Damir Becirevic

Florentin

F Jaffredo

Charged Current Anomalies with Baryons

come

Introduction

Hints of lepton flavor universality violation (LFUV) have been observed in B meson decay by measuring the universality ratios R D ( * ) and R K ( * ) 1,2,3,4 . Being the simplest in terms of valence quark content, mesons are hadrons of choice for the precision flavor physics observables both in experimental studies, due to their abundance in the clean environment of B-factories, and because of the relatively small theoretical uncertainties associated with the relevant hadronic matrix elements.

In what follows, we discuss the possibility of observing other hints of LFUV by looking at the baryon decays. Those decays involve the same b → c ν semileptonic transition and are produced in large quantity at LHCb 5 .

Theory prediction

To describe the charged current mediated semileptonic decay, in addition to the Standard Model (SM) Lagrangian, we also consider the following New Physics contributions (NP):

L NP ⊃ √ 2G F V ij g V (u i γ µ d j )( L γ µ ν L ) + g A (u i γ µ γ 5 d j )( L γ µ ν L ) (1) +g S (u i d j )( R ν L ) + g P (u i γ 5 d j )( R ν L ) + g T (u i σ µν (1 -γ 5 )d j )( R σ µν ν L ) + h.c.
The resulting L = L SM + L NP can describe various flavor transitions; including the semileptonic decays of baryons such as

Λ b → Λ c ν, Λ b → Λ * c ν, Λ c → Λ ν, Λ → p ν.
The main obstacles to the precision computation of decay rates are the hadronic matrix element of the quark currents sandwiched by the two hadronic states. Such matrix elements have been computed to a remarkable accuracy for many processes by means of lattice QCD 6,7,8 . For baryons, the results are expressed in term of 10 form factors, denoted F 0 , F + , F ⊥ , G 0 , G + , G ⊥ , h + , h ⊥ , h + , h ⊥ (resp. 3 vector, 3 axial, and 4 tensor form factors). The tensor form factors are not needed for the SM prediction but they are required in some scenarios of NP. The fact that all of these 10 form factors are available from lattice QCD for the Λ b → Λ c ν decay in the full phase space is a unique situation that makes this study particularly interesting.

By focusing on Λ b → Λ c ν, we can express the decay distribution as:

d 2 Γ(Λ b → Λ λc c λ l ν) dq 2 d cos θ = a λ l λc (q 2 ) + b λ l λc (q 2 )cos θ + c λ l λc (q 2 )cos 2 θ, (2)
where λ c and λ l are the polarization states of the outgoing Λ c and . θ is the angle between the lepton and the Λ c baryon in the dilepton rest-frame. The angular coefficients a, b, and c are functions of q 2 = (p Λ b -p Λc ) 2 = (p + pν) 2 , the NP couplings g i , (i ∈ V, A, S, P, T ), form factors and the polarizations of the outgoing particles. Explicit formulas can be found in Refs. 8,9,10,11,12 , as well as in Ref. 13 , in which we discuss this decay in more detail.

From the explicit formulas, we can see that

b - ± = ±(a - ± + c - ± ). (3)
Contrary to the case of the pseudoscalar to pseudoscalar meson decays where we only have a single spin-1/2 particle in the final state yielding at most 4 linearly independent observables 14 , here we expect 12 observables (4 for each of the a, b, c). Eq. ( 3) reduces this number to 10. Setting all g i = 0, we can compute the SM prediction for Λ b → Λ c τ ν and Λ b → Λ c µν, and combine them to

R Λc = B(Λ b → Λ c τ ν) B(Λ b → Λ c µν) , (4)
in which a bulk of uncertainties on the hadronic matrix elements cancel. Using the lattice QCD form factors 6,8 , we get R SM Λc = 0.333( 13).

(5)

New LHCb measurement

LHCb recently reported on the first observation of the semileptonic decay Λ b → Λ c τ ν. 5 They combine it with the old LEP result for B(Λ b → Λ c µν) 15 , and obtain

R LHCb Λc = 0.242(76). (6)
The error on this measurement is dominated by the systematics coming from the external branching fractions: (i) B(Λ b → Λ c µν) 15 (15%), and (ii) from the normalization channel B(Λ b → Λ c 3π) 16,17 (23%). This result is compatible with the SM prediction, merely ∼ 1σ below. Therefore the anomaly observed with mesons (R Exp

D ( * ) > R SM D ( * ) )
is not yet observed with baryons, which might change once the more accurate LHCb data become available. We remind the reader that the observed value of R D and R D * are respectively 1.5σ and 3.5σ above their SM predictions 3,4,18 . It can be shown that due to the relatively large uncertainty, R Λc has a small effect on the overall constraints on NP.

To illustrate the impact of this new measurement, we consider several scenarios in which NP only involves a single NP coupling (or a combination of two couplings). In particular, we consider scenarios involving g V L = (g V -g A )/2 and g S L = (g S -g P )/2, since they can be observed in several popular extensions of the SM. We also show the results for g T only and for the combination g S L = ±4g T , which is usually evoked in the scenarios involving Low Energy Scalar Leptoquarks 18 . Since g S L and g T are renormalization scale dependant, after running from the matching scale µ 1 TeV down to µ = m b , the above relations become g S L = -8.5 g T and g S L = 8.1 g T 19 . In the latter scenario we consider only the imaginary values of g S L since it has been shown that a real-valued g S L is not compatible with the observed values of R D and R D * 18 .

We extracted the preferred value (1σ limit) of the NP couplings for the aforementioned scenarios, collected in Tab. 1. As expected, the values obtained with R Λc only are compatible with zero, but the inclusion of R Λc in the fit does not sensibly change the values of g i with respect to the values obtained using R D and R D * . The effect of inclusion of R exp Λc in the scenario with complex g S L = 4g T is illustrated in Fig. 1. We see that the solutions for g S L to 2σ, as derived from R exp D and R exp D * (left panel) remain unchanged (right panel). Only the χ 2 min value gets a slight shift. Instead, the 3σ region, after including R exp Λ , allows for a pure real solution to g S L , which nevertheless is not compatible with SM.

Wilson

Angular observables and secondary decay

LHCb measured one of the 10 observables mentioned in Sect. 2, nameley B(Λ b → Λ c τ ν). By a simple inspection of Eq. ( 2), one can immediately see that any physics information contained in b(q 2 ) is lost when integrating over cos θ. To regain access to this information one considers angular observables. In particular, the Forward-Backward Asymmetry (A fb ) defined as (we neglect spin indices when summing over all polarization)

A fb (q 2 ) = 1 Γ tot 1 0 - 0 -1 dΓ(Λ b → Λ c ν) dq 2 dcos θ dcos θ = b(q 2 ) Γ tot , (7)
A fb = A fb (q 2 )dq 2 . (8)
Similarly, we can individually access c(q 2 ) by looking at the so-called "convexity" of the distribution 14 :

A π/3 (q 2 ) = 1 2Γ tot 1 1/2 + -1/2 -1 - 1/2 -1/2 dΓ(Λ b → Λ c ν) dq 2 dcos θ dcos θ = c(q 2 ) 2Γ tot , (9)
A π/3 = A π/3 (q 2 )dq 2 . ( 10
)
Notice that A π/3 is obtained experimentally by counting positively events where the outgoing lepton falls in the cones of radius π/3 with respect to the direction of the outgoing baryon, and negatively otherwise.

One possibility to access many more angular observables is to consider the secondary baryon decay. Restricting ourselves to only 2-body decays, there are 2 possibilities Λ c → Λπ and Λ c → pK S (the branching fraction of which are 1.30(7)% and 1.59(8)% respectively). The small branching fraction of the secondary decay certainly reduces the statistics, since the current measurement is made using the reconstruction channel Λ c → pK -π + for which B = 6.28(32)% 5 .

One can express the total angular distribution by introducing only 2 new parameters: the branching fraction of the secondary decay and an asymmetry parameter α,

α = Λ + π|Λ + c 2 -Λ -π|Λ - c 2 Λ + π|Λ + c 2 + Λ -π|Λ - c 2 , (11)
and similarly for Λ c → pK S . The parameter α has only been precisely measured for Λ c → Λπ, which is why we opt for that secondary decay, even though not every angular observable depends on α.

The full angular distribution for Λ b → Λ c (→ Λπ) τ ν involves two new angles: θ Λ (between Λ c and Λ in the Λπ rest frame) and φ (between Λπ-and ν-planes). The expression similar to Eq. (2) now writes:

d 4 Γ λ l dq 2 d cos θd cos θ Λ dφ = A λ l 1 + A λ l 2 cos θ Λ + B λ l 1 + B λ l 2 cos θ Λ cos θ + C λ l 1 + C λ l 2 cos θ Λ cos 2 θ + D λ l 3 sin θ Λ cos φ + D λ l 4 sin θ Λ sin φ sin θ (12)
+ E λ l 3 sin θ Λ cos φ + E λ l 4 sin θ Λ sin φ sin θ cos θ.
The expressions for the q 2 -dependent angular coefficients A 1,2 , B 1,2 , C 1,2 , D 3,4 and E 3,4 can be found in in Refs. 8,9,10,11,12 . The important points can be summarized as follows:

• We can have a Forward-Backward Asymmetry with respect to the lepton (yielding B 1 ), the baryon (yielding A 2 and C 2 ), or both at the same time (yielding B 2 ).

