Akinobu Shibata
email: shibata.akinobu@nims.go.jp

Takashi Yonemura

Yuji Momotani

Myeong-Heom Park

Shusaku Takagi

Yazid Madi

Jacques Besson

Nobuhiro Tsuji

Effects of local stress, strain, and hydrogen content on hydrogen-related fracture behavior in low-carbon martensitic steel

Keywords: hydrogen embrittlement, martensite, stress and strain, finite-element simulation, digital

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Recently, the demand for high-strength steel has been increasing from economic and environmental viewpoints. As the susceptibility of high-strength steel, particularly martensitic steel, to hydrogen embrittlement is considerably high, one of the issues in the practical application of high-strength steel is hydrogen embrittlement. Several models have been proposed to account for the hydrogen-related fracture [START_REF] Robertson | Hydrogen embrittlement understood[END_REF][START_REF] Dadfarnia | Recent advances on hydrogen embrittlement of structural materials[END_REF]. However, to date, the underlying mechanism for hydrogen-related fracture is yet to be fully understood.

Low and medium carbon martensitic steels comprise several structural units with different length scale: lath, block, packet, and prior austenite grain [START_REF] Morito | The morphology and crystallography of lath martensite in alloy steels[END_REF]. A lath is a single-crystal martensite plate with a thickness of about 0.2 µm, and boundaries between laths are low-angle boundaries with misorientation angle less than three degrees [START_REF] Morito | The morphology and crystallography of lath martensite in alloy steels[END_REF]. A block consists of many laths having nearly identical crystallographic orientation, and a packet consists of several blocks with nearly the same habit planes. As the name suggests, a prior austenite grain is a region corresponding to a single grain at austenite state. Typical modes of hydrogen-related fracture of martensitic steels are intergranular and quasi-cleavage. Intergranular fracture occurs due to a decrease in the cohesive energy at the boundary by the segregation of hydrogen [START_REF] Yamaguchi | Mobile effect of hydrogen on intergranular decohesion of iron: first-principles calculations[END_REF][START_REF] Solanki | Atomistic investigation of the role of grain boundary structure on hydrogen segregation and embrittlement in α-Fe[END_REF][START_REF] Yamaguchi | Multiscale thermodynamic analysis on hydrogen-induced intergranular cracking in an alloy steel with segregated solutes[END_REF]. In the case of martensitic steels, hydrogen-related intergranular fracture often occurs on prior austenite grain boundaries [START_REF] Banerji | Intergranular fracture in 4340-type steels: effects of impurities and hydrogen[END_REF][START_REF] Craig | The structure of tempered martensite and its susceptibility to hydrogen stress cracking[END_REF][START_REF] Wang | Effect of hydrogen and stress concentration on the notch tensile strength of AISI 4135 steel[END_REF][START_REF] Shibata | Characterization of hydrogen-related fracture behavior in as-quenched low-carbon martensitic steel and tempered medium-carbon martensitic steel[END_REF][START_REF] Shibata | Fracture surface topography analysis of the hydrogen-related fracture propagation process in martensitic steel[END_REF][START_REF] Shibata | Microstructural and crystallographic features of hydrogen-related fracture in lath martensitic steels[END_REF][START_REF] Shibata | Mechanical and microstructural analysis on hydrogen-related fracture in a martensitic steel[END_REF].

On the other hand, quasi-cleavage corresponds to a transgranular fracture occurring on non-typical cleavage planes [START_REF] Lynch | A Fractographic Study on Gaseous Hydrogen embrittlement and liquid-metal embrittlement in a tempered-martensitic steel[END_REF][START_REF] Martin | On the formation and nature of quasi-cleavage fracture surfaces in hydrogen embrittled steels[END_REF]. It is well known that increasing the hydrogen content changes the fracture mode from quasi-cleavage to intergranular even in a given material [START_REF] Hagihara | Delayed fracture using CSRT and hydrogen trapping characteristics of V-bearing high-strength steel[END_REF]. Elucidating the critical factors that determine whether quasi-cleavage fracture or intergranular fracture occurs is one of the most important technical challenges in the research field of hydrogen embrittlement. [START_REF] Takeda | Strain controlled vs stress controlled hydrogen induced fracture in a quenched and tempered steel[END_REF] conducted a pioneering study on the effect of strain and stress on hydrogen-related fracture modes. They studied the hydrogen-related fracture behavior of tempered martensitic steels with different impurity contents. The quasi-cleavage cracking occurred when the impurity segregation was small, while the specimen with a large amount of impurity segregation exhibited intergranular cracking. They proposed that the quasi-cleavage fracture corresponded to strain-controlled Mode II fracture and occurred along the planes of maximum shear stress. On the other hand, the intergranular cracking was stress-controlled decohesion at the prior austenite grain boundaries. However, their study also involved the effect of segregated impurities. Wang et al. [START_REF] Wang | Effect of hydrogen and stress concentration on the notch tensile strength of AISI 4135 steel[END_REF] studied the effect of stress concentration on the hydrogen-related fracture behavior of tempered martensitic steel using notched round bar specimens with different notch root radii. The dominant mode of hydrogen-related fracture was intergranular.

Takeda and McMahon

Their conclusion was that the local stress field (maximum principal stress) and the local hydrogen content controlled the intergranular cracking. Novak et al. [START_REF] Novak | A statistical, physical-based, micro-mechanical model of hydrogen-induced intergranular fracture in steel[END_REF] studied the initiation site of hydrogen-related intergranular cracking in tempered martensitic steel and revealed that the local event of intergranular cracking was stress-controlled as the fracture was initiated ahead of the notch root, typically near the elastic -plastic interface, where the local tensile stress was the highest. We previously reported that the quasi-cleavage fracture occurred along the {011} crystallographic planes in both martensitic and ferritic steels [START_REF] Shibata | Characterization of hydrogen-related fracture behavior in as-quenched low-carbon martensitic steel and tempered medium-carbon martensitic steel[END_REF][START_REF] Shibata | Microstructural and crystallographic features of hydrogen-related fracture in lath martensitic steels[END_REF][START_REF] Shibata | Microstructural and crystallographic features of hydrogen-related crack propagation in low carbon martensitic steel[END_REF][START_REF] Momotani | Effect of strain rate on hydrogen embrittlement in low-carbon martensitic steel[END_REF][START_REF] Okada | Crystallographic feature of hydrogen-related fracture in 2Mn-0.1C ferritic steel[END_REF]. Particularly, the references of [START_REF] Shibata | Microstructural and crystallographic features of hydrogen-related fracture in lath martensitic steels[END_REF][START_REF] Okada | Crystallographic feature of hydrogen-related fracture in 2Mn-0.1C ferritic steel[END_REF] definitively determined the crystallographic orientations of the quasi-cleavage facets by two-surface trace analysis and demonstrated that the quasi-cleavage fractures occurred along {011} planes. Several papers [START_REF] Kim | The nature of quasicleavage fracture in tempered 5.5Ni steel after hydrogen charging[END_REF][START_REF] Kim | The influence of precipitated austenite on hydrogen embrittlement in 5.5Ni steel[END_REF][START_REF] Nagao | The role of hydrogen in hydrogen embrittlement fracture of lath martensitic steel[END_REF][START_REF] Pioszak | Hydrogen environment assisted cracking of modern ultra-high strength martensitic steels[END_REF] proposed that the hydrogen-related quasi-cleavage fracture occurred on lath boundaries and / or block boundaries, because the crystallographic orientation of these boundaries are nearly parallel to {011} planes. However, we previously showed the clear evidence that the quasi-cleavage crack propagated inside the lath, not on the lath boundaries [START_REF] Shibata | Microstructural and crystallographic features of hydrogen-related fracture in lath martensitic steels[END_REF].

Moreover, we confirmed that the hydrogen-related quasi-cleavage fracture occurred along {011} planes even in a low carbon steel with a simple ferritic microstructure and proposed that the fracture on {011} planes was an intrinsic characteristic of hydrogen-related quasi-cleavage fracture in steels having body centered cubic (bcc) phases, not related with the complicated martensite microstructure [START_REF] Okada | Crystallographic feature of hydrogen-related fracture in 2Mn-0.1C ferritic steel[END_REF]. Considering that {011} plane corresponds to slip planes in a bcc structure, plastic deformation would have an important role in quasi-cleavage fracture.

To deepen the understanding of the factors that determine the mode of hydrogen-related fracture, whether intergranular or quasi-cleavage, the present study investigated the hydrogen-related fracture behaviors under stress concentration conditions in as-quenched low-carbon martensite steel with different hydrogen content, and discussed the effects of local stress, strain, and hydrogen content on the mode of hydrogen-related fracture by utilizing an FE simulation.

Experimental Procedure

The steel used in the present study was an Fe-0.2C (mass %) binary alloy. The detailed chemical composition was C: 0.23, Si: < 0.02, Mn: < 0.02, P: < 0.005, S: < 0.0005, O: 0.0010, N: 0.0012, and Fe: bal. (mass %). The as-received steel with a thickness of 12 mm was cold-rolled to 1.7 mm thickness. The cold-rolled sheets were austenitized at 900 °C for 30 min in vacuum, followed by ice-brine quenching and sub-zero cooling in liquid nitrogen. The as-quenched specimens exhibited a fully martensitic structure. The heat-treated specimens were mechanically polished on both sides until a final thickness of 1 mm was reached to remove the decarburized layers formed during the heat treatment. Sheet-type smooth and double-notched tensile test specimens with a gauge length of 10 mm, gauge width of 5 mm, and gauge thickness of 1 mm were prepared by spark wire cutting. As shown in the schematic illustration of Figure 1(a), the notch radii are 1 mm and 0.25 mm, and the un-notched ligament length is 2.5 mm. The stress concentration factors (Kt) of the specimens with notch radii of 1 mm and 0.25 mm are 2.1 and 3.2, respectively.

The double-notched tensile test specimens were cathodically pre-charged with hydrogen in an aqueous solution of 3 % NaCl and 3 g L -1 NH4SCN for 24 h at room temperature. The current density of the hydrogen charging was 0.625 A m -2 or 1 A m -2 . A platinum plate was used as the counter electrode. The diffusible hydrogen contents (HD) of the specimens at the current densities of 0.625 A m -2 and 1 A m -2 , measured by thermal desorption analysis (TDA), were 0.41 mass ppm (22.5 at ppm) and 2.21 mass ppm (121 at ppm), respectively. The desorption profiles are presented in Figure 2 (open circles: 0.625 A m -2 , solid circles: 1 A m -2 ). The calibration of the TDA was performed by using a mixture gas of Ar + H2. Here, HD was defined as the cumulative hydrogen content desorbed below 300 °C, where the first TDA peak ended. Slow strain rate tensile tests at a crosshead speed of 5 × 10 -3 mm min -1 were performed at room temperature in air.

