[END_REF] for predictions of the corresponding nuclear modification factors at LHC energies) can provide complementary means to probe the distribution of gluons inside nuclei along the lines of [START_REF] Kusina | Gluon Shadowing in Heavy-Flavor Production at the LHC[END_REF][START_REF] Kusina | Impact of LHC Heavy-Flavour Data on Nuclear Gluon PDF[END_REF]. In proton-deuteron collisions at extreme x F , it can also give us some handle on the gluon distribution in the deuteron at very large x [START_REF] Brodsky | The gluon and charm content of the deuteron[END_REF].

Figure 1 :

 1 Figure 1: (a) NLO P T -differentical cross section in the LHCb acceptance at 13 TeV. The black (green,red) hatched band denotes the scales (mass, PDF) uncertainties; (b) NLO/LO cross-section ratio as a function of P T where only the scale and mass uncertainties on the NLO cross section are shown.

Figure 2 :

 2 Figure 2: (a) NLO P T differentical cross section in the LHCb acceptance at 115 GeV in the fixed-target mode. The black (green,red) hatched band denotes the scales (mass,PDF) uncertainties; (b) NLO/LO cross-section ratio as a function of P T where only the scale and mass uncertainties on the NLO cross section are shown.

Table 1 :

 1 The branching fractions (×10 3 ) of charmonium decays to hadrons and radiative decays to η c γ.

		52 ± 0.16	1.79 ± 0.20	2.9 ± 1.4	unknown 1.09 ± 0.24 0.90 ± 0.26	-	-	5.3 ± 1.8
	J/ψ	2.12 ± 0.03	forbidden 0.83 ± 0.12 0.87 ± 0.09 1.89 ± 0.08 0.97 ± 0.08	unknown	17 ± 4	6.0 ± 0.5
	χ c0	0.22 ± 0.01	0.80 ± 0.07 0.97 ± 0.25 unknown 0.33 ± 0.02 0.48 ± 0.07	0.31 ± 0.12	forbidden	2.1 ± 0.7
	h c	< 0.15	forbidden	unknown	unknown	unknown	unknown	unknown	510 ± 60	unknown
	χ c1 0.076 ± 0.003 0.42 ± 0.05 0.41 ± 0.15 unknown 0.11 ± 0.01 0.08 ± 0.02	< 0.09	forbidden 0.50 ± 0.19
	χ c2 0.073 ± 0.003 1.06 ± 0.09 1.42 ± 0.29 unknown 0.18 ± 0.02 0.14 ± 0.03	0.46 ± 0.15	forbidden 1.32 ± 0.34
	η c	0.07 2	unknown	unknown	unknown	unknown	unknown	unknown	forbidden unknown
	ψ	0.29 ± 0.01	forbidden 0.07 ± 0.02 0.12 ± 0.03 0.38 ± 0.01 0.29 ± 0.01	unknown	3.4 ± 0.5 0.60 ± 0.04

The recent IHEP analysis where λ φ and λ θφ were computed for the first time at NLO[START_REF] Feng | Remaining parts of the long-standing J/ψ polarization puzzle[END_REF] also does not comply with the η c data.

Indirect determination

The FDC (standing for Feynman Diagram Calculation) package have been developed to automated HEP computations. It is based on the LISP symbolic programming language in order to produce FORTRAN codes. The Lagrangian are formed by the code, following the user requirement, from which are derived the corresponding Feynman rules. The package generates all possible Feynman diagrams contributing to a given process up to one loop in a given model. It can in particular deal quarkonium production within NRQCD. The amplitude of the process are analytically manipulated to generate FORTRAN codes of the squared amplitudes up to one loop. Numerical results for the (differential) cross sections are then computed by performing the phase-space integrals using the phase-space slicing method. We refer to[START_REF] Gong | Automatic calculation in quarkonium physics[END_REF] for explanations relevant to quarkonium-production applications.

Acknowledgements

We thank S. Barsuk for useful discussions. The work of YF, JH, JPL, HSS, HFZ is supported in part by CNRS via the LIA FCPPL. JPL is supported in part by the TMD@NLO IN2P3 project. HSS is supported in part by the LABEX ILP (ANR-11-IDEX-0004-02, ANR-10-LABX-63).