• The lepton Forward-Backward Asymmetry A fb has a small positive value in the SM due to the cancellation between the low and high q 2 regions. It can thus be very sensitive to NP, as illustrated in Fig. 2a, because the point at which A f b (q 2 ) = 0 is shifted with respect to the SM value.

• There is no equivalent of the convexity A π/3 for the baryon since there is no cos 2 θ Λ term.

• The spin of the outgoing Λ baryon does not bring any new information beyond α. In other words, the secondary decay encodes information on the polarization into angular observables and it can actually be used to extract α. Forward-Backward Asymmetry before integration over q 2 in the SM (blue) and for the best fit point in the scenario gS L = 4gT (red). In the SM, we observe a cancellation between the low and high q 2 , leading to a good sensitivity of this observable. (b): Fit to the Wilson coefficient gS L in the scenario gS L = 4gT as obtain in Fig. 1. In the background is plotted the value of D4, which exhibits a vertical gradient due to its dependence to the imaginary part of gS L .

Conclusion

In addition to the semileptonic decays of B mesons, Λ b baryons offer interesting opportunities to look for the presence of NP, beyond the SM. In particular, the measurement of the universality ratio R Λc can provide an independent test of LFUV since the relevant hadronic uncertainties are well under control, thanks to Lattice QCD. The first measurement of R Λc by LHCb was an important feasibility test. The reported value is compatible with the SM prediction but also with the previously preferred scenarios of NP selected by R Exp D ( * ) . We updated those scenarios including the new observable. R Exp Λc is currently weaker a constraint than those obtained by R Exp D and R Exp D * , but could quickly become competitive since the sensitivity is held back by the systematics in the normalization channel.

We also explored various angular observables to NP, both for the Λ b → Λ c τ ν decay and when including the secondary decay Λ c → Λπ, and found interesting properties for many of them. If measured, these observables could help disentangling among various scenarios of physics beyond the SM.

Figure 1 -

 1 Figure 1 -Fit to the complex Wilson coefficient gS L in the scenario gS L = 4gT using only the constraints from RD and RD * (left) and with adding the constraints from RΛ c . The constraint from RΛ c alone is also shown at 1 and 2σ in red. The SM is represented by the green point.

•

  In addition to A, B, and C, we have other observables sensitive to φ. In particular, E 4 and D 4 are proportional to the imaginary part of the Wilson coefficients, and thus they are exactly zero in the SM and the scenarios of NP without a new CP-violating phase. Therefore their measurement could be a strong zero-test of the NP phase. An example of a scenario with non-zero D 4 is shown in Fig.2b.The angular observables exhibit a desired features of cancellation of the CKM elements, a partial cancellation of hadronic uncertainties, and not requiring any external branching fraction. They are sensitive to various combinations of NP couplings and form factors and thus could help discriminate among various scenarios of NP. It is also very important to note that one can observe discrepancies in angular observables with respect to their SM predictions even if R Λc is perfectly SM-like.

Figure 2 -

 2 Figure2 -(a): Forward-Backward Asymmetry before integration over q 2 in the SM (blue) and for the best fit point in the scenario gS L = 4gT (red). In the SM, we observe a cancellation between the low and high q 2 , leading to a good sensitivity of this observable. (b): Fit to the Wilson coefficient gS L in the scenario gS L = 4gT as obtain in Fig.1. In the background is plotted the value of D4, which exhibits a vertical gradient due to its dependence to the imaginary part of gS L .

Table 1 :

 1 Fits to single Wilson coefficients based on RD and RD * , RΛ c alone, or all 3 observables.

	Coefficient	R D and R D *	R Λc	Combined	χ 2 min /d.o.f
	g V L	0.084 ± 0.029 -0.15 ± 0.14 0.077 ± 0.035 0.06 → 1.3
	g S L	-1.47 ± 0.08	-0.53 ± 0.54	-1.45 ± 0.11	0.5 → 2.1
	g T	-0.027 ± 0.011 0.13 ± 0.14 -0.026 ± 0.013 1.2 → 1.7
	g S L = +4g T ∈ iIR	±0.49 ± 0.10	0.0 ± 0.39	±0.47 ± 0.13	0.9 → 1.6
	g S L = -4g T	0.16 ± 0.06	0.0 ± 0.39	0.15 ± 0.07	0.7 → 1.0


Valentin Lechner

Christopher Betrancourt

Sebastien Ducruix

Philippe Scouflaire

Nicolas Fdida

Lucien Vingert

Marie Théron

Alexandre Barata

Marie Robcis

INVESTIGATIONS OF LOX/CH4 FLAMES AT VERY LOW MIXTURE RATIO AND HIGH PRESSURE IN ROCKET ENGINE OPERATING CONDITIONS

Keywords: Liquid Rocket Engine, oxygen/methane, transcritical combustion, surface thermometry, soot measurements

whether they are published or not. The documents may come

Introduction

The rising diversity of Liquid Rocket Engines (LRE): Prometheus (ESA/ArianeGroup), Raptor (SpaceX), Archimedes (Rocket Lab), BE-4 (Blue Origin), TQ (LandSpace), Aeon 1 (Relativity Space) … proves a growing interest in liquid oxygen (LOX) and liquid methane (LCH4) as propellants combination. Methane has many advantages, although it offers a lower specific impulse (ISP) than hydrogen. Its liquefaction temperature is close to that of oxygen, facilitating its storage and handling, and its density is six times greater than that of hydrogen, making it possible to use smaller tanks. Methane is a standard fuel used in many applications, can be extracted from natural gas, and its production is well managed. However, the combustion of methane and pure oxygen at high pressure remains poorly understood, particularly under the conditions encountered in the gas generator of LRE.

Understanding the different heat transfer mode contributions is mandatory for the development of a reusable rocket engine, which life cycle highly depends on the accuracy of wall temperature prediction. Moreover, different flame topologies can be triggered by wall temperatures [START_REF] Guiberti | Impact of heat loss and hydrogen enrichment on the shape of confined swirling flames[END_REF] and retrieved by numerical simulations thanks to precise boundary conditions [START_REF] Mercier | Experimental and numerical investigation of the influence of thermal boundary conditions on premixed swirling flame stabilization[END_REF]. Thus, several groups have measured wall heat fluxes in subscale experiments and showed the influence of the injection element design [START_REF] Ahn | Effects of injector recess on heat flux in a combustion chamber with cooling channels[END_REF] and arrangement [START_REF] Marshall | Benchmark Wall Heat Flux Data for a GO2/GH2 Single Element Combustor[END_REF], or chamber pressure [START_REF] Vingert | Heat Transfer Measurements in a Water-Cooled Rocket Combustion Chamber Operated with Oxygen/Methane Mixtures at the Mascotte Test Facility[END_REF][START_REF] Grenard | Experimental investigation of heat transfer in a subscale liquid rocket engine at high mixture ratio[END_REF]. More accurate and time-resolved measurements are still necessary in representative rocket engine operating conditions. Methane being carbon-based and gas generator usually working in fuel-rich conditions, soot particles are expected. Soot is a primary carbonaceous particle resulting from the incomplete combustion of hydrocarbons. Some benefits can be found in the main thrust chamber by the induced protective carbon layer deposing on the wall, thus protecting them from high heat fluxes [START_REF] Schoenman | LOX/propane and LOX/ethanol combustion chamber heat transfer[END_REF]. However, because clogging and damage to the turbine can append [START_REF] Meland | History of the Titan liquid rocket engines[END_REF][START_REF] Hernandez | Carbon deposition characteristics of LO2/HC propellants[END_REF][START_REF] Lausten | Carbon deposition with LOX/RP-1 propellants[END_REF], soot are of great concern in the gas generator, not to mention a significant health and environmental problem. Soot need thus to be characterized to enable a thorough evaluation of their potential impact on engine performance and life duration.

Imaging the flame allows a direct two dimensions comparison with numerical simulations [START_REF] Schmitt | Large-Eddy Simulations of the Mascotte Test Cases Operating at Supercritical Pressure[END_REF] and is frequently implemented in subscale rocket engine experiments. Many studies have thus shown a strong dependence of the flame shape on the injection element geometry and the injection conditions. For example, a better stabilization of the flame is found with a recess in the injection element, enhancing oxygen atomization, mixing, and so the flame opening angle [START_REF] Kendrick | Effects of a recess on cryogenic flame stabilization[END_REF][START_REF] Tripathi | LOx tube recess in cryogenic flames investigated using OH and H2O emission[END_REF][START_REF] Lux | Effect of Recess in High-Pressure Liquid Oxygen/Methane Coaxial Injection and Combustion[END_REF]. However, there are few flame visualizations in conditions where both methane and oxygen are in liquid or transcritical states [START_REF] Theron | Experimental and numerical investigation of LOX/Methane Cryogenic Combustion at low mixture ratio[END_REF]. These challenges are tackled within the Lab. EM2C/ONERA/CNES/ArianeGroup R&D consortium through experimental investigations held at the MASCOTTE test bench [START_REF] Vingert | A Rocket Engine under a Magnifying Glass[END_REF], located at ONERA Palaiseau center. The BHP combustion chamber is equipped with a single coaxial injection element. Advanced diagnostics are implemented simultaneously to understand the behavior of LOX/CH4 cryogenic flames at a low mixture ratio (MR = 0.3) and when LOX and CH4 are injected in either subcritical or transcritical states [START_REF] Zerbs | The influence of wavelength in extinction measurements and beam steering in laser-induced incandescence measurements in sooting flames[END_REF]48, and 60 bar) to mimic engine throttling.