An extensometer was used to measure the displacement of the gauge area. The TDA measurements and the slow strain rate tensile tests were always started 40 min after the completion of the hydrogen charging.

The fracture surfaces of the tensile-tested specimens were observed using scanning electron microscopy (SEM, JEOL: JSM-7800F). To investigate the crack initiation and propagation behaviors, some of the slow strain rate tensile tests were stopped just after reaching the maximum stress. Then, a Ni layer was electroplated onto the tensile-tested specimen to preserve the cracks near the notch root. An aqueous solution of 150 g L -1 Ni2SO4 + 15 g L -1 H3BO4 was used as the electrodeposition solution. Plating was performed at room temperature and at a current density of 100 A m -2 with a plating time of 24 h. Crystal orientation mappings on the mid-thickness sections of the tensile-tested specimens were performed by electron backscattering diffraction (EBSD) using SEM (JEOL JSM-7100F) operated at 15 kV. The specimen surfaces for the EBSD measurements were prepared by electrolytic polishing in a solution of 90 % CH3COOH and 10 % HClO4. The EBSD measurement and analysis were performed with the TSL OIM Data Collection program and the TSL OIM Analysis program, respectively. Elasto-plastic FE simulations were carried out using the commercial Z-set software [START_REF] Besson | Large scale object-oriented finite element code design[END_REF]27] to compute the local stress, strain, and hydrogen distributions around the notch root region. The FE meshes used for simulation are shown in Figure 1(b). Owing to the symmetry, one-eighth of the specimen geometry was meshed with reduced-integration quadratic elements (20 nodes 3D elements with 8 integration points). In order to obtain the material parameters for the FE simulation, the true stress -true strain curve of the uncharged smooth specimen (displacement rate: 5 × 10 -3 mm min -1 ) was fitted according to the following equation:

= + (1 -exp(- + (1 -exp(- Eq. 1
where σ is the true stress, ε is the true plastic strain, σ0, Q1, Q2, b1, and b2 are the fitting parameters.

The fitted values of the parameters were σ0 = 535.63 MPa, Q1 = 296.88 MPa, Q2 = 774.41 MPa, b1 = 47.09, and b2 = 222.18. A Young's modulus of 185.6 GPa (estimated from the stress -strain curve) and Poisson's ratio of 0.3 were used for the FE simulation.

To investigate the local plastic deformation behavior, the DIC technique was used. After hydrogen-charging and electrolytic polishing with the same solution used for the EBSD analysis specimens, SEM images of the specimen surface around the notch root region were taken.

Subsequently, the specimen was tensile-deformed to the maximum stress and SEM images were taken again. The SEM images before and after the tensile deformation were analyzed using the Correlated Solutions VIC-2D software.

Results

Mechanical properties

Fracture surface

The fracture surfaces of the uncharged specimens with Kt We previously reported that the intergranular-like surface with serrated markings originated from the quasi-cleavage fracture propagated in the vicinity of the prior austenite grain boundaries [START_REF] Shibata | Characterization of hydrogen-related fracture behavior in as-quenched low-carbon martensitic steel and tempered medium-carbon martensitic steel[END_REF][START_REF] Shibata | Fracture surface topography analysis of the hydrogen-related fracture propagation process in martensitic steel[END_REF][START_REF] Shibata | Microstructural and crystallographic features of hydrogen-related fracture in lath martensitic steels[END_REF]. Here, we confirmed that the fracture surfaces of all the hydrogen-charged specimens were quite similar to those in Figure 4 and comprised ductile dimple surface, quasi-cleavage surface, and intergranular surface. previous studies reported that hydrogen-related quasi-cleavage fracture occurred on {011} planes [START_REF] Shibata | Characterization of hydrogen-related fracture behavior in as-quenched low-carbon martensitic steel and tempered medium-carbon martensitic steel[END_REF][START_REF] Shibata | Fracture surface topography analysis of the hydrogen-related fracture propagation process in martensitic steel[END_REF][START_REF] Shibata | Microstructural and crystallographic features of hydrogen-related fracture in lath martensitic steels[END_REF][START_REF] Shibata | Microstructural and crystallographic features of hydrogen-related crack propagation in low carbon martensitic steel[END_REF][START_REF] Momotani | Effect of strain rate on hydrogen embrittlement in low-carbon martensitic steel[END_REF][START_REF] Okada | Crystallographic feature of hydrogen-related fracture in 2Mn-0.1C ferritic steel[END_REF]. Although the analysis on Figure 5 was single-surface trace analysis and did not determine the exact orientation of crack plane, we can simply assume that the crack components parallel to {011} plane traces correspond to quasi-cleavage fracture based on our previous studies [START_REF] Shibata | Characterization of hydrogen-related fracture behavior in as-quenched low-carbon martensitic steel and tempered medium-carbon martensitic steel[END_REF][START_REF] Shibata | Fracture surface topography analysis of the hydrogen-related fracture propagation process in martensitic steel[END_REF][START_REF] Shibata | Microstructural and crystallographic features of hydrogen-related fracture in lath martensitic steels[END_REF][START_REF] Shibata | Microstructural and crystallographic features of hydrogen-related crack propagation in low carbon martensitic steel[END_REF][START_REF] Momotani | Effect of strain rate on hydrogen embrittlement in low-carbon martensitic steel[END_REF][START_REF] Okada | Crystallographic feature of hydrogen-related fracture in 2Mn-0.1C ferritic steel[END_REF]. We found that the initial cracks were intergranular-type for both the specimens with Kt = 2.1 and Kt = 3.2. Some of the quasi-cleavage cracks were observed at the regions far away from the notch root surfaces.

FE simulations were performed in order to estimate the local stress, strain, and hydrogen distributions around the notch root regions of the mid-thickness sections. Because clear crack initiations could not be observed before reaching the maximum stress, the applied stress in the FE simulations was the same as the experimentally measured maximum stress (fracture stress). The equations to estimate local hydrogen content have been well documented in previous papers [START_REF] Sofronis | Numerical analysis of hydrogen transport near a blunting crack tip[END_REF][START_REF] Ahn | Modeling of hydrogen-assisted ductile crack propagation in metals and alloys[END_REF], and are briefly summarized as follows. It can be considered that hydrogen at the normal interstitial sites and the reversible trapping sites, such as lattice defects, are always in equilibrium according to the following equation [START_REF] Oriani | The diffusion and trapping of hydrogen in steel[END_REF]:

= exp Eq.2
where θL is the occupancy of the interstitial sites, θT is the occupancy of the trapping sites, WB is the trap binding energy, R is the gas constant (8.31 J mol -1 ), and T is the absolute temperature. The hydrogen atoms at the interstitial sites per unit volume, CL, is expressed as CL = θL β NL, where β is the number of interstitial sites per solvent atom, and NL is the number of solvent atoms per unit volume. NL is given by NA / VM, where NA is Avogadro's number (6.0232 × 10 23 mol -1 ), and VM is the molar volume of the host lattice. Similarly, the hydrogen atoms existing at the trapping sites per unit volume, CT, is expressed as CT = θT α NT (εp), where α is the number of hydrogen atom sites per trap, and NT (εp) is the trap density per unit volume expressed as a function of plastic strain (εp). Under a stress field, hydrogen tends to accumulate in the region where hydrostatic stress (σkk) is large and satisfies the following equilibrium:

= exp Eq.3
where θL 0 is the initial occupancy of the interstitial sites without a stress field, and VH is the partial molar volume of hydrogen. Combining Eqs. 2 and 3, the total atomic concentration of hydrogen, c, can be expressed as:

! = " #" $ = %& ' + ( $ () * $ & Eq.4
Seeger [START_REF] Seeger | On the location of positive muons and solute hydrogen atoms in alpha iron[END_REF] reported that hydrogen was likely to exist at tetrahedral interstitial sites in iron, so that β is equal to 6. Hydrogen might be trapped at lath / block / packet boundaries in martensite structure.

To date, however, there has been no clear evidence that hydrogen is likely to be trapped at these boundaries. Moreover, our previous experiments using hydrogen micro-print technique suggested that hydrogen was not trapped at any boundaries in martensite structure just after hydrogen charging [START_REF] Shibata | Microstructural and crystallographic features of hydrogen-related crack propagation in low carbon martensitic steel[END_REF], and that dislocation movement transported hydrogen to prior austenite grain boundaries during deformation [START_REF] Momotani | Effect of initial dislocation density on hydrogen accumulation behavior in martensitic steel[END_REF]. Considering the fact that hydrogen trapping at boundaries in martensite

structure has yet to be clarified, we simply assumed that most of the hydrogen was trapped by dislocations (WB = 26.8 kJ mol -1 [START_REF] Choo | Thermal analysis of trapped hydrogen in pure iron[END_REF] in Eq.2 and α =1 in Eq.4). According to Taha and Sofronis [START_REF] Taha | A micromechanics approach to the study of hydrogen transport and embrittlement[END_REF], NT is given by:

+ = √ - .
Eq.5

where ρ is the dislocation density, and a is the lattice constant (0.28664 nm of pure iron was used in the present study). From the dislocation density data of the low-carbon martensitic steel measured by neutron diffraction [START_REF] Harjo | Work hardening, dislocation structure, and load partitioning in lath martensite determined by in situ neutron diffraction line profile analysis[END_REF], the relationship between ρ and εp was estimated to be ρ = 3.99872 × 10 15 (1 + 5.159 εp) m -2 . We also assumed that hydrogen diffusion was fast enough so that the local hydrogen content reached the equilibrium value derived from Eq. 4 when the fracture occurred.

However, the total hydrogen content in the specimen obtained using Eq.4 becomes larger than that before applying deformation. According to Wang et al. [START_REF] Wang | Effect of hydrogen and stress concentration on the notch tensile strength of AISI 4135 steel[END_REF], to keep the mass conservation law at the notched area, the hydrogen content obtained by Eq.4 was multiplied by a factor given by:

/ = 0 1 ∑ 3 3 ∑ ( 3 3 0 3
Eq.6

where Vi is the volume of an element, and Hi is the hydrogen content of an element obtained from Eq.4. For the above calculations, the desorption of hydrogen during the tensile test was not taken into account. As shown in Figure 2, the desorption rate of the specimen at a current density of 0.625 A m -2 is almost zero around room temperature, implying that the hydrogen desorption during the tensile test at room temperature was small. When the current density was 1 A m -2 , on the other hand, the desorption rate at room temperature (22.8 °C) is 0.005975 mass ppm min -1 . Simple estimation for hydrogen desorption during the tensile test by using the value suggests that the amount of desorbed hydrogen until reaching the maximum stress is 0.310 mass ppm for the specimen with Kt = 2.1 and 0.191 mass ppm for the specimen with Kt = 3.2. These values are less than approximately 15 % of the total hydrogen content (2.21 mass ppm). Moreover, our previous paper [START_REF] Momotani | Effect of strain rate on hydrogen embrittlement in low-carbon martensitic steel[END_REF] showed that about 10 % of total hydrogen was desorbed from the specimen during the tensile test (alloy composition, strain rate, and fractured strain were almost the same as those in the present study). Based on these facts, the present paper simply estimated local hydrogen concentrations using the Eqs. 1 ~ 6 without considering hydrogen desorption during the tensile tests.

Figure 6 shows the FE simulation results on the mid-thickness sections when HD is 2.21 mass ppm (121 at ppm) for the specimens with (a-f) Kt = 2.1 and (g-l) Kt = 3.2: (a-c, g-i) distribution maps and (d-f, j-l) profiles at the notch root of (a, d, g, j) maximum principal stress (σp), (b, e, h, k) equivalent plastic strain (eq), and (c, f, i, l) local hydrogen content (HL). In addition, the positions of the initial cracks are indicated in each profile. It can be seen that the local hydrogen contents at the notch root regions are higher than the global hydrogen contents (Figures 6 (f,l)). For the specimen with Kt = 2.1 (Figures 6 (a-f)), the peak positions of the maximum principal stress and equivalent plastic strain are the notch root surface. On the other hand, the local hydrogen content is the highest at the position about 50 µm away from the notch root surface. In the case of the specimen with Kt = 3.2 (Figures 6 (g-l)), the peak positions of all the maximum principal stress, equivalent plastic strain, and local hydrogen content correspond to the surface of the notch root. In both the specimens, the plastic strain values are considerably small. We find that the maximum principal stress and the local hydrogen content at the positions of initial intergranular cracks in the specimen with Kt = 2.1 are σP = 1257 ~ 1283 MPa and HL = 143 ~ 144 at ppm, respectively (Figures 6 (d,f)). As shown in Figures 5 (c,d), the intergranular crack was initiated at the notch root surface in the specimen with Kt = 3.2. Though there still remains a possibility that the crack was initiated ahead of the notch and then grew back to the notch surface as deformation proceeded, the highest values of maximum principal stress, plastic strain, and local hydrogen content at the notch-root surface strongly support the crack initiation at the notch root surface. The maximum principal stress and the local hydrogen content at the notch root surface are σP = 1212 MPa and HL = 159 at ppm, respectively (Figures 6 (j,l)). This suggests that the intergranular cracks were initiated under almost the same local stress field and hydrogen content, even though the stress concentration factors were different. the Schmidt factor of the {011} plane corresponding to the deformation band, i.e., parallel to the habit plane / block boundary, is not the maximum among the possible six {011} planes. We confirmed that all of the deformation bands developed nearly parallel to the habit plane / block boundary. Accordingly, we can propose that the concentration of plastic strain along habit planes / block boundaries results in quasi-cleavage cracking. The formation of deformation bands parallel to the habit plane is not a unique characteristic of the hydrogen-charged specimen. We confirmed the same tendency in the uncharged specimen. Previous studies [START_REF] Nambu | Transition in deformation behavior of martensitic steel during large deformation under uniaxial tensile loading[END_REF][START_REF] Michiuchi | Relationship between local deformation behavior and crystallographic features of as-quenched lath martensite during uniaxial tensile deformation[END_REF][START_REF] Morsdorf | Multiple mechanisms of lath martensite plasticity[END_REF] reported that the slip systems, whose slip planes were parallel to habit planes / block boundaries, were preferentially operated in lath martensite structures.

Crack initiation / propagation

The plastic strain values around the notch roots were evaluated in order to confirm the hydrogen-enhanced localized plasticity (HELP) effect [START_REF] Robertson | Hydrogen embrittlement understood[END_REF][START_REF] Birnbaum | Hydrogen-enhanced localized plasticity--a mechanism for hydrogen-related fracture[END_REF][START_REF] Ferreira | Hydrogen effects on the interaction between dislocations[END_REF][START_REF] Robertson | The effect of hydrogen on dislocation dynamics[END_REF]. As the plastic strain distribution determined by DIC was not uniform, as shown in Figure 9, we measured the average plastic strain values along the five parallel lines whose intervals along the tensile direction were 50 µm. Figure 10 shows (a) the profiles of plastic strain values as a function of the distance from the notch root surface (blue: the uncharged specimen, red: the hydrogen-charged specimen, and black: the FE simulation result on the specimen surface), (b, c) DIC analysis results where the positions of the plastic strain value measurements are indicated (b: uncharged specimen, c: hydrogen-charged specimen).

The plastic strain values determined by DIC in the uncharged and the hydrogen-charged specimens are nearly equal to the FE simulation results. We find that the hydrogen-charged specimen exhibits larger plastic strain values compared to the uncharged 20 specimen, although the difference is small. This suggests that hydrogen enhanced the dislocation mobility and increased the plastic strain value under the same stress level. As a result, the DIC analysis confirmed that hydrogen promoted the local plastic deformation, i.e., HELP effect occurred.

Discussion

When the HD was large (2.21 mass ppm (121 at ppm)), the initial cracks were of the intergranular-type at the prior austenite grain boundaries for both the specimens with Kt = 2.1 and [START_REF] Momotani | Effect of strain rate on hydrogen embrittlement in low-carbon martensitic steel[END_REF][START_REF] Momotani | Effect of initial dislocation density on hydrogen accumulation behavior in martensitic steel[END_REF].

Kt = 3.2,
The results suggested that hydrogen tended to accumulate around the prior austenite grain boundaries during deformation, and the high hydrogen content at the prior austenite grain boundary led to intergranular cracking. The present FE simulations used a continuum body and did not consider the accumulation of hydrogen around the prior austenite grain boundaries. We can assume that hydrogen accumulated more locally around the prior austenite grain boundaries, which existed at the position with high hydrogen content in the FE simulations (Figures 6 (c,i)). It is reasonable to consider that the high principal stress and the high hydrogen content can induce hydrogen-related decohesion. Accordingly, we can conclude that the intergranular cracking originated from stress-controlled decohesion at the prior austenite grain boundary.

When the HD was small (0.41 mass ppm (22.5 at ppm)), the quasi-cleavage cracks formed at the surface of the notch root and propagated along the {011} planes (Figure 7). As shown in Figure 8, the maximum principal stress and the local hydrogen content does not exhibit the maximum value at the initiation sites of the quasi-cleavage cracks, i.e., the notch root surface, although these values are relatively high at the notch root surface. It is a common understanding that hydrogen-related decohesion is most likely to occur at the position with the highest principal stress and hydrogen content. The results shown in Figures 7 and8 strongly suggest that the quasi-cleavage cracking was not due to simple decohesion of particular atomic planes and / or boundaries, though several previous papers proposed that quasi-cleavage cracking originated from hydrogen-enhanced decohesion [START_REF] Kim | The nature of quasicleavage fracture in tempered 5.5Ni steel after hydrogen charging[END_REF][START_REF] Kim | The influence of precipitated austenite on hydrogen embrittlement in 5.5Ni steel[END_REF][START_REF] Nagao | The role of hydrogen in hydrogen embrittlement fracture of lath martensitic steel[END_REF][START_REF] Pioszak | Hydrogen environment assisted cracking of modern ultra-high strength martensitic steels[END_REF]. The crystallographic feature of the quasi-cleavage cracks, i.e., {011} crack plane, supports this notion because typical cleavage plane of a bcc crystal is {001} plane. first proposed the HELP effect [START_REF] Birnbaum | Hydrogen-enhanced localized plasticity--a mechanism for hydrogen-related fracture[END_REF], and several studies directly observed the HELP effect using in situ high-voltage transmission electron microscopy deformation experiments [START_REF] Ferreira | Hydrogen effects on the interaction between dislocations[END_REF][START_REF] Robertson | The effect of hydrogen on dislocation dynamics[END_REF]. The DIC analysis, shown in Figure 10, confirms that hydrogen enhanced the local plastic deformation, although the analysis was performed on the free surface of the specimen and the increase in plastic strain value due to the existence of hydrogen was not significant. The first-principles calculations by Matsumoto et al. [START_REF] Matsumoto | First-principles calculation of hydrogen effects on the formation and diffusion of vacancies in alpha Iron: discussion of the hydrogen-enhanced strain-Induced vacancy mechanism[END_REF] suggested that hydrogen reduced the vacancy formation energy and increased the activation energy for vacancy diffusion. Accordingly, one possible explanation for the mechanism of hydrogen-related quasi-cleavage cracking is that hydrogen-enhanced local plastic deformation and hydrogen-induced stabilization of vacancy promote coalescence of micro-voids on the {011} operated slip planes. Morsdof et al. reported that plastic deformation was localized around block boundaries [START_REF] Morsdorf | Multiple mechanisms of lath martensite plasticity[END_REF]. Considering the deformation localization would increase local stress field and local hydrogen concentration around the block boundaries, hydrogen might induce decohesion at block boundaries. Actually, as described above, several papers insisted that the hydrogen-related quasi-cleavage fracture occurred on lath boundaries and / or block boundaries in martensite structure [START_REF] Kim | The nature of quasicleavage fracture in tempered 5.5Ni steel after hydrogen charging[END_REF][START_REF] Kim | The influence of precipitated austenite on hydrogen embrittlement in 5.5Ni steel[END_REF][START_REF] Nagao | The role of hydrogen in hydrogen embrittlement fracture of lath martensitic steel[END_REF][START_REF] Pioszak | Hydrogen environment assisted cracking of modern ultra-high strength martensitic steels[END_REF]. However, we previously showed the clear evidence that the quasi-cleavage crack propagated inside the lath, not on the lath boundaries [START_REF] Shibata | Microstructural and crystallographic features of hydrogen-related fracture in lath martensitic steels[END_REF]. As shown in Figures 7 and8, the present study revealed that the initiation sites of quasi-cleavage cracking correspond to the positions with the largest plastic strain, not likely condition for simple decohesion.

Moreover, one can easily confirm that there are several crack components which are not parallel to but across the block boundaries in Figure 7 (b,d) (though these are two-dimensional observations).

Based on these facts, decohesion of lath and / or block boundaries would not be a common feature of hydrogen-related quasi-cleavage fracture. Compared with the case of the intergranular cracking, the local condition for the initiation of the quasi-cleavage cracking was not simple. The FE simulation results in Figure 8 show that the local values of the maximum principal stress, plastic strain amount, and hydrogen content at the quasi-cleavage crack initiation sites differ depending on the stress concentration factor. Further studies are necessary to understand the local critical condition for the initiation of quasi-cleavage cracking.

Conclusions

In the present study, we investigated the hydrogen-related fracture behavior of specimens with different stress concentration factors through microstructure observation, FE simulation, and DIC analysis. The following conclusions are drawn.