High-speed backlighting imaging, OH* and CH* chemiluminescence detection are coupled and synchronized to observe the structure of the liquid jets and the flame.

Laser

Induced Phosphorescence (LIP) thermometry [START_REF] Brübach | On surface temperature measurements with thermographic phosphors: A Figure 11. Temporal evolution of the normalized internal temperature in the 2 nd module measured by LIP and by K1 thermocouple as a function of pressure, and with or without recess of the LOX post[END_REF] is implemented to track simultaneously the temperature of five surfaces of the combustion chamber during hot firing tests. LIP relies on applying a thermographic phosphor coating on surfaces of interest. A phosphorescence signal is emitted under laser irradiation which is temporally and spectrally temperature-dependent. These properties are used to determine an unknown temperature through an appropriate calibration. A dedicated method developed by Lab. EM2C, the FSF method [START_REF] Lechner | Full spectrum fitting method: a new approach for instantaneous phosphor thermometry in harsh environments[END_REF], allows single-shot measurements in an 800 K temperature range with a unique set of experimental acquisition parameters.

Laser extinction is implemented to determine the soot volume fraction in the burnt gases downstream of the flame. This optical diagnostic relies on measuring the attenuation of an incident light by the absorbing medium [START_REF] Simonsson | Wavelength dependence of extinction in sooting flat premixed flames in the visible and near-infrared regimes[END_REF].

EXPERIMENTAL SETUP

MASCOTTE cryotechnic test bench and operating conditions

Experiments are conducted on the high-pressure combustion chamber (BHP) of the cryotechnic test bench MASCOTTE operated by ONERA [START_REF] Vingert | A Rocket Engine under a Magnifying Glass[END_REF]. As seen in Fig. 1, it is composed of different modules of 50 × 50 mm 2 section. For the first time, two visualization modules are used simultaneously for optical accesses from the injection exit and downstream of the flame. Three silica windows (Corning 7980), providing a 40 × 75 mm 2 view, are mounted on the first module. The two on the side are used for high-speed visualization of the flame and liquid jets, whereas the third one, at the bottom, is used for phosphor thermometry. Two silica windows are mounted on the side of the second module for LIP and laser extinction measurements. A J-type thermocouple is inserted in the bottom blind window made of stainless steel to measure its surface temperature (see Fig. 3).

The 2 nd visualization module is followed by a module including seven ungrounded sheathed thermocouples of 2 mm diameter. Their names, type, and radial positions are indicated in Fig. 3. Gas and soot are sampled downstream of the flame with a special intrusive probe to analyze the composition of burnt gases and the type and morphology of soot [START_REF] Vingert | Experimental characterization of oxygen/methane combustion in rocket engine gas generator operating conditions[END_REF]. This is, however, outside the scope of this paper.

Methane and oxygen are injected into the combustion chamber through the injector head, and a single classical liquid rocket engine sheared injection element. A central tube (LOX post) supplies the oxygen, and the outer tube supplies the methane. Two versions are used, with (WR) or without (NR) recess of the LOX post. Calibration parts are added upstream of the injector outlet to reproduce the pressure drops encountered by the propellants in the engine.

Operating conditions are gathered in Tab. 1. The mixture ratio, defined as MR = 𝑚̇! ! 𝑚̇" # " ⁄ , is kept constant at 0.3. Oxygen and methane thermodynamic states are defined from the value of their critical points 𝑃 $ (𝑂 % ) = 50.4 bar, 𝑇 $ (𝑂 % ) = 154 K, and 𝑃 $ (𝐶𝐻 & ) = 46.0 bar, 𝑇 $ (𝐶𝐻 & ) = 190 K. The injection temperatures are kept constant for all operating points and always below both critical temperatures. Therefore, the reduced pressure 𝜋 ' = 𝑃 $()*+,' 𝑃 $ ⁄ , characterizes subcritical (𝜋 ' < 1) and transcritical (𝜋 ' > 1) states. Three chamber pressures are targeted to reach different combinations of thermodynamic conditions. Chamber pressure is measured by a Kulite sensor in the first visualization module, located in the blind window on the top, next to the H2/O2 ignition torch. geometries. They are designated as the combination of the injection element geometry (WR or NR) and chamber pressure (P26, P48, or P60).

Optical diagnostics 2.2.1. High-speed imaging

Backlighting imaging, OH*, and CH* chemiluminescence detection are implemented simultaneously and synchronized. Fig. 2 shows this imaging setup on the BHP combustion chamber. Some chemical reactions produce radical species in excited states during combustion, emitting a photon at a characteristic wavelength during spontaneous decay. Chemiluminescence is this spontaneous light emission which thus allows the identification of the flame front. OH* and CH* radicals are widely used to characterize oxygen/methane combustion [START_REF] Lux | Effect of Recess in High-Pressure Liquid Oxygen/Methane Coaxial Injection and Combustion[END_REF][START_REF] Singla | Transcritical oxygen/transcritical or supercritical methane combustion[END_REF][START_REF] Lux | Flame Stabilization in High-Pressure Liquid Oxygen/Methane Rocket Engine Combustion[END_REF] and emit light at central wavelengths of 308 nm and 431 nm, respectively.

Intensifiers coupled with high-speed 12-bit monochrome cameras are used to detect OH* and CH* emissions (LaVision HighSpeed IRO and Phantom VEO 710 for OH*, Hamamatsu and Photron SA-X2 for CH*). A 100 mm UV objective (CERCO) opened at f/8 equipped with a filter centered at 310 nm with 10 nm half-width is used for OH* emission detection. A 200 mm visible objective (Nikkor) opened at f/8 equipped with a filter centered at 430 nm with 10 nm half-width is used for CH*. The intensifiers gain and the cameras exposure time are kept constants for all operating points. The gate-opening time of intensifiers sets, however, the effective exposure. They are adapted through the campaign to maximize the signal-to-noise ratio. Both exposure times are synchronized and vary from 400 to 1200 ns for OH* and 1000 to 4500 ns for CH*. These values allow obtaining satisfactory instantaneous images of the flame. A higher exposure time is required for CH* emission detection, so instantaneous CH* images are more time-averaged than OH* images.

Backlighting imaging consists of a pulsed light source placed in front of the flame and the liquid jets [START_REF] Habiballah | Experimental Studies of High-Pressure Cryogenic Flames on the Mascotte Facility[END_REF]. A camera detects the transmitted light on the opposite side. Dense liquid areas transmit less the incident light than gases of lower densities. Liquid structures thus appear in black on the detector. The light source is a pulsed laser (Cavitar Cavilux Smart 400 W) emitting at 640 ± 10 nm. A diffuser and a lens (350 mm focal length) are used to illuminate the whole window. The transmitted light is detected with a 12-bit monochrome highspeed camera (Photron SA-X2) equipped with a 200 mm visible objective (Nikkor) opened at f/8. The exposure time is fixed by the laser exposition, which freezes the flow and varies from 40 to 60 ns. The camera exposure time is set to 5 µs so that the flame emission may be visible and quite similar to CH* images.

The three cameras are focused on the same field of view, with similar resolutions, thanks to dichroic mirrors. The first one separates the UV and visible light, and the second one reflects light above 660 nm.

The sensors are reduced to 496 × 896 pixels to fit the BHP window size and enable fast acquisition leading to a resolution of 11.6 mm/pixel for backlighting and 10.5 mm/pixel for OH* and CH* emission detection.

The three cameras are synchronized to record the entire combustion process from ignition to extinction with a 1 kHz frame rate or record a 3 s film during the stabilized time-step, starting at t = 40 s with a 15 kHz frame rate. About 52 000 images are captured in both situations for each device. For chemiluminescence images, summing up the intensity of each pixel allows tracking the overall intensity of the flame as a function of time to detect the different stages during the run.

Phosphor thermometry

The coating is obtained by mixing Mg3.5FGeO5:Mn phosphor powder (Phosphor Technology, EQD25/NU1) with a commercial binder (ZYP Coatings, HPC Binder) and water. This mixture is coated with an airbrush with ten layers. As seen in Fig. 3, two measurements are performed in the first visualization module and three in the second module. In the first and second modules, silica windows are coated on the internal and external sides. The fifth coating is applied on a blind window in the second module. Each measurement is performed in the middle of the windows.