(1) When the HD was large (2.21 mass ppm (121 at ppm)), both the specimens with Kt = 2.1 and Kt = 3.2 were fractured within the apparent elastic strain regime. The EBSD orientation analysis revealed that the crack formed and propagated around the prior austenite grain boundaries.

Through the FE simulations, we found that the crack initiation sites corresponded to the region with high stress and high hydrogen content. Even though the specimen geometries (stress concentration factors) were different, local states, i.e., stress level and hydrogen content, at the crack initiations were almost the same. This indicates that the hydrogen-related intergranular fracture originated from stress-controlled decohesion at the prior austenite grain boundaries.

(2) The specimen with small HD (0.41 mass ppm (22.5 at ppm)) were fractured after exhibiting some plastic deformation. The quasi-cleavage cracks formed at the surface of the notch root and propagated along the {011} plane traces. The FE simulations revealed that the plastic strains were maximum at the quasi-cleavage crack initiation sites. Moreover, we confirmed that hydrogen enhanced local plastic deformation. Accordingly, we concluded that the quasi-cleavage cracking was closely related to the local plastic deformation on the {011} slip planes. As the local values of maximum principal stress, plastic strain amount, and hydrogen content at the initiation sites of the quasi-cleavage cracks were different depending on the stress concentration factor, the critical quantitative condition for the initiation of quasi-cleavage cracking was not simple compared to that of the case of intergranular cracking.

Figure 3

 3 Figure 3 presents the engineering stress -displacement curves of the specimen with (a) Kt = 2.1

= 2 . 1 and

 21 Kt = 3.2 exhibited fully ductile surfaces with dimple patterns (images not shown). The SEM images showing the fracture surfaces of the hydrogen-charged specimens with Kt = 2.1 and HD = 2.21 mass ppm (121 at ppm) are presented in Figure 4. Figure 4 shows (a) a low-magnification SEM image and (b-d) high magnification SEM images. Three types of surface morphologies are observed: ductile dimple surface (Figure 4(b)), quasi-cleavage surface (Figure 4(c)), and macroscopically intergranular surface (Figure 4(d)). The macroscopically intergranular surface was composed of a relatively smooth intergranular surface (indicated by white arrows) and intergranular-like surface with serrated markings (indicated by black arrows).

3. 3 Figure 5

 35 Figure 5 shows (a, c) SEM images and (b, d) EBSD orientation maps around the notch roots: (a,

Figure 7

 7 Figure7shows the crack initiation and propagation behaviors when the HD was small (0.41

Figure 8 1 and

 81 Figure8shows the FE simulation results on the mid-thickness sections when HD is 0.41 mass

as shown in Figure 5 .

 5 We found that the crack initiation sites corresponded to the areas where both the maximum principal stress and the local hydrogen content were high. Moreover, the maximum principal stress and the local hydrogen content at the crack initiation sites for the specimens with Kt = 2.1 and Kt = 3.2 exhibited similar values (σP = 1257 ~ 1283 MPa and HL = 143 ~ 144 at ppm for Kt = 2.1, σP = 1212 MPa, and HL = 159 at ppm for Kt = 3.2). This suggests that the intergranular cracks were initiated under almost the same local stress field and hydrogen content, even though the stress concentration factors of the specimens were different. Wang et al. [9] also proposed that the local stress and hydrogen content were the key factors for hydrogen-related intergranular cracking. Previously, we studied the local hydrogen accumulation behavior of martensitic steels during deformation by utilizing hydrogen micro-printing technique

Figures

7 and 8

 8 demonstrate that the initiation sites of quasi-cleavage cracking correspond to the positions with the largest plastic strain. Based on these results, we can conclude that the quasi-cleavage cracking was closely related to the local plastic deformation on the {011} planes. Birnbaum et al.

Figure captions Figure 1

 captions1 Figure captions

Figure 2

 2 Figure 2 Desorption profiles of the hydrogen-charged specimens at the current densities of 0.625 A

Figure 3

 3 Figure 3 Engineering stress -displacement curves of the specimens with (a) Kt = 2.1 and (b) Kt =

Figure 4

 4 Figure 4 SEM images showing the fracture surfaces of the hydrogen-charged specimen with Kt =

Figure 5

 5 Figure 5 (a, c) SEM images and (b, d) EBSD orientation maps around the notch roots of the

Figure 6 FE

 6 Figure 6 FE simulation results on the mid-thickness sections when HD is 2.21 mass ppm (121 at

Figure 7

 7 Figure 7 (a, c) SEM images and (b, d) EBSD orientation maps around the notch roots of the

Figure 8 FE

 8 Figure8FE simulation results on the mid-thickness sections when HD is 0.41 mass ppm(22.5 at

Figure 9

 9 Figure 9 DIC analysis result of the hydrogen-charged specimen (Kt = 2.1, HD = 0.41 mass ppm

Figure 10 (

 10 Figure 10 (a) Profiles of plastic strain value as a function of the distance from the notch root

Acknowledgements

This study was financially supported by JSPS KAKENHI Grant Number JP15H04158, JP15H05767, JP19H02459, JP20K21083, the Elements Strategy Initiative for Structural Materials


E Di Salvo

Z J Ajaltouni

Parametrizations of the Spin Density Matrix

Keywords: numbers: Nos.: 23.20.En, 23.20.Js, 23.40.Bw, 24.10.-i

de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The spin density matrix (SDM), which was introduced long time ago [1] to describe a mixture of pure spin states, is an essential tool for various aims, like determining the spin and the parity of the resonances [2][3][4][5][6][7][8][9], singling out some exchange mechanisms in the production reactions [8,[10][11][12][13][14][15][16][17][18][19] and finding possible hints to new physics [20][21][22]. It concerns unstable states -to be denoted as R in the following -which are produced in some reactions, usually of the type

a b → R c or R 0 → R d. (1)
The matrix elements of the SDM of R are inferred through the analysis of one of its decay modes. This matrix is Hermitian, non-negative definite and has trace 1. It is characterized by a set of N pure, orthonormal spin states |n -its eigenvectors -such that [1,5]

ρ = N n=1 |n p n n|, 0 ≤ p n ≤ 1, N n=1 p n = 1. (2)
Here the p n , some of which may be zero [2], are the probabilities of the various eigenstates.

If R consists of a single resonance with spin J, one has N = 2J + 1. However, it may happen to cope with an intermediate state that consists of two spins [23][24][25], of the ρ-and ω-resonances [26,27] or of a resonance and the background [8]; in this case, it results N = k (2J k + 1), J k being the value of each spin.

The number of non-zero eigenvalues is defined as the rank of the matrix, which is denoted by r, with r ≤ N . If the mechanism, which gives rise to R, is of the type (1), an upper bound to r can be fixed [3,28].

In a frame at rest with respect to R, the eigenstates |n are not necessarily eigenstates of the operators J 2 and J z ; as an example, for a spin-3/2 resonance, the SDM may be diagonal with respect to the states |3/2, 3/2 , |3/2, 1/2 , a|3/2, -1/2 + b|3/2, -3/2 , and b|3/2, -1/2 -a|3/2, -3/2 , (3) with |a| 2 + |b| 2 = 1. If R involves a single spin, a given eigenstate of ρ can be reduced to an eigenstate of J z by a rotation of the reference frame, but the unitary matrix that diagonalizes the SDM is not a rotation in the general case [START_REF] Leader | Spin in Particle Physics[END_REF].

The SDM is generally characterized by N 2 -1 real parameters. However, if parity is conserved in the production mechanism [START_REF] Daboul | [END_REF]8], this number is reduced, owing to some relations between the matrix elements; moreover, if parity is conserved also in the successive decay of R, it often happens that the imaginary part of ρ cannot be measured [START_REF] Daboul | [END_REF]8,31]; last, if R is produced in a decay, the constraint of angular momentum conservation [20,21,32] has to be accounted for.

Obviously, all constraints that we have exposed above, including non-negativity [5,6,[START_REF] Daboul | [END_REF]8,28,31] and normalization condition, have to be fulfilled by a parametrization of the SDM. A problem, which experimental physicists have to cope with, is to avoid introducing any external bounds on the parameters. Different solutions were proposed in past years [33,[START_REF] Daboul | [END_REF]8]; however, a simple and sufficiently flexible parametrization seems to be still lacking [31]. Moreover, the suggestion of exploiting the rank condition for reducing the number of parameters of the SDM [5,8] needs implementation; this problem is especially important when higher spins or more than one spin are involved. Last, the parametrizations, which were proposed in the past, concern essentially resonances which are produced in strong interactions, whereas, in the last years, one has to do with structures which arise from weak decays [20][21][22]: a systematic study of such cases has not yet been performed.

The aim of the present paper is to fill such gaps. In particular, we consider the SDM both for parity conserving and for parity violating processes; furthermore, we suggest two different parametrizations that automatically satisfy the above mentioned constraints. The first parametrization, which is based on a theorem, is simple and easily adaptable to different situations; moreover we implement and suggest some methods for recognizing possible null eigenstates and for reducing the number of independent parameters. The second parametrization is inspired by an unusual method [28,15,[17][18][19], which takes into account the mechanism that originates the structures we want to study: indeed, in some cases, it may be convenient to parametrize the SDM as [28]

ρ = U ρ (i) U † , (4)
where ρ (i) is the initial density matrix and U a unitary operator that describes the evolution of the reaction. Moreover, we discuss about the possibility of obtaining the elements of the SDM from data, referring in particular to the case when more spins are involved, or when parity is conserved in the production mechanism. Last, we apply our parametrizations to some reactions of interest.

Sect. 2 is dedicated to the first parametrization, for which we prove a preliminary theorem. In Sect. 3, we show how to exploit situations such that r < N , in part by using a second theorem. In Sect. 4, we introduce the second parametrization. Sect. 5 is devoted to a discussion about the extraction of the SDM elements from the differential decay width. Last, we illustrate a few examples in Sect. 6 and draw some conclusions in Sect. 7.

Parametrization of the Spin Density Matrix -I

The first parametrization is based on a theorem, which we state and prove preliminarily.

Theorem

"Consider a N × N Hermitian matrix, ρ, defined with respect to an orthonormal basis,

ρ = N i,j=1 |i ρ ij j|. (5)
A necessary and sufficient condition for it to be non-negative is that -all of its diagonal elements are non-negative and -the Schwarz inequality [START_REF] Daboul | [END_REF]

|ρ ij | 2 ≤ ρ ii ρ jj (6) holds." Proof a)
The necessary condition is a consequence of the features of the characteristic equation of a non-negative definite Hermitian matrix [8]. However, we give a different argument, which is similar to the one proposed by Daboul [START_REF] Daboul | [END_REF]. If ρ is Hermitian and non-negative, we may set

ρ ij = n U * in U jn p n , n U * in U jn = δ ij . (7)
Here U is a unitary matrix3 that diagonalizes ρ and p n ≥ 0 is the n-th eigenvalue of ρ.