The coated surfaces are excited at 355 nm using the 3 rd harmonic of a pulsed Nd:YAG laser (Continuum, Surelite SL II-10) triggered at 10 Hz. The laser beam is distributed to the five locations with beamsplitters leading to a 64 mJ.cm -2 fluence at each measurement site.

Telescopes composed of two-inch achromatic lenses (𝑓 -= 150 mm and 𝑓 % = 100 mm) are used to collect phosphorescence signals. The light is then focused on a 200 µm multimode optical fiber (CeramOptec, customized). All fibers are joined in a bundle and are aligned vertically, leading to the entrance slit of a spectrometer (Princeton Instrument, Spectra Pro 500i, 𝑓 = 500 mm, grating of 150 grooves mm -1 ). Phosphorescence signals are imaged at 10 Hz with the spectrometer coupled with an intensified CCD camera (Princeton Instrument, Pi-Max, 512 × 512 pixel 2 ) covering a spectral range of 160 nm centered at 𝜆 = 650 nm. The exposure time is set to 30 µs. Each singleshot spectrum is corrected from the dark current and smoothed with a 3-order Savitzky-Golay filter.

LIP calibration was carried out previously in Lab. EM2C by building databases of phosphorescence spectra at each kelvin from 100 K to 900 K, and laser fluences from 11 to 106 mJ•cm -2 at atmospheric pressure. Fig. 4 shows an example of a database at 64 mJ.cm -2 . The Full Spectrum Fitting method (FSF method) [START_REF] Lechner | Full spectrum fitting method: a new approach for instantaneous phosphor thermometry in harsh environments[END_REF] is applied to determine the temperature from the single-shots acquired. A least mean square algorithm is used to find the best agreement between each measured spectrum and one from the database. All the databases at different laser fluences are checked with this procedure leading to the final targeted temperature. This post-processing approach allows being nearly independent of the excitation energy, which can be lower than the calibration one in this high absorbing environment [START_REF] Lechner | Full spectrum fitting method: a new approach for instantaneous phosphor thermometry in harsh environments[END_REF].

In this work, temperatures are presented in the normalized form 𝜃.

Laser extinction

Soot absorbs and scatters light broadband UV to IR. From the Beer-Lamber law, the transmitted light through a homogeneous optically thin medium is:

𝐼 = 𝐼 . 𝑒 /0 # $%& 1 , Eq. 1
with 𝐼 . the incident light intensity at wavelength 𝜆 the extinction coefficient. Extinction is defined as the sum of absorption and scattering by the particles:

𝐾 2 ,34 = 𝐾 2 )+5 + 𝐾 2 5$) . Eq. 2
No molecular absorption is expected at 1064 nm. Within the Rayleigh limit, it is common to neglect scattering, assuming that soot are small spherical particles ;

67 ' 2 < 1<. For most applications in sooting flames, the Rayleigh-Debye-Gans approximation is applied [START_REF] Liu | Effect of aggregation on the absorption cross-section of fractal soot aggregates and its impact on LII modelling[END_REF], giving the absorption coefficient of an aggregate from the absorption for the 𝑁 8 individual particles:

𝐾 2 )+5 = 𝑁 8 • 𝐾 2 )+5,8 . Eq. 3
The absorption coefficient is then given by the Rayleigh theory for isotropic spheres:

𝐾 2 )+5,8 = 𝜋 % 7 ' ( 2 𝐸(𝑚 2 ), Eq. 4
with 𝐷 8 the primary particles diameter and 𝐸(𝑚 2 ) the soot absorption function, which depends on their optical properties and maturity stage [START_REF] Bladh | Optical soot characterization using two-color laser-induced incandescence (2C-LII) in the soot growth region of a premixed flat flame[END_REF]. The soot volume fraction can finally be expressed as a function of the extinction coefficient: .

𝑓 : = 𝑁 8 7 ' ( ; = 20
Eq. 5

𝐾 2 ,34 is determined experimentally from Eq. 1 by measuring 𝐼 𝐼 .

A to determine the soot volume fraction with Eq. 5. In this work, the soot volume fraction 𝑓 : is presented in a normalized form.

TEST SEQUENCE ANALYSIS

Fig. 6 shows the chronograph of a typical run at WRP48 operating point and the evolution of pressure, mixture ratio, gas and surface temperatures, soot volume fraction, OH* and CH* sum of intensities, and instantaneous images taken at different moments. The following paragraphs analyze the successive stages of the run thanks to these simultaneous diagnostics implemented.

Torch ignition

The main flame is ignited thanks to a secondary H2/O2 flame provided by the torch. A spark plug ignites a premix of gaseous hydrogen and oxygen, which is enabled during two seconds and starts at t = 25 s. At t = 26 s, this flame is recorded with high-speed cameras and detected by spontaneous emission of light by OH* radicals. As methane is not injected yet, no CH* radicals are produced. Raising the contrast allows the visualization of this flame. The bottom plot also shows a weak peak in the OH* sum of intensities. Besides, the presence of this gaseous flame results in a slight elevation of chamber pressure and gas temperature. Note that liquid oxygen is already injected into the combustion chamber but not completely vaporized and burnt. Filaments and individual droplets are indeed visible in the backlighting image. There is no increase in surface temperatures yet, but the slow acquisition of LIP measurements (10 Hz) may limit the recording of this event.

LOX/GCH4 flame

Gaseous methane starts being injected through the outer tube of the injection element one second after the opening of the H2/O2 torch. It ignites the LOX/GCH4 flame at t = 27.5 s, as seen in the bottom plot's abrupt temperature rise and OH* and CH* intensities. There is also a pressure rise of 13 bar. In the gas phase, the mixture ratio is thus higher than targeted with liquid methane (but still below the stoechiometric value of 4) for almost four seconds, up to t = 29 s. During this period, oxygen is still in a subcritical state for all operating points. Due to the high velocity ratio between methane and liquid oxygen, liquid ligaments are broken away by shear effect before atomization, mixing, and combustion with gaseous methane [START_REF] Snyder | Analysis of Flame Patterns in Cryogenic Propellant Combustion[END_REF]

LCH4 injection

Liquid methane starts being injected shortly after t = 29 s. It is followed by a decrease in gas and surface temperature resulting from the decrease in the mixture ratio. Up to t = 30.5 s, the pressure rises with the methane mass flow rate increase.

The broad pressure peak corresponds to the injection of gaseous helium near silica windows to protect them from the temperature peak at ignition. The lower plot shows a sudden decrease of OH* and CH* intensities during the beginning of LCH4 injection. The corresponding instantaneous images are presented at their lower intensities, at t = 29.5 s. One can see darker zones surrounding the flame near the injection on OH* and CH* images corresponding to liquid methane jet as seen on the backlighting image. The flame is shorter and tighter, so the apparent surface is smaller, explaining the lower sum of intensities.

The intensities then rise again with the methane mass flow rate increase before reaching a plateau starting at t = 32 s.

LOX/LCH4 targeted flame

The main LOX/LCH4 is stabilized with the targeted mixture ratio and chamber pressure at t = 32 s and maintained for 18 s. The temperature of burnt gases almost reaches a steady state, whereas surfaces temperature continuously increases during the whole run up to t = 52 s. It should be noted that the BHP is an uncooled combustion chamber that functions like a heat sink, which explains the surface temperature increase. OH* and CH* reach the same level of intensity and show the same evolution. However, the intensity ratio between LOX/GCH4 and LOX/LCH4 flames is different for OH* and CH* emissions, indicating changes in the emission spectra. One can see a decrease of intensity as a function of time due to deposits on windows allowing less light to pass through them, degrading the quality of images. Besides, it has been observed for the laser extinction technique that the intensity 𝐼 . , recorded before and after each run decreases by 30 % after the run due to these deposits.

Typical instantaneous images taken during this stabilized time step, at t = 40 s, are presented. They show different properties compared to LOX/GCH4 flames. The intensity is not homogenous: an intense zone is located downstream, whereas the intensity is lower on the left part of the window as liquid methane may mask the flame. This can be seen on the backlighting image: a wide liquid methane jet with widespreading, is hiding the liquid oxygen jet. A bright zone can be seen as well on the right. It corresponds to the flame emission, its shape matching the CH* image quite well. The flame front is thus less distinguishable than with gaseous methane. Anyway, the flame covers the whole window with wider spreading and may be longer. It has been identified indeed that the flame closes outside the window. Again, a dense condensed phase has been regularly observed recirculating around the conic flame near the injection and may be water. It was already observed in past studies [START_REF] Theron | Experimental and numerical investigation of LOX/Methane Cryogenic Combustion at low mixture ratio[END_REF] and often masks the flame anchoring.

The intensity is thus lower in this zone, so it is challenging to discuss the flame anchoring on the injection element.

Flame extinction

Gaseous helium is injected near the windows for five seconds, starting at t = 50 s, rising the chamber pressure, and decreasing gas temperature. The oxygen mass flow rate slowly decreases from t = 52 s, leading to a lower mixture ratio, lower temperatures, and lower OH* and CH* intensities. Before reducing the methane mass flow rate, a gaseous nitrogen purge completely extinguishes the flame at t = 53 s. Pressure and temperatures finally strongly decrease with the flame extinction. The gaseous nitrogen purge is also used to prevent any detonation or backfire. For 53 s < t < 57 s, liquid methane is still injected but not burned. The lack of LIP measurements may be explained by the absorption or scattering of the phosphorescence signal by these dense structures.

OPERATING POINT DEPENDENCE

The flame structure and surface temperature measurements are analyzed separately according to the operating points in the following paragraphs.

Flame structure

Time-averaged images of OH* emission and backlighting are calculated for 35 s < t < 45 s from 10 000 instantaneous images thanks to the acquisition at 1 kHz frame rate for all six operating points. They are shown in Fig. 7. Time-averaged images of CH* emission are not shown here but present the same conic flame shape. As previously seen on instantaneous images (Fig. 6), OH* images show a common similarity: a brighter zone on the right part of the window downstream of the injection. The intensity is lower on the first twothirds of the left part of the window, the flame emission being masked by the liquid jet. The darker zones on the backlighting images correspond with the darker zones of OH* images. Note that these images are spatially convoluted and show a 2D representation of the 3D flame. Besides, more light is accumulated along the line of sight as the diameter of these conic flames increases. Deposits on windows, mask the flame as seen at NRP26 operating point. OH* time-averaged images show that the flame opening angle decreases with increasing chamber pressure with respect to the liquid jet opening angle. The flame seems slightly less open with the recessed injection element at constant pressure.

As observed in past studies in similar conditions [START_REF] Theron | Experimental and numerical investigation of LOX/Methane Cryogenic Combustion at low mixture ratio[END_REF][START_REF] Singla | Transcritical oxygen/transcritical or supercritical methane combustion[END_REF], an ice ring is observed time-to-time in the vicinity of the injection element. Fig. 8 shows an example of instantaneous OH* and backlighting close-ups with a significant ice ring formation. This ice may result from water condensation, recirculating around the flame, in contact with the cold injection plate, and the cryogenic methane injection. It expands over a few millimeters with a cylindrical shape and surrounds the liquid jets, and the flame anchoring is entirely masked. It has been observed that this ice ring grows, not necessarily in a symmetrical way, then breaks away and reappears again. Smaller or larger pieces of ice sometimes break off, significantly influencing the liquid jet and the flame dynamic and stability. Besides, darker narrow stripes are also identifiable on the backlighting image emerging from the ice ring and may result from melting ice. The same stripes are visible on the OH* image, locally reducing the light intensity passing through them. This effect is less visible in the time-averaged images that smooth out the phenomenon. The stripes are still well identifiable at NRP26 and NRP48 operating points. More than two flame fronts can thus appear when performing an Abel deconvolution because these stripes are at a lower intensity, so their interpretation must be made with caution [START_REF] Singla | Transcritical oxygen/transcritical or supercritical methane combustion[END_REF].

Judging flame anchoring is challenging. Although all the flames studied seem anchored on the injection element and no intermittent flame detachments have been observed, the ice ring hides the injection element vicinity and, thus, the exact anchoring location. Moreover, it is impossible to say whether the flames are anchored with the recessed injection element.

Surface temperature

The surface temperatures in the 2 nd visualization module at WRP48 operating point are shown in Fig. 9. LIP measurements on the blind window are in good agreement with the ones from the wall thermocouple. One critical phenomenon captured by LIP measurements is the temperature peak at t = 28.5 s due to a higher mixture ratio during the injection of gaseous methane. It is not wellcaptured and delayed by the wall thermocouple whose response time (∼300-500 ms) is longer. LIP measurements may not give the proper temperature peak because of the acquisition at 10 Hz. Still, single-shot measurements allow detecting the temperature dynamics during a hot fire test, which would not be possible with LIP average signals.

The wall thermocouple systematically underestimates the temperature peak after ignition for all operating points. The temperature peak due to ignition of the H2/O2 torch is even sometimes captured (see Fig. 10 and11).

The internal temperature of the silica window (Mod. 2 int.) is very close to the one measured by K1 thermocouple. However, the temperature peak is lower on the window but delayed with the thermocouple, again due to a slower response time of a few tenths of a second. There is, however, no increase of the temperature in the external face of the silica window, whatever the operating point. This observation allows to model the silica window as a 1D semi-infinite medium. Fitting this model on the measurements allows the estimation of the local convection coefficient and the prediction of the temperature at any depth inside the walls. This is presented in detail in Lechner et al. [START_REF] Lechner | Dynamic characterization of wall temperature in LOX/CH4 rocket engine operating conditions using phosphor thermometry[END_REF], which shows as well that convection is the predominant heat transfer mode and fixes surface temperature downstream the flame. Fig. 10 shows surface temperatures in the 1 st and 2 nd visualization modules at NRP48 operating point. The first difference comes from the increase of external face temperatures, so convection may not be the predominant phenomenon near the flame. The increase of temperature and the main temperature peaks are identical for the internal surfaces. After t = 32 s, the temperature still rises downstream of the flame, whereas it is constant and lower near the flame. The lower temperature of the dense phase observed recirculating around the flame would then result in a lower convection coefficient in the boundary of the internal LIP measurement and thus a lower surface temperature.

Dependence on chamber pressure and injection element geometry is highlighted in Fig. 11 for the internal surface temperature of the silica window in the 2 nd module. Different pressure ramps were reached for the NRP26 operating point to investigate potential low-frequency instabilities. Mass flow rates of propellants were slightly adjusted during this process, thus modifying the mixture ratio and thus the gas temperature. Therefore, these slow oscillations in time should not be associated with the pressure value or the absence of LOX post recess. This phosphor thermometry setup associated with the FSF method showed the ability to track small power modulations [START_REF] Lechner | Dynamic characterization of wall temperature in LOX/CH4 rocket engine operating conditions using phosphor thermometry[END_REF]. A small temperature peak is also noticeable at the end of the run, around t = 55 s for the NRP26 operating point. It results from a short re-ignition of the flame, thereafter prevented with the nitrogen purge. Note that temperature measurements are a bit noisier for the WRP60 operating point due to a lower signal-to-noise ratio of the phosphorescence signal. Anyway, all six plots show globally the same characteristics: the broad temperature peak of LOX/GCH4 flame ignition, again delayed with K1 thermocouple, followed by a continuous temperature rise from 𝜃 = 0.7 up to 𝜃 = 1 concluded by the decrease after the flame extinction. Still, some differences exist at 26 bar. Indeed, the temperature peak at ignition is broader and seems to include the smaller one from the torch ignition compared to 48 and 60 bar operating points for which the first peak is identifiable and separated from the principal. From a thermal point of view, recessing the LOX post shows little apparent effect.

It is, however, challenging to make any conclusions with respect to the operating points in the 1 st module, near the flame. The surface temperature is more dependent on the behavior of the flame and the liquid jets. There are more fluctuations in time, the flame or the jets impacting time to time the bottom window.

CONCLUSIONS

LOX/LCH4 cryogenic flames at typical rocket engine gas generator conditions: very low mixture ratio and high pressure, have been investigated thanks to various optical diagnostics implemented simultaneously on the BHP combustion chamber of the MASCOTTE test bench operated by ONERA.

Imaging the flame with both liquid, or transcritical, oxygen and methane was challenging, the flame being often partially masked by the dense jets. Deposits on windows, progressive along test duration, are another source of poor signal-to-noise ratio. Still, time-averaged images give satisfactory results to judge the operating conditions influence on the flame shape. The flame opening angle appears to decrease with the increase of chamber pressure and with the LOX post recess. An ice ring was identified at the flame foot, significantly affecting the flame behavior and hiding the flame anchoring. A dedicated camera in the visible domain will be implemented in a next campaign for further investigations.

Phosphor thermometry and the Full Spectrum Fitting post-processing method have proven to be particularly well adapted to track transient temporal temperature evolutions in these harsh conditions. In contrast to conventional sensors, single-shot LIP measurements allowed the clear identification of temperature peaks. Besides, measurements near and downstream of the flame showed different thermal transfer modes. These data are crucial information for more accurate numerical simulations. Near the flame, the surface temperature is lower, probably in a zone of recirculating water. Ice formation from the injection element exit and around the flame anchoring suggests a cold injection plate. This will be investigated in the next campaign thanks to additional LIP measurements.

Laser extinction was successfully implemented for the first time on this bench to obtain a dynamic estimation of soot production. However, quantitative results and judging the dependence on the operating conditions are challenging and need more investigation. Therefore, additional axial measurements will be implemented in the next campaign to identify the stage of soot formation. Soot particles were also sampled at each operating point. They will be soon analyzed by Transmission Electron Microscopy (TEM) to identify their type, morphology, and size, which are essential knowledge for the validation of the hypotheses assumed to determine the soot volume fraction. They are coupled with gas samples downstream of the flame [START_REF] Vingert | Experimental characterization of oxygen/methane combustion in rocket engine gas generator operating conditions[END_REF], which analysis by gas chromatography will give valuable information in terms of soot precursors.