Then

ρ ii = n |U in | 2 p n ≥ 0, (8)
which proves the non-negativity of the diagonal elements of ρ. Moreover, Eq. ( 7) suggests to define a set of complex vectors [START_REF] Daboul | [END_REF]:

|V i = n U in √ p n |n . (9)
Then, owing to the first Eq. ( 7), one has ρ ij = V i |V j and the Schwarz inequality for the scalar product implies (6). b) Now suppose that ρ is Hermitian, with non-negative diagonal elements and satisfies the condition (6). Therefore, we may regard each matrix element ρ ij as a scalar product:

ρ ij = W i |W j , (10)
where |W i (i = 1, 2, ... N ) is a set of complex vectors. This set may be fixed in infinitely many ways. Indeed, we may decompose each vector |W i according to the orthonormal basis adopted for defining the elements of the SDM, Eq. ( 5), i. e.,

|W i = N k=1 α k i |k . (11)
Then, Eq. (10) entails the system

ρ ij = N k=1 α k i * α k j , (12)
which is undetermined, as it consists of N 2 real equations and 2N 2 real unknowns. We give in Appendix A one of the infinite solutions to the system. Given any vector

|Y = N i=1 y i |i , (13)
Eq. ( 10) implies

Y |ρ|Y = N i,j=1 y * i W i |W j y j = Z|Z ≥ 0, (14)
where

|Z = N i=1 y i |W i . (15)
This shows the non-negativity of ρ and completes the proof of our theorem.

Parametrization

The results of the previous subsection are now exploited for parametrizing the SDM of the unstable state R, which, as already explained, may consist of more spins.

Since the diagonal elements of the SDM are non-negative, we introduce N real parameters a i , such that

ρ ii = a 2 i and N i=1 a 2 i = N i=1 W i |W i = 1. (16)
Moreover, we observe, analogously to Doncel et al. [START_REF] Doncel | Matrices densité de polarisation[END_REF], that the set of the moduli of the N vectors (9) -which characterize the SDM -can be related to the points of the surface of a hypersphere of unit radius in the N -dimensional Euclidean space R N . Therefore, we introduce generalized spherical coordinates in that space, by means of a number of 'angular' parameters α i . We propose a parametrization for the SDM of R, distinguishing between parity violation and parity conservation in the processes (1).

Parity Violating Processes

We define, in this case, N -1 'angular' parameters:

a 1 = cos α 1 , a 2 = cos α 2 sin α 1 , ... ( 17
) a i = cos α i Π i-1 l=1 sin α l ... a N = Π N -1 l=1 sin α l . (18)
As regards the off-diagonal matrix elements, the Schwarz inequality (6) suggests to set

ρ ij = |a i ||a j | cos γ ij exp(iφ ij ). ( 19
)

Parity Conserving Processes

In this case, we fix a plane, say π, to be identified with the production plane [8] or, in the case of the decay, with the plane which is singled out by the momenta of R 0 and R. Two possible choices are available [START_REF] Chung | Spin Formalisms[END_REF], corresponding to fixing the quantization axis respectively normal to π or lying on it. Here we focus on the latter case, which includes the helicity representation. We shall see two examples in Sect. 6. The matrix elements of the SDM are denoted as ρ JJ mm . Then, fixing the y-axis normally to the π-plane, and defining the reflection operator Π y = P exp(-iπJ y ), where P is the parity operator, one has

Π y |Jm = η exp[-iπ(J -m)]|J -m ; ( 20
)
here η is the intrinsic parity of the state that we consider. Therefore

(Π y ρΠ -1 y ) JJ mm = ηη exp(-iπ∆)ρ J J -m-m , with ∆ = J -J -m + m , (21)
J and J being different spin values and m and m their third components. Parity invariance implies

ρ JJ mm = ηη exp(-iπ∆)ρ J J -m-m . (22)
As regards the parametrization of the diagonal elements of the SDM, we distinguish two cases: a) Odd N , corresponding to an odd number of integer spins:

a 1 = a N = 1 √ 2 cos α 1 , a 2 = a N -1 = 1 √ 2 cos α 2 sin α 1 , ... (23)
a i = = a N -i+1 = 1 √ 2 cos α i Π i-1 l=1 sin α l ... a N /2 = Π N /2 l=1 sin α l , (24)
with N = N -1. b) Even N , in all other cases:

a 1 = a N = 1 √ 2 cos α 1 , a 2 = a N -1 = 1 √ 2 sin α 1 cos α 2 , ... ( 25
)
a i = = a N -i+1 = 1 √ 2 cos α i Π i-1 l=1 sin α l ... a N/2 = a N/2+1 = 1 √ 2 Π N/2 l=1 sin α l . ( 26
)
The off-diagonal elements are parametrized according to Eq. ( 19), taking into account the relationship (22).

If the imaginary part of ρ ij is not measurable, as we shall discuss in Sect. 5, one has to set φ ij = 0 in the parametrization (19); incidentally, we note that the condition ( ρ ij ) 2 ≤ |ρ ii ρ jj |, and therefore the Schwarz inequality -which is necessary to guarantee the non-negativity condition -is automatically fulfilled; moreover, | ρ ij | constitutes a natural lower bound to |ρ ij |, without introducing any external constraint [START_REF] Daboul | [END_REF]. One can also apply the rank condition, according to which

r( ρ) ≤ r( ρ) -r(ρ), (27)
where ρ and ρ are respectively the real and imaginary part of ρ [8].

3 Exploiting the Rank of the SDM

The complete parametrization of the SDM requires, in principle, all of the parameters that we have illustrated in the previous section. However, in some specific situations, considerable simplifications are possible, which are notably useful in the cases of high spins or of more states. Here we examine the case when the rank of the SDM is less than its order, that is, when some of the eigenvalues p n -or equivalently the determinant and some of the principal minors -vanish. Therefore, an important problem is to single out the kernel of the SDM [2, 3, 5, 8]. To accomplish that, one has to proceed according to more steps [8]. At first, one fits the experimental data by means of an order-N SDM, then one determines its eigenvalues, p n , and its eigenvectors, |n . If some of the p n vanish, the N -dimensional space where the SDM acts may be divided into two subspaces, the kernel, of dimension N k < N , and the complementary one, whose basis is constituted by the remaining r = N -N k eigenvectors. As regards the successive steps, we indicate three different methods.

Method 1 Let

|k = N i=1 α ki |i , k = 1, 2, ...N k (28)
be the eigenvectors of the kernel: they are expressed with respect to the basis that has been adopted in Eq. ( 5). The matrix

ρ = U ρU † , (29)
with

U ki = δ ki for 1 ≤ k ≤ r, ( 30
) = α * ki for r + 1 ≤ k ≤ N, (31)
results in

ρ kl = ρ kl = η kl for 1 ≤ k, l ≤ r, 0 otherwise. ( 32
)
Moreover, the r × r sub-matrix η, which is defined by Eq. ( 32), is non-singular; it can be re-parametrized by using the procedure that we have described in the previous section.

Method 2

If the unstable state R has a fixed spin J, a single eigenstate of ρ can be reduced to an eigenstate of J z by means of a rotation of the reference frame [START_REF] Leader | Spin in Particle Physics[END_REF]. Indeed, any normalized vector, say,

|l = m α J m |J, m , such that m |α J m | 2 = 1, (33)
may be re-written as

|l = U [R(φ, θ, 0)]|J, m = exp(-iφJ z ) exp(-iθJ y )|J, m , (34)
where |J, m is a pure spin state and φ and θ suitable values of the azimuthal and polar angle respectively. If |l is an eigenvector that corresponds to a null eigenvalue, the transformed SDM has at least a vanishing row and column, which intersect in the main diagonal.

In this connection, we remark that, if the polarization direction of the resonance may be determined -e. g., through an asymmetry in a weak decay mode -, it is convenient to rotate the reference frame so as to take the quantization axis along such a direction: the transformed SDM is diagonal.

Method 3

The third method for reducing the order of the SDM is based on a simple theorem.

Theorem: "If the Schwarz inequality is saturated for some (i,j)-pair of indices, i = j, i. e.,

|ρ ij | 2 = ρ ii ρ jj , (35)
the rank of the SDM is at least one unit less than its order." Proof We define the 2 × 2 matrix ρ as

ρ11 = ρ ii , ρ22 = ρ jj , ρ12 = ρ * 21 = ρ ij . (36)
The eigenvalues of ρ are 0 and ρ ii + ρ jj ; the corresponding eigenvectors are

|n 1 = N [ √ ρ jj |i - √ ρ ii exp(-iφ ij )|j ] (37)
and

|n 2 = N [ √ ρ ii |i + √ ρ jj exp(-iφ ij )|j ], (38)
where φ ij is the phase of ρ ij and N = (ρ ii + ρ jj ) -1/2 . The matrix ρ is diagonalized by the unitary matrix U such that

U 11 = N √ ρ jj , U 12 = -N √ ρ ii exp(iφ ij ), U 21 = N √ ρ ii , U 22 = N √ ρ jj exp(iφ ij ). ( 39
)
Therefore, defining the N × N unitary matrix U as

U ii = U 11 , U ij = U 12 , U ji = U 21 , U jj = U 22 , U lm = δ lm for l, m = i, j, (40)
it follows that the new SDM ρ = U ρU † has at least a diagonal term which vanishes. But as shown before this implies that ρ ij = ρ ji = 0 for a fixed i and all j and r(ρ) ≤ N -1.

This completes the proof and indicates how to construct, in this case, the unitary matrix for obtaining the reduction of the parameters. This result could be generalized: any null principal minor [8] of the SDM corresponds to at least one zero eigenvalue. In particular, the most trivial principal minor, a zero in the main diagonal, implies the vanishing of the corresponding row and column, as follows automatically from Eqs. ( 16) and (19).

In this connection, it is worth recalling the Eberhard-Good theorem [2,3,5], which provides useful constraints on the spin of a resonance. In the original version [2], one considers reactions of the type (1), such that a boson resonance R is produced; the theorem states that isotropy in the decay of R implies the inequality

J < Q, ( 41
)
where J is the spin of R and Q is the product of the numbers of spin states of the other particles which are involved in the reaction. Successively, Peshkin [3] established, for any J, the more restrictive inequality

(2J + 1) -1 ≤ r -1 ≤ [Trρ 2 ] -1 . (42)
Last, Minnaert [5] obtained even more stringent conditions: if r < 2J + 1, 2J + 1r linear combinations of the quantities Trρ k (k = 1,2, ... 2J + 1) must vanish.

Parametrization of the Spin Density Matrix -II

The procedure, that we have just described, is especially useful in cases when it is difficult to determine a priori an upper bound to the rank (for example, in inclusive reactions [START_REF] Adam | [END_REF]37]), or when this bound is greater than the order of the SDM [28]. If the bound is less than the order, or if the spin structures of the initial and final state are particularly simple [13], it may be sometimes convenient to use an alternative parametrization, based on Eq. ( 4) [18,28,13]. Indeed, in a process of the type (1), this equation yields for the

SDM of R ρ kk = m ll U m kl ρ (i) ll U m * k l . (43)
Here

U m kl = 1 N u A m kl , N 2 u = m kl |A m kl | 2 (44)
and A m kl are the amplitudes of the process; m denotes the spin quantum number of the unobserved final particle (c or d) and l and l indicate either the spin quantum number of R 0 or the pair (l a , l b ) of the initial particles a and b.

If the number of independent amplitudes is N , we parametrize them as b s exp(iφ s ), s = 1,2, ... N , with φ s real numbers, φ 1 = 0 and

b 1 = cos β 1 , b 2 = sin β 1 cos β 2 , ... ( 45
) b m = cos β m Π m-1 l=1 sin β l ... b N = Π N -1 l=1 sin β l . (46)
Then the amplitudes are characterized by 2( N -1) real parameters, to which one has to add those of the initial SDM. Obviously, if parity is conserved, one has to take into account the corresponding symmetry relations.

In order to see whether this method is more convenient than the previous one, the overall number of such parameters has to be compared with the upper bound to the rank of the density matrix of R; this bound is r i × r c(d) , where r i(c,d) are respectively the ranks of the initial state and of c or d. In Sect. 6, we shall compare the two methods in some cases.

Discussion

The number of independent parameters, that one can extract from the normalized differential decay width, is determined by the moments, i. e.,

1 Γ d 2 Γ d cos θdφ = Jm L=0 L M =-L 2L + 1 4π H(L, M )D L * M 0 (θ, φ). (47)
Here the D are the Wigner rotation functions and J m is the maximum spin of R. Moreover, one has H(0, 0) = 1. The other moments can be extracted either by means of a best fit or by exploiting the orthogonality of the D L M 0 functions with different L or M . If R includes only one spin, the number of moments equals the number of parameters of the SDM that can be determined; these are related to the moments by a linear system [8,[START_REF] Chung | Spin Formalisms[END_REF]. Otherwise, the best we can extract are a number of matrix elements and some linear combinations of the remaining ones [8], unless one can reduce the number of the independent parameters of the SDM, by means of some assumption.

If parity invariance holds both in the production and in the decay of R, the imaginary parts of the elements ρ ij of the SDM turn out to be measurable only under very particular conditions, as shown in Appendix B:

-R includes more spins and -there are at least two independent decay amplitudes, with a non-trivial relative phase, owing to T -odd final-state interactions [38].

Such conditions are realized, for example, in the low-energy scattering p p → Λ Λ [13], which involves a few partial waves; the corresponding decay amplitudes have different phases, owing to the T -odd (spin-orbit) term of the interaction between the final hadrons.

Examples

Here we apply our parametrizations to some concrete situations, either with parity conservation or with parity violation.

We have employed 6 parameters in all, one less than required in ref. 39, which is cited in the LHCb analysis of the decay [40]. It is worth noting that, according to the discussion of the previous section, all such parameters can be determined, in principle, by analyzing, e. g., the decay J/ψ → µ + µ -.

Incidentally, the constraint (55) holds true for any vector boson resonance V that comes from a decay of the type

f 1 → f 2 V, (59)
where f 1 and f 2 have spin 1/2. A similar simplification occurs, e. g., for the SDM of ∆(1232) in the decay

∆(1620)1/2 -→ ∆(1232)3/2 + π, (60)
for which parity invariance must be taken into account. Last, in the decay Λ b → Λ J/ψ, a correlation between the decay products of the two final particles may be derived, starting from the direct product ρ Λ ⊗ ρ J/ψ of the density matrices of such particles. One can proceed similarly to ref. 41, where the correlation between the two charged leptons l + and l -, that result from

e + e -→ t t → (b l + ν l ) ( b l -νl ), (61)
is calculated to O(α S ).

t → bW → bτ ν τ

In this decay, which is of the type (59), we employ the alternative method that we have described in Sect. 4. We consider, in this case, the SDM of a W -boson which is emitted in a given direction, adopting a frame at rest with respect to the top quark. It reads as [22]

ρ µµ (θ, φ) = 1 4π λ b λµ ρ ΛΛ (θ, φ)b * λµ . (62)
Here Λ = µλ is the spin component of t along the W momentum and µ and λ the helicities of, respectively, W and the b-quark; moreover,

ρ ΛΛ (θ, φ) = δ ΛΛ + σ i ΛΛ P i , (63)
P ≡ P (cos θ, sin θ cos φ, sin θ sin φ); (64)
P is the top quark polarization, 0 ≤ P ≤ 1, while θ and φ are, respectively, the polar and the azimuthal angle of the momentum of the W -boson. Last, b λµ are the reduced decay amplitudes of the t → W b decay:

b λµ = N -1/2 A λµ , N = |A +,1 | 2 + |A +,0 | 2 + |A -,0 | 2 + |A -,-1 | 2 . ( 65
)
This suggests the parametrization

b +,1 = | cos β 1 |, (66) b +,0 = | sin β 1 cos β 2 | exp(iϕ 10 ) (67) b -,0 = | sin β 1 sin β 2 cos β 3 | exp(iϕ -10 ), (68) b -,-1 = | sin β 1 sin β 2 sin β 3 |. (69)
Moreover, we set

P = cos 2 β 0 . (70)
We have assumed the phases of the amplitudes b ±,±1 to be zero, because, in our case, they can interfere only with b ±,0 respectively. The parametrization of the W -boson SDM results in ρ 1-1 = 0 and ρ11 = cos

2 β 1 (1 + cos 2 β 0 cos θ), ρ00 = sin 2 β 1 (cos 2 β 2 + sin 2 β 2 cos 2 β 3 ), (71) ρ-1-1 = sin 2 β 1 sin 2 β 2 sin 2 β 3 (1 -cos 2 β 0 cos θ), (72) ρ10
= | cos β 1 sin β 1 cos β 2 |[1 + cos 2 β 0 sin θ exp(-iφ)] exp(iϕ 10 ), (73) ρ
-10 = | sin β 1 sin β 2 |(| cos β 3 | + | sin β 3 |)[1 + cos 2 β 0 sin θ exp(iφ)] exp(iϕ -10 ), (74)
having set ρ = 4πρ. As before, the remaining SDM elements are deduced from the hermiticity condition.

It is worth noting that this procedure -whose parameters, again, do not need any bound -could also be applied to the vector boson V in a decay of the type (59); it has the advantage of relating the polarization of the W to that of the parent state.

The SDM of the W -boson in the semi-leptonic decay of the top quark was parametrized also by other authors a few years ago [21,42]. Furthermore, in ref. 42, also the more general quasi-three-body decay t → X b lν l is analyzed, in order to search for possible hints to new physics; in particular, two of the parameters, which are introduced, result to be quite sensitive to contributions beyond the standard model.

Λ

b → Λ c (W * , H * ) → Λ c τ ν τ
The decay involves a virtual W , which includes a scalar component. For decays of this type, a complete list of the 16 matrix elements of the SDM of the virtual W is given in ref. 43. Furthermore, in the leptonic decays of the Higgs boson, the scalar component of the virtual W and Z is added in a gauge invariant manner [44]. See also ref. 22 and refs. 55 to 59 therein.

In the specific decay that we consider, the independent (reduced) decay amplitudes are 8:

b 1 +,1 = | cos β 1 |, (75)
b 1 +,0 = | sin β 1 cos β 2 | exp(iϕ 1 +,0 ), b 1 -,0 = | sin β 1 sin β 2 cos β 3 | exp(iϕ 1 -,0 ), (76) b 1 -,-1 = | sin β 1 sin β 2 sin β 3 cos β 4 |, b t +,0 = 4 k=1 | sin β k cos β 5 | exp(iϕ t +,0 ), ( 77
) b t -,0 = 5 k=1 | sin β k cos β 6 | exp(iϕ t -,0 ), b 0 +,0 = 6 k=1 | sin β k cos β 7 | exp(iϕ 0 +,0 ), ( 78
) b 0 -,0 = 7 k=1 | sin β k | exp(iϕ 0 -,0 ). ( 79
)
Here, the possible new physics is described by the amplitudes b 0 ±,0 . Moreover, analogously to the previous case, we have assumed equal to 0 the phases of b 1 ±,±1 . Then, following the same procedure and using the same notations as in the previous subsection, we find that the matrix elements of ρ read as

ρ1 1 1-1 = 0, ρ 1 1 ±1±1 = ξ ±1 (1 ± P z ), ( 80
)
ρJJ 0 0 = η JJ 00 + ∆η JJ 0 0 P z , ρ 1J ±10 = η 1 J ±1 0 P ∓ . (81)
Here J and J run over 1, t, 0; moreover, P z = cos 2 β 0 cos θ, P ± = cos 2 β 0 sin θ exp(±iφ); (82) last,

ξ ±1 = |b 1 ±,±1 | 2 , η JJ 0 0 = b J +,0 b J * +,0 + b J -,0 b J * -,0 , (83)
∆η

JJ 00 = b J -,0 b J * -,0 -b J +,0 b J * +,0 , η 1J ±10 = b 1 ±,±1 b J * ±,0 . (84)
In this case, we have used 14 parameters in all, whereas the general treatment would require 24 parameters. But according to the considerations of Sect. 5 the number of independent parameters, that can be inferred from the differential decay width, is 8; therefore, the best fit to the data, if performed by using the above parametrization, would present some ambiguities. However, in this case, some parameters can be fixed by inserting the standard model predictions [20,21,42,45]. In particular, in a previous paper [22], we showed the relationship between the tensor, that is usually employed to describe a semi-leptonic decay, and the non-covariant SDM. If at least one of the amplitudes b 0 ±,0 is non-zero, with a non-trivial phase, it causes a T-odd component for the Λ c polarization [46].

Conclusions

We have proposed two methods for parametrizing the SDM of an unstable stateconsisting of one or more spins -that is produced in various reactions. Both methods satisfy automatically all of the numerous non-negativity conditions [START_REF] Daboul | [END_REF]5] and are adaptable to the constraints imposed by parity [8] and angular momentum conservation. The first method is based on a theorem that we have proved and it may be applied in a simple and flexible way. Moreover, we show how to improve the fit to the data when the rank of the SDM is less than its order; we do this, either by implementing suggestions by other authors [5,28], or with the help of a second theorem, which we have proved as well.