A following experimental test campaign will study the injection element design on the flame characteristics.

ACKNOWLEDGMENTS

This work is part of an ongoing joint Ph.D. between EM2C laboratory and ONERA, co-piloted by CNES and ArianeGroup. The authors would like to thank the members of the MASCOTTE test bench for their involvement during the campaign.

Figure 1 .

 1 Figure 1. Schematic representation of the BHP combustion chamber.

Figure 2 .

 2 Figure 2. Schematic of the high-speed imaging setup.

Figure 3 .

 3 Figure 3. Schematic of the phosphor thermometry setup.

Figure 4 .

 4 Figure 4. Phosphorescence spectra database at 64 mJ.cm -2 .

Fig. 5

 5 Fig. 5 shows the laser extinction setup on the BHP combustion chamber. Measurements are performed in the second visualization module at 𝑥 = 292 mm from the injection plate. The incident light is a continuous laser at 𝜆 = 1064 nm (Edmund MIL-III-1064-50mW) with a 1.5 mm diameter beam. A lens of 1 m focal length is used

Figure 5 .

 5 Figure 5. Schematic of the laser extinction setup.

Figure 6 .

 6 Figure 6. Typical run at WRP48 operating point. From top to bottom, evolution of: normalized mass flow rates, chamber pressure and mixture ratio, gas and surface temperatures, normalized soot volume fraction 𝑓 ! , sums of OH* and CH* intensities, and instantaneous OH*, CH* and backlighting images at different instants.

Figure 7 .

 7 Figure 7. Time-averaged OH* emission and backlighting images as a function of pressure, and with or without recess of the LOX post.

Figure 9 .

 9 Figure 9. Temporal evolution of normalized surface temperatures in the 2 nd visualization module at WRP48 operating point.

Figure 8 .

 8 Figure 8. Evidence of an ice ring at the injection element exit at t = 44.65 s at NRP26 operating point. Instantaneous OH* and backlighting images.

Figure 10 .

 10 Figure 10. Temporal evolution of normalized internal and external surface temperatures of silica windows in the 1 st and 2 nd module at NRP48 operating point.

Table 1 .

 1 Operating conditions.

	Name	P26	P48	P60
	Pressure (bar)	26	48	60
	MR (-)	0.3	0.3	0.3
	Tinj (K)	O2 CH4 140 110	110 140	110 140
	𝜋 ' (-)	O2 CH4 0.56 0.52	0.95 1.04	1.19 1.30
	Thermo. state	O2 CH4 sub. trans. trans. sub. sub. trans.
	Six operating points are thus defined with three
	targeted pressures and two injection element


Kazuo Nishimura

Harutaka Takahashi

Alain Venditti

A dynamic theory of the Balassa-Samuelson effect: Why has the Japanese economy stagnated for over 30 years? *

Keywords: Two-sector optimal growth models, optimal steady state, capital intensity, Balassa-Samuelson effect Journal of Economic Literature Classification Numbers: C61, C62, E32, F31, O41

The Balassa-Samuelson effect ("BS effect") has attracted attention as a theory to explain the stagnation of the Japanese economy over the past 30 years. In particular, it has been used to explain the long-term depreciation of the real effective exchange rate since 1995. Furthermore, macroeconomic data show that the BS effect explains well Japan's long-term economic stagnation. However, the BS effect was originally derived theoretically for small open economies, not for large economies like Japan. In other words, the BS effect cannot be theoretically applied to large economies. This is a serious problem in applying the BS effect empirically. In this paper, we embed Balassa-Samuelson's original argument into the optimal growth theory framework. That is, we set up an optimal growth problem for large countries. It is then shown that there exists a stable optimal steady state and that the BS effect is more directly valid in that optimal steady state. In other words, as a long-run property, the BS effect is applicable to large as well as small countries, although, contrary to the small open economy case, it does not depend on the capital shares of the two sectors.

Introduction

The Balassa-Samuelson effect (BS effect, hereafter) is still an important phenomenon in the theory of economic development, as Balassa [START_REF] Balassa | The Purchasing-Power Parity Doctrine: a Reappraisal[END_REF] states, "As economic development is accompanied by greater inter-country differences in the productivity of tradable goods, differences in wages and service prices increase, and correspondingly so do differences in purchasing power parity and exchange rates." Formally it can be expressed as the following BS effect equation.

ṗN pN = 1-β 1-α ȦT A T -ȦN A N (1)
where ẋ indicates a time derivative dx/dt, pN is the relative price of non-tradable goods, where the price of tradable goods is numéraire, α is the capital share of the tradable goods output, β is the capital share of the non-tradable output, and A i is the TFP of the sector i = N, T . 1If α > β and ȦT /A T > ȦN /A N hold, then according to [START_REF] Asea | The Balassa-Samuelson Model: An Overview[END_REF], it implies ṗN /p N > 0. In other words, the relative price of non-tradable goods increases. If the perfect purchasing parity (PPP) holds only for tradable goods, it implies that the real exchange rate will appreciate. Valentinyi and Herrendorf [START_REF] Valentinyi | Measuring Factor Income Shares at the Sectoral Level[END_REF] report that α = 0.37 and β = 0.32 in the US economy.

It implies that (1 -β)/(1 -α) ≈ 1.08 > 1. Furthermore, p N grows in such an economy because the TFP growth rate in the tradable goods sector is expected to be greater than that in the non-tradable goods sector.

Another important property is as follows.

Wages are determined entirely by the factor productivity of the tradable goods sector. [START_REF] Balassa | The Purchasing-Power Parity Doctrine: a Reappraisal[END_REF] Let us call these two properties (1) and ( 2) collectively the Balassa-Samuelson property. In short, the BS property hereafter.

The Japanese economy has stagnated for the past 30 years, especially the real effective exchange rate index (2010=100), which has been declining since 1995 as shown in Figure 1. BS property was often applied to explain Japan's real exchange rate depreciation and economic stagnation problems. In other words, as a result of globalization, major Japanese manufacturers in the tradable goods sector moved their main production facilities to China, Thailand, and other countries, and the remaining production facilities in Japan became less efficient. In addition, deregulation implemented in the service sector increased productivity in the non-tradable goods sector. This fact can be confirmed by the Balassa-Samuelson (BS) effect measures reported in the RPROD database. The rate of TFP growth in the tradable goods sector, as indicated by relative GDP per capita or per worker in Figure 2 above, has declined sharply since 1995, while the relative prices of non-tradable goods, as indicated by the CPI/PPI ratio and the three-and sixsector deflators in Figure 2, especially the three-sector deflator, has declined substantially as expected by the BS property (1) described above. As a result, the real effective exchange rate declined as shown in Figure 1.

Japan's per capita wages have also stagnated due to BS characteristics (2), and as Figure 3 below shows, per capita wages have remained nearly constant during the 1991-2020 reporting period.

As a result, the BS property would seem to fully explain the stagnation of the Japanese economy and the depreciation of the real effective exchange rate. However, there is a major theoretical problem with applying the BS property to a large country like Japan. This is because the BS property was originally proven for small developing countries that are given interest rates in the world market. Therefore, it is important to show that the BS property holds for large countries. This issue is addressed in the framework of optimal growth theory. Source: https : //www5.cao.go.jp/j -j/wp/wp -je22/h06 h z020105.html.

To our knowledge, the BS property has never been formally tested in the framework of two-sector optimal growth theory. Consider two cases. One is the case of a large country that can control interest rates internally, and the other is the case of a small country that cannot set interest rates but can provide access to world capital markets. In a small country, Takahashi and Venditti [START_REF] Takahashi | A Dynamic Theory of the Balassa-Samuelson effect[END_REF] proved that the exact same equation as the BS effect equation (1) holds. In contrast, this paper proves that the following different BS effect equation [START_REF] Benhabib | The Hopf Bifurcation and the Existence and Stability of Closed Orbits in Multisector Models of Optimal Economic Growth[END_REF] holds as a property of the long-run optimal steady state in a large country.

ṗN pN = ȦT A T -ȦN A N (3)
This is in contrast to the equation for the BS effect for small countries shown in [START_REF] Asea | The Balassa-Samuelson Model: An Overview[END_REF].

Since (3) does not depend on the capital intensity ratio term as in equation ( 1), the BS effect shows a more direct relationship with the difference in TFP growth rates. It is important to note that this difference is due to different optimal steady-state conditions, as will be shown below.

The remainder of this paper is organized as follows. Section 2 provides and analyzes the model: Section 2.1 presents a two-sector model comprising tradable and non-tradable goods. Then, in Sections 2.2 and 2.3, we analyze the uniqueness and saddle point stability of the optimal steady state. In Section 3, we discuss the Balassa-Samuelson property in the optimal steady state based on the discussion in Section 2. In Section 4, we provide concluding comments. All the proofs are gathered in a final Appendix.

The model

This section first describes the production structure and the preferences of our 2-sector model.

Then we discuss the intertemporal equilibrium and the steady state. Finally we derive the characteristic polynomial associated with linearization around the steady state.

The 2-sector economy

We consider an economy producing a non-tradable (N) good ỹN , and a tradable (T) good ỹT . Each good is assumed to be produced by using capital k j and labor l j , j = N, T in different proportions via Cobb-Douglas production functions:

ỹN = A N k β N l 1-β N , ỹT = A T k α T l 1-α T , (4)
where A i denotes the total factor productivity of sector i = N, T . Total labor is given by 1 = l N + l T , and total stock of capital is given by k = k N + k T . Let us then denote y N = ỹN /A N and y T = ỹT /A T . We can then rewrite (4) as

y N = k β N l 1-β N , y T = k α T l 1-α T . (5)
A firm in each industry maximizes its profit under productivity-normalized output prices p N and p T , rental rate of capital r, and wage rate w. Choosing the tradable good as the numéraire, i.e. p T = 1, we define from the technologies (5) the following Lagrangian

L = k α T l 1-α T + p N k β N l 1-β N -y N + r [k -k N -k T ] + w [1 -l N -l T ]
with p N , r and w the price of the non-tradable good, the interest rate and the wage rate, all in terms of the price of tradable good. The first-order conditions give

r = αk α-1 T l 1-α T = p N βk β-1 N l 1-β N w = (1 -α)k α T l -α T = p N (1 -β)k β N l -β N (6)
and we thus derive the following input coefficients:

a 00 (w, p N ) = l N y N = p N (1-β) w , a 10 (r, p N ) = k N y N = p N β r , a 01 (w) = l T y T = 1-α w , a 11 (r) = k T y T = α r . (7)
Each coefficient a ij represents the amount of "good" i, that is, labor or intermediate capital good, that it takes to produce one unit of good j -in other words, the non-tradable or tradable good output. Denoting p = (p N , 1) and ω = (w, r) , we can then define the following matrix of input coefficients

A(ω, p) =   a 00 (w, p N ) a 01 (w) a 10 (r, p N ) a 11 (r)  
which can basically be obtained from input-output tables available in national accounting data.

Using the results of Benhabib and Nishimura [START_REF] Benhabib | The Hopf Bifurcation and the Existence and Stability of Closed Orbits in Multisector Models of Optimal Economic Growth[END_REF], and as stated in Lemma 1 and Lemma 2, the factor-price frontier and the factor market-clearing equations depend on this matrix. We derive that, at equilibrium, wage rate and rental rate are functions of the non-tradable output price only, that is, w = w(p N ) and r = r(p N ), while the outputs are functions both of the capital stock and the non-tradable output price, y j = y j (k, p N ), j = N, T .

As can be expected in multi-sector optimal growth models, there is a duality between the Rybczinski and Stolper-Samuelson effects, i.e.

∂y N ∂k = ∂r ∂p N . (8)

Intertemporal equilibrium and steady state

The economy is populated by a large number of identical infinitely-lived agents. Without loss of generality, we assume that the total population is constant and normalized to one. At each period, a representative agent inelastically supplies one unit of labor. Furthermore, utility is derived from consuming the non-tradable good cN and the tradable good cT according to the following Cobb-Douglas specification:

u(c N , c T ) = c θ N c 1-θ T with c N = cN /A N , c T =
ε 00 = -u 1 u 11 c N = 1 1-θ , ε 01 = -u 1 u 12 c T = -1 1-θ , ε 10 = -u 2 u 21 c N = -1 θ , ε 11 = -u 2 u 22 c T = 1 θ . (9)
Profit maximization in both sectors described in Section 2.1 yields the demands for capital and labor as functions of the capital stock and the production levels of the non-tradable good, namely l j = l j (k, y N ) and k j = k j (k, y N ), j = N, T . Considering that at the equilibrium c N = y N , the optimal amount of the non-tradable good is then defined by:

y T = k T (k, y N ) α l T (k, y N ) 1-α = T (k, c N ).
From the envelope theorem, we get:

r = T k (k, c N ) and p T = -T c N (k, c N ).
The intertemporal optimization problem of the representative agent is then given by: max

{c N (t),c T (t),k(t)} +∞ 0 c N (t) θ c T (t) 1-θ e -δt dt s.t. k(t) = T (k(t), c N (t)) -gk(t) -c T (t) k(0) given, (10)
where δ ≥ 0 is the discount rate and g > 0 is the depreciation rate of the capital stock. We can write the modified Hamiltonian in current value as:

H = c N (t) θ c T (t) 1-θ + q(t) [T (k, c N (t)) -gk(t) -c T (t)] .
The necessary conditions, which describe the solution to problem [START_REF] Valentinyi | Measuring Factor Income Shares at the Sectoral Level[END_REF], are therefore given by the following equations:

q(t) = θc N (t) θ-1 c T (t) 1-θ p N (t) (11)
q(t) = (1 -θ)c N (t) θ c T (t) -θ (12) k(t) = T (k(t), c N (t)) -gk(t) -c T (t) (13) q(t) = (δ + g -T k (k(t), c N (t)))q(t) = (δ + g -r(t))q(t) (14)
Taking equations ( 11) to ( 14), we are now in a position to characterize an equilibrium path {k(t), p N (t), } t≥0 and to prove the existence of a unique steady state. Indeed, as shown in Section 2.1, we have r = r(p N ) and

c N = y N = k N (k, c N ) β l N (k, c N ) 1-β which gives c N = c N (k)
, and thus y T = T (k, c N (k)) = y T (k). Using (11), (12), we derive:

c T (t) = c T (k(t), p N (t)) = c N (k(t)) p N (t)(1-θ) θ . (15)
Straightforward computations then yield:

∂c T ∂k = p N (1-θ) θ ∂c N ∂k and ∂c T ∂p N = p N (1-θ) θ ∂c N ∂p N -c N p N . (16)
Considering ( 11)-( 14) and ( 16), the motion equations write:

k = y T (k) -gk -c T (k, p N ) ṗN = p N (t) θ δ + g -r(p N ) . (17)
Any solution {k(t), p N (t)} t≥0 that also satisfies the transversality condition:3 lim t→+∞ e -δt q(t)k(t) = 0 with q(t) as given by ( 11), is called an equilibrium path. A steady state is defined by a vector

(c * N , k * , p * N ) solution of y T (k) = gk + c T = gk + c N (k) p N (1-θ) θ r(p N ) = δ + g. ( 18
)
We get the following result:

Proposition 1. There exists a unique steady state (c * N , k * , p * N , ) > 0 solution of the system of nonlinear equations (18

) with c * N = c N (k * ) and c * T = c N (k * ) p * N (1-θ) θ .
Proof. See Appendix 5.1

Characteristic polynomial

Linearizing the dynamical system around (c * N , k * , p * N ) gives a 2 × 2 Jacobian matrix J which is provided in Appendix 5.2. Let us denote T the trace and D θ the determinant of J . Proposition 2 displays some properties of the eigenvalues of J and the expression of the characteristic polynomial. Proposition 2. If λ is an eigenvalue of the Jacobian matrix J , then δ -λ is also an eigenvalue and thus T = δ. The degree-2 characteristic polynomial is given by:

P θ (λ) = λ 2 -λδ + D θ ( 19
)
where

D θ = ∂y T ∂k -g- ∂c T ∂k ∂r ∂p N θ (20)
Moreover, the two roots are real and distinct.

Proof. See Appendix 5.2.

The results on the structure of the characteristic roots are in line with the conclusions of Kurz [START_REF] Kurz | The General Instability of a Class of Competitive Growth Processes[END_REF] and Levhari and Liviatan [START_REF] Levhari | On Stability in the Saddle Point Sense[END_REF]. Based on Proposition 2, we can further prove the saddle-point stability of the stationary steady state as exhibited in Proposition 1.

Proposition 3. For any α, β ∈ (0, 1) and any δ ≥ 0, the unique steady state (c * N , k * , p * N ) is saddle-point stable Proof. See Appendix 5.3.

The Balassa-Samuelson effect

We focus in this Section on the Balassa-Samuelson effect. The first question to answer is to check whether such a property is satisfied along the optimal steady state. Indeed, we have solved the model through a stationary version of the dynamical equations based on the considerations of the variables y N = ỹN /A N , y T = ỹT /A T , c N = cN /A N and c T = cT /A T . We need now to consider the real variables that are affected by the growth rates of productivities A N and A T . More precisely we need to consider the price of the non-tradable good pN which is linked to the stationary price p N as follows: p N = pN A N /A T . The following Proposition establish that in the optimal growth framework, the BS property is modified as formulated by equation (3). Proposition 4. At the unique steady state, the Balassa-Samuelson property holds, i.e.

ṗ * N p * N = ȦT A T -ȦN A N Proof. See Appendix 5.4.
Building on Proposition 2 showing the saddle-point property of the steady state, we can also conclude that the Balassa-Samuelson property holds not only at the optimal steady state but also along the optimal path.