The second method is a variant of previous parametrizations [3,28] and is particularly suitable under especial conditions, e. g., when the structure to be analyzed derives from some decay.

We have discussed about the possibility of inferring the SDM from the differential decay width. In particular, we have examined the case where a given intermediate state is generated and decays according to strong or electromagnetic interactions, showing that, under very particular circumstances, the imaginary part of the SDM can be measured.

Last, we have illustrated some applications; two of them, which concern the decays t → b W [20,21] and Λ b → Λ c l + ν l [47][48][49][50]22], are very interesting from the viewpoint of the search for new physics beyond the standard model.

As a conclusion, we observe that our suggestions appear efficient for situations where higher spins or newly discovered structures [51] are involved. therefore η is a r × r Hermitian, non-singular matrix, to which we may apply the method described above. We find a set of linearly independent complex vectors |w l , l = 1, 2,...r.

Defining

|W i = |w i for i ≤ r and 0 for r < i ≤ N, (A. 14) the transformed vectors

|W i = U ij |W j (A.15)
give the solution to the system (A.1). This completes the proof.

Appendix B

Here we discuss about the measurability of the imaginary parts of the elements of the spin density matrix (SDM) of an unstable state R that is produced and decays according to parity conserving interactions. We assume this state to include more spins and to have a two-body decay: R → a b. Therefore, under the conditions that we have assumed, the imaginary part of the SDM can be measured only if more spins are involved and at least one of the relative phases between the decay amplitudes is non-trivial.

(B. 1 )

 1 Adopting the helicity formalism, the normalized differential decay width reads as 1 Γd 2 Γ d cos θdφ = JJ mm λaλ b C(J, J )ρ JJ mm D J * mλ (φ, θ, 0)D J m λ (φ, θ, 0)f J λaλ b f J * λaλ b . (B.2)Here C(J, J ) = 1/4π[(2J + 1)(2J + 1)] 1/2 , ρ JJ mm = J, m|ρ|J m is an element of the SDM, D is the Wigner rotation function and amplitudes, with λ = λ aλ b . Parity conservation impliesρ JJ mm = ηη exp(-iπ∆)ρ J-J -m-m , (B.4) ∆ = J -Jm + m , (B.5) f J -λa-λ b f J * -λa-λ b = ηη exp[-iπ(J -J )]f J λaλ b f J * λaλ b , (B.6)as follows from Eqs. (20) to(22) in the text. Moreover,D J mλ (φ, θ, 0) = exp(-imφ)d J mλ (θ), d J -m-λ (θ) = (-) m-λ d J mλ (θ). (B.7)Therefore we may re-write Eq. (B.2) as 1 Γd 2 Γ d cos θdφ = JJ mm λaλ b C(J, J ){exp[-i(mm )φ] + (-) exp[i(mm )φ]} ×ρ JJ mm d J mλ (θ)d J m λ (θ)f J λaλ b f J * λaλ b , (B.8)where = ∆ + J -J + m + m -2λ = 2(J -J + mλ) and the primes in the sums indicate that they are limited to non-negative values of the indices. But is an even number, moreover only ρ JJ mm and (f J λaλ b f J * λaλ b ) are odd under the simultaneous exchange (J, m) ↔ (J , m ), the other terms of Eq. (B.8) being even. Therefore we have 1 Γd 2 Γ d cos θdφ = JJ mm λaλ b C(J, J ) cos[(mm )φ]d J mλ (θ)d J m λ (θ)F JJ mm ,λaλ b , (B.9) with F JJ mm ,λaλ b = ρ JJ mm (f J λaλ b f J * λaλ b )ρ JJ mm (f J λaλ b f J * λaλ b ). (B.10)

As observed by Daboul[START_REF] Daboul | [END_REF], such a matrix is defined up to some given phases, U jn → U jn exp(iζ n )

As a byproduct, it is worth noting that, in this case, all α i i are strictly positive.

Acknowledgments

The authors are thankful to their friend Flavio Fontanelli for useful and stimulating discussions.

6.1 Two Spins: 1 -and 0 ∓ -Parity Conservation

We consider the two cases, in order to simulate a resonance-background interference in strong decays of vector mesons; these were already considered many years ago [8], but with a different approach. One has to take into account the relation (22). a) In the case that R includes the states 1 -and 0 -, we have η = η, therefore ρ JJ mm = (-) J-J -m+m ρ J J -m-m ; (48) this entails the following parametrization:

The remaining terms can be deduced from the previous expressions by taking into account the Hermitian character of ρ. Therefore, we have 7 parameters in all. b) Consider now the case when R consists of the 1 -and 0 + states. Eqs. ( 49) to (51) hold still true, whereas

Therefore, 2 more parameters are needed.

It is worth mentioning that also the SDM of an off-shell spin-1 particle is 4 × 4, owing to its scalar component; however, this case concerns essentially weak decays and will be illustrated in some detail in Subsect. 6.4.

Λ b → Λ J/ψ

We consider the SDM of the J/ψ-resonance. In this case, parity is not conserved in the Λ b decay, but the constraint of the angular momentum conservation has to be considered. Indeed, adopting the helicity formalism, one has

since the third component of the angular momentum of Λ b along the J/ψ momentum assumes a fixed value, either +1/2 or -1/2, but not both simultaneously. The other elements read as

Appendix A

We find an explicit solution to the N complex vectors |W i which appear in Eq. ( 10), i. e.,

This is possible owing to the Schwarz inequality, that we have assumed. The data of the system (A.1) consist of the elements of the Hermitian matrix ρ, defined with respect to an orthonormal basis |k and such that

We assume expansions of the type

for any vector |W i , i = 1, 2, ..N ; furthermore, we establish all of the α i i to be real and non-negative. Then, limiting ourselves to i ≥ j, Eq. (A.1) yields

We show that this equation uniquely fixes all of the coefficients α k i of the expansions (A.3). 1) Assume, at first, that the rank of ρ is equal to N , which implies that all principal minors of the matrix are positive, in particular, ρ ii > 0. Then we prove our statement by induction.

a) For i =1, Eq. (A.4) reads as

which defines |W 1 through Eq. (A.3).

b) Suppose Eqs. (A.4) to be solvable with respect to α k j for all k ≤ j and all j ≤ i < N . This amounts to asserting that all vectors |W j , j ≤ i, (A.6) have been determined. Now we prove that the vector

can be uniquely deduced from the system

a complex system of i + 1 equations, with i complex unknowns, α k i+1 , 1 ≤ k ≤ i, and a real one, α i+1 i+1 . To this end, preliminarily, we show that the i × i ('triangular') matrix A, such that

is non-singular. Consider the i × i submatrix ρ, whose matrix elements ρlm coincide with those of ρ for l, m ≤ i. Eq. (A.1) implies

The assumption of non-singularity of ρ implies the same for ρ, therefore the vectors (A.6) constitute a basis for ρ and are linearly independent. This in turn entails the non-singularity of A 4 . But the system (A.8) can be split into a linear subsystem with i equations and a non-linear equation:

The non-singular character of A allows to solve the linear subsystem with respect to α k i+1 , k = 1, 2, ..i. The solution can be inserted into the non-linear equation, which can be solved with respect to α i+1 i+1 . This completes the proof in the case of non-singular ρ. 2) If the rank of ρ, say r, is less than N , we perform the transformation (29), i. e. ρ = U ρU † .

(A.12)

The result is

 Proposed two methods for parametrizing the SDM of an unstable stateconsisting of one or more spins -that is produced in various reactions. Both methods satisfy automatically all of the numerous nonnegativity conditions and are adaptable to the constraints imposed by parity and angular momentum conservation.

 First method  theorem; appliable in a simple and flexible way. Improvements of the fit to the data when the rank of the SDM is less than its order. Added note on Eberhard-Good theorem.

 Second method: a variant of previous parametrizations; particularly suitable under especial conditions, e. g. , when the structure to be analyzed derives from some decay.

 examined the case where a given intermediate state is generated and decays according to strong or electromagnetic interactions: under very particular circumstances, the imaginary part of the SDM can be measured.

 Illustrated some applications; two of them are very interesting from the viewpoint of the search for new physics beyond the standard model.

Ziad J. Ajaltouni: stress on the importance of the spin density matrix and of non-covariant formalism in data analysis; remarks on the feasibility of an experiment and on the measurability of some parameters.

Declaration of Interest Statement

The authors have been collaborating since 2008 on phenomenology of weak decays and methods of data analysis. The spin density matrix is an essential tool to that aim.


Shenle Pan
email: shenle.pan@mines-paristech.fr

Damien Trentesaux
email: damien.trentesaux@uphf.fr

Duncan Mcfarlane

Benoit Montreuil
email: benoit.montreuil@isye.gatech.edu

Eric Ballot
email: eric.ballot@mines-paristech.fr

George Q Huang
email: gqhuang@hku.hk

Digital interoperability and transformation in logistics and supply chain management: Editorial

Keywords: Digital interoperability, Digital transformation, Logistics, Supply Chain Management, Editorial, Special issue

scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Context

To surf on the wave of digitalization, companies are urging to improve digital interoperability through the digital transformation process. Digital interoperability is the ability to achieve quick, seamless, secure, and reliable data and information exchange between companies [START_REF] Pan | Digital interoperability in logistics and supply chain management: state-of-the-art and research avenues towards Physical Internet[END_REF]. The ability is of particular importance in the area of logistics and supply chain management (LSCM) which stresses the importance of cross-organizational and crossfunctional collaboration for strategic planning and operation management. Although digital transformation is well underway in this area, companies are still struggling to find efficient and effective solutions and approaches to enhance the digital interoperability, especially at large-scale level [START_REF] Leal | Interoperability assessment: A systematic literature review[END_REF].

The practical need for enhancing digital interoperability in LSCM has been particularly witnessed in the COVID-19 pandemic. A large number of issues regarding resilience and viability of the contemporary supply chain have been highlighted, most of which are pointedly related to the problem of digital interoperability, such as supplier's information viability and sharing, logistics service provider's operational information availability, end-toend supply chain mapping and monitoring. At the beginning of the outbreak in early 2020, these issues were even more obvious to the companies (or those having suppliers) located in the quarantined areas. This can be attributed to the presence of information silos, i.e., the isolated information and data owned by individual companies or systems. Due to the lack of fast and reliable communication among the silos, information sharing is very limited. One of the goals of digital interoperability is to interconnect these information silos such that information can be exchanged in efficient and effective way with privacy preservation.

On the other hand, recent research has suggested the need to consider not just supply chains but supply networks. New relevant paradigms include intertwined supply networks [START_REF] Ivanov | Viability of intertwined supply networks: extending the supply chain resilience angles towards survivability. A position paper motivated by COVID-19 outbreak[END_REF] and Physical Internet [START_REF] Montreuil | Toward a Physical Internet: meeting the global logistics sustainability grand challenge[END_REF][START_REF] Ballot | The Physical Internet: The Network of Logistics Networks[END_REF], which advocate horizontal and vertical collaborations based on the interconnection of the existing logistics networks for sharing the logistics resources and services involved. Likewise, in accordance with the Industry 4.0 principles such as plug-and-play and the systems of systems, recent approaches used in digital logistics systems modeling also put emphasis on the interconnectivity of systems (including objects, information entities, etc.). Among them, for example, Cyber-physical systems, semantics and ontology, and digital twins are the most investigated recently [START_REF] Derigent | Industry 4.0: contributions of holonic manufacturing control architectures and future challenges[END_REF]. For all these new approaches and paradigms, digital interoperability is a key to success that makes company-to-company, network-tonetwork, or system-to-system communication possible and reliable.

It is in this context that digital interoperability has gained increasing attention in LSCM, from both researchers and practitioners in the area. Considering the recent challenges placed on today's logistics systems, as well as the new opportunities empowered by the disruptive technologies, new research questions and issues that merit more attention are put forward.

The current research focus

This Special Issue aims to advance the research on digital interoperability in LSCM, by collecting the high-quality and original research works from both researchers and practitioners in the area, and by raising new research questions. The scope is wide enough to incorporate the most relevant contributions including development and application of concepts and solutions, disruptive technologies and techniques investigation, state-of-the-art or survey studies, etc.

The Special Issue particularly encourages the cutting-edge solutions that foster logistics and supply chain collaboration at large-scale level. The solutions and the related issues that are investigated by the papers included in this Special Issues are summarized in Table 1. The solutions can be grouped in four main categories that are hereinafter discussed, as displayed in Figure 1.

Title Authors Solution

An approach and decision support tool for forming Industry 4.0 supply chain collaborations Digital platform: Digital platform can be seen an online platform that consists of information systems, interfaces and engines for the users to collect, exchange and search information. It could be web-based or mobile application-based marketplaces for BtoB, BtoC or CtoC markets (B for business, C for consumer). As per its advantages especially for ecosystem development, digital platform has been considered as a powerful tool of digital transformation, which also assists the interoperability among companies connected to the ecosystem. However, several issues must be dealt with in order to ensure the effectiveness and efficiency, for example, data format and integration, privacy preservation, information system architecture, searching and matching engine and algorithms design. These issues are particularly crucial for large-scale multi-party supply chain and logistic collaboration, as demonstrated by [START_REF] Cisneros-Cabrera | An approach and decision support tool for forming Industry 4.0 supply chain collaborations[END_REF]. From a practical point of view, the authors design a digital platform as a decision making tool for matching demands and supplies from companies in manufacturing. On the demand side, companies post Call for Tenders with specific requirements; on the supply side, suppliers publish their capabilities available. As the platform is ontology based, the devised matching algorithm is able to efficiently and effectively match the demands and supplies. The interoperability is therefore enhanced for multi-party at large-scale.

Data-driven solution design, planning, modeling and control: The decision-making processes in LSCM has been conventionally and primarily supported by Operations Management & Operations Research (OM/OR) fields. Moreover, purely data-driven approaches have also shown the great potential to assist in the processes, such as using real-time information for dynamic planning, exploiting historical data for forecasting, etc. This can be considered as a reorientation of some of the traditionally mathematical approaches to OM/OR. For example, [START_REF] Yee | Synchromodal transportation planning using travel time information[END_REF] suggest to use cargos' real-time travel information in synchromodal transportation planning. The results denote that real-time information may help optimize modal choice by minimizing the total transportation and overtime delivery costs. Likewise, [START_REF] Shi | Digital connectivity in an innovative joint distribution system with real-time demand update[END_REF] investigate the importance of real-time demand update in joint distribution systems, i.e., horizontal collaboration among shippers. They propose an architecture of digital connectivity for the dynamic joint distribution system, in order to collect and communicate synchronously and continuously data from customers, shippers, logistics platforms, etc., for the purpose of route optimization. More recently, the application of Machine Learning technique in logistics industry has further fundamentally altered the conventional approaches, enabling more efficient and visionary decision-making process. In the work of [START_REF] Vanvuchelen | Use of Proximal Policy Optimization for the Joint Replenishment Problem[END_REF], the authors experiment proximal policy optimization (PPO) algorithms in joint replenishment schemes, which is an optimization method in the domain of deep reinforcement learning. The results of their experiments show that, whatever at small-or large-scale, the suggested PPO algorithm outperforms the conventional heuristic methods for joint replenishment decision making. It means the algorithm can eventually facilitate the coordination in cooperative logistics schemes, especially at large-scale. This first work should encourage more research on the applications of AI in the area of LSCM.

Blockchain and API:

To alleviate some critical issues related to data sharing, e.g., data tracking, traceability, trustability, and privacy preservation, Blockchain or API (Application programming interface) based disruptive solutions have been put forward. For example, the blockchain technology is being growingly investigated in LSCM for the traceability of material, responsibility or data, e.g., [START_REF] Wang | Blockchain-Enabled Circular Supply Chain Management: A System Architecture for Fast Fashion[END_REF] use it for tracing the reuse of materials for circular supply chain management in fast fashion, and [START_REF] Suhail | Orchestrating product provenance story: When IOTA ecosystem meets electronics supply chain space[END_REF] suggest IOAT blockchain to ensure trustworthy data for tracing product provenance from raw material to end of life. Further, the technology of Smart Contracts based on blockchain is also receiving increasing attentions, especially for automating processes and exchanges, like services procurement and payment. APIs are also used to simplify and secure data exchange, as shown in the case of track-and-trace capability in Physical Internet ports done by [START_REF] Fahim | An information architecture to enable track-and-trace capability in Physical Internet ports[END_REF]. Besides, API is also the key technology to provides blockchain with information from the outside world, in other words, to connect blockchain to other data bases. As the research on Blockchain in the area of LSCM is still in its infancy, a number of issues still need to be addressed, such as scalability, typology (open, private or consortium), time-and cost-effectiveness, and sustainability of the technology.

Digital twin and Cyber-physical systems: One outcome of the digital transformation in many industries is that physical object and its lifecycle can now be sharped in the digital world, thanks to the advancement of IoT and sensing technologies. The virtual counterpart, namely Digital twin, can then be used to handle the very complex problems, such as product design, real-time status monitoring, maintenance scheduling, new value creation, etc. In addition, cyber-physical systems (CPS) that integrate technologies of sensor, computing and communication are primarily used to intertwine physical and cyber worlds for the purpose of controlling the objects. Based on these technologies, digital interoperability can be achieved among objects (or systems, companies). In the literature, intralogistics is currently the main field of application. The authors in [START_REF] Greif | Peeking into the void: Digital twins for construction site logistics[END_REF] apply digital twin concept on the silos in construction sites; and they prove that the developed digital silo twins can offer new business values as well as supports for silo dispatch and replenishment decisions. Another common application is related to logistics platform, since smart (or intelligent) objects are now being employed in intralogistics systems, for example, industrial wearables, AGVs (automated guided vehicles), AMRs (autonomous mobile robots), autonomous vehicles. Kong et al. demonstrate in their paper [START_REF] Kong | Cyber physical system-enabled synchronization mechanism for pick-and-sort ecommerce order fulfilment[END_REF] that integrating smart objects and CPS modelling may become an efficient and effective solution for pick-and-sort operations in ecommerce fulfilment centers. These works showcase the capability of digital twin and Cyber-physical systems in LSCM, especially to handle the high complexity of decisionmaking and control.

It is worth mentioning that two papers in the Special Issue are putting focus on the digital interoperability research landscape rather than on specific solutions. [START_REF] Pan | Digital interoperability in logistics and supply chain management: state-of-the-art and research avenues towards Physical Internet[END_REF] provide a comprehensive bibliometric analysis on the stat-of-the-arts solutions for digital interoperability in LSCM. The research trends, as well as new challenges raised by recent paradigms such as Physical Internet are discussed. Some future research avenues are also pointed out in order to advance the research. In another paper [START_REF] Hopkins | An investigation into emerging industry 4.0 technologies as drivers of supply chain innovation in Australia[END_REF], the author applies empirical methods to investigate how merging industry 4.0 technologies drive the SCM innovations, the digitalization and the inter-company connectivity.

Discussion

Since digital interoperability is an emerging research topic in the area of LSCM, this Special Issue with the included contributions should encourage more research on the topic. Several research questions can be discussed here.

First, the solutions discussed above could be further industrialized and deployed in real-life world, especially for large-scale multi-party logistics systems. It is foreseeable that some design issues must be addressed beforehand.

Second, this Special Issue also reveals the significance of interdisciplinary approaches to the research topic, involving operations research, data science, computer science, information system management, software and hardware engineering, as well as systems and organizational theory. It is believed that this axis will attract more attention in the future.

Third, as shown by the studies above, data-driven and digital twin-driven approaches have the great potential to enhance system-to-system, object-to-object, or object-to-context interoperability. Data-driven approaches include Big data analytics, data mining, machine learning, graph database etc.; and digital twin-driven approaches are for example semantics and ontology, cyber-physical systems, holonic systems, etc. The research on the application of such approaches in LSCM has just begun.

Last, blockchain and especially smart contract technologies are being considered as an enabler of automated processes and autonomous (self-organizing) systems. But one main issue is the interconnection and dynamic interaction among such processes and systems so that real-time information could considered and communicated. Recent concepts like Oracle (e.g., Chainlink) show the potential to deal with the issue, but more research on the proof of efficiency of applications in LSCM are still appealing.

With the development of the above-mentioned technologies, this emerging research field will also have to consider organizational questions related to adoption and the transformational potential of such technologies implemented at large-scale level by companies involved in supply networks in general.

Table 1 .

 1 Papers included in this Special Issue.

	Data-driven
	planning

Sonia Cisneros-Cabrera, Grigory Pishchulov, Pedro Sampaio, Nikolay Mehandjiev, Zixu Liu, Sophia Kununka Digital platform Digital connectivity in an innovative joint distribution system with real-time demand update Yuan Shi, Meng Chen, Ting Qu, Wei Liu; Yiji Cai Figure 1. The four main categories of solutions presented in the Special Issue.

Acknowledgement

First of all, we should express our sincere thanks to the authors and the anonymous reviewers who have greatly contributed to this Special Issue. We thank the authors for their high-quality and original scientific contributions, and the reviewers for their valuable time and support given to the double-blind peer review process, that are especially precious during the pandemic outbreak in 2020. Last but not least, we are grateful to the Computers in Industry editorial board and office, especially the current Editor-in-chief Professor Bernard Grabot and the current editorial manager for Special Issues Dr. Nick Szirbik, for their precious advice and trust on the guest editors. It is with their support of utmost importance that this Special Issue come to fruition.