The BS effect equation derived in Proposition 4 indicates a sharp contrast to that of the BS effect equation ( 1) presented in the Introduction. Note that this equation does not rely on a capital intensity term as indicated in the equation [START_REF] Asea | The Balassa-Samuelson Model: An Overview[END_REF]. That is, it shows that the rate of change in non-tradable relative prices is exclusively related to the difference in TFP growth rates between the two sectors.

It is interesting to consider why the formulas are different for large and small countries. In the small countries, the interest rate is given in the world market and the wage is determined only in the tradable goods sector. Thus, the allocation of capital goods between sectors is determined by the capital intensity of each sector. Under the standard assumption that the capital intensity of the non-tradable goods sector is lower than that of the tradable goods sector, an increase in capital will increase the output of the tradable goods sector and decrease that of the non-tradable goods sector due to the Rybczynski theorem. This change in output affects relative output prices, which appear as the capital intensity ratio in the BS formula, as shown in [START_REF] Asea | The Balassa-Samuelson Model: An Overview[END_REF]. In contrast, in the case of large countries, since the interest rate is determined domestically based on the productivity of the sectors, the capital intensities of the two sectors do not work directly to determine output prices. Thus, the relative intensity term is removed from the formula in the large country case, as shown in Proposition 4.

Finally, even for the large countries, Property (2) still holds from the following first-order conditions on the wage rate at the optimal steady state,

w * = (1 -α)A T k α T l -α T ( 21
)
This relation clearly indicates that only labor productivity in the tradable goods sector determines the wage rate.

The BS effect in large countries provided in Proposition 4 is shown to be in contrast to the equation for the BS effect in small countries as indicated in [START_REF] Asea | The Balassa-Samuelson Model: An Overview[END_REF]. For large countries such as Japan and the United States, the BS effect was found not to depend on the capital intensity ratio of the non-tradable and tradable sectors, but more directly only on the difference in TFP growth rates of the sectors. Thus, the results more directly support the stagnation theory of Japan based on the properties of the BS effect described in the introduction. An open question remains to study whether our conclusions are related to the fact that we consider specific Cobb-Douglas utility and production functions. The next step will then be to extend our model to general utility and production functions in order to provide a general formulation of the Balassa-Samuelson effect.

5 Appendix

Proof of Proposition 1

From the first-order conditions (6) and using the steady state equation r = δ + g we get:

ω 10 = r w = β 1-β l N k N = α 1-α l T k T r = βp N l N k N 1-β = βp N 1-β β ω 10 1-β = δ + g (22)
We then derive

ω 10 = β 1-β δ+g βp N 1 1-β (23)
Consider again the first-order conditions (6), we get

α l T k T 1-α = βp N l N k N 1-β ⇔ α 1-α α ω 10 1-α = βp N 1-β β ω 10 1-β (24)
Solving for p N using (23) then yields

p * N = δ+g β α δ+g 1-β 1-α β(1-α) α(1-β) 1-β (25)
Consider now Lemma 2, solving A(ω, p)y = x with respect to k using y

T = gk + c T with c T = (1 -θ)p N c N /θ gives k = a 10 + (1-θ)p N c N θ
(a 11 a 00 -a 10 a 01 ) a 00 (1-ga 11 )+ga 10 a 01

Solving A(ω, p)y = x with respect to y N using c N = y N gives:

c N = 1-a 01 gk+ (1-θ)p N c N θ a 00 (28)
Using ( 26) into (28) we find c * N =

1-ga 11 a 00 (1-ga 11 )+ga 10 a 01 +

(1-θ)p * N a 01 θ with p * N as given by (25). We then derive c * T and k * .

Proof of Proposition 2

Linearizing the dynamical system around (c * N , k * , p * N ) gives the Jacobian matrix J :

J =   ∂y T ∂k -g -∂c T ∂k ∂y T ∂p N -∂c T ∂p N 0 1 θ p N ∂r ∂p N   ≡ J 1 J 2 J 3 J 4 ( 29
)
with ∂y N ∂k = ∂r ∂p N . Since the optimization program [START_REF] Valentinyi | Measuring Factor Income Shares at the Sectoral Level[END_REF] has an Hamiltonian structure, and as initially proved by Kurz [START_REF] Kurz | The General Instability of a Class of Competitive Growth Processes[END_REF] and Levhari and Liviatan [START_REF] Levhari | On Stability in the Saddle Point Sense[END_REF], if λ is a characteristic root then δ -λ is also a characteristic root. This is confirmed by showing that T = δ.

Note that from Lemma 1 we can also derive the sectoral demands for capital and labor as functions of the capital stock and the production of the non-tradable good, namely l j = l j (k, y N ), k j = k j (k, y N ), j = N, T , with c N = y N . The optimal amount of the tradable good can be also expressed as:

y T = T (k, y N ) = k T (k, y N ) α l T (k, y N ) 1-α
and from the envelope theorem, we get: r = T k (k, y N ) and p N = -T y N (k, y N ). From Lemmas 1 and 2, we obtain the following derivatives:

Figure 1 :

 1 Figure 1: Japan's real effective exchange rate index Source: Real effective exchange rate index (2010 = 100) -Japan -Data (worldbank.org).

2

 2

Figure 2

 2 exhibits five different BS effect indicators. Since 1995, three of the five series have sharply declined.

Figure 2 :

 2 Figure 2: Japan's BS effect measures Source: CEPII -RPROD.

Figure 3 :

 3 Figure 3: Real per-capita wages (1991:100)

Lemma 1 .

 1 p = A (ω, p)ω and dp = A (ω, p)dω. Lemma 2. Denote x = (1, k) and y = (y N , y T ) . Then A(ω, p)y = x and A(w, p)dy +    ∂a 00 ∂w y N + ∂a 01 ∂w y T dw + ∂a 01 ∂p N y N dp N ∂a 10 ∂r y N + ∂a 11 ∂r y T dr + ∂a 11 ∂p N y N dp N    = dx.

  cT /A T and θ ∈ (0, 1]. Parameter θ measures the share of the nontradable good c N within total utility. The agent's preferences imply properties of interest regarding the (pure) elasticities of intertemporal substitution in consumption goods c N and c T , ε 00 and ε 11 , and the (cross-) elasticities of intertemporal substitution between the two goods, ε 01 and ε 10 :

  (1-β)p N ω 10 δ+g , a 10 = p N β δ+g , a 01 = (1-α)ω 10 δ+g , a 11 = α δ+g .

  Using the input coefficients[START_REF] Levhari | On Stability in the Saddle Point Sense[END_REF] then yields at the steady state∂r ∂p N = ∂y N ∂k = ∂c N ∂k = (δ+g)(1-α) p N (β-α) , ∂y T ∂k = -(δ+g)(1-β) β-α(30)Morever, using c T = (1 -θ)p N c N /θ we have matrix (29), we then deriveT = δ + ∂y T ∂k -(δ + g) + p N ∂c N ∂kand we conclude from (30)∂y T ∂k -(δ + g) + p N ∂c N ∂k = 0It follows therefore that T = δ. We finally conclude that, because of the triangular structure of the Jacobian matrix, the two characteristic roots are real and distincts.Solving for pN using (37) then yields p * N = A T (δ+g)

For the standard derivation of the equation (1) and (2), please refer to Asea et al.[START_REF] Asea | The Balassa-Samuelson Model: An Overview[END_REF] and Couharde et al.[START_REF] Couharde | Measuring the Balassa-Samuelson Effect: A Guidance Note on the RPROD Database[END_REF] for comprehensive explanations.

In detail on the RPROD data base, see C. Couharde et al.[START_REF] Couharde | Measuring the Balassa-Samuelson Effect: A Guidance Note on the RPROD Database[END_REF].

See Michel[START_REF] Michel | On the Transversality Condition in Infinite Horizon Optimal Problems[END_REF] and Kamihigashi[START_REF] Kamihigashi | Necessity of Transversality Conditions for Infinite Horizon Problems[END_REF] for some proof of the necessity of the transversality condition.

The project leading to this publication has received funding from the French government under the "France 2030" investment plan managed by the NRA (reference: ANR-17-EURE-0020), and from the Excellence Initiative of Aix-Marseille University -A*MIDEX.

Proof of Proposition 3

We have already proved that the Trace of the Jacobian matrix satisfies T = δ. Let us now compute the Determinant. Using (30) we get

We then derive

We then derive that if α > β then D θ < 0 for any δ. When β > α, D θ < 0 if and only if

The steady state is therefore a saddle-point for any capital-intensity difference between the tradable and non-tradable sectors and for any δ ≥ 0.

Proof of Proposition 4

Choosing again the tradable good as the numéraire, i.e. pT = 1, we define from the technologies (4) the following Lagrangian

with pN , r and w the price of the tradable good, the interest rate and the wage rate, all in terms of the price of tradable good. The first-order conditions give

Compared to the expression of the Lagrangian L we clearly have the following relationships:

with L = L/A T . Proceeding as in the proof of Proposition 1, we get:

Considering the first-order conditions (35), we get